BIBLIOTECA Digitized by Google Digitized by Google LILA WA TIs OR A TREATISE ON Arithmetic and Geometry BY BHASCAUA ACHARYA, TRANSLATED FROM THE ORIGINAL SANSCRIT A BT I JOHN TAYLOR, M. D. Of THE UO!«'bLK bast INDIA COMPANY’S BOMBAY MEDICAL BSTABLIIHIIENT, BOMBAY: PRINTED AT THE COURIER PRESS BT SAUBEL BANS. 1816. Digitized by GoogI Digitized by Google t TO The Eight Honorable FRANCIS, EARL OF MOIRA, BARON RAWDON IN IRELAND, ^c. ^c. &^c. K. G. GOVERNOR GENERAL OF INDIA, &C. &C. &C. (i * ' V '. THE FOLLOWING WORK, J/' Illustrative af the Mathematical Science of the Hindus^ IS MOST RESPECTFULLY DEDICATED BY » HIS OBEDIENT AND HUMBLE SERVANT, JOHN TAYLOR. « Digitized by Google Digitized by Google Extract from the Minutes of the Proceedings of the Literary Society of Bombay, 27 th Jutie 1815. Mr. Erskinb read to the Society a Translation by Dr. Taylor from the Original Sanscrit, of the Lilawati, a Trea- tise on Hindu Arithmetic and Geometry.* The above being a Work which has been frequently called for by the learned in Europe, and it being desirable for the sake of accuracy, that it should be printed under the eye of the Translator, it is besolved that the printing of the Work shall be immediately undertaken at the expense of 'the Society under Dr. Taylor's superintendance; and that their thanks be returned to him for his valuable labors. Digitized by Google E RR^T^. InlroJuciion page 5, line 7, for Meclewl read Macleod. page 17, lino 2 for 6 X JS, read 9 X W. Page 7 l ine 33 for 317 2S read 17 28. 10 I.'nc 0 for 12 read 20. 13 Line 3 for by double llic sum of this las( figure, read by double lliis liitt figure. do. note = for 7S90 read 7(>!)0 and for 8CJ read Cbl. Page 16 line 27 dele the words “ adding the unit if there be one to the preceding qnofh. cnl figure.” Pa>re 27 dele the first line. The folloryiiig remiirls, which with many other explanatory remarks, was fumishetl me by Li. iilcminli .Macleod and Tate, on the rule lor suiiioiatioii of irnni>o»cd numbers p^ige 135, was omitted to be iiiserled in its projtcr ]>lace. Tlic Irutli of this rule is et blent, for if the whole of the transposed numbers be set down, and added togelbcr, the sum of all the ves'ical columns of units, Kn«, -&c. will be rtpial to each other, and will carli aroomit to the sum of the given digits, muliipiicd by the num* her of transpo‘iiions, ami divided by the nuinlier of these digits; consef|iicnl'y, earli eoluinn being thus sc|>ariit -ly Slimmed up, and set down one place fuiword, the amount ol these sums will give tlie sum of the whole of the trausposed numbers. Digitized by Google CONTENTS INTRODUCTION PART I. AHITIIMRTIC Tablet of Monty, fTeights, Af: Of Jddilion and Subtraction, .. .. Multiplication, .. •— Vidsion, ' — the Square, ^ ^ .. .. .. ■ Square Root, ;_j ^ , . . — - - Cube, , . . , . , . , » (-’ube Root, I. .. .. .. — Fraclioiit, . . « « . . . . — - the Effect of Cipher, . . . . , . — Imersion, ,, ,, ,, — MM Attumed Number, j_, .. , , — Multiplier of the Root, . . .. ,, . Rule of Three Quantities, ^ .. .. , - ■ Fhe, Sre. Quanliliet, .. .. Of Barter, — Mixed Quantities . . . . — Buying and Selling, .. , , — « Computing Gold, .. .. — Pertimlationt, ^ .. ,, — Progretmru, ,, , , .. ,, 1 6 7 11 13 15 17 ao 91 29 SI 39 37 41 43 45 47 51 54 S7 59 Digitized by Google c.; , V i . C/ C i' ; .k i i ^ t % > V. - Of Cilc'ci .. .. CIIAPTKK n. ftCTlOX 1. •• •• •• •• * • • • . • It Of Pondt, lECTiOM II. • • • • . ■ « • f9 Of the Bricks or •':lu astronomers, or rather astrologers; for in this coimtrv these two professions are always oonjoined, and in genera! the former is considered sub- jeruent to the latter. Tlie rules are written in verse in a very concise and even elliptical style, and jwsscss in no slight degree the cliaracteristic obscurity of San- scrit compositions on science and philosophy. Ily dir-etions of the Emperor Acbar, whose liberal promotion of literature and s; iencc added glory to his con.^ucst.s, it was translated into Persian in 1587 by Fyzi, t!)’j brother of Aim Fazil the Einix.*rnr’s secretary. I procured a copy of this ver- sion from Mu'la Firoz, a Ic.anied Parsi, who has studied with much success the astronoinica! .system of the Arabians. Fyzi informs ns that in making his trans- lation he had the “ assistance of men le.arncd in the science, particularly of as- trologers in the Dcocan". IJis translation [lossesses that general accuracy which might be cxjjcctcd from a [xtrson of Fyzi’s talents and knowledge, aided by such eminent mathematicians as his own situation, or the influence of his Uoyal Patron, could obtain. It is, however, often very obscure, and in several places there are consi- dcrnOle omissions, especially towards the end of the arithmetic, and in the geome- trical oficrations which immediately prccc’da the chapter on circles. The chapters on imletcrmiiiate problems and on transpositions are altogether omitted. Besides, the stylo is not only much more diffuse than what iiccc,ss.arily arises from the dif- Ibrcnee of the Persian and Sanscrit idioms, but the m.anncr also of delivering the rules, and of detailing the o(/crations, generally varies in a very considerable degree from that of the original text. This, indeed, is so remarkable as to induce a suspicion, t'l It Fyzi contenleJ himself with writing down the verbal explanation afforded by his ussi:>taiits. Ill the library cf my very learned and estimable friend William Frskinc Esq^ there is a translation of the Lilawati into the language of Marwar which I have ex- amined. It is of a date .so late as l/Gi, and probably was made for the use of the Jaina priests, many of whom jirofcss astrology. In consequence of the close affinity between the Marvvari and Sanscrit languages, the translation is in general xery literal, and also retains all the technical terms employed in the original. In several chapters, however, tliere arc important omissions, and the indeterminate pro- blems and transpositions arc left out altogether. Digitized by Google ( 3 ) A cnrlons account of tlic occasion of writing the I.ila’.vaii is given hv F>;'i in tlic prcf'icc to liis translation, 'i’hc story he relates lias not hecn l onrirmcd to me liy jinv native of this country, nor have I chservctl it nienlioneil hv any Ilimhi author; hut as it mav amuse the reader, I shall line tran=eiil>e a tiam-lation of it made by Mr. Strachev, and |/uliiislicd in Iliittoii’s malhematieal tracts. “ It is said that the comfrosing the Lilawali was occasioned hy tl.c following cir- cumstance. Lilawati was the name of the author’s ( Rlmscara’s ) daughter, concern- ing whom it appeared, from the qualities of tlie Ascendant at her birth, that she was destined to pass her life unmarried, and to remain without ciiiklrcn. The father as- certained a lucky hour for contracting her in marriage, that slic might be firmly con- nectal, and have cliildren. It is said that when that hour approached, he brought his daughter and iiis intended son near him. lie lett the hour cup on the vessel of water, and kept in attendance a time-knowing astrologer, in order lliat wlien tlic cu[i should subside in the water, those two precious jewels siiould be united. E*jt, ss the intended arrangement was not according to destiny, it liap|M:ncd that the girl, from n curiosity natural to children, looked into the cup, to ol)ser\’e the water coining in at the hole ; when by chance a pearl separated from her bridal dress, fell into the cup, and, rolling down to the hole, stopped the influx of tlie water. So the astrologer waited in expectation of the promised hour. VVlien the ojicration of tlic cup juid thus been delayed beyond all moderate time, the father was in consternation, and ex- amining, he found that a small pearl had stopped the course of the water, and that the long-cxpcctcd hour was passed. In short, tlic father, thus disappointed, said to his unfortunate daughter, I will write a book of your name, which shall remain to tlie latest times — for a good name is a second life, and the ground -work of eternal exis- tence.” My object in the following translation is to furnish an authentic document, wliich, by exliibiting not only the actual degree of mathematical knowledge possessed by the Hindus in the I2tli century, but also, by shewing tlieir modes and principles of operation, may lead to a fair conciilusion regarding their pretensions to originality in this dc[wrtment of science. When it is considered that a translation of the pre- sent work has been long a desideratum, and that an unsuccessful attcnifit to translate it was made hy that excellent mathematician Mr. Burrow, it will readily he suppo- sed that considerable difficulties must have occurred in the course of this undertak- ing. It is probable, indeed, that I should have failed entirely, had not most material ■ .assistance been derived from three commentaries which I had the good fortune to Digitized by Google ( 4 ) ohtnin. Two of these were (irociired for me at Nagporc by the kind exertions of Gcor;ie ^ortunity of consulting actually five copiej of the original work ; and all these correspond with a degree of accuracy which I have rarelv found to exist in different co|.ie.s of 'ansiTit books. The chief dif- ference consists in the transposition of a few of the rules ; and in the copy from Chizerat tlierc arc several ishepaka, or iiuerpoluted rules, which are nut contained in the two other copies. l lio' the opportunities now .adverted to were favourable to the siux^'ss of' the im- dertakimj:. 1 am still not so .s.iugmne as to iiungine that tlii.s translation will fully satisfy the wishes and e\|>a:iations of iiiathematieiaiis in l•'llr(•pe. .Some will [jer- liatis he of opinion, that the examples, besides being drawn out at length from the comineniaries, ought also to have been put down in the mathcinatiral language of the western world. It tlid not, liowevv-r, .appear to me that, in n’gard to many of the CMiinpIcs. this was pciuliarly rajui.site iihtr the illuslrat.on afforded by the notrs from the commentaries. Tho’ neither brief- nor elegant, these notes shear the dilferent str|)s of each ojicration in a manner pretty cleiir and intelligible, so tli.it no inalhematiei.an, it is imagined, will cxi>erieiicc any ditfieiiliy or trouble 10 comprehending the rules or cxatuplfs : and thereioie they can occasion no trouble to aiiv per.-oti who is diS|.«)-^ed to translate them into the mathematical language to whieli he has been arcuslometl. being cliielly desirous oi presenliug to the reader the views of the 1 liiidu author, and 1 he train ol thought and ot 0(JCiaLion in the original, without any desire of aecoiiiiiHKjating them to the technical phraseolo- gy of Kiiropcan stience, which in some instances miglit have lixl tlie mind into another scries of ideas rendered familiar to it hy long habit; I in.iy perhaps have appeared sometuies to have cained this conlormity to an excess scaiecly justilied Digitized by Google ( s ) K ‘:rx'.r allows his mind to be led by tli au’thr, < i “ Scholar, and ‘i*^«culty. illustrations, --- - and important, will be found in the notes ‘^“Hous which w-ill be found extremely ingenious illustrations, many of assistance ofniy^oinig friends Lieutenants M ‘he kind it adds n, no small degree to my obligation to t^esT^’!^^ Tate of the Engineers. Uons were written during the short intervals of le7 observa- bonous otticiaJ duties in uhieJ. they were eng^l^^ i ^"“wed them from the very |a. I had not the benetit of tl.eir assistance at aif Llie ^i that A of much more importance than JhaT T ^ undertaking. of b, aL,„. in the Lilawali uiUi that which existed in K ^ hcmatitai knowledge displayed ute Hnnln ,„h» .„d “th^h””' ‘ ■nd of ntodernlia^pe. duel, a eo„,^r e "» -uhool londettid hint eon,peKn^u„^^^“’"’"' tantslator'a ph»M w.th much »eee»i,,u,i.^tpfd,f''^' "«! hme been ,ec„„ ^tato lntte been co...„,^ ! »n.r.m.c. cao» a ^ ,r„l„, il correspooJ. with th. rule ,.bieb I .oe....o..»l to 1-0 a,d ill the translation ; and is an evil of less magnitude than that of adopting which might lead to misinterpretation, and to erroneous conclusions. In the notes taken from the commentaries I have also usually inodes of expression, and of putting down the results. At the .,„. 0 .xhJ,be operation Ina .hor.rr ,pacr, 1 a^ ™,k, o»xl In Euro,, ran noa.,*,. Bu. U .houW b. "tooHoo.od -W ^ - foreign to the Hindu system. I shall h.ave occas.cn to notice immediately, onlv mark cmploved by the Hindus, is that of minus. . u * ir> The arithmetical operations of the are performed on a oar a ou - inches long and 8 broad. A white ground being first formed wit i a in o p^ ' r hiborua ..rovorcawiU.n,„d,or *.(,a.ul.lcl, I. Hour d,cd ot a purpl. f ur ko form, of .1,0 bgun,. or lo...r. arc .raced »l.l, a rvoodon ..ylo r..h„h S=:S“-=-r.:;3=; itTessionttheinte^^^^^ the operation is finished, the ge- neral result only sUnds on the board. ...k The lllmlu method. of opca.lr,g In Iho ‘'""[“"‘‘'"’'"‘f,,™* if tte no4, hu'l , .action, muUipliea,i.n, and dl.laion, an: de.arled a. ^ ^ as these operations are not found in books, and can be learned y y cesses ffZ through, a few more illustrations in this place may not be dee pe fluous. J In a»n.T.on .hey usually begin rowere'nnod.e; „a„l. the lefh in the mm, manner^ .. done E» ^ ^ Digitized by Google ( 7 ) column of figures on the left hand is first added up, and the result put down be- neath. The next column is then added up, and the units being put down, the teitt are carried to the preceding result, which is blotted out, and the new result is set down in its place. The following example will illustrate this mode of operation. S557685 9278346 6346925 8437207 4925624 ?9SMeS7 3148578 The lesiilt of the operation, however, does not stand in this manner on the board i but, as each preceding result is rubbed out, and the new product put down in its place, the total amount standa thus ; 31485787 To prove addition, a line is drawn below the uppermost number, or above the lowermost, which is then supposed to be cut ofF, and the rest of the numbers being added up, the uppermost line, or the lowermost, as the case may be, is added to the •um ; and if the total amount correspond with that found by the first addition, the Operation is considered correct. The Hindus seem to be entirely ignorant of the method of proving addition by rqecUng the nines ; a fact which is deserving of notice, because as this method it familiar to those who follow the Arabian system, it shews how slow the Hindus are to borrow from th»^ neighbours, even after a long period of free and unreserved com- munication. Subtraction is performed either from right to left, or from left to right, the less number in both cases, being placed above the greater. In subtracting from right to left, when the number in the minuend is less than the one above it ia the subtrahend, ten is borrowed upon the next minuend figure (which represents tens, hundreds, &c. according to its place) ; and this operation is denoted by a perpendicular stroke placed opposite the figure upon which the ten is borrow- M ; the subtrahend figure being then subtracted from the borrowed ten, and the re- mainder added to the figure in the minuend, the result is put down in the line be- Digitized by Google ( 8 ) low; nfter which, one is added to the preceding figure in the luhtrahend, and the ■aiue process is repeated as before.* i axAuri-K. 1 1 1 1 • ' ^1)^343 7G52432 73«H8«9 The operation is thus expressed in words ; 3 cannot be subtracted from 2, there- fore borrow 10 upon the next minuend figure 3 ( which here represents tens) ; then subtract 3 from lO, and there remains 7; 7 and 2 sre p, put down g below the minuend, and carry or add 1 to the subtrahend figure 4 ; this makes 5 ; then say 5 cannot be subtracted from 3, therefore borrow ten upon the next minuend figure 4, and subtract 5 from i 0, there remains 5, w hit h added to 3 makes 8 \ put this dovm below the minuend ; and thus proceed thro’ all the figures. This me- thod differs from curs in placing the subtrahend above the mi.nnend, and in sub- tracting the subtrahend figure *rom the borrowed 10, and adding the remainder to the minuend figure, instead of adding the borrowed 10 to the minuend figure, and subtracting the subtrahend figure horn the sum. “ As the method of subtracting from left to right is explained in the note p. “• U will be sufficient in tliis place merely to give an example by way of illustraUun. EXAMPLE. ] 0717338 - ♦•333437 ■ ' iSblLtW 0 07 25000079 Which Is thus expressed ia worih : t from 4, there remains 3 ; 6 cannot fa« flublractcd from 2, therefore take 10 from the preceding lemainder .3 (which re- presents lens, hundreds, &c. according to its place) and adding it to the sub- trahend figure 2, say 6 from 1 2, there remains 6 ; and thus proceed thro’ all fi- ^rc8. • In note B p. 0and7, a inislakc lias occurred in sUuinsr ihU laodc ,.«f aubUaction; but it will b* easily Icotrecled by looking at the opcialion given above. Digitized by Google ( 9 ) Under the be.id of multiplication, the first thing which probably will be noticed, is the w.int of a niiilti|>lication table. The Lible given in European treatises, is con- sidered of higli antiquity, being generally ascribed to Pythagoras, who is supposed by some to have studied arithmetic in India. The omission of such a Uiblc in the Lilawati, may have arisen from that work being designed not for children, but for grown up persons who have received the rudiments of education at school, where •everal aery extensive multiplication tables are taught. No fess than five mcthcals of multiplication are given in the text, and the commentaries add one or two more ; but the most common inetliod is that com- prehended under the first rule. It is peribnned in tlie following manner : Writedown the multiplicand, and below it the multiplier, so that the first figure on the right ol the multiplier shall be immediately below the last figure on the left of the multiplicand ; and multiply this last figure on the left by all the figures of the multiplier ; then hav.ng put down the products according to their places alxive the multiplicand, move the multiplier forward one place, and multiply the second figure of the multiplicand in the same manner as before ; thus continuing to repeat the operation unUl all the figures in the multiplicand are gone through. example. 131 *150 100119 1S0298 *5508691 fmilliplicand .... 53436 .... 280006*4 product uuliipiier . . . . 554 634 534 634 534 The individual remits, however, are not set down in this manner, but the tens in cart, succeeding product are carried to the pneceeding one, which is then rubbed out, and tlic new result » set down in iu place. At tlieend of the operaUon, therefore. Digitized by Google ( >0 ) the whole product stands in one line, with the multiplier under it ; thus 880005124 534 The obliterated figures, and the new results, may be presented in tlic following^ manner : 00 8HH JS9S.0 4?S*S» nuiltiplicand . , 52436 S8000SS4 product nraliiplier .. 534 The figures that arc not crossed out form the product. Though thii method must appear very complicated and confused taanEurob pcan arithmetician, it is the one which is most generally practised by the Jyo- tishis, or astronomers ; ami when we recollect that their manner of writing on tlic board docs not admit of the individual steps being exliihitcd together, the above mcthoil, notwithstanding its apparent perplexity, may perhaps be considered the easiest, and most simple, which could be adopted under these circumstances. To per- ceive this, it is only necessary to attempt the multiplication of the above sum by pro- ceeding from right to left, according to the European motlc ; at the same time blotting out each individual product in succession, and bringing the tens from the succeeding product to form a new result, in the place of the proceeding one, .and thus exhibi- ting at once the whole product, without the intermediate operations. It may be considered a coincidence worthy of notice, that the Greeks performed multiplication from left to right, probably for the same reasons which still influence Hindu arithmeticians. The only method whicli the Hindus h.ave of proving multiplication is by divi- sion. They seem entirely unacquainted with the method of proving it by casting the nines out of the sum of the ligurcs in each of the factors, a method, howevert whicli is given in Arabian treatises, and is denominated tarazu, or the balance. Digitized by Google ( II ) As the method of porfomiing division is piftty fully explained in a note pat^c 1 1 and 12, the reader is referreti to it tor information. 'J he divisor is usually placed below the dividend ; and the quotient is set down indifierciuly on any part of the board which is considered most convenient. In the original work of Bhaseara Acliaryn, it will be observed that no marks are employed to denote addition, multiplii'ation. or division; ntitber arc there any to denote Uie rooU or powers of numbers. The only mark employai is that of minus. It is a small circle or dot placed over the figure w hich is to be subtracted, thus O • 6 or 6. This sign is very generally employed throughout India, and al.so in other parts of Asia. In the common books of accounts kept by the inhabitants of Gu- zerat and of the Mahratta country, items which are discharged, or considered struck out, arc usually denoted by a circle placeei opposite to them, or drawn a« round the figures ; and tliis serves instead of an erasure, or any other remark. For instance, to denote that the item 4 yards of cloth Rs. 40 is discharged or struck out, it would be marked in this manner • 4 yanls of cloth Rs. 40 or thus, 4 y.irds of cloth Rs.fJo' This mode of intimating that any thing is struck out or annulled, appears also lo be employed by some Tartar nation.s. Du Halde, tteating of the Tartarian Ian- guage, informs us, that “ if a word is redundant or ill pla^, instead of blotting it out, an oval is drawn round it ; but should it afterwards on reflection appear to be a good word, two ovals or oo are placed at the side of it to signify that it ought to stand.”* In Sanscrit the wonl shunya signifies a circle, cipher, or vacuity ; and the Arabs, on receiving the numeral notation from India, translated it by the word syfr, which, in their langiwge, also means- emptiness, vacuity, or nothing. The Arabic word tyfr, it is well known, has been adopted into the European languages ; and ciphers, or cyphering, are terms frc(]uently used to denote arithmetic in general, probably on account of the important part which the shunya, syfr, or cipher, performs ia that science. I is stated by a learned Reviewer of the Bija Gannita, that the Hindus denote * Uu UsUle Tol. 4. p. SOj. Digitized by Google ( 12 ) plus or addition by two parallel pcr[)t'ndicular lines. I su.spect however, that thii remark is ineonvet. It is indeed an usual jirdclicc to place one or two such linos hctvvven numiiers that are tube added; tut this stonis merely with a view of se- parating the luiiiibcrs ; fur the same niatk is used in a staieinci.i of numbers which require to be siibtrarted or multiplied. 1 he following examples of addition and Hiultiplication are take-n from a txipy of the llija Gannita in my i>osscssion. llequircd liie remainder of two plus subtracted from three |'lus ; of t*-o minus subtracted from three minus ; of two minus subtracted from three plus ; of two plus subtracted lioin three minus? .Statement. — 3 f 2. 3 H 2. 3 |', 2. 3 [j 2. What is the proeluct of two phis multipliwl by three plus ; of two minus multi- plied by three minus ; and of two plus multiplied by three minus ? .Statement. — 3 |j 2. 3 |‘ 2. 2 [] 3. In translating the .‘■econd rule conuined in page 63 of the following treatise, Fvzi directs that an atamul, or sign of multiplication, l>e put down ; and on this authority Dr. Hutton supposes that the Hindus had a mark for multiplication. la this inslaiKf, however, as is remarked in the note p. 65, Fyai has mistaken the sense of the original. 'J'lie words in the rule are, “ write down multipher,” which the commentators expound by directing to *• write down the word multiplier and ac- cortlinglv, in jicrforming the ojicration, the commentators have written the word multiplier, or its initial letter, in the different places where it isrc.iuircd to be set down. Mr. ^ trachc}' says that the I'indusmark division as we do “ by a liorizonHl line drawn between thediiidend and divisor, the lower quantity being the divisor.” In making this observation he was probab.ly misled either by Hindus who were comcr- aant with the European system of notatioi*, or by the Persian translator Fyzi, wiio, instead of adhering to the modes exhibited in the original, generally jwrforms the operations in the manner practised by the Arabians. On looking .at the Hindu method of perfonning division, it will be observed that the divisor i> pl.iecd uiulcj- the dividend, but without anv horizontal line between tliem. It will al'O be remarked that no line is usetl, even in fractions, to separate the numerator .and denominator ; they are put down simply the one below the other, and the reader has to find out from the question, or operation, whether the figures, thus set down, be a fraction or two integers. Digitized by Google 4 , (; >3 ) , As mixed numbers are exhibited in a fractional form In several of tlie statements, they have been printed in smaller figitres than the whole nnml)crs for the sake ot distinctness, tho’ no such difierence is observed in the Take for example ^statement in page 44. ..... 4 3 JOO s6 * 16 b l?3 9 o T, H'e IS ' Ilpre T •''fc improper fraction of' ,<,? the improper fraction of 5y ; improper fraction of' 3^; and *t‘ the improper fraction of 62 ^. A cipher put at the bottom of the right handicolninn to make’ the number of terms equal to that in the other column.- • - • . . j The mode of .adding the numerator of the fraction to the denominator, land of subtracting it, which occurs in page 26 , shews the exact form in which the ope- ration is done 1 on the hoard ; but as/ it differs from our inctfiod of putting dowq the tame operationj it may be neccssaiy, to notice, tliat X ,+1 = 4 ~ ■ J ' 3 + • 1 *= 4 — - 1 5 1 8 8—1 = 7 1 8 — 1 = 7 Again, in pages 30 and 34, j — 1 s=? j, 1 . I we would put It down 8 + j =3. 1 . \ S ' 1 Also 5 — 1 = 1 ■9—2 = 7 f denote what we would express Urns, I ■ . ' ' > ■ 9 1 * I ' 1 ■' 1 4 2 — 1 = 1 9 — 2 = 7 „4 — 1— 3 is the Irindu method of notation, bu{ I — % a 1 . • — 1 = 3 i I i ^ J i la -!6 = 4 , 1' , n» 10 6 = 4 . After this explanation of the manner in whicli, Jhe Hindus perform the rules of , arithmetic, it will not perhaps be considered altogether irrelevant or unimportant t'p shew the different modes followed by the Arabians in performing the same rulc.s. A, very brief account qf the arithmetical knowledge of the Arabians will also be sub- joined, as it may afford matter for curious comparison, and even lead to interesting conclusions. , f . “ ■ I have-coflsuked- several arithmetical twi.*;... m tiw P«»rri an b ut in tlie subsequent remarks, I shaU cite only theRisala Hisab, and the Khalafatrul- •V-. . . >: .!cV 4 Digitized by Google - ( 14 ) Hisab, wHich ate the two oldest works I have seen, and are also die best «nd of the highest authorit}'. The Kisala Hisab is a short treatise on arithmetic and geometry, written in Per- sian by Kazi Zadeh al Rumi, surnamcd Ali Kushchi, who was one of the cele- brated astronomers employed in the )4th century iliy UJug Beg, to draw out the as- tronomical tables which bear this Prince’s name.* The Khalasat-ul-hisab is in the Arabic language, and was composed by Baha- iid-din, who was bom at Balbec in 1575, and died at Ispahan ia l653.'f' It is reckoned a work of great authority, and has been printed lately under the aus- pices of the College at Calcutta, along with a translation into Persian. An ab- stract of its contents, furnished by Mr. Edward Strachey, will be found in the se- cond vol. ef Hutton's mathematical traOls. Ali KuShchrs treatise gives one method only of performing admtior, and it is fhe same which is followed in Europe. But, besides the rule for this method, Baha- ud-din’s work contains a rule for adding and subtracting from lelt to right, as is prac- tised by the Hindus. This rule is illustrated by three examples, which are thus put down : SXAMPLB ADDITION. 5 3 7 3 J 4 1 7 9 1 >0 6 5 7 » 0 (i 8. 1« 11 52 53 7] 27 9|4 21 I 3 7 I '7 I 9 I 4 i 7 1 9 I )810| KXAUPLB • DBTmACTion. * D’ iirrbolet. Bib. Orient. Art. Ulug Beg and Zig. . t Huitoa’a Mathematical Tracts, ?ol. 2 . p. 180. Digitized by Google ( >5 )’ The lower horifotital liae is called Kiuti Aiahi, or the obliterating line, because it !• supposed to obliterate the result which stands above it This method corresponds in principle with that adopted by the Hindus, who, liowever, do not exhibit all the results at once, but efface the preceding result, ami put down in its place the one obtaiticd by adding the tens of the succeeding column. multiplication occupies a long chapter both in AH Kushchrs work, and that of Buha-ud-din. The latter author’s remarkable definitions of multiplication and di- vision are noticed by Mr. Strachey in Hutton's tracts. Ali Kuslichi docs not de- fine division, but his definition of multiplication is the same which is given by Balia- ud-din : “ Multiplication is the producing a third number, such as shall bear the same “ ratio to one factor, as the other factor bears to unity.” And on the maigin there is written this example. “ 3 X 4 = 12. 3 : 12 : : I : 4. For as 1 is the ^ of •« 4, so is 3 the i of 12. And 4 : 12 1 : 1 : 3. For as 1 » the 4- of so is « 4 the T of 12." AU Kuslichi also delivers a rule which Mr. Strachey has translated from the Kh'.i- , lasat-ul-hisab, and which, as he observes, “ exhibits an application of something resembling the indices of Logarithms." The rule is this : “ As to the multiplica- tion of even tens, hundreds, &c. into one another, multiply the two factors toge- ther without any regard to their places, ( that is, as if they were units ) and write down the product. Then add the numbers of the ranks together, and subtract l from the sum^ the remainder is the number of the ratdL of the product. — Exam- ple ; if I wish to multiply 20 by 400, I multiply the significant figure of 20 by the significant figure of 400, the product is 8. I then add together the numbers of the places of the two factors, the sum is 9 ; from this I subtract l , the remainder is 4 , whici} is the number of the rank of the producL The product, therefore ia 8000.*” The first sort of multiplication mentioned is thstof a single figure by a single fi- * Till rule will answer generally when the product of the significant figures does nne wxceed au unit ; or, when the product exceeds an unit, if it be reckoned only as one digit or place. . Digitized by Google 16’ giire, and is exhibited in the Khalasat-nl-Hisab by the following table : ' ' '2 ’ . ; ' r I 1 1/; ; , ■ 4| 9| ejs ^ ; r Rjje |g g4 • , g'g. gQ “ i:-U) 7!3fi SO 21 18 12 5 , S ' ■ ‘ '''49ftS5 2821 14^7 5648 40 32,24 16,8 -t! . te7‘2i65 3445|S6 27 18 9 - • The Risala H'lsab gives tlie table In this forni ; < j • •'-* ! ;.:.i / . I /.--.I -f'l". • • - • .* ’l ! ’ ■0 1 8| 71 6| 51 41 3| 21 1( 91-81 71 61 51 4| 31 2| J jS 1 16 1 J4 1 12 1 JO 1 8 1 6 I 4 1 2 | 2 27 1 24 1 21 1 18 1 15 1 J2 I 9 1 6 | 3 1 3 36 1 52 1 28 1 24 I 20 1 16 1 12 1 8 1 4 1 4 45 140| 33 1 50.125 1 20,l;l^.y0|j5| 54 I 48 149 I 36 1 SO I 84 1 18 ® l'« 63 1 56 1 49 I 42 1 35 I 28 1 21 I 14 I 7 | 7 72 1 64 1 56 I '48V40 | 39 |'24 | HS f iS | 8 SI 172|C3[ 5f r « T 3C 12 7 -|-18f 9 ii. y / i 'y/-. r> tj ■'o lijba-Kf a.O riily • i - " After dlls there "follow several, ingeniooi modes of maltipliaa^on, of which following may be taken as a specimen : ■ .• I . Multiply one of the factors by ten, and also by the number by which di&rs from the other factor ; and subtract tlie one product from the other. the ten Digitized by Google ♦ ( 17 ) Example.— Multiply 8 by g. « X 13— (9 X iO — 8) ?= 72. 2. Add together the two factor® and subtract ten from' the sum ; mdtiidy the remainder by ten, and to the product add that oftthetwo numbers by \vhicl> ten rliffers from each ol the laotora. I • ;r • , ■ -t « • Example. — Multiply 7 by t. , v ” “ •’ 7 — 10 X JO + (10 X iO — 8) = 56 ‘ Of multiplying units by airy number beticeen l o and 20. x 3 . Add together the two factore, and subtract ten from the sum ; multiply the remainder by .ten, and from the product subtract diat of tlie two numbers by wlui it «ach of the factors differs from ten. , ^ Example. — Multiply S hy 14. ,c : 8^:jrr4~:i:'io x lo — fu — ni x lo — §) = li?. • Of multiplying together factors from 10 to 20. , ' ‘ 4. Add the units of one of the factors to the whole of the otltcr factor, and multiply the sum by ten ; to the product add that of the two numbers by which the Factors differ from ten. •' ‘ • ■ t . ‘ * Example. — Multiply 12 by 13. » • • ' . - • J2 -f- ^ X 10 + (12 — IO.X.J 3 -r 10) = Ji6. Inhere a whole number is muiiplied by 5, or 50, or 500. 5. Multiply half of the multiplicand by 10, if tbe j^rfUltiplierbe 5; or by 100, if 'the muhiplicr be 50 ; or by 1000 , if the multiplier be 500; and it a fraction re- aiilt in halving the rmiltiplicafifl, -tlfcn on account of this fraction, take half of whatever number was taken to multiply half of the multipliuitd ; that is, in the first case lake 3 ; in the second 50 ; and in .the Utird 500. , , . ' < . , ... r. Example. — Multiply by 5. .... 15 V- *® — 150 X ^ OW . ^ y t kr a • ■ '• _ ■' "o r!rr v' r I ;i; '.o . j' t . : Digitized by Google ( >8 ) Example of 2d case. — Multiply 17 by 50. 0 ' ' •: . r J7 -H 2 = 8{. . , , 8 X 100 4- 50 = 850. i Of viuUi ftly in g numbers from 10 to 2 o by numbers from So to loo. 6. Multiply the unit of the less factor by tlie tens of the greater; add the pro« duct to die greater factor, and multiply the sum by ten ; to tlie product add that of the units of the two factors. Example. — Multiply 12 by 26. , . ' ' (2 X '2 +‘ S(i) X 30 + (2 X 6) = 312. If here any whole ttumbet it muhiylted by 15, or 150, or 1500. 7- To the multipliciind add hall of itself, and multiply the sum by 10 in the first case, by 100 in the second, and by 1000 in the third case. If a fraction result, tike on account of it half of whatever may have been ..taken on account ofjho whole number ; that is in the first case take 5, in the second 50, and in the third 500. ■ ■ " E.xamplc. — Multiply 24 by 15. • 21 + V X 10 = SGO. ■ ■ . ^ Example. — Multiply 25 by 150. . . ‘ 2-, + V = 371. 37 X 100 4- 30 = 3750. To multiply together any number between 20 and lOp, where the digits in the place of tens are the same. 8. ' Add the units of one of the factors to the whole of the other factor; multiply the sum by the tens of one of the factors, and having multiplied the product by ten, to this last product add tlie product of the units in the two fectors. I Bnainple.— ‘Multiply 23 by 25. 23 4- 3 X 2 X 10 4- 3~^Tb zzz 575. To multiply any number between 20 and lOO, where the digits in theflaet of tens are different. 9. Multiply tlie tens of the less factor by tlie whole of the greater factor; then Digitized by Google ( 19 ) muUi|)ly the units of the less factor hy ttjc tens of tlic greater factor, and add tliis latter product to the former one ; imdtiply Uie,.-6uin by 10; and to Uiis product add tlic product of the units of the two factors. ^ Example. — Multiply 23 'by 31.'^ I '■ ~ t, X 10 4- 3 X 4 = 78?. - ' ^ 10. Add together the two factors, and multiply half of the Buoi into itself ; then taking hu|f of thedift'crcnce 'of the two factors, multiply it into itself; and subti^art this last product from the funner one; the remainder is the product of die two fac- tors. *. ' . \ U \ ‘ Example. — Multiply 24 by 36. *" '' C f ■ (^+_3S )’_ ggj . . , * s ’ Such modes of multiplic.ation, however, arc never adopted in practice, and are oous'idcrcd merely an ingenious and amusing display of the pro|»erties of numbers. When each of the factors consists of several figures, multiplication is {'crformcd by the figuratc process which is called f>habakh. In Ali Kushclii's work, the opera* tion proceeds from left to right, according to die Hindu mode. ’ The rule is as follows : Draw a square figure, and divide it into a number of small squares equal to the product of the number of places in the two factors ; then divide each of these small squares into two triangles, and place the figures of the multiplicand over each square in tlieir proper order, and the figures of the multiplier ' down die left side of the square in the same manner. Then, agreeably to die suc- ceeding example, muftiply the thousands in die multiplicand first by the hundreds, then by the tens, and lastly by the units, in the multiplier, and put down each product in the triangular spac.es opposite to the multiplier, the units being placed in the low- er triangle, and the tens in the upper one. After multiplying the diousands in the multiplicand by all the figures in the multiplier, proceed to multiply the hundreds, tens, &c. setdng down the results in the manner above directed. When all the fi- gures in the muJtiplicand arc multiplied, sum up die diagonal lines of figures. Digitized by Google ( 20 ). Multiplicand. 17 9 9 8 4 4' prodact The mode of operation «s thus ; . . S X 6 * ^ 4 x 6 , . .. S X 8 . . . , . ■ 5x8 40 . . . .j ■ “4 X 8 32 ' 5 X 7 14' • ■ ’ . • . -35 . I799814 . ' The Khalasat-ul-Hisab, htnve\’er, directs the operation to be performed from right to left., multiplying the units in the multiplicand first by the uniU, then by the tens and lastly by the hundreds in the multiplier ; thus, . ' ' "* 4X6 ** . • ' 4x8 I 4x7 28 , . . 5x6 ' *1 - 5X8 “*2 ^ 6X7 ^ 2x6 J* 2x8 2x7 1700844 Digitized by Google f 21 ) An example of this method is given in Hutton’s tracts, and this learned mathe* matician remarks, that it has some resemblance to the operation by Napier’s bones. This figuratc mode of operation has been communicated to the Hindus, who frequently teach it m their schools, but it is neither contained in Sanscrit ;WOrks, nor practised by the astronomers who follow only the rules of the shashters. Division is exhibited botli in the Risala-Hisab and Khalaaat-ul-Hisab,,ui the following manner ; Draw such a number of parallel lines as sliall form as many compartments as there are places in the dividend ; and then draw a horizonUl line which shall join toge- ther of the perpendicular (lines: Write the dividend under the horizontal line at the top. placing one figure between each two perpendicular lines ; and then set own^the divisor » that Us last place, reckoning fiom the right, shall correspond with the last place of the. dividend,, at the same time leaving sucli a space between toem as will be-sufficient , to contain, the whole operation : Find how often the firat figure on the left of the divisor is contained in the corresponding figure of the divi- dend, and.afier multiplying this figure in the divisor by the number thus found sub- ract the product from the dividend, and drawing a line beneath, write down the remamder ; then moltiplythe next figure of the divisor by the quotient obtained andput de^vnthe result one place forward. Having proceeded thus until all the fi! Muki,,ly oich figure i„ the kinrerby thcnumlrerof linre tlBt the IM (if™ .n dmrer comtined h, ,,be to fig..„ „f ^ ■ ast figure in the divisor is contained in the last figure or fieure. of th. ^ and repeat the former process j proceedi*.r in ihi. ^ ^ ^ dividend. » divided proceeding-m this manner until the whole dividend Digitized by Google ( 22 ) SXAWrLK ]at HCTROD. KXAMPLB Sd HBTROR.' Divide .... 660045 Divide .... 680045 • by 255 by .... 255 After treating of these four fundamental rules, Ali Kushclii and Buha-nd-din give the method of extracting the squ.ire root. The former author in the first place delivers a rule for finding the nearest root of a number whose root is a surd. The rule is not clearly expressed, but I apprehend tliat the following is its meaning: Subtract from the number whose rout is a surd the nearest square number below it ; and make the remainder the numerator of double the root of the nearest square plus an unit ; Then the root of the nearest square together with this fraction, is the nearest root of the number whose root is a surd ; that is, if the mixed number formed by the root of the square and the fraction be multiplied into itself, the Digitized by Google ( 2 * ) product will not be the assumed number whose root is required, but something less. Example ; suppose you require the square root of ten, then from this number subtract 9 , which is the nearest square, there remains one ; call this the numerator of double the root of the nearest square plus one, which is 7 ; then 3y is the nearest .root of 10; that is, if 3| be multiplied into itself, the product is 9 ^ and y of 4, which is less than 10 by 4 of t- — ^T he Persian translator of the Khalasat-ul-Hisab adds, that some arithmeticians after subtracting the nearest square from the num. her whose root is required, make the remainder the numerator of double the root of the nearest square without adding an unit In this case, if the nearest root obtained be multiplied into itself, the product will be more by a fraction than the number whose root is required. According to this rule the square root of 10 would be 34, whidi being multiplied into itself, gives 10 and y of 4 -> which is 4 of t less than 4 of 4 . He states, however, that this rule is not general ; and he in- stances that it will not answer for the number 3. Jn extracting the square root the mode of proceeding is as follows ; taking for example the number^ 128 ) 72 . 358 root J 2 '8 1 7 2 9 "3 3 0 8 2j 5 5 6 6 6 4 j 8 • » '7 i 7 7 ~o 8 6 5 3 t •1 M 'C runaindei Put a mark over the units, hundreds, tens of thousands ; that is, over each ua- even figure. Find a number greater than an unit, such that when multiplied by it- self the product may be subtracted from the last period on the lefl ; the number ii Digitized by Google ( 24 ) 3; write 3 above the left hand mark, and also at the bottom of the diagram ; multiply the upper 3 by the lower one, and write the product g below i the namber whose root is required, so that it shaU be op(osite the multiplier, and subtract it from 12, the remainder is 3; write thisbelory the minuend and subtrahend, and draw a horizontal line; then add the upper 3 to the lower 3, and, after drawing a horizontal line above the 3 at the bottom of the diagram, set down 6 the result one place forward to . the right, in the compartment opposite to which there is no mark: Then find a number greater then one in the same manner as before ; it is 6 ; write tli is above the mark, preceding the last one, and also at the bottom of the * diagram on the right of tire figure 6 ; multiply 5 by 6 the .product is 30; write ■ this below the number whose root is requirerl, so that,tnc units place shall be oppol site the multiplier; subtract 30 from 38 the opposite number, tire remainder is • ' 8 ; write this below the minuend and subtrahend; then multiply the upper 5 by •the 5 in the lower part of the diagram, tire product is 25; write Uiis down, pla- ■cing tlie units opposte the multiplier, and subtract it from its.opposite number which is 81, the remainder is 56; write this below the minuend anji -subtralreiul, and draw a horizontal line ; then add tho upper 5 to 65 in the lower line,. the result is 70; and having drawn a horizontal line ^bovc 6 s, winch is in the lower part of ^ r the diagram, carry the said 70 one place forward on the right, so that the units place may be in that compartment opposite to which there is no timrk ; Then find -another number greater than one m the same manner as before ; 8 is obtained ; write > this above the first markon the right, and also at the under p.-irt of tJie figure on the right of 70 ; Uien multiply this 8 by the 7 in the lower line, and having put down the product 56 below the number whose root is required, so that its units place shall be opposite, the subtrahend, subtract it from its opjwsite number, which is also 56 , nothing remains ; draw a horizontal line below, and multiply 8 into the cypher, the result is cipher ; then multiply this 8 by the 81 in the under line, the product is 64 ; write this below the number whose root is r^uired, so that the units may be opposite the multiplier, and subtract it from die opposite number whicb is 72 ; there remain* 8. The root of the number therefore is a surd. Id order to find the nearest root, • add the figure which stands above the first mark, that is 8, and also add an unit, to the number in the lower fine ; the sum 7 17 is.the denominator, and the remainder 8 is the numerator of this number 7 17/ which is double the root of the near- est square plus an unit. The nearest square root, therefore, of 128172 is 358 • • r'iT' Digitized by Qoogle ( 25 ) The extraction of the cube root Is not contained in the Risala-HIsab or Khalasat* iil-Hisab. Nujin-ud-din says that it was omitted on account of its difficulty, and he supplies this defect at the end of a short treatise on algebra, subjoined to the edition of the latter work, published at Calcutta. His rule is as follows : Draw a judwely or figure, as in division, and in the extraction of the square rooti dividing it by three horizontal lines, and leaving a sufficient sjiace between them for the operation. Call the first of these lines makaab, ot cube line ; the second mal^ or product line, and the third zila, or side line ; and write in order, under the first horizontal line, the number whose root is sought. Then mark with a dot the first, fourth, seventh, &c. places, to the end of the number, passing over two places each time. Then find a number greater than units such, that if it be placed above tlie last mark, and also below it, opposite tliereto at the bottom in the zila line, and the upper figure be multiplied by the lower one, and the product written down in the tnal line, so that its units may be opposite the units of the number placed in the zf/a line, and its tens on its left; and after this, if the upper number be multipU> ed by the numbers placed in the mal line, and the product put down below the ma- kaab line in the same manner as before — that then the subtraction of this producj from the number opposite to it in the haab line, or from that number together with ^hat is on the left, shall be possible : Then when such a number is found, perform the operation in the manner above mentioned ; and after subtracting the last product from the number in the haab line, write down the remainder in the same numeral place, below a horizonatal line : Then in order to the succeeding operation, add the upper figure to the lower figure situated in the ziVa line, and write the result beneath the lower figure, separating them by a horizontal line ; then multiply the upper figure by this result, and add the product to whatever is in the mal line, and set down the result ill the same numeral place below a horizontal line ; after this write the sum of the products, one place to the right in the same mal line ; and in like manner add to the upper figure the sum of the upper and lower figures which is written in the zila line below the horizontal line, and having drawn a horizontal line, in order to the succeeding operation, set down the result in that zila line, two places towards the right. Then seek another greater number such, that if it be written- above the mark preceding the former one, and also below it, in the zila line opposite to the said mark, and the upper figure be multiplied by what- ever is in the zila line, and the product increased by whatever is opposite to it in the mat line j and after this if the upper figure be multiplied into what- Digitized by Google ■ ( 2G ) ever 'is in the mal line— tlwt the subtraction of all the products from the remain- ing numbers opposite to them in the makaab line shall be possible : Then «hen •uch other number is found, perform the operation accordmg as before directed, and after subtracting, write the remainders in tlie mnkaab line below Uie horizontil line ; and in onler to the succeeding operation write down according to the form in the nal line and the i//a line, in the manner formerly mentionetl. And if a number such as above mentioned is not found, write a cipher in its place above the mark, and also write it below in the zita line, aiRl observe the usual form m the n^l line and zHa line, without multiplying. Then seek a number such as above mentioned, in order to write it above each mark that there may e, unli t ic ojw- ration be finished at the first mark. Whenever you bring the preccilmg oiK-ra- tion to this place., the work is done ; aud if no remainder be left the cube root is a rational number. But if any thing roinaiii the cube root is a surd, and the number above the judwel, together with the remainder or fraction, is the nearest cube root. Tlic man- ner of finding the denominator of Uie fraction is this; having performed on the right hand side whatever operation is done in the mat hue and zila line, add an unit to Jlie number .11 the=i 7 « line, and add the result to what is ... the mal line. This last result is the denominator of the Iracliou. The nearest cube root tlierefore 948I8S26 is 4 tTT* Digitized by Google ( 27 ) lixample.— Required the cube root of 9 18 18826? Digitized by Google ( 28 ) Tliese rules for extracting the atjuare and cube roots are much more prolix and con- fused tlian tliose contained in the Lilawati ; for though the mode of extracting the siiiiare root is expressed with greater conciseness in the original Arabic than in the Per- sian translation which I have followed, still it is by no means so concise and clear ai that which Bhascara delivers for performing the same operation. In the Khalasat-ul-hisab the extraction of the square root is succeed^ by along chapter on fractions, w hich, however, docs not seem to contain any thing that re- quires particular notice. The fraction is written below tlie integer in the manner 1 of llic Hindus; thus J, denotes one and two thirds. When there is no integer, its place Is pointed out by a cipher, which [the Arabians write like a dot, and place above the numerator of ilie fraction, thus > two fifths. Compound fractious are written in three different ways. First, the secondary frac- tion is written above the primary one ; or, scxondly, opposite to it on the right hand, with the word of, between it and the primary fraction ; or, thirdly, it is sepa- rated from the primary fraction by a horizontal line ; thus. 1 . . I 2 5 of 1 « 5 6,3 5 6 6 denote one half of five sixths. Two or more fractions are joined by the con- junction and, or are denoted by a perpendicular line drawn between them. Thus, 3 4 and 2 5 3 2 4 5 express two fifths and three fourths. Owing to the Arabian practice of writing from left to right, the positions of the fractions connected by the words q/, or and. are the reverse of what would be given in our treatises. The third Chapter of the Khalasat iil-Hisab treats of proportion, or the Rule of Three. In the first place, it is stated that this rule shews the method of finding an unknown numlier by the operation of a fourth proportional such, tljt the ratio of the first number shall be to the second, as the ratio of tJie third is to the fourth ; that is, if the first be the half of tlie second, tlie third also shall be the half of the fourth ; and the fourth sh.-ill be such, that tlie product of the extremes shall be equal to tlie product of tlic middle terms, The rule is Uien given as follows : Digitized by Google ( 29 )' When one of the two extremes, the first and fourth terms, is unknown, divide the product of the middle terms by the known extreme. And when one of the two middle terms, the second and third, is unknown, divide the product of the extremes by the known middle term. In the first case, tha quotient will be the unknown extreme required, and in the second, the quotient will be the unknown middle term required. The Persian translator observes that this is the rule most commonly used ; but he also gives another, which is thus expressed : When one of the extremes is unknown, divide one of the known middle terms by ihe known extreme, and multiply the quotient by the other middle term ; the product is the unknown extreme required. When one of the middle terms is unknown, divide one of the known extremes by the known middle term, and mul- tiply the quotient by the other extreme ; the product is the unknown middle term required. Chapter fburth shew the manner of finding an unknown quantity by the process of errors. It is, performed thus: Assume any unknown number you please, and calling It the first assumed number, operate upon it according to the conditions of the quesUon. If the result coine out agreeably to the question, the said, assumed number is the number demanded. If, on the contrary, the result contain an error that is, if it be more or less than the required number, call the quantity of excess or deficiency, the first error. After this assume any other number you please, and calling It the second assumed number, operate upon it according to the condiUons of thequestiofi. If the result come out conformably to the question, the number demanded is obtained. If on the contrary, it contain an error, that is, come out more or less than the required numbe-r, call the quantity of excess or deficiency the second error. Thus four numbers are obtained ; the first assumed number, and the first error; the second assumed number, and the second error. After this, multi- ply the first assumed number by the second error, and call the product the first ascertained number ; and multiply the second assumed number by the second error and call the product the second ascertained number ; if both the errors be alike* that IS, if both be greater than tlie required number, or les-s than it, divide the' difference between the two ascerUlned numbers, by the difference between the two errors; but if the two errors be unlike, that is, if the one be greater, and the Digitized by Google ( 30 ) other less, than the required number, divide the sum of the ascertained 'numbers by the sum of the errors ; the product in each of the two cases will be the number demanded. Chapter fifth treats of what is named either tuhlil, resolution, or taakat, inver. aion. It is precisely the same which is given in the Lilawati under the title of Itw version. The following is a literal version of the rule ; “ The process in this case is to operate in a reverse manner to whatever is stated in the question. It' the question require doubling, halve; if it require addition, subtract ; if it require multiplication, divide ; and if it require the extraction of the square root, multiply the number into itself; or if the question require tlie re- verse of all this, then operate in a manner reverse to the conditions of the ques- tion. Thus begin at the ehd of the question, and ^proceed in this reverse manner until the answer is obtained.” Dr. Hutton justly observes that the word algebra, which is compounded of al, the and jrbr, contortion, contumacy, consolidation, appears to have been derived from tliis mode of transposing the terms. The words lyatta and vilom, which are em- ployed in the Lilawati, also signify inversion. . The aritlimetical prt of the Khalasat-al-Misab finishes with the rule of inver- sion. This, however, is considerably more than what is contained in AH Kuschi’s work, which does not go farther than the extraction of tlie square root ; but af- terwards proceeds to explain the sexagesimal numeration. As this system is still used by Arabians in their astronomical calculations, the following brief account of it will perhaps not be unacceptable to the reader. In the sexagesimal system numbers are represented by the letters of the alphabet, which is divided, tho’ not regularly according to the order of the letters, into 9 units I w 3 n } j hi into. 9 tens ^ J ^ O \J^ ' and into 9 hundreds ^ j ^ 3 h ; and ^ which expresses 1000. The compound numbers are formed by uniting the units with these round numbers ; as in the following table : \ Digitized by Google ( 31 ) 1 11 21 0 31 i 41 12 22 32 42 13 i ^ L 33 43 > 14 A 34 Jk 44 15 / 23' J 35 45 1 16 ^ 26 A 3 S 9 - 46 • J 7 - 'J 27 37 • 47 <: c. 16 28 t 38 t 48 k> 1 r> kf' 29 ki 39 L. -•9 And so on with the rest. Several of the letters, when employed to express numbers, undergo a small altera- tion in their form. The dal, for instance, is written in a form which resembles the fi- gure 4 in the decimal notation. The double letters also, which express c«^pound numbers, are, in many cases, so considerably changed, as to retain little affinity to their original make. The reader may recollect that the Hebrew numerals are expressed by dividing the letters of the Alphabet into three classes, the first nine letters representing Q units, the next nine letters Q tens, and the four last letters the numbers lOQ, 200, 300, 40q. I'he final Aaph, mew, nun, pe, &nd jaddi, were employed to express 300, 600,700, 600, 900. Though the Arabian arrangement of the alphabet dif- fers from the Hebrew, the .Arabians express the 9 units, 9 tens, and also 100, 200, 300, 4(X), by the same letters which denote these numbers in Hebrew. The Greeks also expressed numbers by a tripartite division of the alpliabet, vrith the addition of the figures called episemon, hoppa, and tanpi ; and it will be observ- ed, on comparing them, that as far as 80, the letters which correspond with those of the Hebrew and Arabian alphabets, have the same numeral power. When expressed by the letters of the alphabet, numbers are written from right to left, whereas the numeral characters are written from left to right, a fact which, ^ has frequently been remarked, would afiibrd a strong presumption that the de- Digitized by Google ( 32 ) citnal notation was received by the Arabians from a foreign people, were not this universally admitted by themselves. Excepting the figures I, 5, and perhaps 4, the Arabians have adopted the nu- meral figures of the Hindus with very slight alteration of form. The arabic let- ter I alcph, which denotes I in the alphabetical notation, has e|gptly the same form as the figure I in the decimal scale. In the alphabetical notation, also, 5 is represented by the letter which in the numeral figures, undergoes no farther change than in being a little more rounded, so as to have the appearance of a circle or cipher. The form of the figure 4 differs in a very slight degree from that of the letter dal when it represents 4. It may be a question, however, whether this mode of writing the dal has not been adopted since the introduction of the Indian notation. The examples of addition and subtraction are given in signs, degrees,. minutes and seconds. AMPLE. — ADOITIOK. sec. ipin. deg. signs 20 4 8 5 55 41 20 8 15 46 28 13 The process is the following. 20 + 5.s = 75, which is one Go and the re. mainder is 1 5 ; put down 1 5, and carry one to the next number 41. 4 1 1 + 4 = 46, which being less than 60 is put down ; and so also are 20 4. 8 = M,; and 8 -p 5 = 13. The sum then is )3 signs, 28 deg. 46 min. J5 seconds. * BXAMPI.E. — SUBTKACTIO.N. 6 5 2 20 9 1 48 55 ♦ Say as cannot be subtracted from 8, borrow one time 60 from' the preced- ing number 5 ( which represents 5 times 60 ); and 20 being subtracted from 68 there remains 48; then say as 9 cannot be subtracted from 4, one time Go is borrowed from 2 ( which represents twice 60), and there remains 65; then 1 from 1 and nothing remains. Multiplication by the sexagesimal system is performed agreeably to the method cx- lubitcd in page 20, excepting that in the sexagesimal numeration the intersecting lines Digitized by Google ( 33 ) in the diagram are drawn from left to right, instead of from right to left ; and, confor- mably to this, the products arc put down and read from right to left, according to the manner of writing and reading among the Arabians. Another difference is, that in the sexagesimal diagram the number of sixties is put down in the upper triangle, and the remainder in the lower one, which is just the reverse of what is done in the de- cimal diagram ; and in order to correspond with this mode of multiplication, the multiplier is put down on the right hand side. The sexagesimal diagram is drawn in this manner : Multiplicand. 49 0 -11 50 The operation is thus : say 4 times 50 = 200, this contains 3 times 6o, and the remainder is 20 ; put down 3 in the upper triangle and 20 in the lower one ; then say 4 times 11 = 44 ; put this down in the lower triangle ; then 4Q x 4 = 196 = 3 times 60, and the remainder is 1 6 ; proceed in this manner tliro’ the whole diagram. After the whole multiplication is gone through, sum up the products, be- ginning with the last on the left hand, and proceeding towards the right. The ex- cess of sixties is carried to the succeeding number as the excess of tens is carried in common Arithmetic. Division is performed in the sexagesimal s)’stem according to the method adopted by the Arabians in operating by the decimal scale, with this difference, that as the numbers are written from right to left, the operation is begun at the right, and pro- ceeds towards, the left. Digitized by Google ( 34 ) • T«ke the foOowing example: The -process » this : (16 X ») -r fiO = 7, 44 <16 X 1») -r 60 = 3. 88 (16 X 32) -r 60 = 8. S* (59 X 89) 60 = 28. 31 <59 X 13) 60 = 18. 47 (59 X 35) -i- €0 = 31. 88 (22 X 29) -T- 60 = 10. 38 <22 X IS) -i- 60 = 4. 46 ' ’ (g* X 32) ^ 60 = 11. 44 After treating of arithmetic the Risala HUab and the Khuiasat- ul-Hisab pao- ceedto geometry. An abstractor the geometrical part bf these works would, however, greatly ex- ceed the bounds prescribed to these few cursory . remarks ; and it is besides render- ed uimecessary, in consequence of the very excellent account which has been alrea- dy famished by Mr. btrachey, and published in the 2d vol. of Hutton’s mathema- tical tracts. ‘ Digitized by Google ( 35 ) Both treatises set out with definitions of a point, line, surface, solid, rigtil, i- cute, and obtuse angles, circumference, radius, sector, &c. and both also make use of the alphabetical letters in demonstrating problems and propositions. These circumstances which point out the connection between the Grecian and .Arabian geometry, have no resemblance to the .plan of the Lilawati, in whidi no notice it ♦nirpn of angles, but all the geometrical operations seem to be performed by the relati- ons of the three sides in « right angled triangle. It has been already remarked that the Arabians call (he decimal scale of a. rithmetic, Hindati, or Indian arithmetic: a circumstance which cicarlv indi. cates the source from which they consider this manner of notation to have been derived. This astonishing improvement in arithmetic being thus traced to Indi.n it becomes a matter of curious enquiry, whether the decimal notation was disro- vered by the Hindus, or was received by them from a people -who had made great- er advances in the science of calculation. To determine this question with muolt degree of certainty we roust wait for information regarding the state of matheina- •tical science in 4hose cotmtrics which border upon India, and which, from the near resemblance .of their religious and philosophical tenets, appear to have had . a free and large communication with it in every department of knowledge. 3t may be sufficient to observe here, that if the decimal notation did not originate iu Jndiii, it must at ‘least have e.\isted there fron> time immemorial, for no traces whatever are to be found of 4in alphabetical notation. Several ‘hundred -years have elai>scd since the numeral notation was adopted by the Arabians, and tlirough tljem intro- duced among the nations of Europe; but neither in Europe nor Arabia has it vet altogether superseded the use of alphabetical characters to express numbers. Both £uropeans and Arabians still occasionally employ letters for this purpose; and among the latter people, and such eastern nations as have adopted their science, it is considered elegant in noticing an event, to employ a word whose literal powers shall point out the date of its occurence; but 1 mever met with any Hindu who was aware of this use of letters, except thro’ Musselman intercourse ; nor did I ever observe any thing like it in Sanscrit works, or in any books written in the' colloquial dialects of this country. -Objects indeed are employed to represent num. bers, but letters never are. Thus, the earth, or moon signifies one; the eyes pignify two ; yug, and Vedas, four, and so forth : and these symbolical words are almost constantly used throughout the Lilawati. Digitized by Google ( 3S ) • Mr. Strachey’s absli-act of the Bija Gannita and this translation of the Lilawati, are no doubt important documents on the state of mathematics among the Hindus, but still they exhibit only an imperfect and partial view of the subject. Imperfectj however, as these siiecimens are, in no other nation of Asia has any work, been found, which contains nearly the same degree 'of rnatliematical knowledge. The Chinese, it i.s said, possess treatises on arithmetic and geometry : but as no translation, or even an abstract of the contents, of any work has yet been publislied in Europe, we are ignorant how fsr their knowleclge of these subjects extends. All their ordin.iry arithmetical operations are performed by the mechanical contri- vance of the swamp.in. No traces of algebra have l>cen discovered among them t and although tliey pretend to high attainments in astronomical science, it is al- ledged by the best writers on China, that they are unable to calculate an eclipse with any degree of precision. From these facts it .nj'pcars, that the Chinese arc far behind the Hindus in point of mathematical knowledge. » In reference to the opinion which has been entertained by sc'veral men of emi- nence, that the high distinction of priority in mathematical improvement, may be- long to the nations of upper Asia, it is sufficient to observe, Uiat as none of the accounts witli wliich we liave been favoured of the extensive regions ot 'liiibet, afford the least countenance to such an hypothesis, we may be allowed to maintain, that until works of superior antiquity to the X^ilavvati, and Bija Ganiuta, and coiir taining a system of arithmetic and algebra equally extensive and complete, be dis- covered in other eastern countries, it is merely justice to allow that the honour of Itaving invented the decimal notation belongs to the Hindus; and that by means of this wonderful improvement, they had made greater progress m mathematical science at least 700 years ago, than has yet been attained by any other nation in Asia. , That the Hindus are unable to comprehend the demonstrations of the rules em- ployed in their present calculations, is an opinion not uncommon amongst the learn- ed in 'Europe ; and it has consequently been inferred, that they either were not the original discoverers of the rules which they now follow, or that mathematics must have degenerated amongst them to such a degree, that they are now unable to com- prehend tile fundamental principles of the science, as demonstrated and handed down by their ancestors. - Digitized by Google ( 37 ) Without entering into any enquiry regarding the foundation on which this opinion rests, it may be sufficient at present to remark, that I have received information of several works, some of which were composed in modem times, and which are said to contain demonstrations of the rules given in the Lilawati, Bija Ganita, and Surya Siddhanta ; and though little confidence can be placed in such accounts, even when furnished by learned and respectable brahmans, yet, in the present instance, I am in- duced to give them some degree of credit, because the Udaharna, a work in my pos- [ session, contains demonstrations of many of the rules in the Lilawati ; but unfortun- ately my copy is so incorrect and mutilated that no satisfactory use could be made of H.* The operations contained in this work corroborate the verbal information I have received, that the demonstrations, both in arithmetic and geometry arc per- formed by means of algebra ; and that the Hindus never appear to have known or practiced the Grecian mode of analysis. • — Within these two hundred years, however, the mathematical and astronomical scien- ces appear to have been gradually declining among the Hindus. / The Jyotishis of the present day are in general profoundly Ignorant of every branch of mathematics. In- attentive to astronomy as a science, they devote themselves solely to the study of as- trology, and possess no ambition to arrive at a higher degree of knowledge than what enables them to cast up a nativity, or to determine a lucky hour for marriages, and for performing the numerous ceremonies practised by their countrymen. ^ In Poona, which may now be regarded the seat and asylum of brahmanism, it is said that no more then 10 or 12 persons understand the Lilawati, or Bija Ganita: and tho’ there are many professed Jyotishes in Bombay, I have not found one indi- vidual who understands almost a single page even of the Lilawati. Among the brahmans the appellation of learned seems to be confined to gram- marians, logicians, and metaphysical theologians ; and those who devote themselves to any of these studies often exhibit a very fair proportion of ignorance on ever) other subject. Astronomy, in particular, as it relates to gross material objects, is considered beneath their notice, except as the means of developing the purposes of heaven. Those persons even who pursue the study of mathematics and astronomy, are in general very ill informed regarding the opinions of the scientific writers. V Since writing this paragraph I have rcceivctl from I’ooria several valuable and curious tprorki, which 1 hope will throw much light on tliU subject, and an tbc general stale of ^astronomy and mathematics anvong the Hindus. Digitized by Google { 38 ) Few are found who have read the whole of the Lilawati ; for which this very cogent reason is assigned, that whoever actually goes through the whole work, is doomed to a deprivation of his mental faculties, or at least to perpetual proverty. This ridi- culous belief proves tliat a knowledge of the Lilawati is now considered of no easy attainment. The common people, and even learned brahmans who arc unacquainted with ma- thematical studies, relate many absurd stories relating to the wonderful powers which are acquired by a knowledge of this work. It is supposed, for instance, that he who understands it, can, in the twinkling of an eye, tell the number of leaves on a tree, of blades of grass in a meadow, or the number of grains of sand on the sea shore. The Lilawati is in fact considered as a sort of magical production, the understanding of which endows the individual with powers and qualities that command both ad- miration and fear. As the following passages, which are translated from the Siddhnnla Siromani, are both curious in themselves, and shew how much the opinions of men of science differ from the absurd doctrines of the Puranas, I am induced to subjoin them, tho" iKejr are not immediately connected with this work. This globe which is formed of earth, air, water, space, and fire, and which is surrounded by the planets, stands firm in the midst of spiice by its own power^ and hus no support. This globular shaped world has no support, but stands firm in space byitsuvtn power. I shall now answer the objections which have been brought from its being affirmed in the Pnranas that the world has a su])port. If this world has a material supi>ort, Uieii that suiiport must have something else to support it, and this second siqqwrt must also be supported, and so on ; but at fast something must be supposed to stand by its own power ; and why should not this power be ascribed to Uiis world, whidi is one of tlie eight visible forms of Uie deity. As the sun and fire in their own nature^possess lieat, tl>e moon coldness, water fluidity, stones hardness, and the air motion, so is tlie earth in its own nature im- moveable ; for different bodies possess different powers. The earth has an attractive jwwer, by which it draws towards itself any heavy boccinlly in the pr»rin< e of Ueufral. All aulhornamed Misliia gives adifTerent meaning to this paragraph and the first part of the lollowing one : — He says : — “ 20 vansa make I nivartana. Kshelar, i. e. n quadrangular figure, or, literally, a field, is that which is bounded by four sides.”— .And here he is followed by fyzee in bis Persian Traiislaltou : — o 90 vans make I nivartana A space which com- prehends four (last or cubits, so that one dast or cubit may be on each side, is called chefar or field.”— The original will bear this interpretation, but the one I have given appears more pro- bable and correct, and also agrees with that adopted by three commentators. • Derived from ghana and hasta: — fihana moans cube; — the literal meaning therefore is cubic cubit. — lu this sense it is ftcipienlly employed in sulnequcnt parts of this treat ise ; — and sometimes also ills used in the same tense as ghana phal, which means simply cubic result OI centen's. Udoesnotat first appear cvklent what is meant by 19 angles; for according to the defini- tion ghanaliasta is a rube, and coufcquctitly bas9f angles. — It must tlierefnrc lie presumed that the definition relnicsoiily to a hollow cube open oa oac side for the purpose of a measure, arui thill the angles at the bottom alone uie reckoned. « ANOTIIEH TABI.E. Tanka 1 Seer, or. Maiia I Seer or according to some > , oi u sMaiia y 72 I I t Tr Digitized by Google ( 4 ) The Thblcs of Time, he. not here specified, are generally known. * * TABLE OF TIME. 60 Pn’a 1 Gl aliVa CO (»!>a1ika I Diva a or clay, 30 Pars 1 Ma>a or irom'i J2 Months 1 Vaiasiory ar, TABLE OP DIVISIONS OF THE ZODIAC. 60 ViLala 1 Ka'a 60 Kala 1 An-a 30 Ansa 1 liosi orsign 13 Hasi 1 Uu'^ana, or purioil in which the sun movL-s thro' the 12 Digitized by Google ( 6 ) PARTI. CHAP. I. - EVERENCE to Gancsa who is beautiful ns the pure purple lotos, and around whose neck the black curling snake winds itself in playful folds. The places of numbers increase in value in a tenfold proportion. — ^Preceding au- thors, for practical purposes, have given them the following names ; Eka— dasa — shata — saliasra — .lyuta — laksha — pryuta — kotya — arbuda — abja— Uiarva— ~nikhar\a — mahapadina — sanklu — ^jaladhi — antya — madliya — -parardlia , Which are thus put down in figures: — lOOOOOOOOOOOOOOOOO * » The tJdahama, or book of csamplei, states that (he names of these eighteen places are put down on the authority of the Vedas; but it also adds (hat in some books (here are names for 32 places. — The eighteen places and names may be thus put down ; I 00000000000000000 a O tr m C rr o JS ^ ; ■-aiS; 5^' -a — .a ■= i-2^ i -a s e o'S'ag •■po’c'a §1 = c ■: 3 -c = .2'g 2 sa — £ -c = .2 -3 2 i £-1 §•= 2 i „ — — a B-J3 - •'a ee-HaaHaa— s Dr. Wilkins in his Shanscrit grammar has given the names of twenty-two places, several of whichdilfer liom the above. Theotder abo in wbicli some of (hem occur is different. Digitized by Google ( 6 ) S E C T I O N I. or ADDITION AND SUBTRACTION. Figures are added from right to left, * or from left to right ; ■ and also are sub- tracted from right to left, or from left to right, according to their places. Example. — Tell me the sgm of two, five, thirty-two, one hundred and ninety- three, eighteen, ten, and one hundred ? Also what is the result if this sum be sub- tracted from ten thousand ? Statement. — For Addition. — 1, 5 , 32 , 193, 18 , 10, lOO. — Added together the sum is 360. ^ Statement —For Subtraction.— 2, 5 , 32 , 193, 18 , 10, 100. — The sum of these bang subtracted from ten thousand, the remainder is 9640. D ‘ Krama. — Tliis word lilerally sigiiiHcs furwanis or progression. Here it means that the operation is to be iierformcd from rij^ht to left. » Utkrama, — The li'cral significttlion is backnards or retrogression. It means here that (he operation is to be perfonned from left to right. This is done by adding up the column op figures on (he Icithand and putting down the result beneath, then adding the next column and ]>ii(ling down the units, and carrying the tens to (he preceding result, which is (hen blotted out and the new result written down in its place, and so on (o any number of columns, as. 2 examples. 36?0 5735269 .5 8131 3829683 b-i 5629 7258264 193 8765 1537636 IS 10 XtnST 16SaS6S2 100 2615 183S085 ?40 • 5C •Place of units. Place of tens on9^Rnn 6 0 , 0 , 3 , 9 , 1 , 1,0 added . . Place of hundreds 0 , 0 , 1 , 1 , 0 , 0,0 added . . Total sum 3(i0 » The lesser number is placed above the greater, units above units, tens above tens, &c. In subtracting from right to left, if the figure in the ujiper line is greater than that below it, take 10 upon the preceding figure in that line, and mark it by a perpendicular stroke ; Digitized by Google ( 7 ) SECTION II. OP MULTIPLICATION. Multiply the last figure to the left of the multiplicand by the multiplier ; then on then subtract the upper figure from 10, and having added the remainder to the lower figure, and set down the result below, add one to the preceding upper figure over which the pcqiendicular stroke is placed, and sulitracl it thus increased from the one below it, pro- ceeding as before if the ujij'er figure is greater than the lower one. Example. 68387 243675 Thos say, 7 c.mn dbe subtracted from 5, therefore 10 is taken upon 8; mark this fi gure by a perpciidi ;ular stroke, and subtract the upper figure 7 from 10; there re" mains 3; this remauidcr 3 added to the lower figured makes 8; write down this number be- low the 5 ; tlien add 1 to the preceding upper figure 8 over which the perpendicular stroke stands ; thus 1 added to 8 makes 9; as this upper figure 9 cannot be subtracted from 7 the figure below it, take 10 upon the preceding upper figure 3 and proceed u before : and so on thro’ all the figures. The result of the operation is exhibited, thus; ^8387 243675 175288 In order to subtract from IrA to right write down tlie subtrahend above Ibe minuend as before diiected ; and if the subtrahend contains fewer places than the minuend, put ciphers a- bove thy emaining places ol the minuend. Then write down below the minuend line the figures over which (here are ciphers; and after subtracting the first upper figure on (he leA from the responding lower one, set down the remainder below it. If the upper figure is rreM^' than the lower one, take ten from the preceding remainder, and adding it to the lower figurc*^ subtract the upper figure from tbc lower one thus increased. Then blotting out the nreced* ing remainder from which ten is taken, put down the reduced number ui its place- and ccfxJ in this manner thro' all (he figures. Example. ' 0068387 3843675 S185S98 317 28 ■ 3175888 The former method of subtracting from right to leA is considered the eisiest, and accord- ingljr it the One most commonly adopted. Digitized by Google ( -8 ) .moving fonvarci the multiplier one place, multiply the next figure;* thus repealing •the ojK-ration until all the figures of the multiplic^ml are m ultipliecl : » or Write down the multiplicand in as many ditlerent places as the multiplier may have been divided into parts, and by these parts multiply tlie multiplicand, and tlien sum up the products : ^ or Multiply the multiplicand by any figure which divides the multiplier without leav- ing a remainder, and tlieii multiply the product so obtained by the quotient of the -multiplier: ° or . The outlie loft liaml is callrd ante, which sisnifies last or final; (he figure preceili«s it is” called npaotj^a, or that which is next to the anta or final; aU the other ficutesar’f uUeil «rfi, w hich signifies first or Bnlccedcnt. , , . -nuicral however the figures arc classed into anta and ad,, i. e. final and antecedent, excluding the term apa„lj/a, or that which is next to the final; and m this arrangement Uie Kst fi-ure oil the left hand is considered the anta, final, and all Uic others adi, antecedent. Hut these terms being merely relative, the antecedent next to the final becomes itself the final when ill the course of the operation the first auto or final is thrown out; hence each TaTcIil'rxl!MrgZ ‘in ThX^rirml^^^^^ hy 12. The operation according * < .t i* 11-^ anrintti'r : 135 135 i 4? 12 J2 (iO J6 0)20 This is culled saanipa gwtanam; or mulliplicalion by Qic multiplier itself as a faciot. e Thus, 13.a X i Ida X I* 12 540 10»0 1C20 This is called khanda gunanam , or multiplication by comiwncr.t (..tls of the multipUer. » Thus, This is called libhaga gwianam 13,5 X 3 405 X 'I ; or multi plica; ion . . . 405 IG20 by submuliiples of tue muidplier. Digitized by Google { 9 ) Separate the figures of the mnitiplier into their ditferent places; tlicn multiply tlie multiplicand according to the places, and also sum up the products according to tJic places : •' or Multiply by the multiplier either increased or diminished by any number you dioose ; and add to the product or subtract from it, the product which results from multiplying the multiplicand by tlie number chosen. ® Example. — If thou art acquainted with tlie mcthoil of multiplying by tlie multi- plier itself, or by component p;irts of the multiplier, or by subinultipics of the multipli- er, or by the places of the multiplier, then tell me iiiy young girl whose tremulous eyes resemble those of a young I'awm, what is the product of one hundred and thirty- five multiplied by twelve ? Statement. — Multiplicand 135: — Multiplier 12 . According to the rule, multiply the last figure to the left of tlie multiplicand by the multiplier, and so on ; tlie result is 1 620 : or Separate Uie multiplier into the parts, 4, 8 ; and having pm down the tnuUiplicand in two placet, multiply separately by these parts of the multiplier, and add the pro- ducts ; the result is the same, I ti'20 : or Divide the multiplier by 3 ; the quotient is 4 ; then multiply the multiplicand by 3, and multiply the product by the quotient 4 ; ihis gives the same result, 1 620 : or * Thus, 13.5 1.2 13.5 270 1020 This is called St, liana giinanam ; or multiplication according to the places » If the multiplicand is multiplied by the multiplier increased by an assumed number, then from the product obtainixl sulilract the product of the multiplier by the assumed number. If the multiplicand is multiplied by the multiplier diininisbed by an assumed number, then to the product obtained add the product of the multiplicand by the assumed number; thus, £MO]ple where the pioduA ii ad ted 135 X 10 1.350 135 X 2 270 Eiimplc wbcic ibe pioduft ia mbiiieca 135 X 20 2700 135 X 8 lOtiO J02O 1020 Digitized by Google ( 10 ) Separate the^g-urei of the multiplier into their places 1 , 2 ; then multiply the tnul- tiplicnnd by these figures separately according to their places, and also add the producta according to their places ; the result is the same, 1 620 : or Multiply the multiplicand separately by the multiplier minus two which is 10, and also by 2, and add the products together ; the result is the same, 1620. Or mul- tiply tie muhipitcnrtd by the multiplier plus 8, which is 12, and from the product subtract the product of the multiphcand multiplied by 8 ; the same result is found, 1620. • In llic copy from wliicli IIiLs translation is made lliorc is also the following : “ Or, the multiplier being disidixl by 1, the quotient is J ; the iindliplicand mulliplii'd by 4 is ; this nitiltiplicd by the rpioticnt J gira< the product IC20.” — But this appears to be an infer] rial ion, as it is not coiilaint'd in llie other copies. It is also mts-placcd, for it be- loin's Co the third iiielhod. IScidcs these different methods of miilii'plication, another method is adopted when the mul- tiplier contains three or moic figures. In that care, wiitc down the two factors so that the first figure of the multiplier .shall Ixr below the last figure of the mulliplicand reckoning from right to left. Then multiply the la-t figure of the multiplicand by all the figures ol the multiplier be-riaiiing at the hist tigiitc to ll c letl, as directed in tlic retrograde method ot mullipli- calion. After the Imt figure ofthc unitlijirK'aiid is thus multiplied, tlirow itoul, and on moving the nmili)>Hcr tbrwaril one place, multiply the next figure in the multiplicand in the same man- ner as the former one, and thus proceed tliro’ all the figures of the multiplicand, lire result of t!ie dill'eri nl j'roriiicls added together is the whole product. EXAMI’I.E. iMiiUiplicaml 7240 j'> Multiplier The result is shewn, thus. 7 X 829 a X 829 4 X 829 0 X 829 3 X 829 5 X 829 600722415 u o o e last product the sum of the squares of the parts ; the result is the square of the whole number : * or But the operalion will perhaps be more dearly understood by shewing the results in the Callowing manner : 8X2 ♦ 97 X 4 388 9x9 81 7 X 18 196 7x7 49 88909 Ranradhar eivei another me'hod of performing the operaUon, vi*. Having the^square of the la.t figure, multiply the preceding figure by double the sum ol the last Lure • thin square the said preceding figure, and alter multiplying double the sura of the two ]L figures on the left band by this precedii* figure, square the next pieseding figure ;-thu5 0 V ^ , 4 n V.T 4 36 V X 9 ^ ^ 81 < X X a gj e. 2 ... 68 X 7 ... 7 406 49 88809 Bat a. this method is not conformable to die rule it is rejected by the otber commentatom. a ,f a line be divided into any two parts the «,nar« of the parts together with twice the rectangle contained by the parts arc equal to the square o t ew oe ^ ^ ^ ^ » Take for example 9 ; and divide it into the parts 4, 5 ; Then, 4x6 20 X * •a C ft 4x4 6X5 16 25 41 + 40 81 square of 9. Digitized by Google ( 15 ) « Having diminished the quantity and also increased it by any number you choose, multiply the results together, and to the product add the square of the number cho- sen. The sum is the square of the quantity. * Example. — What are the squares of Q, 14, 297, 10005 ? Statement. 9, 14, 297, 10005. These being multiplied in succe»Ion, accord- ing to the rule, the squares obtained are 8 1 , 196, 88^09, 1000100025: or Take the parts of nine, 4, 5; the product of these multiplied by two is 40; to this product add the sum of the squares of the parts ; the result is 81, ike square of 9. Or take the parts of fourteen, 6, 8 ; the product of these multiplied by two is 96 ; to this product add 1 OO the sum of the squares of the parts ; the result is 196, the square of 14. The parU of fourteen, 4, 10, also give the square 196: or The number 297 diminished by three is 294, and increased by three is 300. Tlien 294 and 300 being multiplied together, and the square of 3 added to tlie product, the result is 88209, whicir is the square of 297. SECTION V. OF THE SOUARE RoOT. Subtract from the last uneven period ® the greatest square which it eontaint. Set down double the square root in a separate line, and after dividing by it the next even •period, subtract the square of the quotient from the next uneven period, and also set, down double tliis quotient in the line : Then divide the next even period by the number in the line, and on subtracting the square of the quotient from the next un- even period, set down double this quotient in the line. Thus repeat tlie operation thro’ ^ the figures. The half of the separate or quotient line is the root. * If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is ii^e up of the half and the part produced. Euclid B. S. prop. 6. » The figures in the first, third, fiflli, 8tc. places, reckoning from the right, are called wV- ama or uneven, and are marked by a perpendicular stroke. Those in the fecond, fourth, sixth, &c. places, are called rama or even, and are marked by a horizontal stroke, in the op<'- ration the |>eriod receives its name from the denomination of the first figure on the right hand. \\ heu (he first figure oa the right is uneven, the period is called uneven ; when this first figure Digitized by Google ( 16 ) Example.— Wliat arc the square rooU of 4 , Q, 81, 196, 88:09, ICO100015 ?— They are 2, j, 9, 14, 297, 1005. * ij even, the period is called even. Thus in (he root of 88209, the numbers 48, 122, 4l0, 49, tinevrn. subsequent cxa-TipIe of extracting t!;e square are respectively named even, uneven, even, '._LT'" "e “lua given in the commentary, taking for example . “ Male the marks even and uneven. Here the last uneven figure i.s 8 ; from this subtract 4 which is the square of 2, and there remains of ,he square numlu r 4^^ : Then muhipiv the root ol 1 bv 2, the produel is t ; Mt this doan in a .separate line, amt by it divide the next even jicriod 4.8; the qiioticnl is 9, an.l Ih-rc remains t>/ the square ; subtract 8! which is the square of the quotient 9 from the next uneven period 1^; there remains of the square im, Then multiply the quotient 9 by 2: the product is IS, which l-eing put down ,n the separate Ime b, low 4, one plait foru>..rd, the sum is 58: iJy this number divide thene.xl even j criod 4lJ; the quotient is 7, and there remains 0/ Me square i'J ■, fmm tl.„ unoveu periocl sul, tract 49 whirl. Ls the square of 7 ; no remainder .s Ml r Th. „ muhi,.!, a qitoluiit / \ 2, the pruviuct is 14; pul this down in the fcparate line oii« place lorwaid, and mid toge.Ixr (he dillc.enl products ,hc seimralc Une ; their suiu is 594, and tlie half of this IS 29., which is the .nmt of llie square 882(19.” 1 he rule for exlrar'-tiiig the square root wHl prTliajK be more intell'VIble when expresscti ia the following terms; Subtract from the last uneven period the greatest square in the said period, and wri!e down double the root of this square in a se|varatc line. To the rcm.'i.r..ier bring down the next even figure f.ira dividend, which lieing divided by double tee root, nuJ the next uneven figure an'iexed to the remainder, subtract the scuare of the quotient the |vriad thus fcruicd, and also write down double the quotient in the leparate l.ne one place fc'ivard, adding the unit if there be one to the preceding quotient ti- gu:e. lo the list r-mainJer bringdown the next even figure for a dividend, which bung divi.led *is Iie.’bre by (bo niir.li'.-r contain id in (he separate line, and the next uneven fiiurc annexe, 1 to the renia.iider, fuhd.-ac; tL'.- s.quarc o." Uic but quotient from the period thus formed; and then Digitized by Google ( 17 ') SECTION VI. OP THE CUBE A CUBE Is the product of any number multiplied twice by itself. * or Put down the cube of the final or last ^gitre which is on the left hand, .and mul- tiply the square of this final by the antecedent figure and by three. I'hen having multiplied the square, of the antecedent figure by three and by the final, and also having cubed this .•mteeedent figure, atld up all the reiulls “ each one place forward, and their sum will be the cube. Then taking the two figures whose cube is found call them the final, and rej)cattlie operation. k writedown double this quotient in tlic scparalc line. And thus repeat the process thro' all the figures. i-i-i b«ao9 4 4J 18 , 3G I 1'J‘J 81 68J 110 406 Example SS209 greatest square Separate f.ine even period divided by double the sum of the root, the quotient is 9 root 2x2 quot. 9x2 uneven period square of the quotient 9 quot. 7x2 4 ' IS 58 ft ’ 504 even period divided by the number contain- ed iiitlic separate line, the quotient is 7 The h.ilf of which is 297 the square root. I 49 uneven period 49 Mjuate of the hist quotient 7 * Literally; — “ The same number multiplied Uiriceis called a cube.” » The cube stands in the first place, and the other results or products follow in succession, •ach being put down one place before the other. • I his rule may be stated in the following m.inner: ■Write down somewhere the cube of the last number wbicb is on tbe left hand, and aflet Digitized by Google ( 18 ) The operation for squaring or cubing any number may also be performed by beginning at the first figure on tlic right hand : * or Multiply the number by its two component parts and also by three; then cube the two parts, and add the products. niul;ipljin» ttic squ-irc of Uiis last nimibor by llie next or preceding number and by tliree, put down die product one place forward from (lie cube. 'I'hcn multiply the square of the said precciling number by three ami l.y the last number, and also cube ll.is prccading num- ber, setting dosrn each prmlucl one plate forward. The protliicls being added together ac- cordin' lo"^ their places, their sum is the cube of these two numbers. Then taking the figures whose cube is found, call ihcm the final number, and the next figure the second number; and repeat the operation. Itsvill lie olrservctl that the whole series of figures is divided successively into two numbers, the figures whose cube is found being cunsidcrwl one mimtjcr, and the sucrccdiiig figure the other number. Take for example 12j()7H. After tiudingthc cube of Ihesum formed by tlicfi- n.il number I and its preceding numlicr 2, there two figures are taken for one number which is called the final, nnd the next figure a is the other number. Also alter finding the cube of the sum formed by the two numbers 12 and 5, these three figures are ll.cii taken lor one num- ber and their jircceding figure for the other, .and so on thro’ all the Cgiites. . * This sentence seems t» come in as a no'c or remark. In order to jierlorm the opcialion of squaring or cubing according to this method, li e U- dalmriia first directs the denominations of Uie figures to be changed, calling the final figure on . the left hand the prec eding or aiileicdeni, and the preceding nr aiilct client figure-, (he final. Then to produce the stpiarc of any number proceed agreeably to the rule in jwge 13. Tak* the example 297 given umlcr that rule; thus, % ?<» 7 49 00 1 i 40(j 0 fil o IK 3f> o X 2 4 88209 In order also to produce the cube of any number, proceed agreeably lo the prereiling rule jn the text; observing in addition lo what is there directed, that the prinlue .s imisl be -et down so that in adding up the wboic each product sliuU be one place before its tuclor in tbe Digitized by Google AIw, the cube of the root of a square being multiplied by itself, the product is the cube of that square. Example. — ^Whatare the cubes of 9 , 27, and 125? s Statement. 9 , 27 , 125. The cubes of tliese numbers in succession are 729 , 19683, 1953125: * number. Take for example (be turn 2125 ; thus, 5» 5* X 2 X 3 2* X 3 X 5 2 * 25* X 1 X 3 I* X 3 X 25* 1 * 125* X 2 X 3 2* X 3 X 125 2 * 2125 125 . 150 60 8 15625 . 1875 . 75 . I 1953125 93750 1500 8 9595703125 » Accor Jin!? to the first rule the operation may be exhibited in tbe following manner, faking for e.xainplc the sum 125. — Thus 1 cube of last figure 0 t> left hand 6 square of last figure multiplied by the preceding figure, and the product rauliplied by three 12 square of tlie said preceding figure multiplied by three, and the product mutiplieJ l>y one the last figure 8 cube ot the preceding figure 1728 Then calling 12 the final number, and 5 the preceding number, repeat the process ; Tlius 1728 cube of final number 12 . 2160 square of the final number multiplied by tlic preceding number, and the product iiiultipliinl by three 900 .square of the suiti preceding number mulliplied by three, and the product mulli- plicd by (he final number ‘ 125 cube of the said preceding iiumbet 1953125 cube of 125. Digifized by Google ( 20 ) Or, the nwmber 9 being multiplied by its parts 4, 5, and also by 3, the product is 540; to this number add 189 which is the sum of the cubes of tile parts; the result is 729 , tiiecubeof 9 .* Or, the number 27 being multiplied by its parts 20 , 7, and also by 3, the pitxluctis 11340; to this number add 8343 which is the sum of tlie cubes of the parts; the result is 1 9688, the cute ej 27. Or take the number 4. The cube of its root is 8, the sijiiare of which 64 is the cube of 4. Also tile cube of tlic root of 9 is 27, the square of which 729 is the cube of 9. SECTION VII. OP THK CUBE HOOT. Tlic first place on the right Iiand is called ghona or cube; the two next placet ttghnna or not cube. ® Subtract the cube contained in the final periorl from the said period ; put down the root of the cube in a separate line, and after multiplying its square by three, di- vide the antecedent figure by the. result, and write down the quotient in the separate line : Tiicn multiply the square of the quotient by the preceding numlier in that line and by three, and after subtracting tlie pro ... 64 y 12.5 1S9 189 ■ 729 cube of 9 •The ghana or cube place is marked by a perpendicular stroke ; the atdiana or not cube -I — I place by a korirunlal stroke, as in the extraction of the squaic toot ; thus 190&li. Digitized by Google ( 21 ) Example.— What are the cube roots of 729, 19683, 1953125. The roots are 9, 27, 125 * * » The commentaries shew the operation in the following manner, taking for example i-.i.-i 1053125 : “ Mark the cube place by a perpendicular stroke, and the not cube place by a horizontal stroke. In this example the cube contained in tlie final cube period is 1 ; subtract this num< ber from Uie final, and no remainder is left ; put down the root which is 1 in a separate line, and after multiplying iU square by 3, divide the antecedent 0 by Uic product ; the quotient U S,* and the remainder is 3; put down the quotient in the separate line, thus 1,2; the square of the quotient 2 is 4 ; multiply this by 1 the preceding figure in tlie separate line, the pro- duct is 4, which being multiplied by 3, the product is 12; subtract this from the antecedent 35,+ the remainder is 23 ; the cube of the quotient 2 is 8 ; subtract this from the antecedent 233, Uie remainder is 225. Again the square of Uie root 12 is 144, multiply this by 3, and by the product 432 divide the antecedent SVoT ; the quotient is 5, and the remainder is 91 ; put down the quotient in the separate line, thus 1,2,5; the square of the quotients is 25, which being multiplied by the preceding number 12, the product is 300, and this again multiplied by 3, die product is 900; subtract this from the antecedent 912, the remainder is 12: the cube of the quotient 5 is 125, which being subtracted from the antecedent 125, no remainder in left." The rule for extracting the cube root may also be expressed in th« following terms. Subtract from the last period on the left hand the grwtest cube contained in that period, and set down its root in a separate line. * In this division it is necessary that luch a remainder be left aa, with the antecedent figure annexed to it, will be equal to the square of the quotient multiplied by its preceding figure or figures in the line and by three ; henre 3 is taken twice only in 9, for had it been taken three times, no remainder would hare been left, and 5 the next antecedent figure is leas than 2X3>CIX3 >2. t The term antecedent denotes not only the antecedent figure, but also the number or period form- ed by annexing the antecedent figure to the number which remains after tbe preceding operiiion. Thus the antecedent figure 5 being annexed to 3, the number which remains after dividing 9, makes 3i, which is denominated tbe antecedent In the same manner the next antecedent figure 3 being an. nexed to 23, the number which remaiiia when 12 is sublractcd from 33, fomia tbe anlecedcut number — I or ptriod 233 : and so on; Digitized by Google ( 22 ) KOTE CONTiSVED. Annex llipnrxf ftfrnrc to flic remnimler, and having divided Ui:s number by triple the square •f the root, set ilonn the quoliciit in the separate line. Annex tile next fifrnre to the remainiler, ami from this niirrlier subtract the prmluel of triple (hi- square of the last quotient multiplicJ by (he precwlin^ figure or figures in the separate line. Annex the next figure to the remaimler, ami from this number subtract the cube of the last quolicui. Hi'|ie it this oiMTalion thro' all the figures. The number in the separate or quotient line is the culie rmst. • Agreeably to this enuncialion of the rule the nperalion is cxiut’i'.e;! in the following M .nncr : Example cnlie conlaincil in first period I M]uarc of cube root multiplied by 3 is 3J quolirnt 6 3j square of the quotient 9 is 1 ; this multiplied by 3, and the product by I,’ the preceding figure in tlie, separate liiie, is 12 , 933 cube of the quotient 9 is S square of the root 12 is I U ; this multiplied by 3, is 439 J 99AI (_5 * 2IbO 912 square of lad quotient 5 is 95; this mulltp'icd hy 3, and (he pnnluct by 12, tlic preceding figures in separate line, is 900 195 cube of last quotient 5 is 195 M « I Digitized by Coogle • ( 23 } HOTE COSTISVSO. 0 - The following example is selected merely for further illoatiatioD) and ia not taken from the commentaiies. __i — 1 — 1 — 1 4* 4“ X 3 48J 586 t.9 438 1541 Q* <1 V a. . 1 5695 ** 8 9* Q3 If 49’ X 3 7203J 49663 L6 43818 «* # 64458 s s. 6292 •» 591607 6* •, . 216 SO«* _ 1 _ S . . . . 7S8048J 5913912 L8 • 5904384 95883 8* X 3 X 496 ^ ; 95232 % * 512 8* • ( - Digit ifod by Google ( 24 ) CHAP. ir. SECTION r. OF FRACTION* To reduce Jractions to a common denominator. Multiply reciprocally each numerator and its denominator by the otherdenoml- nators: This will give Aavrfijg- a common denominator, or Reduce the denominators by a common- measure, and multiply eadh naraerator and its denominator by the other reduced denominators. Example. — Required the common denominator of 3 J ; ; a and the sum of these fractions. Also required the common denominator of ; and the difference of these fraiHions. Statement—; ® — ^These fractions reduced to a common denominator are iJ« .’i 5*^ I'he sum of the fractions is Statement for 2d nvethod. s’, These, being, multiplied, according to the ft st rule, by the denominators divided by seven, the two fraaiom nduetd to a com- mon denominator are ,t». T1 1 C difference of the two fractions * The Hindus do not distinguish fractions hy nny niark, but pncrcly write the ftiimersfar above the deaomiiutor. For the sake of ilutinctnirss 1 have employed the common ftactioiud figures in the notes. » 1 he whole number is converted into a fraction by placing an unit for its dcaomiaator. * Thus, , *x5xS = ^ I ‘x1x 3=3 1 >Xlx5=a Tx6x 3=:|5 1 4X1x3 = 16 I 3 X I X i ~Ti * Divide each of the denominators by 7 ; thus 63 4- 7 14 -1. 7 new denominators. Then multiply reciprocally llie original numerator and denoraiaatot of OM ffacUoo by thO ltduced denominator of the other ; lx2_i_ I ±X9_^ 63 X 8 1*6 I 14 X 9 ~ '**1 fbes, _f T_ U6 "" 1*6 — U«. Digitized by Google ( 23 ) Compound Fractions. Multiply the numerators by the numerators, and the denominators by the. dcn That is, when the fraction of a fraction is plus or minus, •e The fi«l rule under this head teaches how (o reduce a mixed quantity to an improper fraction. The second rules gives the method of reducing fractions of a fraction, or compound fracifons to an equivalent sim|ilc one* In tiie commentaries the rule is thus slated t ** hfultiply the denominator of-tlie upper fraction bjr the denominator of the lower fraction. Then add the numerator of the lower fraction to its own denoiuinalor, if the fraction is plus, or subtract it if it is minus; and multiply the numerator of the upper fraction by tlie denominator of the loner fraction thua augmented or diminished by its own numerator. Again, multiply the denominator of the upper fraction by the denominator of the lower fraction, and so on ; repeating the same jirocess (ill you arrive at the upiier-most fraction.” But the operation will perhaps be more easily understood by stating the rule thus : Multiply the upper denominator by the denominators of the lower fractions. Also multiply nhe upper numerator by the denominators of the lower fractions plus their numerators, if tlia fractions are plus, or minus if the fractiou ue miuui.^For iUuttration t^ the three eT^y n plna 'irhich wegiTeainthe (extj ris. -- - - - _ Digitized by Google ( 26 ) Example. — What are the equivalent fractions of two plus one fourth ? Also of tlu-ec minus one fourth ? * 2 I 3 Statement. — , j • Equivalent fractions J [ V ® Example. — W'hat is one fourth pins one third of itself, plus one half of the reJ suit ? What are two thirds minus one eight of itself minus three sevenths of the remainder ? What is one half minus one eight of itself plus nine sevenths of the remainder ? Jst E.xample. — Add i + 4 of il*elf + i of the result ; .thus, i + } "h ~r~ 1 X 4 y 3 = I* _ , ;x3x2 = £4 3 4-1=^ i + 1 = 3 2J Example.— Suhtract from • i of itself, and from]tlic remainder f of itself; thus,' ix7x4=*6_, 3 X 8 4- 7 = lOs — * 1-, = T 3 7 — 3=4 . -*• 4 T 3d Example ; — riz i — f + — ; — T I X 7 X 16 = 'L- = * K 8 X 7 = iiJ 9 7 4- 9 = 16 « In (he original llic minus fraclion is distinguished by a cipher placed above it ; but as this might cre.ale some confusion in reading, I h.ivn here used a dot instead of the cipher. — It may be observed that in staling a mixed quantity the fraction is put Ik-Iow the integer; and fiiilher, that coiiipoimd fractions are arranged in a line below each other, the highest fraction of the scries being at (he head. * Example 1. — Plus fraction. Example 2. — Minus fraction. 2 s 3 , 2x44-1 = 9 numerator 3x4 — 1 = 11 numerator. 4 9 - IL * equivalent fraclion. * * cquivalcut fraction. Digitized by Google ( 27 ) minus one eighth plus nine sevenths ? Statement. MM- Equivalent fractions I Addition and Subtraction of Fractions. Tlic addition of fractions which liavc a common denominator gives their sum ; Uie subtraction gives their difference. •'* When there is a ([uaiuity without a demominator, an unit must be assumed as its denominator, * Example. What is the sum of one fifth, one fourth, one third, one half, and one sixth added together ? And their sum being subtracted from tliree what is tlie re- mainder ? Statement. I i I I their sum is j;. And this subtracted from 3 leaves the remainder J i. ® Multiplication of Fractions. Divide the product of tlie numerators by tliat of the denominators ; tlic quotient is the product in the multiplication of fractioas. c Example. — What is the product of 1\ multiplied by ? And what is the product of ; multiplied by j ? » Tlic commentaries in tlie first place iliroct compound fractions to lie rccluccd to simple ones, and mixed numbers to improper fractions, accorilins to tlie rule for plus and minus frac- tions. TUe text under one rule comprciicnds the tv>o common rules for the adilitioii and siib- traction of fractious; vU. add the nuiucnitors together in the case of addition, and sul)lr.nt tlicin in the cose of subtraction. » Example 1. — Add f- i f t i- — These fractions when rcilucedtna common denominator, are tjj US fiS US fl?! the sum of the numerators added togcllicr is ; and reduced by 3G the fr.iclion is ij. ; Example H . — Subtract • J from 3. Assume an unit for the denominator to 3 ; then *9 X 1 = S3 |- 3 X ‘.M = 60 — 39 = .j; so I e The mixed numbers being first mluced to improper fractions according to the rul^ for plus aud minus fractiuirs. Digitized by Google ( 28 ) 2 2 I * ’ Statement. , , * ’ Reduced to •quivalent fractions, and multiplied, the result* * ’.V 1 i. ‘ ' Division of Fractions. Invert the denominator and numerator of the fraction which is the divisor, and perform the remaining operation according to the rule for the multiplication of frac« tions. Example. — W hat is the result if 5 be divided by 2 J ; and what is the result if I he divided by ; . Statement.— 5 I j ! i- According to the rule, the results ? | * are ob- tained. ■ Of the Square, G?c. of Fractions. Square or cube the denominator and numerator accqsding to the square and cube rules. * V The method of finding the roots is also obvious. Example. — What are the stjuare, square root, cube, and cube root of 3 J ? Statement. — 1. 'J'he square is V ; its root is J. The cube is • J • ; its root is J. « • Example 1. — Multiply 2} by 2}. — The two mixeJ mired numben being reduced to theit improper I'raclioiu, we hate J V ! Then, J- X "V = W Example. II. — Multiply | by J. • In this case also, the mixed numlter is first reduced to its improper fracl'on by the rule of plus and minus fractions; and the whole number is converted into a fraction by placing an Unit for its deriominalur. Aficr this process, the fractions in (he first example are ^ |. Then invert the terms of the dividing fraction, and multiply according to the rule for UieKiiltipU- Cation of fractions ; thus, Example I. — Divide 5 by 2| * X I = V = 2i Example 2. — Divide ^ by f i X i = J = t • In this example likewise the mixed number is first reduced (o its improper fintclion. Digitized by Google ( ) \ • . S E C T. I O N n. V.. ‘ .pHE Effect op Cuhbb;' When a number Is added tor cipher the result is that number, llie square, be. ot dphsr is cipher. A number divided by cipher has cipher for its divisor. * When a number is multiplied by cipher the product is cipher ; ® but in case any operaUon temain to be done, cipher is merely conceived to be the multiplier ; for when cipher is the multiplier, and cipher also becomes the divisor, the number is considered unchang- ed. ^ Also a number is considered unclianged by the subtraction ° or addition of cipher. Example 1. — What is the result of 5 added to cipher? What are the square, aquare root, cube, and cube root of cipher ? What is the quotient if \0 is divided by ■cipher ? * This 1 undersiaml is a kind ol' (echnica) phrase deiioling that in such case the quotient if infinite. The following is the explanation given in the commentaries : “ A quantity divided by cipher is not changed, but obtains the denomination ‘ quantity wills cipher far its divisor.’ When a quantity acquiies cipher for its divisor, tlie quotient is in- finite; for,the quotient is that number by winch, when the divisor is mnltiplied, and the product aubtracted from the dividend, no remainder is left. Hence in Algebra when cipher is the ■divisor, the quotient is considered iirfinite,” In the Udahama it is observed, that “ in proportion as the divisor is small, sots the quo- tient great; b<.ttbe divisor when it is cipher being iafioitely small, the quotient is therefore infinite.” The origiiusl word ananta signifies iofinile or endless. It is an epithet applied to the Deity, space, &c. * When a nnmber is multiplied by cipher, the product is cipher, as cipher has no numerical value. Also when cipher is multiplied by any number, the product is cipher. Cem. « The words ‘in case any operation remain,’ mean that if cipher becomes the multiplier, and more operations remain to be done, cipher is to be considered the multiplier, without how- ever performing the acinal operation. '1 he reason of this is, that after cipher has become the multiplier, should cipher in the subsequent ojreral ion also Ijecoine tlic divisor, the number it unch' nged and sfondtas it is; and as both the multiplier and divisor .ttc ci(i(iers, it is merely required to strike them out, and the work of multiplication and rlivision is done. If, however, A the surcieding parts of the oi>eralion, cipher does not become the divisor, tlic expeit aritli- metuian knows that a number which has cipher fur iU multiplier, will become cipher. Cot't, a In Algebra, liowecer, a nnmber suinraclej from cipher changes ita tleneminalifn. Fut- llicr properties of cipher wiilJse learned from Algebru Lem. Digitized by Google ■ ( '30 ') Example 1 . — Required bucVi a number iHat being multiplied by cipher, and half of tile result added lo the producl, this sum muhiplied by three, and this last fnoduct divided by cipher, the result shall be 03 ? Statement. — For Jrsl example. — O; its augment’s; square of cipher is 0; its root is 0. The multiplicand 5 multiplied by cipher becomes O. The dividend lO ' divided by cipher becomes 'o . Shitement. — Fur second example. — ^The number is unknown ; the multiplier iso* the augment is J ; the multiplier is 3 ; the divisor is cipher; the known number is 63. — Then by the rule for an assumed number, which shall be afterwards given,'* the • quotient obtained is !4, the numlicr required. * This method of reckoning is of great utility in calculating the motions of the hea- venly bodies. ^ _____ — . » .'^co page 3‘3. » .As-iiime the number 4 ; then roTiceive* it imibiptied by cipher, and add 3 wbicb is one half of the nssumc I miinbcr ; lUc result is C ; multiply ibis by 3, the product is 13, which being (livideil by lipl.cr, the nuiiib r remains the same, 18. The answer it then obiained l>y the rule for Three Niimbera. Thus, if 18 gives 4, what will C3 give? Having male this siatciueHl, multiply G3 by 4, and divide the product by J8; the quotient is i4. I'om. Tint it, (r X o -f — ^ ^ Q~' X 3 = 63. Or potting o = o, ax -h 63 ; the soltilinn of wbicb equation gives x — 14. The same commentj -y also sliewa (be question brought out by the rule of Inversion. Fee page 31. “ i irsi prep,aretlie numbers according lo tlie rule ot Inversion. The known quan- -nt is 1 ; to which add the frac- tions of the assumed number, \ \ I, after reducing them to a common denomina- ^ tor ; the result is By this divide »04, which is the product of the known num- ‘ , her 68 by the assumed 3 ; the quotient obtained is 48. “ i t>r 21, ismiilliplicd by 3 , ind the pro.turt 63 being subtneted from 1 47, (here remains 84 ; cividc (In's by 3, (he quniient is 98. Proof, 28 X 3 = 84 ; to this add 63 which is three times the 1 of 84; the result is 147 ; divide this by 7, the quotient is 21, from which t ■ — (14 — 4- 8 I44r 4-8 eO g subtract } of Kscif ; the remainder is J4. Then = — ~ To ~ ® .. llic known number.” The commeiifiry then p'oeerds (o mnbe this remark : “ Tliis example C'-nnot lie performed 4iy (he rule for an assumeil number, bceanse it eonlnins the nxil of a square. When (here is u square, square root, cube, or cube root, (lie opcmtioii cannot lie i>erfermi-d by an assumed xiuiiiber; but exclusive of square, square root, culie, and cube root, all other operations in.ny ' be done by the rule of three qiiantiiics. Thus it is nienlioned in the SUldliaiita.” * The first is theas-siimeJ niimlier represented by an unit, and converted intoa fraction by placing aiiolber unit below it for a denominator. r » The commentaries exhibit the operation in this manner ; “ Multiply an assumed nimdier ' 3 by 5, the protlucl is 15; from this subtract one third of itself, -the remainder is 10; diviile this by 10, the quotient is I : to tliis quotient add on; hidf, one third, and one foiiriit ul the aisumeel number; thus 4 J i which reJuceil to a common denominator, are iJ iJ i] JJ, or, •seduced to their lowest letins, are i ‘ i Hic sum fit* which is ’> the Luawn number 68 Ixting Digitized by Google .( 33 ) % Thus as to every example where a aumber is required to be multiplied by any other number, or divided by it, and fractions to be added or subtracted, and a num- ber is kno'vn — in that case operate upon an assumed number according to the condi- tions of the question, and by the result divide tlie product of the known number multiplied by the assumed. The quotient will be the number required, JFlicrc a certain quantity is known. Example. — One third of a collection of beautiful water-lilies is offered to Ma- hadev, one fifth to Huri, one sixth to tlie Sun, one fourth to Devi, and six which re- main are presented to the spiritual teacher. — Required the whole number of water- lilies ? Statement. — J ; I i : known number 6. Tn this case after assuming a quantity, the result obtained is 120.* Where there are remainders. Example. — A traveller pays away one half at Pryag ; from the remainder he pays two ninths at Benares ; from the remainder he pays one fourth for customs, &c. from the remainder six tenths are paid at Gaya ; and there remain 63 which he brings home. What was the original sum ? Statement. — J J i ,‘o. Here the rule is, divide the product of the denominators minus their numerators by the product of the denominators ; and by the quotient inubiplied by 3, (lie asMiiued, (he product is 2ot ; divide this by 'J, the qiiotivn! is 48, whicll is the number required. — I’ronf: — 48 X 5 = 240 ; ffom this subtract one third of itself, or 80, the rein under is lljO; divide this by 10, the quotient is iC ; to this quotient add a third, half, and lumtIi'Ol 48, tliat is i.j, 24, J2; the result is 08, which is the koowu quantity. Com. * Having assumed one or an unit, reduce it along with the fractions to a common denominator. Thus T T T r reduced to a eomnum dciioiiiiuutnr, and reduual to their lowest terras are ia iff iff ii- si"" “f tbe (our last negative fractions, is subtract these from gS, which is the assumed numlier ennverled into a fraction with the common denominator: the remainder is : Then multiply the known number G by the assumed number 1, the product is Ii, which being divided by the fraction the quotient is = ]20, wlrich is the number of water-lilies. Proof: — One third of 120 is 40; the lifth is S4, the sixth ia SO, (be foutth is SO; Ibe kuown number is C ; then 40-f-S4-f-S0-{-00-i-6 = ISO, Cso. Digitized by Google ( 34 ) « obtained,* divide the product of the known quantity multiplied by the assumed : Thus, The known number is 63. Let I or unit be the number assumed. Then subtract each fraction from the prec:ding remainder, according to the rule for minus frac- tions ; the result is ; and the known number being multiplied by the assumed, and the product divided by the qu.tient is 540. ^ The answer may also be obtained by the Rule of Inversion. Where there is the difference. Example. — One fifth of a hive of bees flew to the Kadamba flower ; one third flew to the Silandhara ; three times the diflbrence of these two numbers flew to an arbour; and one bee continued fljing about, attracted on each side by the fragrant Ketaki and the Malati. What was the number of bees ? Statement . — \ J J. The number of bees is 15." In the same manner other examples may be performed by the rule of an Assumed Number. * Subtract (be niiroeraiors fiom (lie dcnoiniiialurs ; thus, I s I (( i— 1=1 9 — 2 = 7 ; — 1=3 io— G = 4 Then, Ix7x3x 4 = 48 product of dcnoinhiators minus the niimeralors 2 x 9 x 4x 10 = 720 product of original deiiomiiiaturs Reduce by I2; the fraction obtainerl is : tlicn multiply the known number €3 by the assumed number 1, and divide the product by thus X */ = * V“ = ^W. Com. In the Udaharna the o(iemlimi is thus exhiuiled : “ Assumed number 1 ; the half of which being subtracted, there remains two ni'ithi of one half are subtract this from i, the remainder is tV> fm™ this subtract i of itself or the remiiiiJer is ; from which sub- tract ^ of ilscif or 4 ^ 5 * ; there remains ^Vir = s?,- % 'h‘» fraction divide the product of the known quantity multiplied by the assumed ; the quotient which resiills is 640, the number required. Proof: — Half of 540 is 270; two ninths of 270 is 60, whicb being sub- tracted from 270 there remains 210; and the fourth of this, or ' 4 *’ being subtracted from 210, there reraaina ‘}® ; six tenths of this remainder are r/j®, or, reduced by 20 is 'f* ; subtract this from * 4 °, (here remains 63.’’ » Thus t *, being reduced to a common denominator, and the equivalent fraction aub- timcted from the aasumed iinmlier I, (he remainder is — I'r t — Then the known number be- ing multiplied by the assumed, aud the product divided by (be quotient obtained is 15. — P roof t — Number of bees is 15; from wbicb subtract one fifth of itself ox 3, the Digitized by Google ( 35 ) When there are the sum and difference. Add the difference to the sum, and also subtract it from the sum ; then the re- sults halved will be the quantities. Example. — ^The sum is 101 , and the difference is 23. What are the two qwn' tides f Statement.— Sum, lOl ; difference, 25. The two quandties are 63, 38. When there is the difference of the squares. Divide the difference of the squares by the difference of the quantities } the quo-' tient will be their sum. The quantides themfelvet are then found by the last rule; Exampl^-What are those two (quantities the difference of which is 8, and the difference of their squares is 400 ? Statement. — Difference of quandties 8 ; diflierenoe of squares 400. The two quantides are found to be 21 , 29 . Concerning Squares. Muldplytbe square of an assumed number by eight, and subtract an unit from the product ; divide half of the remainder by the assumed number ; and the quod- ent will be one of the quandties ; Add an unit to half the square of this quantity { and the result will be the other quantity: or Divide an unit by the double of an assumed number, and add the assumed number to the quotient : the result is the first quantity : the second quantity is an unit. * teroaioder is 12 ; from tUU subtract 5 which is.one third of 15, the reuiainder is 7 : The diSereace of 3 and 5 is 2 ; multiply this Ixy 3, the product is 6, which being subtracted 0001 the last remainder 7> there remains 1. C^m. * The'^ problems belong to the class called Diophantioe. This case may be selred in the foUowiug muani'r ; ^ “ I to find the vaiae of * and y in terms of any assumed number e. 2d Hule. — If y = 1 then r’+l — 1 = a or x* = ca and X* — 1 — 1 or X* — 2 = ca substituting 2a — x for the side of this square we hare X* — ^2 = 4a*— 4axq-x* 4ac = 4«* + 2 1 * — (according to the lule) = — + » Digitized by Google ( 36 ) The squares of the two quantities being added together, or subtracted from each other, and an unit subtracted/row the remit, there will remain a square. Example. Wliat two quantities are those the squares of which being subtracted from each other, or added together, and an unit subtracted from the result, there will remain a root ? * Tell me this if thou art able to perform an ojMaatiou which is diffi- cult even to the Algebraist. Statement. — In order to find the first quantity, let the number a.ssumed be ; ; the aqiure of which is 1 ; this multiplied by eight, the product is ; s an unit being sub- tracted, there remains ; ; the half of which is J ; divide this by the assumed num- ber ;, ffie quotient is the first qiumtity ;. To half the square of this number add an unit; the result is which is the other quantity. Thus there are produced the two quantities ; J. “ Thus also, if tlie assumed number is one, the quantities found are I : if t'vo is the assumed number, they are V * According to the second mode, let the assumed number be 1; hy the double of which divide an unit, the quotient is ; and the .assumed number bcingadded to this, the result is the fir.-t quantity The second quainlity is the unit itscif. Tims, also, c if two is the n imber assumed we obtain ’ ; ; if three is the number assumed, we obfiiin V I; if one third is the number assumed we ohUiin V Multiply by eight the biquadrate and also the cube of an assumed number, and add an unit to tlie first product. The results will bo ilie two qu.miities. Tl\e operation may be performed either by Arithmetic or Algebra. Let tlie number assumed be I ^ its biquadrate is whuh Ocing multiplied by eight, the product is i ; add an unit to this, the result is which is tlic first q uiiluy. • Again, the cube of the a-ssumed number isl.wliich being rouiiiplierl by eight, llie product is and this is the other quantity ;. Thus there are ni laimsl the two quantities I If one is the assumed number we obtain 9, 8 ; if two is the ns- A Tile word >• bioo ', which sigiiilles root t but here it appear* tu racan a number whose square root enn lie. exlrrie'cd, llial is, a square niunber. 0 ^ 0-1 = ? D’ + {;)-!= Vi 0 - (t) - 1 = : Digitized by Google ( 37 ) nmcd o.mbo, we obuun 1J9. OJi ■<'‘>''ee is (he assaincJ number, we ob(ein 649, ai6,A .. SECTION V. OP MULTIPLIER OF THE ROOT. Df/.—The known quantity is what results after multiplying tlic t^uare root of the required quantity by the multiplier, and subtracting the product from, or adding it to, the required quantity. Rule.— To the known quantity add the square of half the multiplier, and then add to the square root of the sana thus obtained, or subtract from it, half of tlie mul- tiplier; The square of the result will be the number required, * If the quantity has a negative fraction, subtract the fraction from an unit ; if a poMtive fraction, add it to aa unit : By the result divide both the known quantity and the multiplier of the square root ; Then, from the quotients, find the required quantity according to the rule. D When a quantity is observed diminished or augmented by its own square root mul- tiplied by a certain multiplier — to that quantity, so diminished or augmented, add the square of half the multiplier of the square root of the quantity, and to the square root 1 -f- n» l.'ntton’a Tracts, Vol. p . • The object of this paragraph is to explain ific definition and rule. But instead of afford- ing any elucidation, it has a tendency rather to darken and confuse what is staled with suffici- ent clearneaa and precision in the preceding paragraphs. The ‘ quantify diminished or ang. aacjitcd by its own square root multiplied by a certaiu multiplier,’ signifies Uie known qii&nlity. Digitized by Google ( 38 ) required quantity by the multiplier ; but if the known quantity results from adding the said product to the required quantity, then subtract half of the multiplier Jrom the tquare root of the sum. The square of the result is the quantity required. Of the first part, or d^niiion, there are two examples. J. — ffhere the hnotrn number is what remains a ter multiph/mg the tquare toot of the required quantity by the multiplier, and subtracting the product from the required quantity. Example. Half the square root of a flock of geese multiplied by seven was ob- served to march slowly away, and two were seen fighting playfully in the water. Say, what was the number of geese ? Statement.— Multiplier of root is I ; known quantity is 1 . Tlien half of the multiplier is the square of which, being added to the known quantity 2, the sum is 14 , the square root of which is i ; this being added to 4, the half of the multiplier, gives 'f, the square of which » the whole number of geese 16. *. — Where the inoum quantity is the result obtained after muliiph/iHg the square root of the required quantity by the multiplier, and adding the product to the required quantiij. Example. — What is that number to which the product of its square root multiplied by 9 being added, shall Ire 1 240 ? Statement. — Multiplier of root is g ; known quantity is 1240 . According to the rule the quantity is found to be 961 . A Examples er ol bees ? In this example eight ninths, and also the square root of half of the quantity are subtracted from the quantity ; and these minus fractions, and the known quantity two, are halved ; but as a fraction of the half of a quantity is equal to half the fraction of (that quantity, the same fractions are put down. The statement then is, fraction, J ; ^multiplier of root, J,; known number, I. From these tlrere is found by the rule. ‘ In this exunqilcaiid the t«o wliicti follonr, (he fractions are negative. In this cate subtract the fraction , from an unit, (here * remains by (his divide 10 the .nulliplirr of (lie root, aud G (he known quontily ; the quotients are V y ; (hat is, multi- plier of root -y, known quantity y. Then from these find the quantity according lothefirst .part of the. rule, by adding to the known quantity (he square of half the tnulliplaat, and so m. Thus,.mBlli|)lier of root is half of it .is y, the square of which is '}J» ; add this to the .known qiianiity and reduce, the sunalry the coutraon measureseven, (he leauk ia the .square tool of which is y j add this to }f wliicli is half of Jhe multiplier, aod reduce the re- sultby the common rae.isurc seven, wc obtain VV = 18; “ntl 18* = 144 the nno^r reqei- rtil. — Coin. • In this example subtrnottlie fmetion | from an unit, there remains dividing liy this (be • known quantity JO, and also the mnlliplier of the root 4, we obtain •• 4 , which are the known quantity, and the iniilliplier of the root. Tire required quaiUily is then found from .these in the manner already dirccteil. Thus, (he square of half (he multiplier is 16 ; this ad- dcfi to y .the known quantity, the tciuU.is 36, the square root of which is 6 ; this added to 4 which is half of f the multiplier, the. result is 10, the square of which is 100, the number re- quired. ^Proof :— Quantity, 100; and 100* x .4 = 40 ; half of the quantity ij 50 i ♦he kpown quantities are 6, 3, I. Then 40 + 50 + 6 + 3 + 1 = 100. Com. Digitized by Google ( ^0 ) the number 36, which being multiplied by 2, the product is the whole number of bees, 72. * ■ Example. — A certain number being added to its own $(iu3re root multiplied by eighteen, and to one third of itself, the result is 1300. What i&that number .> j . statement. — Multiplier of root 18 { fraction); known quantity 1200. Tlien the fraction being added to an unit, the result is J ; and having divided by this the multiplier of the root, and the known quantity, , we obtain by the rule already staged, the quantity 57 O. • * 111 tlii» cx.-imple the rractiuii is J, the multiplier of root is the known quantity it ?. In order to r dncelhe multiplier o( the root by one half, the known quantity it halved ; tha fractions however remain unchanged :• the statement then is, Iraclion | ; multiplier J ; known qmn'iiy ]. Having sulitracled the fraction from an unit, there remains i ; by this divide (lie known qnaii'iiy 1, nnd also (he imiltiplicr (lie quotients obtained arc J J. Then fro n these find the soiiglil quantity according to the first part of the rule. Thus, muliiplirr known quantity ° ; half of the multiplier is J, (he square of which is f| ; this added to f the known quantity the result is Vs’, the root of which, V, sided to*, which is half of the niiiltiplier, the result is — 6 i the square of this is 36, which is hall of the required number ; and 36 X i! — the number required. Com. As this expluiiaiiuii docs not seem very correct, ve shall add (he following. As it is (ho square root of half die quantity, and } of the whole, (hat arc to be subtraclcd, to Ic.nve a leimiinder of 3, let us find half (he quantity, inst.’ad of the wliole quantity required. Then to preserve llic value of } of the whole, in terms of die half, we must lake it = Hut as vie have siipiposfd Hie rinjiiired quantity halved, we must halve all the rest, in order to preserve the con liliuiis ol the qucition ; this gives the multipii T of the root f, the fraction J, and the knmrn quantity one; then working according to (he rule, we get 36 fur the half quantity required, and multiplying this by 3, the whole quantity required is 73. That is, X — = 2 original statement. Uul as this involves ('^)‘, let us subtiliite ^ for - and (he equation (Inn becomes Dividing by S Then by salving i -y* 8 y y = 1 "C'* ititcraent y = 36 • In this exnmplethe fraction added to an unit is * ; by this divide the known qiiandly 1200, and also the multiplier J8 ; the quoliimts arc 900, >>• From these the required quantity is found, thus; multiplier V. half "f wb'd' »* V 1 ‘1>“ squared is *ii‘, wl'id' being added to * 111 cuusequeace of lUsl properly of frsetioos which i» meniioocil iu the text. Digitized by Google < 41 ■) SECTION VI. HULE OF THItBE ttUAXllTJES^ TJie quantity whose product is given, and the quantity whose product is demand- y the quantity whose product is demanded, the quotient is the required pro. duct. Com. J « The three examples of (his role are too simple to require any illustration from the com* Digitized by Google ( 42 ) RULE or THREE QCA 5 JTIT 1 ES INVERSE. If the quantity whose product is tlemancled increases while its product diminishes ; or if the quantity whose product is dctnandctl diminislics, while its product increases i • that is. If there is a diminution cf iisprotluct while the quantity wliosc product is dc- mtmdcd ifi increased ; or if there is an augmentation of its product while the qiuantity whose product is demandcsl is diminished — then those who arc skillal in the Itulc of Three Quanthits call it Invei-sion. Thisrulcis used in. valuing animals according to llxiir age, in ,w Age-, ». in order .ofied the-- A That U called the greater bide which conlain. the given product. .Transposed Statement I f^rheproductsof the numbers on each side are 100,960 The product of t'-S-tersL b. 4 1“ .Ute^It t reduced to au improper fraction is , . 1 I 12 i I II .gel, r.ae..^»a .dbyt.»d.b..U.W UJ ) } Tb.e.Ueee»U, “ Je .. .. ,>i.™- -p™— - ■" = - «f the lesser side, the quotient is, interest J i- f the lesser sioe, me quo..«.‘ in order to find the time, the transposed statement u ^ 16 And the denominator of 5 the fraction being again carried to the other side, is 4S 16 The product of the groit- 5 er sitle, or that conlamiiiff i the asccitaiiicd pioduct, is 4800; this being divided br 400 In order to find the amount of principal, the transposed st ’ 100 1 5 The pro- niictofthe g^aterside is 4S00 ; thU, being divided by Che prod«; of th: 1^^^ quolionl is 16, th e amount of principal. Com. • 'l*lie je.r reduced to mouibs. The improper fiscliou of V J. Digitized by Google ( ) I Sfcrfjcment, lOO 6 1 6 Quotient, months ! 2. In order to find tlie amount of prin- * I cipat, the statement is too 5 12 o Quotient, amount of principal iff. 4 < » Example. — In one month and a third, the interest of one hundred is five and one Cttb ; -in tlnee months and one fifth wliat will be the interest of sixty-two and a hall f 4 3 Statement — loo iti 16 * 7 '*s Quotient,, interest J O if'here there are Seven Quantities. Example. — Eight pieces of cloth each three cubits in breadth and eight In length, we purchased for one hundred ; What will one pieee tlirec cubits and a half in length wid one half in breadth be purciiased for ? Statement. — 3 e « 100 Quotient, drammas I -I, j.an.'is 9, kakini I, waratakas 6, and fcaetiun ot a warataka j. ^ • It will be obM^rvetl dial all flic mixed iiumbcis are teiliiicd to iui|>ro| er fiaclicns. . • This example shews (lie tmtisposilioii of the di noirim'ors. Alicr recluiing the mixed numbers In improper fractions, and (rans|H)'-iri(r llie dciiomiiiators, the statemrnt is 4 '® The products of (he two sides are V0,(KI0, mid lo.WiCO ; rad the gicalers s 3 15, boot), being rlivi.Icd by (lie lesser, 20,000, the qiurtiviit is 7 having abbre- 100 viated (Ire fraction by 4000. Tlu: (iuie, princii.al, &c. liiay be found us before * directed. Com. » Original Stalement Transposed statement 3 i S 1 8 1 100 1 1 2 7 2 1 I 0 0 The integers are here conver- ted into tractions by placing an tmit under them for a deuutuina* 4or. 3 2 8 2 .8 I 1 1 I 7 1 1 I 100 product of the greater side divided by (hat of the lesser, the quotient is 14, 9, 1, 6}« Con*. Digitized by Google ( -*5 ) Where there are Nine Quantities^ A Example. — Tliirty planks each twelve inches thick, sixteen inches broad, and fourteen cubits long, are purchased for one hundred nisias : What will fourteen planks each eight inches thick, twelve mclies broad, and ten cubits long be purcliased for i Statement . — \2 16 14 30 100 8 13 Quotient, ni.skas l6, drammas 10, panas 10, kakinU 10 2, waratakas 13, and fraction of a waraLaka J. * 14 O Where there are Eleven Quantities. Example. — Tlic first montioned planks were lying at the distance of one league, and to bring them the hire demanded was eight drammas : What will be the hire to bring the second mentioned planks which in length, breadth, and thickness, are less by four, irom tlie distance of six leagues ? Statement. — 12 IS 14 30 1 8 8 1 2 Quotient, hire, drammas 8. • 10 14 6 0 » Tranposed statement 12 16 14 30 0 8 12 10 14 100 Tlmpr^uct of the greater side is 1^44000, Uiat of (be less is 80640 : The greater divided by the less gives 16 oiskas. 10 dnua- mas, 10 pants, 2 kakinis, 13 | waratakas. Cow, a Transposed statement 12 16 14 30 1 Q 8 12 10 14 6 S Digitized by Google ( <6 ) SECTION vin. or BARTER. ' In R-irlcr after transposing the denominator and value, perform the operation as dirc-ctwl in the Rule of Five, &c. Quantities. * Example. — Tliree hundred mangoes are purchased for. one dramma, and thirty jKjmcgran.ntcs for one j«na : How many pomegranates will be got in cxcliange for ten mruigoes ? S tatement. — 1 ft “ | I 300 I 30 Pomegnmates obUiincd ift. ® 10 1 O • AiiolIi.T c<»iiy sinfes tlif rule m»>rc accuracy thus: “ In Harter .atter transposing the va- tne, perform the operation ardirertcil in the Hiile of fire, &c. Quantities'’ The transivwilion ■of tlie dciionunator is dirr’ctcdin the Huh: of five, &c. Numbers; iJie only dilfercnce there- fore between that rule and the rule of Harter is, that the latter requires also the tronspositioa of the value. • The dramma nxlnred fopanas. ® Transposed statement I 300 V 16 ;iO , P''"'!"''*'* •'500. 4S00, and the grealar’ being divided br the iess, Ihc quutieiU b J6* Digitized by Google ( 47 ) CHAP. III. SECTION I. OF MIXED aUANTlTIES. Multiply by the given time the quantity whose product is given, and multiply the interest by the mixed time : Then multiply by the mixed quantity eacli of the pro- ducU put down separately, and divide tlie result by their sum : The quotients are tlie principal and interest, or Find the principal by the Rule of an Assumed Number, and subtract it from the mixed sura ; the remainder is tlic interest. Example. — If at the monthly interest of five per cent, tliere is paid in a year the sum of one thousand principal and interest ; what will be the separate amount of jirincipail and of interest ? I 12 Statement. — ICXJ looo. — ^The qu.mti^- 100 whose product is given, being mul- tipliedby 1 month, the given time, the product obtained is 100 j .md the interest 5 , multiplied by 11 months, the mixed time, the product is 6o. The sum of thes<* |a-oductsis l6o. Then multiply each of the products by lOOO, the mixed quan- tity, and divide the results by tlie sum of the two products ; the quotients are pj-in cip.ll 625, interest 3/5. * ’ 1 • - Or, by the rule of Thrre Numbers. One hundred of principal produces five of interest; then, it an unit IS as€umed,\riiat will it produce? It produces ■ • Thei- if »'o IS the interest for one month, wliat will be the interest of an unit for twXe months ? Interest of one year is J . * Writedown separu.cly (lie proilucts J«) and 60 , anJ multiply each of them hr th^ ,000. ,00.000, 00,000, ,hJ iiriO. 0«.e li urn of the two products; thus, - tw ll-c touooo . . = C2.) j)riuci|)a]. .€•, 0(10 Jbu = JTi interest. f ont. Digitized by Google ( 48 ) Or, accoraratc sums. iixain|)le. — Ninety-four niskas were lent out in thn'C sums ; one sum at five jicr cent. ; one at three per cent. ; and one at four per cent. ; and each of these sums |iroduced an equal amount of interest in seven, ten, and five months respectively. W'hat was the principal lent at each rate of interest? 7 10 5 Statement. — 1 1 1 JOO 100 100. 5 3 4 Tlie sum 0/ the quotients h : tlic mised number is 94. The sums lent at ‘ each rate of Interest, are found to be 14 , 28 , 42 ; “ and by tlie rule of Five Quan- » This rule will jicrhaps lie heller understand when expressed in the following terms: Hy the nnmlier whose interest is known multiply the time in which this known interest if prmiuceil, and divide the product by each rate of interest multiplied by the lime in which, at the particular rale of interest, there is produced the equal interest : Then multiply each of the quotients by the mixed number, oud divide llic products by the sum of the quotients : The re- sults are the principal sums lent at each rale of interest. s In tliis case, having mulliplird 100, the number whose interest is given, by I, Ihetime in which the interest is produced, the product is 100: Divide this by 35, which is tlie in- terest 5 multiplied by the elapsed time 7; and '// reduced bj' 6 is y. Again, divide 100 by 30, which is the interest 3 muUiplietl by the elapsed time 10 ; and reduced by 5 is V ! In the same manner divide 100 by 20 , which is the interest 4 multiplied by the elapsed time ; and *,y reduced liy 20 is | : The sum of the fractions V y is * 7 /, reduced by 2 is y,?. Then muliiply V" by 94, the mixed number, the result is * •/” > “nd divide this by y,? the sum of tlie fractions, the quotient U VsVr = **'c first sum. Thus, also, y and | being multiplied by 94, the products are ; divide these by y,?, (lie quotients are Uie second and third sunu 42; so that the principal SUDU lent at each late of interest are 24, 28, 42. Com,^ Digitized by Google ( -O ) titles, we obtain the equal amount of interest ^ Rule. — Multiply each of the shares by the mixed number, and divide tlie pro- ducts by the sum of the shares ; The quotients will be the amount of each sliarc at tlie division. Example,— If three shares, fifty-one, sixty-eight, and eighty-five, when increas- ed by the proflU of trade make tlie stock three hundred ; what will be the amount of each share at the division ? * Thus, if in 1 nionfli 100 gives 5 of interest, in 7 months what will 2f give i Af- ter making a similarslalement with the tno other quantities, transpose according lo the rule of five quantities r Also reduce llse iinmlicrs when it can be done. Then divide the product ot tlie greater side by that of tlie less i the quotients arc the equal amount of interest : Thus, original statement traus,mscd staUmeut 1 ]00 5 7 17 2i 100 2i 0 0 5 Statement of the numbers reduced. 1 5 0 7 6 1 Then ® ^ I interest 5 4 The two other quantities also give the same amount ; thus, original staUmriit transposed statement Then 1 X 1 X Again, 23 10 1 100 s X 3 10 28 0 1 100 0 10 28 3 Slateracnt of the numbers reduced. 1 1 10 0 3 — = interest lU * And so on. original statement 1 a 100 42 4 0 There is then obtained *’ interest. transposed statement 1 5 100 42 0 ♦ Com. Digitized by Google ( ■ 50 ) Statement. — 51, 68, 83: mixed number, 300. The .cums obtained /jf the d»i. tion arc 75, 100, 125. The original sums subtraeted from these, exhibit the pro- lit. A Or, the sum of the original shares being subtractctl from the mixed number, the remainder is the whole amount of profit, 96 : Then multiply this by each of the sharc.s, and divide the protiucts by the sum of tiic shares ; the results are the se- parate amounts of profit, 2-1, 32, 40. ® Of the time in which a Pond is filed. Divide first the denominators by the nmnei-ators, and then divide an unit by the sum of the denominators. The quotient will be the time in whieh the pond is tilled. Kxaniplc. — .\ pond is filled by oimj stream in one day, by a second in lialf a day» by a third in one third of a d.iy, and by a fourth in one sixth of a day : hi wl.ut time will it be fillet', by the four at once > Statement. — i J J. Time in which the pond is filled, fraction of a day .c Sum « f he * Tims, .5! X •'itX) I53(>0 -i- HOI 75 amount of first share ntllie division. OS X -Xthw t-t)* too «lo. do. Hi X 300 y.i.iOO 4-201 125 do. do. Tlicn 75 .51 tit profit of first share. loo — (iS .32 profit of scron total profit; ami f)(i y 51 4H0i> 4 . 20t St profit on first sli.ire. f!ti X t>'4 5528 4 - 204 .32 profit on fa*< omJ share. f)(j X S5 8100 4 - 204 40 profit on thiid share. c Tints, r r f T » denominators divided by the numerators. j ^2q-3 + 0 .... 12 sum of deiiominitors. Then liy this sum divide an unit; thus, .■‘i , time in whiuh the pond is filled. The following quc.stioiv from Diopliintus, UamJalcd into Latin verse by Bathel, is cited hjr Mouliicia, Itisl. dc Matliemaliques : Tolum imjilrre lat um, UthuUt i quoluor, uno Eat polls isle die. Unis hie, lU tribus tile, (JuiUuor III quarlus : die quo spalio tiinul omnet A question exactly siwilar is also found in the Khulasut-ul-Hlsab. Digitized by Google ( SI ) S E C T I O N m. ■ s I • or BUYING AND SELLING. Multiply the price of each article by its proportion, and divide the produelt respectively by the given quantity of each article : Then multiply the quotients ob- tained, and also the proportions by the mixed sum, and divide the products by the sum of the quotients : The results will be the required cost and quantity of the arti* elc. Example. — ^Three maunds and a half of rice are obtained for one dramma, and eight maunds of mug for the same sum ; than taking two proportions of rice and one proportion of mug, how much of each sort will be got for thirteen kakinis ? 1 1 Statement. — J • 3 1 Mixed sum, kakinis 13. Then the proportion of each article being multiplied by the price of that article, and the product divided by the given quantity of the article, the quorients are * i ; the sum of which is ; J. Tlien multiply J {, and also the parts 1 , 1 , by the mixed sum ii, ^ and divide the products by the sum of the quo- tients; the results are, price of the rice i, * of the mug , J,. “ Also the required quantity of rice is found to be IJ; of mug The cost of the rice, therefore, is kakinis 10, waratakas 13, and fraction of a waraka J, And the cost of the mug is kakinis 2, waratakas 6, fraction of a warataka j . ° The rule in the Lilawati appears oi\Iy to boh! good when the original nuraeraiors are all 1 ; nor do I think that the question from Uiophantus can be solved by it. The rule might also be expressed, “ Invert the fractions, and divide an unit by the sum of the new frac- tions.” This would serve for the above example, but stilt it would not be general. a One dramma 'is equal to 51 kakinis; therefore, by pbeing 6t as a denominator to 15^ there is formed the fraction of a dramma. * Fractious of a dramma. - o In the commentaries the operation is thus exhibited : 6Uicment.->0nunma 1 ; quantity of rice dramma J ; quantity of mug f ; mixed tum Digitized by Google (■ *2 ) Examplc.—One pala of fine camphor costa two niskas, one pala of sandal wood one eighth of a dramma, and half a pala of agaru also costa one eighth of a dram- ma. Then for one niska bring me one part of camphor, sixteen parts of sandal wood, and eight parts of agaru ? Statement. — Drammas^Si ' • 8 s palas I 1 I » i parts 1 l6 8 jl/ued sum, dram maa l6. " The camphor, &c. cott rapeelively 14 0 0 2 8 8 0 9 9 and the quantity of the different articles in palas is } *,♦ >,*. o Rule. — Multiply the gift by the number oj persons; and after subtracting th« product from each number of gems, divide an assumed number by the remainders. The quotients are the values. *2 The product of the remainders multiplied together being divided by these remain* ders, will give the values in integers. Example. — Eight rubies, ten sapphires, one hundred pearls, and five diamond!^ were the respective property of four n’crchants, each of whom in friendship gave on© ii; proportion of rice 2 j proportion of niiig 1. The answer is (ben found, tlius; Tlic price I Ircinij multiplied by the proport'ons J |, the prorluci* arc J | ; divide then by the gnen quantities ’ f, the quotients are f the sum of which is || ; Then 4 f, be- in? multiplied hy the mixed sum, the products are bikI dividing lhe«e by the qiioii ’iiU arc, cost of rice ^ cost of mug yj,. Also the propoitions of the rice and mug 4. I being imilliplied by 4|., the products are ; and dividing these by tlis quotients are, required quantity of rice yj, of mug Then divide the numerator of the fmetion i by its denominator, and the rice is found to have cost, kakinia 10, wraraUkas I34. In the same iniinner after dividing the numerator of the tiactiou yji by its (leaomiualof, the mug is found to have cost, kakinis 2, waralakaa C|. * Two niskas reduced to drammos. a One niska redueed to drammas. « Haring multiplied the prices V -J j respectively by (lie parts ], 16, 8, the products ai« a This rule secius (0 be very partial. Digitized by Google f( .53 ) of his gems to each of the others, by which means the property became equal. Re- quired the values of the gems. Statement. — Rubies 8 ; sapfdures 10 ; pearb 100 ; dbmonds 5. The gift being multiplied by the number o' persons, the product is 4 ; this subtracted from each number of gems, the remainders are 4 6 q 6 I . And an assumed num- rabies pearls diamond* ber being divided by these, the quotients are the values : but as the values might come out fractions if an indiscriminate number were assum^, the skilful as« ■utne such a number as will give in integers : Thus assume g 6 ; the va- lues then found are 24 id I < 96 . rubiei port dUm. Or, the remainders being multiplied together, and the product divided by the le- mainders separately, the quotients are the values in integers, 576 , 384, 24, 2304. • The equal property is drainmas 5592 . V T T ; which being divided by the quantities i i i, the quotienU are J J, tlie sum of which is the divisor 36. Then multiply the quotients y j f by the mixed sum 16, the products are 512, 32, 32 s and divide iliese by 36 the sum of the quotients, the results are cost of camphor, &c. 14} J Also multiply llic parts 1, 16, 8, by the mixed sum 16, and the products are 16, 252, 128 ; which being divided by .36, the results are pains of cam- phor, &c. 4 V V- ^om. * Otte commentator, and the auUior of the Udaliama, find the equal property from these Values ic) tlie following manner : Each person having given one gem to each of the others, the remainders are, rubies 5, sap- phires 7, pearU 97, diamonds 2 ; Then, 24 the value of each ruby x 6 = 120 value of the remaining rubies 10 apphire X 7 = 112 sapphires 1 pearl x 97 = 97 pearls 96 diamond x 9 = 192 diamonds The value of the gem given to each person being added to these numbers respectively, the ttsults are (he equal property 233 ; Thus, + 16 4- 1 + 96 = 233 equal property by gift added to Uie value of the rubiea 112 q. 24 4* 1 + 96 := 233 sappbirea 97 q- 84 q. 16 + 96 = 233 pearb 192 + ^+1 +16=3 233 diamoods • Xhese nninbui U wiU4>e-obsened aieJn the same propouion to each othu os Ui«>preocd« Digitized by Google ( 54 ) SECTION IV. OP COMPUTIKO GOLD. Multiply each quantUy nf gold by its colour,* and divide the sum of the products by the Slim of the 9«on/i7ie» o/" gold; tlte quotient will be the colour of the mixed quantity of gold. The sum of the products of the gold muhipUed by the colour being divided by the quantiiy of refined gold, gives the colour of the refined gold. The sum of the products of (he gold multiplied by the colour being divided by the colour ot the refined gold, gives the quantity of the refined gold. Example. — Of first rule. — ^Ten, four, two, and four mashas of gold of thirteen, • ing. They an- thus /oirnd : 4 X b X %' X J = ^ product of remain len Tlicn 4- 4 =a 676 value of each ruby VMi G = 384 Bajiphire 2304 H- 96 = 24 pearl 2304 J =2304 diuuond From tliese values (he riuul propeity is found as before ; tlins 676 value of eicli ruby x 5 = 2880 value of the remaining rubies 384 X 7 — 2688 24 rj7 — 08 Then add (he value of eidi gift ; 3S4 + 24-1- 2304 + 2880 (value of remaining rubiis) = 5592 equal property by gift added lo value of rubies. The equal property by (he gift a.ldcd to (ho value of (he sapphires, peaij>, and diamonds, may be found in the same maimer. — li will be o'lsctved tuat (be equal piojcdy may be any Ollier number or value wlialever, according lo the value of (lie assumed ua.ubei. a Tlie Hindus judge of tlie purily oi gold by the colour. ' a It is staled in Ayeea Akbory “ that the highest degree of purily of gold is twelve de« grers, called /loro/i barmy, but (hat formerly (he old hurt which is a gold com current in ed number one half, they are I, 2. ^ SECTION IV. OP PERMUTATIONS. Ktile. — By the preceding term multiply the subsequent one, and multiply the next * subsequent by this subsei)uent, in the series one, &c. increasing by one, and set down in an inverse order, and divide by the same series set down in a direct order. This will give the permutations of one, two, three, &c. ® This is a general rule. — ^I’o the learned it is useful in reference to the succession of metres ; it is usej’il also in ascertaining the permutations in the use of windows, the pennutations in the parts of a pile, the permutations in artificers work, and in medicinal preparations. But these will be passed over in a cursory mamier in order .to avoid prolixity. Of Permttlalions in Metres. Example. — In writing the Gayatri tell me how many permutations may be made ■ly the lorig and short syllables contained in a foot, and how many by those centained -jn a verse? Tlic Gayatri foot consists of six syllables. Therefore writing one, two, &c. to ,the last term six, the statement is ® ^ ^ i’ Then agreeable to the rule, 1 3 4 d O .the permutations with one long syllable are found to be 6; with two long syllables, ® il5; with three long syllables, 20; with four long syllables, 15; with five long sylla- » Thus, 16 (liipher ro’oiir) — 12 (rolnur of mixed gold) = 4 j2 (rolour of mixed gold) — 10 (lower colour) = 2 * Tho’ t'le manner of expressing this rule is not very clear, its meaning is made snfficiently .roanitesi by Hieexamplcs which follow. « Thill U, whrn the foot contaios one long syllable and five short syllables, two long syllables and four short, axd so on. Digitized by Google ( sa ) bles, 6 ; and with six long syllables, 1 : -also there is one permutation with all the short syllables: the results then are 1, 6, 15 , 2o, 15 , 6, I. The sum of all these is the number of permutations in one foot, 64. a Thus also tlie number of syllables in four feet being put down as directed, and the permutations with one, &c. long syllables brought out, their sum plus one is the number ol permutations in the Gayatri measure, 16777216. “ • One roinnicntal.ir directs (lie preceding term I lobe mnliiplicJ by llie subccqueiit lerm 6, an.l tlic [iroduc-i (o be divided by (lie unit Mmv : Hiit an nlier comment Uor says 1 1, at the term islands as it is, liecau.se it lias no prtccedirig term. Bo(ii (hen ptoceed lluis : “ Mul- tiply (he suh,equcnl O' by (he preeelmg 5, and divide the product by 2 In the direct seiies • tbeii multiply ihequo i.m 15, which is (he subserpieut, by the preceding term 4, and divide by 3 ill ilie di e. l se.ies ; (hen multiply (lie quo'icnt 20, which is the subsequent, by (he pre- ceding frm 3, anddiv.de (he product by 4 i„ ,hc direct series; then muln..ly the qnob, ut 15, which b (he subsequent, l,y tl.e preceding t rm 2, and divide tbeproduet by 5 in the direct series : (bon multiply ilir qiionenl 6, wliicli is the siilvsequciil, by (lie preceding trim 1, and divide by 6 m die direa senes ; the quoli ni is I. Tl.ere isaNo one periuuUlion w iih all tlieshoit syllables, liiesum ot llie {lerniutaiions in a loot therefore is til.” 'Hie operation then is, - C. ^ * _ ir, ■‘_oo ir, '*X* ’ \ j — •‘-'V 3 — — ■ru, - = 10, — — And this is the same in principle with that given incur books ; viz. ® X ^ ^ L? J. iX-'XJX+XiXf) Ur. iiiuton justly icnmrks that the above “ rule is toon co nprclietifieJ at sigiit, whin cx- piei'cd in our own loni cn enl mode and nutaiiou ; as iii the jiiesuil case, j x x X — ^ &-C’, dcnoles the cumLinatb ns of any number (nj of ih n^s, ItKeu two by two, thifby tliiee, four by lour, &e, ti c series being co..tiiiued to as u.aiiy fictois as there aie things to be coiubtued." — llutioa's tracts vol. 2 ji. 153. * Pennuia'-'iiii of 21 sy fables : ill !3 j so ^ 1 . 9 ' isi 1 ; j 10 j 1 . j M j 13 j l» j II j -1 * ! 4 ^ 1 4 1 3 1 S 1 1 i 1 •s 1 1 1 1 I 1 1 1 1 1 1 1 ^ I-* I'S 1 1 s ! S ! 1 1 1 1 ? 1 3 1 1 p) 1 1 1 1 t 1 ~T msm L-! i 3 1-1 S 1 5 1 S U 1 1 1 i ns 1 1 1 1 1 1 i a 1 3 1 1 n , 1 " 1 1 - 1 - 1 ^ 1 i - 1 * 1 •' 1 I r, 1 - 1 1 MiaglP , •! -1 3 S J 9 j in j 11 j l» j 13 j 14 1 [is 1 i5 1 i '®i .7 !■* 1!^ SI) j j SI j S? j S.7 j S4 Digitized by Google ( 59 ) lh«mmteof |«rmulation. m.jbe .Kcrtalncd u&r»U.c utkrati measure. Example. In Mechanics. — A skilful architect constructed eight windows in the king’s spacimiB and beautiful palace : Required the number of permutations in the use of one, two, three, &c. of the windows ? Also, required the permutations which may be made in the six-flavoured^ condi- ment, composed of sweet, pungent, astringent, sour, salt, and bitter ? Statement. — For the wtndotvs. J2345678. number of permu- . 8 28 50 70 56 28 8 I tions in the use of one, two, three, 8cc. windows, is j ^ 3 ^ ^ 678. Thus the number of permutations in the use of eight windows in the king's palace is 255. Statement for the permutations in the condiment. ^ 2 number of permutations with one, &c. in th e condiment is ^ The sum of which is 03. 4 3 2 3 4 5 15 20 2 3 6 . 15 6 ' I 456. rmp. IV. op PROGRESSIONS. Multiply half the number of terms by the number of terms plus one -, this gives the sum of the scries one, &c. which is called tlie summation. Multiply the sum- mation by the number of terms plus two, and divide the product by three ; this gives the sum of tlic summations. Digitized by Google ( eo ) Example. — ^What are the separate summations of the terms one, &c. as far as nine : And what are the sums of the summations ? Statement. — I, 2 , 3, 4, 5, 6, 7, 8, 9. The summations are I, 3, 6, 10, 15, 21, 28, 3(5, 45. The sums of which are 1, 4, lO, 20, 35, 5t). 84, 12Q, t65. * Itule. — Having multiplied the number of terms by two, and added one to the pro- duct, divide the result by three, and multiply the quotient by the summation : The product is the sum of the squares. Former authors have stated tb.it the sum of the cubes of tlic terms one, &c. is equal to the square of the summation. Example. What is the sum of the squares of the first mentioned series of terms in the last example . And also wliat is the sum of their cubes ? Statement, l, 2, 3, 4, 5 , 6, 7, 8, 9. The sum of tlie squares is 285. ® 1 be » TliesC rules arc, in short, first to find the sum of iiny inmhcr ol' Icriiis (n) in the f.erie* of natural numbers 1, 3, 3, 4, 5, 6, See; .and then lo find the Mini of any niiral)cr«t’ t ic terms arising by llie coniinued addit.oin ol llic Ibrtmr, Ijuingwiiat nc call the triangular num- bers. Thus, 1, 2, 3, 4, 6, 8cc, the natural nnmhers. 1, S, 6, 10, 15, &’C, tlie Iriangiiliir miiiibt.d'.s. Onr rule for the former series is 4» x (n X I) = r ; and for the Int'cr series X (» + ]) X (It + 2) = a X 4(« + 2), wliere Ihc latter form agrees with (hat given in.dicrji.dian tiik:. Exam. In the first scries ; for four terms, I 4 - g q. 3 4 - 4 = { I -f- 4) x • = 10 ; for dx terms, (1 + G) X ‘ = 2J ; for nine terms, (14-9) x ? = 45. In the second seiics ; for (J 4- 2) X_f _ ( 3 4- 2) X- (I 4 - 2 -f 3 ) i X 6 3 — 3 ~ 3 three terras. = iO; for four ^ ^ ^ cv)-. f,r ninclcrms, 3 3 3 ± ^1 X » _ (') + ^) X » The srparalc summations arc flius giren in the cmimeiilarics : The nmnlier of terms I being multiplied by 2, the the product is 2 ; one added, is 3, this divi.led by 3, the qiiolicnl is ] ; this multiplied bjr 1 the sum of the term, the |V0.liict if J, wbirh is tlie sqnare of the number of terms. Again, the number of terms 2 multi- plied by 5, the product is 4, uicre.i5cd by 1 is 5; this divided by 3 is 1, which being mul" Digitized by Google ( 61 ) sum of the cubes is 2025. Rule. — Multiply the dilTercnce by the number of terms minus one, ami to the pro- duct add the first term ; the result is the last term : To the last term add the firsj term, and halve the result ; this gives the mean term : Multiply the mean term by the number of terms ; this gives the sum. * Example. — ^The first day four drammas were given to a brahman, and this sum was increased by five each day. How many drammas were given in fifteen days ? Statement. — First term 4 ; difference ® 5 ; number of terms 15. Here the first term is 4, the mean term is 39, the last term is 74 j the sum is 583. ° Example. — ^Thc first term is seven, th& difference is five, and the number of terms is eight. Required the mean term, the last term, and the sum ? Statement. — First term is 7 ; difference 5 ; number of terms 8. Here the mean term is V ; the last term is 42 ; the sum is 1 96. tiplied by Stlie sum of the number of terms, the result is — 5 , the sum of the squares of the number of terms. Thus, also, the number of terms 3 multiplied by 2 is 6 , one ad- ded is 7, this divided by three is ', which being multiplied by 6 the sum of the number of terms, the product is y = 14, which is the sum of the squares of the number of terms. And soon. Thus there ore obtained 1, 5, 14, 30, 55, 91, 140, 204, 285. Com. This rule is for the summation of series of square and cube numbers, applied to the »e- lies If 4, 9, ] 6 , 25, &c, and ], 8 , 27, 64, 125, 8 rc. Thesum of n terms of the former . ■ 2a + I being — - — X ■», and the sum of n terms of the Litter , where 1 denotes the sum of it terms of the natural series 1 , 2 , 3, 4, &c — HuUon’s tracts vol. 2 p. 156. * Thus rule and Ihe two or three following rules ore for any arithmetical progretsions, * being the first term, tn the middle term, » the last term, d the common diSerenee, and s Uie luro; for which the rules, expressed in our algebraical notation, are these: — l)d d , t . . e 4- n + « = * ; mn : a)-~ + (a — — (a — irf) ... ■ j where the last form is deduced from the preceding one, by means of the solution of a compound quadratic equation, of a rather complex na- ture, by Ihe mode of completing the square, and evincing a considerable degree of expert- ness in the arrangement. — Hutlon's tracts vol. 2 p. 156. 157. » What are here caUed difference, number of terms, and term, ore in the original doo- nunated increase, period, and >nm , . ) Digitized by Google ( 62 ) Tn this example the number of te^ms is an even number ; therefore, in order to find the mean term, add the first and last terms, and halve the result ; this gives, the mean term. Rule. — Divide the sum of the progression by the number of terms, and from the quotient subtract the number of terms minus one multiplied by half the diffirrence ; the remainder is the first term. Example. — The sum of the progression is one hundred and five ; the number of terms is seven ; the difference is three. Required the first term ? Statement. — First term 0 ; ^ difference 3 ; number of terms 7 ; sum of the pro- gression 105. The first term is 6. ® Hule. Divide the sum of the progression by the number of terms ; from the quo- tient subtract the first term, and divide the remainder by the number of terms minus one halved : Tfie quotient is the difference. Example. — A king marched two yojanas the first day, and in order to seise the enemy’s elephants he marched eighty yojanas in seven days to the enemy’s city. At what rate of increase did he marcli ? Statement. — First term 2; difference 0; number of terms 7; sum of the pro- gression 80. The difference is found to be %* . ® Rule. — Multiply the sum of the progression by the difference, and the resulting product by two ; to the last product add the square of the difference of half the dif- ference and first term ; from the square root of tlie result subtract the first term ; to the rcmlander add half the difference, and divide the result by the ditfcrence : Tlie quotient is tlie number of terms. * A ciplict denotes that the number is unknown. » The sum of the progression 105 being divided by the number of terms 7. the quotient is the number of terms minus one is 6 ; imiltijily (his by } half the difl'crence., the product is 0, tills being subtracted fum 15 there remains 6, ntiicli is the first term. Com. r Tlie sum of Itie progression being divided by (he number of terms, the result is ; from (lirlde this by 3, the half of the number the difference. Thus, -f-^ = this subtract (be first term, and there remains Y ; of terms minus one, the quotient is Vi wbicli is 66 ** „ ii = T Digitized by GoogU’ ( 63 ) Example. Three drammas were given the first day, and each day the difference fw two. In how many days were three hundred and sixty drammas given f Statement. — First term 3 ; difference a ; number of terms 0 ; sum of the progres- sion 360 . The number of days or rermr is 18. * Rule. — Add the number of terms to its own square, and by half the result di- vide the sum of the progression. ® Example. — ^The first term and the difference are unknown ; the number of terms is six, and the sum of the progression is also six. Required the first term and tlie difference ? Statement. — First term 0; difference 0; number of terms 6; sum of the pro- gression 6. The first term is found to be ^ , and the difference ^ . Rule.— When the number of terms is uneven, subtract an unit from it, and write down multiplier ; when it is even, halve it, and write down square ; and thus re- peat the operation until the number of terms is exhausted. Then multiply and square beginning at the bottom with the ratio ; from the last product subtract an unit, and ■' divide the remainder by the ratio minus one. -Then multiply the quotient by the first term, and the product will be the sum of the geometrical progression. Example. — ^Two waratakas are given the first day, and it is agreed to increase the gift by doubling it each day. Eow many niskas will be given in a montli ? SUtement. — First term 2 ; ratio (or number of increase) 2 ; number of terms 30. The sum of the geometrical progression is, waratakas 2147483646; equal to niskas A Multiply the sura of the progression by the difference, the product is 72o; and this being multiplied again by 2, the product is 1440. The difference of half the diflerence and the iirst term is S, the square of which being added to 1440 the result is 1444: from the square root of this, which is 38, subtract the first terra, and there remains 33; to this add half (be difference ; the result is 3G. This being divided by (he difference, the quotient b 18| which is the number of terms. | Lorn. J > This rule, which is not contained in the other copies or in (he commentaries, appears to ^ be misplaced in the original, as it comes after the following rule. It will be observed also that it is not general, which may be said perhaps of many of the otlier rules given ; but it an- swers perfectly in the example, and in all arithmetical progressions, whwte first term and dif- , ferenceare the same. It is deduced from one of the preceding rules, — (« _ = «»' if a = (f, it becomes — — (it — J)? = o or s -i- - =: « as above. n If s • Digitized by Google ( 64 ) 104857, drammas 9 , panas 2 ; kakinis 1 ; waratakas 6 . ^ Example.— Two waratakas are given the first day, and each day tlie gift is tri- pled. How many waratakas will be given in seven days ? * The explanation of the rule, and also the method of operation, are thui given in the commentaries ; When the number of terms is uneven, subtract an unit from it, and write down the word multiplier*, but do not write down Uie figures ; when the number of terms U even, halve if, and write down the word sqiiaret, but do not write the figures. He|>cat this operation till the number of terms disappear. Having thus written the words multiplier and square, set down the assumed ratio at the last multiplier place ; then square the rat. u .1 the preceding square place, snd at the multiplier place multiply the proiluct by the ratio; thus continuing the operation to the firsfor liighest place, where the product is ohtjiiied. Subtract an unit from this product, and divide the remaiuder by the ratio less one, ami multiply the quotient by the first term ; the product is the sum of the progression. Example —First term 52; ratio 2; number of terras 30. In this case the number of terms is an even number, therefore halve it, the result is 15; write down the tvord square; again as 15 is an uneven number, subtract one f.om it, Uie r.-maindcr « i4 ; wi.ledown the Ld multiplier; half the even number 14 is 7, square; the uneven number 7 less one « C, multiplier; half the even number 6 is 3, square; the uneven ^ 2, multiplier; half the even number 2 is 1, square; the uiieien number , tiplier. Thus, in succession, the slalcmenl is, ?! square ?! square 3 ; multiplier 7S-- multiplier ?! square ?! square 3 ! multiplier multiplier Tl.cn set down the ratio at the last multiplier place, and square it at the next square place; the result is 4; multiply this by the ratio, the product is 8; its square is 64; this multi- plied by the ratio is 128; its square is 16384; this multiplied by the ratio U its square U 1073741824, which is (he product of muhiplying and squaring to the highest place. From this pr.Kluct subtract an unit, and divide the remainder by 1 which is the ratio less one; the result is 1073741823, which being multiplied by 2 the first term, the pro- * A, IB die example which fulluwi it out- t Aa in the prcaent example. Digitized by Google ( 65 ) Statemeat.— First term 2 ; ratio 3 ; Bumber of terms 7. The sum of the pro gression is 2 1 86. * duct is 3i47t’83c)tC dramrnaii, equal iu ui»ia> Ofcc. Tiie rcsulU arc llius exliibiltil : ^ ! square 1073741824 3: muliiplier .... 32768 square J6384 3 ■ muliiplier ]38 "Bf : square 64 "S' multiplier .... 8 square 4 3 5 multiplier .... 2 Com. It will l>e observed that ranlliplier means the ratio. In this example the ratio hein^ (bo, the nim is alwiysdouhicd at the place of multiplier ; i» tUe ue«t example, as the ratio is three the sum is trijiled 4 t the place of muliiplier. ’ The operation tlien is thus; ratio or muliiplier 2; then 2 = 4, 4 x 2 s= 64 X 8 = 128, 128 — 16384, 1(5381 x 2 = 32768, 32768* = 1073741824, X 2 = 2147483646. 8, 8*= 64, I0737tl8«t— 1 *— I Dr. Hutton remarks that this comes to the same thing as ,onr own rule, thus expressed, r" — t o" — I . , ■ X o = a, or - - X a =it, when the ratio r and the first terra n are equal. As a mark of multiplication is directed, in I'yii’s translation, to be placed above the odd number, and a mark of a square above dhe even number, Dr. Hutton infers lint the Hindus had a mark to denote inulliplic.ation, and another to denote squaring. The 8an- •crit original however does not admit of this supposition. It merely directs the words imiU tiplier and square to be written down to indicate the place for multiplication or squaring. Ac* curdingly these words are some times written at full length in the commentaries, and some times the initial letters only are written, as is seen in the preceding example, where the letters ^ and a are the initials of warga ( square ) and guiiaka ( multiplier. ) I have not observed a mark for multiplication or squaring in any Hindu work. * Thus, as in last example; 7 — 1«= 6 .... multiplier 2187 6 J. 2 = 3 .... square 729 3 — 1 = 2 .... mulii])lier ........ 27 2^2 = 1 .... square 9 1 — 1 ,, 3 Com. Or thus : Ratio or multiplier 3; then 3 = 9, 9 X 3 = 27; 27’= 729, 729 X 3 = ^187, X 2 = 2186. Digitized by Coogle ( 66 ) Rule. — The number of syllables in the foot being the number of terras, and the multiplier (or ratio) 2 ; the product then of multiplying and squaring is the num- ber of like metres. Square this product, and also square' the square ; and from each result subtract its own square root ; the remainders are the numbers of half like metres, and of unlike metres respectivaly. ^ Example. — What is the number of like metres ; the number of half like metres ; and the number of unlike metres, in the anashtupa slock or verse ? " . Statement. — First term 1 , ratio 2 , number of terms The number of like metres * Tlie rule is merely tbi«, that the result of multiplying and squaring gives r* / but Iiere S' = 2, therefore 2" is the rule here given, trow the sum of the series i q. ^ X + " X X ~y~ Ion terms (whicli expresses the combinations of one, two, three, &c. to n tilings, out of n things) it 2’ — I ; but I is to be added, for the additional permutatioa of all the long or all tlie short sylUblcs (aken together, that is 2* = the number of metres. It may not be unnecessary .to remark, that I do not know what word correcliy expresses the kind'of changes alludcti to here ami at pages 57 and 58. It is not exactly comii»tUion, for we count the chunges with one thing, whicli has no combination ; nor is it permutulion, for that would give us a much Jarger result, as it supposes all things different, instead of only two kimls, long ami short. Perhaps, as to this rule, prrmulationi of matrei may serve 4o make some distinction ; and as the expression combimOioHs of one, itco Src. things, shews tliat the changes with one thing are included, it u hoped that no misconception can well Uke place. • The verse here alluded to consists of four ,fee(, each foot contain'wg ciglit qillables. Also like metres mean, that all the four feet are like each>otbcr; half like metres, that only the first and third are alike, and the second and fourth alike: and tiinlike metres mean, that all the four feet are differentd'rom each other. The numbers of metres therefore, evidently correspond with the permutations of metres, pages 57 and.58, and may be found by the rul* there given ; that. is, the like metres are the permutatious pf metres in eight syllables (one fool) = 25S: the halflike metres those imsixteen syllables, (two feet ) minus those in tlic first ciglit syllables, ( as tlie two suecoediiig feat must not correspond with each other ) viz. 65536 — •£56 = 65280-: Aaddhc sinmber of unlike metres, is that of the permnlalionf of metres in thirty two syllables, ( four feet ) raiims rtbe sum of (he numbers of like and half like metresi 8. MK9 OF TUB ABITBMSnC« Digitized by Google Digitized by Google PART II. CHiP.'t. OF GGOMETBICAL OPKBATIOIfS. * Akv line being supposed the base, the other ' line which is perpendicular to it is called the side in a triangle or quadrangle. ® The root of the sum of the squares of the base and side is equal to the hypo- thenuse. The root of the difference of the squares of the base and hypothenuse is eoiml ' to the side. ™ the difference of the squares of the side and hypothenuse U equal The square of the difference of the base and side added to twice their product gives the sum of their squares. ® The sum of the base and hypothenuse multiplied by their difference gives the dif- Icrcnce of their squares. The learned understand this in all cases. Example.— The side. isfour, and the base three; required the hypothenuse ; the -teM and hypothenuse being given, required the side ; and the side and hypothenuse given, required the base ? * Literally, •« of operatioiu ronceming fields”. The word kthelra means a field, or a Ik>It V ot ; also the hody : And in mathematical worka it signifies a geometricar figure. ‘"*“8'“ «'one are Supposed. -A.id the rules are deduced LnTll IK “•* of byporhenusei, «de-ln a quadrangle, the lower side is termH ,he ha^ die side oppos.le io it, is caUed nwikfia, which literally maan, mouth, face. -or fron.i^nd the two other side, are called the sides. ' 5traj.ey’“7er^'''" 'o'- *• »«ch was on the margin of Mr. .howsvei is not Digitized by Google ( 68 ) Statement . — To Jind ihe hypothenuse. See Fig. I, The base and side being multiplied together and by two, the product Is 24 ; Ae square of their diffcrenoc is I, add this to the product, the result is 25, the root of wiiich is the hypothenuse, 5. Statement. — f o Jind the title. See Fig. 2. The sum of the base and hypothenuse being maitipfied by their difForcace, the pro- duct is l6, the root of which is the side 4. Statement.— 7'o>jd the lose. Sec Fig. 3, Thus the base is found to be 3. Example. — ^Thebaseis three and one fourth, and the side Is the same; tequirH the hypothenuse ? SUtement. — Base V; «de, V; tire sum of their squares is * 5*. As the square root cannot be extracted, the surd number VT/ is the hypothenuse. The nearest root is found by the following method ; Assume a large number, and having multiplied by its square lire product of the humerato'r and denomiiutor, divide the root of the result by the denominator mul- tiplied by the root of the square of the assumed number ; the quotient is the nearest root. Thus, hv])Othenu.se (t)'; numerator bi-ing multiplied by the denomimifdr and by 10,000 ^ the product is 13520000, the nearest sqtiare root of which is 3677 ; divide this by flie denoramator 8 multiplied by 100 the root of tlm 4 multiplier ; thus ’^V, the quotient is the nearest root J which is tire hypbthe- iiuse ; ** This is the process in all cases. Rule. — A base being supposed, multiply it by the double of an .nssumed number, and divide the product by tlie ^uare of the assumed number less one ; the quotient h the side ; write down the side in a separate place ; then multiply k by the assumed * That is, the square of the o.ssdnfcd liuinbcr 100. • Tlic reason ef Hits role is evUcut'-, 4»r It o- theniise. Also by knowing the side, tlie base and hypothenusc are found ; and these form a right angled triangle.® Example.— The base being twelve, then according to these two methods what different sides and hypotbenuses will be found, such as make a right angled trianMe > Statement. — Base 12 ; assumed number 2, by the double of whicli multiply the base, and the product is 48; square of assumed number is 4, less one is 3 - and 46 being divided by this number, the quotient is lO, which is the side: Then jnulriply the side by the assumed number, and subtract the base from the product. » To prevent any erroneous inference 1>eiiig drawn, it is necessary to remark here, tliat in the Indian ticoiiielry plane triangles appear to receive their denominations in relation of their sides only, and not in rel.ition of their angles. Tlie three species are named 8nina Tribhuyi — throe skies equal — equilateral triangle. Dwisaina Tribbuja — two sides equal — isosceles triangle. Wisama Tribbuja — three sides unequal — scalene triangle. The words jalj/a trj/asra which 1 have rendered right angled triangle, properly s^ify ge. neric triangle, or that sjiccies of IriaugU: which comprehends the other sorts, and to which dhey may be reduced ; for by Ictliag fall a perpendicular from the vertex to the base, each of the three kinds of triangles is divided into two right angled triangles. The Jyoiishitat a^ronomeis whom I have had an opportunity of conversing with are igno- iranl of the terms large or small, acute or obtuse angles. These terms are employed perhaps in Jliudu astronomical wqyLs,tbo’ 1 have not observed them in my very limited course of reading. »Tlie truth of these nilts is evident; for if 6 = base, and — =s side, then will “ * 4 2bn V , a * I * t » i_ X " = bypoUienaic according lo the idle : iUo if 6 = base, and -i-2= ^ = wide ; then will * >f g* | jb ^ + = bypolheniue according to the rule. Digitized by Google ( 70 ) this gives the h)TX)thenuse 20. If three is the assumed number, the side and liypo. theiiuse are 9 , 15 ; if five is the assumed number, they are 5, 13. Or according to the second method : The supposed base is 12, the square or which is \44i divide this by an assumed number two, tlie quotient is 72 ; the assum ed number being subtracted from this in one place, and added to it in another the results arc 70, 74; these halved are the side and hypothenuse 35 , 37. Or by an assumed number four they are 1 6, 20 ; or by an assumed number six they are 9 J 5 Rule.— Multiply twice Uie value of Uie hypothenuse by an assumed number, and divide the product by the square of the assumed number plus one; the quotient is the side ; put down the side separately, and multiply it by the assumed number • the difference of the product and the hypothenuse will be the base.^ Example.— The hypothenuse is eightyifive ; required such sides and bases as form a right angled triangle. Statement. — Hypothenuse 85; this multiplied by two is 170, and this again by an assumed number two, is 3-10 ; which being divided by the square of the assum- ed mimber plus one gives the side 6s ; multiply this by the a.ssumed number, the product is 136; then subtract the hypothenuse from it, the remainder is the, base 51 ; or if the assumed number is four we obtain 40, 75. Rule. — Divide twice the hj-potheinise by the square of an assumed number ()lus one ; subtract the quotient from the tiypotheniise, the remainder is the side ; then multiply the quotient by the assumed number, the product is the basc,B Example. — ^Tlie same as the preceding. Statement. — Hypothenuse 85, the side and base of which, when the number .as- sumed is two, are 51, 63 ; or by an assumed number four, are 75, 40. ATbatiifetA l>e the bypothenusr ; 6, (he base; and r, (he side; if • —.s ; then will A = 6 according.to,tlie rule foe A* - f-^ y ' * = — A. for 71 2TTm f 2/m fi* Digitized bv Google ( 71 ) The hase and side differ ia name only, not in form. Kule. Multiply two assumed numbers together and I>y two ; this gives the side ; the difference of their squares is the base, and the sum of their squares is the hy- pothenuse, of a ri^ht angled triangle. * Example. — The side, base, and hypothenuse which form a jatya tryasra or right angled triangle are unknown ; required these three ? Hereby the assumed numbers I, 2, the base, side, and hypothenuse are found to be 4, 3, i ” ; or by the assumed numbers C, 3, they are 12, 5, 13 : or by the assumed numbers 2, 4, they are 1 6, 12, 20 And thus as to any others. JVhai the sum of the hypothenuse and side is known, and also the base ; to find the separate values of the hypothenuse and side. Kule. By the staff divide the square of the space between the bottom of the staff and its top ® ; in one place add the quotient to the staff, and in another sub- tract it from the staff ; halve the results; the quotients are respectively the two part* of the staff, or hypothenuse and side. ^ Example. — If a staff' thirty -two cubits high, standing on a level piece of ground» be broken by the violence of the wind, and its top touch the ground sixteen cubits \ X Let m and n be the assumed numbers then 2nu» = J i m* — »* = 6, and m* -p nl = A for (m* -p B*)* = (M* —«•)• + (2 hib)‘ » Thus, 1 X * K 2 = 4 the side g* 1* — 3 the base l*q- 2* =5 ihe hypothenuse 5 2x3x2 = 12 the side 3* — 2* =5 the base 2*q- 3 = ]3 the hypothenuse • 2x4x2 = 16 the sida 4 * — 2* = 18 the base 2* q- 4* =20 the hypothenuse.' ■ That is, the point where the top of the broken staff touches the ground. » That is, if A -p f = n ; and b = the base'; then will (nq. J H-9 = — — = A, and (o - - ) 2 = ^^ = s for A* -p s* = A* ; that is, 6* + ^ * )* = ( - ^ * ) = A» Digitized by Coogle ( 72 ) from its base ; at what height from the bottom is the staflf broken I Statement — See Fig. 5. The upper and lower parts are 20. 1 2, * • When the sum of the base and hypothenuse is inotvn, and also the side / tojind the separate values of the base and hypoiheuiise. ® Rule. — Square the post, and divide by the distance between the serpent and its hole ; from the distance between the serpent and its hole subtract the quotient ; half the remainder is the distance in cubiU from the serpent’s hole where the serpent and the peacock met. Rxaioplc. At the bottom of a post is a serpent’s hole ; on the top of the post it perched a peacock ; the post is nine cubits high, and the peacock descries the ser- pent running to iU hole when distant from it three times the Ijcight of the post ; the peacock flies down obliquely and seizes the serpent when both have passed ovet an equal space ; at what distance from the serpent’s hole did they meet ? Statement. — ®See Fig. 6. The place at whicli they met is distant from the serpent’s hole 1 2. ^ • The example ii thus exhibiled in the Commeatariet : “ Distance brtweeo the bottom of the slafT and place where the (op touches (he ground it 16; the square of which is 256; divide thU by (he staff, the quotient is 8, which being ad- ded to 32, the SUIT, the result is 40, and sulitr.icted from it there remains 24; the result* haired are 20, 12, the upper and lower paru of the staff, or the aide and hypothenuse. Com. See Fig. 5. • Two rules with their examples teem to be transposed at (his place in (he copy from which I translate. I have followed therefore the arrangement of the other copies and the Com- mentaries. « The taller partof the enunciation is taken from another copy. • Let A + & = a : i = side: then will ' — s* O* — f» O* + I* '2a ^ (a--) -~ 9 = - — \ a f 2a for A* -F a* = A*; that is ( A ; and a — 2a -Fs* = A )‘ = A a Or, as inano her copy. — “ Post 9, which is Ike side ; distance Iretwccn the serpent and ill hole 27, which is llie sum of the base and hypothenuse: Then according to (lie rule there is found, base J2 cubits; the remainder is (he hypothenuse 15," •Thus 81 the (Xjuare of the post being diviJed by 27, the quotient is S; mbtract thi* from the ditunce 27, the retziaiader is 24, the Uaif of which is 12. Com. f r Digitized by Google ( 73 ) tfTien the Sfference of the side and hypothenuse is known, and also the base ; to find the separate values of the side and hypothenase. Rule.— Divide the square of the base by the differenoe of the side and hypothe- nuse ; write down the quotient in two places, and in one place subtract the difference of the side and hypothenuse, and in the other add the diftoence ; halve the results ; the quotients are respectively the side and hypothenuse. The intelligent who un- derstand this rule apply it in all cases, a The distance from the water-lily to where it is immersed in the water is the base ; that part of the water-lily which is seen is the difference of the side and hypothenuse ; the stolk is the side, and the water is the same depth as the stalk ; all this being known, required the depth of the water f Baample.— In a Me the bud of a water-lily was observed one span above the wa- ter, and when moved by the gentle breeze it sunk in the water at two cubits dis- tance ; required the depth of the water ? Statement. — See Fig. 7. Depth of the water V J the measure of the bud added to this fa the hypothenase, • 7 a The sum of the hypothenuse and part of the side being known, and also the remainder of the side, ® and the base; to find the separate values ef the hypothenuse and unknown part of the side. DRule. Multiply the height of the tree by two, and to the product add the dis- * The illvstration of this rule is exactly the same a itii Uie two fonacr, only putting A — r = Thus, 'the base is 8, the square of which being divided by j. (ihat part of the water-lily which is seen, a span being equal to half a cubit) the difference of 'the side and hypothenuse, the quotient is 8; ^ subtracted from 8 there remains V ; and i added to 8 the result is V. These numbers being divided by 2, the quotients are, skic y ; hypothenuse y ; and the depth of the water is equal to the side. Com. « The words in lioman are not in the original. • So partial is this rule that it will only hold good, when the given sum of the hypothe- Wise and part of the side, is equal to the sum of the temaioiiig part of the side, and the base. Suppose r = part of (he side required A -j- X = a the given sun d = part of the side g iven 6 = tlie base d -f- jc = s the side Then A* = 6* + s* = 6* + (rf + *)• Digitized by Google { 7* ) tance between the tree and the pool ; by the result divide the heiglU of U\e tree multiplied by the distance between the tree and tlie pool ; the quotient is the ex« tent of the leap. Example. — A tree one hundred cubits high is distant from a well two hundred cubits ; from this tree one monkey descends and goes to the ’^ell ; another monkey takes a leap upwards anil then descends by the hypothenuse, and both pass over aa equal Space ; required tlie height of the leap i Stitcment. — bee Fig. 8. Tlie leap is found to be 50. JFhen the sum of (lie base and si Jets known, and also the hypothennse \ t« frnl the s-parafe values of the base and side. Rule. — Multiply the square of the hypotlienuse by tw», amd from the product subtract the square of the sum of the base and side ; in one p.aix, subtract the root of the remainder from the sum of the base and side, and in another place add it, and halve the results ; the quotients are the measures of tlie base and side. * Example. — ^The hypothenuse is seventeen, and the sum of the base and side is (twenty-three ; required the separate values of the base and side ? Statement. — bee Fig. 9. BOW h = a — Jr; therefore 6* -f (d -I- X)* = (a _ x)» and X — — 77a + general expression for » but the rule snppohes h^-x— a = b + d-, and snbstiliiting this value of a, we have * — — r^7-r according (o the tide. But following the general expression x := ^ * - - — ‘ 2 d 4" '•** T* . I. n* — 4* _ d* . I>' \ . a 1 , , o’ — 4* — d* wc have h = a =(a + d + _) 2 and , = d + — = < a + d -r 2; which arc the true general .values of A and s, and cortcsiioad with a 4” the rule “ by the staff divide," &c. * Let A -f r = a, A = liypotbcause ; (hen will = 6, and 2 ' 2 for (2A* — rt’)’ — ,(9A* — a"+T )^ = A — r = d and — - — = b and ^ 4 Digitized by Google ( ) Hypothenuse 17 ; sum of side and base 23 : the base and side found, are 8, 15. * When the difference of the base and side is known, and also the hypothe- niise ; tofnd the separate values of the base and side.’* Rule. — Subtract the square of the difference of the base and side from twice the square of the hypothenuse ; find the root of the remainder ; put it down in two places, and in one place subtract tlic difference of the base and side, and in the other place add it ; halve the results ; the quotients are tlie separate values of the base and side. Example. — The difference of the base and side is seven, and the hypothenuse is thirteen ; required the base and side separately ? Statement. — Hypothenuse 13; difference of base and side 7; root of difference of the squares 17- The separate values of the base and side are 5, 12.*^ Rule. — Multiply J,i. j; two staffs U^ethcr, and divide the product by the sum of the staffs ; the quotient is the perpendicular let fall from the place of intersection of two cords drawn reciprocally from the bottom of one staff to the top of the other ; multiply each of the two staffs by an assumed base, and divide the products by the sum of the staffs ; the quotients are the segments of the base on eacli side of the perpendicular. Example. — One staff is fifteen cubits high, the other is ten cubits higfi ; the base is unknown ; required the perpendicular let fall from tlie intersection of two cords drawn respectively from the bottom of one sUiff to the top of the otlier ? Statement. — If the Rise which is the distance between the staffs is 5, the seg_ » Sqii.ire of the hypothenuse is 2S9 ; multiply thi* by 2, the result is 578, from which sub- tract 529, the square of the sum of the bsse and side, the remainder is 49, the root of which, 7, being subtracted from the sum of the base and side, there remains 16 ; and again being added to tlie said sum, tlie result is 30 ; then 16 and 50 halved, are the base and side 8, 15. Com. » This problem is not given in any of the other copies. The original al.^ contains an error both in the statement and in the result of the operation. « Thus, 7* == 49 2 X 13» — 7> = 289: and v'289 = 17 IZ ~ l2 — 5 the base 2 17 4-7 , — y — = 12 the tide Digitized by Google ( ) ments of the base are 3, 2 ; If the base is 10, the segments are 8, 4 ; if the ha** is 20, the segments are 12,8; but in all the perpendicular is the same, 6. a Xhus, if a base equal to the distance gives sides equal to thestafis respectively, , what will the corresponding segments of the base give ) ® And in this manner the result is brought out in each case by the Rule of Three Quantities. The figures are thus re- presented. See Fig. 10 and II. Concerning the form of Figures. If a person affirm regarding a figure whose sides are straight lines, that the sum of the other sides is less than one of the sides or equal to it, this must be consi- dered an impossible figure. For example, if it be affirmed that a quadrilateral fi- gure has the sides 3, 6, 2, and 1 2 ; or that a triangle has the sides 3, 6, Q, such must be considered impossible figures. This will appear by placing a straight staff for the base. Of the area of Figures. Rule. — In a triangular figure multiply the sum of the two sides by their differ- i Tlic product of tlie (wo slafls is 150, which being divided by 25, their sum, the quo* ficnl is Ihe perpendicular 6 . Then tusiime the base, or di!.timce between the two uafis, to be 8, and tnuUiply the two by this assumed number ; tlic products are 75, 50; divide these by 25 the sum of the staffs, the quoiienls are tlie segments 3, 2. If tlie distance or base is 10, tlie segments arc t), 4 ; if tlic dislancc is j5, the segments are 9, G; if the dislauce Is 20 , the srgmtnis are 12 , 8 ; but in each case the perpendicular is 6 . lorn. » The following illustration of this rule is given by Strachey in Ids translation of (he hija dannila: — Let ah = 10, oc = 15, un 20. Uy similar Uiangles bd : bp : : nc - re, DO : PD : : BA : fu, whence up : po ; ; ba : DC, tlierefoce bd is divided in p in the ratio of dc to ba. By compo'ition bp -f rn : bp : : da 4 - »c : ba ; but bp + pd =: bd, therefore b a 4. dc and da arc in the ratio of bd to bp ; whcnijc, by the first projjorlion, ba 4 - t t t uc s pn, that is, PO is a fourth proportional to ba 4 - “‘-'s whatever be the length of BD. — Sec I’lale VI. I'ig. 7. Lucas de Hurgo has this proposition, (see his Geometry, p. 56. ) where die lengths arc 4, C, and 8 ; or page 60, where tin y arc 10, 15, and 6 . The same is in I s Lilawati, where the rules arc AS CD CP = — , — -» PD AS 4- CD . and BP = — . Bija Gannila p. 59- ■ AB 4- CU’ AB 4 - CD I Digitized by Google ( n ) ence, and divide the product by the base ; * in one place add the quotient to the base, and in another subtract it from the base ; halve the results ; the quotients arc the a^^ents ; the root of the difference of the square of each segment and its side is the perpendicular; multiply half the base by the perpendicular; the product is the precise area of the triangular figure. Example. — ^The base of a triangular figure is fourteen ; one side is thirteen, and the other is fifteen ; required the perpendicular, the segments, and tiic number of the equal spaces ^ which is called the area ? Statement. — See Fig. 1 2 . The segments are 5, p, the perpendicular is IS ; the area of the figure is 84. c Example. — In a triangular figure one side is ten, the other is seventeen, and tiie i>ase is nine ; required the segment, the perpendicular, and the area ? Statement.— The two sides are J 0, 1/4 the base is g. Here by the rule “ multiply the sum of the two sides,” ** See. there is obtained 21. As this cannot be subtracted from the base, the base is subtracted from it, leaving the remainder 12 ; die half of this is the negative segment. This means that it falls O in a contrary direction. The segments then are 6, 15; the perpendicular as to both is 8, and the areaof die figure is 30. “ The figure is thus shewn. Sec Fig. 13. * f bat is, given a, 6, and e the sides of a triangle .bla4-c: la — cl '* « = difference of (lie segments of tbebase. The rest is Abe same as has been given before ; and. the whole corres(iond< with our own rules. ■ The Hindus cstiinale (he area of a figure by the number of little squares it contains, these being eubils, or any other measure used, in the same manner as is done in Europe. c 1 he sum of the sides 13, 15, is S8 ; multiply this by 9, their difference, the product is 56 ; divillc this by the base 14, the quotient is 4 ; in one place subtract this from the base, and .in another add it; the results are JO, IS, the halves of which arc the segments. The squares A)f the segment 3, and side 13, are 95, 169; the difference of the squares is 144, (he root of a -which is 12: Also the squares ol the segment 9, and side 15, are 81, 225; the difference ■of thesa squares is 144, the root of which is the perpendicular 12. Thu perpendicular be- ing'' multiplied by half the base, or 7, the product is 84, the area. Com. » Sec page 76. Multiply 27, the sum of the sides, by 7 their difference, the product is 189; divide this ily 9, the base, the quotient is 21, from which subtract the base, the remainder is 12; aUo Digitized by Google ( 78 ) Rule.— Halve the sum of all the sides ; put down the result in four places, and subtract from it the four sides respectively ; then multiply the remainders into each other ; and the root of the product will be the approximate * area in a quadrangu-< lar figure, “ and the precise area in a triangular figure. ° Example. — The base is fourteen, the line ^ opposite to it is nine, one of the sides ' is thirteen, the other is twelve, and the perpendicular is also twelve ; Required the area as given by former authors ? Statement — See Fig. 14. The area found by the above rule, is ^ 1 98 OO, which has no exact root, the square root of it being something less than 141 ; this is not the true f area. But by a rule which shall be afterwards given, viz. “ By tlie perpendicular multiply half the sum of ■ *' the base and tiieline opposite to it the true area of the figure is found to be 138.” add 21 to tlip base, the result is 30 ; these two results halved are 6,. 15. In the first case the segment is negative, the base being subtracted frum tliequotient 21, whereas this quotient should be subtracted from it. Thus we have the negative and positive segments 6, 15: Then by the rule “ the root of the differenre of the squares of each segment and it side, &c.’' we obtain the perpendicular 8; multiply this by the base, and divide the product by 2; the quotient is 36, Which is the area. Com. * Asphuta. — Not clear, not evident. Fyzi translates it turhan, wandering, roaming. » I his will give the true area in such a trapezium as can be inscribed in a circle — in other cases it is very vague. c Sphuto. — Clear, evident, apparent. Fyzi translates it durtisl, — just, true, right. D This rule might be more distinctly expressed thus: Halve the sum of the sides, and from it subtract the sides rcsjrcctivcly ; then multiply into each other the four remainders ( fot the quadrangle ) ; and the three remainders and the half sura ( for the triangle ) : the square root of the result will be the approximate area in the quadrangle, and the precise area iq the triangle. » Miikha . — The face, or what is opposite. » fVaslttta. — True, proper. o Page 81. « Thus the sura of all the sides 9, 12, 14, 13, is 48, the half of which is 24 : Pul Uiis down in four places, and subtract tbe sides, thus 24 24 24 24 the remainders are ]5, 12, 9 12 14 13 JO, 11 ; the product of these mullipUeJ together is 19800, the square root of which ii some I Digitized by Google ( ?9 ) Also, on making two divisions of the figure, the same area is obhuncd, 138. See pig. 15. By this rule, likewise, we find that Uie area of a triangle whose sides are 13,',15, .and the base 14, is 64. Reason of the Approximate* Area. As the two diagonals of a quadrangle arc not determinate, » how can the area of that figure be determinate c . Their assumed diagonals which former writers have de- monstrated, are not applicable to other positions of the gmdrangle, because while the sides remain the same, different dUgonals may be obtained, and consequently different areas. Thus, by extending the opposite angles of a quadrangle, the diago> ntil which joins the two angles that go inwards is shortened, but the two other an- gles being drawn outwards increase their diagonal ; therefore it was said that, while the sides remain the same, different diagonals may be obtained. If one of the two what less than 141 ; This is not the true area. But by a rule which shall be afterwards giir«i « by the perpendicular, &c.” the true area is found in tliU manner : The perpondicolar is 12, the base is 14, and tlie line opposite to it is 9 ; these two added make 23, the half of which, V, being multiplied by 12 the perpendicular, the product is ]38, which is the true area. Again, in regard (o a triangle, the area is found by tbe rule » halve the sum of the sides/* Ac. Here the sum of Uie sides IS, 14, 15, is 42, the half of which is 21, which being put doini in four places, and the sides subtracted, thus 21 21 21 21, the remainders arc 8, 7, 6, 13, ]4, 15 21, the product of wbich is 7050 ; the square root of this is 84, the precise area of tlie Irianglei CouJ. s UtrgP— The words and asphula are used in subsequent parts of the work l . denote what is inexact, or merely an approximation ; and the words spashta and skcA- thma, to denote either what is exact and true, or the nearest degree of truth and precisioa ■ thath s been attained. . Amj,ala-'Sol unalterable, not unchangeable; formed of the p.irilire particle « and myata, what is fixed, destined, unalterable. « Aiyato.— Fixed, invariable. Digitized by Google ( 80 ) I^endiculare, or of the two diagonals, be not separately given to people, A the one*, tion is of an indeterminate nature, why then is the determinate area demanded^ uTie person who puts such a question is a demon, and he who pretends to answer it IS an arch demon, and both are ignorant of the indeterminate ( changeable ) positio* of quadrangular figures. • Rule. Having assumed one diagonal in a quadrangle whose sides are equal, sub- •iractiu square from the square of the side multiplied by four ; the root of the re- miunder is equal to the second diagonal. Multiply the unequal diagonals togetlier and divide the product by two ; the quotient is the precise area of a quadrangle whose aides are equal. ® ^ o ""‘I rca,«,nd will, iho^e pnt doua by tl.e>e authors, but will be gre,-er nr less: Tl.ns diagon.ls diflerent from lh.»e demon- strated arc oht nn.-d m the given quadrangle. If ,he diagonal l>e determinate, then tbe an* o c I e will bo dilcrniiiule, but in co/isequence of the diagonal Ixring indeterminate, dif- Tcrenf arcaa arc obi lined. Com. In a quadrangle, while the sides tcmaln the same, the area may vary, as it docs not depend on the sides; but in n liiinglc the area does not vary : Thcrcio e it will not answer to assume a dingonnl, but a delerniinate diagonal must he found or demonstrated. It may then be asked, it the diagonals be demonstrated, to what olijecUon aie they liable. .J leply, that tliey are nut applicable to other cases, for while ihesides remain the same, diil'c.eni nte.is may beoblaincd; tl.eiefnreil issaiil, thatil thc,per|)cnJiciilar or ilisgiuial begivcn, the delermiuale or unchange* able area will lie found. Otherwise, as the measure of die dunmaiion or shoitning is uot known, tlie perpendirnhr and lUsgonil carmot be ascertained. Udaharana. . ^ **'* = o ; the assumed diagonal = A, the unknown diagonal — jr; len as t c sides of the quadrangle are equal, the diag onals bisect each other, and x =; 2 s/fa* — i4’) = y (4o‘ — i») according to tbe rule ; and lif-if + = tf u,e area, 2 i a Digitized by Google ( 81 ) In a quadrangle of eqial sides, and in a parallelogram, when the two diagonals are equal, multiply the side by the base ; thU teill give the area. In other qua- drangles, if the perpendiculars be equal, then by the perpendicular multiply half tlie ■um of the base and side opposite to it ; the product will be the area. Example. — In a quadrangle whose sides are twenty-five, required the two diago. nals ; from the diagonals required the area; also, the diagonals being equal, requioxl Xhe area of the above quadrangle ; and the area of a parallelogram six in breadtli and eight in length, its diagonals likewise being equal. • Statement.— To find the area of the first figure when its diagonals are equal. Sec Fig. l 6. By the rule “ the root of the sum of the squares of the side and base. &c." the two equal diagonals come out an irrational number V 1 250 ; and the area is 625 a Or, assume one of the diagonals to be 30 , the other diagonal will be 40 . The con- struction of the figure then. is thus. See Fig. 17. And the area is 600. * Or, by assuming fourteen for one of the diagonals, then the other diagonal is found to be 48. The construction of the figure is thus. See Fig. 18. And here the area is 336 . < The diagonal it found by the role, the “ root of (he sam «f the squares of the base and side iatbe hypotlienuse.” In ttiis example the sides are 25, 25, and (heir squares are 625, 62S, the snm of which is 1250; and n the roo; of this sum cannot be obtained, the irrational nuinbn' VlSSo stands for each diagonal: then (he area is found by the rule “ in a quadrangle of equal sides, and in a pira. eiogram, when the diagunals are equal,” tcc. thus : The aide is 25, the base is 95; these muliiplied (ogetlier produce 625, which is the area in a qnadraa"Ie when (be diagtmsb are equal. Com. » If SO be assumed a« one du 1 gon. 1 l, the other will be found by the mle, “ having assumed ». -one diagonal in a quadrangle whose sides are equal,” See. Thus the square of tlie side is 625, which being multiplied by 4, the product is 2500 : The square of the assumed diagonal is flOO ; (his being subtracter! from 2500, the remainder is 1600, the root of which is the second diagonal 40; then the area is :onnil by (he rule, “ multiply the unequal diagouals by each •^cr;” thus, the pioduct of the diagonals 30, 40 is 1900, which being divided 2, (hcquuiient •sAQO, the aica ot the quadrangle of equal sides. Corn. Digitized by Google ( 82 ) The construction of the parallelogram is thus. See Fig. 10. And Its area is 48. ^ Example. — ^The line opposite the base is eleven ; tlic base is twenty-two ; one of ' the sides is thirteen, and the other is twenty ; and the perpendicular is twelve ; What is the area ? Statement. — By the rule “ halve the sum of all the sides," &c. the approximate area is found to be 250 •* ; and according to the rule, “ by the perpendicular multi- ply half the sum of the base and of the line opposite to it,” &c. the true area is found to be igS. ^ In order to make this evident, divide the figure into three parts, and find the se- parate area of each part ; then add the three areas, and observe their sum. llie ligure is thus represented. See Fig. 20. And the different parts of the figure are thus exhibited. See Fig. 21, 22, and 23. By the rule “multiply half the base by the perpendicular,” the areas of the right angled triangles are found to be 30, Qti ^ ; and by the rule “ multiply the side and base together,” there is found the area of the parallelogram, 72; lire sum of these areas is 198 , which is the true area. - r Example. — The line opposite the base being fifty-one, the base seventy-five, the side on the left hand sixty-eight, and tlie other side forty ; required the area, diago- nal, and the perpendicular ? Statement. — See Fig. 24. * Here llie squares of the sides 6, 8, are SC, -18, and the sum of Uieae squares is lOO,' tlie .root of which is 10; then the area of the parallelogram is obtained by raolliplying the base and side; thus, 6 x 8 r= 18, the area. Com. » The sum of the sides is CtJ, the: half of which beiug put down in four pl.ice^, nnd the 'sides II, 13, 22, 20 subtractcel, the reinninders are 22, 20, II, 13; these mulliplieel toge- ther produce G2020, the nearest root of which is 250, or the approximate ares. Com, c The sura of the base and line opposite toil is 3.3; the half of which being multiplied by 12 the perpendicular, the pro.luct is = 198, the precise area. Com. B Each of the triangles being half a Jjarallelogram, ihrir areas arc found by raulliplying * ^ X 1 1 the side by the base, and halving the pioJucI, thu.s, — ^ — r- = 30 area -I- = 96 area. Com. Digitized by Google ( 83 ) ' Concerning the area, diagonal, and perpendicular. The perpendicular being known, the diagonal will be obtained ; and the diagonsd being known, the perpendicular will be obtained, and there also the determinate area will be found ; that is, when the diagonal is indeterminate, the perpendicular also is indeterminate. To find the Perpendicular. Rule.— In a triangle inscribed in a quadrangle find the perpendicular as before di- rcctcd. The two sides form the diagonal and side, and the given base is the base of the inscribed .trio ngle. In order to find the perpendicular, assume a diagonal extending from the top of the left side to the bottom of the right side to be 77, by which also a triangle is ibrmed within the quadrangle ; this diagonal is one side of the triangle ; tlie left side is the other side, 68 ; and the base is the same as stateil in the example. The per- pendicular and segments are then found in the manner already directed. Thus the sum of the sides is 143; this multiplied by their difference, the product is 1 303 ; this divided by the base 75, the quotient is ; this quotient being subtracted from the base in one place, and in another added to the base, the results are * J*, • ; • ? these halved give the segments ' J ♦ * ' ‘ ; and by a rule already given we ob- tain the perpendicular ’ J *.® The figure is thus represented. See Fig. 25. The perpendicular being known, to find the diagonal. Rule.— The root of the difference of the squares of the perpendicular and the side which is supported by the perpendicular is called the segment ; subtract the seg- ment from the base, and to the square of the remainder add the square of perpendicular ; the square root of the result is the diagonal. * The following rules in succession gire the methods of finding the perpendienhurt, diago< tuls, and the area of the preceding example. » Thus the squares of the segment and of its side 68, redaced to a common denomi- nator, are difieieace of these iqaaiei is *V/ *» the square loot of which is the perpeodicalar Com. Digitized by Google { «<* ) Suppose the perpendicular let fall from the top of the left side of the quadrangle • to be ’ ; ‘ , the segment will be ' J * : * Then by the rule “ subtract the segment from the base, and to the square of the remainder add the square of the perpen- dicular,” &c. we obtain the diagonal 77. “ To jind the second diagonal. Rule. — First assume a diagonal, and in the triangles situated on each side of tins diagonal, call the diagonal the base, and the two other lines the sides ; then find the perpendiculars and segments. Square the distances between the perpendicular situ- tuated on one side of the assumed diagonal, and to this square add the square of the sum of the perpendiculars ; the square root of the result is the second diagonal. Thus also as to all quadrangular figures. ^ ® Let a diagonal extending from the top of the left side to the bottom of the right side be assumed equal to 77 ; in the figure containing this diagonal line, call The commentnry alM> thews the area brought out according to the rule “ multiply h.iJf (he ba!>e by the periM'miicular,” &c. thus, half the bate being multiplied by the jicr- pendicular 'f*, the product is * Vv”* — 8310 , the area of the figure. t The squares of the assumed perpendicular and the tide 6S which is supported hy it, when reduced to a common denominator, are •*,*r** the difference is tlie root of which is the less segment 'V* Com. s Thus, the less segment being subtracted from the base 75 , afler reducing both to a com- mon denominator, the remainder is the greater segment, the square of which is ‘ V*‘ i this added lo the square of the perpendicular, the result is (be square root of wliich is *^' = 77 the disgonal. Com. e See Plate Vf. Fig. I, Let o, and b, be the perpendiculars; d, the distance between tJie perpendiculars = w -f" * » let je 4- y = the second diagonal. then X = V (n* -b »* ) y = ^{6* + s') ; and x* + = a* + »* + 6* + s* 6* + 6‘H!’ 4. a’ s* + w* s>) blit a : u> : i b : s ; and a* s» 4- 6* ni* = Sabwt therefore 8ry = 2v'('t‘ ** -b Softais 4- »’ i’) = 9 ai + faif then X* 4 - 2 ry 4- = o’ 4- tab 4- 6’ 4 * *s* + 2ws 4 " ** and X 4- jr = v' (a 4- 6 4. » 4- s ) = V (“ + * + rf*) according to the rule. * This paragraph is merely intended as an explanation of the role. Digitized by Google ( 85 ) the diagonal the base of the triangles situated on each side of the diagonal, and call the two other lines the sides ; then the perpendiculars and segments will be found in the manner already shewn. Representation of the figure. See Fig. 36 . The square of the difference of the two segments formed by the falling of the perpendiculars is 169 ; the square of the sum of the perpendiculars is 7056 ; these two numbers being added together, the result is 7335, the square root of which is the second diagonal 85. * Tiius in all cases. Remarks on assuming a Diagonal. The sum of the two less sides which fall upon one diagonal being assumed as the base, and the remaining two sides as the sides, find the perpendicular. In thia case the one diagonal can never be greater than the base of the figure thus form- ed, nor the other diagonal less than the perpendicular. Knowing this the iutcJligCot assume a diagonal. By extending the opposite angles of a quadrangle tlie sides are drawn inwards, and a triangular figure is formed ; •* then the sum of the two less sides which are about one angle being assumed as the liase, and the two other sides as the sides, the per- pendicular will be found according to the rule already given. The shortened diagonal cannot in any case be less than this perpendicular, nor the other diagonal greater than the two sides assumed as the base. Thus it is in regard to both diagonals. Tho’ I have not shewn this, it is unde-rstood by those versed in the science. The sum of the areas of the triangles situated on each side of the diagonal, is the true area of the figure. s After finding the legraenlt 3*3, 49 by the process already exhibited, the perpendiculars are then olitaine<) thus : The squares of the segment 32 and of the side 68 are J024, 4624, and the ililferenee of these squares is 3600 die square root of which is one perpendicular, 60: Abo the .squares of the segment 32 and of the side 40 are 1024, 1600, and the difterence o# these squares is 376, tlierfvore root of which is the other perpendicular, 24. Then the square of 84, the sum of the perpendiculars, is 7036; to this add 169, the square of the difference of (he segments, the result is 7225, the square root of which is 83, the second diagonal. Corn, • The triangular figure is exhibited on the margin, still taking the last ezample, in wllich the sides of the quadrangle are 75, 68, 5J, 40. See Fig 27. Digitized by Google ( 86 ) The areas of the triangles in the figure last mentioned are 924, 2310 ; the sum of these two is the area of the quadrangle, 3234. Rule. — In a quadrangle with equal pcr(>endiciikrs, assume the base minus its op- posite side as the base, and the sides as the sides ; then find the segments as in a triangle, and from the segments find the perpendicular ; subtract the segment from •the base of the quadrangle ; the square root of the sum of the squares of Ute re- mainder and of the perpendicular will be the diagonal. When the perpendiculars are equal, the sum of the side opposite the base, and the greater of the other two sides, will be less than the sum of the base. and the lesser side. Example. — The two sides are !>2 and 3Q ; the side opposite tlie base is 25, and the base is Oo. Preceding authors have said that in this figure the perpendiculars are unequal, and that the two determinate diagonals are fifty-six and sixty-three. I would know two diagonals different from tliese ; also equal perpendiculars, and tlie diagonal which gives these perpendiculars ? Statement. — See Pig. 28. Here, if the greater diagonal assumed be 63, tlie odier diagonal found will be 66. A Or, in place of fifty.six, suppose the diagonal thirty-two ; there will then be ibund, bytlie former rule, two surd parts of the required diagonal v' 2700 , ? 6 » the sum of the square roots of which is the second diagonal, I % * Thus, the two sides are 39, CO, tlieir sum is 99; this muitiplii^l by 81, tlirir dilTcrrnce, •the product is 8079, which being divided by 63,ttfac base, the quotient is 33; in one place subtract this from 63, and in another Mid it; tlie results are 30, 96, these halved are 15, 48, the segments. The tame segments are also found in -the other triangle by performing this operation on (he sides 85, 58. The perpendicular is then found by the rule ‘‘.the squares of the segment and side,” &c. thus, segment 48, side which falls an it 58, their squares are > 2304, 8704, the diOercnce of the squares is 400, -the square root of wliich is SO, the perpen- dicular: Again, , segment 15, side which falls on. it 39, their squares are 885, 1581; difiereiice of these squares is 1896, the *•, Ihe square of which is 'V^*; this addeil to 38016 the square of the perpendicular, the result is ‘ = 8176, the square for the other diagonal ; the nearest square toots of these aie the two dia- gonals, 71.^a, 46if. Com, Digitized by Google ( 88 ) ■ nals are found by Bramha Gupta * and others by the following rule. Let the sum of the rectangles of the sides which fall upon the diagonals be reci- procally divided, and multiply the quotients by the sum of the rectangles of the opposite sides ; the square rooU of the results are the diagonals in an irregular tigure. Statement. — .Sec Fig. 3i, The rectangles of the sides which fall upon one diagonal are ‘2340, 1300, their sum is 3040 ; tlie rectangles of the sides which fall upon the other diagonal are 975, 31 20, their sum is 40p5 ; divide these reciprocally by each other, thus jSjt JSJJ ; or having re95 ; ihe square root of the difference of the squares of these numbers is the segment, here called sandhi, 48 ; the base minus the sandhi is the other segment, which is called pit,k, 252 : “ Thus, also, the otlier perpendicular is 224, the side which falls upon it is 260 ; the sandhi obtained is 132, and the pilji is 168: ® The lower limb of the first perpendicular 1 89 is thus found ; its sandhi is 48, this being multiplied by the other perpendicular 224, and by the diagonal 280, separate- ly, and the products divided by the other pit,h 1 68, the quotients are, lower limb of the perpendicular 64, and lower limb of the diagonal 80. “ disagrees not only wi’h (lie other ro|'iai, but aUo with the definition of the same word which follows, apparently oat of place, at the end of the succeeding paragraph, aod is the one which is here adopted. s Literally, a chair or stool. » The perp«i(UcuIar is 189, the side which falls upon it is 195 ; tlic sandhi is (hen found by the rule, “ t\ie square root of the difference 01' the side an I hy;erpciidicular is 824, (he side which fulb npon it is 260, “ the difference of the squares of these nuiiiters is 17424, the square loot of which b the sandhi 132; subtract this from the base 300, die rciuaiuJer b the J68. Lorn. » The lower limb of the first jierpendirular 189, and (f-e lower limb of the diagonal from when (he hypotlwnase and perpendicular intersect, are thus found t The sandhi of this per- pendicular b 48 ; put this down in two places, and miiliiply it by the otlier per(>endiculac 224, and by its diagonal 280, tlie prorlucis are 10752, J3440; divide this by 168 (he pit,h of the otlier sandhi, (be quolients axe, lower limb of the perpendicular 04, and lower limb of tlie diagonal 60, Com, Digitized by Google^ ■ ( 9t ) Thiw, alsO| the second perpendicular is 224 ; its sandhi is 1 32, which being muU tiplied by the other perpendicular I89, and its diagonal 315, separately, and the products divided by the other pil,h 252, the quotients are, lower limb of the per- pendicular 99, and lower limb of tlie diagonal l65. ^ To Jind the Jower perpendicular from the place of intersection of the two diagonals. Rule. Multiply both the perpendiculars by the base, and divide each by its own pit,h ; the quotients are the two staffs •, then by these staffs are found the perpen- dicular and segments of the base from the iiitcrsection of the two diagonals, accord- ing to the rule formerly given. In this manner we obtain the staffs 225, 400 ; then, according to the rule “ from the intersection of a cord drawn from the base of the one to the top of the o- ther,” &c. the lower perpendicular from the place of intersection of the diagonals is found by these staffs to be 144, and the segments 108, 192. “ Tojind the perpendicular, segments, and sides of the suchi {triangle form^ ed by two sides oj the quadrangle produced.) Def.—~The sandhi of one perpendicular being multifdied by the other perpen- At this placcthere is given (he follewing definilion of the sandhi: “The square root of the difference of the squares of the side and hypotlirnuse is the sandhi." s Thus, also, the sandhi of (he second perpendicular 321 is J33 ; (his being put down in two places and multiplied by u'lo other perpendicular 189, and by i(s diagonal 3]5, (he pro- ducts are 24948, 41580 ; these divided by 252 the pit,h of the other sandhi 48, the quotieols are, lower limb of the perpcnciicidar 99, lower limb of the diagonal J65. Com. • The perpendiculars J89, 224, being multiplied by 300 the base, the products arc 5C700, €7200, divide these respectively by their own pil,h 252, 168, the quotients are tlie two staSa 225, 400; multiply these two staffs together, the product is 90000, divide this by 625, the sum of the two staffs, the quolieitt is the perpend icuLir let fall from the place of intersection of the diagonals, 144 : Also (lie two segments are found by the rule “ multiply the staffs by the base, &c.” Thus, multiply the stiffs 225, 400 by the base 300, the producu are 675000, J20000, divide these by 625 the sum of the two staffs, the quotieats are the segments on each side of the perpendicular, 108, 192. Com. Digitized by Google ( ) dicular, and the product divided by ita own perpendioular, the (jDotient is cafled tama ^ . The sum of the iama and of the other tandhi is called kora * . Rule. — Multiply separately the tama and the other tandbi by the base, and divide the products by the hara ; the quotients are the segn)entB of the tuchi. The pr»* dvct of the other perpendicular multiplied by the base, being divided by the Aarat the quotient is the perpendicular of the tuchi. Multiply each of the two aides by the perpendicular of the tuchi, and divide the product of each side by the perpend dicular of that side ; the quotients are the sides of the tucU. The skilful perform the whole operation by the rule of Three Quantities. Here one perpendicular is 224; its san^u is 132 ; multiply this by the other per- pendicular I eg, and divide the produot by its own perpendicular 224, the quotient is • ; • , which is called the tama ; the sum of this and of the other tandhi 48, is • V ’ i this is called the hara. ® The tama and this other tandhi being severally multiplied by the base 300, and the products divided by the hara, the quotients are the seg- ments of the base of tlie jucW, *;;* ’,*5*.“ The other perpendicular 18g being ntukiplied by the base, and the product divided by the hara, will give the perpendi- cular of the tuchi, * The two sides igs, 260 being multiplied by the per- pendicular of the tuchi, and the products divided respectively by their own perpendi^ • Literally, equal. ■ Uiviter. • Thus, >muUipIy (herondti 139 by 189, (he perpendicular let fall from (he other angle, (lie product ia 94918; divhie tliia by 994, the per|>endicular which fulls U|ion the tondAii 139, the quotient is or reduced by 98 is ; this is called the tamm: add together the other tandhi 48, and the tama 'V, the reauU u ' V ‘ > this is called the hara at divisor. Com. • The raoui *4', and the other tandhi 48, being multiplied by S(X) the base, (he products are 967300, HIOO; divide tlicie by ’y< the hara or divi'or, (he quotienU are “AVi*** ’titt“ 5 (he first leilueed bv 8 is and this again reduced by 75 is ■’J**, which is one segment of the snehi: Also « reduced by 75 is 'tf*, which is the other segment of f fbe tuchi. Com. t The perpend ictilar of the tuchi is found in this manmir : Multiply the perpendicular. 189 by SOO the bnie, the product is 567(X); divide Uiis by * the hara or divisor, and re- .dace the quotient by 75, theresult Js which is the peipendicnliu of (he iitchi. 'Cere. Digitized by Google ( sa ) Oilart 4M, the quotients are the two sides of the «cAi which is formed by two ades of the traptzium produced in thdr own direcUon, »!*• '!,*•> Thus here, and in every instance, by assuming to any particular sum of divison or of multiplication, such a quotient or divisor, as may be proper, the intelligent eoijskler it an operation by the Jtule of Three Qu^tities. ■ * Tbe two sides of the tuchi are found thus > Multiply the two sides 195, SCO of the on- ' ginal figure by the perpendicular of the $vchi, the producto are divide these by their own perpcudiculars 18% 884, the quotieula are *}t* the sida of tbe suchi. Com. ^ i .s In the original tbe first part of this ptragtaph is very obscure, and I doubt whether it be here rightly translated. The meaning however, evidently is, that (he preceding opera* tin os can be performed by the rule of proportion. Digitized by Google ( } CHAP. II. S E C T I O N I. OF CIACLE8. Rule. — Multiply the diameter by 3Q27, and divide Uje product by 1350} tlui gives the more precise * circumference ; Or, multiply the diameter by 32, and divide the product by 7 ; this gives the approximate drcumference, which answers for common operations. Example. — ^The diameter being 7, what is the circumference ? the circumference being 23, what is the diameter ? Statement. — See Fig. 37- 21 The more precise circumference is mt; or the approximate circumference is 22i MSO By making the multiplier the divisor, and the divisor the multiplier, the more /re- 7 cise diameter is found to be ; or the approximate diameter is 7. •9>7 Rule. — ^The fourth of the diameter multipilied by the circumference gives the area of the circle ; this area multiplied by four gives the area of the convex superficies of the sphere which is tike tlie net work of a ball; the superficial area multiplied by the diameter, and the product divided by 6, gives the cubic content of the sphere. Example. — ^The diameter being 7, what is the area of the circle ; also the dia- meter of a sphere being 7, what is the area of the convex superficies which resem- . Sukshma. — LitersTly, small, minute. In note • page 79, it is meniioncd (hat this word is employed to denote either what is exact or true, or the nearest ilepree of (ruth that has hec* Attained. The original does not contain the words which are printed in Italics in the ex- ample. The astronomers whom 1 have consulted, suppose that the number is the precise circumference ; but it is not probable that such a mathematiciiU) as Bhascan sliOoU Itavc coosideied these execllcnt approximatioos as the exact truth. Digitized by Google ( 95 ) blet iJiB net work of a ball ; and what is the aibic content of the splicre ? Ansvper' these three questions if thou understandest the Lilawati well. 38 Statement — The area of the ckde is ,4,, ; the convex superficies of the sphere iooo 153 170 is 117] : the cubic content of the sphere is ,487 t»a IMS. Rule. — Multiply the square of the diameter by 3937, and divide the product by 5000 ; this gives the more precise area of the circle : Or, multiply the square of the diameter by 1 1, and divide the product by 14 ; this gives the approximate area, which is sufficient for common operations. To half the cube of the diameter add its own one and twentieth part ; the result is the cubic content of the sphere. . 38 , . . 38 . Tlie more precise area is . the approximate area is 1 the approximate cubi4 I00« * 1 * 17Q content is i. f Rule.— Multiply together the sum and difference of the chord * (of double the arc) and the diameter, and from the diameter subtract the square root of the product ; half the remainder is the versed sine. ® Subtract the versed sine from the dia- meter, and multiply the remainder by the versed sine ; the sqsiare root of the pro- duct muhiplied by two is the chord (o f double the arc). Divide the square of half the diord (of double the arc) by the versed sine, and add the versed sme to the .quotient ; the result is the diameter of the circle. Example.— The diameter is ten, and the chord (of double the arc) is six ; required the versed sine ; from the versed sine required the chord (of double the arc) ; and fiom the chord (of doable the arc) and the versed sine, required the diameter i Statement— See Fig. 38. The versed sine is 1 by knowing the versed sine we obtain the diord (of dou^ * The string of the bow, or the chord of doable the arc. — In astronomical works gyo always meana onMs half the chord of double the arc, or the lioe. See Phil. Trans, vol. 4. p. 86. As. Ite. vol. *. p. 845, *t icq. a SAara.— An arrow. Digitized by Google ( 96 ) ble the arc) 6 ; by knowing the chord (of double the arc) and versed sine, we obtain the diameter lO. Rule. — Multiply the diameter by 103923, by 84S53, by 70534, by 60000 , by 62055,* by 45922 , by 41031 ;* and divide each product by 120000; the quo- tients will be the sides inscribed in a drcle, from a trigon to a nonagon. Example. — In a circle whose diameter is 2000, required a trigon, tetragon, pentagon, &c. of equal sides ? J732 The diameter is 2000 ; the value of the sides of the trigon is , ; of the 60 tetragon is of the |>entagon is of the hexagon is lOOO; of the •o 0O . . 8(57 , , . 765 - , . 683 heptagon is „ ; of the octagon 1 $ ,, ; of tlie nonagon is : Also different 0n 60 60 chords are obtained by assuming different diameters. These are mentioned in trea- tises on circles. — See Fig. 39 , 40, 41, 42, 43, 44, 45. Short method of finding the approximate Chord. De/l — ^The circumference minus the arc being multiplied by the an^ the product is called prathama . " Rule. — Multiply by five the fourth part of the square of the circumference, and having subtracted the product from the prathnma, ^ by the remainder divide the * Dr. Ilutlon, in his mallir.-rint'cal tmcls, mriit'onj that tlir iiamlier 53055 for the liepineon, and the iiamber 4j03i for tlic nonagon, are erron'oii', and ought to l>e 52070 and 4.043, and he suggests ihni t'lC error has probably ari.en fiom miscopyings. But as ll>c text con- tains the worJs_ which are synibolicad of thoe numbers, as well as tjie n'lniben iheinselves, ft H prob.'ibir that ihere may have been an error in (he original calcubiions. The number for the trigon which U given wrong in Pyzi's translation is here roriectly pot down. r » This word lilerally signifies first, bi:l here it seems to be used in a technical sense. ^ Ad I/a is (he word which occurs in the original, but as it has the same signification as prathnma, we shall retain only the latter, in order to lessen the £>erplexity which arises from tire use of foreign tcrma. Digitized by Google ( 97 ) pr*itkama multiplied by four limes the diameter ; the quotient will be the diord. Example. — The eighteenth part of a circle multiplied separately by one, &c. gives similar arcs ; required the chord of each arc in a circle the semidiameter of which is one hundred and twenty ? Statementj^y ee Fig. 40. The diameter is 240; hence the circumference is 754 ; the eighteenth part of which being multiplied separately by one, &c. similar arcs arc obtained, the chords of which are required to be found. The nearest arcs in integral numbers arc 42, 84, 126, l6s, 210, 252, 2Q3, 335, 377. The arcs are eighteen, but the tenth, &c. being the same as the eighth, &c. in an inverse order, nine only are put down. * The operation is then performed according to the rule, thus : circumference 754, minus the arc, is 7 1 2 ; this multiplied by the arc is 29904 the prallusma ; square of the circumference is 568516, the fourth of which, 142129, being multiplied by 5, the product is 7 10645, this minus theprathaBia is 6SO741, the divisor; the dia- meter multiplied by 4 is g6o ; by this multiply the prathama, the product W 28707810, which being divided by the divisor, wc obtain the cliord 42 : And in this manner the following chords are obtained, 82, 120, 154, 184, 208, 226, 236, 240. In the same manner in regard to all arcs. Or, to shorten the operation, reduce the circumference and the arcs by 42, which is the eighteenth part of the circumference ; the same chords will then be found as before. Thus reduced, the statement is, circumference 18; arcs 1, 2, 3, 4, 5, 6, 7, 8, Q: The circumference minus the arc 1 is 17, tliis multiplied by the arc, the product is 17, which is railed the prathama'. the square of the circumference is 324, the fourtl) part of which is 81 ; multiply this by 5, the product is 405 ; and "this minus 1 7, the pm/Aomo, is 388, which is the divisor. The diameter 240 multi- plied by 4 is 960, this multiplied by the prathama is 16320, which being di- vided by the divisor 388, the quotient is Uie chord. Thus, by rcfieating the ope- » Nolnilhtlimding tliis leiuark, j8 arcs nrr put down increasing in the same rniio ; tlnis, 42, 84, 120, 108, 210, 252, 293, 335, 377, 4i9, 401, 503, 545, 580, 028, 070, 712, 754. 'Ibe other coxites, Itawcver, coiiUua ouly the lirst nine. Digitized by Google ( 98 ). ration in succession, the same chords as before are obtained 43, 83, 130, 154, 184 , 308, 236, 236, 240. The same process answers for any diameter. Rule. — Multiply the square of the drcumference by five times one fourth of the chord, and divide the product by the chord added to (bur times the diameter; subtract tiie quotient from one fourth of the square of the circumference ; and subtract the square root of the remainder from half the diameter ^^s lastre* maiiider is the arc. ^ Example. — Required the arcs of the chords which have just been mentioned ? Statement. — Chords 42, 82, 120, 154, 184, 208, 326, 236, 240. The re- duced circumference is 18; the diameter multiplied by 4 is 960 ; add 42 the chord, the result is 1002; the square of the circumference is 324, this multiplied by a fourth of 43 the chord, is 3402, and this multiplied by 5 is 17010; which being divided by 1002, the quotient is 17; subtract this from 8l the fourth of the square of the circumference, the remainder is 64, the square root of which is 8 ; subtract this from the circumference, the remainder is I , wliich is the arc. In this manner there are found the arcs 1, 2 3, 4, 5, 6, 7, 8, Q , ; these multiplied by the eighteenth port of the circumference give the former arcs. *■ * The rules contained in this section are so fully illustrated by Dr. Hutton, that 1 deem it iufficirnt to quote his words, without any other comincnlary. “ To find the circumference of a circle, multiply the diameter by 3927, and divide ihe product by 1850 ; ot multiply the diameter by 28, and divide by 7. « In this precept we find two approximations; one of them, 7 to 22, the same ratio as had been the result af one of the labours of the prince of the ancient matbemaiiciaiia, Archi- medes ; and the other 1250 to 3927, the very same as J to 3 ■ 14 IG, nearer than tboae of any of the Eurapeans before the labours of Viela. Among otlier rules in measuiation, they hare the following ; i> Iseiog the diameter, and c the csTcamfetence ; then loo = area of the circle; also nr =: surface «f the sphere, and ^D*c z= its solidity. — Other rules are, = area, wliich ii exactly equivalent to 4>ur '78540*, and shows that k is derived from the rstio almvc-mentioned, 1930 to 3927. Anotlier rule for the same is, jjti* = area, wdrich is another of oiir approximations, and derived from the ratio 7 to £2.— Aaother ttule given for the roltdiiy of the sphctc in terms Digitized by Google ( 99 ) • S E C T I O N II, OP PONDS. ^ Hule. After '’measuring' the breadth in several places, add together the differenl breadths, and divide by a figure equal to the' number of places measured ; perform the same operation in regard to the length and depth. Then multiply the area of the pond by the depth ; the product is the cubic content of the pond. Example. — In ctmsequence of the sides being unequal, the length in three places •f the diameter, i* — H — =■ the solidity of the sphere, which must be •wrong printed in vations respects.* Having endeavoured to restore this form to some probability, I have imagined U might be intended to mean the same as ^ X which reduces to | x — -525dJ, and is very nearly the same as our own approximation -SSSCnJ. — Again, ♦o D being the diameter, c the chord, + and v the versed sine of an arch a, then |n — c*) = v, a theorem geometrically correct : from which is derived 2v'(n» — v’) = e, which is correct also.f: — But the two following are only approximations, where c = (he . - . 4od(c— a' circumference, viz, — - — ’ ' Jc’ — (c — o the 58th part in an arc of two degrees, and about the 166th part in an arc of thirty de- grees, being tme to the 4th place of 6gures in the former esse, and to the 3d place in the latter; but it does not appear how they have got this rule. From this however they have derived (he following rule, by resolving a compound quadratic equation, viz, -1 — ^ f^L \ = a. Hutton’s tracts, vol. 2. p. 158-9. a The diagrams of the ponds in tUis section, and of the wall and cut timber in the two next sections, have been omitted, as they afford no kind of illustralian. * The true eipreMion given by Bhssciri i« ^ -f- ^ X ^ which is much nearer the truth thin Dr. Hutton’s conjecture, t In tbcM two cues, e must mean the chord of double the arc. t Dbaictra bai also giren d — — 4" + V — D» = .5233d* 42 Digitized by Google ( 100 ) is 10, n, 12; In three place* the breadth is 6, 5, 7 ; and the depth is 3, 3, 4 j Requii ed tlie cubic content of the pond ? » According to the rule for finding the mean measnre, the mean breadth in cubit* is found to be 6, the length H, the depth 3 ; and the cubic content is 198. Rule, — Divide by six the sum of the several areas of the top, of the bottom, anrf of tire bottom added to the top ; the quotient will be the mean area of the figure } the mean area multiplied by the depth gives the exact cubic content. ^ The third part of the cubic content of the pond is the cubic content of the tuc/ii ( pyramid or rone. ) * Example. — The top ia ten cubits in breadth and twelve in length ; the bottom is one half of these numbers ; and the depth is seven cubiU : Required the cubic content of the pond? Statement.— Area of the top of the pond is 1 20, of the bottom is 30, of the bottom added to the top is 270 ; the sum of the whole is 420 ; and the cubic coa~ tent of the pond is 4QO. Example. — Each of tlie four sides of a pond is 12, and the depth is 9; Required * That is, lot a’ anti A» lx* the lenelh and breadth of the top and r* d' the lenglh •■'nd bie idUi of the bottom then a' ; b' :: c' : d‘ and o* d‘ = A* c> = abrd Now by our rules ~ contrnt of the pond or frustam of a pyra- mid ; but abed = therefore ^ a*tn + 6*c* + - 2ti*4*+2c’4‘+o*(f*+/.«c* S ~ 6 “ 6 solid content .of the pond according to the rule. • The tuebi here means a pyramid or cone, having the same aUiUlde or depth , as the pond, and the area of its base equal to the mean area of the jion.l ; therefore as a pyramid or cone, is one third of a prum or cylinder, of cqital base lud aliiiude, the cubic coutenl «t th« suebi is one tliird that of the pond, Digitized by Google ( lot ) »> »>■ — SECTION V. OP HEAPS. In a heap of large grain the tenth part of the circumference is the depth ; in a heap of small grain the eleventh part, and in a heap of grain with the husk the ninth part, of the circumference is the depti;, c * Here (he wood i.s supposed to he cut loneiludinally. * rite division by 576 is merely to reduce (he square inches to cubits. The result both of this ami (he following rule gives the sum of tlie areas of the vertical sections of (he wood, ami as this .sum expresM-s (he space thro’ which the saw jiasses in cutting, the object seems to be to determine the exi>ence of sawing wood. Digitized by Google ( 103 ) Rule. — By the depth multiply the square of the sixth part of tlic circumference ; the product is the number of solid cubit measures, which are tile kharya of the Magadlia country. ^ Example. — On an even piece of ground there is a heap of large grain, the cir- cumference of which is sixty : Required the number of kharyas in the licap ? Also the number of kharyas in an equal quantity of small grain, and of grain wiUi the husk f Statement. — Circumference 6 o. The depth of the large grain is 6 ; tlic number of kharyas obtained is 6oo. The circumference of the heap of small grain also is 5^5 60 f the depth is and the product in k.haryas is , . Also the circumference II of the grain in husk is 6 o, the depth is *,% and the number of kharyas obtained is Rule. — Multiply respectively by two, four, one and one .third, the circumference of a heap of grain lying against a wall, at the inside comer, and at the outside comer; then find the cubic product according to the -preceding rule, and divide it by the sum by which each heap was multiplied ; the quotients will be the cubic quantity of each heap. " Example.— The circumference of the quantity lying against the wall is 30 cubiU ; of that placed at the inside comer is 1 5 cubits ; and of that at the outside comer is 45 cubits : Required the cubic content of each of the quantities f Statement.— The product of the circumference «f the first heap multiplied by two is 6 o ; of the second multiplied by four is 6 o ; of the third multiplied by one and one third is 6 o. We then obtain the common product 6 oo ; this being divid- ed by each of the multipliers, we obtain the separate quotients 300, 160, 460, s That is, if c be the circomference, and d the depth, as above; (ben trill X d s .083 c* X f be the content ; for wbich our more accurate ruk is .07958 c* X -5 3 ■ These ate evidently i, i, and i of (be entiic heap or cone. Digitized by Google ( 104 ) S E C T I O N VI. Or SHADOWS. Rule. — Square the difference of the two shadows and tlie difference of the two hypothenuses ; and by the difference of these two squares divide the number 576 ; * add an unit to tlie quotient, and find the square root of the result ; then by this square root multiply the difference of the hypothenuses, and in one place subtract the difference of the shadows from the product, and in another place add it ; the re- sults halved are the two sliadows, ® Example. — Tlie difference of the two shadows is 1 9 and the difference of the two hypothenuses is 13. I reckon him who can tell the shadows, a proficient both ia oritlimetic and algebra. * The numtier 57(> is the square of twice the gnomon (which is always understood in tbeso rules to lie 12 ) ; and the shadow is caused by a lamp placed at different distauccs. » Sec Plate VI. I'ig. 3. That is given the gnomon = a = 12 4a‘ = 576 ^ S5 = difference of the shadowt 2 c difference of the bypoihcnusa Let = sum of the shadows Sy = sum of the hypothenuses Then X 4- 6 and x — b are the two shadows and j/ 4 - c and — c are the two liypollienuses We have then a* = (y 4 . r)* — (x 4 - by =: (y — c)* — (x — 6 )» hence cv = bx. nnd x* = • •• also a* sssjr* 4. c» — X* — t* Digitized by Google { 105 ) Statement. — Here the difference of the shadows is 1 9, the difference of the hy- pothenuses is 13 ; and the number 576 being divided by 192, the difference of the two squares, the quotient is 3 , plus one is 4 , .the square root of which is 2 ; by thia square root multiply the difference of the hypothenuses, the product is 26 ; from which the difference of tlie shadows being subtracted in one place, and added in another, 3 and the results halved, we obtain the shadows , , » »■ In this example call the gnomon 12 the side, and the shadows the bases; then 12 2 $ the square roots of the sum of their squares will be the hypothenuses , , p yjg Multiply the gnonaon by the space between the foot of the lamp and the bottom of the gnomon, and divide the product by the height of the lamp minus the or X' • + c' — «• — ^ t'y* — c*jr* _ _ c* + o» r = *1-'*-+-^ X 6*— c* y * * V + ilZT*) V ’i ji— ic» ^ ^ ^ ^ 1 The two slia- therefore x + h = e^(l+ ) d" ^ / +1 X 2 c — 26 V 46»— *e» \ aad» — 6= cV'( 1 + ^ * |»y varying tjie transpositions mc might also get dow* according to tlie rule it , , 7, ^r_ 6 for the values of the 1*0 shadows C — » W „ Digitized by Google ( 106 )■ height of the gnomon ; the quotient is the shadow. * Example. — ^Tbe space between the gnomon and tlie lamp is 3 cnbits, and the height of the lamp is 3^ cubits: fiequired the shadow of a gnomon 12 inches in height i Statement. — Gnomon » ; • distance between the foot of the lamp and the bottom of the gnomon 3 ; this multiplied by the gnomon, the product is • ; this being divided by 3, the height of the lamp minus the gnomon, the quotient in inches is 12. See ' Fi^. 49, 50. c Rule. — Multiply the gnomon by the space between it and the lamp j divide the pro- duct by the shadow, and to the quotient add the gnoinoii ; the sum is the height of the lamp. Example. — ^The space between the lamp and the gnomon is 3 cubits, and the shadow is l6 inches : Required the height of the lamp ; and the height of the lan>p being known, required the distance between the lamp and the gnomon > Statement.— Space between the lamp and the gnomon 3 cubits ; gnomon 1 2 inches . shadow 1 6 inches. The height of the lamp is found to be cubits ? 4* < Rule. — Multiply the shadow by the heiglit of the lamp minus the gnomon, and divide the [>roduct by the gnomon ; the quotient is the space between tl:e lamp ^ and the gnomon. Thus take the numbers from the foregoing example : The hejglit of the lamp Je 2 . . . ' « cubits; the shadow in inches is l6; the gnomon in inches is 1 2. The distance be- * See Plate VI. Fig. 4 Ix-t a be llie gnomon 5 = (he space between the foot of the lamp and the botlom of the gnomon a: = the shadow then a — c:5::c!jt = according to the rule. o— c * This is half a cubit or IS.inche*. V The lamp is placed ou the lop of a long brass rod, wbieh is represented bere by a stiaigli line. :s 1 Digitized by Googic (, ^07 ) tween the gnomon and lamp is found to be cubits 3. * l{„le —Multiply the two shadows, severally, by the space between the ends of the ehadow's, and divide the two products by the difrcrcnce of the lengths of the shadows’; the quotients are the rcsjKCtive s[wcs telween the bottom of the lamp, andtheendt of the shadows; multiply cither of these spaces, and the gnomon, to- gether ; and divide the product, by the corresponding shadow ; the quotient is the height of tlie lamp. All these operations may be performed by the Rule of Three Numbers, which ^rvades all calculations, as Mari, by his own portions, pervades all thing. Example. gnomon of 1 2 inches was observed to throw a shadow of 8 inches ; tite gnomon being placed two cubits forw ard opposite the point of this shadow, was observed to throw a shadow of 12 inches; Required the space between the lamp and the gnomon, •* and also the height of the lamp ? See Fig. 5 1, 52, 63. * See Plate VI. Fig. 6. J.ot o = gnomon ( , and e = the two shadow* d = the space bclivecn the ends of the two shadow* * = the space between tlie bottom of the lamp and the end of the first shadow y = the height of the lamp, then b : a: I j: ty , and czai :x-\-d:y tlicrclorc jmd c — - b X d i I b X JO — ' c — b ■. d ■. I C-. X + d =Z~ vaccording to the rule , ni + "d ! alsojr = -p =— 7— j • All (he copies have this reeling ; but from the subsequent operation, «ie que*Uon ap- pear. 10 be, “ required the space between the lamp and ends of the shadows. If, shadow, besltracted from dii. space, the remainder will be the distance between the lamp sud giiomuii ; aud this perhaps is the object of the question. bd 1 c-i I cd ' Digitized by Google ( 108 ) Here tlie space betw een the ends of the shadows is ; * the shadows ere J * j * the hrstof these ; being multiplied by •.*, and the product divided by the’ «^e between the ends of the shadows, the quotient is the measure of the space ' » ■ this is the space between tile bottom of the lamp and the end of the first shadow Thus, also, the measure of tlte space between the bottom of the lamp and the end ol the second sliadow is Then either space and the gnomon being multiplied together, and the jjroduct divided by the correspondiitg shadow, the height of the lamp is, ill both instancos, found to be the same, Or, tlie measurement of sliadows may be made by the Rule of Three Numbers, thus : If bj- the excess of one shadow above the other, a spaoe is found equal to the ■distance between the ends of the shadows, what will each sliadow give i In this manner we obtain the respective distances between the end of each shadow and the bottom of the lamp, niere is then another statement by the RulcpfThree, thus. If the gnomon is the side, the base lieing the shadow ; ivhat will lie (he side cor- responding to a base equal to the space lietween -the end of tlie sliadow and the hoh* •toih of the lamp ? Thus the height of the lamp is found to be the .same in both cases. In the same manner all the operations for 6, 7, 9, &c. numbers, may be per- •lormed by two, three, &c. -statements of the Rule of Three Numbers . As Vishmi the comforter, -the one seed from which all things spring, pervades the -whole world, mountains, rivers, gods, demons, cities, 8cc. which are only distinc- tions of himself, -so the Rule of Three Numbers runs thro’ all arithmetical ope- * These are fraciions of a cubit. Fii two copies the reiding is (bus ; “ The distance in inches helwmi the ends of the sha.loiis is 5 ^ inches j the shadows are 8, 13 ." The distance between (he ends of the shadows is found in the following mnonet.: Tb* gn'iiiinii is carried 2 ciibitt or 48 indies forward from its first position ; from tliis number siihiract 8 inches, the length of the first shadow, there remains 40; to (his add 12, the ■Irngtli of the other shadow, the sum is 52, which is (he space between the ends of the two ■shadows. Digitized by Google ( 109 ) Tattoni. It may then be asLed, whj’ has so much been written ? The answer is, that tho’ the calculations made jn arithmetic and algebra by means of the processes of multiplication and division,.are merely cases of the Rule of Three Numbers* yet this is perceived only l^y tliose who have acute minds ; but to assist those who, like me, are of slow ui^Tstanding, Uie skilful in the science have made easy, by .^ifierent modes of operation, the various ways in which the Rule of Three Num- bers may by performed. ^ * The mulUin arillimelic, algebra, ambin calciilatiitg (bemoHons of the hraveoly bodies, which are usually oblaiued by (he processes of multiplication and divitioa, are produced also ‘by the Rule of Three Quautiiies; but operations fur the square, &c. do not proceed frooi this rule. C«ai. It is stated in the Bija Gannttn, that with the exception of the square and square root, .cube and cube root, all operations may be done by:tbe Rule of Three Quantities. V Digitized by Google Digitized by Google PART I I I. SECTION I. or TBB KBTAKB.^ IPreparatort to finding the Kutaka ( required multiplier ) reduce first by some number, if it can be done, the dividend, ■ divisor, and augment. If the number which divides the dividend and divisor, do not divide the augment, the question is not solvable. ^ The common measure of the dividend and divisor is the last divisor which is -obtained by their reciprocal division. The dividend and divisor when divided by this « A known quantity being multiplied by an unknown multiplier, and a given augment ad* ded to the pioduct, or tubiracled from it, and the result divided by a given divisor, if no arniaimler be left, the multiplier in this caie has been named by former authors kulaka. Ku- tain, ihereioie, means the particular or individual multiplier. Tbe number multiplied is ta’lfd iiividenJ. The dividend, divisor, snd augment must be raduced by a common divisor V hull it can Ire done ; and for this purpose find any common divisor, except an unit ora fi ciioii. Alierti.is reduciiouof the dividend and divisor, they are called drada or couden- acd, us they cannot be farther reduced. Com. • Dusfila, Literally, ill, bad. A dpawartanam. Abbreviation. * The word thetham whirjh, in the translation, is rendered last divisor, usnally signifles re- mainder. According to this sense, t literal translation of the passage would run thus : “ The canrainder of the two ( dividend and divisor ) reciprocally divided is the abbreviation, by wUcli abbreviation the dividend and divisor being divided, are called drada." The anibigui. Digitized by Google ( 112 ) common divisor are reduced to their lowest proportionals. * Divide the reduced dividend and divisor reciprocally till the remainder in the dividend is an unit. Place the quotients under each other in succession ; below these set down the augment, and below the augment a cipher. Then having multiplied by the last figure but one, the figure immediately above it, add the last figure to the product, and then throw out the last figure. » Repeating the operation in this manner two quantities are found. Divide the upper quantity by the reduced dividend, and call the remainder the quotient; then divide the other, or lower quanti* ty, by the reduced divisor, and call the remainder tlie multiplier. If the number of quotients is an even number, the results q/" these two divisions, are the quotient and multiplier. But if the number of quotienU is an odd number, then the quotient and multiplier thus found being subtracted from their respective divisors, the re- mainders arc the quotient and multiplier. Example. — What is that number by which when 221 is multiplied, and 65 added to ihe product, and the result divided by 195, there shall no feroaindcr ? Statement. — Dividend 221 ; augment 65 ; divisor 195. The last divisor, after the reciprocal division of the dividend and divisor, is 13. The dividend, divisor, and augment, when divided by this number, are tlie reduced di- vidend 17, divisor 15, augments. The quotients which result from the reciprocal division of the reduced dividend and reduced divisor, being placed under each other, ond ty, however, is removed by (he subsequent example, in which it is stated (hat 13 is the last, or remainder, after wliat is termed the mutual division of 221 and 195. Now, if 221 tie divided by 195, and tliis last number by the remainder, and to on, tbc last divisor will be 13. a Drado. Hard, solid, firm, excessive. Here it signifies what cannot be farther reduced -OT divided. » Something seems wanting here; the insertion of the following paragraph would perhnpt supply (he deficiency. “ And by Ihe result multiply the number next above in the line, and to (he product add ihe last remaining number as before." * Takshana. The operation of rejecting tlie quotient and taking the rtOMinder is called tashtai Thus 56 being divided by 17, the quotient is 3, and the remainder is 6i This re- mainder is (he number taken. The number or divisor by which the remainder is obtained is sailed its takshana. Thus 17 is the takshana of 5, because tbc remainder 5 results from the •Digitized by Google ( >13 ) lb« rcauoed »ugm«nt set down Ww them, sod a ciphar Wow the sogmcnt, the lineit * 7 5 O , Then according to the rule " by the last figure but one multiply *e figure hn-' tnriUtely above h. &c.- there are found the two quantities 40, 35. H.v.ng JJJVe|,2 22 4JJ1L8 8 3J HI 3 Digitized by Google ( 115 ) the drntod and angment by ten. The atatement then is, dividend 10 , .u^ent 9t divisoe 63. Divide recip^caHy the dividend and d.v.sor, »»d pla« >l» ,J^enUbelotv each other, the aegment below the <,uouenU, «>d aerpher below the augment ; thus, o 6 3 9 Then according to the fornTer operation, ne obuin the mdtiplier 45 ; but tto Tu^not be Jen-, for the number of quoticnU being an oad numW, the tiplier 45 must be subtracted from its own divisor ; the remainder is the mulUpher IS follows 1 X 1530 + 900 = 2t30 1 X 900 + 630 = 1530 i X + XT« = SOfl X X gVO q. .50 = 6SO St X S» q. so = gvo i X SO + 0 = S» a )ivide Uie uppermost quantity 2430 by the dividend 100 ! lOOJ 2430 L24 430 400 SO remainder, called the quotient Divide the other quotient 1530 by the divisor 63 1 6SJ 1530 L2-1 126 270 253 18 remainder, called the multiplier. Digitized by Google ( "6 ) 1 8. Then multiply the dividend by this number, and having added the augment to the product, divide the result by the divisor ; the quotient is 30. Or, Reduce the divisor and augment by nine. The sUtement then is, dividend lOOJ augment 10; divisor 7. The line is 14 3 lO O and the multiplier obtained is 2. This being multiplied by g, the common measure of the divisor and augment, the s;ime multiplier is found as before, 18. Multi- ply the dividend by this number ; add the augment to the product, and divide the result hy the divisor ; the quotient obtained is 30. *• Thus «bea the augment 90 is affinnative, (lie number requited it 18. For too X 18 + yg _ ^ * One Commentator alto gives tbe following mct'iod : “ First, reduce the dividend and augment by 10. We have dividend 10, augment 9. Again, reduce tliis reduced augment, and also the divisor, fay 9. We have angment 1 ; divisor 7. Divide tbe reduced dividend 10 l>y the reduced divisor 7 : 7J JO LI 7 3J 7 L2 6 1 The line and the two quantities llien oblnined agreeably to ifae rule, are 1 X 2 4- 1 = 3 quotient g 8xl-i-0 = 2 multiplier « These two quantities being Ie« than the reduced dividend, and the reduced divisor, stand as they arc, being called quotient and multiplier. Digitized by Google ( "7 ) ' Or Add to the tnultiplier and quotient their respective divisors multiplied by any assumed number : if one w the nuinter assumed, the multiplier and quotient obtain- ed are 81, 130: if two, they are U4, 230; and so on. Rule —The multiplier and quotient being subtracted from their respecUve divisors, are the multiplier and quotient which result if the augment is negative. Take the last example. When the augment gO is affirmative, the quotient and mul- tiplier obtained are 30, 1 8. These being subtracted from their respective divisors. y the dividends, the quodent is i) ; and Iiaving divided tlie lower quantity 23 by the divisor S, llic quotient is 7. But 7, wbii h is the least quotient, is taken in botli eases, and 5, which is tlx divisor of 46, instead of be- ing mullip'idl by 9, is multiplied by 7, and tlie product 33 being subtracted from 46, tlie remainder is called the quotient, 11. In the same maimer J mnUiplied by 7 is 21 ; tb.s Ixiug subtracted from S3, the remaiiidet is 2, ubieb is called the multiplier, Digitized by GcXlgle ( >'9 ) by the 3i»i And the multiplier and quotient produce*! by an affirmative augment, being sub- tracted from their own divisors, the remainders are the multiplier and quotient pro- O duced when the agument is native ; in this case they arc 1,6. Increase each of these by its own divisor multiplied by an assumed number ; here, each divisor being multiplied by two, and the product added to the multiplier and tirgnih-e • quotient, this quotient becomes alfirm-iiive, and the multiplier and quotient tlien --are 7, 4 . “ And so ou in every case. If the augment be divided by the divisor, 8cc. Uic statement for this method is, dividend 5 ; augment 2 ; divisor 3. When the augment is affirmative, the multiplier and quotient obtained according to a former rule, are 2, 4. These be- ing subtracted from their -own divisors, the remainders ], 1, ere the multiplier and quotient which result when the augpnent is 'negative. The quotient . obtained in dividing the augment being added te the quotient 4, the multiplier and quotient obtained when the augment is affirmative, are 2, II.® When the augment is negative, subtract the quotient l from the quotient obtained in dividing the augment ; the multiplier and quotient then are 1,6: In order to obtain a plus quotient, add to the multiplier and quotient the product of their respective divisors multiplied by 2 ; the multiplier and quotient then obtained arc 7, 4.^ * Thna, liLL+J? = 1| ■ 3 -• Thb ireood method shews how the multiplier and qunlient are otAsined, after dividio,, -•■>rediKiDg -ihe aogmrnt by -the divisor. The number which remains upon this divivion is Digitized by Google ( 120 ) Bulc. — Tf the aiigment be a cipher, or if the augment w^en ditrided by the divisor leave m> remainder, then the multiplier shall be a cipher ;aod in the latter the c|Uotient which results from dividing the augment by the divisor, is the quotient. Example.— What numlier is that by which when 5 is multipUed, and either I ^pher or 05 added to the product, and the result divided by 13, no remainder shall Statement. A_Di\ idend 5 ; augment 65 ; divisor 1 3. The augment being divided by the divisor, no remainder is left ; therefore the multiplier is ciphers and the called tbc augment ; thus, tuenienf divisor 3J 23 ^7 21 - 2 remainder, colled augment Tlic statement (lien is, dividend 5, augment 2, divisor 5. J lien divide llie dividend by (he divinor, &c. and ploce the quotients below each olher, and (lelow them the augment, and below the augmeut a cipher; and find two quanlilies ac- cord iug to the former rule : 1 X 2 + 2 = 4 2 X 2 + 0 =z 2 * a 1 he («o numliers 2, 4, being less tlian the dividend and divisor, are the multiplier and quo- tient. WlicB (lie augment 2o is aflirmalive, add this quotient 4 to the quotient 7 obtained in dividing the augment ; llie resull is 12, which is the true quotient. Again, (lie two mimbers 2, 4 being siililraetcd from their own divisors (dividend and di- visor) the remainders 1, I me tlic miilti;riicr and quotient wliciitlic augment is negative. Sub- tract li. is qiinlieut 1 (ruin the quotient 7 obfiineJ in dividing ilie aitgmeiit; tlic remainder 6 is llic (rue quotient, wliieli in this case is minus. 'I'o obtain a plii« quotient multiply by an srsMimed atiimiH-r 2. Tiic miilliplicr and quobcut found are 7, 4. s file Copy from wliich I liiivc trjnslatcd docs not give an example where a cipher is Uie augment, lint in nnollier copy it is staled tliiis : “ Dividend 5; divisor 13; augment 0. As the augmeut is a cipher, the muiliidicr anil quotient arc 0, 5.” Tliere is a however iii sitting the quo'.ient to be 5. Tiris will ajqjcar by the feU flowing opei;'.iio!i : Digitized by Google ( 121 ) quoti negative ; thus, 15 7 = 8 multiplier, when the augnicnl h nega ive j7 — 8=9 qiioiiin', when the augment U negative. Multiply these I'y an assumed nnniher 5 s 8 X 5 = 40 9 X 5 = 43 Divtle each of the niimhcra 40, 45 by iU own divisor, calling the reinalndera retpectivcly muuipiict and quotient : 15J40L8 SO 10 remainder, called multiplier _ 17 J 45 L9 34 11 remainder, called quotient. Thus the multipher ao’i quo'i nt when the augment ii negalire, arc JO 11. a 1'lie n iml'cr of accomU in a minute. • The spnee ineliide'l in a day of Bramlia a fited period from which eclipses me enl- cnla'e I, a* the beginning of a Aatpa or yugn. Eclipvev are now p nerally calculated from the period of the anauo LaoHowOf which wa$comi>osed about UX) yean ago. Digitized by Google ( >25 ) is seconds, and the ifulliplier Is the numerator of the fraction of a minute. * From thi* multiplier and quotient, find the minutes and the fraction of a degree. Then, from file intercalary month, “ and from the fraction of lost days, c find in the same manner the numlicr of solar and lunar days. Example. — From a fraction of the second of a planet, to find the planet and ahargana. ® Call Go the dividend ; the hudina the divisor ; and the numerator of the fraction of a second the negative augment. Then the quotient number found will be seconds, and the muhiplier will be the numerator of the fraction of a minute. Thus alsoi calling the numerator of the fraction of a minute the negative augment ; the quo- tient obtained will be minutes ; and the multiplier will be tlie numerator of Uic frac- tion of a degree. Call this numerator of the fraction of a degree the negative aug. jnent ; the kudina the divisor ; and 30 >'■ the dividend. The quotient number ob- tained will be degrees ; and the multiplier will be the numerator of tlie fraction of a sign. »Thcn call 12^ the dividend; the the divisor; and the numerator of the fraction of a degree the negative augment ; the quotient number obtained will be sigpas; ° and the multiplier will be the numerator of the fraction of a revolution, w Thai call the revolutions in the ialpa the dividend ; the kudina the divisor ; and the * Vikalla. • Mi. c Ksht/a. o The time or number of days elapsed from the commencement of Bramha's dajr or from the beginning of a ktdpu ; — or from an assumed period. The definitions of kudina and ahargana are given merely npon verbal explanations which I have received ; it is possible, therefore, t* » X *» + » = 4* 1 * a The quotient obtained from these two quantities 830, 50, is 30, and tbe mnltiplier is 11. Bw as the number of quotients in (he line is odd, the multiplier and quotient roust be sniitrnctrd from their respective divisors 60, 13 ; the results are multiplier 10, quotient 2. These, \ r * These small assumed numbers are taken to ihortea the calculations, t Tbenumborof seconds ia a eniaute; Digitized by Google ( 127 ) • In the same manner, call the adi or intercalary months in the kalpa, the dividend, the solar days the divisor ; and the numerator of the fraction of an intercalary monti), the negative augment. The quotient obtained in this case is the number of interca- lary months expired ; and the multiplier is the number of solar days expired. In the same manner cill the thrown out days in the kalpa the dividend ; the lunar days the divisor ; and the numerator of the fraction of thrown out days, the negative bowevrr, are the rauUipUrr antJ quolirnt when the augment is pnsiiive ; but here the ang« ment being negative, they must be siibt'’acte(l frnm llieir respective divisors 60, J3 5 there- ■nits are, quotient 50, multiplier \l; the quotient number 50 is seconds; the multiplier 11 is the numerator of the fraction of a minute. Call the numerator of the fraction of a minute the negaUve augment II ; the dividend 60 ;♦ (heditisor 13. By dividing the dividend by the divisors, &c. the line obtained, and the two quanli:ie« which result from multiplying the numbers in tlie line, are 4 X 55 + 33 = 253 I X S3 -t- 2^ = 5> „ i X + *» = SS .5 4 X « -b n = * X + » = 4* 41 5 The quotient obtained from these (wo qunnli'ies is 13, and (he mnUiplier is 3. But as the tiumlrer of qiio'ientf in the line is o ld, the multiplier andquol rnt must be subtracted from their respective divison 60, 13; the remainders are, qiioii:-nt 47, multiplier 10, when the an«me 't isalhrmalive. But here the ungmenl is negative ; (herefore these must again be sub- t'nrted fiom their respective divisors 60, 13; (he remainders are, quotient 13, multipliers' wlien the .augment is negative. The quotient 13 is minutes, and the multiplier 3 is (he nu- tnenior of (he fracti m of a degree. C ill the divi len 1 30t ; the nameralor of the fraction of a degree the negative augment 3; an I ilic divisor 13. * number »tf miiuitet in a deftrrr, i Tbe uumber of drgrcca in a aigoe Digitized by Google ( 128 ) augment. Tlie quotient then obtained will be the number of thrown out days ex« pircd, and the multijtlier will be the number of lunar days expired. Tiic line and llic two quantities wliicb result, are 2 X S + 3 = 21 « ax3+0=9 t> As the two quantities 21, P, cannot be reduce 1 by their resprstiee divisors 30, 13, (hey ar* ralletl quotient 21, anil multiplier 9. These b. iii^ subiraileil tioin their own divisors 50, 13, the remainilors are qiioiient 9, multiplier 4, when the augment is nagslive. The quotient 9 is degrees, the multiplier 4 is the numerator of the traction ot a sign. ^ ('all T2* the dividend} this iiurncralor of the fraction of a sign tUe negative augment 4 j and (he divisor 13. Here the line and the two qunn'ities aue 0 X 4 + 4 = 4 g I X 4 + 0 = 4 These two quantities being less than their di-sivors, arc the quotient and multiplier; An4 .ubtraelin- them from their respective divisors J2, 13, the remainder, are, quolierU 8, multi. ,,Iicr 9, wdien the autrmenl is negative. The quotient 8 U signs; the multiplier « the numo- inlor of the fraction of a blmgana or revolution. Call (he dividend I It j the numerator of (he fraation of a revolution Ihe negative augmeoft O; and the divisor 13. The line and the two quantities are 0 X M 4- 43 = 45 S 1 X 4 > 4- ^ 5 X 3 + » = 45 9 e The imm1>crs43, 51 being dividi-d by their respective divisors IJ, 1», we obtain tbe rr- anainder 1,2 which are callcil qnotient and multiplier. But as Ihe number qf quotient, m , rhe line is odd, the quotient and multiplier must be subtracted from their respective diviso* • Tlie number of signs in Ihe aoditc; i 'J he number cf bhagaaa assumed in the example. Digitized by Google ( 129 ) SECTION III. or TUB SANSLISTA KUTAKA. Hule. If there be one divisor, and several multiplicands, call the sum of the mu8, tiplicands the dividend ; and call the Mim of the remainders the native augment. This manner of performing the kutaka is called sanshsta kuiaka. Example. — ^What is the number by which when 5 is multiplied, and the pro- duct divided by 63, leaves the remainder 7 : and by which when 10 is multiplied, and the product divided by 63, leaves the remainder 14. In this example call the sum of the multiplicands the dividend; and the sum of the remainders the negative augment. The statement then is, dividend ] 5, aug- ment 21, divisor 63. According to the.former rule, the muldplier .obtained is 14. * 11, ]S; the remairukrs are, quotient 1 0, mulliplicr IJ, when the angroent u positive; And these being again subtracted from their divisors IJ, 13, the remainders are, quotient I, snultiplier 3, when the augment is negative. The quotient 1 ds the number of revolutions, and the uiullipliet S os the Altargaiia. Tims, then, there ai« obtained J rev. 8 signs, 9*, 13', 50”. And the Abargana oie fc" • Thus, dividend 15; divhor C3; augment 21. Aedncc the dividend, divisor, and cug. tnent by 3. The statement then is ; Heduced dividend 5.; .(educed divisor SI ; reduced aug- ment 7. The line and the two quantities obtained are O »s Ox 28 + 7 = 7 4x7+0 =28 V a Divide (he uppermost quantity? by the reduced dividend 5 ; the remainder is 2, whichas called quotient 5J7U 6 9 remainder, called quotle^ Digitized by Google ( 130 ) HOT* COirriKUMB. And divide the lower qoanlity 28 by the reduced di?i»or 21 5 «»« remaiodet ii 7, wbich k called muliipliei 21J 28 LI 2i 7 Femaiader, called muKipner Thui, when Ibe augment ii affirmative, the quotient is 2, and the muKiplier if 7. Sul^- iraei from their own divisors, 6, 21 ; the temaindeti are, quotient 3, multiplier J4, «Ik II t^ic augment is negative, liejecl the quotient; and the multiplier 14 if the lequiied auinhcr. 1 or 6 10 14 14 C3J70 LI (>3 63J 140 L2 126 7 remainder 14 remainder The whole of (he kutaka corresponds with our method for the solution of indeterminate |iro'.ilcins of the first degree, the same rules are gi/en in Strachey's Uija Ganita, pngvs^ and followinir. Tlie process will also be found in Euler's Algebra, vol. 2. p. 17 &c. vrilh which Bhascan's may lie co npared. ]t may not, however, be iiniiiterefUng to solve ena of the exam|>lcs according to our method, to shew how the rules of the kutaka CM« fcspond with ou r, !£!'.+-» = , , = . + 51' +-2? = . + , fljp _ 00 _ . 26p — 00 * = —37— = ^ + — 37 — = /» + 7 37, 4. no _ "9 + 90 _ „ . . P = - 7 + iiu = ? + r 06r — PO 4r — Of) q = — = 2r + p; — = 2r + * 1 1 Ilf -I- 90 r = — y - = & + s. + 00 + 4( — 00 = 2r 4- < 3^=* + S-="*-“ ( c= 3u +, £10 Digitized by Google s. ( 131 ) HOT£ COXTiyVED. then putting w = 0, we have It = 0 — 0 0 « — 3k + 00 = 3 X 0 + 90 90 ■='t t + n 4k 4- OO s = 1 X 90 + 0 — 90 11 2r + t 1 1 K + 270 = 2 X 99 + 90 270 II 2r + » 20 k + 030 1 = 2 X 270 + 90 = 030 P=\ q + r 37« + 900 1 = 1 X 630 + 270 = 900 II w p + q f GJm -f 4530 ^ = 1 X 900 + GSO = 1530 » = | X -f p ^ 100a + 2i30i = 1 X 4530 + 900 — 2430 Comparing .l.e^c columns «ill, note . p. Ill, ween eviden.Iytrncc Rhnscara's rule. A„dn.e« a.e true values of X ami but as have not rejecled a|| tire whole numbers in the roarsft oi the altove opeuio:., they will a.Im.i ol Ireiog reduced (o lower terms: repeating iherc- fotc the solution, and r.jccting all the whole numbers we have, ^ V — 0 =0 r = 3 « + 1 = I » = r + rj — I r =. 2 .r + r —3 9 = 2r + * — 7 /» = 9 + r + l=:jl ^ P + q = J8l t/ — X +/, + according to Bhascara •r by reducing too X I S30 + 90 63 = 2430 s»e have X cq u 3 that is X = ]8, and y = 30 K«Lo appears, that in our soluuon, we Lave one equation more (han Bhwara gives quo. Digitized by Google ( 1*2 ) HOTM COkTlHOKm. drate ; ind it i« eviilent, lliat while (he aagmeiit i* po*i i*e, "B oM number of equMioni V’ — ? BUiiibt» of degrees psused over 13 * i Digitized by Google ( 133 ) jrOTM CONTIirPSO. y" — 9* passed over s *" 4, ^ = fraction of a sign. I2i"' — 4 13 — — whole number of signs passed over y» — g gigsj passed over : =s 9| ^ = fraction of a revoluUra U*"" — 9 13 — y'” — whole number of revolutions passed over y'" — 1 Revolution passed over : sf"' = 2, period or Ahargana required ; and coJ« lectlng all tlie resuUs we have 1 Rev. 8 signs, 9*, ly, 50 " as before given. The Smtiuta Kutaka may be^solved as follows. = w + — 63 ^ 63 and!^=y+ by addition ^ =: w + y + ^ hence 63 15r — 21 03 =: » 4- ^ * srhole munber, from which the solution foUowf I before. eiTD ot Ttu xxrtiXAi X Digitized by Google I Digitized by Google PART IV- OF TRANSPOSITIONS. JVhcre particular fgures are given. The scries of figures one, &c. increasing by one, being multiplied to the last place, the product is the number of transpositions.' When particular figures are given, multiply this product by the sum of the vaUet of the figures ; and having divided this last product by the number of the figures, set down the quotient as many times as there are figures, advancing it one place forward each time, and then add up the whole ; the result is tlie sum ef the valuet of the figures set dou n according to the number of transpositions.'^ Example. — What number of transpositions may be made by 3, 8 ; by 3, 9, 8 ; and by a continued series from 3 to 9 : Also, what is the sum of ike values of the figures set down according to the number of trarupositions. Statement — For first example. — 3, 8. Here the number of places is 3. Mul- tiply one, &c. increasing by one, to the last place, the product is 3. Thus the number of transpositions is 2. Multiply this product by 10 the sum of the fi- gures ; the product is 20, which being divided by 2, the number of figures, the quotient is 10. This quofient being set down as many times as there are figures, ad- vancing it forward one place each time, and the whole added up, the result is J JO, ■ Thai if, the natural aeries of figures being mulliplietl coafinoally (ogetlier up to tbe given number of places, the product is the number of transpositions. ■ This method of finding the sum of the figures set down according to the number of tnuil- poaitious, does not appear to be contsined in any of our ariUimetical books. Digitized by Google ( >36 ) the sum of the transposed fibres. * Statenaent. — for second example. — 3, 9, 8. Multiply tlie figures one, 8tc. in- creasing by one, to the last place; the prcxlnct is 6, the number of transpositions. Multiply this product by 20, which is the sum of the figures; the product is 120, which being divided by 3, the number of places, the quotient is 4U. Tliis i|Uotient ■being set down .as many times as tliere are figures, advancing it fonvard one place each time, and the whole added up, the result is 4440, tlie sum of the transposed figures. “ ‘ . .. j \ Statement. — For third example. — 2, 3, 4, 5, 6, 7, 8, 9. In this cxamiilc the number of transpositions is 40320. Ihe sum of the traiisposetl figun;* is 24t)39999753(5o . ^ \ i F-xamplc. — The net, hook, serpent, drum, cranium, trident, bedstead foot, Spear, arrow', and bow, which are held i'l the ten hands of Mahudev. being chang- ed successively from one hand to another, liow many changes will take place in ilie ». in ^ qiiotiriit set donna* many time* as lUere are figures, aud advsuiced forwurd one • 10 y place each lime ^ • . , . , 1 10 smu of the transposed figures. Tor 110 2 8 2y' J 1 0 % 40 ; 40 ■i '40* • 4410 Slim of the (raiisposcd figure*. 1‘or ; . 1 . • - '•■ijS ^•'“'"her of (rausposiUciBi . ' ’ \ o - ■ > 4440 sum of earancc of the inaage of Mahadev is 3628POO. Statement for second example. — The places are 4. I'Jie number of cliangcs produced in the iinagi^ of Vishnu is 24. « Itule. Divide the number of transpositions which result from all the rtgures, by the sum o4 the transpositions which result from the like figures; the tjuoticnt is the number of transpositions which can be made by the inixc-d figures. Then find the sum of the transposed figures according to the former rule. Example. — Required the number of tran.spositions which result from two, two, one. one ; and the sum of these figures when transposed ? Also, required the number of transpositions wliich result from four, eight, five, five, fiv'e; and the sum of these figures wlsen transposed ? ami the qviolirnt put down as many times are tlierc me figures, advancing it forward one place each time, and tlie wliole added up, the result is llie sum of ibe transposed figures; thus, '’23I7G0 aSl'fiO 2217G0 221760 221760 22l7ti0 221760 221760 J 2463999973360 ium of the transposed figures • Anj/ Sumber of Tilings being given; of which there evre several given Things of one Sort ; and several of another, S(c ; lo find how nianj/ changes cem be made oat of them all.- • Rule. — Tiike tlie series 1 x 2 x 3 X 4, &c, up to the number *f things given, and find the product of all the terras. Take die series I X 2 X 3 x 4, &c, up to (he number of given (liiiigs of the first sort, and the series J X 2 x 3 x 4, &c, up to (be number of given things of the .setuiid sort, &c. Divide the product of all the terms of the first series by the joint product of all the terms «f the remaining ones, and the quotient trill be the answer required. Hutton's Malbeinaucs, vol. 1. p. ■ 1 1 quotient set down at many times ns there are figures in the exam* ^pic, and advanced one jdace ibnvmd each lime. • . h Digitized by Google ( 138 ) Statcm«U._For fir,t exampU.-ii n . The number of transpositions four A. fn e.r ,s 2.,. Then tl.e rule is, “ divide tl,c number of tran,>ositions which result from ^ihe figures, by the sum of the transpositions wl.ich result from the like b-ures-* Thus ilie number of transpositions of tl.e two f.rst like figures is 2 ; am.m the number of trans(K.sitions of the Uo other like figures is 2. Then thc'’number of trans|.03itions, 2-1, being divided by 4, tl.e sun, of the transpositions of the l.ke fip...es, thequofent iati, the number of trans,«,sitio..s whicl. can be made by the n..*ed figures: vu. 22. 2.i2. . 22 ., , 2 . 2 . .,22. 2.2,. Accordmg to the funner rule the sun. of these figures, U.us transposed, is found to l.e qqq^ 5ta,ement._Fa,- sreoud The number of transpositions which re. suit fro.n ttu: nu.nber of figures is 120. Tne numlier of transpositions of th.ee places ,s 0. The for..u*r n..mU*r, I2t), being d.vided by this, the quotient is 20, the number of tn.n<^pnsilions of thr mixed f tuns ' thus, 48355, 84555, 58455, 55485, 65845, 555 18, 555S4, 45855, 54855, 85 155, 54533, 58545, 55458, 55854, 45585, 85545,' 54558, 58554. 45353, 65334. Tl.e sum of the figures, thus transposed, is I ipgg fc8. ® tyiere the par lid, lar figures are not niven, and where they are also unlike. I'he ntUurai series of figures decreafing from the last place by one, to the nun- » The wliolf iiumlier of figures is 5: Titrn 1x2x3x4x6=120. Tlie number of like figures i» 3; Then = 20 number pf Iransposilio* of ibc Uiixtd /inures. M»m ©f pumbtr firisre* rt»iii iirftf Aeurct » Thus, 4-p84'64-6-('6=ii7 x 20 = 540 5 = J08 quolient pat dosm JUS I as uiai.y lime* u lOS yliiere ore figure*, 108 I 11 ml niie place fur. lOSJ ward, &c. 1199988 sum of transposed figures, for 48.555 4 - + 581,55 + 55495 + 5.5845 + 55548 + .5.5584 + 4.5R55 + 64855 + 85455 4 - 54585 + 58545 4 - 55458 + 55854 + 45585 + 85545 + 64558 + 58554 + 45558 + 85554 = II9998S f f i Digitized by Google ( >39 ) her ofjtgure* taken, being multiplied together, the product is the number of changes of unlike figures. ^ Example. — What number of changes can be made by six places of figures, the same figure not being put down twice ? The last figure is 9. And the figures when put down 'in six places, decreasing by one, are 9 , 8 , 7, 6 , 5, 4. These being multiplied into each other, the pro- duct is the number of changes, 60480. Rule. — Subtract an unit from the sum of the figures, and having set down the remainder, decreasing by oae, in as many places less one as the example contains places, divide by one, &c. and multiply tl>e quotients together; the product will be the number of transpositions. This rule is applicable only when the number of places does not exceed nine. To avoid prolixity this is treated in a brief manner ; for the science of calcula- tion is an ocean without bounds. Example. — By figures the sum of which is 13, placed in five places, what number of changes will result ? ® In this case an unit being subtracted from the sum of the figures, the remain- der is 1 2. Set this down decreasing by one in as many places less one, as there are places given, and divide by one, &c. thus The product of the fi. gures in the upper line being divided by that of the figures in the lower line, the quotient is the number of transpositions, 495 . Though neither multiplier, divisor, square, nor cube, has been demanded, yet this specimen of anka pasha (transpositions,) will suffice to humble the most proud. * Anj/ Number of different Things being given / to find horn many Changes can be made oat of them, dy taking a Given Number of Quantities. Rule. — Take a teriet of numben, beginning at the namber of things given, and decreas- ing by 1 to the number of quantities to be uken at a time, and the product of all the terms will be the answer required.” Hutton’s mathematics, vol. I p. JST. • This is a literal translation ; but the following will perhaps be a more cl^r enunciutioit of the example. — Required the number of transpositions that can be made with any five, a a 30 . . . . a . *1* a a 30 ••• a a GO 'I® a a 30 a a a s a a y a 30 1 •• a • a a t a If 60 Number of Changes in (he Fora 495 sum of (be transpaitiona Out of (best eighteen forms there are three, each of which giva fire (ranipotilionf . , 15 two ten 80 fire twenty lOQ four thirty 120 four , sixty 240 ToUI .... 495 which corresponds with the result of Bhascara's rale - x — X — = 493. 1 1 s 4 * The couy from which 1 hare traasbled conclnda with the following iutiniatioB : “ The writing of this copy of the Lilawati was finished on Thursday 15th Kanika Suda^ lathe Samratayear 1729. — May prosperity atiead it.'* The present SamraU or Vicramadiia ye« « 1672, K ttst Utvcapj WM WtittsD 143 yesi^ «g«, ot in tbs ym 1673 «f ou ets.^ r < Digitized by Google Digitized by Google APPENDIX. SHORT ACCOCNT OP THE PRESENT MODE OP TEACHING ARITHMETIC IM HINDU SCHOOLS. _/^RlTIl METICAL scicnci", a> (aaghi in the Lilawati, is confined exclusively to (he Jyo- tisliis or a»(roiioini.'rs. At school childico nru taught little beyond the (bur elementary rules ot' addition, subtraction, miillii>lication, and division, together with one or two examples of the rule of three, and of interest. In the metliod of tcachinglhese rules, however, there is sornctliing peculiar, an account of which may be not altogether uninteresting to those who are lond of oljserving the various motics of calculation practised in different countries. As the instruction received at Hindu Schools is almost entirely confined to arithmetic, a few additional remarks will be sutliciimt to convey a general and pretty accurate idea of the education afforded to Hindu children. The following account being founded chiefly U|ion information received from natives of the Mahratta country and of O'uzerat, and on obser- vations made during visits to schools kept by inhabitants of tlirrse countries, must be re- garded in some me.vsurc as local. At the same time, the conversations which I have had with people from different and remote parts both of Hindustan and the Peninsula, leave lit- tle doubt that, in the general features, it will be found to correspond with the plan adoiited throughout the whole of India. On joining the sclmol the young pupil performs the pati puja, or worship of the writing board, in the following manner. The board is first covered with gulal, on wliicli is drawn llie figure of .Saraswali the goddess of learning ; it is then covered with perfume, rice, Sowers, sugar, beetle-nut and leal, cocoanul, &c. and near it are plated a lighted laiwr of incense, and also a buruing lamp scenleil with camphor, all of wbioli are presented to the master along with a smalt sum of money and a tiirband, or some similar present, suitable to the condition of the parent or relation of the child. The rice, flowers, bretic-niit, tcc. are dis- Iiibuted by the master among the cbildicu of the school. TriHing presents arc also made to such Digitized by GoQgle ( ) bralimain ai may all. ntl upon llie occa<-ion. Tlie scholar (hen prosltatcs himself liefore tlr« ^rilitig board, which is supposed (o represent tlie goddess 5'araswali, and the master wrilci / whole numbers. The fint table of this sort is called Poukt, or Table of Quaiteri. ^ ^ one time one quarter is one quarter two times one quarter are one half P - rs - three limes one quarter are three quarters, and so on to 30 or 100 times. The second table is the A’im/.r, or Table of Halves. = SB ~ 2 ^ is-g I one time one half is one half I t«o times one half are one I I three times one half arc one and a half ami so on to 30 or 100. The third (able is the Pounke, or Table of Three Quarters, i one time three quarteia is three quarters J [ two times three quarters are one and a half 2^ three times tliree quarters arc two and aquarter and so on to 30 or 100. The neat tables are those in which the fractions i j J are joined with whole numbers : The first table of this sort iscalled Sawake, or One and a Quarter Table. one time one and a quarter is one and a quarter. 2;. two times one and a quarter are two and a half 3^ thri'c times one and a quarter are three and three quarters and so on lo 30 or 100 limes. The second is called JJirke, or One and a Half Table. 1 i one time one and a half is one and a half S two times one and a half are three 4} three limes one and a half are four and a half and so on to SO or 100 limes. The third is caller! /irizke, or Two and a Half Table.' Si one time two and a half is two and a half 5 two times two and a half are fire 7| three times two and a half are seven and a half add so on to 30 or 100. Another (able called Oioke, or Three and a Half Table, ii also committed lo memory, one time Uixce wul a half u Unco and n lialf Digitized by Google ( US ) 7 two Times time knil a half are Kven ' lOi three limes three and a half ate ten and a half ond so on to 30 or 100. Two other tables then fallow which exhibit the laulliplicatian of whole niimben. The tint, which is called Akarke, or Jiieveu Table, cahibiu tac untllipUcaliun of the auniban from ] 1 to 20 into one another. 121 eleven times eleven are one hundred and (wenfy-ono 132 eleven times twelve are one hundred and thirly-Iwo 143 eleven tbues thirteen are one hundred and forly-tliree 132 twelve times eleven are one hundred and thiily-two 144 twelve limes twelve arc one hundred and lurty-four 156 twelve limes thirteen are one hundred and filty-six l44 thirteen times eleven are one hundred and forly-three 15G Ihirleen times twelve are one hundred and fifty-six 1 69 thirteen times thirteen are one hundred and sixty-nine and so otK Or, reduced to a tabular form it would be thus : T^BL,E named ^K^RKE. 1 " 12 13 14 1 15 1 16 1 17 18 1 ID 20 11 1 121 132 1 143 154 1 165 1 176 1 187 198 1 209 220 12 1 132 144 156 168 1 180 1 192 1 204 216 1 228 240 13 |143 156 169 182 1 195 1 208 1 221 234 1 247 260 14 1 154 1 168 182 196 1 210 -j 224 1 238 252 1 266 280 15 1 165 180 195 210 1 225 1 240 1 255 270 1 285 300 16 |176 192 20S 224 1 240 1 236 1 272 288 1 301 320 17 1 187 1 204 221 238 1 255 1 272 1 289 306 1 323 340 16 1 198 216 1 234 252 1 270 1 288 1 306 324 1 342 360 19 1209 228 i 247 206 1 285 1 304 1 323 342 1 361 380 20 1 220 240 1 2oO 280 1 300 1 320 1 310 360 1 380 400 TIr oiber table is called Ekotri, or Table of Squares. It coaUtuu the uiulliplicaUeaof Digitized by Google ✓ \ ( 149 ) into iteelf, or the square of each figure, from 1 up to 100 ! tha*, 1X1 = 1 S X 2 = 4 3x3 = 9 and so on to lOO X 100 = 10000 After learning to multiply in this manner, the scholar proceeds to the fables of weight* ami measures. The following tables will be sufficient to shew the order and method adopted in the schools, and to render inlelligiblc the examples which occur tn subsequent parU of these remarks. TABLE OF MONEY. 1 quarter dugani 2 dugani 4 1 dugani 8 1 - aiiied, and, ondniwing another horizontal line, set down the units Irelow it. Carry the lens ohliincd to tlie tens in the next column, lieginning at the bottom, and hav- iiig pul down the sunt op|)o,5ile to the lowermost row, add this sum to the lens in the next lippiT row, and op|rosite to it put down the sum. Proceed in this manner through all the rows ro the iipirermost row. 'I'hcn set down the hundreds obtained below the first bori/.oiit:il line, and the tens ladow the secon 1, and, carrying the him Ire Is to the roliimn of hundreds^ add up this column, and pul down the sum under the second hoii/oiilal line. The nuinber which stands uniter this line is the .sum of the multiples. The three following examples will Lc sullicieiil to shew the nature of this o;H'nilion. EXAUFI.ES. 1 43 17 49 29 2 3 42 34 47 58 17 3 6 50 51 11 . . . . 42 67. 24 4 10 54 t>8 12 .34 116 s'i6 5 15- S8 H5 ! 20“ ■■ • 3.3 11.5 1.3:5 21 102 25 29 171 ] .39 7 v8 90 110 l27 ■ 203 42 s i.hfi 19 .136 1 .36 22 2.32 1 1 +* y ; 45 16 153 1 1 42 19 261 45 10 * 11 170 45 13 2<)0 4 44 44 55 935 1595 The scholar is next taught what is called wtza baki berixf (hat is, AOoiTtoiraad idbtbac* Digitized by Google / X ( 15 ' ,) Tiojf, tliesc two operations being taught at the same time, and by tbe lame exanples, in (he loUowing maaner : EXAMPLI. 546283 €21436 953785 192)704 546283 iub(rac< _1375421 remainder 621436 subtract 953785 remainder In (he mode of |>errorming addition nothing peculiar occurs. Subtraction, likewise, is frequently done in the manner practised in Kuropc; but (he most common method ubya process of addition, according to the following rule : kV hen (he figure in the minuend line is greater than the corresponding subtrahend figure, ^<1 to thesunralienl a number such as shall make it equal to the minuend; and set down Wow (lie subtrahend the numW thus added. M'hcn the figure of the minuend is less than rorre-tpom inj one of (he subtrahend, add ten to the minuend, and also add to the sub- tia.iend a nuo.her such as shall make it equal to the minuend thus increased. Then put down and carry one.to the nex^tsubUa. XXAHPr.B. 2135675 J32I692 1 1 J39S3 ep'Tirf’r^ ""'J V"* » ""d 8 '‘re 17; put down S, and 1 i, obtain 3 are 5; pul down 3; 1 and 2 are 3 ; put down 1 ; 1 and 3 are 4 ; put down 1; 1 and 1 are 2; put down 1. Therefore 1113983 is the remainder. ^ The following is au example of compound addition and subtraclion. Digitized by Coogle ( >52 ) EXAMPLE. qn. annas* 6S4395421 .. 3 .. 2 435S62C43 .. 2 .. 3 C3452I9 .. 1 .. I 112()b032ivt .. 3 2 4)84395421 .. 3 .. 2 subtract 442207893 .. 0 .. 0 remainder 4358li2(>43 . . 2 .. 3 subtract (>345219 .. I .. 1 remainder Gunakar Riiagakar, or Mfl.TiPUCATioM and Division, arc also taught (oge(bcr, as ill iho ca.c ol aJdiuuii and suUUactiuu. In looking at the Ibllovriog example, il should be re- collcvtctl, (liAl the Mubratta niuUiplication table exiciids to 30, and the Gutcratti one to 100; lieiicr, as they can niuliiply by any number bcluir 30, wiliiout separating the figures of the iiiuliiplier iiiiu their places, the result of the operation is exhibited in one line. ]t nill bcobscrv< ed that the multiplier is put don u indiscriaiilulely any where below the multiplicand. EXAMPLE. IdJliCTbinO 15 multiplier lS51»51Sl>bJ0 product 15 divisor l'd^5b7S910 quotient ^Vhen the multiplier is a high numlN’r, the operation is performed in the same way as in lCiiiu)K, piitliiig a mark, however, abore cacli figure in the multiplicand when it is muhipli- e4 and so on. Eacb fi^re in (be muUipIIcand, svlien mnltiplii'd, being marked with a dot, there occur, in this instance, three rows of clots, (wo of which arc above, and one below, tlic mul* (iplicand. The three lines of figures on the right hand are the lens obtained during the mul- tiplication, and are pul down in this manner to assist the memory. The product of (ho multiplication being afterwards taken as a dividend, an I the former multiplier put down any vrherc below it as a divisor, the dividend figures arc marked successively with adotwhmmul. tiplied by the quotient. In tbe case of fractional numbers, the mode of mullipllcition is as follows, tb« muItipliK being 14 rupees, 3 quarters, and 3i annas. EXAMPI.K. 1231i><)789 14..3..3J. 17l«.‘j95046 qrs. ’ 6l7i«394 . . 2 30S(>4197 .. 1 annaa 15432098 . . 2 . . 2 7716049 .. 1 .. 1 « 3858024 . . 2 . . 2* 1929012 ..1 .. H 1849922822 . . 2 . . 2* product 14 ..3 ..31 divisor 123456789 quotient ' Digitized by Google ( 154 ) Thf operation in regird lotlie fractional part* islrrnieJ fdmki, or lialvine, and h fonnded on the regiiUr qnadia^e division of the (ahles of money, weislils, and mejsuies. Tan* the multiplicand being halved afluids 2 qrs. out of the 3 qts. qrs. 61728394 . . 2 Thii again is halved, because the remaining 1 qr. is the half of one half 5()!>etl97 .. 1 This being once more liilvcd affords 2 annas out of (he 3 ann^ 13132093 . . 2 . , 2 Which is again halved, because (he remaining anna is half of (wo annu 771G04D .. 1 .. 1 This again being halvvd yields half an anna out of the 3 qn. of an anna 3503024 . . 2 . . 21 And this is halved, because one quarter anna is half of half an anna 1929021 1 ..II The fractional parts Ireing divisible into fourth parts of the whole, and of each other, this mode of o|M‘ni(iori is l>olh the must c*i>ediiious und least liable to error; os it saves the IroubW vf multiplying by the truclioiial parts, and dividing by a large divisor. CoMPooMD UivitioN is performed in this manner: ■\Vrilcthc divisor any where I>elo\v the dividend, and draw a hne; then find how oflen lh« divisor is contained in as many places of the dividend as nie just iieeessnry, and, having set down these places on one side of the Isoard, multiply the divisor by this number, and subtiaet the product frn.n tliasc places ; to the remainder bring down the ttcxl figure of the dividend; also, if the lemninder contain fr.ictions, tbatjs quarters, annas, or fractions of an anns, multiply (hem by ten, and add the product to die number formed by bringing down the fiiiire; (hen divide tliis niimljer so iiicic.iscd, ia the same manner as iK'forc, placing the qiiotiint under a liurizuntHl line drawn below tbe divisor ; and proccad iu this nutnuer till all (lie figures in the dividend are used. ExajirLB. qr. annu dividend 184D922S22 . . 2 . . 2J divisor ■ 14 ..3 ..34 quotient • • , ■ • • 12343b7b!l Digitized by Google ) ( 155 ) 1* .. 3 .. Si- X 1 JS 11 .. 3 .. o} 3 .. t • * 1 4 brougtil down i> ansa rr- ai{kii‘«4T X 10 Si SI .. o» • t • - J It X 2 li y S 1 2 anaai 3 X 2 1 2 aona i X 2 4 • 2.0 .. 3 .. 3i 51 or 2.0 .. 3 Mr • • 4 . • .. 3 9 bronglit down ii annat maindrr J X 10 .. 3 5 50 .. 3 Q 11 X 3 Ty 3 .. 1 aaaia 3x3 2 .. 1 »nna i X 3 *'« .A 44 .. 3 •• 3i 50 .. 3 3 41 .. 3 .. Si 5 ,, 3 . , and to on diviJeod. Digitized by Google ( 156 ) Whffn fliew funJamenlal rules are tlioroiiclil/ undors(om), the instructor gires a few ex- amples of the Rule ol Three, and of lntere;t. Nospeciil rule of proportion isgiven; but any question, fur iii'slance, what is the amount of wasti for 4 inonlhs and 10 days, at 15 rupees per month ? is answered in this manner : The wages of 4 months are 60 ; — then to find the a iioiiiit for the frnelien of the month the ua;e, for 15 days, or half a month, are llic half of 15, or Tj ; then for 10 ilays, which are tsro thirds of half a mouth, the wages are f of 7i, or 5 rupees. Or they would say at ouce, the wages for one month being 15, the wages for 10 days or 4 of a month are ^ of 15, or 5. One school-master, however, gave me (he following rule of proportion, which he was pleas, ed to call a J.ilawati rule. Having nutUiplied the time by the wages, halve the product, and divide the remainder by fifteen. If iliere he any remainder after this division, reckon it annas, a lixaroplc.— What wUl 12 dayi wages amount to at the rate of 10 rupeta per nionihl 12 lime 10 waget 120 halved 60 15 divisor 4 quotient Altogether, the wages for 12 days amount to 4 rui>ees. * I do not know why the remainder i> reckoned aims.. Be.ulr., it does not hrina out the ai.iwer ex- actly. In the question, what will It days wages amount to at 7 rupeca per moiitli, the auawer is not 2 .. ‘81. bo, 2 .. 9A- , 30 J 7- I'i . . (iO 17 1 8 qrs in a rupee 30J 272 270 2 The perton who gave me the rule Hated very gravely that this difference was of no consequence, it being the practice of the Mahratta kaskuna, or clerks, to pay lha.e employed by their maaleia i. whole uunbers only, and to rclaiu the fraclioua iu their own pockets. Digitized by Google ( 157 ) Example. Attbe rate «f 7 rupees per month, what will 11 da^s wages amount to t 11 days 7 Diunlhly wagea 77 halved 38 .. f 15 divisor 8 .. 8f annas quotient Altogether, two rupees, 8 annas and a half. RULE OF INTEREST. Multiply the turn by the time, and after throwing out the last figure of the proiluct, take the third |>art of the remainder, A and call it kache. Example. — If the sum of 120 ru|vei Ire lent at i per cent, monthly interest ; what will the inteiest amount to in 4 montha and 10 days. 130 time ISO principal sum 1560« 530 third part of 1560 The money being at { per cent, (he interest of 500 rupees is five quarters; and that of 90 dayt it four fifths of an anna for it 100 give 1 qr. monthly interest (ben 50 .... 9 annas do. 85 .... 1 SO .... Having gone thro’ these different rules, the scholar proceeds to learn the method of keep- ing books of accounts, and also to read letters and papers written iu the common Maliratta or Guzeralti character. In books of accounts they employ marks for addition and subtraction. These marks consist of placing at the left hand end of a line drawn beneath the sum to be added or subtracted, the firat letter of the word which signifies addition or aubtractiua ; thus, 83456 34367 a denotes that these two numbers are to be added ; and a This is ibe same as dividing by 30 ihe miinbcr of days of ibe month. a I ke ntiive schoolmaster stated 4 of an anna as the inicrfsl of 20 days. If the reault ba true wilbtn one fourth of an anna, ibey do not seem to aim at a more minute degree of accurary. ^ Digitized by Google ( 153 ) Si567 denotes dial diose numbers an: lo be siiblraned. They filso employ a m:>rk for or receipf, wliich consuls of (wo ttrai'hl line* nniied iti die iiiiil.ltc by a curve. — ll bos b< eu ulreaiiy rcmaikeJ ia tlie iulioJuaion, lUal a circle ia tiseU lu drimle olitileraiio'i ; diiu, 2 I'oumU of sp'.il |»ca4e .... O G i’oands of rire 1 Tlie circle around die number I, in d’e first case, and die circle placed opposi>e (h« IJ'imbcr G, in die m-cond, skew Uwt t'le i'e.n li.ns been disclmrjed. No braneli of messi’iiatio.v i« tmarbt in die schools, lUo* one sclioolmaater fumisbelme With (wo eX'iiiiplei of the measure nent of Gelds, lu order to unJcrstuiJ them it is necessary ta toco.lcct tlic loUuwiii^ table t 5 bicidthsof die baud 1 ruliit G cubits 1 kali 20 kali 1 paod 20 p.iiid 1 Li»:ih £sample 1. — Required the a ea of a field 20 AtUi iu breadth and SO kali in IcnglL? SO 20 600 kali 20 divhsor — kali 30 qiiodciit — pond 20 divisor — pand Ij quotient — bigah In Ibis example, which is a regular figure, tlie area is found bj multiplying the Itngth by the breadth, Kxample 2. — The breadths of an irregular figure, ai three egmdutOHt placet,* being C, 10| 6 kaii, and the whole ieiiglh 25 kali ; required the area? 10 .8 6 24 sum of the different breadth*; and 24 3 = 8, mean breadth i * 'ihe words in Italics are supplied to render the cxsoiple more inlclbgible« Digitized by Google SI lenstli 8 ii.cati brendlb r 9 )■ SOO Inti SO divisor— /a<» JO <]iiotieiil — pnnj, or J Idra'i THs exnmple i» do:ie l\v adding nil tlic ncriwndlrulnr lirfndllis (ojreilifr, rn,! dividing Oicir sum bv tlic iiuinlKT ol' (licm lor llio mean btcadlb, by wliitli is niidli|>licd ll.c leii"lb. Ti e product in tliis case litu’ rot exact, is n iic.tr ajipruxin'.aliuii to the triilli. Sonirtbiiiv like nlial, in Ibe l.iiatrali, is c.illu.l llie riitc of nssume.l number, is a'so occnsi. ondly Iniijlit as a kind of iiiidiiinlicitl tiJ.ilc. 'I'lie sibnlar is de-iicd to lliiiik of, or a-Mm;e any number, then niuliiplr it by taree, nii 1 after lialviK!; Il-.e proilurt, reralions tbe scholar sjvaks in , a loud s.nging tune. An liuiopc.tn would naturally suppose that this pra-. tice must piodin e great coiifusatn, an I distract the mind of eacli scholar. In the Hindu schoohi, however, it doe> not iccii to have this cfTcci ; but, on the contrary, this auddde repetititm appears to keep bp the scholar’s ntd-nlion, and to fix his min 1 firmly on the suhjcct about which he is employ- ed. It also idfottis the teacher means of observing when any one is idle or inattentive to hit Irson ; and by connecting the sound with the thing sigiiilied, the calculator may ix-rform the operation by a kind of mechanicnl process, llc^idcs, it takes aivny tbe idea of mental ex- eition, and converts llie exercises at scbool into a kind of play and amusement. llrlore the .«cholars are dissmisied in the evening, it is usual to repeat thcdilfcreiu muliipU- cal on tables in tiie follow iiig manner ; All the. sc liolars stand lip, when one of (hem, by directions of the master, takes his sta'ion in In nt, anil goes thro’ the different tables with a loud voice, all the other schubars reix-ating Biter him at once. The boy who is the greatest proficient is generally chosen to lake the lead ; blit at other limes the master selects one of the younger boys, in order to ascertain whcltier li-- lie able to go thro’ the tables with accuracy. This proves no small incentive to « ich boy to I make himsell master of these tables, as any failure in this conspicuous situation is accompanied With gre.it disgrace. The luuliiplicatiiin tables being thus daily repealed are fixed indelibly on the mind of the schola.' ; and in this way he acquires a facility of jierforming arithmetical o|>eraliuns ofi' hand, yvhich frequently astoiii.shes an European observer. For instance, I have often he.ird a scric* ul pretty intricate questions, involving fractions and the liule of Three, pul to half a do. leu of boys, ooo question being put to the first boy, another to the second, and so on i« Digitized by Google ( >60 ) •ucrwuion ; and by the lime that a qiieslion had been given to the last boy, the first boy < wcmbi answer (1 e ono wi icli liad been |int to him, iinmodintely after which the sceoiid boy wonkl answer his qiieslion ; and thus it went thro’ the whole; so that in the course of two miniit.s, six diflerent questions would be put to as many boys, and answered by them wicli the utmost rorrectne'S. Tliechildien leirii to write and cipher on aboard covered with sand or brick dost, and the letters or figures are traced with arced, or small wooden style, wliicli the scholar is permitted to hold in whatever msiiiier he finds must conrement. In the inure advanced stages, however, and when tliearilhiuetical operaiions exieiiil to suine length, 1 have observed in ilic sclinoU here, lliat they paiul the board with a black ground, anil tlirn write upon it with a mixture of chalk and water. This occupies much Ics-. room than in writing iipo.i s;irtJ, is less liable to ubhteiulion, and at the same time shews the ligiiri-sin a plant and lii-tinct form. lu the system of cdiiciiiion thus biiefiy det.iiled, s-i era 1 very juditioiis arrangements will be noticed, both in regard to ecunomy, and as to saving of liiiic by writing upon a board covered with sand, llietc is saved the cx|>ence of paper, ink, and pens. Secondly, writing .and reading are taught together, instead of being matic different branchra of insi ruction. While tracing the forms of the letters or figures, the scholirat Ihc same time icpeals Uicir names, u piactice which i. followed also wbni he proceeds to ciphering. Thirdly, Uie scholar Is taught the edecl of placing one or more figiiies before another, and thus learns to duliiiguish between the ualuie of this (losiiiuii and the result of adding numtiers together, a distinction whicli often punlei beginners to whom it has not been carefully puim. cd out. Hut what chiefly distinguishes the Hindu schools is the fi.ak of instroctiok by raa aciioUARs Til Kwss;t. visa. Wlicn a Uiy jo.ns the school, he is iniinedialely tml under (he liii- liuii undcaie ot one who is iiiine .ulvunced in knowleilge, and whose duly it is to give Ibsoiit to Ins young pupil, to assist him ni Itaiiiiug, and to repoit his behaviour and piogio* to the m isler. The scholars aie not classed as with ns, but aie gi iierslly paired off, eacli pair cun* sisiiiig of an instructor and ii pupil. These ['airs are so arranged that a boy less advaiii ed m y til nest to one who has made greater progress, and from whom he receives assistance and iiislruciiuii. When, however, several of the elder boys have made ccmsiderablc and nearly eipisl progress, they are seated together in oi.e line, ami leccive their instructions diteclly from the master. ' This plau of getlii'g (heolder boys, and Ibose who are more advanced, to assist those who are less advanced and younger, greatly lessens the hiirdrn imposed ii|Hm the master, wliosa duly, according to tliis sys'cm, is not to furnish imtruetion to each iiulividual scholar, but to su|>ei intend the whole, and lee that every one do"S his duly. If the younger boy does not learn his lessons with sufliciint promptitude and exactness, his instructor rcjvorls him to the aaasitr, who enquires into the case, order* the pupil to repeat before him what he has kamt, Digitized by Google ( 101 ) and puniihes him if he hu been idle or negligent. As the misfir usmiljr gires lesions tO the elder scholnrs only, he hat suiricient leisure lo exercise a vij^ilaot su|>erinlendaiice Over the whole school, and by c;i' ripline, and keeps every one upon the alert thro’ expsclauon of being calL-d upon to repeat bis lessen. The arithmelicil lessons are written down at fall length. Thus in giving a cue of addU tion, siibtriiciion, mnlliplicatioA, division, or the rule of three, the wlinl? process is set down ill figures, an I the scholar goes over it on atiolher part of the bo ird, repealing the dir* ferent steps in a loud voice us has been alretdy noticed. After each lesson h is bee i gone over till it be committed lo memory, it is rubbed out, and then wriUea down by the setidUr luua* Mlf without any assistance. *s GiOdro Digitized by Google Digitized by Google Digitized by Google Plate !BL. Digitized by Google ■y.r\ I Plate V Litnfi Fij^. *9. ' I I i Vutaun .f Fit:. N Litm/t Fix: .5/ #.w> W Digitized by Google Digitized by Google Digitized by Google Digitized by Google Digitized by Googid