THE PANCH AS1DDHANT1KA THE ASTRONOMICAL WORK oh VARAHA Mill IRA. ^ THE TEXT, EDITED WITH AN ORIGINAL COMMENTARY TN SANSKRIT AND AN ENGLISH TRANSLATION AND INTRODUCTION BY G. THIBAUT, PH. D. A NO mahAmahofAdhyAya sudhAkara dvivedi. PRINTED BY E. J. LAZARClhmr CO., AT THE ffflSfcAL HALL TRESS, BENARES. 188 !). TO F. MAX MULLER K. M. A TOKEN OF ADMIRATION AND REGARD. PREFACE. There is some reason to fear that the feeling of any one who may examine in detail this edition and translation of Varaha Mihira’s astronomi- cal work will, in the first place, be wonder at the boldness of the editors. I am indeed fully conscious that on the imperfect materials at our disposal an edition in the strict sense of the word cannot be based, and that what we are able to offer at present deserves no other name but that of a first attempt to give a general idea of the contents of the Pahchasiddhantika. It would, in these circumstances, possibly have been wiser to delay an edition of the work until more correct Manuscripts have been discovered. Two consider- ations, however, in the end induced us no longer to keep back the results, however imperfect, of our long continued endeavours to restore and elucidate the text of the Pafteh asidd h an ti k L In the first place ve were encouraged by the consideration that texts of purely mathematical or astronomical con- tents may, without great disadvantages, be submitted to a much rougher and bolder treatment than texts of other kinds. What interests us in these works, is almost exclusively their matter, not either their general style or the parti- cular words employed ; and the peculiar nature of the subject often enables us to restore with nearly absolute certainty the general meaning of passages the single words of which are past trustworthy emendation. And, in the second place, we feel convinced that even from that part of the Pancha- siddhantika which we are able to explain more is to be learned about the early history of Sanskrit Astronomy than from any other work which has come down to our time. Imperfect and fragmentary as text and translation are, we may assert at any rate that, in our endeavours to overcome the quite unusual obstacles, which the corrupt and bare text of the Panchasiddh&ntik& opposes to the interpreter, we have spared no trouble. The time and thought, devoted to the present volume, would, I may say without exaggeration, have amply sufficed for the editing and explaining of twenty times the amount of text presenting oidy normal difficulties. This I mention, not of course in order to extol what we have been able to do, but only as an excuse for what we soe ourselves obliged to leave undone. Next to the lamentable state of the text as appearing in the two Manu- scripts at our disposal, the greatest disadvantage under which we laboured was the absence of a Commentary. Commentaries can be hardly done with- out in the case of any Sanskrit astronomical work ; much less so, when the text, as that of the Pahchasiddhantika, describes many mathematical pro- VI PREFACE. cesses more or less diverging from those commonly employed. Commentaries probably existed formerly, and possibly exist even now ; but we have failed to procure any. The Commentary published in the present volume is an entirely original composition by my Collaborator. A mere translation of the text with notes would, indeed, have sufficed for the European reader ; we however, wished to make the results of our labour accessible to Pandits also who understand no English. And a full tika giving full demonstrations in the ordinary Hind A style will, in many cases, be useful to the European student also. The right hand columns of the text give the emended text ; the left hand columns the text of the better one of our two Manuscripts which we thought advisable to exhibit in extenso. Some remarks on the Manuscripts and the mode of emendation of the text will be found at the end of the Introduction. As this preface is signed by myself only, I may, I think, here acknowledge — in a somewhat more explicit way than the mere association of names on the title page is capable of doing — the great obligations under which I am to my collaborator Pandit Mahamohopadhyaya Suclhakara Dvivedl. His constant assistance was altogether indispensable to me, and all the more welcome as among the Jyautishas of my acquaintance I know of no other, fully equal to work of this kind and at the same time equally ready to devote himself to a task which in certain aspects is so entirely unremunerative. I may express the hope that the Pandit, who is already so well known for his efforts to spread a knowledge of modern higher Mathematics among his countrymen, will continue to devote a part at least of his learning and talents to the elucidation of the ancient history of science in this country. I further wish to express my best thanks to the Bombay Government and to Professor R. G. Bhandarkar, who with great liberality have allowed me the use, for lengthened periods of time, of all those Manuscripts in their charge which I required for the present edition. Nor must I omit to record my obligations to Professor G. Buehler to whose activity, when in charge of the search for Sanskrit Manuscripts in parts of the Bombay Presidency, we are indebted for the discovery of the two Manuscripts on which this edition is based. AllIhabJLd : 16 th December , 1888 . G. THIBAUT. INTRODUCTION. The PaiichasiddhAntikfi, by VarAba Mihira occupies a marked position of its own in Indian astronomical literature. As a rule works treating of that branch of science claim either to be directly revealed, as/, i. the Siirya SiddMnta in that form which has come down to our time ; or else to base in all essential points on some older work of divine origin, as / i. the Siddhantas by Brahmagupta and Bhaskar&cluirya, both of which are reproductions, however greatly amplified and improved, of an old Paitamaha Siddhanta. One of the consequences of this is, that these works claim for themselves direct or derived infallibility, propound their doctrines in a calmly dogmatic tone, and either pay no attention whatever to views diverging from their own, or else refer to such only occasionally, and mostly in the tone of contemp- tuous depreciation. The latter attitude is assumed f. i. by Brahmagupta who indeed devotes a special chapter to the task of reviewing those astronomi- cal systems which were opposed to the teaching of the Brahma Siddhanta, but who would have rendered that part of his work much more valuable and interesting, had he been less anxious to criticize and ridicule than to impart information. The astronomical writers, it is true, therein only exemplify a general mental tendency which displays itself in almost every department of Hindi! Literature ; but mere dogmatic assertion appears more than ordinarily misplaced in an exact science like astronomy, and the absence of all appreciative reference to the views of preceding authors is particularly vexatious, when we have to do with a branch of Hindi! Learning which shows clear traces of having been remodelled under the influence of Greek teaching. To the general rule the PaiichasiddhAntikA forms a striking exception. As far as we can judge at present, Vara, ha Mihira was the only one among Hindi! writers on astronomy wlio thought it worth while to give an exposition of all the more important forms of astronomical doctrine which were current at his time. Not that he was unable to judge of the relative value of the systems which offered themselves to his examination ; for, as we shall see further on, he knew very well in what order of merit the five Siddhantas whose teaching he summarizes are to be arranged. But he seems ready to acknowledge that even inferior systems deserve a certain amount of atten- tion, as long as they continue to occupy in certain circles a position of authority ; and he appears not to be altogether incapable of taking a purely intellectual interest in examining the various, more or less perfect, methods which may be applied to the solution of scientific problems. At the same time he seems to have no hesitation to acknowledge the connexion of the VIII INTRODUCTION. modern phase of Hindd astronomy with Greek science. Although not directly stating that the Hindds learned from the Greeks, he at any rate mentions certain facts and points of doctrine which suggest the dependence of Indian astronomy on the science of Alexandria ; and, as we know already from his astrological writings, he freely employs terms of undoubted Greek origin. The Panchasiddlidntikd thus becomes an invaluable source for him who wishes to study Hindi! astronomy from the only point of view which can claim the attention of the modern scholar, viz. the historical one. Regarding its form the Panchasiddlidntikd belongs to the class of the so-called karnnagranthas i. e. compendious astronomical treatises which do not set forth the theory of the subject at comparative length as the Siddhantas do, but merely supply a set of concise — and often only approxi- mately correct — rules which suffice for the speedy performance of all the more important astronomical calculations. It however contains a few chapters whose contents lie outside the limits of a mere karana and resemble the cor- responding chapters of the best known Siddhantas ; notably the chapter which describes the general constitution of the universe, and the 15 th chapter called Jyotishopanishad. And it of course decidedly distinguishes itself from all ordinary karanas by the fact that it does not base on any one particular Siddhdnta, but undertakes to reproduce the more important doctrines of five different Siddhantas. These five Siddhantas, named by Vardha Mihira in the first chapter, are the Paitamaha, Vasishtha, Romaka, Paulisa and Saura Siddhantas. Vardha Mihira there also states his view as to their order in importance, assigning the first place to the Sftrya Siddhanta, placing next the Romaka and Paulina Siddhantas as about equally correct, and declaring the two remaining works to be greatly inferior to the three mentioned. In agreement with this estimate very different amounts of space are allotted to the individual Sid- dhdntas in the body of the work, the Shrya Siddhanta and Paulina Siddhdnta being treated at some length, next to these the Romaka, and very little atten- tion being paid to the Paitdmaha Siddhdnta, and, although this is a point somewhat difficult to decide, to the Vasishtha Siddhdnta. In addition to the general character of the five Siddhdntas, this differ- ence of treatment is owing to a special cause, mentioned by Vardha Mihira in the first chapter viz. his wish to devote the Pahchasiddhdntikd chiefly to the task of setting forth the calculation of solar eclipses, the most difficult problem attacked by Hindfi astronomers. The Paitdmaha Siddhdnta at any rate was altogether incapable of furnishing any rules to that end ; and so perhaps also the old Vasishtha Siddhdnta. INTRODUCTION. IX I now proceed shortly to discuss the teaching of each of the five Siddhantas as represented by Varaha Mihira. This, however, requires the preliminary settlement of two questions. In the first place we must attempt to ascertain with accuracy which chapters of the Panchasiddh&ntikd are devoted to each of the five works in question. — This is a task beset by considerable difficulties, as we have no commentary to assist us, and as the indications to be met with in the text as well as in the colophons of the chapters, as exhibited by the two Manuscripts at our disposal, do not, in all cases, enable us to arrive at definite conclusions. I begin with those chapters, fortunately constituting the majority, which allow themselves to be referred to their respective sources with confi- dence. — The very short twelfth chapter is, in its colophon, called Pait&maha Siddhtlnta, and is in its first stanza declared by Varaha Mihira himself to base on the teaching of Pitamaha ; it is the only chapter in the whole work which is concerned with that Siddhanta. — The eighth chapter treats, according to its colophon, of the calculation of solar eclipses according to the Komaka Siddhanta ; and that this really is so, we again have no reason to doubt, as the first stanza refers to the Romaka by name, and as, moreover, the contents of the chapter agree with the statements made in the first chapter about the yuga and the ahargana of the Romaka Siddhanta. — The ninth, tenth and eleventh chapters undoubtedly summarize the doctrines of the Shrya Sid- dhanta, as is stated in the colophon, indicated in the first stanza of chapter IX, and borne out by the general agreement of the contents of the three chapters with the Sfirya Siddhanta as known at present. The sixteenth chapter contains, according to the colophon and to stanza 1, the rules of the Sftrya Siddh&nta for finding the mean places of the planets ; and the seven- teenth chapter which teaches how to calculate their true places we may without hesitation refer to the same Siddhanta. Among the remaining chapters of the work I at first single out those in which Varilha Mihira apparently does not intend to reproduce specific features of one particular Siddhdnta, but rather to summarize doctrines held by all the more advanced astronomers of his time, and most probably set forth, with greater or less variations, in three of his five Siddhantas, viz., the Shrya, Paulina and Romaka Siddh&ntas. To this class of chapters, in which we discern more of the individual Var&ha Mihira than in the remainder of the work, I feel inclined to reckon three or perhaps four sections. In the first place the thirteenth chapter, designated in the colophon as ‘trailokya- samstMna \ which gives a popular exposition of the sphericity of the earth and the different aspects of the celestial sphere which are due to difference of X INTRODUCTION. terrestrial latitude. The mode of treatment of these questions is no doubt Varaha Mihira’s own, as also the interesting criticisms passed on some astronomical schools. In the same way the fourteenth chapter, which is chiefly engaged in showing how certain results may be obtained not only by calculation but more directly by observation and the inspection of certain mechanical contrivances, appears, on the whole, to be Var&ha Mihira’s own, although the more scientific of his five Siddh&ntas no doubt treated of those topics in a similar manner. The same remarks apply to the fifteenth chapter which is even more distinctly individualistic, and contains interesting re- ferences to other astronomers. I am more doubtful about the position of chapter IV. which in the colophon is merely counted as such, without any special designation. The matter of the chapter corresponds to what in the best known astronomical works is set forth in the so-called tripra3nMhikara, with the addition, however, of rules for calculating the table of sines (which ordinarily are given in the spashtddhik&ra). It is not improbable that here also Var&ha Mihira sums up, in his own fashion, whatever he found of value in the corresponding chapters of the Romaka, Pauli 3a and Siirya Siddhan- tas. On the other hand, as the fourth chapter follows and precedes chapters specially devoted to the Paulina Siddh&nta, it is not impossible that its contents are meant to sum up the teaching of that Siddhanta only. The decision in this case is however of no very great importance, as the rules given in the fourth chapter on the whole closely agree with the general Siddh&nta doctrine. Among the chapters not yet discussed we first notice the sixth chapter which the colophon states to treat of solar eclipses according to the Paulina Siddh&nta. I see no reason for rejecting this statement ; for although the text of the chapter itself does not refer to the Paulina Siddh&nta, it most probably actually bases on the teaching of this latter work, since the two other chapters (VII and VIII) which teach the theory of solar eclipses cer- tainly refer to the Sfirya and Romaka Siddh&ntas. From this again it follows with great probability that also the sixth chapter, which treats of lunar eclipses, represents the teaching of the Paulina Siddh&nta ; and if so, then likewise the fifth chapter merely designated as S'aiidarsanam. These assumptions are confirmed by the fact that these three chapters treat only of the calculation of eclipses in the narrower sense, to the exclusion of all preli- minary operations, such as the ascertainment of the mean and true longitudes etc. of sun and moon, so that an introductory chapter setting forth those latter topics is required. Now, a chapter of this nature is supplied by the thir d one of the Panchasiddh&ntik& which gives rules for finding the mean and true places of the sun (and of the moon ?) and for similar operations, and INTRODUCTION. XI which, in its colophon at least, is said to represent the teaching of the Paulina Siddh&nta. The relation, however, of the third chapter to the one immediately preceding is puzzling. The second chapter is, in the colophon, merely desig- nated as “ nakshatr&diehchheda,” but its contents comprise firstly a rule or set of rules for finding the mean (and perhaps also true ?) places of the moon (stanzas 1 — 7), and, secondly, a set of rude, approximative rules for calculat- ing the length of the day at an} r time of the year, the length of the shadow of the gnomon, and, from the latter, the mean place of the sun, and the lagna (and vice versa ; stanzas 8 — 13). The chapter concludes with the words “ This is the (calculation of the) shadow according to the concise Vasishtha Siddh&nta.” The question now is, whether this whole chapter has to be viewed as epitomizing the Vasislifcha Siddh&nta, or whether that work is represented only by its latter part. The rules contained in stanzas 8 — 13 are of a very rough character, and can, for that reason, hardly come from the Paulina Siddh&nta ; their character, on the other hand, agrees very well with the criticism passed by Vara ha Mihira, in the first chapter, on the imperfec- tions of the Vasishtha Siddh&nta. It is more difficult to arrive at a conclu- sion regarding the rules embodied in stanzas 1 — 7. If they do not belong to the Vasishtha Siddh&nta, it would follow that the Panchasiddh&ntikd, which after all promises to render us acquainted with the doctrines of all the five Siddhantas, however imperfect some of them may be, does not even inform us how the place of the moon is calculated according to the Vasishtha Siddh&nta, while it yet gives the. corresponding rules from the, certainly not more ad- vanced, Pait&maha Siddh&nta, very concisely indeed but yet with sufficient fulness. On the other hand there appears to be some reason for tracing the rules to the Paulina Siddh&nta. The third chapter, which, as we have seen above, we may connect with the Paulisa Siddh&nta with a very high degree of probability, gives in stanzas 1 — 3 the required rules for finding the mean and true places of the sun, and then continues, in stanzas 4 — 9, to give cer- tain rules about the moon. Now these rules have unfortunately remained obscure to us ; but yet so much appears certain that they are somehow connected with the rules concerning the moon given in the former half of chapter II, constituting, as it seems, a kind of continuation, or more accurate version of the latter. But again, on this latter hypothesis no reason is apparent why the two sets of rules should be separated from each other by the altogether heterogeneous matter treated of in the latter half of chapter II. I therefore see myself obliged to leave this point undecided, and only wish to suggest, as a third not impossible alternative, that the method for calculating the places of the moon which is set forth in chapter II belonged, in its essen- tia] features at least, to the Paulina as well as to the Vasishtha Siddh&nta. XII INTRODUCTION. and that stanzas 8-13 of the third chapter add certain details which were peculiar to the former of the two Siddhantas. It is greatly to be regretted that the introductory stanza of chapter II, which possibly would throw some light on the position of the chapter, has remained altogether obscure to us. There now remain for adjudgment only the first and the last chapters of the Panchasiddhantika. The latter I shall discuss further on. The position of the former is altogether clear ; it contains, subsequently to some intro- ductory stanzas, a rule for calculating the ahargana according to the Romaka Siddhanta, an exposition of the principles according to which the intercalation of lunar months and the omission of lunar days are managed in the Paulina, Romaka and Surya Siddhantas, and finally a set of rules for calculating the so-called Lords of the year, month etc., which rules were most likely given in each of the three Siddhantas last mentioned. The second question, which must be touched upon before we can review the teaching of the individual Siddhantas, is whether the Pahchasidd h&n tik & represents the teaching of the five astronomical works, on which it is pro- fessedly based, with absolute accuracy, or rather allows itself certain modi- fications of the doctrines summarized. This question is one of considerable importance ; for before we have settled it one way or other, we are unable to judge of the historical position of the five Siddhantas, and to compare the account, given of them by Yardha Mihira, with what we know about them from other sources. We have, in this part of our investigation, to occupy ourselves almost exclusively with the Sfirya Siddh&nta, because that treatise is the only one of the five Siddhdntas which has come down to our time, and thus allows of our comparing it with what Varaha Mihira tells us about the Sftrya Siddhanta as known to him. Now a cursory survey of those chapters of the Panchasiddhantiksl which treat of the Shrya Siddh&nta shows at once that the treatise of that name known to Varaha Mihira agreed with the modern Sftrya Siddhanta in its fundamental features. The methods of the two treatises are essentially the same and, on the other hand, sufficiently different from those of the other Siddhantas summarized by Var&ha Mihira, to ensure to the Sfirya Siddh&nta in its two fold form a distinct position of its own. At the same time we cannot fail to notice that in certain points the teaching of the old Sfirya Siddh&nta (by which name I shall, for shortness sake, designate the Sdrya Siddh&nta known to Var&ha Mihira) must have differed from the correspondent doctrines of its modern representative. If we, for instance, observe that the old Sftrya Siddhanta assigned to the mean diameters of sun and moon the values 32' 5" and 30' 54" (P. S. IX. 15. 16), while 32' 3."6 and 32' are the corresponding values according to the modern INTRODUCTION. XIII treatise ; or if we notice the values assigned in XVII 1. 2 to the epicycles of the apogee which altogether differ from those stated in the modern Shrva Siddh&nta ; we are driven to the conclusion that in these and similar points the treatise used by Var&ha Miliira really differed from the modern one known to us. For we are altogether unable to imagine any reason why Varaha Mihira should have changed, in the details referred to, the doctrines of the book which he aims at epitomizing. There is however a series of other cases in which the decision is not quite so simple. While, as remarked above, the mathematical processes prescribed in the old Siirya Siddhanta agree on the whole with those of the modern treatise, it at once appears that Vartllia Mihira whose intention it is to write a karana considers himself entitled to represent the teaching of his original in a somewhat condensed form, facilitating the quick despatch of the required astronomical calculations. What he f i. says, in the first chapter, about the yuga of the Siirya Siddh&nta, clearly is an abbreviated statement of the corresponding doctrines of the old Siirya Siddhanta, and we therefore have no reason to doubt of the old Siddhanta, as well as the modern one, having taught that 4320000 years constitute a great age, and that one thousand such great ages go to a kalpa. The fact is that for all the merely theoretical part of a Siddhanta there is no room in the karana, and that hence the latter does all that is required if, instead of describing the great periods of the world, it stales the smallest possible aggregate of years comprising an integral number of lunar months and natural days. So far we have no reason to hesitate in accepting Varaha M'hira’s statements as a faithful, though somewhat modified, rendering of the moaning of the old Siirya Siddh&nta ; the question however assumes a somewhat different aspect when we compare the number of natural days contained, on the one hand, within the mahayuga of the modern Slirva Siddhanta, and, on the other hand, within the corresponding period according to Varaha Mihira. The modern Shrya Siddli&nta teaches that a mahayuga of 4320000 years comprises 1593336 intercalary months and 25082252 omitted lunar days, whence it follows that the number of silvan a days contained within the same period amounts to 1577917828. Varaha Mihira on the other hand, following his Siirya Siddh&nta, states that a period of 180000 years comprises G6389 intercalary months and 1045095 omitted lunar days, so that a mah&yuga ( = 24 x 180000 years) con- sists of 1577917800 s&vana days, i. e. 28 days less than according to the modern Siirya Siddh&nta. Here it certainly appears possible that Varaha Mihira should have slightly diminished the number of the savana days of the maha- yuga, and implicitly the length of the solar year, in order to be able to reduce XIV INTRODUCTION. that number, as well as the number of the years of the yuga, by twenty-four and thus to arrive at figures more easy to manipulate ; all the more as the inaccuracy involved in that change would affect to an almost insensible degree only the comparatively short periods to which the rules of the karapa grantha are meant to be applied. But in spite of this undeniable possibility I am in- clined to think that in the present case also Var&ha Mihira proceeded with strict accuracy, and that his Sftrya SiddhsLnta actually assigned to the great yuga twenty-eight days less than the modern treatise does. For in addition to the general consideration that there are several other items in which the old and the new Siddliftntas differed beyond any doubt, we have in the pre- sent case two special reasons viz. firstly that it would have sufficed to diminish 1577917828 by four (instead of twenty -eight) in order to make it divisible by twenty-four ; and secondly that the estimation of the length of the solar year implied in the statement of the old Sffrya Siddh&nta agrees exactly with that value of the length of the Solar year that results from the elements of that Paulina SiddMnta about which Bhattotpala’s commentary on the Brihat Samhit& and Prithfidaka Sv&min’s commentary on Brahmagupta’s sphuta Brahma Siddh&nta furnish some information. As we shall see at once. Var&ha Mihira’s Sftrya Siddhanta agreed with that Paulina Siddh&nta in several other points also, and it therefore is not improbable that the two Siddhantas were at one also concerning the length of the solar year. If this is so, the most important item by which hitherto the Silrya Siddhanta was considered to be distinguished from the Paulisa Siddh&nta (as reported by Bhattotpala etc.) would vanish; which clearly shows that an accurate inves- tigation of the degree of strictness with which Yarflha Mihira reproduces the doctrines of his Siddhantas cannot be dispensed with. Similar to the case just discussed is that of the mean revolutions of the planets, as reported, according to the Shrya Siddh&nta, in the 16th chapter of the ParichaslddMntiku,. As appears from the notes to the translation and the latter part of this Introduction, the periods assigned to the mean re- volution by the old Sfirya Siddhanta differed more or less from the correspond- ing values stated in the modern treatise. There, however, the hypothesis of Var&ha Mihira having for some reason or other modified the elements of the work with which he had to deal seems altogether excluded. If he had chosen to state the length of the revolutions of the planets in the ordinary form i. e. by establishing periods within which the planets perform integral numbers of com- plete revolutions, he might possibly have had reason to manipulate the tradi- tional numbers to a certain extent, so as to reduce them to more manageable terms. But in the case under discussion he follows another plan viz. of at first stating the time of one revolution in round numbers, and then directing INTRODUCTION. XV us to apply a certain correction, in order to make up for the inaccuracy in- volved in the employment of those round numbers Now it is easy to see that, if Var&ha Mihira’s Shrya Siddh&nta had exhibited the same figures as the modern Siddh&nta, the amount of the corrections would differ from that actually stated by him, and we therefore are entitled to conclude that regard- ing the revolutions of the planets also the old Sftrya Siddhftnta actually differed from the modern one ; a conclusion moreover made more acceptable by the circumstance that several of the values assigned to the mean revolu- tions by Var&ha Mibira’s Siddhanta agree with the teaching of the Paulina Siddh&nta known to Bhattotpala, and with that of Aryabhata. That the difference, observed between the numbers of the natural days of the yuga as stated by the two Sfirya Siddhantas, is due to a real discrep- ancy of the two books, is further confirmed by the rule given in Chapter X 2 and 4 for finding the mean place of the moon. This rule is based on the elements of the yuga as stated in chapter I, but for the sake of greater facility of calculation employs reduced numbers. Instead of multiplying the given ahargana by (the numerator of which fraction are the side- real revolutions of the moon during the period of 180000 years, and the denominator the sdvana days comprehended within the same time), it directs us to employ the expression and thereupon — in order to make up for the error involved in this substitution — to deduct from the mean place of the moon thus found ;i ’]h for each revolution. In other words, Var&ha Mihira is unwilling to allow to pass an error in the mean position which amounts to no more than one sixtieth of a second of space for each revolution. But if he, on the other hand, had purposely, for mere convenience of calculation, lessen- ed the length of the mail sly uga by twenty-eight days, he would thereby have reduced the length of each sidereal month by about four hundreths of a second of time, which in its turn would have implied an error in the moon’s mean place amounting to about one fiftieth of a second of space for each revolution. So that, while anxious to correct one small error, he would have allowed another greater one to pass ; an assumption which we have absolutely no right or reason to make. The investigation of special cases thus certainly favours the conclusion that the changes which the old Sftrya Siddhanta has undergone in Varaha Mihira’s representation are purely formal, and that convenience of calculation is held by bim to be a consideration of altogether secondary importance. We therefore, and this is the most important conclusion to be drawn from the preceding enquiry, may hold ourselves entitled to look in the same light upon Var&ha Mihira’s rendering of the other SiddMntas which we can XVI INTRODUCTION. check neither by means of the originals nor with the assistance of modem recasts. There also we must hold Var&ha Mihira to have closely followed the elements and methods of the authors of the Siddh&ntas, and to have permitted himself only minor changes, such as facilitate calculation with- out affecting the fundamental character of the rules. General principles, enabling us to judge with certainty how far those changes may extend, can however not be laid down ; wc rather must judge each given ease on its own merits. When we f i. find that the yuga of the Romaka Siddh&nta com- prised, according to Var&ha Miliira, only 2850 years, we may raise the ques- tion whether this yuga is the true yuga of the Romaka, or only represents a subdivision of the true yuga, analogous to the 180000 years of the Stlrya Siddh&nta which, as we have seen above, must be considered as the smallest fraction of the mah&yuga with which the calculation of the ahargana can be effected. But we shall without much hesitation decide in favour of the former alternative, in the first place because the yuga of the Romaka Siddh&nta is expressly called a yuga of the sun and moon, for the formation of which a comparatively small number of years was sufficient, and in the second place because Brahmagupta, in a passage to be quoted later on, testifies that the Romaka Siddh&nta did not conform to the traditional views concerning the large periods of time. If, again, we find that according to the Paiicha- siddh&ntikft the Paulina Siddhanta made no use of yugas of any kind to the end of calculating the ahargana and the mean positions of the planets, but employed for those purposes a peculiar system of its own, we certainly must conclude that system to have been actually taught in the original Paulina Siddhanta, and not constructed, as indeed it might have been, by Varaha Mihira on the elements of the Paulisa Siddhanta. For why, we must ask ourselves, should he have transformed in that way the elements of the Paulina Siddhanta rather than those of the other Siddh&ntas which without any difficulty might have been thrown into the same form ? And, to single out one further point, if we find that the Pafichasiddh&ntik& gives a rule how to calculate, according to the Sftrya Siddhanta, the equation of the centre of sun and moon for any given anomaly, while it represents the Paulina and Romaka Siddhantas as merely stating the amount of those equations for a certain series of anomalies, without teaching us how to calculate the equations for the intervening anomalies ; we must again suppose that Var&ha Mihira faithfully renders characteristic features of the original Siddhantas as he found them ; for if he had held the opinion (which as the writer of a karana he indeed might have held) that the practical astronomer knows enough, if be can assign the equations of the centre for, let us say, each fifteen degrees of anomaly, he would no doubt not have given the general rule from the Sfirya INTRODUCTION. XVII Siddhftnta, but calculated from it the amounts whose knowledge he considered indispensable, and inserted them ready calculated in his text. We therefore arrive at the conclusion that Vara ha Mihira has in no case obliterated the characteristic features of the Siddhantas he had to deal with, and that whatever distinguishes those works from one another in the text of the Pafiehasiddhantika really distinguished them in their original form. We may note in conclusion that there is one interesting circumstance which furnishes a kind of counterproof to this conclusion. According to VII. 1. and VIII, 9 the Paulisa and Romaka Siddhantas calculated the parallax in longitude at a solar eclipse in exactly the same manner. Now Varaha Mihira accentuates this agreement of the two works by stating the rule each time in exactly the same words. But an author, who is so evidently desirous to mark the points in which the different authorities on which he draws are at one, may certainly be supposed to bo no less scrupulous in stating the details in which they diverge. After having thus cleared the way, I proceed to give short summa- ries of the doctrines of the five Siddhantas, beginning with that one which, owing to the existence of a modern recension, is best known, viz. the Surya Siddhanta. According to I. 14 the Sftfya Siddhanta of Varaha Mihira taught that sarya si month of the solar year ; here too we miss a general rule enabling us t.o calculate, the true motion for any given time. The estimation of the length of the revolution of the moon’s node given in III. 2G. differs hut little from that made in the Stirya Siddhanta and tic.', great majority of Hindu astronomical works.— The greatest latitude of the moon is put equal to 270’ (III. 31). Hut here also — as above in the ease of the Komaka Siddhauta — we meet with another rule which appears to pre- suppose a greatest latitude of 240' only, riz., the rule given in VI. 5. for calculating the duration of total obscuration during a lunar eclipse. A simi- lar rule on the other hand which likewise seems to belong to the Paulisa Siddhauta. ( V I . (!) again seems to he based on the assumption of the moon’s greatest latitude amounting to 270'. The processes prescribed in the Paulisa Siddhauta for the calculation of lunar and solar eclipses are of a very rough nature and much less accurate than the corresponding operations according to the .Romaka and Surya Siddhuiitas. The author of the Paulisa apparently aimed at nothing further than the establishment of convenient numerical formulas, gave no exposition of the general theory of the subject, and, for the sake of ease of calculation, introduced merely approximative values. No rule is prescribed for ascertain- ing the true (apparent) dimensions of sun, moon, and shadow at the time of an eclipse From VI. 4. it seems to follow that the mean value of the moon’s diameter is supposed to amount to 34', and that of the shadow to 7G'. Some- * In the note on the translation of III. 2, 3. the remark is made “The reason jrjbabty i* nt«." Hut I think it would have been more appropriate to say that it certainly is so. XXXVI INTRODUCTION. what different values seem however to result from VII. 6, which stanza also appears to imply a statement of the sum of the diameters of sun and moon. It is difficult to speak on these points with assurance, as the details of the rules cannot in all cases be explained with full confidence, and several entire stanzas have remained altogether obscure to us. The stanza VII. 1. gives the ordinary rule for calculating the parallax in longitude. We are unable to point out any rule for determining the parallax in latitude. Among other items of doctrine which appear to be based on the Paulisa Siddhanta there may be noted the interesting statement of the differ- ence in longitude of Yavanapura on the one, and Ujjayim and Benares on the other hand, which agrees fairly well with the assumption of Yavanapura being Alexandria.* Next the rule (III. 14) for calculating the difference in longitude of two places whose distance in yojanas is known ; an approxima- tive rule which would give tolerably correct results for places not very remote from each other. — The same chapter (21) also contains the famous, often quoted, statement regarding the former position of the solstice, in which the word ‘right’ (yukta), as we now see, can only be explained from the context in which stanza 20 stands with the preceding one. We may also mention in connexion with the Paulisa Siddhanta the table of sines given in the fourth chapter, although it is doubtful from which Siddhanta it was borrowed by Varalm Mihira. It may have been common to the three scientific Siddhdntas ; the values assigned in it to the different sines are employed by VarAha Mihira throughout the Pahehasiddhantika, whom- ever such are required. The most interesting feature of the table is that it liases on a subdivision of the Radius into 120 parts and of each of those 120 parts into 60 ; instead of subdividing the Radius, in the ordinary Indian fashion, into 3438'. It thus closely follows the Greek fashion of expressing the values of sines, only preferring to divide the Radius into 120 parts instead of sixty. In the majority of cases the agreement of the stated values of the sines with those given by Ptolemy is as close as possible, taking into account that the latter writer subdivides his hundred and twentieth parts of the diameter into minutes and seconds, while the table in the Panchasiddhantika subdivides into sixtieth parts only. There are however a small number of cases in which the agreement is not complete. The Sanskrit text had in a few places to be emended to a considerable extent, and it thus is not impossible that a more correct text would exhibit an even closer agreement with Ptolemy. * Compare on this poiut the Journal Asiatic Society of Bengal 1884. p. 265. INTRODUCTION. XXXVII On the other hand the circumstance that the amount of each sine is not stated directly, but has to be arrived at by the summation of the amounts of all the preceding sines favours the presumption of the emendations made being the right ones on the whole ; for if they were not, the table as it now stands might be expected to show greater, and more regularly progressing, de- viations from the true values of the sines than it actually does. It may be noted that, in ease of the table of sines basing on a Greek prototype, the plan of subdividing the Kadius — and not the diameter — into 120 parts would have enabled the borrower to take over, without any change, the amounts assigned in the Greek table to the chords of the angles, and to insert them in his own table as the values of the sines of half those angles. We finally have to compare the information about the Paulina Sid- dhanta given in the Panelmsiddliantika with what we know from other sources about the same work, or at any rate a work of the same name. The sources alluded to are principally Bhattotpala’s commentary on Varalia M ihira’s Brihatsanihita, and Pyithftdaka Svamin’s commentary on Brahmagupta’s Sphuta Brahmasiddhanta, and from them Colebrooke already extracted the most noteworthy points. It appears that the Paulisa Siddhanta, known to the two Scholiasts referred to, was a work following the same general methods as the Sftrya Siddhanta, Aryabhata and all the later astronomers ; at any rate it agreed with the great majority of Hindi! astronomical works in es- tablishing a mahayuga which contains an integral number of savana days etc. and of revolutions of the planets. The data proving this are to be found in Colebrookc’s essays vol. II. p. 3G5 ; from them it also follows that the length of the year, according to the Paulisa Siddhanta, amounts to 3G5' 1 G h 12 m 3G". Now, in spite of the corruption of the text of those passages in which Varil'lia Mihira treats of the Paulisa Siddhanta, it is clear that the work of that mime whose contents are summarized in the Panchasiddhantiki\ followed a plan altogether different from that of the Paulina Siddhanta referred to by Bliattotpala. I need not restate here in detail what has been remarkea above about the methods employed by the Paulisa of the Panchasiddhantika : it will be remembered that in calculating the ahargapi, and the mean as well as true places of the planets, it employs peculiar methods of its own, which sharply distinguish it from the Paulina of Bliattotpala and works constructed on analogous lines. The length of its year amounts to 3G5 d 6 h 12“ while, on the other hand, the length of the year of the Paulisa known to Bliattotpala is the same as that of the year of the SiVya Siddhanta known to Varalia Mihira. And if any of the rules, given in the last chapter of the Paftcha- siddluiutikil for calculating the places of the planets, base— as not improbably XXXVIII INTRODUCTION. they do — on the Paulisa Siddhanta, they supply an additional feature mark- ing off the two Paulina Siddhantas from each other. We are thus led to the conclusion that the Paulin v Siddhanta also has in the course of time undergone recasts, and that that form of it which was known to Bhat^otpala widely differed from its original form, so widely indeed that there is some reason for wondering that the later work could go by the same name as the earlier one. Viwisbiiw Siddimuu About the Vasishtha Siddhanta the Paftchasiddhantildt. as exhibited in the two Manuscripts at our disposal, gives only very scanty information. Vaiiha Mihira, iudeed, places the Vasishtha Siddhanta, together with the Pai tamaha Siddhanta, in the lowest rank of the works whose tenets he repro- duces ; yet it appears strange that its teaching should not he treated at some- what greater length. If anything, it is the small space allotted to the Vasishtha Siddhanta in the text of the Panchasiddhantika, as here edited, which might induce us to suppose that certain parts of the work are missing in our Manuscripts. It is indeed —as I already have remarked above when discussing the allotment of the different chapters of the Panchasiddhantika to the difierent Siddhantas-— a somewhat difficult task to ascertain, how much of the matter contained in the Panehasiddhantikit in its present form has to he traced hack to the Vasishtha Siddhanta. With confidence we may re- cognise its teaching only in the rules given in the second part of the second chapter which are, at the end, designated by Varuha Mihira himself as being founded on the Vasishtha Siddhanta, and which moreover are of a quite peculiar character, marking them off very decidedly from the doctrines of the other Siddhantas. The rule, given in stanza 8 for calculating the len gth of the day at any time of the year, is similar to that of the Pa i ta- malia Siddhtoita in so far as assuming an equal daily increase ; but it differs, from it regarding the length of the shortest and longest days. The rules, given in stanzas 9—13 for finding the length of the shadow, the mean longitude of the sun and the lagna are also of a very primitive nature; but yet superior to what a work of the typo of the PaiUmaha Siddhanta could supply. We may infer from them that the Vasishtha Siddhanta no longer oper- ated with nakshatras, but subdivided the sphere into signs, degrees and minutes ; and we also notice that it knew about the so-called lagna i. e. the ecliptic point which is on the eastern horizon at any given time. But, apart from this, the methods are so rude, and so completely omit to distinguish be- tween mean and true astronomical quantities, that the Vasishtha Siddhanta can hardly be included within Scientific Ilindd Astronomy. INTRODUCTION, XXXIX That the rules concerning the motions of the moon contained in the former part of the second chapter are of an entirely different character, has been remarked above already (p. XI). In spite of this it is, as said in the same place, not altogether impossible that they should base on the teaching of the Vasishtha Siddhanta, which may have borrowed various rules and methods from different sources. The point must he left unsettled for the pre- sent. — The Vasishtha Siddhanta is not referred to in any other part of the Pancliasiddhantikfi, with the exception of a colophon in the last chapter, about which something will he said later on. There yet remains the question as to the relation of the Vasishtha Siddhanta, known to Varaha. Miliira, to that Siddhanta of the same name about which we know from later commentators, chiefly those repeatedly referred to in the course of this introduction. From them Colebrooke already has im- parted some information about a Vasishtha Siddhanta which he ascribes to VisluTio Chandra, on the authority of the last line of the passage quoted above (p. XXVI) from Brahmagupta. The right interpretation of that line clearly depends outlie way in which the preceding lines concerning Srishena’s treatise are explained. Following the version of them given by me above we have to translate “and by Vishnu Chandra, taking the same (astronomical elements) the Vasishtha (Siddhanta) (was disfigured)” v. c. Vishnu Chandra, by borrow- ing various items of doctrine from different sources, gave to the original Vasishtha Siddhanta the same variegated, incongruous appearance which S'risheij.a, acting in an analogous manner, had imparted to the original liomaka Siddhanta. To the existence of a Vasishtha Siddhanta, older than the recast by Vishnu Chandra, there moreover testifies the seventh line of the passage from Brahmagupta, which apparently says that SVlsherta borrowed the (num- bers of the) elapsed years and (planetary) revolutions of the yuga from a VsLsishtha (Siddhanta) ; and the second half of that line further seems to intimate that that Vasishtha was composed by, or at any rate somehow con- nected with, Vijayanandin. An astronomical writer of that name is mention- ed by Brahmagupta in another place also, and — which is of some importance as more definitely indicating his lime — also by Varaha Mihira himself who, in the last chapter of the PauchasiddhAntikil (XVIII. 02), refers to Vijayanandin as having given rules for the calculation of the places of the planets. Neither Vijayanandin’s nor Vishnu Chandra’s treatises appear to have come dowu to our time. The (Laghu) Vasishtha Siddhanta which we possess* does not evince any relationship either to that Vasishtha Siddhanta which was known to Varaha Mihira, or to the work of Vishnu Chandra, with some details of which we are rendered acquainted by Brahmagupta and later commentators. ~~~~ * Published by P. Viudhycsvari Prnsdda Dube, Bonnies 1881. XL INTRODUCTION. Having thus taken a survey of the chief doctrines of the five Siddlnintas as exhibited in chapters I — XYII of the Panchasiddhfuitika, and before exam- ining in detail the contents of the last chapter, a few remarks may be added about what Vara, ha Miliira states in XV. 33 — 38 about the longitudes and latitudes of certaiu stars. What authority he follows therein, we are unable to say. The statement referred to gives the longitudes and latitudes of a small number of junction-stars of nakshatras. How the longitude and latitude are measured, the text does not define ; we can only presume that the Siddhanta which Varaha Mihira here extracts followed the usual Indian method viz. of referring the stars outside the ecliptic to the latter circle, not by latitude- circles, but by declination-circles, so that the quantities stated are what Whit- ney, in his translation of the Surya Siddhanta, calls polar longitudes (dbruva) and polar latitudes (vikshepa). The degrees of (polar) longitude are, as in the Silrya Siddhanta, taken from the beginning of the nakshatra ( = twenty- seventh part of the ecliptic) to which the junction star belongs. The latitudes are, in a peculiar manner, expressed not in degrees, but hastas, which we are enabled to turn into degrees by remembering that one hasta comprises thirty- four ahgalis, and that the diameter of the moon — here assumed equal to 34 minutes — is divided into fifteen angulis. We give the results, in a tabular form, adding at the same time, in order to facilitate comparison, the corres- ponding statements made in the Sfirya Siddhanta (Whitney p. 321 .) Name. | Acc. to Pan chasiddhan tika. Acc. to Silrya Siddhanta. i Position of star in its nakshatra. Polar 1 Longitude. Polar Latitude. Position of] star iti its nakshatra. Polar Longitude. Polar Latitude. Krittika ■■ 32° 40' 3’ 10' N. 10’ 50' 37° 30' 5° N. Roliigi 8° 48° 4° 59' S. 9° 30' i 1 49° 30' 5° S. Punarvasu 8“ * 88° 7° 15' N. S. 13’ 93° 6° N. Pushya 4° 8" 10' N. 12° 40' 100° O ASlesM ... 1° 107° 40' 54' S. N. 2° 20' 109° j 7° S. Magha 6° 126° O 9° 129° O Chitr& r 30 ' | o o o 00 rH 2° 43' S. 6° 40' 180 ° r s. INTRODUCTION. XLt Why Varilha Mihira should have confined himself to stating the longi- tudes and latitudes of seven junction stars only, remains unaccounted for. Possibly the Manuscripts are defective just at that place. The details given concerning those seven stars (supposing our interpretation of the text to ho correct) di verge so widely from the corresponding statements of the Surya Siddhanta that, considering the incompleteness of the material, a critical comparison of the two appears hardly feasible at present. — About the rule concerning the heliacal rising of Canopus (XIV. 39 — 41) a few remarks will be made further on. I now proceed to the examination of the last chapter of the Paneha- siddMntika, which is one of the most interesting but at the same time most perplexing of the whole work. The text itself apparently does not state to what Siddhanta the rules about the planets, constituting the matter of the chapter, have to he traced. To some indications given in the colophons I will refer later on. A survey of the contents of the chapter shows that it consists of two different sets of rules, of which the former extends from stanzas 1 — GO, while the latter, preceded by some personal remarks on the part of Varaha Mihira, comprises stanzas GG up to the end. I begin by analysing, as far us feasible, the former set of rules.* The plan generally followed in this part of the work is to state at first for each planet a rule enabling us to calculate how many (heliacal) risings of it have taken place within a given ahargana, or — in other words — how many synodical revolutions it has performed. To that end we arc directed to divide the ahargana by the duration of the different synodical revolutions. In Venus’ ease the rule proscribes (stanzas 1 and 2) to divide the ahargana by 584, and to add to the remaining days the eleventh part of the quotient of the division ; i, e, in other words, the length of Venus’ synodical revolution is estimated at 584 - ^ days, In the same xvay it follows from stanzas G and 7 that one synodical revolution of Jupiter is estimated at 39!) - * days ; from stanzas 1 4 and 15 that one revolution of Saturn amounts to 378 J days ; from stanzas 21 and 22, that one revolution of Mars is estimated at 780 days minim 161 yim'ulikas ( = , about, of a day) ; and from stanzas 36 and 37, that cue synodical revolution of Mercury is supposed to amount to 115 days plus 52 n&clikfts plus 43 vinftdikas (for the rules given there imply that 8 synodical revolutions occupy 927 days plus 2 n&dikils). * jt, w iH bo observed that the English translation of tfcwfct chapter confines itself to a literal rendering of the Stanzas. The following, systematic, discussion of tho contents of the chapter is meant to supply the place of explanatory notes to the translation. XI.II INTRODUCTION. In the second place the rules teach us, how to calculate the amount of sidereal motion performed by the planets within the given number of synodi- cal revolutions. — Stanza I says that during one period of rising, i. e. during one synodical revolution Venus passes through 7 B 5° 30', and this is indeed the approximate motion of Venus within 584— .1 days, if we suppose its mean motion to be the same as that of the sun, and reject the entire revolution performed. In Jupiter’s case we are directed (stanza 7) to multiply the number of synodical revolutions contained within a given ahargana by 36, and to divide by 391 ; which shows that 391 synodical revolutions are supposed to be about equal to 36 sidereal revolutions. This is indeed nearly the case ; for if, basing on the duration of a synodical revolution of 399 days (not 399 — g days) and on the assumption of 364220 sidereal revolutions of Jupiter being performed within, a mahayuga of 1 5.779 J 7800 days, we calculate the sidereal revolutions performed during the lapse of 391 synodical revolutions, we find 36 revolutions plus about four degrees. And for these four degrees also an allowance is made, vis. in stanza 10, where we are told to multiply the num- ber of risings by five, to divide by eight, and to take the result as minutes ; for is nearly equal to The rules for calculating the sidereal motions of the other planets are of a similar nature. Assuming the sidereal revolutions of Saturn within one mahayuga to amount to 146568, the quantity of sidereal motion within a period comprising 256 synodical revolutions of 378 days each is equal to nine revolutions minus four degrees + eight minutes. We are hence directed (st. 15) to multiply the risings of Saturn by 9, and to divide by 256 ; and (st. 17) to multiply the risings by 3, to divide by 32, and to look on the result as minutes to.be deducted ; for 4° + 8' = 248', and =■ For Mars we have the assumption (st. 22) that fifteen synodical revolutions comprise eighteen sidereal revolutions ; hence we multiply the number of the risings by eighteen and divide by fifteen. The quotient which indicates the entire revolutions is rejected ; the remainder, multiplied by 12 and divided by 15, gives the mean place of Mars in signs and so on In Mercury's case, finally, 389 synodical revolutions aro supposed to be equal to 123 sidereal revolutions ; hence st. 37 directs us to multiply the given number of synodical revolutions by 123, and to divide by 389. So far the rules contained in the last chapter can, bp satisfactorily explained. We now however are met by a series of rules whose rationale we are unable to, assign. Hitherto, we have been told, how to calculate firstly INTRODUCTION. XLItl the full "number of synodical revolutions of a planet which have taken place within a given ahargapa, and secondly the full number of sidereal re- volutions within the same period ; and, for the purpose of ascertaining the amount of synodical as well as sidereal motion which the planet has accom- plished in addition to complete revolutions, we have at our disposal the remainder of the two divisions whose quotients indicate the number of the full revolutions. Now, from these remainders — which represent in both cases a certain number of days — the mean motion and the mean position of the planet at a given time can be calculated without difficulty ; but what is requirod, and wliat the rules of the text apparently mean to supply, are directions enabling us to determine, roughly at least, the true place of the planet at the end of the given ahargana. We are, however, unable to point out the general principles on which the rules are constructed. I begin with Saturn in whose case the rules appear to be simpler than in that of the other planets (setting aside Venus about which later on). Stanza 15 directs us to multiply the number of the synodical revolutions by 9, and to divide by 256, the quotient of the division indicating, as explained above, the number of completed sidereal revolutions. The remainder, the text goes on to say, is the ‘pada.’ The remainder — which alone is required for the calculation of the true place of a planet — is thus denoted by the same term which, as we have seen above, was employed in the Paulina (or Vasishtha) Siddh&nta to denote the remainder of days from which the true place of the moon is ascertained. The text then goes on to distinguish three different amounts of the pada, viz. 30, 127 and 99 ; and it will be observed that the sum of the three fractions |||> ~ amounts to i. e. one entire revolution. There are further mentioned certain additive, or subtractive, quantities, which have to be applied in case of the padas reaching the mentioned amounts, and it certainly appears probable that those quantities are, in some way or other, connected with the equation of the centre, which has to be added or sub- tracted in order to render the planet’s place true. But we are quite unable to demonstrate, by a satisfactory explanation of the three amounts, the truth of this hypothesis. The text finally (in 18, and 19) states certain amounts of motion which Saturn performs within the three khandas, i. e apparently, the three fractions of a revolution | 5 ®) which were mentioned above. But here again we meet with a difficulty ; for while the text states for the two first khandas the following amounts ; I s 15° 51' and 5* 28“ 35', a calcula- tion of the mean motion proceeding on the data of the text gives 1® 12° 11' for — th of a revolution, and 5® 27° 34' for y 01 of a revolution. It is indeed by no means unlikely that the discrepancy may be otoing to the circumstance of the text wishing to state the amount of true— not mean— motion ; but XTJV INTRODUCTION. we again are unable to prove this by calculation. (The amount of motion during the third pada stated above viz . , ~ th of a revolution is, according to calculation, equal to 4 s 19° 13', not to 4 s 7° 30' as, by mistake, stated in the commentary p. 101. The emendation of the text — tnunayo liptai chatur- gunah sapta, stanza 19 — which bases on the erroneous quantity stated in the commentary must on that account be rejected. Owing to the obscurity of the meaning it is hardly feasible to propose another emendation). The text finally gives (stanzas 19 and 20) a statement of the different phases of Saturn’s motion in the course of one synodic revolution, dividing that period into seven minor periods, and mentioning the amount of motion during each. We have similar rules in Jupiter’s case. There too (st. 7) wc are told to take the remainder of the division whose quotient indicates the completed sidereal revolutions, and to term it pada. Again three different padas are distinguished (st. 9), whose sum amounts to one complete revolution -j- + !- = 39 1), and for each of these padas there is stated a certain additive or subtractive quantity of obscure meaning. And again there are stated three different amounts of motion which do not very widely differ from the amounts of mean motion that Jupiter performs within each of the three fractions of a revolution stated above ; the amounts given in the text being 5 s 9" 30'; 0 s ; 13°, while the calculated amounts arc 5 s 15° 43'; 5 s 29° 33'; 14° 44'. Here too we can only conjecture that what is aimed at is a statement of the true motion within the given intervals. But, in addition to these rules which so far are strictly parallel to those concerning Saturn, we have, in stanza 8, some additional directions concerning mean and true quantities whose sense lias remained quite obscure to us, so that even the attempted literal render- ing of the words of the text — which has been given in the translation — must be looked upon as altogether tentative and provisional. — Stanzas 12 and 13 exhibit a statement of Jupiter’s apparent motion during each of nine minor periods into which the entire synodical period of the planet is subdivided. With regard to Mars again another plan is followed, of which the details however have likewise remained obscure to us. In Mars’ case the text, after having given the rules, explained above, for finding the elapsed synodical and sidereal revolutions, proceeds (in 24) to make a . statement re- garding the difference of the planet’s mean and true places, it being obscure, however, in what way the true place is supposed to be found, and then declares that the course (cMra) of Mars will be stated according to the dif- ferent motions (gati). Tb judge from stanza 35, these motions appear to be eight ; and as the text mentions the terms vakra, anuvakra, anugati, ativakra INTRODUCTION. xuv aml sighra, wo may assume that, wc have to do with something like the eight kinds of motion enumerated in Surya Siddhanta IT. 12, among which we meet witli vakra, anuvakra and sighra. But most of the details we must leave unexplained, owing partly to the corruption of the text, and partly to the obscurity of the subject. Stanzas 25 (conclusion) and 26 state the periods intervening between the time of the heliacal setting of Mars and its conjunction (when it becomes niramSa), and between the conjunction and its heliacal rising. The meaning of stanzas 27 and 28 is obscure. Stanzas 29 — .22 treat of the motion of Mars in three of its gatis (cp. the ‘ traye’ in 32), of which the ‘ vakra’ i. c. no doubt the retrograde motion is the first ; the nature of the two other motions we are unable to explain, the text exhibiting only the vague term, ‘anugati’ (in 29). In st. 33 we meet indeed with the terms vakra, ati vakra and anuvakra, and we hence suppose that those three motions are referred to in 29 — 32 also ; but. considering the various senses in which the terms ati vakra and anuvakra are used by different authorities, we do not attempt to assign the meaning they bear in our text. For Mercury the text, after having given the general rules for syno- dical and sidereal motion, at once states the amount of supposed true motion for eight periods of days whose total aggregate amounts to 380 days, the amount of true motion being 3G0 degrees. 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II ngn^fci nm: are ^feumtRi fam 1 *215 fan fawra*?Rwrcmimmii€l tw m tfmn ow^rontafs* fxngire s^faur « Rj)gm5p:^r5m f m uffa^Tfrraframfam onijn 1 TRANSLATION CHAPTER I. Introductory. 1. Having at the outset done devout reverence to the various most excellent Munis among whom Surya and Vasishtha aro foremost, and to my J'at her and teacher by whom I was instructed in this Sastra, 2. I shall endeavour to state in its entirety, according to the opinions „f earlier teachers, that correction of the planetary motions (bija) which is most excellent, easy, clear and a wonderful mystery. The term bija signifies any correction applied to astronomical elements which aims at bringing about an agreement between the celestial phenomena such as calculated and such as actually observed. Varalia Mihira seems anxious to show that the bija applied by him is not his own invention but bases on the authority of former teachers. 3. There are the following Siddhantas : The Paulina, the Romaka, the Vasishtha, the Saura and the Paitamaha. Out of these five the first two have been explained by Latadeva. 4. The Siddhanta made by Paulina is accurate, near to it stands the Siddhanta proclaimed by Romaka; more accurate is the Savitra (Saura) ; the two remaining ones are far from the truth. 5. That subject which is the greatest mystery, which perplexes the minds of the writers of astronomical works, viz., the eclipses of the sun 1 am going to explain in this work, dismissing all jealousy. 6. Moreover, there are contained in this work the (rules for the calculation of the) direction, the duration, the period of total obscuration. ( 4 ) the hypothenuse, the time of the measures (i. e., beginning, middle, etc., of eclipses) ; the eclipses or (eventual) non-eclipses of the moon ; the con junctions and obscurations of the stars and planets ; the means of finding the difference in longitude ; 7. The prime vertical ; the rising of the moon ; the construction of astronomical instruments; the shadow of the gnomon; other useful matters ; the sine of terrestrial latitude ; the sine of colatitude ; the declination and other topics. 8. Deduct the S'aka-ycar 427 (i. e., deduct 427 from the number of' that Saka-year for which the ahargana is wanted), at the beginning of the light half of Chaitra, when the sun has half set in Yavanapura, at the beginning of Monday. 9. Turn (the number of solar years remaining after the deduction of 427) into (solar) months, add the months (i r., the elapsed lunar months of the current year), set the result down in two places ; multiply it (in one place) by seven and divide by 228 ; add the resulting adhimasas (to the number of solar months obtained above) ; multiply the sum by thirty ; add the tithis (i. e., the elapsed tithis of the current month) ; set the result down in two places. 10. Multiply it (in one place) by eleven, add 514 and divide by 703 ; deduct the quotient (from the number of tithis found above). The final result is the (savana) ahargana, according to the Romaka Siddhanta. So it is also according to the Paulisa which is not much older (?) The three preceding stanzas contain a concisely stated rule for the calculation of the savana ahargana (i. be calculated by a proportion founded on the motion (of the moon); the time of the current tithi is to be found from a proportion founded upon the differ- ence of motion of sun and moon. 17. The (daily) motion of the sun amounts to sixty (minutes) minus three, three, three, three, two, one ; plus one, one, one, one ; minus nought, one, in turn. A rough statement of the daily amount of motion of the sun during the twelve months of the year beginning with Chaitra. The amounts stand as follows : 57', 57 ', 57 ', 57 ', 58', 59'; 61', 61/, 61', 61', GO', 59'. 18. In the light half of the month six degrees have to be subtracted and in the dark half as many have to be added ; the minutes of the longitude ( 18 ) of the moon lessened by the longitude of the sun have to be divided by 300 ; the result is the karana. The remaining operations are to be performed as in the case of the tithi. 19. The fixed karanas begin from the middle of the fourteenth titlxi of the dark half; they are called S’akuni, Chatushpada, Naga and Kimstuglma. The other karanas are moveable. (All) karapas occupy half (of a tithi). The above two stanzas teach how to calculate the karanas, i. 0, i. the number of minutes of are constituting one karana; the quotient then indicates the number of the elapsed karanas. In order however that the counting of the karanas may uniformly begin from Bava the first moveable karana, wo, before division, deduct from the difference of longitude six degrees in the case of the light half of the month and add six degrees to it in the ease of the dark half. Compare the table of the karanas given in Burgess- Whitneys Surya Siddhstnta, p. 287. — The calculation of the elapsed liadikas of the karana is analogous to that of the tithis. 20. When the sum of the longitudes of the sun and the moon plus ten naksliatras amounts to six signs, it is called Vaidhrita ; when it amounts to a complete circle it is called Vyati pitta; the time (of those conjunctures) is to ho ascertained by means of the degrees passed through by the sun and the moon. 21. When the return of tire sun towards tire south (i. c., the summer solstice) took place from the middle of Asleshii, then the ayana was right; at the present time the ayana begins from Punarvasu. 22. When the degrees of the ayana are in the apposite direction (i. c., when the precession is retrograde), and the quantity to be added to the longitudes of sun and moon amounts to (as much as) the degrees of the sun’s greatest declination (i. e., when the degrees of procession amount to 24); then the Vyatipata takes place when the surp of the longitudes of sun and( moon amounts to half a circle. ( 19 ) The Vyatip&ta — aspect takes place when the sum of the longitudes oi sun and moon amounts to 180°; the vaidhriti-aspeet when it amounts to 400". This definition, without any qualification, applies to the time of the Siddhiintas when there was no precession, or when — as stanza 22 expresses it— the ayana had retrograded 24 degrees from the point which it formerly occupied, viz., the middle of Asleshu. At the time however when the solstice occurred at this latter jaunt, there were 24 J — or more strictly — 24" 20 / precession (reckoned from the beginning of niravana mesha) and hence, in order to express the condition of Vaidhrita and Vyatipata we should have to say : Vaidhrita occurs when longit. sun — 24' 20',+ longit. moon — 24' 20'= 12" ; and Vyatipata, when longit. sun — 24° 20' + longit. moon — 24° 20' = 6* . Kearianging the terms of these two equations of condition we have longit. sun + longit. moon = 12’ +40 40' and longit. sun + longit. moon = (3’ +40° 40'. Substituting naksbatras for signs, the right side of the former equa- tion =4, ( nakduitras (the whole revolution being rejected) ; and the right side of the latter equation = 1 7 naksh. Now wo may write for the termer equation longit. sun + longit. moon + 10 naksh. = 3^ naksh +10 naksh. = 6’; and for the latter one longit. sun + longit. moon +10 naksh. = 1 7 naksh. + 10 naksh. = 12*. Thereby the rule given in stanza 20 explains itself. When that rule applied, then the ayana was right, i. e., agreed with the rule given in stanza 20 for the determination of Vaidhrita aud Vyatipata, 24. The equinox (vishuvat) occurs at the beginning of Aries and Libra; the Shadasitimukha within the degrees of that sign which precedes Libra (i. c. Virgo) ; those (degrees or) solar (days) which remain from the tShadasitimukhas are the days of the fathers. 24. The Shadasitimukha is placed at the fourteenth degree of Virgo, the eighteenth degree of Gemini, the twenty-second degree of Pisces and the twenty-sixth degree of Sagittarius. ( 20 ) The Shadasi ti mukhas are divisions of the ecliptic into arcs of 86°, begin- ning from the first point of Libra. — The sixteen degrees or solar days remain- ing- between the fourth shadasitimukha and the first point of Libra are said to belong to the fathers, because whatever is given in them to the fathers is imperishable (SO. Si. XIV. 0). 25. The northern progress of the sun begins from the first point of Capricorn, so also the solar seasons beginning with S'isira. Each season ex- tends in time over two signs. The southern progress of the sun begins from Cancer. 26. The minutes of (the diameter of) the sun’s disc, multiplied by sixty and divided by the sun’s daily motion, give the number of niidikas which constitute the holy time on the occasion of the sun entering a new sign ; half before (the sun’s actual entrance) and half afterwards. The samkraoti actually takes place at the moment when the sun’s centre enters the new sign ; but the whole time from the entrance of the sun’s eastern limb to’ the entrance of the western limb is considered holy. The proportion employed for the calculation of the length of* that time is easily intelligible, 27. When the sun rises touching the end of a tithi and another day, the conjunction is the one, ‘ which touches three days’; while in the case of one day touching three tithis (we have the conjunction touching three tithis). An explanation of two terms connected with tithivriddhi and tithihrasa. The former term refers to the case of a tithi beginning before the sunrise of a certain day and terminating after sunrise on the next day, so that it ‘ touches’ three sayana days. The latter term applies to the opposite case of one savana day touching three tithis, 28. Multiply the ahargmja by 8 and divide by 151 ; the quotient indicates the degrees of Rahu (i. e., the moon’s node), to which as many minutes have to be added as there are complete revolutions. The above rule for calculating the place of the moon’s node bases on the assumption that the node moves very nearly eight degrees in 151 days, whence it would follow that it performs a complete revolution in 6795 days. We are however told that for each full revolution performed within the ( 21 ) period of the ahargana one minute has to be added to the place resulting from the rough process. Taking this correction into account we find that the accurate period of one revolution of the node is 0704'' I <>" 27 / 2D". 29. [A stanza stating a certain correction to be applied to the place of the moon’s node as found according to the rule given in stanza 28. Apparently twenty-five minutes have te be deducted from that place. We do not know what is meant by the ‘ vrischika-bhagah ’ of Rahu.] 30. When the moon is more advanced in longitude than the head of Rahu and less advanced than its tail, she moves to the north of tin* ecliptic ; when she is less advanced than the head and more advanced than the tail, she moves to the south of the asura, i. a., Rahu. 31. The moon is ninety degrees distant from Rahu when her latitude is at its maximum, (and the latitude then amounts to) 270 minutes ; (the latitude) for other places is to be found by proportion. 32 — 37. [Six stanzas of obscure import. A few lines indeed lend themselves to translation and a few emendations are obvious ; but we are un- able to elicit from the text a connected meaning.] CHAPTER IV. 1. The square root of the tenth part of the .square of the eireii in- ference — which comprises 3G0 parts- is the diameter. Having assumed the four parts of the circle (/. c. having divided it into four parts) the sine of the eighth part of a sign (i. c. of 22a') (has to be ascertained). 2. Take the square of the Radius and call it the ‘constant’ (dhruva); the fourth part of it is (the square) of Aries (/’. e, of the sine of one sign.) The ‘constant’ square is to be lessened by the square of Aries (i. <>. one sign). The square roots of the two quantities (i. c. tin; square of Aries and the ‘constant’ lessened by the square of Aries) are the sines (re. of ,10° and 00° respectively). 3. In order to find the remaining desired (sines), take the double of the are (whose sine you wish to find), deduct it from the quarter (of the circle), diminish the Radius by the sine of the remainder, and add to the square of half of that (viz. the Radius so diminished) the square of half the sine of double (the original) arc. 4. The square root of that sum is the desired sine. The ‘constant’ square lessened by that sum (is the square) of the remaining quantity (i. c. of the cosine of the given arc). Half of the ‘constant’ square is called (the square of) one and a half (viz. signs i. e. 45°). — Another method also is taught here. 5. Lessen the Radius by the sine of three signs (i. c. Radius) from which (three signs) double the required arc has been previously deducted, and multiply the remainder by sixty ; the result is the square (of the desired sine). By deducting that square from the square of the Radius you obtain the square of the cosine. The above four stanzas teach how to calculate the sines generally employed in Indian astronomy. — The formula expressing the relation of the diameter to the circumference is the usual one. — The sine of 30° is half of the chord of 60° which is equal to Radius ; the sine of 60° is found by taking ( 23 ) tlic square root of the difference of II s and the square of sine 60°. — The method by which the other sines required are found will bo understood with the help of figure 1. Of the arc the sine of which is required — let us say ah — double is taken — ae. The sine of this double arc — cf and the sine of the complementary arc (ed) — ef. Ry deducting ef from the Kadi us we get af and by drawing from h (i. <•., the point marking the centre of the chord ael the line hg parallel to ef we halve the line af and thus obtain the value of ag. The line hg again is half of cf, and ah <■., the desired sine = y/hg* + ag\ — Tire sines of (10° and 30° being given, the above method can of course be employed for finding tins sine of 15° and so on. — The cosine of the given are - Stanza 4 goes on to say — is found bv deducting the square of the sine from the square of the Radius and taking the square root of the remainder. The sine of 45° is equal to t/ 1 *’ (the square of the chord of 90° being 2R* and consequently the square of half that chord v. <\, the sine of 45 being equal . 2IU R'-\ to 4 The so-called different method described in stanza 5 is. not essentially different from the method described before. We are directed to multiply the difference of the Radius and the sine of the complement of twice the given are by 60, the product being equal to the square of the desired sine. Now jn the above diagram ah 3 = hg 3 + ag 3 . Now ag s = (!!)'= p- and ce 2 — (ae — a/)* 2 ae \ a f - af 3 -"4"" 4 Adding the expressions thus found for a g s and hg 3 we obtain i o af s 2a e x nf — af- no x af -go a f (ae = R = 120). 6. The sines in Aries are 7 ; 15; 20 + 3 = 23; 20 + 11 = 31; 20 + 18 = 38 ; 45 ; 50 + 3 = 53 ; 60 minutes ; ( 24 ) 7. (To which have to he added in succession) 51 ; 40 ; 25 ; 4 ; 34 ; 5G ; 5 ; 0 (seconds). 8. The sines in Taurus are 6 ; 13 ; 19 ; 24 ; 30 ; 35 ; 39 ; 43 minutes. 9. The seconds in Taurus are 40 ; 3 ; 7 ; 51 ; 13 ; 13 ; 4G ; 5G. 10. The sines from the end of the second sign (i. e. the sines in the third sign, Gemini) are, in minutes, 3 ; G ; 9 ; 12 ; 13 ; 15 ; 15 ; 1G. 11. The seconds are 42 ; 57 ; 42 ; 0 ; 47 ; 4 ; 49 ; 5. Stanzas G — 11 contain the table of sines calculated for the Radius 120. As generally in Indian astronomy, the arcs for which the sines are calculated progress by 3° 45', so that twenty -four sines have to be computed for the quadrant, each sign containing eight sines. By the minutes of the text we have to understand one hundred and twentieths of the Radius, by the seconds sixtieth parts of those minutes. The amounts stated for the sines of Taurus have to be added to the last sine of Aries; so likewise the; amounts stated for Gemini to the last sine of Taurus. The complete table of sines stands as follows : — No. Arcs. Sinks. No. Arcs. Sinks. 1 . 3" 45' 7' 51" 13. 48" 45' 90' 13" 2. 7” 30' 1 5' 40" 14. 52' 30' 95' 13" 3. 11° 15' 23' 25" 15. 5G J 15' 99' 4G" 4. 15° 31' 4" 1 6 . G0° 10 3' 5G" 5. 18° 45' 38' 34" 17. G3° 45' 107' 38" G. 22* 30' 45' 5G" 18. G7° 30' 110' 53" 7. 26° 15' 53' 5" 19. 71° 15' 113' 38" 8. 30 c GO' 20. 75° 115' 56* 9. 33° 45' G6' 40" 21. 78° 45' 117' 43" 10. 37° 30' 73' 3" 22. 82° 30' 119' 11. 41° 15' 79' 7" 23. 8G° 15' 119' 45" 12. 45° 84' 51" 24. 90° 120' 1" 12. In Aries the minutes are 7, in the last sine 6; in Taurus they are 6 ; 6 ; G ; 5 ; 5 ; 5 ; 4 ; 4 ; in Gemini they are 3 ; 3 ; 2 ; 2 ; 1 ; 1 ; 0 ; 0. ( 25 ) 13. In Aries the seconds are 51 ; 49 ; 45 ; 39 ; 30 ; 22 ; 9. 14 15. In Gemini they are 42; 15; 45; 18; 47; 17; 45; 16. Stanzas 12 — 15 state the differences of the tabular sines. There is manifestly a lacuna after the first part of 13 extending over the end of 13 and the former part of 14, and the preserved part of 14 seems to be hopelessly corrupt ; but this does no great harm, as the table of the differences of the sines can easily be drawn up from the table of sines as given above. It stands as follows : No. Sines. Differences. No. Sines. Differences. l. 7' 51" 7' 51" 13. 90' 1 3" 5' 22" 2. 15' 40" 7' 49" 14. 95' 13" 5' 3. 23' 25" 7' 45" 15. 99' 46" 4' 33" 4. 31' 4" 7' 39" 16. 103' 56" 4' 10" 5. 38' 34" 7' 30" 17. 107' 38" 3' 42" 6. 45' 56" 7' 22" 18. 110' 53" 3' 15" 7. 53' 5'' 7' 9" 19. 113' 38" 2' 45" 8. 60' 6' 55" 20. 115' 56" 2' 18’ 9. 66' 40" 6' 40" 21. 117' 43" 1' 47" 10. 73' 3" 6' 23" 22. 119' 1/ 17" 11. 79' 7" 6' 4" 23. 119fr45" 45" 12. 84' 51" 5' 44" 24. 120' 1" 16" 16. 17. 18. [Three stanzas apparently referring to the moon’s latitude, which we are however unable to explain satisfactorily.] 19. By means of the shadow’s entering into — and passing out of — a circle, whose diameter comprises as many ahgulis as the gnomon is high, the west-east direction is ascertained ; and thereupon the north-south direction is found by means of figures shaped like barley corns. The above process for ascertaining the different points of the compass is well known and requires no special elucidation. The customary height of the gnomon is twelve ahgulis. After the west-east line has been found, we describe two arcs of circles, taking the points where the line cuts the circle for centres ; the line joining the points of intersection of those two arcs is the ( 26 ) north-south line. The figure formed by the two intersecting arcs is by Varaha Mihira likened to a barley corn ; its ordinary name “matsya” bases on its re- semblance to the outline of the body of a fish. 20. To the square of the midday shadow of the gnomon on the day of the equinox add 144 and take the square root of the sum. By this square root divide the product of 120 and the equinoctial shadow ; 21. The quotient is the sine of latitude ; the arc corresponding to that sine is the latitude. — Or else on any given day (after having gone through the process described) add to the result the sun’s declination in the six signs beginning with Aries, and deduct it in the six signs beginning with Libra. The result is the latitude of the given place. The two well known methods for ascertaining the latitude of a given place. — The first method avails itself of the equinoctial shadow and of the proportion Equinoct. hypothenuse : equin, shadow— 120 i. e. Radius: sine of latitude. The second method dispenses with the knowledge of the equinoctial shadow and starts directly from the midday shadow of any given day. It thus ascertains the sun’s zenith distance (the natamsas), and by adding to it — or subtracting from i% in the southern hemisphere — the sun’s declination obtains the desired latitude. 22. Deduct the sun’s declination from the latitude (in the northern hemisphere), and add it (in the southern hemisphere); divide the sine of (the remainder or the sum) by the square root of the difference of the squares of the sine (found by the above subtraction or addition) and the Radius, and multiply by twelve. The result is the length of the shadow at noon. A rule for finding the length of the shadow at noon on any given day from the latitude of the place and the sun’s declination. By either deducting the sun’s declination from the latitude or adding it to the latter we obtain the sun’s zenith-distance. We then establish the proportion Square root of difference of squares of Radius and sine of zenith-distance : Sine of zenith-distance = 12 (i. e. length of gnomon) : x (i. c. length of shadow). ( 27 ) 23. The (sine of the) colatitude is found by taking the square root of the difference of the squares of the Radius and the sine of latitude. — The day- diameter is found by doubling the square root of the difference of the squares of the Radius and the sine of declination. The former of these two rules teaches how to find the sine of colatitude (lamba) or, what is the same, the cosine of latitude. — The latter rule teaches the calculation of the day-diameter, i. e., in this place, the diameter of the circle described by the sun on any day for which the sun’s declination is given. 24. The sines of declination of the sun at the end of Aries, Taurus and Gemini are twenty-four, forty-two and forty-eight plus twenty-four, fifteen and forty-eight sixtieths respectively. 25. The day -diameters (at the same places) are two hundred thirty- five, two hundred twenty-four and two hundred nineteen ; the diameters of Taurus however and the last one (i. e. the day-diameter at the end of Gemini) are to be increased by forty and fifteen sixtieths respectively. The two above stanzas apply the rules given before for finding the sun’s declination and day-diameter to the calculation of the said quantities at the end of the different signs. The calculation is made with the Radius 120, the greatest declination being assumed to amount to 24°. Its result is exhibited in the following tabular statement in which the whole numbers represent unities of which the Radius contains 120, and the fractions sixtieths of those unities. Aries. Taurus. Gemini. Sine of Declination. 24 24— ,,15 42 — 00 j -f 1 3 GO GO GO Day-diameter. 235 224— | to | 219— to 26. Multiply the sine of latitude by the diameter and by the sine of declination, and divide the product by the sine of colatitude and the day- diameter. Divide the minutes of the arc corresponding to the result by three. The quotient indicates the vinadikas of the ascensional difference. A rule for finding the ascensional difference for any given latitude and ( 28 ) any given declination of the sun, and therefrom the respective length of the day and the night. For this purpose the so-called earth-sine (kujy&) has to be found, which is done by dividing the product of the sine of latitude and the sine of declination by the sine of colatitude. Thereupon the sine of the ascensional difference is found by dividing the product of the earth-sine and the Radius by the day -radius, i. e., half of the day -diameter found previously. Of this sine of the ascensional difference the arc is taken. As six minutes of arc go to one vinftdikd of time, the minutes, in order to be turned into vin&dikfis, would have to be divided by six, but as, in order to determine the true length of the day, the double ascensional difference has to be added or deducted (according to circumstances), the divisor three is substituted for the divisor six. 27 — 28. Multiply the day -diameter by the sine of half the ascensional difference and divide by two hundred and forty ; the result is the earth-sine (kujyii). Take the square root of the sum of the squares of the earth sine and the sine of declination, and divide by that square root the product of earth sine and Radius. The quotient is the sine of latitude. Deduct the degrees of latitude from ninety ; the sine of the remainder is the sine of colatitude. A rule for finding the latitude and colatitude from the given ascen- sional difference. At first the earth-sine is found from the ascensional differ- ence ; the divisor is 240 i. e. 2 x 120= 2 R, since the multiplier is not the day- radius but the day-diameter. The earth-sine and the sine of declination are the two sides of a right-angled triangle whose hypothenuse (the measure of amplitude ; agr&) is therefore known. A proportion is then established Measure of amplitude : earth-sine = Radius : sine of latitude. The complementary sine of the sine of latitude is the sine of colatitude. 29. Of the signs beginning with Aries take the square of the sine and of the sine of declination ; by the square root of the difference of those squares multiply the diameter and divide by the day -diameter. Multiply the arc (of the sine thus found) by ten ; the result are the vin&dik&s of the rising of the signs. 30. Thus there are found beginning from Aries (i. e. for the three signs beginning with Aries) 278, 299 and 323 (as the vin&Jikiis of rising); further on (i. e. for the three signs Cancer, Leo, Virgo) the same quantities ( 29 ) are valid in reverse order (323, 299, 278) ; and the six quantities taken in reverse order are valid for that half (of the ecliptic) which begins with Libra. Stanza 29 gives the rule for finding the times occupied by the in- dividual signs of the ecliptic in rising in the right sphere. At first there is found — from the sine of the sign of the ecliptic (reckoning from the beginning of Aries) and the sine of declination of the end point of the sign — the sine of that arc of the day-circle described by that end point, which, if reduced to terms of a great circle, furnishes the ascensional equivalent of the sign on the equator. In order to effect the reduction, the sine is multiplied by the dia- meter (instead of the Radius) and hence divided by the day-diameter (instead of the day-radius). The degrees of the arc thus found are divided by 6 in order to ascertain the nadik&s (360°= 60 n&diMs), and then multiplied by 60 in order to give the vimtdik&s of rising ; hence the text directs us to multiply the arc by ten. — Stanza 30 states the numerical results of the pre- ceding rule for each sign of the ecliptic. ' 31. Three (i. e., the times of rising in the right sphere of the three first signs) are to be lessened by half the time of ascensional difference ; three (i. e., the corresponding times for Cancer, Leo, Virgo) are to be increased by the same (amounts of ascensional difference) in the reverse order. In the same time which a sign of the ecliptic occupies in rising, the seventh sign from it goes to its setting. A rule for calculating the time occupied by the signs of the ecliptic rising in the oblique sphere, i. e., in any given latitude. 32 — 33. Multiply the sine of the given degrees of northern declination by one hundred and twenty, i. c., Radius and divide by the sine of latitude. The time resulting from the arc corresponding to the sine (found by the above process) indicates the moment when the sun reaches, after sunrise, the prime vertical. The same time being deducted from half a day indicates when the sun in the western hemisphere (again reaches the prime vertical). The rule does not apply when the sun is in the six signs beginning with Libra. A rule for ascertaining the time when the sun is on the prime verti- cal. — The samamaiydalaSahku, t. e., the sine of the sun’s altitude when the sun is on the prime vertical is found by comparing two similar triangles, the former of which is formed by the Radius, the sine of latitude and the sine of colatitude; while the sides of the latter represent the sine of the sun’s ( 30 ) altitude when the sun is on the prime vertical, the sine of the sun’s decfinaticm, and that part of the sun’s day circle which is intercepted by the projections of the prime vertical and the six o’clock circle. On the supposition of the sun’s declination not changing within one day, the sine of altitude thus calculated will enable us to fix that moment also when the sun again reaches the prime -vertical in the western hemisphere. 'The rule has of course no application in the case of the sun having south-declination. 34. Multiply the sine of declination by 240 and by the sine of latitude, and divide by the sine of colatitude. Divide thereupon by the day-diameter. The entire sixth part of the corresponding are is the quantity by which the day is to be increased. A rule for finding the ascensional difference, basing on the proportion sin colat. : sin latit. — sin declin. : kujyfi. The result is -thereupon reduced to terms of Radius by another proportion. 35. Multiply, in the -northern hemisphere, the sine of the sun’s longi- tude by the sine of his greatest declination, and divide by the sine of latitude ; the result are the so-called minutes of the £anku (i. e. of the sine of the altitude of the sun when the latter is on the prime vertical) ; from them the shadow of the £anku is to be found. A rule for finding the sine of the altitude of the sun when the latter is on the prime vertical. — The first step is to find the sun’s declination for the given time ; this is done by means of the proportion Radius : sin greatest declin. — sin sun’s longit ; sin declin. Thereupon the ganku is found by comparing the two similar triangles of which the one is formed by the Radius, the sine of latitude and the sine of colatitude ; and the other by the 4ahku, the sine of declination and that part of the sun’s day circle which is intercepted by the prime vertical and the six o’clock circle. The proportion .established is : sin latit. : Radius * sin declin. : 6ahko. . ■>' ( 31 ) Combining the two proportions we get 6 nfi Vn Had x sin great, d eel in. x sin longit. sin great, decliu. x sin longit. sin latit. x Had * sin latit. * Therefrom the shadbw of the gnomon at the moment when the sun is on the prime vertical may be found without difficulty. 36. That person, who determines the moment at which the sun crosses the prime vertical and who is able to produce general confidence in his calculations, thoroughly understands the theory of the sun.. 37. “ If the sun performs one complete revolution in a year, how much does he accomplish in a given number of days ? ” Such like calculations even an ignorant fellow can easily perform by means of lines drawn with a piece of chalk. 38. When the shadow of the gnomon completely coincides with the east-west line in a circle in which the directions have been marked, then the sun is on the prime vertical. 39. Multiply the Radius by the given- sine of declination, and divide by the sine of colatitude ; the degrees of the result indicate the distance from, the east-west line at which the sun. rises or sets (on the given day). A rule for calculating the sine of amplitude (the so-called agrd), founded on the proportion sin colat, : R = sin declin. : sin amplit. 40. Multiply tile sine of declination by 120 and divide by that (i. by the sine of amplitude as found in 114) ; the result is the sine of colatitude; Deduct the arc corresponding to that sine from 90 ; tlie remainder are the degrees of terrestrial latitude. O A rule for finding the terrestrial latitude, the sine, of amplitude being given. The proportion is sin ampl; sin' deck — R : sin. oolat: 41 . Take the twentieth part of the vinddik&s to which the ascensional’ difference at the given time amounts, and put it down in two places. Ac- cording an the sun has northern or. southern, declination* deduct that twentieth. ( 52 ) part from — or add it to — the given n&dikAs (i e. the n&dik&s which at the given time have elapsed since sunrise) multiplied by six. 42. The sine of the remainder (resp. the sum) is to be increased or diminished by the sine of the twentieth part (mentioned above), according as the sun has nothern or southern declination. Of the nAdik&s multiplied by six take the sine without applying (to the said nadik&s) any correction. 43. Multiply the sine thus calculated (i. e. the sine of the remainder, increased or diminished in the way taught in 42) by the product of the day- diameter and the sine of colatitude, and divide by 28800. The quotient indicates the minutes of the sun’s altitude. 44. Deduct the square of those minutes from 14400, and take the square root of the remainder. Multiply the square root by 12, and divide by the minutes of the altitude. The result is the shadow. A rule for finding the length of the shadow at a given time. — The first step is to find the degrees of the ascensional difference which above was determined in terms of vinadfikAs. The vinadik&s divided by 60 give the n&dikAs, and as the ascensional difference had above been taken double (in order to determine the length of the day), 120 is substituted as divisor for 60. Six degrees going to one n&dikA, the final expression for the degrees of the ascensional difference is ; Vin&i x 6 Vin&d. 2 x 60 ~ 20 ‘ The given nadik&s (i. e. the nadikas which have elapsed from sunrise) are thereupon multiplied by six, whereby the number of corresponding degrees is found, and the degrees of ascensional difference as found above are deduct- ed from them in the oase of northern declination, and added in the oase of southern declination. The sine of the arc so determined is technically called * shtra ’ (op. Siddh&nta Siromani, Tripra4nAdhikAra, 54). This sfttra increased by the sine of ascensional difference in the northern hemisphere — and diminished by it in the southern hemisphere-* gives the sine technically called ishta antyA, From it is found the so-called ishta hriti, i. & , the perpendicular from the sun to the uday&sta sfitra, i. e., the line joining the points in whioh the sun rises and sets. (The ishta hriti represents the sum or difference of the sines of ( S3 ) two arcs, viz., that arc of the day circle which is intercepted by the sun and the six o’clock circle, and that arc which is intercepted by the six o’clock circle and the horizon). The proportion employed is as follows : Rad: antyu=Rad. of day circle: hriti. . i ti.: antyfi, x dyujyft _ »atyax 2 dyiijjA . . BOT — 120 ~ 240 In order to find therefrom the sine of the sun’s altitude the following O proportion is employed Rad : sin colat. = hriti : sin alt. • Qi'n «lf — autya x 2 X iii.i colat .. sill bhuja found above), and divide by the hypothenuse of the shadow. Of this result there has to be taken the difference (from the result found above, viz., the base of the sun’s altitude), if there is sameness ( t. e., if the bhuja of the 6anku and the base of the sun’s altitude are both measured in the same direction) ; while the sum has to be taken, if there is difference (i, if one of the mentioned quantities is to be measured from the north to the south, and the other from the south to the north). The result is the sine of the sun’s amplitude. 5fi. Multiply the latter sine by the sine of colatitude, and divide by the sine of greatest declination ; the result is the sine of the sun’s longitude. — According to the rules here givon for the sun the corresponding calculations have to be made for the other planets also, The above rule for calculating the sun's longitude from the observed ko^i ( — bhuja) i, c., the distance of the point of the shadow from the east- west line at a given moment is simply the reverse of the preceding rule. — The expression for the sine of the surfs longitude is as follows sin longit, sin ampl. x sin colat. , Sad. Rad. sin a m pi. x sin colat. sin great. decl> Bin great, declin. CHAPTER V. On the Moon’s becoming Visible. 1. Take on the one hand the sum, and on the other hand the differ- ence, of the distance of sun and moon and the difference of their declinations ; multiply the two quantities thus obtained and take the square root of the product. Divide by that product the difference of the declinations, which previously has been multiplied by the latitude of the moon. 2. The result has to be deducted from the interval of sun and moon, if the moon’s latitude has the same direction as the difference of the declinations ; in the opposite case the result has to be added. In the ease of the morning-twilight the directions as to adding and subtracting have to be reversed. 3. If that (corrected interval of sun and moon) rises in two nadikus, which we infer from the seventh sign of the ecliptic — counting from the sun ; then, the sky being clear, the moon is seen by men. A rule for determining the distance from the sun at which the moon becomes visible after new moon. Let (fig. 2) E S’ represent a segment of the Ecliptic, in which S is the place of the sun and E the point in which the moon’s circle of latitude cuts the ecliptic. The moon’s place is in. E B is a seg- ment of the day circle — parallel to the Equator — described by the point E of the Ecliptic; B S a segment of the sun’s circle of declination ; md a segment of the moon’s circle of declination. Treating all the arcs as straight lines, we have in the right angled triangle EBS ( 39 ) EB = v 'ES^TS i =v'(ES + B SfjES^BS) Now BS and m d being parallel, the angles E S B and m d E are equal, ami hence the two right angled triangles are similar. We thence find by proportion Ed = By the amount of Ed so found the distance of the moon from the sun in the ecliptic has to he corrected. Ed has to be deducted in case of the moon’s latitude and the difference of the declinations of the two bodies having the same direction ; while it has to be added in the opposite case. — The moon is first seen when Iter corrected distance from the sun amounts to so much that it sets — or, what is the same, that an equally large piece of that sign of the Ecliptic, that is seventh from the sign in which the sun is at the time, rises — in two midikas. 4. (The text and meaning of the first half of this stanza is doubtful ; it possibly contains a remark of an astrological character, as explained in the Sanskrit— Commentary). The twelfth part of the hypothenuse represents the illumined part of the moon ; it is to be laid off in the direction of the base. By the hypothenuse we have to understand the interval between sun and moon as in figure 2. The moon’s diameter is supposed to be divided into fifteen angulis all of which are illumined when the moon is in opposition, i. c., at the distance of 180° from the sun. Hence the formula Illui nined part = — -f|\ 5. The perpendicular is constituted by the difference of the declina- tions (of sun and moon) and of the latitude ; of which two quantities the sum has to be taken if their direction is the same, otherwise the difference. The hypothenuse is the interval between sun and moon. The square root of the difference of the squares of those two sides is the base. 6. Let the perpendicular be laid off in that direction in which the sun is with regard to the moon, one degree being supposed equal to one ahguli ; in the same way the base and the hypothenuse are to be laid of! with angulis. ( 40 ) 7. First lay off the hypothenuse from the centre of the moon ; then from the hypothenuse the perpendicular ; then the base in the direction of the moon’s centre. This base is called aksha (axis) on the moon’s circumference. Measure out the illumined part from the middle (i. e., the hypothenuse which divides the moon into two halves) ; it has the shape of a bow. ‘ Directions how to draw a Fig. 3. diagram of the illumined part of the moon, which are immediately intelligible from figure 3, in which S represents the sun’s place and m the centre of the moon. The line mh, which ordinarily is called koti, Var&ha Mihira calls bhuja, and h S which ordinarily is called bhuja he calls koti. The illumined part of the moon is abc ; the angulis which determine the amount of the illumination are measured off on the dia- meter bin from the point b. 8. Multiply (the moon’s) southern or northern latitude by the equinoctial shadow, and divide by 12. The resulting degrees are, in the case of the moon’s rising, to be added (if the latitude is southern), and to be deducted (if the latitude is northern) ; the reverse has to be done at the time of setting. 9. From the longitude of the sun lessened by that of the moon (ascertain the number of the intervening signs) ; according as they are less or more than six, the rising of the moon has to be determined as taking place during the day or during the night, owing to the rising of the sun. 10. Having thus (i. e. in the way indicated in stanza 8) performed the subtraction or addition, deduct the moon’s longitude, lessened by that of the sun, from half the circle ; in the time (reckoned from sunrise) equal to the time of the rising of the remaining signs the moon goes to her setting. The above are rules for finding the time of the moon’s rising and setting. — First there is given a rule for the so-called drik-karman, which however neglects the dyana-drik-karman, and limits itself to the aksha-dpik- ( 41 ) karman, i. c., that operation by which there is determined that point of the ecliptic, together with which a planet having latitude rises in a given terres- trial latitude. In figure 4 E L B re- P presents an arc of the Ecliptic, L m H the horizon, m the moon’s true place, E its place in the Ecliptic to which it jj is referred — in the Indian fashion— by means of a circle of declination, L that point of the Ecliptic which rises to- gether with the moon. In order to determine E L, we look upon the triangle Em L as a plane one, in which the angle E m I j is equal to the terrestrial latitude, and its complement m L E to the colatitude. We then establish the proportion Fig. 4. EL: sin latit. = E m : sin colatit. EL sin lat, x E in ein culat. we finally obtain •/ Substituting Equin. shad , e* gin hat. 12 ( —gnomon) ^ Bincolat/’ EL moon’s lat. x equin. shad. 12 " The amount so found is added to, or deducted from, the moon’s longitude, according as the moon has southern or northern declination. — Stanzas 9 and 10 present no difficulties. CHAPTER VI. Lunar Eclipses (according to the Paulina Siddhanta.) 1. From the nadik&s of the tithi, which, taking sunrise as the starting point, have elapsed or are yet to come, the corresponding minutes of motion of the moon have to be calculated. They are to be deducted from the moon’s place (in case of the full moon tithi having terminated before sunrise), and we thus obtain the place of the moon at that time (i. e., the time of the termination of the full moon tithi, i. e., the time of opposition) ; while in the opposite case the minutes have to be added. There is some corruption in the last line of the text of the stanza ; but there can be no doubt about the general meaning. The longitude of the moon had hitherto been calculated for sunrise ; now, when the discussion of lunar eclipses begins, it becomes requisite to find it for the moment of opposition, i. e., the end of the tithi of full moon. According as that moment precedes or follows the sunrise for which the moon’s longitude had been found, the latter has to be diminished or increased by the amount of the moon’s motion during the interval. 2. Deduct from the longitude of Rahu twenty-six minutes, and thereupon take the degrees intervening between Rfthu and the moon. If these degrees are within thirteen, there is an eclipse; if within fifteen, there is the shadow of an eclipse. 3. Deduct the square of the minutes of the moon’s latitude from the square of fifty-five, and take the square root of the remainder. Doubling it and operating upon it in the manner of the tithi, we find the time of the duration of the eclipse. 4. Lessen thirteen by the degrees of the interval between the moon and Ra.hu, and multiply by five ; the result taken as vin&dikfts is to be added to the duration of the eclipse in the case of Rahu being more advanced in longitude than the moon ; otherwise it is to be deducted. ( 43 ) The former half of stanza 2 refers back to the rule given in III. 29 concerning the place of the moon’s node. Its latter half states the lunar ecliptic limits. By the expression ‘shadow of an eclipse’ we apparently have to understand a quasi-eclipse, i. t\, the diminution of the moon’s light by the penumbra (?) Stanza 3 calculates the duration of the eclipse from the line 1 C (fig. 5) which itself is determined as one of the sides of a right angled triangle of which ml i. c., the lati- tude of the moon at the moment of contact with the shadow is the other side, and mC i. e., the sum of the radii of the moon and the shadow the hypothenuse. Seventeen minutes are taken for the mean value of the moon’s radius, and thirty-eight for the mean value of the radius of the shadow ; the sum of these two quantities amounts to fifty-five. The time which the moon takes in passing — by means of the difference of the motions of sun and moon — along 1 C is half of the total duration of the eclipse. This time is found by a proportion similar to that employed for finding the nadikas, past and to come, of a tithi, viz. Minutes of diff. of motions : 60 nad. = double the minutes of 1 C : duration of eclipse. To the duration so found stanza 4 directs us to apply a correction whose rationale we are however unable to assign. 5. By five, diminished by the degrees of interval (between the moon and her node), let ten be diminished and multiplied, then multiply by four, take the square root and multiply the latter by twenty-one ; the fifth part of the result indicates the minutes of total obscuration. A rule for calculating the time of complete obscuration during a total eclipse of the moon. — The quantity to be calculated first is the latitude of the moon at the moment when complete obscuration begins. For this purpose ( « ) the sine of the degrees intervening between the moon and the node is calculated, according to the formula : sine= (21 being the sine of 10 degrees if the Radius = 120, and ten degrees being introduced into the formula, because in the case of a total eclipse the interval between moon and node must be less than 10 degrees.) The sine of latitude is then calculated on the assumption of the moon’s greatest latitude amounting to 240', whence the proportion Radius : sin great, latit. = ?— -J l3g£ff ; sin lat. 240 x 21 x degrees 21 x degrees^ • • Sln ~ 120 x 10 5~ ' ' Having thus found m'l' (%. 5) we have l'C = v/ E'C*^T» = ^(Ct-tmO’-^xde^ 5 = V/ (38 — 17)’ — x degr.’ = 'x' / 25 — degr.’ = (5 - degr.)(5 + degr,) - = - degr. j [10 — (5 -degr.)] Multiplying this last expression by two, in order to find the whole duration of total obscuration, we finally have : Minutes of total obscuration -~l- x 2V / [5 - degr.] [10 - ( 5 -degr.)] = V/ 4 [5- -degr.] [ 1 0 - ( 5 - degr. )] 6. Within a time equal to the difference of half the duration of the eclipse and half the period of total obscuration the shadow swallows the whole moon The direction of the contact and the separation is to be determined by means of the degrees of the interval between the moon and the node. 7. That direction is opposite to the direction of the moon’s latitude. Divide the quadrant of the moon’s circumference into thirteen parts, and from the east-point lay off the degrees of the direction of the eclipse (i. e., the degrees of the valana or deflection). At the point thus determined the parvan, i. e., the beginning of the eclipse takes place. 8. Multiply the degrees of terrestrial latitude by a quarter of the moon’s circumference, and again by the degrees intervening between the moon and the zenith, and divide by 8100; the result is the deflection (valana) which is north in the eastern hemisphere, south in the western one. ( 45 ) A rule for calculating the soealled aksha valana, i. e., the deflection due to latitude (the ay ana valana being altogether neglected). The rule given in stanza 8 for the aksha valana is based on the proportion : If 90'' zenith-distance give a deflection equal to the terrestrial latitude, how much do the given degrees of zenith distance give? — This is combined with the further proportion : If 90“ give an are equal to one quarter of the moon’s circum- ference, how much do the calculated degrees of deflection give ? — The resulting formula for the arc of deflection to be laid oft’ on the moon’s circumference is ^ Tin zenith dist. x latifc, x \ circumf. of moon c * Do x yoTsioa “ " We do not know the reason for the direction — given in stanza 7 — to divide each quarter of the circumference into thirteen parts. 9. During a total eclipse dark yellow is to be declared the peculiar colour of the moon ; her colour is dusky in the ease of eclipses taking place during the rising or setting of the moon ; and waterish in the case of partial eclipses. 1 0. [A stanza of doubtful import ; see the Sanskrit Commentary.] 11. Describe in one place ( i . e. from one centre) three circles, one, vis. f the moon’s circle with a string measured by seventeen minutes, a second one, viz., the circle of the shadow with a string of thirty-eight minutes, and a third on c,viz., thecircleof duration with a string equal inlengtli to the sum of the two quantities. 12. (Ascertain) by means of the degrees (of deflection) to be laid oft in the direction stated above the east and west-line of Lanka, and draw on both sides of it thirteen long lines, at equal distances from one another. 1 3. This is a concise description of the projection of a lunar eclipse, the details of which are to be known from explanation. The first contact, the complete obscuration and the duration are there seen through appropriate constructions. 14. In a lunar eclipse the moon is seen to touch the shadow of the 0 arth from the west-side ; in the same way she touches, in a solar eclipse, the sun from the west-side ; hence the eastern part of the moon is first touched, but not that of the sun. Stanzas 11-13 contain some rather general directions how to draw the projection of a lunar eclipse, on the ground of the aksha valana previously calculated. Compare the Sanskrit Commentary. CHAPTER VII. Solar Eclipses, according to the PauliSa Siddhant 1. As many nadikfts as there remain until midday (from the time of conjunction), or as many as have elapsed since midday, have to be multiplied by six. The thirtieth part of the corresponding sine is the “ bending” (displacement) of the tithi (i. e. the parallax in longitude). A rule for calculating the parallax in longitude. We ascertain the hour angle (multiplying the given nadikas by 6, six degrees going to each nadika), and establish the following proportion : Radius : greatest parallax = sin hour angle : desired parallax. The greatest parallax being assumed equal to the mean motion during four nadikas, we thus have : Desired parallax = 2 — 4. [Three stanzas of doubtful import.] 5. Deduct twenty-six minutes from the longitude ofRjthu, and take the degrees intervening between Rahu and the moon. If they are within thirteen, there takes place an eclipse of the moon ; and an eclipse of the sun, if they are within eight. A stanza agrreeing, as far as lunar eclipses are concerned, in contents and enunciation with stanza 2 of chapter VI. 6. Deduct, in the case of the moon, the square of, the degrees of the interval from 169, in the case of the sun from 64 ; the square roots of the remainders minus their fourth parts indicate the duration of the lunar and solar eclipses. ( 47 ) If we calculate the moon’s latitude at the moment, when complete obscuration begins, in the manner taught in stanza 5 of the preceding chapter, with the one difference of making the greatest latitude equal to 270 ( not 240) minutes, we obtain latitude = 270 x ^ x degrees of interval 120 = aw J c^x jte= »xj SE!? (approximately ). By means of the latitude so found, and assuming half the sum of the diameters of moon and shadow to be equal to 58 minutes, we find — in the customary manner Measure of half duration of eclipse = V / 58 5 — latitT * .--v/ 58 J _ -|l (degrees of interval)' = | ^ ( 5? y- 2 ) ! —degrees ! — 13* — degrees' (approximately). In order to find from the latter expression the whole duration of the eclipse in terms of nadikas, we multiply it by 2 x GO, and divide by the difference of the daily motions of sun and moon, when we obtain : duration of eel ipso =^ 574 ° ^169 — degrees' = \/ 169 — degrees' = ^169~^degrees*. In the case of a solar eclipse w'e find by an analogous process, and assuming half the sum of the diameters of sun and moon to be equal to 35 minutes : Measure of half the eclipse = ^ 35' — j- x degrees* = i m ~ degrees’ = 3^6 4 — degrees' (approximately). And for the duration of the whole eclipse in nadikas Duration = Y 64— degrees* = | \/ 64— degrees' (approximately). CHAPTER VIII. 5olar Eclipses, according to the Romaka SiddhAnta. 1. Multiply the ahargana by 150, deduct 65, and divide by 54787 ; the result is the mean longitude of the sun in due succession ( i . e. revolutions, signs, etc.), according to the Romaka-SiddMnta. This rule follows immediately from the nature of the yuga acknowledged by the Romaka Siddhanta, as described in I. 15. Instead of -~~j the reduced fraction is employed. Sixty-five is the kshepa — quantity enabling the calculations to start from the epoch chosen. 2. The operation of finding the true places of the sun and the moon is performed by means of the quantities (about to be stated) measured for half signs of the anomaly of sun and moon, and arranged in direct as well as inverse order. Of the sun (the mean longitude) has to be deducted from half of Gemini, i. e., from two and a half signs. The Romaka Siddhanta calculates the equation of centre of the sun and moon for each half sign, i. e., progressing from 15 to 15 degrees. The six quantities thus obtained for the first quarter of the circle are employed for the second quarter in the inverse order and so on. — In order to obtain the kendra, i. e., the anomaly of the sun we have to take the difference of his mean longitude and of the longitude of his apogee, which is estimated at signs =75°. 3. Twenty increased in succession by fifteen, fourteen, ten and four, and diminished by six and fourteen are the minutes (which added up in succession give the amounts of the equation for 15° 30° 45° etc). Eighteen and five seconds have to be deducted (from the first and second quantities) ; (to the four others) two, ten, sixteen, eighteen seconds have to be added. « The six quantities indicated in the above stanza stand as follows : ( 49 ) 34' 42"; 33' 55" ; 30' 2" ; 24' 10" ; 14' 16" ; 6' 18" ; which added up give the following equations Anomaly i.v 30° 45° 60° 75° 90' Kquafcion j 34' 42" j 68' 37" 98' 39" 122' 49" 137' 5" 143' 23" 4. Multiply the ahargana by 38900, deduct 1984, and divide by 1040953 ; the quotient indicates the mean longitude of the moon. This rule follows immediately from the constitution of the yuga assumed in the Romaka Siddhanta (excepting the kshepa quantity). 5. Multiply the ahargana by 110, add 609, and divide by 3031 . the quotient gives the position of the moon’s kendra at sunset in Avanti. The assumption here is that the kendra of the moon performs 1 1 0 revolutions in 3031 silvana days or, in other words, that the latter period contains 110 anomalistic months. 6. One degree plus 14, 11 and 2 (minutes) ; four times eighteen (i. e., seventy-two) lessened by eight times three (24); five times six (30); and sixty minus eight times six ( 1 2). The last two quantities are to be lessened by one. A statement of the differences of the moon’s equation of the centre, taken from fifteen to fifteen degrees. By adding up the stated quantities we obtain the following table Anomaly 15° 30° 45° 60° 75° 90° Equation r 14 ' 2° 25' 3° 27' 4° 15' ' 4° 44' 4° 56' 1 7. The daily motion of the moon amounts to 790 minutes, that of the moon’s anomaly to 784. By taking the difference of the true motions (on two consecutive days) we obtain the past true daily motion, as weR as the one during the coming night (i. e. nychthemeron). ( 50 ) 8. Multiply the ahargana by 24, add 56266 and divide by 163111 ; the result is the successive position (in revolutions, signs, etc.) of the head of RS.lm (i. e., the moon’s ascending node), reckoning backwards from the 'end of Pisces ( = first point of Aries). Twenty-four revolutions of the node — whose movement is retrograde — are supposed to take place in 163111 .savana days. 9. As many nadikas as there remain until midday, or else as many mldikas as have elapsed since midday, are to be multiplied by six. The thirtieth part of the corresponding sine is the displacement of the tithi, (i. e., the parallax in longitude), A rule literally agreeing with the corresponding rule of the Paulina Siddh&nta (VII. 1), 10. And from the nfulikas which have elapsed since sunrise calculate the orient ecliptic point ; from the latter plus nine signs (i. from the point called vitribha or tribliona) ascertain the degrees of declination. A rule for finding the highest point of the ecliptic, the so-called tribliona or vitribha, whose longitude is less by three— or more by nine — signs than that of the orient ecliptic point. — In order to find the zenith distance of that point we have at first to determine its declination. 1 1. Multiply the sine of the difference of the orient ecliptic point, of three and of the node by two, and divide by sixty. The result in degrees is to bo deducted from the domination (calculated above) if the directions of the two are opposite ; while the two are to be added, if the result (and the declination) have the same direction, 12. If northern, (the declination so corrected) deducted from the terrestrial latitude — and, if southern, added to it — is to be considered as the southern (zenith distance of the tribhona) ; while if northern and greater than the terrestrial latitude it is to be viewed as northern (zenith distance). A process preliminary to the calculation of the parallax in latitude. What we require — and what generally is ascertained in Hindu Astronomy to that end — is the zenith distance of that point of the ecliptic which has the greatest altitude (which point is called vitribha or tribhona) ; the siue ( 51 ) of which distance is tech- nically called drikkshepa (T Z in figure (> ; in whicli figure A C is the Ecliptic, a c the moon’s orbit, and V Z T 1) the projection °f a great circle passing through the pole of the Ecliptic, the zenith and K the tribhona T, and cut- ting the moon’s orbit in t)- S The Romaka Siddh&nta however makes the amount of the parallax of latitude to depend not on the drikkshepa, but on the sine of t Z. In order to calculate this latter quantity it at first calculates t T. There having been ascertained d t 1 i. e., the interval between the tribhona and the moon’s node, the following proportion is established Rad : sin greatest latit, of moon = sin d T : sin t T. The greatest latitude being assumed equal to ‘240', we have , m 240 x*indT 0 rp sm tT= — i-jy — = 2 sm a 1. The resulting minutes are turned into degrees by being divided by sixty. To t T so found T Z ought to be added in order to find the zenith distance of t ; but instead of T Z there is taken LZ which is known from the declination of L and the terrestrial latitude. 13. Multiply the daily motion of the moon by the sine of that (zenith distance as found above), and divide by 1800 ; the result is the parallax in latitude. The mean measure of the sun is thirty minutes, that of the moon thirty-four. Fig. 6. ( 52 ) 14. Multiply the sine of the distance of the moon, which (at the moment of conjunction) has the same longitude (with the sun), from the node by twenty -one and divide by nine ; take the sum of the result and of* the parallax in latitude, if there is sameness of direction ; and the difference of the two, if there is opposition. Rules for finding the parallax in latitude and the moon’s true latitude. The rule for finding the parallax is founded on the supposition that the greatest parallax is equal to the fifteenth part of the moon’s daily motion ; we therefore have the proportion Radius :: — 1 — j“ — - = sin zenith distance of tribhona : parallax Parallax = daily motion x sin zen. dint. ~ 15x120 In order to find the latitude of the moon we at first establish the proportion Radius: sine of greatest latitude ( = 27 O') — given sine of moon’s distance from node : desired latitude. We therefore have T 270 X sin distance JLdltlt — I2Q 27 x sin diat. 21 x ein dist. 3x4 3x4x21 21 x sin di»t./ •. . , . 3x3 (approximately ) . By increasing or diminishing the latitude so found by the parallax found above we obtain the true latitude. 15. Multiply the true motion of the sun and the moon by their mean measure, and divide by the mean motion ; the result is the true measure, in minutes, of the two bodies at the given time. 16. Deduct the square of the avanati (i. e. the latitude of the moon as corrected for parallax of latitude) from the square of half the sum of the measure of sun and moon. From the double square root of the remainder determine the time (of the eclipse), as in the case (of the calculation) of the elapsed portion of a tithi. ( 53 ) A rule for determining the duration of the eclipse. We at first take the right angled triangle of which half the sum of the diameters of sun and moon forms the hypothenu.se, and the moon’s corrected latitude the perpendi- cular ; tlie base of this triangle represents half the time of the duration, expressed in minutes of are. We then establish the proportion Minutes of difference of sun’s and moon’s motion : 60 n&f which expression only the former term is employed for the general rule, while to make up for the neglect of the latter a correction is stated in the next stanza. 4. Multiply the revolutions of the moon by 51, and divide by 3120 ; deduct the result taken as seconds. —Also multiply the revolutions of the moon’s uchcha by 10 and divide by 297 ; the resulting seconds are to be added to the moon’s uchcha. This stanza states the corrections referred to above, which have to be applied to the mean places of the moon and her apogee as calculated according to the general rules embodied in stanzas 2 and 3. —A calculation (about whose details see the Sanskrit Commentary) shows that, in order to make up for the terms neglected in the general rule, seconds for each completed revolution have to be deducted from the moon’s mean place; and that on the other band seconds for each revolution have to be added to the place of the moon’s apogee. 5. Multiply the ahargana by 270, add to it , and divide by 1 834582 ; the result is the place of Raliu in revolutions, etc. 6. Deducting this from 12 signs we have the head of R&hu (i c. t the moon’s ascending node) : while by adding six signs we have the tail of RfLhu (i. e„ the descending node). The minutes of arc intervening between R4hu and the moon (serve to calculate) the moon’s latitude ; 270 minutes (are the greatest latitude). ( 56 ) The above rule for calculating the place of the moon’s node requires the assumption of 232226 revolutions of the node in a mah&yuga ; for in that ease we have the proportion i 57 ^i^oo~ > which may be reduced to The line stating the kshepa quantity of R&hu is too corrupt to lend itself to emendation ; we may however in this place state the principles on which all the kshepa quantities given in the preceding stanzas of this chapter are calculated. They of course all have the purpose of throwing the rules for finding the mean places into a shape enabling us to begin our calculations not from a very remote period — which would oblige us to use inconveniently large multipliers and divisors — but from the epoch of the Pahchasiddhantika, viz . , 427 S'aka. From tentative calculations it appears that the kshepas exhibited by Varaha Mihira are calculated, not from the beginning of the Kaliyuga or the MahAyuga, but from the beginning of the Kalpa. We therefore have to form the ahargana from the beginning of the Kalpa down to 427 S’aka (cp. the Sanskrit Commentary and also Jour. A.siat. Soc. of Bengal vol. LIII. 1884 p. 268 IF), and to treat that ahargana according to the general rules for calculating the mean places. We thus find that the place of the sun at the epoch of the Panchasiddhantika was 2 ^ 2 §oV‘ as tl> e Sftrya Siddhanta makes all its calculations for midnight, while Var&ha Mihira declares that the rules given by him are adapted to midday, we have to deduct a further amount equal to the sun’s motion during half a day (= “ 2 x 292267 * — - 29 &T> 7 ~ ) an< ^ thus finally obtain “ 2022 ^ 7 ’ which agrees with the kshepa stated in the text. — Calculating the moon’s position at the epoch in an analogous manner, and again making an allowance for the moon’s motion between midday and midnight, we obtain for the kshepa— 670197, which differs by an inconsiderable amount only from the kshepa stated in the text, mz . — 670217. — And finally applying the same process to the moon’s uchcha we find for its kshepa 2260355, which differs by one only from the quantity stated in stanza 3. 7 — 8. By deducting eighty degrees from the sun’s longitude the anomaly is found, and the moon’s anomaly by diminishing her longitude by that of her apogee. Multiply the sine of the anomaly by fourteen in the sun’s case, and by thirty-one in the moon’s case, aud divide by 360 ; take the arcs corresponding to the results. Put the arcs down in two places. The result which follows from the calculation for the moon is subtractive in the first half of the circle and additive in the second. ( 57 ) The longitude of the sun’s apogee is assumed to amount to 80°, and the circumferences of the epicycles of sun and moon to 14° and 31° respect- ively. —The direction to put down the result in two places seems due to the circumstance that it is employed also in the calculation taught in the next stanza. —What, in the latter half of the second stanza, is said about the moon’s equation, is valid for the sun’s equation as well. 9. Multiply the sun’s equation, which had been put down separately, by the sun’s daily motion, divide by 21600, and treat the result in the case of the sun as in the previous process ( i . ., the parallax in longitude) as calculated for the beginning of the eclipse and of the half dura- tion (gives the true half duration of the eclipse) ; if the eclipse takes place in the western hemisphere, the parallax has to be added. We proceed in an analogous manner with regard to that half of the duration which precedes the separation. CHAPTER X. Lunar Eclipses, according to the SOrta SiddhAnta. 1. Multiply the true distance of the sun by 90, and divide by 276 ; the result is to be employed as the divisor of the true distance of the moon multiplied by 36. Deduct the result from 36. 2 a. Multiply the remainder by 120, and divide by the true dis- tance of the moon ; the corresponding arc is the diameter of the shadow. A rule for calculating the diameter of the true shadow. Let S of the path of the shadow, m n half the diameter of the true shadow which — D m being drawn parallel to F n — is equal to F D. E u being drawn parallel to F t, we have the two similar triangles wSE and E D m, in which S u is equal to the Radius of the sun minus the Radius of the earth, e. (the abridged diameters of the sun and the earth being put equal to 146 and 36) to 73-18 = 55 ; hence E D = ^ ' true chat, of etui true diet, aun x 90 55x90 . ... — — - x true dist. moon A/O 90 x true diet, aim ~~ 276 18 x true dist. moon 90 x true dist. aun 276 The quantity so found being deducted from eighteen, t. e., the Radius of die earth, we obtain the value of D F. — The process prescribed in stanza 2 has the purpose of expressing the half diameter — so far found in yojanas — in minutes of arc. — In order to double the result (so as to find the whole diameter), JJ6* is substituted for 18 in tho above expressions, ( 64 ) 2 b. Divide the sum of the diameters of the moon and the shadow by 2, and take the square (of the quotient) ; 3. Deduct therefrom the square of the latitude, and take the approximate square root. Multiply the latter by 1 20, and divide by the difference of the motions of sun and moon ; the result are the n&JikS-s of the duration of the eclipse. 4. Calculate therefrom the latitude of the moon at the moment of first contact, and therewith again, in the manner described above, the duration of the eclipse. Repeat this operation again and again, until there is no longer any difference with regard to the duration. A rule for calculating the duration of a lunar eclipse. At first we calculate the socalled bhuja, which is one of the sides of a right angled triangle, of which the hypothenuse is formed by the sum of the Radii of moon and shadow, and the other side by the latitude of the moon at the moment of contact. As however that latitude is not yet known, the moon’s latitude at the moment of conjunction is substituted for it. Thereupon a proportion is established Minutes of difference of motion of sun and moon: 60 n4dik&s = minutes of bhuja : half duration of eclipse Half duration 3^*3, and Duration = 120 x bhuja minutes of dill. As we now know the longitude of the moon at the time of the first contact, we calculate the latitude for the same moment, and therewith again determine the bhuja and the duration. This process is repeated, until the results no longer differ. 5 . Multiply the difference of the motions of sun and moon by the given n&dik&s, and divide by sixty. Take the difference of the minutes of that (result) and of the minutes of duration (as calculated above), and further take the latitude of the moon at the given time ; ( 65 ) 6. Deduct the square root of the sum of the squares of those two quantities from half the sum of the measure, expressed in minutes, etc., of the moon and the shadow. The remainder indicates the part of the moon or the sun which is obscured at the time. A. rule for finding how large a part of the moon (or sun) is obscured at a given moment. — We calculate for the given moment the bhuja and the latitude of the moon ; the square root of the sum of the two quantities, is the distance of the centres of shadow and moon at the given time. The difference of this latter quantity and the sum of the Radii of the two bodies indicates the amount obscured. — Mutatis mutandis the same rule may be used for calculating how large a part of the sun is obscured at a given moment of a solar eclipse, 7. Take(l) half the difference of (the diameters of) the eclipsed and the eclipsing bodies, and (2) the latitude ; multiply the square root of the difference of the squares of those two quantities by two and treat it in the manner of the tithi. The result is the time of complete obscuration of the sun and the moon. The above rule for calculating the time of complete obscuration is analogous in every way to the rule — given above — for finding the time of the duration of the eclipse, and therefore requires no special comment. CHAPTER XI. (On the projection of eclipses according to the Scry a Sipdhanta). 1. By moans of a staff, on which the ahgulis are marked, describe a first circle with a Radius equal to the sum of half (the diameters of) the obscuring and obscured bodies, and mark on it the different directions. Describe a second circle with half the diameter of the obscured body. 2. Multiply the sine of terrestrial latitude by the versed sine of the degrees intervening between the moon and the zenith ; add the degrees corresponding to the 120th part (of that product) to the north or south respectively in the eastern and western hemispheres. 3. Add three signs to the longitude of the moon, and treat the corresponding degrees of declination according to their direction. Thus the east-west direction is ascertained ; the north and south points are to be determined therefrom by means of a fish-figure. 4 — 5. Draw a line, representing the moon’s latitude, in the direction opposite (to that latitude), which extends up to the east-west line and cuts the second circle. The first contact takes place at the point where a line drawn from the centre cuts the other circle (viz. the circle representing the eclipsed body). The point of separation also is to be determined analogously, in the opposite way ; in ascertaining the direction one has to start, in one’s considerations, from the moon at the moment of separation. Rules for calculating the valana (deflection), aksha as well as ilyana, and for drawing a projection of the eclipse. The rules altogether agree with those of the modern Sftrya Siddh&nta. For the details see the Sanskrit Commentary. 6. On the horizon two minutes (of the diameter of some heavenly body) go to one ahguli ; while three go to it (when the heavenly body is) in the zenith. If the body is placed between (the horizon and the zenith), a proportionate calculation is to be made, to the end of bringing about agreement between observation and theory. CHAPTER XII. PaitAmaha Siddhanta. 1 . Am >rding to the teaching of Pitamaha five years constitute a yuga of the sun ami moon. The adlmniisas are brought about by thirty mouths, and an omitted lunar day (avama) by sixty-two days. 2. Lessen the time of the Saka king by two, and divide by five ; with the remaining years form the ahargana, which begins with the light half of Milgha. The ahargana begins with the day, viz., from sunrise. In the quinquennial lunisolar yuga of the PaitAmaha siddhanta one adhimasa is comj>rised within each period of thirty solar months, and one avama, i. omitted lunar day within each period of sixty-two days. According to Stanza 2, which directs us to deduct two from the num- ber of elapsed Saka year >, a new yuga began with 2 S'aka elapsed. 3. If the ahargiimi is increased by its own sixty-first part, the result are the tithis. If it is multiplied by nine and divided by 122, the result is the nakshatra of the sun. Multiply the ahargana by 7, divide by 010, and deduct (the result from the ahargana); the result is the lunar nakshatra, counting from D banish tha. Rules for calculating the tithis contained within a given ahargani, and the nakshatra s in which the sun and the moon are at a given time. — The yuga comprising 1830 savana days, and at the same time 1860 tithis, the number of tithis of a given ahargana = — ~ — ^ - ahargana + — ; gp~ . As the sun revolves in one yuga five times through the twenty- seven nakshatras, the nakshatras through which he passes during a given . *27 x 5 x ahargana 9x ahai-pf. ahargana = I5F“ As the moon passes in one yuga through 27 x 67 nakshatras (the yuga comprising 67 sidereal revolutions of the moon), she passes within a given ahargana through = 2 Ii^ = 6 -^=almrg. ( 08 ) The nakshatras are to be counted from DhanishtM, in which sun and moon are in conjunction at the beginning of the yuga. 4 a. [See the Sanskrit Commentary]. 4b. Multiply the ahargapa by 12, and divide by 305 ; the result are the vyatipatas. A rule for finding how many of the yogas called vyatip&tas have oc- curred within a given ahargana. There are 27 yogas which are calculated by dividing the sum of the longitudes of sun and moon by 27. At the begin- ning of the quinquennial yuga sun and moon are in conjunction at the begin- ning of D banish th a or — which is the same — at the end of SVavana. The longitude of each therefore amounts to 22 nakshatras — if we count in the ordinary way from Asvini — , and the sum of their longitudes to 44 nakshatras. Forty-four being divided by 27, the remainder (=17) indicates that the yoga at the beginning of the yuga is the 17th of the series, i. e., Vyatipata. Now in one entire yuga the accumulated longitude of the sun amounts to 5 x 27 nakshatras, and that of the moon to G7 x 27 nakshatras ; hence the sum of the two to 72 x 27. Dividing this sum by 27, the quotient 72 indicates how many vyatipatas take place in one yuga. Hence the proportion 1830 ( = days of yuga) : 72 = given aharg. : x — 72 x aharg. 12 x aharg. x— — 305 . 5. Add to 732 those days of the northern progress of the sun which are passed, and in the case of the southern progress those days which are yet to come ; multiply by 2 and divide by 61 ; the result is the measure of the day minus twelve. A rule for finding the length of any given day of the year. The supposition being that the length of the shortest day is twelve muhhrtas, and that of the longest day eighteen muhurtas, and each ayana comprising 183 days, the length of any day of the year is found by adding to twelve the product of six and the number of the day, divided by 183. In the case of the uttar&yaija the number of the day is counted forward from the winter solstice, while in the case of a day in the dakshijjayana it is counted back- ward from the same point of time . We then transform the expression for the length of the day in the following manner 1 9 J. 6x g iven d “y _ 1 9 . 2 X dftjr _ 9 . 2 x day , n _ 24 x 61 +2 xday + 183 — 1x5 + -«r — + ~61 14 61 = -£ r (l2 x 61+day)— 12 = gf(732 + day) - 12. CHAPTER XIII. On the constitution of the Universe. 1. The round ball of the earth, composed of the five elements, abides in space in midst of the starry sphere, like a piece of iron suspended between magnets ; 2. Covered on all sides with trees, mountains, towns, groves, rivers, oceans and other things. — In its middle there is Sumeru, the abode of the gods. Below (i. e. at the pole opposite to Meru) there are placed the Asuras. 2. As the reflections of men standing on the brink of water are seen with the faces downwards, so the gods consider the condition of the Asuras to be ; and those on their part deem the gods to be below. 4. Just as here in the region of men the flame of the fire rises up- wards into the air, and as a heavy object when being thrown falls to the earth ; so it also happens below there in the region of the Asuras. 5. Straight above Meru in space one pole is seen ; the other pole is seen below, placed in space. Fastened to the poles the sphere of the stars is driven round by the pravaha wind. 6. Others maintain that the earth revolves as if it were placed in a revolving engine, and not the sphere ; if that were the case, falcons and other (winged creatures) could not return from the ether to their nests. 7. And, to mention another argument, if the earth revolved in one day, flags and similar things would, owing to the quickness of the revolution, stream constantly towards the west. If the earth, on the other hand, moves slowly, how docs it revolve (once within 24 hours)? 8. If, in agreement with the doctrine of the Arhat, there were two suns and moons rising by turns, how then is it that a mark made in the polar constellation by means of a line drawn from the sun revolves within one day ? ( 70 ) The Jaina doctrine is that there are two suns, two moons, etc, rising on alternate days. But if we at sunset draw a line from the sun to the constel- lation of the polar fish, we observe that that point of the constellation which the line reaches, i. e., a point on its western side is on the next morning reached by a line drawn from the rising sun ; hence we conclude that the constellation us well as the sun performs one complete revolution within 24 hours. 9. For the gods the rising sun, when at the first point of Aries, revolves to the right, moving in the horizon ; at Lanka he then revolves right overhead ; and in an opposite direction ( viz. to the left in the horizon) for the Asuras. 10. At the end of Gemini the sun revolves for the gods at the height of 24 degrees above the horizon, while at Avanti he then moves right overhead. 11. In the same place there is then no shadow at noon, while the shadow falls towards the north for all who dwell to the north of Avanti, and to the south for the inhabitants of the countries south of Avanti. 12. Those who have maintained that for the gods dwelling on Meru it is day as long as the sun stays in Aries, Taurus and Gemini, but night when he is in Cancer, Leo and Virgo, to them reverence is due indeed ! 13. In the very same places, in which the sun goes to the north from Aries, he moves when returning from the north ; how then should he be visible at one time and invisible at another, while all along he is in the same region ? A criticism on those — unknown — authors who were ignorant of the fact that the sun while in Cancer, Leo and Virgo describes the same day circles, in reverse order, which he had described while in Aries, Taurus and Gemini. 14. In the visible half of the sphere there are three signs, extending from the middle of the sky (down to the western horizon) ; they contain ninety degrees. The same divisions are to be assumed from the rising (i. e, the eastern horizon, up to the zenith). 15. One degree answers to nine yojanas minus one ninth of a yojana (on the earth) ; this fact is manifest to those, who dwell on the same meridian, from the deflection (of the heavenly bodies) from the zenith. ( n > 16. Thus there is effected by ninety degrees (a difference of) eight hundred yojanas. What is sunrise for one observer is noon (for another observer) in a place distant to that extent. The circumference of the earth being equal to 3200 yojanas, 9 — ’ yojanas correspond to one degree, and 800 yojanas to 90 degrees. 17. Ujjayini is near to Lanka, being situated to the north on the same meridian ; hence the noon of the two places occurs at the same time ; but their days are unequal (in length) with the exception of the equinoctial days. 1 8. Three thousand two-hundred yojanas are the measure (of the circumference) of the earth. The sun when at the equinoctial point revolves round so much of the earth from Meru as centre. The sun when moving in the celestial equator revolves round the terrestrial equator, of which Meru is the pole. 19. Going 5 8 6 § yojanas north from Avan ti we reach the middle of the earth (Meru) ; so also by going 800 yojanas to the north from Lanka. Ujjayini has 24° northern latitude, is therefore 66° from Meru. The distance in yojanas is then found by a simple proportion. 20. Tn any country, by as many degrees as the pole is raised above the horizon, by so many degrees the sun is depressed from the zenith to the south on the day of the equinox. 21. If we go 373’, yojanas north from Ujjayini, this whole sphere (as described above) comes to an end. 22. There the sun is once seen for 60 nadik&s after his rising. The farther (we proceed towards the north), the longer (the day becomes), until at Suuieru we have a day six months long. We find by an easy calculation that going 373J yojanas north from Ujjayini we reach 66° northern latitude. In that latitude the last point of Gemini whose northern declination amounts to 24° — revolves in a day circle which is raised in its whole extent above the horizon ; and the sun, when he ( 72 ) has reached the end of Gemini, hence remains visible for a whole day of 60 nadikas. In that latitude therefore the sphere becomes different from the sphere described hitherto. At Mcru six signs revolve constantly above the horizon, and the sun is therefore constantly visible as long as he stays in those six signs, i. e., for six months. 23. If we go 403 j yojanas to the north of Avanti, the two signs Sagittarius and Caper never become visible. 24. If we go from the same place (i. e. Avanti) somewhat more than 482 yojanas to the north, Scorpio, Caper, Aquarius and Sagittarius never rise. 25. And if we go 586 — J; yojanas (to the north of Avanti), the latter half of the sphere (Libra to Pisces) never rises, while the former half (Aries to Virgo) never sets. The above statements base on the assumption that Sagittarius and Caper do not rise above the horizon in 69°24' northern latitude; Scorpio and Aquarius in 7 8° 15' northern latitude; and the whole latter half of the ecliptic in 90° northern latitude. The corresponding yojanas of distance from Avanti are easily found by proportion. 26. The people at Lankii see the polar star in the horizon ; those on Meru in the zenith ; those dwelling between see it between (the horizon and the zenith). 27. For those who dwell on the back of Meru the sun once risen remains visible for six months, while he moves in the six signs beginning with Aries ; for the Asuras he is visible as long as he is in the latter (half of the ecliptic). 28. For them (viz. the gods) the first point of Aries constantly is the ecliptic point on the horizon ; and the dreshk&pa as well as the triipg&rpga, the (so-called) ninth part and twelfth part all belong to Mars. Mars being, in astrological parlance, the Lord of Aries, all the sub- divisions of Aries also belong to him, i. e., are ruled by him. ( ^3 > 29. Beneath the equinoctial circle is Lankil ; there the sphere is right. Day and night there are always of the same length, viz., 30 nadikAe. 30. Having, by means of water, levelled a raised surface, on which the directions are marked, and having placed on its southern side a gnomon of the same measure as the surface ; 31. The observer, placing his eye at the base of the straight gnomon,, is to incline it in such a way, that the top of the gnomon is in the straight line joining the eye and the pole star.. * 32. At LankA, this observation is performed with a gnomon lying flat on the surface, on Sumeru with one standing upright, and in the intermediate- regions with one inclined more or less. 33. The perpendicular (from the end of' the gnomon to the surface)- represents the sine of latitude ; the distance of that perpendicular from the base of the gnomon — which indicates the north-south line — represents the sine of colatitude. 34. By such means the learned confidently determine the centre of the earth or the measure of the whole earth, just as we ascertain the taste (of all salt), by drinking a little water mixed with, salt.. A method for determining the elevation of' the pole above the horizon,, i. c., the terrestrial latitude,, and similar matters.. 35. 0f the moon whieh. is. constantly placed below the sun one half is illuminated by the sun’s rays, while the other half is obscured by the moon’s own, shadow ; as is. the case with, a jar standing in the sun light. 36. The rays of the sun,, being reflected from rile moon which con- sists of water, destroy the dhrkness of the night, just as the rays of the sun. Ming on. the surface of a mirror destroy the darkness inside a house.. 37» As the- moon daily changes her position, with, regard! to* the sun,, her illumined part increases,, just as in, the afternoon mote and. more of the western side of a jar is lit up. 38. This holds good, from the end of' the- dark fortnight, while the dark part of ‘ the. moon increases from the end of the light fortnight.— Those ( 74 ) who live on the back of the moon see the sun during one half of each lunar fortnight, since there is no shadow (obscuring the sun to them) to the extent of three signs on each side (of the conjunction). 39. Above the moon there are Mercury, Venus, the Sun, Mars, J upiter and Saturn (in succession), and then the stars. All planets move to- wards the east with the same velocity, each in its own orbit. 40. As the interstices between the spokes of an oil-press are small on the nave and large on the outer circumference, so the distances of the signs from each other become larger and larger, the higher^we ascend from the earth. 41. The moon which is placed (farthest) below the sphere of the stars revolves quickly in her small orbit ; Saturn which is placed highest above revolves in his large orbit with the same velocity, 42. The planets arranged in the ascending order upwards from the moon are the Lords of the months (in succession) ; in their descending order downwards from Saturn, they are the Lords of the hours ; if we take each fifth member of the ascending series we have the Lords of the days. The Lords of the years are clear (as explained in the first chapter). CHAPTER XIV On* Astronomical Instruments, Observations and the Like. 1. Draw upon the ground a level circle with a diameter one hundred and eighty angel is long, anti mark upon its circumference the signs (degrees, etc.) at equal distances, and also the degrees of declination (of the signs). 2. Further describe from the centre (of the first circle) three other circles, taking for their Radii the strings running (at right angles) from the string, which marks the north-south line, to those points on the circumference of the first circle where the degrees of declination (of the signs of the ecliptic) are marked ; and mark those circles with the degrees, as you did with the first one. 3. Thereupon draw a line from the centre towards the latitude (i. e. that point of the first circle which marks the latitude of the given place), and lengthen it up to the sphere. Take that piece which is due to the declina- tion (of a given sign of the ecliptic) and is intercepted by the line of latitude and the north-south line ; 4. Double it and mark it off on the circle belonging to that sign ; multiply half the degrees of the corresponding arc by ten. The result represents the vin&dik&s of ascensional difference in the case of the first sign ; in the case of the two other signs the vinadikas come out mixed. ( 76 ) The above stanzas teach how to find the ascensional difference for any given latitude without calculation, by the mere inspection of a kind of diagram. Fi S • 9 - — In diagram 9NESW represents the first circle whose circumference is sup- posed to be divided into 360 degrees. We further mark off on the circumfer- ence the degrees of declination of the first three signs of the ecliptic, the de- clination of Aries= Wa' that of Taurus = Wf, that of Gemini = Wg' and then draw the perpendiculars a' a, t' t, g' g on NS. Taking these perpendiculars as Radii, we describe from the centre C three circles (of which the diagram however represents only one, viz. , that one whose Radius = a a'). We then lay oft" from N the degrees of the latitude of the given place = N P, draw the perpendicular P 1 and join P C. Comparing the similar triangles PCI and d C a we establish the proportion Cl (= sine of colatit. given place) : P 1 ( = sine of latitude) = C a ( = sine of declination of Aries): ad (= ascensional difference for Aries). Doubling a d and applying it as a chord (d' d") to the day circle of Aries, we take — from an inspection of the degrees marked on the circumfer- ence the degrees of arc corresponding to half that chord and, in order to turn them into vin&dik&s, multiply them by ten (one degree = £ nddikd=10 vinSdik&s). — In the case of Taurus and Gemini the vinSdikSs come out mixed ;. e > the result gives us the sum of the vin&dikis for Aries + Taurus, and for Aries + Taurus + Gemini. An appropriate subtraction then furnishes the desired result. 5. Multiply the given n&4ik&s by six ; they are thus turned into degrees. Take the versed sine of those degrees and deduct it from the Radius ; there remains the shadow. Thus there is found the shadow up to noon. In order to find the n&dik&s, lessen the Radius by the shadow. ( n ) 6. Take the sixth part of the degrees of arc corresponding to the sine of the interval between the end of the shadow and the horizon ; that sixth part represents the elapsed n&dikds in the eastern hemisphere, those to come in the western hemisphere. The above stanzas teach a rough process for ascertaining, from the sun s altitude, the length of the shadow, and vice versa. In the first place the sine of the sun’s altitude at a given moment is roughly identified with the sine of that arc of the day-circle, which the sun has described up to that moment ; hence the direction to turn the elapsed nddikds into degrees. If we Fl S- 10 - then construct an artificial sphere, as in figure 10, in which c e is the height of the gnomon, we know in the triangle b c f the side f b= sine of sun’s altitude; and by de- ' f \ ducting a b — which is the versed sine* corresponding to b f — from the Radius we have b e. To this b c the line c d i. e. the length of the desired shadow is now assumed to be equal, the two hypothenuses c f and d e — which in reality meet in the sun — being looked upon as parallel. 7. Take that line which constitutes the east-west line corresponding to the oblique line, and the north-south line representing the degrees of decli- nation ; the degrees of the corresponding arc, multiplied by ten, give the vinddikfis of the rising of the signs in succession. A rule for finding, by the inspection of a diagram or globe, the time of the rising of the individual signs of the ecliptic on the equator. The rule is rather obscurely worded ; it however appears that by the oblique line we have to understand the sine of one, two, etc. signs of the ecliptic, while the north-south line is the sine of the declination of the end point of the respective sign. The east-line then is the other side, at right angles to the north-south line, and represents the socalled udayajyfi, which is a segment of the day circle described by the end point of the sign of the ecliptic. Each degree of the corresponding arc represents £ nddikd= 10 rind- dik&s ; the degrees multiplied by 10 therefore give the vinddikds of rising. ♦ The conteit obligee us to take 1 * in stansa 6 in the sense of 'versed aine/ ( 78 ) 8. A stanza of doubtful meaning. — If the restoration of the text attempted by us is right, the stanza directs us to ascertain the zenith, from the fact of the gnomon throwing, at a certain time, no shadow Cp. the Sanskrit Commentary. 9. 10a. Lay off the shadow (from the centre) towards the north, and place the gnomon at the other end of the shadow. Draw from the centre a line parallel to the hypothenuse up to the circumference. The distance of that (i. e. the point of the sphere thus marked) from the zenith is the latitude- In the same way the shadow may be determined from the latitude. A reference to diagram 10 will render the above processes intelligible. 106. 11. Having ascertained, for the given day, the declination of the sun — which is either greater than the terrestrial latitude or less than it — place its sine between the ecliptic and the equator ; in which degree that sine intersects the ecliptic, to that degree know the sun to he equal in longitude, according to that part of the sphere (in which he is at the time). A rule for finding the longitude of the sun from observation. The declination of the sun is ascertained from the observation of the sun’s zenith distance at noon, to which the terrestrial latitude is added if the zenith distance is north, while it is deducted from it, if the zenith-distance is south. The longitude of the sun is thereupon calculated as the hypothenuse of a right angled triangle, in which there are given one side, viz. , the sine of the sun’s declination and the opposite angle, viz . , the inclination of the ecliptic = 24°. The resulting degrees are to be taken as they are in the first quarter, in the second quarter they have to be deducted from 1 80° and so on. 1 2. By means of an observation, made by means of two staffs equal to Radius and placed in the centre, ascertain the degrees intervening between sun and moon. The twelfth part of those degrees represents the elapsed true tithis. From this (observed interval) again the next tithi is to be ascertained. 13. Adding to the degrees (intervening between sun and moon) the longitude of the sun as ascertained by observation (stanza 10), we obtain the longitude of the moon at the given tir pe by mere observation. A rule for finding the number of elapsed true tithis and the longitude of the moon by observation. — When the distance of the moon from the sun amounts to 12 s , one tithi has elapsed, and so on. ( 79 ) 14. 15. Mark three times, from the centre, the end of the gnomon’s shadow, and then describe two fish-figures. Thereupon describe a circle, taking for Radius a string, that is fastened to the point in which the two strings issuing from the heads of the fish-figures intersect, and that is so long as to reach the three points marked. On the given day the shadow of the gnomon moves in that circle, without departing from it. 16. The line joining the centre of that circle and the base of the gnomon is the south-north line; and the interval in the north direction (between that circle and the gnomon) is the midday shadow. ♦ A rule for finding the path described by the extremity of the shadow, based on the (erroneous) assumption that that path is the arc of a circle. We mark the extremity of the shadow at three different moments of the day, and thus obtain the points a, b, c (fig. 1 1 ). In order to lay a circle through these three points, Ave describe from a and b the fish-figure d e, and from b and c the fish-figure f g ; the point h in which the cords d e and f g meet is the centre of the desired circle. — It is then easy to show that the lino joining h and g (i. e. the foot of the gnomon) is the meridian line, and g 1 the midday shadow. Fig. 11 . 17. ‘Horizon’ we call that (circle) in which the sky is joined as it were to the earth ; we draw in it even east-west and north-south lines. 18. The interval between the pole and the horizon is the terrestrial latitude ; and the difference of the latitude and ninety (degrees) is called colatitude, which declines from the zenith towards the pole. The day-circle is what intervenes between (the sun s etc.) rising and setting. 19. Make, for the purposes of observation, a circle with marks (indicating the n&tfMs of rising of the signs of the ecliptic) in reverse order, and indications of the directions. Mark the centre and place the circle on even ground in such a way as to raise its axis to the amount of the given latitude. ( 80 ) 20. (Mark the place touched by) the shadow of the crossing of the two strings, (i. e,, of the centre of the circle.) Add to the place of the sun the degrees which the sun has passed through (at a given hour of the day) ; you thus obtain the sign which at that hour rises in the east. And also the elapsed n&dik&s of the day (may be calculated from those degrees); The circle is to be placed in the plane of the celestial equator. The observation of the places at which the shadow of the centre pin touches the circumference of the circle at different hours of the day informs us through how many degrees the sun has passed in the interval ; whence calculate the ecliptic point on the eastern horizon, and the time of day. 21. 22. Take a circular hoop, on whose circumference the 360 degrees are evenly marked, whose diameter is equal to one hasta, and which is half an anguli broad. In the middle of the breadth of that hoop make a hole. Through this small hole made in the circumference allow a ray of the sun at noon to- enter in an oblique direction^ The degrees, intervening on the lower half of the circle between (the spot illumined by the ray and) the spot reached by a string hanging perpendicularly from the centre of the circle, represent the degrees of the zenith-distance of the midday sum An easily intelligible rule for finding, by observation,, the sun’s zenith- distance at noon. 23. Make, of some material, a delicate sphere well rounded on all sides. On its periphery mark (the end points of) two lines called the lines of the portions of time (kiiabhoga), which answer to the points where the sun stops (viz. at the solstices). 24. Mark (between those two lines), according to observation, on both sides of the first point of Aries, the degrees of declination (of the different signs), by means of which oblique observations are to be made. ,< , 25. Baise this globe towards the north, to the extent of the degrees of latitude of the given locality, and take the nadiksis intervening between'the (point of) oblique observation and the (corresponding point on the) horizon. Those are the elapsed n&dik&a of the day ; multiplied by six they give the degrees. ( 81 ) The above is a somewhat free rendering of what we suppose to be the meaning of the text, the construc- tion and phraseology of which are, at the best, particularly involved and vague. The accompanying diagram shows a projection of the sphere on the plane of the meridian (H PZH' F), H II' being the H horizon and P P' the poles of the sphere. We ascertain on the cir- cumference of the sphere the points B and B', in which the sun rises and sets on the days of the two solstices, and draw the day-circles in which the sun moves on those days. Between those two day- circles we then draw the day-circles in which the sun moves when entering each sign of the ecliptic (and, possibly, as many more intervening day- circles as there is room for). We then, at any given moment, ascertain, by observation from the centre, the place of the sun in his day -circle,/ i. S and take the nadikas or degrees — which are to be considered as marked on each day -circle — intervening between the place of the sun, and the intersection of the day-circle, and the horizon (/ i. between S and T). A mere inspection of the globe thus will furnish the time of day. 26 . As long as, in the rising sphere, there rises that part which be- gins with the east point (i. c. the first point of Aries), there takes place an increase of the day ; while in the opposite ease a decrease takes place. — From what has been explained, the remaining points (not expressly referred to) may be understood. 27. The fundamental arrangements of all instruments depend on g - n g Sj water and bits of earth. By means of them one may make on a level surface instruments shaped like a tortoise, a man and so on. 28. The teacher is to communicate those things (only) to a pupil of Steadfast mind ; and the pupil after having learned them is to make his mechanical contrivances in such a way as to keep them secret from his own Fig. 12. z son even, ( 82 ) 29. HO. Let the astronomer observe (the position of the moon) in a given locality, (at that time when, according to calculation, full moon takes place on the prime meridian on the equator). Let him divide the observed difference of degrees in the manner of the tithi. The resulting time is to be deducted (from the time of full moon at Lanka) in the ease of the degrees being more, and to be added in the case of their being less. Let him increase the result by the time of ascensional difference, when the sun is in the six signs beginning with Aries, and diminish it by the same, when the sun is in the six signs beginning with Libra. In this way he ascertains the difference in longitude. While there can be hardly any doubt about the purport of the above stanzas, their wording and construction are obscure, and the translation there- fore merely aims at rendering the general sense. — The rule teaches how to ascertain the longitude of a given place by an observation of the moon at the hour when — according to astronomical calculation — full moon takes place at Labka. We are directed to find at that time the degrees of interval be- tween the moon and the sun, according to the method taught above in stanza 12 ; if those degrees are more than 180, the given locality is to the west of the prime meridian; if less, it is to the east. From the degrees of interval the difference of longitude in time is calculated proportionally ; and finally an allowance made for the ascensional difference. 31. The sixtieth part of so much water as within a nychthemeron escapes (from a vessel) through a given aperture fixes the duration of one nadika ; or else one hundred and eighty respirations of a man. 32. Make a copper vessel shaped like the half of a jar, and pierce a hole in its bottom. Place it in a basin filled with pure water ; when it has become full of water, a nadiM has elapsed. On account of the smallness of the bottom (?), the. hole has to be made in such a way that sixty immersions take place in one nychthemeron. — Or else a n&dika may be measured by the time in which sixty S'lokas, each consisting of sixty long syllables, can be read out. The above three stanzas are clear, with the exception of one line in 32 which is not very perspicuously expressed. — It will be observed that stanza 32 consists of 60 long syllables, and thus constitutes a Sfloka such as — accord- ing to Var&ha Mihira — may be recited in the sixtieth part of a nadika. 33. Having ascertained the latitude of the moon and observed the distance of the moon from the fixed star, and having made the requisite calculations one may then declare the conjunction of the moon with the star. ( 83 ) 34. (The yogatara, i. e., junction-star) of krittika is at the end of the sixth degree (of the nakshatra), and three and a half hastas to the uorth of tile ecliptic ; that of Kohiq.1 is at the end of the eighth degree, and five and a half hastas to the south of the ecliptic. 35. The two stars of Punarvasu are at the eighth degree, and to the north and south of the ecliptic at an interval of eight hastas ; the star of Pushya is at the fourth degree, three and a half hastas to the north. 36. Of A'tflesM the southern star is at the first degree, one hasta (south of the ecliptic) ; so also the northern star. Of Maglia the conjunction (with the moon) takes place in its own field (i. e., in the ecliptic), at the sixth degree (of the nakshatra), 37. Of Chitr/t (the yogat&ra is) at seven and a half degrees, three hastas to the south,— The angulis are to be calculated from the centre of the moon, 38. Deduct seventeen from the latitude (of the star), multiply the remainder by fifteen, and take its thirty-fourth part ; the result is to be taken as the measure in angulis. The time (of conjunction) is to be calculated from the difference of the moon’s daily motion. The above stanzas contain the statement of the longitudes and lati- tudes of the socalled yogatanis of some nakshatras. The longitude has to be reckoned from the beginning of each nakshatra, i. e„ twenty-seventh part of the ecliptic. The latitudes are expressed in angulis of which twenty-four go to one hasta, while the diameter of the moon— whose mean value is thirty- four minutes -is divided into fifteen angulis. In order to express the dis- tance of the star from the edge of the moon in angulis, seventeen, i. c., the moon’s radius is deducted from the distance in minutes (taken from the moon’s centre), and the remainder turned into angulis by means of the proportion supplied by the dimensions of the moon, 39. Multiply twenty-five by half the equinoctial shadow ; take the corresponding arc to which fifteen is to be added, multiply by ten, and add twenty-one times the equinoctial shadow ; the result are the vin&dik&s, 40. 41. By means of the latter calculate the ecliptic point on the eastern horizon from the beginning of Cancer ; when the sun stands at that ( *4 ) point, the rishi Agastya becomes visible — by means of instruments constructed on mathematical principles — beautifying tiro southern region as a mark beautifies a lady’s brow. — Gratified are the minds of men by this divine knowledge based on time. A rule for calculating the time of the heliacal rising of Agastya ( Canopus} in a given latitude. In order to ex- plain its ra - tamale, we at first ap- ply tin; longitude' and lati- tude of A g a, s t y a given in the m o (I e r n entry a Sid - dhanta, viz. yo° longit. and 80" s o uthe r n latit. Let H IP be the hori- zon, P the pole of the equator, E E' an arc of the equator, A L an are of the Ecliptic, C the place of Canopus on the horizon, G the last point of Gemini (on first point of Can- cer) which has the same longitude as Canopus, viz. 90° ; P G C’ C a circle of declination (on which according to Hindd practice the latitude is measured). The segment C' S' of the equator represents the time which Canopus must rise before the sun in order to' be visible in the morning. It divides itself into three parts, viz., C'B representing the ascensional difference of Agastya, BG" representing the ascensional difference of the last point of Gemini, and G" S' representing the amount to which the sun must be below the horizon in order that a star may be visible at its rising. This latter amount the Hindtis ( 85 ) always measure on the ecliptic (instead of measuring it, as would be proper, on a vertical circle). The modern Surya Siddlnmta assumes it to be equal to I but Varaba M,h.ra-as we understand him-to 1 5 . J„ order to fmd B<\" we make use of the statement made above in 1IJ. l| »«/;,, that at the end of Gemini the ascensional difference amounts to 21 palas in those places where the equinoctial shadow is equal to one. We thus obtain the following expression Kequired ascens. difi. — 21 x equinoct. sliad. Dividing this result by ten we obtain the degrees correspond! ny to tin palas (tiO palas going to one nadika which answers to 0 degrees). In order, finally, to determine C' B, we at first ascertain the value of D C, the socailed earth-sine (kujyii) of Cain .pus, by means of the proportion 12: cquin. shad. = sin deelin. of Canopus: kujya. If, as said above, CG ( = latitude of Canopus) is supposed equal to «7 (07 = sin 01° = cos 50°) ♦•quin, rthad. x 990 __ 1980 x equin. slnd. 07 ^ " 67x2 1980 67 x equin. ah ad. 2 30x'*quin. ahad. Instead of this value of C'B the text however has * 5xc<1 T' s, ' 4, S wherc- from it appears that Vara ha Mihira estimated to latitude of Canopus at less than the modern Surya Siddhftnta does. By an easy inverse calculation we find that a latitude of 75° 30' satisfies the expression given in the text. CHAPTER XV. The Secrets of Astronomy, 1. Following those, who possess the knowledge of the relative posi- tions of sun, moon, stars, and earth, I give the following explanations.— -An eclipse of the sun takes place constantly ; in consequence of the difference of position it becomes visible ip some locality. 2. In the minds of those also who are ignorant of the relative posi- tions knowledge may be engendered, just as — (see the Sanskrit Commentary.) 3. In those places, for which the sun is covered by the moon owing to the straight line drawn from the eye to the sun (i. e. for which the moon is in the straight line drawn etc.) an eclipse of the sun takes place ; and such a place is every day somewhere (in space). 4. The Fathers dwelling on the moon see the sun, if once obscured, for a half-month (in that condition) ; and they also see him non-obscured for a half-month. The middle of the eclipse js at full moon. The text of this stanza appears to be correct, but its meaning is obscure to us. — Might we, perhaps, have to understand by the * eclipse’ of the sun referred to merely his being invisible for a lunar half month to those beings which live on the side of the moon turned away from the earth ? — But a statement of this nature would have its proper place, not in the present chapter, but in the 1 traiiokya-samstMna ’ ; and such a statement is jn fact actually made there (xiv. 38). Moreover, the terms ‘ grasta ’ and 4 graha ’ would appear to be rather misapplied in such a connection. 5. The beings that live on Meru or near Meru never see an eclipse of the sun, owing to the fact of sun and moon being (for them) not high above the horizon, ( 87 ) G. Those living on Meru or close to it never see sun and moon in a line ; therefore they always see the two bodies separated by an interval. Of these two stanzas also the text seems to be correct. But in that case their contents are incomprehensible. 7. When an eclipse takes place at sunrise or sunset, then the sun stands for us low (?) and the moon high ; the latter therefore becomes the cause of the (obscuration of the) sun. 8. If, for us, an eclipse of the sun takes place at sunrise, it does not take place at the same moment for those with whom the sun is about to set. nor for those with whom it is midday, 9. For, on one and the same day, the eclipse may be past for those with whom it is sunrise, future — to the extent of two kshanas — for those with whom it is sunset, and actually occurring for those with whom it is midday. 10. In the Sanihita, in the beginning of the chapter on Rahus course, 1 have fully explained to what causes, apart from Rahu, solar and lunar eclipses are due. Stanza 7 is not fully intelligible. Stanzas 8 and 9 refer to the effect of parallax in accelerating or retarding the moment of an eclipse of the sun. 1 1. For Meru there exists no distinction of directions, because there the eastern direction is not indicated by the sun ; for as long as the sun is risen there, he revolves round the earth (the horizon). 12. Should it be said that the distinction of the eastern direction may be based on tbe observation of the first small part (of the sun when he rises) ; we reply that after he has moved for half a year (above the horizon) he again sets at the same point ; does that point then indicate the caste » n or the western direction ? 13. For those who live on Meru its being day depends on the sun’s declination, not, as for us, on his daily revolution. For us sixty n&Jik&s constitute a nychthemeron, while a year is a nychthemeron of the gods. ( 88 ) 1 4. Each year comprises a day and a night of the gods and the Asuras, it being day for the former while it is night for the latter, and vice versa. A nychthemerori of the fathers (whose abode is the moon) lasts one month, while one of men comprises sixty nadikas. 15. The gods see the sun moving at a distance (from the horizon), equal to so much as he rises above the horizon in two muhurtas ; they never see him higher up. The gods on Meru never see the sun higher above the horizon than 24 degrees (to which height he rises when at the end of Gemini). The sun performing 360 degrees in thirty muhurtas passes through 24 degrees in two muhurtas. 16. The succession of the Lords of hours and days is not the same on Meru as with us ; because the nyohthemeron there does not consist of sixty nadikas. 1 7. The rule about the days of the week is not everywhere the same, As no (decisive) reason can be assigned for it, the astronomers disagree con- cerning this point. 1 8. 19. The day of the week is to bo determined from the ahargaija ; the ahargaija itself depends on connexion with place and time. According to the teaching of Lafcacliarya the ahargaija is to bo reckoned from sunset at Yavatiapura ; according to Simluiohurya from sunrise at Lartka ; while it is to be reckoned from the moment when ten muhurtas of the night of the Yavanas have passed, according to their guru (i. e. the teacher — or master — . of the Yavanas), a 20, Aryabhata maintains that the beginning of the day is to be reckoned from midnight at Lanka ; and the same teacher again says that the day begins from sunrise at Lankl 21. If, there having been applied the correction for difference of meridian, the result does not agree with the actual circumstances of a given place ; the following statement (at any rate) as to the correspondence of time has been made by the same teachers (mentioned aba ye), in agreement with traditional science. ( 89 ) 22. The sun, when rising in the Bharatavarsha, at the same time makes midday in the region of the Bhadrasvas, sunset in that of the Kurus, midnight in Ketumfda. 23. What is sunrise in Lankfi is sunset in Siddhapura, midday in Yamakoti, midnight in Iiomaka-country. 24. The intercalary months, the omitted lunar days, the days of the planets, the lunar days, the days, Aries, the sun, the moon, the half-years, the seasons, the motions of the stars, the nights ; all of them begin together at the beginning of the yuga. 25. The difference in longitude when taken from Koruaka-country is not the same as when taken from Yavanapura ; and there is a difference be- tween (reckoning the beginning of the day) from midnight or from sunrise at Lanka. 20. And if we determine the Lord of each day by (counting the ahargana from) the moment when the sun has half set, we have in our favour neither any traditional authority nor reasoning of any kind. 27. Owing to the (various positions of the) sun, it is twilight in one place, day in another place, and night in another place. A small difference of place thus sullices to entangle the question as to who is the Lord of the day. 28. The question as to the boras is in the same predicament ; for the first bora belongs to the Lord of the day. If, then, the latter is not fully determined, how can the Lord of the hour be so ? 29. Ordinary people, as a rule, proceed in their business according to the days of the week (as known from tradition), without reflecting on such questions. The learned, ou the other hand, declare such (assumptions) to be right as result in the proper determination of the true lunar day. CHAPTER XYI. The Mean Motions or the Planets, according to the SOrya Siddhanta. 1. The determination of the (mean places of the) smaller planets for midnight at Avanti is, according to the Surva Siddhanta, as follows. — Mercury and Venus have the same motion as the mean sun. 2. For Jupiter multiply the ahargana by 100, and divide by 433232. For Mars multiply the ahargana by one, and divide by 687. 3. For Saturn multiply the ahargana by 1000, and divide by 10766066. The quotients are the entire revolutions; from the remainders the mean places of the planets are ascertained in signs, degrees and so on. 4. For each revolution of Jupiter ten tatparas (i. e., sixtieth parts of seconds) have to be deducted. Fourteen tatparas have to be added for each revolution of Mars ; five have to be deducted for each revolution of Saturn. 5. Four signs, two degrees, twenty-eight minutes and forty-nine seconds have to be added to the mean place of Saturn. 6. Eight degrees, six minutes and twenty seconds constitute the additive quantity for Jupiter. For Mars that quantity amounts to two signs, fifteen degrees and thirty -five minutes. 7. For the Sighra of Mercury multiply the ahargana by 100, and divide by 8797. Add the product of the (accomplished) revolutions and four and a half tatparas. 8. For the S'ighra of Venus multiply the ahargana by 10, and divide by 2247. Add ten and a half seconds, multiplied by the revolutions. ( 91 ) D. Twenty eight degrees of Leo (?'. e. four signs plus twenty-eight Negroes) and seventeen minutes are tlie additive quantity for the S’ighra of Bud ha. h rom (the Sighra of) Venus 332901 seconds are to be deducted. The above nine stanzas contain rules for calculating the mean places ot the planets. — The complete calculation for each planet is subdivided into three distinct operations. W e are at first taught how to find the mean place by means of a rough calculation resting on the assumption that the planet performs an integral number of revolutions within an integral number of savana days. Next we are informed how to make up for the mistake involved in the above assumption. And finally we are told what quantity has to he added to — or deducted from— the result found hy means of the previous two processes, in order to enable us to start in our calculation, not from the beginning of the kalpa or yuga, but from the epoch, of the kararia. The process by means of which we evolve from the rules of the text that duration of the sidereal revolutions of the planets which was assumed by Varalia Mihira’s Sut-ya Siddlianta is simple, and the same for all planets. We at first calculate the revolutions according to the approximative rules given in stanzas 1 — 3, and the first halves of stanzas 7 and 8. We thereupon deduce — from the corrections stated in stanza 4 and the second halves of 7 and 8 — the amount of the modifications to be applied to the rough results- found previously. We thus obtain the following numbers of revolutions within one muhayitga of 1577917800 days. Saturn — 14050-1 Jupiter — 304220 Mars — 2290824 Venus — 7022388 Mercury — 17937000 The correctness of these figures we are finally enabled to test by means of the kshepa-quantities stated in the fifth, sixth and ninth stanzas- p\ u - if aH we had done before in the case of the sun and the moon — we calculate, on the ground of the number- of revolutions stated' above, the moan positions of the planets at the epoch of the km ana, we find that they aorec down to seconds with the mentioned kshepa-quantities. As in the case of sun and moon, the calculation has to be made from the beginning, not of the mahayuga, but of the kalpa. ( 32 ) 10. Seventeen seconds are to be added for each year to the mean place of Mars ; ten to be deducted from that of Jupiter ; seven and a half to bo added to that of Saturn. 11. Forty-five are to be deducted from that of Venus ; one hundred and twenty to be added to that of Mercury. Fourteen hundred seconds are to be deducted from the mean place of Jupiter. These two stanzas state certain corrections to be applied to the mean places of the planets as found by means of the preceding rules. No other reason for those corrections appears to be assignable, but that they tended to effect an agreement between the rules of the Surya Siddhanta and the observations made by Varalia Mihira or the astronomers of his time, CHAPTER XVII. The true Motions of the Planets. 1. Of the other planets beginning with Mars the sun is the so-called Sighra. Thirty-five, fourteen, sixteen, five, twenty-four — each multiplied by two — are the degrees of the epicycles (of the Apsis) ; 2. Six, eleven, eight, four, twelve, — each figure multiplied by twenty and ten being deducted in the case of Mars— are the degrees (of the longitude) of the Mandochchas of Mars, Mercury, Jupiter, Venus, Saturn. We therefore have the following epicycles and longitudes of the apogees of the five planets Mars. 1 Mercury. Jupiter. Venus. Saturn, Degrees of epicycle . . . 70 28 1 32 14 60 Longitude of Apogee no to o 160 80 240 3. The degrees of the epicycles of the conjunction (sighraparidhi) of Mars, etc. (Mars, Mercury, Jupiter, Venus, Saturn) are 234, 132, 72, 260, 40. 4. If, the mean planet being deducted from the Sighra, the remainder is within three signs, the sines of those parts (of the three signs), which are passed through and yet to be passed through, are the base sine (bhuja) and the perpendicular (koti). If the remainder exceeds three signs, it is to be deducted from six signs, and after that the same method is to be followed. ( 94 > а. The (bhiijajy& and kotijya found according to stanza 4) are to bc- multiplied hy the degrees of tire epicycle of the planet, and divided by 360 ; the results are those two (sines) reduced (to the terms of the epicycle; viz.,. the so-called bhu jajya phala and kotijya phala).. Tlie koti plvala is to be added to the Radius in the half circle beginning with. Capricorn, and to be deducted in the half circle beginning with Cancer. б. Take the square root of the sum of the squares of the Radius so- increased or decreased and the bhujaphala, and divide thereby the bhujaphala multiplied by 1 20. Half of the corresponding arc is to be deducted’ from the mandochcha (if the anomaly is in the half orbit beginning with Aries), and to be added to it (in the half orbit beginning with Libra). A rule for finding the equation of the conjunction which in all points agrees with that given in the modern SOrya Siddhanta. — By half the equa- tion found the place of the mandochcha is to be corrected, and the manda equation to be calculated from the new place. 7. The mandochcha, having thus been made true, is deducted from the mean planet, and the sine (bahu) (of the resulting arc) is reduced t<> terms of the epicycle (by being multiplied by the degrees of the epicycle, and divided by 360) ; the corresponding arc is either added to the mandochcha (in the half orbit beginning with Aries), or deducted from it (in the half orbit beginning with Libra). 8. (The mandochcha so corrected) is again to be deducted from the mean planet, the corresponding sine is to be taken and to be reduced to terms of the epicycle, and the corresponding arc to be added to (in the half orbit beginning with Aries)' — or deducted from (in the half orbit beginning with Libra) — -the mean planet. 9. The planets, having thus been made what is called ‘true-mean’ (the mean planet having been made true as far as- the mandochcha is con- cerned), are to be deducted from their sTighrochehas. They are then to- be treated according to the former method (i. y 584 ; the quotient indicates the risings of Venus. The portion (passed through by Venus dur- ing that time) is five degrees of Scorpio (i. e. seven signs plus five degrees) together with the third part of a degree. 2. Having proceeded, by means of the degrees of time, for twenty- six days, Venus goes to its rising in the west. Add to the days the eleventh part of the risings, and therefrom (calculate) the motions. 3. In three periods of sixty days each, Venus passes through seventy degrees, increased respectively by four, three, and two ; thereupon in eighty- five days through seventy -seven degrees; then in three days through one and a quarter degree. 4. 5. Thereupon becoming retrograde it passes in fifteen days through two degrees; sets after five days in the west; rises after ten days in the cast ; becomes anuvakrin after twenty days, having moved four degrees (within each of the last mentioned three periods) ; passes through 250 degrees in 232 days, and sets in the east ; passes through 75 degrees in 60 days, and rises in the west. 6. In the case of Jupiter deduct from the ahargana thirty-four days and as many n&dikas, and divide by 399 ; the quotient are the risings of Jupiter. Put down separately the (remaining) days. 7. And add to them the ninth part of the number of the risings. Multiply the number of risings by 36, and divide by 391 ; the remainder to ■which 18 is added is called pada. g p u £ down in two places. (Calculate the equations) by means of ( 08 ) the mean and the true quantities. If on the deduction of those two quantities (from each other) the true quantity is smaller than the mean one, add (as many days as the result comprises degrees) to the days (as found above) ; in the opposite case (so many days) are to be deducted. 9. 10. As long as the pada is within 180, there is a positive quantity of 1 450; up to L 1)5 there is a positive quantity of 1205; if (the pada) is within 10, the third quantity is negative, amounting to 1480, — The risings of Jupiter being multiplied by five and divided by eight, give the minutes (of the mean place of Jupiter) from the first rising. 11. In the case of the first quantity there are 9), degrees of Virgo (L e. 5 s 9° 00') ; in the case of the second one there is half a circumference ( = 0 s ) ; and in the case of the last one there are 10 degrees, 12. Jupiter passes within sixty days through twelve degrees ; after that in forty days through four degrees ; then in twenty-four days through two degrees ; thereupon he becomes retrograde within fifteen days, and passes in sixty days through six degrees, and again in sixty days through six degrees. 13. Then, becoming anuvakra, he passes in eighty days through twelve degrees, and in forty-five days through nine degrees. Thereupon he sets, and having remained in that state for one month he again rises in the following month. 14. For Saturn, deduct from the ahargapa 150.) (days), divide (the remaining days) by 378 ; the result are the days etc. constituting one rising, 15. Deduct from the days the tenth part of the risings, Multiply the risings by nine, and divide by 25G ; the remainder is the pada ; add to it 89. 1 G. If the padas of Saturn are within thirty, there is an additive quantity of 2406 ; >vhile there is a subtractive quantity of 2509, if the pada amounts to 127, 17. There is an additive quantity of 2307, if (the pada amounts to) 99. Multiply the risings of Saturn by 31, and di vide by 32 ; (the resulting minutes) are to be added to the negative quantity, 18, In the case of the first aggregate we have sixteen degrees of Taurus minus nine minutes (= l 8 15° 51'); in the case of the second aggregate we have five (signs) plus twenty-seven degrees plus thirty-four minutes. ( 99 ) 19. In the ease of the last aggregate we Lave seven degrees plus twenty -eight minutes. — Saturn passes in sixteen days through three degrees, thereupon in fifty-six days through 232 minutes ( = 3° 52'). 20. Thereupon lie, within fifty-five days, becomes retrograde. He then passes in 08 (lavs through throe degrees, and in GO days through four degrees. After that, becoming anuga, lie passes through eight degrees in 105 days (and then sets). Thereupon lie passes through three degrees in 30 days, (and again rises). 21. Deduct from the ahargana 250, and 14 nadikas, and divide by 780; the result are the risings of Mars. 22. Multiply the number of risings by I 01 and add the result, taken as vinadikas, to the days (forming the remainder of the division prescribed in 21). Multiply the risings by eighteen, divide by fifteen and take (the re- mainder) of that division. 23. Lessen the remainder by its own fifth part ; you thus obtain the mean position of Mars, expressed successively in signs and so on. — Thereupon you must calculate the succession of its true courses. 24. The degrees of difference of the true and mean positions are to he added to the days (calculated above), if the mean position (exceeds the true one); and to ho deducted, if the mean position is less. — I will now describe the motion of Mars according to its different courses (gati). 25. ? * 2G. Having then passed through 13 degrees, Mars becomes niramsa (<’. c. without degrees ; t. e. having the same longitude as the sun), and having thereupon passed through twenty degrees he goes to his rising. I will now state the succession of days occupied by the courses (gati) of Mars. 27. 28. ? 29. When Mars is retrograde in Pisces, Scorpio, Aries, or Sagittarius, he passes through nine degrees within 5G days, then through seven degrees within 42 days; and thereupon, passing through sixteen degrees in GO dajs, he becomes anugati. ( 100 ) 30. In Taurus, Gemini*, Libra, and Virgo Mars passes in forty days through seven degrees, again in forty days through ten degrees, and in sixty- three days through seventeen degrees in succession, beginning from (?) the retrograde motion. 31. In Cancer and Leo, Mars passes in forty-four days through seven degrees, in forty days through six degrees, and in sixty days through eighteen degrees, in his courses beginning with tiro retrograde one. 32. In Aquarius and Capricorn, Mars passes through six degrees in thirty-two days, through nine degrees in thirty-nine days and through fifteen degrees in fifty-seven days, in his threefold course. 33. ? 34. Add two to one, five, eight, eleven, fourteen, eleven, nine. In the quick course (Slghragati) you have forty diminished by one, four, four (?). 35. Thirty-six increased by two, three, nine, twelve, nine, three, zero are the days. The motion in the eighth course is the same as in the seventh. 36. 37, 38. Add to the ahargapa twenty-eight and a third. Multiply by eight and divide by 927 ; the result are (the risings) of Mercury. Take the eighth part of the (remaining) days, and deduct from those of Mercury as many nadikas as the fourth part of the risings amounts to. Multiply the risings by 123, and deduct forty -three; diyide the remainder by 389; you thus find the mean place of Mercury. — Mercury passes within six plus five ( = eleven) of the (days forming the) remainder through eight true degrees ; with- in thirty (days) through thirty true degrees. 39. Thereupon he passes in eighty-one days through sixty degrees ; in eighty -eight days through a hundred degrees ; in fourteen days through twelve degrees ; in thirty days through thirty degrees. 40, In one hundred and four days through ninety-seven degrees ; in thirty -one days through twenty-three degrees. These are the true motions of Mercury, 41, Deduct from the days the degrees of difference of those two (viz. of the mean and true places of Mercury), in case of the true place being in excess (of the mean place) ; add those degrees in case of the mean place being in excess (of the true). The course of the true Mercury is as follows. ( 101 ) 42. In Aries, Mercury passes in thirty-six days through thirty-five degrees, in thirty-six days through forty-two degrees, in twenty-nine days through twenty-one degrees, and again in twenty -nine days through thirty degrees. 43. Tn Taurus he passes in forty-five days through forty-four degrees, in twenty -three days through seventeen degrees, again in twenty -three days through thirty-nine degrees; in forty-nine days through forty-three degrees. 44. In Gemini lie passes in forty-five days through forty-eight de- grees ; in twenty days through fourteen degrees ; in twenty-six days through twenty-seven degrees ; in forty-seven days through forty degrees. 45. In Cancer he passes in forty-two days through twenty-six de- grees ; in eighteen days through twelve and a half degrees : in thirty days through twenty-six degrees ; in forty -six days through twenty-five degrees. 40. In Leo he passes in thirty-four days through twenty-five degrees ; in sixteen days through eighteen degrees ; in thirty-two days through twenty- seven degrees ; in forty-five days through twenty-eight degrees. 47. In Virgo he passes in twenty-six days through twenty-seven de- grees ; in thirty-eight days through forty-five degrees; again in thirty-eight days through fifty-six degrees, and again in thirty-eight days through sixty- two degrees. 48. (In Libra) he passes in forty-two days through thirty-seven de- grees, in forty days through thirty-two degrees, in thirty-four days through sixty -four degrees, 40. In Scorpio he passes in eighteen days through twenty degrees, in forty-five days through fifty-two degrees, in thirty-four days through forty- five degrees, in forty-eight days through seventy -four degrees. 50. In Sagittarius ho passes in forty days through thirty-nine do- greos ; in sixteen day. through eleven degrees; in forty-two days through forty-throe degrees ; in thirty-two days through tlurty-hve degrees. 51 In Capricorn he passes in twenty days through nineteen degree, ; in thirteen days through fourteen degrees ; in thirty-eight days through tinny- nine degrees ; in thirty-two day, through hity-e.ght degrees. ( 102 ) 52. In Aquarius he passes in twenty -three days through twenty -two degrees; in twenty-two days through twenty-five degrees; in twenty-four days through twenty degrees ; in thirty-two days through sixty degrees. 53. In Pisces ho passes in twenty -four days through twenty -six de- grees ; in twenty-five days through tliirty r degrees ; in twenty-nine days through thirty degrees ; and in twenty-seven days through forty-nine degrees. 54. 55. 5(5. ? 57. Multiply the time of the ascensional difference by the (sine of the planet's) latitude — which is calculated according to the method of sines — and take the 480th part of the product. The result is to be deducted from — or added to — the planet’s longitude according to the direction (of the latitude). 58. This operation having been performed, the setting and rising of the planets is to be calculated by means of the degrees intervening between them and the sun. These degrees are, for the moon and the other planets in succession, twelve, fourteen, twelve, fifteen, eight, fifteen. 59. Multiply those degrees by three-hundred, and divide by the vina- dikas of rising ; from the resulting degrees the true setting and rising of the planets is to be determined. GO. Mercury, Venus, Mars, Jupiter (and Saturn ?) (become visible in the east when they are) less advanced in longitude than the sun by the amount of the planetary degrees (as calculated above); in the moon’s case the reverse takes place. — Having thus ascertained (all requisite items) from the latitude, the astronomer may make declarations regarding future planetary occurences. Stanza 57 contains a rule for the so-called Aksha drikkarman, i. c. the correction for apparent longitude which depends on the planet’s latitude at the given time. The rule is based on the rough proportion Sine of greatest latitude of ecliptic ( = 48) : time of greatest ascensional difference = sine of planet’s latitude : time of planet’s ascensional difference. Dividing by ten (whence the divisor 480) the vinAdikas of the result are turned into degrees. — Stanza 58 thereupon states the number of degrees to which the difference of the longitude of the sun and the single planets must amount in order that the latter may become visible. — Stanza 59 finally teaches ( 103 ) how tho.se mean degrees are rendered true by the introduction of the time of rising of the ecliptic in which the sun and the planet are at the time (•>00 vinadikas being the mean time of the rising of the signs). 01. I or the benefit of Ins pupils Varalut Mihira of Avauti has com- posed this short treatise on the smaller planets, which effects an agreement between observation and theory. 02. He whose efforts are frustrated by the (theory of) Mars of Pra- dyunma, and by Jupiter, Saturn as made (calculated) by Vijayauandin, and by Mercury, let him honour this very accurate treatise. 03. Which has been composed (? drishta) by Varaha Mihira, easy to understand 04. II e who, although knowing the faults of others, yet does not men- tion them, even when an opportunity offers, hut rather proclaims their good qualities; to that good man let honour he paid as to a benefactor of bis kind ! 05. Free from jealousy Varaha Mihira gives this excellent short trea- tise on the smaller planets, comprised in eighteen ary as. 00, Take the degrees of the sun’s longitude (at the time of the last conjunction anteccding the given time), and add to them the days connected with the degrees of the (planet’s) motion. If the degrees, thus derived from the days, are more (than 300), they are to be taken from a full revolution (t. e. 300 is to be deducted from them). The longitude of the sun is found by adding to bis longitude at the last conjunction with a planet as many degrees as there have elapsed days of the planet’s eluirii. 07, Lessen (the ahargana) by 0321), multiply by four, and divide by 3075. Divide (the remainder) again by four; the result are the days (which have elapsed) since Mars was without degrees (t, e. had the same longitude as tile sun), 08. (Mars) becomes visible when less (in longitude than the sun) by 15°, within thirty-six days; then (passes) in one hundred and eighty-eight days (through) sixty degrees ; in one hundred and eight days (through) sixty degrees ; within seventy-two days (through) ninety degrees. ( 104 ) 69. In sixty-eight (lays (through) fifty degrees ; in two hundred and forty days (through) seventy degrees. Then it sets ; (passes) thereupon in fifty-six days (through) fifteen degrees and becomes nirain&i. 70. Lessen (the ahargana) by 14681, multiply by 29 and divide by 3312; divide (the remainder) by the multiplier ( = 29); you thus obtain the days of Mercury. 71. Mercury having fallen behind (the sun) by twelve degrees — which takes place within ten days — rises in the east; thereupon he falls behind by ten degrees in fourteen days. (Advancing thereupon) nine degrees within eighteen days he sets, and again rises (in the west), having advanced thirteen degrees within thirty days. 72. Then he advances nine degrees within eighteen days, and then, falling behind eight degrees within sixteen days, he sets in the west. After that, falling behind nine degrees in eight days, he again becomes niramsa. 73. Deduct (from the ahargana) ?, multiply by seven, and divide by 2752. Divide the remainder by seven, the result are the days of Jupiter, taken from the niramsa position. 74. Which are to be deducted (counted) from the place of the sun. Having fallen back twelve degrees within sixteen days, Jupiter rises in the east. He then passes through forty-four degrees within fifty-four days; through sixty-four degrees in seventy days. 75. Through one hundred and twenty degrees in one hundred and nine days; through seventy -six degrees in eighty -eight days; through thirty- two degrees in forty days. Thereupon he sets, and then passes in sixteen days through twelve degrees (when he becomes niramsa). 76. Lessen (the ahargana) by 11122, multiply by two, and divide by 1151. Divide the remainder by two. The result are the days of Venus counted from the niramsa position. 77. Falling behind nine degrees within five days Venus rises in the east. It then falls behind twenty-one degrees within fifteen days ; after that fifteen degrees within two-hundred and eight days ; after that it advances five degrees (?) within twelve days (?), and sets. ( 105 ) 78. Then it advances ten degrees within forty-eight (?) days, and be- comes niramsa. After that it moves in the opposite direction, and rises in the west within the time which it had taken to go to the niramsa position (?) ; and again moving in the opposite direction sets (in the west). 70. Lessen (the ahargarm) by 10518 (?), multiply by three and mul- tiply by lllS. 1 .Di vide (the remainder) by three. (The result are. the days since the last conjunction) of Saturn. In eighteen days he falls behind the sun sixteen and a half degrees, and then rises in the east. 80. Then he falls hack ninety and a hall degrees within ninety-eight days; then thirteen degrees within fourteen days. Then one hundred and twenty degrees within one hundred and thirteen days; then ninety-one de- grees within ninety-eight days. 81. Then in thirteen days twelve degrees and a half. Then he sets, and, passing within twenty-one days through sixteen and a halt degrees, he becomes uirarnsa, being always behind the sun.