LIBRARY OF WELLESLEY COLLEGE FROM THE FUND OF EBEN NORTON HORSFORD 5V^^ BIJA GANITA, S^c, §-c. S^c, IJA GANITAs OB THJB ALGEBRA OF THE HINDUS. EDWARD STBACHEY, OF THE EAST INDIA COMPAiNY's BENGAL CIVIL SERVICE. LONDON: PRINTED BY W. GLENDINNING, 25, HATTON GARDEN; AND SOLD BY MESSRS. BLACK, PARRY AND CO. BOOKSELLERS TO THE HON. EAST INDIA COMPANY, LEADENHALL STREET. ^- CONTENTS. PREFACE 1 INTRODUCTION 13 CHAPTER. I. On Affirmathe and Negative , ibid. Sect. I. On Addition and Subtraction ibid. Sect. II. On Multiplication , , 14 Sect. III. On Division ibid. Sect. IV. On Squares ibid. Sect. V, On the Square Root 15 CHAPTER II. On the Cipher. ibid. Sect. I. On Addition and Subtraction ibid. Sect. II. On Multiplication ibid. Sect. III. On Division 16 Sect. IV. Oh Squares, ^c , ibid. CONTEXTS. . CHAPTER III. . On Unknown. Quantities ....>..........> 16 Sect. I. On Addition and Suhtraction ibid. Sect. II. On Multiplication If Sect. III. On Division. ibid. Sect. IV. On Squares .,...., IS Sect. V. On the Square Root .....,,, ibid. CHAPTER IV. 0/7 Surds 1» Sect. I. On Addition and Subtraction ibid,. Sect. II. On Multiplication 20 Sect. III. On Division < 23 Sect. IV. On Squares 24 Sect. V. On the Square Root 24 CHAPTER V. On Indeterminate Problems of the first Degree £9 CHAPTER VI. On Indeterminate Probktns of the second Degree 36 ^CONTEXTS. BOOK 1. Page. On Simple Equations m BOOK 2. On Quadratic Equations 6) BOOK 3. On Equations involvi'ig indeterminate Quesliom of the 1st, Degree ... 70 BOOK 4. On Equations invoicing indeterminate Questions of the 2d. Degree .... 77 BOOK 5. On Equations invoking Rectangles .«.,,........ 8?* ••ce9i^^iee»«« Mm. Davis's Notes. On Chapter 1st. of Introduction go On Chapter 3d 92 O'l Chapter 4i/( 93 On Chapter 5th 95 On Chapter 6th 102 Oi Bunk 1 104 On Book •> 1 0 j On Book 3, 4, a»d 5 1 1 0 Extracts from a modern Hindoo Book of Astronomy ioid. _Explanution of Sanscrit Terms , 1 1 7 Mr. Davis's Authentication of his ^otes 119 ERRATA, Page 5, line 4, for Heilbionnen Teai Heilbronner. b, 1, for uncUstinguished read undistingu'ishing. 11, 6, after notes read at the bottom of the pages. 13, 1, deU the inTerted commas at the begianing of the line. 17, 5, (at the end), for number read colour. ax •\- c , 29, 25, Suppose — = y where a, b, and c are known and x and y unknfiwn, 93, 9, 10, and 11, the character prefixed to the numbers 1 and 2 is here r^) which ii the first letter of the word Loheet (see opposite page) ; but it should be a different character, viz. the first letter of the word Hoop ST. ^o)=h^ — b*- b being given to find h and i in any number of ways; let/ be any number; then ^——^z=s, and ps—bz=.h, C = s, and !—— — = ft. 2 2 A being given, rrqTi ~ ^' *°^ pi--'h=^. PREFACE. 3 always considered the Indian astrology and astronomy as different from theirs and the Greeks. We hear of Indian astronomy known to them in the time of the Caliph Al Mamun. (See d'Herbelot). Aben Asra is said to have compared the Indian sphere with the Greek and Persian spheres. (Heilbronnen Hist. Math, p. 456). We know that the Arabians ascribe their numeral figures to the Letp and j be any numbers; then 2pq = s, p^ — ?' = ^1 and p* + 3* = //. A' i" a — — a -t- — Giren a = hdzs; then =1 s, and — = h. a— 1/(2;;'— a') _ Given a =:b + si then , „ There are also rules for finding tlie areas of triangles, and four-sided figures; among others tjje rule for tije area of a triangle, without finding the perpendicular. For the circle there are these rules (c being the circumlerence, d the diameter, c die chord, » tlie versed sine, a the arch,) c : D :: 22 : 7; and c : d -.: 3927 : 1250. (Also see Ayeen Akbery, vol.3, p. 32.) p— t/((D +c)(d — c)) _ 4ao(c-.) ^,,,„af-./f"-^'M^, 4 * Also formula: for the sides of the regular polygons of 3, 4, 5, 6, 7, 8, 9 sides inscribed in a circle. There are also rules for finding the area of a circle, and the surface and solidity of a spliere. It will be seen also that IMiascara is supposed to have given these two roles, viz — the sine of the sum of two arcs is e(|ual to the sum of the products of the sine of each multiplied by the cosine of the other, and divided by the radius • and the cosine of the sunn of two arcs is equal to the difference between the products of tlieir sines and of their cosines divided by radius. Is it to be doubted that the Hindoos applied their rule fur the construction of the sines, to ascertain the ratio of the diameter of a circle to its circumference?— thus the circumference of a circle being divided into 360 degrees, or 21G00 minutes, the sine of 90 degrees which is equal to the radius would be found by the rule 343 S. This would give the ratio of the diameter to the circumference 7:21 ^^ and fso • 30 g^^^ 5"-5 ~ ■ ■ 573' and assuming, as the Hindoos commonly do, flie nearest integers, the ratio would be 7 : 22 or 1250 : 3927. It is not to be denied that there are some remarkable coincidences between the Greek and the Hindoo science • for example, among many wliich might be given it may be suggested that the contrivances ascribed to Antiphon and Bryso, and tliat of Archimedes, for finding the ratio of the diameter of a circle to its circumference mi»ht have been the foundation of tlie Hindoo method; that Diophantus's speculations on indeterminate problems might be the origin of the Hindoo Algebra. But there are no truths in the history of science of which we are better assured than that the Surya Siddhanta rule for the sines, with the ratio of the diameter of a circle to its circumference 1250 : 3927 ; and the Bija Ganita rules for indeterminate problems were not known to the Greeks. Such are the stumbling blocks which we always find in our way when we attempt to refer the Himloo science to any foreign origin. A 2 4 TREFACE. Indians; and Massoudi refers Ptolemy's astronomy to tlicni. (See Bailiy'b preface to his Indian astronomy, where is cited M. de Guignes. Mem. iVcad. Ins. T. 36, p. 771). Fyzee, who doubtless was conversant with Greco-arabian learning, and certainly knew the Hindoos well, has never started any douljt of the originality of what he found among them. The preface to the Zeej Mahommed Siiah}-, or .Astronomical Tables, M-hich were published in India in 1728, speaks of the European, the Greek, the Arabian, and the Indian systems as all different. Tliat work was compiled with great learning by persons who were skilled in the sciences of the West, as well as those of the East*. JNIore examples might be given — but to proceed. The Bija Ganita is a Sanscrit treatise on algebra, by Bhascara Acharya, a- celebrated Hindoo Astronomer and IMathematician. Fyzee t, who, in US?} translated the Lilavati, a work of his on arithmetic, mensuration, &c. speaks of an astronomical treatise of Bhascara's, dated in the 1105th year of the Salibahn, which answers to about 1 183 of our tera ; but Fyzee also says, it was 373 years before 995 Hegira, which would bring it down to A. D. 1226. So that Bhascara must have written about the end of the 12th century, or beginning of the 13th. A complete translation of the Bija Ganita is a great desideratum ; so it has been for more than 20 years, and so it seems likely to remain. It will be seen however that we have already means of learning, with sufficient accuracy, the contents of this work. I have a Persian translation of the Bija Ganita, which was made in India in 1634-, by Ata Allah Rusheedee. The Persian does not in itself afford a correct idea of its original, as a translation should do; for it is an * See Asiatic Rescarclies, 5th vol. on the Astronomical Labours of Jy Singh. t I will heie translate a part of Fyzee's preface:—" By order of king Akber, Fyzee translates into Persian, " from the Indian language, the book Lilavati, so famous for the rare and wonderful arts of calculation and " mensuration. He (Fyzee) begs leave to mention that the compiler of this book was Bhascara Acharya, whose •' birth place, and the abode of his ancestors was the city of Biddur, in the country of the Deccan. Though " the date of compiling this work is not mentioned, yet it may be nearly known from the circumstance, that the " author made another book on the construction of Almanacks, called Kurrun Kuttohul, in which the date of " compiling it is mentioned to be 1105 years Irom the date of the Salibahn, an xra famous in India. From " that year to this, which is the 32d Ilahi year, corresponding with the lunar year 995, there have passed " 373 years." As the Ilahi began in the Hegira (or lunar) year, 992, (see Ayeen Akbery) the date 32 of Ilahi is of course an error. It is likely too that there is an error in the number 373. Mr. Cokbrooke, in the 9th vol. of A'^iatic Researches, gives, on Bhascara's own authority, the date of his birth, viz. 1063 Saca. la li05 Salibahn (or S.ica) that is, about A.D, 1183, be was 42 years old. preface: 5 undistinguished mixture of text and commentaiy, and in some places it even refers to Euclid. But to infer at once from this that every thing in the book M-as derived from Greek or Arabian writers, or that it was inseparably mixed, would be reasoning too hastily. A little patience will discover evidence of the algebra which it contains, being purely Hindoo*. The fjllowing paper consists of an account of this translation, and some notes which I shall now mention : ]\Ir. Davis, the well-known author of two papers on Indian Astronomy in the Asiatic Researches, made, many years ago, in India, some abstracts and transla- tions from the original Sansciit Bija Ganitaf, and it is greatly to be regretted that he did not complete a translation of the whole. The papers which contain his notes had long since been mislaid and forgotten. They have been but very lately found, and I gladly avail myself of -Mr Davis's permission to make use of them here. The chief part of them is inserted at the end of my account of the Persian translation. To prevent misconception about these notes, it is proper for me to observe that in making them Mr. Davis had no other object than to inform himself generally of the nature of the Blja Ganita; they were not intended probably to be seen by any second person ; certainly they were never proposed to convey a perfect idea of the work, or to be exhibited before the public in any shape. Many of them are on loose detached pieces of paper, and it is probable that, from the time they were written till they came into my hands, they were never looked at again. But nevertheless it will be seen that they do, without doubt, describe accurately a considerable portion of the most curious parts of the Bija Ganita ; and though they may seem to occupy but a secondary place here, they will be found of more importance than all the rest of this work together. They shew positively that the main part of the Persian translation is taken from * The late Mr. Reuben Burrow in one of his p.-ipevs in the Asiatic Researches says, he made translations of the Bija Ganita and Lilavati. Those translations he left to Mr. Dalby. They consist of fair copies in Persian of Ata Allali's and Fyzee's translations, wiih the English ot each word written above the Persian. The words being tluis translated separately, without any regard to thC meaning of complete sentences, not a single passage can be made out. It is plain, from many short notes which Mr. Burrow has written in tlie margin of bis Bija Ganita, that lie made his verbal translation by the help of a Moonshee, ami that he had the original Sanscrit at hand, with some opportimily of consulting it occasionally. lam much obliged to Mr. Dalby for allowing me the use of Mr. Burrow's copy w hich has enabled me to supply deficiencies in mine ; and it is otherwise interesting, because it shews that Mr. Burrow had access to the original Sanscrit (probably by means of a Moonshee and a Pundii) and compared it with the Persian. t It is to be remarked that they were made fiom the Sanscrit only. Mr. Davis never saw the Persian translation. the SaiibCiit work, and that the references to Euclid are interpolations of the Persian translator they give most of the Hindoo Algebraic notation* M'hich is -wanting in the Persian, and they shew that the Astronomy of the Hindoos was connected with their Algebra. I must however confess, that even before I saw these notes the thing was to my mind quite conclusive. For I found (as will be seen) in this Persian trans- lation of 1634, said to be from the Sanscrit, a perfectly connected structure of science, comprehending propositions, which in Europe were invented successively by Bachet de Mezeriac, Fermat, Euler, and De La Grange f. * The Hindoos have no mark for +, they only separate the quantities to be added by a vertical line thus I or II , as they separate tiieir slocas or verses. Their maric for mimis is a dot over the quantity to be subtracted. Instead of a mark for multiplication they write the factors together as we do, thus, «S for a x l>. Division they mark as we do by a horizontal line drawn between the dividend and divisor, the lower quantity being the divisor. For unknown quantities they use letters of the alphabet as we do. They use the first letters of tlie words signifying colours. The known quantity (which is always a number) has the word roop (form) or the first letter of the word prefixed. The square of the unknown quantity is marked by adding to the expression of the simple quantity the first letter of the word which means square, and in like manner the cube. The sides of an equation are written one above another ; every quantity on one side is expressed again directly under it on the other side. Where there is in fact no corresponding quantity, the form is preserved by writing that quantity with the co-efficient 0. The methods of prefixing a letter to the known number, and using the first leUer of the words square and cube are the same as those of Diophantus. I mention it as a curious coincidence ; perhaps some people mav attach more importance to it than I do. + The propositions which I here particularly allude to are these: — V. A general method of solving the problem — ^^^ =.y, u, h and c being given numbers and x and y indeterminate. The solution is founded on a division like that which is made for finding the greatest common measure of two numbers. The rules comprehend every sort of case, and are in all respects quite perfect. 2. The problem am' + \ = "', (a being given and m and « required) w'ith its solution, 3. The application of the above to find any number of values of a«'+ b=:y' from one known case. 4. To find values of* and y in c*' + b=zi/' by an application of the problem — '^^ r= y. It is un- necessary for me here to give any detail of the Hindoo methods. The first question about this extraordinary matter is, what evidence have we that it is not ;ill a forgery ? I answer, shortly, that independently of its being now found in the Sanscrit books, it is ascertained to have been there in 1G34 and 1587, that is to say, in times when it could not have been forged. The following extract from a paper of De La Grange, .in the 24t!i volume of the Memoirs of the Berlin Academy, for the year 17fi8, contains a summary of that part of the history of Algebra which is now alluded to. As for the 4lh of the points abovementioned, the method in detail (however imperfect in some respects) is, as far as I know, new to this day. The first application o' the principle in Europe is to Le sought in the writings of De La Grange himself. rUEFACE. 7 To maintain tliat the Bija Gaiiita rules for the solution of indeterminate pro- blems might have been had from any Greek or Arabian, or any modern European writers before the INIathematicians just named, would be as absurd as to say that the Newtonian Astronomy mi.o-ht have existed in the time of Ptolemy. It is true that Bachet wrote a few years before 1634, but this is no sort of objection to the argument, for that pait which might be questioned as a mere copy of Bachet's method, namely, the rules for indeterminate problems of the first degree, is closely connected with matters of latter invention in Europe, and is in Mr. Dalby's copy of F} zee's translation of the Lilavati, which I have before said was made in 158? ; and Mv. Davis's notes shew that it is in the Sanscrit Bija Ganita, which was " La pill part des Geometres qui out cultive I'analysc de Diophante se sont, a rexemple de eel illiistre inven- •• tiier, uiiiqueiiient appliques a eviltr Ics valeurs iirationelles; et tout Tartifice de leurs methodes se rcduit i " fuiie en sorte que les grandeurs inconnues puissent se determiner par des nombres commensurables. " L'art de resoudre ces sortes de questions ne demande oueres d'autres principes que ceux de I'dnalyse " ordinaire: mais ces principes deviennent insulli^ajit lors-qu'on ajoute la condition que les quantitfe clier- " cliei-s soient nou seulenient commensurables mais encore egales a des nombres entiers. " M. Bacliel de Mezfriac, auteur d'un excellent commentarie sur Diophante et de dilftrens autres ouvrages " est, je crois, le premier quit ait tante de soumettre cette condition au calcul. Ce savant a trouv^ une nretjjode " generale pour resoudre en nombres entiers toutes les equations du premier degr6 a deux ou plusieurs incon. " nues, mais il ne paroit pas avoir ele plus loin; et ceux qui apres lui se sont occupes du nieme objct, ont " aussi presque tous borne leurs recherches aux equations indeterminees du premier degre ; leurs elT'orts se " sont reduits a varier ks methods qui pcuvent servir a la resolution de ces sortes d'equation>, et aucun, 5i " j'ose le dire, n'a doime une methode plus directe, plus generale, et plus ingenieuse que celle de M. Bachet " qui se trouve dans ses recreations mathematiqucs intitueles ' Problems plaisa7is et dileclahles qui se font par les " nombres.' II est a la verite assez surprenant que M. de Fermat qui s'etoit si long terns et avcc tant de " succes exerce sur la theorie des nombres entiers, n'ait pas cherch^ a resoudre gen^ralenient les problems " indetermines du second degre, et des degres superieurs comme M. Bachet avoit fait ceux du premier dcgr6 ; " on a cependent lieu de croire qu'il s'etoit aussi applique a cette recherche, par le probleme qu'il proposa " comme une espece de den k M. Wallis et a tous les Geometres Anglois, et qui consistoit a trouver deux " carres entiers, dont I'un etant niultipli^ par un nombre entier donne non carr6 & ensuite retranch6 de I'autre, •' le reste fut ctre egal a I'miile, car, outre que ce probleme est un cas particulier des Equations du second " degre a deux inconnues il est comme la clef de la resolution giSnerale de ces equations. Mais soit que " M. de Fermat n'ait pas continue ses recherches sur cette maliere, soit qu'elle ne soit parvenue jusqu'ft nous, " i! est certain qu'on n'en trouve aucune trace dans ses ouvrages. " II paroit mcme que les Geometres Anglois qui ont resolu le probleme de M. de Fermat n'ont pas connu '• toute I'importance dont il est pour la solution generale des problemes indeierininfis du second degre, dil " moins on ne voit pas qu'ils en ayant jamais fait usage, et Euler est si je ne me trompe, le premier qui ait '• fait voir comment a I'aide de ce probleme on peut trouver une infiniie de solutions en nombres entiers de " toute equation du second J6gi6 a deux inconnues, dont on connoit deja une solution. " II rfesulte de tout ce que nous venous de dire, que depuis I'ouvrage de M. Bachet que a paru en 1613, ♦• jusqu'a present, ou du moins jusqu'au memoire que je donnai I'r.nnee pass^e sur la solution de problems " indetermines du second d^gre, la theorie de ces sortes de problemes n'avoit pas a propremcnt parler, ^te " poussee au dela du premier degre." PREFACE. written four centuries before Bachet. Though we are not without direct proofs from the original, yet, as even the best Sanscrit copies of the Bija Ganita, or any number of such copies exactly corresponding, M'ould still be open to the objection of interpolations, it is necessary in endeavouring to distinguish the possible and the probable corruptions of the text, from what is of Indian origin, to recur to the nature of the propositions themselves, and to the general history of the science. Indeed we have not data enough to reason satisfactorily on other prin- ciples. We cannot rely upon the perfect identity or genuineness of any book before the invention of printing, unless the manuscript copies are numerous, and of the same age as the original. Such is the nature of our doubt and difficulty in this case, for old mathematical Sanscrit manuscripts are exceedingly scarce ; and our uncertainty is greatly increased by a consideratii)n of this fact, that in latter times the Greek, Arabian, and modern European science has been introduced into the Sanscrit books. Yet, in cases precisely parallel to this of the Hindoos, we are not accustomed to withhold our belief as to the authenticity of the reputed works of the ancients, and in forming our judgment we advert more to the contents of the book than to the state of the manuscript. When the modern Europeans first had Euclid, they saw it only through an Arabic translation. Why did they believe that pretended translation to be authentic ? Because they found it contained a well connected body of science ; and it would have been equally as improbable to suppose that the Arabian translator could have invented it himself as that he could have borrowed it from his countrymen. There are principles on which we decide such points. We must not look for mathematical proof, but that sort of proba- bility which determines us in ordinary matters of history. Every scrap of Hindoo science is interesting ; but it may be asked why publish any which cannot be authenticated ? I answer, that though this translation of Ata Allah's which professes to exhibit the Hindoo algebra in a Persian dress, does indeed contain some things which are not Hindoo, yet it has others which are certainly Hindoo. By separating the science from the book w'e may arrive at principles, which if cautiously applied, cannot mislead, which in some cases will shew us the truth, and will often bring us to the probability when certainty is not to be had. On this account I think the Persian translation at large interest- ing, notwithstanding it contains some trifling matters, some which are not in- telligible, and others which are downright nonsense. I have said that Mr. Davis's notes shew a connexion of the algebra of the PREFACE. 9 Hindoos with their astronomy. Mr. Davis informs me that in the astronomical treatises of the Hindoos, reference is often made to the algebra; and particularly he remembers a passage wliere Bhascara says " it Avould be as absurd for a person " ignorant of algebra to write about astronomy, as for one ignorant of grammar " to write poetry." Bhascara, who is the only Hindoo writer on algebra whose works we have 3'et procured, does not himself jjretend to be the inventor, he assumes no character but that of a compiler*. Fyzee never speaks of him but as a person eminentlv skilled in the sciences he taught. He expressly calls him the compiler of the Lilavati. I understand from Mr. Davis, and I have heard the same in India, that the Bija Ganita was not intended by Bhascara as a separate unconnected work, but as a component part of one of his treatises on astronomy, another part of which is on the circles of the sphere. I have found among Mr. Davis's papers, some extracts from a Sanscrit book of astronomy, which I think curious, although the treatise they were taken from is modern. Mr. Davis believes it to have been written in Jy Sing's time, when the European improvements were introduced into the Hindoo books. Two of these extracts I have added to the notes on the Bija Ganita. The first of the two shews that a method has been ascribed by Hindoo Astronomers to Bhascara of calculating sines and cosines by an application of the principles which solve indeterminate problems of the second degree. This suggestion is doubtless of Hindoo origin, for the principles alluded to were hardly known in Europe in Jy Sing's timet- I think it very probable that the second extract is also purely Hindoo, and that the writer knew of Hindoo authors who said the square root might be extracted by the cootuk ; that is to say, the principle which effects the solution of indeterminate problems of the first degree. From this, ami from what is in the Bija Ganita, one cannot but suspect that the Hindoos had continued fractions, and possibly some curious arithmetic of sines. On such matters how- ever, let every one exercise his own judgment. :]: * " Almost any trouble and expence \vould be compensated by the possession of the three copious treatises ■ on algebra from which Bhascara declares he extracted his Bija Ganita, ami which in this part of India are '. supposed to be entirely lost," — As. Res. vol. iii. Mr. Davis " On the Indian Cycle of 60 years." t Jy Sing reigned from 1 C9i to 1744. i. Mr, Reuben Burrow, who, by the bye, it must be confessed is very enthusiastic on these subjects, in a paper B JO PEEFACE. We must not be too fastidious ia our belief, because we have not found the works of the teachers of Pythagoras ; we have access to the wreck only of their ancient learuins:; but when we see such traces of a more perfect state of know- ledo-e, when v.e see that the Hindoo algebra 600 years ago had in the most interestino- parts some of the most curious modern European discoveries, and when we see that it was at that time applied to astronomy, we cannot reasonably doubt the originality and the antiquity of mathematical learning among the Hindoos. Science in remote times we expect to find withiu very narrow limits indeed . its history is all we look to in such researches as these. Considering this, and comparing the contents of the Hindoo books with what they might have been expected to contain, the result affords matter of the most curious specuiittion. Mav 1 be excused for adding a few words about myself. If my researches have not been so deep as might have been expected from the opportunities I had in India, let it be remembered that our labours are limited by circumstances. It is true I had at one time a cop3" of the original Bija Ganita, but I do not under, stand Sanscrit, nor had I then any m.eans of getting it explained to me. Oihcial avocations often prevented me from bestowing attention on these matters, and from seizing opportunities when they did occur. Besides, what is to be expected in this v.av from a iticre amateur, to whom the simplest and most obvious parts onlv of such sultjects are accessible? E. S. The following account of Ata Allah's Bija Ganita is partly literal translation, partly abstract, and partly my own. The literal translation is marked by inverted commas ; that part which consists of my own remarks or description will appear by the context, and all the rest is abstract. I have translated almost all the rules, some of the examples entirely, and in the appendix of lie Sd toI. of the As. Ees. speaks of the Lilavati and Bija Ganita, and of the mathematical knowledge or the Hindoos : He savs, he was told by a Pundit, that some time ago there were other treatises of slgebra, &c. (See the paper.) PRKFACE. 11 others in part; in short, whatever I thought deserving of particular attention, for the sake of giving a distinct idea of the book. Perhaps some of the translated parts might as well have been put in an abstract ; the truth is, that having made them originally in their present form I have not thouglit it worth while to alter them. The notes are only a few remarks which I thought might be of use to save trouble and to furnislt necessary explanation. B 2 BIJA GANITA. " A FTEU the usual invocations and compliments, the Persian translator begins thus : " By the Grace of God, in the year 1044 Hegira" (or A. D. iGSi) " being the " eighth year of the king's reign, I, Ata Alia Raslieedee, son of Ahmed Nadir, " have translated into the Persian language, from Indian, the book of Indian " Algebra, called Beej Gunuit (Bija Ganita), which was written by Bhasker Acharij " (Bhascara Acharya) the author of the Leelawuttee (Lilavati). In the science of " calculation it is a discoverer of wonderful truths and nice subtilties, and it con- " tains useful and important problems M'hich are not mentioned in the Leelawuttee* " nor in any Arabic or Persian book. I have dedicated the work to Shah Jehan, and " I have arranged it according to the original in an introduction and five books." INTRODUCTION. " The introduction contains six chapters, each of which has several sections." CHAPTER I. On Possession (JLc)* and Debt ((^^ti). *' Know that whatever is treated of in the science of calculation is either " affirmative or negati\'e ; let that Mhich is affirmative be called 7nal, and that '■' which is negative (kin. This chapter has five sections." Sect, I. On Addition and Subtraciion, that is, to encrease and diminish. " If an affirmative is taken from an affirmative, or a negative from a negative, " the subtrahend is made contrary; that is to say, if it is affirmative suppose it " negative, and if negative suppose it affirmative, and proceed as in addition. " The rule of addition is, that if it is required to add two affirmative quantities^ * Most of the technical terras here used arc Arabic. li IXTROUUCTlOy. " or two negative quantities together, the sum is the result of the addition. If " they are afiirmative call the sum affirmative: if negative call the sum negative. "If the quantities are of different kinds take the excess; if the affirmative is " greater, the remainder is atitirmative ; if the negative is greater, the remainder " is negative; and so it is in subtraction." (Here follow examples). Sect. IL On Multiplication *. '* If affirmative is multiplied by affirmative, or negative by negative, the product '' is afiirmative and to be included in the product. If the factors are contrary •' the product is negative, and to be taken from the product. For example, let us " multiply two affirmative by three affirmative, or two negative by three negative, " the result will be six affirmative; and if we multiply two affirmative by three " negative, or the contrary, the result will be six negative." Sect. III. On Division. " The illustration of this is the same as what has been treated of under multi- •' plication, that is to say, if the dividend and the divisor are of the same kind " the quotient will be affirmative, and if they are different, negative. For " example, if 8 is the dividend and 4 the divisor, and both are of the same kind, " the quotient will be 2 affirmative ; if they are different, 2 negative." Sect. IV. On Squares-\. *' The squares of affirmative and negative are both affirmative; for to find the * In ihe Persian translation the product of numbers is generally called the rectangle. f I had a Persian treatise on Algebra in vhich there was this passage — " Any number which is to be multiplied " by itself is called by arithmeticians root ( itAs^J, by measurers of surfaces side (xX/^^, and by alge- " braists thing (/ ^j*ij. And the product is called by arithmeticians square f i luXcsVcJ, by measurers " of surfaces square (Xj-^j, and by algebraists possession (jj"-^}. (jL«« is also used for plus, and its opposite debt ( IvrJ^^J 'of minus. These terms, all of which are Arabic, are used in the Persian translation of the Bija Ganita, the geometrical more frequently than their corresponding arithmetical or algebraical ones. INTRODUCTION. jg " square of 4 affirmative we multiply 4 affirmative by 4 affirmative, and by the rules " of multiplication, as the factors are of the same kind, the product must be l6" " affirmative, and the same applies to negative." Sect. V. On the Square Root. " The square root of affirmative is sometimes affirmative and sometimes nega- " tive, according to diflerence of circumstances. The square of 3 affirmative or " of 3 negative is 9 affirmative; hence the root of 9 affirmative is sometimes 3 " affirmative and sometimes 3 negative, according as the process may require. " But if any one asks the root of 9 negative I say the question is absurd, for there " never can be a negative square as has been shown." CHAP. IT. On the Cipher. " It is divided into four sections." Sect. I. Oti Addition and Subtraction. " If cipher is added to a number, or a number is added to cipher, or if cipher " is subtracted from a number, the result is that number : and if a number is^ub- " tracted from cipher, if it is affirmative it becomes negative, and if neo-ative it " becomes affirmative. For example, if 3 affirmative is subtracted from cipher " it will be 3 negative, and if 3 negative is subtracted it will be 3 affirmative." Sect. II. On Multiplication. •' If cipher is multiplied by a number, or number by cipher, or cipher by cipher, •■' the result will be cipher. For example, if wt multiply 3 by cipher, or con- " versely, the result will be cipher" IQ IN'TRODUCTION. Sect. III. On Division. " If the dividend is cipher and the divisor a number the quotient will be cipher. *=' For example, if we divide cipher by 3 the quotient will be cipher, for multi- " plying it by the divisor the product will be the dividend, M'hich is cipher: ^^ and if a number is the dividend and cipher the divisor the division is impossible; " for by whatever number we multiply the divisor, it M-ill not arrive at the divi- " dend, because it will always be cipher." Sect. IV. On Squares, ^c. " The square, cube, square root, and cube root of cipher, are all cipher ; the *' reason of which is plain." CHAP. III. On Colours. " Whatever is unknown in examples of calculation, if it is one, call it thing, " ( / u*)-, and unknown ( J^^s^^) ; and if it is more call the second black, " and the third blue, and the fourth yellow, and fifth red. Let these be termed " colours, each according to its proper colour. This chapter has five sections. Sect. I. fO/j Addition and Subtraction of Coloui^s. " When we would add one to another, if they are of the same kind add the ■" numbers* together ; if they are of two or more kinds, unite them as they are, " and that will be the result of the addition." Here follows an example. *' If we wish to subtract, that is to take one from the other, let the subtrahend " be reversed. If then two terms of the same kind are alike in this, that they are " both affirmative or both negative, let their sum be taken, othenvise their dif- " ference, and whatever of the kind cannot be got from the minuend, must be * Meaning here the co-efEcients, I>fTRODUCTlOy. 17 " subtracted from cipher. Then let it be reversed, and this will be the result " exactly." (Here follows an example). Sect. II. O/i AlultipUcatlon of Colours. *' If a colour is multiplied by a number the product will be a number*, v X 3: " will be J*, whether the number is the same or different, and the product multi- " plied by ,r will be .i-'. If the colours are different multiply the numbers of both " together, and call the product the rectang'le of those two colours.'' The following is given as a convenient method of multiplying : + 3.V + 2 + 5,r + IJ.r' + lOx — 1 - 3.V 0 Product + IJ.r' +7j: — 2 which shews the product of (5.r — l) X (Sr 4. £,) (Here follow examples). Sect. III. On Division of Coloui's. " Write the dividend and divisor in one place, find numbers or colours or both, " such that when they are nniltiplied by the divisor, the product subtracted from " the dividend will leave no remainder. Those numbers or colours will be the " quotient." * In the Persian tianslation there is no algebraic notation, I mean to^ translate " the unknown" by r, " the black" by y, and so on. And in like manner f have used the marks of multiplication, &c, insteadof writing the words at length as thi»y are in the Persian. IS INTRODUCTIOK. Sect. IV. On the square of Coloitrs : " That is to say, the product arising from any thinu' iiiultipHed by itself." Examples. Sect. V. O/i the S(]uarc Root of Colours. " To know the square root of a colour, find that which when it is multiplied by " itself the product subtracted from the colour whose root is required, M'ill leave no " remainder. The rule is the same if there are other colours or numbers with. " that colour." Example. Required the square root of i6.r- + 36' — 4S.r. The roots of ]6,r* and 56 are 4.r and 6, and as 48.i' is — these two roots must have different signs. Suppose one + and the other — , multiply them and the product will be — 2kr; twice this is — 48,r which was required. The root then is +4.r— (5, or +6— 4.c. Another Example. Required the square root o^ 9x'^ + A^^'^ + z''-{-\^xlJ—6.iZ— Ayz—6x—^y+^z+^. Take the root of each square ; we have 3x, ly, z, and 1, Multiply these quantities and dispose the products in the cells of a sc^uare. 3x 1y - 1 Sx 9.v' 6xy 3xz 3x 2j/ 6xy 4y^ oyz "^ii - Sxz 2yz -' s 1 . 3x 2i/ ~ 1 To find M'hat sort of quantities these are: The product of .r and j/ is +, there- IMTRODUCTIOK. 19 fore the factors are like, suppose them both — . The product of ,v and z is — , therefore the former having been supposed — the latter must be + because the factors must be different. 3.v is the product of 3j.' and 1 ; and a.' being — , 1 must be +. The sorts thus found are to be placed in the cells accordingly. The sum of the products is the square whose root was required. If .v had been supposed + the sorts would have been contrary, the reason of which is phiin. CHAP. IV. On Surds. Containing five sections. Sect. I. On Addition and Subtraction. To find the sum or difference of two surds; \/a and \/b for instance. Rule. Call a + b the greater surd ; and if « X /> is rational call ^\/ab the less surd. The sum will be \/(« + Z>4-2\/«Z>) *, and the difference \/{a-\-b — ^\/ab). If rt X 6 is irrational the addition and subtraction are impossible. Example. Required the sum of y/Q. and \/S ; 2 + 8 == 10 the greater surd. 2X8 = 16, \^\6—4:, 4X2 = 8 the less surd. 10 + 8=18 and 10—8 = 2. v/i8 then will be the sum and \/2 the difference. If one of the numbers is I'ational take its square and proceed according to the rule, and this must be attended to in multiplication and division, for on a number square with a number not square the o))cration cannot be performed. Another Rule. Divide a by b and write \/ j in two places In the first j)lace add 1, and in the second subtract 1 ; then we shall haxex/ ((\/ - + l)',X b) — \/a + s/b and \/ ((y J~ ^)''>^b)^z\/a— V b. If r- is irrational the addition can only be made by writing the surds as they are, and the subtraction by writing the greater number + and the less — . * Fax ^{a + b±i2^ab)z=y/a-±.^b. c 2 20 INTRODUCTION, Sect. II. On Mull iplicnl ion. Proceed according to the rules already gi\ en ; but if one of the factors has numbers as dirliems or dinars, take their squares and go on with the operation. Exanipk. Multiply \/^-\-5 by \/2 + v/3 + v/8. As 5 isof the square sort take its square, and arrange in a table thus : v/2 v/3 v/8 v/3 v/6 ^9 \/24 v/£5 ■v/5o \/7-5 \/200 Product 3 + v/o4 + \/-i5Q + \/75 In sammiag the terms of the product, if any square number is found, take its root. Here 9 is found and its root is 3. The rest of the terms being irrational, add such as can be added. v/6+ V'24=v/54. If this last were a square number its root should be extracted. Again, v/5O + v/20O=:V'450. No further addition is possible; the complete product therefore is 3 + V'54' + v/450+ v^75. Another rule to be observed is, if any of the terms which compose the factors can be added, take their sum and write it in the table instead of the terms of which it is formed. Thus in the last example v/2 and \/8 may be added. Write V^3 8 which is their sum in the table, and we shall have JNTRODUCTI©N. 21 v/3 v/18 v/3 v/9 \/j-* x/25 v/7.5 \/4jO 3 + v/7o+v/J4+v/450 and the result is the same as before. Another Example. Multiply \/3 + \/i2o by \/3+v/l2 — 5. Instead of \/3 and v/12 write their sum \/'2.7. Take the square of 5 it is 25, and this is negative notwithstanding the rule which says that wliether the root is negative or affirma- tive the square shall be affirmative. Here the square must be of the same sort as the root. Multiply \/t7—V^5 by n/3+\/25. +n/27 — v/25 + v/3 +v/8l — v/75 + \/2o + v/675 -n/625 -I6 + V/3OO \/81 =9 and \/6!£5=25. 25 being negative and 9 affirmative their sum is — 16, and the sum of + v/675 and - v/75 is + n/SOO. Therefore (\/3+\/25)X {y^ + ^n-5) =-l6 + v/300. INTRODUCriON. Sect, III. On Division. Divide tlie dividend by the divisor, and if tlie quotient is found without a re- mainder the division is complete. When this cannot be done proceed as follows : When in the divisor there .are both affirmative and negative terms, if there are more of the former make one of them negative ; if more of the latter make one of them affirmative. ^Vhen all the terms are affirmative make one negative, and when all are negative make one affirmative. When the number of affirmative terms is equal to that of the negative, it is optional to change one of them or not. Multiply the divisor (thus prepared) by the original divisor, and add the pro- ducts rejecting such quantities as destroy each other. Multiply the prepared divisor by the dividend, and divide the product of this multiplication by that of the former the result will be the quotient required. Example. Let the dividend be that which was the product in the first example under the rule for multiplication^ viz. 3 + \/5^ + \/i50-\-s/75, and the divisor v/]8+\/3. f =25,^^ = 25,3-9,1 =3, f|=3, v/25=5, the quotient then is 5 + v^S. Jmther Example. Divide v/9 + v/54 + v/45() + \/75 by 5 + \/S. Make x/3 negative, and multiply 5 (^or v/25) — \/3 by the divisor v^25 + v/S. + ^^25 -v/3 + '/3 +v/75 -v/9 + v/25 +\/625- ->/75 I.MTRODUCTION. 23 v/75 occurring twice with opposite signs is destroyed. \/6t5—25, %/i)r: — 3 '25 — 3=22=:\/484. Multiply \/Q5 — \/3 by the dividend and we have + v/9 +\/54 +V/4JO +V/75 1 -v/3 -v/'27 -v/lG'2 — \/l350 — n/225 + n/2j + v/22J + n/],:350 + / 11250 +v/l875 Here \/-25 and \/\350 are rejected. Find the sum of \/27 and ■v/1875 in thJ-^ manner, 1S75 + 27 = 1902, 1875 X 27 = 50625, v" 50625 — 225 225X2:^450, 1902 — 450=1:1452, 1/1452 ^in/I 875 -•v/27. Next find the sum of v/l62 and v/l 1250. 162 + 11250—11412, 162 X11550IZ 1822500, v/l 822500= 1350, ]350X2=:2r00, 1 1412 — 2700:^8712, v/S712=:v/'l 1250— v/162. By the multiplication of the dividend we have found v/l 45 2 and v/8712. Divide these by v/484 which was the result of the multiplication of the divisor, and we shall have v/lS, and v/3 for the quotient required. If \/3 is retained as correct, and ■/18 is considered as incorrect, instead of -v/l 8 other numbers may be found by the following rule *. Divide the incorrect number (meaning the number under the radical sign) by any square number which will divide it without a remaindei', and note the quotient. Divide the root of that square number into as many parts as there are numbeis required. Take the squares of these parts ; multi])ly them by the quotient above * To resolve ^a into several parts, divide a by any square iS and let b be resolved into as iHaiiy parts, c, d, e, &c. as may be required. Tlicni/a=; Y^|^ci)+ ^J (^-^,h) j^ \(~e*}kc. whidi may be ]>roved by adding the quantities. 04 INTKODUCTIOW found, and the roots of the several products will be the remaining parts of the quotient required. ~ z= o^ v/9 = 3, 3=1 + 2, r =: 1, 2' = 4, 1 X 2 =: 2, 4 X 2 = 8. \/2 and v/8 are the remaining parts of the quotient. Sect IV". On the Sgiiaixs of Surds. Multiply the surds by themselves. — (Here follow examples). — The squares arc found by multiplying the surds in the common way. Sect. V. On finding the Sgnare Roots of the Squares of Sards*. " If the square is of one surd or more, and I would find its root ; fust I take the " square of the numbers that are Avith it, and subtract these squares from it. " Accordingly after subtraction something may remain. I take the root of what- " ever remains, add it in one place to the original number, and in another sub- " tract it from the same. Halve both the results, and two roots will be obtained. " I then re-examine the squares of the surds to knoM' whether any square remains * Leta+ \/b+ \/c+^/d, &c. be the square of a multinomial surd, a the sum of the squares of the roots and yi + y'c + \/d + &c. the product of the roots taken two and two. The number of roots being n, the num- ber of terms in the square will be ?/% of which n will be the number of rational terms, and ?;^ — ?i the iiumber of Tt ^ Ti surd products, if we call the double products single terms, ^ will express the number of surd terms, and considering the sum of the rational terms as one term, the proposed square may be reduced to the form (x + ^ + z + &c.) + (Cv/Jti- + 2x/xz + &c.) + ( v.*-- + &c. &c.) where i/x + \^i/ + y/z + &c. is the root of the square, and the surd •^rms of the square arc divided uito periods of n — 1, n — 2, n — 3, &c. as directed in the Beej Gunnit. Supposing * + ^ + £ + &c. := Q ^ + ;j + &c. = R 2 -1- &C. = S &:c. &c. ■/(a* — 4iR ) v- v- v- 2 : v/(r'_ 4j/s)_ = V'.r or v'r. Vy or v's ^—^ ■' = ^/z or &c. and so on. INTRODUCTION'. 2 8+2=10, 8-2 = 6,^ = 5, and - =r 3. All the terms of the square having been brought down, the complete root is Another Example. Required the root of 72 : 72'=alS4, (Y — Q, 5184 — 0=5184, v/5184 = 72, 72+72 = 144, 72 — 72=0, — ^ = 72, - = 0, v/72 then is the root. If instead of one term three terms are required, find them by the rule given in the section on division ; divide by 36 whicli is a square number, i| = 2, y-36-6, 6 = 3 + 2 + 1, 3'-Q, 2'=4, 1^=1, 9X f^^- '3, 4X ^ = 8, 72 1 X — =2; therefore \/72=v/ 18 + \/8 + -v/ 2. If three equal terms had been required, the root of the divisor must have been divided into three equal parts. Another Example. It is required to find the difference of v'3 and \/7. The rule not being applicable to this case, suppose %/? affirmative, and \/3 negative, the square of these numbers is 10— v/84. To determine the root of this, suppose 84 to be positive; 10'= 100, 100—84=16, v/l6 = 4, 10 + 4 = 14, 10—4 = 6, 14 6 . ~ ~7, ~^::-3. ^"Ve have then \/7 and \/3, one of which must be mmiis because \/84 was mimts. Another Example. Whether the root is +v^2+v/3— •v/5 or — \/2 — \/3+s/5 the square will be the same, viz. 10+\/24 — v/40— v/60. Let the root of this square be determined. 10' = 100, 100 — (40 + 60) = 0, v''0=0, 10 + 0=10, 10-0=10,^ = 5,^ = 5. As v^24 remains, 5^ = 25, 25 - 24 = 1, v/1 = 1, 5 + 1 =6, 5—1=4, ^ = 3, ^ = 2. If 24 + 40 is sub- . tracted from 100 there remains 36, \/ 36 = 6, 10 + 6 = 16, 10—6=4, — = 8, - = 2. As v/60 remains 8^=64, 64— 60=4, \/4=2, 8 + 2=10, 8-2=6,— zz5,~ = 3 If 24 + 60 is subtracted from 100 there remains 1 6. ■s/l6=4, 10+4 = 14, "14 6 10-4=6, — = 7, - = 3, v/40 yet remaining, 7' = 49, 49-40=9, \/9zsz3, 10 4< 7 + 3=10, 7-3 = 4, Y = 5, - = 2. The terms of the root are v/2 and IN'TRODUCTIOy. 27 \/3 and v^5. If 2 and 5, or 3 and 5 are both negative or l)oth aflirniative the oper- ation will be the same ; the only difference will be in the signs. " If the root consists of one term* only, its square will be of the kind of " number; if of two terms, its square will be number and one surti ; if the root " has three terms, the square will have one number and three surds ; if it has four, " the square M'ill have one number and six surds; if five, one number and ten " surds ; and if six, one number and fifteen surds. The rule is, add the numbers " in the natural scale, from 1 to the number next below that which expresses " the number of terms in the root, the sum will shew the number of surds. For " the use of beginners is annexed a table in which the first column shews the " number of terms of the roots; the second column shews the number of surd "terms in the squares; and the third the number of rational terms in the " squares, from 1 to 9. No. of terms in the root. IVo. of surds in the square. No. of rational terms in the squares. 1 0 2 1 3 s 4 6 5 10 6 15 7 21 8 28 9 36 ' * The number of surd terms in the square being — — - , is = the sum of the numbers in tb€ natural scak from 1 to the number next below n, T> 2 :2S INTRODUCTION. " For mimbeis consisting of more terms than 9 the number of surds in the " squares may be found by the rule which lias been given. If in the square there " are three surd terms, first subtract two of them from the square of the numbers " and afterwards subtract the third. If there are six surds, first subtract 3, then " 2, and so on ; if there are 10 surds, first subtract 4; if 15, first 5 ; if 21, first 6; " If 28, first 7; if 36, first 8 ; and in general the number of surds of the square " will be found in tiie table in the column of roots next above the number "of its root. If they are not subtracted in the regular order, the result will be " wrong. The test of the operation of this : if either of the two numbers found " by tlie rule is multiplied by 4, and the number which was subtracted from the " square of the rational term is divided by the product, the quotient will be the " other number found, Avithout any remainder. If either of those two numbers " is a correct term of the root, and the other the sum of two roots, the least, or " that which is the correct term, M-hether in number it be more or less than the " number of the sum of two roots, must be multiplied by 4, and every quantity " that has been subtracted must be divided by the products, the quotient will be " the numbers of the required roots from the second number. If, after this divi- " sion, there is any remainder the operation is MTong. " The squares of all moofrid numbers* are made up either of rational numbers " alone, or of rational numbers and surds, as has been seen in the examples of the " section on squares. " If a surd occurs there must be a moofrid number with it, otherwise its root "cannot be found. If a surd is divided into two: — For example, if \/l8 is " divided into v/2 and ■\/8, its root will have one term more than it would have " had regularly ; and if two surds are united the root will have one term less. " These two operations of separation and union must be attended to and applied " whenever they are possible." Example. Required the root of 10 + v^32 + \/24 + \/8. From the square of 10 which is 100, subtract any tM'o of the numbers under the radical signs, and the remainder will be irrational: the case is, therefore impossible. If we proceed contrary to the rule, by subtracting at once the three terms from 100, we shall 1 /^ t have 56 the remainder, then v'36 = (), 10 + 6= 16, 10—6=4, — = 8, - = 2. "We * Moofrid means simple as opposed to compound, but in the language of this science it is generally used to r by b and call the remainder .r. ^Multiply a by the value " of .r and to the product add c. Divide the sum by b and the quotient will be " ]/ without any remainder. And if to the first remainder we add a again and ** again, and to the second remainder b as many times, we shall have new values " of .r and 3/. " This rule is applicable only when the number of quotients is even; when it is * Tlie rules given in this chapter are in effect the same as those which have been given by the modern European Algebraists for the solution of indeterminate problems of the first degree. Compare them with the process by continued fractions. 30 INTRODUCTION'. " odd proceed as follows. Having pcifoniied the operations diiected above, siib- " tract the value oi' y from a and that of .r fjoin k If a number cannot be found '' to divide a, b, and c, without a remainder, but a number can be found to divide " fl and c without a remainder, (supposing the reduction of these two instead of " that of the three which was directed b_y the foregoing rule) .v will be bruught " out right and v/ wrong. To find t/ right, multiply its value now found by the " divisor of a and c, and the product will be the true value of 3/. If c and b only " can be reduced by a common divisor, the value of ,r must be nmltiplit-d by the ' common divisor, and the quotient M'ill be the true value of,!. ^Mieu c is — sub- " tract the value of .r from b, and that of j/ from a.'' " If the subtraction is possible let it be done, and the question is solved ; if it " is impossible suppose the excess of the subtrahend above the minuend to be " negative. Multiply the minuend by a number, so that the prouuct may be " greater than the negative quantity. From this product subtract the negative " quantity, and the remainder will be the number required. " ^Vhen ah — the same rule is to be observed ; that is, subtract the values of .p " and 1/ from b and a. If c is + and greater than b reject b, and its multiples from " c till a number less than b remains. Note the number of times that b is rejected " from c; if there mIU be no remainder after rejection it is unnecessary to reject. '' Go on with the operation, add the number of rejections to the value of j/ and " the sum will be its true value. The value of a: will remain as before. If cis — *' subtract the number of rejections from the value of j/. If a and c are greater " than b reject b (or its multiples) from both; call the two remainders a and c " and proceed ; .r will come out right and_y wrong. If there is no augment, or " if c divided by b leaves no remainder, ,v will be = 0, and 3/ the quotient. If '• the numbers are not reduced, but the quotients are taken from original num- " hers, s and 3/ will always be brought out right. If the numbers are reduced, " .r and 3/ will be brought out right only when both are reduced, and but one of " them will be brought out right when both are not reduced." Example, a ■=. 2'21, cs=z65, b =. 195, dividing these numbers by 13 we have, a' = 17, c' = 5, b' — 15. Divide 17 by 15 (as above directed) continu- ing the division till the remainder is 1. The quotients are 1 and 7, write these in a line with c' below them, and 0 below c, thus : Multiply 5 by 7 the product is 35, add 0 the sum is 35. Mul- tiply 35 by 1 the product is 35, add 5 the sum is 40. The two last numbers then are 40 and 35. From 40 throw out 17 twice, 1 40 7 35 6 0 INTRODUCTION. ^j () leinains ; from 35 throw out 15 twice, 5 remains ; therefore .v = 5 and v = G. '221 X 5 + 65 ^ i^j.«_oo- 1 £• 1 Yg^ = o- 17 + 6 := £3 IS a new value of _y, and 15 + 5 = 20 a corres- ponding value of .r, C X 17 + 6 = 40 is another vahie ofy, an dsx 15 + 5=35 a value of ^r. In like manner we shall have 3X17+6=57 and 3X15+5=50 new values of j/ and .v, and so on without end. Another Example. fl'3:100, b — 63, czzyo ; c heing + or -. Although in this case 10 is a common divisor of a and c, yet as the reduction would o-ive a wronff value of y, \vn\.Q a, b and c as they are, and proceed. We find the quotients ], 1, 1, 2, % 1. Arrange them in a line with c below the last, and 0 below c, in this manner: 1 1 1 2 ft 1 90 0 We have then 1 X 90 +0 =90 2 X 90 + 90 = 270 2 X 270 +90 = 630 1 X 630 + 270 = 900 1 X 900 + 630 = 1530 1 X 1530 + 900 = 2430. The two last numbers are 1530 and 2430, divide the former by 63 and the latter by 100; the remainders are 18 and 30, therefore .rssrlS and y = 30 100 X 18 + 90_3p 63 By another method. Divide 100 and 90 by 10, then c' = 13, b :=. 63, c' = 9* The quotients are now found 0, 6, 3, write them in a line with c' and 0 below • we have 3X^ + 0 =: 27 6 X 27 + 9 = 171 0 X 171 + 27 = 27. 32 INTRODUCTION'. The two last numbers are 27 ^"'^ ^71. From 27 thiow out 10 twice, 7 re- mains; from 171 throM' out 6'3 twice, 45 remains. The number of quotients being odd, subtract 45 from 63, the remainder 18 is the value of .r. 7 sub- tracted from 10 gives 3 for 2/, which is not the true value. To find 3/ correct, multiply 3 by the common divisor 10, the product 30 will be the true value ot'j/. Another way of solving the same question is this, find a common divisor of b and c, for example, 9- Dividing b and c by 9 we have a = 100, b' zz 7, c'~ 10. Perform the division and arrange the quotients in a line with c and 0 below, the quotients will be found 14 and 3, then 3 X 10 4- 0 = 30 14 X 30 -I- 10 = 430. From 430 throw out 100 four times, 30 remains. Here we have found a true value of ^ and a wrong value of t. Multiply 2 by the common divisor 9, and the product 18 is the true value of .r. This question may also be solved by first taking a common divisor of « and c, and afterwards a common divisor of b and c, as follows : Reducing a and c we have a = 10, c^ — 9, and b =. 63. Reducing b and c we have a = 100, c' = 10, b' = 7. Unite the reduced numbers thus; a zz 10, b' = 7 ; biit c having undergone two reductions *, take the difference of the numbers arising from the tM'o operations ; then a = 10; b' =-7, c' — 1, divide and arrange the quotients with c' and 0, as above directed, and we shall have 2X1+0 = 2 1 X 2 4- 1 =r 3. 3 and 2 are now found for .r and y, but they are both wrong, for c was re- duced both with ^ and «. 2 must be multiplied by 9 the common divisor of i and c, and 3 must be multiplied by 10 the common divisor of a and c; the true values will be i' =. 18, j/ rz 30 ; and new values of ^ and x may be had by adding a and b again and again to those already found. — dr - i^ — - * Let — ^p- =7, divide a and c by p, then £__£ = ^^ -(Thence — ^ ^ = x, now divide 6 and - o b p a p X — M- — by y, then ^ " ~ a ' '^^''"'E '^* differencs is only true in this case, liecause pq — c, and p — y = ^r INTRODUCTION. 33 What has been said is applicable only when c is -f , When c is — , subtract 18, which is the value of a-, from 63, the remainder is 45 ; subtract 30, which is the valueof 3/ from 100, there remains 70. We have in this case >r = 45, j/ = 70, By adding a and ^ as above, new values of ^r and j/ may be found. Afiother Example. Suppose c i= — 60, h — \3, and c r: 3 + or — . Without making any reduction, divide, and place the quotients with c and 0 as before, we have IX 3 + 0 = 3 IX 3 4-3= 6 IX 6 + 3 - 9 IX 9 + 6 = 13 4 X 15 + 9 = 69. The last numbers are 69 and 15. From 69 throw out 60, 9 remains ; from 15 throw out 13, 2 remains. The number of quotients being odd, subtract the value of .r from 13, and that of y from 60, the remainders are 11 and 51. As 60 is — the subtraction must be repeated, by wliich means we have as before s — 9, and y •= 9- If c is — subtract the value of .r from b and that of y from a, and we shall have again ,t' :r: 1 1 and 3^ =: 5 1. Another Example, a — \%, b — \\, c — — \0. Divide and arrange the quo- tients as before, we have 1 X 10 + 0 n 10 ^ 1 X 10 + 10 = 20 ^ 1 X 20 + 10 = 30 1 X 30 + i;0 = 50. From 50 reject 18, and from 30, 11 ; the remainders are 14 and 8. c being — • subtract 8 from 11 and 14 from 18 ; whence x r: 3 and j/ = 4. Another Example, a zz 5, b =: 3, c zz '-23. Proceeding as before, we shall have 1 X 23 + 0 z: 23 1 X 23 4- 23 = 46. As 3 can be rejected but 7 times from 23, reject 5, 7 times from 46, the re- mainders are 2 and 11. If c is — subtract 2 from 3 there remams 1, ana ll from 5 there remains — 6. Here twice 5 must be added to — 6, the sum 4 is the value of j/ : and that the numbers may correspond add twice 3 to 1 ; the sum 7 is the value of .r, If c is greater than b, reject b from c. Throw out 3 seven E 34 INTRODUCTIOK. times from 2.3, there remains 2, Make c' = 2 and place it with 0 under the line of quotients, we find 1 X S! + 0 rr 2 1X2 + 2=4. 2 is the true value of .r, and 4 which is found for the value of j/ is wrong. Add 7 the divisor of c to 4, the sum 11 is the true value of ?/. If c is — subtract 2 from 3, and 4 from 5, and we shall have 1 for the value of .r which is right, and 1 for the value of 3/ which is wrong. Subtract 7 from the value of j/, the difference is — 6 ; add twice 5 to — 6, and we shall have 4 the true value of?/. That the numbers may correspond, twice 3 must in like manner be added to 1, and 7 will be the true value of x. Another Example, a = 5, b = 13, c =: 0, or c — 65 ; the quotients are 0, 2, 1,1; place them in a line with c and 0 below, we shall have 1X0 + 0 = 0 1X0 + 0 = 0 2X0 + 0 = 0 0X0 + 0 = 0. Add 5 to 0, which stands for the value of j/, and 13 to that which stands for the value of .i^, we have then j/ = .5 and .r =13. In the second case «=5, ^=13, c= 65. As b measures c, .z' will be found = 0 and y ■=. 0. To the value ofj/ add 5, which is the number of times b is rejected from c, and this will give a correct value of y, for -— = 5. Adding 13 to 0 which is the value of j7 13 ° X, we shall have x = 13, and adding 5 to 5 which is the value of j/, 3/ = 10, forl>Lll±i2 = 10. 13 Another method is to suppose c = 1, and proceed as above directed. Multiply the values of x and y, which will be so found by f, rejecting a from the value of?/ and b from that of .?^, the remainders will be the numbers required. Example, a = 221, ^= 195, czz65; dividing these numbers by 13, their common divisor, we have a' — 17, h' -=: 15, c — 5. For 5 write one, and finding the quotients as above, arrange them with 1 and 0 below, then 7X1+ 0.= 7 1X7+1=8 Multiply 8 and 7 by 5, the products are 40 and 35 ; rejecting 17 twice from 40. INTRODUCTION. 3J and ]5 twice from 33, the remainders are 6 and 5-, whence .r rr 5 and ^ ir 6. Ifcis — subtract 7 fi'om 15 and 8 from 17, 8 and 9 remain. Midtiply these numbers by 5, the products are 40 and 45; 15 and 17 being twice rejected, .1'= 10 and J/ — \l. By subtracting 6 from 17, and 5 from 15, tlie same numbers will be found. " Know that the operation of the multiplicand is of use in n^any examples*, " as, if by the rule I shall have brought it out and any one destroys it, and some " remains ; by the operation of the multiplicand, I can determine the numbers " which have been destroyeqj from that M'hich remains. " In the operation of the multiplicand of a mixed nature, the multiplicand is •■' of another kind, and it is called the multiplicand of addition, and that relates " to determining the value of an imknown number, which being multiplied by " a known number, and the product divided by a known number, tliere will re- " main a known number: and again, if the same unknown number is multiplied " by another number, and the product divided by the former divisor, the re- " mainder after division will be another number. Call the numbers by which the " unknown is multiplied the multiplicand, and that by which it is divided the " divisor, and that which is left after division the remainder. Here then are two " multiplicands, one divisor and two remainders. The method of solution is as "follows: add the two muL^licands together and call the sum the dividend. "Add the two remainders and call the sum the augment negative; leave the " divisor as it is ; then proceed according to the rules which have been given: " but the values of .r and j/ must be subtracted from b and a, x will be found " right, and j/ wiong." Example, a = 5, c n 7, c( i= lO, c rz 14, i iz 63, -7- := w + c, and -1. =: " b = z + c, A + fl = 15, c + c r: 21, we have now a' = 15, b' =z 63, c' = * At this place Mr, Burrow's copy has " and besides this it is of great use in determining the signs and " minutes and seconds." And in the margin there is an example by the same commentator, apparently thus : " I give an example which comes under this rule; a star makes 37 revolutions of the heavens in 4i) days and " nights ; how many will it make in 17 days?" Then the writer goes on to say the answer is 12 rev., 10 s,, 1°, 13', 28" ,Sj, which is got I suppose by proportion. Now he adds, " if all this were lost except ^\ it might be re- stored by the rule. He then gives the equation *g~- =^ from which x is found = 23 and 5- = 2S", go x' 23 then from — — =:>/' he finds a' and/, and so on till the whole is had. E 2 36" IJfTRODUCTIOW. f». _21,wefind0and4j^ ^^ + ^ ^ ^^, 7 - 5=2, 2a - 21 =7, 5-2 = 3=3/ wron"-; 21 — 7 = 14 =>»' right; for multiplying 14 by 5 tlie product is 70, which being divided by 63 leaves the remainder 7 ; and multiplying 14 by 10, and dividing the product by 63, the remainder 14 is obtained. CHAP. vr. " On * the operation of multiplication of the square; and that relates to the " kiiOAving of a square, such that when it is multiplied by a number, and to the " product a number is added, the sum will be a square. " In this question then there are two squares, one less and the other greater, " and a multijilicand and an augment. From the multiplicand and augment "known, the two unknown squares are to be found. The method of solution " is this : Assume a number and call it the less root ; take its square and mul- " tiply it by the multiplicand, and find a number which when added to it or " subtracted from it will be a scjuare ; then take its root and call it the greater ** root. Write on a horizontal line these three, the less and greater roots, and the *' number which was assumed as the augment. And again write such another " line under the former so that every number may be written twice, once " above and below ; then multiply crossways the two greater roots by the two " less ; then take the sum of the two and call it the less root ; then take the " rectangle of the two less roots and multiply it by the multiplicand, add the * The rules at tlie beginning of this chapter for the general solution of av'+b^^' are, as they stand in the Persian, to this purport : Find A/^+g =S''» where /, g, and g may be any numbers which will satisfy the equa- tion. Make x' -^/g + /g and y = Aff+gg, and 0' = (30. Then ax''^ + 0' =:y ; and making ;i" = x'g -j- y'/, and y" = ax'/+ y'g, and 0'' = g'|3^ or x'' = x'g — y'/ and y" = y'g — Ax'f, we have a;ic'" + 0'' = y" '. If (3 ' 7 B then — = b, and if 0" ^ b, then B"p'= b, but in the first case the values of a"and^" must be divided, and in the second case midtiplied by/i. In this way, by the cross multiplication of the numbers, new solutions are had for ax^ + b-^ y'^. When g =: 1 and 3' = B the rule is the same as Fermat's proposition, which first was applied in this manner by Euler for finding new values of x and y in the equation ax* -^b =z y^. (See the investigation of this method in his algebra.) If A;t'+ b =? 3;^, then « = -^ , r being any number; this expression is true only when B = I. In that case ^\ ^^ ) +'— (^V^~) which is the same as Lord Brouncker's soktion of Fermat's problem. • INTRODUCTION. 37 m " product to the rectangle of the two greater roots, the result will be the greater " root, and tlie rectangle of the two augments will be the augment. " And to find another square in the same condition write on a horizontal line " the less root and the greater root, and the augment, which have been found, " below the less and greater roots, and the agument which were assumed, Per- " form the same operations as before, and what Mas required will be obtained. " And another method in the operation is, after multiplying crossways to take " the difterence of the two greater roots, it will be the less root. And having " mulfiplied the rectangle of the two less roots by the multiplicand, note the pro- " duct; then take the rectangle of the two greater roots, and the difference of *' these two will be the greater root. " And know that this augment, that is, the augment of the operation, if it is " the same as the original augment, is what was required. Otherwise, if it is " greater, divide it by the square of an assumed number, that the original aug- " ment may be obtained. If it is less multiply it by the square of an assumed " number, that the original augment may be obtained. And that they may corres- '* pond in the first case divide the greater and less roots, by that assumed number, " and in the second case multiply them by the same number. " And a third method is this : Assume a number and divide its double by the " difference of the multiplicand and its square, the less root will be obtained. " And if we multiply the square of it by the multiplicand, and add the augment " to the result, the root of the sum will be the greater root. Example*. " What square is that which being multiplied by 8, and the pro- " duct increased by 1, will be a square. Here then are two squares, one less and " one greater, and 8 is the multiplicand and 1 the augment. Suppose 1 the less " root, its square which is 1 we multiply by 8 ; it is 8. We find 1 which added " to 8 will be a square, that is y. Let its root which is 3 be the greater " root. Write these three, that is to say, the less and greater roots, and the * Tofind*and vsothatS*' +l=j^=. Siippose/=l, andS/«+g=Q. Let @ = 1, tlu-u 8/'+ 1=S=3* ; 3X1+3X1 = 6=*. 1X1X8+3X3=17=3', ixi = l the augment; 1 being the original augment there is no occasion to carry the operation farther, b x 36 + I =289 = 17 \ For new values, 3 X 6 +- 1 x 17 = 35 = *, 1 X 6X8 + 3x17= 99 =_j'. 1 X 1 = I the aiignieat. 8 X 35 ' + 1 = 9S01 =99^. In like manner more values maj be found. 3S INTRODUCTION. " augment on a horizontal line ; and write these numbers below in the same " manner, thus : Less Greater Augment 1 3 1 1 3 1 " Multiply the two greater roots crossways by the two less, it is the same as it " was before ; add the two, it is 6 ; and this is the less root. Take the rectangle of " the two less roots, it is 1. Multiply it by 8, it is the same 8 ; add it to the rect- " ano-le of the two greater roots, that is 9 ; it is 17, and this is the greater root. *' Take the rectangle of the two augments ; it is I. As it is according to the ori- ** o'inal there is no occasion to work for the original augment. The square re- " quired is 36, which multiplied by 8 is i288 ; adding 1 it becomes 289, and this *' is a square whose root is 17. Again, to find a number under the same con- " ditions. Below the less and greater roots and first augment, write the less and ** greater roots and augment M'hich have been obtained by the operation, thus : Less Greater Augment 1 3 1 6 17 1 " Multiply crossways the two greater and the two less roots, that is 3 by 6, and 17 by 1, it is thus: Less Greater Augment 1 18 1 6 17 1 m INTRODUCTION, 39 " Add the two greater roots; it is 3.5, and this is the less root. Take the rect- " angle of the two less roots ; it is 6. Multiply it by 8, the multiplicand ; add the " product \vhich is 48 to the rectangle of the two greater roots 3 and 17, which "is5j', it is 99, and this is the greater root. Take the rectangle of the two " augments; it is the original augment ; for when the square of 35, which is 1225, " is multiplied by 8, it will be 9800 ; adding 1 it will be a square, viz. 9801, the " root of which is 99- In like manner if the two roots and the augment are " written below the two roots and the other augment; like 6 and 17 and 1, and " the operation is performed we shall find what we require, and another number " will be obtained. J nother Example. ''What square is that M'hich being multiplied by 11, and " the product increased by 1, will be a square *? Suppose 1 the less root, and " multiply its square, which is 1 by 11 ; it is 1 1. Find a number which being " subtracted from it, the remainder will be a square : Let the number be 2 ; this " then is the negative augment, and 3 which is the root of y is the greater root. " Write it thus : Less Greater Augment 1 3 o 1 3 2 * 11j:»+ 1=5/'. Suppose/= 1 and I)/' — 0= Q. Lete = 2. 11x1 — 2=9=3». 3x l+3X 1= 6=.t;. 4 IX 1 X n+3x 3 = 20=y. — 2 X — 2 = 4- 1. — =z 1 the original augment. Therefore — = 10 = ^. - := 3 =: x. For 1 1 X 9 + 1 =: 100 = 10'. Anotherway. Suppose/= 1 and ll/»+l3= Q. Let 3 = 5. 1 1 x 1 +5 = 16 = 4-. J x 1 + 4 x I = 8 = j. 1X1X11 + 4X 4=27=> 5 x 5 = 25 the augment. - =.v, — =„. ii x f- V + 1 = --= (I?)* 3 5 \ b J 25 \ 3/ For new values, 10 x j + 3 X ^ = ^ = *• 3 X ^ X H + 10 x y = ^-|^ =i^. 1X1 = 1, the augment. .. ^ /''^'V-. I - S85l5g /'531V J, . , ,, , SI 80 1 2-0 24 6 11 X (—T') + 1 = — T- — = \~7-) By the second method, = - = x. 11 x = — = v \ 3 / ^3 ^ b J ' 555 5 53''' 1 the augment. 1 1 x (- j + 1 = — = (- j . In the same way other values may be found. 40 INTRODUCI'ION. " Multiply crossways, and add the two greater, it is 6 ; and this is the less "root. Take the rectangle of the two less roots; it is 1. Multiply by 11; it is 11. " Add it to the rectangle of the two greater which is 9; it is 20, and this is the " greater root. Take the rectangle of the augments, it is 4 affirmative. Now " we have found a number such that when we divide this number by the square " of that, the quotient will be the original augment. We have found 2 and per- " formed the operation ; 1 is obtained. And we divide the greater root which " is 20 by 2, 10 is the greater root. And we divide the less root, 3 is the less " root. For if the square of 3 which is 9 is multiplied by 11, it will be 99, and " when we add 1 it will be 100, and this is the square of 10 which was the " greater root, " Another method is, suppose 1 the less root, and multiply its square by 11, " it is U. We find 5 which being added to it will be a square, that is 16; its root " which is 4 is the greater root, thus : Less Greater Augment 1 4 5 1 4 5 " After multiplying crossways, add the two rectangles; it is 8, and this is the " less root, and the rectangle of the two less ; which is 1 we multiply by 11, it is " II ; add it to the rectangle of the two greater which is l6; it is 27, and this is " the greater root. And from the rectangle of the augments, 25 augment is " obtained. We have found an assumed number 5, such that when the aug- " ment is divided by its square the quotient will be 1. And for correspondence " we divide 8 by 5 ; 8 fifths is the less root. And we divide 27 by 5 ; 27-fifths " is obtained for the greater root. For multiplying the square of 8-fifths, that is " 64 twenty fifth parts by 11, it is 704 twenty fifth parts ; add 1 integer that is *' 25. It is 729 of the abovementioned denomination. And to find other " numbers under the same conditions, write the two roots and the other augment " below these two roots and augment, and that is on the supposition of 3 and 10 " and 1, which were obtained before, thus : INTROPUCTION. 41 Less Greater Augment 8 5 27 5 1 1 3 10 1 "After multiplying crossways, add the two rectangles, it is l6l-fiftlis, and " this is the less root. Multiply the less rectangle, which is 24-fiftiis, by 11, it is " 264; add it to the rectangle of the two greater, which is 270-fitths, it is 534- *' fifths, and this is the greater root. Take the rectangle of the augments, it is 1. " The operation is finished, for multiplying the square of l6l fifths, which is "25921 twenty-fifth parts, by 11, it is 285131 twenty-fifth parts ; add 25 that " is 1 integer, it is 285156, and this is the square of 534-fifths. " And by the second method. After multiplying crossways it is Sl-fifths and " 80-fifths ; the difterence is 1 -fifth, and this 1 -fifth is the less reot. Multiply the " rectangle of the two less, which is 24-fifths, by 11, it is 264-fifths ; and the rcct- '* angle of the two greater is 270-fifths. Take the diflFerence ; it is 6-fifths ; and " this is the greater root; and 1 integer is the augment. The square of 1 -fifth, " which is 1 twenty-fifth part, multiplied by 11, is 11 twenty-fifth parts; add 25, " it is 36 twenty-fifth parts, the root of which is 6-fifths; and in like manner " any number which is wanted may be obtained. Ed-ample. " Let the first question be solved by the third method * : Suppose " 3 the less root, and take the ditference of its square and the multiplicand which " is 8 ; it is 1. Divide twice 3 by 1 ; it is 6 ; and this is the less root. For niul- " tiplying the square of this, which is 36, by 8, it is 288 ; add 1, it is 289, and " this is a square, the root of which is 17, and this is the greater root. " Another method is what is called the operation of circulation f. To bi ing * To solve the first ciuestion by the third metliod. Suppose r = 3, ' =6=;», S x 36 + I z= i!89 =s U-. 17 =y. t _» ocXJ *° '"^'•s to go round, from . ^^ to go round. This rule is, supposing \x- -J- b =j/-. a and b feeing given to find x and i/ Ijy the operatiou of circulation. Find /, 3 and g so that */' +6zzg'. Support F 42 INTRODUCTION. " out that which is required by the rule of the multiplicand. It is thus : After " supposing the less and greater roots and the augment, suppose the less root the " cHvidend, and the augment the divisor, and the greater root the augment. " Then by the rule of the multiplicand which is passed^ bring out the multipli- " cand and the quotient. If that number by Mhicli the questioner multiplied the " square can be subtracted from the square of this multiplicand, let it be done; " otherwise subtract the square of this multiplicand from that number of the mul- " tiplicand. If a small number remains, well; if not increase the multipli- " cand thus: add the divisor again and again to the multiplicand as before " explained, till it is so that you can subtract the number of the multipli- " cand from the square of it, or the square of it from the number of the mul- " tiplicand. Whatever remains we divide by the augment of the operation of " multiplication of the square, and take the quotient which will be the augment "' of the operation of multiplication of the square. If then we shall ha\ e sub- " tracted the multiplicand from the square, let the quotient remain as it is : and " if we shall have subtracted the square from the multiplicand it will be contrary, " that is, if negative it will become affirmative, and if affirmative negative ; and " that quotient which was obtained by adding the dividend to the quotient, as " many times as the divisor was added to the multiplicand, will be the less root ; -^— — -^ = V and from tlie known numbers/, g, 3, find x and y by the rules which have been given. If x- be > A take X- — A, or if not take a — x'-. If a small number remains it is well, otherwise take nuiltiples ofP, and add them to the found value ofx for anew value, till we have (m3 + x)= — a, or A — {)ii0+x)-: divide this by (3, and if the square has been subtracted change the sign of the quotient. If instead of x tlie value V10 + X lus been used a corresponding value of y, m/+ Y must be taken: by substituting these values as . 1 X" ^ A (w|3 + X)- M A , , . , . r ,,■ follows: Y, orwy + Y = «, and — — or '—--^ = p., we have the solution of this equation A.v'- + (3' = y-. If 5' is neither == a nor to hp" nor to — . proceed as before. Let a/'- + 0' = g'- be a solution f'x' + m of Ajc'- + |3':=y- where/', (3' and ^' are known. Suppose" -;r--- ^ y'; proceed as before, and solutions ^^ n n n will be had for ax"- + 3' =:>"-, and in like manner for ax- + ^ = ^- till g is found = b or Bp- or — . The trutli ofthis is plain, for as x'=~^ ^ and S' = ^- ~"^^■, we have ax'H 0 = a ( ^ ) + "~"— which p p \ p / p i, = (A/^+g)x^+2A^^x + A(g°-g) . ,^^^ ,y.^^^ g.^ 3„d ^,. _ g ^ ^y,^ ,„j therefore a.' + |3' = r'x^ 4- 2 A/;ifX + aV"- ,. , . (gX+A/\- „ rr-i ■ , , , ■ . . "— -r. whicn IS = \^° — — i~j ^^ ya. i ins rule, though in some respects imperfect, is in principle the same as that for solving the problem in integers by the application of continued fractions, which was first given in Europe by De La Grange. INTRODUCTION'. 43 *' and from the less root and the augment bring out the greater root. If then this " augmeut shall have been found a square *, the operation is finished ; for find " a number by the square of which, if we divide this augment, the result will he " the original augment, when this augment is greater than the original augment ; " or otherwise, if we multiply by it, the result \vill be the original augment, in " the same manner as before. And if it is not a square pertorm the operation " again in the same manner ; that is, supposing the less root the dividend, and " the augment the divisor and the greater root the augment ; and work as be- " fore till the original augment or the augment of the square is found. Example. " What square is that which being multiplied by 67, and the pro- " duct increased by 1, will be a square f. Let, us suppose 1 the less root, mul- * I suppose it should be 'Bp- or — ; . I think it likely that this does not form a part of the original rule which P' seems to relate to integer values only, t 6T;i- 4- 1 =y-. Suppose/=; 1 and Z-=. — 3, then 67 x 1- — 3 = n = 6t = S'-, we have now A/-4-g=^«, 1 ^ „ 1 „ c fy- A- e , . I X + 8 vhere/= 1 and 0 = — o, and ^ = S. bupposc ~^ = v ; tiiat is to say, — - — == y, reject twice S from g we have 8 — 2 x 3 = C and work for x and Y in ■ " 7" ■" = "^ • Divide 1 by 3, as directed in tlie last chapter. The quotient is 0, write under it 2 and 0, we find x = 2 and Y = 0. The number of (|U0tients being odd, sub- tract the value of ;t from 3 and that oiy Uamf. . 3 — 2= i = x, 1 — 0 := 1=:y. As 0 was rejected twice from g, add 2 to tlie value of Y. 1 +2 =3 ^ Y we have now x = I and y = 3. As we cannot subtract 67 from 1 = and as a greater number will remain if we subtract 1- from 67, add twice 3 to x for a new value of x, 2x3 + l=7=x, and for a conespondihg value of y add twice/to y. 2xI+3=5 = y. a — x- or 67 7-=: is. IS — - = — 6. As we have taken A — x- we must change the sign of — 6, it becomes + 6 = |2', and \ -z^x we have now AX - + 3'=:^'-, where a = 67, 2'=6,and a^=5, whencey=t!. Since b = I and 3'= 6 we pro- ceed to find 3 = K. Let a/'- + 3'=^'^ where/'= 5, S'= 6, and ^' =41. Make ~±ii — y', we shall 6 find x'= II and y' = -I 1. Subtract /' C times from the value of y', -1 1 — 6 x 5=11= y', and subtract S' the same number of times from the value of x'. -H — 6x6=:5 = x'. a — \'- or 67 5' = 4"- — =: 7 = 5" Ac ' ' 6 ■ -" wg^Iiave takeu.A — x'- the sign of 7 must be changed, and — 7=2", and ll = Y"=;i"; therefore a V- -<- 3' ;:; y""-, and y" ■=. 90, 3' not being = u we must proceed as above. Let a/''- + 3' = g"-, whcrey"=:: 1,3"= 7 and ^" = £)0. Make ' ^ = y", reject 7 twelve limes from 90, 90 — 81 = 6, we sliall find x'= 12, and y"= 19. .Subtract/" from the value of y" and 3' from tliit of x", 18 — 11 = 7=y', 12 7 = 5=*''. The number of quotients in the divi^ion of il-by 7 being odd, subtract tiie value of y" from/' and that of x" from 3", 11—7 = 4 = y", 7 — 5=2 =x". As we cannot, take 67 from 2'^ and as a greater numbej; remains, if we subtract 2- from 67 , add 3" once to x' for a new value of x", 7 -J- 2 = 9 = x"; x"= _ a, or 9= — 67 = i j, 14 __ _ , „ Z:^^ — — ^ — ^- As 3" was rejected 12 times Irom/', 12 must be added to the value of y, 4 -{- 12=; 16 = ■»'■. 44 INTRODUCTION. " tiply it by 67, it is 67. Find 3 the number of the augment, which sub- " tracted from 6? will leave a square, that is 6^, the root of which is 8, and this " is the greater root. 1 then is the less root, and 8 the greater root, and 3 the " augment negative. If we wish to bring it out by the operation of circulation, let " us suppose 1 the dividend, and 8 the augment, and 3 the divisor As rejection " of the divisor from the augment is possible, reject it twice, 2 remains. Sup- " pose this the augment, take the numbers of the line, cipher is obtained. Write " under it 2 tlie augment and cipher. Perform the operation, the multiplicand is " found 2 and the quotient cipher. The number of the line being odd, subtract *' the multiplicand and the quotient from the divisor and the dividend, 1 and 1 " are obtained. As we rejected the divisor which is 3 from the augment which is " 8, add 2 to the quotient, the quotient is 3 and the multiplicand 1. As we '• cannot subtract 67 which is the multiplicand of the operation of multiplication " of the square, from the square of this multiplicand, and if we subtract the " square of this from 67 a greater number remains ; from necessity we add the " divisor, which is 3, twice to the multiplicand 1, it is 7 ; add the dividend to "the quotient it is 5. Subtract the square of 7, which is 49, from 67, 18 re- " mains. Divide by the augment of the operation of multiplication of the " square, which is 3 negative, 6 negative is the quotient. As the s([uare has " been subtracted from the multiplicand the negative becomes contrary ; it is *' 6 affirmative, and this is the augment ; and 5, which was the number of '* the quotient, is the less root. Then bring out the greater root, from the less " root and the augment, and the multiplicand 67, it is 41. Write them in order. " As 6 is the augment of the operation and 1 is the original augment, perform the " operation again to find the original augment : that is to say, suppose 5 the " dividend, and 6 the divisor, and 41 the augment, and perform the operation of " the multiplicand, the multiplicand is found 41 and the quotient also 41. Sub- " tract 5, the dividend, 6 times from 41, the quotient, 11 remains; and subtract " 6 the same number of times from 41, the multiplicand, 5 remains ; take its And as e' was added once to the value of x" add/" to that of v", II + 16 = 27 = y'' = a:''. Now a/'"" ^. 3" = g"'\ because «"'= £7, andS"' = — 2, therefore y"'= 221. Let a/""^ +£'" =:g"'^, where/"'=21, 0'" = — 2, and g'" = 221. Having now found 6", which, multiplied by itself will be the augment of the square, (meaning, Isuppose, = 3/)=) apply the first ruleof this chapter. i""=2/"'V'= U934, ^"'' =g"'^ + A/"''^=016Si, B"'-= i, we find p =: 2 such that —zzassl. Dividing a/"* + e'" = /"^ by p\ we hate ( ^ y) + ^ = (4" )'. and 67 X 5967« + 1 = 48843". INTRODUCTION. 45 *' square, it is 25 ; subtract it from 67, 42 remains. Divide it by 6, the augment, " 7 is tiie quotient, and this is the augment. As we subtracted the square of the " multiplicand from 67, it is contrary ; 7 then is the augment negative ; and 1 1 *' which is the quotient, tlie less root; bringing out the greater root, it is 90. " In this case too the original augment is not obtained. Again, perform the oper- " ation of the multi])h"cand ; 1 1 is the dividend, 90 the augment, and 7 the di- " visor. The divisor can be rejected 1-2 times from the augment ; reject it ; 6 " remains. Take the line, and perform the rest of the operation ; 18 is the " quotient and 12 the multiplicand, thus: 1 1 1 6 18 12 6 "Subtract 11, the dividend, from 18, and 7, the divisor, from 12; 7 and 5 " are obtained, the quotient and the multiplicand. As the number of the line '* was odd subtract 7 from 1 1 and 5 from 7 ; 4- is the quotient and 2 the multi- " plicand. As we cannot subtract 67 from the square of 2 ; and after subtract- *' ing the square of 2 from 67 a greater number remains, add once the divisor " which is 7 to the dividend ; it is 9. Subtract 67 from its square which is 81 ; " 14 remains. Divide by 7 the augment negative, 2 negative is the quotient, " and this is the augment. Again, as we rejected 7 twelve times from the " augment, add 12 to the quotient which is 4 ; it is l6. And a* we added 7 to ** 2 the multiplicand, add 1 1 to 16 the quotient; it is 27, and this is the less root. " Find the greater root; it is 221. As an augment is obtained which, after being " multiplied into itself, will be the augment of the square, we Avrite this line " below that, and multiply crossways in both places. From one cross multiplica- •' tion it is 5967, add these two; 11934 is obtained the less root. And the *' greater root is 97684, and the augment is 4 affirmative. We ha\e found 2 an "assumed number, by the square of which, if we divide this augment of the ** operation, the quotient will be 1, which is the original augment. In like 46 INTRODUCTION. " manner we divide 11934 by 2, 5967 is the less root, and 48842 is the greater " root. Another Example. "What square is that which being multiphed by 61, and " the product increased by ], will be a square*. Let 1 be the less root; 8 is the " greater; and 3 the augment^ affirmative. Applying the operation of the multi- " pUcand, it is thus : Dividend. Divisor. Augment. 13 8 '* Reject the divisor twice from the augment, 2 remains ; and after the operation " 2 tbe multiplicand, and cipher the quotient are obtained. As the line is odd " we subtract cipber from the dividend and 2 from the divisor. It is 1 and 1. " As we rejected the divisor twice from the augment, we add 2 to the quotient. "The quotient is Sand the multiplicand 1. If we subtract the square of the " multiplicand which is 1 from 6l, a greater number remains. We therefore add " twice the dividend and the divisor to the quotient and the multiplicand. The * 6U»+I=y. Let A/H(5=t^ "■here/=I, (3=3, g-=8. Make ^^ ^ ^ z=. v tliat is ' ^ ^ ^ = v, reject |3 twice from g, 8 —2x3=?, we shall find x:=2 and v=:0. The number of quotients in tlje division of 1 by 3 being odd, subtract the valueof Y from/, and that ol x Irom S, 1 — 0 = l=Y, 2—1 =l=x. As & was rejected twice from g add 2 lo the value of v, 1 -(- 2 = y. If we take a — x' a greater number remains ; add twice y to the valueof Y, and twice 13 to that of X. 3 -h 2 X 1 = 3 = V. l-t-2x3=7=x. Take a — x^ C) — 12 7''== 15. Divide by S, —" = 4, which becomes — 4=0', and 3 := y = a;'. Now Aa'- + B' = 3''-, wiience y = 39. Let A/'- + ^' = ^'* where/' =, 5, p' =: i,g'=z39. As|2'isnot= e, we find a number 75:=2, sucli that — = — I. Divide x' and u' by p, and we have _ = - := x^' ^ /", and - =z. '— =:i"="", and — =: — 1 = .J J J r' /> 2 -^ p 2 ^ -^ ' p 5 " 1 $'. As B = -t- 1, apply the first rule of this chapter, S' x P" = — I X — 1 =+ 1 =: B. '-/"&" = 2 x 77 x -p = -^- = x, and|:'^-fr a/'-= ^— j +61 X \,j) z=-~=y, — = -^, -j- = -^ and 61 x (^-^j + 1 ss (-ii-)'. If AX- + u = j/'^ where B = — I, or 61 x- — 1 = 7/-, then ^ =_/",'— = 5', and — 1 =(3. Multiply a/' -}- 0 := g:'' crcsswaxs with a/'- -t- 0 zz: g'^, where/' = — '- ani.] B'zzz + I, in older that we may haveg3'= — I. Thena'=3S05, and^=29718, and b-= — 1 ; 6i x (3S05)-— I = (:9718)'^ For new values, where b = + 1,-fliiiltiply a/^ + $ z= g^ crossways with the vahns of x and j/, which we ha\e ji st found, 61 X 226153980= + 1 =: 1766319049=. If B is — take Uie product of two augments which have unlike signs, and if B is + that of two which have like signs. INTRODUCTION. 4.f "quotient is 5 and the niiiltiijlicand 7- Subtract the square of 7 from 6l ; 12 " remains. Divide by the augment of the operation of multiplication of the " square which is 3 affirmative ; 4 affirmative is the quotient; and after reversion " it is 4 negative ; and this is the augment; and the quotient which was 5 is the " less root; 39 then will be the greater root. As 4 is not the original augment, " we have found 2 an assumed number ; and by its square we divide this augment. " ] the augment negative is the quotient. We also divide 5 and 39 by 2. These " same two numbers, with the denominator '2, are the quotients. As our " question is of the augment affirmative perform the operation of cross multipli- " cation. When we multiply the augment negative by itself it will be affirma- " tive. The less root will be 390 fourth parts ; the greater root 3046 fourth " parts ; and the augment 1 affirmative. Reduce the less and greater roots to " the denominator 2. The less root is IQo second parts ; the greater root 1523 " second parts; and the augment 1 affirmative. And if, for example, the question "was of the subtraction of the augment, the answer would be as above; 5 " second parts being the less root, and 39 second parts the greater root, and l the " augment negative. And besides this, if we would obtain another case, let this " be multiplied crossways with that in which 195 second parts is the less root; " for multiplying affirmative by negative, negative is obtained. The less root " then is 3805, and the greater 29718, and the augment 1 negative; and this is " the answer to the question. " To find another case with the augment affirmative write this below it and "multiply crossways, 226153980 is the less root, and 1766319049 the greater " root, and 1 the augment affirmative. And in like manner Avherever the aug- " ment is required negative, we must multiply crossways two augments of dif- " ferent sorts ; and if affirmative two of the same sort. Rule. " If the multiplicand of the question is the sum of tvv'o squares, and "the augment 1 negative; it maybe solved by the foregoing rules*, and if " wished for, it may be done in another way, viz. Take the root of those tM-o * InA.v-+ B=j/-, if A =//- -1- J-, and us:— 1, a=— and a =- ; for (p- + ?') X [-) — 1 = ^- ) , 48 INTRODUCTION. "squares, and divide the augment by each, the two numbers wliich are found " will both be the less root ; what was required may be obtained from each. Example. " What square is that which being multiplied by 13, when 1 is sub- " tiacted from the product, a square will remain *. 13 then is the sum of 4 and " 9, arid 1 the augment negative. Take the roots of 4 and 9, they are 2 and 3. " Divide the augment by these two, the quotients are - and -, both these are " the less roots. What is required may be had from either. For multiplying " the square of - which is - by 13, it is 13-fourths ; and subtracting from it ], " which is 4, 9-fourths will remain ; and this is the square of ]-. Multiplying ** the square of 7 which is -by 13, it is 13-ninths; and subtracting 1 integer o y 4 1 • • " which is 9, - remains; and this is a square." Here follow solutions of the same question, by the former methods: I omit them because they contain nothing new, and are full of errors in the calculation. Another Example. AVhere 8.r* — 1 =3/* is solved by the last rule, is omitted, because it is immaterial. jinother Example. " What square is that which being multiplied by 6, and *' 3 added to the product, will be a square. And what number is that M'hich " being multiplied by 6 and 12 added to the product will be a square f- The *' operation in the first case is thus. Suppose 1 the less root, and multiply by 6", " it is 6 ; add 3, it is 9 ; and this is a square. And for the second case thus : " Multiply 1 by 6, it is 6 ; and find a number which added to it will be a square ; * ISx'— 1 =y», here a = 13 = 9 + 4 =p2 + 9', p = e, g = 3i x = - and a:= -; for 13 x (~)' — 1=1=(„)'> a„d,3x(iy-I=l = (|/. \ 6x' + 3 =^', and 6*' + 12 =y=. First suppose * = 1 and b = 3, then 6 x 1 + 3 =9 = 3', ^ = 3. Second, 6 X 1 = 6, find S such that 6 + 0 = Q. Let 0 = 3, 6 + 3 = 9 = 3% 3 being not = b, but less than it, find p such that 0/)- = b, ^ = 2, 3 x 2= = 12 = B. Now if 0' = 3, i' = 1, and y =. Z, multiplying ax'- + 0' :=; y'''' by p", we liave Ax'p^ + 0^"^ =zt/-p'', and making x = x'p, y = y'p, and b = S'p'^, we have * = 2, 2/=6andE = 12. ex22+12=6«. INTRODUCTION. 49 " we find 3. As this is not the original augment, but is less, find by the rule " which was given above, a number by the square of which when we multiply " this augment the original augment will be obtained; We have found 2. Mul- " tiply 3 by its square which is 4, it is 12 ; and this is the original augment, " Then that they may correspond, multiply the less and greater roots together ; *' also by tliat number, which is 2. The less root is 2, and the greater 6, and "the augment 12. Multiply the square of 2 by 6, it is 24 ; add 12, it is 36; " and this is a square tiie root of which is 6." Here follows another example where, in A.r^ + b =_y*, b — 7'>, and a =s 6. The solution of this question is like that of the first part of the preceding: jr(in aJ'' + (i =: g'-) is assumed — 5 and (i zz 75. Another Example. * " 300 being the augment the less root is 10 ; its square " which is 100, we multiply by 6, it is 600. Add 300, it is 900; and this is a " square, the root of which is 30. And know that when the augment is greater " you must bring out what you require by the operation of circulation f, that " the augment may be less. And if you wish to obtain it without the operation " of circulation call to your aid acuteness and sagacity. And when you have " fountl one case, and the augment is 1, you may find others without end, by " cross multiplication. For however often you multiply 1 by itself, it will still " be one ; and the less root and the greater will come out different. RiileX- " If the multiplicand is such that you can divide it by a square with- " out a remainder, divide it ; and divide the less and greater roots by the root of '• that square, another nund^er will be found. And if you multiply it by a square •' and multiply the less and greater by its root, the numbers required A\'ill also be *' found. Exaynplc. " What square is that which being multiplied by 32, and 1 added to * fl-t- -4- 300 = ij". Let .r = 10, then 6 X 10- + 300 = 900 = 30- f When 3 is a greater number find &', 0'', &c. less by tlie rule of circulation. Solutions of these problems witliout tlie rule of circulation, are to be had only by trials judiciously made. When one case of a/" -f- 1 =^' is known any number of cases may be found by cross multiplication ; for 1 X 1 = 1. and different values of x and^ will be found at every new step. X I suspect that this is incorrectly translated; the example does not illustrate the rule. Perhaps it should be^ d in Ax» + B = y'', A = \'p"; then a',x» + 1 = / i^ y. If a s= ^, then k'x'- -t- Bp'^ = {i/p)'. ^0 f ^•TKODUCTI0N. " tlie product, will be a square*. The less root then, is -, the square of which ^ ''■ nmltiplied hy 32 M'ill he 8 : add 1, it is ,9 ; and this is a square. If we suppose '■' 2 the less root and divide 32 the multiplicand hy 4, 8 will be the multiplicand ; " and dividing the less root by tlie root of 4 which is 2, 1 is the less root. For "multiplying by 8 and adding 1, it is 9, which is a square, the root ofAvhich *' is 3. link. " If tlie multiplicand is a s(|uarct, divide the augment by an assumed " number, and write the quotient in two places ; and in one place add to it, and " in the otlier subtract from it the assumed number, anil haUe them both ; the " greater number will be the greater root. Divide the less by the root of the *' multiplicand, the quotient will be the less root. E.vample. " What square is that which M'hen multiplied by Q, and 52 added " to the product, is asquare;]:. AVhat other square is that which when multiplied " by 4, and S3 added to the ])roduct, is a square In the first case divide 52 b}- 2, " 26 is the quotient ; write it in two places and add and subtract 2, it is 28 and *' 24 : the halves are 14 and ]2 ; 14 then is the greater root. And diviile the less " number which is 12 by the root of the multiplicand which is 3, 4 is the quo- " tient, and this is the less root: for when the square of 4 which is l6 is multiplied •■' by 9, it is 144; add 52, it is igS, which is the square of 14: and in the *32X*'+l = n- Letx = ~. 32 X {~y + l = 9 = 3K If Ar = 2. ^ = a', ^ = 8,-*= 1. »=I; for S X 1 + 1 = 3'. B , B B ^ + n n 71 '+ If AA-+ B =_y= and A =:p*. take n any number; and we have -— — =:^ ; and — - — = xj for P P B B __ ,(» n B + w' (b ?i-\2 ^BH-«'\* ^ — and =: — ^'^— . But ft' x \ / + B = V — ;:; ) i wlience the rule. 52 2ft 24 J ?jr' + 52 = ^' and ix' + 33^^-^. First y=26, 26+2=28, 26 — 2 = 24. -^ = 14 ;—= 12. ^ = 12 33 14 H, — =4 = jc. 9 X 4' + 52 = 196= 14*. Second, — = 11, 11 + 3= 14, U — 3 = 8, —- = 7 = ^, -=:4, ■;- = 2=;«', 4X2'+ 33 = 49 ^7'. Values of xaQd^/ might have been foundby takingn = 4 in the lirst case, and « = 1 in the second. INTRODUCTION'-. 51 " second case divide 33 b}'' 3, 1 1 is the quotient : after adding and subtracting 3 it "is 14 and 8 : after halving, the greater root is 7. Divide 4 by 2, 2 is the quo- " tient; and this is the less root: for multiplying 4 by 4 and adding 33 to the " product, it is 49; and this is the square of 7. And if at first we divide 52 by 4, " and 33 by 1, what is required will be obtained " Another Example, when the multiplicand is equal to the augment. What " square is that wliich being nmltiplied by 13 and 13 subtracted from tlie [)ro(luct, " and in another case added to it, will be a square *. In the first case suppose " 1 the less root, its square which is also 1, we multiply by 13, it is 13: subtract " 13, their remains cipher, the root of which is cipher. And in the second case " suppose 3 the less root, take its square, it is 9 ; take the diti'erence between it " and the augment, 4 is the augment. Divide by it the assumed root wliich is " 6, it is 6-fourths, that is 1-, and this is the less root. The square of thu " which is 9-fourths we multiply by 13; it is 117-fourths, We see that addinp- •' 1 integer, that is 4-fourths to this, it is 121-fourlhs; and this is a square, the " root of which is 11 -second parts: the less root then is 3-second parts, and the " greater root is 1 1-second parts ; and the augment is 1 affirmative. As the " original augment is 13 affirmative, perform the operation of cross multi- " plication with the former which was 13 negative, thus : First multiply 3-second "parts by cipher, it is ci])her; and U-secoud parts by I, it is the same. • 13j:'— 13=5/», and 13*-+ 13 =3^*. Let i = 1, 13X 1 — 13 =0. For 13x^+13=^% \et x'=i5, x'-=9, (Here are two or three errors in the Persian : A case of aV + 1 = y'- is found by the rule — '— = x). ~=-^ = *'> 1' X To) ■*" ' ~ "i~= ( o) • ■^'='7'/= 7;,andp'=l. As B = 13. multiply cross- ways with the former case where 13 X 1 [ — 13 =2 0. 2 x 0 4- — x 1 = — =*''• 3 X 1 X 13 + ~x 0— ^^■=y"- 13X— 1 =— 13=S'', but B = + 13. Suppose tiicn *"•= ~ and 3"=— i, I2x(r)'—i—' = (liy. Multiplycrossways,-:^ + f = l.UlS = .. And iil + "ii = !2l := 55 = ^. ,„j ,3 ^ «. Or by the rule a//' - gg' =y, and/;- -/^' = *, |_ ^= £ = .-, 112_ ii2 = ?J -y. ,3x (jj' + 15 = ar G 2 5Q INTRODUCTION, " add them together, it is 1 1 -second parts ; and this is the less root. Multiply " 3-second parts by 1, it is the same: multiply it by 13, the multiplicand, it is " 39-second parts; add it to the rectangle of the two greater roots which is cipher, " it is the same; and this is the greater root ; and 13 is the augment negative; " as it is not the original augment, for 13 affirmative is required ; again, suppose •' the less root- and the augment 1 negative ; and multiply — Avhich is the square, " by 13 ; it is 13-fourths. Subtract 1, that is 4-fourths, the augment negative, there " remains 9-fourths, the root of which is ]-. By this we multiply crossways, " thus : 11 39 — i: — ] " the less root is 72-fourths, which is 18 integers, and the greater root is 260- " fourths, M^hich is 65 integers, and the augment is 13 affirmative. " If Ave would perform the operation of cross multiplication take the dif- " ference of the two, which are 39-fourths, and 33-fourths, that is 6-fourths ; " 1- is the less root ; take the difference of the two less, after multiplying by " the multiplicand, and the rectangle of the two greater, it is £6-fourths, that *' is 6-; and this is the greater root and 13 is the augment affirmative. Another Example. " What square is that which being multiplied by b nega- " tive, and the product increased by 2 1 Avill be a square *. Suppose 1 the less " root, and multiply its square by 5 negative, it is 5 negative : add 21 affirma- " tive, it is l6 ; 4 then will be the greater root. In another way. Suppose 2 the * — 5 X »' +21 =^'. Suppose«= 1; — 5 X 1 + 21= 16; y = 4. Or, suppose .t = 2, — 5 x 2»-i- 31 = 1, ^ = 1. By multiplying crossways when B = 1, new values may be found. INTRODUCTION. 0,3 *Mess root and multiply its square by 5 negative, it is 20 negative; add 21 " affirmative, 1 affirmative is obtained, the root of which is I ; the less root then " is £, and the greater I, and the augment 2L And if in the place of the multi- " plicand there is 3, and the augment is 1 affirmative, multiply crossways and " numbers without end will be obtained. " And this which has been written is the introduction to the Indian Alo-ebra. ♦' Now by the help and favour of God we will begin our object." END OF THE INTRODUCTION. BOOK 1*. OX THE EQUALITY OF UNKNOWN WITH NUMBER, iVNOW tliat whatever is not kiionn in the question, and it is required to bring " it out by a method of calculation, suppose the required number to be one " or two unknown, and with it whatever the conditions of the question in- '* volve, and proceed by multipHcation and division, and four proportionals and " five proportionals, and the series of natural numbers, and the knowledge of the " side from the diameter, and the diameter from the side, that is the figure of the " bride-f, and the knowledge of the perpendicular from the side of the triangle, " and conversely, and the like, so that at last the two may be brought to equa- " lity. If after the operation they are not equal, the question not being about " the equality of the two sides, make tliem equal by rejection and perfection, and " make them equal. And that is so, that the unknown, and the square of the " unknown of one side is to be subtracted from the other side, if there is an un- " known in it; if not subtract it from cipher: and subtract the numbers and " surds of the other side from the first side, so that the unknown may remain " on one side, and number on the other; the number then, and whatever else is *' found, is to be divided by the unknown, the quotient will be the quantity of " the unknown. " If the question involves more unknown quantities than one, call the first " one unknown, the second two unknown, the third three unknown, and so on. " And the method is this. Suppose the quantity of the lower species less than "that of the higher, and sometimes suppose—, and-, and- of the unknown 2 'o 4 * There are many parts of the rules given in tlie rest of the Work wliich are unintelligible to me; tlicy are obscured probably by the errors of transcribers and of the Persian translator. — I translate them as exactly as I can from the Persian. + The Arabs call the -Wth proposition of the first book of Euclid, " the figure of the bride." I do not know why. OF EQUATIONS. Ji " and the like ; and sometimes suppose the unknown to be a certain number, and " sometimes suppose 1 unknown and the rest certain numbers." The shortest metliod of solving- the (jucstion is directed to be observed, and the whole at- tention to be given to what is required. Tlie first example is, " A ])erson has 300 rupees and 6 horses; and another "person has 10 Loises and luO rupees debt; and the property of the two is " equal; and the price of the horses is the same; what then is the value of each? " Or, the first person has two rupees more than the property of the first person " in the first question, that is 3 horses and 152 rupees; and tlie second has the " same as he had before, and the property of both is equal ; what then is the " price of one of the horses? Or, in the first question, the property of the first " person is three times tlie value of that of the second, wliat then is the value of one " horse r Tlie operation in the first question is this : I suppose the price of a horse " to be the unknown ; 6 horses are six unknown. The first person's property then "■ is 300 rupees affirmative and 6 unknown ; and the property of the second is \0 " unknown and ]00 rupees negative. As by the question both these sides are " equal there is no occasion for the operation of rejection and perfection. I " make them equal in this manner; + 300 Rupees (u' — 100 Rupees lO.r " First I write them both, above and below, and I take 100 rupees negative from " 300 rupees affirmative, it is 400 rupees affirmative ; and I take 6 unknown from " 10 unknown, there remains 4 unknown. 400 rupees is equal to 4 unknown. I " divide the first by the second, 100 rupees is the quotient, and this is the price of a horse." The other questions produce also simple equations, hi which nothing remarkable occurs. The second example has three unknown quantities Mith only one equation ; it is solved first by assuming the unknown quantities in the proportion of 3, '2, and 1 ; and secondly by assuming them as 1, 5, and 4. In the third example the iVIussulman names, Zeid and Omar are introduced. The fourth and fifth examples contain nothing worthy of notice. The sixth is as follows : " A person lent money to another on condition that he o6 OF EQUATIONS. " should receive 5 per cent, a month. After some montlis he tcok from him the " principal and interest, and having subtracted the square of the interest from " principal gave the remainder to another person, on condition that he should " receive 10 per cent, and after the same time had passed, as in the former case, " he took back the principal and interest, and this interest was equal to the first " interest ; what sum did he lend to each person, and what was the time for " which the money was lent* ?" The first principal is supposed unknown, and the number of months during which il was lent is supposed 5. The question is solved by the rules of propor- tion and a simple equation. Another way is given for working this question, viz. *' Divide the interest of the second by that of the first, call the quotient the " multiplicand, and suppose a number the interest for the whole time and take its "square, and from the multiplicand subtract 1, and divide the square by the " remainder ; the quotient will be the amount of the second sum, and the second " sum multiplied by the multiplicand, or added to the square of the interest of " the whole, will be equal to the first sum." The next question is like the preceding, and is solved by means of the rule. I pass over several other examples, which contain nothing new or remarkable. A question in mensuration comes next, " There is a triangle, one side of which is 13 surd, and another side 5 surd, *' and its area 5 direhs ; how much is the third side ? I suppose the third side " unknown; the side 13 is the base. It is known that when the perpendicular " is multiplied by half the base, or the base by half the perpendicular, the pro- " duct will be the area of the triangle. Here the base and the area are known, " and the perpendicular is unknown. I divide 4 which is the whole area by half " of 13 surd ; the quotient is the perpendicular. I perform the operation thus : " As 4 is a number I take its square 16, for the division of a number by a surd " is impossible. I take half of 13 surd thus: I square 2, which is the denomi- " nator of-, it is 4. I divide 13 by it. The quotient is 13 parts of 4 parts. I * Let P, p, be the principal, i, /, the interest ; r, r, the rate, and N, n, the number of the months. If prn = i, PRN =: I, P =p — z% n =: «, and i =i; we have (p — ■ ?'). rn = i ■=.prn; or ps.n — j'-rn = firn; whence p« X (R- — '') = »-R>f> and r — r = but this is equal to — , and P = = - — ^ ' ' r ^ ^ ' p,i ' 1 p ' jt. __ y R r ' which is the first part of the rule ; the rest is evident. OF EQUATIONS. J7 "divide 16 by 13 parts of 4 parts ; it is 64 parts of 13 surd; and this is tlic " perpendicular. I then require the excess of tlie scjuare of 5 surd above 64 parts " of i3 surd : First I take the square of 5 surd ; it is 5 number ; take its square " it is 2j surd ; the root of which is 5. I then take the square of 64 parts of 13 " surd, as above. I take the excess thus : I make 5 of the same sort; it is 65 ; " I take the excess of 65 above 64 ; it is one part of 13 surd ; and this is fronr " the place of tlie perpendicular to the angle formed bj^ the side 5 and the base." The other segment of the base is f amd by subtracting this troni the whole, by a rule \\hich was given in the 4th chapter of t!ie introduction, for finding the difference of two surds, viz. ^a—^/b= \/[\/W — ^Y X ^)- The square root of the sum of the squares of this segment anil the perpendicular gives the quantity of the unknown side of ihe triangle. In the next question, tlie sides of a triangle being given, its area is required. One of the segments of the base made by a perpendicular, is supposed unknown. From two values of the perpendicular, in terms of the hypothenuses of the two } iglit-angled triangles, and their bases, an e([uation is formed ; from which the unknown quantity is brought out. The equation involves many surds, and they are reduced by the rules laid down in the introduction. The perpendicular is then found b}' taking the square root of the difference of the squares of a segment of the base, and of the adjacent sides of the triangle. The operation is here con- cluded, lu a marginal note are directions to f;nd the area, as in the foregoing case. i'lie next is, " What four fractions are those whose denominators are equal, and •' whose sum is equal to the sum of their squares. Also what four fractions are •' those, the sum of whose squares is equal to the sum of their cubes.'' For the first part of the question : " Suppose the first fraction one unknown, the second '•■ two unknown, the third three unknown, and the fourth four unknown, and " below each write 1 for the denominator. The sum of the-fbur is 10 unknown. " Their squares are 1 and 4 and 9 and l6, whose sum is 30 square of unknown, " and these two quantities are equal. Divide both by one unknown ; the quotients ''■ are 10 number aiul 30 unknown. Diside 10 by 30 unknown, the quotient is I ^ ] o 3 " — of unknown. The fiist fraction tlien is -, the second -, the third — , and ;> 3 3 3 " tlie fuuith ■-; and the scpaares of these fractions are —and — , and ^~aud — ; and 3 9 .V S/ S 3S OF EQUATIONS. " the sum of these four is — , and this is equal to -— ." In the same manner the 9 >^ 369 12 the other fractions are found to be — , -— , -— , and — . Tiie next is to find a right-angled triangle, " the area of which is equal to its *■ hypotlieouse;" and to find a right-angled triangle, " the area of whicli is equal " to the rectangle of its three sides." For the first part of the problem, one side of the triangle is assumed equal to 4 unknown, and the other side equal to 3 un- known ; the hypothenuse is found equal to .').v, and the area equal to 6.r^ ; the equation 5.v — 6.v'^ being reduced, gives the value of .r. For the second part, the sides are assumed as above, and the value of .r is deduced from the equation 60,v^ = 6.v\ The next problem is, to find two numbers of which the sum and the difference shall be squares, and the product a cube. The numbers are supposed 5x' and 4.r^ and the cube to which their product must be equal JOOO.r, whence .r is found. The next is to find two numbers such that the sum of their cubes shall be a square, and the sum of their s([uares a cube. One number is sup])osed ,v'; and the other 2.v% and the cube \25d'^*. In the solution of this the following- passage occurs : " The cube of the squaie of unknown, which in Persian algebra " is termed square of cube." In the margin is this note : " Here is evidently a " mistake ; for in Persian algebra the unknown (^Jj(?'yv-j) is called thing (, ^), "and its square ( " and when the square of cube is multiplied by thifig, the product is called cube "of cube (v_.vji/ u-^-x/O) "'^t square of cube. For example, suppose 2 thing <' 4 is its square, 8 its cube, 16 its square of square, 32 its square of cube, 64 its " cube of cube, not its square of cube, although it is the cube of the square " (^j^ V^^^^x)) or the square of the cube ((.^^oi^=a^ Xj^)." In the next example the three sides of a triangle are given, and the perpen- * Sum of the cubes = x' + Sj;* = 9a-^ (a square) ; and the sum of squares = x-' •+- 4x* = 5x', assume this = 125x', or 5*'' = \2Jx\ whence 5x= 125, and « = 25 ; therefore 625 and 1250 are the numbers. OF EaUATIONS. 5f) tlicular is required. It is found in the same way as the perpendicular was found in one of the former questions, when the sides heing given the area of tlic triangle was required. The three foUowing are different cases of riglit-angled triangles in which the parts required are found hy the principle of the square of the hypothenuse bein"- equal to the sum of the squares of the two sides, and simple equations. In the first the ba«;e and the sum of the hypothenuse and theother side are given. In the second the base and the diiFerence of the hyptohenuse and the other side arc given ; and in the third the base, part of one side, and the sum of the hypothe- nuse and the other part of that side, are given. The first book ends with the following example: " Two sticks stand uprio-ht " in the ground, one is 10 direhs in height and the other 15 direhs, and the •' distance between the two is 20 direhs. If two diameters are drawn between " them, what will be tlie distance from the place where they meet to the ground * ? " Suppose the perpendicular unknown; it is known that as 15 to 20, so is the " unknown to the quantity of the distance from the side 10 to the place where the " unknown stands. We find then by 4 proportionals, 4 thirds of unknown is the said " quantity. In like manner we find the second quantity 20 parts of 10, that is *' 2 unknown. Take the sum of the two, it is 1 0-thirds, and this is equal to 20. " Divide GO by 10-thirds, the quotient is 6; and this is the quantity of the un- " known, that is of the perpendicular. From the place where the perpendicular " stands on the ground, to the bottom of the side 15, is 12 ; for it is 2 unknown. " The second quantity is 8 direhs ; for it is 1 unknown and a third of unknown. " And know that M'hatever the distance is between the two sticks, the quantity " of the perpendicular will be the same ; and so it is in every case. We can also * Let AB = 10, r>c = 15, bd = 20. By similar triangles ed ; bp :: dc : pg, BD : PD :; BA : pg, whence bp : PD :: ba : dc, theivfore bd is divided in p in the ratio of dc to ba. w ~ By composition bp 4. pd : bp : i ba + dc : ba ; but bp + pd = bd, therefere ba + 0c and ba are io the ratio of bd to bp ; wiience, by the first proportion, b a + dc : ba : : dc : po, that is, pg is a'fourtb proportional to b 4 -h t.c, ba, and DC, whatever be the length of bd. Lucas de Burgo has this proposition, (see his Geometry, p. 56.) where the lengths are 4, 6, aad S^ or page €0 ■wliere they are 10, 15, and 6. Tiie same is in Fy^ee's Lilavati, wliere the rules are AB + CD* AB -|- CD* AB+CD' H 2 go 01^ EQUATIONS. " asceitalu these two quantities by another method, and that is the ratio of 25, ' (that is the sum of the two sides) to 20, is Hke the ratio of 15 to the unknown ; ' tliat is the quantity towards the side 15. INIultiply 15 l)y 20, it is 300. Divide ' 300 by 25, it is 12. The ratio of 25 to 20, is like the ratio of 10 to the unknown ; ' the result is 8, and this is the quantity towards the side 10. By anotlier * method, bv four proportionals, we lind that the ratio of 20 to 25, is like the ' ratio of 8 to the unknown ; 6 is the result. In like manner the ratio of 20 to ' 10, is like that of 12 to unknown ; again 6 is the result. Another method is, ' divide the rectangle of the two sticks by the sum of the two, the result is the ' quantity of the perpendicular, and the quantity of the ground we multiply by " each side separately, and divide both by the sum of the sides. The two quotients ' will be the quantities from the place of the perpendicular to the bottom of the 'sticks; accordingly divide 150, which is the rectangle of tlie two sticks, by ' 25, the quotient is 6. Multiply 20 direhs, which is the quantity of the ground ' by both sticks, the products are 300 and 200. Divide both by 25, the quotients 'are 12 and 8. In this manner the figure may be found by calculation as " correctly as if it were measured. " }:NO OF THC FIRST BOOK.. BOOK 2. " Un the interposition of the unknown : where the square of unknown is " ecjual to number, and that is rejected with the unknown." (Or divided by the tmkiioxcn (J»^sz^.^ .\"£—,,\j I j]). (I do not know xchat he means here, perhaps there is some error.) " It is intitled, ' Intcrjjobition of Unknown,' ( Ji^s:^ ia^y) ; because " that which is rtquircd is brought out by means (ja^l.) of the unknown. It is " called Interposition (Lawj «.j) ; and jMudhum Uhrun (unknown means) in *^' Hindee is to be so understood. Its method is this : The square of unknown " being equal to number, multij)ly both, or divide both by an assumed number, " and add a number to the two results, or subtract it from them that both may " be squares. For if one side is a square the other also will be a square; for they " are ecjual, and by the equal increase or diminution of two equals, two equals " will be obtained. Take the roots of both, and after equating divide the num- " ber by the root of the square of unknown, that is by unknown ; the result will " be what was required. And if there is equality in the cube of the unknown, " or the square of the square, that is after the operation in the thing and cube, " and square of square, if the root cannot be found, nor be brought out by rule, " in that case it can only be obtained by perfect meditation and acuteness*. " And, after equating, if the two sides are not squares, the method of making " them squares is this. Assume the number 4, and multiply it by the number of " the square of the first side, and multiply both sides by the product. And in the * From this place to the end of the rule Mr. Burrow's copy is as follows : " And if in the side which lias tlie.' *' unknown there is a number greater than the unknowji, if the number is affirmative make it negative, and if " negative, two numbers will be found in tlie conditions required, and the way to find the assumed number by " which the two sides should be multiplied, and the number to be added, is extremely easy ; for multiply the " multiplicand of the number of the square of the unknown by 4, and let the square of tlie numbers oftheuji- " known, of the side in which there is the square of the unknown, be the number added.'* 6<2 OF EQUATIONS. "■ ])Iacc of niimber increase both sides by tlie square of the tiling of tb.e unknown, " wl^-ich is on that side; both sides will be squares. ' Ta.--.> tlie roots of both and " equate them, and the quantity of the unknown will be found. Example. " Some bees Mere sitting on a tree ; at once the s(]uare root of half " their nundier flew away. Again, eight-ninths of the wno^e flew aw.iy the " second time ; two bees remained. How many were there ? The method " of bringing it out is this : From the question it appears that half the sum has " a root ; I therefore suppose 2 square of unknown, and I take J unknown, that " is the root of half. And as the questioner mentions that two bees remain, 1 B IG " unknown and -- of G square of unknown, tliat is — of ] square of unknown, " and 2 units, is equal to 2 square of unknown. I perform the operation of " equating the fractions in this manner, I multiply both sides by 9, which is the " denominator of a uinth ; l6 square of unknown and 9 unknown, and 18 units, " is equal to 18 square of unknown. I equate them tlnis : X subtract \6 square of " unknown of the first side from 18 square of unknown of the second side ; it is *' 2 square of unknown affirmative; and in like manner I subtract 9 unknown of " the first side from cipher unknown of the second side ; 9 unknown negative " remains. Then I subtract cipher the numbers of the second side from i8 units of " the first side ; it is the same. Tiie first side then is 2 square of unknown altirma- " tive and 9 unknown negative, and the second side is 18 units affirmative. In this " example there is equality of square of unknown, and unknown to number ; that " is equality of square and thing to number. As tbe roots of these two sides can- " not be found, suppose the number 4, and multiply it by 2, which is the siumber " of the square of the unknown, it is 8. I multiply both sides by 8 ; the first side is '< \6 square of thing, and 72 unknown negative; and the second side is 144 " units, I then add the square of the number of the unknown, which is 81, to tlie " result of both sides; the first side is 16 square of unknown, and 72 unknown *' negative, and 81 units ; and the second side is 225 units. I take the roots of " both sides : the root of the first side is 4 unknown and 9 units negative ; and *' the root of the second side is 15 units affirmative. I equate them in this ~' manner : I subtract cipher unknown of the secontl side from 4 unknown of the ■"first side; and 9 units negative of the first side from 15 units affirmative of " the second side •, the first is 4 thing, and the second side is 24 units affirma- " tive, I divide, 6 is the result, and this is the quantity of the unknown ; and OF EQUATIONS. 6S " as we supposed ^ square of unknown, we double 36 ; the whole number of bees " then was 72." In the next example arises the equation t- +4.r+ lOzz.r^ ; then a*'" — 8.r=20, 4a'' — 32x + 6-if — -i X -0 + 6i =144, S.r — 8 = 12 ; .r =r 10*. The next example is, " A person gave charity several days, increasing the ' gift equally every da}'. From the sum of the days 1 being subtracted and ' the remainder halved, the result is the number of dirhenis which he gave the ' first day; he increased always by half of that number: and the sum of the ' diihenis is equal to the prothict of these three; that is to say, the number of ' days, the number of dirhems the first day, and the number of the increase, ' added to of the product." Let the number of days be 4.1' + 1, 2^' is the ' number of dirhems given the first day, and .r is the number of the increase , , 1 ] . n . l(^-«'' + f> + i'' (4.i' +1) X 2.r X .r = 2.i'' + 8a", add - of this = the sum of the dirhems. Then by a rule of the Leelawuttee, ((4.r + l) — l) .r + Q.v ~ the number given the last day ; half the sum of what was given the first and last days = what Avas given the middle da}-. Multiply this by the number of days; it is 8i'' + 10a- + 2a', which is = "T —. 8a'' — 54a- = 14. Mul- tiply by the assumed number 8, for in this case as the co-eflficient of ,r is even, assume the co-efticicnt itself of .v', and add the square of half the co-efficient of .V, 64a- - 432a' + 729 = 841 ; 8a' - 2/ = 29, ^v = '^ ^ ~'^ = 7. The next is to find .v in the equation (77- + tt) X 0 z= 90. The solution is, " I suppose, what is required to be t/ii>?g, I divide it by cipher; as the quotient " is impossible, thi/ig is obtained, whose denominator is cipher. Its square, which " is the square of tiling, whose denominator is cipher, I add to t/ii/ig, which is the " root. It is the scjuare of t/ii/ig and tiling, whose denominator is cipher. I ** multiply by cipher; it is the square of thing and thing. Cipher is thrown out " by a rule of the Leelawuttee, M'hich says, that when the multiplicand is cipher, " and the multiplier a number whose denominator is cipher, the product will be * Pafl of this example, and most oi the rest in this book, are wanting in Mr. Burrow's copy. D^ OF EQUATIONS. " that iiuml-ier, and cipher will be rejected." "Whence the equation d' -f- .r ~ QO which is solved in the common way. The next is; a value oi'.v is required in the case ((.r+ ~J X O)'' + G Ci' + ' 1 X 0 = 15. It is brought out in a manner similar to that of the foregoing. The next example is of a cubic equation, viz. .r' + 12,i' — G.r* + 35. The terms involving the unknoMai quantity being brought all on the same side, 8 is added to complete tlie cube. " I take the cidie root of the second side 3, and I " write the terms of the first side in the arithmt^tical manner, tlius : 8 units nega- " tiv'e, and 1'2 unknown afitirmative, and 6 square of unknown negative, and I " cube of unknown. First, 1 take the cube root of the last term, it is 1 un- " known. I square it and multiply it by 3, and I divide the term Avhich is last " but one by the product ; 2 units negative is the quotient. Its square, which is 4 " affirmative, I nmltiply by the term first found, viz. 1 thing ; it is 4 thing. I " multiply it by 3, it is 12 unknown. I subtract it from the third term which is " after the first, nothing remains. After that I subtract the term i2 negative from " the first term, nothing remains. The cube root then of the first side is found 1 " thing affirmative and 2 units negative." Whence :c — 2 n 3, which is re- duced in the usual wa}'. In the next a biquadratic is found, .r* — ^OO.r — 2,i^ r= 9999- To solve this iOQ.v + 1 is directed to be added to each side ; the equation is then .r'*- — 2.i'' + 1 zr 10,000 4- 400,r. The root of the fn'st side is .r' — 1, but the root of the second side cannot be found. Find a nuiiiber which being added, the roots of both sides may be found; that is 4.i'^ 4- 400a' + 1. Tliis will give .r* + 2.r'' + I =: 10,000 -f- 4a-' + 400.r ; and extracting tlie sf[uare root, a-" + 1 =: 100 + 2.r, which is reduced by the rules given in this chapter. At the conclusion of the ex- ample are these words ^ "The solution of such questions as these depends on " correct judgment, aided by the assistance of God. In the two next examples notice is taken of a quadratic equation having two roots. " When on one side is tJi'mg, and the numbers are negative, and on the " otlier side the numbers are less than the negative numbers on the first side, " there are two methods. The first is, to equate them without alteration. The " second is, if the numbers of the second side are affirmative, to make them nega- " tive, and if negative to make them affirmative. Equate them ; 2 numbers will •* be obtained, both of which M'ill probably answer." The next example is, "The style of a dial 12 fingers long stands perpcndi- OF KQUATIONS. Gr> " ciilar on the grouiul. If from its shadow, a third of the hypothenuse of these "two sides, viz. the style and the shadow, is subtracted, 14 lingers will remain. " What then are the shadow and the hypotheuuse ?"' In the right-angled triangle -' _^ , a beins; — 12, and b — -= 14 ; c and b are required. The cqua- tion {3b — 42)'- = // + 144 arises, and is reduced to U — 63 = 27. The two values + 27 are taken notice of. First, + '27 gives b — 22- , which is declared to be right. From — £7, b is found =: 9 ; " hut here," il is observed, " 9 is not correct ; for, after subtracting a third of the hypotheuuse, 14 does " not remain." In opposition to this, some one speaking in the first person (the Persian translator, I suppose) says, " I think that this also is right," and goes on " to prove that in this case the hypotheuuse will be zi: — 15." The next problem is to find four numbers such that if to each of them 2 be added, the sums shall be four stpiare numbers whose roots shall be in arithmetical progression ; and if to the product of the first and second, and to the product of the second and thud, and to tire product of the third and fourth, 18 be added, these three sums shall be square numbers; and if to the sum of the roots of all the square numbers 1 1 be added, the sum shall be a square number, viz. the square of 13. It is here observed, by way of lemma*, that, in questions like this, the " aug- " ment of the products" must be equal to the square of the difference of the roots, nmltiplied by the " augments of the numbers ;" otherwise the case will be im- possible. The following is an abstract of the solution : (Let zv, .r, y, z be the four num- bers required, and r, s, t, v the four joots which must be in arithmetical pro- 1 8 gression). By the lemma we find the common difference \/~^ — 3. The first root being r, the second will be := r -f 3, the third = ;- + 6, and the fourth = r + 9. Nowr*— 2=:v/(r£U'4-18), and sl—o—^y{:iy+]S), and tv—2,-=:V{i/z + lS). * In a marginal note, which I suppose to be written by the Persian translator, the application of the Lemma to the problem is illustrated thus : Let a, b, c be three numbei-s ; {a — byXc+ {a- — c) x (i- — c) = {ab — ty. In this case we have a — i = 3, f=2 and 6 and a two successive roots; and as it = r- — 2, an«t*=i''— 2, &c. *be reason of the rule is plain. G6 OF EQUATION'S. We havfMio\v;r + 2z=r' ; .r + 2 = (/' + 3)' ; y + 2-{r+6y- ; z + 2=:(r+9y-; And z(\v+lS = {n- — '2y; ji/+ \S:^{st — '2)\ iuu\ 7/z + lS = ((v — £.y. Making ?^+s + t+v + {rs—2) + (st—'2) + (tv~'2.)-\-U~\3';ii quadratic equa- tion arises, which heing reduced /• is found — '2, whence zti — 2, .r — 23, _y— (52, and » rr ] 19. Some questions about right-angled triangles occur next; the first is, "Given "the sides of a right-angled triangle 15 and 20; required the hypothenuse. " Although by the figure of the bride the hypothenuse is the root of the sum of ■' the squares of the two sides, tlie method of solution b_) Algebra is this : In this " triangle suppose the hypothenuse unknown, and then divide the triangle into " two right-angled triangles, thus: Suppose the unknown hypothenuse the base " of the triangle, and from the right-angle di aw a perpendicular; then 15 is the " hypothenuse of the small triangle, and 20 that of the large one. By foiir pro- " portionals I find, when the least side about the right angle, whose hypothe- " nuse is 1 unknown, is 15 ; how much will be the least side about the right *' angle whose h^'pothenuse is 15." In like manner the otlier segment is to be brought out, whence .v — 25. "If I Mould find the quantity of the perpen- " dicular, and the segments of the hypothenuse at the place of the perpendicular, " it may be done in various ways ; first by four proportionals," &c. They arc found on the same piinciple as above. '• And another way which is written *' in the Leelawuttee is this; The difference of the two containing sides, that is " to say 5, I multiply by 35, which is the sum of the two sides; it is 175. *' I divide by 25 that is the base ; the quotient is 7. I add this to the base, it "is 32. I halve it, l6 is obtained; when I subtract 7 from the base, 18 " remains. I halve, 9 is the smaller segment from the place of the perpen» " dicular. Rule; " The square of the hypothenuse of every right-angled triangle is equal " to twice the rectangle of the two sides containing the right angle, with the " square of the difference of those sides. As the joining of the four triangles •^ abovementioned is in such a manner that from the hypothenuse of eachj the " sides of a square will be formed, and in the middle of it there will be a S(juare, " the quantity of whose sides is equal to the difference of the two sides about the " right-angle of the triangle ; and the area of every right-angled triangle is half " the rectangle of the sides about the right triangle. Now twice the rectangle " of the two sides containing that is 600, is equal to all the four triangles ; and OF EQUATIONS. Q.^ " u'hen I add 23, the small square, it M'ill be equal to the whole square of the " hypothenuse, that is 6^25, which is ecjual to the square of thing ; and in many "cases an effable root cannot be found, then it will be a surd; and if we do " not suppose thing, add twice the rectangle of one side into the other, to the " square of the difference of the sides, and take the root of the sum, it will be " the quantity of tiie hypothenuse. And from this it is known that if twice the " rectangle of two numbers is added to the square of their difference, the result " will be equal to the sum of the sqiuu-es of those tM-o numbers."* The next is in a riglit-angled tiiangle ^\. . Given \/(ab — j) — 1 z= ac — B'- -i^C Bc, required the sides. " First, I perform the operation of contrariety and op- " position : let AC — liC be supposed '2. To this add 1, it is 3; take its square, " its is 9; add 3, it is 12. This is the quantity of the less side ; its square which " is 144 is zz Ac'' — bc"; here then the differences of the two original numbers^ " and of the two squares are both known ; and the difference of the squares of two "numbers is equal to the rectangle of the sum of the two numbers, into their " dift(?rence, Tiierefore when Me divide the difference of the squares by the " difference of the two numbers, the sum of the two numbei-s will be the quotient; " and if we divide by the surn, the difference will be the quotient: because the " square of a line has reference to a four-sided equiangular figure whose four " sides are equal to that line ; for example, the square of 7 direhs is 49. If I " subtract the square of 5 from it, 24 remains; and the difference of 7 and 5 is " 2, and their sum 12, and the rectangle of these two is 24, which is the number " remaining. Then it is known that the rectangle of the sum of the two numbers " into their difference, that is 12 multiplied by 2, is equal to the difference of the " squares of the two that is 24," &c. On this principle the sum and difference being found, the numbers themselves are had " by a rule of the Leelawuttee," viz. a + h , a — d . a + h a ~ h — - — + — ; — n a, and — h. 2 2 2 2 By supposing other numbers besides 2 for the difference, and proceeding in the above manner, triangles M'ithout end may be found. As objection is here made (I suppose by the Persian translator), that the above is not algebraical. It is then stated that the translator has found out an easy way of solving the question by Algebra. He directs that the difference j^c — BC may be assumed n: 2, as before; and making bc h: .c, ac will be I 2 bS OF EQUATIONS. z= X + £, and ab being rr 1'2, the value of .v may be found from the equation x' + 1^2' = (.r + ':)■-. Rule. " The difference of the sum of the squares of two num'jers and the " square of their sum is equal to twice the rectangle of the two numbers. For " example, the squares of 3 and 5 are 9 and 25, that is 3-i, and their sum " is 8 and its square 6i, and the difference of these is 30, Avhich is equal to twice " the rectangle of 3 and 5 that is by the 4th figure of the second book thus." In the copy Mhich I now have, the figures are omitted. In Mr. Burrow's copy it is 6i 9 15 13 •^5 Then follows another rule : Aab — (a + by = {a — by, which may be easily understood by this figure. There is no figure in Mr. Burrow's copy, nor in m\- present copy, but I had one in which there was a figure for the demonstration of the 8th proposition of the second book of Euchd. Next come two examples : The first of them is, what right-angled triangle is that " the sum of whose 3 sides is 40, and the rectangle of the two sides about " the right angle 1'20? "The method of solution is this: By the first rule take twice 120, it is C-iO, " and this is equal to the difference of the sum of the squares of the sides about " the right angle, and the square of their sum that is the hypothenuse. Then " the difference of the squares of the two numbers, one of which is the sum of the " two sides and the other the hypothenuse, is 240; and the sum of both is 40. " In the method of finding out the triangle, it was before known that the dif- " fereace of the squares of two numbers is equal to the rectangle of their sum and ** difference ; when the difference of the two squares is divided by the differ- " ence of the two numbers, the quotient is the sum of the numbers ; and if it is " divided by the sum, the quotient is the difference. Let then 240 be divided •^ by the two numbers, which together make 40 by the question, the quotient OF EaUATioys. G9 " is 6, and this is the difference of the hypothenuse and the sum of the two sides " about the right angle ; then add 6 to 40, and take its half, it is 23 ; this is tlie " sum of the sides ; subtract 6 from 40, and take its half, it is 17, and this is the " hypothenuse, for the sum of the two sides is always greater than the hypothe- " nuse by the asses proposition*. It Mas stated in the second rule that the " difference of the square of the sum of two numbers, and 4 times their rectangle, " is equal to the square of their difference. Take then the squares of 23, it is " 529, and 4 times the rectangle of the two sides, it is 480; their difference is " 49, which is equal to the square of the difference of the sides, that is 7 : then " add 7 to 23, and subtract it from the same, and the halves, air 15 and 8 tlie " two sides." The next example is, ^^' [^ required r, y, z, such that j:- + 7/ -\- z =. 56, z and ayz — 4200. " I suppose the diameter (//«? hi/potlieniisc) unknown ; take its " square it is x^ : This is equal to the sum of the square of tlie two sides about " the right angle, by the figure of the bride; and as 4200 is the product of the " rectangle of the two sides multiplied by the hypothenuse, I divide 4200 by the 4200 " uiikiiown, the quotient is the rectangle of the two sides. And it was *' stated that the excess of the square of the sum of the numbers above the sum " of their squares is equal to twice the rectangle of the two numbers. The sum " of the two sides is 06 ~ x ; I take its square, it is x"- — 1 12.r +3136; and the " sum of the squares of t!ie two sides is x^, for that is the square of the hypothenuse «' which is the same. 1 take the difference of the two — Wis + 3136, and this " is equal to twice the rectangle of the two sides, that is — ;— ," &c. The equation is reduced in the common way : the square iu the quadratic, which arises, being completed by adding the square of 14, which is half the co- efficient of ,r. In this Avay the hypothenuse, and thence the other sides are brouc'ht out. * Meaning I)y tlie asses proposition the 2Ctli of the first book of Euclid, which we are told was ridiculed b» (he Ej)icureans as clear even to asses. These passages are only interpolations of the Persian translator. END OF THE SECO^'D 300K. BOOK 3. "EXPLAINING THAT MANY COLOURS MAY BE EQUAL TO EACH OTHER.' J. HE rule in this case is to subtiact the unknown of one side from the un- " known or cii'her of the other side, and all the otiier colours and the nuuiljers of " the second side from the first side, from which the unknown was subtracted, •' and divide those colours by the unknown. If, as may happen, the denonii' " nators are one quantity, perform the operation of the multiplicand ; and if the " deuonilnators are different unknown quantities let them be unknown. Sup|)ose " the (luantity of every one of these unknown the denominator, and put it below " the colours of the dividend, and reduce the fractions and reject tlie denomina- '• tors • then the unknown will not remain on any side. After that subtract the " black of one side from the other side, and subtract the rest of the colours and " the munbers from the side from which the black was subtracted, and perform '■' the same operations as were directed for the unknown, and the quantity of the " black will be obtained : and in like manner the rest of the colours, and all " the quantities of the multiplicand will be obtained. Then perform with it the opera- " tion of the multiplicand ; and the multiplicand and quotient will be obtained. The " multiplicand will be the quantity of the dividend, and the quotient the quantity of " the divisor. And if in the dividend of the operation of the multiplicand, two " colours remain ; as for example, black and blue, suppose the second in order, *' which is blue, the dividend, and suppose black a number, and add that to the " auo-ment, and perform the operation ; and when the quantity of the two last " colours is obtained, we shall known by the method which has been explained " an + ly + Sz -|- Izo, all equal to each other; and the values of ^r, y, z, and xv are required. From the first and second is found 'J.r z=: 5y — 6z — Gxv ; from the second and third 3.v ■= 3t/ + z — w; and from the third and fourth Q.r :=. 3y — is + w. From the two first of these three equations 9_y=: 20.:r +l6ry, and from the two last oil — 8.Z — 3x0; whence 12a = 93xv ; and dividing — '- — — — iz z ; " and " above, where the rule of the multiplicand was given, it was said that Avhen the *' augment is cipher, the multiplicand will be cipher, and the quotient the quotient " of the augment divided by the divisor ; here then the multiplicand and quotient " are both cipher."' Then adding 31 for a iiew valu^ of ,■:;, and 4 for ^ new j^o OF INDETERMINATE PROBLE.^IS. value of w, 3] =. z and 4 =■ a', and the other qnantitjes are hrouglit out in the usual manner. The next example gives 5x + 7j/ + 9- + 2rv = 100, and 3.v + 5j/ + 7? + OwzzlOO. From these comes Ay = — 8; — SGio + 200, and for the operation of the multiplicand j — = j/. Suppose zv = 4, tiicn — Sz will be the dividend, and + 56 the augment, and 4 the divisor. As 4 measures .56, 14 times without a remainder, the multiplicand will be =: 0, and the quotient — 14 : adding — 8 to ll, and 4 to 0, j/ — 6 and z zzi i. The other quantities are found in the same way as in the former examples. Another method, not materially different from the foregoing, is also prescribed for the solution of this question *. A "Tcat part of the next example is not intelligible to me. Mhat I can make cut is this. To find .r so that —-. — — y, — : — = .: Avhole numbers. Taking values of x in these equations the following are found 6y = 55—1, oz^z \v — 1, and 4t' = 3;^ — 1 ; from this last w = 3 and v = 2, 7 but these numbers giving r a fractional value of z., new values must be sought for w and v. Then after some part which I cannot understand, the author makes ic = .3 -f Au, and says u is found = 4 ; then a- = 19, i- = C + 3«, r — 14. After 4p \ more, which I cannot make out, he finds - — : — = 11 =1 r by means of t winch ' .o he adds to v and finds zn 15. After more, which I can make nothing of, he finds 7/ = 9 and x — 59- The next example is, what three numbers are those which when the first is multiplied by 5 and divided by 20, the remainder and quotient will be equal ; and when the second is multiplied by 7 and divided by 20, the remainder and the quotient will be equal, with an increase of 1, to the remainder and quotient of the first ; and when the third is multiplied by 9 and divided by 20, in like manner, the remainder and quotient will be equal with an increase of 1 to the remainder and quotient of the second? The first remainder is called x, the second x + 1, and the third, X + 2, and these are also the quotients. Let the first number be y. By the » From Uiis place there is a great omission in my copy as far as the question 7a.' + Si/^ =z Q, and Ix'' — 8^* — 1 = D> in the next book. Mr. Burrows's copy, however, being complete in this part, I shall proceed to supply the omiKion in mine from his. OF INDETEllMINATE PROBLEMS. 73 ciuestion ^^ — x + — , whence x ~ — . Let the second number be z, then '-- z= A- + ] H , whence x — -■ "■-. Let the tliird number be v, then 20^20 - 1 Qv X + 2 t)t' — 4'2 — :r ,v + 'i H ^rr~' ^vhenceA■ — ^^ — — . From the first and second values 7; o[ _ _ Px' 2 1 of ,1; is found — ^- n ?/, and from the second and third '■ ^ — z. From J ^ 7 tliis last is found by the operation of the multipHcand .: = 6 and » = 7, and 9 is called the augment of:, and 7 the augment of v ; as this value of ;: does not give y integer, other values must be sought. The augment of z is directed to be called tc, and the value of zv is to be sought; zv is found rz 3, and its augment 5 ; 33 is found by multiplyins: 3 by 9 and adding 6 ; at last the required numbers are found 42, 33, and 28. ■\Iost of this example after that part where ;: is found — 6, is unintelligible to me. It appears only that new values of - are found from 6 + 9a;, and that to and its values arc found zo — 3 and 3 + 5u, and from zv zz 3 the numbers are found. I suppose the question is solved much in the same way as such questions are now commonly done. X — 1 g' — 2 ^ x~3 m ^ ^- • X— 1 X — (1 X ~ 3 ~ '~3 " ~5 ^ Ihe next question gives -5 2 ' 3 ' 5 to iind .V so that all these numbers shall be integers. Let the number required be x ; let the first quotient be 2j/ + 1, this multiplied by the divisor 2 will produce for the dividend 4j/ + 2, and 1 being added for the lemainder ^ ir 4j/ + 3. In like manner the second quotient being assumed 3c + 2, ()z + 5 zz 4i/ ; from this last, by the operation of the multiplicand, find z zz 3 and 3' = S, and the augment of :: is 4r, and that of ?/ is 9v ; then x= 4j/+ 3 n 3 + 4 X (8+9r;) =:: 35 + 36v. As the value of y, 8 will not answer for w in the third condition, proceed thus: Let the third quotient be 5u + 3. Multiply by 5 and add 3, 25m + 18 ::: *, this is = 35 + 3Gv, hence 25« — 17 =: 36v; then by the operation of the multiplicand uzz 5 and v zz 3, 36v =z 108, 25 X 5— I7 + 35 = I43 =z X, and as « =: 5 + 3^x0 and v =: 3 + 25a', the augment of ,v is 9OU because 25 X 3*^) zz 900. The next question is to find two numbers r and s such that , ~ — 5 6 .i -i , , and — - — are integers, lo hud other numbers K /^ OF IXDETEKMINATE PROBLE.MS, besides 6 and S. Let the first number be o.v + 1, and the second 6.v + 2, the difference is *• + l. Divide by 3 ; suppose tlie quotient y and the remainder 2 ; then X + I — 3 1/ + 2, and x — 3j/+], and i.r + 1 the first niunber — Ijij + 6; 1 , ,1 1 • ■ , . , T 33?/ + 14 and (jx + 2 the second number — lSi/+8 ; then- sum is 3b y+ 14. Let — ;: ■ — ;: -) : then ^^ = i/ zz , — : from this is found y iz 3 and z =: 12, or ?/ = 0 + Set; and z zz \ -^ \\w; hence 5«+l:=45w + 6, and 6a; + 2~54r<74-8. " As the product of these taken according to the question involves zc^', and would '' be a long work," suppose ^5xo + 6—51 and let tlie second number be as it was; throw out 7 from both, the remainders are 2 and 5xo -\- I; take their product, it is lOro + 2. Divide by 7 ; suppose u the quotient and 6 the re- . , 10a) + 2 & ^ 7^^ + 4- „ , . . mainder; =: u -\-— lOro + 2= 7?< + o, — — — zz xv\ from this is found rrr=6 ov 6"-l-7r. The second number being 54w + 8, it is 54X6 + 8 — 332, and its augment is 5 !• X 7u — 3781^. As the first number is \5xo + 6, and -was supposed — 51, its augment is 45 X I'c. The next is, what number is that which being multiplied bv £) and 7 and the two products divided by 30, the sum of the two remainders and two quotients will be 26. " Suppose the number x, multiply it by 16, it is l6'x, for if 1 had " multiplied separately by 7 and 9, by the first figure of the second book, it would " also be \Qx\ Let the quotient of \Qx divided by 30 be y, l6z— 30j/ is the re- mainder, add the quotient 7/ ; l6.t; — 29j/ =26, and ~^^;^~" = '?". The augment being greater than the divisor, subtract S from 26, it is 10. By the operation of the multiplicand, the quotient is found 90 and the multiplicand 50. From 90 subtract the 29S and from 50 the \& ; 3 and 2 remain. Take 3 from 29 and 2 from 16, 26 and 14 remain. As 16 was once rejected from the augment, add 1 to 26, X — 27, and the quotient is 14 and the remainder 12. No new values can be had in this cas(; by the augment, for then the quotient and remainder would be greater than 27. The next is, what number is that M'hich multiplied by 3, 7, and 9, and the products divided by 30, and the remainders added together and again divided by 30, the remainder will be 11. Suppose the number x; let 19^ be divided by 30, and let the quotient be 3^, then \', and then .1' will be =: rj/ and z =pj/, or apply the rule given at the end of the 6th chapter of the introduction for the case, when a zz □• Required* ,£■ and j/, such that 7.i'' + 8j/' — Q, and 7.t'^ — 8?/' + 1 iz Q At this place my copy comes in again. 30 OF I-VDETERJIIXATE PROBLEMS. Yx"- + St/- is supposed =: 2\ The operation of nuilti,plicatinn of the square is directed to be performed, 7^i'^ beins,- the imdtiphcand, and Sj/' tlie augment: " I " suppose 2 the less root, and multii)ly its square wiiich is four by 7 the niuUi- " plicaud; it is 28 ; add S, the square is 3() ; 6 then is found the greater root ; " gblaek then is tlie quantity of blue, and 2 black tlic (pnintity of the unknown.'' Thus z is found —dj/nndx = 2j/. The second condition is 7.i'' — Sj/'' + 1 = D> whence by substituting 2j/ for .r; 2Sj/' — Sij"- + 1, or 20//' + ! — □=: a\ N(>u' by the operation of multiplication of the square, supposing the less root, and 20 X 2''+ 1 =81 —zv''; whence zc — 9. Tlierefore .c — 4 and ij zz 2, supposing 36 the less root, .r will be n 72 and j/ — 30". In the next Example .r and ?/ are recjuired such that .r' +j/'= D- ^'I'l- ^i- _l_ J/ — Q. The multiplicand being a square let the augment be divided by i/. Then by a rule of the 6th chapter of the introduction" — ^ — .r*. Let' — ^^ = w\ then j/^ +_(/ = 2wS TVIultiply by 4 and add 1, 4i/ + 4// + 1 —Su-- + 1. The root of the first side of the equation is 2j/ + 1. Find the root of the second side by the operation of multiplication of the square, supposing 6" the less root, 17 will be the greater; now 2j/ + 1 =17; whence j/ = 8 and x =z 28 Other values of 1/ and a- are 4y = j/, and ] 176 = .r. Another method of solving this question is given. Supposing one of the numbers e.r' and the other 7..T' ; the sum is p.x', which is the square of 5x. The square of the hrst added to the cube of the second, is 8x^ + 4£).r' ; let this l^e = j/'; divide by x^, the quotient is 8x' + 4J). Perform the operation of multiplication of the square, supposing 2 the less root, 8 X 4 + 49 = 81 =9'. Therefore x =r 2, and the first number 2^;' is = 8, and the second 7^'' is=28 ; and supposing 7 the less root, 21 will be the greater root ; then x = 7, and the first number will be 98 and the second 343. Ru/e. " If a square is equalf, the root of which cannot be found ;[;, and in Viz, If A=/)» (supposing Ax' + B=^-), then ■ ■ '■ = j/ ; and ■ t Here seems to be an omission. % If the number can be reduced to the form (a:» + 7;y/)' +rj*, it become; rational hy makhg ax + mj/ z^ — - — 1/, for tlien ( — -^-S/) + rj^' = V — 9^). '" ^^'' Burrow's copy this rule begins, " If there are two " fides, the root,'' &c. OF INDETERMINATE PROBLEMS. .81 "which there are two squares of two colours, and the rectangle of those two " colours; take the root of one square, and find from the second square a root, so " that from the two squares that rectangle may he thrown out. " For example: In the second side is 3(5 square of unknown, and .'36 square of " hlack, and 36 rectangle of unknown and black. Take tiie root of 36 square of " unknown, 6 unknown ; and from 36' square of black, take the root of 9 square " of black, 3 black. When we take twice the rectangle of these two roots, the " rectangle which is also 36 will be thrown out ; and from the squares 2/ square " of black will remain. Divide whatever remains by the colour, of which this is " the square ; and from the number of the colour of the quotient, having sub- " tracted one, halve the remainder, make what is obtained equal to that root " which has been found. After dividing the second by the first, the quantity of " the first colour will be obtained." In the next example x and ^ are required such that ^ + ?/^ + o-j/ = D, and V'(.i' + j/' + .ij/) X {-v -]- 1/) + I — n. "The first e(}uation being multiplied by " 36 gives 36,i''' + 36j'i/'^ + 36j:i/ — 36.3\ The root of one square and part of the " second square, the rectangle having been thrown out, are found 6 unknown, and " 3 black : there remains 27 square of black." Then applying the rule, .r is ,. , 5y , . <25ir , , , Q5i/ . ^ , 5i/ 49y* tuuntl n ~, whence a" = —^, and .z- + y + .rj/ — — — + j/ + -?rj/ — —^ = (J1), andv/Cr^+y-l-.j/)x(.+i/)+l=f (f + 3/) + 1 = ^^^ make this —'U!\ then 56j/- +9r:9?i;\ Then root of Qxt)'' is 3cv, and by the oi)eration of multiplication of the square, making 6 the less root, 45 will be the greater root. For 56 X 36 + 9 = 2025 =: 45"; therefore y — 6 and .r — 10; or making 180 the less loot, j/ =z 180 and .v zz 300. The next question is: Required .r and 3/ such that '-^^ — = ^, and ,z'' +3/* = n, and .?• + 3/ + 2 = P, and *■ — 3/ + 2 =r a, and .i'* — y- + S — D, and ' ^'^t^ + ^^''' + -^'^ + ^^''' + i/ + -) + ^(■'' - y + 2) + v/(.r'-3/^ +8) — n. It is plain that 6 and 8 will answer the above conditions. Pass them and find two others. It is required to find them by means of one unknown quantity only. Suppose the first number//— 1, and the second 2;). Then ■'■V + 3/ _ (P^~ 0 X^p + Qp _ 2/>^ - 2/7 + Qp _ ^,^ ^^^i ^,. ^ ,^^ _ j^, _ iiS OP IXDETEUMINATE PKOBLEMS. op' + 1 + 4// -p'' + 2/+ \—ip+ \y. And .r + ?, + 2 = (/>* - l) + 2/) + 2 rr /;' + 2/^ + 1 = {p + O'- And .r' — J/' + 2 =(/? — !)'. And .r'— ^'- + 8 = //— 2p'+l— 4/>* + 8=//— 6/>'+9 = (// — 3)^ ; and the sum of the roots is equal to P + (/•''+ n + (p+1) + (/>-!) + (/>'-3) = 2/ + 3/> - 2. As the root of this cannot he found, make it equal to 9': then ^p^ + 3p = y- + 2. Multiply ])y 8 and add 9 ; K^/J' + 24p + 9 = B9' + 25. Find the lOot of the lirst side v/(l6/>' + 24/; + 9) = 4/> + 3. For the root of the second side perform the operation of multiplication of the square. Suppose the less root 5, tlie greater root will i)e 15 ; for 8X 25 + 25 n 225 = ]5\ Make the root of this ecjual to that of the lirst side 4/> -f 3 = 15, whence p =3. In this case x = 8 and _?/ = 6 ; 41 1677 making the less root 30, the greater will be 85 ; p will then he — and x =: ■ -. — and?/ =:: 41 ; or making the less root 175, the greater will be 495, /; z= 123 .f =: 15128, and y n 266. Or .r may be supposed = p' -\- 2/>, and ?/ — 2. Or .1' = p' — Qp, and y — Qp — 9,. Or .r — p"- + 4/) + 3, and y ~ ^Ip + 4. And the numbers required may be brought out in an infinite number of wavs besides the above. Here follows an observation, that in calculation, correctness is tlie chief point ; that a wise and considerate person uill easily remove the veil from the object ; but that where the help of acuteness is wanting, a very clear exj)lication is necessary. *' And so it is when there is such a queition as this : What two numbers are those, " the sum or difference of which, or the sum or difference of the squares of " which, being increased or lessened by a certain number, called the auguient, " will be a square. If examples of this sort are required to be solved by one " colour only, it is not every supposition that will solve them ; but first suppose " the root of the difference of the two numbers one unknown, and another " number with it either affirmative or negative. Divide the augment of the " difference of the two squares, by the augment of the sum of the numbers, and " add the root of the quotient to the root of the supposed difference aI)ovemen- " tioned ; it will be the root of the two numbers. Take then every one, the " square of the root of the difference of the numbers, and the square of the root " of the sum of the numbers, and write them separately. Afterwards, by the way " of opposition add and subtract, the augment of the difference, and the sum of " the two numbers aforementioned, as is in the example, to and from the squares of " the two, which, by the question, were increased or diminished. The result of " the addition and subtraction will be known, and from that the two numbers OF INDETERMINATE PROBLEMS. 83 " may be found in this manner, viz. by the rule ~ ■ ^^ ~ ^' — j. ^nd The next is to find .v and 3/ such that x + y + 3 zi □, and ,r _ j/ -f. 3 — □ and .r' +f — 4= Q, and .r -/ + 12 = □, and ^ + j/ = g^, and the sum of the roots + 2 = Q- Exclude 6 and 7, which it is plain will answer. \/(.t' — 3') is supposed — p — 1, then x is made equal to />' — 2 and j/ = 2/\ wherefore x+ y + S — (// — 2) +2/> + 3 =: (/; + j)^ and .r — j/ + 3 = (// — £) -2/) + 3= (/>-])', and .r+j/'-4=: QA-^p' + A) +4//-4= (/;=)', and .r^-j/' + 12 = (;,*_4/ + 4) -4/+12 = (//-4)^ and ~ +y ~ (/^ "^"■^ '^■' +'2p:=p\ and the sum of the roots +2 = {p+\) + (p—'i) +/>'+ (/*"— 4) +p+Q—2p^ + 3p—2 ; make this =: q^ ; 2// + 3j!>— 2=<7^ and 2p''-h3p = ij^-\-2. Multiply by 8 and add 9. l6p^ + '24'p+9 = Sq'-\-^5. The root of the first side is 4/;+ 3. Find the root of the second side by the operation of multiplication of the square ; making th^* less root 175, the greater root will be 495. Tlierefore 4p + 3 — 495, and/;— 123, and .1'::= 1.3 127, andj/zz24f). The next is : Required .v and 1/ such that x''-—i/^ + 1 = D- ^"f' ^' +j/'+ 1 — '-'. Let .v^=oJ5''— 1 and j/=' = 4/;', a-^— j/' + l =;/ and x^+f+ izz{3p)\ The root of 4/>'' is 2/;. Find the root of 5/?'— 1 by the operation of multiplication of the square. Su])posing the less root 1, the greater will be 2. Supposing 17 the less root, the greater will be 38. Or if x^ + y^ ~ 1 ;r n, and x'^ — y'- — i ~ q, Jet A''' =: 5yy^ + 1 and y' r: 4/;* ; and so on as in the first case. Ride. " When the root of one side is found, and on the second side there is " a colour, whether with or without a number, equate that side with the square " of the colour which is after it and one unit. And bring out the quantity of " the colour of the second side which is first in the equation ; and bring out what " is required in the proper manner." Example. To find ,r and y such that 3x + 1 — n , and ox +1 = 0. Let 3x + \ = {3z + ly, then x =z 3z^ + 2z, let 5(3.s'- + Sz) + 1 = w'; Avhence 15-^+10; — a;*—l ; multiply by 15 and add 25 ; 225=' + I50z + 25 r= 15r, then X — 10. In like manner any number whatever being put for y the value of X will be found. The next is to find le, x, y, z, such that (za+x+y+ z) 20 =. uwyz. Suppose the first ?£', the second 5, the third 4, and the fourth 2; then 20u' + 220 :=40ci:;, and a' z= 11. Other values of tr, .?■, y, z, are taken notice of. The next is to find .i- and y in integers such that \/{x +y + xy + x'' + y') -{- X + y ~ 23, or = 53. In the first case, suppose the first number x, and the second 2, then v^Cc' + 3,r + 6) + x + '2 — 23, and \^ (.r* + 3x + G) ~ 21 — x, 29 and .r' + 3,r + 6 rz x'- — 42i' + 4il ; whence x will be found z= ~ ; this not being an integer, let the operation be repeated. Suppose y — 3 tlien x will be SS OF IN'rETZUMIXATE PR0DLEM3. fouad ^ -~7 ; this roo being a rraction, suppose y := 5 ; tnea x wili bo = 7. in the second case a number is put for v, aud a fractional valr.e of s is found. " And if v.e suppose the second number 11, the quantity ei' the unknown will " be 17, and this is contrary; for if the second number is supposed 17, the quau- *• tity of the unknown vr'nl be 11 ; and if one is suppjseu a coiour and the otiier a " certain number, it is probable that the unknown will be brought oat a fraction; " and if a whole number is required, it may be found by much searv.h. And if " both are supposed coloui-s, and the question sohed by this rule, a whole '' number will easily be found," Jiule*. •• When two sides are equal, the method of equating them is thus : •' subtract the rectangle of one side from the other side, and besides that what- " ever is on the second side is to be subtracted from the fiist ; then let both sides " be divided b\- the rectangle ; and on tiie side where there are colours let those " colours be multiplied together. And let a number be supposed, and let the " numbers which are on that side be added to it ; and let the result be divided '' by the supposed number ; and let the quotient and the number of the divisor " be separately increased or lessened by the number of the colours which were '' before multiplication, whichever may be possible. Wherever the unknown is " added or subtracted there will be the quantity of the black ; and wherever the " black is added or subtracted there will be the quantity of the unknoMn. And " in like manner if there is another number, and if both addition and subtraction " are possible, let both be done, and two different numbers will be found. Also " if the number of the colours is greater, and cannot be subtracted, subtract the " quotient an I the number of the divisor from the colour if possiijle, what was •'• required will be obtained/" Example, x and u are required sucli that 4.i + S?/ + 2 n: ly. Multiply 3 by * Thjs rviie is very ii expressed ; it miist raean — The equilion being reduced to ex -i- bv -^ c := xj^, c 4- ' will he:= y and i -J-_; ^ i. Becauje ax-j-!'y-\-c=zxy, c =: ij. — ax — ty, add ab to both s'.de?, thent^-J-c = iL — ju — ffj -f '^ = (^ — ^} 0 — '')■ 3nd making p=:x — -',> — a will be = . Therefore jr = i -J- c, and jr = e -| . More ibnnulz inay be had bv resolving ai-\-c into different factors. or IXDETERJIIVATE PROBLEMS. §9 4, and rulcl a to tlieprotluct 3X4 + 2=14. Suppose]. Divide 14 bv 1. Add 4 to the iiun.ber 1, and add 3 to the quotient 4+lz=.5=j/, and 3 + — = 17 = j\ Or 4 + — = IS =r J/, and 3 + 1 = -ix. " And no other case is possible." Di- viding by 2, the quotient will be 7 ; j/ = 1 1 and x = 3. And by another method j/ will be fouml = 6 and j- r: ]() *. To find .r and 1/ so tiiat lOx + Hi/ — 58 = 2xi/. After reducing the equation to ox + 71/ — 29 — -ly. By the rule above given, assume divisors of 5X7 — 29 ; 1 being the divisor, 6 is the quotient. o + 1 =: G = y and 7 -\- 6 — 13 — .r, or 5 + 6 = 1 1 = j/ 7 + 1 = 8 — X, or 5 — \ — i ~ y 7 — 1 = 6 — x\ 2 being the divisor, 3 is the quotient. 3/ = 8, y — 7, y — % .!'=:£), a; z= 10, j; 1= 4; and no otliers are possible. 3 bein^r the divisor 2 is the quotient, and the quan- tities are as above. It is added that these two examples may be proved by geometiical ligures as well as numbers. * Ip. Mr. Burrow's copy there is another example which is wanting in mine. It is as above. THE END. Mi% Davis's Notes, -"•«09»»— J. HERE put together all I have been able to make out of Mr. Davis's notes of the Bija Ganita. What I have extracted literally is marked by inverted commas ; the rest is either abstract, or my own remarks or explanations. I have preserved the divisions of the Persian translation for the coiivenience of arrangement and for easy reference. Mr. Davis's letter to me, authenticating these notes, is annexed. Chapter 1st of Introduction. JL he manner in which the negative sign is expressed, is illustrated in the notes by the addition and subtraction of simple quantities, thus: " Addition —When ** both affirmative or both negative, &c. When contrary signs, the difference *' is the sum. 3 3 3 3 4 4 4 4 *' Subtractiott. 3 3 3 3 2 2 2 2 NOTES. 91 •' Multiplication. " When both are affirmative or both negative the product is affirmative. Q X 3 = 6, '^ X S = 6, 2 X 3 = 6, BX 3 = 6. " Why is the product of two affirmative or two negative quantities always " alifirmative ? The first is evident. With regard to the second it may be ex- " plained thus : Whetlier one ([uantity be multiplied by the other entire, or in *' parts, the product will always be the same, thus: " 135 X 12 = 1(320 " 135 X 8 = JOSO " 135 X -i = -540 1620 '■ Then, let 135 be X by 4, but 12 — 4 = 16 and 135 X 4 = -540; 135 X 16 '• =2160, and 540 + 2l60 =: 2700, Mhich is absurd: but'540 + 2l60 = 16:^0." Mr. Davis remarks to me that there are here evidently some errors and some omissions, and he thinks that the meaning of the last part of the passage must have been to this effisct: 12 may be composed of \6 added to 4. Let 135 be multiplied by 12, so composed 135 X 16 = 2160 135 X 16 = 2l60 135 X 4 = 540 15 X 4 = 540 135 X 12 = 2700 This is absurd : but 135 X J2 = I6'r' + 3^' + 3. Also " The multiplication of unknown quantities. " To multiply 3j72 I into 2 we have ^i. q7 by qT gives its square or Sff ^' ''^"'^ *^^'^ multiplied by 7^, gives 3jj r^, &c. Also this example of multiplication. "3TT3 "^2 ^5 7^3 ^1 ^1 ^]0 which is the product of (5.r — l) X (3>r + 2). ^2 ^TT^Ii I ^7 \ ^2 Chapter 4. « OF THE CARNI OR SURD QUANTITIES.' " Example of two numbers, 2 and 8. " 2 + 8 iz 10, the mahti carni. " 2 X 8 = l6; its root is 4, and 4X2=8 the laghoo carni. " The mahti carni .... 10 " Laohoo carni added . . 8 " the sum of their roots. 1 8, the sum of these carnis. This 1 8 is the square of 54 XOTES. And thei-e is another example with the numbers 4 and 9, and the following o o theoiem, " 2)8(4., its root is 2, + 1 — 1 3 1 "3X3=9 9 X Gi = 18 sum. "1X1 = 1 1X2=2 difference." Also this : " The carni 18 is found ; its root is the sum of the roots of the two "given numbers; but if there be two roots there must be two squares, the " difterence is the square of the difference between these squares." And the following examples in multiplication : " To multiply the square roots " of 2, 8, and 3, by the square root of 3 and the integral number 5. " These are surds, therefore take the square of the sum of the square roots of 2 and 8, and multiply by the square of 5. " Square of sum of square roots of 2 and 8 is 18. 18 I 25, 3 I 450, 54 I 3 I 25, 3 j 75, 9 I root of 9 is 3 roop. " Example second 25 27 Sqrs. Sqrs. Sqrs. 450 54 75 — roop 3. 3nd. Foop. Carni. Carni. " Multipliers 5, 3, 12. 25, 3 6'25, 75 25 roop 25 roop 675 25, 3 675, 81 9 roop 9 \6 roop 75 300 "The product therefore is 16 roop, 300 carni." The square of a negative quantity being made negative is here taken notice of as in the Persian Translation : In division the following rule is mentioned. *' The carni divisor : reverse of each terra, its sign, and multiply both divisor " and dividend." Carni which here means surd, means also the hypothenuse of a right-angled triangle. NOTES. gs. Chapter 5. Vf HAT is that number by which when 221 is multiplied and 65 added to the '* product, and that product divided by 195, nothing will remain. " The dividend bhady, divisor hur or bhiijuk, the number added or subtracted " is called chepuk. The bhady is here 221, the bhujuk 195 ; when divided the " quotient is 1, this is disregarded ; the scke or remainder is 26, by which 195 " divided the quotient is 7 disregarded, tlie remainder is 13, by which divide "221, the quotient IS 17, the remainder is 0. The quotient 17 is the true or " dirt-bhadif. *' Then i95 divided by 13, the quotient is 15 ; the remainder 0. Tliis quotient " is named dlr I- bhujuk. " Then divide 6"5 by 13, the quotient is 5 ; the remainder 0 ; the quotient is *' the dirl-chepuk. " They are now reduced to the smallest numbers. "17 d ill- bhady. " 15 dirl-bhujuk. *' 5 dirlchepuk. The quotients are found and arranged as in the rule with 5 and 0 below, " 1 thus : 7 0 this is called biillee ; the cipher is called unte or the latter ; the next (5) *' is called upantea. Multiply this by its next number (7) and add the next below '* 5, this being 0, the product will be 35. Multiply this by the uppermost number " (1) and add the next below (5) the amount is 40." 96 NOTES, Then 40 and 35 are directed to be divided by the dlrl-bhady and bhujuk. 17)40(2 34 6 tliis is called lubd. }5) 3.5 (2 30 5 this is called gootmk, and it is the number sought. — — ^ ~ 6, and directions are 2,'iven for findins: new values of .r and i/, (supposing ^^ — ^^ — i/) by adding rt (in its reduced state) and its multiples to the value of j/ ; and b and its multiples to the value of .r. The next question in the notes is also the same as that in the Persian. " Bhady 100, bhujuk 63, and chepuk 90. " OPERATION. " These numbers cannot be all reduced to lower proportionals, " 100 divided by 63, the quotient is 1, the remainder 37 ; by this remainder " divide 63, the quotient is 1, the remainder is 26 ; by this divide 37, the quotient " is 1, the remainder 1 1, Divide again ; quotient 2, remainder 4. Divide " again ; quotient 2, remainder 3. Divide again ; quotient 1, remainder 1 ; this re- 1 " mainder 1 is disregarded. The several quotients write down thus : 1 1 2 1 90 the cliepuk. 0 NOTES. 57 " Multiply and add from the bottom as in the former example, 90X1 +0— 90, "90x2+90 = 270, 270x2+90=630, 630X 1+270=900, 900Xl+630=15oO, '* 1530X1+900 = 2430. " The two last are the numbers sought ; then " 100)2430(24 this is disregarded. 200 430 400 30 Seke or remainder is the lubd. "63)1534(24 126 270 252 18 this is the goonuk." 100 X 18 + 90 _ ^.. 63 -■^''' The method of reducing the bhady and chepuk is noticed, and the values of X zz 171 and y = 37, being first found the true values are found, thus : 63)171(2 and 10)27(2 126 20 45 7 63 — 45 = X and (10 - 7) X 10 = 3/. The several methods of proceeding: first, by reducing the bhady and chepuk; second, by reducing the bhujuk and chepuk ; third, by reducing the bhady and ehepuk j and then the reduced chepuk and the bhujuk are also mentioned. The following explanation of these reductions is given : *' The bhady 27, bhujuk 15 ; " these are divided each by 3 .... 9 and 5. " Write 27 in two divisions 9 anrl 18 " these again divided by 3 3 and 6 " these two add 3 + 6=9; thus the parts added, how many so ever arc, always " equal to the whole, thus therefore they are reduced to save trouble, and there- " fore all these numbers are so reduced ; but the goonuk is as yet unknown. Let N gS NOTES. " it be supposed to be 5, by wliicli multiply the parts of the lihady 9 and 18; '«g X o = 4J, 18 X 5 :-- 90, w^iich added aie 135, and the bliady 27 X 5 = " the same 135 ; tliis divided in two parts, 60 and 75, and added again, are 135. " The lowest terms of 27 and 15 above, are 9 and 5 ; the conimun measure 3, " mulliplied by 5, 3X5 = 15 and .9 X 15 = ]S5. " Thus too the chepuk must be reduced, and when they are all reduced to the " lowest, the lubd and gounuk will be true; and if their numbers are not reduced " to tlieir lowest terms, the work will be the gieater." The principle on which the chepuk is reduced is explained thus : " OF THE CHEPUK." " The bhady 221, bhujuk ]95, chepuk 65; the goonuk was found 5, lubd 6. 001 X 5 = 1105 195)] 105(5 lubd. 975 130 seke, M'hich deduct from the bhujuk 195 — 130 1= 65 equal " to th.e chepuk, which divide by the bhujak 195)l9o(l. The lubd is 5, to " M'hich add 1 ; 6 = the original lubd." In another example the bhady — 60, bhujuk ir 13, and chepuk =z I6 or — 16, By the bullee are found the numbers 80 and 36S ; then 368 — 60x6 = 8 the lubd, and 80 — 13X6 = 2 the goonuk ; 60 — 8 = 52 the lubd corrected, and 11 1 ^j 60X11 + 16 _ 60X2—16 13 _ <2 = U the goonuk corrected. — =52, and — - — —8. 1 y 13 " Note in the text : The product by the two uppermost terms of the bullee " when divided by the bhady and bhujuk respectively, have hitherto always " quoted the same number, as in the last example 6 the quotient, and the hike " also in the foregoing examples, but when it happens otherwise, as in the fol- " lowing : When the bhady is 5, the bhujuk 3, the chepuk 23 affirmative or " negative, what will be found the goonuk ? 3)5(1 1 — 1 2)3(1 — / Bullee. 2 23 1 seke disregarded 0 23 X 1 = 23, + 0 = £3 5 ) 46 ( 9 3 ) 23 ( 7 23 X 1 = 23, + 23 = 46 45 21 1 2 goonuk NOTES. 99 " The two quotients being different numbers they must be taken the same ; " thus instead of 9, take the quotient 7. 5)46(7 35 11 " therefore the goon is 2, the lubd 1 1. ^ ^ ^ "^ ^^ r: 1 1. " Next, when the chepuk is negative, or to be deducted, the rule directs to " subtract the hibd from the bhady, but here it cannot be done : the rule is *' reversed, thus 11— 5:^:6, which is the lubd for the negative chepuk; next for "the goon of the rhin chepuk S — Q—\ ; therefore the goon and lubd for the " rhin chepuk are 1 and 2 ; iX 1 :=:5 ; but from this the rhin chepuk cannot be ** taken; therefore take it from the chepuk 23 — 5 = 18. "3)18(6 the lubd." 18 Other cases are mentioned for the negative chepuk, and for the chepuk re- duced, and for new values of the goon and lubd. The examples — — — and ■ ' — , which are in the Persian translation, are also stated here, but no abstract of the work is given, only the lubd is said to be 5 and the goonuk 0, which applies to the last of the two only. " The seke in bekullas is termed sood, meaning that it is the chepuk ; the " bhady, let it be 60 The coodin or urgun is the bhujuk, from which the lubd *' will be found in bekullas, and the goon will be the seke of the cullas, which " must be taken as the chepuk ; making the bhady again 60, the bhujuk will be " the urgun, the lubd of this will be in cullas, the seke is the seke of the ansas, "which seke must be taken as the chepuk; the bhady being taken 30, the " bhujuk is still the urgun, the lubd is in ansas, the seke is the seke of the signs, V 2 300 NOTES. "which seke take as the chepuk ; making the bhady 12, the-bhujuk will be still " the coodin, the lubd here will be signs, the seke is the seke of bhaganas, " revolutions, wliich seke must be taken as the chepuk ; the lubd will here be " in bhaganas, the seke the urgun." Example. " Let the calp coodin or urgun be 19, the bhaganas 9, the " urgun 13." " Then by proportion if 19 gives 9, what will 13 give?" This is found to be 6 rev. 1 sign^ 26°, 50', 31", with a fraction of ] 1 ; then from — '—— = j/, .r and j/ are found Jc—\0,]/ — 3l ; then from -— — = y\ y' — 50 and .v' zz 16, QOr" \6 12.1'"' — 17 from ^"-"^^ = /', /' = 26, x" = 17, from :^j-^ =/", y-=l, x"'-3 qx"" 3 from —y"", y""z=.6 and a?'"' r: 13, which is the urgun. In another Example. Seke bekullas =: ll", bhaganas =r 49, calp coodin or urgun =149, Jeist urgun =97. The quantity is found by the rule to be = 23 rev. 10 signs, 18°, 23', 3l", the remainder 11. " The addy month 1, is the bhady ; the coodin 195, the bhujuk ; the seke of ** the addy mouth 95, is the chepuk. " 19^)1(0 0 1 seke disregarded 95 bullee. •' 0 X 95 + 0 = 0 Raas^ *' 1)0(0 19^)95(0 0 lubd 95 goonuk, «« 95 X 1 - 9o = 0 195)0(0 0 NOTES. 101 ** The che tits 26 is the bhady ; coodin 2'i5, is the bhujuk ; abum seke 220, " chepuk. " 225)26(0 0 Raas 660 26)225(8 208 17)26(1 17 8 1 I 5720 26)660(25 10 seke the lubd. 9)17(1 9 8)9(1 8 1 220 0 225)5720(25 95 seke the goonuk. 1 dis regarded. 26 X 95 - 220 _ j^ 225 " Hence the chandra days are BS." The last rule of this chapter is taken notice of as follows : " OF THE SANSTIST COOTUK." " By what number may 5 be multiplied and divided by 63, the remainder will " be 7 ; and that number so found, when multiplied by 10 and divided by 63, " the remainder will be 14. " The two goonuks are 5 and 10, the sum is the bhady ; the two sekes are " 7 and 14, the sum is the chepuk; the bhujuk in both is the same or 63." The question is solved as before; it ends "Thus, however numerous be " the goonuks given, let them all be added for the bhady ; and the same with " respect to the given sekes for the chepuk ; the bhujuk will be always the *' same." > :: E ]02 NOTES. Chapter 6. " THE CIIACRA BAL.V.- X HE multiplication of the square is a cliacia bala. Tiiere are six cases : " The first quantity assumed is called hiirsiia (the smallei) ; its square must be " multiplied by the pracrit, and then must be added tlie chcpuk; that is sucli a " chepuk as will by addition produce a square, and this chepuk may require to be " affirmative or negative, which must be ascertained. The root of this square is " the jm/ ; these three, the canist or hursua, jeist, and chepuk must be noted " down and again written down." The distinctions of samans babna and anter habna are given as follows : " OF THE SAMANS BABNA"" " When the jeist and canist are multiplied into each other (budjra beas) " the sum is the bursa or canist. It is called budjra beas from its being a tri- " angular multiplication ; the upper, or jeist, or greater, being multiplied by the " lower, smaller, the canist; and the canist multiplied by the greater or jeist ; " the two products added is the burs. " The two canists multiplied together, and multiplied again by the pracrit, " then the product of the two jeists— added altogether, produces the root of the "jeist; the product of the two chepuks then becomes the chepuk." The anter babna is described thus: " The difference between the two products " or budjra beas, produces bursa or canist. The product of the canists nmltiply " by the pracrit, and the diiference between [this and) the product of the two "jeists is the root of the jeist, and the product of the two chepuks is the •' chepuk." The rest of this is very imperfect, but the cases of /J/J = b/>* and /33 =-i, and the rule a \-i J + 1 =:n) ^'^ plainly alluded to. (See notes on the Persian KOTES. 103 translation.) " Thus" (it is added) " the root of tlie canist and jeist n^.ay be in a " variety of cases found." After this there are examples the same as in the Persian translation, and worked the same way as far as the " Operation of Circulation ;" and, atVer the examples. " Hence, how various soever the 1st, from the somans babiia and anter babna may " be produced canist, jeist, and chepe ; and hence it is called the chacra bala." I find no abstract of the rule for the *' operation of circulation," but there is tlie first example, viz. 67.i'^ + 1 i= □, as follows : " Roop 1 is the canist, 3 is " the chepe; then the pracrit 67, canist 1, jeist 8. Hurs 1 is the bhady, chepe is " the bhujukj jeist 8 is the chepuk ; then by the cootuk gunnit " Bha. 1, che 8, " Hur 3 ; hence the goonuk 1. "then the square of 1 is 1, 67 — 1 ::^ 66, but this is not the smallest; then " 3 + 3=:6, 6+1=7; its square 49, deduct from pracrit 67—49 = 18; 3)18(6; "but the negative must be made affirmative 6; and 5x5=25, and ^5X67-=. " 1675, and 1675 + 6 = 1681 its root 41 ; then by the cootuk gunnit "Bha 5, che 41, « Hur 6 ; " then 5X5=25 and 61 — 25=42, 6)42(7* ; the lubd is the canist 11; 11x11 = "121, 121X67 = 8107; chepe is 7, 8107 + 7 = 8100 its root is 90, which is the "jeist; then by the cootuk " Canist is bhady 11, che 90, " Hur 7. "Here the goon is 2, che 7; 7 + 2=9 the second goonuk; its square is 81 " 81—67 = 14 ; 7)14(2 the other chepe." " The canist 27. This is made jeist 221. " Ca. 27. 221 jeist, che 2. " Ca. 27, 221 jeist, che 2. " Ca. 1 1934, jeist 97684, che 4. " Ca. 5967, jeist 48842, che 1. " The square ca. 35605089, which multiply by 67, and 1 added, the sum will " be 2385540964, 'and its root is 48842." BOOK 1. J. HE unknown quantities, &c. must be clearly stated, and then the equation " must be reduced in the manner hereafter shewn by Xj by -f-, by the rule of " propoition, by progression, ratios, by f^^^ ; still maintaining the equality. " When tliey are otherwise, add the difference ; then sodana the quantities ; the *' same with respect to roots. In the other side of the equation the roop must be " sodanad with the roop. When there are surds tliey must be sodanad with " surds ; then by the remainder of the unknown quantities division, the roop " must be divided ; the quotient is the quantity sought, now become visible.'" " Then the quantity so found must be utapanad, in order to resolve tlie " question." It will be remarked that the Persian translation has " thefgure of the bride," for that expression which is represented by P^'-v^ in the above abstract. Mr. Davis tells me that the original had nothing like a reference to Euclid, and that this part related simply to the proportions of right-angled triangles. There follow abstracts of the seven first questions of this book, with their solutions, Avhich are the same as those in the Persian translation. The first part of the firet example is : " One man had 6 horses and 300 pieces of <' silver, and the other had 10 horses, and owed 100 pieces of silver; their pio- " perty was equal. Qucere, the value of each horse, and the amount of the pro- •' perty of each person. Here the unknown quantity is the price of one horse. " Ja 6, roo 300 " Ja 6, roo 100 these are equal. " Ja 6, roo 300 " Ja 10, roo 100. Sodan, that is transpose. " Ja6 + 300 = Ja 10 - 100 • " Ja 4 = 400 " Ja =100 NOTES. 105 The third example, where the Persian tianslator has introduced the names Zeid and Omar, is in Mr, Davis's notes thus : " One man said to another, if you give me 100 pieces of silver I shall have " twice as many as you ; the other said give me 10 pieces and I siiall have si?: ' times as many as you. Qucere, the number each had. " Ja 2 roo 100 " Ja 1 roo 100 " Ja 12 roo 660 " Ja 1 roo 110 " DiflF. Ja 1 1 roo 770 " Ja roo 70 BOOK 2. J. HE square root of the sum of the squares of the bhoje and cote is the carna. " Explain the reason of this truth. Cel ^Vi^ / " The carna is ka ja ; the figure thus, ■"'L__\^ • Divide this by a perpen- " dicular ^?LA^^<»>«<«»»»» Extracts from Mr. Davis's Notes, taken from a modem Hindoo Treatise on Astronomy. " By the method of the Jeisht and Canist from two jyas* being found, others " may be computed by those who understand the nature of the circle (the bow * Jya or jaw; sine. — The modern Europeans acquired their knowledge of the sine from the Arabians ; and it is obvious that Ihey used the term sinus only, because the word jeeb ( V.^.>vOfe), by which the Arabians called the line in question, is translated si?ius indusii. The radical meaning of (c•^Jl.^.) is to cut, and it denotes the bosom of a garment only, because the garment is cut there to make a pocket; accordingly we fiod that t.^A-s. NOTES. J 1 I " and arrow), and thus, by the addition of surds, may the sum and the " diirerente of the arc and its sine be computed whether that arc be 90 deorees " more or less. does not mean bosom, hut that among tlie Arabians it signifies that part of their dress wlicre Ihe pocket is usually placed, and in s^.me languages which abound i[i Arabic words, as the Persian and the Hindoostanee, it is the common term, not only lor a pocket in thi- bosom, but for any pocket wherever it mav be. In all Arab c dictionaries this word is explained as above, and in some, though not in all, (it is not in the Kushfool Loghat) the line j/e call sine is given as a second meaning. The Arabs call the arc Adm (^^jj), which signifies cioa' ; the cord -xutr ( -J «), which is U\c 6ou-iiririg ; itid the versed sine suhum (j^j^), which is tlie arroa-. But the sine they express by a word which has no connexion whatever with the bow. The Mathematical history of the Arabians is not known enough for us to speak positively about the first use of sines among them, but there seems to be reason to suspect that they had it from a foreign source, probably from the Indians. The Sanscrit word for the chord isyau-, or more properly j'l/a amijiva. (For those terms see Mr. Davis'j paper in the second volume of the Asiatic Researches; the literal explanation of the words lias been given me by Mr, Wiikins,) and the sine is calletlj^a ardhi, or hall cord ; but commonly the Hindoos, for brevity, use /ya for the sine. They also apply the word in coiupjsillon as we do; thus, they call the cosine coliji^a, meanino the sine, the side of a right-angled triangle; the sine (or right sine) bhvjjya, meaning the j^w, the base of a right* angled triangle, and cramajya the iine moved; the versed sine they call oolcruniajya, or lhe.H«e moved up* wards; the radius they call friWy/a, orthe.v;«e of three, (meaning probably three signs.) In their term for the diameter yj-o/u'rji/a, or whole y^a, the word is used in its proper acceptation for chord, and not io\- jya ardhi, or sine. It seems as if i,.rf\X2!» andjya were originally the same word. Mr. Wiikins (the best authority) assures me tliat^i^a, in the feminiueyi'Dfl, is undoubtedly pure Sanscrit, that it is found in the best and oldest dictionaries, and that its meaning is a bow-string. The Arabians in adapting a term to the idea of chord, had reference to the thing which it resembled, and called it _J » or the bow-string; but having so applied this term, they liad to seek another for sine; then they would naturally refer to the name of the thing, and call it by some word in their own language, which nearly resembled that under which it v^'as originally known to them. This mode of giving a separate designation to the sine was evidently more convenient than that of the Hindoos, so I conjecture that L-va2^ lor sine is no other than the Sanscrit word 751a or yi'oa. It is remarkable that the Sanscrit terms for the sides of a right-angled triangle have reference to a boiv : Ihey seem to be named from the angular points which are lorrn«d by the end of the bow, the arm whicli holds it, and Uie ear to which the string is drawn; thus the side is called irl — Reduced. ChepuJi or citepe — Augment. lluUec — Chain or series. Vnte — Last. Upanlea — Last but one. J.nbd — Quotient. Giionuk or G^oii — Multiplicand. Rhiii — Minus — Literally decrease. Coodiii — An astronomical period. L'rgan — Number of days elapsed. llekuUas — Seconds. Cullas — Minutes. A'isa< — DeuTCPs, 118 NOTES. Bhaganas. — Revolutions. Calp — The great period, Raas — Literally a heap, a sum total, a constellation. ox -\- c CootuJc — The principle On which problems of this form -^—^ — rr y are solved, Sanstist — Ditto of -7^ = v + c and -^ = ; + e. 0 "^ 0 Chacra-bala — Ditto of ex* -\- b =. ?/*— Literally strength. Hursiia, hurs, hursa — x in the above form — Literally small. ^ Pracrit — a in Ditto — Literally principal. Jeist or Jeisht — y in Ditto — Literally greatest. Canist — x in Ditto — Literally least. Samans babiia — If av'" + ^ = y'^ and a/^ -I- /3 = g^, then the rule x" — x'g f y'f Is called samans babna — Literally contemplation of equal degrees. Anter bab/ia — In the ahoveiovm, when x" — A'g — y'f, it is called anter babna — Literally contemplation of diflerence. Badjra beas — Cross multiplication which produces the above forms —Literally cross diameter. Cootuk gunnit or cutuca ganita. — Cootuk Calculation. Sodana — Reduce — Literally purify, I7ia/)a.«a— Brought out. Bhoje — Base of a right-angled triangle — Literally arm. Coie— Side of Ditto — Literally end of a bow. Carna — Hypothenuse — Literally ear. Lumb — Perpendicular— Literally length. Ahada — Given. 1st — Assumed. Jaw or Jya — Sine or chord — Literally bow-string. Bojeja w — S ine. Cotejaw — Cosine. rn^/ato- Radius — Literally sine of three ; perhaps meaning of three signs or 90 degrees. Addy — Intercalary. Che-tits fCshaya tithi) — Diflerence of solar and lunar days. Ahum — ? For bhumi savan — solar days. Chandra — Lunar. For the literal explanation of these terms, as far as they could be made out, I am obliged to Mr. Wilkins. Most of the words are written here according to their com-mon pronun- ciation in Bengal. NOTES. ll§ Dear Strachey, Having just laid my haads on a parcel of papers of notes, containing abstracts and trans- lations from the Bija Ganita, made by me, with the assistance of a Pandit, as long ago as when I was stationed at Bhagulpore*, I send them to you with full liberty to make any use of them. Ever since my removal to Burdwan these papers have lain unnoticed, and might have continued so had it not occurred to me that you are occupied in such researches. There may be trifling inaccuracies in some places, the translations having been made carelessly and never revised ; but their authenticity may be depended on, as they were made from the original Sanscrit Bija Ganita, which was procured for me at Benares, by Mr. Duncan. I send also a book of memoranda, containing chiefly trigonometrical extracts from a modern astronomical work in Sanscrit, which I suppose to have been written in Jey Sings time. I am very sincerely your's, Fortlatid Place, Jan. 1812. S. DAVIS. THE END. * About the year I7P0. Gkndinnmg. Printer, Ilalton Garde", ton/lor Some Ohfcrvations on the originality^ extent, and importance of the Mathematical fcience of the Hindoos; with Extracts from Perfian Tranflations of the Leelawuttee and Beej Gunnit — By Edward Strachey, of the Bengal Civil Eflahlifhment. THE chara6ler which Sir William Jones has given of Perfian tranflations from the Sanfcrit is enough to deter men from the labour of examining them. To difcovcr the full extent of Hindoo learning the Sanfcrit originals fliould be ftudied. Nevertheless fomc of thefe tranflations have their v.iluc. If examin- ed attentively and without prejudice, they will, on many points, give an infight into Hindoo fcience without hazard of deception, although they arc juftly open to a general objedlion of confufion and inaccuracy. The tranflator feldom diflinguiflics the text from his own additions, but he fomctimes introduces matter which he muft have been incapable of fupplying, if he had not had accefs to fome extraordinary means of infor- mation, which means I conclude to be the original work he pretends to tranflatc, or fome other Hindoo books of fcience. In works which arc avowedly tranflated from ancient or obfcure authors, but are fufpefled to abound with interpolations, we cannot pronounce any propofition to be original, without a previous confidcration of its nature, and of the circun:)ilances of the tranflator and the reputed author. ( -" ) If the propolition in queRion is thought to have been invented at a period later than the age of the perfon naw faid to be the author, and is to be found in books to which t-he tranflator undoubtedly had, or poffibly might have had, accefs, we may well fufped that it is not ge- nuine. If, however, it is direfliy deduced, or deducible from, or in any way intimately conned;ed with unfufptdled matter, wc may perhaps be difpofed to admit its claim to originality ; at all events wc Ihall hefitatc in pronoun- cing it fpurious. But if the propofition could not be derived from any of the common fources of knowledge which were extant in the time of the tranQator, wc mull riecelTarily fuppofe the exigence of fome other fource : and the difcovery of a well conneded feries of fuch propofitionSg would afford ample proof of the cxiftence of a fund of faience, to which the world has not hitherto had accefs. Let thefe principles be applied to an examination of the Perjian tranfla- tions of the Beej Gunnit znA Leelawuttee ; and many new and curious fa£ls will be afcertained, illuftrative of the early hiftory of Algebra and of the flatc of Mathematical knowledge among the Hindoos. The BeeJ Gunnit and Lelawuitee, were both written by Bhasker ; AcHARlj, a famous Hindoo mathematician and aflronomer, who lived about ^ the beginning of the 13th century of the Cbrijlian eri : the former of the two freatifes is on Algebra with, fome of its application; the latter is on Arithmetic and Algebra, and Menfuration, or Praftical Geometry. The Bet] Gunnit v/as tranflatcd into Ferji-an in 1634 A. C. by Utta Ulla ( 3 ) RosHEEDEE, {it Agra o{ DeMi jpTohihly), and the Lfclawuttes 1111587 A, C. by the celebrated Fyzee. Before any opinion is formed of the extent to which the tranilators might have interpolated, a view of the fources of their knowledge fh'ould betaken. It is well known that the only Ftrjian fcicncc is Arabian, and that the Arabs had much of their mathematical knowledge from the Greeks ; It is certain that they had their Arithmetic from the hdians, and mofl likely their Algebra was derived from the fame fource ; but the time and other circum- flances refpe6ling the introduflion of thefc fcienccs among the Arabs is unknown *. As vc arc not informed of the full extent of Greek knowledge among the Arabs, nor of the Greek and other books, which may have been tranfla- ted into Arabic, and which Utta Ulla Rusheedee and Fyzee, might have had accefs to; and as we cannot be certain that they had not the means (through travellers for instance) of becoming acquainted with the modern Eurcpejn difcoverics, vv'c muH: fuppofe that they might have had the benefit of all the Greek, and all the Arabian, and all the modern European learning, which was known in their time -f-. * The firlt account of any Indian mathematical fcience among the Arabs, is, I believe, of Indian aftro- nomv, which was known in the reign of Al Ma moo if. In later times, many Mohammedans have had accefs to the Hindoo books ; There are accounts of feveral in the Aytin Ackhery, and in D'Herbelot. Abul Fuzl gives a lift of Sanfcrit books, which were tranf- lated into Perfian in Akber's time. The Leelawoultet is the only mathematical work amongft them. + Bernur, who arrived in India in 165^, is faid to have tranflated or explained the pliilofophy of Gas- rjNDi and Des Cartes for his patron at the Mogul's court, but it is likely that only the metaphyfics of thefe writers arc alluded to. Bernier. had a good opportunity of making refearches into the learning of the Hindoos, and he has given fome account of it ; but his information appears to have been very circum- fcribed ; he was entirely ignorant of the extent of their mathematical fcienccs, and knew fcarcely any thing of their aftronomy. In the beginning of the eighteenth century, Jve Si no had accefs to the modern Euro' plan aftronomy and mathematics, and introduced a great deal cf them into India. (See Mr. Hukteh's paper in the fifth volume of the A/a:ick Refearchc?. ( 4 ) It is however probable, that, cxcludve of what they might have had from the Hindoos, 151 r a Ulla Rusheedee and Fvzee had no means of acquiring fcience except through Arabic hooks. From the tranflation of the Leela.'wuttee {liould be rcjeded, as not ccr- tsinly genuine, all fuch propofitions as were known to the Greeks or Arabs, or to the modern Europeans, till the end of the fixtecnth Century j and from the tranflation of the Beej Gunnit, all fuch as were known till the beginning of the feventcenth century, except fuch as arc intimately conncdcd with others whofe originality cannot be doubted. It is very dcfireable that fome pcrfon properly qualified for the tafis fhould compare the Algebra of the Arabs, and that of the Greeks and that of the modern 'Europeans with the Perjian tranflatjons of the Beej Gunnit and Leeiawuttee,* I HAVE no doubt that the rcfult of the comparifon would fliew; That the Algebra of the Arah is quite different from that of Dia- PHANTUS, and that neither of them could have been taken from the other, though both might have been taken from one common fourcc : That if the Arabs did learn from the Indians, (as they probably did) ihcy did not borrow largely from them ; * The only known book of Gretk Algebra, it Diophantu:. Of Arabian Algebra, no full account, as faf as I know, has hithei to been «iven, bat it is certain that the firft European treatifes were taken from it entirely. Many Arabic Algebraicgl works are extant, and well known in India ; fome of them have been tranilated into Ftrjiatw ( 5 ) That the P^-zj/fuv; tranflatlons of the Bccj Gunnit ^n^ Leelawut tee con* tain principles, which are fufficient for the folution of any propofitions in the Arabian., or in the Diophantine Algebra : * That thefc tranflations contain proporuions, which arc not to be folved on any principles, which the Arabian or the Diophanline Algebra could fupply : That the Hindoos were farther advanced in feme branches of this * Perhaps this may^be going too far ; it is not eafy to fay.vvhat principles are neceflary for the develope- ment of the Diophantim Algebra, which is merely a coUeftion of difficult quellions, (hewing more ingenuity than mathem.uical knowledge.. The Biej Gmmit will be found to differ much from Diophantus's work. It contains a great deal of knowledge which the Grttks had not ; fuch- as the ufe of an indefinite number of unknown quantities and the ufe of arbitrary marks to exprefs them; a good arithmetic of Surds ; a perfeft theory of indeterminate problems of the firft degree ; a very extenfive and general knowledge of thofe of the fecond degree ; i knowledge of quadratic equations, &c. The arrangement and manner of the two works will be lound as tffentially different as their fubftance. The one conftitutes a body of fcience, the other does not. The Beej Gunnit is well digelled and well conneftcd, and is' full of general rules, which fuppofe great learninn- ; the rules are illuftrated by examples, and the folutions arc performed with Ikill. Diophantus, though not entirely without method, gires very few general propafitions, and is chiefly remarkable for the ability with ■which he makes affumptions ia view to the folution of his queftions. The former teaches Algebra as a fcience, by treating it fyftematically ; the latter (harpens the wit, by folving a variety of abftrufc and com- plicated problems in an ingenious manner. Theaurhor of the Bei^j Gunnit goes deeper into his fubjeft, and treats it more abftrac"ledly, and more methodically, though not more acutely, than Diophantus. The for. mer has every charafteriftic of an afliduous and learned compiler; the latter of a man of genius in the infancy of fcience. This comparifon however, it muft be admitted, is made from a view fomewhat partial and fuperficial. The Beej Gitunit is feen through a very defecflivc medium. - It may have been improved, or it may have been deteriorated by the tranflator : and the account here given of Diophantus, is not the rcfult of deep ftudy of that very difficult author, but of a hafty review of hi$ work, and an examination of a i-\f only of his material propofuions. ,"; It may be proper to advert to the following point, which may be thought important : " Whether it ij ■« poffible, or, if poflible, v.! ether it is probable, that the Algebra of the Hindoos is nothing more than a « branch of Greek fctencc long loft, but now reflored ?" I am perfuaded that it is not : but a good difciiilion cf this queftion is neceffary,, before the originality of the Indian fcience can be eftabliflied on folid grounds. The intent of this p:^per is to throw out hints, for inquiry into the mathematical fciences of the Hindon. 1 cannot by any means pretend to embrace the fubjeft in its full extent f my opinions therefore will bs re- ceived with caution; tliey are fubmittcd with deference, - - ( 5 ) fciencc than the modern Europeans, with all their improvements, till the middle of the eighteenth century. Annexed, arc a few fpccimcns of Arithmetic, Geometry, and Algebra, taken from the tranflations of the Beej Gunnit and Leelawuttce, On SerieSf with Extrads from the Leelawuttce. In the tranflation of the Z,(?^/<;w«//^^, is a chapter on combinations, and another on progrejiions, as follows : Combinations. " To find a number froui the mixture of different things," " If it is required to add different things together, (o that all the corH" "* binations arifing from their addition may be known, this is the rule : " First, write them all with one, in order, and below write one " the lafl oppofite to the firft in order. Then divide the firft term of ■** the firft line, by the number which is oppofite to it in the fecond *' line. The quotient will be the number of combinations of that " thing. Multiply this quotient by the fecond term of the firft line, " and divide the prod udt by the number which is oppofite to it in the " fecond line, the quotient will be the number of combinations of that " thing; and multiply this quotient by the third term, and divide by " that which is below it, and add together whatever is obtained belovi^ " each term, the fum will be the amount of all the combinations of •• thefe things." ( 7 ) Or fhortly : The rule for combinations of any nuiTibcr of things (n) taken 2 by 2, 3 by 3, 4 by 4, &c. is fx^x^x^&c. coiitinued to as many h£tois as there arc things to be combined. Example. " The fix flavoured, called in Eindee Khut Rus, contains ift a fvveet, ** 2d a fait, 3d a four,~ 4th a foft, 5th a bitter, 6th. a (harp. I would " know the number of different mixtures which may be had by adding " thefe together. Write them thus 'I ^1 3I 4l 5' 6 6| 5l 4l 3l *l ■ 1 2 . 3 *" -y = '^' ~-^' -F='' the fum is &j. The number of mixtures then "•' of 6 things is 63." Progression. " Of numbers encreafing. This may be of feveral kinds; Firfl-, with ** one number, that is, when each number exceeds the preceding by i. The " way to find any number is ; Add i to that ntimber, and multiply by " half that number. To find the fum of the added numbers j Add 2 to " whatever number you fuppofe the laft, and multiply by the fum of " that number ; Divide by 3, the quotient will be the fum of the added *' number, or this number which was fuppofed the lafl." The rule is illuftrated by thefe examples. In a feries encreafing from 1 , to find the fum of 4 terms. 1+2+34-4 = 7+^^-^ = 10, In the fame feries to find the fum of 6 terms i,+6x— = 2,, the fum of 9 terms TT9X-^ = ^5. ( 8 ) This is when the encrcafe is by r. To find the fura of the added terms " ^vith 3." -TTT x^T^3 = io with 4, 1+7x7+^^== 20, with 9, 7^ H-2+3+4+5+6-r7-r^-|-9 " I 6^. 3 " If a pcrfon gives away at this rate, ift day i, 2d day 2, 3d day 3, 4th ♦* day 4, and foon, encrcafang by i to 9 days. According to the i ft rule by '^ the 3d day he will have given 10, by the 4th 20, by the 6th 45, by the 9th " 165." Thefe riiles .are fhortly ' + « ■><—-= sj where n is the num- ber of terms, and s the fum of the feries i, 2, 3, 4i 5, &c. and (g + n) X ^, the fum of the trianguiars. 3 The next rule is for tlie fiimmation of feries of fquarc and cube numbers, applied to the feries i, 4, 9, 16, 25, &;c. and i, 8, 27, 64, 125, &c. The fum of n terms of the former will be ^ — '-^^ and the fum of n terms of the latter will be s*. Rules are then given for arithmetical progrefllons, viz. (fuppofing a to reprefent the firft term, m the middle term» z the laft term, d the common difference) A RULE for fumming a geometrical progreflion comes next. •* A PERSON gave a number the firft day; the 2d day he added that '• number multiplied by itfelf, and fo on for feveral days, adding every *• produyfl multiplied by the firft number. The rule for finding the fum ( 9 ) ^ " is this; firft, obferve whether the number of the time is odd or even : " if it is odd, fubtra(3: i from it, and write it forecwhexc, placing a ** mark of multiplication above it. If it is even, place a mark of a fquare ♦' above it, and halve it till the number of the time is finiHied : after " that, beginning at the bottom with the number of encrcafe, multiply, " where there is a mark of multiplication, and take the fquare, where •' there is a mark of a fquare : fubtradl i from the produd, and divide " what remains by the number of cncreafe lefs i, and multiply the •' quotient by the number of the iirll day : the produd will be the fum '• of the progreffion." Example, where the number of terms is even, 2, 4 &c. to 30 terms — Example, where the number of terms is odd, 2, 6, 18, 54 &c. to 7 terms. In thefe examples the refult is given without s^ny detail of the application of rule to the cafe. Probably there is fomc error in the rule ; it fhould be In! x ^ = «• From want of books one cannot in this country have the means of afcertaining precifely the time when each particu- lar mathematical rule was invented in Europe, but I have no doubt it will appear, that all the rules which are in thefe two chapters were not known in Europe till after the fixteenth century. 'I'he formula ^ x ^ x ^ &c. is the fame as that for finding the coefficients in the binomial theorem, which was in £Mr<5/)e firft taught by Briggs about the year 1600. It does not appear whether this ufe of it was knowi> to the Hindoos. Mr. Reuben Burrow publiftied in the Afiatick Refearches a queftion folved by the laft mentioned rule, inferring from it, that the Hindoos had the binomial theorem. Peletariub in his Algebra, v hich was printed in "iSS^^ §^^^ ^ ^^^^^ °^ ■ ( 10 ) fquare and cube numbers, remarking, among other properties of thefc numbers*, " that the fum of any number of the cubes taken from the be- ginning, always makes a fquare number, the root of which is the fum of the roots of the cubes j" which is the fame thinj as the Leelawuttee rule. It does not appear whether the Hindoos had any knowledge of figurafe numbers further than what is here given. Qn the menfufation of the Circle and Sphere, with extraEls from the Leelawuttee. The rules in the Leelawuttee for the mcnfuration of the circle and of the fphere are thefe ; , To find the circumference of a circle, multiply the diameter by 3927,, and divide the produ6l by 1250; or multiply the diameter by 2a, and divide by 7. And (fuppofmg D to be the diameter and C the circumference") ^^5= / 4- area. D x C = furface of a fphere. ^= folidity of a fphere. Other rules given are the id 50C0 "** : verfed fine, D X II D area — : chord ;c_a) -tC-a) a - 21 of an = c and - folid arch V - ity < a). 3f a fphere, and v)v=^c c the , 4»D( 4 {y being ^-•(D+cjCD — <=) = v ai 2 X ^ (D- 4 ^-^^=a. 4 4D+C Quoted from Hutton's Didionaiy, Art. AWhu ( n ) Rules for finding the fides of regular figures inrcribed in a circle*. „ „„„ 0x131325 ~J 0 no, 00 6 gon 4 gOT>°^'^^-" 7 gon 5 gon ''^ '""^ 8 gon 9 gon Dx6f>oor» 120OCO 120000 DX4592a 120000 Vx 41031 It is to be lamented that ihe method of deducing thefe rules is not ftated. The ratio for that of the circumference of a circle to its diameter 22 : 7 is the fame as that of Archimedes ; that of 3927: 1250 is, I believe, a nearer approximation than was known in Europe before Viet A. This is mod probably (if not certainly) an Hindoo invention, and therefore fhould give the credit of originality to the other rules which are connefted with it. On Oiiadratic Equations, with extraHs from the Leelawuttee, the Beej Gunnit, and Diophantus. In the tranflation of the Leelawnties, the rule for quadratic equations is, " When the number, by which the root of the number thought of is " multiplied, is given; and, the fum or difference of the product: when " added to, or fubtra£ted from, the number thought of, is alfo given; the " rule for finding the number is : Take half tije multiplier and fquare it, " add what remains to it ; take its root; add, or fubtraft, half the multi- " plicand to the root, according as the queflion is of fubtradion, or addi- * Some of thefe numbers appear to be wrong. ( 12 ) •* tion i take the fquar^ of the aggregate, and that will be the number thought of J." A RULE is given, for clearing the higher power of fractional coefficients. Thus : " To add to the number thought of, or to fubtrait from it, a fra£lion of •' that number, the rule is: Add, or fubtradl, the frad:ion from i j divide " the multiplier and the remainder by the fum or the difference, and pro- " ceed according to the foregoing rule, v/ith the quotients of the multiplier " and remainder [|." Ih the tranflation of the BeeJ Gunnit, the rule is j " The fquare of unknown being equal to number, multiply both, or " divide both, by an affumed number, and add a number to the two ** refults, or fub trail it from them, that both may be fquares. For " one fide being a fquare, the other will alfo be a fquare; for they are •« equal and by the equal increafe or diminution of two equals, " two equals will be obtained. Take the roots of both, and, after " equating, divide the number by the root of the fquare of un- " known : the refult will be what was required." t X ± a v* X = b, a and b are given, i is required. Rule [ \L.~ , , ' ^ ^__ . UK X ± ;^±av'x-b:- ( 13 ) Again, " After equating, if the two fides are not fqiiares, ths mj- •• thod of making them fquares is this : Adume the number 4, and mul- ♦' tipljr it by the fquare of the unknov^ n of the firfl: fide, and multiply " both fides by the produdl, and in the place of number encreafe both " fides by the fquare of the thing of unknown which is on that fide. " Both fides will be fquares. Take the roots of both and equate them, " and the quantity of unknown will be found *" This rule is not very accurately expreffed, but there can be no doubt of what is intended by it. The following example will ferve to illuf- trate the rule, and fiiew the manner in wich the equation is reduced, " Some bees were fitting on a tree; at once, the fquare rcot of half •* their number flew away ; again, eight-ninths of the whole flew away " and two bets remained. How many were there? The method of " bringing it out, is this : From the queftion it appears that half the fum " h.as a roor. I therefore fuppofe 2 fquare of unknown; and I take " one ui. known, that is ths root of half; and ^ j and as the queftioner •* mentions, that 2 bees remain; 1 unknown and — of two fquare of " unknown that is '^ of i fquare of unknown, and 3 units is equal to '• % fquare of unknown. I perform the operation of equating the frac- " tions ii "this manner. I multiply both fides by 9, which is the deno- '^ minator of a ninth, 16 fquare 6f unknown and 9 unknown and iS " units is equal to 18 fquare of unknown. I equate ihem thus: — I fub- '= tracL 16 fquare cf unknown of. the firfl fide from 18 fq-nre of cn- " knov>n of the fecond fide. It is 2 fquare of unkno^vn aSrmafiv.', * Shc'jLl bf, It" a X ^ + 'b x = c multiply both fidss by 4 a, and sdd h '; then tstraft t];e roc? end reoivrc the ec'j.iiion. in) '' and in like manner I fubtraft 9 unknown of the firft.fiJe from cipher " unknown of t'-e fecond fide, 9 unknown negative remains. Then " I fubtraifl cipber the numbers of the fecond flJe from i8 units of the " firfl: fide. It is the fame. The firfi: fide then is 2 fquare of unknown " .affirmative and 9 unknown negative, and the 'fecond fide is 18 units " affirmative. In this example there is equality of fquare of unknown^, «' and unknown, to number; that is equality of fquare and thing to " number. As the roots of thefe two fides cannot be found, fuppofe «' the number 4, and multiply it by 2 which is the number of the " fquare 01 unknown ; it is 8. I multiply both fides by 8. The firOt " fide is 16 fquare of thing, and 72 unknown negative; and the fecond '' fide is 144 units," 1 then add the fquare of the number of the un- " known, which is 81 to the refult of both fides. The firil fide is j6 " fquare of unknown, and 72 unknown negative, and 81 units; and •' the fecond fide is 225 units. I take the roots of both fides. The " i'oot of the firfl: fide is 4 unknown, and 9 units negative; and the root '« of the fecond fide is 15 units affirmative. I equate them in this man- *' ner. I fubtra£l cipher unknown of the fecond fide from 4 unknown «' of the firft fide; and 9 units negative of the firll fide, from 15 units " affirmative of the fecond fide. - The firft is 4 thing and the fecond fide '* is 24 units affirmative. I divide; 6 is the refult, and this is the " quantity of the unknown, and 36. is the fquare of the unknown. " i\nd as we fuppofed 2 fquare of unknown, we double 3$; the whole " number of bees then was 72." . Farther, " When on one fi.de is thing and the numbers are negative, " and on the other fide the numbers are lefs than the negative numbers on " the fir.^ fide, there are two methods. Thefirftis to equate them without " alteration — The fecond is, if the numbers of the fecond fide are afiirma- ( 15 ) " tivc, to make them negative; and if negative, to make them affirmative. " Equate them J 2 numbers will be obtained, both of which will pro- *' bably anfwer." From this it will appear that in the tranflations of the Leelaitnitte and Beej Gunnit, the knowledge of quadratic equations is carried as far as it was among the Arabs 2Si^ the modern Europeans b'efore CaRdan, and no farther. Button* fays of Lugas de Burgo, whofe book was printed about the end of the 15th century, * He ufes both the roots or values of ' the unknown quantity, in that cafe of the quadratics which has two *, pofitive roots, but he takes no notice of the negative roots in the other ' two cafes.' Bachet in his commentary on the 33d queftion of the ift book of DioPHANTUs, takes a view of the rules for the folution of quadratic equa- tions which were known in his time — And he thinks that Digphantus had fimilar rules : He judges io, becaufe in the 30th and 33d quefticns of the 1 ft book, Diophant us has thefe propofitions, which he gives as the conditions of his queftions, (^)— x y = D and 4 x y + (x-y)^ =n, and fays of each of them ??' SfTsToxAaff/iaT/Hov. Bachet is of opinion, that thefe words indicate that rules for quadratic equations may be deduced from the proportions, and he accordingly deduces rules, calling them Dio- PHANTUs's method. Digphantus however had no fuch method. He always affumes his numbers fo as to avoid quadratic equations j He no- where folves quadratic equations, and therefore it may be prefumed that he did not underfland them. Im the 45th queftion of the 4th book, however, he has a procefs which • Mathematical Diftionary, article Algebra. ( 16 } is very like that of the folution of a q^uadratic equation " *6 N + 12 *' minus funt quam 2 Q-6. Adjiciantur quae dcfunt utrimque unitates, ♦' erunt 2 Q majores quam 6 N + 18. In asquatione autem hac expli- *• canda, dimidium numerorum in fe ducimus, et fit 9. Ducimus etiam " quadratos in unitates, ct fiunt 36. et addito 9 fiunt 45, cujus latus •' non minus eft 7 adde fsmiffim numerorum, et divide per quadratos, " fit I N minor 5." And, in queftion 33, book 4, Diophantus, having the equation 3X+i8 = 5X% proceeds thus: " et non efl rati- •' onalis atqui 5 Q efl quadratus unitate au£lus. Oportet itaque hunc duc- •' lum in unitates 18 et adfumentem quadratum femiffis 3 N nimirum 2^ " facere quadratum" &.'c. (fee queflion and Backet's commentary) but no quadratic equation is folved. It may be prefumed from what has been faid above, that the knowledge which the ^r^^/^z.'zj had of Algebra, as far as regards quadratic equations, was not derived from the Diophantine Algebra, but either that it was in- vented by themfelves, or, which is moft likely the cafe, that it was had from the Hindoos. , On Indeterminate Problems of the 2d Degree, with extrads from Diophantus and the Beej Gunnit. The i6th Queflion of the 6ihbook of Diophantus is as follows : " Datis duobus numeris, fi aliquo quadrato in unum eorum du6lo, ct " altero dc produdto fubtrado, fiat quadratus: invenietur et alius qua- " dratus major quadrato prius fumpto, qui hoc idem prasftet. Dentur " duo numeri 3 et 11, et quadrato aliquo, puta a latere 5 dud:o in 3 et * Bachet's tranflation. ( 17 ) " a produtflo detrafto ii fiat quadratus a latere S. oporteat invenire alium " quadratum majorem quam 25 qui hoc ipfum praeliet. Efto latus qua- " drati 1 N + 5 fit quadratus i O + ioN + 25 — Hujus triplum dempto 11 " fit 3 0 + 30 N + 64 asquale quadrate; fit ejus latus 8-2 N et fit N, 62. *' Eft ergo latus 67 quadratus 4489, qui prajftat imperata." And in the next example, referring to this, he fays, by purfuing this method an infinite number of fquares may be found. In the BeeJ Gunnit this problem is folved very generally and fcicnti- fically, by the afliftance of another, which was in Europe firft known in the middle of the 17th century, and firfl applied to queftions of this nature by Euler in the middle of the iSth century. Abstracting from the fign, the EeeJ Gunnit rule for finding new values of a X ° + b = y " is: Suppofe a f " + b = g " a particular cafc- find m and n fo that a n " + i = m ' then x = n g + m f and y = m g + A n f. General methods according to the Hindoos for the folution of inde- terminate problems of the ift and 2d degrees will be found in the 5th and 6th chapters of the Bee] Gunnit'. POSTSCRIPT. Since writing the above I have had accefs to Englijh tranflations of the Leelawuttee and part of the Beej Gunnit ; aad I have the fatisfadion to find that in all the mofl: material points they agree with the Pcrfian tranflations. '^'^ . ■ I'- III. Oji the eaf^ly History of Algebra. By EDWARD STRACHEY, Esq. If it were as generally known as it is certainly true, that there is a fine field for oriental research in the mathematical sciences, and that it is easy of access, the subject would not be so much neglected as it is at present. Four years ago I printed at Calcutta, some observations on the ma- thematical sciences of the Hindus. In that tract I proved, that an exten- sive and accurate knowledge of the Algebra of the Hindus might be had, by means of translations, extant in the Persian lano-uao-e, of certain Sanscrit books. As the Persian language is understood by most of the Company's civil servants in Bengal, I conceived that a consideration of the fact might induce persons who were competent to such studies, to direct their attention to them. Of the Bija Ganita, or Hindu Algebra of Bhascara Acharya, I have sent home a full account, which I suppose must have been published by this time. In that account (derived en- 2 ON THE EARLY tirely from a Persian translation) it is proved, that the Hindus had made a wonderful progress in some parts of Algebra ; that in the indeterminate analysis they were in possession of a degree of knowledge, which was in pAirope, first communicated to the world by Bachet and Fermat, in the seventeenth century, and by Euler and De La Grange, in the eighteenth. It would be very curious to push these inquiries into the Hindii indeter- minate analysis, as far as possible. They might, perhaps, shew that the Indiiins had a knowledge of continued fractions, and possibly speculations in physics and the higher geometry, that we know nothing of : for the foundation of the indeterminate analysis of the Hindus i^ directly ex- plicable on the principle of continued fractions. And there are branches of natural philosophy and mathematics, where equations will arise, which can be solved only by the rules of the indeterminate analysis. In the in- troduction to the Bija Ganita, where the first principles are given, a method is taught of solving problems of the form w4a:-f fc=D- This, simply considered, may be thought only a vain speculation on numbers; but, in the body of the Bija Ganita, the rule is applied to the solution of equations. It is true, that these equations arise from questions purely numeral ; yet it appears, nevertheless, that the application of the rule was understood. But whatever may be thought of this argument, it is, at all events, interesting, to ascertain the progress which has been made in the sciences, by different nations, in distant times. A GOOD comparison of any of the mathematical sciences of the Greeks, the Arabs, and *he Lidians, would be exceedingly valuable ; and every inforniation, uhichwill serve to illustrate the subject, is of importance to the early history of science. We know but very little of Algebra, in its infancy and first progress. HISTORY OF ALGEBRA. 3 It was introduced into Europe, from Arabia, towards the beginning of the thirteenth century; and the work of Diophantus became known about three hundred years after. From the difference between his Algebra and that of the European writers, there was reason to beheve that they were- not of the same origin. Some learned persons thought that Diophantus was the inventor; But the more received opinion was, that his writings bore internal evidence of the contrary ; and that Algebra must have been known long before his time. In 1579, BoMBELLi published a treatise of Algebra, in which he says,, that he and a lecturer at Rome, whom he names, had translated part of Diophantus, adding, " that they had found that in the said work the " hidian authors are often cited ; by which they learned that this science " was known among the Indians before the Arabians had it." (Hutton's Dictionary.) Dr. Hutton has adopted the opinion, that the Arabians had their Al- gebra from the Greeks. In his dictionary (article Algebra) we find, " the " Arabians say, it was invented amongst them, by MahoxMEt* Bin- " Mu's A or son of Moses, who it seems flourished about the eighth or " ninth century." It may be observed, by the way, that no Arabian writer has been cited in support of this. It does not appear on what foundation the assertion stands ; I imagine it is taken from Wallis. The learned Mus/emdns in India, certainly consider the science as having originated among the Indians; and the arithmetic, which in their treatises always precedes Algebra, is undoubtedly Indian. t}J J'"" -^"^'f '^-^^N-Mo sa-ul-Kha-rezmi, according ? -^f ",^ ''"'-^ ''^''' ^ ''■* °*' astronomical tablss, wliicli were highly esteemed, befoie NAs'iiuDDis Tu si published his. . B 4 ON THE EARLY Dr. Hutton goes on : " It is more probable, however, that Mahomet " was not the inventor, but only a person well skilled in the art ; and it " is further probable, that the Arabians drew their knowledge of it from " DiopHANTUs or other G/r^^' writers; and, according to the testimony " of Abulpharagius, the arithmetic of Diophantus was translated into " Arabic, by Mahomet-Bin-Yahya-Baziana." This I suppose is taken from Pococke's translation,* but the word which he has explained by " interpretatus est " is 'Ja meaning he commented on, rather than he translated. Surely, this is not sufficient to give rise to a probability, that the Arabians derived their Algebra from the Greeks. The x\lgebra of the Arabians bears no resemblance to that of Diophantus, the only Greek writer on the subject who has ever been heard of. Inquiries have been made, in different parts of India and Persia, for the supposed transla- tion of Diophantus ; but without success. In the five first propositions of the 13th book of Euclid, and in the 10th and 1 ith propositions of Ar- chimedes' book on spiral lines, and in the 9th proposition of the 2d book of his Isorropics, Wallis thought he saw traces of Algebra ; and it is to be presumed, that no farther evidence of its existence, among the- ancient Greeks, is discoverable ; for, except the above, I do not know that any authors have been directly quoted, in proof of the argument ; although there has been much assertion, in general terms, that the works of cer- tain writers do contain traces of Algebra. If there were any undoubted marks of it, in the writings of the ancients, they could not have escaped the notice of so learned and so indefatigable a Scholar as Wallis. "What he says on this subject, appears to result from a prejudiced conviction of the antiquity of the science, and not from an unbiased search for truth. * Dioplianti librum ile Algebra interpretatus est. HISTORY OF ALGEBRA. 5 If the analysis of the five first propositions of the 13th book of Euclid were (as is believed) by Theon, they could not well be adduced in proof of the ancient Greeks having a knowledge of Algebra ; because Theon is supposed to have been nearly contemporary with Diophantus. He could jiot have been long before him, if it is true, that his daughter Hypatia commented on a work of Diophantus. But, be this as it may, the ana- lysis of the propositions in question is not at all Algebraical. It is the common analysis of the ancient geometers, which is quite different from Algebra; the former being geometrical and the latter arithmetical. Wallis's reasoning, on the three propositions of Archimedes, to which he refers, amounts to no more than this. The demonstrations, as they now stand, are difficult ; they might have been done by Algebra with ease ; therefore, it is probable they were done by Algebra. We know of no Greek writer on Algebra, but Diophantus ; neither he, nor any known author, of any age, or of any country, has spoken, directly or indirectly, of any other Greek writer on Algebra, in any branch whatever ; the Greek language has not even a term to designate the science. The instance of DioPHANTus's treatise, with some indirect and disputable arguments, drawn, by inference, from works on other subjects than Algebra, is not sufficient. It is unlikely that the ravages of time and the depredations of barbarians should have destoyed all the direct and indisputable proofs. Such causes might account for the deficiency of our information on cer- tain particulars, but will not authorise forced constructions, to argue the existence of a complete science, from its supposed demolition. The ge- neral extent of the literature of the Greeks, especially in mathematics, is well known ; and that they had Algebra, can be established only by clear and positive evidence. For the different arguments which have been used, and the authorities which have been quoted on this question, see 6 ON THE EARLY on one side Wallis's Algebra, Chap, i, 2, 75. &ic. with the authors he refers to ; and, on the other side, the French Encyclopedie Art. Algehre, Application, Diophante by D'Alembert, and Analyse by De Castillon. See also Montucla- Though Bhascara Acha'rya, who is compara- tively a modern writer, could not have been one of the authors whom DioPHANTUS is said to have quoted, it is by no means improbable that some Alexandrian merchant, trading to India, might have learned a little Algebra from the Bramins, and instructed some of his countrymen ; or DioPHANTUS might have learned from Indians at Alexandria. If there is doubt of the Diophantine Algebra being of Greek origin, it is worthy of remark that its author had opportunity of communicating with persons from whom he might have drawn materials for his work, and whom there is evidence of his having actually cited. It is objected that Bom- BELLi is the only person who has taken notice of Diophantus' reference to Indian authors, and that no such reference is now to be found in his work. But the authority of Bombelli, on this point, cannot be overset, till it is ascertained that the manuscript of the Vatican, which he par- ticularizes, does not contain the citations. One would think that Bom- EELLi's assertion must have had some foundation, that it is not a mere fabrication. Though it does not appear that any Sanscrit works on this science, of greater antiquity than the Btja Ganita, have yet been dis- covered, we are not to conclude, therefore, that there are none ; for the author of the Bfja Ganita expressly says, his work is extracted from three copious treatises. These books have not been found ; we know nothing of their contents nor their dates. The following was the result of a general comparison of the Bij a- Ganita with Diophantus.* " The * From " observations." &c. above referred to. HISTORY OF ALGEBRA. 7 " Bija Ganita will be found to differ much from Diophamus' work. " It contains a great deal of knowledge which the Greeks had not ; such •' as the use of an indefinite number of unknown quantities, and the use " of arbitrary marks to express them ; a good arithmetic of surds ; a " perfect theory of indeterminate problems of the first degree ; a very «' extensive and general knowledge of those of the second degree ; a " knowledge of quadratic equations, &c. The arrangement and man- " ner of the two works will be found as essentially different as their " substance. The one constitutes a body of science, which the other " does not. The Bija Ganita is well digested and well connected, and " is full of general rules which suppose great learning : the rules are *' illustrated by examples, and the solutions are performed with skill. " DioPHANTUs, though not entirely without method, gives very few ge- " neral propositions, and is chiefly remarkable for the ability with which *' he makes assumptions in view to the solution of his questions. The " former teaches Algebra as a science, by treating it systematically ; the *' latter sharpens the wit by solving a variety of abstruse and complicated *' problems, in an ingenious manner. The author of the Bija Ganita " goes deeper into his subject, and treats it more methodically, though *• not mdre acutely, than DioPHANtus. The former has every charac- *' teristic of an assiduous and learned compiler; the latter of a man of " genius in the infancy of science." The Greek Algebra may be seen in Diophantus, who is the only Greek writer on the subject who has ever been heard of. The Indian Algebra may be seen in the Bija Ganita, and the Lilavati (by the author of the Bija Ganita,) and as the Persian translations of these works contain a degree of knowledge, which did not exist in any of C 8 ON THE EARLY the ordinary sources of science, extant in the time of the translators, they may be safely taken as Indian, and of ancient origin. To give some idea of the Algebra of the Arahiajis, whereby we may be enabled to judge, whether, on the one hand, it could have been derived from Diophantus; or, on the other, that of the Hindus could have been taken frotn them, the work entitled Khuldsat-til-Hisdb, may be taken as a specimen ; especially because, as will be more particularly stated in another place, there is a part of this book which marks the limits of Algebraical knovr^ ledge, in the time of the writer. We have seen, that the first Eiiropeaji Algebraists learnt of the Ara<- hians, but no account has been given of the nature, the extent, and the origin of Arabiafi Algebra. No distinct abstract or translation of any Arabic book, on the subject, has appeared in print; nor has it been esta- blished beyond controversy, who taught the Arabians. The Khuldsatr- ul-Hisdb is of considerable repute in India; it is thought to be the best treatise on Algebra, and it is almost the only book on the subject, read here. I selected it, because I understood, that as well as the shortest, it was the best treatise that could be procured. Besides general report, I was guided by the authority of Maulavi Roshen Ali, an acknowledged good judge of such matters, who assured me that among the learned Muslemdns it was considered as a most complete work ; and that he knew of no Arabian Algebra beyond what it contained. In the SuldfaU ul-Asr, a book of biography, by Niz'am-ul-din-Ah'med, there is this ac^ count of Bah'a-ul-din, the author of t\\e Khuldsat-ul-Hisdb. " He was " born at Bdlbec, in the month D'hi'lhaJ, 953 Hijrl, and died at Isfahan " in Shazvdl, 1031." Mention is made of many writings of Baha-ul-di'n' on religion, law, grammar, Sec. a treatise on astronomy, and one on the HISTORY O? ALGEBRA, • 9 astrolabe. In this list of his works, no notice is taken of his great trea- tise on Algebra, the Behr-ul-Hisdb, which is alluded to in the Khuldsat- ul-Hisdb. Maulavi Roshen Ali tells me the commentators say, it is not extant. There is no reason to believe that the Arabians ever knew more than appears in Baha-ul-dIn's book, for their learning was at its height long before his time. From what has been stated it will appear, that from the Khiildsat-ul- Hisdb, an adequate conception may be formed of the nature and extent of the Algebraical knowledge of the Arabians; and hence I am induced to hope that a short analysis of its contents will not be unacceptable to the society, I deem it necessary here to state, that possessing nothing more than the knowledge of a few words in Arabic, I made the translation, from which the following summary is abstracted, from the viva voce in- terpretation into Persian of Maulavi Roshen Ali, who perfectly under- stood the subject and both languages, and afterwards collated it with a Persian translation, which was made about sixty years after Bah'a-ul- Din's death, and which Roshen Ali allowed to be perfectly correct. The work, as stated by the author in his preface, consists of an in- troduction, ten books and a conclusion. The introduction contains definitions of arithmetic, of number, which is its object and of various classes of numbers. The author distinctly ascribes to the Indian sages the invention of the nine figures, to express the numbers from one to nine. Book i, comprises the arithmetic of integers. The rules enumerated uiiderthis head are Addition,- Dupla:. ion. Subtraction, Halving, Muni- plication, Division, and the Extraclioi, of the Square Root. The method of 10 ON THE EARLY proving the operation by throwing out the nines is described under each of these rules. The author gives the following remarkable definitions ef multiplication and division, viz. •« Multiplication is finding a number " such that the ratio which one of the factors bears to it shall be the " same as that which unity bears to the other factor," and " division is " finding a number which has the same ratio to unity as the dividend has " to the divisor." For the multiplication of even tens, hundreds, &c. into one another, the author delivers the following rule, which is remarkable in this res- pect, that it exhibits an application of something resembling the indexes of logarithms. " Take the numbers as if they were units, and multiply them together *' and write down the product. Then add the numbers of the ranks to- " gether, (the place of units being one, that of tens, two, &c.) substract " one from the sum and call the remainder the number of the rank of " the product. For example, in multiplying 30 into 40, reckon 12 of the " rank of hundreds ; for the sum of the numbers of their ranks is 4, and " three is the number of the rank of hundreds, multiplying 40 into 500, " reckon 20 of the rank of thousands, for the sum of the numbers of the " ranks is 5." The following contrivances have sufficient singularity to merit par- ticular mention. I. To multiply numbers between 5 and 10. Call one of the factors tens, and from the result, subtract the product of that factor by the dif- ference of the other factor from ten. For example, to multiply 8 into 9. Subtract from 90 the product of 9 by 2, there remains 72. Or add the HISTORY OF ALGEBRA. H factors together, and call the excess above lo, tens. Multiply together the two differences of the factors from lo, and add the product to the former number. For example, to multiply 8 by 7, add to 50 the product of 2 into 3. II. To multiply units into numbers between units and 20 ; add the two factors together, call the difference of the sura from 10, tens. From this result, subtract the product of the difference of the simple number from 10 and of the compound number from jo. For example, to mul- tiply 8 by 14. Subtract from 120, the product of 2 into 4. III. To multiply together numbers between 10 and 20 ; add the units of one factor to the other factor and call the sum tens : add to this the product of the units into the units. For example to multiply 12 into 13, add 6 to 150. IV. To multiply numbers between 10 and 20 into compound num- bers between 20 and 100 ; multiply the units of the smaller by the tens of the greater, add the product to the greater number and call the sum tens. Add to it the product of the units in both numbers. For example, to miiltiply 12 into 26, add 4 to 26 and call 30, tens. Finish the operation, it is 312. V. To multiply numbers between 20 and 100, where the digits in the place of tens are the same ; add the units of one factor to the other and multiply the sum by the tens, call the product tens, and add to it the product of the units multiplied by the units. For example, to multiply 23 by 25, multiply 28 by two. Call the product 56 tens, finish the operation; 575 is obtained. D 12 ON THE EARLY VI. To multiply numbers between lo and loo, when the digits in the place of tens are "different. Multiply the tens of the smaller number into the larger number; add to the result, the product of the units of the smaller number into the tens of the greater ; call the sum tens ; add to this the product of the units into the units. For example, to multiply 23 into 34, add 9 to 6S, add 12 to 770. VII. To multiply two unequal numbers, half the sum of which is simple ( Mufrid,) take the sum of the two and multiply half of it into itself. From this product, subtract the square of half the difference of the two numbers. For example, to multiply 24. by 36. From 900 sub- tract the square of half the difference of the numbers, that is 36. There remains 864.. For multiplying numbers consisting each of several places of figures, the method described by this author, under the name of Shabacah or net work, and illustrated by the following example, may have suggested the idea of Napier's bjnes. Multiphj 62374 ly 207. 6 2 3 7 4 /2 /4 /6 yA / / / / / 4 / /4 \/ /4 2/ 4/ 2/ /8 HISTORY OF ALGEBRA. 13 On the other rules, nothing is deUvered differing scmuch from those contained in our common books of arithmetic, as to require specific men- tion. Book second, contains the arithmetic of fractions ; and book third, the rule of three, or to find an unknown number by four proportionals. Book fourth, delivers the rule of position, or to find an unknown number by assuming one once or twice, and comparing the errors. Book fifth, gives the method of finding an unknown number, by reversing all the steps of the process described in the question. The sixth book, treats of mensuration. The introduction contains geometrical definitions. Chapter I. treats of the mensuration of rec- tilinear surfaces. Under this head the two following articles are deserv- ing of notice. I. To find the point in the base of a triangle where it will be cut by a perpendicular, let fall from the opposite angle. Call the greatest side the base ; multiply the sum of the two lesser sides by their difference; divide the product by the bass, and subtract the quotient from the base ; one half the remainder will shew the place on the base, where the perpendicular falls towards the least side.* * Let a be the base, or longest side, b the middle, c the smallest, and x the distance of the perpendicular from the least side. Then i * = a * + c ^ — 2 n.r (Eud. 13. 2.) 2a.r=a' +c' — b ' a b^—c^ 2a But 6 ^ — c * = 6 + c X "6 — c 14 ON THE EARLY 2. To find the area of an equilateral triangle. Multiply the square of a quarter of the square of one of the sides by three : the square root of the product is the area required.* Chapter second, treats of the mensuration of curvilinear surfaces. For the circle the rule delivered in many common books of mensuration is given : viz. multiply the square of the diameter by ii, and divide the product by i4,.-f Chapter third, on the mensuration of solids, contains nothing of sin- gularity sufficient to merit particular notice. This chapter concludes with the following sentence. " The demonstrations of all these rules are " contained in my greater work, entitled Bahr-2il~Hisdb (the ocean of " calculation,) may God grant me grace to finish it." Book seventh, treats of practical geometry. Chapter first on levelling, a b + c -X. b Therefore x = — — 2 a Sec the geometrical demonstration in the elements of pianc trigonoEHC-tiy, annexed to SiMso>i'-s Euclid, prop. 7. * Let a side of the triangle be a and the perpendicular x. a v The area is — - a * 3 a ^' But .r » = (x ^ 7- = ~T~ a X _ a I 3 a" _ / 3a* "5' — o ^ 4 —^ 16 + This is founded on the rough proportion of the diameter, to the circumference as 7 : 22. Bhascara, in the Lilavati, assigns 1250:3927, ^vhich is 1:3.1416 and differs only 0.000007 from the most accurate computation hitherto made. HISTORY OP ALGEBRA. B 'for'the.-pnt'pose of making canals. In this are described the plumme'r -fevel, and the water level on the same principle with our spirit level. •ChaptejR second, on the mensuration of heights, accessible and inaccessi- ble. Under the forrtier of these heads are delivered the common methods, 1)y bringing the top of a pole whose heiglit is known, in a line between the "eye and the top of the height required ; by viewing the image of the top in a horizontal mirror; by taking the proportion between a stick of known length, set up perpendicular to the horizon and its shadow; and by taking the length of the shadow of the li-eight when the sun's altitude is 45 degrees. The last method is this, " Place the index of the astro- " labe at the mark of 45 degrees, and stand at a place from wlience the " height of the object is visible through the sights, and measure from the *' place where you stand to the place where a stone would fall from the " top; add your own height, and the sum is the quantity required." For the mensuration of inaccessible heights the following rule is de- livered, " Observe the top of the object through the sights, and mark on " what shadow line (division) tlie lower end of the index falls. Then " move the index a step forward or backward, and advance or recede " till you see the top of the object again. Measure the distance between *' your stations, and multiply by 7 if the index is moved a Dhil-Kjddm, " and by 12 if it is moved a Dhif-Asbd,^ according to the shadow lines " on the Astralabe. This is the quantity required. * This part of (be ii&trolabe^spnsists oT fwo squares put togctiier laterally ; the index of tliC'inslrument being at t!ic {joint of the adjacent angles above. One square has seven, and the other, twelve'di visions: the former called G/iil-i-Kmlnni, the laUer Dhil-i-Asbd. Tbe squares are graduated on tlie outer sides'from (lie top, and at (he bo(toni from the point of the adjacent angles. The divisions on (he upright sides are those lines which Chaucer, in Lis treatise oa the astrolabe, calls Umbra-recta ; those oa the horizontal he calls Umbra-verses- E .16 ON THE EARLY Chapter third. On measuring the breadth of rivers and the depth of wells, ist. Stand on the bank of the river, and through the two sights look at the opposite bank ; then turn round and look at any thing on the land side, keeping the astrolabe even. The distance from the observer to the object is the same as the breadth of the river. 2d. Place something Chaucer's astrolabe had only one square, Dhil-i-As'bd, being divided into twelve parts. The Umbra-recta is called Dhil-3Justawi, and the versa, Dhil-Mdcus. The rule in the text is very inaccu- rately delivered ; for the only case in which it will apply is when at the first station the index coincides with the dia- gonal of the square, and being afterwards moved one division on the horizontal side, the observer advances towards the object, till the top is again seen through the sights. For let ^ C be the height required, B the first station, D the se- cond. As the angles at A and B are equal, A C = B C. But at the second position A C : D C -..l : Q. There- fore A C= 7 B D. But suppose at tlie first station B, the index falls on the fourth division, Dhil- Kadam, on the vertical side ; and tliat, by retiring from the object to D, it is brought on the third ; then it is evident that B C:yf C::7:4, and DC : A C :: 7 : 3. Therefore DC = ^4-^ = ^ B D. Consequent- ly 7:3:: 4i3D:JC 12 BD HISTORY OF ALGEBRA. 17 over the well which shall serve for its diameter ; from the center of this diameter drop something heavy and shining till it reach the bottom, and make a mark at the center ; then look at the heavy body through the two sights of the astrolabe, so that the line of vision may cut the diameter. Multiply the distance from the mark on the diameter to the place where the line of vision cuts it, by your own height, and divide the product by the distance from the place where the line is cut to the place where you stand. The quotient is the depth of the well.* Book eighth. " On finding unknown quantities by Algebra, In this i)Ook are two chapters. «' Chapter first. Introductory. Call the unknown quantity Shai (thing,) its product into itself Mdl (possession,) the product of Mdl into Shai, Cab (a die or cube,) oi Shai into Cab, Md-Mdl ; of Shai into Mal- i-Mai, M.d-Cdb ; Shai into Mdl-i-Cdb, Cdh-i-Cdb ; and so on, without end. For one Cdb write two Mdl, and from these two Mdls one becomes Cab ; afterwards both Mdls become Cdb. Thus the seventh power is * The irnpossibilify of attaining accuracy in either of Ihese operations is abundantly ob- yious. The first depends on the principle, ihat on a level plain, two places, which with a given height of the observer's eye haye the same dip below the horizon must be at equal distances. The second is thus : let the body drop from a to e ; and let the observer at c ^ observe it in the line d e which cuts a c iu &* Then b c : c d ■.: ab : a e = — b c vl8 ON MiE ^ARiY Mdl-i-Mdl-i-Cdb, and the eighth Mdl-i-Cdb-i-Cdh, m XhQvim'CaCab-i-Cah- 'Cab, and so on. All these powers are in proportion, either ascending or descending. Thus the ratio of Mdl-i-Mal to Cab is like the ratio of CdS to MdL, Mdl to Shai, and Shn to one, and one to one divided by Shai ; and one divided by Shai to one divided by Mdl ; and one divided by Mat to one divided by Cab ; and one divided by Cab to one divided by Mdl-i~ Mdl. To multiply one of these powers by another, if they are both on the same side, (viz, of unity) add the exponents of their power's to- gether; tl>e product will have the same denomination as this sura. For example, to multiply Mdl-i-Cdb by Mdl-i-Mdl-i-Cdb, the first is the 5tk power and the 2d the 7th. The result then is Cdb-i-Cdb-i-Cdb-i-Cdb or four Cabs, which is the 12th power. If the factors are on different sides, the product will be the excess 0:1 the side of the greater. The product of one divided by Mdl-i-Mdl into Mdl-i-Cdb is Shai; and the product of one divided by Cdb-i-Cdb-Cdb into Cdb-i-Mdl-i-Mdl, is one divided by Mdl: and if the factors are at the same distance (from one,) the product" is one. The particulars of the methods of division, and extraction of roots and other rules, I have given in my greater book. The rules of Algebra which have been discovered by learned men are six, and they relate to number and Shd and Mdl. The following table will shew the products and quotients of these, which are here given for the sake of brevity. HISTORY OF ALGEBRA. 19 Multiplier. I I a 1 a a ' 1 a a' « ' a* a * -d 5 1 3 a 1 1 a 1 a I a a-- fi^ a a ^ 1 1 1 , a a I 1 0^ - ! - 1 a I a 1 n ^ 1 1 7^ I 1 « 1 1 « ' a 1 1 a 1 Divisor. " The use of the table is this : multiply the co-efficient of one of the two quantities by that of the other ; the result is the co-efficient of the product, which is of the denomination contained in the square where the lines from the two factors meet. If on either side there be a substrac- tive (negative) quantity, call the minuend plus or affirmative, and the subtrahend minus negative. The product o^ plus into plus and tninus into minus are ho\.\\ plus, and the product of different kinds are minus. Mul- tiply the quantities together, and subtract the negative from the affirma- tive. For example, the product of lo and one Shai into lo all but one Shai, is loo all but Mdl. The product of 5 all but Shai, by 7 all but Shai is 35 and one Mdl all but 12 Shtii. Another example. The product of 4 Mdl and Q all but 2 Shai, into 3 Shai all but 5, is 12 Cab, and 28 ShaizW but 26 Mai and 30. In division, find a number which multiplied by the divisor will produce the dividend. Divide the co-efficient of the dividend by that of the divisor, the quotient is the co-efficient of the quantity which is opposite to the dividend and divisor. F 20 ON THE EARLY " Chapter second. On the six rules of Algebra. To find unknown quantities by Algebra depends on acuteness and sagacity ; an attentive consideration of the terms of the question, and a. successful application of the invention to such things as may serve to bring out what is required. Call the unknown quantity Shai, and proceed with it according to the terms of the question, as has been said, till the operation ends with an equation. Let that side where there are negative quantities be made per- fect, and let the negative quantity be added to the other side : this is cal- led restoration (Jebr.) Let those things which are of the same kind, and equal on both sides, be thrown away : this is opposition ( iMukdbaluh.) Equality is either of one species to another, which is of three kinds, called ( Mufriddt ) simple ; or of one species to two species, which of three kinds, called ( Muktarindt) compound. " Case the first. Mufriddt. Number is equal to things. Divide the number by the co-efficient of the things, and the unknown quantity will be found. For example; a person admitted that he owed Zaid looo A A and one half of what he owed Amer; and that he owed Amer lOOo all but one half of what he owed to Zaid. Call Zaid's debt Shai. Then A Amek's debt is looo, all but hilf of -S/u?/. Then Zaid's is 1500 all but a fourth of Shut. This is equal to Shai. After Jebr, 1500 is equal to one A Shai iind a quarter of Shai. So for Zaid is 1200 and for Amer 400." " Case the second. Multiples of Shai equal to multiples of MdL Divide the co-efiicient of the things by that of the Mdi ; the quotient is the unknown quantity. Example. Some sons plundered their father's inheritance, which consisted of Dinars. One took 1, another 2, the third S, and so on increasing by one. The ruling power took back what they had plundered; and divided it among them in equal shares. -Then each HISTORY OF ALGEBRA. SI received 7- How many sons were there, and how many Dinars? Sup- pose the number of sons Shai, and take the sum of the extremes, that is to say, 1 and Shai. Multiply them by half of Shai. This is the number of Dinars. For the product of the sum of any series of numbers in arithmetical progression, is equal to the product of the sum of the two extremes, into half the number of terms. Divide the number of the Dinars by Shai, which is the number of the sons; the quotient, according to the terms of the question, will be seven. Multiply 7 by Shai, which is the divisor; 7 Shaiis the product, which is equal to ^ Mdl and ^ Shai. After Jebr and Milkahalah, one Mdl is equal to 13 Shai. Shai then is 13 ; and this is the number of the sons. Multiply this by 7. The number of Dinars will be found 91." " Questions of this sort may be solved by position. Thus, suppose the number of sons to be 5 ; the fii'st error is 4, in defect. Then suppose it to be 9, the second error is 2 in defect. The first MuhfudJi is 10 :ind the second is 36 ; their difference is 26 ; the difference of the errors is 2. Another method," which is easy and short, is this. Double tlie quotient, (the number 7 in the question) subtract one, and the result is the num- ber of sons. " Case the third. Number equal to Mdl. Divide the number by the co-efficient of the Mdl ; the root of the quotient is the unknown quantity. For example. A person admitted that he owed Zaid the greater of two sums of money, the sum of which was 20 and the product 96. Suppose one of them to be 10 and Shai, and the other 10 all but Shai. The pro- duct, which is 100 all bat Mdl, is equal to 96" ; and after Jebr and Mukd- haldh, one Mdl is equal to 4, and Shai equal to 2. One of the sums ihen is 8 and the other 12, an.l 12 is the debt of Zaid. 22 ON THE EARLY «' First case of Muktarinat. Number equal to Mai and Shai. Com- plete the Mdl to unity if it is deficient, and reduce it to the same if it ex- ceeds, and reduce the numbers and Shai in the same ratio, by dividing all by the co-efficient of the Mdl. Then square one half the co-efficient of the Shai, and add this square to the numbers. Subtract from the root of the sum half the co-efficient of the Shai, and the unknown will remain. Example. A person admitted that he owed Zaid a sum less than lo, so much that if the square of it was added to its product by |- what it wants of lo, the sum would be is. Suppose the number Shai, its square is Mdl; half the remainder from lo is 5 all but half of Shai. The product of Shai by this is 5 Shai all but -j of I\Idl. Therefore i of Mdl and 5 Shai are equal to 12. One Ma/ and 10 Shai are equal to 24. Subtract half the co-efficient of the Shai from the root of the sura of the" square of ^ the co- efficient of the 5"^^/ and the numbers. There remains 2, which is the number required. Second case. Shai equal to numbers and Mdl. After completing or rejecting, subtract the numbers from the square of half the co-efficient of the Shai, and add the root of the remainder to half the co-efficient of the Shai ; or subtract the former from the latter ; the result is the unknown quantity. Example. What number is that which being multiplied by half of itself and the product increased by 12, the result is five times the original number. Multiply Shai by half itself, then half of Mdl added to 12 is equal to 5 Shai. One Mdl and 24 is equal to 10 Shai. Subtract 24 from the square of 5, there remains one, and the root of one is one. The sum or difference of 1 and 5 is the number required. Thiud Case. Mdl equal to nuniber and Shai. After completion or rejection, add the sqrare of half the co-efficient of the Shai to the numbers. Iiistohy of algebra; q-j and add the root oT tlie sum to half the co-eiTicient of the Shai Tiiis is the unknown quantity. For example. What number is that which being subtracted from its square, and the remainder added to the square, is lo? Subtract Shai from Mdl and go on with the operation, 3 Mdl all but Shai is equal to 10; and after Jebr and Radd, hldl is equal to 5 nnd \ of Sluii. The square of half the coefucient of the Shai and 5, is 5 and half an eighth, and its root is sf . To this add ~^, the result is 2^, which is the number required. EooK ninth, contains twelve rules regarding the properties of numbers, viz. 1st. To find the sum of the products of a number multiplied into itself and into all numbers below it: add one to the number, and multiply the iwm by the square of the number ; half the product is the number required. 2d. To add the odd numbers in their regular order: add one to the last number and take the square of half the sum. sd. To add even numbers from two upwards : multiply half the last even number by a number greater by one than that half. ^th. To add the squares of the numbers in order: add one to twice the last number, and multiply a third of the sum by the sum of the num- bers. 5ih. To find the sum of the cubes in succession : take the square of the sum of the numbers. 6th. To f;nd the product of the roots of two numbers : multiply one by the other, and the root of the product is the answer. G 24 ON THE EARLY 7th. To divide the root of one number by that of another: divide one by the other, the root of the quotient is the answer. 8th. To find a perfect number : that is a number which is equal to the sum of its aliquot parts, (Euclid, book 7, def. 22.) The rule is that delivered by Euclid, book 9, prop, 36. 9th. To find a square in a given ratio to its root: divide the first num- ber of the ratio by the second ; the square of the quotient is the square required. 10th. If any number is multiplied and divided by another, the product multiphed by the quotient is the square of the first number. iith. The difference of two squares is equal to the product of the sum and difference of the roots. 12th, If two numbers are divided by each other, and the quotients jnultiplied together, the result is always one. Book tenth, contains nine examples, all of which are capable of solu- tion by simple equations, position, or retracing the steps of the operation, and some of them by simple proportion ; so that it is needless to specify them. The conclusion, which marks the limits of algebraical knowledge in the age of the writer, I shall give entire, in the author's words. " Conclu- sion. There are many questions in this science which learned men have to this time in vain attempted to solve ; and they have stated some of these questions in their writings, to prove that this science contains dil. culties, to silence those who pretend they find nothing in it above their abihty, to warn arithmeticians against undertaking to answer every ques- HISTORY OF ALGEBRA. S5 tion that may be proposed, and to excite men of genius to attempt their solution. Of tliese I have selected seven, ist. To divide lo into 3 parts, such, that when each part is added to its square root and the sums are mul- tiplied together, the product is equal to a supposed number. 2d. What square number is that which being increased or diminished by 10, the sum and remainder are both square numbers? 3d. A person said he owed Zaid 10 all but the square root of what he owed Amer, and that he owed Amer 5 all but the sqLiare root of what he owed Zaid. 4th. To divide a cube number into two cube numbers, 5th. To divide 10 into two parts, such, that if each is divided by the other, and the two quotients are added together, the sum is equal to one of the parts. 6th. There are three square numbers in continued geometrical proportion, such, that the sum of the three is a square number. 7th. There is a square, such, that when it is increas- ed and diminished by its root and 2, the sum and the difference are squares. Know, reader, that in this treatise I have collected in a small space the most beautiful and best rules of this science, more than were ever collect- ed before in one book. Do not underrate the value of this bride ; hide her from the view of those who are unworthy of her, and let her go to the house of him only who aspires to wed her." It is seen above that these questions are distinctly said to be beyond the skill of algebraists. They either involve equations of the higher or- der, or the indeterminate analysis, or are impossible. It does not appear that the Arabians used algebraic notation or abbre- viating symbols ; that they had any knowledge of the Diophantine Alge- bra, or of any but the easiest and elementary parts of the science. We have seen that Baha'-ul-din ascribes the invention of the numeral fig- ares ill the decimal scale to the Indians. As the proof commonly given 2o ON THE EARLY of the Indains being the inventors of these figures is only an extract from th3 preface of a book of Arabic poems, it may be as \<:£\\ to mention that all the Archie and Persian books of ariihmetic ascribe the invention to the Ind'ans. The following is an extract from a Ptiyiun treati.'^c of ariih^ne- tic \t\ my possession. . " The lu.'Iiiin sage;?, ^vish!i^g to exprr-?s' nnmhcrs convcn:ent]y, iiiYcnt- *' cd these nine figures | i' T P' ^'! v a s. The first figure on ihe right hand " they made stand for units, tlie second for tens, the third for hundreds, " the fourth for thousands. Thus, after the third rank, the next follow- " ing is units of thousands, the second tens of thousands, tha third hun- *' dreds of thousands, and so on. Every figure tlierefore in the first rank " is the number of units it expresses ; every figure in the second the num- " ber of tens which the figure expresses, in the tliird the number of hun- " dreds, and so on. "When in any rank a figure is wanting, write a cipher " like a small circle o to preserve the rank. Thus ten is written lo, a " hundred lOO ; five thousand and twenty-five 502^." Of the Indian Algebra in its full extent the Arabians seem to have been ignorant ; but it is likely they had tlieir Algebra from the same source as their Arithmetic. The Arabian and Persian treatises on Algebra, like the old European ones, begin with the' Arithmetic, called in those treatises the Arithmetic of the Indians, and have a second part on Algebra ; but no notice is taken of the origin of the latter. Most hkely their Algebra, be- ing numeral, was considered by the authors as part of Arithmetic. Though part only of the Khuldsat-ul-Hisdh is about Algebra, the rest« relating to arithmetic and mensuration, must be thought not wholly un- connected with the subject. It is to be hoped that ere long we shall HISTORY OF ALGEBRA. 27 have either translations from the Sanscrit of the Bija Ganita and Lila- zvati, or perfect accounts from the originals ; and that other ancient Hindu books of Algebra will be found, and made known to the world. But as there is no immediate prospect of these desiderata being realized, the translations into Persian will be found well deserving of attention. Only let them be examined without prejudice. There are principles which will safely lead to a distinction of what is interpolated from what is original; and it is the neglect of these princi- ples, and not any fair examination of the translations, that may lead to error. DI S S E RTATION ON THE CONSTRUCTION AND PROPERTIES ARCHES. BY G. ATVVOOD, ESO. F. R. S. LONDON : PRINTED BY W. BULMER AND CO. CLEVELAND-ROW, ST. JAMEs's. FOR LUNN, OXFORD-STREET, AND ECERTON, WHITEHALL. i8oi. PREFACE. J\.N arch being formed (according to the usual modes of construction) by the apposition of wedges, or sections of a wedge-like form, the pro- perties of arches seem to be naturally derived from those of the wedge, on which principle the inquiries in the ensuing Tract are founded. By considering the subject on this ground, it appears that tlie theory of arches may be inferred from geometrical construction, depending only on the known properties of the wedge and other elementary laws of mechanics, without having recourse to the more abstruse branches of geometry in explaining this practical subject, to which a more direct and obvious method of inference seems better adapted. A geometrical construction for adjusting equilibration on these principles, extended to arches of every form, with the various con- sequences arising from, or connected with it, are the subject of the ensuing pages, in which rules are investigated, in the first place, for establishing the equilibrium of arches on two distinct conditions, namely, either by adjusting the weights of the sections accoiding to the angles which are contained between their sides, supposed tube given quantities ; or, secondly, by supposing the weights of the weJ-es or sections to be given, and investigating what must be the angles con- tained by their sides, so that the pressures on them, may be an exact counterpoise to the weight of each section, due regard being iiad to its place in the arch. In the case when the arch is designed to support ar^ horizontal plane or rcrad, on which heavy weights aie to be sustarncs;;. iv PREFACE. the intermediate space between the arcli and the horizontal road way, ought to be filled up in such a manner, as not' only to afford the sup- port required, but also to add to the strengdi and security of the entire fabric. If this should be effected by columns erected on the arch, and acting on the several sections by their weight in a direction perpendi- cular to the horizon, rules are criven in the followino- -^acres for estab- lishing the equilibrium by adjusting the angles of the sections to their several weights, including the weights of the columns superincumbent, so that the pressure on the sides of each section, may be a counterpoise to its weight, taking into account the place it occupies in the arch. But in structures of this description, the columns of masonry which are erected upon the arches of a bridge, as a support to the road way, cannot be expected to act on the sections of an arch according to the exact proportions required, which are assumed as data in the geometrical pro- positions, for determining the equilibration, as these proportions would probably be altered either by the differences of specific gravity which may occasionally be found in the materials used, or by differences in the cohesive force, which would prevent the columns from settling and pressing on the several parts of the arch with their full weights, such as the theory requires. Perpendicular columns of iron would not be liable to this objection : by adopting supports of this description, the weights of the columns, added to the weight of the road, would press on the interior arch, to be sustained in equilibrio, by adjusting the angles of the sections to the superincumbent weight, according to the rules determined in the pages which follow. But perhaps the space be- tween the interior arch and the road might be more effectually filled up, by other arches terminated by circular arcs, drawn from centres situated in the vertical line which bisects the entire arch, so as to be- come united in the highest or middle wedge. The sections of these arches may be adjusted, by the' rules here given, so as to become PREFACE. V distinct arches of equilibration, M'hich when united, will constitute a single arch of equilibration, similar in form to that which is expressed in the plan of an iron bridge, of one arch, which has been proposed to be erected over the river Thames,* as it is represented in the eno-ravincf inserted in the Third Report of the Committee of the House of Com- mons, for the further Improvement of the Port of London. According to this plan of construction, each part of the edifice would partake of the properties of equilibration, contributing additional strength and security to the whole building. In the course of this inquiry, exclusive of the general principles which have been here described, sundry other properties are investi- gated, which, it is presumed, may be of use in the practice of architec- ture, in the construction of arches of eveiy kind, as well as in ex- plaining some particulars relating to the subject, which have not hitherto been accounted for in a satisfactory manner. Some propositions of this kind are comprised in six general rules, inserted in page 19, which are expressed in simple terms, and are easily applicable to practical cases. Supposing an arch to consist of any number of sections or wedges, adjusted to equilibrium ; this arch resting on the two abutments, may be considered analagous to a single wedge, the sides of which are in- clined at an angle equal to the inclination of the two abutments : the forces therefore which would be necessary to sustain such an arch or wedge when applied perpendicularly to the sides, ought to be equal to the reaction of the pressures on the two abutments ; this principle is found on examination to be verified by referring to the tables annexed ;+ whether the arch consists of sections, without, or with the load of super- incumbent weight, and whether the angles of the sections are equal or • Designed by Messrs. Telford and Douglass, t Appendix, Tables, I. II. III. IV. V. vi PREFACE. unequal : For according to all these tables, the weight of the semiarch is to the pressure on the corresponding abutment, or the reaction there- of, as the sine of half the angle between the two opposite abutments, is to the radius ; which is a proportion equally applicable to the wedge, and to the arch, when adjusted to equilibrium. From the second of these rules it appears, that the lateral or hori- zontal pressure of any arch adjusted to equilibrum depends wholly on the weight and angle between the sides of the highest, or middle sec- tion : If therefore the weight and angle of this highest section should continue unaltered, the lateral force or pressure will be invariably the same, however the height, the length, the span, and the weight of the whole arch may be varied. This lateral force is called, in technical language, the drift or sJioot of an arch, and the exact determination of it has been considered as a desideratum in the practical construction of arches. When the dimensions of the sections composing the rectilinear or flat arch appeared to follow from the general construction for deter- mining equilibration, the author was inclined to suspect, from the apparent paradox implied in this inference, that some mistake or misapprehension had taken place, either in the general proposition, or in the deductions from it : but finding from trials on a model of an arch of this description, that the sections formed according to the dimensions stated for the flat arch in page ^5 of this Tract, Fig. 15, were supported in equilibrio, without any aid from extraneous force, he was convinced that the properties of equilibration deduced from the principle of the wedge, are no less true when applied to practice, than they are in theory. On inquiry it appears, that this species of arch has been long in use among practical artists; the dimensions of the wedges having been formed according to rules established by custom, but without being referred to any certain principle. PREFACE. vii A few observations may be here added, concerning the principles as- sumed in this Tract, as trudis to be allowed. It is supposed, that the constituent parts of an arch are portions of wedges, the sides of which are plane surfaces inclined to each other at an angle. Each wedge is considered as a solid body perfectly hard and unelastic, in respect to any force of pressure which can be applied to crush or alter its figure when forming part of an arch, the equilibrium of which is established by the pressure and gravity of the sections only, independently of the ties or holdings, which are applied for the purpose of prevent- ing the extrusion of the wedges by the force of any occasional weight which may be brought to press on the arch. These fastenings supply in some degree the place of the natural force, by which the parts of solid bodies cohere till they are separated by artificial means. When the weights of the sections are not very great, a defect of equilibrium to a certain extent may subsist, without producing any material change in the fio;ure of the arch, or endanoerinw the security of the fabric. But if heavy massive blocks of stone or iron should be placed toge- ther in the form of an arch, without being well adjusted, any con- siderable defect of equilibrium would cause a stress on the fastenings, which would overbear the weak alliance of cement or the mechanical ties and fastenings, that are applied to prevent the separation of the sections. In other cases, when sections of less weight are used, the cohesion which takes place between the surfaces of blocks of stone, with the aid of cement, and fastenings of various kinds, may impart a considerable degree of strength to edifices; insomuch, that although many arches have been counterpoised according to rules which produce rather a fortuitous arrangement of materials for forming the equilibrium, than an adjustment of it, according to correct principles, the cohesion of the parts have, notwithstanding, preserved them from falling; or from experiencing any considerable change of form. But this power viii PREFACE. has been sometimes too much relied on, especially by the architects of the twelfth, thirteenth, and the centuries immediately succeeding, who, in particular instances, entertained the bold idea of erectino lofty pillars, subject to a great lateral pressure, without applying any counterpoise. The consequence has been a distortion of figure, too evi- dently discernible in the pillars which support the domes in most of the old cathedral churches. However great the force of cohesion may be, which connects the parts of buildings, every edifice would be more secure by having all the parts of it duly balanced, independently of cohesion or mechanical fastenings, by which means, that distortion of shape would be prevented which the want of equilibrium in structures, must always have a tendency to create, whether the effects of it should be sufficient to produce a visible change of figure, or should be too small to be discernible by the eye. When arches are not perfectly balanced, and a change of figure ensues, the only security for the preservation of the fabric from entire disunion, is the excess of the cohesive force above the force tending to separate the parts of the building, arising from the want of counterpoise; and as cohesion is a species of force, which can- not be estimated with exactness, where the circumstances of an edifice are such as may weaken this force, or render the effects of it precarious, the more attention is necessary to establish a perfect equipoise between the weights and pressures of the several parts. DISSERTATION ON THE CONSTRUCTION OF ARCHES. As the exterior termination of an arch always exceeds the interior curve (usually called the curve of the arch), the sections or wedges of which it is composed will partake of a similar dispro- portion, the length of the exterior boundary in each wedge always exceeding that of the interior. A consequence of this wedge-like form is, that the weight of each section by which it endeavours to descend towards the earth, is opposed by the pressure the sides of it sustain from the sections which are adjacent to it. If the pressure should be too small, the wedge will not be supported, but will descend with greater or less obliquity to the horizon, according to its place in the arch. If the pressure should be too great, it will more than counterpoise the weight of the section, and will force it upward. The equilibrium of the entire arch will consequently depend on the exact adjustment of the weight of each section or wedge, to the pressure it sustains, and the angular B C 2 3 distance from the vertex. This equiUbrium is understood to be estabhshed by the mutual pressure and gravity of the sections only, independent of any aid from friction, cohesive cement, or fastenings of any kind. "When an arch is erected, fastenings are necessarily applied, to prevent the extrusion of the wedges by any force of weight which may be occasionally brought to press upon the several parts of the arch. The ensuing geometrical constructions are in- tended, in the first instance, to adjust the equipoise of the wedges or sections which are disposed according to the direction of any curve line that may be conceived to pass through the extremities of their bases, requiring only, as conditions to be given, the weight of the highest or middle wedge, and the angles contained by the sides of the several wedges or sections. From these data, the weight of each section is to be inferred, so that when the whole are put together, they may balance each other in perfect equipoise in every part. Since, according to this construction, the weights of the sections are dependent on the given angles between their sides, the exterior termination of the sections, or the weights superincumbent on them, will usually take some form which cannot be altered, without a change in the conditions given : to effect this change, when requisite, other considerations will be necessary, which are the subject of the latter part of this tract. For these reasons it appears, that the principle of establishing the equilibrium of an arch, by inferring the weights of the sections from their angles, not requiring a determinate form to the exte- rior boundary, is best suited to the construction of those arches which are erected either to connect the several parts of an edifice, or for their support and ornament, to which they so eminently contribute in many of those ancient monuments of skill and C3 ;] magnificence, which still remain to be contemplated with delight and admiration. According to this principle of construction, the architect is not restrained to curves, observing any particular law of curvature, but may adopt any that require mechanical delineation only, for describing the forms of arches he may wish to erect; to this ad- vantage is added, that of having each arch balanced within itself, and the pressure on each part, as well as on the abutments, exactly ascertained. And, as the angles of the sections may be varied at discretion, by properly altering the adjustment of the equilibrium, according to the rules here given, the force of pressure on the abut- ments may be made to take any direction which best contributes to the strength of the edifice, so far as the limits of those rules will allow. In addition to the principles which have been the subject of the preceeding observations, the following case is next to be con- sidered. It has been already observed, that when the weights of the sections are inferred from the angles between their sides, the heights of the masses added to the sections, or making a part of them, will be terminated by some line depending on the dimen- sions of these angles. But arches are often constructed for the purpose of supporting considerable weights, the terminations of which are required to be of particular given forms. In the case of bridges, a wall of masonry is usually erected on the arches, as a support to the road-way, which is always horizontal, or nearly so; the superincumbent weight of this wall, by adding pro- portionally to t'.ie weights of the sections, must require a stronger force of pressure on their sides as a counterpoise to it; this is effected in consequence of two alterations, by which the loaded arc differs from the arc constructed only for supporting its own B 2 c 4:1 weight. 1st. The weight of the highest or middle section, being augmented, increases the pressure, and the reaction between the surfaces of all the sections ; and to compensate fur the different degrees of weight which are superadded to the other sections, the angles contained between their sides, and the pressures upon them, are to be increased in due proportion. For as the angles of the wedges are increased, a given force of pressure acting on their sides, will have the greater effect in supporting the inter- mediate wedges. The heights of the wall built on the several sections proportional to the weights superincumbent on them, are supposed to be given quantities, so that the upper extremities may terminate in an horizontal, or any other given line: instead there- fore of inferring those heights, or weights, from the angles of the sections considered as given, according to the principle of construc- tion, which has been described in the preceding pages, we are to consider the heights of the wall, or weights on the several sec- tions, as given quantities, and to infer from them, what must be the magnitudes of the angles contained by the sides of the sec- tions, so that the weight of each, including the weight superin- cumbent, may be an exact counterpoise to the pressure on the sides of it. The curve line, which passes through the bases, may be of any form, without affecting the equilibrium established according to these principles. For the counterpoise of gravity and pressure between two or more wedges, is wholly independent of the line which may be drawn through their bases. If it should be objected that the more nearly an arch approaches to a right line, the less weight it will securely bear, it may be re- plied, that this insecurity is caused by circumstances which are quite independent of the equilibrium. If the materials of which an arch is constructed, were perfectl}^ C53 hard and rigid, so as not to be liable to the smallest change in their form, and the abutments were immoveably fixed, an arch, when the sections have been adjusted, although but little deviating from a right line, would be equally secure, in respect to equilibrium, V. ith a semi circular, or any other arch. From these general observations, the object of the ensuing tract appears to consist principally in the solutions of two statical problems, which may be briefly expressed in the following terms; ist, from having given the angles contained by the sides of the wedges which form an arch, together with the weight of the highest or middle section, to infer the vs^eights of the other sections ; and conversely, from the weights of each wedge given, together with the angle of the first section, to determine the angles between the sides of the other sections, so as to form an arch perfectly balanced in all its parts. In the construction of circular arches, the joinings of the sections, or sides of the wedges, are usually directed to the centre of the circle. In the following constructions, the sides of the wedges are directed to any different points ; but thei'e is no reason to suppose, that the equilibrium of the arch would be altered, or that the construction would be less secure, from this circumstance. Considering that an arch supports the weights which press upon it, and preserves its form in consequence of the wedge-like figures of the sections ; the principle of its construction and properties seem naturally to be referred to those of the wedge, which prin- ciple has been adopted in the ensuing disquisition founded on plain geometry and statics, or the doctrine of equilibrium or equi- poise, as established by Gallileo and Newton. Fig. 1. KCGA, DBG A, DBFE, represent three of the C 63 sections or wedges which form an arch, the lower curve of which passes through the points C, A, B, E, &c. The wedges are also, for brevity, denoted by tlie letters B, A, and C respectively. The highest wedge of the arch is G A D B, which (being here considered isosceles) is terminated on each side by the lines D B, G A inclined to each other at the angle BOA, which is termed, for the sake of distinction, the angle of the wedge or section. The termination of this wedge on the lower side is the line B A, the extremities of which coincide with the curve of the arch, and on the upper part, by the horizontal line D G parallel to B A. If therefore D G is bisected in the point V, a line V O joining the points V and O, will be perpendicular to the horizon. In like manner, the inclination of the sides K C, G A forms the angle of the wedge C, and the inclination of the two sides D B, F E is the angle of the wedge B. In the construction of arches, the angles of the sections are commonly made equal to each other, Tjut in a general investigation of the subject, it will be expedient to consider the angles of the sections of any magnitude, in gene- ral, either as quantities given for forming the equilibrium of the arch, by the adjustment of their weights, or as quantities to be inferred, from having the weights of their sections given. Fig. 1. The wedge A when unimpeded, endeavours by its gravity, to descend in the direction of the line V O, but is pre- vented from falling by the pressure of the wedges on each side of it, acting in the direction of the lines PO, K I, perpendicular to the surfaces D B, G A respectively. By the principles of statics, it is known, that if the force P Q, or its equal K I, should be to half the weight of the wedge, in the same proportion which the line O D bears to V D, that is, jn the proportion of radius to the sine of half the angle of the wedge V O D, the weight of the wedge will be exactly counter- poised by these forces ; and conversely, if any wedge is sustained in equilibrio by forces applied perpendicularly to the sides, these forces must be to the weight of the wedge in the proportion which has been stated. It is to be observed, that all pressures are estimated in a direc- tion perpendicular to the surfaces impressed ; for if the direction of the pressure should be oblique, it may be resolved into a force perpendicular to the surface, and some other force, which neither increases nor diminishes the pressure. Fig. 2. If the wedge A when unsupported, should not be at liberty to descend freely in the direction of the vertical line V O, but should be moveable only along the line G A, considered as a fixed abutment,* the force P Q singly applied will sustain it in equilibrio, the reaction of the abutment supplying the necessary counterpoise. For produce P Q (Fig- 2.) till it intersects the line G A in the point X, and in line X P, take M X equal to P Q ; the force P Q, considered as applied perpendicular to the surface DB, will have no effect in impelling the wedge toward the point O in the direction D B, or in the opposite direction B D, but the same force M X acting in an oblique direction on the line GA, may be resolved into two forces, M R perpendi- cular to G A, acting as pressure on it, and the force R X which impells the wedge directly from the point O in the direction of. * To prevent repetitions and unnecessary references, it is to be observed in the fol- lowing pages, that the lower surface of each section is considered as a fixed abutment, on which the weight of the section, and of all the sections above it, are supported. For this reason, the angle between the lower surface of any section and the vertical line, is termed, for brevity, the angle of the abutment of that section. C 8 3 the line A G. I'hrough any point A, in the line A G, draw A a parallel to the line V O representing in quantity and direction the weight of the wedge A ; through the point a draw a H per- pendicular to A G ; then will H A represent the force by which the wedge endeavours to descend in the direction of the line G A, considered as a fixed abutment. If then the line H A should be proved equal to the line R X, the contrary directions of these equal forces will balance each other, and the wedge so impelled will remain at rest in equilibrio. The proof that the lines H A and X R are equal is as follows. By the construction, the line A a represents the ^^'eight of the wedge A, and the angle H AiZ is equal to the angle V O G, or half the angle of the wedge. And because the line M X is per- pendicular to B D, and M R perpendicular to G A, it follows that the inclination of the lines G A, D B is equal to the inclination of the lines MX and MR, or the angle XMR is equal to the angle DOG. By the construction, and the properties of the wedge MX : 4r Aa :: radius : sine of V O G and RX : MX:: sine XMR: radius Joining these ratios RX :iAa :: sin. ofXMR : sin.VOG or because X M R is equal to D O G RX: Aa :: sin. DOG: 2sin.VOG also by con-' 'JAa: H. ^ . , ^A :: radius : cos. HA^ or VOG struction J Joining these ratios RX : H A : radius x sin. DOG : : 2 x sin. VOG x cos. VOG. But because the angh DOG is double to the angle VOG, from the principles of trigonometry it follows, that sin. DOG C9 3 X radius is equal to 2 sin. VOG x cos. VOG: therefore since the line R X is to the line AH, as sin. D O G x radius, to 3 x sin. G O V X COS. G O V, it follows that the line R X is equal to the line AH. All the successive sections or wedges which form the arch being by the supposition balanced and sustained by their gravity and mutual pressure, independent of any other force, if the whole of the arch is considered as completed, whatever force P Q ( Fig. 1 and 2.), is necessary for sustaining the wedge AGDB in equi- librio will be supplied by the reaction of tlie wedge adjacent to the surface D B. And in every part of the arch, when perfectly ba- lanced, whatever force of pressure is communicated to any section from that which is contiguous to it, the force of reaction will be precisely equal between the two sections. It appears, there- fore, that the equilibrium of the wedges will depend on the due adjustment of their successive weights to the pressures sustained by the sides of the sections. To effect this, the several sections are to be successively balanced ; first, the wedge A alone ; consider- ing it as moveable along the line G A, as a fixed abutment. It has been shewn, that if the force (Fig. 1.) P O is to half the weight of the wedge A, as the line DO is to the line V D, this force P O acting perpendicularly against the surface D B, and communicating an equal pressure M X obliquely on the line G A will sustain the wedge A from descending along the line G A. In the next place, let the wedge B (Fig. 3.) be adjusted to equipoise conjointly with the wedge A ; and let it be required to ascertain the weight of B, in proportion to the weight of A, so that they may continue in equilibrio, when moveable along the fixed abutment K B. (Fig. a and Fig. 3.) To effect this, produce M R till it intersects the line K B in the point V, and in the line R M pro- duced, take M N equal to the line Ha, In V N take V ^ equal C C lO to RN ; and from the point Q draw Q W perpendicular to the line K B produced if necessary, intersecting it in the point W: from any point B in the line K W, take B H equal to W V, and through the point H draw Hz (indefinite) perpendicular to the line KB. Through B draw Bo: (indefinite) parallel to the line V L O, intersecting the line H z in the point b ; then, the line B b will represent by construction the weight of the section B, when the line A a denotes the weight of the section A, and the wedges are balanced in equilibrio, although freely moveable in the direc- tions of the lines G A, K B. , For the pressure P O or M X, which impels the wedge A upward along the line A G with the force R X, is perfectly counterpoised by the force of gravity A H referred to that direc- tion, because it has been proved, that the line R X is equal to AH. If the pressure Ha or MN* on the line G A, arising from the weight of the section A, be added to the pressure M R, the sum or R N, will be the entire pressure on the surface A G, equal to V O by construction, or the oblique pressure on the line B K ; that isRM+H«=:RN = OV; QVis resolved into two forces, O W, perpendicular to K B produced, and W V in the direction of that line. The force O W, acts as pressure on the line B K, and the force W V impels the wedge in a direc- tion contrary to gravity along the line B K. The weight of die section B, is by the construction denoted by the line B b, and being resolved into two forces, H B acting in the direction of that line, and H b perpendicular to it ; the force H b acts as pressure, and the 'force H B is that part of the weight which impels the wedge B to descend in the direction of the line K B. But, by the construction, the hne H B, is equal to the line W V, * The lines H a MN, represent th? quantity and direction of these pressures, but arc not to be understood as determining the jjoint or place where the pressures arc applied. c 11 : these equal and contrary forces will therefore balance each other ; and the wedge so impelled, will remain at rest, so far as regards the direction of the line B K. In respect to the forces Q W, H b, which act in a direction perpendicular to tlie surface K B, they are perfectly balanced by the reaction of the abutment B K, or the reaction of the wedge C, when the forces in the direction of the hne F C have been adjusted to equilibrium. The result is, that when the weights of the sections A and B, have been adjusted according to the preceding construction, each of the forces both of pressure and gravity is exactly coun- teracted by an opposite force which is equal to it. (Fig. 3.) The weights of the wedges C and D are adjusted to equilibrium by a similar construction. Produce Q W till it intersects F Z in the point X. In W O produced, set off Q 6 equal to H b, and in the line X b take X Y equal to W 6 ; through the point Y draw Y Z perpendicular to F C produced, and from any point C in the line C F, take C H equal to Z X ; through H draw the inde- finite hne H z perpendicular to C F, and through the point C draw the indefinite line C x parallel to V L O ; the intercepted line C c will represent the weight of the section C. The weight of the section D is constructed on the same principle, b}^ making the line DH equal to the line FB, and drawing Dx parallel to VO, and H z perpendicular to I D, cutting off the line D d, which is the weight of the section D. If the sections D, C, B, A, then adjusted, are placed contiguous, and the opposite semiarch is completed, when the extreme sections D, D, are supported on the two abut- ments, the whole will remain in equilibrio, although freely move- able in the direction of the lines I D, F C, K B, G A, D C, &c. These and the remaining weights having been thus adjusted, so as to form an equilibrium, the lines A a, Bb, Cc, Dd, to which they are proportional, might be determined by lineal construction, according to the rules which have been given ; but as the correctness of such graphical delineations depends both on the excellence of the instruments employed, and on the skill of the person who uses them, to supply the want of these advantages in any case that may occur, as well as to view the subject under a different form, it may be expedient to ex- press the several weights and pressures which have been hitherto represented geometrically, by analytical and numerical values. From the preceding observations, it has appeared, that the weights and pressures depend in a material degree, on the weight of the highest or middle wedge A, which is bissected by the ver- tical line VO. (Fig. i, 2, and 3.) The weight of this section has been denoted by the line Aa in the construction, and is represented in the analytical values, by the letter w ; all other weights being in proportion to it. The initial pressure acting obliquely against the side of the wedge A G, represented by the line P Q = M X, has been found = — r-^,, „ ,, =p; and because the line M X 2 sin. V O O -" is perpendicular to D C, and M R is perpendicular to G A, it follows that the inclination of the lines DC, G A, is equal to the angle X M R ; or if A° is put to represent the angle contained between the sides of the wedge A, it will follow that A° = XMR = AOB. Since therefore p = ^ ^ ^-^^ ^ ^o =MX = PQ, the direct pres- sure against the surface G A, that is, M R is = M X x cos. A" =j^ X cos. A°. And as the additional pressure arising from the weight of the wedge Ais = Ha=Aax sin. |- A° = w x sin. i A°, the entire pressure on the surface G A =/> x cos. A°-\- w x sin. |- A" ; or since p = ^-j-^, the entire pressure on G A=; M R -f- Ha < wxcos-A° , . , .0 COS. A" + 2 siH.* -I A" •, = — —. — taT + w X sm. i A'= ~ I .0 ^ — X w, or because 2 X sill. I A° ' 2 ,2 sin. i A° ' 2 sin.^'i A''= the versed sine of the angle A°, it follows that c 133 MR + Ha = VQ= ^ ^ ^,^ ^ ^„ = /). But, because the line Q V is perpendicular to G A, and Q W is perpendicular to K B, tiie inclination of the sides K B, G A of the wedge B is equal to the angle VOW, which may be denoted by B''; and because O V =/>, it follows that QW =/» x cos. B°, and V W =^ x sin. B°; which is equal to the line H B, by the construction. If, therefore, the angle HBb, or the angle at which the abutment K B is inclined to the vertical line V O, or B 6 should be represented by V*, since VWorBH=/»x sin. B°, H b will he = p x sin. B" x tang. V*, and the line B6 will be/) x sin. B° x secant V* x which is the weight of the section B ; the pressure on the next section, or C, is Q W 4- H 6, which is =/> x cos. B° + /> x sin. B° x tang. V* : let this be made = q ; then the weight C c of the section C will be found in like manner to be = ^r x sin. C° x sec. V% and the pres- sure on the next section D = q x cos. C° -f ? x sin. C x tang. V', and so on, according to the order of weights and pressures which are here subjoined. It is to be observed, that A", B°, C, D°, &c. denote the angles of the sections A, B, C, &c. V" signifies the angle of inclination, to the vertical, of the line G A, on which the section A rests, = to the angle G O V or H A a. In like manner V* is the inclination to the vertical of the line K B, on which the section B rests = to the angle HBb: V' is the inclination to the vertical of the line F C, on which the section C rests, = to the angle H C ^, and so on. The initial pressure p = ^%-to. ^ •» 2 X sill, i A° Sections. Weights of the Sections. Pressures on the Sections next following. A zu p X COS. A°-{- p X sin. A° x tang. V" = ^ B px sin. B" X sec. V* /> x cos. B°-|-^ x sin. B° x tang. V* == ^ C q X sin. C X sec. V' q x cos. C° -\- q x sin. C° x tang. Y'=r *D r x sin. D° x sec. V'' r x cos. D° + '' x sin. D° x tang. V'' = 5 E 5 X sin. E" X sec. Y' s x cos. E° 4- 5 x sin. E° x tang. V' = / c: lO Or the weights of the sections and pressures on them may be expressed somewhat differently thus : the weights of the succes- sive sections being denoted by the letters a, b, c, d, respectively. Sections. Weight of the Sectiens. Pressures on the Section next following. K a =:zu p X cos, A° -f ^ X sin. V" = p B b =p X sin. B" x sec. V* p % cos. B° -f 6 x sin. V* = q C c = q \ sin. C° x sec. V' q x cos. C° -{• c x sin. V = r D d = r X sin. D° x sec. V'' r x cos. T)° -{■ d \ sin. Y'^ = s E e = s X sin E° x sec. V' s x cos. E° -j- ^ x sin. Y' := t In the following illustration of these analytical values, the angles of the sections are assumed equal to 5° each, and the weight of the first section is put = 1 : consequently, the initial pressure or /> = r'—- 5 — ; = 11.46279 = the pressure on the second section or B : for inferring the weight of the second sec- lion, we have p = 11 .46279, sin. B° = .087156, V* = 7° 30', and sec. V*= 1.008629; wherefore the weight of B =p x sin. B°x sec. V* = 1.00767. The pressure on the section next following, or C, = p X cos. W -{• p X sin. B° x tang. V*= 11.550708 = q, and therefore the weight of C = ^ x sin. C° x sec. V' = 1 .031 15, and the pressure on the next section D = ^ x cos. C° -f- ^ x sin. C° X tang. V' = 11.72992; and thenceforward, according to the values entered in the table No. 1 . When any number of sections have been adjusted to their pro- per weights on each side of the vertical line V O, the whole being supported on the abutments, on the opposite sides, will remain in equilibrio, balanced by the mutual pressure and gravity of the sections only; so that if the contiguous surfaces were made smooth, and oil should be interposed between them, none of the sections would be moved from their respective places. But if the wedges should be put together without adjustment, the weight of the sections near the abutments, if too great. C 15 ] would raise those which are nearer the vertex, and would them- selves descend from their places, in consequence of their over- balance of weiglit; contrary effects would follow, from giving too small a weight to the lower sections. In either case, they would not retain their places in the arch, unless the fastenings applied in order to prevent them from separating, should be stronger than the force created by the imperfect equilibrium, impelling the sections toward a position different from that which they ought to occupy in the arch. But a distinction is to be made between the deficiency of equi- librium which is inherent in the original construction, and that which is caused by an excess of weight, which may occasionally be brought to press on an arch. In the case of occasional weight, such as loads of heavy mate- rials which pass over the arches of a bridge, the stress on the joinings is temporary only ; whereas the stress which arises from a want of equilibrium in the construction, acts without inter- mission, and in a course of time, may produce disturbance in the fastenings, and in the form of the arch itself, which might resist, without injury, a much greater force that acts on the several parts of it during a small interval of time. When the wedges which form an arch have been adjusted to equi- librium, the whole will be terminated at the extremities by two planes coinciding with the abutments, and the entire arch will in this respect, be similar in form to a single wedge, the sides of which are inclined to each other, at an angle equal to the angle at which the planes of the abutments are inclined, (Fig. 4,.) IVC, represents an arch adjusted to equilibrium, and termi- nated by the curve lines IVC, F B D. The extremities of the arch are placed on the abutments I F, C D, which lines when produced, meet in a point O, If the arch be bisected in V, a Hne joining the points V and O will be perpendicular to the horizon. Let the line K L be drawn perpendicular to the surface I F and MN W perpendicular to C D, The arch acts by its pressure in a direction perpendicular to the abutments, the reaction of which is equal and contrary to the pressure, and in the direction of the lines K L, M N. From the principles of statics before referred to, it appears, that if forces are applied in the directions K L, M N, perpendicular to the surfaces IF, CD, considered as the sides of a wedge, those forces K L, M N, will sustain the wedge, provided each of them should be to the weight of the semiarch, as radius is to the sine of the semiangle of the wedge, that is as radius to the sine of V O C. If therefore, the abutment should be considered as removed, and two forces or pressures equal to the forces K L, M N, should be substituted instead of them, each being in the proportion that has been stated, the wedge or arch will be sus- tained in the same manner as it was by the abutments. The force that sustains the arch or wedge, is the reaction of the abutments, which is exactly equal and contrary to the force of pressure upon either of them, and has been ascertained in the preceding pages, when an arch consisting of any number of sections is adjusted to equilibrium as in the table. No. i . If, there* fore, the forces K L, or M N, be made equal to the pressure on the abutment, determined as above, the following proportion will result : As the force of pressure on the abutment is to half the weight of the wedge or arch V B C D, so is radius to the sine of the inclt- nation of the abutment to the vertical line, or is as radius to the sine of the angle V O C. Thus, referring to the order of weights and pressures, stated in page 13, let the semi-arch consist of any number, suppose five sections, including half the first section, or i A : then the weight of the semi-arch will be^A-|-B-|-C + D-|-E. The pressure on the abutment which sustains the section E is = s y. COS. E" -{- s -x. sin E° x tang. V' = t. E° is the angle of E, the fifth wedge = 5°, and V* is the angle of inclination to the vertical of the abutment on which the section E rests : or V' = 22°. 4,0'; then according to the proportion which has been stated, as the sum iA-j-B-|-C-|-D4-Eisto the pressure t, so is the sine of the angle V' to 1, which may be verified by referring to the numerical Table I. according to which, iA-j-B-j-C-j-D-j-E = 4.7436 ; V = V O N = 22° 30' ; and the pressure on this abutment appears, by the same table, to be = 12.3954 : the sine of 22° 30' = .382G8, which being multiplied by the pressure 12.3954, the product is 4.7434, scarcely differing from 4.7436, as entered in the table. In general, let the letter S denote half the weight of an arch, when adjusted to equilibrium : and let Z represent the pressure on the abutment, the inclination of which to the vertical is V" ; then S = Z x sin. V". Fig. 4. I F B D C represents an arch of equilibration, which is bisected by the vertical line VO; CD is one of the abutments in- clined to the vertical line in the angle VOC : through any point N of the abutment draw the line M W perpendicular to C D, and through N draw the line TQ perpendicular to the horizon. In the line N W set off" N E representing the pressure on the abutment CD, and resolve N E into two forces, E A, in the direction parallel to the horizon, and AN perpendicular to it. Then, because the angle A N F is equal to the angle N O V, or the inclination of the abutment to the vertical, denoted by the angle V% it follows, that the angle N E A is equal to the angle A N F or V O C = V^ from D whence the same proportion is derived, which has been otherwise demonstrated before, namely, as NE : N A : : 1 : sin. VOC ; that is, the force of pressure on the abutment is to the weight of the semi- arch, as radius is to the sine of V'' = N O V. It is observable that the additional weight of wedges, by which the weight of the semiarch is increased, reckoning from the vertex, or highest wedge, always acts in a direction perpendicular to the horizon ; but neither in- creases nor diminishes the horizontal force, which must therefore remain invariably the same, and is represented by the line EA. But the initial force of pressure which has been denoted by tlie letter^, is not precisely horizontal, being in a direction perpendicular to the surface of the first wedge A, although it is very nearly parallel to the horizon, when the angle of the first section is small, and might be assumed for it as an approximate value : a force which is to the initial pressure, orp = - — ^rj^' ^^ ^^^ cosine* of ^ A° (Fig. 5,) to radius, will approximate still more nearly to the constant force, the direction of which is parallel to the horizon, and is therefore For the sake of distinction, let this ■w X COS. i A° 2 X sin. i A° 2 X tang. { A°' force be represented by p' = - — ^^rr-^, to be assumed as an approximate value, which will differ very little from the truth when the angles of the sections are small, and in any case will be sufficiently exact for practical purposes. Then, because E A is denoted by />', N E = Z (Fig. 4.) represents the pressure on the abutment, and N E is the secant of the angle N E A, or V O C, to radius E A, it appears that Z=p' x secant V^ : by the same construction, A N is the tangent of the angle N E A to radius E A ; also A N is the sine of the angle N E A, or V O C, to the radius N E. • Note in the Appendi.x. C 19 3 The following general rules are derived from the proportions, which have been inferred in the preceeding pages : Rule i. The initial pressure is to the weight of the first section, including the weight superincumbent on it, as radius is to twice the sine of the semiangle of the middle, or highest wedge, or . __ w P 2 X sin. I A°' Rule h. The horizontal force, which is nearly the same in every part of the arch, is to the weight of the first section, as ra- dius is to twice the tangent of the semiangle of the first section, or p' = T-r-o- r 2 X tang. {- A° Rule hi. The horizontal or lateral force is to the pressure on the abutment, as radius is to the secant of the inclination of the abutment to the vertical, or Z = />' x sec. V^ Rule iv. The horizontal force is to the weight of half the arch as radius is to the tangent of the inclination of the abutment to the vertical, or S = ^' x tang. V". Rule v. The weight of the semiarch is to the pressure on the abutment, as the sine of the said inclination of the abutment is to radius, or S = Z x sin. V~. Rule vi. The horizontal force is to the pressure on the abut- ment as the cosine of the inchnation of the abutment is to radius, or/)' = Z X COS. V^ By these rules, the principal properties of the arch of equilibra- tion are expressed in simple terms, and are easily applicable to practical cases. Rule 3d. The horizontal force, or p', being the weight divided by twice the tangent of the semiangle of the first section, deter- mines the pressure on any abutment of which the inclination to the vertical line is V* ; the pressure being = />' x secant V*. Ds Rule 4th. The weight of the semiarch, when adjusted to equih- brium, is found by the fourth rule to be = ^' x tang. V^; or the horizontal pressure increased, or diminished, in tlie proportion of tlie tangent of the vertical distance of the abutment to radius. From this property, the reason is evident, which causes so great an augmentation in the weights of the sections, when the semiarch, adjusted to equilibrium, approaches nearly to a quadrant, and which prevents the possibility of effecting this adjustment by direct weight, when the entire arch is a semicircle. Rule 5th. The fifth rule exemplifies the analogy between the entire arch when adjusted to equilibrium, and the wedge. For let the angle between the abutments be made equal to the angle of the wedge, the weight of which is equal to the weight of the arch ; and let Z be either of the equal forces, which being applied per- pendicular to the sides of the wedge, sustain it in equilibrio : then by the properties of the wedge, the force Z is to half the weight of the wedge as radius is to the sine of the semiangle of the wedge, which is precisely the property of the arch ; substituting tlie angle between the abutments instead of the angle of tiie wedge, and the pressure on either abutment instead of the force Z. Rule 6th. The lateral pressure, or the pressure on the abutment, reduced to an horizontal direction, is nearly the same in all parts of the arc; being to the weight of the first section, as radius is to twice the tangent of the semiangle of the wedge. The force of pressure on tlie abutment is therefore at every^ point resolvable into two forces ; one of which is perpendicular to the horizon, and is equal to the weight of the semiarch ; and the other is a horizontal or lateral force, which is to the weight of the first section, as radius is to twice the tangent of the semiangle of that section. c 21 :i These conclusions are the niore remarkable from the analogy tliey bear to the properties of the catenary curve, altliough they have been deduced from the nature of the wedge, and the principles, of statics only, and without reference to the catenary or other curve, and will be equally true, when applied to the sections of an arch, which are disposed in the form of any curve whatever. Many cases occur, in the practice of architecture, in which it must be highly useful to form an exact estimate of the magnitude, and direction of pressure, from superincumbent weight, both on account of the danger to be apprehended if such pressure is suf- fered to act against the parts of an edifice without a suitable counterpoise ; and from the consideration, that when the extent of the evil to be provided against is not certainly known, it is probable, that more labour and expense will be employed in making every thing secure, than would have appeared necessary if the pressure to be opposed had been exactly estimated. The Gothic cathedrals, and other edifices built in a similar style of architecture, whicii very generally prevailed in this and other countries of Europe, during several centuries, have been constructed on principles to which the preceding observations are not intirely inapplicable. The striking effects for which these structures are remarkable, seem principally to be. derived from the loftiness of the pillars, and arches springing from considerable heights, which could not be securely counterpoised without making great sacrifices of ex- ternal appearance, and bestowing much labour and expense in imparting sufficient stability to the wliole fabric : while, there- fore, the eye is gratified by the just proportions and symmetry of design, displayed by the interior of these edifices, with an apparent lightness of structure, scarcely to be thought compatable with. C 22:1 the use of such materials, the external building presents the unpleasing contrast of angular buttresses, with their massive weights, which are indispensably necessary for the preservation of the walls, as a counterpoise to the lateral pressure from the ponderous arched roofs. That the security provided has been perfectly effectual, is evident from the solidity and duration of these buildings ; but whether any part of their heavy supports might have been spared, or whether the form of applying them might not have been changed, without diminishing the security of the walls, is a ques- tion which would require much practical experience and informa- tion to decide. An estimate of the lateral force from arches of this description, may be readily obtained by referring to the rules given in the preceding pages ; from v/hich it appears, that the lateral or horizontal force arising from the pressure of any arch is always to the weight of the highest or middle wedge, as radius is to 2 X tangent of the semiangle of the wedge. Some of the highest wedges, in roofs of this description, are said to weigh two ton ; the angle of the wedge may be taken (merely to establish a case for illustrating the subject) equal to 3°: the tangent of half this angle, or 1° 30', is .02618, and the lateral force, or pres- sure from any part of the arch, will be ^ ^ '^^^^g = 38.3 ton, which weight of pressure, acting on walls of great height, must certainly require a very substantial counterpoise for their support ; and it is for this purpose, that a buttress is erected against the external walls corresponding to the key-stone of each arch. In this estimate, the arch is supposed to h^ve reference to one plane only; which passes through both the buttresses and the key- stone : but in the case of groined arches, or such as are traversed by other arches crossing them diagonally, on which the same i: 23 3 key-stone acts, the lateral pressure in any one plane will be less than has been found according to the preceding estunate. In the preceding geometrical constructions, tlie angles of the several sections Iwe been assumed equal to each other, or as given, although of different magnitudes, from which data the weights of the sections have been inferred when they form an arch of equilibration. Let DC V D, &c. (Fig. 6.) be the curve-line passing through the bases of the sections which form an arch : because the wedges increase as they approach the abutment ; the exterior line a b c d, &c. will take a foi'm not very dissimilar to that which is re- presented in the figure, (Fig. 6.) : the arch being here adjusted to equilibrium in itself without reference to any extraneous weight or pressure. Although in many cases, it is immaterial what may be the form of the line a b c, &c. yet it often happens, that the termination of the sections, or exterior boundary, must of neces- sity deviate greatly from that which is represented in the figure; particularly in the case of bridges, over which a passage or road- way is required to be made, which is either horizontal, or nearly so. Let E D L D E (Fig. 7.) represent an arch by which an horizontal road, P a O, is supported. For this purpose a wall of masonry is usually erected over the arches ; the weight of which must press unequally on the several sections, according to the horizontal breadth and height of the columns, which are super- incumbent on them : through the terminations of the wedges A, B, C, D, &c. (Fig. 7.) draw the lines Ka, Bb, C c, &c. perpendicular to the iiorizontal line P Q, and in the lines ^ A, i B, c C, d D, &c. produced, if necessary, take the line 6 B of such a magnitude, that it shall be to the line a A in the same proportion, whicsh the weight of the wedge B, with the weight above it, bears C 24 3 to the weight of the wedge A, with the weight above it, and so on ; each line cC,dD, &c. being taken proportional to the weight of the respective wedges, including the weights superincumbent on them. In the next place we are to ascertain from these con- ditions, together with the angle of the first section or A°, the other angles B°, C°, D', &c. so that the entire are, when loaded with the weights, denoted by the lines Aa,Bh, Cc, Sec. may be equally balanced in all its parts. Admitting that the angle of any section D°, can be ascertained from having given the weight of the section D, and the pressure on it, together with the inclination of the abutment of the preceding section, or the angle of the abutment C to the vertical, it follows as a consequence, that, by the same method of inference, the angles of all the sections will be successively obtained from the angle of the first section, and the initial pressure, which are given quantities in the construc- tion of every arch. Suppose, therefore, any number of the sections A, B, C, to have been balanced by the requisite adjustments. It is required to determine the angle of the next section or D°,. on the following conditions. 1st. That the direct pressure on the abutment F C of the section D, from the preceding sections shall be given, equal to the oblique pressure on the line I F, denoted by the line S B, wbich is drawn perpendicular to the line F C produced. (Fig. 8. ) ad. That the weight of the section D, including the weight superincumbent on it, shall also be given : let this weight be represented by the line D d. 3d. The angle 6- C H being the inclination of the line F C, or the abutment of the preceding section C to the vertical, is also a given quantity. C^5l Referring to figure 3, we observe, that the adjustment of the angle D* depends on the equahty of the line H D representing the force arising from gravity, and urging the section to descend along the line ID, and the force B F, which impels it in tlie op- posite direction D I (Fig. 3.) In this case, F B is the sine of the angle * B S F = D° to a radius = B S, and H D is the cosine of the angle H D ^ to a radius = D d. Through the point D draw D A parallel to C F ; so shall the angle A D ^ be equal to the angle HCc, which is given by the conditions : moreover, the angle I D A is the unknown angle of the section D', which is required to be determined, and the inclination of the line I D to the vertical line Dc/, when constructed, will be equal to the sum of the angles AD^+IDA=ID^. BS represents the pressure on the section D (Fig. 8.) From these conditions the angle required D° or I D A, and the inclina- tion of the abutment I D ^ are determined by the following con- struction : through the point D, which terminates the base of the section D, draw the line Dd perpendicular to the horizon and equal to the given line which represents the weight of the section D : with the centre D and distance D d describe a circle : through the point D draw the line D A parallel to C F, cutting off an arc A d, which measures the angle A D ^ equal to the given angle c C H. Through the points D and d, draw the indefinite lines DZ dY perpendicular to the radius Dd, and through the point A draw the indefinite line AW parallel to d Y. In the line AW setoff AN equal to the given line B S, and through the points N and D draw the line D N, intersecting the circle in the point H : * Because S B, S F are by construction perpendicular to the lines F Z, I F respectively ; consequently the inclination of the lines F Z, IF, that is, the angle of the section D°, will be equal to the inclination of the lines SB, S F, or the angle B S F. E C 26 3 from the point A set ofFan arc iVI, equal to the arc GH, and through the points I and D draw the line I DQ. The angle I QF or IDA will be the angle of the section D, which was required to be de- termined by the construction, and I D^ will be the inclination of the abutment D I of the section D to the vertical hne D d. The demonstration is as follows : through the point S draw S F per- pendicular to I Q, and S B equal to the line denoting the given pressure on the line F I, and perpendicular to F O ; also through G draw GO perpendicular to DG, and through H draw HP per- pendicular to DG; also produce DN till it intersects ^Y in L: produce N A till it intersects D ^ in the point F, and through d draw d H perpendicular to D I. It is to be proved that the cosine of the angle ID x cos, B''^- 6 xsin,V^=g C c tang. C= ///;;:„^;, V'- = V'' + C° ^ x cos. C° + c x sin. V— r D d tang. D°= -i^^°-t^ V''=V'+ D° r x cos. 0°+ ^/xsin.V''=5 o r-f dx sin.V ' ' E e tang.E°=^-l^J^ V'=V''+ E^ 5 x cos.E° + e x sin. V' = ^ Eq C 28 3 The application of these analytical values will be exemplified by referring to the case of an arch formed by sections, which are disposed according to the figure of any curve, when the columns which are built on the arches, terminate in an horizontal line ; the given weights of the sections A, B, C,- &c, (including the weights of the columns built upon them), denoted by the lines A a, Bb, C c, &c. being as follows. Aa = the weight of the first section, (Fig. 7.) is represented by the number 2; B6 = 2.76106, Cc = 5.03844,, and so on. The weight of the first section, which is denoted by the num- ber 2, may be taken to signify 2 hundred weight, 2 ton, &c. all other weights being in proportion to it ; the angle of the first or highest section is 5° = A°, and w = 2. The initial pressure or - = 23.9225 =i p : making b = 2.76106, we obtain 2 X sin. 2° 30 from the preceding theorem, the tangent of B° = ^ ^ J.^ y^ = tang. 6° 49' 31", wherefore the angle of the section B or B° = 6° 49' 31"; this added to 2° 30' will give the inclination of the abutment of the section B to the vertical line =: 9° 19' 31" = V* ; also since/* = 22.9255, the pressure on the section C° =/> x cos. B° 4- & X sin. V*= 23.21305 = q, and thenceforward, according to the successive angles and the pressures on the sections next following, as they are entered in the Table No. II. entitled, A Table shewing the Angles of the Sections, &c. calculated from the given Weights, &c. The lines a A, bB, cC, representing the given weights of the sections and the weights superincumbent on them, (if the line La = 2 be subtracted from each) are nearly proportional to the versed sines of the arcs of a circle, increasing by a com- mon difference of 5°; the curve of the arch will therefore be scarcelv different from the arch of a circle. The third column L 29 3 of tlie second Table shews the several angles of the sections, which will create a suflScient force of counterpoise to the weight of the sections, together with the weights of the columns built upon them. It appears by inspecting this Table, that the angles of the sections first increase, reckoning from the highest or middle wedge, till the semiarc is augmented to about 55^; and afterwards decrease ; plainly indicating that part of the arch which requires the greatest aid from the increased angles of the sections, as a counterpoise to the weight above them. If therefore the angles of the sections were constructed equal, as they usually are, the form of the arch being circular, and if a wall of solid masonry should be built upon it, terminating in an horizontal line or plane, it is clearly pointed out, what part of the arch would be tlie most likely to fail, for want of the requisite counterpoise of equilibrium ; and although the fastenings should be sufficient to prevent the form of the arch from being immediately altered, the continuance of its constructed figure would depend on the resistance opposed by the fastenings to the stress arising from a defect of equilibrium, which acts incessantly to disunite the sections ; a preponderance of this force, to a certain degree, would probably break the arch somewhere between 50° or 60° from the highest or middle section; In adjusting the equilibrium of an arch, it is observable that the lengths of the bases which form the interior curve, usually termed the curve of the arch, are not among the conditions given, from which the weights or angles of the sections are inferred. A cir- cumstance which renders the solution here given of the problem for adjusting equilibration, very general. "Whatever, therefore, be the figure of the interior curve, the bases of the sections which are disposed in this form, may be of any lengths, provided the weights and angles of the sections are in n so :3 the proportions which the construction demands : observing only, that if the lengths of the bases should be greatly increased in respect to the depths, although in geometrical strictness, the pro- perties of the wedge would equally subsist, yet when applied to wedges formed of material substance, they would lose the powers and properties of that figure : this shews the necessity of preserv- ing some proportion between the lengths of the bases and depths of the wedges, to be determined by practical experience, rather than by geometrical deduction. The following constructions and observations will further shew how little the equilibrium of an arch depends on the figure of the curve line by which it is terminated. All the properties of arches being (so far as the preceding constructions and demonstrations may be depended on) the consequences of the weights and pres- sure of the sections, acting without relation to the figure of any curve, so that arches may be constructed which terminate in a circular, elliptical, or any other curve, retaining the properties of equilibration indifferently in all these cases. To exemplify this principle by a simple case, let all the sections which form any arch be of equal weights, the angle of the first wedge, or A^ being = 5"; and let it be required to ascertain the angles of the other sections, so that the pressures may be a counterpoise to their weights in every part. Assuming, therefore, as conditions given, (Fig. 9.) the angle of the first or highest wedge A° = 5°, and the weights of the several sections = i=Aa = B6 = Cc=DJ, &c. we have for the construction of this case the initial pressure = r ; = 6 = M X, the ansrle 01 the abutment A or V" 2 X sin. 2° 30 ■* ° = 2° 30'. From these data the angles* of the sections will be • If the weiglies of the sections A, B,.C, D, &c. which are denoted by the lines a, b, c, d. Sec. were made equal to 1.C077, 1.031Z, 1.07 19, &c. as they are stated in the Table C 31 3 constructed according to the solution in page 27, from which the following results are derived : ^^^".^ y, = tang. B° = 4° 57' 40", therefore B°= 4,° 57' 40", which being added to 2° 30', the sum will be equal to the angle H B 6 = V^ 7° 27' 40". C is found by the same theorem to be = 4° 51' 10", and V'= 12" 18' 50", and so on, according to the statements in the Table No. 3. 01 VIO (Fig. 10.) is the arc of a circle drawn from the centre O, and bisected by the vertical line VO. PWQ (Fig. ii.) is the arch of a circle drawn from a centre any where in the line VO, produced if necessary; T Y S (Fig. 12.) is the arch of an ellipse, the lesser axis of which coincides with the line VO. WXZ (Fig. 13.) is a catenarian or any other curve which is divided by the vertical line VO into two parts, similar and equal to each other. These three curves form the interior figures of the three arches, the exterior boundaries of which are of any figures which make the semiarches on each side of the vertical line V O similar and equal. In the next place, the circular arc OI VIO is to be divided into arcs, by which the angles of the sections in the three interior arches are regulated. For this purpose, from the point V on either side thereof, set off an arc VG = 2° 30'; set off also from G the arc G K = 4' 58', omitting the seconds, as an exactness not ne- cessary : and the subsequent arcs K F, F I, &c. according to the dimensions in the schedule annexed, extracted from the 3d Table, to the nearest minute of a degree. No. I. The points O, R, Q__, P, would all coincide in the point O ; if the weights of the sections should be assumed greater than they are stated in the Table No. I, the points R QJ" would be situated between the points O and V. 4 58 4 51 4 41 4 21 4 13 3 55 3 39 3 23 C 32 3 Sections. Arcs, Angles. iA VG 2 s'o B GK C KF D FI E IM F MN G NO H OO I OR The arcs (Fig. lo, et sequent.) VG, G K, K F, &c. having been thus set off, according to the angles of the sections A, B, C, &c. through the points G, K, F, I, in the circular hne OI VIO, &c. draw the lines GO, K O, F O, I O, &c. dividing the three arches into sections or wedges A i, B i, C i, &c. A3, B2, C2, and A3, B 3, C 3, &c. Which wedges, when formed of ma- terial substance, will become so many arches of equilibration, if the weights of each section be equal to the weight of the highest or middle wedge A. This construction is not intended to point out any practical mode by which the forms of wedges, that constitute arches of equihbration may be delineated, but merely to shew, by a very simple case, and at one view, how much the curves of arches may be varied, while the properties of equilibrium still remain the same, in a geometrical sense; although in the practical constructions of arches, the greater curvature of an arch allows greater latitude for the unavoidable errors in exe- cution, and for those which are the consequences of the imper- fect nature of the materials used in the construction. The figures in which the sections are here disposed have been adopted for tlie purpose of shewing in what manner the wedges in the several i n 33 3 arches Ai, B i, C i, &c. A 2, B 2, Cs, &c. are adjusted to equi- librium by means of the division of the external arc, according to the angles inferred from the solution of this case, (page 30) in which the weights of the sections are all equal, producing the angles which are entered in the Table No. III. and in the Schedule, page 32. Admitting, then, for the sake of establishing a case, that the specific gravity of the sections should be capable of adjustment, so that their weights may be equal, although their magnitudes be different; admitting also, that in increasing or diminishing the volumes of the sections, the angles are continued invariably the same ; the lengths of the bases may be lessened or augmented in any proportion that is required, the equilibrium of the section remaining unaltered. Thus, if it should be proposed, that any number of the sections in the arch (Fig. 13.) A3, B3, C3, shall occupy a length denoted by the curve (Fig. 14.) ivi, and that the joinings of the sections should intersect the curve in the points ifkggkfi, as represented in the figure. Through the point g draw the line g G parallel to G O, and through k draw ^K (Fig. 14,.) parallel to K O, and through/ draw/F parallel to FO, and so on : the sections A, B, C, &;c. in Fig. 14, admitting their weights to be equal, would form an arch of equilibration : the same consequences will follow if this con- struction is applied to the rectilinear or flat arch, (Fig. 15.) if it be allowed to use that term, meaning the figure terminated by two arched surfaces when their curvature is diminished to nothing, and coinciding with tv/o plane surfaces parallel to the horizon ; sucli is the rectilinear figure P O, PO, Fig. ij. Suppose this figure to be divided into wedges that have the properties of F c 34 n equilibration, by which the force of pressure impelling them up- ward is counterbalanced by their weight ; and suppose the join- ings of the wedges are required to pass through the points M, N, O, O, &c. Through M draw T M parallel to G O, and through N* draw RN parallel to K O and through O draw- WO parallel to F O', &c. If wedges of these forms are disposed on each side of the vertical line V L, the extreme sections being supported by the abutments P O, P Q ; the whole will be sustained in equili- brio, on the condition that the weight of each section is equal to the weight of the section A. But supposing the wedges to be formed, as is usually the case, of solid substances which are of the same uniform specific gravity, to make their weights equal, the areas of the figures must be adjusted to equality ; which requires the solution of the following problem: Having given the angles of any of the wedges as above stated, and having given the area of the wedge A = —^ — ^^^-^ , to ascertain the lengths of the upper surfaces RT, RW, P W, and of the bases M N, NO, O Q, of the wedges B, C, D, &c. So that the areas RNT M, WORN, PQWO, may be equal to the area T ^ Mm, with the condition that the angles of each wedge shall remain unchanged ; that is, the lines T M, R N, WO, &c. shall be parallel to the lines G O, K O, F O, I O, &c. respectively : to obtain the lengths of the lines R T, M N, which terminate the wedge B, according to these conditions, make the perpendi- cular distance V L = r, the area TMtm = A ; cotang. M N R — cotang. LMT = D; then RT = ^^\"l''^, and MN 2 A — r"^ X D zr ' . » Fig. 10. I I 35 '2 As an illustration of this theorem, let the angle of the middle wedge A (Fig. 15.) be assumed = 30°: if the weights of all the other sections are equal to that of A, the successive angles as de- termined by a preceding construction, in page 24, &c. will be as follows, B° = 23° 47' 38": and in hke manner, the angles C°, D°, and E° are found to be as they are stated in Table IV; the wedges being constructed according to these angles will give the follow- ing results : the angle L M T = 90° -}- V" = 105°; M N R = 90' + 38° 48' : = 128° 48' = 90 -f V^ ; N O W = 90° + 53° 16' == 143° 16' = 90 + V', and so on. From hence we obtain the lengths of the lines T R, M N. When the area R N T M is equal to the area T t M m ^ A. Let the perpendicular distance V L = r = 2 feet ; and suppose the radius = 5 feet, then the angle TO^ being = 30°, Tt = 2.679492, and Mm = 1.607695, and the area of the section A = 2.679492 -j- i-6"o7695 x — = 4.287187 = A. Cotang. 105° — cotang. 1 28° 48' by the Tables = 5360714 = D ; wherefore the line T R = -^—^ == 2.679492, and M N 2 A _ r'^ D .. _ = — = 1.607522. The dimensions of the sections C, D, &c. are determined from the same rule, and are as underneath. 0 .• Lengths of the upper Leniflhs of the ^,,. Sections. *> s^fj^^s ' ^ g^^^^ Oblique Lines, or Secants. A Tt =2.679492 M;/?=: 1.607695 VL =2. B TR =2.679665 MN =: 1.607522 TM= 2.0705524 C RW = 2.679548 NO = 1.607639 RN = 2.5662808 D WP= 2.679077 OP = 1.608110 WO = 3.3439700 E PX =2.680363 PY= 1.606824 PQ =4.2508096 According to the geometrical construction for adjusting the F2 C 3^ ] equilibrium of an arch by the angles between the sides of the sec- tions or wedges, the architect will be enabled to distribute the mass of materials, whether they consist of stone or iron, of which the arch is intended to consist, among the sections, in any propor- tion that may best contribute to strengthen and embellish the entire fabric, establishing the equilibrium of the arch at the same time. This principle of construction would be of use, more par- ticularly where circumstances may require that the equilibrium should be adjusted with great exactness. Supposing that accord- ing to the plan of the structure, the angles of the sections are equal to each other; if the mass or weight which the adjustment of the equilibrium allots to the sections near the abutments, should be diffused over too great a base ; or may be, for other reasons, independent of any consideration of equilibration, judged too weak to support the superincumbent weight with security, this inconvenience would be remedied by adding such a quantity of materials to the weaker sections, as may enable them to sup- port the weights or loads they are required to bear, and after- wards adjusting the angles of these sections, so as to form an arch of equilibration, according to the rules which have been given, (page 27). Or perhaps it might be expedient to arrange, in the first instance, the quantity of materials which ought to be allotted to the several sections of the entire arch, and afterwards to adjust the angle of each section, so as to form the equilibrium : suppose, for instance, the form of the arch be such as is repre- sented in the figure 16 : V A B C D, &c. is a circular arc drawn from the centre O and with the radius O V. Let the bases of the sections be terminated by the arcs A B, B C, C D, each of which subtends, at the centre O, an angle of 1°. Through the points n 37 3 A, B, C, D, &c. draw the indefinite lines A a, Bb, Cc, perpen- dicular to the horizon, and suppose the masses or weights allotted to the sections would give a sufficient degree of strength to the entire structure, when the weight of the section contiguous to the abutment is three times the weight of the first section, and the intermediate weights are increased by equal differences, from i to 3. If, therefore, the number of sections should be fixed at 49, so as to make the angle of the arch when viewed from the centre O = 4,9°, the weight of the highest or middle section being assumed equal to unity, the weight of the section B or B 6 = 1.083333, Cc = 1 . 166066, &c. ; and finally the weight of the section next the abutment, or Z = 3 ; as they are stated in the Table No. V. in the column entitled. Weights of the Sections. Since, therefore, the angle A° is by the supposition = 1°; the initial pressure, or ^ X sin. 4 A° = 57-29%9 = P- And because V" = 30', and b is the line denoting the weight of the section B = 1-083333, accord- ing to the rule for determining the angles of the sections, so as to form an arch of equilibration, r^rj—-- — tf: = the tan- gent of 1° 34,' 58" = B° ; which angle being added to 30' or Y", the sum will be = 1° 34' 50" = V*, or the inclination of the abutment to the vertical, of the section B ; from whence we ob- tain the entire pressure on the next section = p y. cos. B° -{- b X sin. V" = 57-31593 = (1' and ^ 11 '^'2'.v^ = ^^"g- i°9' 54"; therefore the angle of the third section C°= 1° 9' 54", and so on. The angles of the sections D°, C°, and the corresponding angles of the abutments are entered in Table V. From the angles of the abutments determined by these calculations, the practical de- c 38 : lineation of the sections will be extremely. easy: having described the arc of a circle with any radius O V, and the several arcs A B, B C, CD, &c. being set off equal to i°, the positions of the suc- cessive abutments which terminate the sections on each side will be found by taking the difference between the inclination of the abutment to the vertical, and the angle subtended by the semi- arc at the centre of the circle, if this difference be put = D, that is, to exemplify for the abutment F Y, if the difference of the angles V P F — V O F, or P F O be made = D ; then the line O P = ^-^ — J^ : consequently the length of the line O P being ascertained, through P and F draw the line PF Y, which will be the position of the abutment on which the section F rests. And a similar construction will determine the positions of all the lines EX, D W, C O, &c. ; when the sections form an arch of equili- bration according to the conditions given. By this rule the lines OT, OS, OR, &c. (Fig. 16.) are found, according to the fol- lowing Table, O V = O F being put = radius = 1000. C 39 3 Sections. Distances from the Centre O. Sections. DistJnces front the Centre O. A 00 = 0 0 = 338.00 B C D OT= 52.651 0 S = 90.009 0 R = 123.22 P = 351-10 = 3% 17 = 374.12 E F G H 0Q= 15338 OP = 181.90 == 207.48 = 230.95 s T U V = 383.21 = 393-25 = 401-33 = 408.62 I K = 252.44 = 273.36 w X = 41513 = 420.78 L M = 290.93 = 307-9.5 Y Z == 426.15 = 429-96 N = 323 53 The Tables No. I. II. III. IV. V. subjoined to these pages, h?ive been calculated rigidly from the rules in pages 14 and 27; a column containing the weights of the semiarcs, or the weights of I A 4- B 4- C -f D, &c. has been added to each of the Tables, for the purpose of comparing them with the general rules for approximating to the correct values inserted in page 19. The calculations in the Tables are expressed to five or six places of figures, and the results of the approximate rule 5, which is S = Z X sin. V-, coincides with the correct values in the Tables to four or five places, including the integers ; the calculations made from the other rules, which include the horizontal force, approximate to the true values the more nearly, as the angles of the sections are smaller; but in any, except very extreme cases, they are sufficiently correct for all practical purposes ; which will make the use of these approximations preferable to the troublesome C 40 ] calculations which are required for inferring the correct values from the original rules. That a judgment may be formed of the errors to which these approximate values are liable, the Table No. VI. is added, contain- ing the comparative results therein stated. The examples in the Table No. VI. have been taken from the Table No. II. in which the angles are calculated to the nearest second of a degree; and the numbers to be 6 or 7 places of figures: an exactness not necessary, except for the purpose of comparing the results arising from the different rules for computing. c 41 n Table No. I. Shewing the weights of the several sections or wedges which form an arch of equilibration, when the angle of each section is 5°; and the weight of the highest or middle wedge is assumed = 1. Also shewing the pressures on the lowest surface of each section, considered as an abutment. The initial pressure The lateral or horizontal pressure = 2 X sin. 2° 30' 2 X tang. 2° 3 = 11.4628 =/» -,=z 11.4,519 =p' Weights of the Semi- Seciions. Anglesofthe Sections. .\ngles of Inclina- tion to the vertical Line of the Surface, on which each Sec- Weight of each Section. Pressure on each successive Wedge considered as an arcs, being the suc- cessive Sums of the Weights in the 4th Co. lumn, deducting from tion rests. Abutment. each Sum the Weight ofiA=.5. A 5 2 30 1. 11.4628 •5 B 5 7 30 1.0077 11-5507 1-5077 C 5 12 30 1.0312 11.7300 2-5389 D 5 17 30 1.0719 12.0076 3.6108 E 5 22 30 I.I32S 12.3954 4-7436 F 5 27 30 1.2180 12.9098 5.9616 G 5 33 30 -^■334^-^ ^3-5775 7-2957 H 5 S7 30 1.4916 14-4346 8.7873 I 5 42 30 1.7064 155325 10-4937 K 5 47 30 2.0038 16.9508 12-4975 L 5 52 30 2. 42 68 18.8116 14-9243 M 5 . 57 30 30515 21.3136 ■^7-975^ N 5 62 30 40530 24.8284 22.0288 0 5 67 30 5-6547 29.9582 27.6835 P 5 72 30 8.6830 38.1254 36.3665 g 5 77 30 15.3661 52.9822 51.7326 R 5 82 30 35-4777 87-9547 87.2103 S 5 87 30 1757425 263.1952 262.952S C 40 Table No. II, Shewing the angles of the sections which form an arch of equih- bration, calculated from the given weights of the sections, in- cluding the weights of the columns built upon them, terminating in a right line parallel to the horizon ; the weight of the first section, or j^ = 2 = zv. The initial pressure = p =z -^ ^-Ta° ^^ 22.92558 The horizontal force or pressure* =/>'= — taTo- ^a° ^^ 22.903766 Weights of the Semi- Angles of Inclina- arches, being the suc- Sections. Given Weights of Angles of the tion to the Verti- cal of the lower Total Pressure on the Abutment, or cessive Sums of the W^cights in the ad the Sections. Sections. Surfaces of each Section, consider- ed as Abutments, on the Section next following. Column, deducting from each Sum the Weight of i A. 0 A 2.00000 500 2 30 0 22.90558 1.00000 B 2.76106 6 49 31 9 19 31 23.21305 3.76106 C 5-03844 11 41 23 21 0 54 24-53843 8-79950 D 8.81484 16 32 41 37 33 3^' 28.89592 17-61434 E 14.06148 16 34 7 54 7 43 39.09063 31.67583 F 20.73844 12 15 55 66 23 38 57-19849 52.41426 G 28.79492 7 51 16 74 14 54 84-37357 81.20918 H 38.16960 4 53 24 79 8 18 121.55373 119.37878 I 48.79112 3 6 ig 82 14 37 169.71989 168.16990 K 60.57864 3 2 16 84 16 54 229.88977 228.74854 » The invariable horizontal force, or pressure, is called, in the technical phrase, ti>e drift or sboot of an arch. 1143 3 Table No. III. Containing the angles of the several sections, with the angles between the vertical line and the abutments, calculated from their weights, when they are assumed equal to the weight of the first section, the angle of which is given = 5° ; the weight of the first section, or A = i. The initial pressure = 2 X sin. i A' = 11.46279 The horizontal force 2 X tano -= 11.451! Angles of Inclina- tion to the Vertical Total Pressure on Sections. Angles of the of tlie lower Sur- the Abutment, or Weights of tho Sections. faces of each Sec- on the Section next Semiarches. tion, considered as following. Abutments. 0 0 / / A 500 2 30 0 11.46279 .50000 B 4 57 40 7 27 40 11.54977 1. 50000 C 4 51 •« 12 18 50 II. 72169 2.50000 D 4 40 40 16 59 40 11.97492 3.50000 E 4 27 20 21 27 0 12.30440 4.5OOCO F 4 12 10 25 39 10 12.70422 5.50000 G 3 55 30 29 34 40 13.16809 6.50000 H 3 38 40 33 13 2° 13.68938 7 50000 I 3 21 40 36 35 0 14.26180 8.50000 K 3 5 40 39 40 40 14.87946 9.50000 L 2 50 20 42 31 0 15-53697 10.50000 M 2 36 20 45 7 20 16.22973 1 1.50000 N 2 23 20 47 30 40 16.95286 12.50000 0 2 1 1 20 49 42 0 17.70315 13.50000 P 2 0 20 51 42 30 18.47722 14.50000 ^ I 50 40 53 33 >o 19.29421 15.50000 R I 41 40 55 14 50 20.10753 16.50000 S I 33 30 56 48 20 20.93711 17.50000 T I 26 30 58 14 50 21.78074 18.50000 U I 20 0 59 34 5° 22.63742 19.50000 V" 1 14 0 60 48 50 23.50502 20.50000 W I 8 50 61 57 40 24.38295 21.50000 X I 4 0 63 I 40 25.27003 22.50000 Y 0 59 40 64 I 20 26.16520 23.50000 Z 0 55 40 64 57 0 27.06773 24.50000 A 0 52 Q 65 49 0 27.97699 25.50000 B 0 48 40 66 37 40 28.91436 26.50000 C 0 45 40 67 23 20 29.83490 27.50000 D 0 43 0 68 6 20 30.76038 28.50000 E 0 40 30 6S 46 50 31.69023 29.50000 F 0 38 10 69 25 0 32.62449 30.5OOCO G2 n 44 3 Table No. IV. ' Shewing the angles of the sections, and the inclination of the abutments to the vertical line, calculated from the weights of the sections, when the angle of the first section is assumed = 30^ and the weight of each section is equal to the weight of the first section = tt* = 1 , The initial pressure p = ^ ^ ^-'^_ ^ ^, = 1.931853 The horizontal force p' = - — ^^^ ^^^ = 1.866025 Sections. Angles of the Sections. Angles of Inclination to the Vertical of the lower Surfaces of each Section, considered as Abutments. Total Pressure on the Abutments, or on the Section next following. Weights of the Semi- arches, deducting .5. 0 A 30 0 0 15 0 0 1.931850 .500000 B 23 47 38 38 47 38 2.394167 1.500000 C 14 28 4 53 15 43 3.119616 2.500000 D 8 40 25 61 56 8 2-9^^257 3.500000 E 5 32 31 67 28 39 4.871542 4.500000 F 3 46 33 71 15 32 5.807912 5.500000 G 2 43 23 73 58 56 G.yG^SOQ 6.500000 H 2 2 46 76 1 41 7.7286 7.5000 I 1 35 23 77 S7 4 8.7024 8.5000 K 1 16 8 78 53 12 9.6815 9.5000 L 128 79 55 21 10.6645 10.5000 M 0 51 38 80 46 59 11.9504 11,5000 c: 45-3 Table No. V. Shewing the angles of 49 sections, forming an arch of equilibrium, calcu- lated from given weights of the sections, when the angle of the first section is 1 degree = A° ; and the weight thereof is denoted by unity, the weights of the successive sections increasing by equal differences from 1 to 3, which is the weight of the 25th section of the semiarch Z. The initial pressure p = ■ 2 X sin. 4 A'= The horizontal force p' = Z X tang. \ A° = 57.29649 = 57.29432 Sections. Weights of the Sections. Angles of the Sections. Angles of the Abutments. Pressure on the Section next fol- lowing. Weights of the Semiarches. A 1. 000000 I 0 0 0 1 // 0 30 0 57.29649 .500000 B J-083333 I 4 58 I 34 58 57-3'593 1-583333 C 1. 166666 I 9 54 2 44 51 57.360C2 2.750000 D 1.250000 I 14 44 3 59 36 57-43352 4.000000 E 1-333333 I 19 28 5 19 4 57.54176 5'333333 F 1.416666 I 24 3 6 43 8 57.69030 6.750000 G 1.500000 1 28 28 8 II 37 57.88512 8.250000 H 1-583333 I 32 41 9 44 18 58.13190 9-833333 I 1.666666 I 36 35 11 20 53 58-43571 1 1.500000 K 1.750000 I 40 19 '3 I »3 58.805 1 1 13.250000 L 1-833333 1 43 39 14 44 52 59.24522 15083333 M 1.916666 I 46 38 16 31 30 59.76187 17.000000 N 2.000000 I 49 13 18 20 44 60.36220 19.000000 0 2.083333 I 51 22 20 12 6 61.04895 21-083333 P 2.166666 I 53 4 22 5 10 61.83064 23.250000 QL 2.250000 1 54 18 23 59 29 62.71133 25.500000 R 2-333333 I 55 4 25 54 33 63,69574 27-833333 S 2.416666 I 55 21 27 49 55 64.78815 30.250000 T 2.500000 I 55 II 29 45 7 65.99240 32.750000 U 2-583333 I 54 33 31 39 41 67.31184 35-333333 V 2.666666 J 53 3« 33 33 12 68. 74902 38.000000 w 2.750000 I 52 4 35 25 17 70.30633 40.750000 X 2-833333 1 50 16 37 '5 33 71.98551 43-583333 Y 2.916666 I 48 10 39 3 44 73.78788 46.500000 Z 3.000000 I 45 46 40 49 3' 75.71422 49.500000 C ^(^1 Table No. VI, For comparing the pressures on the abutments, the weights of the semi- arches, and the invariable horizontal or lateral force, calculated from the direct rules in the pages 13 and 27; with the pressures on the abutments, weights on the semiarches, and horizontal forces respectively, deduced by approximation, according to the general rules inserted in page ig. In this Table, the weight of the highest or middle section is equal to the number 2, and the angle of the said section is 5° ; the weights of the several sections as they are stated in the Table II. By the approximate Rule I, the initial pressure/* = ^^j-^^ = 22.93558. By the ap- proximate Rule II, the horizontal force p' = - — ^^ — ^ =: 22.90377. S is the weight of the semiarch, and Z is the pressure on the abutment, the inclination of which to the vertical is = V^ Angles of the Abutments, or Rule III. Pressures on the Abutments. Rule IV. Weights of the Semia rches. Sections. Values of the Values of Z by Pressures Z, the approximate entered in the Rule, Z = / Table No. II. X Sec. V^. Differences. Given Weights of the Semi- arches, entered in Table II. Weights by Ap- proximation S=/-'XTang. V-. Differences. A 2 30 0 22.92558 22.92550 .00008 1. 00000 r. .0 B 9 19 31 23.21305 23.21052 .00253 3.76106 3.76107 .oooor C 21 0 55 24 53843 24.53560 .00283 8.79950 8 79896 .00054 D 37 33 36 28.89592 28.89120 .G0472 17.61434 17.61204 .00230 E 54 7 43 39.09063 39.08690 •00373 31.67582 31-67351 .00231 F 66 23 38 57.19849 57.19580 .00269 52.41426 52.40850 .00576 G 74 14 54 84-37537 84.37180 •00357 81.20918 81.20260 .00658 H 79 8 18 121.55373 121.54790 .00583 119.37878 119.36970 .00908 I 82 14 37 169.71989 169.71090 .00899 168.16990 168.15830 .01160 K 84 16 54 229.88977 229.88290 .00687 228.74854 228.73900 .00954 c: 47 n Table No. VI. continued. Rule V. 1 Rule VI. Weights of the Scmiai ches. Invari ble horizontal Force. Sections. Given Weights Weights by Ap- Invariable hori- tlorizontal Force of the Semi- arches, entered in Table 11. proximation S = zXSm. v. Differences. zontal Force by Approxima- tion^ =2 XCos. V-. Differences. ~ = XTaiis-2°3»' A I.OOOOO I. .00000 22.90377 22.90311 .00066 B 3.76106 3.76147 .00041 22.90377 22.90626 .00249 C 8.79950 8.79990 .00040 22.90377 22.90625 .00248 D 17.61434 17.61473 .00039 22.90377 22.90723 .00346 E 31.67582 31.67647 .00065 22.90377 22.90592 .00215 F 52.41426 52.41092 •00334 22.90377 22.90483 .00106 G 81.20918 81.20665 .00253 22.90377 22.90514 .00137 H 119.37878 II9.37512 .00366 2290377 22.90499 .00122 I 168.16990 168.16712 .00278 22.90377 22.90514 .00137 K 228.74854 228.74630 .00224 22.90377 22.90444 .00067 N. B, The rules for approximating to the weights of the semiarcs, pressures on the abutments, and the invariable horizontal force, when applied to the Tables I. III. IV. and V. will be found to give results for the most part as exact as in the above calculations, which are formed from the Table No. II. The conditions on which this Table is founded being rather more complicated than in the other Tables, it was considered, on this account, to be the most proper test for examining the correctness of the approximate rules. APPENDIX; CONTAINING NOTES AND CORRECTIONS. Page 2 , Line I . j4fteR — ** angular distance from the vertex," — add measured by the inclination of the lowest surface to the vertical line. Page 1 1 , the three last Lines. The weights are supposed to have been adjusted by geometrical proportions, but not mechanically determined with exactness. Page 12, Line 25, and in several other Places. In all the numerical computations of sines, cosines, and other lines diMwn in a circle, tlie radius thereof is assumed equal to unity. Pai'e 17, Line 4. The semiarch is understood to be that part of any arch which is comprehended between the vertical line and an abutment on either side. Jfote, Page 18, Line 15. Fig, 5 . Let BOA = A° represent the angle of the highest or middle section, so that the angle VOA shall = i A° : through any point I in the line OA draw the line K I perpendicular to O A, and supposing the weight of the wedge to be = w, let I K = . — r-ZT., or the initial ° ° 2 X sin. I A°' pressure : resolve I K into two forces, namely, I O parallel, and D K perpendicular, to the horizon. By the similar triangles V O A, I D K, as H I K : 1 D : : radius to the cosine of K I D or V 0 A : it will follow that ^ ^ — Z X siu%'A° ~ 2 X taTg. i A° ' "^^^^^ ^^ 'he measure of the inva- riable force, the direction of which is parallel to the horizon. Page 26, Line 6, al ihe word " Through." As the point S has not been yet determined by geometrical construc- tion, instead of — " throu2;h the point S," Sec. — insert through any point B in the line I Q^ draw the line B S perpendicular to the line C Q,, and equal in length to the given line B S, which represents the pressure on the section D : and through the point S draw the line S F perpendicular to the line F I, and through G, &;c. Page 29, Line 6. After — " till the semiarc is augmented to about 5 5°" — add (estimated bv the inclination of the abutment to the vertical line). ERRATA. Page 7, line i8, for the line X P, read in the line X P. ■ g, zo, for Fig. i, read Fig. i and 2. . II, 24, for then, read thus. - II, 28, for direction, read directions. ■ 12, 1 1 , /or bissected, rearf bisected. I 23, 22, for over, read upon. . 24, 6 et alibi, for arc, read arch. 29, 2 and 9, for weight, read weights. JO, — — 1 2, after terminated insert ; - 30, in the note, for weighes, read weights. 31, line 4, read V' — y° 27' 40". 32, 5, fot 4° 21', re. d 4° 27'. 35, 17, /or TTR = 2.679492, read 2,679665. 37, 19, for 1° 34' 58', read 1° 4' 58". 37, 20, for 1° 34' 50", read 1° 34.' 58". 38, 10, for D F, read O F. c 51 ;] The angles entered in the Tables I. 11. III. IV. and V. are expressed to seconds of a degree, in some cases to the nearest ten seconds of a degree. These results will probably be found on examination, in most cases, correct to the degree of exactness here stated. Some errors may be expected to occur in the course of the long and troublesome compu- tations which are required for forming these Tables. On a revisal, the undermentioned errata have been discovered, which the reader is requested to correct, together with any other which his own observation may have pointed out. Section Table III. Page 43. for 16° 59' 40", read 16" 59' 30". for 5 1 42 30, read 5 1 42 20. Table IV. Page 44. for 53° '5' 43". read 53° 15' 42'. for II 9504, read 11.6504. Table V Page 45. for 2° 44' 51", read 2° 44' 5 2". for 3 59 36, read 3 59 35. The angles opposite the sections F, G, K, &c. are affected by similar errors of 1", which will appear by adding the angle of any section to the angle of the abutment preceding. The sum ought to be the angle of the abutment of the section. c D E K S /or 90.009, read go. 127. for 123.22, read 124.26, for 153.38, read 154.03. for 273.36, read 272.48. for 383.21, read 384.10. D P C M Table II. Page 42. C D for 37 33 3^, read 37 33 35. D Printed by W. Bulmer and Co. Cleveland-row, St. James's. /7 ! /vy /. fi,,. /rvi O N il i Fll^.lff A SUPPLEMENT TO A TRACT, ENTITLED A TREATISE ON THE CONSTRUCTION AND PROPERTIES OF ARCHES, PUBLISHED IN THE YEAR 1801 ; AND CONTAINING PROPOSITIONS FOR DETERMINING THE WEIGHTS OF THE SEVERAL SECTIONS WHICH CONSTITUTE AN ARCH, INFERRED FROM THE ANGLES. ALSO CONTAINING A DEMONSTRATION OF THE ANGLES OF THE SEVERAL SECTIONS, WHEN THEY ARE INFERRED FROM THE WEIGHTS THEREOF. TO WHICH IS ADDED, A DESCRIPTION OF ORIGINAL EXPERIMENTS TO VERIFY AND ILLUSTRATE THE PRINCIPLES IN THIS TREATISE. WITH OCCASIONAL REMARKS ON THE CONSTRUCTION OF AN IRON BRIDGE OF ONE ARCH, PROPOSED TO BE ERECTED OVER THE RIVER THAMES AT LONDON. PART II. • BY THE AUTHOR OF THE FIRST PART. LONDON : PRINTED BY W. BULMER AND CO. CLEVELAND-ROW ; SOLD BY T. EGERTON, WHITEHALL; AND J. TAYLOR, AT THE ARCHITECTURAL LIBRARY, 59, HOLBORN. 1804. PREFACE. J\. PLAN for constructing an Iron Bridge of one arch, to be erected over the River Thames, designed by Messrs. Telford and Douglass, and proposed to the Committee of the House of Commons for the further Improvement of the Port of London, has excited consider- able attention, both from the novelty and magnitude of the design, and the evident advantages to navigation which would attend such a structure ; yet as some doubts arose respecting the practicability of erecting such an edifice, and the prudence of attempting it, the Committee judged it necessary for their own information, as well as to furnish the House with some grounds by which an opinion might be formed, to propose the following Queries, which were therefore transmitted, together with the engraved designs of Messrs. Telford and Douglass, and the explanatory drawings annexed, to such persons as were supposed to be most capable of affording them information. The following are the juries that were drawn up and transmitted to the Persons zvhose hames are undermentioned. (See Page vi.) Q_U E R I E S. I. What parts of the arch are to be considered as wedges, which act on each other by gravity and pressure, and what part merely as weight, acting by its gravity only, similar to the walls and other loading com- monly erected on the arches of stone bridges ; or does the whole act a 2 iv PREFACE. as one frame of iron, which cannot be destroyed but by crushing its parts ? Qiiery II. Whether the strength of the arch is affected, and in what manner, by the proposed increase of its width towards the two extremities or abut- ments, when considered both vertically and horizontally ; and if so, what form should the bridge gradually acquire ? III. In what proportion should the weight be distributed, from the centre to the abutments, to make the arch uniformly strong ? IV. What pressure will each part of the bridge receive, supposing it divided into any given number of equal sections, the weight of the middle section being known ; and on what part, and with what force, will the whole act upon the abutments ? V. What additional weight will the whole bridge sustain, and what will be the effect of a given weight placed on any of the fore-mentioned sections ? VI. Supposing the bridge executed in the best manner, what horizontal force will it require, when applied to any particular part, to overturn it, or press it out of the vertical position ? VII. Supposing the span of the arch to remain the same, and to spring ten feet lower, what additional strength would it give to the bridge ; or, making the strength the same, what saving may be made in the mate- rials ; or, if instead of a circular arch, as in the Print and Drawings, the bridge should be made in the form of an elliptical arch, what would be the difference in effect as to strength, duration, and expense ? VIII. Is it necessary or adviseable to have a model made of the proposed bridge, or any part of it, of cast iron ; if so, what are the objects to which the experiments should be directed, to the equilibration only, or to the cohesion of the several parts, or to both united, as they will occur in the iron work of the intended bridge ? IX. Of what size ought this model to be made, and in what relative proportion will experiments on the model bear to the bridge when executed ? X. By what means may ships be best directed in the middle stream, or prevented from driving to the side, and striking the arch ; and what is the probable consequence of such a stroke ? XI. The weight and lateral pressure of the bridge being given, can PREFACE. V abutments be made in the proposed situation, for London Bridge to resist that pressure ? Query XII. The weight of the whole iron work being given, can a centre or scaffolding be erected over the river, sufficient to carry the arch, without obstructing those vessels which at present navigate that part ? XIII. Whether would it be most adviseable to make the bridge of cast and wrought iron combined, or of cast iron only ; and if of the latter, whether of the hard and white metul, or of soft grey metal, or of gun metal ? XIV. Of what dimensions ought the several members of the iron work to be made, to give the bridge sufficient strength ? XV. Can frames of iron be made sufficiently correct to compose an arch of the form and dimensions as shewn in the Drawings No. i and 2, so as to take an equal bearing in one frame, the several parts being con- nected by diagonal braces, and joined by iron cement, or other substance? N. B. The Plate XXIV. in the Supplement to the Third Report, is considered as No. i. XVI. Instead of casting the ribs in frames of considerable length and breadth, as shewn in the Drawings No. i and 2, would it be more ad- viseable to cast each member of the ribs in separate pieces of considerable length, connecting them together with diagonal braces, both horizontally and vertically, as in No. 3. ? XV^II. Can an iron cement be made tliat will become hard and durable; or could liquid iron be poured into the joints ? XVIII. Would lead be better to use in the whole, or any part, of the joints ? XIX. Can any improvements be made upon the Plans, so as to render the bridge more substantial and durable, and less expensive ; if so, what are those improvements ? XX. Upon considering the whole circumstance of the case, agreeably to the Resolutions of the Select Committee, as stated at the conclusion of their Third Report,* is it your opinion, that an arch of 600 feet span, * The Resolutions here referred to are as follow : That it is the opinion of this Committee, that it is essential to the improvement and accommodation of the Port of London, that London Bridge should be rebuilt, on such a construction as to permit a free passage, at all- times of the tide, for ships vi PREFACE. as expressed in the Drawings produced by Messrs. Telford and Douglass, on the same plane, with any improvements you may be so eood as to point out, is practicable and adviseable, and capable of being rendered a durable edifice ? Query XXI. Does the Estimate communicated herewith, according to your judg- ment, greatly exceed, or fall short of, the probable expence of executing the Plan proposed, specifying the general grounds of your opinion ? After paying every attention to the subject which the importance of it demanded, it appeared for many reasons absolutely necessary, for furnishing satisfactory answers to the above Queries, to inves- tigate the properties of arches from their first principles. The substance of these properties is comprised in a Tract, entitled a Dissertation on the Construction and Properties of Arches, pub- lished in the year 1801, and continued in the present Treatise, now offered to the Public as a Supplement to the former Tract. The of such a tonnage, at least, as the depth of the river would admit at present, be- tween London Bridge and Blackfriars Bridge. That it is the opinion of this Committee, that an iron bridge, having its centre arch not less than 65 feet high in the clear above high-water mark, will answer the intended purposes, with the greatest convenience, and at the least expense. That it is the opinion of this Committee, that the most convenient situation for the new bridge \vill be immediately above St. Saviour's Church, and upon a line leading from thence to the Royal Exchange. * Answbrs BV I. Dr. Maskelyne, 10. Mr. Rennie, 2. Professor Robertson, II. M. Watt, 3- Professor Playfair, 12. Mr. Southern, 4- Professor Robeson, 13- Mr. Reynolds, 5- Dr. Milner, 14. Mr. Wilkinson, 6. Dr. Hutton, 15- Mr. Bage, 7- Mr. At wood. 16. General Bentham, 8. Colonel Twiss, 17- Mr. Wilson. 9- Mr, Jessop, * See the Report from the Select Committee upon the Improvement of the Port of London. PREFACE. vii reader will perceive that most of the propositions in these Dis- sertations are entirely new, and that they have been verified and confirmed, by new and satisfactory experiments, on Models con- structed in brass by Mr. Berge of Piccadilly, whose skill and exactness in executing works of this sort are well known to the Public. Considering the importance of the subject, and the di- versity of opinions which has prevailed respecting the construc- tion of arches, and the principles, on which they are founded, it seems requisite, that the final determination of the plan for erecting the bridge of one arch in question, should be subjected to a rigorous examination, in order to discover if any, and what, errors might be found in them. The best means of effecting this ap- pears to be by a publication, in which the propositions recom- mended for adoption being fairly stated, every person, who is of a different opinion, may have an opportunity of explaining his ideas on the subject, and of suggesting any different modes of construc- tion, that are judged to be less liable to objection. To persons interested in these inquiries, it may be satisfactory to be informed, that the properties of arches, which are comprised in this latter Tract, have been found, on a careful and minute examination, and comparison, in no instance inconsistent with those, which are the subject of investigation in Part the First, but rather appear to strengthen and confirm the theory before published, allowing for the differences in the initial force or pressure, expressed in page 2, and in Figs. 1 and 2, inserted in this Tract, representing the diffe- rent dispositions of the key-stones, from whence conclusions arise very different from each other, although all of them are strictly consistent with the laws of geometry and statics. It is particularly observable, that the deductions of the weights and pressures arising from a supposition of a single key-stone, do not exhibit conclusions X PREFACE. more secure from disunion. The effects therefore of similar or other impediments, such as may be supposed to take place in the construction of real bridges, will have ■ a much o-reater effect when they consist of iron braces and fastenings of various kinds ; by which all efforts to disunite the sections are immediately counteracted. The effects of this will be not only to prevent the separation of the sections by any casual force, tending to disunite them, but will likewise secure the edifice from the more silent, but not less destruc- tive assaults of time : for when the sections of an arch are not duly balanced, every heavy weight which passes over the road-way, even the motion of a lighter carriage, must create a tendency to separate the sections by degrees, and at length entirely to disunite them ; an evil to be remedied only by a requisite equilibration of parts of the bridge. On a review of the whole, whether the subject is considered theo- retically, as depending on the laws of motion, or practically, on the construction of models erected in strict conformity to the theory, it would seem difficult to suppose, that any principle for erecting a bridge of one arch would be adopted, that is very different from those, that have been the subject of the preceding pages : never- theless, as the most specious theories have been known to fail, when applied to practice, in consequence of very minute alterations in the conditions; and as it is scarcely possible to frame experiments ade- quate to the magnitude of the intended structure, the Author of this Treatise thinks it incumbent upon him to state freely the doubts which remain upon his mind, respecting the construction of the bridge intended ; suggesting, at tlie same time, such ideas, as have occurred to him, which probably may contribute to remove or to explain those doubts ; particularly by causing an arch to be erected, PREFACE. xi the span of which is from 20 to 50 feet, the expense of which would be of little moment in the case of its success ; and, on a supposi- tion, that the experiment should fail, the important consequences that would probably arise from the observation of such a fact would, in the opinion of many persons, amply compensate for its failure. A doubt occurred during the construction of the flat arch,* whether the angles at the summit were most conveniently fixed at St 38' 0", or whether those angles should not subtend 5°, 10°, 15°, or any other angles, which might better contribute to the strength and stability of the entire structure. Since the materials, of which the Models are formed, are of a soft and elastic nature, which yields in some degree to the force of pressure ; this circum- stance, joined to that of making the angle subtended at the centre of the circle no greater than 2° j8' 0", prevents these sections h^om having much hold on the contiguous sections above them, and creates some difficulty and attention in adjusting the Model No. 2, - to an horizontal plane, suggesting the necessity of forming the angles of the first or highest sections at 5°, or some greater angle, by which the holdings would be more effectually secured ; but it is to be remembered, that this source of imperfection could not exist if the sections were made of materials perfectly hard and unelastic; and the Model having been constructed as aii expe- riment, it seems proper that the angles of the first sections should be formed on the smallest allowable dimensions, in order to ob- serve more distinctly the advantages wliich would arise from making the angles larger in any subsequent experiment, if any should be approved of, previously to a final determination of the plan to be adopted for erecting the iron bridge. It is to be * The Model No. 2, so called to distinguish it from the Model No. i, in the form of a semicircular arch. xii PREFACE. observed, that no imperfection of the kind which is here spoken of, takes place in the Model of the arch No. 2, after it has been care- fully erected : but a larger angle seems to be preferable for the angles of the first sections, from the difficulty which subsists, at present, in adjusting the Model of the arch No. 2, to the true horizontal plane, so much exceeding the trouble and attention in adjusting the Model No. 1. Many thanks and acknowledgments are due to Mr. Telford and several other engineers, who have had the goodness to favour the Author with their able advice and assistance, in answering such questions as he had occasion to propose to them, respecting the ori- ginal plan of this Treatise, and subsequently concerning the prac- tical experiments, accounts of which are contained in it. G. A. London, agth Novemier, 1803. A DISSERTATION ON THE CONSTRUCTION AND PROPERTIES OF ARCHES. PART II. 1 HE sections or portions of wedges which constitute an arch may be disposed according to two several methods of construction, which are represented by Fig. i and Fig. q. In Fig. i the highest section, or key-stone, is bissected by the vertical plane VO, which divides the entire arch into two parts, similar and equal to each other. In Fig. 2, tzvo highest sections A, A, similar and equal to each other, are placed contiguous and in contact with the vertical line VO. The former plan of construction has been before the subject of investigation, in a tract on arches, and published in the year 1801. It remains to consider the properties which result from disposing the sections according to the last-mentioned plan in Fig. 2. The first material circumstance which occurs is the difference in the direction of the initial pressure, which in the former case, Fig. 1, was inclined to the horizon in the direction EQ perpendi- cular to AB ; whereas, according to the latter disposition. Fig. 2, of the key-stone, the initial pressure is parallel to the horizon in B C2 -2 the direction QR. In any arch of equilibration in which two equal and similar sections occupy the summit of the arch, the initial force or pressure is parallel to the horizon, and is to the weight of the first section as radius is to the tangent of the angle of that section. For let the two highest equal sections, A, A, be represented by Fig. 3, when they form a portion of an arch of this description ; let Yz' Ta represent one of these equal highest sections. Through any point Q, of the line VO, draw QP perpendicular to the line TO, QR parallel, and PR perpendicular, to the horizon ; then will the three forces, by which the wedge A is supported in equilibrio, be represented in quantity and direction, by the lines QP, QR, and PR ; of which, QP denotes the pressure between the surface TO and the surface of the section B, which is contiguous to it. QR is the force which acts in a direction parallel to the horizon, and is counterbalanced by the reaction of the other section A, similar and equal to the former : and PR measures the weight of the section A. Because PQR is a right-angled triangle, the follow- ing proportion will be derived from it : as the horizontal force QR is to the weight of the section A, or PR, so is radius or QR to PR, The tangent of the angle PQR = VOT, whicli being equal to the angle contained by the sides \v, aT of the wedge A, may be de- noted by A": finally, if the weight of the section A be put equal to w, we shall have the horizontal force at the summit of the arch = — -— TB = ty X cotang. A°, radius being = 1 ; from this deter- mination the following construction is derived : having given the several angles of the sections A°, B°, C, D°, together with the weight of the first section A, to ascertain by geometrical con- struction, the weights of the successive sections B, C, D, &c. when the arch is balanced in equilibrio. AA, AB, BC, CD, &c. Ls'2 represent the bases of the sections in Fig. 4 : through the points A, B, C, D, &c. draw the indefinite lines A a, B 6, C c, D ^, Sec. perpendicular to the horizon ; through any point X, in the line AF, draw the indefinite line XZ parallel to the horizon ; let A a denote the weight of the section A; and through the point a draw a z, at right angles to AF ; and in the line XZ take a part XM, which shall be to the line A a as radius is to the tangeU of the angle VOF or A"; so shall XM represent, in quanaty ani direction, the pressure between the first section A and the vertical plane VO ; or, when both semiarches are completed, the line XM will represent the pressure between the contiguous vertical sur- faces of the two highest sections A, A. Through the point M draw MRV perpendicular to OF; and in this line produced, take MN = to za ; and make QV = to RN, which will be to radius, as radius is to the sine of VOA or A°. For because the line XM is to A a as radius is to the tangent of VOxA. or A° ; if the sin. of A° be put = s, and the cos. A°= c: to radius 1, this will give RM = iiLiLf!; and because MN = za = Aa x 5, and KM KM + MN or RN = ^112i£+J: = VO = ^^, which quantity is to radius, as radius is to the sin. of VOA or A°: and VQ or RN = — — is the measure of the entire pressure on the abut- ment OF. To construct the weight of the section B, and the pressure on the next abutment OG, through the point Q, draw KT perpendi- cular to GS, and from any point B, in the line BG, set off Bz =: to VS : through the point z draw zb perpendicular to G 6 cutting off B6, which will be equal to the measure of the weight of the sec- tion B; from the point Q in the line KT produced set ofFQT = to B 3 c 4. :] zb : also in the line KT, make Ks = to ST, then Ks or ST is the measure of the pressure on the abutment OG of the section C. On the same principles, the weights of the sections C and D, as well as of the sections following, are geometrically constructed, Cz being set ofF= WK, and Dz = to nl ; from this construction, when completed, the general expressions for the weights of tlie sections are inferred, which are inserted in the 13th and 14th pages in the former tract, except that the initial pressure, arising from a diffe- rent disposition of the key-stone, represented in Figs. 1 and 2, in consequence of which the initial pressure is p' z= w x cotang. A°, instead of ^ = — ^-ttt, hi the former tract. In this manner the weights of the several sections and pres- sures on the abutments, are found to be as underneath. Sect- Weights of the sections on the ions. vertical abutments. p'=iw X cotang, xV A=:iy p=p\ COS. A° + p'x sin. A° x tang. V" B =p X sin. B* X sec. V* q=p x cos. B" -f- /> x sin. B° x tang. V* C = 5^ X sin. C° X sec. V' r =z q ■><. cos. C° -f sin.-A-i-t! cos A + B - cos.B-J-sin.B xsin. A-{-6 A + B = COS. B 4- sin. B x r^, = ttt: ' ' cos.A-)-B COS. A-fB COS. A -1- B — B COS. A therefore cos. B + sin. B x tang. A4- B COS. A + B cos.A+B ' ° = : multiply both sides by sec. A, the result will be : sec. A COS. A xsfc. A X cos. B + sec. A X sin. B X tang. A+ B = '-^^^4^1^ = sec. A+B. ° ' cos.A + B * C8 3 This proposition may be extended to ascertain, generally, the proportion of the successive secants in an arch of equilibration, by supposing an angle of an abutment M° to consist of the angles of several sections, such as A°, B°, C°, D°, E° = M°, if an additional section F° is next in order after E°; so that the whole arch may consist of sections, the sum of the angles of which = M° + F", then it is to be proved that the secant of M°, is to the secant of M° -f F°, as 1 to cos. F -f sin. F° x tang. M" -|- ¥% or sec. M° X cos. M° + sec. M" x sin. F° x tang. M' + F° = sec. By the elements of trigonometry, cos. F -f- sin. F x tang. M -[- F rp _, • rp sin. M + F cos. F x cos. M + F + sin. F x sin. M + F COS. M -f F COS. M + F 1^ , • T^ - TJ — 1 — T^ COS. M -|- F — F COS. M or COS. F -f sm. r x tang. M -j- F = — COS. M + F COS. M,+ h" Multiply both sides of the equation by sec. M, the result will be sec. M x COS. F -f sec. M x sin. F x tang. M + F = — — tt-^t-p— ' o 1 COS. M -f F = sec. M + F. Thus the relation of the successive secants of the angles be- tween the vertical line and the lowest surface of each section in any arch of equilibration is demonstrated, in general, and the measure of the pressures on the abutments proved to be equal to the weight of the first or highest section x cotang. A° x sec. of the angle of that abutment : and, in general, any sec. of an angle of an abutment is shewn to be to the sec. of the angle of an abutment next following, in the proportion as i is to cos. of the angle of the section -|- sin. of the same angle x tang, of the sum of the angles from the summit of the arch to the abutment. The ensuing geometrical proposition is intended to investigate the weights of the individual. sections in an arch of equilibration ; C 9 3 also to infer tlie sums of the weights of the sections which fortn the respective semiarches. A, B, C, D, Fig. 6. is a circular arc drawn from the centre O and with the distance OA. The arc AB = A", AC = B°, AD = C° ; AG is drawn a tangent to the circle at the point A ; through the centre O and the points B, C, D draw the lines OBE, OCF, ODG: then the line AE will be a tangent to the arc AB, AF will be a tangent to the arc AC, and AG will be a tangent to the arc AD ; through the points B, C, D draw the lines BH, CI, DK perpendicular to the line OA ; then will BH be the sin. and OH the cos. of the arc AB = A°, CI and OI the sin. and cos. of the arc AC = B° and DK = the sin. and OK the cos. of the arc AD = C° through C draw CM perpendicular to OE, so shall CM be the sin. of the arc CB. The following proposition is to be proved : the difference of the tangents of the arcs AC and AB, or the line FE, is to the line CM, or the sine of the difference of the same arcs, so is i to the rect- angle under the cosines of AB and AC, or OH x OI : the demonstra- tion follows, radius being = i ; the tangent of the arc AB = ^^^"" ^^, and tansr. of the arc AC = ,,, ; therefore the difference of the O COS. AL t. r\-n J \ r^ sin AC sin AB sin. AC xcos. AB— sin. AB xcos. AC tang, of AB and AC ^ ■ 'cos. AC COS. AB COS. AB x ^os. AC but the sin. of AC x cos. AB — sin. AB x cos. x\C = sin. AC — AB = the sin. of the difference of the same arcs = CM ; therefore the difference of the tangents EF = 7-5 rr^; which equa- o COS. AJB X COS. AC ^ tion being resolved into an analogy, becoines the following pro- portion : as the difference of the tangents FE is to the sine of the difference of the arcs sin. AC — AB, so is radius 1 to the rectangle under the cosines OI and OH, which is the proposition to be proved. C C 103 Since it has been shewn in the pages preceding, that the pres- sure on each abutment is xv x cotang. A°x sec. of the angle of that abutment, the pressures on the several sections will be expressed as follows : Pressure on the vertical abutment VO = w y. cotang. A° = zv X p' sec. v. Pressures on ihe lowest surface of each section. A p = zv X cotang. A° x sec. V" B q =w X cotang. A° x sec. V^ C r = za x cotang. A° x sec. V'' D s = zii x cotang. A° x sec. V"^ &c. &c. Let CB be an arc which measures the angle of any section, so that OF may represent the secant of the angle AOF, and OE = the secant of the angle of the abutment AOE : the difference of the tangents FE = ^ rr^ = sin. B° x sec. of the angle o COS. AB X COS. AC ^ AOB, X sec. of the angle AOC, or, according to the notation which has been adopted, the difference of the tangents FE = sin. B° x sec. of V" X sec. V*, radius being = 1 . The weight of the section B, by page 6,=p % sin. B° x sec. V*, but by the table in page above inserted, p =: zu x cotang. A° x sec. wherefore the weight of the section B = w x cotang. A° x sin. B° X sec. V" X sec. V" : on the same principles the weights of the several sections will be expressed as underneath. Sections. W^eights. A = m; X cotang. A° x sin. A° x sec. V° x sec. V" B = zt; X cotang. A° x sin. B° x sec. V'' x sec. V' C = w X cotang. A° x sin. C° x sec. V* x sec. V" D = TO X cotang. A° x sin. D° x sec. V^ x sec. V^^ E .i:= TO X cotang. A° x sin. E° x sec. V'' x sec. V' F = TO X cotang, A° x sin. F° x sec. V' x sec. V^ &c. . &c. C 11 3 Because the lines Fig. 7. AE, EF, FG represent the weights of the several sections AB, BC, CD, the sum of those hnes, or AG, will denote the sum of the weights of the sections A + B + C. And in general, if the angle of an abutment in an arch of equilibration should = V^ and the angle of the first section = A°, and its weight = zv, the sum of the weights of the sections when adjusted, will = It; X cotang. A° x tang. V*. On this principle the weights of the sums of the successive sec- tions, or the weights of the semiarches, will be as they are stated underneath. Sums of the weights of the sections, or weights of the semiarches. A . . . . :z w X cotang. A° X tang. V-" := w X cotang. A° X tang. A A-fB . . . =w X cotang. A° X tang. V* =: TO X cotang. A» X tang. A -f B A-fB + C . =TOX cotang. A° x tang. V' zzw x cotang. A° x tang. A + B -f C A+B + C-i-D=:wx cotang. A° ^ tang. V'' = w x cotang. A° x tang. A -J- B -f C + D &c. &c. &c. The method of fluxions affords an additional confirmation of this proposition : suppose an arch adjusted to equilibrium to be composed of innumerable sections, the angles of which are evanescent ; to as- certain the weight of the sum of these evanescent sections included within a given angle from the summit of the arch to the lowest abut- ment V' ; since the angles of the sections are evanescent, the quan- tity V' = V': and for the same reason, the sin. of the angle D° will ultimately = D. Wherefore, the evanescent weight of the section D = r x sin. D x sec. V'= r x D x sec. V'. Let the tangent of the angle Y' = x to radius 1 ; then the sec. of V' = \/ 1 -f- -^^ ; and because V' == V'', it follows that V x V"^ = 1 -f- x'' : the weight therefore of the evanescent section D =:tc; x cotang. A°x D x 1 -|- .r''; C 2 C 12 D which is the fluxion of the weight of the arch equal to the fluxion of the angle D° sec/ V' x w x cotang. A". But the fluxion of an arc x into the square of its secant is known to be equal to the fluxion of the tangent of the same arc, when both quantities vanish together: therefore the integral or fluent, that is, the weight of the arch, will be equal to the tangent of the arc X into constant quantities ; that is, the sum of the evanescent sections, or the weight of the entire arch, from the summit to the abutment = w x cotang. A x tang. V'. On the Model, No. i,for veryfying the Construction of an Arch, in ivhich the ^''eights of the Sections A, B, C, D, &c. are inferred from the Angles given in the present Case = 5- each. Although the various properties of arches described in the preceding pages, respecting the weights and dimensions of the wedges, and their pressures against the abutments, require no fur- ther demonstration than what has been given in the preceding pages ; yet, as it has been remarked, that philosophical truths, al- though demonstrable in theory, have often been found to fail when applied to practice ; in order to remove every doubt of this sort, concerning the theory of arches, which is the subject of the preceding and present Dissertation, a model of an arch was con- structed according to the conditions in Table I. in which the angle of each section = 5°, the weight of the first section = 1, the weights of all other sections being in proportion to unity. This arch, like most arches which were erected previously to the 16th century, consists of two semiarches, similar and equal and resting against each other, in the middle of the curve, as described in figure 2 : the summit of the arch is occupied by two equal wedges A, A, resting against each other when coincident with the vertical plane VO ; according to the construction of this propo- sition, the weight of the wedge A being assumed = 1, the weight of B appears to be 1.0154^2, and the weight of the wedge C = 1.04,724,. These weights being applied in the form of truncated wedges, supported upon immoveable abutments, sus- tain each other in exact equilibrium, although retained in their places by their weights and pressures only, and independently of any ties and fastenings which are usually applied in the case when the structure is intended for the purpose of sustaining superincumbent loads. The pressure between the two first sec- tions in a direction parallel to the horizon =^'= 11.24300, ^the pressure against the lowest surface of the first section =p = 11.47371 : the pressure on the lowest surfiice of the second section, or B = g = 11.60638: on the lowest surface of C, the pressure is = r = 11.83327. The intention of this model is not only to verify the properties of equilibrium of these wedges, acting on each other, but also to examine and prove the several pressures on the lowest surface of the sections to be in their due proportions, according to the theory here demonstrated. And it ought to be re- membered that these pressures being perpendicular to the surfaces impressed, the reaction is precisely equal and contrary ; for this reason, each of the surfaces subject to this pressure will have the effect of an abutment immoveably fixed. The most satisfactory proof that the pressure on any abut- ment has been rightly assigned is, by removing the abutment and by applying the said force in a contrary direction ; the equili- brium that is produced between forces acting under these circum- stances, it is a sufficient proof that the reaction of the abutment is precisely equal to the force impressed upon it in a contrary direction. After the weights of the several wedges in an arch of equilibra- tion have been determined, in proportion to the weight of the first wedge A assumed to be = i, some difficulty occurs in forming eacii wedge of proper dimensions, so that their weights shall be correspondent to the conditions required. A wedge being a solid body consisting of length, breadth, and thickness, of which one dimension, namely, the thickness, or depth, remains alwa3's the same; the weight of any wedge will be measured by the area or plane surface in each section, which is parallel to the arch ; that is, if the thickness or idepth of any section K (Fig. 7.) be put = li, the solid contents of the section K will be measured by the area KttS multiplied into 1^; put the angle SOT = 5°, the sin. of 2° 30' o" =: s, cos. 2° 30' o" = c; also let O^ = x; then we find, by the principles of trigonometry, that the area O^^^ = x^'sc, and the area OTS = r'^sc, and the area T//S = x^sc — fsc. Let the area corresponding to the weight of the section proposed = k, so that X' — r^sc = k; and x'' = -^Llif : wherefore X = \X ' ' ; and T^ or S^, the slant height of the section K = \/ - — — — — ?'• This being determined, the breadth of the section it = ^ s x ot = qsx, making therefore the radius OV = 11.4,6281, with the centre O, and the distance OV=: 11.4,6281, describe the circular arc VABC; and in this arc from V set off the several chords Vx\, AB, BC, &c. = 1 inch, in consequence of which the angles VOA = AOB = BOC, &c. &c. will be 5° each. The slant height and the breadth of each section will be computed by the preceding rules. C 15 D 0)1 the Model, No. 9., for illustrating and verifying the Principles of the Arch, zvhen the Angle of each Section, after the first SectioJi A°, are inferred according to the Rule in Page 27 of former Tract, from the Weights of the other Sections. In the propositions which have preceded, the several angles of the sections A°, B°, C°, D°, &c. have been considered as given quantities, from which the weights of the corresponding wedges have been inferred, both by geometrical construction ar.d b}^ cal- culation, when they form an arch of equilibration. The next inquiry is to iiwestigate the magnitudes of the angles from having given the weights of the several sections ; but as the construction and demonstration would not in the least differ from that which has already appeared in page 27 of the former Tract on Arches, it may be sufficient in the present instance to refer to the former Tract, both for explaining the principles of the construction and the demonstration, inserting in this place only the result, which is comprised in the following rule. Having given A° the angle of the first section, and the weight 6 = 1.25 of the section B next following, together with the angle at which the lower surface of A is inclined to the vertical, called the angle of the abutment of tlie section rS., or V', and the pressure on it =^p, to ascertain the magnitude of the angle B°, in an arch adjusted to equilibrium : in the proposition referred to it is proved, that on the conditions stated, tang. B" = ——-. '—r- radius be- ing = 1. The model constructed to verify the principles of equilibration, consists of a circular arc drawn to a radius = 21.7558 inches, VA, AB, BC, &c. are chords = 1 inch each, and subtend at the centre of the circle angles of 3° 38' o' : as the angle of the first C i6 3 section A° =± 2° 38' o", the angle of the abutment, or the angle con- tained between the vertical and the lowest surface of the section A = ¥"= a'gS' o": the pressure on the lowest surface of A°=^ = TTa* = 21.765553, and according to the rule inserted in page 12, tano'. B° = ■■ '■-?>^'^"--^ 3 o __ o ^g, „^ Wherefore the anffle of " /i -f- 1.25 xsin. 2 3b o 00 o the abutment contained between the lowest surface of B and the vertical line = A° + B° = 2° 38' o" + 3° 16' 29" = 5° 54' 29" =V*. By the same rule, the angles of the successive sections C°, D°, E°, &c. &c. and the angles of the abutments corresponding, are com- puted as they are stated in the columns annexed, in page 17. Let the arch to be constructed be supposed such as requires for its strength and security, that the weight or mass of matter contained in the lowest section R, shall be five times the weight of the hrst or highest section A, and let the arch consist of thirty-four sections, seventeen on each side of the vertical plane : on these con- ditions, the weight of the successive sections will be as follows : A = 1, B = 1.25, C = 1.50, D = 1.75, E= 2.00, F= 2.25, &c. as stated in Table IX : by assuming these weights for computing the several angles B°, C°, D°, &c. according to rule in page 1 2, they are found to be as in the ensuing columns, and the succes- sive sums of the angles are the angles of the corresponding abut- ments. By considering the drawing of this model, it is found to contain the conditions necessary for calculating the areas required for estimating the weights of the voussoirs. For the inclination of each abutment to the abutment next following, is equal to the angle of the section which rests on the abutment ; thus, the in- clination of the lines li, Hz, is equal to the angle of the section I = Hf I ; also the inclination of the lines Hh, Gh, forms the angle of the section 11 = Hb G, aiid so on. Model No. 2. DimensioJis of an Arch of Equilibration : the Angle of the first Section, or A° = 2° 38' o", and the Aiigles of the other Sections, and the Angles of the Abutments, are asfollozv : Angles of the Sections. Angles of the Abutments. A' = 2 38 O V- = 2 38 O B°=3 16 29 V*= 5 54, 29 C°=3 52 39 V'== 9 47 8 D° = 4 24 36 V'' = 14 1 1 44 E°= 4 50 9 V'=i9 1 50 F°= 5 71^ V/=24 9 9 G°=5i4 4i V5=29 23 50 H°= 5 12 14 V'^ = 34 36 4 r = 5 1 8 V' = 39 37 12 K° = 4 43 23 V* = 44 20 S5 L° = 4 21 27 V'=:48 42 2 M° = 3 57 33 ¥"=52 39 35 N° = 3 33 26 ¥" = 56 13 1 0° = 3 10 21 - V* = 59 23 22 P" = 2 49 o V^ = 62 12 22 Q° = 2 29 42 V^ = 64 42 4 R°= 2 12 31 V^ = 66 54 35 Geometrical Construction for drazving the Abutments, in the Model for illustrating Equilibrium of Arches, when the Magnitudes of the Afigles are inferred from the Weights of the several Sectiojis. VABC, &c. represents the portion of a circular arc, which is drawn from the centre O, with the distance OV: VIO (Fig. 8.) is a line D r 18 n drawn perpendicular to the horizon, dividing the entire arch into two parts, similar and equal to each other: the radius 0V= 2 1.7598 inches :. from the point V, set off the chord VA = 1 inch, and tlie chords AB, BC, CD = i inch each ; the angle of the first section will therefore be = 2° 38' o": for as one half : 1 : r the sin. of |- A% or sin. 1° 19', to radius, which is, consequently, = 21.759S inches : the semiarch V R consists of seventeen sections, the weights of which increase from 1 to 5, which is the weight of the lowest or last section ; and from these conditions it is inferred, by the rule in page 15, that A' = 2° 38' o", B° = 3° 1 6' 29", C'=3'52'39", &c, the successive sums of these angles, or the angles of the abutments, A° = 2° 38' o" = V^ A° -f B° = 5° 54' 29" = V^ A + B -{- C = 9° 47' 8" = V^, &c. as stated in Table IX. The direction of the line must next be ascertained, determining the position of the abutment on which either of the sections, for instance the section I, is sustained : from the point O draw the line 01 : it is first to be observed, that the angle contained be- tween the line I and VO, or the angle VII = 39° 37' 12", according to the Table IX. and the angle VOI = 2° 38' 0" x by 9 = 23° 4,2' o": make therefore the following proportion: as the sine of 39° 37' 12", is to the sine of VII —VOI = ^5° 55' I's.", so is radius OV, or 21.7598 inches to OI = 93597 inches ; this being determined, if a line ilt'is drawn through the point I, the line so drawn will coincide with the abutment on which the lowest sur- face of the section I is sustained ; and by the same principle the directions of all the abutments are practically determined. Also it appears that the successive abutments I/, H/, include be- tween them the angle Hz I, which is therefore equal to the angle of the section I ; therefore to find the solid contents measured by the area of the section I, the triangle iss, being made isosceles C^9l the area tss will be = "'^""'"^ ; * from which if the area 1/ H be subtracted, the remaining sum will be equal the area of the section I : put either of the lines is = x, then by the proposition which has been above mentioned, the area iss = '^ ^ ^'"" ^'\ and by the same proposition, the area H/I = '" ^ '^ ^ sin.H^p ^q^^^q_ quently, is being put = x, we shall have ^ ^'"' H/ x 1/ sin. P J . ii 4. a -j-zl -f il X AH X sin. I" , X —^ = 1, It appears that x = ' — - — ^r^^-^ , and conse- quently X = \/^—-^ — x&in.i . i^y ^j^g same rule the weights and dimensions of all the sections K, L, M, &c. are determined. By the principles stated in the preceding pages, the weight of either of the highest sections in any course of voussoirs, together with the angle of the said section, regulates the magnitude of the horizontal thrust or shoot, and the perpendicular pressure on the ultimate or lowest abutment and the direct pressure against the lowest surface of any abutment will depend on the cotang. of the angle of the highest section and the sec. of the angle of the abutment jointly. PROPOSITION. • The area contained in a right-lined triangle ABC, Fig. lo, is equal to the rectangle under any two sides x \ the sine of the included angle. Let the triangle be ABC ; AB and AC the given sides, including the angle BAC, between them. Through either of the angles B draw BD perpendicular to the opposite base AC : by the elementary principles of geometry it appears, that the area of the triangle ABC — the rectangle under the base AC, and half the perpendicular height BD, or -. But when BA is made radius, BD is the sine of the angle BAC: consequently, the line __ BA X sin. A« . , , „ _ ,„ ,_ lin. CAB AC X AB sin. A» , . , BD = — ; and the area ABC =: AC x AB x ■ ■=. ' which 2 1 ) is the proposition to be proved. t I, here, means the weight of the section I. Da C 20 I] In consequence of these properties, since each course of vous- soirs stands alone, independent of all the voussoirs above and beneath, the strength of an arch will be much augmented by the degree of support afforded to the voussoirs situated in the course immediately above, as well as to those underneath, which may be connected with the former. Moreover, the inconvenience is avoided which obviously belongs to the principles, that are sometimes adopted for explaining the nature of an arch, by which the whole pressure on the abutment is united in a horizontal line, contiguous to the impost ; whereas the magnitude of the horizontal shoot, and the perpendicular pressure on the ultimate or lowest abutment has appeared by the preceding propositions to be proportioned to the weight of the highest section in the semiarch, and to the sec. of the angle of the abutment jointly ; and consequently, the pressure on the different points of the abutment may be regulated according to any proportion that is required. Whatever, therefore, be the form intended to be given to the structure- supporting the road-way, and the weight superincumbent on an arch, no part of the edifice need to be encumbered by su- perfluous weight ; on the contrary, such a structure, consisting of the main arch and the building erected on it, is consolidated by the principle of equilibrium, into one mass, in which every ounce of matter contributes to support itself, and the whole building. The equilibration ..of arches being established by theory, and confirmed by experiment, it becomes a further object of experi- ment to ascertain, amongst the varieties of which the constructions of arches is capable, what mode of construction will be most ad- vantageous, in respect to firmness and stability, when applied to- any given case in practice. A simultaneous effort of pressure c ^1 n combined with weight, by which the wedges are pressed from the external towards the internal parts of an arch, being the true principle of equilibration, the wedges by their form endeavour to occupy a smaller space in proportion as they approach more nearly to the internal parts of the curve. It has appeared by the obser- vations in page 29, Part I. that the bases of the sections may be of any lengths, in an arch of equilibration, provided their weights and angles of the wedges be in tlie proportions which the construction demands, observing only that if the lengths of the bases should be greatly increased, in respect to the depths, although, in geome- trical strictness, the properties of the wedge would equally subsist, yet when applied to wedges formed of material substance, they would lose the powers and properties of that figure ; this shews the necessity of preserving some proportion between the lengths of the bases and depths of the wedges, to be determined by prac- tical experience rather than by geometrical deduction. With this view, a further reference to experiment would be of use, to ascertain the heights of the sections or voussoirs, when the lengths of the bases are given, also when the angles B°, C°, D°, &C' are inferred from the weights of the sections considered as given quantities, to ascertain the alterations in the angles B°, C°, D°, &c. from the summit of the arch, which would be the consequence of varying the angle of the first section A°, so as to preserve the equilibrium of the arch unaltered : by referring to Table VI. we observe, that when the weights of the sections are equal to each other, or A = B = C = D, &c. and the angle of the first section = 5°; then to form an arch of equilibration, the angle of the second section, or E° must = 4,° 55' 30", the angle of the third section C° = 4° 46' 53", &c. And it becomes an object of expe- rimental examination how far the stability and firmness of an arch C 22 3 will be affected by any alterations of this kind, and to judge whether in disposing a given weight or mass of matter (iron for instance) in the form of an arch, any advantage would be the consequence of constructing the sections so that the first section will suDtend an angle of i°, 2°, 3°, 5°, or any other angle at the centre of the arch, all other circumstances being taken into ac- count. When the angle of the first section = 5°, and the weights of the successive sections = 1 , the angles of the abutments will be severally V" = 5° o' o", V* = 9° gg' 30", V''= 14° 42' 23", and so on, as stated in Table VI. By referring likewise to Table VIII. we find the angle of the first section assumed =: 1°, and the weight of each of the subsequent sections being = 1, the angles of B°, C°, &c. are severally B° = 1^4' 57", C° = 1° g' 51", D" = 1° 14' 39"; consequent!}', the angles of the abutments will be as follows : V^ = 1°, V^ = 2° 4' 57", V^ = 3° 14' 48", V^ = 4° 29' 28", &c. which give the dimensions of the sections when they form an arch of equilibration. It has been frequently observed, by writers on the subject of arches, that a thin and flexible chain, when it hangs freely and at rest, disposes itself in a form which coincides, when inverted, with the form of the strongest arch. But this proposition is with- out proof, and seems to rest on some fancied analogies arising from the properties of the catenary curve, rather than on the laws of geometry and statics, which are the bases of the deductions in the two Dissertations on Arches, contained in the preceding pages ; if it should be proved that an arch built in the form of a catenary or other specific curve, acquires, in consequence of this form, a superior degree of strength and stability, such proof would super- cede the application of the properties demonstrated in these Dis- sertations. Concerning the relative Positions of the Centres of the Abutments* and the Centre of the Circle. When the angle of an abutment is greater than the correspond- ing angle at the centre of the circle; in this case, the centre of the abutment falls above the centre of the circle, as in Fig. g. When the angle of the abutment is less than the angle at the centre of the circle, the centre of the abutment falls beneath the centre of the circle, as represented in Fig. 9. When the angle of the abut- ment is equal to the angle at the centre, this case will coincide with that which is stated in pages 4 and 5 preceding, in which V" = A", V* = 2 A°, V'- = 3 A°, &c. V^ — A°, Y' =2 A°, V* = A°, &c. &c. and consequently the centre of the abutment coincides with the centre of the circle, -f Further Observations on the Courses of Voussoirs. A, B, C, D, E; &c. terminating the letter F, denote the sections which form the first course of voussoirs in a semiarch of equili- bration, of which A° is the first, or one of the highest, sections : if the weight of the section A be = iv, and the angle of the abut- ment VOF = V'. then it has appeared, by the preceding pages, that the pressure against the lowest or ultimate abutment = w X cotang. A x sec. V". edly. Let B° be the angle of the first secticjn in the next course of voussoirs, terminated on each end by the letter L, and let y be the weight of the first section, the • The point in which my abutment intersects the vertical line is called, in these pages, the centre of that abutment. •J- Let VO be a line drawn through V, the middle point of the arch passing through the centre of the circle O ; on this arc the angles of the sectiun:. and tne angles of the abut- ments arc measured : p, the point wher^ any abutmeat, for instance I, continued intersects the vertical vO, is called the centie of the abutment. C 24 3 pressure on the last or ultimate abutment = x x cotang. B x sec. V'. Moreover, let z be the weight of the first section C, in the tliird course of voussoirs, which is terminated by the letter P. It follows that the proportions of pressure on the ultimate abutment denoted by the letters F, L, P, will be w x cotang. A° x sec. Y' -]- x X cotang. B" x sec. V", and y x cotang. C^x sec. V' respectively, and according to these quantities, the respective pressures on the several parts of the abutment, will be regulated according to any law that may be required.* The principles of arches having been established according to the preceding theory, and confirmed by experiment, described in the experiments No. i and 2 ; in the first of these, the angles of each section are constructed = 5°, and the weight of the section A having been assumed = 1, the weights of the sections B°, C, D°, &c. are inferred as stated in Table I, from the angles B°, C'', D°, Sec. considered as given quantities. In No. 2, the angle of the first section is assumed = 2° 38' o". The remaining angles are inferred from the given weights by the rule in page 15, A = 1.00, B = 1.25, C = 1.50, &c. to Z = 5, which is the weight of the lowest or ultimate section. It has appeared in page 29, in the for- mer Tract, that whatever be the figure of the interior curve corres- ponding in an arch of equilibration, the bases of the sections which are disposed in this form may be of any lengths, provided the weights and the angles of the sections are in the proportions which the construction demands. CORRECTION OF THE ENGRAVING FIG. 6. • That the engraving of the .Figure 5 may correspond with the text, the summit of the first course of voussoirs ought to be marked A, the first section of the second course should be marked B, and of the third the first section = C, and so on ; this will make the text correspondent with the Figure 6. Z^5l A further reference to experiment would be of use in practical cases, to ascertain how far the strength and stability of an arch would be affected by altering the proportion between the lengths of the voussoirs and the heights thereof; for instance, when the lengths of the wedges are given to ascertain the alterations in the stability of the arch when the depths or heights of the sections are three, four, or five times the length. Let the fol- lowing case be also proposed ; the entire weight of an arch being supposed known, what part of this entire weight must the first section consist of, so as to impart the greatest degree of strength to the structure ; also to decide whether the angle of the first section ought to be made i°, 5°, 10°, &c. or of what ever magnitude would contribute to the same end. To these may be added the following cases to be discussed; when the angles of the several sections are inferred from the weights thereof, to investigate what must be the proportion of the said weights, so as to make the arch uniformly strong throughout. FURTHER CONSIDERATIONS CONCERNING THE CON- STRUCTION OF THE MODELS No. 1 AND No. 2- Dimensions of a Model No. 1, of an Arch of Equilibration. Radius = 0V= 11.4,6281, tJje Angle of eac/j Section = 5°, the Chord of each Arch = 5° = 1 Inch. ( Fig. 7. ) The first section is a brass solid, the base of which = KV = 1 inch, and the sides Vv, Kk, or the slant height of the section A = .961, and the depth or thickness of each section = i| inch, the breadth of A or vk = 1.084. The weights of the sections, as they are calculated according to Table No. I, the first section being assumed as unity. E Rule for making the brass voussoirs equal to the weights which are expressed in Table No. I. Let the sine = 2° 30' o" = s, the cosine c when radius = 1 ; then making the radius = r, the area of the triangle vOk = r^ X sc, and the area VOK = VO sc ; from whence Yv, or the slant height of the section A, when the weight = 1, is found to be s/^——~ ~ OV = .()6i, the breadth vk = 2s x.Ovz= 1.084, = VX'. Thus, by the same rule, the slant height of the section B = \/ — ^~— — }'== •9749» and the breadth // = 1.08^, in all the sections entered in Table I. are calculated. Model, No. I. Weights of the sec- Weights of the sections Weights of the 5? ions as they are cal- when made of brass, sections when Sums of the p. culated in Table Slant height of the Breadth of the specifically heavier made of brass, weights in lbs. 2 No. 1. in the Tract sections. sections. than water, in propor- in lbs. avoir- avoirdupois. on Arches. dvoirdupois. dupois. A = 1. 00000 Ka" = .961 vk = 1.084 A = 6.9444 0.43403 0.43403 B = 1.01542 L/ =: .974// =i.o85|B = 7.0515 0.44072 0.87475 C= i.o4,724JM/?z= 1.004,'mw = 1.087IC = 7.2725 0 45453 1.32928 D = i.o9752N/i = i.o^onn =i.o92JD= 7.6315 0-47535 1.80564. E = 1.16972 Oo = 1.11600 =i.oc)7E= 8.1230 0.50769 3.31333 F = i..6:^2. Pp = i.oQjpp =i.io5F = 8.8140 0,55087 2.86421 G = 1.40427 Qil = 1.329 qq = 1.1 16G = 97518 0.00949 3-47370 H = i-£«75i Rr = 1.492 rr =1.130 H = 11.024 0.69903 4.16274 I = 1-83910 S5 = 1.713 ss =1.149 I =12.771 0.79822 496096 K = 2.19175 T^ = 2.016 tt =1.176 K = 15.220 0.95128 591224 L --= 2.70196 Wv = 2.444 vv = 1.213 L = 18.764 1.17419 7.08643 M --= 3.473t6'iU/^ = 3.067 nil = 1.264 M = 24.122 1.50619 8.59263 N = 4 7i|l«^,''''V"^'= 4-098 TC'W= 1.357 N =32.738 2.04618 10.63882 0 = 6.89iyL.!Xx = ,^55s TX = 1.48.1 0 =: 47.860 299130 13.63013. P = 11.^.537^ Yy = 8.279 yy =1.722 P =78.150 4 88443 1851455 Q = 22.1655-. Zz =13.836 zz = 2.207 Q = 153-92 9 62045 28.13500 R = 65.8171 Aa =29.056 aa =3.534 R = 450.96 28.57000 56.70500 Z ^7 1 On the Constructio7i of the Model No. 2, m a7i Arch of Equilibration, in zvhich the Angles of the several Sections are bf erred from the Weights thereof, according to the Rule in Page 15. In this model the arc A, B, C, &c. is a portion of an arc of a circle : the first section A subtends an angle at the centre of the circle A° = 2° 38' o", the chord of which = 1 inch = to the chord ofBC, CD, DE,&c. radius = 0V= 21.7598 inches: the weight of the first section being assumed = 1, the weights of the sections B, C, D, &c. are considered as proportional to the weight of the first section when it is = 1 ; if the weight of the seventeenth section or R = 5, the weights of the intermediate sections will be B =^ 1-25, C = 1.50, D = 1.75, &c. as statid in Table IX : and since A* the angle of the first section = 2° 38' o", by applying the rule demonstrated in page 27 in former Dissertation, and referred to in page 15 of this Tract, the angles of the several sections are found to be A" = 2° 38' o", B° = 3" 16' 29", C" = 3' 52' gg", and the corresponding angles of the abutments, or successive sums of the angles of the sections, are 2° 38' o" -|- 5° 16' 29" = 5° 54' 29" = V*. Moreover, A° + B° -|- C° = 9' 47' 8 " = V, and thenceforward according to the same law of progression. The next object of inquiry is, to ascertain from what point I in the line OV the line Oil must be drawn, so as to coincide with the lowest surface of the section I, when inclined to the vertical at the given angle VII. The angle subtended by the semiarch VI at the centre O is measured by the angle lOI, and the difference of these angles, or VII — lOI = no. The radius lO being denoted by the same letters which distinguish the line 10, the different meaning will be determined by the context. From the principles of trigonometry, the following proportion is inferred ; as lO : OI : : the &in. of IIO Ea n 28:1 to the sin. of on or VII; consequently, the hne 10= ^^-i^. As an example, let it be required to ascertain the inclination of the abutment to the vertical, on which the section I is sustained when it form.s a portion of an arch of equilibration, and the angle of the abutment VII = 39' q;]' 12": the angle VIO subtended by the semiarch VI at the centre of the circle = 23° 42' o", which being subtracted from the angle of the abutment 39° 37' 12", leaves the angle lOI = 15° 55' 12, and the distance required from the centre, 01 = OV X ^111' r, or because OV= 2 1.7598 inches, 01 = 9.35978 inches ; making, therefore, the line 01 = 9.35978 inches, through the points II draw the line I, I, t, which will be the position of the abutment on which the section I rests, the angle of which, V' = VII, is the inclination of the abutment V' to the vertical : for the same reason VHH = the angle of the abutment V' = VHH, the difference of these two angles VII — VHH = G/jH, or the angle of the section H°: making, therefore, the line G/j = a, H/j = h, the properties of trigonometry give the area of the triangle G/jH :=.ah x^-^^^; on the same principle, the area of the triangle sin H** Hz I = Hil = Hz X 1/ X — — ' ^i"id thus the areas of all the triangles will be measured, from having given the sides of the triangles and the angles included between them. The sides of the triangles may be measured by a scale of equal parts, as stated in Table I. and in this manner the sides of all the triangles were correctly measured by Mr. Berge, so as not to err from the truth by more than an unit in the fourth decimal place. This measurement was essen- tial for computing the distance of the vertex from the base, so as to form the dimensions of the brass wedges, correctly and inde- pendently of their weights, in each triangle. For instance, /O being put = a and /R = 6, this will give the area of the triangle C 29 3 rQRz^ab x-^-^; and if the triangle raa is made isosceles, or ra = X, the area of the triangle Raa = '^ ^ ""' ^' '— the area si^^ — ; or if the difference of the areas is put = zv, the re- sult will be flili'llil! _ the area ' ^ ^^ ^ ^'"- ^' = zv, or x = ra R" ^^.., + thearea.x/,xsin.Ro ,,,herefore Qa =v/^" + ayj><^^^ sui. K ^^ '^ sin. R. = 28.5777 - ;-Q and Ra = v/^ "' + " .^ ^L^^H^ - ?-R. Thus, by actual measurement, a = 23.9248 inches, and /; = r> 1 J.^ L sin. R° ^ ^ /zw + a x b x sin 24,.305(j, and the area ab x — ^ = 1 0.73670 =v^ — ^HTF — - Or, and Ra ^ ^yjZIZI^Eji^E — rR: the area raa = - — —- — == 1573670, or the area raa = 15.73670; the result is, that the area aaR 0 = aa — rQR = 5 square inches : and since every square inch of area is occupied by a weight of a section = 6.94,4,^ oz. avoirdupois, we arrive at the following con- clusion, that the weight of the section R r= 5 x 6.944,4 = 34, 7222 • 1 • r> /zw 4- a X h x sin. R'^ o oz. avoirdupois, cecause v^ ^^- ^z = ra =28.57770, this determines both the greater and lesser sides of the section R ; namely, the greater side being = ra — rQ =4.6529; and the lesser side being = ra — rR = 4.2721 inches; in this way, the Table is formed, shewing the greater and lesser sides of the several sections. According to this mode, the dimensions of all the brass wedges were formed ; the investigation of the angles of the wedges from the weight thereof is the subject of investigation in page 27 of tiie First Part of the Tract, entitled a Dissertation on the Construction and Properties of Arches ; and it appears that if the angle of the first section is given == A°, together with the weight thereof = a. assumed to be = i, the weights of the other secticn B = b= 1.25, the weight of C = ^ = 1.50, of D = f/ = 1.75, &c. The prin- ciple of equihbrium is estabhsheci, by maki;ig the tang, of the angle B'=4^i^^, o ^ -J- a X sin. A -, also the tang, of the angle C° = h X COS. D° as they are stated in Table IX. which contains the conditions, founded on supposing that the strength and security of the arch are such as require that whatever weight should be contained in the first section, the weight of the seventeenth section R shall be five times as great: making, therefore, the weight of A = 1, the weight of B = 1.25, and C == 1.50, and the weight of the seven- teentii section or R = 5, &c. Thus the angle of the first sec- tion A° being assumed = 2° 38' o", and the initial pressure on the lowest surface of A = / = ^'^■76555, ^nd the vs^eight of the first section z= a = i : from these data the following results are obtained a X COS. V° p -i- a X sm. V = 2° 38' o" tang. B° = b X COS. V' p + b X sin. V = 3° 16' 29" tang. C° = ^^"^x's'in.V = 3° 52' 39". &c. &c. accord- ing to the statement in Table IX. The Dimensions of the Sections, according to the Rule in Page 29. Lesser Sides. A = 0.97827 B = 1.20676 C = 1.4.1568 D = 1.61833 E = 1.81718 F = 2.00676 G = 2. 20080 H = 2.4,2850 I = 2.62584, Greater Sides. 0.97827 1.21126 1.44928 1 .666g^ 1.90118 2.13656 2.36920 2.63910 2.88134 Lessor Sides. K = 2.73754 L = 3.14849 M = 3.36S00 N = 3.64620 O = 3.87463 P =4.16762 O = 4- 437^8 R = 4 27210 Greater Sides. 3.12824 3.47069 3.71700 4.01460 4-254<7^ 4-55142 4.82038 4.65290 rstn o 1^ f oco toci^^Cit^tfi ri o G0'>o c^ iH o r^ t^ '-' " O O^ 0-.\Z> 00 CO 30 OiOO 'O r- CO '-' <£> t^ -** 0-5 CO Oi pi^ -oco CO CO O GO Oi'O O GO CO OO 1-^ TTi COOO i-0 CO !-< CO J^ !>. CO^ 00 t~^00 tv. s o CO i>.cr) to *o «o io *o 'o iocd CO t^oo di d S- il II II II II 11 II II II 1! 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O Ci OiCO CiCO - '^OO CO 1:^ CO'-O rH O 0) r- 01COCO rH'^" 0IC1C0 01W "^GO u 00 CO 01 '^ o ' CO i-t *0 01 to t^Tj(01 tH GO^ t^COO CiOiO* rtr-' -*01iOC0rH-^e(l-t > ■^ oOl eOGO'^^iO^T^^O'^TftGOGOGOCI 0^ 0» II II 11 jl II II 11 II 11 11 11 jl 11 11 II 11 II < °- CO o iO'X) CO t^ C^.OO iO '^CO d fr?CO 01 C?i Oi Oi '-' to loco 01 a^ o\ >jri T- 01 o -^-^o Cioir^rf* t^co o '^ cr^co Ci ^ 01 'f* (y-joo ^i co co oo ■#• > g j i-^ r^ 01 oi oi 01 01 N 0) 01 GO '^CO 00 O 01 lOCTiOICO O l-O 01 01 01 01 COGOCOGOtJ<^1jO^ ii II II II II II II II II 11 1! 11 11 II II 11 11 li £ 3 •^ !3-< ^ t^ t-i &sBH?^^?«-Cl.o 01 tot^o ^ a r^ rt i-^ r-i oi oi oi oi GO CO GO CO '^ '^ TJ. 4< i^ O'B II II [1 II II 11 II II II 11 II II II II II II II ^3 ~a '-J ~^3 'Ij ^^. c,:~^ -w-ss--^ g s o-a,0 5>.' g II II II II II II II II 11 11 II II II II !l II II " -^ Da rj O a :^ a ffi ^ ^ wj § ?5 o cu o?s^ 2 _^ -o , G O O O O o o o o o O Vj o ^o o O 01 '-0 !>. O oooooooooooo OOOOOOOOOOOO iOO «00 iOO'-OO 'OO iOO 01 'iO t^ O OI *0 1^ O « 1-0 i^ o Oi M^ = rii « I-' rs ci ci ci oi CO CO CO CO ^■< -^ " vvhich is a cubic inch and half a cubic inch. In Model No. 2. the dimensions in the first section of the semiarch : the base, or the chord of 2° 3S' o', to a radius of 21.7598 — i inch, the slant height are as follows : the area of the first section parallel to the plane of the arch r: i square inch ; this multiplied into the dcprh or thickness, which is li inches, the solid contents of the first section be- comes ^ I X I X li, or the solid contents of the first section :r if cubic inches — .9782, and the breadth =: 1.044^9, which makes the solid contents of the section iz: i| cubic inches, the weight of which zz 4-3027 parts of an avoirdupois pound. G c 42 ;] By this means, another method of computing the weight of any section P is obtained, by putting the sum of the weights of all the sections from the summit to the section P ; that is, the sum of all the weights from A' to P° = Sp, and the sum of the weights of all the sections from A to O = So, the weight of the section P will be = S/) — So, for the rule in page lo, Computition for S^, Computation for So. Log. TO - =: 9.6375176 Log.tr - r: 9.6375176 L.cotang. 5°~ 1.0580482 L.cotang. 5*=i.05Rg482 L.tang. 75°= 0.5719475 L. tang. 70" = 04389341 L. S^ z= 1. 2675133 S^ = 1S.514 L. So — 1.1344999 So — 13.630 Sp ziz 18.5 14 So zz 13 630 Sp — So zz weight of Ihe section P — 4.884, as before determined. The computations of the dimensions (Fig. 7) of the brass sections in the Model No. 1. are much facilitated by the use of logarithms, particularly in finding the slant height Ot from the centre O of any section (K,) and the height of the section itself, or St = Tt. Computation of the slant Height OT of the Section K. It is first necessary to ascertain the area of the surface OST comprehended between the radii OS, OT, and the chord ST. Since the radius OS = 11.4,6281 and the angle SOT = 5°, half SOT = 2° 30' o", the Sin. of 2° 30' o" or 5 = 8.6396796 Log. r = 1.059291© Cos. 2 30 o or c = 9-9995^^5 2 L. sc = 8.6392661 L. r' = 2.1185820 L. -^ = 1 .3^07339 L. sc = 8.6392661 Log. of the area OST, or L, sc y.r''^ 0.7578481 n 43 J The area OST = 5.72595 The weight K = 2.19175 K-\-r'sc= 7.91770 L. K 4" ^^ -^^ = 0-8985990 L-^= 1-3607339 T K + r* sc L-— ,^— = 2.2593329 L. v/ -—7-^ = 1-1 296664 0/ = 13-47928 Radius OS, or=:r = 11.46281 Height of the section K = tt =z 2.01647 Similar Computatioti for the Section L. L, r^= 2,1185820 L. sc= 8.6392661 Log. of the area OTV = 0.7578481 area OTV = 5-7^595 L= 2.70196 L -f ^*-^<^= 8.42791. L. L -|-r'5c= 0.9257199 L. ^= 1-3607339 L. L^_ 2.2864538 L. X L + r' sc r> V -~ — = 1.1432269 Or; = 13.90679 r= 11.46281 Height of the section L = ft; = 2.44398 G 2 For the Section M. L. r'^= 2.1185820 L. sc = 8.6392661 Log. of the area OVU = 0.7578481 area OVU = 5.7Q595 M= 347366 M4"^*-^^= 9-19961 L. M -{- r^ sc =z 0.9637694 L. ■^= 1.3607339 L.— ^^^ — = 2.3245033 i^,yE+ZE^ 1.1622516 Ou = 1452953 r = 11.46281 Height of the section M=uu= 3.06672 . r /ztSD -{■ ah X sin. L° Computation oj v/ ^mTL" ' L. ^= 1.0827423 L. 6= 1.0941566 L. sin. L° = 8.8806960 L. a 6 X sin. L" = 1 .0575949 ab X sin. L°= 11.4181a 2 w= 7. L. 2w + afc X sin. L°= 18.41813 C 45 3 L. 2 w + u6 X sin. L°= 1.2652453 L. sin. L° = 8.8806960 sin.L° = 3.3845493 ZZIEZZZ^ 1.1922746 y/2 Ki 4- « 6 X sin. L° ^ v^ — nns — = 15-56949 See page 19 and page 29, in which the computation is inserted r ^i i'i X 2 w + ab X sin. R* of the quantity w = v^ — ^^r; • Computation for M°. L. «= 1.1263101 L. 6= 1.1375598 L. sin. M°= 8.8391355 L. a6 X sin. M°= 1.1030054, ab X sin. M''= 12.67667 2W= 7.5 sw -f ^6 X sin. M° = 2o.i7667 h. azv -{• db X sin. M° = 1,3048496 L. sin. M''= 8.8391355 X 2w 4- aZ) X sin. M° ^ ^- Ti^Tivr = 2.4657141 T /z w -\- ,ih X ''in. 1V]° r. ^•v^ — nnvi^ = 1-2328570 yz w -f .,b ' s n. M° V^ iihTM^ = 17-0945 Breadth of the Section L. Log. slant height from the centre = 1,1922561 L. 2 = 0.3010300 L. sin. I V = 8.5799524 Log. breadth of L" = 0.0732385 Breadth of L°= 1.1836 Breadth of L in the drawing = 1.1838 2 error. Breadth of M. Log. slant height from the centre = 1.2328570 L. 2 =0.3010300 L. sin. I M° = 8.5385170 L. breadth of M" = 0.0724040 Breadth of M°= 1.1814 Breadth of M by the drawing = 1.1814 Explanatory Notes on the Propositions in Pages 13 and 14 in the First Part of this Tract, in which A" = 5°, B° = 5°, =. C° = D", &c. according to the Explanation in Page 12. The initial Pressure = - — ^tj;, or putting IV =.\, the initial Pressure orpz=.~x cosecajit 2° 30' o". L.p= 1.0592904 L./) = 1.0592904 L. cos. A^z^r 9.9983442 L. sin. A° = 8 9402960 L. tang. 2° 30' o" = 8.6400931 L. p y. COS. A" :=: I.O576346 p X cos. A°=: 11.41917 L.p X sin. A" x tang. 2" 30' o" =z 8.6396795 V" X sin. 2''3o'o" = .04362 p x sin. A° x tang. 2 30 o = .04362 11. 46279 =/> It appears from this computation that p x sin. A* x tang. V'' is equal a x sin. V, when the weight of the first section, oc a — i. C47n The Weight and Pressure on the lozvest Surface of the Section B. L./>= 1.0592904 L,/> = 1.0592904, L. COS. B° = 9.998344.3 L. sin. B° = 8.9402960 L. tang. V* = 9. 11 94291 L./> X COS. W =. 1.0576346 p X COS. B° = 11.41917 L./>x sin. B°xtang.V*=9. 1190155 6xsin. V*= .13153 /»x sin. B°x tang. V*= .13153 •■ — ^ X COS. B° = 11.41917 1 1 .55070 =zq px COS. B-+ p X 1-0592904 8.9402960 L. sec. V*= 0.0037314 L.p== 1.0592904 sin. B°xtang.V*= 11.55070 L. sin. B" =: 8.9402960 L./Jxsin.B'xsec.V^^: 0.0033178 ^xsin.B''xsec.V'^= 1.0076 ERRATA. Page 5, line 5, for cotang. A x sec. A, read cotang. A° x sec. A°. 6, — 19, for that part of weight, read that part of the weight. 10, — 20, for p zz 7a X cotang. A° x sec. read w x cotang. A° x sec, V'. 14, — 12, /or area K is, read T Its. 14, — I'jtforx^ — r^sczzk, readx'^sc — r^sczzk. ^3' — 5> for Fig. 9, read Fig. 8. 24, — I, for X X cotang. B, ready x cotang. B. 24, — 5, for X X cotang. B*, read z x cotang. B". 24, — 16, for in No. 2, read in the JVIodel No. 2. 28, — 9, for 01, read OV. 28, — 12, for tiic pjiat II, read through the points II. In Table No. IV. in the weight of the section I, insert 0.654983. In Table No. X. for OVS read OV taken at 21.7598. Printed by W. Buliner and Co. Cleveland-row, St. James't, Table No. I. Shewing the weights of the several sections or wedges which form an arch of equilibration, when the angle of each section is 5*; and the weight of the highest wedge is assumed = 1. Also shewing the pressures on the lowest surface of each section, considered as an abutment. The weights of the two first sections A in each semiarch = i. TA^ lateral or horizontal pressure = p' = n .430052, S^ = the sum of the four successive weights = A-fB-|-C4-D, &c. &c. 1' Angles of Sec- Angle, of ttiL- Abut- Weights of each Section. Weights of the Semiarchei. Entire Pressures on the lowest Surface of each tions. ments. Section. A 5" V" 5° 1.00000 Sa = 1.000000 11.47371 =/> B 5 V 10 1.01542 S b = 2.015426 11.60638 = q C 5 V' 15 1.04724 Se = 3.062673 11 83327=7- D 5 v-^ 20 1.09752 Sd = 4.160196 12.16360 =s E 5 V' •■^5 1.16972 Se — 5.329920 12.61165 = t F 5 v/ 30 1.26922 S/ = 6599144 13.19829 =v G 5 v^ 35 1.40427 S^ = 8.003420 '3-95351 =« H 5 \h 40 1-58754. S /6 = 9-59^9^^ 14.92087 =11) I 5 V' 45 1.83910 S / = 11.43006 16.16453 =x K 5 v^ 50 2.19175 Sk = 13.62181 17.78200 =y L 5 v 55 2.70196" S / = 16.32377 19.92768 =:Z M 5 V- 60 3-m^^ Sni = 1979743 22.86010 =a N 5 V" (^5 471440 S7i = 24.51183 27.04880 = b 0 5 yo 70 ^■•^9199 S 0 = 31.40382 3341923 = ^ P 5 V^ 75 11.2537 S/) = 42.fc''5753 44. 16234 = ^^ Q 5 V? 80 22.1655 S 9 = 64.82305 6r) 82304 =3 e R 5 v 85 65.8171 Sr = 130.6401 131.1450=/ 1 a; c; ^ c < ) i~. o fi o -a a C- CO fli Ui 4-* u o CO c .2 CO OJ •S 0) -t-j 0) " T3 O o > S o o ^ "^ t—, fcJO "« < CO a; o n n C ^ t^ _o c o o a> QJ c cu .c > o CO ■*-* oC Cu CO r- -; 0) -s o o; . rt 5 £ , ^ ^ +-1 Z o o t*- CO « ^ f^ o s; h 0) r. OJ -H 3 I ^ ri ■*^ c t*-i 1) "3 • »— « o , ^ (1> H »0 C5 O n o Oh a; o CO i. ' ^ rH o 0) fe Trt »-t CO OJ r, II ^ o N r- tM O (*-i c O o Sm 1^ '"' ^ o s 0( rt ^ E P bJD ^ o , c; _o 0") Cj — ^ 't' o C o )-.' c3 th JZ ^ ^ CO 0) 4^ rt G. II c ^ 1 ~ ^ ] ■g 1! "i o CO GO CO iO iO -*< o o o CO CO Ci o CO GO ^ en CO CO . Ci o Ot GO i r^ oi •^ c^ Ci 01 ci CO 00 iO S II II II II II II II II II II ii c <; •«^ ^ ^ ■*^ ti «N* PJ H ^ bh 3 i o — ^^— — — _ >. ^ •^ :; ^ :; i "b "co ~bi t^ ~CO 00 ^ o -* Ci g iO rH c?t ^ CO *o '"" '"' B "o "'-0 ~C1 "b^ rH o^ t^ "■^ 00 c> -^ 'i- ^ Cl '-' '" '"' J °iO 0 "co Ci "iO o "-^ °^ o 01 °^ ■^ l-H 0( ^o 'o 1^ t-^ 00 00 s II II II if II II II 11 II II c < > > > > > ^ > > > > o "o i-VN "Ci ~iO cb "vj CO "-* ^ *o "H . iO CI ^ ^** 01 "o ^ *o "t,-^ 'O V- "o» 00 V- r- "go 1H 2 5 ^ c« «o *o Tf GO "^ < °^ Sd 0 "'O °t? o °f ^ o °co O 01 . °< M U Q k L o X "^ k ■ 2 ^ bJO bb bb b'n bh bb bb ^ bb o c .c c C c ZZ c C3 c 1) 3 C3 a CS rt > > > > > > > > > > > > > > > > > > > 1 1-: X : X 5 X c X o X o X § X c X o X c X ° X -3 X -a X o X ■« X X ■■*^ X oo X -^ X + X + c <3 + ^ + + + . •^ ^ Ci ° C o 00 O CO ^ 25 o Ci CO S t o CO lO '^ q CO GO 00 CO 01 .5?^ rH ci 4^ t^ 6 '*' Oi 'j' d ^ •-< "-I '- 01 CO II II II II II a II II II II c Q -o ^ ^S <» ^ ^ -Q ••* -Itf w < : c: > C ) C :5 (I J u i4 C J n »- . U Table No. III. In which the angles of the sections are i", 2°, 3°, &c. making the angles of the abutments 1"*, 3°, 6°, 10°, for inferring the weights of the successive sections and the sums thereof, with the pres- sures on the lowest surface of each section, as computed from the general rules in page 15, as they are inserted in the 5th, 6th, and 7th columns of this Table. Angles between ih'; Angles of lowest surface of the sec- each section and Weights of the suc- Weights of the succes- tions. the vertical, or angles of the abutments cessive sections. sive semiarches. Pressures on the abutments. A 1° V" 1° 1.000000 1.00000 /» = 57-29869 B 2 V* 3 2.002440 3.00244 q = 57.s6S5g C 3 v^ 6 3.018978 6.02141 r = 57.60508 D 4 Y' 10 4.080347 10.10176 ^=58.17374 E 5 w 15 5-249031 15-35079 i =59-31090 F 6 v/ 21 6.640753 21-99154 V = 61.36580 G 7 v^ 28 8.470050 30-4^' 159 u = 64.8848s H 8 v^ s(^ 11.16197 41.62356 lu = 70.81421 I 9 V' 45 -^5-^66^5 57 28991 X = 81.02014 K 10 V* 55 24.52854 81.81845 >'== 99.88185 L 11 ¥•' 66 46.85674 128.6751 z = 140.8525 M 12 v« 78 140.8525 269.5276 ^^ = 275.5490 Table No, IV. in this Table the angle of the first section A*'= 5', and the angles B°, C, D°, &c. are assumed of any given magnitude, taken at hazard = 6°, 8', 1 2°, &c. making the angles of the abutments = 5°, 11°, 19°, 31°, and p = 11.4737, ^c* ^^^ initial pressure P'= 11,43005- Entire pressures on the Angles contained Weightsof the semi- lower surface of each sec- ? between the lower arches, lound by tion, considered as an ?. Angles .urfaccof eachsec- calculating from abutment, found by cal- § of the uon and the verti- Weights of the lections. ihevaluesinserted culations from the values sections cal line. in paoc 14 of the Dissertation on Arches. for the pressures inserted in page 14 of the Disser- tation on Arches. A 5 V— ^» a = 1 .000000 1.00000 11.47371 =p B 6 V*=ii 6 = 1.221776 2.22177 11.64392 =q C 8 V^=,9 c = 1.713895 3-935^7 12.08864 = r D 12 V'^si d = 2.932180 6.86785 13-33465 =•' E 10 V' = 4i e = 3.068 11 7 9-935^<(i 15.14492 ==:t F 9 V/=5o / = 3.685800 13.62176 17-78193 =^ G 4 V^ = 54 g = 2.110300 15.73206 19-44585 = « H 3 ¥^=56- h = 1.213626 16-94569 20.44014 =w I 1 V^ = 57 i= .654983 1760067 20.98633 =x K 7 V^ = 64 ^ = 5-834303 23-434^8 26.07^73 =y L 4 V'==.6-S / ~ 4.855258 28.29023 30.51193 — s M S V"'=7i fn ~ 4.904875 3315^511 35-10776 —a N 5 V = 76 72 == 12.64806 45-84317 47.24652 = b 0 12 V = 88 0 = 281.4682 327-3113 337.5108 =c P 1 V^=89 /)== 327.5107 654.^220 6549206 z=d V 0) V- ^ a ♦3 w TJ u r: o PS o CO CJ GO J-l »^ CO II (U -G •^ *s OT o -G ^ JZ (U o -C o 'c3 .r. o ^ CO c f <:o v> on U c GO > 6 a. I! n rS .5 11 H 3 _c c o o -^ CJ -TT o ■»-' bO dJ 4-. '*^ -X OJ D c "bi -5 rt Ol u hn • (^ fo •to fs a g H II 11 II II II II II II ^ o CO o i>^. 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CO 00 CO 00 t^ GO to CO Cl '^ j2 « t^ '^ ca q Oi 00 t^ 5 CO ^ ""j* »c ^ !>. t^ CO 05 s 11 s •«^ ^ II o < 3 6 ^ a> « CO «• O Ok U ^ ^ ^ ^ ^ ^ o « t^ "■•«?• ''I '^ 1> *o 00 1 a 1-t CO ,?* 0» rH b %H "K "o^ "lO "K to "Ci OO JJ i-" '^ ^ I-t *o »o 0« ^ o CO CO ?o °co o 00 r-l °^ o "i '-1 0^ vO '^ »o *o to to to II 11 II 11 II 11 11 11 11 c -» -* •<3 ^ <<^ >» -ii • w < > > '> > > > > > > ^ ^ ^ ^ ^ ^ ^ -C o 0» iO r^ CO OJ CO CO CO " ^ T-t c» o< CO 01 »o SI "b T-< & 1^ cb "^ s "cO Oi -^3 T-t CO I-" T-4 ^ '* w c 3 o o o e < 'O CO o 00 ;o '^ CO CO 0» 11 II II II II II II II 11 ^ o o o o o o 0 o .9 < « o Q W fc O K >-* 2 bb bb ip ^ ^ bJO U3 bJO re 4^ t> E c JS S c« re 5 5 5 re re «^ ■+-* •*-' *-* o 11 II T I 1 II G ^ ^ ^ ■^ ^ N> -c ■« • « > > > ^ > > > X > _^ > > ^ > .n > c > .= > 5 > a > .- > ■^ > •^ K» C fcb o o X o < 6 X o X c X o X o X S X 8 X X,« X -o X u X "^ X -a X "-^ X b« X -Q X + « + j; + u + 1= ■1- =! + — 1-^ ■«H ■" -i^ 13 o 03 4-1 •— . ^ c <: CJ '^ bJC CO « " _• rJ o 00 <» CO 3i 30 tN. lo F-l 0 — ^ 5 s r-- CO 01 o^ CTi 01 CO ;o -^ '- -f* +-> m GO t-. o^ ^ c^ CO oc I^ CO <-l-l *~ 4J t^ o T-< o r-. o o vj -f CO ;o o "S -c '^ ;d 00 ■f* ^ '^ c^ *o IH CO B ;:^ ^ oi ci . oi CO CO ^ *o u3 3 i i 11 II 11 H i II II 11 ii i II V3 " s ■^ O- K «> •s^ pj s s H ?N N *-> S ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ i "o "b "go "^ >5 Cb "« "K 05 r~t iH O £ CO 0< CO rf< 1-1 »o ^ OJ B "o "^ ~b< "K V- -^ c^ "o^ "bi "o "■-* „c ^ ^ -* ^co o^ 01 iO *o 4-J 0 o c o _^ iO °ai ^ Oi °G0 V. 1^ %f 00 r* GO • c IN 01 CO GO GO Tf -f" , o II II i it II 11 II II 11 11 II Ih "5= -y ■-^ bj, -^ -^ -^ -V. 3 < > > > > > > > > > > > 0-) jr o "b "go "ci b =^ 'O "'O "oi "bi b (U GO iO o 01 T-* I-l r-i '^ ^ tH "o g ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 4_l S '"- o iO CO 'f o '^ t^ o CO i^ CO CJ g=^ ^ -* CO d ^ CO Tl lO '^^ S-i < o o ^ 0 , o o 0 .2 ^o ^ "^ -* '^ '^ CO CO CO d d 2i II Jl II II II II II II II II II *-> ^ Q Q o o Q o o 0 o -. ■73 5 < h U Q w i^ o ffi >— ( Ui u C3 i bJO ^ bJD bJD bb bn bb ^ ^ bh bb u c C E c c E i*H c G .- r^ -r; ci 3 C3 rt a cS ns rt C3 Cj rt iO CO •^ '*-' ■4-1 '*"' 4J ■'"' +-• •*-rf ■I-l ■M o r^ o ^ a II II II 11 i T II 11 I "^ r^ "w 0 -« ^ ^ s . -s ,^ ^ - !! 1 > > I> > > > > > > > > > ^i ^ > > > > > > > jl '^^o ti o X O X o X O X c X o X 8 X d X o X o X c X c X 5! X -3 X (o X ■« X »J X ■-^ X b/j X -Q X •- X ■« X •-» "o cj « + jv + ^ « ■" B ^ s H ?s III H 1-1 IH TH T-l r-t 1-1 1-1 iH T-< 11 J ^^1 II II II II II II II II II II II CO <3 ~Ci U ■^ ^ ^ ^ <5 < 99125 = 7 ,10140 = r ,25714 = 5 .459S6 = t 23 23 23 •7-93 : 7.221530 : 8.336290 : 9.487050 : 10.68020 ■■ 11-92394 = 13-22350 = 14-59130 : 16.03738 ^17-57468 = 19-21854 = 20.98743 : 22.90377 '■ H-99507 ■ ^7-^9565 29.84877 32.70989 •35-95171 39.67048 ■ 43-99772 49.11730 55-29^57 62.92757 72.64146 ; 85.47800 .71172 = V 01526 = u 37368 = w ,79086 = 0: 27151 =y .82129 = z •44699 = ^ ■15674=6 .96033 = c .86g^6z=: d ,89874 — e 0^533 —J 39081 —g 90187 = b 63193 = ^ 6^355 — k 93149 = ^ 62755 = m 80753 = « 60224 = ^ .19492 =/» 85041 = q 9651 1 = r 11813 = s 88.49336 = i Table No. VIII. Shewing the angles of fifty sections, forming an arch of equilibration, calculated from given weights of the sections wlien the angle of the first section is one degree =: A° ; and the weight thereof is denoted by unity ; the weights of the successive sections encrcasing by equal differences from i to 3, ■which is the weight of the twenty-fifth section =:Z in each semiarch. The initial pressure parallel to the horizon / = cotang. A" — 57.28996 : the pressure on tlie lowest surfaci; cf the first section is =: J> z= 57.29868 =: cosecant A°. Sect ons. Weights of the sections. Tang, of the angles of the sections. Angles of the sections. Pressuies on the Angles of the abutments. lowest surface of each section. A a =: 1.000000 aXcos.V _ 1° 0' 0' V" = 1° 0' 0" /> = 57.29868 ^Z + ^XMnV ""^--^ - P ft = I.C83333 iXcos.V. _^^^^„ p,_ J° 4'57".457 V'= 2° 4'57"..457 q ~ 57-327^2 /I + A X Sin. V.^ — " c c = 1.166666 .Xcos.V* _^^^^„ ^„_ 1° 9' 5i",204 V'=z 3°i4'48",66i r= 57.38205 y + c X s.n. V* — ^ " D d = 1.250000 45i^ = .,n,.D.= i°i+'39".795 V"^ = 4° 29' 2S',456 s - 57 46639 £ e = 1-333333 < X COS. V-' _ £0 _ 1° i9'2i",558 V' = 5° 48' 5o",oi4 i - 57-58614 . + .Xs.n.V^ -tang.li _ F J = 1.416666 <+/Xsin.V. 6 jo 23' 54",634 V/= 7°i2'44",648 V - 57-746S4 G g = 1.500000 ^ X COS. V/ _ Qo _ 1° 2 8' i6",987 V.' = 8^41° i",638 u - 57.95427 „+,^Xsin.V,'— '""S-^ - H b = "•583333 AX COS. v.. _^,,^„„o_ i°32'26",4i7 V* = 10° 13' 28",055 w = 58.21435 « + A X s.n. Vz — ° I i = 1.666666 ^Xcos.V. _ _ 1° 36' 20", 646 V' r: 11° 49' 48", 70 1 j: = 58.53326 I, + , X MM. V- — K k = 1.750000 ^Xcos.V. _j^^„j^o_ i°39'57".36s V" =: 13° 29' 46.",o66 y — 58.91692 x + yiXs.n V- -"" = -^ - L I - 1-833333 1° 43' 14", 297 V'=z 15° 13'. o",363 z - 59-37'54 M m = 1.916666 ^Xcos.V _j^,^„^^o_ i°46' 9",294 V'" =: 16° 59' 9", 66 7 a - 59.90315 ^ + 7^ X COS. V — ° N H n 2.000000 A cos. V — tantr N" — I "48' 4o",404 V" = i8»47' 5o",o7i h =: 60.51760 . + „XMn.V.. - i" 0 0 = 2083333 .Xcos.V" _^.^,^„no_ 1° 5o'45",954 y. _ 20= 38' 36", 07 1 c =: 61.22067 i + flX sin.V — P P = 2.166666 ^XcosV. _t, po_ 1° 52' 24", 6 1 1 \jf _ 22° 31' o",7i5 d =: 62.01767 CL 1 - 2.250000 q X cos. Vf _ ^0 _ 1° 5^'35",6ii V? ■=. z.\" 24' 36",326 e zz 62.91365 - w w zz 2.750000 n. Xcos. Vi' _. -ryo- 1° 51' 10", 121 V- = 34°45' 17", 1 37 ^ = 70-59525 ^+^.+sin.V. -""- '" - X X =: 2-833333 xxcos.v- -r^n- V°- 1° 49' 2 2", 000 V'= 37° 34' 39", 137 "' = 72.28737 /+..Xs.n.V-- ° — Y y = 2.916666 ^Xcos.Vx _ y„_ i°47'iS">273 V/ = 39°2i'54',4io 71 = 74.10210 m + vXs.n.V.«- «• - Z z r: 3.000000 - X cos. V.i — 20 — i«44' 52'',429 V* =; 41" 6' 46", 83 9 0 = 76.04024 . + iXMn.V>-""^-^ - 1 Table No. IX. ;:!ontaining the angles of thirty-four sections or wedges, constituting the model of an arch, No. 2, the weights of which increase regularly in each semiarch, from 1, which is assumed as the weight of the first section, to 5, which is the weight of the lowest or seventeenth section from the summit : the angle of the first section A° = 2° 38' 0", and B, C, D, &:c. are inferred by the rule in page 15, from the weights of the said sections. The initial pressure parallel to the horizon = cotang. 2° 38' = 21.74,25 ==/)': the pressure upon the lowest surface of the section A, co- secant A° = 2 1.76555 =/), 1 tetioiu. Weights of the sections. Tang, of the angles of the section!. Angles of the sections. Angles of the abutments. Pressures on the lowest surface of each section. 1.00 a X COS. V- _ ^ ^0 __ 2^38' 0', ¥"= 2° 38' 0" p = 21.76555 p' -\- a X. sill. V- ° 6 = 1.25 b X COS. V" . T^o 9 ■■ — Tz = tanij;. B =r /> + /) X sin V'' 0 3° 16' 29" V*= 5° 54' 29" q = 21.85867 Cz= 1.50 c X COS. V _ ^0 _ 3° 52' 39" V^= 9^47' 8" r=z 22.06356 9 + c X sin. V " b d= 175 d X COS. V^ _ ^ po _ 4° 24' 36" V''^ 14° 11' 44," s = 22.42739 r + rf X sin. \" °' I e — 2.00 . ' X ^°'- ^' — tantr E° — 4° 50' 9" V' = i9° i'5.S" t = 22.99973 s -{. e X sin. V' 0 ? f = 2.25 /XCOsV _ p_ t +/ X sni. V' 0 5° 7' 16" V/=24' 9' 9" t'^ 23 82853 ^ g = 2.50 g-XCOS.V _^^j^„<-o_ 5° 14' 41" V^=29°23'5o" w = 2495590 V -{- g X sin. V/ ^ H h = 2-75 b X COS. V^ _ ^^^^^ p^o _ 5° 12' 14" V'^=34°36' 4" ty = 26.41465 « -f /j X sni. \'..' ^ [ i = 3.00 w + t X sin. V" ^ 5° 1' 8" V' = 3.9°37'i2" X = 28.22645 k k = 3-25 k xcos.V _j^n"- K° — 4° 43' 23" V*= 44° 20' 35" ^ = 30.40220 a + kx sin. V' ^ L / = 3-50 /Xcos V Lo — >■ + / X sin. V^ ^ 4» 21' 27" V'=48''42' 2" - = 32 94376' I. — 3-75 ^ + m X sin. V " 3° 57' 33" V'"=52<'39'35" ^ = 35S4656 N n = 4.00 "xcos.v^^ _tang.N^ — a + n X sin. V" ^ 3° 33' 26" ¥' = 56=13' 1" 6 = 39.10209 0 0 = 4-25 P X cos. V" _ ^^^^ n° — 3° 10' 21" V' = 59'23'22" c = 42.69902 t ^- 0 X sui. V" '=' ? p = 4-50 /.xcos.v ._tang. P^ — c +;> X sm. V" '=' 2" 49' 0" V' = 62''l2' 22" d =z 46.62917 2 9 = 475 <,xcos.V'^ -tang.O" — 9 + rf X SHI. V/ » -^ 2° 29' 42" V?=6V42' 4" 3 2i 0 GS Si 03 in hn 0 " 0 0 ■53 (1) 0 Q b 2 r; K! w ^ C) W3 0) ■r; 0 ■M -G O 03 g > GJ 5: 0) 0 r^ 0 to 0 CO > bO 0 4J «u ■^ Iz; rs ci "H bD 0 0) c w -C T3 a> >-] Si c 0 M 1J «j-. rr! -< 0 0 H QJ t/3 00 ^ +-I _o CO r- D U en 11 (J co 0 n C O "^ CO S -Z = ^ t^ O O ^ 0 0 '^ ^ ^--J c^ ii-\ 0 OS ! o\ , • •J-. - 00 °ll 0 C 0 \o -+ - U-\ r<-) N 0 0 so 0 so ca .^0 r^ 00 ^ ■-0 d 0 so en ■^ <^ ^y-^ C\ § "ON^ - ro ir, q ? 1 ro ro 0 - OsCJ °ll 0 0 0 «u-> r^ " 0 0 'l- ^ DO r-. ■j-> C^ r<-, r^ 0 ■J-ivO 0 M t^ '" SO '^i- N en c~^ >) 0 r^!0 -i- 0 "s ^J 'n 00 ~-+- ?^^ ■^ ! Tl- LO -i- « CsO 0 ' 0 0 0 II r^. 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M. « M 0\ _, o ^o -+ ^ M V.'n' r^ r^ °0 "^ "'^ 1+ II o If 0 Th -^- fi - *^ N f^ e^. r-. .^ o^ 0 ^ y O p; N - 00 0 o\ o - o\ O Th ■^ if II o If O to ro J^ 1^ rv, t-1 o po ci ^o":^ "pj O * On O " on O m H- . 1+ If 0 11 Q O o oo ^ O 00 oo vO O M t^ N r«-i 'J °'":^ 6 N 0 If If i^ -h II o o O CO sC Tt-\0 NO 00 rv. c^O t(- f^ t^ O - t^ CD 0 O O M M r^ =a -""i;; ■*- - t^O 00 o - o\ Vf 11 O ^ . , . • . ■ • ■ ■ , o < -Q-O "b 1 1 1 ' , • ' ' • ■ voui o , • . • . 1 + 1+ II II II ^ 1 c o ,1 2o '-5.2 o 2 ' ■ , e i f= ?. s s =^ ^^S •S 3 , w -^ 5 jj = y o 1 ^ C SI ■Z -° K S .B " ^ 'ot ?> s Ml C cH^is "m ° OX) b/1 ... . «.i . *: M OOO C.SrJOO o« 'cn'^ ^ 1 U-, N — .^ r^ r^ - ON O N 1-1 o * •^ « On 0 O H N - ON " :2 ^^r^ o If If lO ^ II o sO *^ a\ OT oT M .^^ O u-» u-^^o C\ =b'- o\ !o«r l^ — « m t*^ rv-i o 0\ ..>» o o ON — d\ d O N « '-' Ch - o ^ oo If If II O o rs. r^ L/^ o o « OS *^ ^ ^o-V; 5^ -> OnO O - - « ON If" 0 On If ri- CO II o NO O ON CO s^^^? ON CO N N NC -<- O NO "" _ tOl^ o 00 On O K -""S "o - CO 0 O - - - On O CO o If If o ^ Tj- m II K o ^ 'o "n CO - Th NO r-l^ -hco tj^ r^ p4 O - - O r^ 00 00 O "^ 2" O - 00 o O "- - ^ On ONO O O J, If If ' . . ' . II o o J= I + If II II 11 o , 1 o 3 £ ° I w C ■ . s ^ £ " ' , nj m K j= 3 .2 . . ::-s 5 oj i S ° •£ 2 li'i des Equations numeriques de tous les degres , par J. L. Lagrakoe ; Lecons » du meme auteur aux Ecoles normalesi >'. A PARIS, Chez CoURCiER, Iraprimeur-Libraire pour les Matheraatiqiies, quai des Augustins, n° Sy. ANNEE 1807. A L'EMPEREUR ET ROI. Sire, TanDIS que les Muses qui president h la Po^sie et k I'Eloqueace s'empresseut , a I'envi , d'offrir leurs tributs k VoTRE M\JESTE, la Muse des Hautes Sciences pourroit-elle demeurer en retard ? Les Sciences et les Arls dolvent surtout I'hommage de leurs d^couvertes k un Prince qui joint au pouvoir de les proteger , I'avan- tage d'etre , par ses vastes connoissances , un juste appreciateur de leurs progres. Vous le savez , SIRE , les inventions dans I'Analyse algebrique sont des plienomenes assez rares. Peut-etre aussi VoTRE Majeste jugera- t-elle que la Methode que j'ai eu le bonheur de decouvrir , n'est pas sans quelque utility. Quels resultats , en efFet , n'a-t-on pas droit d'en attendre dans ces recherches physico-mathe- matiques , ou Ton est conduit k des equations d'un degre tant soit peu eleve, qui jusqu'a ce jour d^concertoient les plus savans calculateurs , et dont la resolution , par la nouvelle Methode , sera d^sormais I'ouvrage des arithmeticiens les moins verses dans les profondeurs de la Science. Par ce double motif, j'ose esp^rer que V. M. dai- gnera me permettre de Lui dedier ce produit d'une longue meditation. Je suis , avec le plus profond respect , SIRE, DE VOTRE MAJESTE Le tres-humble , tres-obeissant et tres-fidele Sujet , F. D. SUDAN, D. M. P. AYANT-PROPOS. v^ET Ouvrr.ge tralte d'une mafi^re sur laqnelle se sont exerc^s les plus c^lebres Analystes , depuis Viete jusqu'a M. Lagrange ; c'est-i-dire , depuis le premier age de I'AIgebre jusqu'a nos jours. Avant les Merits de M. Lagrange sur la rt^solution des equations num^riques , les travaux multiplies de ses pr^de- cesseurs n'avoient abouti qu'a des m^thodes incertaines , et rebutantes dans la pratique. Celle qu'il a publi^e est exempte d'incertitude , mais on convient g^n^ralement que la pratique en est encore assez rebutante. Elle ne permet- troit certainement pas de remplir le voeu de cet illustre G^om^tre , qui voudroit qu'on enseignat , dans lArithm^tique m^me , les regies de la resolution des equations num^riques. C'est done pour noua conformer a son desir que nous avons cherche une methode d'une th^orie plus simple , qui fut en meme temps sure et vraiment usuelle , susceptible , en un mot , d'etre pratiqu^e par les commen^ans eux-raemes. Cette methode simple et facile , nous sommes parvenus a la d^couvrir ; et nous avons ainsi couronn6 assez heureuse- znent , ce semble , les travaux de deux si^cles sur cet objet. II a paru conyenable de presenter d'abord une histoire abr^g^e de ces travaux : on pourra , d'apr^s cette notice , juger de I'importance attachee par les plus grands G^ometres, au probleme de la resolution des Equations numeriques. Nous donnons ensuite un algorithme qui fait trouver , par de simples additions et soustractions , tous les termea des transformees en ( x — i ) , {x — a) , etc. , dune Equation donnee en x. Cet algorithme a re9u , le 23 mai i8o3 , I'appro- bation de la premiere Classe de I'lnstitut. Puis , apr^s avoir rappel^ diverses notions fournies par I'Algebre , concernant les equations numt^riques , nous ex- posons successivement les trois parties dont se compose la nouvelle Methode. Nous faisons voir quels sont les cas dans lesquels la premiere partie suffit toute seule a la resolution de I'equation ; quels sont ceux dans lesquels il faut joindre la seconde a la premiere ; et dans quels cas , enfin , Ton est oblig6 de recourir a la troisieme pour d^- couvrir les limites des racines incommensurables. Cette derniere partie sert aussi a approclier , jusqu'a telle d^cimale qu'on voudra , de la valeur exacte des racines dont on a dijk des limites. Get ^crit est terming par des Notes contenant des details qui nous ont paru dune assez grande importance pour nous faire desirer qu'elles soient lues avec la meme attention que le corps de I'Ouvrage. La premiere partie de notre Methode a obienu , le 3i oc- tobre i8o3, 1'approbation Je la premiere Classe de llnstitut, qui a reconnu , dans ce nouveau precede , ii?ie Methode generale , directe et sure , pour r<5soudre une (Equation num^rique , dans les cas oii Ton sait que toutes ses racines sont r^elles. Des circonstances particuli^res ont empecli6 de presenter a cette meme Classe la suite de notre travail ; mais nous ne craignons pas d'avancer que les deux autres parties compl^tent I'ouvrage commence dans la premiere. ERRATA. Page 3o, lignes i , 3, i5 et iG , au lieu de 7, mettez 4- Ibid., ligne 17, au lieu de o, mettez 3. Jbid., ligne 18, au lieu de est — 1 , mettez est entre o et —i. Jbid., avant-derniere et derniere lignes, : au lieu de. . . i-j- q + i4 — 1 14-12+354-20 , mettez 1 + 9+20 — 1 1 + 12+41 + 29. Page 3S , lig. 4 ^n remontant, au lieu de permanence, mettes. permanences . Page 09, lig. i3, au lieu de +6 — 12; mettez +6x— 12, Jbid. , lig. 19, au lieu de 32, mettez 24. Page /^4> I'S"^ ^' '^^ ''^" *^^ — °' mettez =0. Page 45, ligne 4> ^^ ''^^i ^^ ^' ' '"'^""^ •^'• //!»/£/., ligne 14, a la fin, au lieu de 1, mettez 8. Page 46, ligne 2, au lieu de iGiooo, mettez iGoooo. Page 4? J ligne i5, au lieu de nest pas , mettez est. Ibid. , lisne 3 en remontant , au lieu de — , mettez a:'". ' ° 10 Page 48, lisne q , au lieu de -, mettez -; ,„ ,,. Page 54 , ligne 7 , au lieu de valeurs , mettez racines. Page 56, ligne 4> ^^ I'^u de entre et 1 , mettez entre zero et it Ibid. , ligne 1 o , apres prendre la somme , mettez n'^'"'. Pase 65, liane 8, au lieu de — , mettez — , Ibid., ligne 17, au lieu de p" , mettez pW. Page 66, ligne 2 , avant Am—jX, mettez +. Page 72, ligne i5, au lieu de x^ , mettez a.-\ Page 84 > ligne 6, au lieu de — , mettez — . NOUVELLE MfiTHODE POUR LA RESOLUTION DES EQUATIONS NUMERIQUES DUN degr:e quelconque. GHAPITRE PREMIER. Histoire ahregee des travaux entreprls sur cette maiierc -pendant les deux deniiers sieclcs. I. LjE probleme de la E.^solution des Equations nume« riques pent etre regard^ , suivant Tillustre successeur d'Euler , comme le point le plus important de toute I'Analyse. La raison qu'il en donne est que la solution de tout probleme determine conduit a une ou plusieurs Equations numeriques , c'est-a-dire , dont les coefEciens sont donnas en nombres ; que tout le calcul qu'on a fait est en pure perte, si Ton n'a pas les moyens de resoudre ces equations ', que des le troisi^me degr6 r expression algebrique des racines est insuffisante pour I ( ^ ) faire connoitre, dans tous les cas, leur valeur numd- rique j qu'a plus forte raison le seroit-elle, si on parve- noit enfin a I'obtenir pour les Equations des degr^s sup^rieurs • et qu'on seroit toujoiirs forcd de recourir k d'autres raoyens pour determiner, en nombres , les valeurs des racines d'une equation donnee j determina- tion qui est, en dernier r^sultat, I'objet de tous les pro- blemes que le besoin ou la curiosite ofFrent k resoudre. [Seances des £ coles JSoivnales, torn. 3, p. 463, 476.] 2. Independamment d'une autorite aussi grave sur ce point, I'importance de ce probleme est assez d(^montree par les efforts multiplies d'un grand nombre d'analystes celebres des xvii^ et XViir sifecles , pour obtenir une m^thode gen^rale , directe et sure , propre a faire d^- couvrir toutes les racines reelles d'une equation nume- rique donnde. Nous allons presenter une l^gere esquisse des travaux de ces analystes , en prenant pour guide I'illustre auteur deja cite. 3. Vi^te qui, le premier, s'occupa de la resolution des equations numeriques d'un degr^ quelconque , y employa des operations analogues a celles qui servent k extraire les racines des nombres. Harriot, Ougtred, etc., ont essaye de faciliter la pratique de sa methode. « Mais » la multitude des operations qu'elle demande, et I'incer- » titude du succes dans un grand nombre de cas , I'ont » fait abandonner entierement, avantlafin duxvii'' si^cle». [ De la Resolution des Equations numeriques , par M. Lagrange , pag. 1 . ] 4. La methode de Newton a succ^d^ a celle de Vifete. Ce n'est proprement qu'une methode d'approximation j (3 ) qui suppose qu'on connoit deji la raclne clierclide , k une quantite pr^s , moindre que le dixieme de cette racine. c< Elle ne scrt , comma on voit , que pour les » equations numeriques qui sont deja a-peu-pres resolues ; » de plus , elle n'est pas toujours sure ; elle a encore Tin- M convenient de ne donner que des valeurs approcliees » des racines memes qui peuvent etre exprim^es exacte- » ment en nombres , et de laisser en doute si elles sont » commensurables ou non ». [ De la Kesolutioii des Equations yiinneriques , p. 3. ] 5. La methode que Daniel Bernoulli a deduite de la consideration des series recurrentes, et qu'Euler a exposee dans son Introduction a I'Analyse infinit^simale , n'offre aussi qu'un moyen d'approximation. « Cette methode » et celle de Ne%vton, quoique fondees sur des principes >3 difFerens, reviennent i-peu-pr^s au meme, dans le fond , » et donnent des resultats semblables ». [jDe la Resolu- tion etc. , p. i52. ] 6. Ce fut Hudde qui trouva qu'en multipliant chaque terme d'une equation donnee par I'exposant de I'in- connue , et en egalant le produit total a zero, on ob- tient une equation qui renferme les conditions de I'egalite des racines de la proposee. RoUe , de I'Academie des Sciences, decouvrit ensuite que les racines de I'^quation ainsi formee sont les limites de celles de I'equation proposee. Ce principe est la base de sa methode des Cascades , publiee d'abord sans demonstration , dans son Traite d'Algebre en 1690. Cette methode a cte ainsi nommee, parcequ'elle fait dependre la determina- tion des limites de chacune des racines de I'equation (4) proposee , de la rdsolutioa de difFerentes Equations snccessives , qui voiit toujours en baissant d'un degrd. « La longueuL' des calculs que cette methode demande, » et I'incertitude qui nait des racines iraaginaires , I'ont » fait abandonner depuis longtemps » [ De la Hesohc- tion, etc. p. i66 ]. Rolle , dans ce meme Traitd d'Al- gebre, assigne pour liuiite de la pkis grande valeur de I'inconnue, le plus grand coefficient negatif de I'ecpa- tion, augmente d'une unite ; le coefficient du premier terrae etant r. 7. La melhode de Stirling, pour determiner le nombre et les limites des racines r^elles du troisieme et du qua- trieme degr^, a €ii generalisee depuis par Euler, dans son Traite du Calcul differentiel. « Elle revient dans le » fond a celle de E.olle». {De la Resolution etc., p. 166.] 8. En 1747, le celebre Fontaine donna, sans de- monstration, une nouvelle methode. Je la donne, disoit-il , ■pour V analyse en entier , que Von cJierche si inutile- ment depuis Vorigine de VAlgehre. Cette methode sup- pose que Ton pent toujours, par la substitution des nombres i, 2, 3, etc., au lieu de I'inconnvie , dans les equations qu'elle emploie, trouver deux nombres qui donnent deux resultats de signes differens : k ce qui n'a » lieu , dit M. Lagrange , qu'autant que ces equations » ont des racines positives, dont la moindre difference » est plus grande c[ue I'unite -^{ou , pour parler plus » exactement , qiCautant qu^il y a de ces racines qui ne '> sent pas comprises ^ en nonibre pair, entre deux nombres » entiers consecutifs ). D'apres cette consideration , il est :■) facile de trouyer des exemples oii la methode de Eon- (5) » talne est en defaut». [De la Resolution etc., p. 162.] 9. Ce defaut avoit lieu egalement dans toute methode qui emploie les substitutions pour determiner les limites des racines reelles et inegales d'une equation num^- rique, lorsque M. Lagrange publia, dans les Memoires de I'Academie de Berlin j>our I'ann^e 1767, un nou- veau procede , le seul jusqu'ici qui ait offert un moyen direct et sur d'obtenir cette determination. Son Memoire contenoit aussi une methode pour approcher, autant qu'on veut et en employant I'expression la plus simple, de la valeur exacte d'vine racine , lorsque Ton connoit le plus grand nombre entier compris dans cette valeur. Le procede du a M. Lagrange, consiste a substituer successiveraent, a la place de I'inconnue, dans requatioii debarrassee des rucines egales qu'elle pent avoir , les termes d'une progression arithmetique o, D, 2_D, 3Z), etc., dont la difference D soit moindre que la plus petite difference des diverses racines de cette equation. La grande difficulte etoit de trouver cc nombre D : le genie fecond de TiUustre geometre lui fournit trois manieres d'y parvenir. 10. La premiere, qu'il proposa en 1767, exige le calcul de I'equation qui a pour racines les differences entre les racines de I'equation proposee. ffMais, dit » M. Lagrange, pour pen que le degre de I'equation » proposee soit eleve , celui de I'equation des differences » monte si haut , cpi'on est effraye de la longueur du » calcul necessaire pour trouver la valeur de tons les )) termes de cette Equation 5 puisque le degre de la pro- » pos(?e etant ?w, on a — ^ coefficiens a calculer. » [ Par exemple , pour une equation du dixierne degre f » la transfonnee seroi't du quarante-cinquierne ]. » Comrne cet inconvenient pouvoit rendre la me- « tliode generale presque impraticable dans les degres » un peu eleves, je me suis longtemps occupe des » moyens de I'afFranchir de la recherche de I'equation » des differences , et j'ai reconnu en efFet que , sans cal- » culer entierement cette equation , on pouvoit nean- » moins (rouver une limit e moindre que la plus petite >3 de ses racines j ce qui est le but jjrincipal du calcul de » cette meme equation ». [Z)e la BJsolution etc., p. 124.] 11. La seconde maniere de trouver le nombre ID est consignee dans les lecons que I'auteur donna aux Ecoles Normales , en 1796. Ella demande le calcul d'une Equa- tion du meme degrE que la proposee, ajant pour ses racines les differentes valeurs dont est susceptible le coefficient Y de I'avant-dernier terme d'une equation en (^ — «) ; a etant une racine reelle quelconque de la propoJee, dont x est I'inconnue. « Mais cette equation » en Y , dit M. Lagrange, pent encore etre fort longue » h. calculer , soit qu'on la deduise de I'elimination , soit » qu'on veuille la chercher directement par la nature » meme de ses racines ». \De la Resolution etc., p. 127.] 12. Ce coefficient I^etant une fonction de .r, I'auteur a fait depuis reflexion qu'on pouvoit toujours Eliminer I'inconnue x du produit du polynome Y , multlplie par un polynome ^ k coefficiens indetermines , procedant suivant les puissances 7«— i , m — 2, etc., de x ', en ( 7 ) ^ fai'sant disparoitre du produit Y^, au fiioyen de la pro- pos^e, toutes les puissances desc plus hautes que x'^~~^ , puis egalant a o chacun des multiplicateurs de a-, ce qui donne la valeur des coefEciens indetermines de ^, et reduit le produit Y^ a son terme tout connu repre- sentd par K, d'oii Y=—. Par suite de ces operations, les coefficiens de I'equation inverse de celle aux dilFt^- rences , qui etoient divises par Y, ne sont plus afFectes que d'un diviseur indi^pendant de x, et la recherche de D en devient moins penible. Ce troisieme precede , publie en 1798 , est moins rebutant que les deux autres j neanmoins son auteur reconnoit qyCilpeut entrainer dans des calculs assez loyigs. [De la Resolutioji etc., p. 228.] 1 3, K Le nombre D [ troiwedhme de ces trois manieres^ >i pourra etre souveut bcaucovip plus petit qu'il ne seroit » necessaire pour faire decouvrir toutes les racines ; mais, » dit M. Lagrange , iln y a a cela d'autre inconvenient que » d'augraenter le nombre des substitutions successives a » faire pour x dans la proposde » [ Seances des Ecoles Normales, tome 3, p. 466]. Get inconvenient paroit encore assez grave dans la pratique, car il peut , en cer- tains cas, donner lieu a des miUiers, et meme a uu nombre indefiniment plus grand, d'operations superflues. Du reste , I'auteur la consid^rablement diminue , en donnant le raoyen d'operer , par de simples additions et soustractions , les substitutions de nombres entiers qui suivent celles des ?/i premiers nombres i , 2 , 3 , etc. , dans une equation du degre m. 14. II semble done que la ra^thode de la limite de la ( 8 )^ plus petite difference des racines, qui d'ailleurs porte rempreinte du genie de son immortel auteur, ne reponde pas, en tout point, a I'objet qu'il s'est projjose, qui est de « determiner les premieres valeurs a substituer » pour X , desorte que , d'un cote , 07i nefasse pas trvp » de tdtonnemens inuiHes , et que , de I'autre , on soit >3 assur^ de decouvrir , par ce mojen , toutes les racines » reelles de I'equation » [Seances des Ecoles JSomiales , tome 3 , p. 477 ]. Nous ferons voir dans les chapitres suivans qu'on pent, a beaucoup moins de frais et sans recourir a cette longue et penible recherche de la limite de la moindre difference des racines, se procurer tou- jours cette assurance. 1 5. En outre , le desir du celebre auteur ^tant que les regies de la resolution des equations num^riques soient donn^es dans I'arithmetique , sauf k renvojer a i'algebre les demonstrations qui dependent de cette der- niere science , ne peut-on pas dire que ce vceu ne se trouve point rempli par une methode dont la th^orie esc trop corapliquee, et la pratique trop difficile pour des commencans? 1 6. II restoit done encore a glaner dans ce meme champ ou M. Lagrange a recueilli line si abondante moisson. Nous avons cherch^ a realiser sonprojet, en de- couvrant une methode nouvelle d'une ih^orie simple et d'une application facile. Nouspresentonsaux jeunes eleves im aliment de facile digestion, dont peut-etre ils nous sauront quelque gre. Nous n'osons nous flatter d'obtenir le meme accueil des personnages consommes dans la science: suivant un ancien adage , les mouches ne sont (9) point la pature des aigles, aquila non capit mitscas. On voudra biea cependant observer que les mcthodes des an- ciens , lesquelles supposoient un grand travail, une grande force de tete, out cede la place, dans I'enseignement, h. des methodes modernes plus a la portee du vulgaire ; nous esperons que cette consideration preservera d'un superbe dedain les proced^s aussi faciles a pratiquer qu'a concevoir, que nous ofFrons en ce moment au public. 17. A cette consideration il en faut joindre une autre, tiree du besoin que Ton a d'une metliode qui soit pra- ticable et vraiment usuelle pour la resolution des Equa- tions numeriques, si Ton veut que I'algebre puisse s'appli- quer convenableraent aux arts et aux besoins de la soci^te. Nous rappellerons, a ce sujet, ce que disoit I'academicien RoUe , lorsqu'il publia sa metliode des Cascades. ccLors- » qu'on a envisage toutes les conditions qui sont neces- » saires pour le succes d'une entreprise, on pourroit sou- » vent s'aider de Talg^bre pour y reussir ou pour en x connoitre rimpossibilitE ; mais on aime mieux chercher » d'autres conditions , ou tenter I'execution par difFerens » moyens, que d'avoir recours a cette science, et, encela, » on a eu cjuelque raison ; car si I'on veut se servir de » I'algebre dans I'invention d'une machine ou pour quel- » qu'autre recherche , en n'employant d'ailleurs que les » experiences des physiciens et lesprincipes des g^ometres, » on arrivera a des egalites (equations) irrationnelles d'un » degre fort eleve , et il est plus difficile d'dviter ces ega- » lites dans cette application , que d'eviter les fractions » quand on pratique I'arpentage. Cependant les regies 5> quona donnees jusqu'ici pour resoudrc ces Egalites , ne 2 ( 10 ) ^ » sont ni scieiitlfiques nf g^nerales, et il sufEt de les eprou- » ver ponr en etre rebute ». On a aiijourd'hui , k la veiite^ des regies scientifiques et generales j n\^\% quel est celui qui, les ayant essaj^es, pourra dire qu'elles ne sont pas rebutantes ? 1 8. Si dans cette esquisse des travaux de deux siecles ^ concernant la resolution des Equations num^riques , Fim- mortel Descartes semble avoir ^t^ oublie, c'est que nous nous somraes rdserv^ d'en parler ailleurs. Comment au- rions-nous pu oublier sa fameuse regie des variations et des permanences de signes, publi^e pour la premiere fois en 1637, et qui, longtemps ndgligee, revolt dans notre m^thode une application nouvelle, et, en quelque sorte^ une nouvelle existence ? ( II ) GHAPITRE ir. PeobL]6me preliminaire : Etant donnee line equation numeriqueen x d'lm degre quelconque ^ troiwer ^ par de sifnples additions et soustractions , les coefficient de sa transformee e?i (x — i ) ,- et generalement , de sa transformee e/z ( x — n ) , n etant un nomhre entier ou decimal. 19. ixVANT que de donner la solution de ce probleme, nous expliquerons ce qu'il faut entendre par les sommes premieres , secondes , troisiemes , etc. , d'une suite de termes. Lorsqu'une suite de termes quelconques etant donnee, on forme une autre suite somrnatoii'e de la premiere , c'est-^-dire , qui a pour loi que son /z'^'"^ terme soit la somme des n premiers termes de la suite donnee , cela s'appelle prendre les $ommes premieres , ou simplemeut les sommes de la premiere suite. Ce mot somme doit s'entendre dans le sens alg^brique ; il exprime Texcedant de la somme des termes prdc^des d'un des signes -H ou — sur celle des termes precedes du signe contraire. Prendre ensuite les sommes de ces sommes premieres, cela s'appelle prendre les sommes-secondes de la suite ( lO donnee. De meme , les sommes de ces sommes-secondes s'appellent les sommes-troisiemes de la premiere suite , et ainsi du reste. Voicf un exemple de ces diverses sommes : Suite donn^e i . . . i . . . i . . . i . . . i. Sommes-pi-emieres i . . . 2 . . . 3 . . . 4 . . . 5. Sommes-secondes 1 . . .3. . . 6. . .10. . .i5. Sommes-troisiemes i. . .4. . .10. . .20. . .35, etc. etc. Les suites dont on s'est servi dans ce premier exemple^ appartiennent a celles des nombres que les Geometres appellent nombres figures , lesquelles out generalement pour i^r terme , I'unite ; pour 2^ terme , un nombre entier /?? ; et pour terine tz'^'"^, un nombi-e exprim^ par I. a (« — i) Autre exemple, dans lequel la suite donnee est com- posee de termes pris arbitrairement, les mis positifs, les autres n^gatifs : Suite donnde 2-+-" 5 — 3-|- 4 — 3-1- o — i, Sommes-premi^res . . 24- 7-+- 4-1- 8-h 5~\- 5-\- 4. Sommes-secondes. . .2-+- 9-hi3-|-2i-|-26-h 3iH- 35. Sommes-troisiemes. . 24-1 i-r-24-f-45-}-7H-io2-t-i37. etc. etc. 20. Voici maintenant deux propositions d'ou rdsulte la solution demand^e. (Pour leur demonstration j i^ojez ci-apres /e,? NoT E S. } ( i3 ) Pt'emLt;7''e proposition. La somme w'^'"'* des n premiers termes d'une suite quelconque, dgale la somme de ces termes multiplies respectivement , mais en ordre inverse , par les n premiers nombres figures de I'ordre m , c'est-a- dire , apparterjaot a la suite dout le second terme est m. Ainsi la somme m'^'"^ des 7i premiers termes de cette suite est egale &.... ii,_.-h ?/^^_^- ^m....(im-\-n--^) ^ Par exemple , la somme-troisieme de ces quatre termes 2+5— 3 + 4 est (i X4 — 3x3-f-6x5-i-ioX2)=45, de meme qu'on I'a reconnu plus haut en prenant leS sommes et les sommes de sommes. Seconde proposition. Un polynome quelconque , pro- c^dant suivant les puissances entieres et positives d'une quantite x , depuis le degre in jusqu'au degre z6-o , se transforme en un autre polynome d'egale valeur , proce- dant suivant les memes puissances de (.r — i } , dont les coefficiens respectifs , h. commencer par celui du dernier terme , sont , 1°. La somnie-premiere de tous les coefficiens du po- lynome donne. 2.°. I^a somme-seconde de tous les coefficiens , horm.is le dernier. 3°. La somme-troisieme de ces coefficiens , excepte les deux derniers. Et ainsi de suite. ( 14) Soit , par exemple , ce polynome Coefficlens donnes ...2 — 3^-5 — 3. Sommes-premiferes . . . 2 — 14-44-1. Sommes-secondes. . . . 2-4-i-}-5. . . Sommes-troisiemes ... 2 -f- 3. ... , Sommes-quatriemes. . . 2 r Ainsi les coefEciens du polynome en (.r — i)sont..,.. 24-34-5-1-1 ; et Ton a a (;i; — I )^ 4. 3(a;— I )»+5(j:— 1) 4- I =2a;'--3.r»4- 5^— .3. Cette equation a lieu , quelque valeur qu'on donne a x. S'il manque dans le polynome propose quelque puis- sance de :r , il faut la mettre en evidence , en lui donnant z6ro pour coefficient. Soit, par exemple, x"^ — 7:1; 4- 7. Coefficiens donnas. . . . i 4- 0 — 74-7. 14-1 — 64- r. i4-2 — 4. i±3. . . I Sommes-premieres . Sommes-secondes . . Sommes-troisifemes . Sommes-quatriemes. On a done... ^3 — 7:r4-7=(:r — 1)3 4-3 {x — i)^ — 4(0; — 1)4- r. 21. II est Evident qu'une equation dont le premier membre est egal h zero , ofFre pr^cis^raent le meme cas que le polynome de la proposition precedente. Ainsi Talgorithme par lequel on oblient la transformee en (x — i) d'une Equation donnee en x, consiste dans le ( 15) meme procedd employ^ pour la transformation d'un po- lynome d'lme valeur quelconque. Etant done donn^e I'dquation.... ^3 y^^, _J_ r, = o ^ les coefficiens de sa transformde en (^ — i) sont..., I -H3 — 4-1- I. Pareillement , pour I'dquation.... x3 — 2X — 5 = o , les coeflSciens de sa transform^e en{x — i ) sont.... I -h 3 H- I — • 6. 22. Par le meme algorithme, on passera de la trans- form^e en {x — i) a celle en (a: — 2); de celle-ci a la transform^e en (x — 3); et ainsi de suite indefiniment. On obtiendra done trcs promptement les coefficiens de ces diverses Equations. Coefficiens des Equations , dans le i^exemple dun° 21... enx i-f-o — 7-+: 7 en(.r — i). . . i-h3 — 4-}- i en(:r — 2}. . . H-6-H 5-+- i en (.2; — 3). . . i -f-9-f-2o-i- i3 etc. etc. Coefficiens des Equations , dans le second exemple.... en(.r — i ). . . i-h3-h i — 6 ea(a; — a). . . i-i-6-i-io — i en (.r — 3). . , i-hp + ^S-t- 16 etc. etc. ( I6 ) 23. II est aise d'observer que par ces transformations, on finit par avoir des cgefficiens qui sont tous de meme signe. Obser^'ons aussi que si I'equation proposee n'est que du troisieme degi^e, on peut obtenir les coefficiens de ses transformees successives par uu moyen encore plus rapide que ralgoritlime general. Nos lecteurs le devine- ront sans peine a la simple inspection des coefficiens re- presentes dans le n° precedent. Dans ce cas , le calcul des transformees s'opere instantauement , sans avoir besoin d'ecrire d'autres cliiffres que ceux qu'on voit ici. 24. Le meme algorithme fonrnit le moyen d'obtenir les transformees en (vT — 10), (x — 20), ''sc — 3o},etc.5 celles en (x — 100), (a: — 200) , (x — 3oo), etc.5 etc. II faut , pour cela , substituer dans la proposee une inconnue x' qui soit, respectivement , dix fois, cent fois, etc. moindre que x. Les coefficiens de cette equation en X s'obtiennent , comme Ton sait , sans calcul , par le placement convenable de la virgule qui indique les de- cimales. On se procure ensuite les transformees en (:t-' — 1 ) , (^x — 2), {x' — 3) , etc.; ou, ce qui revient au meme, /X ION /X 20\ (X — 3o\ , . /jc — ioo\ /x — 20o\ fx — 3oo\ . , ' , r. ., ( ) , ( ) , ( ), etc. ; selon qu on a lait \ 100 / \ 100 / \ 100 y^ ' -^ x'= — , ou x'. = , etc. 10 ^ 100 ' II ne s'agit plus que de rendre les inconnues de ces transformees, respectivement , dix ou cent fois, etc. aussi C 17 ) grandes ; ce qui s'opere par le deplacement conVenable de la virgule dans leurs coeflGciens. Soit , par exemple , I'equation a;3 — 4^- -i- 3x — 6=0, dont ondemandeles transform^esea (x — lo), (oj— 2o),etc. On fera x — lox; d'ou.... ^'^ — o,^x'^ ■+■ o,o3a;'— 0,006 = o. Coefficiens des (Equations.... en x' I — 0,4+ o,o3 — 0,006 en ^ — ^^j ou (x' — I } . . . I + 2,6 -t;- 2,23 H- 0,624 ^" (~li~) °^ ^"^' — 2), . .i-h 5,6-}- 10,43 -f- 6,464 etc. , etc. Et par consequent on aura , pour les coeflaciens des Equations.... en (x — I o ) . . . I -+- 26 -h 223 -f- 634 en (x — 20). . . I -+- 56 -H 1043 -H 6454 etc. etc. On voit aisement comment , par une marche ana- logue, on se procureroit les transformees en (x — tt)? (x — -7^), etc. ; cellesen (x — 7^0 ) > (^ — r3-o)> etc. ; etc. 25. Si I'on veut avoir I'equation ou I'inconnue de la propos^e est diminuee d'un nombre de plusieurs chifFres, par exemple , I'equation en (a; — 3i2) ; on se procurera d'abord I'equation en (a/ — 3), en faisant.v=iooj,'; etpar suite, celle en (x — 3oo), comme il vient d'etre indique. Puis on fera x — 3oo=io:r'j on obtiendra I'equation en (x'— 1); et par suite , celle en {x — 3io}. De cette ( i8 ) . dernifere, on passera a celle en (^ — 3ii); et de celle-ci k Tequation demandee en {a: — 3 12). Voici, par exemple, la marche qu'il faudroit sviivrC; si la propos^e etait.... x^ — 4x^'-{'3x — 6=0. Soit ^=iooa;';d'ou :r'3 — 0,04a;'' H- 0,ooo3.y' — o,0000o6=0. Coefficiens des Equations en X I — 0,04 -t- 0,OOo3 — 0,000006 en (x' — i). . .i-f- 2,96-1- 2,9203 -f- 0,960294 en (.r' — 2). . .i-::i- 5,96 -H 11,8403-+- 7,840694 en (a;' — 3). . . i 4-8,96 -f- 26,7603 ■4-26,640894 Ainsi les coefficiens de requation en (a: — 3oo) sont.i.. I 4- 896 4- 267603 4- 26640894. Falsant ensuite x — 3oo=io:t', on a les coefficiens des Equations... en x' 14- 89,6 + 3676,03 -4- 26640,894 en ( x' — I ) ... I 4- 92,6 ■+■ 2858,23 -jr 29407,524. Or a^' — I = ; il s'ensuit qu'on aura les coeffi- ciens des Equations.... en (a: — 3io). . .1 4-9264-2858234-29407524 en (x — 3i I ).. .1 +9294-2876784-29694274 en {x — 3i2).. .1 + 932+2895394-29982882. II est ais^ de voir comment on obtiendroit I'^quation oul'inconnue de lapropos^e seroit diminu^e d'un nombre decimal de plusieurs chififres ; par exemple , I'equation en (x — — -) : nous ne nous arreterons point k cos details. ( 19 ) ^6. En consld^rant le tableau des operations par les- quelles on passe d'un polynome en x k son equivalent en (:r — i ) [ ao] , on n'aura pas de peine k rQconnoltre comment on pent passer r^ciproquement d'un polynome en (x — I ) k son Equivalent en x ; et parcons^quent , de celui en x h son Equivalent en (x-i- i) , et ainsi de suite. Dans le premier cas , on a pris des sommes ,• dans le second, onprend des differences. Choisissons pour exemple, le polynome en x du n® 20, dont les coefficiens sont.... et son Equivalent en {x — i), qui a pour coefficiens.... 2-h3-t-5-f- I. Pour passer de celui-ci a I'autre , on Ecrit ses coefficiens et on procede comme il suit : Coefficiens du polynome en(:r— i)...2H-34-5«-f-i Suites dans cliacunedesquellesle i i^'^. . . .2 4- i -f- 4 — 5 jjiime terme est la difference du 1 ai^me, . .2 — i -f- 5. . . . terme qui le precede au tertue ^ S'*'^^. . .?, — 3 • jiieme x''- — 0^461 = 0; la limite en plus de x etant 1,0451 , celle de x est 10,45 r ; et cette derniere est bien plus resserree que 462, limite indiquee par le plus grand coefficient negatif de I'equa- tion en x. Cette limite plus resserree peut se reconnoitre a la seule vue de la proposee,par une simple operation mentale. Le terme tout connu de I'equation etant divise par la somme de ce terme et du plus grand coefficient de signe contraire, prise sans egard pour les signes^ le quotient est plus petit que la plus petite racine positive que I'equation puisse avoir j il en est une limite. 36. L'Algebre fournit le mojen de preparer une equa- tion, de maniere que son premier terrae n'ait d'autre C ^5 } coefficient que I'unit^, et que les autres coefficiens Soient tous des nombres enti'ers. II en rdsulte que les dquations k resoudre peuvent toutes etre consid^r^es comme rame- n^es a cette forme. L'^quation ainsi pr^paree ne peut avoir pour racines r^elles que des nombres entiers ou des nombres fraction- naires irrationnels. En general, ces racines irrationnelles ne sont susceptibles d'etre determindes que par approxi- mation. 37. L'Algebre donne aussi le moyen de d^barrasser une Equation des racines egales qu'elle peut avoir , ensorte que les racines multiples n'y subsistent plus que comme racines simples. Ainsi les equations a rdsoudre peuvent etre con- sider^es comme u'ayant que des racines in^gales. 38. Une equation ne peut avoir plus de racines reelles positives , qu'il n'y a de variations dans la succession des signes de ses coefRciens ; ni plus de racines reelles nega- tives , qvi'il ne s'y trouve de permanences de signes : telle est la fameuse regie de Descartes. Ainsi , dans le cas ou toutes les racines de I'equation sontrdelles , il y a precis^ment autant de racines positives que de variations de signe , et avitant de racines negatives que de permanences. Quand un des coefficiens de I'equation est z^ro, et que les coefficiens du terme precedent et du suivant sont de meme signe , I'equation a necessairement des racines ima- ginaires. On peut reconnoitre si une equation a toutes ses racines reelles ou non , au moyen de I'equation dont les racines 4 (26^ sont les qnarr^s dcs differences des racines de la propo- sde. Dans le premier cas , cette Equation aux quarrds des diif(^reuces n'a que des variations de signe ; tandis qu'elle a necessairement des permanences, si ]a proposee a des racines imaginaires. Mais le calcul des coefficiens de cette Equation est en general tellement penible, qu'on n'est gueres tente d'employer ce moyen. 89. On pent deduire de la regie de Descartes , les deux propositions suivantes : 1°. Une equation en a: , dont toutes les racines sont reelles , a autant de racines comprises entre z^ro et ^ , qu'il y a de permanences de signe dans la transformee en- {x — p) , de plus que dans I'equation en ^. 2°. Une Equation de cette esp^ce ne pent avoir , soit une, soit deux , soit 7i racines comprises entre zero et^ , si sa transformee en(.r — p) n'a pas, respectivement, une, ou deux , ou 71 permanences de signe, de plus que I'equa- tion en x. Nous avons meme de fortes raisons de croire que la seconde proposition est applicable h une equation quel- conque. (^7) CHAPITRE IV. Exposition de la noiwelle Methode. Premiere Pariie. Cas oil Von ?ia hesoin que de cette partie de la Methode. 40. 1\| OUS allons malntenatit exposer successlvement les divers procddes qui constituent notre Mdthode , en ren- voyaat aux nos du cliapitre precedent , ou sont contenus les principes qui servent de base aux r^sultats que Ton obtient par ces procddes. Pour concevoir le rapport des uns aux autres , il suffit au lecteur qui ne seroit jDoint assez avance dans I'Algebre , de tenir les principes pour d^montres , sans chercher k en connoitre la demonstra- tion ; et s'il ne veut que poss^der le mecanisme de la Methode, il n'a besoin que de savoir op^rer les ti-ans- formatioas , conformdment a I'algorithme du second cliapitre. 41, Etant done donnee une Equation en x du degre m , on se procurera ses transformees successives en {x — i ) , {x — 2), (a; — 3), et ainsi de suite, jusqu'a ce qu'on parvienne a une transformee en (a; — u) , dont les coeffi- ciens soient tous de meme signe. Cette derniere transformde ne pouvant point avoir de racine positive, le nombre entier u est une limite de la plus grande valeur positive de rinconnue. (28) S'il arrive que la proposee elle-meme n'oifre que des permanences de sigae , il ne reste a cherclier que les i-acines negatives qu'elle peut avoir , et on procedera comrae il sera dit plus has [44]. 42. Lorsque le dernier coefficient d'une equation qui a pour inconnue {x — p) , est egal a zero , I'equation en x a une racine ^gale au nombre p ; et plus generalement , si n coefficiens consecutifs de la transformee , a compter du dernier , sont egaux cliacun a zero , la proposee a 71 racines ^gales, chacuue^ kp [34]. Par cettecirconstance, I'equation en ( x — p ) se trouve abaiss^e de n degres, i- A ralson de cet abaissement^ il peut y avoir quelque avantage a ne debarrasser I'equation de ses racines egales, qu'apres avoir opere les transfoi-mations du n° 41. -' f I! - 43. Lorsque le dernier coefficient d'une Equation en (x — p) est de signe contraire a celui de la transformde en (x — p — I ) , la proposee a une on plusieurs racines en nombre impair , dont la valeur est comprise entre p et ^; -h I. Car les coefficiens dont il s'agit , expriment preci- sement les resultats que doune la proposee, quand on y met successivement ^ et p -{- 1 kla place de :i' [3i ].~ 44. Les racines negatives de la proposee e'tarit , au signe pres^ ^gales aux racines positives qu'auroit cette Equation si les sigues de ses termes pairs etoient tons changes, on fera ce cbangement , puis on op^reracommeci-dessusf4i] , et on dbtiendra des resultats analogues. ' : ' - 45. Par cette premiere partiede la M^thodej on trouve. ( 29 ) en certains cas , foutes les racines reelles de I'equation, soit exactement, soit approximativement , a moins d'une unit^ pres. Un premier cas est celui ou la proposee n'a ni racines imaginaires , ni plusieurs racines reelles comprises entre deux nombres entiers/7 etp-\-i. Un second cas est celui ou Ton sait d'avance que toutes les racines de la proposee sont reelles, encore que parmi ces racines, il y en ait d'incommensurables comprises, en tel nombre que ce soit, entre deux nombres entiers consecutifs. Un troisieme cas a lieu , lorsqu'on sait que Te'quation n'a qu'une racine reelle , positive ou negative, ou bien qu'elle en a deux , I'une positive , et I'autre negative , ainsi qu'il arrive dans des equations de cette forme , ^l.mip A = O. 46. Premier exemple. Soit I'equation.... ^~4 — 10^3 4- 2)6x^ — 54:f-H 27 — o. Coefficiens des Equations.... enx I — 10-4-36 — 64 -f- 27 en (.r — i}.,.i — 6-f-i2 — 8-f- o en(.r~2)...i— . 3+ 3— i en (a.- — S)...!-^ o-f- o-f- o. Les racines de cette equation sont done i et 3 ; cette dcrniere racine est triple [42], c'esL-a-dire que la proposee est divisible par (.f — 3)3. (30) Second exemple.. .. x3 — 5x' + ^ + 7 = 0. Coefficiens des equations en X. . T — — 5+ H- 7 en (a:- -I). . . I — 2 6-i- / I en ix- -2). ..i-h I — 7 — 6 en (X- -3). ..i-h 4— 2 II en (x- -4). .,1-4- 7 + 9— 8 en i^- -5). . .i + io4- 26~i- 9- La proposee a done deux racines positives incommen- surables , dont les valeurs sont respectivement comprises entre i et 2, et entre 4 et 5. Pour avoir ensuite la racine negative , on change les signes des termes de rang pair dans la proposee [44] , et Ton a.... x3-|- 5x^-t-x — 7 = 0. Coefficiens des Equations en X = — X . . I -j- 5 + i — 7 en (x — i). . . I -h 8 H- 14+0. Done la racine negative de la proposee est — i. Troisierne exernple.... x^ — 70^-1-7 5=0. Coefficiens des equations.... en X i-f-o — 7 + 7 en {x — i). . . I -+-3 — 4 -H I en {x — 2). . . 1-^6 4-5-1-1. Coefficiens des Equations.... en x = — X. . . I — o — 7 — 7 en (x — I ). en (x — 2}. en (x — 3). en ( X — 4 }. i-f- 3— 4— i3 i-t- 6+ 5— - i3 I -t- 9 -h 14 — I I -h 12 4- 35 -h 23. ( 3i ) L'(?qiiatIou proposee etant de celles qu'on sait avoir toutes ses racines rdelles , il en resulte que non-seule- ment elle a une raciiie negative dont la valeur est entre — 3 et — 4 [43 ] , mais aussi qu'elle a deux autres racines positives comprise entre i et 2 , parceque la transform^e en (^ — 2) a deux permanences de signe de plus que celle en (x — i) [Sg]. Telle est, dans ce cas , la conse- quence de la r^gle de Descartes. Quatrieme exeniple.... x'^ — I745 = o. Coefficiens des equations..., en X iH- o-f- o — 1745 en {x — i).-.i-f- 3-{- 3 — 1744 en (.r — 2).'--i-H 6-\- 12 — 1737 en (^x — 3) ..I-+- 9-H 27 — 1718 en (x — 4)---n-i2-f- 48 — 1681 en {x — 5)>- -i-hiS-t- 76 — 1620 en {x — 6}- • -i-H i8-|- 108 — iSag en {x — 7). .•H-21 + 147 — 1402 en {x — 8). •.!■+- 24 +192 — 1233 en {x — 9}.- -I -h27-{-243 — 1016 en (a;— io)--"i-f-3o-4-3oo — 746 en {x — 1 1).. -1-1-33-1- .363 — 414 en {x — i2)...iH-36-H432 — 17 en {x — i3_)-..i-f-39-f-5o7-{- 462. Done la racine de I'equation est entre 12 et i3. 47. Nous avons suivi dans ce dernier exemple, la marche la plus longue ; car il est aise de voir que x devant etre un nombre entier,exprim€par deux chiffres, ( 32 ) on pouvoit d'abord se procurer les transformees en (.V — lo), (x — 20), etc., par le proced^ indique plus haut [24], en faisant d'abord a: = 10^', ce qui changeoit I'equation en x'i — Ij745 = 0. — Coercions des Equations.... en :t' I -f- o -{- o — 1,745 eu { ^ J ou (jc' — I )...!-+- 3+ 3 — 0,745 en r^~~^ \ ou (x — 2).". i -i-6-f- 12-4-6,255 Et par consequent.... en (a; — lo}- • • i + 3o -f- 3oo — 746. en (a: — 20). • • i 4- 60 -4- i20o + 6255. Done la racine est entre lo et 20. II ne reste qu'^ se pro- curer les transformees successives apres celle en (x — 10}, jusqu'a celle en (^x — 19) tout au plus. Coefficiens des Equations.... en (x — io)---i-i-3o-4-3oo— 745 en (x— ri)...H-33H-363— 414 en (x — 12)... I -f- 36-4- 482 — 17 en (x — 1 3).. -I -J- 39-^-607-4- 452. Et I'on conclura , comme plus haut , que la racine 3'i^nie ou cubique de 1746 est entre 12 et i3. La nouvelle Methode offre done un moyen d'ex- traire , par des additions et soustractions , la racine YYiieme ^ cxactc OU approchec , d'un nombre quelconque. Si Ton veut comparer cette methode avec les anciens ( 33 ) precedes , nous lalssons a juger lequel des deux moyeus inerite la preference. 48. Le precede que nous avons employe dans le n° precedent , n'est pas applicable seulement aux equa- tions ci deux termes ; on pent aussi I'employer dans une Equation quelconque , toutes les fois qu'on aura sujet de penser , d'apres I'examen des coefficiens de la pro- posee , que le plus grand nombre entier , faisant partie de la plus grande racine positive , pent elre exprime par plusieurs chiffres. Dans ce cas , il pourra etre plus con- venable de faire x = lox', ou x = 100.1', etc., et de se procurer d'abord les transformees en ( .r — lo) , (a: — 20 } , etc. ; ou en (^x — 100) , {x — 200) , etc, etc, ; ou bien encore , de resoudre I'equation en x' , a I'aide des transformees successives en ( x' — i), (a;' — 2}, etc. j puis d'en deduire les valeurs de x. Ces remarques detruiront sans doute cette objection que I'irreflexion pourroit opposer a notre Methode; savoir : -;\_ ■ — i ne peut elre positive, \ nioins que le d^nominateur f^ -}- (p ne soit plus petit quey- ce qui n'a lieu qu'autant que^ et

^ —

^i|ucnt (j: — 2) ne pent avoir plus de deux valeurs eaire o et i , ni >c' plus de deux valeurs entre o et 10: les transforuiees successives faisant ici connaitre ces deux va- leurs , 11 est inutile de calculer ies coUateraies. Pour la verificatiou des racines negatives qui peuvent etre comprises eiiLre o ct i , on fera 10 x = x', et par les transformees successives en ( x' — ij,(x' — 2)jetc.j oa trouvera deux valeurs pour x', comprises respectivement entre 4 et 5, et entre 7 et 8. D'ouil suit que les racines negatives de x soat — 0,4 et — 0,7 , a moins d'un dixieme pres. 64. Les Equations en (.r' — p'J ct en (z'^^ — i ) peuvent n'etre pas suffisantes pour determiner Tadmission ou le rejet de la totalite des racines douteuses. Alors on a recburs aux equations ea (a;'— p'jet eu (z'^ — i), qu'on obtient (47) en faisant lo (^x' — p') = x', , , = z'p", et en proce- dant comme on a fait ci-dessus pour les equations en (.t-' — p') et en (z'f — i ). Par ce moyen on approche , jusqu'a la seconde d^ci- male inclusivement , des racines dont Texistence est deja reconnue^ en raeme temps que Ton decouvre les racines reelles jusques-la douteuses, qui ^tant comprises entre Cp-\~-^\etfp-^- V n'ont point pour communes limites (p^^^-B!-) et fn 4-^4- £l±i.V les diffe- V 10 100/ V^ lo 100 /' rences de ces racines entr'elles pouvant d'ailleurs etre indefiniment moindres que -r~-. On detruit aussi , par cemerae moyen, le soupcon qui auroit et^ maintenu dans I'equation en (x' — p") par les imaginaires y + 1/' — 100 jaloux de la gloire que la France s'est acquise dans les s Math«5mathiques, vieut renouveler cette accusation ridicule, » on est en droit de ne le point croire , puisqu'il parle sans y> preuves. M. Hiidde , holiandois, qui n'est point suspect, puis* » qu'il n'avoit aucun interet a soutenir I'honneur des auteurs » Francois, est bien plus equitable dans le jugeraent qu'il porte > de M. Descartes ». C52) Sur le CHAPITRE II. CQ J_j E s deux propositions, dont depend rAlgorithme dix cliapitre second , nous avoient parii nouvelles. Mais nous ne devons pas taire que M. Legendre , dans son rapport sur nne partie de notre travail, en a jnge auirement. Suivant lui, « ces » deux theoremes cpie I'Auteur regarde oomrne nouveaux, ne » sont que Penonce de proprietes dtja conniies , relatives a » la soramation des suites, et ce qui lui apparlient se reduit y> a I'AIgorithrae propre a operer les transformations ». Si Ton considere que le second de ces theoremes et I'AI- gorilhrae ne sont (pi'une seule et meme chose , on aura sans doute quelque peine a comprendre que I'un appartienne a I'Auteur, si Taufre ne liti appartient pas. Peut-eire le Rap- porteur se seroit-il exprime avec plus de ju'^tesse et deyV/;5//ce , S'il eut dit que ces deux propositions, jiisqn'ici inconnucs, sont des consequences si faciles a dc'duire des priucipes deja recus , qu'il petit paroitre etonnant tpi'on ne s'en soit pas avise plus lot. Peul-etre, du mojtis, auroit-il mieux valu cjue M. Le- gench'e , en niant la nouveaule de ces propositions , ne se fufc pas borne a cette simple negation , et qu'il eiit hien voulu iudicpier en quel ouvrage , elementaire ou non, eles se trouvent consignees. Quoi qu'il en soit , d'apres I'nnposante auiorile du savant Rapporteur , on concoit qu'il est inuiile de s'arreter ici a pronver des pruprielea conniies. (D) L'Algorilhme indiqu6 an n° 26, pourroit eire employ^ a la recherche directe des racines negatives-, raais il nous a paru p^us simple et plus commode de raraener cette recherche, comme on a coutnmv_> de faire , a celle des racines positives efc d'emplojer; a cat ejQetj iiotre Algorithme ordinaire. C 53) (F,) L'Algorithrae du second chapifre, en perdant un pen de sa simplicite , pent s'^tendie au calcul de la ti-ansformee en(j: — ']), d n'etant plus seulement une puissance de lo , mais un nombre entier quelconque. 11 faut , pour cela, faire a; = -, , et , pour avoir les coefficiens de I'equation en x , mul- tiplier respectivement ceux de Tequation en :i; , a compter de celui de Ji"", par d" , d\ d"",... d"'. Ensuite, par de simples additions et soustraclions, on se procure [a"'* 22 , 24, 25] la transformee en a' — /I, dont les coefficiens , a compter de celui de la plus haute puissance, respectivement divises par d" , d' , d" ,.,. d"' , deviennent ceux de I'equation en (x — ^ j. Par ce precede, le nombre des multiplications et divisions est diuiinue; autant qu'il se peut. ( 54) Sui- le CHAPITRE III. (F) vJn a vu [n°35] cnmraent on peut determiner une limite J en moins, de la plus petite valeur positive, et luia limite , en plus , de la plus grande valeur que puisse avoir I'inconnue d'une equation. Maisil estunereraarque qui n'a pas encore ete faite, c'est qn'on peut determiner deux iimites sem- blables pour les valeurs reelles qu'une equation peut avoir eutre zero et un •, et voici comment. Pour obtenir luie limite moindre que la plus petite racine , on use du nieme precede qu'au n° 55 j c'est-a-dire , on prend le quotient du dernier terme divise par la somme de ce nieme ferrae et du plus grand coefficient precede d'un signe con- traire : ce quotient donne necessairement une fraction pour 1 i mite de la plus petite racine positive. I. a limite de la plus grande racine qui puisse etre comprise entre zero et un , se decouvre a I'aide de la transformeeen {x — i), apres qu'ona change les signes de ses coefficiensde rang pair : ie plus grand coefficient de cette equation , ainsi modifiee , de signe contraire a celui de son dernier terme, ^tant divise par la somme de ce coefficient et du dernier terme , le quotient est une frac- tion dont la valeur surpasse celle de la plus grande racine que la proposee en x puisse avoir entre o et i. Cette fraction est le complement, a I'unile , de celle qui exprime la limite de la racine la plus voisine de zero , que I'equalion en (j: — i ) puisse avoir entre o et — i. Avec un pen de reflexion, on appercoit aise-. ment la raison de ceci. On jugera , par la suite de ces Notes , de quelle importance peut 6tre cette remarque. (55) Sur le CHAPITRE IV. (G) IT ARMi les cas snsceptibles d'etre resolus par la premiere parlie de la nouvelle Methode , on a compte celui oii ia pro- pos^e n'a ni racines imaginaires , ni plusieurs racines reelles comprises entre deux nonibres entiers p et yP + i. II pent nean- moins se presenter alors line difficult^ , provenant de la pre- sence des racines commensurables dans I'eqnation : en voici un exemple avec Ic mojen d'j obvier. Soit I'equation . . . x^~\- x"" — 3^ + I = o ; on a les coefficiens des equations. . . . en X 1+1 — 3 + 1 en (:i;— i) i +4+2 + 0. Dans cette circonstance ou la propos^e a I'unitd pour racine , il se pourroit qu'il y eut vme autre racine entre zero et un , dont rexistence ue seroit point manifestee par le dernier terme. Si cette racine existe en efFet, on s'en assurera en prenant la somme des trois premiers termes de la proposee i + 1 — 3 , laquelle somme egalant — i , est de signe contraire au troisieme terrae + 2 , de la transformee en ( a: — i ) ^ ^t P^"^ consequent , atteste I'existence d'une racine entre o et i. La raison de ceci est que , dans ce cas, I'equation du deuxieme degre qui resulte de la division de la proposee par x — i , a pour ses coefficiens respectifs les soraraes-premieres des coefficiens de la proposee , a commencer du premier jusqu'au troisieme. Ces sommes etant i, 2, — i , I'equation du second degre est. . . . x"" -\-ix — 1 = 0, dont la transformee en (a; — i ) est (a;~i)'+4(^~i) + 2 = o. ( 56 ) Cetie operation serolt inuiile , si ron savalt d'avance que la proposee n'a point de racines imaginaires , la simple comparai- son des signes decelfe equation aveccenx de satransforniee ^tanfc alors snffisanle pour manifesterl'existencedela racine antra et i. Quoique I'exemple employe dans cette note, soit celni d'une equation en a; dutroisieme degre, dont la transformee en (x — i ) n'a que son dernier terme egal a zero, le precede est general. La proposee etant du degre w , et sa transformee en (a: — i ) ajant ses 7Z derniers termes egaux , chacun , a zero , il faut alors prendre la sorame des 7?i -}- i — n premiers coefficiens de I'equation proposee-, cette somme est la valeur du dernier terme dfi I'equation en x du degre ( 772 — n ) , qui est le meme degre auquel la transformee en ( a; — 1 ) se trouve abaissee par I'ega-. lite a zero de ses n derniers termes, (H) Nous n'aurions peut-etre pas du faire mention , au n° 4^,' de I'objection opposee a la nouvelle Methode ; mais nous savons que cette objection a ete faite dans les propres termes que nous avons rapport^sj et des lors il a bien fallu en montrer la frivo- lity. Quelles sont d'ailleurs cesMethodes ordinaires qu'on puisse dire plus expdditit^es , et en meme temps aussi silres , aussi. generales que la notre ? (57) Sur le CHAPITRE V. (I) Outre le criterium que nous avons fait connoitre [n" 53] , il en existe plusieuis autres qui , sans avoir tous lesavanlagesdu premier , peuvent souvent en tenir lieu. Un second criterium consiste dans ce corollaire aussi important par son utility que facile h. deduire de la remarque que nous avons consignee dauf la note (F) : Une equation ri'a point de racine entre zero et un , lorsquG la limite , en mains, de sa plus petite racine , est egale ou superieure a la limite, en plus , de la plus grande racine qu^elli puisse apoir entre o et i. Soit, pour exeraple , la mftrae Equation du n" 53. . . j x^ — 4'^* -\-Zx — 6 = O4 Coefficiens des Equations. . . . en X 1 — 44-5 — 6 en (x — 1) I — 1 — 2 — 6. Ici la plus petite valeur que x puisse avoir entre o et i , doit etre superieure a | ou f; et la plus grande doit etre au-dessous de I ou -j", la contradiction qui se rencontre entre ces deux con- ditions fait voir I'impossibilite qu'il y ait des valeurs positives de a; au-dessous de I'unit^. (K) A I'aide du criterium que nous avons indiqu^ dans la note precedente , on pent souvent resoudre une equatioa num^rique , sans avoir besoia de recourir aux trausformees coUat^rales. ( 58 ) Prenons pour exeraple la meme Equation.... x^ — 4'^" + 3:c — ^6 = 0, Coefficiens des equations en a? i — 4+ ^ — ^ en ( x — i)....i — I — 2 — 6 en(:i: — 2). ...1+2 — i — 8 en (:c — 3) 14.5+ 6 — 6 en (x — 4). ...1+8+ 194-6. On a dejavu que x ne peut avoir de valeuu entre o et 1. La plus petite racine de I'equation en ( ^ — i ) doit surpasser f ^ et la plus grande racine positive , inferieure a i, que cetle equation puisse avoir , doit etre au-dessous de -^ ou f . La con- tradiction est evidente. Done I'equation en (x — i) n'a point de racine positive entre o et i 5 et par consequent, celle en xn'en a point entre 1 et 2. De merae , les fractions qu'on voudroit admettre comme racines de I'equation en (x — 2), devroient etre en meaie temps au-dessusde ^ou|, et au-dessous de-^-, conditions incorapatibles. Done I'equation en (x — 2) n'a pas de racine entre o et 1 5 et par consequent celle en x n'en a point entre 2 et 3. L'equation en (x — 3) n'a qu'une racine positive qui est manifestee entre o et i •, par consequent x a une valeur positive entre 5 et 4- Et la proposee n'a pas d'autre racine reelle , vu que n'ayant point de permanence de signe , elle n'a point de racine negative , et que I'absence des variations de signe dans latransforraee en (x — ■ /^) etablit le nombre 4 pour limite de la plus grande racine positive de la proposee. (L) Un troisierae criterium s'ofFre encore a nous : Une equa- tion n\i point de racine entre zero et un, lorsque la suite formee par les sommes-premieres de ses coefficiens pris a rehours , ne presente point de variation de signe. Cette propo- sition est une consequence de notre Algorithme [n° 2o]j cat: ( % ) il est evident que I'absence des variations de signe dans cefte suite , entraine cette meme absence dans la fransformee colla- t^rale. Ainsidanslameme equation qui vient de nous servird'exerople, les coeificiens pris a lebours etant. . . . — 6+3—4 + 1, les sorames-premieres sont ... — 6 — 3 — 7 — 6 ; d'ou il suit que I'equation en x n'a point de laciue entre o et i . Ce criteriiwi s'applique pareilleraent aux deux transformees de cette equation en {x — i ) et en (x — 2). L'operation qu'il exige peut souveut se faire meutalement , et meme d'un coup- d'ceil , comme cela se trouve dans le cas piis pour exemple; ce qui rend ce criterium tres-commode. (M) II est encore d'autres circonstances ou Ton peut se dispenser de calculer les transformees collaterales. Lorsque les transformees successives en (x' — i)^ (a; — 2), etc. ont fait decouvrir autant de racines positives que la proposee a de variations de sigue, on voit que les collaterales en ( z — i ) , (i:, — I ) > ( ~^ — ^ ) > ^'^^' deviennent inutiles. C'est done sura- bondamment que ces dernieres ont ett? emplojees au n° 54 , dans la recherche des racines positives de I'equation x"^ — IX — 5=0 ; et au n" 55 , dans celle des racines negatives de I'equation a:+ — 5.c^ + 5a;' + 6:c — 12 = 0. Dans la premiere equation de ce meme n° ^^ , la seule regie de Descartes rendoit inutiles toutes les transformees collaterales, a I'exception de celle en ( z^ — i ). II suffit , pour s'en convaincre , de Jeter les jeux suir les signes des coefficiens de cette equation et de ses transformees successives. On en peut dire autant par rapport a I'equation en x du n° 62 , et a ses transformees succes- sives. En general , il ne faut point perdre de vue cette regie de Descartes, dont les applications se presenlent fr^quemment dans la nouvelle Methode. ^ (6o) Lorsqu'on esf parVeou a d^couvrirTn — 2 racines r^elles d'une equaiion du degre m , on ne oeutsupposer que les deux restantes aient des valeurs reelles comprises entre deux nombres entiers ■p Qi p -\- \ , ?,\ I'equation en (x — p — i) n'a pas au moins deux permanences de signe de plus que celle en (,r — yy). Ainsi dans I'equalion du n° 4^, x^ — y-c -{-7 = 0, lors raeme qu'on igno- reroit que toutes ses racines sont reelles , la seule vue des transformees successives apprendroit qu'il ue faut chercber les deux racines positives de la proposee qu'entre i et 2. (N) Le criterium qui est indique au n° 53 , et que Ton doit cousiderer corarae le plus important, peutetre generalise ainsi : line equaliun en x ne pent avoir plus de racines comprises entre zero et u , qiCil ny a de variations de signe dans I'e- quation en ( z — ' - ) ; u representujit une valeur positive quel- conque,etz egalant-. Sur quoi , 11 faut observer qu'en faisant 2' == i/z , on a les uieraes variations de signe dans I'equation en ( z' -^ 1 ) ou (- — 1 j que dans celle en T z j ou i j ; ensorte qu'il suffit, sous ce rapport, d'obtenir la premiere. Ainsi la proposee en x n'aura point de racine entre zero et u, lorsque la transTorm^e en ( x:' — i ) ou f - — i j n'aura que des permanences de signe. Cette uniformit(5 des signes de la transformee aura cons- tarament lieu , lorsqu'il n'y a aucune valeur de x entre o et z^ , si ce n'est quand la proposee a une ou plusieurs couples de racines imaginaires de la forme f'^ \/ — ■ o qu'autant que le nombrey est positif et plus petit que u , et que

divise par ID""*. Par consequent , le dernier terme d'une transformee en ^^w — ^w^ ^ divise par lo""', est egal au resultat que donne le nombrefp -{-—+... .4- -^J substitue a x daus requation proposee. (a) II resulte de ce qui precede que les transformees en (jc — p) , (x' — p) ) etc. , sans autre operation ulterieure de Galcul que le placement convenable de la virgule indicative des decimales , donnent arithmetiquement les valeurs de I'ordon- nee y , correspoiidant aux valeurs entieres et decimales de 9 (66) I'abscisse x , dans la courbe qui a pour Equation. . . .- AcX"-\-A,x"'~^ + A„,_,x' -f- A„,oc' =y. On ne s'arretera point ici a montrer comment la cousidi?- ration de ces valeurs numerique* de j peuvent contribuer a la resolution d'une equation numerique. (b) Une autre consequence est que las coefficiens de Tequa- tionen (^"^"^ — lo), respectivement divisespar io% 10% io%...io"', deviennent ceux de I'equation en (x'-''~'^ — p^"~^^ — i ) •, c'est-a- dire , Pequation en (x' — 10 ) , ainsi modifiee , devient celle en (x — p — i)j I'equation en (x" — 10) devient pareillement celle en (x' — ;»'— i ) ^ et ainsi de suite. Cela se prouve genera- leinent par I'equation 10 (.-c*^""'^ — p'^"-'^ ) =z x'^"^ ; d'ou x^"^ — 10= 10 (x'^""'^ — p'^"'"'^ — I ). Or celte consequence merile quelque attentioti , en ce qu'elle fournit au calculateur un contrule , ou , comrae on s'exprime en Arithmetique , luie preui'e de la justesse des calculs relatifs aux transforuiees successives. Et par cette raison , lorsqu'on attacbe quelque importance a ^viter les erreurs , et que les metnes operations ne se font point concurremraent par deux calculateurs qui se servent mutuellement de controle , il cou- vient de continuer les transformations jusqu'a celle en (x*^"^ — 10), quoique , sans ce motif, on filt souvent dans le cas de s'arreter plus tot. Prenons pour exemple I'equation du cinquieme degre du n° 63, On a pour les coefllciens de ses trausforraees. . . . en (.r — '?/)..., \ -{- 7+1S — 3 — 16-J-6. en (a: — 5) i + 12 H-5i +88-|-5o + 8. On s'est trouve dans le cas de faire 10(0; — 2) = x', et de calculer les equations en (x — i), (x' — 2), etc. , jusqu'^ celle en ( a;' — 8 ) •, mais pour s'assurer qu'il n'j a point d'erreur de calcul dans ces transformations , il faut les continuer jusqu'a i'equation en ( x' — 10 ). (C^7 ) Coefficlens des transformees en (x' — 8). . . .i+iio4-4i8o+6o200-f2o544o-j-ii3o88 en (x' — 9). . . . i + i i5-f-4Sjo-f-7 j4io~(~5^S825-j-583oig en (x' — 10). . . . i+i2o-|-5ioo+88ooo4-5ooooo+8ooooo. I-es coeflficiens de la transformee en (x' — 10) respectivement divis^s par 10% 10', lO^ . . . . 10^, deviennent I + 12 -f- 5i + 88 + 5o H- 8 ; ils se reduisent , comine cela devoit etre, a ceux de la transfor- mee en ( X — 3). (V). On a vu , dans le Cliapitre YI, comment une meme methode nous sert a approcber davantage d'une racine deja manifestee, a raoins d'uue unite pres entiere ou deciraale, efc a operer simultaneraent la verification et rap>prosimation des racines qui restent encore a determiner. Cette unite de methode a ete prescrite par la nature meme de la chose , dans le dernier cas •, et il a paru convenable de la conserver dans le premier, autant pour ne pas deroger a la simplicite des moyens , que pour ne point multiplier les methodes sans necessite , et pour conserver dans tons les calculs I'espece de p?-ein'e ou de co7itrdle iaentionn.6 dans la note precedenfe. Voici neanmoins un nouveau precede d'approximation que nous proposons pour le premier cas , c'est-a-dire pour celui ou al'aide de deux transformees successives eu (^ — tt) et (^ — tt — i), et en cas de besoin, de la collaterale en (^ — i }, ou a reconnu I'existence d'une seule racine comprise entre o et i , pour I'equation en ( ^ — tt) , et par consequent d'une seule comprise entre tt et •tT + i , pour I'equation en ^. ( a ) Soit ^ = tT -{-?,, ou ^ — TT = 0, i soient respectivement Tf, et n, les limites , en moins et en plus, de la valeur de ^, comprise entre o et i ,determinees conformement a ce qui a ete (68) dlf ,plus liaui [note (F)]. On pent prendre > oil 'ti',, ou n, potir deuxieme valeur approchee de ^, , les premieres etant zero et un ; et par consequent ^r-f- "^i j ou tt + ^^ , pour deuxieme valeur approchee de ^. Supposons d'abord qu'on veuille approclier de la racine par des valeurs de plus en plus convergentes , qui soieut toujours inferieures a la valeur esacte. On fera 0, = -tt, + 0^ , ou 0, — '7C,=^^', on passera de I'equa- lion en ^, a celle en^^ [vojez la note (E)] , et Ton determinera lalimite, en moins, de la valeur de ^^ comprise entre o et i. Cette liraite etant representee par 'z^, la troisieme valeur appro- chee de 0 sera tt -H'^fi + 'Tri. On 86 procurera ainsi successivement les equations en; ?3j ?4i- • .fvjet l'onaura'7r-H^,+7r^ + . ..+7r, pour la (i+i)'^"'e- valeur approchee de 0, Supposons maintenant qu'on veuille approclier de la racine par des valeurs de plus en plus convergentes, mais toujours supe- rieures a la valeur exacte de 0. Ou passera del'equation en 0, a celle en (^, — IT, )j puis faisant ^, = rfj — Z, ou S = — (0, — n,) , on obtiendra I'equatioa eu H, en changeant les signes des termes de rang pair dans I'equation en ( |, — IT,). II ne restera plus qu'a obtenir des valeurs de plus en plus convergentes , mais toujours inferieures a S ; de sorte que la valeur de plus en plus approchee de (n, — H) ou ^, , et par consequent celle de 0 ou ■7t-j-t,,de~ meureront toujours plus grandes que la valeur exacte. Ces approximations vers la valeur de H se feront de la meme maniere que dans la premiere supposition. (b) Prenons pour exemple cette equation que nous avons deja xesolue [ 54 ] j a moins d'une unite pres. . . . x^. — 20; — 5;^ 0, On a frouv^ pour les coefficiens de ses transform^es. . . . en (x — 2)....i+G+io — ^i ea (x — 3). . . .1 + g + 25-|- i6. II resulte de ces transformees que I'equation eii (x — 2) a une racine comprise entre o et i , dont les premieres valeurs approchees, I'une en moins , I'autre en plus, sont, respective- ment , o et i. Mais , en outre , ces deux transformees fournissent immediatement les secondes valeurs approchees de cette racine , qui sont — ; — ou — pour la valeur en moins , et ■ - , ,, ou -— ^ 1 + 10 11 t^ 25+ib 41 pour la valeur en plus. [ Voyez la note (F).] On voit done , en se bornant aux valeurs approchees en moins^ que les deux premieres sont, pour la proposee. . . . a et 2 ^ , ou — ; ou bien 2^0909090909 Ou reconnoitra ci-apres que cette derniere valeur est exacte dans ses deux premieres decimales. II faut maintenant, en faisant x — 2 = ^,, passer de I'equa- lion en ^, a celle en 0, ou ( ^, j. On peut employer a c€t efFtt TAlgorithme modifie [note (E)] , de la maniere suivante. Soit 11?,= ^\ : on a pour les coefficiens des equations. . . . en^', i-|-6.ii4-io.ii° — i.n^ ou I 4-66 +1210 — i35i en (^'. — I )....! 4-69 4-1545 — 54. Substituant a 0', — i sa valeur 11?, — i, ou 11 ( f , ~j' et faisant 0, = 0a> on a pour les coefficiens de I'equation.... en0,....ii' 4-69.11^4-1^45.11 — 54. ou 1 33i 4-8349 4-^4795 — 54« ( 70 ) La limite , en moins , de ?. est ^^^^ ou -^ = _^ .. limife a laquelle on peut substituer , pour pins de simplicity , — ? = r = o.oo36363636 '2-jo 11. a5 Ainsi la troisieme valeur approcliee de x est... - "^ 77 ~f" 17^ ' °^ 5^ ' °" ^^^" 2,0945454545 .... Nous ferons voir que cette valeur est exacte jusqu'a la qua- frieme decirnale inclusivement. Oil passe ensuite , de la raerae raaniere , de Tequation en ^. a celle en 03 ou T^, ^j.Faisant 11 .250,=|',, etsubslituant dans I'equation en 0, , on a pour les coefficiens des equations . . .' en %\ I +6g.25 + i545.25' — 54-25' ou I -f- 1725 -4-840625 — 843750 en (?'s — i)... .1 + 1728 +844078 — 1399. Substituanta^'i — i sa valeur 11. 25?^ — \ , ou 11 .25^^^ — TTri)' et faisant t^ = 03 * on a pour les coefficiens de I'equa- tion. en 0s..-. 11^.25^+1728. II '.25+844078. II .25 — 1399, ou 20796875+13763750 +232i2i45o — 1399» La liinite , en moins, de 03 est.... 1599 1399 _ 1 . i399+232i2i45o ' 232122849' 1^^920 ■n"^' Mais on peut , pour simplifier , lui substiluer. . . ♦ . 1 1 iGSgaS 25.6637' (70 Aiost ]a quatrieme valeur de x , approch^e eu moins , est. ou 2,094545454545. . .-{-0,000000026819. . . , ou bien.. . .2,094551481 504- . . ., et cette derniere valeur est exacle jusqu'a la neuvieme deciraaie iaclusivement , corame on le verra plus bas. (c) Les valeurs approchees de cette nierae racine , calcu- lees par Newton , suivant le precede qui lui appartient, sent. . . . 2 2, 1 2,0946 2,09455147 M. Lagrange a aussi calcule , suivant son precede, les valeurs approchees de cette racine , en fractions continues , alternafi- vement plus petites et plus grandes que x. Les resultats sont. . . . 2 21 2,5 44 111 i55 57S 75 1 i3o7 iS4i5 I ' 10 ' 11 ' 21 ' 53 ' 74 ' 275 ' 349 ' 624 ' 7837 ' La dixieme de ces valeurs, ■ Jl , qui est approchee en plus , ^lant reduite en decimales , devient 2,09455 14865. Les valeurs approchees en moins , trouvees suivant le nou- veau precede que nous indiquons dans cette note , etant. . . . .1 20 11 57S ,1.1, T - ' 11 11 ' 11 ' 273 275 ' 11 ' 270 ' 160925 OU bien 2,09096909 2,09454545 2,094551 481 564. • •* on voit que ce precede a donne des r(5suhafsun pen plus exacts que celui de Newton, et qu'il les a donnes plus proniptenient qu'on ne les obtient par le precede de 1\L Lagrange. En outre, ce precede est general et sur , et la methode de Newton n'a pas ccs avantages. « En general , I'usage de celle » methode n'est sur , dit M. Lagrange , que lorsqne la valeur » approchee est a la fois ou plus grandc ou plus petite que ^ ( 70 > cliacune des raclnes reelles de I'eqiiation , e'l que chacune des » parties reelles des racines imaginairesj et par consequent, cette » raethode ne peut etre employee sans scrupule que pour trouver » la plus grande ou la plus petite racine d'une equation qui » n'a que des racines reelles ou qui en a d'imaginaires , mais » dont les parties reelles sont moindres que la plus grande » racine reelle , ou plus grande que la plus petite de ces » racines en regardant, comme on le doit, les quan- 5> tites negatives comme plus petites que les positives , et les » plus grandes negatives comme plus petites que les moius 5> grandes. ( De la Resolution des Equations numeriques , » page i4i' ) » Si Ton emploie , au lieu du precede de Newton, la metliode d'approximation tiree des series recurrentes , on trouve , pour les valeurs approchees de x , dans I'equation x'^ — 2X — 5 = o.... 3,089. . . .2,09467. . , .2,094549- . • .2,0945515. . .etc. On ne pourroit , ainsi que I'a prouve M. Lagrange, emplojer generalement cette methode d'approximation pour chacune des racines reelles d'une equation quelconque , qu'autant que Ton connoitroit d'avance une valeur approchee de cette racine , telle que la difference entre cette valeur et la vraie valeur de la racine filt raoindre en quantilo, c'est-a-dire , abstraction faite des signes , que la difference entre la meme valeur et cliacune des autres racines , et en meme temps moindre que la racine quarree de chacun des produits des racines imagi- naires correspondantes , s'il j en a , diminuees de la meme valeur. Autrement , cette methode ne sert qu'a trouver la plus grande et la plus petite des racines reelles; encore faut-il que le quarrd de la plus grande- ou de la plus petite racine cherchee soit en meme temps plus petit que chacun des produits rdels des racines imaginaires correspondantes , et qu'on ait quelque niojen de s'en assurer. \^De la Resolution elc, pag. 147, i5i]. ( 73 ) ( d ) Nous avons indique plus haut comment on pourroit se procurer une suite de valeui'S approchees de I'inconnue , conver- gentes en plus. Mais pour eviter descalculs inutiles, on peut , au mojen de quelques operations ajoutdes a celles qui ont donne les equations en 0, , ^^ » f 3 > • • • ?v , obteuir une liraite , en plus , de ?, , et par consequent de toutes les valeurs approchees de ^ , depuis la premiere jusqu'a la v''""^. Nous prendrons d'abord un exeraple particulier, et nous traitcrons eusuite ce sujet d'una raaniere generale. Dans I'exemple qui nous a servi, on a frouve —7^ — r ou - ,.,.-^ ,' pour la valeur de tTj , c'est-a-dire, de la limite , en raoins, de ?3. Faisant done ^3 = - ,^ ^'3, et calculant les equations en ^'3 , ( t'i — I ) > ( ^'s — 2 ) , on trouve pour les coefficiens des equations. . . . en (;'3 — 1)... 13314-138772104,9+4089138623104907 — 12049769755 en(;'3 — 2)...i33i + i3877a3c42+4o9o°i398548998+4o898796io6752i.' Puis on fiut 10" (^'3 — \^ = ^"~^; etl'on obdent les coefficiens des equations. . . . on t'i 1 53 1 + 1 08772 1 o4900o+4°S.99S62oi 04907000000 — 1 2049759755000000000 €n(j''3 — i)...i33i-ri38772iG52993-f-4°S99S^2588o34gioi993-f-39694886473662£o5o3ji. Done la limite , en plus, de ^"3 est.... 4:Bq98 ^^40899^ 408998 4-595948 805897 On pent done faire ^\ < g^; et par consequent. . . .. '' ^ "^ "^ 8o5coo ' -^s *-^ ib5i)i!5 "^ 165925. boSooo' D'une autre part, on a ?3 > j 55^35- Done , en se tenant a cette derniere valeur , I'erreur est moindre que —r-. — 1^%—- ou o.ooooooooBoCa. . . . 1 16:^920. boocco ' 10 ( 74 ) - Bien plus , dans I'exemple qui nous occupe, il sufEt de Jeler les jeux sur ies coefficiens de i'equation eu ^"s pour reconnoitre qu'on a ?"3 < ^ » et par consequent ^'3 < 1 +-^ > et . . . ? <" ^ - _i_ ' ' . done I'errenr est naoindre que 0,0000000006026819.... II suffisoit m^me de I'equation en (^'3 — i ) pour s'assurer d'uii Icl resultat , puisqu'a la seule inspection des coefficiens decelte equation , on pent reconnoitre que Ton a ^'3 — i ^-i-7T^^>^i h> 41000 ' '-■"'■ ^ iGSgaS '^ ib'SgaS. 41000 ' ou ;> 0,000006026819. . . -f- 0,0000000001469. . . . ou bien > 0,00000602696 •On a de I'autre part ff<_i |_ \ , ^■^ ib'SgaS ibSgaScooo OU <; 0,000006026819. . • .+ 0,0000000006026. . . , ou bieii <; 0,00000602742 .... Ici la difference des deux limites est 0,00000000046. . . . Done , si I'on prend une des deux limites pour la valeur de ^3, I'erreur ne pent avoir lieu qu'a la dixienie decimale. Ainsi la valeur exacte de x , dans I'equation x^ — 2.x — S:=:.Of est entre 2,09455 1 48 1 3 et 2,09455 1 4844 et raeme entre. . . 2,0945514815 et 2,0945514819. .... En prenant le premier de ces nombres, ou I'uu des deux derniers , pour la valeur approchee de .r , on est assure que eette valeur est exacte jusqu'a la neuvieme decimale, inclusi- venient, comme nous I'avous aunonce plus haut. /■yS ) On est egalemeut assure par ce moyen, que les deuxieme et troisieme valeurs approchees en raoins , que nous avons trouvees ci-dessuspouro;, sont, respectivement , exactes jusiju'ala seconde et a la quatrieme decimale , inclusivement. La dixierae approximation , suivant le precede deM. Lagrange, a bien donne la huitieme decimale exacfe , raais I'exactitude de cette decimale n'est pas assuree par le precede meme, vu qu'il indique pour liraite de I'erreur , 0,00000001 63. . .; d'ou il resulfe que la valenr de x est comprise entre.. .. 3,0945514702 et 2,og455i4865 et que I'exactitude de la valeur approcliee n'est garautie que pour les sept premieres decimales. (e) Voici maintenant comment on peat proceder d'uue ma» iiiere generale. Soient ^, = X;^,= ^; et KX = X'. X n'ayant qu'une valeur entre o et 1 , X' n'en a qu'une seale entrc o et K. On se procurera done les deux transformees en (X' — P') et (X' — P' — i) dont les termes tout connus sent de sigties contrairesj P' etant -< K. Ces deux transformees fournissent deja une double limite , en plus et moins , pour ( X' — P') , et par consequent pour X. Mais pour avoir des limites plus resserr^es , on fera 10" (X' — P')=X''j etcomme (X' — P') n'a qu'une seule valeur entre o et i , X" n'en a qu'une seule entre o et 10^. On se procurera dene les deux transformees en ( X" — P") et(X'' — P' — i) dent les termes tout connus sont de signes contraires; P" etant ■< iC. Ces deux transformees donneront une double limite , en plus et en raoins , de (X" — P") , et par consequent , de X' et de X. ( 7^ } Soit la limife en moins = -^•, et la Hmite en plus On aura X" — P"> -^ , et < ~; et X'-P'>-^+-4-, et <^ + -I--, 10 10- K io 10 /v d'ou X'ou KX>P'+— 4--i-, et ^r- ' et X< 1^ io''K io"KK."' P' . P" . 1 1^ lo^K lo^KA" Si I'on prend pour X uue de ces deux Umites , I'erreur sera moindre que leur difference, qui est ( — , „ , "- ). ^ ' io=^K V 1^ A" J Soit k'+ I le nombre de chiffres que renferme le nombre entier K. II est evident que I'erreur sera <1 — -, ", d'ou il suit que si on veut obtenir, par ce proc^de , une valeur approchee, exude jusqu'a la 7£"^™« deciraale au moins, il faut , pour cu efre geueralement sur, prendre N = « — n'. C'est ainsi que dausl'exempledonton s'est servi , K cu iCSgaS ^tant compose de six chiffres, d'ou n' = 5, on a du faire n = 5, si i'on a pretendu avoir une valeur exacte jusqu'a la liuilierae decimale, Dans ce raeme exemple , P" s'est frouve =o , et P'= i ; ce qui a rendu le calcul tres-expecJitif. En pareil cas , si Ton s'en tient a tt- pour la valeur de X approcliee en moins , I'erreur est toujours moindre que : — -; et par consequent ■< — ^^,, ic^K-A" lo (f) Ce procede pourroit eire etendu a la rechercbe de plu- sieurs racines comprises eutre deux nombres entiers consecutifs, et I'on tireroit pour lors un assez bon parti du criterium que nous avons generalise dans la note (N). Mais il nous paroit ( 77 ) inutile d'enfrei- dans ces details. Nofre principal objet dans cet Ouvrage , a cte de presenter, pour la resolution des equations numeriques , une Methode qui fiit praticable , comme mecani- quement ; la science du calcul pouvant , de nieme que les arts, avoir ses manomriers , et en tirer , dans des travaux en grand, de notables avantages. Smis ce rapport, notre Methode generale sera peut-etre preferee au precede particulier que nous donnons ici , surlout si Ton ne vent avoir des racines exactes que jusqu'a la secoude ou troisieaie decimale. (g) L'evaluafion des racines en fractions continues , suivant le precede de M. Lagrange , est particulierement recomman- dable , lorsqu'elle fait connoiti'c les facteurs coramensurables du second degre dans un polynome qu'oo se propose de decom- poser en facleurs de ce degre. Mais quand il ne s'agit que de la resolulion , pioprement dite , d'nne equation numerique , ce procedc ne nous paroit pas preferable a Tapproximalion en nombres decimaux , salt pour la commodite des calculs , soit pour la rapidiie de I'approximation. De plus , la methode de M. Lagrange et la notre etant de telle nature qu'on y precede simultanement a la verification et a I'approximation des racines, il semble que c'est surtout a ces deux Methodes que devalua- tion des racines en fractions continues ne sauroit etre generale- ment convenable. Par exemple , dans celle de I'illustre Geo- metre , si le nombre D , ou la limite de la plus petite diffe- rence des racines , etoit un millieme , il est (Evident que chaque racine seroit tout a la fois reconnue et appreciee , a moins d'un millieme pres. Or il paroit infiniraent dur , lorsqu'on a obtenu par des milliers de substitutions, une valeur aussi ap- procliee , d'etre force de retrograder jusqu'a la valeur du plus grand nombre enfier contenu dans celte racine , pour chercher une nouvelle evaluation en fractions continues. C'est par un semblable motif, joint a quelques aufrcs , que nous ii'avous pas cru devoir adapter ce precede a notre Mctliode ; et ce motif semble plus decisif encore dans la Metbode de M. Lagrange , qui exige un bien plus grand nombre d'ope- rations pour la manifestation des premieres limiles des racines. En eiFet , lorsque D est , par exemple , nu uiillieme , on ne peut , suivant la methode dont ils'agit, decouvrir deux racines moindres que I'unite , telles que 0,920... et 0,921..., qu'aii mojen de neuf cent vingt-deux substitutions, taudis que le nombre des transformees esigees pour le meme objet dans la nouvelle Methode , egale seulemeut 10 + 3 -j- 2, c'est-a-dire, i5. En uu mot , s'il faul decouvrir une racine ajant n deciuiales , la recherche de ces deciraales n'esige au plus que 10.7? trans- formations , tandis qu'elle peut exiger jusqu'a 10" substitutions , suivant la progression o, D, 2D, etc. (h) La comparaisou que nous venous de presenter , con- cernant le nombre des operations , dans la nouvelle Methode des tran^'^formees , et dans la Methode des substitutions suc- cessives , telle qu'elle a i^te perfectionn^e par M. Lagrange , a donne lieu a une observation qu'il est bon de rapporter ici, ne fut-ce que pour empecher qu'elle ne soit desorraais repro- duite. « Si la resolution des equations, a-t-on dit , exigeoit » I'emploi d'une pareille Methode ( ce//e de M. Lagrange), » assuremeut celle de I'Auteur, quoique tres-longue, meriteroit » encore la preference. Mais, pour I'ordiuaire, on ne procede » pas ainsi. La Methode de Newton , qui est la plus usitee , » suppose qu'on connoit , soit par la voie des substitutions , » soit par des constructions geometriques , ime premiere va- » leur de x , qui approche au moius dix fois plus d'une racine » de I'equation que de toute autre racine ; et d'apres cette » valeur, on en trouvera facilement une autre dont I'erreur » n'est qu'environ le quarre de la premiere , savoir -^ , si la » premiere valeur est 7^. Une seconde operation qu'on peut » faire par la merDB forajule , reduit I'erreur du centieme k C 79 ) > son qnarre , qui est d'environ ts^sz t etainslde suite. D'oit T> I'on voit que I'approxiraation continuelle est beauconp plus T> rapide par cette Methode que par celle que propose I'Auteur. y> [Ils'agit de celle que nous avons expose'e au Chapitre Vl\^y Une pareille observation prouve que son Auteur n'a nuUe- Bient compris I'etat de la question. Nous nous sommes propose de comparer deux Metbodes qui, I'une et I'aufre , precedent simultauement a la verification et a I'approximation desracines, et qui jouissent, toutes deux, de I'avantage de resoudre gene- raleraent et avec certitude , une equation numerique , dans des cas oil toutes les Methodes precedentes ecbouent , ou n'abou- lissent qu'a des resultats faux ou douteux : et I'on vient nous opposer le procede de Newton , qui n'est pas menie une me- thode de resolution proprement di(e , et qui d'ailleurs , comme nous I'avons dit plus haut d'apres M. Lagrange , n'a pas raeme le merite d'etre generalement siir ! Ce procede , fut-il aussi sur qu'il I'est peu , est evidemraent insuffisant pour I'ap- proxiraation des racines qui, etant par exemple, 8, i... 8,2. ..8,3..., ont une meme premiere valeur connue 8 , tandis que , dans rotre Methode , il ne faut que quelques instans pour decouvrir la decimale de chacune de ces racines. Et c'est un procede aussi incertain qu'incomplet qu'on a pretendu opposer a une Me- thode generale et sure \ Encore une fois , nous en appelons a la pratique. Qu'on se donne la peine de resoudre les equations numeriques paries diverses Meihodes , et I'on verra qu'absfraction faite du pro- cede approximatif indique dans cette note , notre Methode generale , meme en ne faisant decouvrir , qu'un a un , les chiffres de la racine , est encore celle qui, dans son ensemble, se trouve en meme temps , la plus sure et la plus expeditive. Bien plus, dans certains cas, on sera force de reconnoitre qu'elle est la seule praticable. II faut I'avouer , cette observation , que nous avons rapportee iextuellement , et robjection citee au n^ 4'^ , reunies a quelques ( So ) aiilres indices , ont paru pi'ovenir d'une disposition d'espritpcu favorable , et nous ont rappele la pensee de Pascal au sujet de ceux qui inventent. II faiit sansdoiUe, suivant son conseil, que cehii qui a rencontre quelques inventions , ne se pique point de cet avantage. Quand on considere un objet sous toutes ses faces, avec une attention perseverante , il est difficile qu'il ne se presente pas a I'esprit quelques vues nouvelles ; et , ce qui semble reduire a peu de chose cette gloire a laquelle on croit pouvoir pretendre par des inventions scientifiques , c'est que la science nieme a ses hasards, et quesouventles inventions s'offrent comme fortuileraent a I'esprit , a I'instant meme ou ses recher- ches le portoient ailleurs. Mais il pent, du moins , etre perrais a I'auteur d'une decouverle utile, de desirer que la commu- nication qu il en doune , soit accueillie avec quelqne bien- veillance. (X) Quoique Ton puisse tirer quelque parti de la nouvelle IMelhode pour la determination des racines imaginaires , nous rie nous arreterons point a cet objet , qui appartient au pro- bleme de la decomposition d'un poljnonie en factenrs reels du second degre , plutot qu'a celui de la resolution des equa- tions nuraeriques •, I'objet essentiel de cette resolution elant de trouver les valeurs reelles qui peuvent etre attribuees a I'inconnue. C'est cequ'areconnu M. Lagrange, lorsqu'il a donne des moyens de trouver une limite dela plus petite difference des racines, sans recourir a I'equation aux quarres de leurs differences. Cependant I'illustre auteur a cru pouvoir surabondamment se servir de cette equation , qui donne la valeur de la quantite B precedee du signe — sous le radical dansles racines imaginaires, pour determiner , au moins par approximation , la parlie reelle de ces racines. Pour cela on substitue A+ \/ — B ax, dans la proposee , et on en tire deux equations en A , dont I'une a tous ses termes reels , et dont I'autre a tons ses termes multiplies Bar \/— B, facleur commun que Ton fait disparoitre ; ce ( 8i ) qui rend les termes de la seconde Equation fous reels , pares qu'ils ne dependent alors, aiusi que ceux de la piemiere , que du quarre de V — B, c'est-a-dire , de — B. Ensuite, procedant a la recherche du plus grand comraiin di- viseur de ces deux equations, on s'arrete au resfe ou A n'est plus qu'audegre n et au-dessous , n etant le uombre des valeurs egales que I'equation au quarre des difFdrences a lo rnies pour B. Ce reste etant egale a zero, on j substitue a B sa valeur exacte ou approchee, et cette equation ^tant aiusi devenue numerique , on en tire les valeurs reelles de A. Cette resolution a, comma Ton voit, rinconvenient d'exiger la formation de I'equation aux quarres des differences ", mais eu outre , il serable qu'on puisse douter qu'elle soit generalement exacte dans les cas ou le plus grand commun diviseur est de plu- sieurs dimensions. Car la substitution de la valeur a()prochee de B ue donnant aux coefficiens de Tequation en A qu'une valeur approchee, ne peut-il pas arriver que cette alteration, meme tres- legere, change la nature des racinesde I'equation, en substituant; des racines imaginaires a des racines reelles , et vice versa ? Lorsqiie le reste egale a zero est seulemeut du premier degre , et qu'on a ainsi determine la valeur de Aenfonction deB, il serable qu'en y donnant a B deux valeurs respectivement ap- prochees en plus et en raoins , il en doit piovenir deux limites entre lesquelles se trouve la valeur exacte de A. Cependaut le resultat meme obtenu par M. Lagrange , dans la resoluiion de I'equation x^ — 7.x — 5=o, est evidemment faulif. D'apres ce 1 5 1 '3 resultat, la valeur de A seroit comprise entre 7: et ^. \ Dc ' f^ lb 18 ^ la Resolution etc., pag. 3g]. Or on a vu plus haut que la racine positive de I'equation est bien cerlainemeni 2.og45...", et son second terme ayant zero pour coefficient , il s'ensuit avec la meme certitude , que A egale la moitie de cetle racine posi- tive , precedee du signe — ; on voit done que la valeur exacte de A est comprise entre — l,Ql^'].,, et — 1,048... Ce resultat H (B2) n'est nullement d'accord avec le precedent ; ce qui vient appa- remment de quelque errenr de calcul. L'observation que nous avons faite siir le cliangement possible de la nature des racines d'une equation par une legere alteration dans la valeur de ses coefficiens , peut aussi inspirer quelque doute sur la legitimite de la resolution de deux equations a deux inconnues x et j , lorsqu'apres I'elimination de x , on obtient pour j^' une valeur seulement approch^e , dont la substitution ne peut produire que des coefficiens d'une valeur approcb^e pour I'equation en x. La meme difficulte se rencontre dans la decomposition d'un poljnome en facteurs du second degre. D'une autre part , il seroit extremement facheux de ne pou- Voir , en aucun cas, se fier aux resultats qu'on obtiendroit d'une equation numerique , propre a resoudre un problerae pbjsico- matbematique , lorsqu'on n'a point la valeur rigoureuse de ses coefficiens. II est done a desirer que I'on trouve quelque regie certaine qui fasse connoitre quelles sont les equations dont les racines ne cbangent point de nature; malgre ralteration produite dans leurs coefficiens. (Y) On auroit un grand embarras de moins , si Ton pouvoit dt5couvrir toutes les valours reelles de I'inconnue , sans de- pouiller I'equation des racines egales qu'elle peut avoir. On a vu que , dans notre Methode [4^] , la presence des racines (5gales coramensurables ne forme point un obstacle a la reso- lution d'une equation-, et il est aise d'appercevoir que la presence des racines Egales iraaginaires n'en forme pas davantage. Bien plus , lorsqu'on sait d'avance que toutes les racines de I'equa- tion sont reelles , on peut la resoudre , sans qu'elle ait et^ prealablement d^pouiliee de ses racines multiples , meme de celles qui sont reelles incommensurables. Car une fois qu'on sera parvejui a reconnoitre I'existence d'une racine, au moins^ entre deux limites qui ne difierent que d'une unite decimale de I'ordre que Ton veut , ou qu'exige le problerae dont la so- lutioa depend de reqnation a resoudre ; il est indifferent, pouE ( 83 )^ la pratique , que la taleur trouvee appartlenne a une ou a plusieurs racines, soit absolument <5gales enfr'elles, soit 6ga\e$ seulement jusqu'a tel chiffre demande. L'essentiel est que Ton connoisse foutes les valeurs reelles qui representent , jusqu'au degre I'equis d'exactitude , celles dont I'inconnue de I'equation est susceptible. D'apres cette consideration , on pourroit meme, dans (ous les cas , laisser subsister les racines egales d'une equation num^- rique , si I'ou avoit le moyen de connoitre une limite, enmoins, de la valeur de (p , e'est-a-dire, de la plus petite valeur que pourroit avoir la partie des racines imaginaires representees par A d= y/ —

S]. Or il paroit qu'en substituant \/ — (P al'inconnuede I'equa- tion Gn (x — p — — . . . — ^)' °" P^^t determiner une li- mite de la plus petite valeur dont (p soit susceptible, la valeurde la partie reelle A etant supposee exister entre... (B4) L'objection qui resulte de la legere alteration des coefficlens, se represente ici , raais moins grave que dans la note precedente. II seiiible bien que cette alteration , toute legere qu'elle puisse etre , suffit pour rendre positif , dans I'ecjuation en

'.. ^. priori ordinem altiorem tribuemus quam pofteriori, fi fit vel a> a\ vel a — a'et /3 > p', vel a — a', ^ — ^' et y>y' vel etc. i. e. fi e difFerentiis a — a', 3 — /3'. 7 — y' etc. prima, quae noii euanefcit, pofitiua euadit. Quocirca quum termini eiusdem or- dinis non differant nifi refpectu ccefficienlis M., adeoque in ter- minum vnum conflari poITint, (ingulos terminus funclionis q ad ordines diuerfos p^rtinere fupponemus. .r| itt a^ fc-rs os'l' lara obseruamus, fi Ma^b^cy Gt ex omnibuS terminis functionis p is, cui ordo altidimus competat, neceffario a effe maiorem, vel salt«m nonminorem, quam /3. Si itiftb ~ elTet 'p> a, ft CAROL. FRID. GAVSS /3 ^ ct, terminus Ma'^&'C ....... quem foncuo-p, v-tpote , fy^vme-, trica , cjuoque inuoluet, foret ordinis aUipris quam Ma='bPcv..^..f^ contra hyp. Simili modo (B erit maipr vel faltem noii; .minoB: quam y; porro y non minor quam exponens fequens ^etc. : proia^ Gngulae differentiae a — (o, p — y, y — <5^ etc,..erunt,integii.non, negatiiii. .o(;i i ,a ^v.' mijf Bjfrt; '^-\r 'hpi. fng Secundo petpendamus, fi e quotcunque funclionibus integris indeterniihatarum a, b, c elc. pioductuiii confleOjr, Imius termi- num altilTimum neceftario efTe ipfum productum e terminis altis- llmis illorum factorum. Aeque tftanifefium eft, terminos altia* fimos funclionum X', V, X'" etc. refp. effect, ab, ahc etc. Hiiic' colligitur, terininum altiffimum e producto p = MX''-" X'^r^-y X"'y-'' prodeuntem effe Ma^b^ c'^ ; quocirca ftatuendo g — p zz g\ terminus altiflimus funclionis g* certo erit ordinis inferioris quam terminus altiflimus functionis g. Manifefio autem p, et proin etiam g, fiunt functiones integrae fyrametricae ipfarum a, b, c etc. Quamobrem p' perinde tractata, vt antea g, discerpetur in p' -|- ^'', ita vt p' fit productum e poteftatibus ipfarum XU X", X'" etc. in coeflicientem vel determinatum vel faltem ab a, b, c etc. non pen- dentem, g" yero functio Integra fymmelrica ipfarum a, b, c etc. talis, vt ipfius terminus altiffimus pertineat ad ordinem inferiorem, ouam terminus altiflimus functionis g'. Eodem modo continuando, manifelto tandem j ad formam p -)- P' + P"+ P'" *tc. redacta, i.e. in functionem integram ipfarum X', X", X'" etc. transformata erit. Theorema In art. praec. demonftratum eliam fequentl modo cnuiiciare poffum^us: Propofita functione quacunque indetermina- tarum a,. &, c etc. inlegra fymmelrica g, alTignari poteft functio integra tpf^dem aliatum indeterminatatum i^, l'\ I'" etc. talis, quae per DEMONSTR. NOVA ALT. THEOR. OMN. FVNCT. ALGEBR. etc. g per fubrthutlones l' = \', I" — X" , V" — X'" eic. transeat in g. Fa- cile infuper ofienditur, hoc vnico tnntwn modo Jieri pojje. Sup- ponamus enim, e duabiis fiinclionibus diuerlis iodeterminatariim /', I", I'" etc. pula turn ex r, turn ex r' poft fubflitiuiones I'^zX', I" ■=.\", I'" ziX'" etc. refultare oandeni functionem ipfarum a, h, c etc. Tunc itaqne r — r' erit functio ipfarnm Z', I", V" etc. per fe Kon euanescens, fed quaeidentice deflriiitiir poft illas fubftitntiones. Hoc vero abfurduin effe, facile perfpiciemus , fi perpendamus, r — r' necelTario compofitam elTe e certo numero pattium forniae Ml"" I'"'' l"'y qiiaruni coefHcientes M non euanefcant, et quae fingulae refpectu exponentium inter fe diuerfae fint, adeoque terminos altiflimos e fingulis iflis parlibus prodeuntes exhiberi per 31 n' j<-/>ji etc. ^yti* etc. et proln ad ordinfs diuerfos referendos effe, ita vt terminus abfo- lute altilTimus nuUo mode defirui pofTit. Ceterum ipfe calculus pro huiusmodi transformationibus plu- ribus conipendiis infigniter abbreuiari pofTet, qulbus tamen hoc loco non immoramur, quum ad propofitum nolirum fola transfor- mationis pofBbilitas iam fufficiat. 6. Confideremus productum ex m (?« — i) factoribus {a — b) (a — c) (a — d) - ■ X (b — a) (b-c) {h-d) X (c-«) (c — fe) (c-d) y.Xd-a)id-b){d-c) etc. quod per it denotabimus, et, quum indeterminatas a, h, c etc. fymmetrice inuoluat, in formam functionis ipfarum X', X", X'" etc. redactum fupponemus. Transeat haec functio in p, fi loco B ipfa- jQ CAROL. FRID. GAVSS ipfarum X^ K", \"' etc. refp. fubflituuntur V, I", V" etc. His ita factis, ipfam p vocabimus determinanteiii funcrionis y — x'n — Z'x"' — * -J- Z"x'" — 2 — Z"'x"» — 3 .-{- etc. Ita e.g. pro 7n rz 2 habemu3 p - — Z'2 4- 4Z" Perinde pro m-=zz inuenitur p = — Z'-i"2 -I- 4Z'3Z'" + 4Z"3 _« x^VVn"! + 27Z'" = Deteiminans functionis y itaque eft fnnctio coefliGientiu.Ti V;l", Z'" etc. talis, quae per fubftitutiones Z'=r A.', Z" — A.", V" = }J" eKc. tranfit in productum ex omnibus differentiis inter binas quanliia- tum a, b, c etc. In cafu eo , vbi inz=.i , i. e. vbi vnira tanluni indeterminata a habetur, adeoque nullae omnino adfunt diffeicn- liae, ipfiiin numeium i tamquam determinantem functionis y adoptare conueniet. In llabilienda notione determiaantis , coelTicIentes functionis y tamquam quantitates indeterminatas fpeclare oportuit. Deter- minans functionis cum coefEcientibus determinafis F = X™ — L'x'»-i -\- L''x™-2 — L'^'x-n-S -\- etc. erit numerus determinatus P, puta valor functionis p pro I'zzL' I" — L'\ I'"— U" etc. Quodfi itaque fupponimus, Trefolui poffe in faclores finiplices Y — {x — J) (x~B) {x — C) fiue Y oriii ex V =: (x — fl) (x — h) (x — c) . . r . . ftatuendo a-:=: A , hz^B, crrC etc., adeoque per easdem fubflitu- tiones X', X", X'" exc. refp. fieri L', L", L.'" etc., manifefld P ae- qualis erit producto e factoribus ^A-B) {A-C) {A-D) X {B~A) {B^C){B-D) X (C-^, (C-S. {C-D) ..... X (D-^j (£»-£; (D-C) «'c. Patet DEMONSTR. NOVA ALT. THEOR. OMN. FVNCT, ALGEBR. etc. j j Patct itaque, fi fiat P = o, inter quanlitates A, B, C etc, duas fakem aequales reperiri debere; contra, fi non fuerit P = o, cun- clas A, B, C etc. necelTario iaaequales efTe. lam obferuamus fi dY ^ - fiatuamus ■::— z=. Y', fiue ox Y' — 7/2X"'-^ (;7i 1 ) L'x™-2 -f- (7tt— 2) L' — -'! > — f ^i — 1 ', - - "^ - I i-*-/x-. Cx — ^.. (x — i:;..r — r_x — cj.... J a--. — J") — , _ f I ^ — r^ -!- ::x — EipriEssmu; brecitads caolTa /x.(y-3,-,x.— ^^ OT»e eft fnnctio iotegia indetermitataixm x, 2', I", I"' etc per F ,x, 2'. 2", Z'" etc.) Tsde arit idectice 1 - -x . (. — 2' - 3 X . — ^ = 1 — Fix, >/, X", X'^'ett) ■ a X Habebimns itaque aeqiulion^ identicaj [ i ] Z a.{ji — h) {a — c) (a — c). . .. = i -rF{a, '/.', '/J', '/."' etc) ^ & . rfe — c) (&— O (6— «0 •••■=» -r f C&. >.'. >-". X'" eta) ^c.(c— fl)(c— J)Cc — d).... = 1 -f FCc, X', X", X'^etc) Sappo- DEMONSTR. NOVA ALT. THEOR. QMS. FVNCT. ALGEBR. etc. 15 Supponendo itacne, productuni ex omnibus - 1 -f F(a, I', I", I'" etc.) 1 4- F(&, /', /", I'" etc.) 1 -r r(c, /', I", I'" etc.) etc. quod erit functio Integra irdelerninatarurn a, Z', cetc. , 1,1' ,1 " etc. et quldem fanctio rytumeliiza rtf^ectu ipfaruni a, h, c etc., ex- hiberi per ■^ (>.', //', >J" etc. , /', I", l"< etc.) e itiultiplicaticne cunctarum aequationum Til refnitabit aeqnalio idenlica noua [2] 7T;:a.^b.Zc.... = -^ (X', A", >.'" etc. , X', X", >.'" etc.) Forro patet, quura productum Z a, Zb . Zc . . . . indetern^ir.atas a, h, c etc. fymmelrice iruoluat, icueniri poITe functicncm ir.te- gram indeterniinatarum V, I'', I'" etc. talem , quae per fubltiluuo- nes r=X', /" = X'', Z"'=:X"' etc. transeat in ^fi.Xo.pc Sit t ilia functio, eritque etiara identice [-5] pt=-^{l', I", l"'elc., V, I", I-') quoniam haecaequalio per fubftitutioces l'=.\',l ' = /.'',! " :=\"' elc. in idenlicam [2] transit. lam ex ipla def.nitione fuccticriis F fequitur, ideniice haberi F(x, L', L", L'" etc.) = o Hinc etiam idenlice erit 1 4- FCa. i', i.", L'" etc.) = \ 1 -j- f (&, L', L-', L'" etc.) = I 1 4- F (c, L', L", L'" etc.) = 1 etc. et proin eiit etiam identlce ■^ (X', X% X'" etc. , L', U', V" etc.) = i adeoque etiara idenlice [4] ■4, [I', I", V" etc. , L', L", L'" etc.) = \ Quam- I5 CAROL. FRID, GAVSS Quamobrom e combinatione aequallonuni [3] et [4], et fubRi- tuendo V — L', V'^L", I'" =.L"' tic. habebimus [5] pr- I Ii per T denotamus valorem functlonis t illis fubflitutionit)us re- fpondentero. Qui valor qmmi necelTario fiat quantitas finita , P cerlo nequit effe =: o. p. E. D. 10. E praecedentlbus iam peifpicuum e!t , quamllbet functionem intcgram 2' vnius indeterminatae x, cuius determinans fit =: o, decomponi pofTe in factores , quorum nullus habeat determinan- tem o. Inueftigato enim diuifore communi altiffimo functionum 2'et — , ilia iam in duos factores refoluta habebltur. Si qnis Ax hornm factotum*) iterum habet determinantem o, eodem modo in duos factores refoluetur, eodemque pacto continuabimus, donee T in factores tales tardem refoluta habeatur , quorum nullus ha- beat determinantem o. Facile porro perfpicietur, inter hos factores, in quos J^refol- vitur , ad minin\um vnum reperiri debere ita comparatum, vt in- ter factores numeri, qui eius ordinem exprimit, binarius faltem non pluries occurrat, quam inter factores numeri in, qui exprimit ordinem funclionis T: puta, fi fiatuatur mz=.k.2.i^, denotante k riumerum imparem , inter factores funclionis T ad minimum vnus reperietur ad ordinem A'. 2" referendus, ita vt etiam A' fit impar, atque vel v^^^jj., vel p ^ f^. Veritas huius alTertionis fponte fequi- tur *) Reuera quidetn non niii factor ifte, qui eft ille/diuifor communis, de- terminantem o habere poteft. Sed demonftratio huius propofitionis hocce loco in quasdam ambages perdueeret; neque etiam hie necefl'aria eft, quum factorem alterum, fi et huius determinans euanefcere poffet, eodem modo tractare, ipfuraque in factores refoluere liceret. DEMONSTR. NOVA ALT. THEOR. OMN. FVNCT. ALGEBR. etc. j^ tar inde, quod m eft aggiegatum numerorum, qui ordlnem llngu- lofum faclorum ipfius 2' exprimunt. II. Antequam vlterius progrediamur, expreflionera quandam ex- plicabimus, cuius intioducUo in omnibus de functionibus fynime- tricis disquifltionibus maxiniam vtilitatem affert, et quae nobis quoque peropportu.na erit. Supponamns, M efTe functionem qua- rundam ex indeterminalis a, b, c etc., et quidem fit /^ niultiuido carnm, quae in expreflionem M ingrediuntur , nullo refpectu ha- bito aliarum indeterminatarum , C quas forte implicet jpfa M. Permutalij illis ju indeterrainatis omnibus quibus fieri poteft modis turn inter fe turn cum m — ^ reliquis ex a, b, c etc., orientur ex 31 aliae expreffiones ipfi JI Gmiles, ita vt omnino habea'ntur 7/2 (in — i) (in — 2) (in — 3) (in — ;u+i) expreffiones , ipfa Minclufa, quarum coiuplexum fimpliciter dice- mus coinplexuin oiiiuiuTii M. Hinc fponte patet, quid fignificet aggregatiim omnium M, productum ex omnibus M etc. lia e. g. Tt dicetur prodnrtnm ex omnibus a — h, y productum ex omnibus V bus X — a, v' aggregatum omnium etc. Si forte M eft functio fymmetrica refpnctii quarundam ex ^ indeterminatis, quas continet, ifiarum permulationes inter fe fun- ctionem M non variant, quamobrem in complexu omnium ikf qui- libet terminus pluries, et quidem 1.2.3.. ...j; vicibus reperietur, C I' eft muhiiudo indeterminatarum, quarum refpectu M eft fym- metrica. Si vero M non folum refpectu v indeterminatarum fym- metiica eft, fed infuper refpectu v' aliarum, nee non refpectu v" aliarum etc., ipfa M non variabilur fiue binae e primis j^ indeter- minatis inter fe permutentur, fiue binae e fecundis v', fiue binae e tertiis v" etc., ita vt femper 1.2.3 V. 1.2.3 v'. 1.2.3 »'"etc. C permu- jq CAROL. FRID. GAVSS permutationes terminis indenticis refpondeant. Qnare fi ex his terniinis identicis femper vnicum tantuiu relineamus, omnino habebimus m (m — i) (m — s) (m — .'^) ("^ — M + O 1.2.3 v. I.e. 3 v\ 1.2.3 J'" etc. termlnos, quorum ccniplexum dicemus coinplexum oinniwn M exclufis repetitionibus , vt a coniplexu otnniuTa M odniijjls repeti- tionibus difiinguatur. Quolies nihil expreffis verbis moniiiim fuerit, repeliliones admitii femper ftibintelligemus. Ceterum facile perfpicielur, aggregatnm omnium M, vel pro- ductum ex omnibus M, vel generaliter quamlibet functionem fytii- metricam omnium M femper fieri functionem fymmetricam in- determinatarum a, b, c etc., fiue admittaritur repetitiones, fiue excludantur. 12. lam confiderabimus, denotantibiis Ji, x indeterminatas, pro- ductum ex omnibus u — {n-\-h) x-\-ab, exclufis repetitiocibus, quod per ^ defignabimus. Erit itaque ^ productum ex | rn (in — 1) factoribus his u — (a-\-b) X -\- ab u — {a -\- c) X -\- a c u — (^a -\- d) X -\- a d etc. u — (b-^ c) x-\-bc u — {!?-[- d) x-\-bd etc. u — (c -\- d) X -\- c d etc. etc. Quae functio quum indeterminatas a, h, c etc. fymmetrice impli- cet, alTignari poterit functio Integra indeterminatarum u, x, Z', l'\ 2' 'etc. DEMONSTR. NOVA ALT. THEOR. OlIN. FVNCT. ALGEBR. stc. |q I'" etc., per z denotanda , quae tranfeat in ^, fi loco itideterminau< rum /', /", /'" etc. fubftituantur A.', X", "k"' etc. Denique deGgne- mus per Z functionem folarum indetciniinatarum u, x, in qnnm z tranfit, li indetermlnalis Z', V, V" etc. liibuamus valores detern.i- natos L', L", L'" etc. Hae tres functiones ^, z, Z cotifidernri pofTunt tamquam fun- cllones integrae ordinis | m (in — i) indeterminatae u cum coeffi- cienlibus inuctermiiiatis , qui quldcni coefllcientes erunt pro ^, fucctiones indeterniinaiarum x, a, b, c etc. pro z, functiones indeterminatarum x, l',l", I'" etc. pro Z, funcliones folius indelenuinatae a;. Singuli vero coelTicientes ipfius z tranfibunt in coenicientes ipfius ^ per fubiiilutiories I' — X', l":=.\'', I'" —\"' etc. nee non in coef- ficientes ipfius Z per rublliuuioaes I'zzL', V'^zL", V" — L,'" etc. Eadem , quae modo de coeflicientibus dixinnis, etiam de deternii- nantibus functior.um ^, z, Z valebunr, Atque in bos ipfos iani propiws ioquiremus , et qiiidem eum in finem, vt demonftretur Theorep.ia. C)uoties non eft Pno, deienninans functionls Z certo nequit ejje identice z=. o. n- Peifacilis quidem effet demonftrelio huius theorematis, 11 fupponere liceret, 2' refolui pofl"e in factores fimplices {x — A) (x — B) (x — C) (x — D) ..... Tunc enim cerium quoque eflet, Z effe productum ex omnibus ii_(^^B) x-\-AB, atque determinantem functionis Z productum e dlfferentiis inter binas quautitatum {^■\-B) x—AB (^ + C) x — AC iA + D) x^AD etc. Cs C^ + C) 2Q CAROL. FRID. GAVSS (B + C) x — BG iB^D) x~ED etc. {^C + D) x — CD etc etc. Hoc vero productum identice euanefcere nequit, nifi aliquis facto- rum per fe identice fiat = o , vnde fequeretur, duas quantilatum ^, B, C etc. aequales effe, adeoque determinanleni P functionis 2" fieri —o, contra hyp. At fepofita tali argumentalione, quani ad inftar art. 6. a pe- tllione principii proficisci manifeftum eft, flaiim ad demonlha- tionem Uabilem theorematis art. la. explicandam progredimur. 14. Detetminans fnnctionis ^ erit productum ex omnibus dlffe- rentiis inter binas (^a-{-b) x — ab, quarum differentiarum tnul- titudo eft I 7n (in — 1 ) (i '" ('" — 1) — *) — ^ ('" +0 '" ('" — ^ ) ('" — 2) Hie numeriis ilaque indicat ordinem determinantis functionis ^ re» fpectu inJeterniinatae %. Determinans functionis z qnidem ad eiin- dem ordinem pertinebit: contra determinans functionis .Z vtique ad ordinem inferiorem perlinere poteft , quoties fcilicet quidam cocflicientes inde ab altiffmia potefiate ipfius x euanefcunt. No- firxim iam eft demoiifirare, in determinante functionis Z omnes certo coefRcientes euanefcere non poffe. Propius coniideranJo differentials illas, quarum productum eft determinans funciioRVs ^, depr* hendemus, partem ex ipfis (puta differentials inter binas (a -f h) x — ah tales, quae elementum com- mune habcntj fuppeciltaie productum- ex omnibus (a — £} (jc — c) e re- DEIVIONSTR. NOVA ALT. THEOR. OWN. FVNCT. ALGEBR. etc. 2 1 ei-eliquis vero (puta e differenliis inter binas (a -\- b) x — ab tales, quarum elemenla diuerfa funt) oriri productum ex omnibus (a-j-fc — c — d) x — ab-\-ccl, ex- clufis repetilionibus. Productum prius factorem vniimquemque a — imanifeftom — 2, vicibus continebit, quemuis factorem x — c autem (m — 1) {m — a) vicibus, vnde facile concludimus , hocce productum fieri _- T^m — 2 ^(m— 1) (nz— 2) Quodfl ita productum pofierius per § defignamus, determinans functionis ^ erit — ^ m — 2 i; (m - 1) ( ni — 2) p Denotando porro per r functionem indeterniinatarum x,l',l",l"' f-^c. earn, quae iranfit in § per fubltitntiones I' — \',l" z^X",l"' — X'" etc. nee non per R functionem folius x, earn, in quam IranfiLr per fubflitutiones l'=.L\ l" = L", l"'zzL'" eU., patet determinan- tem functionis z fieri — p m — 2 y ( j)i — O ( "^ — 2) r determinantem functionis Z autem _ pm-2. 2"( m - 0 C "» — 2) R Quare qunm per hypothefio P non fit rz o , res 5am in eo verti- tur vt demonltremus, R certo identice euanefcere non polle. 15. Ad hunc finem adhuc aliam indeterminatam w introduce- mus, atque productum ex omnibus {a-{-h — c — d)w-y{a — c){a — d) excluGs repetitionibii? conGderabinms, qnod quum ipfas n,b,ceic. fymmetrice inuolnat, tamquam funciio integra indeterminata- rum It;, >/, /C", >."' etc. exh^beri poterit. Denotabiiuus banc C 3 funciio- 22 CAROL, FRID. GAVSS funciiosiem per / (lu, \', X", X'" etc.) MuJiiludo illorum faclo- rum {a -\- h — c — d) w -Y {a — c) {a— d) erit — ^7;i(;«— i) (?« — 2) (7/i — 3) vnde facile colligimus fieri /(o^ X', X", X'" etc) = Tf On - OO" - 3) et proin etiam /(o, V, I", V" etc.) =: p0"-O0"-5:) nee non /(o, L', L", L"' etc) =r pO"-2:)0«-3) Functio /(to, U, L", L'" etc.) generaliier quidem loquendo ad ordinem i m (ill — i) (ni — o) (711 — 3) referenda erit: at in calibus fpecialibus vtiqiie ad ordinem infe- rioiem pertinere poteft, fi forte conlingat, vt quidam Goefficien- tes inde ab altilEma poteftate ipfius w euanefcant: inipojlibile autem eft, vt ilia farictio tota fit identice r^ o , quum aequdtlo modo inuenta doceat, functionis falteni terminum vltimum non eiianefcere. Supponemus, terminum allifEmuni functionis f(u:, L', U', L'" etc.)» qui quidem coefficientem non euahefcen- tem habeat, effe A"iu". Si igitur fubftituimus w=:.'x — o, patet j (x — o, L.', L", U" etc.) efTe functionem integram indetermi- natarum x~, a, Cue quod idem eft, functionem ipfius x cum coef- ficientibus ab indeterminata a pendentibus, ita tamen vt termi- nus altifTimus fit A^x' , et prbin coefficientem delerminatum ab a non pendentem habeat, qui non fit -zzo. Perinde /(x — &, U, L.", U" etc.), J (x—c, U, L", U" etc.) erunt functiones integrae indeterminatae x, tales vt fingularum terminus altifll- mus fit A'x" , terminorum fequentium. autem coefficientes refp. a t, c etc. pendeant. Hinc productum ex m factoribus fix — a, L', L.", U" etc.) f(x — b, L.\ L", L'" etc.) f (x — c, U, L", L'" etc.) etc. erit DEMONSTR. NOVA ALT. THEOR. CJIN. FVNCT. ALGEBR. etc. ^5 erit functio Integra jpfius x, cuius terminus altiflimus erit N'^x^" , diim terminorum fequentium coefficientes pendent ab indeteroii- natis a, b, c etc. ConGderemus iam porro productum ex m factoribus his f (x^a, I', l'\ V" etc.) f{pc — h, I', I", I'" etc.) f (^x — c, I', I", I''' etc.) etc. quod qunm fit functio indeterminatarum, x,a,b,cetc.\ V, I", I"' etc., et quidem fymnietrica refpectu ipfarum a, h,c etc., exhiberi poterit tamquani futictio indeterminatarum x, X',X",X"' Glc. l',l",l"' etc. per (p (x, X', X", X'" etc., I', I", I'" etc.) denotanda. Erit itaque (p {X, X', X'', X'" etc., X', X", X'" etc.) productum ex factoribus f (x— a, X', X", X'" etc.) f(x — b, V, X", X'" etc.) / (x — c, X', X", X"i etc.) etc. et proin in:) ? etc. erunt (?, Q' , Q" etc. funciiones integrae indeterniinatarum m, x, a, 3. V. *'. .3'. y'. «". P"' V" etc. atque *) Vel nobis ron monentlbus qakqne videbit, figDa in art. praec. intro- ducts reCiringi ad iilam fo!um arricuium, et proin flgnificationem cha- racterum ^ , w praefentem non effe confundeDdam cam priftina. DEMONSTR. NOVA ALT. THEOR. OMN. FVNCT. ALGEBR, etc. jC d(p(u,x) ^ an = VV + ict' + e>'u + y'x). ^- O X etc. a u etc. Subftitutis hisce valoribus in factoribus , e quibus coriflatur pi'oductum 12 , puta in ax 0 2/ ^ ^" + "'^ d^' ''~"'- d^' ^'' ^"' ^'" ^'''•^ — z-^ {u, X, u), I', Z", Z-" etc.) nee non dZ d2 / (u + ro , -,- , X — zi; . -r- , i', i", L'" etc. d X d M = Z.// (m, X, !i), Z.', L", L'l' etc.) Quodfi itaque functionem Z Hinpliciter exhibemus per F {u, x) ita vt haheatur /(«, X, L', L", U" etc ) = F{u, x) erit DEMONSTR. NOVA ALT. THEOR. OMN. FVNCT. ALGEBR. etc. 2'7 erit identice ^ dZ dZ\ IS. Si itaque e %'aIorlLu3 determinatis ipfarcm ii, x, pnta ex u-zzU, X := X prodire fupponirr.us d Z AZ ax Q« eiit identice F(U-\- IV X', X — 7^ U') - F(L', .J) . ^(U, X, 10, L', L'\ U" etc.) Quoties U' con euiinefciE, itaiuere licebit _ .X — ^ '« — ^., vnde emcriiit XX' X'x ■ ,. , X — :>; quod etiam ita enuncla:e licet: ';' W-il'stfr' ;■' Si in fuaclione Z ualuilur u rr U -{ jtj~ — Tt" * ireJiii- bit ea iu F{JJ, X) . ^1/ (L^ X, -~~, L!, L", L'" etc.) 19- Quum in cafu eo, vbi non el't Pzro, dcterminaas funciio- nis Z fit functio indeterniinatae x per fe non euanefcer.s, m.ini- fefio niultiludo valorum determinatotuni ipfius x, per quos hie determinans .valorem o nancifci poteft, erit numerus finitus, ita vt infinite multi valores determinati ipfius x afilgnari poffinf, qi)i determinanti illi valorem a o diuorfum concilient. Sit X talis valor ipfius :ic { quern infuper realeni fuppcnere Iice|.). Erit itaque D c dtter- gg CAROL. FRID. GAVSS determinans functionis P (u,X) non =o, vnde fequitur, per theo- rems II. art. 6 , funcllones dF(u, X) ^C".X)et— ^^^ habere non polTe diuiforem vllnm communem. Supponamns porro, exftare aliquem valorem determinatum ipGus u, puta U (flue realis fit, fiue Imaginarius i. e. fub forma g^liV — i con- tentus), qui reddat F(u,X)= o, i. e. effe F {U, X) r= o, Erit itaque u — U factor indefinitus functionis F(m, X), et proin functio certo per u — Z7 non diuifibilis. Supponendo d u d F(u, X) itaque, banc functionem nancifci valorem U' , u ua- ^ au tuatur u^zU, certo elTe nequit U'=. o. Manifefio autem U' etit va- . . d Z lor quolientis dilterentialis partialis — — pro u^:U, x=:X: quoJfi itaque infuper pro iisdeni valoribus ipfarnm z;, x valorem quo- . . . . dZ tientis difFerentialis partialis 3 — per X' detiotemus, perfpicuum eft per ea quae in art. praec. demonflrata funt, functionem Z per fubftitutionem XX^ X'x identice euanefcere, adeoque per factorem X' y XX*\ indefinite elTe diuilibilem. Quocirca fialuendo u=.xx, pater, F{xx,x) diuifibilem effe per X' /- XX'N adeoque obtinere valorem o, fi pro x accipiatnr radix aequatiouis X' /- XX'\ DEMONSTR. NOVA ALT. THEOR. OMN. FVNCT. ALGEBR. etc. 29 i. e. fi Itatuatur ^ - at/' qaos valores vel reales efTe vel fub forma g-\-hV — i contentos eonitat. Facile iam demonfiratur , per eosdem valores ipfins x etiam functionem 2^ euanefcere debere, Manifefto enim (^ (xx, x, X', X", V etc.) eft productum ex omnibus (x — '«) (x — b) exclulis repetitionibus , et proin zru"*"*: Hinc fponte fequitur ^ (xx, X, I', I", V" etc.) — y"^--'-

.58 3ac. Riccati p. 57 Saverien ibid. Ravioli p. 60 Sctunidt P- 59 Schultz p. 6e Wolf p. Q7 Ibid, ibid, ibid, ibid, p. 60 ibid. P- 59 p. 67 ibid, ibid, p. 60 p. 69 p. 67 Errata. 2 liir. 25 legatur praestet pro praestare 6 - — 13 — quern — qiiam . . (AB)^ 8—11 loco commatis inter et AG ponenclum est si- AD gnum aequalitatis — 25 legatiir CE pro CB — moniusse pro monuisse 1 a 2"- an AQ pro AO — CB. sin ^' pro CB. sin ^ — primum pro milii — AI pro AB — clato pro data — 2AD pro AB, AC — IG pro AG — AI pro AB 51 — 11 — est pro sit — — AB, AD pra AF, AG lin. 26 — hVi: sin. RAQ r sin. PAR : sin PAQ pro AQ : : sin RAQ : sin QAP lin 28 — ex praecedentibus pro praecedentibus — Iin.28et29 All: AI: AG pro AI AH:: AG — , lin. 50 deleatur : sin. PAR 56 signtftn # e linea 10 transponendum; est in lin. 9 inter AD et B'C — lin. 9 inter B'C et AR ponendum est signum |) p. 8 — 25 p.. 12 — 30 P'i3 — 13 p. 15 — ult. p. i6 — 24 p. 24 — ult. p. 25 — 21 P'27 — 2 1 F 29 — 10 — 31 p. 59 sub fin. legatur dividenda pro divenda. p. 41 lin. 9 Rin_i +^m pl^O I^m_i-Ra __ lo -- . ^'^-'R„,_i;pro^"-R^_. p. 42 — 1 — pit) p. 45 — 2 — niCm.pro MCni p. 48 — 2 -- quum pro quain p. 51 _i8 _ Be— «/K:pro Bca— V^I"^ p. 60 — 11 — MH pro MB p. 61 — ult. — 1804 pro 1814 p. 62 — 6 — constiiutae pio constituae Cetera, quae le\-ioris momenti sunt, bejievolus lecior ipse a.^noscat atque corrigat. PRAEMITTENDA QUAEDAM. Quum omnes fei'e hominiun artes, omnisque doctrlna ac disciplina ad varias humanae naturae necessitates praecipue spectent, vel quum, ut animi nostri vires excitentur, irritainen- ti?, iisque saepissime exteruis, indigeanius, eas potissimum scien- tias, quae plurima ad ea, quae cuiqiie maxime sunt necessaria, praebenda et praestanda conferunt, tractari atque excoli, per se saiis apparet. Jam sexcenties matheseos, quod niaximam horai- nibus utilitatem adtulerit, prosperrimo successu mullis rebus in vita communi obviis, immo toti fere physicae adplicata fuerit. laudes sunt praedicatae. Mechanica, sen scientia de niotu cor- porum, jure meritoque ad potiores matheseos, quani vocant, adplicatae partes refertur. Itaque jam inde ab antiquis tem- poribus multi deinceps viri, iique saepe ingenii praestantissimi pertractandae atque excolendae huic scientiae cperain navarunt Incredibile eft dictu, quantopere Archimedis, Galilei, Neutoni FAileri, Lagrangii alioruraque virorum opera Iiujus scientiae exactissima cognilione progressi simus. .. §• ^- Natura animi liumani ita comparata eft, ut, si quis primam curam et cogitationem in rem aliquam conferat, in singulis ac sppcialibns, quae ei offeruntur, perpendendis acquiescat. Tem- poris demum progressu , his singulis varie inter se confe- rendis atque nectendis, ad rerum proprietates magis universas ac gpnerales accediL, Item animo omnino exciiltiore opus eft, ut quis rerum, quas jam diu fortasse notas habet, caussas explo- rare studeat. Hinc intelligitur, quomodo factum sit, ut plura tnechanicae theoremata multum antea cognita sint, quam cui- A 2 PRAEMITTENDA. dam leges, ex quibus haec omnia deduci, immo secundum qnas- ampliari atque dilatari po^^sint, univcrsas explorare in nientem inciderit. Quibus de mechanicae principiis siunmi demum viri, Leibnitius, Wolfius, Bernoullius, Lambertu':, Fdccatus , Alem- bertus, Eulerus, Lagrangiiis, Kaestnerus, Kar'^tenius, Salimbe- nus, Ferionus aliique magni geometrae disquisitiones in^titue- rnnt. Multum de his principiis, sed multo niagis de niotuum caussis, 'sen de viiibus inter geometras, et praesertim inter Ita- los, est disputatum. Quorum quidem omnium expositio idonea atque apta, quantumvis essetutilis, partial tamen ad propositrun nostrum baud conveniret, partira, ut ingtnue faleavj tironis viies longe excederet. §■ 3- Quocunque autem mudo geometrae de his rebus inter se dissideant, ejus tamen mechanicae partis, quae de virium aequi- librio agit, tres potissiumm propositiones esse fundamentales, inter omnes fere constat. Quavis earum, tanquam fundamento, cetera omnia staticae theoremata superstrui possunt. Sed ne ab lis, quae hie prosequimur, nimimn aberremus, in unam tan- tummodo ex his tribus, in illam scihcet, quam compositioein vel parallelogrammura virium vocant, mquirere nunc possumus. Quae quidem doctiina, una vel altera denominatione in libris staticis et dynamicis fere omnibus obvia, consislit in eo, ut, ^-iribus, quotcunque libuerit, una corpus quoddain soUici- tantibus, unam omnino esse vim, quae sola eundem, quern illae conjunctim agentes, effectum praestare, suinatur. In de- terminanda autem cum directione, tam quantitate ejusmodi potentiae omnis haec doctiina versatur, ^- . 4- Quum omnes autores statiri pimctum id, in quod ^uae yires una agant, secundum diagonalem moveri, ideoque pun- ctum ad eundem, five vires actiones suas conjunctim, sive sin- gulatim exseruerint, locum ferri contendant, haec libenter uti vera accipimus. Veri atque longrui quid, ut ita dicam, in eo con- spiciinus, quod duae ejusmodi vires, etsi altera alterius actio- nem iiUHunuit, euiidem taxuen effectum praestant, h. e. quod PRAEMITTENDA. 5 pnncTum coniuncta ipsariim acrione ad eundem lociun, ad quern singulis deinceps agentibus latum fuisset, movetur. Hinc iheo- rema nostrum satis diu, etsi non rigida demonstratione miuni- tum, non solum notum es?e, sed etiam ad alias ejus ope pro- positiones probandas adhiberi potuisse intelligitur. §• 5- Quod attlnet ad originem propositionis nostrae, secimdum ea, quae summus geometra a) refert, prima ejus vestigia apud Galileum b) reperiuntur. Idem vevo Italus geometra aliique ne suspicabantur quidem, quanti momenti theorema nostrum sit, et quam facile ab eo plures, ni omnes staticae propositiones de- deduci possint. Ipsius plani inclinati theoriae, quae proxime ad Composilionem viriiun accedit, exponendae tunc temporis theo- rema nostrum baud adhibitum est. INIulto satis post Petrus Varingouius omnia certe ea staticae theoremata, quae ad aequi- libiium macliiuarum pertinent, hac ipsa nostra propositione su- perstrui posse ostenditj multoque serius Dan. Bernoullius primus demonstrationem ejusdeni theorematis magis rigidam conficere tentavit. Cujus quidem \iri exemplum aliis secutis, plures ejusmodi demonstrationes accepiuxus, quas omnes mox cogno- scemus. §. 6. Quaedam nunc de his demonstrationibus, generatim specta- tis, praecipue vero de Kantiana disserere milii licitum sit. Ma- thesis adplicata, uti nomen satis indicat, ea est scientia, quae de mathesi variis naturae externae rebus, quatenus ipsae aut ad calculum, aut ad constructiouem geometricam revocaii pos- sunt, adplicanda agit. Quam quidem disciplinam a mathesi pura distare ac differre, per se satis apparet. Paucis vero, qua- a) Lagrange mecanique analytique; ab init. Vcrbotinus totus hie locus iterum l^gitur in libro : Montucla histoiie des mathemat. Tojn III- p. 608- sq. b) Galileo opere. Bologna 1556. Tom II; Dialoghi p. igo, A3 4 PRAEMIT'TENDA. nam in re hoc discrlnien positnm'sit, exponere, hand inconve*' niens essemihi videtur. iMatheseos nimiruni purae principia vel axiomata, quibus omnia superftruuntur, ita comparata sunt, ut ex ipso nostro intellectu, eoque solo prodierint, vel ut mens noslra, cujusque alius rei ratione plane nulla habita, secundum ipsius tantummodo naturam atqne indolem haec sibi finxerit et eonstituerit. Itaque non solum liaec ipsa principia, sed etiam omnia ea, ad quae iisdem rite adhibendis pervenitur, intellectui nostro ita accommoddata esse, ut contrarium eorum, quae ibi pronuntianttir, ne cogitari quidem possit, omnino necesse est. riinc ilia perspicuitas atque evidentia, qua omnia gaudentj quae matliesis nos docet. Ex his ipsis vero jure colligendum est, hanc evidentiam ve» ritatemque necessarian! non amplius locum habere, qunra prae- ter principia vere raathematica simiil et alia spectentur, quae non a sola mente nostra statuta, seu potius ab ea sola, ut ita dicam, genitasunt, vel, quod idem eft, cum mathesis externae naturae rebus adplicetur. Turn enim et universarnna naturae legum, quibus ejusmodi res vel corpora obnoxia sunt, et varia- rum, quibus gaudent, proprietatum multorumqne alioruni) quae, licet plane non geometrica sint, aut philosophia, aut ex- perientia nos docet , ratio omnino est habenda. Turn proposi* tiones, quae neque calculo, neque consructione geomtMrica de- monstrari po^sunt, in sub«idium vorari debejit. Ouodsi enim V. c. in propositionein hanc: " Vis duarum aeqiialium media, si ejus directionem respicias, angulum, quern illae coinprehenduntj hifarlara secat. " si, inquam, in hanc propositionem diligentius inquirimus, mox neque ip'^um axioma esse vere geometiicum, neque ex his deduci, vel sola eoriun ope d'='monstrari posse vide- bimus. Attanien omnes eos auctores statico'; , qui hanc propo- silionem, tanquam axioma acceperunt, minimp reprehendendos esse puto. Ibi enim omnia, quae ad determmandam resultantis directionem aliqiiid conferre possunt, utrimque ita inter se sunt aequalia, vit nulla reperiri queat caussa, qua ip';a haec directio propius unam, nuam alteram viiinm componontium anedat. Eadem, quae de hac proposilione diximus, de uiultis aliis, non PRAEMITTENDA. 5 vere matlif matlcis , sed taraen in mathesi adplicata accipieildis; did possunt. Ex his auteni malheseos adplicatae propositiones, universe spectatas, non eodein, quo matheseos purae iheore- mata, sensu ac niodo, vel non plane geonietrice demonstrari poss?, patet. Est igitur omnino , lit Kanlio, conipositionem nio- tuuni rigore plane vereque geometrico a se demonstratam esse, dicenti, non statim fidem praebeamus. Postqnam infra autem ipsum Kantii argnmentum exposuerim, spero, fore lit, banc demonslrationein ^ejusmodi rigore haud gaudere j ostendere possim. Panca quaedam denique jde discrimine inter compositionem •viriuni et conipositionem motnum adjecerim. Multi auclores statici, has duas notitiones haud inter se distare, sumsisse videntur. Itaque, una cum compositione nio- tuuni demonstrata virium quoque conipositionem probatam esse, existimamnt. Quantum autem equidem video, res non ita se habet. Quum enim sine dubio phires vires, secundum diversas directiones in corpus vel punctum quoddam una agentes concipi possint, mprito quoque, quaenam sit corporis, hoc modo ad motum sollicilati, directio et celeritas, disquiritur. Unusquis- qiie autem, si quis eidem corpori eodem tempore plures secun- dum diversas directiones motus tribueret, hoc plane absurdum esse, intelligit. Quodsi igitur de compositione motuum sermo est, hoc tantumraodo quaeri potest, quinam sit motns ille, quo solo corpus ad eundem perveniret spatii locum, ad quern pkiri- bus singulis deinceps motibus latum fuisset? Quae vero tunc re'^pondenda sint, per se ita apparet, lit nullis disquisitionibus, rrllaque demonstratione opus sit. Hoc autem ipso composi- tionem virium nondnm esse deiiionsrataiii; non minus darum atque perspicuum est. PRINCIPIA, Pars commentationis prior. Demonstratlones eae, qiiavum aiiciores coinpositiouem vi- riitin', ut jmidamentalem magis staticae, qjiant mecha- nicae propositionem pt'-ol^andam sibi snmunt, ideoque motiis , a vlribus adhibitis efftcieiidi^ saepissime nullam fere rationem habent. Liber primus. Antoref^ qni theorema nostrum sine titla alius staticae pyopositionis, ope denionstrare student. :. ^- ^- .. . Principia vel axiomata, qnibus alils aliae harum demon- stratiuiium nituntur, haeccine sunt: i) Qnarumvis potentiarum loco aliae, illis aequivalentes, svxbstitui possnnt. 2) Plures potentiae, plane inter se conspirantes, uni poten- tiae, summae earum aequali, aequipoUentes habendae sunt. 5) Potentia duarum aequalium media angulum, quam hae comprehendunt, bifariam secat. 4) Duae vires nonnisi aequales ac plane repugnantes sunt inter se in aequilibrio. 5) Item duae potentiae, in aequilibrio positae, aequales et inter se repugnantes sint necesse est, 6) Aequilibrium, quod inter plures constat potentias, aut addendis, aut auferendis et ipsis in aequilibrio viribus, baud toUitur, 7) Aequilibrium, in quo plures versantur potentiae, etiam inter omnes illas vires, quae, secundum similes directiones agen- tes, quodammodo illis sunt proportionales locum habere oportet. quibits demonstrationes , in libr. I obviae , nituntur. 7 8) Artio cujnsdam potentiae semper sibi constat, quocun- ^ue directionis suae loco ipsa adplicata concipiatur. 9) Vires omnino in aequilibrio esse nequeunt eae, quae cor- pus seu punctnm, in quod agunt, omnino omnes ad eandem spatii partem*) movere tendunt. Sectfo prima, Demonstrationes geoimtrkae ^ atque geometricO' analijticae. C 1 a s s i s prima. Demon stratknes pvinap'ns vet axicmatibiir** ) 1. 3. 5. 4. 6 sfipersrucfae. Cap. I. Argiimentum BernouUianum ***)♦ §. 10. Theor. Qnodsi plnres vires, pnnctuni quoddam sollicitan- tes, inter se in aequilibrio versantur, potential etiauieas, qnae, qiiodamniodo illis proportionales, secundum easdem directioues agunt, in aequilibrio esse necesse est. *) Ad eandem spatii partem plures v\xes corpus quoddam movere dicnntur, dum omnes, sicuti v. c AB, AD, AC etc. (lig. i.)ab eadenilineae leciae C F parte sunt adplicatae. ♦♦) Auctores ipsi haec axioEiata mode rite pronuntiaverunt, modo tacite suppleii volueie. *♦♦ Dan. Bernoiillii examen piincipioium meclianicae; v. Comment. Acad. Petrop 1728 Tom I. p. 12(3 sqq. g Class. L Cap. I. Demonstr. Quum enim vires, diiplo, triplo etc. deinceps auctas concipiamus , his sane incrementis , et ipsis inter se aequi- librantibus — sit venia verbo — aequilibriuni tolli plane ne- qxiit. Eadem omnino, durrt ab liis \iribus, quarum quamvis in tantasdem partes aequales divisam ponamus, ejusniodi par- tes deinceps auferuniur, valere debent. §. 11. Theor. "Vis ea AD (fig. i.), quae duabus aequalibus rectan- gulisque AB, AC, aeqnivalet, si ejus quantitatem respicias, per diagonalem quadrati exhibetur. Demonstr. Ducatur EF perpendicularis ad AD fiatque (AB)^ EA = -, AG = AF. Quum verositBAD =DAC=:4g°, ideoque EAB z= BAD = CAF, potentia AB duabus AE, AG, itemque AC viribus AG, AF aequivaleus habenda est. Itaque potentia AD viribus his: 2AG, AE, AF aequipolleat necesse e*tj vel cum AE, AF semet ipsas invicena destruant, AD = 2AG = -i:^ — i-,. et hinc (AD)^ = 2(AB)» AD ^ ^ ^ Coroll. Potentia igitur duabus aequalibus ac rectangulis ae- quivalens, tam quantitatis quam directionis respectu per diao-o- nalem exprimitur. §. 12. Theor. Ouum potentia CF (fig. 2) duabus rectangulis, sed Inaequalibus CD, CE aequipolleat, aequationem hanc, (CF)^ = (CD)* -\- (CE)^ locum habere necesse est. Demonstr. Ducatur et nuncBA perpendicularis ad CF; deinde (CD)» (CR)^ CD.CE sitCG= ■— ^-, etCH=^^, deniqueCB=CA = . His CF Cb • CF . ita positis, CD duabus CG, CB et CB aliis duabus CH, CA aequivalere necesse est; unde patet potentiam CF hisce aequipollere poten- tiis; CH, CG, CB, CA, vel, quum CA, CB, sint inter se aequa- Argumentum Beruoullianum. 9 (CVj)^ , (CF,)2 les ac repugnautes CF =::; 1 , seu (CF)* — (CD)* hfH(CE)^ ^^^ ^^^ $■ 15- Theoy. Quodsi vis quaedam AD (fig. 5.) duabus aequalibus AB, ACaequivalet, atque AE, AF duae aliae vires sunt aequales, quae, et ipsae polentiae AD aequipoUentes. angulos BAD, CAD bifariam secant (posilis AB = AC z=i a, AE =: AF = x, AD = b) I a . I. aeqnationem hanc x :=: b y ( ; — ) prodire necesse est. 2a-|-b Demonstr. Posito enim AG=AH=AL=:— n AE duabus b' AG, AH, et AF aliis AH, AL aequipollet ; porro sit AM = buAG x^ ♦ -r = — •) Iiinc ergo AM viribus AG, AL, ideoque AD poten- r a a ••TV • '^^ 1 ^'^^ tus AM, 2AH aequivalet, habebimus itaque b=: — -{- ——7 a ^ b a . vuide elicimus x = b y( Quum AB, AC (fig. 4.) duae sint vires aequales ac rectan- gulae, ducta AD perpendiculari ad BC, binas vires, AG, AH, item AF, AF, etc. angulos BAD, GAD etc. bifarianl secantes, eideni potentiae 2 AD, cui AB, AC aequipollent, aequivalere necesse est. - Demonstr. Postquam enim ex theoremate precedcnte va- lor virium AG, AH determinatus fuerit, habebimus a 1/2 AG = AH = — — V— TT \/(2+l/2) Idem vero valor geometricus duabus AG, AH, angulos ABD, ADC bisecantibus, statuendus est; ergo potentiae AG, AH po- tentiae 2 AD aeqitivalent. Eadem facile pro AE, AF aliisque binis ejusmodi viribus probari possunt. B 10 Class. I. Cap. I. CoiOil. Binis igitur viribus aeqnalibns, uno horumce an- guloruin: iR, |P\, ^^R, -|R etc. in punctum quoddam agenti- bus, vis earum media, tam quantitatis, quam dii'ectionis respe- dlu habilo, diagonalem exaequat. §■ 15- Onodsi vires aequales AB, AC (fig. 5.) potenliae BF, per lineam AC in duas partes aequales BL, LF divisae, aequivalcnt, lineis DE ad CF, et AD, CE ad DE perpendicirlaribus duclii, BA duabus BD, BL et BC duabus BE, BL aequipolleant ne- cesse est. Dcmonstr. Quodsi enim res non sic se haberet, BA dua- bus aliis V. c. Bd, Bl, et BC viribus Be,. Bl aequipollentes po- namus ; hinc ergo BF viribus Bd, Be, 2B1 aequivalere sequi- tur^ quod quidem absurduinj alteram igitur viriiun potenliie B\ aequipollentiura, vim BL aequai e necesse estj hiuc aulcm alteram esse potentiam BD jure colligiuius. Coroll. In omnibus igitur rpctangulis iis, quae lineis aut AB, aut AG, aut AE (v. fig. ad §. 14.) tanquam diagorialibu^ cir- cumscripta, et quorum alteruin latus est AD, duin per ea dem lineas vires exliibentur, vis inediae et quanlitas et diretliu de- teriiiinata est. §. 16. Theor. Viribus aequalibus BA, BE (fig. 6.), itr-m-ue aliis aequalibus BC, BD uni potentiae 2BL aeqnivalputibus, vires quoque BE, BG aequales, angulos ABM, NBE bifariain secan- tes, eidem potentiae 2BL aequipollere necesse est. Demonstr Productis lineis BC, BD, ita ut sit BM=:BN= BE, MN, PR ipsi AE et AP, MO, QN, RE, ipsi BS parallelae ducantur. Ex prececentibus potentiam 2BS viribus BM, BN ae- quivalere patet5 itaque si BU duabus BA, BM, et BW duabus BE, BN aequipollere ideoque angulos ABM, NBE bisecare poni- tur, duas BIT, BVV uni potentiae 2BL-|-2BS aequivalere neces- se est. Resolutis autem BM in duas BO , BS , et BA in duas BP, Ar o^umentum Bernoullianum. ii D BL aequipollentes, hanc habemus aequationem BU = i/((BP -fBO) »+(BL-f LS)»). Quum vero, uti vidimuf, potentiae BU, B\V uni potentiae 2BL-I-2BS, aeqiiivaleant , et si earum directionem respicias cum duabus BF, BG coincidant, vim duarum. BF, BG mediam esse= X posito, = — (1) esse debet. Qua quidem ae~ ^ 2BL-I-2RS X quatiune quantitas X deterininari potest. Quern in fiiiem AB i=a, BC = b, AC=:c statuamus. Ex ipsa constructione nostra stalim hae prodeunt aequationes: a' — ab- — ac* BO = MS = - BP=:AL: 2 be a2 — b^-j-c^ 2C , 2b*c^-}-2a^b^4-2a*c* — a" — b* — c* BL=-l/C ■ r ) ^^ ^^^^ 2bc v'Ca' b + ab' 4-2 a^ b^ — abc*) BU — - — — — ~- -' b Dum autem BAF et BFZ angulos inter se aequales reddamus, ex trianguloruin BAF et BZF similituduie aequatio haec BF=: i/'ra'b + ab'-faa^b^-fabc^) ,. ■ • LI ! ■ deducitur : substitutis autem m a-fb aequatione (1) quantitatum BU , 2BL -|- 2BS et BF valoribus supra repertis, hanc habebimus aequationem l/f2b*c^ + 2a*b*4-2a2c^ — a^* — b* — C*) X — i-^ ■ ■ = 2BL. Theor. Duae vires aequales AG, AH (fig. 7.), quopiam an"'ulo in punctum quoddani A concurrentes, uni potentiae, quae taui quantitatis, quani directionis respectu habito, diagona- lem aequat, aequivaleant oinnino necesse est. Demonstr. Quum enim supra, duas vires AB, AC, dum " ideoqiie 00 -{- (d' -\- u)" ^^ 180° esse ponatur. Ceterum iisdem de- notationibus retentis, erunt aequationes C/3 = CB cos (p, CS =1 CD cos ^' Cb rr CB sin " ^^ 180° esse patet^ Argumentum Lamberti. 17 Etsi igitur

" trianguli angu- los aequant; ergo quaevis potentia, cum duabus aliis in aequi- librio versans, omnino diagonalem exaequat. §. 36. Quae pro viribus rectangulis demonstrata stmt, facile, uti supra vidimus, aliis quibuspiani viribus accommodari possum. C A p. III. Argttmentv.m ab Eytehvino aUatnm *). §• 27. Potentia quadam duabus inter se rectangnlis P, Q aequi- pollente, aequationem liaiic R^=;P^-[-Q^ nobis esse necesse est. In demonstrando hoc tlieoremale noster Bprnoullium omni- no sequutus est v. Supr, §. 12. §. 28. Theor. Quodsi angulo director! w angulus hypotbenusalis O5 respondeat, ejusniodi angulo 2 m angulura hyp. quoque 2 (J) respondere necesse est. *) E}telwein's Grundlehreii der Statik nnd Meclianik fester Korper, Tom. I. ab init. Gilbert's Annalen der Pliysik, XVIII. Th. p. igi sqq. — Qaod argumentuni ab iis, quae jam exposuimus, eo quidein diaciepat, quod piopositionem, quani auctores illi in piimis tlieore- matibus evincere conati sunt, tanquam axionia pronuntiaverit. Ni- hilominus tamcn banc nostri demonstrationem jure ad banc classem referri puto. c i8 ' Class. I. Cap. III. Denionstr. Potentiani P (fig. 15.) vitn mediam, earn que in dvias vires rectangulas p, q, secundum directiones GF, GM agentes, ita, ut sit DGM=:w, ideoque CGM = 2m, .resolutam concipiamus. Simili modo Q in duas p', q' vires resolvatur. .^ His ita positis, sponte hae prodeunt aequationes P^ P. Q PQ Q^ Duarum autem virium P, Q, loco potentiis quatuor p, q, p', q' sutstitutis, dum p — q' = P' et q -{- p' = Q' posuerimus, has haLebimus aequationes P'=R(cos (p^ — sin (p^) = R cos. 2(p, 0'=:2R cos (p. sin (p =R. sin 2(p. , §. 29. Theor. Quodsi ea, quae modo pronunciavimus , 'pro an- gulis to,

QP'-J-PQ' R ' ^ R pp/_ QQi Est autem =: R (cos (f). cos nCp — sin (p. sm nDC, resultanti non eadem, quae et diagonali^DB rectanguli CA, esset directio, ab alterutra hujus diagonalis parte sitam esse illam , h. e. avit secundum Dc, aut secund\im Db agere necesse esset. Quum prius auteni illud suraamus, hoc iis, quae aiitea demonstrata sunt, oranino re- pugnat. Quum enim, viribus tanquam aeqirahbus assumtis , harum resuitantis directionem quoque cum diagonah coincidere, ideoque corpus nostrum, duabus viribvis una agentibus, inde a puncto D versus rectam AB aeque, ac si potentia DA sola egis- set, auferri videremus; quum ergo hoc respectu potentia DC, quamvis ex ejus actione corpus secundum directionem, a DA diversam, moveri nititur , alterius DA actionem haud imminu- ere dici possit, hanc actionis imiiiinutionem mullo minus tunc,- quum sit DC <; DA, locum habere necesse est. Itaque corpus et nvinc non minus, quara. spatio DA, inde a D versus AB amo- veri oportet. Sed hoc ipsum non fieret, si corporis directio es- set Ac. Sed non magis corpus, hoc modo sotlicitatum, secun- dum directionem Db moveri potest. Quum enim iunc corpus majori, quam DA, spatio, versus AB moveretur, ideoque ma- jori, quam si potentia DA sola egisset, tanquam caussa hujus rei potentia DC spectari deberet. Quod quidem onmino absur- dum. Itaque duarum inaequalium DA, DC, resuitantis dire- ( AI esse debet^ sit Ao =:AI' itaque Ab his duabus Ao, AI aequipolleat necesse est. Item Ac aequivaleat duabus Ao, AK^ itaque Ab, Ac vivi- bus his AI, AR, 2A0 aequipollent. Quoniam vero AI, AK po- *) D'AIemlext opuscules mathemaiiques, Tom. I. ^j, i6g sqq. D 2 6 Class. L Cap. YI. Argumentum Alemberti. " tenliae 2Ai aequivalent, Ab, Ac iini potentiae 2Ai-|-2Ao ae- qnivaler^ oportet. Piaeterea antem, cum sit lb !> AI, etiam IG > Ai esse debet; atqni etiam Al > Ap, ergo 2Ai-|-2Ao x, idtuque etiam X > x esse debet. §. 44. Theor. Dato quopiam angulo BAG (fig. if).)? alius omnino BAC — q. ' — — 5 tanto magis igitur k > BAG — 211 -" , 120' _ , , ratione luiueinus 120° 120 120 — BAC < (q 4- 1) ^ q 120° 2'V ergo(q+i) ^^ — BAC AH diago- nalem poneremus, atque duarum virium AG, AD, angulo quo- T). 12 0^ dam — < BAC concurrentium , vim mediam diagonalem 2" minus quam linea HR superare conciperemus, duarum AG, AD resultans major quam virium AB, AC potentia media esset. Quod quidem praecedentibus plane repugnat. Simili ratione AR «< AH esse nequire probat^^r j ergo omnino AR = AH. A r gum en turn Alemberti. 29 §. 46. Quae hue usque pro viiibiis iis, quae sunt inter se aequa- les, probavitnus, facile ad potentias quodauimodo inaequales transiVni possunt. Ouum eniiu vires AG, AD, AH, AD (fig. 20.) esse pona- mus , quarum binae angulo recto punctnm A sollicitant, et quae ita inter se comparatae sunt, ut AH = AG sit, vim omnium mediam 2 AD esse, satis apparet. Secundum ea, qiaae Bernoul- lius demonstravit, vires binarum ejusmodi mediae diagonalea AB, AC exaequant; sint autem earum directiones x\E, AF. Tunc omnino AE, AF eundem, quern et AB, AC, affectum produce- rent; quod quid em fieri nequit, nisi puncta D et K coincidant; ila autem AE, AB, et AF, AC coincidere necesse est. Quomodo omnia theoremata, pro viribus rectangulis demon- strata, quibuslibet aliis potentiis accommodari possint supra vidimus. E p i c r i s i T. Ouodsi auctor ipse banc suam demonslrationem facilem at- qne simplicem praedicat, pro mea sententia, haec laus hactenus tantmn ei tribuenda est, quatenus non nisi aliquot planimetriae theoremata in subsidium vocat. Quum enim in ejusmodi sim- plicitate recte dijudicanda ambitus quoque argumenti rationem habeas, quae, me judice, omnmo est habenda, ab hac parte simplicitas atque facilitas demonslrationis nostrae laudari nequit; nulla alia -certe earum, quae saepius nimiae prolixitatis acru.^atae sunt, ipsa praeccllit. — Quantum equidem video rigori geo- metrico hand alienum fuisset, si noster in §.41, figuram ALbl esse rhombum, demonstravisset. Quum enim lineae bL, ipsi AG parallelae, eo, quod punctum b transire sumebatm-, plane esset directio determinaia, insuper AL = bL esse, probandum fuit. Quae auctor eadem theorematis hujus parte coUigit, scili- cet rectam AG, cum lb = AL > AB sit, majorem quam Ai esse, ut essent vera, lineam li = bG, aut li > bG esse noster evi^icere JO Cl-ass. n. debuisset. Quod vero, pro niea sententia non solum probari ne- quit, sed etiam saepius vere non locimi habet. Praecipue autem ea nostro, quae in corollario primo, theoremati (§. 42.) adjecto, pronuntiavit, adsentire non possum. Nervus enim theorematis illius proband! sine dubio in eo consistit, ut Ab, Ab' latera sint rhombi, cujus diagonalis AB. Itaque non nisi tunc, cum vires in hac ratioue inter se sint positae, minime vero cum, uti noster dicit, vires quaepiam sint, ea, quae autor hoc corollario contendit, valent. Hoc autem corollario primo, qiiod doleudum^coroTlarinm secundum, inio tota demonstrationis pars posterior tanquam fundamento superstructa est. Quo quidem labefiente tota demonstratio corruere debet. Classis secunda. Demonstvationes axiomatihus 4. 5. 6. 8 snjjersruc/ne. Caput u n i c u m. Argumentiim Salimbenianuin *). g. 48. Quamquam omnia, q^uae noster in his staticae elementis proLulit arctissime inter se cohaerent, haec tamen hie exponi plane nequeunt. Itaque tantummodo ea, quae maxime ad com- positionem virium pertinent, reieram. Po.^tquam niiniruin auctor ligore vere Fuclideo plura stati- cae iheoremata, praecipue autem hoc: „quum iude a puncto *) Leonardo Saliniboni Saggio di un nuovo corso di elementi di statica V. Momorie di iiiatematica e fisica della Sccieta Ital. Torn. V. p. 426 s'-1 1- Argil men Luna Sa limL enianum. ' 51 quodam D, in directione unms triani inter se aequilibrantium viriiun P, O, R, v. c. R sunito (fig. 21.)? ceteris viribus ducan tur parallelae DB, DC5 analogiam banc R:0: P: : x'\D: AC: AB locum habere nccesse est" nitniuui prolixa denionstratione osten- derit, hoc fere niodo per git. §• 49- Ouodsi triuni virium, in pxmctimi quoddam A (fig. 22.) con- currentium, atque in eodem piano sitaruni, quaevis ad earn spatii partem, quae illi, ad quam ceterae punctum sollicitant, oppo«ita est, corpus vel punctum urget, et si, duni inde a puncto, in directione unius illarum sito, ceteris ducantur pa- rallelae, sit, r:0:T:: AF: AC: AG, liae vires in aequibibrio sint necesse est. Demonstr. Quum enim vires P, Q una punctiim A per- ineent, quaepiam vis, quae, idem punctum transient, cum his aequilibratur, 'c^se debet. Quae quidem potentia sit T. Sed non securidum directionem AH, sed secundum aliam quandam V. c. AE agere earn ponamus. Product! HA ad G, ductaque GF ipsi AC parallela, ex praecedentibus est. P : Q : T : : AF: AC: AG Sed per hyp. est-P : Q :; AB : AC, ergo AB = AF.; quae quidem aequatio locum habere nequit, ni«;i AH et AE coincidant. CorolL Eadem , quae pro potentia , cum illis aequilibrante demon- travimus, pxo ea quoque, quae iisdem illis aequipollet, valent. §. 50. Quum tres vires AP, AO, AR (fig. 25.), in pimctum A con- currentes, inter se aequilibrentnr, hancce analogiam AR : AQ : AQ : : sin. RAQ : sin QAP locum habere necesse est. Demonstr. Ductis enim GI et GH ipsis AP, AQ parallelis, praecedeiitibus erit: AP : AQ : AR : : API ; AI : AG: est auteui Al AH : : AG : : sin. c : sin b : sin a ergo etiam AH • AI ; AG : : sin QAR : sin PAR : sin. PAR : sin PAQ. Coroll. Quodsi igitur potentia AS secundum directionem AK aequipollet, banc esse analogiam AP : AQ : AS : : sin. QAK : sin PAK : sin PAQ jure hinc concluditur. ^2 Class. II. . §•51. Quum ad directiones trium virium AP, AQ, AR. (fig. 24.), inter se aequilibrantium , ducantur perpendicular es, quae satis productae triangulum EGF constituunt, hanc analogiani EG: GF : EF : : AP : AQ : AR esse contenditur. Demonstr, Ductis inde a puncto D, libere sumto DC, DB ipsis AP, AQ parallelis AP: AQ : AR : : AB : AC : AD esse debet. Quoniain vero BAD = GEF, atque DAC = EFG, etiam EGF 1= ACD esse oportet, itaque AP : AQ : AR : : sin EFG : sin GEF : sin EGF ideoque AP ; AQ : AR ; : GE ; GF ; EF esse oportet. §. 52. Quuni, datis trium, quae inter se aequilibrari debent, vi- rium directionibus BK, CH (fig. 23.), DE, ipsaruin autem viriuni una tanlummodo nobis sit nota, ceterae duae ita sumendae sunt, ut, si IG, IH lineis DE, CH, parallelae ducantui- sit analogia haec : AI : AG : AH : : AR : AP : AQ. Demonstratio nostri theorematis sponte ex iis, quae supr, §• 49. probavimus, fluit. Probl. Duabvis AP, AQ (fig. 26.), quae punctum A fran- seunt, viribus datis, reperiatur vis ea AR, quae cinu iliis aequi- libretur. Solut. Constructo parallelogrammo ARDC , AR in directio- ne KD ita siimatur, ut sit AD : AB: AC : : AR: AQ: AP. Eadem vis AR quemadrnbdum cum potentiis AP, AQ aequilibratur, ita potentia AD, ipsi aequalis sed repugnans, iisdem illis aequi- polleat necesse est. Ouani quidem soiutionem esse ratam ex precedentibus intelligitur, §•04- Probl. Datis tribus potentiis AP, AQ, AR (fig. 27.), qua- rum binai'um sumnia major, quam tertia est, inveniantur di- rectiones, secundum quas liae vires agentes, inter se aequili- brentur. Argumentum Salimb enianum. 55 Solut. Constructo parallelogrammo AHGI, ita ut sit AR; AQ; AP:: AG:AH:Al5 potentiae AR secundum AD, AQ secun- dum AH et AP secundum AI agant. His ita constitutis , vires ilias aequilibrari ex prioribus patet. §. 55. E p i c r i s i I. Nosier in his staticae elementis summa opera Id asit, ut methodum vera Euclideam sequatur, omnesquc suas propositio- nes satis rigide demon-tret. Ipsis igitur praemittenda in difini- tiones, postulata et axiuuiata dividit. Ceteriun puto liypothe- sin, quam attulit, earn: „vires, una in corpus quoddam adpli- catae, non inter se aequilibrari possunt, si omnes ad eandein spatii partem corpus vel punctum sollicitant" non majore frui evidentia, qnam plures propositiones, quas noster denionstran- das sibi sumsit. Laudanda omnino est nostri diligentia, cura, ac perspicuitas, quam in his suis demonstrationibus, quae non nisi aliquot planis geometriae theorematibus superstructae sunt, consectatur. Eorundum autem elementorum ambitum multo esse majorem, quam debeat, quisque satis intelligit. Tanta enim est prolixitas, ut non sine labore satis 'magno ad fineni perveniri possit. Quod attinet ad rigorem, quantum equidem video, nihil contra has demon^traiiones nioneri potest- 34 Class, m. Classis tertia. Demonstrationes axiomatibus i. 2. 3. 8 nitentes. Caput unicum. Argttmentum a viro doEdffimo Duchayla adlatum *). §. 56. Tlieor. Quodsi vis duarum media non solum cum. hae vi- res per p et m, sed etiam ciun per p et n exhibeamur, diago- nalem parallelogrammi, sub his construcli, exaequat, idem, dum vires laterales sint p et m-|-n, easdeinque obtineant dire- dliones, locum habere necesse est, Demonstr. Si (fig. 28.) AR = p, AE = m, et EC =r n esse poniraus, vires p, et m-|-n per rectas AB, AC exhibentur. In- columi antem potentiae m-j-n actione, fieri potest, ut poten- liain n vel EC puncto E adplicatam esse concipianins. Quum autem potentiani, duabusni, p aequivalentem, directionis suae respectu, cum diagonali AF coincidere sumserinius, hanc ipsam vim, a puncto A ad F trari'^positam, secundum directionem FK agere concipi potest. Resolvatur nunc, quod omnino conceden- dum est, potentia AF in duas FH, EG, ita, tit sit Fil = AB et FG =: AE j alteram earum FH puncto E adplicatani esse po- namus. Sed cum duabus p, n aequivalentem cum diagonali ED coincidere sumatur, vires ED, FG inde a punctis E, F ad pun- tftum D transferri possunt; duabus igitur ED, FG, vel tribus EF, EC, FG aequipollentem punctum D transire necesse est. Hae autem vires omnino potentias AB, AC, vel p, m-|-n exhi- bent , itaque his quoq^ue aec^uivalens idem, punctum D permeare *) Legitur haec demonstrafio in hisce litris : Poisson tiaite de Mecani- que, Par. ign. Tom. I. p. 475 sq. — Francceur elemens de Stati- que 1812, ab init — Correspoudance de I'ecole polyteclmique. Demonstratio a Duchayla tradita. 35 debet; quum ilia autem pimctum quoque A transeat, cum dia- gonal! coincidere omnino debet. §• ST- Ex his vero, quae modo probavimus, et ex axiomate tertio, potentiam duabus viribus , ita inter se comparatis, ut sit P:Q:: m:n (m et n nummeroj quospiam integros denotent) ae- q^uivalentem, diagonalem omnino aequare facile colligi potest. Ponamus enim esse p=n=i, ac deinceps na = 1 , ni = 2 etc. Primnm igitur demonstrata quum pro viribus aequalibus i et i ■valiant, eadern pro viribus quoque a et 1 , 5 et i , et universe N et 1 locum habere oportet. Itaque si nunc p = N esse ponitur, quoniam, quae contendimus, pro potentiis, N et 1 valeant, etiam pro N et 2 , N et 5, et generatim pro N et M valere ne- cesse est. §• 58. Quae pro viribus commensurabilibus sic demonstrata sunt, facile et incommensurabilibus accommodari possunt. Potentiam enim, duabus ejus modi viribus aequivalentem, cum diagonali non coincidere ponamus; sit igitur ejus directio V. c. AD (fig. 29.). Linea D'B' ipsi DB parallela ducta, AC in par- tes aequales, quarum quaevis •< BB' dividatur, ita, ut si in re- dla AB inde a puncto A ejusinodi partes sumantur, unum certe punctorum, partes illas determinantium, intra B et B' situna sit. Quum vero potentiae, duabus commensurabilibus AE , AC ae- quivalenti, directio secundum AF omnino sit vindicanda, et quo magis altera virium lateraliuni decrescat, dum eadum maneat altera, eo propius earum resultantem directionis respectu majo- rem accedere necesse sit, potentiam, duabus AR, AC aequiva- lentem , secundum directioncm AD' agere omnino non posse in- telligitur. Eadem vero ratione non aliara quandam ei esse iii- rectionem probatur ; itaque ipsam cum diagonali coincidere oportet. §• 59- At quantitatis quoque respectu habito vim duarum mediam diagonalem aequare, demonstrari potest, E 2 jG Class. III. Demonstratio a Ducliayla tradita. Sint enim (fig. 50.) AB, AG duae vires componentes, fiatque parallelograniinum ABDC. Quuni ex praecedentibus i-esultan- tem cum diagonal! coincidere pateat, vis quaedam AR,-secuja- diim directioiiem DAD' agens, cmn duabus AB, AC aequilibrari debet; immo quaevis tiium virinm, circa punctum A in aeqnili- brio versanlium, aequalis sed repugnans potentiae, ceteris diia- biw aequipollcnti, esse debet. Itaque si AB' = AB, et BAB linea recta, AB' duarum AC, AR media est habenda. Qumn autem sit AB',#DC, etiam AD, B'C ideoque H'C, AR est. Dmn duca- tur BTD' ipsi ACparallela, 9^ ACB'D' pai allelogrammum hoc mo- do prodit. Quum autem vim, duabus AR, AC aequipollentem, secundum direclionem AB' agere necesse sit, AD = AR omnino esse debet, Itaque, cum sit AD' — AR =B'C=: AD , vis dua- rum media omnino diagonalem cxaequat. §. 60. E p i c r i s i J. Quae qtiidem nos'ri deiTionst ratio tarn facilis, tarn bre-^is^ tamque expedita est, ut ab hac parte permullis aliis antecellat. Inprimis auctor in determinanda restiltaiitis directione methodiuu multo inagis brevem atque expeditani, quam omnes fere alii, iisurpavit. Tanto magis eo dolco, quod in ipsa hac demonslra- tioiiis parte priore non omnia, ut mihi quidem videtur, rigore gpomelrico satis munita sunt. Equidem enim nostrum axioma 7. nimiuin extendisse, eoque abusuin fuisse puto. Nam axiomate illo nihil aliud exprimitur nisi id: „actio cujusdam potentiae eadem omnino manet, quocunque directionis suae loco vel pun- c^o ipsaui adplicatam esse concipiamus." Quodsi nimirum ex. gr. pondus quoddam filo vel re quadam alia corpori alicui nexum est, iJque vi quadam ad niotum solli- citaty actionem hujus potentiae, cujuscunque longitudinis filum illud sity semper sibi constare contenditur. Omnino igitur no- stro, potentiam, duabus AB, AE aequivalentem, quopiani di- rectionis suae AFK loco adplicatam poni, et ubique ejusmodi loci in duas, ex quibus composita est, vires iterum resolvi, mi- nime vera potentiam HE a puncio F ad punctum E transferri posse concedendum est. Hac enixa transpositione potentia HF in Class. IV. Argumentum ab Araldo tra'dltum. 55 pvmctum plane aliud quam antea^ agere incipit; quod quidem axiomati 7. omnino repuguat. Nostri igitur argumentum omni- bus nummoris esse absolntum dici non potest. Classis quart a. Demonjlrntiones axiomatibus /, 2 ct j superstructae^ Cap. T. Argumentum a cl. Araldo traditum *}. §. 61. Duarum virium aeqnalium rectangularunique media quan- titatis quoque respectu per diagonaleni exhibetur. Sit enim. duarum AB, AC media major, quatn diagoualis earum quadrati AD. Eandem rationem, dum et AB,» AC in binas aequales atque re- dlangula'? resolverentur, locuin habere necesse essetj ergo AB, AC aequi\alerent potentiis minoribus, quam latera ipsarum qua- drati, V. c. AM', AO', AN', AO', vel potius aAO'j quod quidem hypothesi nostrae, qua duarum AB, AC media major quam diago- ualis sit, omnino repugnat. Eadem ratione quum resultantem non niinorem, quam diagonalem esse detnonstraii possit, utramque inter se aequalem esse oportet. §. 62. Quae pro potentiis aequalibus modo probavimus in vires quoque inaequales DA, DC facile extendi possunt. Angulo enim DAC (fig. 15.) linea DE bifai-iam secto, et non solum ad banc *) Araldi snl principie dell' equipoUenza v. Memorie dell' institute aa- zionale Italiano , Tom, I. P. 1, Bologna 1806. p. 415. 58 Class. IV. ipsam DE recta OH, sed etiam ad utrainqne lineis AO, CTi et AT, Ce perpendicularibus ductis, potentia DA diiarum DO, DT, altera autem DC virium DH, De resultans haberi potest^ ducan- tur BF, Bf perpendicnlares'. Quuni autem sit Df=DT-|-De et £)P r=: DO — DH virium DF, Df media plane eadem, quae et duarum DA, DC esse debet. Itaque quum utrique rectangulo Ff, AC una eademque sit diagonalis, hoc tantuinmodo probanduin est, han-c duarum potentiis DA, DC et DF , Df aequivalentium, aeaualitatem nullo modo locum habere posse, nisi hae vires me- diae per diagonalem exhibeantur. §. 63. Quodsi vero etiam illas cum diagonali non coincidere suma- mus, ab eadcm tamen parte diagonahs esse sitas necesse est. Sit earum directio inter lineas DB, Df. Dum ergo duo virium paria DF, Df, et DA, DC primum singula tim, deinde una secundum directiones DD', DC agere ponamus (fig. A), ex hypoth.esi nostra duarum DF, Df mediam inter DB, Df, virium autem DA, DC mediam inter DB', DC sitam, angulosque, ab utraque parte cum diagonalibus DB, DB' efformatos aequales esse oportet; vis igitur ea, quae duabus his aeqrupollet, secundum ipsain diagonalem directa est. P'iat autem DC=FC etDA = fD'-, ex hac ipsa construclione DB, B'B'' duoium rectangulorum omnino aequalium simt diagonales; ergo DB#:B'B"; idcoque rectangulo CD', et rhombo B'B eadem est diagonalis DB" ; resultans igitur quatuor virium DF, Df, DA, DC secundum diagonalem ejus rectanguli, cujus alterum latus DA-|- Df, alterum vero DF-j-DC, directa est. Ex ipsa igitur nostra hypothesi, qua duarum DA, DC mediam non cum diagonali coin- cidere ponimus, duas tamen alias esse vires DD', DC, secundum easdem directiones agentes, quarum resultans secundum diagona- lem directa sit, rite colligere possuraus. §. 64. Quodpianr par virium DD', DC (fig. A) nobis sit datum j diagonali illarum rectanguli rhombi infiniti, quorum unus DBB"B',circuinscripti concipiantur. Omnino autem potentias has Argumentum ab Araldo traditum. 59 DD', DC tanquam e binis viribus Df, DA, et DC, DF compo- sitas spectare, iino potentiani DA cum altera DC, viinone Df cum vi DF conjugare licet. Ex hoc ipso autern necessario harura. quatuor virium resultantem, id^oque etiam duarutn DD', DC rnediam secundum diagonalem DB" directam esse sequitur. §. 65. Quantitatis quoque respectu duarum ejusmpdi mediam exae- quare diagonalem simili plane, quo modo antea, ratione demon- stratur. — Quomodo, quae pro viribus rectangulis probata sunt, facile cpubuspiaui aliis accomodentur, saepius jam vi- dimus. Epicrisis, §. 66. Quod quidem argumentum pro mea sententia multis, quae inter magnam illam ejusmodi demonstrationum turbam repe- riuntur, praeferendnm est. Primum enirn quod attinet ad ejus ambitum atque tenorem, non nimia sane prolixitate premitur, et non nisi aliquot planimetriae theorematvuu auxilio utitur. Deinde si ad ejus rigoieai respicias hoc quoque respectu, pluri- ma, tanquam bene munita placent. Atlamen, quantum equidem video, rigori geometrico hand alienum fuisset, si auctor nos do- ciiisset, quomodo quaevis duarum potentiarum quarumpiam DD', DC in binas alias vires divenda sit, ita ut non solum inter sin- gulas has novas potentias, sed etiam inter omnes in universum ratio ilia quaesita intercedal. ^^ Class. V. Classis quint a. Denionsirationei principiii vet axiomatibus /, 2, s, 7 superslructae. C A p. I. Dcnionjiratio a CI. Wachtero tradita *). §• 67. Quemadmodunj duaruiii potentiarum aequalium P', P", quavum direcdones o et x , media R' secundum directionem ~ x, ila potentia 1 V , duabus, alteri P' aequalibns P'" , P'^, quarnm direcliones y et z, aequivalens, seciuidum directionem y -|- i: z aoit. Loco igitur potenliamm, P', P", P"', P^^' (dum y = ij^ 7]) vis ipsis aeqviipoUens R'-{-R" substitui potest. Duaruni eio;o, viribus P', P'" et P", P'^ aequivalentmm R'", R^^, quae inter se aequalcs, quarumque alteri directio i (x — z), alteri vero i (>^ "h ^) ~1~ ^ (^ — ^^' ^^'^^i'^' quae et quatuor virium P', p// p^// , P'"^ est resultans R'-j-R". Qua ex niutua viriuni ha- rum ratione suppeditantur nobis aequationes, quae virium, R', R" R'", quarum directiones notae, quantitati determinandae inser^ire possunt. Quodsi nimirum effectum duarum virium, quae, uuitali aequales, angulo quodam a in punctum agunt, per 2 CD («) = R- exliibeamus, et brevitatis caussa P' = P" = P"^ = piv__i ponamus, has naucisciitiiu- aequationes: I. 2 ® (x) = R'i ni. 2 (p i (x — z) = R'" IL 2 (?) (z) = R" i IV. R'". 2 (p X (x-f- z) = R'-f R". Dum vero x = 180° = •tt, et z == o essesumamus, inde R'=::o et R"=r2 prodire quisque intelligit. Quibus valoribus in aequatio- nes nostras (III.) et (IV.) substitutis, hae aequationes emergunt: 2(7)('')=R"'^R,,etR'". 2;p(-) = R.. 2^(-) = R,. R, = 2. 2 1 ^ ^ *') Frid. Tau^ov. Wachter cominentatio de elemenlis, quae ad corpoiuai coelest.Lim revolutionem spectant etc. GcEttingae 1815. p. 55 sqq. Demonstratio Wachteri. 41 Accepto z= o Gt x = - 5 habebimus, R^^. 2 (^ ^ — ^ =R,- R^ = 2-|-Ri =2-\-\f2 , et sic ulterius piogrediendo 2 (p (—7) .... 2 2; ( — ) = R'" eliciiintur. Pof-ito esse z = o et x = ^ \.i"'y 2'"-' hinc obtiuemns R,,. • 2 (p ( — ^j=Rm.Rm==:2 -]-Rm.- i. Ad in- veniendam poteiuiam -" + ^Bm ^ , , duabus aliis, angulum ~ — . 2'" "*" « constituentibus, aeqnivalentem, tribuamus quantitati z valorem "^ , . , , ■ , '^ 27r mr — ^ aiteii A'ero X iios demceps A'aiores: n ... Hoc modo suppeditantur nobis hae aequatioiies : Rm^ X. 2(p (~^J =R,.^i. 'Rm^., =«„,_.- B., et perinde r(2n4-i)7f-1 + B,n(A). Qiium autem lilteris m, n ouaios diversi valoves inde a o usque ad co tiibui possiut, angulum — , in infinitum eumbise- cando, minoi'em quam quempiam angulum datum reddere possu- (2n 4- 1) TT mus , quo deinceps contlnuo multiplicando, angulus ad quenilibet datum x quam proxime accedens haberi polest. Tta- que qnaiumlibet duarum poteniiaruiu acqualium, queuilibet an- £,ulum inter j^e constituentiuui, reslilUiniis ofiectus in nosti-a au- quatione (A) exhibetur. Est autem 2 cos. x. cos z = cos. (x-}-/) ^ cos (x — z) ergo, (2u — Ott (2n-\-\)TT ('in — i)'7r V — et z = esse posito, 2 cos. ♦ ^— om-^» 2'"^-' '- 2"'+' 4» Class. V. Cap. 11. (2n-|-i)Tr ^^ , "^ /TN c J 2 COS. ■ = 3 COS. h 2 COS. — (I.) Sed quum mo- 2'"-t- ' 2™-*- ' i" do antea habereinus "" 'Rn,^ i. ="■*■ 'R,n+ , = "Rm_, +Rm (H-) (2n-f-i)'7r iiinc ^°"+'Rj„ ^1 = 2 COS. . 1 = 2 COS. x (HI.) esse jure me- ritoque colligiiiius. Resultans igitur duariim aeqnalium virium diagdnali rhonibi, sub illis coiistructi, omnino aequalis est. §. 68. , E p i c r i f i s. Quae quidem demonstratio, duni ad ejus ambitum teno- remque spectes, sine dubio laudanda est. Quod autem ad ligo- rem spectat, pro mea sententia non omnia ita coniparata sunt, ut totum arguinentum omnibus nunimeris absolutuni dici possit. Quum enini, uti auctor ipse dicit, m, n nonnisi nummeros inte- (2n-|-i)'7r eros denotent, formula ilia omnes omnino angulos a o usque ad co niinime complectitur. Itaque a nostro probata non tarn late patent, quam ipse ea patere sunisit. Ceterum fortasse eliara sunt, qui nostro, ex aequationibus (I.) et (II.) necessai-io flurre ac deduci aequationem (III.) qui contenderit, non adsen- tianc. Caput IT. Argnmentum a v'lro clariffmo Foncenex *) traditum. %• 69. Sit (fig. 30.) mCM angulus ita comparatus, ut sit = — 90° V (v quendam numerum integrum denotet)j anguli ACM, BCM ♦) Snr les principes fontamentaux de la mechanique v. Melanges de philos. et d£ mathem, de la Soc. roy. de Turin. T.U. p.404sq. 1760—61. Argumentuni a Foncenex adlatum. 43 sint inter se aequales, iique = n. MCm; deinde sit MCm = ACa = aCa'=:BCb = bCb'. Nunc vim duarum mediam, angulo MCtn concurrent iiun, quarum quaevis = a, esse:=:kaj potentiam iis- dem duabus, angulixm vero niCm comprehendentibus, aequiva- lenteni = paj porro, cum eaedeni BCA angulo concurrant, vim earuni mediam = p"a; si ejusmodi angulus aCb, aequipoUentem potentiam = p°"** 'a, denique duarum ejusmodi potentiarum, an- gulum a'Cb' inter se constituentium, mediam =p""*" -^a esse po- namus. Ouum autein a'CA, fiCb', mCin inter se aequales sint, vires a'C, AC uni potentiae, quae secundum directionem Ca agit, et quaez=:ka, aequipollent j eademque pro viribus CB, Cb' valent. Viribus autein Ca, Cb potentia p"-*-'a aequi valet j itaque bina- riun virium ka vis media estr^ikp"-** 'a. Quum ergo duabus CA, CI? potentia = p"a, viribus autem Ca', Cb' potentia = p" + ^ ae- quivaleant, erit nobis haec aequatiu; p" •*• *a -|- p"a = kp" ■•■ 'a, et hinc p" ■*■'■ — kp° + • >{h p" =: o. Itaque universe habebimus p" = Dx"-{-Ey", dam scilicet x, y duae sunt radices hujus'aequationi': u* — ku-}-i=:o. Erit ergo p"=Dri+V(!il-,)]" + EQ-v'(^ -.)]". L^ 4 --i 4 -J Ouum vero k = 2 cos « esse ponamus, hinc iionciscimur [cos K-j-V^'Ccosa* — i)]" = cosn«-j-V^ — ^- sin n« et [cos a, — y^(cosa* — 1)]" z=cos. na — y' — 1. sin. n«. ; unde pn=(D4HE) cos.n«-|-(^ — E,) ]f — i.sinna., vel pn = F cos n«-|-G \/ — 1. sin. n« Si autem n = o esse ponitur, hinc prodit p" = 2, ideoque ctlam F = 2 ; dum sumas n= 1 , erit (p. hyp.) p" = k = 2 cos a itaque 2 cos » = 2 cos a -|- G sin a ergo G = o, ideo([ue p" = 2 cos. n« esse debet. Ceterum dum n = v esse poiiimus h. e. dum angulus, cui resultans n. MCm respondet nn: i,^o"^, omnino p^' rrz o esse debet} ergo 2 cos. va nr 0 unde accipitur Iff TT . HTT ACB v« zzz' — , ergo a :^ — id-eoque p*^ — 2 cos — in; 2 cos ♦ ■ 4 * 4v ■ 4v 2 F 2 44 Class. VL Cap. I. Demonstratio Alembertana prior. §• 70- Epicrisis. Quodsi ambiliim ac tenorem hujus demonstrationis respicias, omnino laudanda est. Quod attinet vero ad rigovem geometri- cuni inerito quaeri potest: quonam jure noster k ^^^ 2 cos a esse sumserit? Equideiu non video, quomodo haec aequatio ex ce- teris, 'quae auctor adtulit, sit deducenda. S e c t i o s e c 11 n d a» Denioiistrationes analyticae, Classis sexta. ^rgiimeuia, quae axiomatibus i. 2. 5. niiunhir. . C A P U T I. Demonflratio Alembertana prior *). §. 71- AB, AC (fig. 52.) sint duae vires, punctum A una sollicitan- tes. Ponatur esse AC r=: a, AB nr b, vim illarum mediam AF — z, FAC:ziUy RAG z= a. Ductis lineis AD, Ac ita, ut sit pars auguli CAc dimidia CAD z: m, duas vires Ab, Ac, quarum, *■) Demonstration de la composition des forces par d'Alemtert v. Me- nioiies de I'acad. des Sc, de Paris annee i76g» Class. VI. Cap. I. Demonstratio Aleinbertana prior. 45 eoclem nngnJo, quo AR, AC, concurrerttinm, altera Ac =r a, al- tera Abzizb, una cum viribus AB, AC agere concipianius. His ita positis^ unusquisque non solum vires, utrique pari aequipol- lentes, omnino aequales esse, sed etiaui viui duarum AF, Af mediam, secundum directionem AD agentem, euiideni praestitu- ram esse elTectimrj quem resultantes virium AC, Ac etAB, Ab, secundiun eandem directionem punctum A sollicitajites^ produ- cant, facile intelligit. Est igilur aeqaalio a. (p (m) -j- b. (p (« hjn in) = z (p (u -f m). Qncjdsi vero viiinm (fig. 55.) AC, Ac coincidere directiones, simulqnc duas vires AB', Ab', ita inter se comparatas, ut sit AB' zzz Ab' = AB zzz b, et BAB' — — bAb' =zr 2m' praeter alias pun- dtiun A soUicitare ponimus , vis omnium haruur media , quae secundum directionem AE a^at necesse est, ita exhiberi potest: 2a4-bCp(m'). (p(a4-m'). Quum autem sint AF, Af resultantes vicium AB, AC et Ab, Ac, viies, AF", Af et AB -^-' lotenliis aequivalentes, has esse necesse est: 7. (p (11) -\-h (p (ci-\- 2 ni'). Habebimus igitur banc ae- quationem: 2 a-f-b ip (m'). ;) (^ 4- m') = (p (m'). (p (§) Quae quidem aeqnatio non nisi hac conditione, ut sit (P (S— m') + (p (§-[- m') ^rr g « X/A , g — a l/A locum habere potest. •" Ouum autem, .si a : go°, eiquesola conditione, sit (p («) o hinc \fA r ]/ — 1 , et (p («) 2 cos « esse rite col- ligitur. . 46 Clas>. VI. Cap. I. §. 74- Ex praecedentibus autetn est z -— - — - — r et ® (m) =: 2 cos m. Itaque, cum quantitati m successive alium aliuuique valorem tribuas, primnm vero m r:z= o statuas, liaec a. cos ni -1- b cos (a 4- m) a -}- b. cos « eiitaequatio: L_ ^J :== — , uncle fa- ^ cos (u-f-ni) cos u. b ■ sin II cile elicitur - -zzz - — ; r ♦ a siu (a — n) Ouae quiilem aequatio quum locum habere riequeat nisi duarurn virium a, b, media secundum directionem diagonalis aoat, ea ipsa direclio liujus potentiae estdeterminata. Potentia a b. cos a . . b. sin a ipsa cum sit = ■ -| » hinc facile deducitm': z = — ; ^^ COS. u cos u ^ sin u Itaque potentiam quarundam, duarum media diogonalem omnino aequare necesse est. §• 75- E p 'i c r i s i s. Qiuim doctrinam de compositione virium jam in primis sta- ticae elementis obviani esse necesse sit, demonstratio bujus theo- rematis satis idonea atque apta, me judice, ita comparata esse debet, ut ab iis, qui his elementis priraam operam dant, intel- lio^i possit *). Hoc igitur nostri argumentum, ceteraque omnia, ad hanc classem referenda, ab hac parte, quod analysin sublimio- rem in subsidiiun vocant autojes, non omnino absoluta judicari possunt. Quod attinet ad brevitatem, ceteris autoris, eandem propo^itionem demonstrandi, pliuibusque aliorum virorum ejvis- niodi conatibus, hie nostri est praeferendus. Itaque, quum au- tori ipsi, haec sua demonstratio hand satisfecerit, immo quum *) Ipse nostor in libro suo : Uaite de Dyiw.miqne i'45, (licit: un Prin- cipe (conipositicn des mouvcmens.) , qui etant l^an des preniieis de la INIechanique doit necessairemem etie appuye sin- des preuves sim- ples el faciles. ■;^- Demonstratio Alembertana prior. 47 non inulto post aliam, et ipsarn analyseos sublimioris llieoreuia tibus snpeislructain, earaque hiulto prolixioveni tradiderit, banc nostrani non satis e;eometrico rigore niunitani esse existinianduui est. Quantum enuidem video auctor in hunc rigoreni peccat, duui ex eo, quod, si a, = 90°, (p («) = o sit, necessario esse oportere ^f 1^^ rrr y" — 1, igituique !p (a) =^1 cos a, colligi posso con- tendit. Quiuii nimirum anguli recti cosinus sit = o, jure qui- dcin, si, dnm a =1: qo^ ,

')^ dg>(n) __ ^. / ^ + "Y '+^-^ [r^ DAB. Ceterum quantitatem cp (z) tain diu futuram esse po-ilivani, quam diu ipsa z posiliva sit, satis appa- ret, Dum autern z= i esse snmiiTius, eliam (p (z) = i esse de- bet; tunc enini est b rr^ a h. e. \is media angulum BAG bifariani secat. Quibus conditionibus vit satisfiat in aequatiouibus supra repertis (i) et (2) aut a 3= 1 aut a = 1 -|- 4r sumi debet. Nullo cniui alio modo, dum z=: 1 , etiam ^(z) = 1 esse potest. Quodsi igitur a = i-{-4i'? simulque angulum, cujus tan- 90° gens z, pavillo majorem quam — — , minorem vero, quam 90° angulum vero \- a itacumparatum, ut (1 -l-4r) «<27o", i4-4r esse ponaraus, tangentem anguli hujus I — V — T« J (1 4^ 40 i"F4r quantitatem esse negativam omnino necesse est; ita igitur, dum z augeretur, (p (z) neque cresceret, neque positiva esset; ergo aequ'atio a=i4^4r locum habere nequit^ ergo a^i, hinc b vero (2)(z) = z ideoque tg x= z = — esse,h. e. vis duarmn media, quoad directionem omnino cum diagonali coincidere debet. §• 79- E p i c r i s i T. In demonstranda virium compositione earn praeclpue par- tem quae de directione resultantis determinanda agit, difficul- tatibus premi , ideoque merito, ceteris ommissis, banc solam par- tem tractari, uti noster fecit, jam alias monitum est. Num ve- re et ciuomodo noster banc directionem determinaverit nunc vi- deamus! Prinium quod attinet ad usum calculi differentialis at- que iinteTralis jam ad alteium nostri argumentum verba feci. Ceterum ipse noster, banc suam demonstrationem non tanta, quanta debeat, simplicitate ac facilitate gaudere, ideoque alteri, quam modo antea cognovimus, demonsirationi suae posthaben- go Class. \1. Cap. IIL dam esse dicit. Praeterea omnia debito rigore demonstrata esse, equidem contendere non ausim. Conditiones enim, quas noster collocat, nimirum dum angalum, cujus tangens z paulo majorem oo . , qviam — • -, mmorem. autem quam go°, porro, angulum. 1 1^ 4r oo (■ '4^ '^'^ (i'T^40 ^°^ majorem quani 270° esse sumit; mi- iHh4r J . , . nirae esse necessarias, ideoque nostri demonstrationem non ne- cessario veram, miusquisque inficias ire nequiu c A p. in. Argumentum a geometra Foueenex traditum *). §. 80. Qnmn potentia z duabus aequalibus rr: a, quae angulum oc inter se constituunt, aequipolleat, summi posse zma(p(a) satis- adparet. Sit aequatio (p(a) zry. Cujus quanlitatis y ad inveniendum valorem lineae Cm, Cm', Ca, Ca', Cb, Cb' ita ducantur (fig. 51.), nt, si ACB =. a , mCM zn m'CM — ACa r=: BCb zz: aCa' — bCb' dot = — 5 ergo etiam ACa' zz: BCb' zz. mCm' :=z §(», atq^ue aCb =: a^J^Sa a'Cb' HZ c4i^2S« esse oportet. Dum autem angulus ACB in aCb, y in yi^dy, et dum aCb in a'Cb', y)^ dy in y 1^ 2dy i^ d*y mutatur. *) Daviet de Foueenex sur Ics principes fondamentaux de la mecbani- que V. Melanges de philos. et de mathem. de 1ft Soc. loy. de Tuiin. Torn. II. a75o — 61. p. jgg sq^c^. Demonsti'atio a Foucenes adiata, 31 Itaque si y 4^ dy =:: y', et y 1^ 2dy i^d^y ~ y", deinde dy ±: udfis, et, dum « n o, ii ^ziY esse poniraus, si « in m'Cm ma- tatiir, yrz2(^Vda esse debet. §• 8u Quodsi igitnr quatiior \ires Ca', CA, Cbf', CB una pun- dtuni C sdllicitant, vim eatum niediam, secundum directionem CM agcntem, esse m: a(yHj-(y") necesse est. Quvim autem in locian ^^lium a'C, AC, et CB, Cb' duae aliae, quarum quaeli- bet r:rr a (2 i^Yd«), quibusque una potentia, secundum dire- dlionem CM agens, — zr ay' (2 hJ^ Vd«) aequivalet, substitui pos- fint, hinc facile aequatio haeccine y-j-y"=:=y' (2-j-Vd«) de- ducitur, et suffectis in locum y'j y" valoribus supra statutis d^y riir yVd« -|- Vdy. d« ergo quoque d*y rnr yVd« nanciscimur. Quum autem y quantitas sit finita, sine dubio V infinita, eaque ejusdein, cu]us et dct, ordinis esse debet. Itaque, si V— — kd« esse ponimus, d^y == ykd«^ (1) esse debetj cujus aequationis integralis, universe spectata haec est: ergo d y n /K (A e «^K + Be«->^R) d « (3) Quum nunc in aequationibus (2) et (5) « rr: o esse suma- mus, inde nobis erit A-j-B := 2 et dyr=/(k). (A — B) du. -Verum, si a^=^ o, est quoque, iiti vidimus, dyrz:Vd« IZKda-, ergo v/'K. d«m A — B, er^o etiaui A — Brr o. Hinc autem A ~B, ideoquey:=re«V^^^ + C— « ^'^^^ = 2 cos (os /— k). w . If Dum autem cc= - esse pommus, est y=o, ergo cos - / — k = 0 5 hinc quoque - V" — k = - (2_u-f-0 If^ quendam nummeram iu- 2 4 •2fJ.-\-l tegrum denotat]; unde patet, fore/— k = ergo (2 y,-{- IN . - — — ■ — a )* Liquet autem, z, ideoque etiam y, dum 2 -^ a<2R, quanlitates esse positivas; quae conditio omnino locum G a 52 Class. VL Cap. III. habere nequiret, nisi m = o,- itaqtte est 2 cos — = (p(a,) "—. y. Coroll. Quodsi igitur tarn quantitatem , quani directionerrv potenliae duabus acqiralibns aequivalentis respicias, ipsam dia- gonalem aequare necesse est. §. 82. Eadenij quae pro \iribus aequalibus inodo demoiistravimus, etiam de inaequalibus iisque primum rectangulis valerej, sic ostenditiir. Sint ejusmodi vires CA, CR (fig. 55.), earumque resultantis directio CD5 recta FCE ita ducta, ut FCB z^BCD, ideonue etiam ECAr=3ACD, potentia CB in dua.s alias aequales, secundum di- rectiones CD, CF, itemque CA in duas ejusmodi secundum CE, CD resolvatur. Ex praededentibus quaevis potentiarum, quibus BC BC aequipollet = —5 illarum autem, quarum yis me- 2 cos.DCB AC dia est AC, quaelibet nzr ■ esse debet. Ita quatnor no- 2 cos ACD bis sunt vires, quarum duae secundum directionem CD conspi- rant, duae avitem plane sibi repugnant. Quum duarum CA, CB mediae directionem CD esse a nobis positum sit, duas illas vires repugnantes inter se aequcdes esse omnino necesse est. Qua ex AC BC re patet , fore ^ -^ et ^ 2 COS. ACD 2 cos.DCB AC , BC =rCD. 2 cos. ACD ' 2 co&DCB Dum in locum quantitatis BC, ejus valorem, e priori aequatione CA elicitum, in posteriori sufficias, prodibit CD ~ cos. ACD et hinc nobis erunt hae analogiae ' COS. ACD ; sin ACD : ,• AC : AB et sin. DCB : 1 : : AC : CD quarum ilia directio, hac vero quantitas resultantis determina- tur. — DemonstratioaFoncenexadlata. 55 Quae quideni demon? trata facile viribus, quodauimodo in aequahbus acconiinodari possunt. §• 83- E p i c r i s i s. INIethodum fere novam calculi differentialis atque integralis ad denionstrandaiTL virium composiuoiiem adhibendi hie iioster iisurpavit. Qua in re ipsum magnara soHertiam ingeniique acumen praebuisse nemo inficiari potest. Nibilominus tamen non eine satis magno, ut ita dicam adparatu, atque sumtu, no- strum priori demonstrationis parte ad id, quod petit, pervenisse concedi debet. Ceterum — siea, quae equidera sentio, ad alios transferre licet — uoster lectori non plane satisfacit, dura, quod A- — B^V^k. d«, A — B :::^ o esse contendit. Desideramus sal- tern atque optamus, ut auctor alia, quain liic factum est via, banc aequationem, quae maximi est momenli, invenerit. Denique quod attinct ad rigorem nostrae demonstrationis geometricum , quaedam contra eum monenda esse milii videntur. Nimirum ex eo, quod y =zr 2 cos \^ -a J, atque y, dum 2 «<2R, quantitas esse debeat positiva, nonduni /a = o esse ne- cessario seqnitur. Inter infinitum enim valorum numerum, qui quantitatibus ju. et a tribui possunt, permulli sane ita comparati 2 u-f- 1 sunt, ut — a< 90° 2 u -\- 1 .... ideoque • « quantitas sit positiva. Quum autem in liac 2 aequatione jtt = o, nervus probandi positus sit, atque pars de- monstrationis po'iteriur hac piiore omnino superstructa sit, Toti hinc ar»umento absolutum rigorem vindicari non posse videmus atque intelligimus. 54 Class. VL Cap. TV. Cap. IV. Argumentum a fummo geometra Laplacio traditum *). §• 84- Ouum X, 3"^ duae vires inter se rectangulae sint, quae una quoddam puiictuni sollicitant, quarumque resultans est z, trium Jiai'um potential um quantitateni multifariani variari posse, dura directio potentiae z eadera maneat, per se satis patet. Universe ioitur, cum angulus, quera vis media' et componentium altera V. c. X inter se comprehend unt, sit =: «, ponere possumus X =:=: z (p (a,) et y = z (p (-1 tt — a). Quodsi vero potentiam x, tanquam resultanteiK duarum rectan- gularum x', x^', quarum ilia cum potentia z coineidit, specta- x^ mus, eadem, qua autea, ratione nobis erit x'=x(p(«)= — z /Tr "N X. y ■K"z=.x(p\ a 7 = — 2 Z Altera potentia simili niodo resoluta, habebimus ji — y(p(- — «) ^^ — •2, 2. y//=: y(P(«) = ^ Itaque in locum potentiarum x, y, vire<5 liae; x^ y^' X. y X. y — - — 5 •) , Z ' Z Z Z 9 - erjTo Ills et illis eadem est resultans. Sed ex Ipsa nos-tra constru- X. y X. y dtione putentias duas aequales — ^ — — plane sibi repu^nare, ideoque inter s€ aequilibrari satis liquet, unde erit nobis aequa- tio z* :=zx^-|"y^ ^- ^- '^"^^ duarum x, y media quantitatis re- spectu habito diagonalem rectanguli x. y aequat. *) Laplace mccanique celeste. Tom. I. abinit. Arguinentum Laplacii. 5^ §. 85- Ad cieterniinancTam resultantis direcLionem, x in qtiantilatem x-|-dx, duui y sibi constet, variari eoncipiamus. iLa qiiidem an- gitiiis a, quant itate act decrescit. Resoluta autem potentia dx, in duas rectangulas dx', dx", quarum ilia secundum divectionem patentiae z agit , duae potentiae z-}-dx' et dx" rectangulae, qua- rum resultanteni z' quantitate innviamus, una punctum idem sol* licitant; liabebinius ergo dx" = z' (p V. da J* Itaque quan- titatLs dafunctio infinite parvum quid exliibere, ideoque liu^ jiTS forniae — kda esse debet; ergo = — kdcj [k quantita- z' tern quandam constanteni, liaud ab a, pendenteni innuit]. Quum autem vires dx", dx, anguhun = v^ aj compre- liendant, atque z' non nisi infinite parvo quodam a quantitate z differat, erit aequatio . /tt "N rdx ydx dx" =^ dx (p \ ay iz '■ , ergo da= — ■ 3 z k z ^ Simili ratione, quiun y iny-|-dy varietur, dum x eundem valo- X. dy rem servet, liabebinius dy" = dy (p («) = — p-^ d z /tt "N X d y X. d y era;o d V »y =: ; — 5 ideoque da = ° 2 kz kz^ Itaque quum x, y una variabiles esse sumatur j erit aequatior dy — ydx kda = :^+y^ Integranclo aecipimus tang, (k « -(- ^) =: - [^ quampiam quanti- aatem constanteni innuit], vel x = z. cos. (k« -1-^) • Ilac in aequatione quum ad determinandas k, et g quantitates, y = o esse ponamus, habebimus x=iz et « = o itaque cos P=i ideoque x = z. cos (k«). 36 Class. VI. Cap, IV. Argumentnm Laplac'ii, Quodsi vero x = o esse sumimus, inde erit z=ry et « = ^ir; quum vero tunc -cos (ka) = o esse debeat, quantitati k hanc esse ionnara 2 n -]- i , necesse est. Ilabebimus ita X = o , dum ; — '=055 cum autem, uti 2 n -|- 1 viJiaius, etiam sit x = o, si azzriw, omnino k =: 1 , ideoque X =^ z cos « esse opprtet. Itaque potentia duabus rectangulis ae- quipolleiis, et si ejus directionem respicias diagonalcui aequat. §• 86. Jure meritoque igitnr vis quaclibet in duas alias, quae per latera rectanguli, illi tanquam diagonal! circumscripti, exliiben- tur^ itemque quaevis potentia in trcs alias, quibus id parallele pipedum,cuius ipsa diagonalis est, deterniinatur, resolvi potest. Quodsi igitur a, b, c coordiuatae rectangulae sunt, potentiae, cujus exordium cura exordio coordinataruni coincidat, respon- dentes, per quantitatem -}/'^(a^-|-b^ -|-c^) ipsa potentia exbibe- tur^ dum per a, b, c Tires indicantur, in quas ilia resolvi pot- est. Quum autem a'-, b', c' sint coordinatae alius ejusmodi po- tentiae, a -\- a' , h -\- h' , c -j- c' coordinatas h arum du arum re- sultant! respondentes esse omnino necesse est. Eaedem -vero coordinatae qutnii exhibeant vires eas laterales, in quas potentia ilia media resolvi potest, ipsa diagonalem parallelepipedi sub il- lis con^tructi aequare debet. Quo quidem modo tarn quantitas, quam direclio quarum- piam virium determinari potest *). *) CI. BuFXKHARDT in vcvsione lilDri Laplaciani „Mecaniquc celeste" p. 6 et ipse demonstrarionem adtulit, quae fere liaec est : Quum latio, quae tribus viiibus x , y , z (coraponentibus et com- positae) intercedit, constaus sit, atqiie angulorum, qvios illae inter sese' constituunt, quantitas ab liac niutua ratione pendeat, ratioiiem bi-, naruin omiiino virium , functione angu]i, ab ipsis constituti, datam nobis esse satis elucet. Angulis igilur, quibus vis media versus com- ponentes inclinatur, per littoras 5-7 et \ denotatis , liaec nobis erit aequatio z=: X (p (S) ~\- y (p(K) Class. VI. Cap. V. Demonstratio Poissonii. 57 Cap. v. Argiimentum a Poijfonio adlatum *). §. 87. Quvnn duae vires aeqnales, una in pnnctiim qnoddam agen- tes, angulum 2x inter se comprehendant, utramque vero = P, ii'^que aequipoUentem = Resse ponamus, aeqnationemR=rP^(x) prodire jam supra vidimus. Lineis mA'^ niA", mB', mB" ita ductis, utAmA'znAmA" BmB' rr: BmB" = z, potentia P in duas aequales vires secun- dum direciionrs mA', mA" agentes resolvaturj sit xitraque ejus- nrodi potentia :^^0-^ unde erit aequatio P "ZZ Q(P(z). Siiiiili niodo et altera potentia P resoluta, quum duaLus se- cundum mA', mB' agentibus aequipoUentem = Q', et aliarum duarum, secundnm mA'', mB" sollicitantium, mediam rz: Q" esse positum sit, liasce habemus aequationes: Q' r=p?)(x — z) Q"=r Q(p(x-fz) R = Q' -f O" et liinc CP' (x). (p:pedita csset, ceteris omnibus sine dubio palmam prae- ripcret. .Sed ab ipso hoc rigore, quantum equidem video, baud pa- rum abest. Ex iis enim quae auctor sumsit , omnino hae duae ae- quationes z = X (p (S') prodeunt; quomodo autem inde aequatio illd z = x(p(.9)4-y(?)(A) doduci possit, nemo sane intclligit. *) Poisson traite de Mecanique. Tom. I. ab init. Ti 33 Class. VI. Cap. V, Ex hac aequadone valor quantitatis (^(x), 'vel, quo idem adsequimur, (p(z), ut eliciatur, Iiaec fieri oportet. Evolvantur quantitates (p (x-|-z) et (p (x — z) fecundum theorema Tayloria- nuuT., ita ut sit r , d^(p(x) z- d^(x) .7.^ -1 (B (z) = 2 1 4- - — -• \- -♦ \- etc. I ^^' L ^dx^(p(x) i.2^dx*.(p(x) 1.2.5.4^ J Quuni autein Cp(z) quantitatem x non coniinere necesse sit, ean- d^(p(x) d'*(p(x) dem in quantitatibus 5 etc. non occurrere dx^. (p(x) dx'*. (p(x) omnino oportet, Hae igitur quantitates constantes esse, vel hand a variabilibus x, z pendere debent. Statuta igitur aequatione '- — — = b (p (x), inde identidem dii'ferentiando derivatur dx* d^(p(x) d-(p(x) _ - , , , ,. _ , zz: b =r b-(p(x)etc. etbmc, dura b ::= — a- es- dx* dx^ se ponamus, p a^z* a^.z* 3*^2^ ~1 ffl (7.) = 2 1 ! 4- etc. = 2 COS. Caz) L 2 2.3.4 is.5-4-.-j-(^ J unde, dum in locuui quantitatis z sufficitur x, erit (p(x) =:= 2 cos. (ax) ideoque R 2 P. cos. (a x) Ad determinanduni valorem quantitatis a, ipsarn ab x haud pen- dere memineris. Posito autem esse x = go", eritRzuo, ergo cos (a. 90°) zz o. Quae quidem aequatio a quendam nummerum inparem denotare satis indicatj simul vero aHIi esse contendi- 00° tin-. Nam si a > 1 , v. c. a :z: 5 esset , R = o , dum x = 9 5 esse deberet. Quod quidem fieri nequit; ergo R = 2 P cos x. Quomodo liaec, pro viribus aequalibus demonstrata, ad vires in aequales extendantur jam alias vidimus *). *) Demonstiationem theorematis nostri earn , qnam attulit Francoeur in Jibi-o suo: Elemens de Statique. Paris i8i3-j plane eandem esse, quanj uiodo cognovimus , auctox ipse fatetur. Jure igitur hie omitti potest. DemonstratioPoissoriii. 59 §. 89. Epicrisif. Quae duae demonstrationes a multis aliis, vel potlus ab om- nibus, quas hucusque expojuinius, baud parum differunt. Me- thodi, quas autores in denionstrando nostro theoremate calculi differeutialis etintegralis ope. sequuntur, plane sunt novae. Utra niethodus sit praeferenda, multi fortasse quaerunt. Quamquam cuique suae sint virtutes, equideni tamen earn, quam Poissonius usurpavit, Laplacii methodo praeferre velim. Utriusque autem demonstrationis auctores summuiu ingenii acumen simimamque sollertiam ostenderunt. Demonstrationes ipsae inter omnes, quas cognovimus, praecipue rigore geometrico eminent. Liber secundus- Demonjlrationes aliis Jiatlcae ikeorematibus firperjlruciae. Classis septima. Argnmenta^ quae vectis theoria iiitttntKr. Cap. I. Argumentum Kaeflneriannm. %■ 90. Postquam auctor ope axiomatis: ^vectis, cul paribus inde ab hvpomochlio dista'ntiis aequales vires adplicatae sunt, aequilibra- tur " baec tbeoremata probare tentavit: Item ea, quae Lagrange, Piony aliique geometrae de compositione viiium scripserunt , ommitti debeie puto. Quae quidem doctiina in hoiuin viromm libris arctius cum principiis, quibus totam mechani- cam supersuueie student, cohaeiet , quam ut hie rite explicaii possit. Go. Class.- VII.. Cap. L Argumentuni Kaesttierianum. • 1) Quilibet vpctisDB, velDC (fig. 57 et 58.)7 siP.DA=rQ.AB vel p. DA = Q.AC, in aequilibrio versalui'; eademque de vecte angular! (fig. 39.) valent. 2) Vectem ACB, quum\duae vires P, Q (fig. 40.),^ in ejns bracliia oblique agentes, ita comparatae sint, ut, ductis CE, CD perpendicularibus, sit: CD :CE: : Q :P,aequiIibrari oportet, — ubi, iaquaiTL,. haec piobare conatus est, hoc fere niodo pergit: Quodsi potenlias, P, Q vecti ACB (Hg. 40.), eique mobili, adplicalas, tanquam inde a puncto M, qiio earum directiones concurrunt,^ agentes spectamus, punctuni C secundum directio- nem CM moveri omnino pater. Ductis ergo CG 1 1 MQ et CHIJMP, erit: P:Q::MG:MB, h. e. si potentiae MG, MH una in punctum M agunt, vis earurn medke directio cum diagenali parallelogramnii coijicidit. Potentiaigitur ea, quae lutic potentiae . aequalis est, ipsi vero repugnat, cum duabus MG, Mil aequili- bretur necesse est. Quodsi igitur MT, MV, Mc (fig. 41.), tres sunt vires, inter se aequilibrantes, et directionem potentiae Mc cum diagonalis parallelogrammi MTCV coincidere sumitur, etiam Mv tanquam duabus MT, Mc aequivalens spectari potest; itaque Mv secun- dum directionem diagonalis par. McvT agere debet. Hoc autem modo quuin sit MC = Mc, vim, duabus aequipolieateni, omninoi diagonalem aequare oportet.. §• 91- Inter alios,, qui eandem methodum sequuti sunt, anctores memorandi sunt: Karstex*), Pasquich**), Lorenz ***), Fer- RONif), Peyrard if ) , ScHMroT ff f ) aliique. Omnes autem ho- ruuL virorunr demonstrationes exponere partim locus non per-^ *) LehrBegr. der gesammten Math. ITT. p. 40 sq.- **) Opuscula statico -meclianica. II. p. 55 sq. ♦+♦) Giundiifs der ges. Math. IL p. 3X sq. f) Memoiie di M.. et di F. della Soc. ItaL Tom. X. P. II. p. 481 sq. •J^-j-), Statique geometrique v. Bezout cours. de mathematiques. IV""" par- tie, p. 30 sqq. tff) Anfangsgr. der math. Wissensch. Frankf. 1806. Th. II. ab init. Class. VII. @ a p. IT. A r g u m e n t u m a M a r i n i t r a d i t u m. 6 u mittit, partim hand necessarium est. Unam igitur instar om- nium cognovisse sufficiat- §• 92. Omnia haec arjrumenta laudanda sunt, quod non nisi ali- quot geometriae theoremata satis plana in subsidium vocant. Pluia antem eorum, maxinieque Fenouianum, maxima prolixi- tate premnntur. Ceterum niethodum earn, qua, plures certe auctores, ilia, quae pro viiibus vectisque brachiis inter se com- mensurabilibus probavere, ad res easdeni inconmiensurabiles ex- tendant, non es=;e geometricam, vei rigori geometrico non con.- seutaneain, negaii nequit.. Caput II. Ar^iimenUim a viro doctiffimo Marini traditiim *). §• 93- And or totam lianc suam demonstrationem liocce theoremate supersLruit: quodsi vecti rectilineo DB (fig. 57.) duae vires P, Q, ab eadeni parte norraales adplicantur, ita ut P.DA = Q. AB, punctoque A vis L, quae et ip.sa nonnalis secundum directio- nera plane contraiiam agit, atque est = P-}-0, huic vecti iieque luotus rotatus, neqne progressivus tribui potest. Ex hoc theoremate facile hae propositiones deducuntur: 1) Potentiae L actione sublata, vectis secundmn directio- nam virium P, Q moveretur. Itaque directio duarum harum mediae punctum. A, circa quod ipsae aequihbrantur, permeat. 2) Quum alia quaedam potentia S puncto- vectis cuidam adplicetur, haec cum altera P in aequilibrio versatur, dum CA. S =:PA.P. Potentiae igitur S, Q relate ad punctum A eun- *) Aloysii Maiini tentamen de motu composite. Romae 18 14- 62 Class. VII. Cap/II. dem omnino effectum praestant^ sed resultans duarum, circa A aequilibrantium, modo per P-}-Q, modo per P -^ S exhibetur, prouti aut unam aut alteram adplicatani concipias. 5) Binae vires, quanun altera secundum directioiiem, vecti normalem, altera secundum obliquam agit, dum in ratione per- pendiculariuni, a quopiam vectis puncto in earum directiones de- nii.isaium, inversa constitiiae sunt, in acquilibrio esse debent. 4) Eadem ilia ratio duas ejusmodi vires subeat necesse est, si utraque secundum directionem, vecti obliquam, adplicata est. 5) Duae vires aequales P, O, quae, vecti GS (fig.**) in puncto F adhibitae, cum vi quadain normali Q aequilibrantur, angu- los SFO, PFC aequales efformant. Quavis ear^mi in duas laterales U, N et U-, n rcsoluta, quarum altera cum vectis directione coin- cidat, altera ad banc sit perpendicularis, ex ipsa rerum natura liae duae sunt aequatfones: U = uv et N = n; ideoque U, u semet ipsas invicem destruere oportet. Ceteras N, n, quum sit N-{-n = 2N, cum polentia 2Q circa C aequilibrari necesse est^ 'hinc autem N. FCrz: Q. CG = P. CA. Sed bis ipsis aequationibus ratio indicatur, quae inter vim compositam alteramque compo- nentium intercedere debet. Simulac enim P inutatur in 2P, 5 P etc. etiam N, non mutata ejus directione, 2N, 5N etc. evade- re necesse estj duae igitur potentiae in ratione directa sunt po- sitae. Quura vero, quantitate manente eadem, dugulus, quem P, N inter sese constituunt, mutetur, simul et potcntiam N mutari oportet. Itaque N tam a quantitate potentiae P, quam ab angulo, quo versits P inclinatur, quemque niincupemus x, de- pendet, meritoque igitur poni potest j N = P(|)(x) ' U:=P(9o°— x) §• 94. His propositionibus praemissis, corpus quoddam, in puncto R versans (fig.***), in quod duae potentiae rectangulae G, M itna agant, quarum mutua ratio per rectas BD, BC jindicatur, hoc, inquam, corpus secundum diagonaleui ferri BA rectanguli DC probare possuaius. Argumentnm Kaestnerianiini. 63 1) Corporis moti directio eacleni est, quae diagonalis BA. Ducta enim recta TU alteri BA perpendiculaii, potentiisque diiabus P, Q, quae, viribus G, M aequales, secundum dire- dtiones omnino contrarias agunt, in punclis F, E adplicatis, has quatuor vires vel potius duas binarum resullantes inter se aequi- librari oportet. Est autem BD.DA = CA.BC, vel G. DA = M. AC, ideoque P. DA=: Q.AC 5 potentiae ergo P, Q circa punctu.m A in aequilibrio versantur itaque quum duarum G, M, media pun- dtuin B transeat, liaec ipsa secundum BA directa omnino esse debet. 2) Sed et quantitas resultantis per diagonalera exliibetur. Ouavis enim duarum P, Q in duas alias, U, S, et T, L rectangulas resoluta, duarum P, Q media, tanquam quatuor vi- rium S, U, T, L resultans spectari potest 5 ergo, postquani CBA = x resultantemque =:i R esse posuerimus, liasce habebimus aequationes : M = R. (p (x) T = M. CP (90 ° — x) G =11. (p (90° — x) -U= G. Cp (x) L =M.(p(x) S = G. 9(90°— x) unde, eliminando quantitates (p (x) et (? (90° — x), hae nobis aequationes oriuntur: _M^ _M.G _I\T.G __G- ^R~' ~"r~' """rT"' ""r" et liinc facile R^ = INP-f G^ 95- E p Quae quidem demonstratio pro moa sententia ceteris Iiujus classis omnibus, permultisque aliis ejusmodi argumentis praefe- renda est. Quod enim primum adtinet ad ejus ambitum atque tenorem baud ea, qua multa alia, prolixitate premitur, et non nisi quaedam matheseos elementaris theoremata in subsidiuni vocat. Sed, quod maximum est, rigore sane geometrico haec nostri demonstratio gaudet. Ea enim, quibus auctor totum ar- gumentum superstruit, ab aliis geometris probata fuerej ipse 64 Class. Vin. autem qua« probanda sibi sumsit, re -vera illoriim ope demon- stravit. Pars commeiitationis posterior. Bemonjlrationes eae, hi qmbiis moius ^ a viribiis adhibi- tis efficiendi, ratio omuino habetiir. C 1 a s s i s o c t a V a. Jl\-aummia , quorum anctcres theorema nojlrum if a demoiijlrare Jlndent, lit vires .ear unique attioncs nobis, ut itn dicmn, ante oculos ponant-, V. c. fidihiis elajlicis adkibitis. JDemouJlratio -a Vincentio Rlccato tradita *). §•96. ^ ■ Auctor his tbeorematibus: 1) Qiuun fides quaedam eiastica AS (fig. 42.) vi qnadam AB, . corpvis A, nonnisi secundum directaonena AD mobile, sollicitet, potentiam AH, quae linea perpendicular! determinatur, poten- tiae AB aeqmpoUere necesse est.' 2) Ouuni AS, AT (fig. 45.) duae una in corpus, secundum directioaem AD lantummodo mobile, adplicatae sint fides, eun- deiaque, quem vires AB, AC, praestent effectum, linea DK =;AH assunita , tota potentia AD duabus his chordis aequipollet. *) Vin-cenzo Riccati dialoghi delle force vive etc. Bologn. 1749. P- 2o3sqq. — Commentariorum Boiioniens.-T. II. P. I. p, 573 sq. Ceteium multa ' quoque de hac le disputavit : J. Bapt. Scaiella in libio siio: Physica genoralis. Tom. I. p. 454 sq. , quae tamen hie satis explicaii atque dijudicaii nequeuiit. Demonstra tio a Vincentio Pvlccato Iradita. 65 CoroU. Ductis CF, EB (fig. 4.4.) lineae AD et EA, AF al- ter! EH parallelis, sine dubio per AN := AF — AE, vis exprimitur ea, quae corpus inotum a directione AD detorqnere nititur. Ita- que si AE = AF punctual A sine ulla resislentia tubuui vel cana- leai AD perineal. His, iiiquam, propositionibns compositionis virimn demon- strationeni hoc fere niodo noster superstruere conatur: Quodsi duae fides elasticae AS, AT (fig. 44.) una in punctum A agunt, lineis BD , CD ita ductis, ut fiat parailelogi'ammuni ABDC, potentiani, duabus AB, AC aequivalentem, tani quan- titatis, quani directionis liberae respectu diagoiiiileni AD aequare necesse est. Sioiulac enini punctum, viribus AB, AC una in id agenti- bus, secundum certain directioneiu moveri non est coactum, viain usiu-pabit omnino earn, in qua vires, quae illud a dire- di«ne antea necessaiia amovere, uti vidimus, conabantur, in- vioem sese destruunt, b. e. ubi BH ^=r CK. Item actionem hujus- modi potentiae AD singularum actionum,_a viribus AB, AC praestitaiTim, summae aequalein esse satis liquet. Utrique au- tem condition! omnino non satisfieri potest, nisi sit ABCD pa- rallelogrammum ejusque diagTonalis AD. §• 97- K p i c r i s i s. Ipse noster declarat, in hunc potissimum finem lianc compo- sitionis ac resolutioiiis motus demonstrationeni a sese esse tradi- tam , ut principium illud naturae primarium , aequalitatem scili- cet inter caussam et elTectura*), nostro hoc theoremate, dum- niodo res ea, qua debeat, ratione tractetur, hand tolli pi'obet atque ostendat. Quodsi enini, inquit, vis duarum media, uli omnes auctores static! ac mechanic! inter se conscntiunt, diago- nalem parallelogramm! aequat, ejusmodi compositione ac reso- *) Non sine caussa ipsis nostri verbis : „aeqiialitas inter caussam et effc- (ftuin," usus sum. I 6S Class. \m. lutione virium, dnm dno parallelogrammi latera ejusdem dia- gonaleni omnino supercnt, aequalitas inter caussam et effectum miniine servatur. In plei'isqiie igitur tlieorematis nostri demon- strationibus , gravitcr hue usque peccatum est. Staiim autem quaedam monere niihi liceat. Quantum equidem enim videa, verba nostri, „aequalitas inter caussam et effectum," sunt am- bigua. Primum enim ita intelligi possunt, ut caussa quaevis effectui, quein gignit, omnino aequalis, "vel potius res plane eadem, quae ipsius effectus, esse debeat. Sed haec veiba, sic intellecta, principium, vel legem naturae universae primariain innuere, quis est qui dicat? Quum autem verbis nostri hunc tribuamus sensuin: effectus, quos caussae , aequales praestant, ceteris paribus aequales esse necesse est" et plures alias compo- sitionis virivnn denionstrationes huic principio non repugnare equidem credo. Nimirum denominatio „compositio virium" ^non ita urgenda est, ut vim duarum niediam ex his vere esse coni- positam, h. e. potentiarum singularum summae aequalem esse putemus. Quod quidem nuUus fere *) omnium, qui hanc rem tractarunt, dixit,; immo dum vim duarum mediam diagonalem aequare contcnderunt, his verbis nihil aliud intellexerunt, nisi hoc: si duae vires quopiam angulo ima punctum quoddara solli- citant, hoc reriim Jlatii eundein, quern potentia una, per dia- gonalem exhibita, effectum praestant. Quid igitur de quorun- dam sententia, fides elasticas rem esse unicam , qua loco virium adhibita, earum coinpositio rite possit demonstrari, judicanduiH sit, facile iiitelligi potest. Immo hunc ejusinodi chordarum iisura singularibas premi difficult atibus, equidem puto. Nam si duas fides una in punctum A agere ponitur, ipsas, dum contra- hantur, corpus secundum directionem diagonalis movere noster dicit. Quae quidem fidium contractio cum maxima sit, dum corpusculum m eo directionis puncto, quo fidium directiones cuin ilia angulos constituunt rectos, versatur, uf corpusculum alterius moveatur, ipsae chordae extendantur necesse est. Ita autem, secundum ipsa ea, quae noster dixit, punctum A vi, qua- ♦) Demonstrationes eorum, qui revera hoc contendeiunt, raox cognos- temus. Ciaes. IX. 67 cum moveri nititur, privatur, nee spatium, quod noster statuit, percurrcre potest. Deinde quamvis negari nequit fidis elasticae actionem, universe spectatani, ab ejus contractione pendere, hinc tamen, quantum equideni video, duarum chordarum AB , AH, particu- lis aequalibus Ap, A a contractaruni, actiones, duni AB. Ap = AH. Aa, oinnino esse aequales liaud necessario sequitur. Cete- rum aequatio AB. Ap = AH. Aa non nisi tunc vera est, cum Aa, Ap lineae sint infinite parvae. Noster enini arcum circularem ap tauquam lineam rectam spectat. Quod quidem, pro luea &ententia, demonstrationis rigorem omnino impedit. NoTA. Argumentum Jacobi Riccati *) , qui partim Beinoullium, partim Vincentium Riccatum sequutus est, ideoque nihil plane novi at- iiilit, jure ommitti liic potest. Classis non a. ^lutorei u, qui coniposithiiem virinm ef cowpnsiL mofninn v.rinvi fan- deuique rem esse ivrbitrantur. §• 98- Satis .nota sunt ea, quae Neutoniis multique cum eo alii ad propositionem ; „punctum A, a duabus viribus AB, AC (fig. 45.) una sollicitatura, per diagonalem AD fertur" probandam protu- lerunt, Innummerabiles fere sunt demonstrationes — si modo hac voce hie uti licet — in banc classem referendae. Gravesande, Belidoi , MdTssehenbrceck , Wolf, Pascoli, Saverien, de la Caille, Mako, Ferguson, Hennert, INTartin, d'Antoni, Brisson, Al aria- no Fontana, multique alii, quorum nomina referre hie non li- cet, hanc methodum, tanquam facillimam sequuti sunt. Sed praecipue notandus est Savioli, qui propositionem, sane inex- pectatam, ipsi demonstrationi praemittit. Ope nimirum theore- *) Jacopo Riccati opeie. Tom. II. p. 5S5 sqq- I 3 68 Class. IX. inalis hujus: „si poteittiae ciiidam AD (fig. 46.) altera AR, quae ciini ilia angiilum constitnit r'pctvim, supervenif, punclum A baud secus versus lineam DC [| AB movebiLur^ quiam si potentiae AB actionem plane non expertus fuisset" noster corpus^ dnabus viri- bus AB, AC (fig. 47.) soUJdtatum, secundum diagonalem AD nioveri probare conatur. Quid de omnilius hujusmodi argumentis sit judicandum,^ duo viri clarissimi, Bernoullius *), atque Lambertus **) satis osten- derunl. Itaciue nunc longa verborum serie in eorum rigorem aliaque inquircre llias sane post IIornerunT esset. C 1 a s s i s d e c i m a. Argument a eontin^ qui conipofitioncin fnnfnmniodo viotmim, niinime vero viriiim demon/lrari pujfe dicunt. Cap. I. ArgumenUim Kantianum ***).. %• 99' . Theor. Motuum duorum compositio ita tantummodo in- telligi potest, lit alterum eorum in spatio absolute corpus ipsuin re vera sequi , alterius autem loco spatium relativuiu secundum directionem oppositam moveri ponamus. Nam etsi etiam corpus utrique motui satisfacere sumamus, lii tamen non ad lineas AB, AC (fig. 49.), sed ad alias tantum, *) Commentar. Acad. Petiopol. Tom. I. p. 127. **) Beytiage zum Gebrauche der Mathem. II. Th. p. 448 sq. =>**) Metaphysische AnfangsgrLinde derNaturwissenschaft. 1786. p. I3sq. Argument mil Kantlanum. 6q -his ipsis parallelas, fieri possum. Itaque altcium moLum alte- rius mulationenl, h. -e. deflecticmem a directione data efficere ponendum esset. Quod quideni compositiunis virium noliorii repugnat. At si raotuni corporis secundum AC vere locum habere, simul vero- spatium relativum celeritate AB, secundum directio- hem hiiic oppositam moveri ponamus, hnea AC, in aliquot • V. c. tres partes aequales divisa, dum corpus per spatium AE, ipsum spatium relativum et una cum eo punctum E per Ee ' = MA fertur, item, dum corpus per totam lineamAC, punctum C per Gc =: AB uiovetur. I la autem idem oumino , effectus gignitur, ac si corpus A tribus variis tempusculis per spatia Em, Fn, CD =: AM, AN, AB latum fuisset. Klapso igitur tempore statute quum corpus A in puncto D versetur, anteaque singula deinceps diagonalis pun- dta otcupaverit, hac ipsa diagonali tarn quanlilas quam directio motus compositi exhibetur. Scliol. Argumenlum Schultzianum *) plane non discrepat ab eo, quod aiodo legimus; ommilli igititr hic potest^ E p i c r i si s. De principiis, quibus totnm hoc demonstraudi genus nititm> jam supra (§§. &sq.) quaedam diximus. Ilaque nunc videamus, ulrum auctores principiorum illorum ope vere ea demonstra- rint, quae, probanda sibi sumserunt. Quum celeritas, uti om- nes, ni fallor, consentiunt, nihil aliud sit, quam facultas, qua corpus quoddanir ceilo tempore per certum spatium moveri pos- sit, et quum celeiitas eo major habcatur, r[uo breviore tempore per idem spatium corpus feratur, equidem non intelligo , quo- nrodo noster, eo, quod corpus eodcm tempore per duplum, quam antea, spatium moveatur, celeritatem ejus duplo auctam esse,, necessario non sequi, dicere potuerit. Sed haec etsi *) Scliultz Anfangsgv. der reinexi Mechanik. Konigsb. igo). 70 Class. IX. Cap. IT. Argumentum Alembertanum. nostro concedantur, eo tamen ejus demonstratio nondvtm omni- ums difficultatibus liberatur. Ipse enim quantitates eas, qua- nim aequalitas constructioiae geometrica deinonstrata sit, orani- no tam similes, quam aequales esse debere ait 5 et ejusmodi conslructione compositionem motuum a se demonstratam esse -piitat. Motum igitur, ex duobus compositiun., motibus c-oiupo- nentibus esse omnino aequalem oportereL Haberemus ergo ae- quationem AD = AB+^ BQ haec ipsa aiiteui aequatio, uti jam in prirais geometriae eleiiientis demonstratur , locum habere nequit. Itaque ea, ad quae noster liac ipsa sua demoastratioiLe pervenit, ceteris, ab eo statutisj repugnant. c A p. n. Argumentum Alembertanum *). §. 101. Auctor SO, TP (fig. 50.) tanquam columnas mobiles, inter quas planum NCDM moveatur, spectat, et simultaneae duarum virium AB, AC, actloni omnino satisfieri, si pro AB columnis SO, TP motus secundum directionera BA atque celeritate AB, pro AC auteaa piano NCDM niotus secundum CA celeritateque AC tribueretur, contendit. Facillime igitur noster hoc raodo, corpus a duabus viribus sollicitatum , secundum diagonalem nio- veri, probavisse putat. — Hoc aulem non re vera factum esse ex multis aliis, quae jam ad alias ejusmodi jdemonstrationes re- preliendimus, praecipue autem e corollario prirno satis adpareL Quid enim judicandum sit de argumento, cujus auctor, ut quae in coroUariis proponit, noniis, in ipso argumento pronuntiatis, repugnent, parallelogrammum construere omnino coactus est, CUJUS diagonalis baud minor sit, quam duo ejus latera constitu- *) Traito de dvTiamique. Paris 1743. p. 22 sq. Class. XI. Denionstratio a Fontaine adlata. 71 tiva, quid, inquam, de hujusmodi arguniento judicanduni sit, omnes sane, et non nisi priniis geoinelriae elemeiitis irabuit, sciunt. C 1 a s s i s u n d e c i m ff. Argnmentum a viro cL Fontaine adlatum *J. Ea, quibus argunientum, tanquam prin'cipiis, nidtur, haec fere sunt: 1) Corpus quoddanr M (fig. 51.), secundum directioneux BC, celeritate AC latum, ut inde a puncto A non ad punctum C, sed celeritate AD ad punctum D maveatur, viin quandam, cujus directio lineae CD est parallela, iai corpus, duui in ptm- dlo A versatur, agere necesse est. 2) Quum vis ea, qua corpus ejusmodi mutationi resistit, partim ab hujus materiae quantitate, partim a discrimine dua- rum celeritatum — alterius nimiium, qua ab initio corpus fe- rebatm-, alteriusque, quam mutatione accepit — pendeat, cor- poris M, modo memorati, potentiam, quae hoc corpus secun- dum directionem AD moveatur haudsinit, hac quantitate M. CD exhiberi ponere possumus. 3) Quodsi igitur haec potentia, M. CD vi alia, huic aequali, sed contraria ciestruitur, corpus M inde a puncto A ad D niove- bitur. Fieri enim non potest, quin corpus nostrum, ubi in puncto A aclionem potentiae M. CD expertum luerit ^ secundum AD ferri p on a in us. ♦) Fontaine tiait^ de calcul differentiel et integral, rails J7;""o. p. 306 sc^q. 72 Class. XI. D emonstratio a Fontaine adlata. §• 103. His aiitem ita constitutis , corpus quoddam M, celeritate quadam AC sectinduni AO praeditum, in quod auteni eodein temporis momeiito, vis altera M.AE (fig. 52.) agat, setauKkun direclionem AD, diagonalem parallel-ogrammi ACDE movevi, facile deraonstrari potest. Pro natura enitn rei nunc corpus, cui inodo antea celerita- tem AC tribuimus, quiescere quideni, sed eodeni, quo m A ver- satur, niomento poteniia^n M. AC in id agere sximere oiunino possunius. Conjunctis igitur harum duarum viriiun M. AC eL ^I.AE actionibus, corpus «ecundnna diagonalem ierri necesse est. Quod quidem arguuientum eo laudanduni est, quod auctor ejus \iam, antea nonduni tentatam, ineiuido, argunientum non nimis prolixum coiupositionis virium conficere tentavit. Sed et in hoc tentamine sicuti in mvtltis aliis , quae hue usque cogno- vinius, hanc ipsani conipositionem magis temere sunitam, quam ricvide demonstratam esse, non inficiandum. Unusquisque enitn totius argunicnti cardinem in eo versari, quod directio potentiae (fig. 51.), cujus actione corpus secundum AD movealur, lineae CD sit parallela inteiligit. Sed hanc ipsam propositionein sine ulla demonstratione nost-er accepit. Itaque t'oti arguinento rigor eeometricus minime vindicandus est. x \ DATE DUE 0EC2 . 1.995 1 CAVLCD ""•^•° "•"■•■*■ a/ELLS.B!r;r^ vMLTHAM. (v., I)A32.B4513 3 5002 00031 9660