oe १ a ‘ ~ we. Id vw wwasy, Nae Se dh tee ० _BIBLIOTHECA INDICA; |‘ ` CT 1 |) {| Es TAT 9 ION OF ORIENTAL WO ^ | ` रण्ये UNDER THE (ध. धष OF THE ++ ; aa २, eg \ क ४1 १ i vee क kt : 1 Nuw 388, No, 18.11 i १९: ~, a ihe £ र I. +", क Ohad iy ‘ >t ‘ ५ र 4 ५, चू Sei + । ~ \ ५५ e te ४८ me । [च्‌ + 3 (4 - c ५ चै eh al = Ss + 1 ष ङ ~ ~ 7 + tak च नि +“ ; ~ = oe यै I> त ` " Sel ०८५ ९ र्ब र. > भवय > न वीक 0 RS द HINDU’ ASTRON RAIS i क १.4 te [१ “9 THE SIDDHANTA.§/1RO Ds ४, ths. re + 9४ ‡ † TRANSLATED FROM.'T म ९ ॥. ४५६. ९५ . REVISED BY PANDIT Bird De | qo ® Ren 9 नौर a Se eat sz ai THE SUPERINTENDENCE OF THE VEN'BLE ९ et ats + चि ७ ४५ x 4 +~ 45 FASOIOULUS 1. ` ? 7 al 9 vets + Ap % >. ~" Ma) pe ty ५ ww : ट ह 5 bet’ 8 +; wt af ४ 4. 11.99९ eR RS COR Ce ४५ ६ 2.2 > ह ८ # + । oy = + ॥ + 4 Ri % | et aS it Ose 1 A a ie! १ | og VR OALOUTTA YS 4 Y % : wy व ५; + x 4 fee > i त + ¶ ae (4 Pe ie ¢ + 9 » ‘ ^ प PRE: A ४ 4 “आङ 2 °च व; a Sh +) 4. ( | , PRINTED BY ©. ४. प AT THE BAPTIST MISSION PRESS. | 4 1 : Pe - _ | 4 ~ (१ © od we थ sy < An c १ । 3 ध, ५ 01801 व; 4 ) ह 4 9 # । + 3 4 4 J _ | ty +^ + | a 1 « <2 * — * ५ $ १44 “> | + भ ind व 1 प ~ ie 1. ह y b Arch- 110 a Siddhdnta. Jaya N १ . क ध i‘ > ## ४ क | = - w ४ त . ce of ara ae 6.and Digitized by G oogle AY 5 4 ct hee ५ te ¥ *. ae | ज eed + ras त 7 + ek Pra न a] . ~ >. ५ # 2 । 8. = य अः 3.8 he aa r é | 2% = o 8G 9 243 1 न, 4 a2 2,2 8 Pe ez. <~3! ie ८] कु ^ = कवी ग्र 5 8 i = ^ २4 ~ metas १५८. -2 a | ; उ. me ’ । = # 11 ५1 + nat wae ॥ ह १ ह san क) oy ह क व = = $ * et es प mesh hope! | od ae ^" Be , = ate 48 = els #- षे = ! Wis xa = २4 १९ हि ५ > # ~— * = क - कै + न ` +. जै ध $ <~ ड - a) a ५1 4 tae a ae — ४ Jas” > ox 4 a ^? $= ar 4 ~ \ ‘ty - Bg 4 2 १ = ब. * TRANSLATION OF TIE SIDDHANTA SIROMANL Digitized by G oogle TRANSLATION SURYA SIDDHANTA PUNDIT BAPU’ DEVA SASTRI, SIDDH AN TA SIROMANI (4 RCRA ८44८2 a 4.१ BY THE LATE LANCELOT WILKINSON, ESQ., ©. 8., REVISED BY PUNDIT BAPU DEVA SASTRI, FROM THE SANSKRIT. CALCUTTA : PRINTED BY C. B. LEWIS, AT THE BAPTIST MISSION PRESS. 1861. Digitized by G oogle CONTENTS. (^ एह I.—In praise of the advantages of the study of the Spheric, ०००००००० cos see renee ० ०००००००००७०००७० ०७००७ CiapTer II.—Questions on the general view of the Sphere, .. (^+ एषः, III.—Called Bhuvana-kos/a or Cosmography, ......... Cuaprer 1V.—Called Madhya-gati-vdsané; on the principles of the Rules for finding the mean places of the Planets,... Cnarter V.—On the principles on which the Rules for finding | the true placcs of the Planets are grounded, १११०० १०१.००.००.० CuarterR VI.—Called Golabandha; on the construction of an Armillary Sphere, 1 Cuarter VII.—Called Tripras’na-vésana ; on the principles of: the Rules resolving the questions on time, space, and directions, .. 42 rere Crarter VIII.—Called Grahana-vadsand ; in explanation of the cause of Kclipses of the Sun and Moon, ® Ciarter 1X.—Called Drikkarama-vésané; on the principles of the Rules for finding the times of the rising and setting of the heavenly bodies, .......... sees Cuapter X.—Called S’ringonnati-vdsané; in explanation of the cause of the Phases of the Moon, ........... errr CitarterR XI.—Called Yantrédhydya; on the use of astronomical instruments, ... ४७६ ०७००००७० ५००००००० (11147 हए XII.—Description of the Seasons,.. var ass (^ एषाः XIII.—Containing useful questions called Pras’n4- UN VRS, sxe cin Seu tcaeghsuasisscaswensenessen cus sebeveuaesaroescsceveues PERPRIPPPPP PPP PEP PPP PPP POAPPPREPPPPPLP SD Page 105 107 11 2 127 135 160 176 196. 206 209 228 281 Digitized by G oogle Ni: ie By ne De BES / 2 क ~ ८८ ८ (१८ ‹ { 7८८ +" ८ .. | * e 4 a 9 “3 { „1 न ERM GOST OF) १. : ~^ ale TRANSLATION OF THE GOLADHYAYA OF THE SIDDHANTA-STROMANT. a el CHAPTER I. In praise of the advantages of the study of the Spheric. Sulutation to Gangsg ! 1. Having saluted that God, who when called upon brings all under- takings to a successful issue, and also that Goddess, through whose benign favour the tongues of poets, gifted with a flow of words ever new and with elegance, sweetness and playfulness, sport in their mouths as in a place of recreation, as dancing- girls adorned with beauty disport themselves in the dance with elegance and with every variety of step, I proceed to indite this work on the Sphere. It has been freed from all error, and rendered intelligible to the lowest capacity. 2. Inasmuch as no calculator can hope to acquire in the assemblage of tho learned a distinguished reputation 98 an Astronomer, with- out a clear understanding of the principles upon which all the calculations of the mean and other places of the planets are founded, and to remove the doubts which may arise in his own mind, I therefore proceed to treat of the sphere, in such 9 manner as to make tho reasons of all my calculations manifest. On inspecting the Globe they become. clear and manifest as if submitted to the eyc, and are as completely at command, as the wild apple (कष्णे held in the palm of the hand. f B | Invocation. Object of the work. \ 106 Translation of the [I. 3. ais | 8. As a feast with ubundance of 1 ignorance all things but without clarified butter, and as a kingdom without a king, and an assemblage without eloquent speakers have little to recom- mend thom ; so the Astronomer who has no knowledge of the spheric, commands np consideration. 4, Asa foolish impudent disputant, who ignorant of gram- mar (rudely) enters into the company of the learned and vainly prates, is brought to ridicule, and put to shame by the frowns and ironical remarks of even children of any smartness, so ho, who is ignorant of the spheric, is exposed in an assemblage of the Astronomers, by the various questions of really accom- plished Astronomers. Object of the Armillary 5. The Armillary sphere 18 said, by a ai the wiso, to bo a roprosentation of tho celestial sphere, for the purpose of ascortaining the proofs of tho positions of tho arth, the stars, and tho plancts: this 18 a species of figure, and hence it is deemed by the wise to be an object of mathematical calculation. 6. It is said by ancient astrono- mers that the purpose of the science is judicial astrology, and this indeed depends upon the influence of the horoscope, and this on the true places of the planets : these (true places) can be found only by a perfect knowledge of the spheric. A knowledge of the spheric is not to be attained without mathematical calculation. How then can a man, ignorant of mathematics, comprehend the doctrine of the sphere &c. ? Who is likely to under. 7. Mathomatical calculations aro take the study with effect. of two kinds, Arithmetical and Alge- braical : he who has mastered both forms, is qualified if he have previously acquired (a perfect knowledge of) the Grammar (of the Sanskrit Language,) to undertake the study of the various branches of Astronomy. Otherwise he may acquire the name (but never the substantial knowledge) of an Astronomer. In praise of mathematics. IT. 3.1 Sidhanta-siromani. 107 8. He who has acquired a perfect knowledge of Grammar, which has been termed VEDAVADANA i. 6. the mouth of the Vepas and domi- cile of Saraswati, may acquire a knowledge of every other science—nay of the Vepas themselves. For this reason it is that none, but he who has acquired a thorough knowledge of Grammar, is qualified to undertake the study of other sciences. The opinion of othors on 9. 0 learned man ; | if you intend tad 0 to study the spheric, study the Treatise of it. of Bufsxara, it is neither too concise nor idly diffuse: it contains every essential principle of the science, and is of easy comprehension ; it is moreover written in an eloquent style, is made interesting with questions ; it im- parts to all who study it that manner of correct expression in learned assemblages, approved of by accomplished scholars. End of Chapter I. In praise of Grammar. CHAPTER II. Questions on the General view of the Sphere. Questions regarding the 1. This Earth being encircled by — the revolving planets, remains sta- tionary in the heavens, within the orbits of all the revolving fixed stars ; tell me by whom or by what is it supported, that it {918 not downwards (in space) ? 2. Tell me also, after a full examination of all the various opinions on the subject, its figure and magnitude, how its prin- cipal islands mountains and seas are situated in it ? ; 3. Tell me, O my father, why the Pennie regarding धव place of a planet found out from well taining plancte’ true places calculated AHARGANA (or enumeration and their causes. , * of mean terrestrial days, elapsed from B2 108 . Translation of the [II. 3. the commencement of the Katea)* by applying the rule of pro- 9 4 Kata is that portion of time, which intervenes betwoen one conjunction of all the planets at the Horizon of LanxA (that place at the terrestrial equator, where the longitude is 76° ¥., reckoned from Greonwich) at the first point of Aries, and a subsequent similar conjunction. A Kapa consists of 14 Manus and their 16 sanDuIs; each MANU ¢ ing botwoen 2 84 ता 18, Each MANU contains 71 इ ए048 } oavh yuaa is dividod into 4 yuaANnaunis viz., (1) Tusta’, DwApana and Katt, the length of cach of these is as the numbers 4,8,2and 1. The beginning and aad of each yuaa’NauHRIs being each one 12th part of it are respectively called its 8८7 एप ६ and Sanpuya’Nsa, The number of sidereal years contained in each yuaa’Naurl, &c. are shown below ; Kat, 1 Hab eieteeneseesueuss 432,000, DWA PARAS: icciscvdeccecisncvscvcparsecsedcenstecedesuveumcexs 864,000, BTA, coves ७9७७९७०७ ०७० ०७०७००७ ७७७७०७० ७०७ ००७०००० ०७०७ ७७१९०७० १०० ००१७०७४ 1,296,000, Keita, ०७७ ०७७०७००९७१ ७९७०००७ ७७७ ७७७०७०० ००७७ ०७००१७०७ ०००5७० ५७० ०७७००७००००७९ 1 BY 28,000, Yuaa, 000000000 OOF cee FS ७०७७७06 HEE OOH HOF ०७०७०७० ०००००७०००९७० ०७१५०७७ 4,320,000, 11 >< YUGA = MANU, ,,,,०००,१००१०००००००००९०००० ०००१०००० 4, 294,080,000, 1 9 {९.1 Se en Ao Pa CE Pn oR oe APTS YA 806,720,000, 16 Manu sanpHIs each egual toa एषा ए एतज, 25,920,000, Kawpa,.. wees 4,320,000,000, Of the present Kazea 6 manus with their 7 sawpnHis, 27 vvuGas and their three yuGa’NGHari, ९. Krrta, Treva, and Dwa’Panra, and 3179 sidereal years of the fourth yua@a’nounr of the 28th Yuaa of tho 7th manu, that is to say, 1,972,947,179 sidereal years have elapsed from the beginning of the present Katpa to the commencement of the Sa’Liwa’HaNa era. Now we can easily find out the number of oe that have elapsed from the beginning of the present Karra to any time we like. By astronomical observations the number of terrestrial and synodic lunar days in any given number of years can be ascertained and then, with the result found, their number in a Katpa or Yuaa can be caloulated by the rule of proportion. By this method ancient Astronomers found out the number of lunar and ter- restrial days in a Kaxpa as given below. 1,602,999,000,000 (synodic) Junar days } inaK and 1,677,916,460,000 terrostrial days ES eee With the foregoing results and a knowledge of tho number of sidercal yoars contained in a Katpa aswell of those thut have passed, we can find out tho number of mean terrestrial days from the boginning of a Kara to any given day. This numbor is called Auanaana and tho mothod of fluding it is givon in Gayitipuy(ya by Bua’sxana’oua’nya. By the daily mean motions of the planets, ascertained by astronomical observa- tions, the numbers of their revolutions in a Kaupa are known and are given in works on Astronomy. To find tho place of a planet by the number of its revolutions, the number of days contained in a Katra and the Auaraaya toa given day, tho fullowing pro- portion is used. As the terrestrial days in a Karp, : the number of revolutions of a planet in a 7 4124 : the AHag@aya : : the number of revolutions and signs &. of the planet in the AnaRaaya. By leaving out the number of revolutions, contained in the result found, the remaining signs &०, indicate the place of the planet. Now, the intention of the querist is this, why should not this be the true place of aplanet? In the GayiripHyiya. Budsxaniona’Rya has stated the revolutions in a Kapa, but he has here mentioned the revolutions in a yu@a on account of his constant study of the S’isnya-puivkiDDNIDA-TANTRA, 9 Troa- (द Astronomy by [44114 who has statod in it the revolutions in a YuGa.— = णी = = = eee TN Ne कः ` चक = + = = क ककः = II. 4.] Sidhanta-siromant. 109 portion to the revolutions in the Yuaa* &c. is not o true one ? (i. ©. why is it only a mean and not the true place) and why the rules for finding the true places of the different planets are not of the same kind? What are the Desantara, UDAYANTARA, BaousdntTarA, andC#ara corrections?+ What is the ManpocucHat (slow or Ist Apogee) and S/faHrocucHa§ (quick or 2nd Apogee) ? What is the node ? 4. Whatis the Krenpraj| and that which arises from it (i. 6, the sine, cosine, &c. of it)? What is the Manpapnazal|| (the first equation) and S‘fanrapHatag (the 2nd equation) which depend on the sine of the Kenpra? Why does the place of n planet become true, when the ManpapHaa or S’fGHRAPHALA # (It may bo proper to give notes explaining concisely the technical terms occurring in theso questions, which havo no corresponding terms in English, in order that the English Astronomor msy at once apprehend these quostions with- out waiting for the explanation of them which the Author gives in the sequel.— t [1 find the place of a planet at the time of sun-rise at a given place, the several important corrections, i. e. the Upaya NTARA, BHUJA’NTARA, Des’ANTARA, and Cuara are to be applied to the mean place of the planet found out from the Ananaana by the fact of the mean pee being found from the AHaRrGawa for the time when a fictitious body, which is supposed to move uniformly in ह ui- noctial, and to perform a complete revolution in tho samo time as the Sun, 08 the horizon of Lanxa’. We now proocced to cxplain tho corrections. The Upaya’nTaka and Bnvusa’ntTara corrections are to be applied to the mean placo of a planct found from the Amargaya for finding tho place of the planot at tho truco time whon tho Sun comes to the horizon of Lanxa’ arising from thoso two portions of the equation of time respectively, one due to the inclination of the ecliptic to the equinoctial and the other to the unequal motion of the Sun in the ecliptic. The Des’a’nTaRna and Cara corrections are to be applied to the mean place of a planct applied with the Upava’nrara and Bausa’wrara corrections, for finding the place of the planet at the time of sun rise at a given place. The Dzs’a’wraRa correction due to the longitu de of the place reckoned from the meridian of Lanka’ and the Cara correction to the ascentional differ. ence. 1. D. t sige cia is equivalent to the higher Apsie. The Sun’s and Moon's Manpooncitas (higher Apsides) arc the samo as their A pogecs, while the other plancts’ Manpooncias are equivalent to their Aphelions. B. D. § (8’1’eurocuona is that point of the orbit of each of the primary planets (i. ©, Mars, Meroury, Jupiter, Venue and Saturn) which is furthest from the Earth. B. D. || (Kewpra is of two kinds, one called Manwpa-KENDRA corresponds with the anomaly and the other called 8’r'onRa-KENDRA is equivalent to the commutation added to or subtracted from 180° as the 31614 -7 8 74 is greater or less than 180° B. D. q ध is the eame as the equation of the centre of a planet and 8’ GuRa-PHALA is equivalent to the 1 arallax of the superior planet ; and the clongation of the inferior planets. B. 9. 110 Translation of the {II. 5. are (at one time) added to and (at another) subtracted from it ? What is the twofold correction called Drixxkarma* which learned astronomers have applied (to the true place of a pla- net) at the rising and setting of the planet? Answer me all these questions plainly, if you have a thorough knowledge of the sphere. Questions regarding the 5. Tell me, O you acute astrono- length of the day and night. 6, why, when tho Sunis on tho northern hemisphere, is the day long and the night short, and the day. short and the night long when the Sun is on the south- ern hemisphere ? क 6. How is it that the day and len gth of the day and night night of the Gods and their enemies 2 Gods ATTEAS,EITRIS Oarryas correspond in length with the solar years? How is it that tho night and day of the Pirgis is equal in length to a (synodic) lunar month, and how is it that tho day and night of Braumd 18 2000 yuaast in length ? Queslions: xegatding: ‘the 7. Why, 0 Astronomer, is it that periods of risings of the the 12 signs of the Zodiac which are signe of the ee all of equal length, rise in unequal times (even at the Equator,) and why are not those periods of rising the same in all countries ? Questions astotheplacesof == 8. Shew me, 0 learned ono, tho the Drusrd, the Kusxa',&. places of the Dyusyi (the radius of the diurnal circle), the Kusy4 (the sine of that part of the arc of the diurnal circle intercepted between the horizon and the six o’clock line, i. e. of the ascensional difference in terms * [7 एता 42244 isthe correction requisite to be applied to the place of a pla- net, for finding the point of the ecliptic on the horizon when the planot reaches it. This correction is to be applied to the place of a planet by means of its two portions, one called the Ayana-DRIkxkaRMA and the other the AkaHa-DRIK- ^ ४44. The place of a planet with the AvYANa-DRIKKARMA applied, gives tho point of the ecliptic on the six o’olock line when the planet arrives at it: and this corrected place of the planet, again with the AksHa-DRIKKARMA applied, giyes the point of the ecliptic on the horieon when the planet comes to it. B. D.] ¶ The Karta, Trera/, Dwa’rara and Kati aro usually called Yuoas: but the four together form only one Yu@a, according to the Sippia’nta system, each of these four being held to be individually but a Yuaa’nauar, L. W. 11. 10. | Sulhunuta-siromani. 111 ofa small circle), and show mo also the placcs of the declination, Sama-s4nku,* Agra (the sine of amplitude), latitude and co-latitude &८, in this Armillary sphere as_ these places are in the heavens. 0 If the middle of a lunar Eclipse tain differences in the times takes place at the end of the Tirni aie of colar audlunar = (क the full moon), why docs not tho middle of the solar Kelipse take place in hke manner at the change? Why is the Eastern limb of the Moon in a lunar Eclipse first involved in obscurity, and the western limb of the Sun first eclipsed in a solar Kelipse ?+¢ Questions regarding tho 9. What, O most intelligeut one, Peraliaxes: is the LambanaAt and what 18 the Nati? why is the Lamsana applicd to the Tirut and the Nati applied to the latitude (of the Moon)? and why are these corrections settled by means (of the radius) of the Earth ? Questions regarding tho 10. Ah! why, after being full, docs + the Moon, having lost her pure bright- ness, lose her circularity, as it were, by her too close associa- tion, caused by her diurnal revolution with the night : and why again after having arrived in the same sign as the Sun, docs she thenceforth, by successive augmentation of her pure = [Sama-sa’NxU is the sino of the Sun’s altitude when it comes to the prime verlical, ॥!. 1). ] + {An Eclipse of tho Moon is caused by her entering into the Marth’s shadow and as the place of the Earth’s shadow and that of tho Moon is tho samo at the full moon, the conjunction of tho Earth’s shadow and the Moon must happen at the same time ; and an Eclipse of tho Sun is caused by tho intorposi- tion of the Moon between the Earth and tho Sun, and the conjunction of the = Sun and Moon inv like mannor must happon at the new moon, as then | the place of tho Sun and Moon is the same. As this ie the case with the eclipses of both of them (i. ०. both tho Sun and Moon) the querist asks, “Ifthe middle of a Junar eclipso &c.” It is scarcely necessary to add that tho assumption that tho midd le of a lunar eclipse takes place exactly at the full moon, is only approximately correct. B. D.] ‡ (The Lampawa is equivalent to tho Moon’s parallax in longitude from tho Sun reduced into time by means of the Moon’s motion from the Sun: and tho Nar! is tho samo as tho Moun’s parallax in latitude from the Sun. B. D.J 112 Translation of the (III. 1. brightness, as from association with the Sun, attain her circu- lar form ?* End of the second Chapter. CHAPTER III. Called Bhuvana-kos'a or Cosmography. The excellence of the 1. LheSupreme Being Para Bran- Bupreme Being. , ua the first principle, excels eternally. From the soul (PurvsHa) and nature (Prakriti,) when excited by the first principle, arose the first Great Intelligence called the Manarrarrwa or Buppxitatrwa: from it sprung self-con- sciousness (AHANKARA :) from it wore producod tho Mthor, Air, Fire, Water, and Earth ; and by tho combination of theso was mado tho univorso Braumsnpa, in the contro of which is tho Earth: and from Branm{ Cuaturdnana, residing on the sur- face of the Earth, sprung all animate and inanimate things. 2. This Globe of the Harth form- ed of (the five elementary principles) Earth, Air, Water, the Ether, and Fire, is perfectly round, and encompassed by the orbits of the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn, and by the constellations. It has no (material) supporter; but stands firmly in the expanse of heaven by its own inherent force. On its surface through- out subsist (in security) all animate and inanimate objects, Danvsas and human beings, Gods and Darryas. Description of the Barth. * This verse has a double meaning, all the native writers, however grave the subject, being much addicted to conoeits. The second interpretation of this verse is as follows : Ah! why does the most learned of Brahmans, though distinguished by his immaculate conduct, lose his pure honour and influence as it were from his mis- conduct caused by derangement? It is no wonder that the said Brahman alter having met with a Brahman skilled in the Vxpas, and by having recourse to him, thonooforth booomes distinguished for his ०01000४ good conduct by gradual augmentation of his illustriousnces. L. W. III. 6,] Sidhianta-s’iromani. | 113 8. -It is covered on all sides with multitudes of mountains, groves, towns and sacred edifices, as is the bulb of the Nauclea’s globular flower with its multitude of anthers. Refutation of the supposi- 4. If the Earth were supported by : ध hae 5५० any material substance or living crea- ture, then that would require a second supporter, and for that second a third would be required. Here we have the absurdity of an interminable series. If the last of the series be supposed to remain firm by its own inherent power, then why may not the same power be supposed to exist in the first, that is in the Earth? For is not the Earth ono of tho forms of the cight-fold divinity i. ©. of S’1va. Rofutation of tho objcc- 5. As heatis an inherent property ॥ Heald 1 of the Sun and of Fire, as cold of the Moon, fluidity of water, and hardness of stones, and as the Air is volatile, so the earth is naturally immoveable. For oh! the properties existing in things are wonderful. 6. Tho* property of attraction is inherent in tho Earth. By this property the Earth attacts any unsupported heavy thing towards it: ‘l'ho thing appears to bo falling [but it is in a state of being drawn to the Karth]. The etherial expanse being equally outspread all around, where can the Earth fall ? Opinion of the Bavp- 7, Observing the revolution of the DAB. constellations, the BauppHas thought that tho Marth had no support, and as no heavy body is soon stationary in the air, they asserted that the eartht goes eternal- ly downwards in space. 8. ‘The Jamnas and others main- Opinion of the Jatnas. । eyes tain that there are two Suns and two * (It is manifest from this that neither can the Earth by any means fall downwarde, nor the men situated at the distances ofa fourth part of the circum- ference from us or in the opposite hemisphere. B.D + [He who resides on the Earth, is not conecious of the motion of it down- wards in space, as 8 man sitting on a moving ship does not perceive its motion, B. D.] © 114 Translation of the (TIT. 9. Moons, and also two sets of constellations, which nse in con- stant alternation. To them I give this appropriate answer. Refutation of the opinion 9. Observing as you do, O Bavp- of the Baupomas. DHA, that every heavy body projected into the air, comes back again to, and overtakes the Karth, how then can you idly maintain that the Earth is fallng down in space? [If true, the Earth being the heavier body, would, he imagines*—perpetually gain on the higher projcctile and never allow its overtaking it. |] Rofutation of the opinion 10. But what shall I say to thy of the Jamras. folly, O Jaina, who without object or use supposest a double set of constellations, two Suns and two Moons? Dost thou not see that the visible circumpolar con- stellations take a whole day to complete their revolutions ? Rofutation of the eupposi- 11. Ifthis blessed Earth wore level, tion that the Earth islevel. = 6 8 plane mirror, then why is not the sun, rovolving abovo at a distanco from tho Barth, visiblo to men as well as to the Gods? (on the Paurdyrxa hypothesis, that it is always ‘revolving about Meru, above and horizontally to the Earth. 12. If the Golden mountain (Merv) is the cause of night, then why is it not visible when it intervenes botween us and the Sun? And Meru being admitted (by the PaurantKas) to lie to the North, how comes it to pass that tho Sun 11368 (for half the year) to the South ? ee ee 13. As the one-hundredth part of pearance of tho plane form the circumference of a circle is (scarce- + ly different from) a plane, and as the Earth is an excessively large body, and a man exceedingly small (in comparison,) the whole visible portion of the Earth consequently appears to a man on its surface to be perfectly plane, * (This was BrtaskaRa’s own notion ;—but oyon on the more correct principle, that all bodivs fall with equal rapidity, the argumont holds good, B. D.] IIT. 18.] Sidhanta-s'tromani. 115 . 14. That the correct dimensions of roof of the correctness of alleged circumferonce of the the circumference of the Earth have + been stated may be proved by the simple Rule of proportion in this mode: (ascertain the différ- ence in Yujanas between two towns im an exact north and south line, and ascertain also the difference of the latitudes of those towns: then say) if the difference of latitude gives this distance in YUJANAS, what will the whole circumference of 360 degrees give f To confirm the eame cir- 15. As itis ascertained by calcula- eumilerence! of the mart: tion that the city of Ussayrnf is situated at a distance from the equator equal to the one-sixteenth part of tho whole circumforonco: this distance, therefore, multiplied by 16 will be the mensure of the LHarth’s cir- cuinference. What reason then is there in attributing (as the Paurdyikas do) such an immense magnitude to the earth ? 16. For the position of the moon’s cusps, the conjunction of the planets, eclipses, the time of the risings and settings of the planets, the lengths of the shadows of the gnomon, &c., are all consistont with this (estimate of the extent of tho) circumference, and not with any other ; therefore it is declared that the correctness of the aforesaid measurement of the earth is proved both directly and indirectly,—(directly, by its agreeing with the phenomena ;—indirectly, by no other estimate agreeing with the phenomena). 17. [^ क्र is situated in the middle of the Earth: Yama- KOTI 18 situated to the ast of LANKA, and RoMAKAPATTANA to the west. ‘The city of SippHapura lies underneath Lanx(. Sumenu is situated to the North (undor the North Pole,) and VapavAnaa to the South of LanxA (under the south Pole) : 18. ‘These six places are situated at a distance of one-fourth part of the Karth’s circumference each from its adjoining one. So those who have a knowledge of Geography maintain. At Menu reside the Gods and the Sippuas, whilst at VaDAVANALA are situated all the hells and the Dairyas. ५४ 116 Translation of the [11. 19. 19. A man on whatever part of the Globe he may be, thinks the Earth to be under his feet, and that he is standing up right upon it: but two individuals placed at 90° from each other, fancy each that the other is standing in a horizontal line, as it were at right angles to himself. 20. Those who are placed at the distance of half the Earth’s circumference from each other are mutually antipodes, as & man on the bank of a river and his shadow reflected in the water: But as well those who are situated at the distance of 90° as those who are situated at that of 180° from you, main- tain their positions without difficulty. ‘They stand with tho same ease as we do here in our position. Positions of the Dwfras 21. Most learned astronomers have and Seas. stated that Jamptpwfpa embraces the whole northern hemisphere lying to the north of the salt sea : and that the other six Dwfras and the (sevon) Seas viz. those of salt, milk, &c. aro all situated in the southern homis- phere. 22. To the south of the equator lies the salt sea, and to the south of it the sea of milk, whence sprung the nectar, the Moon and the Goddess [^ एप्त, and where the Omni- present VAsuprEva, to whose Lotus-feet Branmf and all the Gods bow in reverence, holds his favorite residence. 23. Beyond the sea of milk lie in succession the seas of curds, clarified butter, sugar-cane-juice, and wine: and, last of all, that of sweet Water, which surrounds VaDAVANALA. The 47414 Loxas or infernal regions, form the concave strata of the Earth. 24, In those lower regions dwell the race of serpents (who live) in the light shed by the rays issuing from the multitude of the brilliant jewels of their crests, together with the multi- tude of Asuras; and there the 77848 enjoy themselves with the pleasing persons of beautiful females resembling the finest gold in purity. 25. The S’dxa, 9 419 ^1५/, Kaus‘a, Kefuncua, Gomepaka, and III. 31.] Sidhanta-s’tvomani. 117 PusnkarA Dwfras ore situated [in the intervals of the above mentioned seas} in regular alternation: each Dwipa lying, it is अभणत्‌, botween two of these seas. Positions of the Moun 26. ‘To the North of Lanxé lies aa 0 the HimfLaya mountain, and beyond caused by the mountains. that the Hemaxtta mountain and beyond that again the NisuapHA mountain. These three Mowntains stretch from sea to sca. In like mannor to the north of SippHaeura lie in succession the S’Rincgavin S‘uxLa and Nita mountains. 10 the valleys lying between these mountains the wise have given the name of VarsHas. 27. This valley which we inhabit is called the BaArata- varsia; to the North of it lies the KINNARAVARSHA, and boyond it again the Hartvarsua, and know that the north of अप+ एए in like manner are situated the Kuru, H1rray- MAYA and RaMyAKA VARSHAS. 28. To the north of Yamaxortt lies the Matyavdn mountain, and to the north of Romaxapratrana the GANDHAMADANA mountain. These two mountains are terminated by the NfLa aud NistaAbitA mountains, and tho space botwoen these two is called the ILAvrita VarsHa. 29. The country lying between the MAtyavin mountain and the sea, is called the Buapras'wa-varsHa by the learned ; and geographers have denominated the country between the GanpuamMApana and the sea, the KEtuMALA-VARSHA. 30. The I aAvprra-varsHa, which is bounded by the Niswapia, Nfta, GANDIAMADANA and MALYAVAN mountains, is distinguished by a peculiar splendour. It is a land rendered brilliant by its shining gold, and thickly covered with the bowers of the immortal Gods. Position of the mountain 81. In the middle of the ILAvrrra Mnev in Indvetra. Varsua stands the mountain Meru, which is composed of gold and of precious stones, the abode of the immortal Gods. Expounders of the Purdnas have further described this Meru to be the pericarp of the earth- lotus whence Brana had his birth. 118 Translation of the (TIT. 32. 32. The four mountains Manpara, SUGANDHA, सा+ and Sup{rs wa serve as buttresses to support this Meru, and upon these four hills grow severally the Kapampa, Jambi, Vata and एए. trees which are as bannors on those four hills. 33. From the clear juice which flows from the fruit of the JamBG springs the JamMBO-Nnapf; from contact with this juico earth becomes gold: and it is from this fact that gold 18 called sAMBONADA : [this Juice is of so oxquisite a flavour that] the multitude of the immortal Gods and Srppuas, turning with distaste from nectar, delight to quaff tlis delicious boverage. 84. And it is well known that upon those four hills [the buttresses of Mgru} are four gardens, (lst) CarrraraTHAa of varied brilliancy [sacred to Kusera}, (2nd) Nanpana which is the delight of the ^ 0841148, (3rd) the Dunit: which gives refreshment to the Gods, and (4th) the resplendent VAIBHRASA. 35. And in these gardens are beautified four resorvoirs, viz. the Aruna, the Mdnasa, the ^ (प्र? and the S’/weta- JALA, in due order: and these are the lakes in the waters of which the celestial spirits, when fatigued with their dalliance with the fair Goddesses, lovo to disport thomsolves. 36. Merv divided itself into three peaks, upon which are situated the three citics sacred to शशाक, Brand and S’/iva [denominated VarkuntHa, BranMapura, and Kariasa], and beneath them are the eight cities sacred to Inpra, Aani, Yama, Naregita, Varuya, Viyu, 81481, and 8.8, [1. €. the regents of the eight Dixs or directions,* viz., the east sacred to ॐ (As the point where the equator cuts the horizon is the east, the sun therelore rises due east at time of the equinoxes but on this ground, we cannot determine the direction at Merv (the north pole] because there the equator coincides with the horizon and consequently the suu moves at Manu under the horizon the whole day of the equinox. Yet the ancient astronomers maintained that the direction in which the yamMaxkorft lies from Menu is the east, because, according to their opinion, the inhabitants of Menu saw the sun rising towurds the YamaKo7yI at the beginning of the Katpa. In the same manner, the direction in whioh LanKA lies from mount Muu 18 south, that in which Romakaeatrana lies, is wost, and the direction in which Sippua- IIT. 40.} Sidhanta-s’iromani. 119 Inpra, the south-east sacred to Acni, the south sacred to Yama, the south-west sacred to Narraita, the west sacred to Varuna, the north-west sacred to Vayu, the north sacred to S‘as’{ and the north-east sacred to Is’a.] 37. The sacred Ganges, springing from the Foot of Visunu, falls upon mount Merv, and thence separating itself into four streams descends through the heavens down upon the four VisuKaM- BHAS or buttress hills, and thus falis into the four reservoirs [above described]. 38. [Of the four streams above mentioned], the first called Sfta, went to BHaprAs’wa-VARSHA, the second, called ALAKANANDA, to DButkrata-varsiA, the third, called Cuaxsuu, to KeromAta-varsua, and the fourth, called Buapra to Uttara Kuru [or North Kuru] 39. And this sacred river has so rare an efficacy that if her name be listened to, if she be sought to beseen, if seen, touched or bathed in, if her waters be tasted, if her name bo uttored, or brought to mind, and hor virtuos be colobrated, sho puvifics in many ways thousands of sinful men [from their sins]. 40. And if a man make a pilgrimage to this sacred stream, the whole line of his progenitors, bursting the bands [imposed on them by Yama], bound away in liberty, and dance with joy ; nay even, by a man’s approach to its banks they repulse the slaves of Yama [who kept guard over them], and, escaping from Naraka [the infernal regions], secure an abode in the happy regions of Heaven. Some peculiarity. &c. are situated in the east, south &o. from > हए respectively Note on verses from 21 to 43 :—Buaaskara’cna’rya has exercised his ingenuity fn giving a locality on tho carth to the poctical imaginations of Vyra’sa, at the samo timo that. he lias preserved his own principles in regard to the form and dimensions of the Earth. But he himself attached no credit to what he has described in these verses for he concludes his recital in his commentary with the words, PuRA lies from Mxnv is north. The buttresses of Menu, BD) यदिद्मृक्त तत्‌ सव पुरारात्रितम्‌। * What is etatod here rosts all on tho authority of the PunAyas.” As much as to say “ crodat Judwus.” L. W 120 Translation of the (III. 41. Te Serre rr Cer LC Here in this BHARATA-VARSHA xuticuatas of Bua’zata- are embraced the following nine KHAN- VaRSHA. ° ° pas [portions] viz. Arnpra, Kas’‘eru, TAMRAPARNA, GaBHASTIMAT, Kumarika, Nxaa, Saumya, VARUNA, and lastly GANDHARVA. 42. In the ई एण {पाह ¢ alone is found the subdivision of men into castes ; in the remaining KHANDAS are found all the tribes of Antyasas or outcaste tribes of men. In this region [Bufrata-varsna] are also seven KULACHALAS, Viz. the MAHEN- pra, 9 एतन, Mauaya, Riksnaka, PduryArra, tho Sanya, and Vinpaya hills. Arrangement of the seven 43. The country to the south of ORAS worlds: the equator is called the Butrioxa, that to the north the Buuvatoxa and Merv [the third] is called the Swartoka, 702४ is tho MawarnoKka in the JVouvons beyond this is the Janatoxa, then the Tarotoxa and last of all tho 94161014, ‘Thoso 1oKas aro gradually attained by increasing religious merits. 44. When it is sunrise at [^+ प्रा, it is then midday at Yamaxkofi1 (90° east of 144 प्रह), sunset at SippHapura and midnight at RomMaKapaTraNa. Boiss «Be lie. aes 45, Assumo the point of tho why Meru is due north of horizon at which the sun risos as a ea tho cast point, and that at which lo sets as the west point, and then determine the other two points, i. e., the north and south through the martsya* effected by the east and west points. ‘lhe line connecting the north and south points will be a meridian line and this line in whatover placo it is drawn will fall upon tho north point: hence षठ lies due north of all places. ‘A ourious fect is rehearsed. 46. Only Yamakofr lies due east Geographical Anomaly. from Ussayinf, at the distance of 90° @ (From the east and west points, as centres, with a common radius describe two arcs, intersecting each other in two points, the place contained by the 9708 is called Mateya “ae fish” and the intersecting points are the north and south points. B. 7. III. 51. ` Sidhanta-s'iromani. 121 from it: but Lanka and not Ussayinf lies duo west from ‘YAMAKOTI. | 47, Tho same is the case everywhore ; no place can lie west of that which is to its east except on the equator, so that cast and west are strangely related.* 48. A man situated on the equa- tor sees both the north and south polos touching [the north and south points of] the horizon, and the cvlestial sphere resting (as it were) upon the two poles as centres of motion and revolving vertically over his head in the heavens, as the Persian water-wheel. 49, As a man proceeds north from tho oquator, ho observes the constellations [that rovolve vertically over his head when seen from the equator] to revolve obliquely, being deflected from his vertical point: and the north pole clevated above his horizon. The degrees between the pole and the horizon nro the degrees of latitude [at tho place]. These degrees aro causod by tho Yosanas [between tho equator and tho Right sphere. Obliquo sphoro, plico]. Ilow tho dogrocs of Iati- 50. Tho number of Yosanas [in cee 4 4९ tho are of any torrostrial or colostial vico रशा, circle] multiplied by 360 and divided by [the number in Yosanas in] the circumference of the circle is tho number of degrees [of that arc] in tho carth or in tho planctary orbit in the heavons. Tho Yosanas aro found from the degrees by reversing the calculation. 51. The Gods who live in tho Parallel sphoro. mount Merv observe at their zenith [* As the sun or any heavenly body whon it roachos the Prime Vertical of any place is called duo east or west, 80 according to the Hindu Astronomical language all the places on tho Karth which aro situated on the circle corresponding to the Primo Vertical arc due cast or west from the place and not those which aro situated on the parallel of latitude of the place, thatis the places which havo the angle of position 90° from any place are duo cast or west from that placo. And thus all directions on the Kartlh aro shown by weans of the angle of position in the Hindu Astronomical works. 1). 1.1 7 129 Translation of the [TII. 52. tho north pole, णात tho Dairyas in Vapavdnata tho south pole. But while the Gods behold tho constellations revolving from left to mght, to the Darryas they appear to roevolvo from right to left. But to both Gods and Dartyas the equatorial constellations appear to revolve on and correspond with the -horizon. Dimensions of the Earth’s 52. The circumference of the earth श has been pronounced to be 4967 Yoyanas and the diameter of the same has been declared to be 1581, Yosanas in length: the superficial arca of tho Marth, like the net enclosing the hand ball, is 78,53,03- squaro Yosanas, and is found by multiplying the circumference by the diameter.* The error of Lalla is ex- 58. The superficial area | of the Ae in १०४ एः (1.1 Earth, like the net enclosing tho Clai aroa 9 0 war ° : hand ball, is most erroneously stated by Lanta; tho truo aroa not amounting to ono hundredth part of that so idly assumed by him. His dimensions are contrary to what is found by actual inspection: my charge of error therefore cannot be pronounced to be rude and uncalled for. But if any doubt be entertained, I beg you, O learned mathomaticians, to examine well and with tho utmost impar- tiality whether the amount stated by me or that stated by him is the correct one. [Tho amount stated by Latta in his ® (The diameter and -the ciroumference of the Earth here mentioned are to ech other as 1250 ; 8927 and tho demonstration of this ratio is shown by Bafsxakicuhrya in the following manner. Take a radius equal to any large number, such as more than 10000, and through this determine the sine of a smallor arc than even the 100th part of the circumference of the circle by the aid of the canon of sines (Jyorrartt,) and the sine thus dotermined whon multiplied by that number which represents the pert which the aro just taken is of the circumference, becomes the longth of ciroumference because an aro smaller than the 100th part of the ciroumference of a circle is [ecarcely different from] a straight line. For thie reason, the cir oumference equal to the number 62832 is granted by ARYaDMaTTaA and the others, in the diameter equal tothe number 20,000. Though the length of the circum- ference determined by extracting the square root of the tenfold square of the diameter is rough, yet it is granted for convenience by SripuandcHa RY, Bran- 246 00774 ond tho others, and it is not to bo supposed that they wore ignorant of this roughness.—B. D.] IIT. 59.] Sidhanta-s‘tronani. 128 work cntitled Dufvgrppmipa-TantRA is 285,63,38,557 squaro Yosanas, which he appears to have found by multiplying the square contents of tho circlo by the circumference. ] Shows the wrongness of 54. If a piece of cloth be cut in ane Henle airen’ ty Lalle: a circular form with a diameter equal to half the circumference of the sphere, then half of the sphere will bo (entirely) covered by that circular cloth and there will still be some cloth to spare. 55. As the area of this piece of cloth is to be found nearly 24 times the area of a great circle of the sphere: and the area of the piece of cloth covering the other half of the sphere is also the same ; * 56. Thereforo the area of the whole sphere cannot be more than 5 times tho area of tho great circle of the sphere. How thon has 10 multiplied [the arca of the great circle of tho sphere] by the circumference [to get the superficial contents of the sphere] ? 57. As the area of a great circle [of the sphere] multiplied by the circumference is without reason, the rule (therefore of 1५11, for tho suporficial contonts of tho sphoro) is wrong, and tho superficial aroa of the Earth (given by him) 18 conse- quontly wrong. 58, 59. Suppose the length of the [equatorial] circumference of the globe equal to 4 times the number of sines [viz. 96, there being 24 sines calculated for every 3°, which number multiplied by 4 == 96] and such oblong sections equal to the number of the length of the said circumference and marked with the vertical lines [running from pole to pole], as there are seen formed by nature on the Anwif fruit marked off by the lines running from the top of it to its bottom. ® Let the diameter of a sphere be 7: the circumference will bo 22 nearly. The area of a circle whoso diameter is 7 will be about 383; that of a cirelv whose diameter ia 11 (4 circumference) will bo about 89१ this 89$ is little less than 24 times 383. DL. W. D2 Otherwise. 121 Translation of the (TIT. 60. 60. If wo dctermino the superficial arca of onc of theso sections by means of its parts, we have it in tlis form. Sum of all tho sines diminished by half of the radius and divided by the same.* * The correctness of this form is thus briefly illustrated by Baa’skana’cua’RYA in his commentary. Let g aha, g be thoscction in which ab, 3 0, ० d &९, and 01९3; 5,¢,, ¢,d,, &o. are each equal to 1 cubit and also aa, aro equal to 1 cubit: then bd,, ९८९), dd,, &e. are proportional to tho sines m ©, # ©, ० d, &c. and are thus found, mb. If ka or rad: give, aa 1 (= 1) : : mb: 88, = —— Rad, 20 If Rad: 1:: no: ce, = Rad od again Rad: 1 : : od: dd, = re &e. Now aa,, 028, ०00; &0. being found, the contents of onch of aa, 5,5, bb, 0,0, ००, dd, &o. tho per of tho section is found by taking half the sum of aa, & bd, Lb, & co,, co, & dd, &०. and multiplying it by ad 4 (which is equal to oash of do, ed, &o.) horo ab iv assumed ns 1 and tho whole surface cach of aa,b,b, ९९,०,० a8 & plano, for an arc of 3°} is scarcely dilfuront from a plane. Now to find the sum of aa,},}, bb,c,c &o. we have aa, + ९९१ 66, + ce, ce, + did, a ee eo — >€ 1 + ke, adding these and saa | out 1 multiplier, we have aa, + bb, + cc, + dd, + ko, Substituting the values of aa, bb,, &o, st a ४, nd 20 t+ तु + त नी = + &o. 80 on for tho assumod sinca +2 R +2 but = — = -- - — R By substitution we get + smb ne 4R 2 eo ee R R+ mb + १० + 0d + ko .. —4 2 SD R It is evident from this that the sum of all the sines diminished by the half of the Radius and divided by the Radius is equal to the contents of tho upper half of the section, therefore by dividing by $ Rud we get the whole section instead of ouly the upper half of it. sum of all the sines — 4 2. $2 i. 6. contente of the whole seotion = = A. IIT. 65.] Sidhanta-s’iromani. 125 61. As tho superficial arca of ono section thus dctormincd is equal to tho diamotor of the globe, tho product found by multiplying tho diamoter by tho circumforenco 108 thoroforo beon asserted to be the superficial contents of a sphere. The grand deluges or dis- 62. The earth is said to swell to वः the extent of one Yosana equally भा around [from the centro] in a day of [^ पक by reason of the decay of the natural productions which grow upon it: in the Briuma deluge that increase is again lost. In the grand deluge [in which Baaum« himself as well as all nature fades away then] tho Harth itself is reduced to a state of nonentity. 63. That extinction which is daily taking placo amongst created beings is eMlled tho Datnanptna or daily oxtinction. Tho Britta ox- tinction or dclugo takes placo at the end of Braumd’s day: for all created beings are then absorbed in Brahmé’s body. 64. As on the extinction of Brauma himself all things are dissolved into nature, wise men therefore call that dissolution tho PR&KRITIKA or resolution into nature. Things thus in ao alate of oxtinction having thoir destinios soverally fixed aro ngain produced in scparate forms when nature is excited (by tho Creator). 65. The devout men, who have destroyed all their virtues and sins by a knowledge of the soul, having abstracted their minds from worldly acts, concentrate their thoughts on the Aro four-fold, ba by substituting the valucs of tho 24 81166 etatod in the Gayrra’pia’Ya wo have A = 803§ viz. the diameter of the globe where the circumference = 96. L. W. [५५ the demonstration of tho rule (multiply tho superficial area of tho sphero by tho diameter and divido the product by 6) for finding out the solid content of the sphere is shown by Bma’skaRa Cua RYA in the following manner. Suppose in the sphere the number of pyramids, the height of which is equal to the radius and whose bases aro squares having sides equal to 1, equal to the niuuber of the suporficial arca of tho sphoro, then Tho solid contents of overy pyramid = ॥ R. =4 diameter and the number of pyramids in the sphoro is cqual to tho numbor of tho superficial contents of tho sphiro. .. Tho solid content of the sphore = 3 diamoter > suporiicial area.—B.D.] 126 Translation of the (IIT. 66. Supreme Being, and after their death, as they attain the stato from which there is no return, the wise men therefore denomi- nate this state the AtyantrKa dissolution. Thus the dissolu- tions are four-fold. 66. The earth and its mountains, the Gods and Danavas, men and others and also the orbits of the constellations and plancts and the Loxas which, it is said, are arranged one abovo the other, are all included in what has been donominated the BranmAnpa (universe). The universe. Dimonsions of tho Bran- 9: 0mo astronomers have asscrt- Ma‘NDA. ed the circumference of the circle of Heaven to be 18,712,069,200,000,000 Yosanas in length. Some say that this is the length of the zone which binds the two hemispheres of the Branminpa. Some Paurdyixas say that this is the length of the circumference of the LoKALoKA PAaRVATA.* ` ® Vide verses 67,68,69, Bua’sxaRa‘ona BYA Goes not answer the objection which these verses supply to his theory of the Earth being the centre of the system. The Sun is here made the principal object of the system—the centre of ४16 BranMa’Npa—the centre of light whose boundary is supposed fixed: but if the Sun moves then the Hindoo Branma’Npa must be supposed to be constantly changing its Boundaries. Subbuji Bapdi had not failed to use this argument in favour of the Newtonian system in his S’iromani Praka’s’s, vide pages 65, 56. Bua’sKaRa’OHa’RYa however denies that he can futher tho opinion that this is the length of the circumference limiting the BaanMa‘Npa and thus saves him- 8५1 from 8 diMculty. L. भ, (Mr. Wilkinson has thus shown tho objoction which Subbajo Bapd mado to the assumption of the Sun’s motion, but I think that the objection is not a udicious ono. Because had the length of the circumforence of tho BRamMa’NDA een changed on account of the alteration of the boundary of tho Sun’s light with him, or had any sort of motion of the stars boon assumed, as would havo been grauted if the oarth is supposed to bo fixed, then, tho inconvenience would havo occurred ; but this is not tho caso. In fuct, as wo cannot fix any boundary of tho light which issucd from tho sun, tho statod lungth of tho cireumforonco of tho BuaumAnpa is an imaginary one. Vor this ronson, BudsKaudcuARya doos not admit this stated length of the oircuwmforunco of the 81८५1 प्र ^ त ५५. ऽ stated in his GaniTa‘DuyaYa’ in the rings alee on the verse 68th of this Ohapter that ^“ those only, who have a perfect knowledge of the Beauma‘Npa as t Ley have of an a’nvata’ fruit held in their palm, can say that this length of the circumferonce of the BuauMma’npa is the true one ;” that is, as it is not in man’s power to fix any limit of the Branma’mna, the said limit is unreasonable. Therefore no aaa can be possibly made to the system that the Sun moves, by assuming such an imaginary limit of tho Branudnna which is littlo loss impossible than the existence of the heavenly lotus.—B. D.] IV. 8.1] Sidhanta-s'iromani. 127 68. Thosc, however, who have had a most perfect mastery of tho clear doctrine of the sphere, have declared that this is tho length of that circumference bounding the limits, to which the darkness dispelling rays of the Sun extend. 69. But let this be the length of the circumference of the BraumMdypa or not: [of that I have no sure knowledge] but it 18 my opnuon that each planct travorses a distance correspond. ` ing to this number of Yosanas in the course of a Katra or a day of BrauMA and that it has been called the Kuaxaxsui by tho ancients. Ind of third Chapter called the Buuvana-xkos’a or cosmo- graphy. CHAPTER IV. CaLLED MapHYA-GATI-VASANA, On the principles of the Rules for finding the mean places of the Planets. 11१68 of tho sovorad 1. Tho sovon [grand] winds havo wines: thus been named : viz.— 18४. The Avaha or atmosphero. 2०१. The Pravaha beyond it. 8rd. The Udvaha. 4th. The Samvaha. 5th. The Suvaha. Gth. Tho Parivaha. 7th. The Pardvaha. 2. The atmosphere extends to tho height of 12 Yosanas from the Earth ; within this limit are the clouds, lightning, &c. The Pravaha wind which is above the atmosphere moves con- slantly to tho wostward with uniform motion. 3. As this sphere of the universe includes the fixed stars and plancts, it theroforo being impelled by the Pravahn wind, is carriod round with the stars and plancts in a constant revolution. 128 ' ‘Translution of the [[४. 4. An illustration of the 4. Tho Planots moving oastward mohon:ot the planets: in the Heavens with a slow motion, appear as if fixed on account of the rapid motion of tho sphere of the Heavens to the west, as insocts moving roversely on 9 whirling potter’s wheel appear to be stationary [by reason of their comparatively slow motion]. Sidorcal and torrestrial 5. Ifastar and the Sun rise simulta- स cher lengths. neously [on any day], the star will rise again (on the following morning) in 60 sidereal अप्र ^ वृणा (8 : the Sun, however, will rise later by the number of asus (sixths of a sidereal minute), found by dividing the product of the Sun’s daily motion [in minutes] and the asus which tho sign, in which the Sun is, takes in rising, by 1800 [the number of minutes which each sign of the ecliptic contains in itself). 6. Tho timo thus found added to tho 60 sidorcal anatixis forms ao true ‘terrestrial day or natural day. ‘The longth of this day is variablo, as it doponds on tho Sun’s daily motion and on tho time [which different signs of the ccliptic take] in rising, [in different latitudes: both of which are variablo elements] .* @ [18१ the Sun moving with uniform motion on the oquinoctial, the cach minute of which rises in cach asc, the number of asus cyual to the number of tho minutes of the Sun’s duily motion, being added to the 60 sidereal @uaqixas, would have invariably made tho oxact longth of the truo terrestrial day as Lanna and othore say. But this is not tho ९५) 0०७१५५५ | tho Sun moves wit # unequal motion on the ecliptic, the equal portions of which do not rise in equal times on account of its being oblique to the equinoctional. Therefore, to find the exact length of the true terrestrial day, it is necessary to determine the time which the minutes of the Sun’s duily motion take in rising and then add this time to 60 sidereal Quatica’s. For this reason, the turres- trial day dotermined by Latia and others is not a truo but it is a menn, Tho difforence betweon tho obliquo ascension at tho beginning of any given day, and that at the ond of it or ut tho boginning of the next day, is the time which the minutes of tho Sun’s notion at the day above alluded to take in rising, but as this cannot bo easily determined, tho anviont Astronomors having determined the periods which the signs of the ०५11६४८ take in rising at a given place, find the time which any portion of a given sign of the ecliptic takes in rising, by tle following proportion. If 80° or 1800’ of a sign: take numbor of tho asus (which any given sign of tho ecliptic tukos) in rising at a given place ; : what time will any portion of tho sign above alluded to take in rising P Tho calculation which is shown in tho Sth vorso deponds on this proportion.— 1. 7.1 IV. 10.1 Sidhanta-s'tromani. 129 Rovolutions of the Sun in ` 7. A sidorcal day consists invari- ® year are lees than the ably of 60 sidereal aHaTIKAs: a mean revolutions of stars by one. ren day of the Sun or terrestrial day consists of that time with an addition of the number of asus equal to the number of the Sun’s daily mean motion [in minates]. Thus the number of terrestrial days in 9 year is less by one than the number of revolutions made by the fixed stars. 8. The length ofthe (solar) year is $65 days, 15 auatrxfs, 80 paLas, 22} viraLas reckoned in BuGmi SXVANA or terrestrial days: The tsth of this is called o saura (solar) month, viz. 30 days, 26 GHATIKAS, 17 PaLAs, 31 viraLas, 524 pRavipatas. Thirty sXvana or terrestrial days make a s{vANA month.* Length of lunar month 9. The time in which the Moon ca acois [after being in conjunction with the San] completing a revolution with the difference between the daily motion and that of the Sun, again overtakes the Sun, (which moves ata slower rate) is called a Lunar month. It is 29 days, 31 Gnaqixas, 50 raas in length.t Tho reason of additivo 10. An ApuimAsa or additivo month months callod ApmrMa’s48. == which is lunar, occurs in tho duration of 324 saura (solar) months found by dividing the lunar month by the difference between this and the saura month. From Longth of solar year. © [Here a solar year consists of 365 days, 15 ©प् + 68, 80 Paxas, 22% VIPADAS, i. 0. 865 d. 6 4. 12 m. 9 ॐ. and in SGnya-eippaa’wra the “He of the year is 866 d. 15 g. 81 p. 81. 4 ०. i. €. 865 d. 6 A. 12 m. 36. 66 2.—B. D.] it That lunar month which ends, when the Sun is in Mrsua stellar Aries is called onaiTRA and that which terminates when the Sun is in vaisHabaa stellar Taurus, is called प्र 18 4 ए ५ and 80 on. Thus, the lunar months 0 to the 12 stellar signs षडा} (Aries) ए दाशा ^ णा ¢ (Taurus) Mitnona (Gemini Karxa (Oancer), Sinna (Leo), Kanya’ (Virgo), Tuna’ (Libra), ए 218 OHIKA "स; io), 77 ^ दए (Sagittarius), Makara (Capricornus), एषा (Aquarius) and Mua (Pisces), aro Curatrra, ए 41874 "र्ता +, ठ र दञाकृा ^) A’stta’pua S’Ra’vana, Bua’prapapa, A’s’wina, (५५) Ma’raas’{xsua, Pavstta, Ma’oua, and Pratauna. If two lunar months terminate when tho Sun is 0015 in one stellar sign, the socond of theso is callod Anmrma’sa an additive month, Tho 80th pert of « lunar month is called Tithi (a lunar day).—B. D.) _ 130 Translation of the (IV. 11. this, the number of the additive months in a KALPA may also be found by proportion.* 11, As a mean lunar month is shérter in longth than o mean sauRA month, the lunar months are thereforo moro in number than the sauRA in a KALPA. The difference between tho numbor of lunar and sauraA months in a katra is callod by astronomers the number of AputmAsas in that period. The reason of subtractive 12. An avamMA or subtractive day ५ which is sXVANA occurs in 64,!, TITTIIS (lunar days) found by dividing 30 by tho difference betwoen the lunar and sfvana month. From this, the number of avamas in a yuaa may be found by proportion.t 18. Ifthe Apsim4sas are found from saura days or months, then the result found is in the lunar months, [as for instance in finding the Amaryana. If in the saura days of a KALPA: aro @ [After tho commencement of a yuGa, a lunar month terminates at tho ond of AMAvVa'sya’ (now moon) anda saURa month at tho mean vurriswabita- BANKRA’NTI (i. ©. whion tho mean Sun entors tho socond stellar sign) which takos place with 54 $. 27 p. 31 ०, 52) p. after the new moon. Aftorwards a second lunur month ends at the 2nd new moon after which the MITnUNA-SANKRA‘NTI takca place with twico the Ghatis. &०, above mentioned. Thus the following San- KRa/NTIS Karka So. take placo with thrice four times Ko. those Guatis, &c. In thie manner, when the Sanxkra’NTt thus going forward, again takes place at new moon, the number of the passed lunar months exceeds that of the saura by ono, This one month is called an additive month: and the savrRa months which an additive month requires for ils happening can be found by the proportion as follows. As 64 ghatis, 27 p. &o. the difference betweon a lunar and a saura : Ono saura month : 3 29, 81, 60 the number terrestrial day &o, in a lunar month : 82, 16, 831, &c. the number of saura months, days, &0.—B. D.] t ead the beginning of a KaLPpa or a दए, the terrestrial and lunar days simultaneously, but the lunar day being less than the terrestrial day, terminated before the end of the terrestrial day, 1, 6, bofore the next sun-rise Tho interval between the end of the lunar day and the next sunrise, is called 4 ४५.१4 6 -8^28 प्त ^ the remainder of the subtractive day. This remainder increases every day, thorcfore, when it is 60 Guayikds (24 hours), this constitutes a AvaMa day or subtractive day. The lunar days in which a subtractive day ocours, are found by the following proportion. If 0 4.28 g. 10 p. the difference between the lengths of terrestrial and of a lunar month. : 1 lunar month or 80 tithis :: 8 whole terrestrial day : 64-y,tithis nearly.—B. D.] 4 The objects of these two verses scema not to bo more than to assert that the fourth term of a proportion is of the same denomination as the 2nd.—L, W. IV. 16.] Sidhantu-s’tromani. 181 so many Apmimasas : : thon in given number of इणेण days; how many Apuimisas?}] If the Apsimfsas are found from lunar days or months, then the result is in sAukA months, and the remainder is of the like denomination. 14. (In liko manner] tho Avamas or subtractive days if found from lunar days, are in sfvana time: if found from s{vana time they are lunar and the remainder is so likewise. 15. Why, O Astronomor, in find- ing the AnaragaAna do you add 8^ णाः months to tho lunar months Cuarrrea &c. [which may havo elapsed from the commencement of the current year]: and tell me also why the [fractional] remainders of ApHImAsas and Avama days aro rojocted: for you know that to give a truo result in using the rule of proportion, remainders should be taken 1160 account ? A question. Reason of omitting to in- 16.* As the lunar month ends at 0 711 the change of the Moon and tho 1. ^“ 8०५२५ month terminates when the Sun enters a stcllar sign, the accumulating portion of an ADHIMASA always lics aftor cach now Moon and bofore the Sun नामाह tho sign. श [The meaning of these 4 verses will be well understood by a knowledge of the rule for finding the ^ छ 4264 त 6, we therefore show the rule here. In order to find the ^ घ ५४6८ त 4. (elapsed torrestrial daye from the commence- mont of the Kanpa to the required time) astronomers multiply the number of BAURA years expired from the beginning of the Kaira by 12, and thus they got tho numbor of saura months til 1 tho last Mzsua Sankranti (that is, tho timo whon tho Sun cntera the Ist sign of tho Zodinc called Arics.) ‘Yo these months they add thon the passod lunar months Citarrra &o., considoring thom as SAURA. ‘hese 840RA mouths become, up to the time when the Sun enters the sign of tho Zodiac corresponding to the required lunar month. They multiply then the num- ber of these months by 80 and add to this product tho number of the passed TiTH18 (Junar days) of the required month considoring them as sauna days, ‘The number of 84 ए४८ days thus Found becomes greater than that of those till the end of the required दका by the apHiMasa 8’eTHA. ‘To mako theso sauRA days lunar, they determino tho clapsod additivo months by tho proportion in tho following manner As the nnmbor of sauna days inn Kara : the number of additive months in that period 2: tho number of sauna duys just found : the number of additive months vlupsed ‘2 132 Translation of tho [1V. 17. 17. Now tho number of rirmis (lunardays) olapsed sinco the change of the Moon and supposed as if saura, is added to the number of saura days [found in finding the Anaraaya] : but as this number exceeds the proper amount by the quantity of the Apaimasa-s’EsHA therefore the ApHImAs-sESIIA is omit- ted [to be added]. 18. [In the same manner] there is always a portion of a Avama-s'asHa between the time of sun-rise and the end of the [preceding] ऋतपा, By omitting to subtract it, the AHARGANA is found at the time of sun-rise: if it were not omitted, tho AHARGANA would represent the time of the end of the कपा [which is not required but that of the sun-rise]. 10 19, 20, 21 and 22. As the true, called tho Uparixrara terrestrial day is of variable length, the भ Anaraana has been found in mean torrostrial days: tho placos of tho plancts found by this Aaraaya when roctifiod by tho amount of tho corroction owllod tho Upaydyrara whothor additive or subtractive will bo found to be at the time of sun-rise at Lanki.* The ancient If these additive months with their remainder be added to the 84 ए 24 days above found, tho sum will be the number of lunar days to the end of the sauna days, but we require it to the end of the ध uired TITHI. 41) 83 tho remainder of the additive months lies between the end of the TITHI and that of its corre- sponding 84 ए24 days, they therefore add tho whole number of ^ णा - र {848 just found to that of the saura days omitting the remainder to find the lunar Jaye to the end of the required शका, Moreover, to make these lunar days terrestrial, they determine Avama subtractive days by tho proportion euch as follows. . As the number of lunar days in a (41.24 : the number of subtractive days in that period :: the number of lunar da ye just found : the number of Avama elapsed with their remainder. If these Avamas be subtracted with their remainder from the lunar days, tho difference will be the number of the Avama days elapsed to the end of the requir- ed TITH!; but it is required at the time of sun-rise. And as the remainder of the subtractive days lies between the end of the काक्या and the sun-riso, they therefore subtract the Avamae above found from the number of lunar days omitting their ee BD] AvVaMa-s’ssHs. ‘Thus the Amaraana itself becomes at the sun- 2198699 [| ® {If the San been moving on the equinoctial with an equal motion, tho terrestrial day would have been of an invariable length and consequently the Sun would have reached the horizon at Lanxi at the end of the AWana@ana which is an enumeration of the days of invariable length that is of the mean terrestrial daye. But the Sun moves on the ecliptic whose cquul parts do not IV. 23.] Sidhanta-s'iromani. 133 Astronomers havo not thus rectified the places of the planets by this correction, as itis of a variable and small amount. The difference betweon the number of asus of the right ascension of the mean Sun [found at the end of the Auarcana] aud the number of asus equal to the number of minutes of the mean longitude of the Sun [found at the same time] is the difference between the true and mean AHARGANAS.* Mul- tiply this difference by the daily motion of the planet and divide the product by the number of asus in a nycthemeron.t The result [thus found] in minutes is to be subtracted from the places of the planets, if the asus [of the right ascension of the mean Sun] fall short of the xaLfs or minutes [of the ` mean longitude of the Sun], otherwise the result is to be added to the places of tho planets. Instead of the right as- cension, if oblique ascension be taken [in this calculation] this UpayAntara correction which is to be applied to the places of the planets, includes also the cHARA correction or the correction for the ascensional difference. Reason of the correction 23. The places of the planets callod tho Dus‘a’ntaRa. = Which sro found boing roctifiod by this UpaAYANTARA corroction aot the time of sun-riso at LANKA may be found, boing applicd with tho Desaxnrara correction, at the time of sun-rise at a given place. This DesanTaRa correction is two-fold, one is east and west and the other rise in equal periods. For this reason, the Sun does not come to the horizon at Lamxa’ at the end of the AHarGana. Therefore the places of the planets doterinined by the mean AgARGaANA, Will not be at the sun-rise at Lanka’. Henoe @ correction is necessary to be a pplied to the places of the planets. This correction called UpayAnrara has been first invented by BaXsHaracuAeya who consequently abuses them who say that the places of the planets determined by the mcan AnaRGaya become at the time of the sun-rise at Lanxd.—B. D.] ®= The difforence between tho mean and truo AmArGANAS is that part of the equation of time which is due to the obliquity by the eoliptic—L. भ, ¢ [This calculation is nothing clsc than the following simple proportion If the number of Asus in a nyothempron : daily wotion of the planet :: the difference between the true aud moan AHAUGANAS givo.—B. )).] 184 Translation of the (TV. 24. is north and south. This north and south correction is called CHARA, 24. The line which passes from Lankd, Ussayinf, Kurv- KSHETRA and other places to Meru (or the North Pole of the Earth) has been denominated the MapnyarEekHA mid-line of the Earth, by the Astronomers. The sun rises at any place east of this line before it rises to that line: and after it has risen on the line at places to its west. On this account, an amount of the correction which is produced from the difference beo- tween the time of sun-rise at tho mid-lino and that at a givon place, is subtractive or additivo to tho places of tho plancts, as the given place be east or west of the mid-line [in ordor to find the places of the planets at the time of sun-rise at the given place]. 25. As the [small] circle which is described around Meru or North Pole of tho Marth, at the distance im Yosanas rockon- ed from Meru to given placo and produced from co-latitudo of the place [as mentioned in tho verso 50th, Chapter 111. | 18 called rectified circumference of the arth (parallel of latitude) [at that place] therefore [to find this rectified circumference], the circumference of the Earth is multiplied by the sine of co- latitude [of the given place] and divided by tho radius. End of 4th Chapter called Mapuya-Gatr Vasana. ® This amount of correction is determined in the following manner The yosanas between the midline and the given place, in the parallel of Jatitude at that place, which is denominated SpasutTa-PaRIDaI ara called Drs’a’NTABA YOSANAB Of that place. Then by the proportion As the number of yosanas in tho Spasura-Paniput: GO auatiKas: : Desa’n- TARA 204 47548 : the difference between the time of sun-rise at midline and that at a given place, This diffurence callod Des’a’NTaRa GiaTIKa’s is tho longitude in time east or west from Lanxa’, Again As 60 anatixa’s: daily motion of the planct : ; DrsantTa’na QUATIEA’s the amount of the correction required Or this amount can be found by using the proportion only once such as follows As the number of yovanas in the Spasura-PaniDur: daily motion of tha planet: : Das‘AntTapA.YoJaNas: the same amount of the corrvction ubove found. — 1 V. 2.] Sidhanta-s‘tromani. 135 CHAPTER V. On the principles on which the Rules for finding the true places of the Planets are grounded. 1, The planes of a Sphere are intersected by sincs of BHusJA ond KoTI,* as a [01666 of cloth by upright and transverse threads. Beforo describing tho spheric, I shall first explain the canon of sines. . 2. Take any radius, and suppose it the hypothenuse (of a right-onglod trianglo). ‘Tho sino of biusa is tho base, and the sinc of एणा is tho squaro root of the difference of the squares of tho radius and tho base. The sines of degrees of BHUJA and KOTI subtracted separately from the radius will be the versed sines of KofI and BHoJA (respectively). On tho canon of sines. [* Tho pirvsa of any given arc is that arc, less than 90°, the sine of which is cqual to tho sine of that given arc, (tho consideration of the positivencss and negation of tho sino ia hero negleotod). For this reason, tho mitusa of that oro which terminatce in tho odd quadrants i. ९. tho 18४ and 8rd is thnt part of tho given arc which falls in tho quadrant whero it torminates, and tho 711१५ of tho are which ends in tho ovon quadrants, 1, 6, in the 2nd and 4th, is te aro which is wanted to complete the quadrant where the givon aro is ended. The koft of any arc is the complement of the BHUJA of that arc. Ch 7 Py Let the 4 quadrants of a circle ABCD be successively A B, B C, © Dand D A, then t he BWUJAS of 110 ११८४ AP,, ^ 1 19, ^ 15; ADV, \५।॥ 0५ ALP, (15 © 09, A P, and the complements of {11680 nutsas aro thoarcs BD P,, BP, DP,, 1) 1५ respectively. — D.)} 8 e 136 Translation of the [V. 3. 3. The versed sino is like tho arrow intersecting tho bow and the string, or the arc and the sine.* The square root of half the square of the radius is tho sine of an arc of 45°. The co-sine of an arc of 45° is of the samo length as the sine of that arc. * These methods are grounded upon the following principles, written by Boa‘sxara’omarya, in the commentary VasaNa’-BHA’SsHYA. (1) Let the aro A ए = 90° and AO = 45e AD (== ॐ A B) == 910. 45°; and let OAorOB=the radius (R) thon A LB? == 0 A'-+0B? = 20 A? = 2 RB? ०० AB= A/2 B' and AD =} AB = ^ 2 orsin, 45° == JF * (2) It 18 ovidont and statod also in tho [न ५०४५, that tho sido of a rogulae hexagon is equal to the radius of its circumscribing circle (i. ०, cli. GO° = KR). Hence, sin. 80° = ‡ R. (3) Let A B be the half ofa given aro A P, whose sine P M and versed sine A M ure given. Then AP=A/P 2४ + ^ M? and} 4 2 =^ ति == 11. AB sin. A 8 = $ ^ ` - ^ ४ (4) The proof of the last method by Algebra cos = R — versed sino ५०, cos? = R'— 2R.v0-+ > subtracting both sides fromR’, € द R*—cos* = 2 R. ० —v* or sin = 2R. 0 — v* adding v* to both 91068 8111. ~ v’= 2R. % and 4 (sin.? 4 ९) == ‡ 7२. % oxtraoting tho squaro root, + a/ vine न ९४ = 7 ॐ ४.० but by the preceding method ¢ 9 ~~ + sin.? +. »* = the sine of half the given aro; „^, sin, } aro = ८/१ ४.१४. DJ V. 7.) Sidhdnta-s'tromani. 187 4. Half the radius is the sino of on arc of 80°: The co-sine of an arc of 30° is the sine of an arc of 60°. Half the root of tho sum of the squares of the sine and versed sine of an arc, is the sine of half that arc. 5. Or, the sine of half that arc is the square-root of half the product of the radius and the versed sine. The sines and co-sines of the halves of the arcs before found may thus be found to any extent. 6. Thus a Mathematician may find (in a quadrant of a circle) 3, 6, 12, 24 &c., sines to any required extent.* Or, in a circle described with a given radius and divided ‘into 360°, the required sines may be found by measuring their lengths in digits. _ Reason of correction which 7.¢ As the centre of the circle of 1 ue the constellation of the Zodiac coin- planet. cides with the centre of the Earth: * (When, 24 sines are to be determined in 8 quadrant of a circle, the 8 sines, १, €. 12th, 8th and 16th, can be easily found by the method here given for finding the sines of 45°, 30°, and the complement of 30°, i. 6. 60°. en by means of theso threo sinos, the rest can bo found by the method for finding tho sine of half an arc, as follows. From tho 8th sine, tho 4th and the co-sino of tho 4th i. €,» tho 20th sine, can bo detorminod. Again, from the 4th, tho 2nd and 23nd, and from tho 2nd, tho lst and 23rd, can be found. In liko manner, the 10th Mth, 6th, 1th, 7th, 17th, 1 1४), and 18th, can aleo bo found froin the 8th sine. From tho 12th again, the Gth, 18th, 8rd, 2let, 9th and 15th can be determined, and the radius ie the 24th sine. Thus all the 24 sines are found. Several other methods for finding the sines will be given in the sequel.— 8. D.] [+t Bra’sxaRa’cua’kya maintains that the Earth isin the centre of the Universe, and the Sun, Moon and the five minor planets, Mars, Mercury 9 &e. revolve round the Earth in circular orbits, the centres of which do not coincide with that of the Earth, with uniform motion. The circle in which a planet revolves is culled PRatTIVRITTA, or excentrio circle, and a circle of the same size which is supposed to have the same centre with that of the Earth, is called KAKSHA VRITTA 07 concentric circle. In the circle, the planet appears to revolve with unequal motion, though it revolves in the excentric wit i equal motion. The place where the planet revolving in the excentric appears in the concentric is its true place and to find this, astronomers apply a correction called ManDa- PHALA (Jat equation of the centre) to the mean place of the planet. A mean planet thus corrected is called ManDa-sPasntA, the circle in which it revolves MANDA-PEAT{VRITTA (1st excentric) and its farthest point from the centre of the concentric, MANDOCHCH (1st higher Apsis). As the mean places of the Sun and Moon when corrected by 1st equation become true at the centre of the Earth, this correction alone is sufficient for them. But the five minor planots, Mare, Mercury, &c. whon corrected by the 1st equation are not true at tle centre of the Earth.but at another place. For this reason, astronomers having assumed F 138 Translation of the [V. 8. and the centre of the circle in which the planet revolves does not coincide with the centre of the Earth: the spectator, therefore, on the Earth does not find the planet in its mean place in the Zodiac. Hence Astronomers apply the correction called BHUJA PHALA to the mean place of the planet [to get the true place]. | । Mode of illustration of 8. On the northern side of a wall १0 running due cast and west, lct the teacher draw a diagram illustrative of the fact for tho satis- faction of his pupils. A verse to द ९6 those 9. But this science is of divine who may be disposed to de- oe . t + spond in consequence ofthe Origin, revealing facts not cognizable difficulties of the science. = by the senses. Springing from the the concentrio cirole as second excentrio of these five planets, take another circle of the same size and of the same centre with the arth as concentric, and in order to find the place where the planct revolving in the 2nd excentric appears, in this concentrio, they apply a correction called s‘faitra-PiulaLa, or 2nd equation of the contre, to the mean place corrected by tho lst equation, Tho MANDA-SPASIUTA planct, when correctud by the 2nd equation is cnilud s’vasiura, or true planot, the 3ndexoentrio, s'famna-PRATIVRITTA, and ils furthest point from the centre of the Earth, s’tenRoogon the 2nd higher Apsis. If a man wishes to draw a ding ram of the arrangoment of the planets accord- ing to what we have briefly stated here, he should firat describe the excentrio circle, and through this excentrio the concentric, and then he may determine the place of the manpa-sPasHTa planet in the concentric thus described. Again, having assumed the concentric as 2nd excentric and described the concentrio through this 2nd excentric, he may find the place of the true planet. This is the proper way of drawing the diagram, but astronomers commonly, having first described the concentric, and, through it, the oxcentrio, find the corrected mean place of the planet in the concentric, After this, having described the 2nd exoentrio through the same conocntric, they find tho true placo in the concentric, through f 16 corrected mean place in the same. These two modes, of constructing the diagram differ from each other only in the respect, that in the former, the concentric is drawn through the excentrio circle, and in the latter, the excentrio is drawn through the concentric, but this can easily bo undorstood that both of these modes are equivalout and produce tho same result, In order to find the 1st and 2nd equations through a different theory, astro- nomers assume that the oontre of a small circle called nicnhoououa-VRiTTA or epicyole, rovolves in the concentric circle with tho mean motion of the planot and the planet revolves in tho epicycle with a revorso motion equal to the moan motion. Bua‘sKana’oua’Bya, himself will show in the sequel that the motion of the planet is the same in both these theories of excentrics and 3 ricycles. It is to be observed here that, in the oase of the planets Murs, Jupiter and Saturn, the motion in the excentric is in fact their pro per revolution, in their orbits, and the revolution of their s 1@HROOHCHA, Or quick a pogee, corresponds to a revolution of the Sun. But in the case of the planets Mercury and Venus, the revolution in the excentrio is performed in the sume timo with tho Sun, and the revolutions of their s'taHRrocuonas are in fact their proper revolutions in their orbits.—B. D.] ४, 12.) Sidhdnta-s'tromani. 139 supromo Brana himself it was brought down to tho Barth by VasisutHa and other holy Sages in regular succession ; though it was deemed of too secret a character to be divulged to men or to the vulgar. Hence, this is not to be communi- cated to those who revile its revelations, nor to ungrateful, evil-disposed and bad men: nor to men who take up their residence with its professors for but a short time. Those professors of this science who transgress these limitations imposed. by holy Sages, will incur a loss of religious merit, and shorten their days on Earth. 1 2 10. In the first place then, de- gram to illustrate the ex- scribe a circle with the compass opened न to the length of the radius (8438). This is called the KAKsHAvRITTA, or concentric circle; at the centre of the circle draw a small sphere of the Earth with a radius equal to ,',th* of the mean daily motion of the planet. 11. In this concentric circle, having marked it with 360°, find the place of the higher apsis and that of the planet, counting from the Ist point of stellar Aries; then draw a (perpendicular) diamoter passing through the centre of the Earth and the higher apsis (which is called ucHcHA-REKHA, tho linc of tho apsidcs) aud draw another transverse diameter [perpendicular to the first] also passing through the centre. 12. On this line which passes to the highest apsis from the centre of the Earth, take a point at a distance from the Earth’s centre equal to the excentricity or the sine of the greatest equation of the centre, and with that point as centre and the radius [equal to the radius of the concentric], describe the PRATIVRITTA or excentric circle; the UCHCHA-REKHA answers the like purpose also in this circle, but make the transverse diameter different in it. = All the Hindu Astronomers seem to coincide in thinking that the horizontal paralinxs PARAMA-LAMBANA of oll tho planets amounts to a quantity equal to rgth of their daily motion.—L. W. ह 2 140 Translation of the [V. 13. 18 and 14.* Where the ucHcHA-REKHA perpendicular dia- meter (when produced) cuts the excentric circle, that is the = Fig. 1. aN oN | L ¢: [ In Bg. 19४ let & be the centre of the concentric circle A 8 0 D, ¥ the place of the etellar Arica, A that of tho highor apsis, and M that of the moan p (त in it: then HA will bo tho vouona-unKita (tho lino of the apsides). Again let EO bo the oxoontricity and H FQ tho oxcontrio which has O for its centre; thon H, # P, will be the places of the higher apsis, the stellar Arios and the planet respectively in it. Hence H P will be ६ he KENDRA; P K the sine of the kenDRA; P I the co-sine of the KENDRA. The kENDRA which is more than 9 signe and less than 8 is called MRIGADI i. e. that which terminates in the six signs beginning with Capricornus) and 4 9६ which is above 8 and less than 9 is called KanxyaDI (i. ७, that which ends in the six signs beginning with Cancer). Thus (Fig. 1) that which terminates 118 H Fis अपता ददद, and that which ends in F L G is Kanxya‘p1.—B. D.] ए. 15. Sidhdnta-s'iromant. 141 placo of tho highor apsis in it also. From this mark the first stellar Aries, at the distance in degree of the higher apsis in antecedentia: the place of the planet must be then fixed counting the degree from the mark of the 180 Aries in the usual order. । The distance between the higher apsis and the planet is call ed the kenpra.* The right line let fall from the planet on the UCHCHA-REKILA 18 the sine of suusa of the KenpRA. The right line falling from the planet on the transverse diameter is the cosine of the KENDRA, it is upright and the sine of BHUJA is a transverse line. The ] rinciple on which 15. As the distance between the 1 (1 diameters of the two circles is equal is based. to the excentricity and the co-sine of the KENDRA is above and below the excentricity when the ` KENDRA is MRIGADI and KARKYADI (respectively). ® The word Kznpra or centre is evidently derived from the Greek word xeytpov and means the true centre of the planet.—L. W. + [In (Fig. 1) P K is the इमाए ^ Kort and 7? 7 tho kanya (tho hypothonuso) which cuts the concentric ot ‘I’. Lence the point T will be the apparent place of the planet and T M the oquation of the oentre. This os ha can be detorminod as follows. Draw M »# perpendicular to E ‘I, it will be the sine of the equation and the triangle P M # will be similar to tho triangle P 9 K. ००७ Pp E (1 E K = P M $ M % 3 PM.EK ——— = sino of the equation ; PE EO.EK PE Now, let & = KENDRA, a = the distance between the centres of the two circles excentrio and concentric, « = sine of tho equation, and # = hypothenuse : then the sPHUTA KOTI <> cos. & + a, according as the KunpRa is MRIGADI or KARKYADI, and ¢ = o/sin.* ‡ (cos. & ‡ ०); hence by substitution hence M 2 = + 9 2 ४ = 1 == ४0 ©, 91). & a.ein. & aa h (नर्क (on FE a) sin.? £ ~+ (oos. & & a)? 142 Translation of the [V. 16. 16 and 17. Therefore the sum or difference of the co-sine and excentricity (respectively) is here the spHuta KOT! (i. e. the upright side of a right-angled triangle from the place of the planet in the excentric to the transverse diameter in the concentric,) the sine of the शण [of the KENDRA] is the BHUJA (the base) and the square-root of the sum of the squares of the sritufa ण्य and BHUJA is"called Karna, hypothenuse. This hypothenuse is the distance between the Earth’s centre and the planet’s place in the excentric circle. The planet will bo observed in that point of tho concentric cut by the hypothenuse. The equation of the centre is the distance between the mean and apparent places of the planet: when the mean place is more advanced than the apparent place then the equation thus found is subtractive; when it is behind the true place, tho equation 1s additive.* Tho roason for assuming 18. Tho moan planot moves in its the MaND-sragitya planet es Wanpa-prarivgitta (first excentric) ; 8 mean in finding the 2nd equation. the MANDA-SPASHTA planet (1. 6. whose mean place is rectified by the first equation) moves in its S IGHRA-PRATIVRITTA (second excentric). The MANDA-SPASHTA It also follows from this that, when cos. & is equal to a in the KARKYADI KENDRA, then & will be equal to sin. &, otherwise ¢ will always bo greater than sin. & and consequently ॐ will be less than a. Hence, whien ¢ is equal to sin k, w will then be greatest and equal to a, i.e. the equation of the centre will be greateat when the hypothenuse is equal to the sine of the KENDRa, or when the planet reaches the point in the excentric cut by the transverse line in the concentric. Therefore, the centre of the oxcentric is marked at the distance ual to the exoentricity from the contre of the concentrio (as stated in the v 12th.)—B. D.] @ (Thus, tho mean planet, corrocted by tho Ist equation, becomes MANDA-SPASHIPA and this process is called the maNnDa process. After this, the MaNDA-SPasHTa when rectified by the sl’anra PraLa, or 2nd equation, is the spasata planet, and this 2nd process is termed the s’IGHRa process. Both of these processes, MANDA-SPASHTA and SPasHTa are reckoned in the VIMaN- paLa or the orbit of the planet as hinted at by BuaskanacHarya in the commentary called VasaNa-BHAsHYA in the sequel. These places are assumed for the ecliptic also without श any correction to them, because the correction required is yery small.—B. D.] V. 23.) Sidhdnta-s'tromani. 148 18 therofore hore assuined as tho mean planct m the second process (i. e. in finding the second equation).* Tho reason for the inven. 19. The place in the concentric tion of the higher apsis. in which the revolving planet in its own excentric is seen by observers is its true place. To find the distance between the true and mean places of the planet, the higher apsis has bcen inserted by former Astronomers. 20. That poimt of tho oxcentric wlich is most distant from the Earth has been denominated the higher apsis (or एला ¢) : that pomt is not fixed but moves; a motion of the higher apsis has therefore been established by those con- versant with the science. 21. ‘The lower apsis is at o distance of six signs from the higher apsis: when the planet is in either its higher or lower apsis, then its true place coincides with its mean place, because the line of the hypothenuse falls on the mean place of the planet in the concentric. 22. As the planet when in the higher apsis is at its greatest distance from the Earth, and when in the lower Tho cauro of varintion of S8psis ab ils loast dishanco, therofure its apparent sizeof planets ५७०, ise appears small and Jargo accord- ingly. Jn liko manner, its dise appears sinall and Jarge accord- ingly as the planet is near to and remote from the Sun. 23. To prevent the student from becoming confused, I have separately explained the proof of finding the equation by the Prarivrirra ^ परजा of the diagram of the excentric. I shall now procced to oxplain the same proof in a different manner by the diagram of a NfcnocucHa-vRitta (epicycle). * [For this reason, having assumed the Manpa-6Pasnta planet for the mean, which MANDA-8Pa8uTA can be determined in the concentric by describing the excentric circle &c. through the menn planct and mMaNnnocncya, make the place of the stellar Arics from the MANDA-SPABUTA place in the reverse order of the signs and then determine the place of the s1amrocHowa in the order of the signs. Through the places of the stellar Aries and s’1anrocncua describe the 2nd excentric circlo ८. ` in the way mentioned before, and then find the place of tho truc planet in tho concentric.—B. D.] 144 Translation of the [V. 24. त ce aie 24. Taking the mean place of the gram to illustrate the theory of planet in the concentric as the centre, + with a radius equal to the excentricity of the planet, draw a circle. This is called nfciocncua VRITTA or epicycle. Then draw a line from the centre of tho Marth passing through the mean place of tho planct [to the circumference of the epicycle]. 25. That place in the epicycle most distant from the centre of the Earth, cut by the lino [joining the centre of the Earth and mean place of the planet] is supposed to be the place of the higher apsis: and the point in the epycicle nearest to the Earth’s centre, the lower apsis. In the epicycle draw a transverse line passing through the centre of it [and at right-angles to the above-mentioned line which is called here UcHCHA-REKH(]. 26. As the mean planet revolves with its KENDRA-GATI (the motion from its higher apsis) in the Ist and 2nd epicy- cle marked with the 12 signs and ४60 १५८०३ towards thie reverse signs, and according to the order of the signs respec- tively from its higher apsis. 27. Mark off therefore the places of the first and second KENDRAS or distances from their respective higher apsides in the manner directed in the last verse: the planet must be fixed at those points. [Here also} The (perpondicular) line from the planet to the UCHCHA-REKHA is the sine of the BHUJA of the KENDRA: and from the planet on the transverse line 18 the cosine [of the KEnDRA].* (See note next page.) To find the hypothenuse 28 and 29. The BHUJA PHALA and and the equation of centre, = cons pata of tho KENDRA which aro found [in the Ganir{pHy(ya] are sine and cosine in the epicycle. As the KofI PHALA is above the radius (of the concentric) in MRIGADI KENDRA and within the radius in KARKYADI-KENDRA, the sum and difference, therefore, of the KoTI PHALA and the radius is here the spHuTA-KoTI (upright line), the BHUJA PHALA is the BHuJA (the base) and the kakya hypothenuse (to complete भ. 27. ] Sidhénta-s'tromani. | 145 the right-angled triangle) is the line intercepted between the centre of the Harth and the planet. The equation of the centre is here the arc [of the concentric] intercepted between ॐ Note on verses from 24 to 27. [In fig. 2, let ABC D be the concentric, f the place of the stellar Aries, E the centre of the Earth, M the mean place of the planet in the doncentric, A f 2 g, the Epicycle, # the place of the higher apsis in it, EB & the UOCHCHA-REKHA’ ए the place of the lower apsis, P that of the planet, 4 P the KeyprA, P & the sine of tho KENDRA and P ¢ tho cosine of it. The sine and oo-sine of the KRNDRA in the escontric, reduced to their dimensions in the opicycle in parts of the radius of the concentric, are named BHUJA-PMALA aud ROTI-PHALA respectively in the GayitApHyAya. That is As the radius or 860° of the concentrio : the sino and cosine of the KgNDRA in the excentric :: excentricity or the periphery df the epicycle : BHUJA-PUALA and KOTI-PHALA respectively. Therefore the BHUJA-PUALA and KOTI-PHALA must be equal to the sine and cosine of the ए 8724 in the epicycle.—B. D © 146 Translation of the [V. 30. the mean place of the planet and the point cut by the hypo- thenuse. The equation thus found is to be added or subtracted as was before explained.* 30. The planet appears to move forward from MANDOCHCHA, Construction of the mixed OF L8t higher apsis, in the excentric dingrams of the excentrio circle with its KENDRA-GATI (the mo- and epicycle. ‘ : ‘ tion from its MANDOCHCHA) and in the order of the signs and to the East: From its s/aHrocucna, 2nd higher apsis, it moves in antecedentia or reversely, as it is thrown backwards. ` 31. When the epicycle however is used, the reverse of this takes place, the planet moving in antecedentia from its 18४ higher apsis and in the order of the signs from its 2nd higher apsis. Now as the actual motion in both cases is the same, while the appearances are thus diametrically opposed, it must be admitted therefore that these expedients are the mere inventions of wise astronomors to ascertain the amount of equation. ® In (Fig 2) E & is the spnuta-xopr, 2 E the hypothenuse, T the apparent place of the planet in the concentric und T M the equation of the centre. This equation can also be found by the theory of the epicycle in the following manner. Draw T # perpendicular to E M, then T # will be the sine of the equation; let it be denoted by ॐ, the KENDRA in the excentrio by &, the excentricity by a, and the hypothenuse by 4: then R:snk=a: P& tho pwusa-PHata asink |, Now, the triangles 7 T » and E P & are similar to each other ०० the BHUJA-PHALA = ० BP: P&kE=ET: TH or ¢$ : P&E=—B ३ ॐ ९& ॐ 2 ०१ &= ` h that is, the BHUJA-PHALA multiplied by the radius and divided by the hypothe- nuse is equal to the sine of the equation. asin k But P & = ०० by substitution asink 2 asin & ॐ ~~ = R h BD) by the theory of the excentric in the note on the verses 16, 16 and 17.— (| ° » the sine of the oquation as ४. 87.] Sidhdnta-s'iromani. 147 32. If the diagrams (of the excentric and epicycle) be drawn unitedly, and the place of the planet be marked off in the manner before explained, then the planet will necessarily be in the point of the intersection of the excentric by the epicycle. 33. [In illustration of these opposite motions, examine an oil-man’s screw-press.] As in the oil-man’s press, the wooden press (moving in tho direction in which the bullock fastened to it goes) moves (also itself) in the opposite direction to that in which the bullock goes, thus the motion of the planet, though it moves in the excentric circle, appears in antece- dentia in the epicycle. 84. As the centre of the Ist epicycle is in the concentric, Explains why the 6 minor let the planet therefore move in the 1. 0 tte concentric with its mean motion: In their true places. the concentric [at that point cut by the first hypothenuse] is the centre of the s1'oHRA NicHocHCHA, vgiTTa or of the 2nd epicycle: In the second or इल ^+ epicycle is found the true place of the planet. 35. The first process, or process of finding the 18४ equa- tion, is used in the first place, in order to ascertain tho position of the centre of the sfoHRA NfcHOCHCHA VRITTa or of the 2nd epicycle, and the 2nd process, or the process of the 2nd equation, to ascertain the actual place of the planet. As those two processes are mutually dependent, it on this account becomes necessary to have recourse to the repetition of these two processes. 36 and 37. Some say that the hypothenuse is not used in plains: ccaseit of Gules the lst process, because the difference sion of hypothenuse inthe (in the two modes of computation) MANDA process. er 5 18 inconsiderable, एप others maintain that since in this process the periphery of the first epicycle being multiplied by the hypothenuse and divided by the radius becomes true, and that, if the hypothenuse then be used, the result 1s the same as it was before, therefore the hypothenuse is a 2 148 Translation of the | (V. 38. not employed. No objection is to be made why this is not the case in the 2nd process, because the proofs of finding the equation are different here.* 88. As no observeron the surface of the Earth sees the planet moving in the excentric, de- flected from his zenith, in that place of the concentric, where an observer situated at the centre of the Earth observes it in the eastern or western hemisphere, and at noon both observers see it in the same place, therefore the correction called ति ^^ ^ ए is declared (by astronomers). The proof of this is the same as in finding the parallax.t+ Reason of NaTaxkarMa. © (The BuUsA-PHALA, determined by means of the sine of the first KENDRA of the planet (i.e. by wultiplying it by the peri phery of the lst epicycle and dividing it by 3609) has been taken for the sine of the Ist equation of the centre : and what we have shown in the note on tho ए, 28 and 29, that the एर र PitaLa, whon inultiplied by tho rudius and dividud by the hypothonuse, becomes the sine of the equation may be understood only for finding the 2ud equation of tho fivo minor planets and not for detormining the lst oquation. 8५५1५ say that the omission of tho hypothenuso in the let process has no other ground but the very inoonsidorable difference of the result. But bRanMa- GUPTA maintains that tho periphery of the lst epicycle, varies according to the hypothenuse ; that is, their ratio is alwaye the same, and the periphery of the 188 opicycle, mentioned in the @anrréADHYAYa, is found at the instant when the hypothenuse is equal to the radius. For this reason, it is necessary at first to find the true periphery through the hypothenuse and then determine the lst equation. But, he declares that by so doing; also the sine of the equation becomes equal to the BHUJa-PHALA as follows. As RB: lat periphery = the hypothenuse: the true periphory PK4 ` the true periphery = ——— , and consequently the buUJA-PHALA in RB Px A’ sink the true epicyole = ——-——- , ——;_ R 860° PXA sink R % — X —and abridging = 260० ` ० the sine of the lst equation = ॐ .sin & whioh is equal to the BHUJa-PHaLA. Hence the hypothenuse is not 860° used in the 1st process. BrauMacorta’s opinion is much approved of by Bua’sxapa’oma’rya.—B. D.} + But this is not the case, because the NaTaKatM which Bua‘skama’cHa RYA has stated in the GanivAvuydyva has no connoction with the fact stated in this 81701८4 und therefore many say that this s’toxa does not belong to the text.—B. D.]} Digitized by G oogle V. 40.] Sidhdnta-s’tromani. 149 39. The mean motion of a planet is also its truo motion 1 when the planet reaches that point in and truo motions of all the the excentric cut by the transverse म diameter which passes through the centre of the concentric: and it is when the planet is at that point that the amount of equation is at its maximum. [LaLa has erroneously asserted that the mean and true motions coincide at the point where the concentric is cut by its ex- centric. ] # 40. Having made the excentric and other circles of thin Manner of observing the pieces of bamboo in the manner ex- retrogression &c. of Planets. Finined before, and having changed the marks of tho placcs of tho planct and its s‘futrocucna 2nd higher apsis with thoir daily motions, an astronomer may quickly show the retrogressions, &c.t * Tho ancient astonomers Lax, S’atratt &८, say that the true motion of a planet equals to its mean motion when it reaches the point of interaection of the concentric and excentric. But Bia’skara’Cnarkya denying this, says, that when the planet reaches the point when the transverse axis of the concentric cuts tho oxcentric and when the amount of equation is a maximum, the truo motion of + planet becomes equal to its moan motion, For, suppose, 209 yg, 25, &., are tho mean places of » planet found on successivo days at sun-riso w hen the planet proceeded from its higher or lower apais and ¢,, ०9) ०9) &c. are the amounts of oquation, thon p , + ¢,, po, £ ९४, 28 + ९३, &c. will bo the truo places of the planet, <^ Pa—P, ¬ (९9-९1), Ps— 79 + (^ - ^9), Pa—Ps 2: (०५--९3), Se. will be the true motions of the planet on successive days. Now, 85 the difference between tho true and mean motions is called the GaTipHaLa, by cancelling therefore, Pe—P: Par --7 9, Ke. tho parts of the trne motions which are equal to the mean motion, the remaining parts ¢s—e,, ¢,—e, &०. will evidently be the GATIPHALAS that 18 tho diflerences betweon two successive amounts of equation are the GaTIPHALAS. ‘Thus, it is plain that the @atipuata entirely deponds upon the amount of equation, but as the amount of equation increases, so the GATIPHALA is decreased und therefore when it 13 a maximum, the @aTipaaLa will indifintely be decreased i. e. will be equal to nothing. Now as the amount of equation becomes a maximum in that placo where the transverse dinmeter of the con- contrio circle cuts the excontric, (sce the note on verses 15, 16 and 17) the GATIPHALA, therefore becomes equal to nothing at the same place, that is, in that very place, the true motion and mean motions of a planet are equal to cach other. Ulaving thas shown a proof of his own assertion, BItAsKARA‘OWARYA says that what the ancient astronomers stated, that the ध पठ and mean motious of a planet are eqnal to each other when the plunet comes in the intersection point of the concentric aud excentric circles, is entirely ungrounded. —B. D.] + According to the method above mentioned, if the place of the higher apsis and that of the planet be changed, and the planet's place be marked, the motion of the planet will be in a path like tho dotted line as shown in the diagram. See Diagram facing this page. 150 Translation of the [V. 41. 41. The word KENDRA (or xevrpov) means the centre of a The reason of the inven. Civcle: it is on that account applied tion of the appellation of to the distance between the planet and KENDRA. ; , higher apsis, for the centre of the NICHOCHCHA-VRITTA or epicycle, is always at the distance of the planet from the place of the higher apsis. 42. The circumference in yosanas of the planet’s orbit Spnupa-Kaxsua or core being multiplicd by the s‘iqmRa-KARNA ५ (or 2nd hypothenuse), and divided by tho radius (3438) is sruura-KaKsud (corrected orbit). Tho planet is (that moment) being carried [round the carth] by the PRavaHA wind, and moves ata distance equal to half the diameter of the spHUTA-KAKSHA from the earth’s centre. 43. When the sun’s MANDA-PHALA i. e. the equation of the Reason of Busa'ntaxa centro is subtractive, the apparent or eee real time of sun-rise takes place before the time of mean sun-rise: when tho cquation of tho centro 18 additive, the real is after the mean sun-rise, on that account the amount of that correction arising from the sun’s MANDA- PHALA converted into asus* of time has been properly declared to be subtractive or additive. 44, Those who have wits as sharp as the sharp point of the inmost blade of the DORBHA or DARBHA grass, find the subject above oxplainod by diagrams, १ matter of no difficulty whatover : but men of weak and blunt understanding find this subject as heavy and immovable as the high mountaint that has been shorn of its wings by the thunderbolt of Indra. End of Chapter V. on tho principles on which tho rules for finding the true places of tho planets aro grounded. It is to be observed here that when the planet comes to the places a, a &ec. in the dotted line, it is then at its higher apsis, when it comes to the places ©, © and ©, it is at its lower, and whien it comes to 5, © &o. it appears, stationary : and when it is moving in the upper aro ba ©, its motion being direct appears quicker, and when in the lower aro © ¢ 6, ite retrograde motion is seen.— B, D) ® (These asus are equivalent to that part of the equation of timo, which is due to the unequal motion of the sun on the ecliptic.—B. 1.1 + Mountains are said by Hindu theologians to have originally had wing. ASD VI. 3.] Sidhdnta-s'iromant. 151 CHAPTER VI. Called GoLABANDHA, on the construction of an Armillary Sphere. 1. Leta mathematician, who is as skilful in mechanics as in his knowledge of the sphere, construct an armillary sphere with circles made of polished pieces of straight bamboo; and marked with the number of dogrces 10 the circle. 2. In the first place, let him mark a straight and cylindri- cal DHRUVA-YASHTI, or polar axis, of any excellent wood he pleases: then let him place loosely in the middle of it a small sphoro to represont tho carth [so that tho axis may move freely through it]. Let him then firmly secure the spheres beyond it of the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn and the fixed stars: Beyond them let him place two spheres called KHAGoLA and DriccoLa unconnected with each other, and fastened to the hollow cylinders [in which tho axis is to bo insortod].* [Description in detail of the fact above alluded to. ] 3. Fix vertically the four circles and another circlo called nis 00 horizon transversely in the middle of meridian and the Koyavait- them, so that one of those vertical द circles called SAMAMANDALA, prime vertical, may pass through the east and west pvints of the horizon, the other called yAmMyorrara-ve{traA, meridian the * The sphere of the fixed stars which is mentioned here is called the BHAGOLA starry sphere. This १८ ७01,4 is assumed for all the planets, instead of fixing a separate sphere for cach planet. This sphere consists of the circles ecliptic, equinoctial, diurnal circles, &c. which are moveable. For this reason, this ephero is to be firmly fixed to the polar axis, so that it may move freely by moving the axis. Beyond this sphere, the xHaGoxa celestial sphere which consists of the prime vertical, meridian, horizon, &. which remain fixed in a given latitude is to be attached to the hollow cylinders. Having thus separately fixed these two spheres, astronomers attach, beyond these, a third sphere in which the circles forming both the spheres KadaGOLA and BHAGOLA are mixed together. For this reason the latter is called priagora the double sphere. And as the spherical fingers are well seen by mixing together the two spheres KnaGoLa and BHAGOLA, the third sphere which is the mixture of the two spheres, is separately attached.—B. D. 152 Translation of the [VI. 4. north and south points, and the remaining two called KoNA- verrras the N. 19. and S. W. and N. W. and 8. E. points. 4. Then fix a circle passing through the points of tho The UNMaNpata or six horizon intersected by the prime verti- ००००८1०. cal, and passing also through the south and north poles at a distance below and above the horizon equal to the latitude of the place. This is called the UNMANDALA, or six o’clock line, and is necessary to illustrate the increase and decrease in the length of the days and nights.* 5. The equinoctial (called nipf-vavaya), marked with 60 ghatis, should be placed so as to pass through the east and west points of the horizon, and also to pass over the meridian at ४ distance south from the zenith equal to the latitude, and at 9 distanco north of tho nadir also oqual to tho latitudo of tho placo [for which the sphere is constructed]. 6. 1५6 tho azimuth or vortical circlo bo noxt attachod within the other circles, fixed by a pair of nails at the zenith and nadir, so as to revolve freely on them: [It should be smaller than the other circles so as to revolve within thom]. It should be capable of being placed so as to covor the plunct, wherover it may happen to bo. | 7. Only one azimuth circlo may be uscd for all tho planots ; or else eight azimuth circles may be made, viz. one for each of the 7 planets and the 8th for the nonagesimal point. ‘The azimuth circle for the nonagesimal point is called the णा SHEPA-VRITTA. The equinoctial. Azimuth or vertical circle, * The circle of declination or the hour circle passing throngh the east and west points of the horizon is culled UNMaNDALA in Sanskrit; but I am not acquainted with any corresponding term in English. In the treatise on astronomy in the Encyclopedia Metropolitana the prime vertical is named the wix o'clock line. ‘This term (six o’clock line) should, I think, be applied to the UNMANDALA, because it is always six o’clock when the sun arrives at this circle, tho UNMaNpaLA. Tho prime verticnl or the sama-MaNpsLa of tho Sanskrit cannot, with propriety, be called tho six o’vlock lino; because it is ouly twice a 041". it is six o'clock when the sun is ut this circle, tho prime vertical.— VI. 11.] Sidlhdntu-s’ trunani. 153 8. Let two hollow cylindors projoct boyond tho two poles north and south of the KHAGOLA ce- lestial sphere, and on these cylinders Jet the skilful astronomer place the pgiaaoLA double sphere as follows. 9. When the system of the हप ^ 6014, celestial sphere, 18 mixed with the ecliptic, and all the other circles forming the BHAUOLA (which will bo presontly shown) it is thon callod DRIGGOLA, double sphere. As in this the figures formed by the circles of the two spheres KHAGOLA and BHAGOLA are seen, it is therefore called pgiaaoLa double sphere.* THE BHAGOLA. 10. Let two circles bo firmly fixed on the axis of the poles answering to the meridian and horizon (of the KHAGOLA) ; they aro called the Apufza-vRittas, or circles of support: Let the equinoctial circle also be fixed on them marked with 00 ghatis like the prime vertical (of the kHAGoLA). 11. Make the ecliptic (of the same size) and mark it with 12 signs; in this tho Sun movos: and alyo in ib 10४०1१७३ tho Marth’s shadow ata distanco of 6 sigus from tho Sun. Tho Kvanti-rhva or vernal cquinox, movos in it contrary to the order of the sigus : The spasnta-P4ras [of the other planets] have a like motion: the places of these should be marked in 1४. Tho 0166014. Tho Keliptie, * Sco the note on 2 Verse. + [The Sun revolves in tho ocliptic, but the planets, Moon, Murs, &. do not revolve in that circle, and tho planes of their orbits are inclinod to that of tho ecliptic, Of the two pointe where the planctary orbit cute the plano of the ecliptic, that in which the planct in its revolution rises to the north of the ocliptic is called its Pa’Ta or ascending node (it is sla called the mean 2474) and that which is at the distance of six signs from the former is called its SASHADBUA PATA or descending node. The pa’ta of tho Moon lies in its concentric, becuuse the plane of its orbit passes through the centre of the concontric, i. ०, through tho centro of tho Earth; but tho 24148 of the othor plancts aro in their second exeentric, because tho planes of their orbits pnss through the centros of their 2nd exvantrics, which centres lio in the planc of the ecliptic. When the planet is at any other place than its nodes, the distance botween if and tho plano of the ccliptic is called its north or south latitude as the planct is north or south of the ecliptic. Whon the planct is at the distance of 3 signs forward or backward from its ra’Ta, it is then at the greatest distance north or south from the ecliptic: Thies distance is its greatost latitude. Thus, H SESE ESS Sl पी 151 Translation of the [VI. 12. 12. Let the ecliptic be fixed on the equinoctial in the point of vernal equinox KRAnfiI-PdATA and in a point (autumnal equinox) 6 signs from that: it should bo so placed that tho point of it, distant 3 signs eastward from the vernal equinox, shall be 24° north of the equinoctial, and the 3 signs westward shall be at the same distance south from the cquinoctial. 18. Divide a circle called KsHEPA-vRiTTA representing the orbit of a planct into 12 signs and mark in it tho places of the sprasuta- pAtas, roctified nodes, as has been before prescribed [for tho ecliptic]. Then this circle should bo so placed in connection with the ecliptic as it has been placed in connection with the equinoctial. 14. The ecliptic and the xsHEPa-vritra should be 80 placed that tho latter may intorsoct tho formor at tho [rectificd] ascending and descending nodes, and pass through points distant 3 sigus from tho asconding node cast und west at ४ distance from the ecliptic north and south equal to the rectified greatest latitude of the planet [for the time]. 15. The greatest (mean) latitudes of the plancts being multiplied by the radius and divided by the siaHRa-KARNA Planet's orbit. the latitude of the planct begins from its 2474 nnd becomes extremo at the distunco of 8 signs from it, therefore, in order to Mud the latitude, it ia necessary to know the distance between tho planct and its vata, ‘This distance is equal to the sun of the places of the planet and its va’ra, becuuse all 2414148 wove in antecedentia from the stellar aries. ‘This sum is called the viksuBPA-KENDRA or the argument of latitude of the planet. As the 76/74 of the Moon lies in her concentric, and in this circle is her true place, the sum of tlicse two is her VIKXSHEPA-KENDKA, but the 4.74 of any other planet, Mara, &. 1168 in ites 2ud excentrio aud its MANDA-SPasHTA place (which is cquivalent to its heliocentric placo) is in that circle, therefore its Viksitkea-KENDRA is found by adding tho placo of ite Pa’ra to 1८५ MANDA-srasiita pluco, ‘Tho seasiuva-ra'ta of tho planet 13 that which being added to the true placo of the planet, equals its VIKSUBVA-KENDRA for this renvon, it is fuund by reversely applying the 2ud equation to ite mean Pa’Ta. As ० SPASHTA PA'TA + true place of the planet, == VIESHEPA-KENDRA, == place of the ManDa SPASHTA planet -}- mean PA’Ta, = p, of the m. ॐ. p. -+= 2nd equation + m. 2 = 2ud equation, == truo place of the planet + moan 24/74 2nd equation, ० SPABUTA 74 4.4 = mean Yara F ५५ equation. ‘Ihe place of this 82481774 Pa’Ta is to be reversely marked in the ecliptic from the stellar ०11९8. - ए. D ] VI. 15.] Silhdnta-s'tromani. 155 second hypothenuse becomes srasnfa, rectified. ‘The Ksitera- vgitra, or circles representing the orbits of the six planets, should be made separately. ‘I'he Moon and the rest revolve in their own orbits.* ® [As the 24/74. of the Boon and her true place lie in her concentric, the sum of these two, which is called hor VIK8SHEPA-KENDRa or the arg ument of latitude, must be measured in the same circle, and hor latitude, therefore found through her VIXSHEPAa-KENDRA, will be as seen from the centre of her concentric i. e. from the centro of the Earth. But the ra’ta of any other planet and its Manpa- 87481114 place (which is its heliocontric place) lie in its 2nd excentric, therefore its latitude, determined by mcane of its VIXSHEPA-KENDBA, which is equal to the sum of its manDa-sPasHTa placc and 24/74 and measured in the same circle, will be such as ecen from the centre of its 2nd excentric and is called its mean latitude (which is equivalent to the heliocentric latitude of the planet). As in Fig. 1, let N E be the 0 quarter of tho ochptic, N 0 that of tho 2nd excontric, N the node and VP tho planct. Suppose O Ki and P }› (parts of great circles) to bo drawn from ^ and ¬› porpendiculerly to the plano of the ecliptic: then O E will be tho greatest lati- tude and P p the latitude of the planet at P, by which a spectator at the centre of the 2nd excentric and not at the contre of the Earth, will sce the planet distant from tho celiptic. This latitudo, therefore, is called 9 mcan latitude which can be found as follows, sinNO: sinOE:: sinN P: sinP p, _ ० 1४. श 17 = 9101. भदन; consequently, in order to determine 7 p, it is noccssury to know proviously 0 E, tho grentost lntitiulo and NP, (11५ distanoo of tho plnco of tho planet from tho node, which distance ix evidently equal to tho vixsn1era-KENDEA that is, to tho sum of tho MANDA-sPasitra place of the planet and the mean place of the node. Now the latitude of the planet as seen from the centre of the Earth is called its true latitude. This true latitude can be found in the following manner, “ # Let E bo the centre of the earth, O that of the 2nd excentric, P the MANDA 8487114 place of the planct init: then Jo 2 will be the 2ud hypothenuse which is) suppored) to ent tho concentric at A: then A will be tho true place of the planet in the concentric, Again let P क be a circle with the contro O, whose plane is perpendicular to the eclip- tic plano and A © another circle with the centre 10 whose placo is also perpendicular to the samo piane, then Pg will be the mean latitude of the planet and A 6 will bo the true. Let Pp and Aa lines be porpendicularly drawn to tho plano of tho celiptic, these lines will also bo at right angles to the line Ep: then P p will be the sine of the mean latitude P क and A a that of the true lati- tude Ab. Now by the similar triangles E P p and EK A a, EP:Pp::EA:Aa; EA.Pp ee A @ = = ne 3 EP N ~ 156 Translation of the [V1. 16, 16. The declination is an arc of a great meridian circle: ti ere “oh ] क्त cos the equinoctial at right ang es, and continued till it touch the ecliptic. R % sine of the mean latitude aD or the sine of the truo latitude = h sinOE. sin N P R R 810८. sinNP the sine of the true latitude == —— x १ | sin OF . sin N P but, the slne of the mean latitude = ०० by substitution As the latitude of the planet is of a smaller amount, the are of a latitudo it, therefore taken in the SippHantas instead of the sino of the latitude. O E.sin N P Hence, the true latitude = h that is, the sine of the argument of latitudo multiplied by the greatost reg and divided by the 2nd hypothwnuse is equal to the true lutitude of the planet. Now in the Bragora, a circlo should be so fixed to the ooliptic, that tho former imny intorsect the Intter at tho srasuta-rhva and tho point six signs from it, and whose oxtreme north and south distance from the ecliptic may be such that the distance between the cirole and the ecliptic at the place of the true planet may be equal to the true latitude of the planet. This circle is called the VIMANDALA Or VIKSHEPA-VRITTA and its extreme north and south distance from the ecliptic is called the true or rectified extreme latitude of the planet which can be found as follows. Let N be the spasHtTs- Pata, N P the vixsHEpa-xEn- > pra, P p the true latitude, . | EO the true extreme latitude: च ¢ then sin N ०: sin 7 O:: sin N 29020 sin NO. sin Pp sin EO sin 2 R.Pp or EO= 3 sin N P L.sin NP but if L be taken for the mean extreme latitude the ए p = — R L.isinNP R.L ००७ E O — > 4 —- = > sin N P h 2 This is the mean extreme latitude stated in the Ganitapuyfya multiplied by the radins and divided by tho 2nd bypothenuso equals the true or rectified extreme latitude.— B, D.} , VI. 21] 1/0 1 157 celestial latitude is in like manner an arc of a great circle (which passes through the ecliptic poles) intercepted between the ccliptic and tho KSIEPA-VRITTA. The corrected declination [of any of the small planets and Moon] is the distance of the planet from the equinoctial in ४ circle of declination. 17. The point of intersection of the equinoctial and ecliptic circles is the xRANTI-PATA or inter- 860९019 point for declination. The retrograde* revolutions of that pgint in a Kanra amount to 30,000 according to the author of the SGrya-sippHANTA. 18. Theo motion of the solstitial points spoken of by Mun- sXuA and othors.is the same with this motion of the equinox : according to theso authors its revolutions are 199,669 in a Katra. 19. The place of the Kranti-pMta, or the amount of the precession of the equinox determined through the revolutions of the krAntTI-p{Ta must be added to the place of a planet; and tho declination thon ascertained. ‘Tho ascensional differ- enco and periods of rising of the signs depend on the declination : hence the precession must be added to ascertain the ascensional difference and horoscope. 20. Thus the points of intersection of the ecliptic and the orbits of the Moon and other planets are the KSHEPA-PATAS, or intersecting points for the KsHEPA celestial latitude. The revolutions of tho Ksitkra-P4Tas are also contrary to the order of the signs, hence to find their latitudes, the places of the KSHEPA-rATAS must be added to the places of the planets (before found). 21. As the manpa-spasHTa planet (or the mean planet cor- rected by the Ist equation) and its ascending node revolve in tho s’IGMRA-PRATIVRITTA or 2nd excentric, hence the amount of the latitude is to be ascertained from (the place of) the MANDA-SPASHTA planet added to the node found by calculation. * The motion of tho KnAuti-patTa is in a contrary direction to that of the order of the signs.—L. W. Precession of tho equinox. 158 Translation of the (VI. 22. 22. Or the amount of the latitude may bo found from the spasHTa planet added to the node which the s‘iaHra-PHaLA 2nd equation is added to or subtracted from accordingly as it was subtractive or additive.* As the Moon’s node revolves in the concentric circlo, the amount of the latitude, therefore, is to be found from the true place of the Moon added to the mean node. 23. The exact revolutions of the nodes of Mercury and Venus will be found by adding the revolutions of their s’fanra- KENDRAS to the revolutions of their nodes which havo boen stated [in the Gayrrdpnyfyal]: if it be asked why these smaller amounts have been stated, I answer, it is for greater facility of calculation. Hence their nodes which are found from their stated revolutions are to be added to the places of their 8'faitRA-KENDRAS [to got tho oxact placos of tho nodes] .+ 24. To find the kunpra [of any of tho planets] tho placo of the planct is subtracted from tho s’faurocucia: thon tako * [See the nodes on V. 11, and ए. 18, 14, 15. —B. D.} ¶ [In all the original astronomical works, the sum of the २4/44 and s’fanmRocH- cua of Mercury and Venus, is assumed for their VIKSHEPA-KENDEA, and through this, their latitude is determined. But the latitude thus found would be at the place of their s’fanRooncna and not at their own pluco, because their vee are different from those of their s’famroononas. To remove this iMvulty, Bra’sxra’ona’Rya writes. “The exact revolutions &.” But the difficulty arises in the supposition that, the earth is stationary in tho centre of the universe and all the ॥ revolve round her, because we are then bound to _ that the mean p of Mercury and Venus are equal to that of the Sun, and henoe their places will be different from those of thoir s’iaHRocucHas. But no inconvenience occurs in the supposition that, the Sun is in the centre of the universe and all the planets together with the earth revolve round him. For, in this case the places of the s’faHroonoHas of Mercury and Venus are their own heliocentric places, and consequently the sum of tho places of their e’faHROCHOHAS and 24748 will be equal to the sum of their own places and thoso of their Pa’ras, that is to their VikamkrakkeNDuA. For this reason, their lutitude found through this, will be at their own places. Now, it isa curious fact that, the revolutions of the 8188 of Mercury and Venus, statcd in the original works, are suoh as ouglit to be mentioned whien it is supposed that the Sun is in the middle of the universe and the planets revolve round him, and not when the Karth is supposed to be stationary in the centre of the universe. From this fact, we oan in be that the original Authors of the As- tronomical works knew that all the planets together with the Earth revolve round tho Sun, and conscquontly thoy stated tho smallur amounts of tho revolutions of the Pa’ras of the Mercury and Venus. Whiou thie is the case, why ia it supposed that all the planets revolve round the Earth, because the oo oan more easily bo understood by thia supposition than by the other.— ए. 27.] . Sidhdnta-s'tromani. . 189 the KENDRA with tho pAta added [to get the exact amount of the p4ra or node] and let the place of the planet be added theroto, [we thus gct tho VIKSHEPA-KENDRA or the argument of the latitude of Mercury or Venus]. Therefore from the s’‘fenrocHcuas of these two planets with the 24148 added, their latitudes are directed by the ancient astronomers to be found.* 25 and 26. ‘The ratas or nodes of these two planots added to the s‘icurocHcuras from which the true places of the planets have been subtracted, become spasuta or rectified. It is the s’PASHTA-PATA which is found in the BHAGOLA (above described), In the sphere of a planet, take the ecliptic above described as the concentric circle, to this circle the second excentric circle should be attached, as was explained before, and a circle representing the orbit of a planet (and which consequently would represent the real second excentric) should be also attached to the latter circle with the amount of latitude detailed for it. In this latter circle mark off the mean places of tho nodes of tho (superior) planots, and also mark in it the mean placo of tho nodes of Mercury and Venus addod to their respective 8 {GHRA-KENDRAS.f 27. Next the AHORATRA-VEITTAS or diurnal circles, must be Diurnal circles calleq made on both sides of the equinoctial AUOB‘ATRA-VRITTAS. [and parallel to it] at every or any degree of declination that may be required :—and they must all be marked with 60 aHatis: The radius of the diurnal circle [on which the Sun may move on any day] is called prusvf. * (Let, ¢ = s’fenroonona or the place of 2d higher apsis k = the s’faitna-KenDra ॐ = the place of the planet % = Pa'Ta or the place of tho ascending node and XY, = the exact pa’ta then £ == A—pjadsk=—k ~+ * = ¢}. -? +n; " VIKSUEPA KENDRA or argument of latitude of Mercury or Venus = + p=h—pt+a+p=h+ *-8.7.] † [3०९ the note on verses 13, 14 and 15 :- 2, 2D.) 160 - Translation of the {VII. 1. 28. From the vernal equinox mark the 12 signs in direct order, and then let diurnal circles be attached at the extremity of each sign. 29. On either side of the equinoctial, threo diurnal circles should be attached in the order of tho signs: these again will answer for the three following signs. The BHAGOLA has thus been described. This is to be known also as the KHECHARA-GOLA, the sphere of a planct. 30. Or in the plane of the ecliptic bind the orbits of Saturn and of the other plancts with cross diametors to support them, but these must be bound below (within) the ecliptic in succes- sive circles one within the other, like the circles woven one within the other by the spider. $1. Having thus secured the BHAGoLA on the axis or YASUTI, after placing it within the hollow cylinders on which the KHAGOLA is to be fastened, make the BHAGOLA revolve :— it will do so freely without referenco to the KIfAGOLA as its mo- tion is on the solid axis. The KHAGOLA and DRIGGOLA remain stationary whilst the BHAGOLA revolves. End of Chapter VI. on the construction of an armillary sphere. CHAPTER VII. Oalled Trrpraswa-visan< on the Principles of the Rules for resolving the quostions on timo, space, aul directions. The ascensional difference 1. The time called cHara-knuNDA and its place. or ascensional difference is found by that arc of a diurnal circlo intercepted between the horizon and the six o’clock line. The sine of that arc is called the KUJYA in the diurnal circle: but, when reduced to relative VIL. 5.] Sildhdnta-s'iromunt. 161 value in a great circle, it is called cuarasy4 or sine of as- censional dilference.* 2. Tho horizon, as seen at the equator, or in a right sphere, is denominated in other places [to the north, or south of the equator] the UNMANDALA six o’clock line: but as the Sun appears at any place to rise on its own horizon, the difference between the times of the Sun’s rising [at a given placo and tho oquatorial region undor the same meridian] is tho ascensional difference. 3. When the sun is in the nor- Determination of ४18 d : : question when the owaRA thern hemisphere, it rises at any correction ie ,seditive and place (north of the equator) before it does to that on the equator: but it sets after it sets to that on the equator. Therefore the correction depending on the ascensional difference is to be subtracted at sunrise of a given place from the place of the planet [at sunrise at the equator] and to be added at sunset to the place of the planet [as found for the sunset at the equator}. 4. When the Sun is in the southern hemisphoro the reverse of this takes place, as the part of the UNMANDALA in that homisphere lies bolow the horizon. The halvos of the sphero north and south of the equinoctial are called the northern and southern hemispheres. Cause of incroaso and decreaso 5. [And it is in consequence of in length of days and nights. = , this ascensional difference that] tho days are longer and the nights shorter (than they are on the ॐ (The times found by the arcs intercepted between the horizon and the six o'clock line, of tho three diurnal circles attached at tho end of the first 8 signs 1. e. Aries, Tauras and Gemini are called the omaRa-Ka’La8 or the ascensional differenoes of these signs, and the differences of these OH ARA-Ka’LAS are called the OtARA-KiANDAS of ४ fioss throo signs. As, whore the 2474 714. is 6 digits or tho Iatitudo is nearly 224° north, tho ns- consional differonoes of the 3 first signs are 297, 641 and 642 asus, and the dif- foronces of those i. 6. 297, 244 and 101 are tho onaRa-KHANDAS of thoso signs, Theeo aro again tho onaRa-KlaNpas of the following three signs inversely i. o. 101, 244 and 297 asus. Thus the chaua-Kuanpas of tho firet six signs answer for the following six signs.—B. D.]} I 162 Translation of the [VIT. 6. equator) when the Sun is in the northern hemisphere: and that the days are shorter and the nights longer when the Sun is in the southern hemisphere. For, the length of the night is represented by that arc of the diurnal circle below the horizon, and the length of the day by that arc above tho horizon. 6. But atthe equator the days and nights are always of the same length, as there is no UNMANDALA there except the horizon fon the distance between which, the variation in the length of days and nights depends}. A circumstance of peculiar curiosity, however, occurs in those places having a latitude greater than 66° N. viz. than the complement of the Sun’s greatest declination. Determination of plese and 7. Whenever the northern declina- te of porpetual dey and tion of the Sun oxcceds tho complo- mont of tho latitude, thon thero will be porpotual day for such timo as that oxvoss continued ; and whon the southern declination of the Sun shall exceed the comple- ment of the latitude, then there will be perpetual night during the continuance of that excess. On अभा, therefore, day and night are each of half a year’s length. 8. To the Celestial Beings [on MERU at the north pole] the equinoc- tial is horizon : 80 also is to the pairyas [at the south pole]. For, the northern and southern poles are situated respectively in their zeniths, 9. The Celestial Beings on uerv behold the Sun whilst he is in the northern hemisphere, always revolving above the horizon from left to myght: but parryas the inhabitants of the southern polar regions behold him whilst he is in the southern hemisphere revolving above their horizon from the right to the left. Place of MERU. a 10, Thus it is day whilst the Sun Definition of the artificial . .. : | „ day and night and the day 18 visible, and night whilst he is in- न visible. As the dotermination of VII. 15.] Siddhdnta-s'iromani. 163 night and day is made in regard to men residing on the sur- face of the Earth, 80 also is that of the एकाह or deceased ancestors who dwell on the upper part of the Moon. 11. Asfor the doctrine of astro- Pees pear sa et logers, that it was day with the Gods professors or 8५ धाक 48. at आहार whilst the Sun was in the vut- TARAYANA (or moving from the winter to tho summer solstice) and night whilst tho Sun was in the DAKSHINAYANA (or moving from the summer to the winter sol- stice), it can only be said in defence of such an assertion, that it is day when the Sun is turned towards the day, and it is night when turned towards the night. Their doctrine has reference merely to judicial astrology and the fruits it foretells. 12. By the degrees by which the Sun proceeds in his nor- thern course to the end of Gemini, he moves back from that sign: entering also the same diurnal circles in his descent as he did in his ascent. Is it not therefore that the Sun is visible in his descent to the Gods in the place where he was first seen by them in his ascent ? 13. The pirris reside on the upper ध a part of tho Moon and fancy tho foun- tain of nectar to be beneath themselves. They behold the Sun on the day of our amAvAsy4 or new Moon in their zenith. That therefore is the time of their midday. 14. They (1. 6. the ritRis) cannot see the Sun when he is opposite the lower part of the Moon: it is therefore, midnight with the rireis on the day of the एषषा or full Moon. The Sun rises to them in the middle of the KRISHNA PAKSHA or dark half of the Moon, and sets in the middle of the s UKLA PAKSHA or light half of the Moon. This is clearly established from tho context. 15. As Braumf being at an im- ह rc of adey 0686 distance from the Earth, always sees the Sun till tle time of the pra- £aYA or gencral deluge, and sleeps fur the same time, therefore 12 164 Translation of the [VII. 16. the day and night of Brana are together of 2000 manAyuaas in length. 16. As the portion of the ecliptic 7 sine taken by etch which is more oblique than the other, rizon. rises and sets in a shorter timo and that which is more upright takes a longer time in rising and setting, hence the times of rising of the several signs are various [even at the equatorial regions]. 17. The (six) signs from Capricorn to Gemini or ascending signs which are inclined towards the south with their respec- tive declinations whilst they rise oven at the cquator aro still more inclined towards the south in the northern latitudes (on ` account of the obliquity of the starry sphere towards the south) ; hence they arise in still shorter times than they do at the equator. 18. At the equator, the [six] signs from Cancor or do- sconding signs incline whilst they riso to the northerly diroc- tion, but they will have upright direction in consequence of the northern latitude, hence they rise in longer times [than they do at the equator.] The difference between the period of the rising of a sign in a given latitude, and at the equator under the same meridian, is equivalent to the cHARAKMANDA of that sign. 19. Kach quarter of the ecliptic risos in 15 amas or 6 hours to those on the equator: and the 6 signs of the northern as well the 6 of the southern hemisphere appear to rise each in 12 hours or 80 auaris in every or any latitude. 20. The three signs from the commencement of Aries to the end of Gemini, 1, 0, the first quarter of the (नो, pass the unMANDpALA in 15 auatis; but the horizon [of a place in north latitude] is below the uNmaNpALa, they therefore pre- viously pass it in time less than 15 auafis by the cHARAKHANDAS. 21. The three signs from the end of Virgo to the end of Sagittarius, i. e. the 3rd quarter of the ecliptic, pass the. uNMAN- PALA in 15 qHaqis; but they pass the horizon of a place VII. 23.] Siddhdnta-s’tromani. 165 afterwards which is above the unmMANDpALA [in north latitude] in 15 aHatis added to the CHARAKHANDAS. 22. The three signs from the end of Gemini to the end of Virgo, i. e. the 2nd quarter of the ecliptic or those from the end of Sagittarius to the end of Pisces i. e. the 4th quarter of the ecliptic, pass the horizon in the time equal to the remainder of 30 aHatis diminished by the time which the first or third quarter takes to pass tho horizon respectivoly. For this reason, the times which the signs contained in the 1st and 4th quar- tors of the ecliptic, or ascending signs, and those contained in the 2nd and 3rd quarters, or descending signs take to pass the horizon at a given place are found by subtracting the CNARAKIANDAS of tho signs from and adding them to tho times which those signs take in rising on the equator respectively.* 23. Having placed the Ist Aries in the horizon and set the sphere in motion, the tutor should show the above facts to the * The times taken by the several signs of the ecliptic in rising at the equator and in northern latitu des will be scen from the following memo. according to the SippHanrTa. 2 -2 & = > 8.4 3 = (338 (P28 2 .8 ॐ 2 ० ~ .६ 2 ६ < §.8 (82 ~ _ (£ £3 = | ss ट Py BS ¢ ६ =€ (825 5laas 8 € ०“ 4 env A 3S ASUS. | ASUS. 4808. । = 11686 8 and tho lost 411९8) ,०१०,०,००००००००.००००.| 1670 207 | 1373 8 signs take less time to Tauruz, 000 000 608 O60 ०७७७०. 1793 — 244 1549 rise in north latitude Gemini, ,,०,०००००,००,००००.| 1937 | —101 | 1886 than at the equator. Cancer, ०,.| 1987 +101 | 2038 Leo, 1793 न 244 | 2087 These 6 signs take a Virgo, ,,१,.०.०००००००००००००.| 1670 | ~+ 297 | 1967 longer time to rise in Libra, ,,,०००,००,००,०००००,०, | 1670 + 297 | 1967 north latitudo than at Scorpio, ... ०००,०,००,०००..| 1793 | ~+ 244 | 2037 the equator. Sagittarius,...............] 1937 | =+ 101 | 2088 Capricorn, 9०७०७०७ ७9०७७००५ 1987 + 101 1836 ‘ Aquarius, ..........0....] 1793 | — 244 | 1549 Pisces, ७०७००७०१ 008 688 1670 166 Translation of the [ VII. 24. pupils, that they may understand as well what has been ex- plained as any other facts which have not been now mentioned. 24. In whatever time any sign rises above the horizon [in any latitude] the sign which is the 7th from it, will take exactly the same time in setting: as one half of the ecliptic is always above the horizon [in every latitude]. 25. When the complement of latitude is less than 24° (i. e. than the extreme amount of the Sun’s declination taken to be 24° by Hindu astronomers) then neither the rising periods of the signs, nor the ascensional differences and other particulars will correspond with what has been here explained. The facts of those countries (having latitudes greater than 66°) which are different from what has been explained on account of their totally different circumstances, are not here mentioned, as those countries are not inhabited by men. 26. That point of the ecliptic which is (at any time) on Ftymology of the word tho custorn horizon is callod tho LAana ० or horoscope. This is expressed in signs, degrees, &©, reckoned from the first point of stellar Aries. That point which is on the western horizon is called the asTa-Lacna or setting horoscope. The point of the ecliptic on the meridian is called the MapHya-LaGNa or middle horoscope (culminating point of the ecliptic).* ® [When the place of the horoscope is to be determinod at a given time it is necessary at first to ascertain the height and longitude of the nonagesimal point from the rig ht ascension of mid-heaven, and then by adding 8 signs to the longitude of the nonagesimal point, the place of the horoscope is found: but as thie way for finding the place of the horoscope is very tedious, it has been determined otherwise in the 79 49148. As, from the periods of risings of the 18 signs of the ecliptic which are determined in the Siddhantas, it is very easy to find the time of rising of any portion of the ecliptic and vice versa, we can find a portion of the coliptic corresponding to the given time from sun-rise through the longitude of the San then determjned and the given time. The portion of the ecliptic which can be thus found is evidently that portion of the ey intercepted between the place of the Sun and the horizon. Therefore by adding this portion to the lace of the Sun, the place of the horoscope is found. Upon this principle, the following common rule which is given in the 8ippHantas for finding the place of the horoscope is grounded. Find first the true place of the Sun, and add to it the amount of the procession of the equinox for the longitude of the Sun. Then, from the longitude of the Sun, the sign of the ecliptic in which the Sun lies and the degrees of that sign VII. 27.) Stddhanta-s‘tromani. 167 27. If when you want to find the Lacna, the given GHATis स emule are siVANA-GHATIS, a they will be- exact place of tho Sun at Come sidereal by finding the Sun’s the time of question in order instantaneous place i. e. the place of the Sun for the hour given. The times which he has passed, and those which he has to pass, are known. Thus the degrees which the Sun has passed, and those which he has to pass, are called the BARUKTANS a8 and BrHoGyANs’As respectively. Now the time which the Sun requires to pass the nmo@yAns‘as is called the pHogya time, and is found by the +: proportion. 800 : the period of rising of the sign in which the Sun is 2: BHOGYANS a8 : BHOGYA time. In the same manner, the BHUKTA time can also be found through the DNURTANBAS, Now from tho time at the ond of which the horoscope is to be found, and which is called the 1877८ or given time, subtract the bHoaya time just found, and from tho remaindor subtract the periods of soar of the next successive signs to that in which the Sun is as loug as you can. Then at last you will find the sign, the rising period of which being greater than the remainder you will not be able to subtract, and which is consequently called the as’cppua sign, or the sign incapable of being subtracted, and its rising period, as‘UDDHA rising. From this it is evident that the as’UppHa sign is of course on the horizon at the given time. The degrees of the as’ UDDHa sign which are above am horizon ahd therefore called the BHUKTA or passed degrees, are found as ollows. If tho rising period of the as’uppm sign : 800 १ १ the romainder of the given time : the passed degrees of the as’uDpiA sien. Add to these passed त thus found, the preceding signs reckoned from the 1st point of Aries, an fron the Sum, subtract the amount of the procession of the equinox. The remainder thus found will be the place of the horoscope from the stellar Aries. If the time at the end of which the horoscope is to be found, be given before sun-rise, then find the BHUXTA, or passed time of the sign in which the Sun is, in the way above shown, and subtract it and the rising periods of the pro- coding signe from the given time. After thie find the degrees of the 4807701 4 sign corresponding to the remaiudor of the given time which will evidently be the BHOGYA deyrees of the horoscope by proportion as shown above, and subtract the sum of the BHoara degrees of the horoscope, the signs the rising ge of which are subtracted and i 16 BUUETA degrees of the sign in which the Sun is from the Sun’s place and the remainder thus found will be the place of the horoscope. + Thus we get two processes; one when the given time at the end of which the horoecope is to be found, is after sun-riso, aad the other when that time is given before sun-rise, and which are consequently called KraMa, or direct, and VYUTKRAMA or undirect processes respectively. 1t is plain from this that if the place of the Sun and that of the horoscope be known, the given time from sun-rise at the end of which the horoscope is found can be known by muking the sum of the BHOooYa time of the sign in which the Sun is and the BHUKTA time of the horoscope and by adding to this sun the rising periods of intermediate signs.—B, VD.) 168 Translution of the [ ४11. 27. of rising of the signs which aro siderecal must be subtracted from these auafis (of the question) reduced to a like deno- mination. When tho hours of tho question are already sidereal, there is no necessity for fiuding the sun’s real place for that time.* * 111 it be asked whether the time at the end of which the horoscope is to be found is terrestrial or sidereal time; if it be terrestrial, how it is that you subtract from that the rising periods which are of different denomination on account of their being sidereal, and why the sun’s instantaneous place i. e. tlio place determined for the hour given is used to ascertain the एप 0७३५ time, the givon time is reckoned from sun-rise and the bHoGYa dogrocs of tho sign ` 111 which tho sun is, riso gradually abovo tho horizon aflor sun-risc. 11५११८५ the Buoaya degrees of tho sign of the Sun’s longitude, determined at tho timo of eun-rise, should be taken to find the place of the Horoscopo, otherwise tho place of the Horoscope will be greater than the real 016, As for example, take the time from sun-rise, at the end of which the Horoscope is to be found, equal to GO sidereal Guatis and 44 asus when the Sun is in the vernal equinox at a place whore the Patauua is 6 digits or the latitudo is 22°} nearly, and ascertain the place of the Horoscope through the instantancous place of tho sun, ‘Thon, tho San of the Ioroscope thus found will be greater thun tho placo of tho Sun ound at tho time of next sun-rise, but this ought to bo equal to it, and you will not be able to make this equal to the place of tho Sun dotermined at the time of next eut-rise, unloss you determino this through the place of tho sun asccrtainod at suo-rieo, and not through the Sun’s instantanvous place. [1९066 it nppears wrong to ascertain the pluce of the Horoscope through tho Sun’s instantaneous place. But the answer to thie 18 as follows. The aHafis contained in the arc of the diurnal circle intercepted between that point of it where the Sun is, at a given time and the Horizon are the sAvaNa or terrestrial @HaTiIs, but the GHaTis contained in the aro of the diurnal circle in- tercepted between that point of it where the Sun was at the time of sun-rise and the Horizon are the sidereal, Gaatis. Thus it is plain from this that if the Bun’s place determined at the time of sun-rise be given, the time between their place and the Horizon reckoned in the diurnal circle will sidan | be tho side- real time and consequently tho place of the Horoscope determined through this will be right. But if the instantaneous place of the Sun be given, the tino givon must be the sAvaNa time, because let the instantaneous place of tho Sun be assumed for the Sun’s place determined at the time of sun-rise, then the time between this assumed instantaneous place of the Sun and the Horizon, which is shvana, will evidently bo the sidereal time. Hence the fuct as stated in the verse 27th is right, - Thereforo if tho Sun’se instantaneous place and the place of the Horoscope be given, the time found through these will be tho s{vana time, but if the placo of the Horoscope aud that of tho Sun dotormined ot the timo of sun-rige bo given, the time ascertained through thicso will bo the sidereal time. And if you wish to find the sivaNa timo through the place of the Horoscope and that of the Sun determined at the time of the sun-rise assumed the eidereal time just found as a rough sivamwa time and determined through this the instantaneous place of the Sun by the following proportion. If 60 anatis : Bun’s daily motion : :thoso rough sAVaNa GTATIS : tho Sun’s motion rolating to this time; and add thon this rosult to tho place of the Sun found at the time of sun-rise. ‘I'he sum thus found will bo the instantaneous placo of the Sun nearly. Find tho timo again through this VII. 32.] Sildhdnta-s’iromani. 169 28. In those countrics having a north latitude of 69° 20’ the a signs sagittarius and capricornus are Detormination of lntitudes a : sc in which different signs are never visible: and the signs gemini | and below the and cancer remain always above the horizon. 29. In those places having a northern latitude of 78° 15’, the four signs scorpio, sagittarius, capricornus, and aquarius aro nover 80611, and the four signs taurus, gemini, cancer, and lco, always appear revolving above the horizon. 80. On that far-famed lull of gold षष्ठ which has a lati- tude of 90° N. the six signs of the southern hemisphere never appear above the horizon and the six northern signs are always above the horizon. 81. Lauta has declared that when the asus of CHARA- KHANDA [in any latitude] are equal to the time which any sign takes to rise on the equator, then that sign will always remain visible above the horizon : but this assertion 18 without reason. Were it so, then in places having a latitude of 66°, the whole twelve signs of the ecliptic would always be visible, and would all appear at onco on all occasions, as the times of their rising on tho equator are equal to the asus of their CHARA-KHANDAS: but this is not the fact. 32. Lata has also stated in his work on tho sphere that where the north latitude is 66° 30’, ee gross error of sopittarius and capricornus aro not y visible, and also that in north latitude 75°, scorpio and aquarius are never there visible: but this also is an idle assertion. How, my learned friend, has he managed to make so gross and palpable an error of three degrees ?* An error of Lata exposed. instantaneous place of the Sun, and through: this time ascertain the instantancous place of the Sun. Thus you will get at last the exact sX{vana time from sun-rise to the hour given by the repetition of this process. As the Sun is taken here for an example, you can find the sMvana time of any planet or any planetary time from A planet’s rising to the hour given by the repetition of the aforesaid rocess.—B. 2 . ॐ [BirisKarnAonanya means here that Latta mentioning the degrees of lati- tudes, hns committed s grand mistake in omitting 3 degrees, bocause he has K 170 Translation of the [VII. 33. 33. ‘The altitude of the polar star and its zenith distance as found by observation, give respectively the latitude and tho LAMBANSA or complemont of the latitude. Or tho zcnith distance and altitude of the Sun at mid-day when on the equi- noctial give the latitude and its complement. 84. The unnata the time found in that arc of the diurnal circle which is intercepted between the eastern or western horizon and the planet above it, is sMvana. This is used in finding the shadow of the planet. The sine of the UNNaTA which is oblique, like the akstta-xarna, by roason of tho lati- tude, is called cHnzDaKa and not 8/4 पए because it is upright.* 85. In order to find the shadow of the Moon, the upiTA (the time elapsed from the rising of a planet) which has been found by some astronomers by means of repeated calculation is erroneous, for the upita, (found by repeated calculation) is not savaNa. The labour of the astronomer that docs not thoroughly understand mathematics as well the doctrine of the stated in his work that sagittarius and capricornus are always visible in a place bearing a latitude 66° 30’, and scorpio and aquarius at 75° N., whereas this is not the case, those signs are always visible in the places bearing the latitudes 69° 30’ and 78° 15’ respectively as shown in the verses 28 and 29.—B. D.] # [When the Sun is abovo the Horizon, the shadow causod by a gnomon 12 digits, high, is called the उपा" shadow according to the stppuANTa langusges. and having at firet determined the sine of the Sun’s altitude and that of it eomplement through his uDITa time, astronomora asccrtained this by tho follow- ing proportion. As the sine of the Sun’s altitude : the sine of its complement :: gnomon of 12 digits : the shadow caused by the gnomon. Thus they determine the shadow of all planots, Moon, &c., and that of the fixed stars. Though the light of the five small planets, Mars, &c., and the fixed stars is not so brilliant, tik ce that of the Sun and Moon, as to mako their shadow visiblo, yet it is nocessary to dotermine tho shadow of any hoavenly body in order to know the direction in which the body may be. Because, if the length and direction of the shadow of the body be known, the direction in which it is can be ascertained by spreading a thread from the end of its shadow through that of the gnomon. For, if you will fix a pipe in the direction of the thread thus spread, you will see through that pipe the body whose shadow is used here. The time given for determination of any planet’s shadow must be the 8 +त 4 timo, because it ie necessary to determine the degrees of altitude of a planet to know its shadow, and the degrees can be determined through the time contained in that aro of the diurnal circle intercepted between the ^ and horizon. But the time containod in this arc cannot be other than the sAVaNA time.— ए. D.) VII. 39.] Sidldhdnta-s'tromant. 171 aphore, in writing a book of instruction on tho scicnco is uttor- ly futile and usoloss.* 36. Tho degrees of altitude are found in the DRINMANDALA or vertical circle, being the degrees of व of sisKU Glevation in it above the horizon ; the degrees of zenith distance are (as their name imports) the degrees in the same circle by which the object is distant from the zenith or mid-heaven of the observer : the s’anxu is the sine of the degrees of altitude: and the priasyf is the sine of the zenith distance. 87. When the Sun in his ascent arrives at the prime verti- cal, the 8^^ पए found at the moment is ११00 SAMASANEG, ०4 the SAMA-8’ANKU : the s'ANKUS found at the moments of his passing the Kowa- vgitta and the meridian are respectively termed tho Kona- s/ANKU and MADIYA-8/ANKU. 38. One-half of the vertical circle in which a planet is क Rae oe observed should be visible, but only of parallax to the sincof alti- one-half less the portion opposite the ५ radius of the Marth is visible to observ- ors on the surface of the Earth. Therefore क part of tho daily motion of the planet observed is to be subtracted from the sine of altitude or from the s’anxu to find the shadow: [inasmuch as that amount is concealed by, or opposite to, the Earth]. 39. The aari (the sine of amplitude) is the sine of the arc of the horizon intercepted between the ee and prime vertical and the planet’s diurnal circle in the east or west i. ©. between * (In order to determine the Moon's shadow at a given time at full moon, some astronomers find her एणा + time i. 6. the time elapeed from her rising to tho hour given by the repeated calculation, through her instantaneous place and the placo of tho horuscopo dotorminod at the givon hour, But thoy groatly orr in this, becauso the timo thus fuund will not be tho s’avana time and consequently they cannot use this in finding the Moou's shadow. Their way for finding the . UDITa time by the repeated caloulation would bo right, then only if the given placo of tho Moon would bo such us found at tho time of her rising and not her instantaneous place. Because her ए 7174 time found through her instantancous aad becomes s’aVaNa at onco without having a recourse to tho repeated calcu- ation, as it is shown in the note on the verso 27 of this Ohapter.—B D.] Kk 2 172 Translation of the [\।।. 40. the east or west point of the horizon, and the point of the horizon at which the planet rises or sets. Tho lino connecting the points of the extremities of the east and west aarf is = called the upayasta-sutRa, the line of rising and setting. 40. The s'anku-TaLa or base of the s’aNnxku stretches dur- ing the day to the south of tho upydsTa-surra; because tho । diurnal circle have during the day a southern inclination (in nor- thern latitude) above the horizon. But, below the horizon at night, the base 1168 to the north of the upayAsta-suTRA as then the diurnal circles incline to the north. ‘Tho s’anxu- taLa’s place has thus been rightly defined. _ 41. (The s‘anxu-rTata lies to the south of the extreme point of ^५ when that aarf is north and when the 4084 is south, ` the s’aNKu-TALA lies still to the south of it. The difference and sum of the sine of amplitudo and s’aNnKu-Tata has boon denominated the एदाए or BHUJA; it is the sine of tho degrees lying between tho prime vertical and thio planct on the plano of the horizon. 42. [Taking this 84प्रए as one side of a right-angled triangle.] The sine of the zenith distance being the hypothe- nuse then the third side or the हणा being the square root of the difference of their squares will be found : itis an east and west portion of the diameter of the prime vertical.* I now propose to explain the triangles which aro created by reason of the Sun’s varying declinativun : and shall then proceed to explain briefly also the latitudinal triangles or those created by different latitudes. (The formor are called KRrANTI-KSHETRAS and the latter AKsHA-KSHETRAB. | * Vide accompanying dia- gram. a being place of the Sun : द ite place of rising in the horizon : 4 ¢ the upaYa'sta-s0TRa d 7 the acura’: ab the 8’aNKuU-rTaLa: then ag is the Ba‘HU and the triangle @ ॐ g is the one here represented to.—L. W. VII. 45.] Siddhdnta-s‘iromani. 173 43. In the Ist trianglo of declination. Ist. The sine of declination == BHUJA or baso, tho radius of diurnal circlo cor- | = K - responding with the declination eee pendicular, above given and radius of large circle == hypothenuse. 2nd. Or in a right sphere. Tho sine of 1, 2 or 3 signs = hypothenuso : The declination of 1, 2 or 3 signs पशम | __ उक्र) 8. o’clock line responding with the declination above given ‘Those sines being converted into terms of a large circle: and their arcs taken, they will then express the times in asus which each sign of the ecliptic takes in rising at the equator 1, 6. the mght ascensions of those signs or the LANKODAYAS, that is the 2nd will be found when the Ist is subtracted from two found conjointly, and tho 8rd will be found when the sum of tho Ist and 2nd is subtracted from three found con- jointly. 45. In the nght-angled trianglo formed by the s’anxu Triangles arise from lati- OF gnomon when the Sun is on the 44. Sines of arcs of diurnal circles cor- | = KOTIS. tude. equinoctial.* Ist. The s’anxu of 12 digits == the णण, Tho ratanité or tho shadow of s‘anKu or guomon | == tho 7० and the AKSHA-KARNA ॥ ५ or 2nd. The sine of latitude == BHUJA. The sine of co-latitudo - = KOTI and radius = hypothenuse This triangle is found in the plano of the moridian. [* Tho right anglo triangles atatcd in tho five verscs from 45 to 49, aro clearly scen by fastoning sumo diumetrial threads withiv tho armillary spliere. As 174 Translation of the [VII. 46. 46. Or the sine of declination reckoned on the UNMANDALA from theeast and westline (एण) the sine of ascensional difference in the diurnal circle of the given day } == KOTI. } = BIUJA. Let & G ति H be the meridian of tho given place, 8 A H the diamoter of tho horizon, 2 the Zenith, P and Q the north and south poles, EF ^ 7 the diameter of the equinoctial, P A Q thut of the six o’clook lino, Of D that of one of the diurnal circles, aud E B, f& the perpendiculars to @ Hl. Then it is cloar from this that 2 ¥E or H P = the latitude, A B = the sino of it, KB = the oo-sine of it, A f = tho declination of a planet revolving in tho diurnal cirelu whose dinmoter is 0 D, end .. A g = tho a@na or tho sinv of umplitude, ^ 4 = the kusya’, Ae = the saMa-8a'NKU or the sine of the planet’s altitude when it reaches the prime vertical. eg = the TavDuURITI, ef = the TADDHRITI—KUSYaA’, fA = the UNMaNDaLa s’anku or the sine of the planct’s altitude whon it reaches tho six o'clock 1091) A & = the aGera’DI-kitanwa or tho let portion of the sine of amplitude, aud &g = tho aGra’@ua-Kuanpa or tho 2nd portion of thowine of aniplitude ; VII. 48.] Siddhdnta-s’iromani. 175 The sine of amplitude in the horizon = hypothenuse This is a well known triangle. 47. Or the sama 8’ANKU in the primo vor- } tical being ae The sine of amplitude == BHUJA The rappuRIT! in the diurnal circle = hypothenuse Or ` Taking the sine of declination == BHUJA and the SAMA-s ANKU = hypothenuse TADDHRITI minus KUSYX = KOTI. 48. The UNMANDALA 8 ANKU being == BHUJA The sine of declination will then be == hypothenuse And ^61401 KHANDA or 1st portion of the } a earl sine of amplitude will be Therefore, with the exception of the first and last the other six triangles stated in the verses are these in succession. AK 8, Agf,Aeg, Acs, ASfh andg f hand the ee you will get by dividing tho three sides of the triangle A E 8 by —— and for the last see the note on the verse 49. 12 It. is cloar from tho abovo described diagram that all of theso triangles are similar to each other and consequently they can be known by means of propor- tion if any of thom bo known, Tho 8111011 (418) having thus producod sovornl trinnglos similar to thoso original by fastoning tho threads within the armillary 1 find answers of the several questions of the spherical trigonometry. Some problems of the ` spherical trigonometry can be solved with greater facility by this Srppnanra way than the trigonometrical way. As Problem. The zenith distances of a star when it has reached the prime vertical and the meridian at a day in any place are known, find the latitude in the place. (1 way for finding the answor of this problem according to the शण +त 74 is as follows. Draw Ce 1 A 2, (See the proceeding diagram) then O ८ e will be a lati- tudinal triangle. Now, let a = C © the sine of zenith distance, b = Ac, the co-sino of Zc, © == Ae, the 8AMa-8'ANKU, and @ = the latitude. Then 0 ० = A/a? + (b—0)’, and Oe:Oc:: AE: AB, or n/a 4 ~+ (bB—c)’: @ : : rad: sin # } aX Rad ० sine = o/s" + (b—c).—B. 7.] 176 Translation of the [VITI. 49, Or Making the UNMANDALA SANKU = KOTI the acRAGRA-KHANDA or 2nd portion of the ; | = BMUJYA sine of amplitude is ; the Kusya then becomes = hypothonuso 49.* The s’anxu being = KOT! and the 8 ANKU-TALA == BHUJA Then the CHHEDAKA or HRITI = hypothenuso Those who have a clear knowledge of the spherics having thus immediately formod thousands of triangles should oxplain the doctrine of the sphere to their pupils. End of Chapter VII. on the principles of the rules for resolving the questions on time, space and directions. Cuarrrr VIII. Called Granana VAsani. In explanation of the cause of eclipscs of the Sun and Moon. 1. The Moon, moving like a cloud in a lowor sphoro, The cause of the direc. OVertakes the Sun [by reason of its tions of the beginning and quicker motion and obscures its shin- ond of the solar eclipes. ing dik by ita own dark body :] hence it arises that the western side of the Sun’s disk is first obscured, and that tho eastern side is the last part relieved froin tho Moon’s dark body: and to some places the Sun 18 eclipsed and to others is not eclipsed (although he is above the horizon) on account of their different orbits. * This triangle differs from the 1st of the 47th verse only in this respect that tho buse of the triangle in the 47th vorse is oqual to tho sine of the whole ampli- tude while the base found when the Sun is not in the prime vertioal, will always be moro or less than the sine of amplitude and is thoroford gouerully called BANKUTALA.—L. W. VITI. 6.1 Siddhdnta-s'tromoni. 177 2. At the change of the Moon it often so [णाइ that an The causo of the parallax observer placed at the centre of the in Jongitade and that in Warth, would find the Sun when far from the zenith, obscured by tho intervening body of the Moon, whilst snother observer on the surface of the Harth will not at the same time find him to be so obscured, as the Moon will appear to him [on the lugher elevation] to be depressed from the line of vision extending from his eye to the Sun. Hence arises the necessity for the correction of parallax in celestial longitude and parallax in latitude in solar eclipses in consequence of the difference of the distances of the Sun and Moon. 8. When tho Sun and Moon are in opposition, the Earth’s Tho reason of tho correc. ®H%dow envelopos the Moon in dark- tion of parallat not being ness. As the Moon is actually enve- "त loped in darkness, its eclipse is equally seen by every one on the Earth’s surface [above whose horizon it may be at the time] : and ag the Earth’s shadow and the Moon which enters it, are at the same distance from tho Earth, there is therofore no call for the correction of the parallax in a lunar cclipse. 4. Asthe Moon moving eastward enters the dark sha- “The canse of the direo- dow of the Earth : therefore its eastern tions of the beginning and side is first of all involved in obscurity, i and its western is the last portion of its disc which emorges from darkness os it advances in its course. 5. As the Sun is a body of vast size, and the Harth insigni- ficantly small in comparison: the shadow mado by the Sun from the Earth is therefore of a conical form terminating in a sharp point. It oxtends to o distance considerably beyond that of the Moon’s orbit. 6. Tho length of the Earth’s shadow, and its breadth at the part traversed by the Moon, may be easily found by propor- tion. 178 Translation of the (VIII. 7. Tn the lunar eclipse tho Earth’s shadow is northwards or _ southwards of the Moon when its latitudé is south or north. Hence the latitude of the Moon is here to be supposed invorso (1. e. it is to be marked reversly in the projection to find tho centre of the Earth’s shadow from the Moon.) 7. As the horns of the Moon, when it is half obscured form The determination of the पशप obtuse angles : and the duration coverer in the eclipse of the of a lunar eclipse is also very great, + hence the coverer of the Moon is much larger than it. 8. The horns of the Sun on the contrary when half of its disc is obscured form very acute angles: and the duration of a solar eclipse is short: hence it may be safely inferrod that tho dimensions of the body causing the obscuration in a solar oclipse are smaller than and difforont from tho body causing an oclipse of the Moon.* 9. Those learned astronomers, who, boing too oxclusivoly devoted to the doctrine of the sphere, believe and maintain that 1६८ घए cannot be the cause of the obscuration of the Sun and Moon, founding their assertions on the above mentioned contrarieties, and differences in the parts of the body first obscured, in the place, time, causes of obscuration &c. must be admitted to assert what is at variance with the ^ पपा १५, the Vepas and Pourénas. 10. All discrepancy, however, between the assertions above referred to and the sacred scriptures may be reconciled by understanding that it is the dark Rauv which entering tho Karth’s shadow obscures ‘tho Moon, and which again ontoring the Moon (in a solar eclipse) obscures tho San by tho powor conferred upon it by the favour of BranMa. ® (Had the San’s coverer been the same with that of the Moon, his horns, when he is half eclipsed, would have formed, like those of the Moon obtuse angles. For the apparent diameters of the Sun and Moon are nearly equal to each other. Or the Moon when it is half eclipsed would have represented its horns, like those of the San, forming acute angles, if its coverer hat! beon tho same with that of the Sun. But as this is not the case, the coverer of the Moon is, of course, different and much larger than that of the Sun.—B, D.]} VIII. 15.1 Siddhanta-s'iromoni. 179 11. As tho spectator is clovated abovo tho contro of tho What is tho cause of paral- earth by half its diameter, he there- lax, and why it र भ fore 8068 tho Moon doprossed from its न nent seat! place [as found by a calculation made for the centre of the Earth]. Hence the parallax in longitude is calculated from the radius of the Earth, as is also the parallax in latitude. 12. Draw upon a smooth wall, tho sphoro of the carth द तन reduced to any convenient scale, and to ० the cause of the orbits of the Moon and Sun at a proportionate distances: next draw a transverse diameter and also a perpendicular diameter to both orbits.* 7 13, 14 and 15. Those points of the orbits cut by this diameter are on the (rational) horizon. And the point above Fig. 1, = In Fig. 1, 1९६ 8 be the ह र centro of the earth; A a SS from tho vision line in the vertical circle is her parallax from the Sun. spectator on her surface; C D, इ G the vertical circles passing through the Moon M, and ६ ७ 801 8; D, G tho points of tho horizon cut by When the Sun reaches the zenith I, it is evident that the Moon also will then be.at O and the vision line, and the line drawn from the centre of the Earth will be coincident. Hence thero is no parallax in the zenith. Thus the parallax of tho Moon from the Sun in the vortical circle is hero shown by means of a di os BT which becomes equal to the difference between the tho vortical circlea C J), ४ in which the Moon lies always parallaxcs of the Sun and Moon ree found in the vertical circle as stated by i at the time of conjunction,and A 8 the vision line drawn from tho spectator A to the Sun. Tho distance at which the Moon appears depressed 0} and 0, tho zenith in tho Moon's sphero, and ¥ in that of the Sun. Now, let EMS एप 4/8 ^ 84.014. + in the chaptor on eclipses in the commentary va/saNa/BILA - siya and tho theories and methods aro also givon by him on the parallaxes of tho Sun and Moon. This parallax in the vortical circle which arises from the be a line drawn from the centre of the Earth to the Sua zonith distance of the planet is callod the common parallax or tho parallax in altitudo. L 2 180 Translation of the (VIII. 15, cut by the perpendicular diameter will represent the observer’s zenith; Then placing the Sun and Moon with their respective zenith distances [as found by a proportional scale of sincs and arcs,| let the legrned astronomer show the manner in which Vig. 2. As in Fig. ®, 10४ A be a sppcta- A ४ tor on the earth’s surface; Z the senith ; and Z 8 the vertical circle passing through the planet 84 Let a circle Z’ m $ be described with centre A and radius E 8 which cute the lines A Zand A 8 produced in the points 2’ andr: Let a line ® m be drawn rallel to EZ, then the aro Z’ = will be equal to the 270 Z 8. Now the planet 8 seon from E has a zenith dis- tance Z 8 and from A, a zenith distance Z‘ # greater than 2 8 or Z’ m by tho aro mr, hence the apparent place r of the planet is depressed by mrin the vertical circle, This aromr is there- fore the common perallax of the planet, which can bo found as follows, Draw m # ५1 to Ar and rotoAZandlt P=ESorAr; : &=EAormS; p=mr the paral- Jax 3 @ = 2 8 orZ’m thet ` ६४७ zenith distanco of tho planet ; aad ^, 2 + p= 2/ ^ the ap (न zenith distance of the planet, Then m pr sin p and r o = sin (d + 2). Now by similar friangles A r 0, 5S mn. Ar: ro=Sm: mn, or ॐ 3 sin (4 + p) =4: sinp; AX sin (2 + p) B e Henpe, it is pyident Sa this that when the sin (द +p) = Rord$p = 90°, then the parallax will be greatest and if it be denoted by P, sin P > sin (4 4 p) ००० sin P = sin P = A and „१, sin p = R Now, tho parallex is gonoyally so amall that no sousiblo orror is introducod by making sin p = p and sin P = P; ; P X sin (2 +p) ee + = Again, for the rpason jnst mentioned sin d is assumed for sin (द - 2) in the फक क+म a . ein op R that is, the common parallax of planet is found by multiplying the greatest rasta by ५ sine of the zenith distance and dividing the product by the Ya oe e e VIII. 20.1 Siddhdnta-s'iromoni. 181 the parallax arisos. [107 this purposo] let him draw 006 lino passing the centre of the earth to the Sun’s disc: and another which 18 called the DRIKSUTRA or lino of vision, let him draw from the observer on the Earth’s surface to the Sun’s disc. The minutes contained in the arc, intercepted between these two lines give the Moon’s parallax from the Sun. 16. (At the new Moon) the Sun and Moon will always appear by a lino drawn from the centre of the earth to bo in exactly the same place and to have the same longitude: but when the Moon is observed from the surface of the Earth in the pgixsTRa or line of vision, it appears to be depressed, and hence the name LAMBANA, or depression, for parallax. 17, (When the new Moon happens in the zenith) then the lino drawn from the Earth’s centre will coincide with that drawn from its surface, hence a planet has no parallax when in the zenith. Now on a wall running due north and south draw a diagram as above prescribed; [i. e. draw the Earth, and also the orbits of the Sun and Moon at proportionate distances from the Earth, and also the diameter transverse and perpendicular, &c. | 18. Tho orbits now drawn, must be considered as DRIKSIIE- PA-vgitTTas or the azimuth circles for the nonagesimal. The sine of the zenith distance of the nonagesimal or of the latitude of the zenith is the pgixsHEPa of both the Sun and Moon. 19. Mark the nonagesimal points on the DRIKSHEPA-VRITTAS at the distance from the zenith equal to the latitude of the points. From these two points (supposing them as the Sun and Moon) find as before the minutes of parallax in altitude. These minutes are here Nari-Katfs, i. ९ the minutes of the parallax in latitude of the Moon from the Sun. 20. The differonco north and south between the two orbits 1, e. the measuro of their mutual inclination, 1s the samo in evory part of tho orbit as it is in the nonagesimal point, hence this difference called Nati is ascertained through the DRIKSUE- pa or the sine of the zenith distance of the nonagesimal.* [* When the planet is depressod in tho vortical circle, its north and south 182 Translation of the (VIII. 21. 21. The amount by which the Moon is depressed below the Sun deflected from the zenith [at the conjunction] wherever it be, is the cast and west difference betwoen the Sun and Moon in a vertical circle.* distance from its orbit caused by this depression is called Natt or the parallax iu latitude. Vig. 3. As, in Big. 3, let 2 be the zenith; N the nona- ~ gesimal; 2 N P ite vertical circle; N s r the ecliptic; P its polo; 2 ॐ ६ the vertical circlo passing through the true place 8 and tho depress- ed or apparent place ¢ of theSun; P ¢ r a secondary to the ecliptic passing through tho apparent pluce ई of the Sun ; i en $ 9 18 tho 8748114५ LAMDANA or the parallax in longitudo and ¢ # tho Natt or tho parallax in latitude which can be found in the foliowing manner according to the sippHANTas. Let ZN be the zenith dietance of the nona- gesimal and 2 8 that of the Sun; then by the triangles ZN 8, ter sin 28:90 ZN = sin et: sine é, sine’ K sin ZN | sin 2 8 Now, 9 ६19 taken for sin sé, and ¢ ¢ for sin १४ on account of their being very emall st > sin ZN sin ZS but according to the sippHANTas ९. 91028 & ई == ~~ ००, 810 # és ॐ rt ~> (see the preceding note).. (1) P.sin ZN ow {भ्र R ०० 00 00 00 90 00 cc cece 00 00 ne (2) that is, the Narr is found by multiplying the sine of the latitude of the प्रभा gesimal by the greatest parallax and dividing the product by the radius. ` _ It is clear from this t fiat the north and south distance frem tho Sun depressed in the vertical circle to the ecliptic wherever he may be in it, bocomes equal to the common parallax at the nonagosimal, and henco the NaTI is to be dotermined from the zonith distance of the nonagesimal. . For this reason, by subtracting the natr of the Sun from that of the Moon, which are soparately found in the way sbove montionod, the parallux in latitude of the Moon from tho Sun is found ; and this bocomos equal to the difference between the mean parallaxes of the Sun and Moon at the nonagesimal. ‘Ihe same fact is shown by Buisxapfouinya through the diagrams stated in the verses 12th &, At the time of the eclipse as the latitude of the Moon revolving in its orbits is very small, the Moon, therefore, is not far from the ecliptic; and hence tlie parallax in longitudo and that in latitudo of the Moon is hero determined from hor corresponding plavo in tho ocliptic, on account of the dillcronoe being vory emall.—B. 1). : _ * [According to the tochnicality of the Siddhantas, tho distanco taken in any circlo from any point in it, is called the cast and weat distanco of the point, and VIII. 27. Siddhanta-s’iromoni. 183 22. For this reason, the differenco is two-fold, beg partly east and west, and partly north and south. And the ecliptic is hore cast and west, and the circle sccondary to it is north and south. (It follows from this, that the east and west difference lies in the ecliptic, and the north and south differ- ence in the secondary to it.) 23. Tho difforenco east and west has been denominated LAMBANA or parallax in longitudo, whilst that running north and south is parallax in latitude. 24, The parallax in minutes as observed in a vertical circle, forms the hypothenuse of a right angle triangle, of which the NATI-KALA or the minutes of the parallax in latitude form one of the sides adjoining the right angle then the third side found by taking the square-root of the difference of the squares of the two preceding sides will be sPHUTA-LAMBANA-LIPTA or the minutes of the parallax in longitude.* 25. The amounts in minutes of parallax in a vertical circle may be found by multiplying the sine of the Sun’s zenith distanco of tho minutos of tho oxtromo or horizontal parallax and dividing tho product by tho radius. Thus tho nati will bo found from tho prixsmera or tho sino of the nonagesimal zenith distance.t 26. The extreme or horizontal parallax of the Moon from the Sun amounts to ,', part of the difference of the Sun’s and Moon’s daily motion. For कष part of the yosanas, the distance of which any planet traverses per diem (according to the श7- pHANTAS) is equal to the Harth’s radius 27. The minutes of the parallax in longitude of the Moon from the Sun divided by the difference in degrees of the daily tho distanco taken in tho sccondary to that circlo from the samo point, is callod the north and south distanve of that point.~-B. D * [Sce Fig. 3, in which by assuming tho triungle r ° £ 88 a plane (व triangle, r ¢ = base, ® ¢ = hypothenuse and ड r = perpendioular, therefore sr=/et—r t?.—B, D.] + [This is clear from the equations (1) and (2) shown in the precoding large note.—B. D.] 184 Translation of the (VIII. 28. motions of tho Sun and Moon will be converted into aHATIS fi. 6. the time between the true and apparent conjunction] .* If the Moon be to the east [of the nonagesimal], itis thrown forward from the Sun, if to the west it ia thrown backward (by the parallax). 28. And if the Moon be advanced from the Sun, then it must be inferred that the conjunction has already taken place by reason of the Moon’s quicker motion; if depressed behind the Sun, then it may be inferred that the conjunction is to come by the same reason. | Hence the parallax in time, if the Moon be to the east [of the nonagesimal] is to be subtracted from the end of the TITHt or the honr of ecliptic conjunction, and to be added when the Moon is to the west [of the nonagesimal]. 29. Tho latitudo of the Moon is north and south distanco between the Sun and Moon, and the nati also is north and south. Henco the sara or latitude applied with the Natt or the parallax in latitude, becomes the apparent latitude (of the Moon from the Sun). VALANA or variation (of the ecliptic). [The deviation of the ecliptic from the eastern point (in reference to the observer’s place) of a planet’s disc, situated in the ecliptic is called the Vatana or variation (of the ecliptic). It is ovident from this, that tho variation is cquivalont to tho arc which is the measure of the angle formed by the ecliptic and the secondary to the circle of position at the planct’s place in the ecliptic. It is equal to that arc also, which is the * It is clear from the following proportion. If difference in minutes of daily motions of Sun and Moon. : 60 aGuatis — what will 3 : given Lampawa—Ka‘Las or minutes of the parallax give ; 60 x given minutes of the parallax diff. in minutes of Sun’s and Moon’s motions given minutes of tho parallax == accoloration or dolay of con- diff. in ०९६१००३ of Sun’s and Mooun’s motious junction arising from parallax.—L. W. ४111. 29.] Stddluinta-s'tromone. 185 measure of tho angle at the place of the planet in the ecliptic formed by the circle of position and the circle of latitude. It is very difficult to find it at once. For this reason, it is divided into two parts called the AKsHA-vVALANA (latitudinal variation) and tho AyaNa-vALANA (solstitial variation). The AKSHA-VALANA is the arc which is the measure of the angle formed by the circlo of position, and the circle of declination at the placo of tho planct in tho ocliptic, and the AYANA-VALANA is the arc which is the measure of the angle formed by the circle of declination and the circle of latitude. This angle is equivalent to the angle of position. From thesum or differ- once of these two arcs, the arc which is the measure of the anglo formed by the circle of position and the circle of latitude 18 ascertaincd, and hence it is sometimes called the s'PasuTa- VALANA or rectified variation. Now, according to the phraseology of the SippnAntas, the point at a distance of 90° forward from any place in any circle is the east point of that place, and the point at an equal distanco backwards from it is the wost point. And, tho right hand point, 90° distant from that place, in the secondary to the former circlo, is tho south point, and the left hand point, is the north point. According to this language, the doviation of the east point of the place of the planet in the ecliptid, from the east point in the secondary to the circle of position at the planet’s place, is the vaLANA. But the secondary to the circle of position will intersect the primo vertical at o distance of 90° forward from the place of the planet, and hence the deviation of the east point in the ecliptic from the éast. point in the prime vertical is the VALANA or variation, and this results equally in all directions. When the east point in the ecliptic is to the north of the east point in the prime vertical, the variation is north, if it be to the south, the variation is south. The use of the vaLANA is this that, in drawing the projec- tious of the eclipses, after the disc of the body which is to be eclipsed is drawn, and the north and south and the east and M 186 Translation of the [VIIT. 29. west lines are also marked in it, which lines will, of course, represent the circle of position and its secondary, the direction of the line representing the ecliptic in the disc of the body can easily be found through the vatana. ‘This direction being known, the exact directions of the boginning, middle and the end of the eclipse can be determined. But as the Moon revolves in its orbit, the direction of its orbit, therefore, is to be found. But the method for finding this is very difficult, and consequently instead of doing this, Astronomers deter- mined the direction of the ecliptic, by means of the Moon’s corresponding place in it and then ascertain the direction of the Moon’s orbit. The vALaNna will exactly be understood by seeing the follow- ing diagram ट स Let E P C be the ecliptic, P the place of the planet in it, A # ए the equinoctial, V the vernal equinox, D ॥ F the prime vertical, ॥ the point of intersection of the prime vertical and VIII. 29.] Siddhanta-s'iromoni. 187 tho cquinoctial, henco & tho cast or wost point of the horizon and D h equivalent to the nata which is found in the ए, 36. Again, 160 ५ 7? ¢, % Pb and dP f be the circles of latitude, declination and position respectively passing through the place of the planet in the ecliptic. Then, the arc f ¢ which is the measure of Z b P f = the Axsna- VALANA : thie are 0:6: 43 Z.cP b= the fyana- VALANA : and the arc fc १,१,००००१००००००.०००० Z.¢ Pf =the spasuta- VALANA. Or according to the phraseology of the SippHAntas ए the east point of P in the ecliptic ; 0 chaise, 1 ,०, the equinoctial ; 1) sisiionseenaduanesussteinass the prime vertical ; hence, the distance from D to A or arc D A or fb = the (शा VALANA : 4 ००... A to 1 ग 86 A 1 or ¢ ८ = the AYANA-VALANA: भित ......00 1 to E or arc J) 1 or 7 ^ == tho अशृ १॥ 1५ प or rectified variation. , These arcs can be found as follows Let, ८ = longitude of the planet, e = obliquity of the ecliptic, d, = declination of the planet, L = latitude of the place, १ = NATA, 2 == AYANA-VALANA, y == AKSHA-VALANA, and Z = rectified VALANA. Then, in the spherical triangle A V E, sin EA V:sn AVE = श EV: sin AB, or cos (| : sin ¢ = cos l : sin 2, m2 188 Translation of the [VITT. 30. 811 ¢ , ९08 1 , ,०, sin zor sine of the AYANA-VALANA = (A) cos d See ४. 32, 33, 34. This vaLana is called north or south as the point 1 be north or south to the point A. And, in the triangle A h D. sn DAA:snAhD=sinDh:snD A; here, sin DA h = sin E A V = cos d, sin Ah D = asin J, and sin Dh = sin 9. cos d: sin L = sin १ : sin y, sin L. sin 2 .* sin y or sine of the AKSHA-VALANA = (1) cos d Seo V. 37. The AxsmA-VALANA is called north or south as the point A be north or south to the point D. And the rectified vaaana D E=DA+ ^ 2, when the point A lies between the points D and I, but if the point A be beyond them, the rectified vauana will be equal to the difference between the AksHA and AYANA-vVALANA. This also is called north or south as the point E be north or south to the point D. Tho ancient astronomors Latta, S’/rfrart &c. used the co-versed sin { instead of cos! and the radius for the cos व in (A) and the versed sin » in the place of sin १ and radius for the cos d in (B) and hence, the vaLanas, found by them aro wrong. Budskaracit&rya therefore, in ordor to convinco the people of the said mistake mado by Lata, 3 पता, &e. in finding the vaLaNas refuted them in several ways in the subsequent parts of this chapter.—B. D.] 30. In either the Ist Libra or the Ist Aries in the equi- noctial point of intersection of tho equinoctial and ecliptic, the north and south lines of the two circlos 1. e. their secondaries aro different AYAWA-VALANA, VIII. 36.] Siddhdnta-s'iromoni. 189 and arc at adistanco* of tho extremo declination (of tho Sinn 1. €. 24°) from each other. 81. Henco, tho Ayana-vaALaNnA will then be equal to the sine of 24° :—The north and south lines of these two circles however are coincident at the solstitial points. 82, 33 and 84. And the north and south lines being there coincident, it follows as a matter of course that the east of thoso two circles will bo tho same. Lenco at tho solstitial points there is no (AYANA) VALANA. When the planet is in any point of the ecliptic between the equinoctial and solstitial points, AYANA-VALANA is then found by proportion, or by multiplying the co-sine of the longitude of tho planct by tho sino of 24°, and dividing the product by tho करण or tho co-sino of the declination of tho planct. 11118 AYANA-VALANA 18 called north or south as the planot bo in the ascending or descending signs respectively. Thus in like manner at the point of intersection of the primo vertical and equinoctial, the six o’clock lino is tho north and south hno of the equinoctial, whilst tho horizon (of the given place) is the north and south line of the prime vertical. The distance of .these north and south lines is equal to the latitude (of the place). 89. Hence at (the east or west point of) the horizon, the AKSHA-VALANA 18 equal to the sine of the latitude. At midday the north and south line of the equinoctial and prime vertical is tho same. Hence at midday thore is no AKSIA-VALANA. 36. For any intervening spot, the AKsHA-VALANA 18 to be fonnd from tho sine of the natat by proportion. First, the degrees of NATA are (nearly) to be found by multiplying the’ time from noon by 90 and dividing the product by tho half longth of day. AKSIIA-VALANA. * [By tho distance of any two great circles is hero meant an aro intorceptod betweon them, of a great circle through the poles of which thoy pass.—B. D.] + (Here the wata is the arc of tho prime vertical intercepted between the zenith and the secondary circle to it passing throug: the place of the planet.— B. D. ] 190 Translition of the | VIII. 37. 87. Then tho sine of the nata degroos multiplicd by tho sine of latitude, and divided by the co-sine of the declination ofthe planet will be the AKsHa-vALANA. If the nata be to the east, the AkSHA-VALANA is called north. If west, then it is called south (in the north terrestrial latitude). The sum and difference of the AYANA and AKSHA-VALANAS must be taken for the sPASHTA-VALANA, viz. their sum when the Ayana and AKSHA-VALANAS are both of the same denomination, and their difference when of different denominations i. 6, one north and the other south. 38. When the planet is at either the points of the inter- section of the ecliptic and prime vertical, the sPASHTA-VALANA found by adding or subtracting the AYANA and AKSITA-VALANAS (as they happen to bo of tho same or different donominations) is for that time at its maximum. 89, But at a point of the ecliptic distant from tho point of intersection three signs either forward or backward, there is NO SPASHTA-VALANA : for, at those points the north and the south lines of the two circles are coincident. 40. However, were you to attempt to show by the use of the versed sine, that there was then no SPASHTA-VALANA at those points, you could not succeed. The calculation must bo worked by tho right sine. I repeat this to impross the rulo more strongly on your mind. 41. As all the circles of-declination meet at the poles; it Another way of refutation i8 therefore evident that the north SPASHTA-VALANA. of using the versed sine. and south line perpendicular to tho east and west line in the plane of the cquinoctial, will fall in the poles. 42, But all the circles of celestial latitude meet in tho pole of the ecliptic-called the Kapampa, 24° distant from the equinoctial pole. And it is this ecliptic pole which causes and makes manifest the VALANA. 43. In the ecliptic poles always lies the north and south VIII. 49.] Siddhanta-s'iromoni. 191 linc which is porpendicular to tho cast and wost line in tho plane of the ecliptic. To illustrate this, o circle should bo attached to tho sphcro, taking the equinoctial pole for a centre, and 24° for radius. This circle is called the KADAMBA-BHRAMA-VRITTA or the circlo in which the KADAMBA revolves (round the pole). The sincs in this circle correspond with tho sincs of tho declination. All the secondary circlesto the prime vertical meet in the point of intersection of the meridian and horizon, and this point of intersection is called sama i.e. north or south point of horizon Now from the planet draw circles on the sphere so as to meet in the saMA, in the equinoctial pole and also in the ecliptic pole The three different kinds of vaLANA will now clearly appear between these circles: viz. the Axsua VALANA is the distance between the two circles just described passing through the SAMA and oquinoctial pole. 2. ‘Tho AYANA-VALANA 18 tho distanco betweon tho circlos passing through tho ocliptic and equinoctial polos. 3. ‘The sPAsiTA-VALANA 18 the distance between the circlos passing through the saMA and KADAMBA. These three VALANAS are at the distance of a quadrant from the planet and are the same in all directions. 48 and 49. Or (to illustrate the subject further) making Second mode of illustrat. the planet as the pole of ४ sphere, ing the SPaspya-valana. = draw a circle at 90° from it: then in that circle you will observe the AKsHA VALANA—which, in it, is the distance of the point intersected by the equinoctial from tho point cut by tho prime vertical. The distance of the point cut by the equinoctial from that cut by the ecliptic is tho A£yaNa—and the distance between the points cut by the ecliptic and prime vertical the srasuta- VALANA. 192 | Translation of the [ VIII. 50. 50. In this case the plane of the ecliptic is always east and west—cclestial latitude forming its north and south line. Those therefore who (like s’rfratr or Latta) would add the s‘ara celestial latitude to find the vatana, labour under a grievous delusion. 51. The 18४ of Capricorn and the ccliptic polo reach the meridian at the same time (in any latitude): so also with regard to the Ist Cancer. Hence at tho solstitiul points thero 18 NO AYANA-VALANA. . 52. As the Ist Capricorn revolves in tho sphere, so tho ecliptic pole revolves in its own small circle (called tho Ka- DAMBA-BHRAMA-VRITTA round the pole). 53 and 54. When the 1st of Aquarius or the Ist of Pisces comes to the meridian, the distance in the form of a sine in tho KADAMBA-BIIRAMA-VRITrA, between tho ecliptic pole and the meridian is the Ayana-vaLana. ‘This VALANA corresponds with tho Krantisy{ or the sino of declination found from tho degrees corresponding to the time elapsed from the Ist Capri- cornus leaving the meridian. 55. As the versed sine is like tho sagitta and the sine is the half chord (therefore the versed sine of the distance of the ecliptic pole from tho meridian will not expross tho proper quantity of vALANA as has been asserted by Latta &c.: but the right sino of that distance docs 80 prociscly). ‘lho 4१५१५ VALANA will be found from the declination of the longitude of the Sun added with three signs or 90°. 66. Those people who have directed that tho versed sine of the declination of that point three signs in advance of the Sun should bo usod, have thereby vitiatod tho whole calcula- tion. AKSHA-VALANA may be’ in like manner ascertained and illustrated : but it is found by the right sine, (and not by the versed sine). 57. He who prescribes rules at variance with former texts and docs not shew the error of their authors is much to be blumed. Hence I am acquitted of blame having thus clearly exposed the crrors of my predecessors. VIII. 64.] Suddhdnta-s'iromont, 193 58. (16 inapplicability of the versed sine may be further Another way of refutation, Jlustrated as follows. Make the eclip- of using the vorsod sine tic pole the centre and draw tho circle called the J1nA-vRITTA with a radius equal to 24° 59. Then make a moveable secondary circle to the ecliptic to revolve on the two ecliptic poles. This circle will pass over tho equinoctial poles, when it comes to the end of the sign of Gomi. ; 60. By whatevor number of degrees this secondary circle is advanced beyond the end of Gemini, by precisely the same number of degrees, it is advanced beyond the equinoctial pole, in this small srna-vgitTa. The sine of those degrees will be thore found to correspond exactly with ond increase as does the sino of the doclination. 61. And this sine is the AyYANA-VALANA: This VALANA is tho VALANA at the end of the pynyyA. For the distance between the equinoctial pole and planet is always equal to the arc of which the pynsya is the sine 1. e. the cosine of the declination. 62, But as tho valuo of tho result found is roquired in terms of the radius, it is consequently to be converted into those terms. As the jtNa-vgiTTA was drawn from the ecliptic pole as centre, with a radius equal to the greatest declination, so now, making the saMA centre draw a circle round it with a radius equal to the degrees of the place’s latitude. (This circle is called AKSHA-VRITTA.) 63 and 64. To the two samas or north and south points of the horizon as poles, attach a moveable secondary circle to the prime vertical. Now, if this moveable circle be brought over the planet, then its distance counted in the AKSHA-VRITTA or small circlo from tho equinoctial polo will bo oxactly equal to that of the planct from tho zenith in the prime vertical. Tho sino of the plauot’s zenith distanco in tho prime vertical, will, when reduced to the valuo of the radius of AKSHA-vRITTA represent the AKSHA-VALANA. | N 194 Translation of the [VIII. 65. 65, As in tho XYANA-vALANA 80 also in this AKSHA-VALANA, tho result at the end of the prnsyv is found ; this therefore must be converted into terms of the radius. From this illustration it is evident that it may be accurately ascertained from the zenith distance in the prime vertical. 66. Iwill show now how the (xsHa-vaLana may be also ascertained from the time from the planets being on the meridian in its diurnal circle. [The rule is as follows.] Add or subtract the s‘anxuTAta [of a given time] to and from the कः ५१ Chap. VIE sine of amplitude according as thoy verso 41, Chap. $ “iT are of the same or of different deno- minations (for the एए or BHUJA). 67. The sine of the latitude of the given place multiplied by the sine of the asus of the time from the planet’s being on the meridian, and divided by the square-root of tho differenco between tho squares of the nnusa (above found) and of the radius, will be exactly the {xstta-VALANA.* ® This rule and the means by which it has been establiahed by BuAskaBiouA- nya require elucidation. ए (82५0 ५/४ २८ first directs that the 247ए or एष्व एर be found for the time of the middle of the eclipse and that a circle parallel to the primo vertical, be drawn having for its centre a point on the axis of the prime vertical distant from the centre of the prime vertical, by the amount of tho Ba’nv. From this as centre and the ण्म equal to = 4८४००8५0 as radius draw acircle paral- Jel to the prime vertical. This circlo called an UPavertra will cut the diurnal circle for the time on 2 points equally distant froin tho moridian, Connect those inte by achord. ‘The half of this chord is the NaTaamaTrsrA as well in tho diurnal circle as in the UPAVEITTA, but as these 2 circles differ in the magnitude, these sines will be the sines of a different number of degrees in each circle. Now the waraauatrsrx is known, but it 13 in terme of alarge circle. Reduce them to their value in the diurnal circle. 1. {कणर : waTasyA : : DYNJYA' : sine of diurnal circlo. Thies sine in diurnal circle is also sine in UPAVBITTA. 2. If UPA-VRITPA-TRIYA : this sino: ; TRIJ¥a cqual to AKSUAJYA, $ pyvaya’: this result : : TRIs¥A : sine of AKsUA-VALANA now canoe! and there will remain the rule above stated waTasya >€ AXSHAJYA’ ; ——— = BNO of AKSHA-VALANA. UPAVBITTA-TRIJYA’ Hore our author makes use of the diurnal circlo and uravrtrra in torm of the equator and prime vertical, whose portions determino tho vataNa. ‘Tho amaller circles being parallel to the larger, the object sought 18 cqually attained, —bL. W. ; VIII. 74. ] Sildhanta-s'iromone. 195 68. Or the AksHA-VALANA may be thus roughly found. Multiply the time from the planet’s being on the meridian and divide the product by the half length of day, the result are the nata degrees. The sine of these Nata degrees multiplied by the sine of the latitude and divided by the DYNJYA or the cosine of the declination, will give the rough AKSITA-VALANA. 69. Vlaco the disc of tho Sun aot tho point at which tho diurnal circle intersccts tho ecliptic. The arc of the disc intercepted be- tween these two circles represents the AYANA-VALANA in terms of radius of the disc. 70. This vaLana is equal to the difference between the sine of declination of the centre of the Sun and of the point of intersection of the disc and ecliptic; and it is thus found ; multiply the radius of disc by the suoaya-xHanpa of the BHUJA of the Sun’s longitude and divide by 225. 71. Then multiply this result by sine of 24° and divide by tho radius : tho quotiont is the difforonco of the two sino of declination. ‘This again multiplied by the radius and divided by tho radius of Sun’s disc will givo tho valuo in torms of tho radius (of a great circle). 72. Now in these proportions the radius of the Sun’s disc and also radius are in one case multipliers (being in third _ places), and in the other divisors (being the first terms of the proportion) thereforo cancel both. There will then remain rule, multiply the Sun’s एप्०७१+ KHANDA by sine of 24° and divide by 225. 78. And this quantity is equal to the declination of a point of ecliptic 90’ in advance of Sun’s place. Thus you observe that the vALANA is found by the sino of declination as abovo alleged, (and not by the versed sine). Abandon therefore, O foolish men, your crroneous rules on this subject. 74, The disc appears declined from the zenith like an umbrella ; but the declination is direct to the equinoctial pole : N 2 Further illustration. 196 Translation of the (1X. 1. the proportion of the pynsyA or complement of. declination is therefore required to reduce the vaLana found to its proper value in terms of the radius. End of Chapter VIII. In explanation of the cause of eclipses of tho Sun and Moon. CHAPTER IX. Called DRIKKARAMA-VASANA on the principles of the Rules for finding the tines of the rising and setting of the heavenly bodies. 1. A planet is not found on the horizon at the time at Objeot of the correction Which its corresponding point in the वा requisite vo be tp ecliptic (or that point of the ocliptic lied to the place of the having the same longitude) reaches planet, for finding the point । ५ त of the ecliptic onthehorizon the horizon, inasmuch as it is clovatod when the planet reaches it. shove or depressed below the horizon, by the operation of its latitude. A correction called Drik- KARAMA to find the exact time of rising and setting of a planet, 18 therefore necessary. * 2. When the planet’s corresponding point in the ocliptic reaches the horizon, the latitude then does not coincido with the horizon, but with the circle of latitudo. ‘The olovation of the latitude above and depression of it below the horizon, is of two sorts, [one of which is caused by the obliquity of the ecliptic and the other by the latitude of the place.} Hence the DRIKKARAMA is two-fold, 1. 6. tho AYANA and tho AKsHWAJA or AxsHa. The detail and modo of performing these two sorts of the correction are now clearly unfolded. 8. When the two vaLanas are north and the planet’s corresponding point in the ecliptic is in the eastern horizon, the planet is thereby depressed below the horizon by south latitude, and elevated when the planet’s latitude is north. 1, १4१३.) 6 IX. 9.] Siddhdnta-a'tromoni. 197 4. Whon the two kinds of vanana are south, then tho reverse of this takes place ; the reverse of this also takes place when the planet’s corresponding point is in the western hori- 2011. [And the difference in tho times of rising of the planet and its corresponding point is called the resultant time of the DRIKKARMA and is found by the following proportions. } If radius: AYANA-VALANA :: what will celestial latitudo give ? $. And if cosine of the latitude of the given ; AKSHA-VALANA placo ‡ : what will srasipaA 8 ARA givo ? Multiply tho two results thus found by theso two propor- tions, by the radius and divide the products by the pyusyf or cosine of declination. 6 and 7. Take the arcs of these two results (which are sines) and by the asus found from the sum of or the difference between these two arcs, the planet’ is depressed below or elevated above the horizon. The Laena or horoscope found by tho direct proccss (as shown in the note on the vorso 26, Chapter VII.) when the planet is depressed and by the indirect process (as shown in the same note) when it is elevated, by means of the asus above found, is its UDAYA LAGNA rising horoscope or the point of the ecliptic which comes to tho castern horizon at the same time with tho planet. When the planet’s corresponding point is in the western horizon, the LacNA horoscope found then by the rule converse of that above given, by means of the place of the planet added with 6 signs, is its Asta LAGNA setting horoscope or tho point of the ecliptic which is on the eastern horizon when the planet comes to the western horizon. 8and9. For tho fixed stars whose latitudes are very considerable the resulted time of the DRIKKARMA is found in a 198 Translation of the [IX. 9. different way. Find the ascensional difference from the mean declination of the star, i. e. from the declination of its corre- sponding point in the ecliptic, and also from that applied with the latitude, i. €. from the true declination. The asus found from the sum of or the difference between the ascensional differences just found, as the mean and true declinations aro of the different or of the same denominations respectively, are the asus of depression or elevation depending on the AKSsHA DRIKKARMA. (Find also the time depending on the 4२५५ - DRIKKARMA): and from the sum of or tho difference between them, as they may be of the same or different denominations, the UDAYA LAGNA or ASTA LAGNA may be ascertained as above found (in the 6th and 7th verses).* * Tot AD BO ho tho moridian; OF D tho horizon, A tho sconith; FB tho east point of tho horizon; ¥ 1 च the equinoctial; K tho north polo; 7, tho south; P tho planot ; p its corresponding point in the ocliptio; 1 P 2 J the 80001 09४ to tho ecliptic passing through tho p'anot P, and hence p P tho latitude, Lot f P g the diurnal circle passing through tho planet P and henco ॐ B the rectified latitude. Now, when the corresponding place of the planet is in the horizon, it is then evident from the acoompanying figure, that the planet is clevated above or depreasod below the horizon by its lutitude p P ani as it ia vory diflicult to find the elevation or depression at once, it is thorefore ascertained by means of its two parts, the one of which is from the horizon to the circle of declination, i. ७. Qto R. This partial clevation or depression takes place by the planet’s rectifiod latitude p R. And the other part of the elevation or dupression is from the circle of declination to the circle of latitude; i. e. from R to P and this occurs by the planet’s mean latitude p P. From the sum or difference of these two ts, the exact elevation of the planet above tho horizon or the depression below it, can be determined. When the terrestrial latitude, of the given place 18 north and the planet's 0 lace 10 the ecliptic is in the eastern horizon, the a’ksHa-VaLaNa 18 then north and tho circle of declination is elevated above the horizon to the north. For this reason, when the +^ +. VaLawa is north, tho planet will be elovatod above the castern horizon if its latitude be north, and if it be south, tho planet will bo depressed below the horizon. But the revorse of this takos placo when tho a’KsilaA-VALANA is south which occurs on acoount of tho south latitudo of tho given place, i, ©. whon tho a’KSMA-VaLaNa is south, the circlo of declination is depressed below the horizon to the north and hence tho planet is depreesod bolow it, if ite latitude bo north, and if it be south, the planet is elevated above the horizon. Again, when the planet’s longitude terminates in the six ascending signs, it is evident that the Ayawa-vaLana becomes then north, and the north pole of tho ecliptic is elevated above the circle of declination passing through the planet. Hence, when the a’YaNa-VALANA 18 north, the planet is elevated above or depressed below the circle of declination by its mean latitude, as it is north or south. But the reverse of this takes place, whon tho aA’yaNa-VALANA is south, i. €. the planet is depressed below or elevated above the circle of declination, as its latitude ia north or south. Bocause when the a’yaNa-yALaNa is south 1X. 9] Siddhdnta-s'iromoni. 199 the north pole of the ecliptic lies below the circle of declination and the south above it. Again, whon tho planet ia in the western horizon, the circle of declination | passing through tho placo of the planot in tho ecliptic lics to the north abovo tho horizon, but the aksi1A-vaLANA, becomes south and henco the reverse takes place of what is said about the elevation or depression when the planet is in the enetern horizon. But as to the AyvaNa-vaLaNna, it becomes north when the longitude of the planet terminates in the ascending six signs and the north pole of the ecliptic lies below the circle of declination. Hence the depression of the lanet takes place when its latitude is north and the elevation when the latitude is south. But when the longitude of the planet terminates in the discending six signs, the AYANA-VALANA becomes then south and the north pole of the ecliptic lies sbove the circle of declination. For this reason, the elevation of the planet takes place when ite latitude is north, and the depression when it is south, ‘I'hus in the western horizon the elevations and depressions of the planet are opposite to those when the planet is in the eastern horizon. Now, the time elapsed from the planet's rising when it is elevated above the horizon and the time which the planet will take to rise when it is depressed below the horizon, are found in the following manner. 200 Translation of the >. 10. 10. The [Aspasu7a] 3484 or true latitude [of the planet] To find the value of co. Multiplied by the Dyusya or cosine lestial latitudo in terms ofs of declination of the point of the eclip- circle of declination, to ren- ; ह der it fit to be added to or tic, three signs in advance of tho bt fi ination. i i pubérsoted: rom dectnayon planot’s corrosponding point and di- 866 tho figure above described in which the angle Q K B or the equinoctial arc Q’ p’ denotes the time of elevation of the planet from Q to R, and the time of elevation of the planet from Rto P is denoted either by the angle P K R or by the equinoctial aro P’ p’, Out of these two times Q’ p’ and P’ »’, we show at first how to find P’ p’, In tho triangle P p BR, P 2 = the latitudo of the planct, 2 P 2 R = tho. A’YANA-V4LaNa and <= P 7६ p = ~, and "=: snPpR=sin Pp: sin RP; or 11 radius : sin of a’YaNa-VALANA == the sine of latitude : sin ए ९. । Again, by the similar triangles K P R and K P’ p’ sin K P: sin R P = sin K P’: sin P’ p’, here, sin K P = cosine of declination and K = R, R X< sin R P es 8171 Pp’ p’ = cos of declination Now, the time p’ Q’ is found as follows. In the triangle p RQ, p 2४ = the spasta-s’aRa which can be found by the rule given in the ए. 10 of this chapter, ~ R p Q = aKSHA-VaLaNa and <. EQ p = oo-latitude of place nearly and ..sinp QR:sin Rp Q::sinp B: sin BQ or, if cosine of latitude, : sine of AKSHA-VALANA, = SPaBUTA-8/aRa : ५ les K Q R, K Q’ p’ again, by tho triangles p’, ran K Q: sin Q BR = ain K Q’: sin p’? तः} here, sin K Q = oosine of declination and sine K Q’ = R, RB xX sinQR ००७ sin ४ Q’ = ‘i cos of declination. If both of these times thus found, be of the elevation or both of the depres- sion, the planet will bo elevated above or depressed below the horizon in the time equal to their sum, and if one of these be that which the planet takes for its elevution and the other for its depression, the planet will bo elovated ubove or depressed below the horizon in, the time equal to their difference as the remainder is of the time of elevation or of that of the depression. The sum or , difference of the two times just found is called the resulted time of the Drix- KaRMa in the S’IppHANTas. That point of the ecliptio which is on the castern horizon whon the planot reaches it, is called the UDaya La@wa rising horoscope of the planet. As it is necessary to know this UDayA La@Na for finding the time of the planct’s rising, we are now going to show how to find the rising horoscope. If the planet is depressed by the resulted time above mentioned, it is evident that whon the planct will como to the castern horizon, its corresponding placo in the ˆ —s: BIBLIOTHEOA INDICA; A COLLECTION OF ORIENTAL WORKS PUBLISHED UNDER THE- PATRONAGE OF .THE pt $^. Hon, Court of Directors of the Last Envia Contpany, ‡ AND THE SUPERINTENDENOR OF THR ^. / ve et Y i ak ee ASIATIC SOCIETY OF BENGAL, 1. | eB (2 \ (र hs eee ae! त्र ¥ € P = = । < ~+ ‡ + 4 ट # (4 : १ a ” eb ‘ ‘x > > ९. SS 1 # > + oh ४. rea i om 5. SANSKRIT WORKS IN PROGRESS tani a ee 3 5 ON पय OLDS SERIES; as ८) 91; The ExrxMryts or Portry, by KAmanpaxrt ५ Edited by; 4 ६ 1 .(गदप77.41.414 Mitra, Already published, Fascioulus 1. being No. 19. ee ५८.५2.६8 ne CRMC + ay +^ 1; > >: गष 1, The Laxira 1874144, or Memoirs of the Lifé’and Doctrines of SAkYA SinwA, Edited by Babu RAsknpRratdta Mrrrar Already’ published,’ Faséiouli 1, II 717 IV. and ४, Nos 51 73 1438 144 and 145. ५५; बै; wv ४ ue Sh i ae wa A ‘ The Prikrita Grammar of Kramadfs’wara.’ : 10:66 by Babu RAsenpratira Mitra. + PAK ant AT. L9G SA 4.14, Tat | । । i hoe RM PERE TE BE कः An English translation of the Cunhxpoo¥a UPantsnan of the Sama Veda, by Babu RAsenpranAnsa Mirra, ` Already published, Fasciculus I, Noi 78; 0 (< The ए 7५ Su’rras, Commenced by Dr. 1687, and dontinued by Pandita 16018 Narayana Vidyaratna, Published, Fasoi os ठ Illy N 4 wel ‘~ ` | io? ¥ ध iy *s . and BIB 0 प (4 क; The Tarrrrefya Sannrri of the Black Yajur-Veda d by Dr... ष, Seta, = and E. B. (20 षत, M. Ai ` Published, Faseiculi 1. 11.11.18; ए ^ १) } 11. VIII. eae v IX. >. XI, XII. and XIII, Nos. 92, 117, 11 a la 91, 189, 34.137, 149, 167; 4. ^ 160, 161, 166 and 171. _ . (6 ९ 9 Sec Nea Py ty Sth st ४ पै Loree sa The TairtmriyA BrinmAna of thé i Voda, १५४ | i ५/9 Mibu ~. RAsENDRALALA Mirra. Published, Faacicull aut IViV. VI. VIL 7111. and | rh er: AX. Now: 126, 126, 147, 160, 161, 168, 169164१1. oe The Ma‘RxANDBYA Puna’NA, Edited by the Rey. K; M: published, Fasciculi J, 1I, 11], and 1V, Nos, 1 14, 127 AG ahd १4 Bal Bik be An English Tratslation of the SAuiryA Darrana by Dix Batxaniyya. 1 ४ ह ae TTS Ey et "9४ * For a list of thé Persian and Atabio works in progress, see ध 180 of the 2%). Bibliotheca Lidica, ण ¢ 84 (4 धः ५५ च“ । `, ` . Price. a “of ‘the’ Sanhita’ of ‘the Rig Véda,. । ४७२४ Edited by Dr E. ठ्न, — नी ate 9 ०5. 9९ se ee ee ee १५१५ 4 0 0 । 2 8 0 panishad, with the Commentary | wf of S’ankara,Achérya,> and ‘the Gloss .of Ananda Giri, $ ४२४; Edited by Dr. 7. 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Fi 7 8 0 Pe franslated by J iF . 141 ५११४११११ ४१०११. 000 ५ ] 140 Epitome of .the क. -4 । all, ११५०६१७१ ०६ ०१५०००७ 000 2 80 ts Com- न ॐ # °, ,. ११४ ८०६६ ५१११११११ १५१११५ 000 #1140 = ३ न= 9" > क en ११ ere? ht oe So) ae ) "छे as ous ५ Digitized by G O OS le ° BIBLIOTHECA INDICA; COLLECTION OF ORIENTAL WORKS| | ASIATIC SOCIETY OF BENGAL. | New Seniss, No. 28. { ~ eet) de ४। ४ [ । 2 [1 ११८ "(द ee EE ~~~ र्व व्य | é | | ष | > = -~ * 5 eh, ©. 5 तोति es NS Unis As 7 ro = अ= > (=) 4 न द." र" त्र ey A | ष HINDU ASTRONOMY II. THH SIDDHANTA &IROMANI. TRANSLATED FROM THE SANSKRIT BY THE LATE LANCELOT WILKINSON, ESQ,, 0. 8. AND Revisep By Panpit Bed Deva S’dstefh, UNDER TIE SUPERINTENDENCE OF TUB VEN'BLB AROHDEACON FASOIOULUS II, CALCUTTA : PRINTED BY 0. B. LEWIS, AT THE BAPTIST MISSION PRESS. 1862. ~erw PPR PPPOE PPD PPL PPL PP LPP PPP PLD PP PLP LPP PDP PPL PD PP PPP PPP I PPI AAA AAAI AAPL PALA AAPA AANA LAA A OT PPP SB SANSKRIT WORKS PUBLISHED, 1N THE NEW. SERIES. The Vais’eshika Sitras, with Commentarics, by Pandita Jaya Narayana Tarkapanchanana. Complete in five Fasc. Nos. 4, 5, 6, 8 and 10. The Sandilya Sitras with Swapnes’wara’s Commentary. Edited by Dr. J. R. Ballantyne, LL. D. Complete in one Fase. No. 11. The Kaughitaki-Brshmana Upanishad with S‘ankarénanda’s Com- mentary, edited with a translation by E. B. Cowell, M. A. Complete in two Fasciouli, Nos. 19 and 20. A translation of the ath a Siddhénta and Siddhadnta S‘iromapi, by Pandita B&pi Deva Sastri, under the superintendence of Arch- deacon Pratt. Nos, 1, 18 and 28. (Se SANSKRIT WORKS IN PROGRESS. The Das’a Rapa with the exposition of Dhanika, Edited by ए. E. Hall, D. C. L. Fase. I. II. Nos. 12, and 24. ae पि Pancharétra. Edited by Rev. K. M, Banerjea, Fase. » No. 17, ८ IX. 11. Siddhénta-s'iromoni. 201 vided by the radius becomes [nearly] the srasuta or rectified latitude, [i. e. the arc of the circle of declination intercepted between the planet’s corresponding point in the ecliptic and the diurnal circle passing through the planet]. This rectified latitude is used when it is to be applied to the mean declination and also in the AKSHA DRIKKARMA.* = 11. Tho colostial latitude is not reduced by Branmaaurr ecliptic will be elevated above it by the resulted time. For this reason, having assumed the corresponding place of the planet for the Sun, find the horoscope by the direct process through the resulted time and this will be the rising horoscope. But if the planet be elevated above the horizon by the resulted time, its corresponding place will then be depressed below 1४ by the same time when the planet will come to it. Therefore, the horoscope found by the indirect proccees through tho rosulted time $ will bo the rising [व of the planet. That point of the ecliptic which is on the eastern horizon when the planet comes to the western horizon, is called the asta LaGna or setting horoscope of the planet. As it is requisite to know the setting horoscope for finding the time of setting of the planet, we therefore now show the way for finding the setting horoscope. If the planet be depressed below the western horizon by the resulted time, it is plane that when the planet will reaches it, its corresponding place will be elevated above it by the resulted time and consequently the corresponding place of the planet added with six signs will be depressed below the eastern horizon by the same time. Therefore, assume the corresponding place of the planet added with six signs for the Sun and find the horoscope by tho indiroct process, through tho resulted time and this will bo tho asta LAGNA sotting horoscope. But if tho planot bo depressed bolow the westorn horizon, its corresponding place addod with six signs will then be elevated abovo the castorn horizon by the resulted time and hence tho horoscope found by tho direct process will thon be tho asta LAGNa ectting horoscope. Now tho time p’ Q’ which is determined above through the nine’? f R Q, is not the exact one, because, in that triangle the angle ॐ Q ‰ 18 assumed equal to the co-latitude of the given place, but it cannot be exactly equal to that, and consequently the time p’ Q’ thus determined cannot be the exact time. But no 0 erable error is caused in the time p’ Q’ thus found, if the latitude be of a planet, as it is alwaye small. As to the ster whose latitude is considerable, the ave प Q’ thus found cannot bo the exact time. The exact time can bo found as follows. Beo the preceding figure and in that take R for a star and p the intoreccting point of the ecliptic, and the circle of declination passing through the star R then p p’ is called the mean declination of the star, R p, the rectified latitude and Rp’ the rectified declination. Now, find the ascensional difference E p’ through the mean declination p p’ and the ascensional difference E Q’ through the rectified declination R p’ or QQ’. Find the difference between these two ascensional differences and this difference will bo oqual to p’ Q’i.e. B Q’—E p’ = p’ Q’. But it occurs then when p and R are in the same side of the equinoctial F G and when p is in one side and R in the other of the equinoctial, it is evident that p’ Q’ in this caso will be equal to the sum of the two ascensional differonces.—B. D. | ^ * This rule is admitted by BufsxanMonf{rya to be incorrect; but the error being small, is neglected. Instead of using the pyusy4, the yasuy1 should have been adopted. 0 202 Translation of the (IX. 12. 7 and other early astronomers to its value Omission of the last , : : ६ mentioned correction or ree 11) declination: and the reason of this duction of Celestial latitude «+ ह + to its value in declination, O™Mssion, seems to have been its 1 ४०५ =gmallness of amount. And also it is the uncorrectod Iatitnde which is used in finding the half duration of the eclipses and in their projections &c. 12. As the constellations aro fixed, thoir latitudes as given in the books of these early astronomers are the srasHaf- 84148, i. ©, tho reduced values of tho latitudes so ४8 to rendor them fit to be added to or subtracted from the declination ; and the paRovas or longitude of these constellations are given, after being corrected by the AYANA DRIKKARMA 80 as to suit those corrected latitudes that is, the star will appear to rise at the equator at the samo time with longitude found by the ~ correction. Let ad be equinootial and P the cquinoctial pole, @ b = Ecliptio, b $ = Celestial lutisude, 2 o = Celestial latitude reduced to its value in declination is KoT!, $ © == BHUJ: being aro of diurnal circle ¢ s g ¢ © == & 2 portion of diurnal circle of the planct’s longitude at 5. : The triangle so b or s 2 18 assumed to be a Dia- इ 41414454 THYABRA. The engle ® © o = AyaNa-vaLaNa or the angle of the inclination of s& which goes to ecliptio pole with dc ‘which goes to equinoctial + ~ ae ole. Hence this (0 ¢ © is called DIG-VaLANasé < - कृष्ट र8४८, the angle sc varying with the Ayana-vatawa. 16 were at the 196 Cancer, then the north line a © ¢ which goes to the pole would go also to the a tio polo, ence the aSPAsuTA s8a’Ba, and spasuya 8474 of a star of 90° of latitude being both represented by 5 ¢ would be the same. ‘Lo tho longitude of ॐ stur being 270°, ite aspasufa and spasHTa Bika would be tho same.—-L. W. (The rule stated in this verse is founded upon the following prise le. Assuming the triangle s @ ० as a plane right-angled triangle an the ange $ 2 ०, a8 the deolinatiun of the point of the ecliptic three sigue in advance of the plunet’s corresponding place, because this declinatiun is nearly equal to the AYANA-VALANA, We havo, sinsob:cossbe==bs:b0e; or ह ; yasHtt or nearly the cusine of the declination of the planet’s place 9°0 -+ = Celestial latitude : rectiflud latitude.—B. D.) IX. 18.] Siddhdnta-s'iromont. 203 13. ‘Those astronomers, who havo mentioned that celestial वाः latitude is an arc of a circle of de- Brta’8KARA CHARYA € ae . poses the incorrect theory clination, are stupid. Were the ce- of certain of his [6] latitude nothing more than an tice which is irreconcilable rc of 9 circle of declination, then why with their own theory. should they or others have ever had recourse to the AYANA DRIKKARMA at all? (The planets or stars would appear on tho six o’clock line at the timo that the corresponding degree of the ecliptic appeared there.) 14. How moreover have these same astronomers in deline- ating an eclipse marked off the Moon’s latitude in the middle of the eclipse on sPASHTA-VALANA-8UTBA or on the line denoting tho secondary circle to the ecliptic? and how also have they drawn perpendicularly on the vaLANa-sUTRA or the line repre- senting the ecliptic, the latitudes of the Moon at the com- mencement and termination of the eclipse. 15. How moreover, have they made the latitude KoT!,1i. e. perpendicular to the ecliptic and thus found the half duration of the eclipse? If tho latitude were of this nature, it would never be ascortained by the proportion (which is used in finding 1४). 16. A certain astronomor has (first) erroneously stated the Censure of the astrono- DRIKKARMA and VALANA by the versed न erronoously uscd sine. This course has been followed KARMA and VALANA. by others who followed him like blind men following cach other in succossion: [without scoing their way]. ` 17. Bransacurta’s rule, however, is wholly unexceptionable, but it has been misinterpreted by his followers. My observations cannot be said to bo presumptuous, but if they are alleged to be 80, 1 have only to request able mathematicians to woigh thom Praise of BRAHMAGUPTA. with candour. 18. The pgixkarma and vALANA found by the former astro- o 2 204 Translation of the [IX. 19. nomers through the verscd sine are erroneous: And 1 shall now give an instance in proof of their error. 19 and 20. In any place having latitude less than 24° N. An instance in proof of Multiply the sine of the latitude of the the error. place by the radius and divide the product by the sine of 24° or the sine of the obliquity of the ecliptic and take the arc in degrees of the result found. And find the point of the ecliptic, the degrees just found in advance of the 18४ Aries. Now, if from this point the planet’s corresponding point on the ecliptic three signs backwards or forwards, be on the western or eastern horizon respectivoly, then the ecliptic will coincide with the vertical circle, and the horizon will consequently be secondary to the ecliptic. Hence the planet will not quit the horizon, though it be at a distance of extreme latitude from its corresponding point in tho ocliptic [which is on the horizon], as the celestial latitude is perpen- dicular to the ecliptic.* 21. In this case the resulted times of the DRIkKaRMA being of exactly the same amount but one being plus and the other minus, neutralize each other [and hence there is no correc- tion]. Now this result would not be obtained by using tho versed sine—hence let the nght sine (as prescribed) be always used for the DRIKKARMA. © (It is evident that the longitude of this point is equal to the aro through which it is found, and as the point of the ecliptic 8 signs backwards or forwards from this point is assumed on the horizon, this point therefore will at that time be the nonagesimal, and as the longitude of that point or nonagesimal is less than 90° the declination of this point will be north. This declination equals to the latitude in question. For R X asin latitudo sin 24° ० Tho sine of the latitude of the point = (by the as- sumption) * ein latitude = clination. ० The declination of that point or no 0 equal to the latitude of the place. And 1169008, if the latitude be north the nonagesimal will be in the zenith. For this reason the ecliptic will coincide with the vertical circle.—B, D.] sin 24० >€ sin longitude of the point 9 but this = ein de- Radius IX. 26.| Siddhanta-s'iromont. 205 22. Again here, in like manner, it is from the two VALANAS having different denominations, but equal values, that they mutually destroy each other. By using the versed sine, they would not have equal amounts, hence the vaLanas must be found by the right sine. [In illustration of the fact that the vaLana does not corre- spond with the versed sine, but the right sine BuaskardcHARYA gives as an example. | 23. When the Sun comes to the zenith [of the place where the latitude is less than 24°], and consequently the ecliptic coincides with the vertical circle, the sPASHTA VALANA then evidently appears to be equal to the sine of the amplitude of the ecliptic pomt 90° in advance of the Sun’s place in the horizon. Ifyou, my friend, expert in spherics, can make the SPASHTA VALANA equal to the sine of amplitude by means of the versed sine, then I will hold the vanana found in the Dafvaip- DHIDA TANTRA by Latta and in the other works to be correct. [To this Bufsxaracudeya adds a farther most important and curious illustration :] 24. In the place where the latitude is 66° N. when the Sun at the timo of his rising is in Ist Aries, 18४ Taurus, 1st Pisces, or in Ist Aquarius, he will then be eclipsed in his southern limb, because the ecliptic then coincides with the | horizon. Therefore, tell me how the 8748 प्त {^ VALANA will be equal to tho radius by means of the versed sine |! [In the samo mannor the DRIKKARMA calculation as it depends on the vaLANA, must be made by the right sine and not by the versed sine and for the same reasons. | 25. Even clever men are frequently led astray by conceit Cause of error ३ 1,८7.4 12 their own quick intelligence, by anc chars abated. their too hasty zeal and anxiety for distinction, by their confidence in others and by their own negligence or inadvertence, when it is thus with the wise, | what need I say of fool? others, however, have said :— 26. Those given to the service of courtezans and bad poets, 206 Translation of the [X. 1. are both distinguished by their disregard of the criticisms and reflections of the world, by their breach of the rules of time and metres, and their destruction of their substance and of their subject, being beguiled by the vain delight they feel towards the object of their taste. End of Chapter IX. called DrixkaRMA-VASANA. CHAPTER X. Called S'ntnaonnati-vdsani in explanation of the cause of the Phases of the Moon. 1. This ball of nectar the Moon being in contact with rays of the Sun, is always illuminated by her shinings on that side turned towards the Sun. ‘The side opposite to the Sun dark as the raven black locks of a young damsel, is obscured by being in its own shadow, just as that half of a water-pot which is turned from the Sun, is obscured by its own shadow. ; 2. At the conjunction, the Moon is between us and the ` Sun: and its lower half which is then visible to tho inhabit- ants of the earth, being turned from the Sun is obscured in darkness. ` That half again of the Moon when it has moved to the distance of six signs from the Sun, appears to us at tho period of full Moon brilliant with light. 3. Draw a line from the earth to the Sun’s orbit at a distance of 90° from the Moon, and find also a point in tho Sun’s orbit (in the direction where the Moon is) at a dis- tance equal to that of the Moon from the earth. When the Sun reaches the point just found, he comes in the line per- pendicular at the Moon to that drawn from tho earth to tho ` Moon. Then the Sun illumines half of the visible side of the X. 5.1] Siddhdnta-s'tromoni. 267 Moon. That 18 when the Moon is 85°. . 45’ from the Sun east or west, it will appear half full to us.* 4, Tho illuminated portion of the Moon gradually increases as it recedes from the Sun: and the dark portion increases as it approaches the Sun. As this sea-born globe of water (the Moon) is a sphcre, its horns assume a pointed or cusped Qppearanco (varying in acutcucss according to its distance from the Sun). 5, (To illustrate the subject, a diagram should be drawn Diagram for illustrating ४8 follows). Let the distance north and the subject. south betweon the Sun and Moon re- present the BHosa, thé upright distance between them the Koti and the line joining their centres the hypothenuse. The Sun is in the origin of the BHUJA which stretches in the direc- tion where the Moon is, the line perpendicular at the end of the BHUJA 18 KOTI at the extremi‘y of which is the Moon and the line stretching (from the Moon) in the direction of the Sun is the hypothenuse. The Sun gives light (to the Moon) through tho direction of the hypothonuso. @ This is thus illustrated. Let a repre- sent the Karth, 6 ९ द the orbit of the Sun, e f = do. of the Moon. . Then it is ob- vious that half of the sido of the Moon visible to us will be illuminated whon the Sun 18 at cand not at d, when the Sun 18 at d it will illumine more than half of the Moon's disco; ¢ c is less than a quadrant by the arce d, the sine of which (=) aeoreg in terms of the radius of the Sun’s orbit, equals to the Moon’s dis- tance from the earth. L. W. [The arc ¢ © can be found as follows :— In tho trinngle क ee right angled ate, ae = 61666 yosanas, ac == 689377 — @' g @ YOIaNAS according to the Sippua’sTas. ae 51566 ; ध Then, cos ८ ¢ ८ = - == ——— = , 0748 = cos 8०५० .. 43 : ac 689377 ० arc bc = 85?.. 45’ ncarly.— ए. D.] 208 Translation of the [X. 5. [For instance Let 8 be the Sun and m the Moon, then a S = sauja, am = Kofi,m § = hypothenuse. Then f g a line drawn at right anglos to extremity of hypotenuse will roprosont lino of direction of the enlightened horns and the angle h m d opposite to BHUJA will be equal to <~ g mc = the amount of angle by which the northern cusp is elevated and southern depressed,— were the Moon at k, there would be no elevation of either cusp either way. For the hypothenuse will also bisect the white part of the Moon. If the Sun is north of the Moon, the north लाश) of tho Moon is elevated: if south tho southorn cusp. i. W.] {Mr. Wilkinson has extracted the following two verses from the GaNnITADHYAYA. I. When the latitude is 66° N. and the Sun is rising in Ist Aries, then the ecliptic will coincide with the horizon; now suppose the Moon to be in Ist Capricorn, thon it will appear to be bisected by the meridian and the eastern half will be enlightened. | But according to Branmagurta this would not occur, for he has declared that the Kort will be equal to radius in this case whereas it is obviously “nil,” and it is the कणौ which is equal to radius when there is no north and south difference XT. 2.] Siddhanta-siromoni. 209 between the Sun and Moon then the हणा would be equal to the hypothenuse or radius and the BHUJA would be “ nil.” II.* And the Moon’s horns are of equal altitude when there is no BHUJA, whilst they become perpendicular when there is no Kofi. That the एणा and BHusA shall at one and the same time be equal to radius is an obvious incompatibility. But what business have I with dwelling on the exposure of these errors? Braumacupra has here shown wisdom indeed, and I offer him my reverent submission !] 6. I have thus only briefly treated of the principles of the subjects mentioned in the Chapters on Mapuyaaati &c. fearing to longthen my work; but the talonted astronomer should understand tho principlos of all the subjects in completion, becauso thus 18 the result to be obtained by a complete know- ledge of the spheric. End of Chapter X. called S’ginconnati-vasani. CHAPTER XI. Called YanTRADHYAYA, on the use of astronomical instrwments. 1. As minute portions of time elapsed from sun-rise cannot be ascertained without instruments, I shall thoreforo briofly detail a fow instruments which are of established use for this purpose. 2. The Armillary sphere, NApf-vaLaya (the equinoctial), the yasuTi or staff, the gnomon, the ल्वा or clepsydra, the circle, the semi-circle, the quadrant, and the rHALAKa: but of all instrumonts, it is “ wwaxnuity” which is tho best. Object of the Chapter. ® Bufsxankon{rya is here vory severo on BranMaGueta who of all his pre- dcccasors is cvidontly bis favorite, but truth seemed to require this condemn- ation. 11९ at tho samo horo docs justice tv Anya-BuaTra and the author of the Suayva-sippua/nTa, ‘Choy both justly concur in saying there is no एणा in this case.—L. W. P 210 Translation of the [XI. 3. 8 and 4. (This instrument is to be made as bofore de- scribed, placing the Buagota starry sphere, which consists of the ccliptic, diurnal circles, the Moon’s path, and the circles of declination &c. within the KHaAGoLA celestial sphoro, which consists of the horizon, meridian, prime vertical, six o’clock line, and other circles which remain fixed in a given latitude). Bring the place of the Sun on the ecliptic to the eastern horizon: and mark the point of the equinoctial (in the Buagota) intersected by the horizon, viz. east point. Having made the horizon as level as water, turn the BHAGOLA westward till the Sun throws its shadow on the centre of the Earth. The distance between the mark made on the equinoctial and the now eastern point of the horizon will represent the time from sun-rise. 5 and 6. ‘Iho Laana or horoscopo will thon bo found in that point of the ecliptie which is cut by tho horizon. Tuko a wooden circlo and divido its outor rim into 60 aua- qikis: Then place the twelve signs of the ecliptic on both sides, but instead of making ‘each sign of equal extent, they must be made each with such variable arcs ४8 shall correspond with their poriods of rising in tho placo of obsorvation (the twelvo poriods are to be thus marked on cithor side, which aro to bo again each subdivided into two 7101748 (or hours), throoe एड KANAS, into NAVANS’AS or ninths of 3°. . 20’ each, twelfths of 2°. . 10’ and into reins‘ana‘ds or thirtieths. These ore called the 8 ^ ए१^ 864 or six classes). These signs, however, must be inscribed in the inverse order of the signs, that is Ist Aries, then Taurus to the west or right of Aries and so on. Then place this circle on the polar axis of the KHaGoLa at the centre of the Earth (the polar axis should be elevated to the height of the pole). Now find the Sun’s Jongitude in signs, degrees, &c. for tho sun-rise of the given day (by calculation) and find the same dogree in the circle. Mark thoroe the Sun’s place, turn the Use of Armillary Sphere. The Na’pi-vaLaYa. XI. 8.1 Siddhanta-siromoni. 211 circlo round tho axis, so that the shadow of the axis will fall on tho mark of the Sun’s place at sun-rise and then fix tho circle. Now as tho Sun rises, the shadow of tho axis will advanco from the mark made for the point of sun-rise to the nadir and will indicate the hour from sun-rise, and also the Lacna (horos- cope) : the number of hours will be seen between the point of sun-rise and the shadow: and the LAGNA will be found on the shadow itsclf. [Whilo tho Sun goos from east to west tho shadow travels from west to east and henco the signs with their periods of rising must be reversed in order—the arc from W to Laona represents the hour arc: and the Laana is at the word Laana in the accompanying figure.—L. W.] 7. Or, if this circle marked as above, be placed on any axis elevated to the altitude of the pole, then the distance from the shadow of the axis to the lowest part of the circle will repre~ sent the time to or from midday. 8. A auati made of copper like the lower half of a water- pot, should have » large hole bored in its bottom. See how often it is filled and falls to the bottom of the pail of water on which it is placed. Divide 60 auatis of day and night by the quotient p 2 The लानत or clepsydra. 212 Translation of the [XI. 9. and it will give the measure of the clepsydra. (If it is filled 60 times, then the aHafTi will be of one aHafika; if 24 times it will be of one hour or 24 GHafIKAs.) 9. For a gnomon take a cylindrical piece of ivory, and lot it bo turned on a latho, taking caro that the circumferonce be equal above and below. From its shadow may be ascertained the points of the compass, the place of observer, including latitude &c. and times (as has been elsewhero explained). 10. The circle should be markod with 360° on its outor circumferonce, and should 6 sus- pended by a string or chain moveable on the circumference. The horizon or Earth is supposed to be at the distance of three signs or 90° from the point at which it is suspended: the point opposite to that point being tho zenith. 11. Through its centre put o thin axis: and placing tho circle in a vertical plane, so as to catch the shadow of the Sun: the degrees passed over by the axis from the place deno- minated the Earth, will be altitude : 12. And the arc to the point denominated the zenith, will be that of the zenith distance. | Some former astronomers have given the following rule for making a rough calculation of tho time, viz. multiply tho half length of day by the obtained altitude and divide the product ` by the meridian altitude, the quotient will be the time sought. 13. First let the circle be so held or fixed that any two To find the longi itades of Of the following fixed stars appear to planets by the ci touch tho circumforonco, viz. Ma- GHA (a Leonis, Regulus), Pusuya (8 Cancri), Revarf (¢ Piscium) and S‘aratXrak{ (or A Aquarii). [These stars are on the ecliptic and having no latitude, are to be preferred.] Or, that any star (out of the Cuirea or a Virginis Spica &c.) having very inconsiderable latitude, and the planet whose longitude is required and which is at a considerable distance from the star, appear to touch the circumference. 0110100, The omakRa or circle. XI. 17.) Siddluinta-siromont. 213 14 and 15. Then look from the bottom of tho circle along its plane, so that the planet appear opposite the axis ; and still holding it on the plane of the ecliptic, observe also any of tho above mentioned stars. The observed distance between the planet and the star, if added to the star’s longitude, when the star is west, and subtracted when east of the planet, will give the planet’s longitude. The half of a circle is called a ciara or semicircle. The half of a semi- circle is called rurfya or a quadrant. 16. As others have not ascertained happily the apparent time by observations of altitudes in a vertical circle, I havo therefore labourcd myself in devising an instrument called PHALAKA YANTRA, the uses of which I now proceed to explain perspi- cuously. It contains in itself the essence of all our calculations which are founded on the true principles of the Doctrine of the Sphero. 17. I Brfsxara now proceed to describe this excellent instrument, which is calculated to romovo always the darkness of igno- rauco, which is morcover the delight of clever astronomors and is founded on the shadow of its axis: it is also eminently serviceable in ascertaining the time, and in illustrating truths of astronomy, and therefore valued by the professors of that science. It 18 distinguished by having a circlo in its centro. I procecd to describo this imstrument after invoking that bright God of day, the Sun, which is distinguished by the epithets I have above given to the instrument viz. he is eternal and removes obscurity and cold: he makes the lotus to flower and is ever shining: he easily points out the time of the day and season and year, and makes the planets and stars to shine. He is worthy of worship from the virtuous and resides in the centre of his orb.* Somi-circle and quadrant. PuaLaAKa-YANTRA. Addresses to the Sun. * This vorse is another instance of the double entendre, in which even the 214 Translation of the [XI. 18. 18.. Let a clover astronomer make a PHALAKA or board of a plane rectangular and quadrilateral form, the height being 90 digits, and the breadth 180 digits. Let him halve its breadth and at the point thus found, attach a moveable chain by which to hold it: from that point of suspension let him draw a per- pendicular which is called the LaMBA-REKIIA, 19. Let him divide this perpendicular into 90 equal parts which will be also digits, and through them draw lines parallel to the top and bottom to the edges: these aro called sines. 20. At that point of the perpendicular intersected by tho 80th sine at the 80th digit, a small holo is to be bored, and in it is to be placed a pin of any length which is to be con- sidered as the axis. 21. From this hole as centre draw a circle (with a radius of 30 digits: the circle will then cut the 60th sine), 60 digits forming the diameter. Now mark tho circumforenco of this circlo with 60 amaqis and 860 dogrcos, coach deprco boing subdivided into 10 vatas. 22. Let a thin वृषा or index arm with a holo at one end be made of the length of 60 digits and let it be so marked. [The breadth of the end where the hole is bored should be of one digit whilst the breadth of the whole patfiKs be of half digit. Let the rafrrKi be so suspended by tho pin above mentioned, that one side may coincide with the LaMBA-REKHA. The accompanying figure will represent the form of the २५ वाह. The rough ascensional difference in 41.43 determined by the KHANDAKAS OF parts, being divided by 19, will here become the sine of the ascensional difference (adapted to this instru- ment.*) best authors occasionally indulge. All the epithets given to the instrument apply in the original also to the Sun. This kind of duuble meaning of course १०९४ not admit of translation.— L. W. @ The sines of ascensional difforence for each sign of tho ecliptic were found by the following proportions. XI. 26.1 Siddhdnta-svromoni. 215 23. ‘The numbors 4, 11, 17, 18, 18, 5 multiplicd soverally by the aksHa-KakyA and divided by 12, will be the KHANDpAKAS or portions at the given place; each of these being for each 15 degrees (of sHusA of the Sun’s longitude) respectively. 24. Now find the Sun’s true longitude by applying the precession of the equinoxes to the Sun’s place, and adding togother as many portions as correspond to tho एम of the Sun’s longitude sbovo found, divide by 60 and add tho quo- ticnt to AKsHA-KARNA. Now multiply the result by 10 and divide by 4 (or multiply by 24). Tho quotient is here called the yasHT! in digits and the number of digits thus found is to be marked off on the arm of the एकवृष counting from its hole penctrated by the axis. 25. Now hold the instrument so that the rays of the Sun shall illuminate both of its sides (to secure its being in a vertical circle) : the place ig the circumference marked out by tho shadow of the axis is assumed to be the Sun’s place. 26. Now place the index arm on the axis and putting it over tho Sun’s placo, from the point at tho ond of tho yasirf! set off carefully above or below (parallel to the Lampa-REKHA) on the instrument, the sino of the ascensional difference above found, setting it off above if the Sun be im the northern ^ sign or 2 or 8 signs, give. or ee 12: Panabya : Kusya’ of 1, ४ or 8 "6 2. If cosino of declination : this result : : what will radius ; sine of ascon- sional diflvronco in ऋ ^1,48. + The aro of this will give ascensional «difference. This is the plain rule: but Bra’sxana cua’eya had recourse to another short rule by which the ascensional | differences for 1, 2 and 8 signs, for the place in which the 2414987 6“ was 1 digit, were 10, 8, 8} Patas. These three multiplied by PaLaBHa’ would give the ascensional differences with tolerable accuracy for a place of any latitude not having 9 greater PaLabita’ than 8 digits. Now take these three PaLATMAKAS 10, 8, 8 and multiplied by six, then the Pazas of time will be reduced to asus. These are found with a radius of 3438 : to reduce them to the value of a radius of 80 digits say, 60 x 30 As 8438: 10 x 6 = 60’: : 80 digits : 1. If cosine of latitude : sine of Int. : s what will sine of declination of 1 = quantity of omara for 1 sign in this instrument, but instead of multiplying tho 10 by 6 >< 80 or 180 and dividing by 38488, the author taking 180 = + part of 3438, divides at once by 19.—L. W. 216 Translation of the [XI. 26. hemisphero, and below if it be in the southern hemisphero. The distance from the point where the sine which meeting the end of the sine of tho ascensional differonce thus sot off, cunts the circle, to the lowest part of tho circlo will represent tho anaTis to or after midday.* ® In the accompanying diagram of the ruaLaka Yantra, 0 is the centre of the circle a © ० and the ‘line ० *# passing through o is called MaDHYasYa’ or middle sine. If the shadow of the pin touches the circumference in 8 whon the instrument is held in the vortical circle passing through tho Sun, 8 3 will then be the zenith distance ofthe Sun, From this the time to or after midday can bo found in tho following manner, Lot a = altitude of the Sun, d = declination, A = ascensional difference, i= north latitude of the place, ॐ = degrees in time to or after midday. Thon, we have the equation which is common in the astronomical works, °. sine = BR. sind. sind 008 p = 3 cos 2. 008 ध B 3. sina tanZd.tand ER EEL 3 coe ¢ , cos d R here, when the latitude is north, the second term becomes minus or plus as the declination is north or south respectively. tan ~.tand But ——— = ain A or sine of ascensional differonce. B® . sina es cos P Se eee 0०७ ¢ , cosa + sin, A. TS: etic on oo ee Se eee er Oe Oe ee eae XI. 27.] Siddhdnta-s'iromani. 217 27. Set off the time from midday on the instrument To find the placo of the Counting from the LAMBA-REKHA ; from ehadow of axis from time. the end of the sine of this time, set off the sine of ascensional difference in ® line parallel to the Now, cos a R= ४ # 1. 6, aksHaxaRna (See Chapter VII. v. 45.) or —— ome —nwe cos J 12 28 h RBR.sine ० COS p =——, sin A 12 cos व sin @ h RB? --- + sin A, when y == ——. » which is called 12 YasuTI and can be found as follows. 9 R h 12R gy _—_-, = 12 cosd 12 12 coed । R h 12 versed <== ~---- . 7 12 ~+ ~ 12 12 cos d F ; Madi BuvJa of the Sun’s lorgitude is 15, 80, 45, 60, 76, 90, the value of 2 verse is 4, 15, 32, 60, 68, 68 sixtieths respectively. The differences of cos @ ॥ these values are 4, 11, 17, 18, 13, 5 which are written in the text. Multiply those difforenees by 4 or tho aAKsitaKanna, divido tho products by 12 and the quotionts thus found aro called the xmanpas for the given place. By sssuming tho एर + of the Sun’s longitude as an argument, find the result through the KIlANDAS and tako r for this rosult, e e But in this instrument R == 80 10 r = ys § ^+ + —) which exactly coincide with the rule given in the 4 0 text for determining the yasutt. The value of the yasutr will certainly be more than 80, because the value of tho aKsHAKARWA or & is more than 12, Now, (see the diagram) suppose m is the end of the yasnt1 in the PayfIKA or indox ० m which touches tho circle in 8, then, in tho trianglo o mn Riom=sinmon: mn; or RB: y=sine 2M; y Kaine ००० पह 8 == — ;5 R and hence, cos p = mn + sin A, Q 218 Translation of the (XI. 28. LAMBA-REKHA, but below and above according as it was to bo set off above or below in finding the time from the shadow, (this operation being the reverse of the former). The sine met by the sine of ascensional difference, thus set off, is the new sine across which the PATTIKX or indox is now to bo placed till the yasHpI-cnhINHA or point of yasnfr falls on it. This position will assuredly exhibit the place of the shadow of the axis. : 28, 29 and 30. Having drawn a circle (as the horizon) with a radius oqual to radius of a great circle, mark cast and west points (and the line joining these points is called the pRAcHYAPARA or east and west line) and mark off (from them) the amplitude at the east and west. Draw acircle from the same centre with a radius equal to cosine of doclination i. o. with a radius of diurnal circle, and mark this circle with 60 anatis. Now tako tho yasiqi, oqual to the radius (of tho grout circlo) and hold it with its point to the Sun, so that no shadow be reflected from it; the other point should rest in the centre. Now measure the distance from the end of the amplitude to the point of the yasHt! when thus held opposite to the Sun. ‘This distance applied as a chord within the interior circle will cut off, if it be before midday, an arc of the number of aiaqikds from sun- rise, and if after midday an arc of tho timo to sun-sot.* The Yasurt or Staff. that is, the sine of the ascensional difference is subtracted from or added to ms the distance between the end of the xasuy1 and tho middle sine, as the Sun be in the north or the south to the equinoctial. Again, by taking m r equal to sin A we havo, csprma ty 80) & == mat mr, =nsrortt’, = 009 0 $, ४ । 2 ==> @ é—B. D.] * [It is plain from this, that the distance from the point of the staff to the end of the amplitude is the chord of the aro of the diurnal circle passing through the Sun, intercepted between the horizon and the Sun. For t his reason, the arc subtended by the distance in quostion in this interior circlo doscribed with a radius of the diurnal circle which is equal to the cosino of the declination, will denote the time after sun-rise or to sun-sct.—B. D.] XI. 33.] Siddhdnta-s'iromani. 219 31. Tho perpendicular 1९6४ fall from the point of tho yasuti To find the patanua with 18 the 8 ANKU or sine of altitude: the tho rasuqt. place betwoen the s’ANKU and contre is equivalent to DRIGYA or sine of zenith distance. The sine of amplitude is the line between the point of horizon at which the Sun rises or sets, on which the point of the र+या will rost at sun-rise and sun-set, and the east and west line the PRACHYAPARA, | 82 and 33. The distance between the s’anKu and the UDAYASTA-SUTRA, multiplied by 12 and divided by the s’anxu, will be the PALABHA. | Take two altitudes of the Sun with the yasuTr: observe the s’anKus of the two times and the BHUJAs. Add the two प्रण ॥8, if one be north and the other south, or subtract if they be both of the same denomination: multi- ply the above quantity (whether sum or difference) by 12 and divide by the difference of the two s/ankus, the result will be the ra.aBHi.* ‘The differenco between the east and west line and tho root of 8 anku is called Brusa. # [Let O bo tho cast or west point of tho horizon O a, 2 the zenith, a eS the Kal diurnal circlo on which 8 and ¢ aro the Sun’s two places at different times and 8 # ond + # {16 8. ANKUS or tho sinos of altitudes of the Sun, then O m, an will be 1110 puvsas, # m or ॐ p tho difference between the Buusas and 8 p the diflerence betwoen the s’aNkUS. ५४ 220 Translation of the (XT. 34. If the s’anxu be observed three different times by the To find patanua’, decline YASHTI, then the time, declination &c. 1 may be found (by simply observing observations by the Yasuqi, of three 6’aNxus, the Sun). 84. First of all find three s‘ankus: draw a line from theo top of the first to the top of the last ; from the top of the second g’ANKU, draw a line to the eastern point and a line to the western point of the horizon, so as to touch the first line drawn. 35. A line drawn so as to connect these two points in the horizontal circumforenco will be tho upaydsra svrra. ‘Tho distance between it and the centro will give the sine of ampli- tude. The line drawn through the centre parallel to the upy- asTa-s0TrRa at the distance of the sine of amplitude is the east and west line.* 80. Find the 741. ए as before (and also tho aKsHa- KARNA). Now the sine of amplitude multiplied by 12 and divided by axsHa-KARNA will be the sino of doclination. This again multiplied by the radius and divided by the sine of 24° or the sine of the Sun’s greatest declination, will give the sine of the pHusa of the Sun’s longitude. 87 and 38. Which converted into degrees is Sun’s longi- tude, if the observation shall havo boon mado in the Ist quarter of the year. If in the second quartor, the longitude will be found by subtracting tho degroos found from 6 signs: if Now as the triangles = ठ » and S क = are the latitudinal triangles, the trjangle 8 ¢ p is also the latitudinal ० Sp: ®? == 18 : ParaBHat 12 ep ०, Patabua’ = It is when 8, * two “ae of the Sun are both north or botl) south to tho prime vortical, but when one place is north and othor is south, the sum of the BuUsas is taken.— B, D. * [As it is plain that ४ © tops of the three s’anxkus are in the plane of the diurnal circle, the line therefore drawn from the top of the first s‘anxu to that of the last, will also be in the samo plane and hence tle two lines touching this line, drawn from the top of the middle s‘anxu one to eastern and the other to western point of the horizon, lie in thig plane. Therefore, the line joining these two points of the horizon is the intersecting line of the plane of tho diurnal circle and that of the horizon, and consequently it is the upara’stTa 30794. B, D XI. 41.] Siddhdnta-s'tromani. 221 in the 3rd quarter, 6 signs must be added: if in tho fourth quarter of the year, then the degrees found must be subtracted from 12 signs for the longitude. The quarters of the year will be known from the seasons, the peculiarities of each of which I shall subsequently de- scribe. It is declared (by some former astronomers) that the shadow of the gnomon revolves on the circle passing through tho ends of the three shadows made by the same gnomon (placed in tho contro of the horizon), but this is wrong, and conse- quently the east and west and north and south lines, the latitudes &c. found by the aid of the circle just mentioned are also wrong.* 39. Whether the placo of the Sun be found from tho shadow or from tho sino of tho amplitude, it will be found corrected for precession. If the amount of precession be subtracted, the Sun’s true place will be found. If the true place of the Sun be subtracted, the amount of precession will bo ascertained. 40. But what does a man of genius want with instruments 1 about which numerous works have called pifYaNTRA or genius treated? Let him only take a staff in व his hand, and look at any object along it, casting his eye from its end to the top, there is nothing of which he will not then tell its altitude, dimensions, &c. if ib bo visiblo, whethor in the heavens, on the ground or in tho water on the carth. . Now I proceed to explain it. 41. He who can know merely with the staff in his hand, the height and distance of a bamboo, of which he has observed the root and top, knows the use of that instrumont of instru- monts—genius—(the pufyantTza) and tell me what is thero that * Tho existence of such gross error in the principles of a calculation as aro hore referred to as oxisting in the works of Baa’sxaka’s predecessors would seem to indicato that the ecience of astronomy was not of more recent cultiva- tion than Mr, Benticy and othore havo maiutained.—L. क्र, 222 Translation of the (XT. 42. he cannot find out. [Here the ground is supposed to be perfectly level. ] 42. Direct the staff lengthways to the north polar star ; let drop-lines fall from both ends of staff, when thus directed to the star. Now the space between the two drops is the Buusa or base of a right angled triangle, when the difference between the lines thus dropped is the Kot! or perpendicular. 43. The Kofi multiplied by 12 and divided by tho पाए gives tho PALABIIA.* Having in the same way observed tho root of tho bamboo ; [and in so doing found the BHusa and Koq!], multiply the BHuUJA by the height of the man’s eye. 44 and 45. And divide the product by the एनय, the result To find the distanco and 18, you know tho distanco to tho rout height of a bamboo. of the bamboo. Having thus observod the top of tho bamboo (with tho staff, and ascertained the pHuJa and णय), multiply the dis- tance to the root of the bamboo by the एणा, and divide the product by the suusa, the result is the height of the bamboo above the observer’s eye: this height added with the eye’s height will give the height of the whole bamboo.t For instance, suppose the staff 145 digits long, the height of obsorvor’s oyo 68 digits; that m making the lower observation the BHUJA = 144 digits = 6 cubits, and एणा = 17 digits ; that in making the observation of tho top of the bamboo, tho muuwa = To find pataBBf, Example. @ 1. 8. Ifthis BaUsA: gives the Koy! : ; 12 digite of gnomon: gives the PaLABHA’, ~~ ¶ Tho obsorver first dircots a 2 his staff to d, the root of the tree: Tho staf XI. 46.] Siddhdnta-siromani. 223 116 digits ond एना = 87 digits. Thon toll me the height of bamboo and the distance of it. As, 68 x 144 17 576 x 87 and —————~ = 432 height of tree above observer's eyo, 116 = 576 digits or 24 cubits distance to bamboo ; 68 add tho oyo’s hoight, 500 height of tree. Let a man, standing up, first of all observe the top of an object: then (with a staff, whether it be equal to the former or not in Iength), lot him observo again the top of the samo object whilst sitting. 46. Then divide the two दणड by their respective BHUJAS: take the difference of these quotients, and by it divide the difference of the heights of observer’s eye—this will give the distance to the bamboo: from this distance the height of the bamboo may be found as before.* 9 kh ई © is furnished at eithor end with drop lines ah, 5k:5k—ah=be= sin of ८. bac. Then say 7 be:ac::be:de=fob, © then observes ths top of object and finds g f, which is easy, as f © has ben found.—L.W. ^ ५ Be eee * BuAsx aka four.ds this rule on the followivg algebraic process. 224 Translation of the (XI. 47. 47. There is a high famous bamboo, the lower part of which being concealed by houses &c. was invisible: the ground, however, was perfectly level: If you, my friend, remaining on this same spot by observing the top (first standing and then sitting), will tell me the distance and its height, I acknowledge you shall have the title of being the most skilful of observers and expert in the use of the best of instruments DHfYANTRA. The observer, first standing, observes the top of the bamboo and finds tho पाए, withtho first staff, to be 4 cubits or 96 digits : he then sits down and finds with another staff the BHusa to be 90 digits. In both cases the णका was one digit. Tell me, O you expert in observation, the distance of observer from the bamboo and the bamboo’s height. 48. So also the altitude may be observed in the surface of smooth wator: but in this caso the height of observer’s eye is to be sub- tracted to find the true height of the object:—Or the staff may be altogether dispensed with: In which last case two heights of the observer’s eye (viz. when he stands and sits) will be two Kofis: and the two distances from the observer to tho Question. Example. Observation in water. Let ॐ = base, distance to bamboo. Then say ® ` @ if96:1:: 2: ——: then —— + 72 = height of bamboo. 96 96 @ 2 By second observation 90: 1:: 2: —— , then —— -++ 24 = height of bam- 90 90 boo. @ @ @ 62 Then 72 ~+ —— = 24 ~+ —— ; ~ ~ —— = 48, or —— = 48 90 90 96 .*. & = 69,120 digits = 2880 cubits, 2 (| That is ———- — —— == 72 — 24 90 96 73 ~ 24 or z= that is difference of observer’s height—difference of two KoT1s pomvs divided by their respective buusa‘s.—L. W. AI, 49.] Siddhdnta-siromani. 225 placcs in the wator tvlcre tho top of the object is roflected, the BHUJAS. 49. Having secn only the top of a bamboo reflected in’ water, whether the bamboo be near or at a distance, visible or invisible, if you, remaining on this same spot, will tell me the distance and height of bamboo, I will hold you, though appearing on Earth as a plain mortal, to have attributes of superhuman knowledge. An observer standing up first observes (with his staff) the reflected top of a bamboo in water: The ण्म == $ digits and BHUJA = 4 digits. Then sitting down he makes a second observation and finds the snusa = 11 digits and दण = 8 digits. His eyo’s height standing = 3 cubits or 72 digits, arid sitting = 1 cubit or 24 digits. ‘Tell me height of bamboo and its distance.* a. Question. Example. me eae क mea ~~ ~ ~ ~ ---- -~ - र - -— * Let d f= fo = height of bamboo = 4 6 then 3 @ or y = height of bamboo and man’s height together. Let ९ ८ = breadth of water = x then by 8८86 observation R 226 Translation of the [XI. 49. A man standing up secs the shadow of a bamboo in the water—the point of the water at which the shadow appears is 96 digits off: then sitting down on the same spot he again observes the shadow and finds the distance in the water at which it appears to be 88 digits: tell me the height of the bamboo and his distance from the bamboo.* Example. 49 4:8::z:yor32—=4yo0rz=>—— by 2nd observation 11:8:: 2:9 — 48 digils 11 # — 528 or 8 x = 11 y — 528 or = —-———- 8 4y 11 y — 528 thus « = —— and zs = — 8 ay 11 $ — 528 39 # — 1584 ०", ~-- = ---- or ¢+ ¢ ~~ — 8 or 82 y = 38 y — 1584, or $ = 1689 .*. 1684 — 72 = 1512 dig 1४6 = 63 cubits = height of bamboo. 2ud part. ‘To find width of water or x 4y 1584 > 4 2 == a = ———— = 212 digits = 88 cubits. —L. W. 8 * Let oe = 96 digits cd = 88 oc= 72 9 © == 24 a let ॐ == distance from observer to bamboo. Nowce:ac=jh:ja 720” 82 or 96: 7४ == > : yo=-—=— 96 4 8 2 Thon छ — 8 = height of bamboo Agsned: be::j4:79b ॐ ॐ or 98 : 2 ; : ॐ ¦ # -- 46 = — 83 8 11 thon ए — 1 == height of bamboo XI. 55.] Suldhantu-siromani. 227 50 and 51. Mako a wheel of light wood and in its circum- 1 अ ference put hollow spokes all having ment or swaYaNvalta yaN- bores of the same diameter, and Ict ae them be. placed at equal distances from each other; and let them also be all placed at an angle somewhat verging from the perpendicular: then half fill these hollow spokes with mercury : the wheel thus filled will, when placed on on axis supported by two posts, revolve of itself. Or scoop out a canal in the tire of the wheel and then plastering leaves of the 7414 tree over this canal with wax, fill one half of this canal with water and other half with mer- cury, till the water begins to como out, and thon cork up the — orifice left open for filling the whcel. ‘The wheel will thon revolve of itsclf, drawn round by the water. _ Make up a tube of copper or other metal, and bend it into the form of an anxus’a or elephant hook, fill it with water and stop up Description of a syphon. both ends. 54. And then putting one ond ito a resorvoir of wator, let the other end remain suspended outside. Now uncork both ends. Tho water of tho roservoir will bo wholly suckod up and fall outside. 55. Now attach to the rim of the before described self- revolving wheel a number of water-pots, and place the wheel and these pots like the water-wheel so that the water from tho lowor end of tho tube flowing into thom on ono sido shall sot the whol in motion, iinpelled by the additional weight of the pots thus filled. The water discharged from the pots as they reach the bottom of the revolving wheel, should be drawn 8 8a 8z 82 @ ~~~ — 1 ~~ — — 8or2= ~ ~ ~ <= ~ 11 4 4 11 44 ०, ¢ < 44 ॐ 2=88 82 3 X 88 Then y = - = न्न = 8 x 22 = 66, height of bamboo. 4 228 franslation of tho [XI. 56. off into the reservoir before alluded to by means of a water- course or pipe. 56. The self-revolving machine (mentioned by Latta &c.) which has a tube with its lower end open is a vulgar machino on account of its being dependant, because that which mani- fests an ingenious and not a rustic contrivance is said to bo a machine. 57. And moreover many self-revolving machines are to be met with, but their motion is procured by a trick. ‘They are not connected with the subject under discussion. I have been induced to mention the construction of these, merely because they have been mentioned by former astronomers. = ‘End of Chapter XI. called YanrrApuy(ya. OHAPTER XII. Description of the seasons. 1. (This is the season in which) the xoxttas (Indian black birds) amidst young climbing plants, thickly covered with gently swaying and brilliantly verdant sprouts of the mango (branchos) 11५४ ing their sweet but shrill voices say, “Oh travellers! how are you heart-whole (without your sweethearts, whilst all nature appears revelling) in the jubilee of spring cuatrRa, and the black bees wander intoxicated by the delicious fragrance of the blooming flowers of the sweet jasmine |” | 2. The spring-born MALLIKA (Jasminum Zambac, swollen by the pride she feels in her own full blown beautiful flowers) derides (with disdain her poor) unadorned (sister) MALATY (Jasminum grandiflorum) which appears all black soiled and without leaf or flower (at this scason), and appears to beckon ‘her forlorn sister to Jeaye the grove and garden with her Spring. XII. 7.] Siddhdnta-stromant. © 229 tender budding arms, agitated by the sweet broezes from the fragrant groves of the hill of Manaya. 3. In the summer (which follows), the lovers of pleasure The orisuma or mid-sum- and their sweethearts quitting their = stone built houses, betake themselves to the solitude of well wetted cottages of the kus’ak4s'a grass, salute each other with showers of rose-water and amuse them- selves. 4. Now fatigued by their dalliance with the fair, they proceed to the grove, where Kama-peva has erected the (flowering) mango as his standard, to rest (themselves) from tho glare of the fierce heat, and to disport themselves in the (svell shaded) wators of its nowgss (or large wells with steps). 5. (The rainy season has arrived, when the deserted fair one thus calls upon her absent lover :) Why, my cruel dear one, why do you not shed the light of your beaming eye upon your love-sick admirer? The fragrance of the blooming mA.atf and the tuybid stato of ovory passing torrent proclaims the soason of tho rains and of all-powerful love to have arrived. Why, theroforo, do you not havo compassion on my misorablo lot ?* 6. (Alas, cries the deserted wife, alas!) the peacocks (delighted by the thundering clouds) scream aloud, and the breeze laden with the honied fragrance of the KADAMBA comes softly, still my sweet one comes not. Has he lost all delight for the sweet scented grove, has ho lost his ears, has he no pity—has he no heart ? 7. Such are the plaintive accusations of the wife in the season of the rains, when the jet black clouds overspread the sky :—angered by the prolonged absence of him who reigns over her heart, she charges him, but still smilingly ond sweetly, with being cruelly heedless of her devoted love. Rainy season. ® This is one of those verses in which a double or triple meaning is attempted to be supported : to effect this, soveral letters however aro to be read differently. —L. Ww. 230 Translation of the [XIT. 8. 8. The mountain burning with remorse at the guilt of The skraTKa’La or season having received the forbidden em- of early autumn. braces of his own pusHpavatf daughter, forest appears in early autumn through its bubbling springs and streams sparkling at night with the rays of the Moon, to be shedding a flood of mournful tears of penitence. 9. In the HEMANTA season, cultivators seeing the earth smiling with the wide spread harvest, and the grassy ficlds all bedecked with the pearl-liko dow, and tooming with joyous herds of plump kine, rejoice (at the grateful sight). 10. When the 8087186 season sets in what unspeakable 81781४4 or close of win- beauty and what sweet and endless bs variety of red and purple does not tho ^ kacirndv’ grove uncoasingly present, when its leaf is in full bloom, and its bright glories are all expanded. 11. The rays of tho Sun fall midday on the carth, henco in this 81818 season, they avail not utterly to driveaway the cold : HeManra orearly winter. मै * * मै 12. Here, under the pretence of writing a descriptive account of the six seasons, I have taken the opportunity of indulging my vein for poetry, endeavouring to writo something calculated to please the fancy of men of literary taste. 13. Where is the man, whose heart is not captivated by tho ever sweet notes of accomplished poets, whilst they dis- course on evory subject with refinement and taste? or whoso hoart 18 not enchanted by the blooming budding beautics of the handsome willing fair one, whilst she prattles swectly on every passing topic :—or whose substance will she not secure by her deceptive discourse ? 14. What man has not lost his heart by listening to the pure, correct, nightingale-like notes of tho genuine pocts? or who, whilst he listens to the soft notes of the water-swans on ` Sweets of poetry. XIII. 2.1 Siddhdnta-siromani. 231 tho shores of large and overflowing lakes well filled with lotus flowers, is not thereby excited ? 15. As holy pilgrims delight themselves, in the midst of the streams of the sacred Ganges, in applying the mud and the sparkling sands of its banks, and thus experience more than heaven’s joys: so true poets lost in the flow of a fine poetic frenzy, sport themsclves in well rounded periods abounding in displays of a playful taste. Ind of Chapter XII. CHAPTER XIII. Containing useful questions called pras’NXDHYAYA. 1. Inasmuch as a mathematician generally fails to acquire Object of tho Chaptor and distinction in an assomblego of earnod ite praise. men, unless well practised in answor- ing questions, I shall therefore proposo a few for the entor- tainment of mon of ingenuity, who delight in solving all descriptions of problems. At the 096 proposition of the questions, he, who fancies in his idle conceit, that he has attained the pinnacle of perfection, is often utterly discon- corted and appallod, and finds his smiling chooks dosortod of thou colour. 2. ‘These questions have been already put and have been duly answered and explained either by arithmetical or algebraic processes, by the pulverizer and the affected square, i. e. methods for the solutions of indeterminate problems of the first and of the second degree, or by means of the armillary sphere, or other astronomical instruments. To impress and make them still more familiar and easy I shall have to repeat 8 few. | 233 Translation of the (XIII. 3. 8. All arithmetic is nothing but tho rulo of proportion : Praise of ingenious per- ®nd Algebra is but another name for क ingenuity of invention. To the clever and ingenious then what is not known! I, however, write for men and youths of slow comprehension. 4. With the exception of the involution and evolution of the square and cube roots, all branches of calculation may be wholly resolved into the rule of proportion. It indeed assumes many shapes, but it is universally prevalent. All this arith- metical calculation donominatod Pf aanira, which has beon composed in many ways by the wisest of former mathema- ticians, is only for the enlightenment of simple men like myself. 5. Algebra does not consist in the letters (assumed to represent the unknown quantities): neither are the different processes any part of its ossential propertios. But Algobra is wholly and simply a talent and facility of invention, bocauso the faculties of inventive genius are infinite. 6. Why, O astronomer, in finding the attaraaya, do you add saura months to the lunar months cnaiTRA &c. (which may have elapsed from the commencement of the current year): and tell me also why the (fractional) remainders of Apmimasas and AVAMAS are rejected : for you know that to give a true result in using the rule of proportion, the remainders should be taken into account, Question ist. 7. Ifyou have a perfect acquaintance with the mis‘ka or allegation calculations, then answer this question. Let the place of the Moon be multiplied by one, that of the Sun by 12 and that of Mars by 6, let the sum of these three products be subtracted from three times the Jupiter’s place, then I ask what aro tho revolutions of tho planct whose placo when added to or subtracted from the remainder will give the place of Saturn ? Question 2nd. XIII. 9.1 Siddhdnta-siromani. 233° 8and9. In ordor to work this proposition in tho first place proceed with the whole numbers of revolutions of tho several plancts in the KALPA, adding, subtracting and multiplying them in the manner mentioned in the question: then subtract the result from the revolutions of the planet given: or subtract the revolutions of the given planet from the result, according as the place of the unknown planet happen to be directed to be added or subtracted in the question. This remainder will represent the number of revolutions of the unknown planet in the KALPA. If the remainder is larger than the number from which it is to be subtracted, then add the number of terrestrial days in 9 KALPA, or if tho remnindor excced tho number of torrestrial days in the KaLPA, thon reduce it into the remainder by dividing it by the number of days in the KALPA.* Ruz. न Bra‘sxaBa ona'RYA himself has given the following example in his com- mentary va’SANA-BILA BILYA Suppose Moon to have 4 revolutions in a KALPA of 60 days Sun, 94 ७9 oe ce @@ ee eee ०७ Mara, ५0 648 saws ०००७ 00 Oe 08 08 00 08 00 08 Oe Jupiter, cece 7 0000 00 00 ००७०००७० 00 ७७०००७०० ०७०५ Saturn 9 @ ॐ 9 @ 9 @% ® @ @@ @ @ @ @@ 00 02602008 ® @ ® @ 480 Then + ॐ 1 ~+ 3 9 18 ~+ 6 % 6 == 70० १११7 ‰ 8= 21. As 70 cannot be subtracted from 21 add 60 to it == 81, Subtract 70 remainder ll: let p == revolutions of the unknown planet, then by the question 11 — p == 9 ० 11 -- 9 = 2 == 2 but 11 ~+ 7 == 9 or p= 9 — 11 == 60 + 9 — 11 = 58 It thus appears that tho unknown planet has 2 or 58 revolutions in the KALP Now let us 866 if this holds true on the 28rd day of this 74126 3 revolutions signs for Moon, 160 : 4 :: 28: 6 .. 12°thisx 16 .. 12° 8 : 28: 1 .. 24 thisx 19 == 9.. 18, : 23: 14 .. 0 भह 6==6.-. 0 signs 10... 0 subtracted ee 13 6 this ॐ 8= 0O.. 18 from ee 6 this sub. from 2 .. 18 remainder 9.. 6 € 00 or २ 00 : 9 :: 2: Jupiter, 60:7 :: 2: 60:2 : : 28 corresponding with Suturn, 6 .. 12 234 Translation of the [XIIT. 10. 10. The algebraical learned, who knowing the sum of tho additive months, subtractive days elapsed and their remainders, shall tell the number of days elapsed from the commencement of the KALPA, deserves to triumph over tho student who is puffed up with a conceit of his knowledge of the exact pulverizer called sau’sLisHta united, as the lion triumphs over the poor trembling deer he tears to pieces in play. 11. For the solution of this question, you must multiply the given number of additivo months, subtractive days and their remainders, by 863374491684 and divide by one less than the number of lunar days in a KALPA i. 6. by 1602998999999, the remainder will be the number of lunar days elapsed from the beginning of the 41.24. From these lunar days tho terrestrial duys may be readily found.* Question Srd. - Bore. 8 or if, 60:68 :: 23:2: 24 Then?.. 24 added to2.. 18 still gives Saturn’s place 5 .. 12 When p = 9 — 11, then as 11 cannot be subtracted from 9 the sum of 60 is added to the 9. The reason for adding 60 is thut this number is always be denominator of the fractional remainder in finding the place of the planets ; for the proposition. If days of KaLPa: revolutions :: givon days give: hore tho daya of KALPA aro assumod to bu 00 honce 60 is added.—L. W. * (When the additive months and subtractivo days and their remainders are given to find the aHARGANA. Let 2 = 1602999000000 the number of lunar days in a KALPA. € = 169300000 the number of additive months in a KaLva. @ = 25082550000 the numbor of subtractive days in a KaLPA. A = additive months elapsed. A’ = their remaindor. 7 = subtractive duys clapsed. B’ = their remainder. > a = the given sum of the elapsed additive months, subtractive days and their remainders. । and ॐ = lunar days elapsed ; & thon say As?2?:2@::2:A+4 हन ; B’ Asl:d:: 2:8 + a ; ATI. 12.] + | Siddhanta-siromani. 235 12. Given tho sum of tho clapsod additivo months, sub- tractive days and their remainders, equal (according to BRAILMAGUPTA’S system) to 648426000171; to find the anarcanaA. He who shall answer my question shall be dubbed a “ BRAHMA-SID- DHANTA-VIT” i. e. shall be held to have a thorough knowledge of tho BRAIIMA-8IDDNANTA.* Examplo. A’ + B’ A’ + B’ ० Asl:etd::2:A+B + or ¢ ~+ ^ ७१००4 0 + १०१ + B ० by addition, (e+ dad “ - (८ - 1) $= ^ +B + A+ B’ by substitution, 26675850000 2 — 1602998999999 # =a now let, 26675850000 x’ — 1602998999999 $ then wo shal] have by the process of indeterminate problems 374491684. Again, 16४ m= ८ + ९ १०१ » = ८ - 1 hen MOo—8Y = ०) and ॐ” —ay=1; 4 @ # > -- @ # ‰” =a, and mnt—mnt =0 , # (@ 2” -- # ‡) -- # (८ ॐ -- %# £) == 2 : which is similar to (1) ; : | ० & == ० ---*८ = 868374491684 a — (1 — 1) ६. Hence the rule in the text.—B. D.] ® Solution. Tho givon sum = 648426000171 and ¢ he Junar deys in 8 KALTA = 1602999600000 : 648426000171 2८ 863874491684 ho eee ——— = 849241982886 1602998999999 and 10800 remainder : -. 10300 these are lunar days elapsed T’o reduce them to their equivalent in terrestrial days says 161 _—_subtraotivo Iflunar daysin] | Number of sub- } Lunar days a-({ , days and romain- ® KALPA * tractive days ` 0०१९ found ˆ der amounting 7426000000 From 10800 Lunar days subtract 161 Subtractive days remainder 10139 Terrestrial days or ANARGANA. Now to find additive months clapsed. If lunar days] , additivo months \ , , lunar days] , 10 additive months and in a KALPA | ˆ of KALPA aoe: | remn. 381000000000. 10 additive months = 300 lunar re .. 10800 — 300 = 100,00 sauna days elapsed Henco 27 कः 9 months and 10 days elapsed from the commencement of KALPA.—L. s 2 236 Translation of the ` (ROT, 13. - 13 and 14. Given the sum of the remainders of the revo- lutions, of the signs, degrees, minutes and seconds of the Moon, Sun, Mars, Jupiter, the s’fanrocucuas of Mercury and Venus and of Saturn according to the pufvrippAIDA, including the remainder of subtractive days in finding the anaraana, abraded (reduced into remainder by division) by the number of terrestrial days (in a yuaa). He who, well-skilled in the management of SPHUTA KUTTAKA (exact pulverizer), shall tell me the places of the planets and the anaraaya from tho abradcd sum just mentioned, shall be held to be like the lion which longs to make ita seat on the heads of those elephant astronomers, who are filled with pride by their own superior skill in breaking down and unravelling the thick mazes and ‘wildernesses which occur in mathematical calculations. 15. Ifthe given sum abraded by the number of terrestrial days in a ४०६५, on boing divided by 4, leaves a remainder, then the question is not to he solved. It is then called a KAILA or an “ impos- sible” question. If, on dividing by 4, no remainder remain, then multiply the quotient by 298627203, and divide the product by 894479375. The number remaining will give tho aHaRGaNA. Ifthe day of the week doos not correspond with that of the question, then add this amaraana to tho divisor (394479375) until the desired day of the week be found.* Question 4th. Rous, © [According to the DH{VEJDDHIDA TANTRA Of TaLLa the terrestrial days in a yuGa = 1577917500 and the sum of all the 96 romaindcrs for one day = 118407188600068 : this abraded by the terrestrial days in a yuGda == 259400968 Let 2 = anarGaya then say As 1 । 259400968 59400968 K 2 This abraded by 1677917500 the terrestrial days in a yuGa will be equal to 1491227600 the given abraded sum of the 36 remainders, now let $ = the quotient got in abrading 259400968 ॐ by 16779176500, then 259400968 ® — 1677917500 y = 1491227 It is evident from this that as the coefficients of x and yare divisible by 4, the given remainder 1491227500 also must be divisible by 4, otherwise the question will be impossible as stated in the text Hance, dividing the both sides of the above question by 4, 60242 ॐ — 8944793875 # = 87280687 » (A) aad let 64850242 se 894479876 — 1; © 00 ००७७ ०७००००७० ०००७ (B) XIII, 17. ] Siddhdnta-siromant. 237 16. Tell mo, my friond, what is tho AHARGANA when on a Thursday, Monday or Tuesday, the 35 remainders of the revolutions, signs, degrees, minutes and seconds of the places of the planets, (the Sun, the Moon, Mars, Jupiter and Saturn and the s'fanrocucuas of Mercury and Venus) together with the remainder of the subtractive days according to the pufvRIp- DHIDA, givo, when abraded by tho number of terrestrial days in 8 YUGA, & remainder of 1491227500.* 17. The place of the Moon is of such an amount, Question 5th. that The minutes ee 2 Example. + 10 = tho soconds tho minutes — scconds + 3 = degrees the degrees 2 *, @ = 293627203 by the processes of indeterminate problems, Now let a == 64860242, b = 894479876, and ० = 872806875 ; र we have the equations (A) and (B) in the forms = signs. and @ ॐ -- ® $ = 1, ‘, a= ¢ 2’ -- ¢ ६ (seo the procoding noto) == 293627203 ० — 894479876 £ : as stated in the text.—B. D.] ® Solution. Tho givon sum of the 36 remainders in a yuaa = 1491227500 according to the DH{VRIDDHIDA TANTRA. *. 1491227600 ~> 4 = 872806875 : 872808875 > 293627203 894479375 and र, <== 277495471 and remainder 10000 i. e. AHARGANA. 1 = 1428 — 4 remainder, 1, c. 10000 = anaraaya on a Tucaday, for 7 the yuGa commenced on Friday. : This would be the aHaRnGana on a Tuesday. To find the +^ 8644 on Monday, it would be necessary to add the reducod terrestrial days in ® yuaa to this 10000, till the remainder when divided by 7 was 3. 10000 ~+ 89447093875 X 2 788968750 = न ~~ = 112900821 — 3 remaindor = 7 ॥ Monday : 10000 "+ 394470376 9 3 1188448126 ~ eee = 1659064017 — 6 remainder or = 7 Thursday.—L. W. and 238 Translation of the (XIII. 18. And the signs, degrces, minutes and seconds together equal to 130. On the supposition that the sum ofthese four quantities is of this amount on a Monday then tell me, if you are expert in rules of Arithmetic and Algebra, when it will bo of the same amount on 8 Friday.* 18. Reduce the signs, degrees and minutes to seconds, adding the seconds, then reducing tho terrestrial days and the planet’s re- volutions in a KALPA to their lowest terms, multiply the seconds of tho planct (such as ‘tho Moon) by tho torrostrial days (reduced) and divido by tho number of scconds in 12 signs: then omitting the remainder, take the quotient and add 1 to it, the sum will be the remainder of the BHaaanas revolutions. Ruiz. ® Let ® = minutes ०, # == 68 minutes, 58 + 22 2 58 — 39 -+- 8 = 22 १०५६००७. 22 — = 11 signs. 2 == 89 seconda. Hence the Moon’s place = 11s. .. 220 .. 58’ .. 89”. + The mean place of the Moon = 11s. .. 220 .. 68’ .. 89“ == 1270719” Tho numbor of seconds in 18 signs = 1296000. Torrestrial days in a KALYA = 1577916450000 | Theeo divided by { 956313 1650000 become Dri- Revolutions of Moon == 57768300000 DMA or reduced. 85003. 3111. 20.] Siddhanta-siromani. 239 19. Tho remainder before omitted subtracted from the divisor will give the remainder of seconds: if that remainder of tho seconds is greater than the terrestrial days in 9 KALPA, then the question is an “impossible -one” (incapable of solution and the planet’s place cannot be found at any sun- rise): but if less it may be solved. Then from the remainder of the soconds tho anARGANA may bo found (by the KUTTAKA pulvorizor as given in the नीरद and bfya-aayira) Or, : 20. That number is the number of anarcana by which the reduced number of revolutions multiplicd, diminished by the remainder of the revolutions and divided by the reduced number of terrestrial days in the Kara, will bear no remainder. 116 reducod number of terrestrial days in a Katra should be added to the AHARGANA such a number of times as may make the day of the week correspond with the day required by the question. | Now when the mean place of the Moon was sought, the rulé was | As the Terrestrial) | Revolutions ina Given days or + 0११8 {118 14774. | : । ११, | ध 101 } : Revolutions. If ony romninder oxistod, it, when multipliod by tho number of 8060148 in 12 signs and divided by ए +, terrestrial days gave the Moon's mean placo in seconds. We now wish to find the pmaGana-8 Bsa or the remainder of revo- lutions, from the Moon’s given placo in seconds: we must therefore reverse tho operation Moon’s place in seconds > KALPA terrestrial days ० ~~ ----- = BHAGANA-8 BSHA. seconds in 13 signs The torrestrial days, however, to be used, must to be reduced to the lowes; terms to which it, in conjunction with the KALPA-BHAGANAs or revolutions in a KALPA can bo reduced: the lowost terms as above stated wero of tho torrestrial days = 966313, of tho Moon’s KaLPA-nitAGaNas == 36002. 1270719 K 656313 1215205099047 ws = ———— = 937658 quolicnt — remainder 1296000 1 331047. 937658 quotient 1 adding one gives 937659 for the BHAGANA-B’ ESHA. Tho reason for adding one is, that wo have got a remainder of 881047, which wo never could have had, if the original remainder had been exactly 937658, it must have been 1 more. This is therefore added : but the remainder of seconds may now be found—for it will be 12963000 — 831047 == 964953. This romainder 964958 being greater than the terrestrial days reduced to lowest torms, viz. 956313, the question does not admit of being so fred. —L. W.. 240 Translation of the [XTIT. 21. 21. Ifthe Moon’s puagava-s'/EsHa or the remaindor after finding the complete revolutions admits of being divided by 1650000, without leaving any remainder, the question may thon be solved: the reduced BHaGANA-8 EsHA on being multiplied by 886834 and divided by 951368, then the remainder will give tho AHARGANA. The divisor should be added to this remainder till the day of the week found corresponds with that of the question.* 22. The mean place of the Moon will never be at any sun-riso, equal to 0 signs, 5 dogroos, 36 minutes and 19 seconds. 28. When will the square of the apnim{sa-s'ksHA remainder of the additive months, multiplied by 10 and the product increased by one, be a square : or when will the square of the ADHIMASA-8 ESITA decreased by one and the remainder divided by 10 be a square? Tho man who shall toll mo at what poriod of tho Kara this Impossible question. Question 6th. 9 {2९ find the aHaRGaya from the Moon's sHaGawa-8’ESIA. ¢ R = BHAGANA-8’ ESHA, T = 1577916450000 terrestrial days in a इ 47.24; M = 57758300000 the Moon’s revolutions in a KALPA, ॐ = aAHARGANA, R Then, as T: M :: #: revolutions + a or y + = : ००७ M # ~ T gy — BR : In this equation as M and T are divisible by 1650000, R must be divisible by the same number, otherwise the question will be KHILa or “impossible,” as stated in the text. ० Dividing both sides of the equation by the number 1650000, we have 35002 a» — 956318 y = R’ or “2 — T’ y= RB’: Now let M’ a’ —T’ y= 1 : or 85002 x — 956318 y’ = 1 : honco we havo 2 = 4; and 2 = Ra’ -- ¶ £ (see the note on tho verse 11th) == 886834 R' — 956313 ¢. Henoo the rule in the text. And, as the reduced BHaGayabs’£8HA == 987659 (seo the preceding note) hence 937659 X 886834 == 881547881606 : This divided by 956318 will give as quotient 869555 (i.e. ४) leaving a re- | mainder of 267151 whioh should be the ^ घ्र ^ 2७4 त 4, but as the BHAGANAS’ESHA 1. ०, 937659 does not admit of being divided by 1650000 (tho numbers by which tho terrestrial days woro roduood) it ought to have boon Kita or insoluble question: but BudsxanAonAnya horo still stated this numbor to be the truo alakGaNa,—B, D.]} XIII. 24.) Siddhdnta-s'iromani. 241 will tako placo—will be humbly saluted even by tho wise, who generally speaking, gaze about in utter amazement and confu- sion at such questions, liko the bee that wanders in the bound- less expanse of heaven without place of rest. 24. (In working questions of KuTfaKAa pulverizer, the aug- Remark on the preceding ment must be reduced by the same question. number by which the sufsya dividend and IfARA divisor aro roduced to their lowest terms, and when the augment is not reducible by the same number as the BUAJYA and WARA, tho question is always insoluble.) But here, in working questions of KuTTAKA, those acquainted with the snhjcect should know that the given augment is not to bo reduced, i. €. it belongs to the reduced BudsyA and HARA, otherwiso in somo places the desired answer will not bo obtained, or in others the question will be impossible.* ® [The questions in the 23rd verse are the questions of the VARGA-PRAKRITI or tho affected square, i. €. questions of indeterminate probleus of the second degree. 18४ question. Lot a = the apuimAsa-s’esita : then by question 10 2? + 1 = ¢. In auch questions the coefliciont of = is called rraxrrtt, tho valuo of ¢ Ka- Nisutita, that of the augment kaicsea and that of y Jyssuriva, Now assume y = m = + 1, then 10 2* + 1 = (mz ~ 1) = ५2 ~° + 22 + 1, 2m ०० = $ 10 — m? Hence the rule given by 2 पडा ^ 9.40 ^ ४८4 in his Algebra Ch. VI., verse VI., for finding the KANISHTHA where tho KsHEPA is 1; is “ Multiply any assumed number by 2 and divide by the differenco between the square of the numbor and tho PRakuittl, tho quotiont will be the Kanisnfita where tho Xsugea is 1.” 2m 2x3 Now assume m = 3, thon «= ————- = - = 6: and = You ५ 10 2? + 1 = r/ 361 <= 19: 4 णापर + 8/8 == 6. From two sets, whethor identical or otherwise, of Kanisurita, Jyesnpaa and 7811 24 belonging to the same PRaxkRIni, all others can be derived such as follows. Lot @ == rraknitt, and | es A । ध ; ५ tho two sote of KANISHTUA, JYESUTUA and KSUEPA, thon 2 3 wo havo” ax? + 8१ = y?; ari + 09 = 4#ई: b ॥ = ¥ ~~ x? ry b, = ३ — ari: T 242 Translation of the (XIII. 25. 25. Tell me, 0 you competent in tho spheric, considcring it frequently in your mind for awhile, what is the latitude of the city (A) which is situated at a distance of 90° from vssayinf, and bears Question. and .°. 8, x by == (४१ — ० >?) (y3 —a 29), yt ५४ — arty} — ० $१ + ०52०३) ^“ ००१४६ ००2४१ + b, bg yi ys + a? zi 2G: adding +% a2, 2 $; Yq to both sides a atyZt%ax, wey, #9 ~+ ० ०2 yt +b, 99 == 44 ६ 9 az, 299, 99 + ०° 2} 25. or a(t, ¥9 ¬ 9 91) > +5, 09 = (४, $#9 £42, 29) °: thus we get a new set of इ + पाह ५) JYRSHTHA and KSHEPA: i. ५. new KANISUTUA = 2, ¥g + =); wow JYESUTHA = & , ४४ + 92, 29; and new 81128124 = b, by: Hence the Rule called BuavaNd given by BuAskandcnaRya in his Algebra Ch VI. vorses IIT. & IV. Now in the present question ॐ, == 6, #। =19and6,=1, and also @, == 6,9, == 19 84 ९» == 1 : ५९ Now KANTBUTHA == 0 >< 721 + 228 x 19 = 4326 ~+ 43323 = 8058 ; now उ ४४811104 == 731% 19 + 10 K 6 >< ४४६ = 1५699 +4 ॥५७०५ = 17379 ; and now kauRkea == । K1=1. Thus ~+ = 8658 &o., nocording to tho 711६१४१४ ५४७५1१५५. ‘Mio sovoud quustion is ॐ -- 1 == $$ 10 or ot—10y*+1., Here then we have an equation similar to the former one, but ॐ is now be in the pluce of y* and $> in the place of >, ६ ॐ will 06 = 19), or = 721 &. Now given apuimasa-sesua as found by the first case = 6. Tho proportion by which this remainder was got, was if KALPA SaUMA Guys : KALPA-ADUIMasAS : : ॐ or clapsed 84 एए + days 6 KaLPa 8aUxa days ० KALPA-ADHIMASAS > ॐ == KALPA 84074 १९९४ X y + 6 KaLPa-aDHIMasas K x — 6 or # == ~ ---- ~ --- ° KALPA BAURA days From this we got a now question: “ What aro tho integor values of ॐ and ॐ in thes equation?” whioh question is ono of the questions of koyyAKa ond in which the coeflicient of the unknown quantity in the numerator is callud BHasYA or dividend, the denominator Haka or divisor and the sugment KsHEPA. It is olear that in this equation, if the augment be not divisible by the same number as the dividend and divisor, the values of ॐ and y will not be integers, and hence the question will be insoluble. But here in order that no question should be insoluble, the author has stated that the dividend and divisor should always be tuken, reduced to their lowest terms, otherwise the question will be insoluble. : As in tho present question, if the dividend xatpa-aDnIMasas and the divisor KALPA ३५४४५ duys be tuken not reduced to their lowest terms, 1, 0, not divided by XIII. 25.] Siddhduta-s’ iromani. 243 duc cast from that city (रौरव) ? What is tho latitudo of the place (B) distant also 90° from the city (A) and bearing due west from it? What also is the latitude of a place (C) also 90° from (B) and bearing N. E. from (B): and of the place (D) which is situated at a distance of 90° from (C) and bears S. W. from (C) ?* tho number 300000, the question will bo an impossiblo ono, becauso tho augment 6 is not divisible by the 88116 number. For this reason the dividend and divisor must be taken here reduced to their lowest terms 1693300000 Hence, dividend =: reduced KALPA-ADIIMASAS = ———-——- == 6311; and 800000 1565200000000 divisor == reducod KALPA SAURA days == ———-—-—— = 6184000 . 6311 « — 6 , By substitution, $ == —— which gives 2 = 826746 the elapsed 84 24 da or 2276 years 6 months and 6 days.—B. D @ Let a = the azimuth degrees, d = the distance in degrees between the two citios, p == PataBua’ at the given city, k == AK8SHA-KARNA, and v = tho latitude of the othor city. 8111 व XK cos @ cosd +< *) 19 Thon sine = - + a Rad 12 k Now in tho Ist question, क = 90°, द = 90°, p = ५ digits, the ranauna’ at 3438 XO OX5 12 sin 2 = ( + ) ५५ -- , 12 13 = (0 + 0) X 3 = 0 x == 0 = 1१६४५१० of (A) or of Yamaxort (2). In the second question, a = 90°, d = 90°, p = 0 digits at yamaxoy!, and ,*, k= 12 t 12 + ~ - 3438 12 = (0 0) 3 = 0: ¢ == 0 Latitudo of city (8) or LANKA. 3438 x¥0 OxX0 12 ,० sne— ) (3). In tho 3rd question, a = 45°, d = 90°, p = 0 at nana and &k = 12: 3438 % 2481 ० % 0 12 sin 2 -- ~+ -- ॥ OX 8438 12 12 = (2431 4+ 0) > 1 == 2431: ¶ 24.4 Translation of the [XITI. 26. 26 and 27. Convert the distance of yoyawas (between the two cities, one is given and the other is that of which the latitude is to be found,) into degrees (of a large circle), and then multiply the sine and cosine of these degrees by the cosine of tho azimuth of tho other city and raLaBHA at the given city, and divide the products by radius and 12 respectively. Take then the differ- ence between these two quotients, if the other city be south of east of the given city ; and if it be north of that, the sum of the quotients is to be taken. But tho roverso of this takes place, if the distance between the cities be more than a quarter of the earth’s circumference. The difference or sum of the quotients multiplied by 12 and divided by axsHaxarya will give the sine of the latitude sought.* Rune. , ॐ == 45° Latitudo of city (©). We, In the 4th question, a == 45°, d= 90°, ॐ = 12 at © and .. k= 12 , ^: ° ee 3438 yx ° / 3 0X12 12 ) x 124/ a $488 1 8438 Vv ) ^ 4८2 ~ ए 12 3438 ॐ = 30° Latitude of D.—L. W. * (Let Z bo the Zenith of the given city bearing a north Intitude, Z WN © the Meridion, @ A प tho Horizon, P the north pole, 8 the Zenith of the other city, the latitude of which is to be found and 2 8 N the azimuth circle pase- ing through 8. Then the arc Z 8 (which is equal to the distance in. degrees between the two cities) will@ be the Zenith distance of 8; the aro If ©, tho arc 1 the given azimuth degrees, and 8 4 which 18 equal to the declination of the point 8, the latitude of the other city which can be found as follows. Let a = Ii g the given azimuth degrees, = Z 8 the 01911166 in degreos between the two cities, ॐ = PaLaBHa, & = AXstta-Kauya XIII. 28.] Siddhdnta-s'vromani. 245 28. 'Tcll mo quickly, O Astronomer, what is tho latitude of a place (A) which is distant इ of the earth’s circumforonce from the city of pHARA and bears 90° due east from it ? What also is the latitude of a place distant 60° from pHARA, but bearing 45° N. E. from it? What also is the latitude of a place distant 60° from pHARf and bears S. E. from it? What also are the latitudes of threo places 120° from छा and boaring respectively due — east, N. E., and 8. E. from it ?* Question. and «= 8 # the declination of the point 8 i. ९, the latitude of the other city Then say, As sine Zg: sine Ag:: sine 2 §: the Baus i. 6. the sine of distance from 8 to the Prime Vortical. or R:coea:: sind: BDHUJA cos ¢ sin द ०० BAVA = ~~ + R And by similar latitudinal triangles, 12 : p : : cos d: s’ANKUTALA, , P XK cos ० SANKUTALA = —————. 12 Now when the other city is north of east of the given city, it is evident that the एप एर will be north and consequently tho sino of amplitude = BnUJA + s’ANKUTALA : but whon tho othor city is south, tho nuvsa also will be south and then, the sine of amplitude == BHUJA ^+ S'ANKUTALA, coaxsind pooed R 12 or the sino of amplitude = And by latitudinal triangles k: 12 : sine of amplitude : sine of declination i. e. sin cosa ‰ 81) ८ p x cos -) = GSD 12 ( 12 X sine of amplitude R 12 चि ४ ०० in ॐ त $ hence the rule iv the text If the distance in degrees between the two cities be more than 90°, the point 8 will then lie bolow the Horizon, and consequently the direction of the Buusa will bo clunged. Therefore the revorse of the sigus £ will take place in that case.—B D.) ® Here also sin « = ( 811) @ ॐ 608 @ 608 @ >< ¢ 12 ॐ ) x—. 12 k (1.) In tho first question, « == 90° द == ७५०, p = 6 digits tho raLabita of Ditaka and 977 9 0 1719 $ 6 12 ae rin 2= ( )x= 3438 12 13 1719 x & 8896 9 = < -- = -~ — 663 — 12 18 13 13° 246 Translution of the (XIII. 29. 29. Tell me, my friend, quickly, without being angry with me, if you have a thorough knowledge of the spheric, what will be the paLabHA of the city where the Sun being in the middle of the arpRA NAKSsHaTRA (i. e. having the longitude 2 signs 13° 20’) rises in the north-east point.* १ @ == 11°,.15’..1” Latitude of city due east from DBARA. Question. (2). In the 2nd equation, a = 45°, d= 60°, p=5&.. k= 18: (— 9 2431 1719 x *) 12 =-= > -- ; 3438 12 19399109 1913 = ~ = 2604 —-: 7449 4१449 ,, 2 <~ 49° .18'..24” Latitude of city bearing 45° N. E. from pHaBa. ००, शा) # ~ (8). In the 870 question, a == 45°, d = 60°, p = 6 and & = 13. (— > 2431 1719 x ^) 12 eS 1॥ "| ०० 810) ॐ == > - 8438 12 19 9549239 7070 ~~~ ooo ~ < 1281 ———— 4 7449 7449 | ०, ॐ == 21° . 64/.. 84“ Latitude of city bearing the 3. E. from णा ५४५. (4). To find latitude of place 120° from DHaRa and due east. Here, sin d= sin 120° = sin 6८० = 2977, ००४ d == cos 1200 = — sin 80° = — 1719 cos a = 0, 2 = Sand & 18: 29779८0 1719 x 5 12 ०, 811) ॐ == ( +——) K—-;3 3438 12 9 = 662 ~ ; 18 ० ® or latitude = 11° . 16’... 1”. The latitudes of the places 120° bearing N. 7. & 8. E., will be the same as the latitudes of those places distant 60° and bearing 8. & N. 9. Hence the latitudes are 21° +54’..84/ and 49° 18’ 24”.—L. W. ® Ansr. Suu's amplitude = sine of 45° = 2431’, the sine of longitude of middle uf ^ ४774 = sine of 2 signs 18° 20’ = sin 73° 20" == 3292’..6" 40°” and tho sine of the Sun’s greatest declination = sin 24° = 1897’. Then say: As Rad: sin 24°:; sin (73° 20’): sine of declination, and ns sine of amplitude : sine of declinution : : Rad : cos of latitude, * eine of amplitude : भण 24° ; ; ain (78° 20’) : oos of latitude, sin 24? >< sin (73°..20/) 1897! >< (3292’..6” . 40") <2: aE =a °, cos of latitude = sine of amplitude 2431/ = 1891’ 650’ 48’ = sine of 33° 23’ 87" : whence latitudo will bo 66° 36’ 28’ , sine of latitude == 2870’ 13”. ‘then sy: As cos of latitude: sino of latitude: : Gnomon : oquinoctial shadow 1891^.. 61“ : 2870° 18“ :: 12 12 >< (2870. . 13’) 13 ०, equinoctial shadow = = 18 — digits. —TL. W. GU 1891“ 61’ 11. 32.] Siddhdnta-s'iromani. 247 30. Tell mo the sovoral Jatitudes in which the Sun remains above the horizon for one, two, three, tion. Question four, five and six months before he sets again.* 31. If you, O intelligent, are acquainted with the resolu- tion of affected quadratic equations, then find the Sun’s longitude, observ- ing that the sum of the cosine of doclination, tho sino of docli- nation, and the sine of the Sun’s longitude: equal to 5000 is (the radius 18 assumed equal to 3438.) 32. Multiply the sum of the cosine of declination, the sine of declination, and the sine of Sun’s longitude by 4, and divide the product by 15, the quotient found will be what has been denominated the Apya. Next square the sum and double the square and divide by 387, the quotient is to be substracted from 910678. Take the square-root of the remainder. That root must then be subtracted from the apya above found: the remainder will bo tho declination, when tho radius is equal to 3438. From tho declination the Sun’s longitude may be found.t Question. Ruz. # Ansr. When tho Sun has northern doclination hoe remains abovo tho horizon for ono month in 67° N. 1 two 0101४18 in Gye three months 7३० four months 78° five months 84° six 10011118 90° These are roughly wrought: for BrasxanfcmArya’s rule for finding these Tntitudes ece the TRIPRAS’NADHYaYas of the GoLaDU Yaya and aleo the GaNITA- 1011 $^ ६ ॥.--[,. W. [Let a = the given sum, ॐ = tho sine of the Sun’s extreme declination ॐ = the sine of the Sun’s declination Then the cosine of declination will be A/ Ra and the sine of the Sun's Re longitude == —— : P reer Rea ० by question R?—2? + > + - =a: ति or Pp /R— >° + (R +p) > =ap, and p fst == ० -- (7 + 2) 2 ; 2 ‘== at p* — 209 (RB + p) z+ (£ + 2 Rp + p*) 2; +2Rp + 2p") = —2ap (R + p) 2 = - (० — BR) 22; 2.48 Translution of the [>411., 33. 33. Given the sum of the sines of the declination and of tho altitude of the Sun when in the prime vertical ; the rappiiritt, the एरर and sine of amplitude equal to 9500, at a place where the paLabni Question. 2ap (R + p) (a* — R*) 2 or न~ - “~ -- - R? + 2? + 22 ° + 2Rp+ 2 a! ; 2ap (R + p) ० p* (R + 2) * completing the square, z* -- r+ + ——_ —— R? +2 Rp + 2p (ए + 2 Rp + 2p’) * a? p* (R + p) ° (a? — R*) p (1 + 2९6 + 28) 1 + 21२8 + 22 1४* p* + 2 R* p® + 2 R* p* — द pt (R* + 2 Rp + 2 p?)* R? 2४ a* pt ४ + 288 422 (R? Rp + 2 72) ap (R + 2) JS ५२ p cro — न= | ढक १" + 252 + 2p 1 + 27२2 + 22 (R°+2Rp+2p%)* ap (R + 2) १ a* pt Te ors= + eee a Bie ake te R'4+2Rp+2p" (BR? 4+2Rp+42p%)? Now here ‰ = 8438 and p = 1397, ०2 (1 + p) ap (£ + 2) ax 1897 X 4895 6734495 ० [11 1 [मै `" ° + 28 + 27 (7 + 8) * + 2* (4835) » + (1897) * 25328834 4 — a nearly = ADYA; 6 a pt 3808777688881 a* 2 a® —— = — = — nearly; (R?+2Rp+2p%)*% 641549831799556 337, Rp 280677 13928996 R? + 2Rp + 22 25328884 thor has taken the number 919678 and = 910729, in place of this the Au- but of these, the positive value is excluded by the nature of tho case, because the sine of declination is always less thau 1397. Hence the Rule in the text. Solution. The given sum = 5000, 000 x 4 ,. ADYA = ————— = 13833’ 20” and इड 7 a* = 148367! 57” 9.“ 15 *, sino of declination == 1333’ 20’/ — 49 10678 — 148367! 5¶ 9’ == 1833’ 20/7 — 873/ 6" 1937/4 == 460’ 13” 47५“ : from which we havo tho longitude of he Sun = 0*,,19°..14’ 36” or §* 10°..45/ 24 or 6 19¢ 14/.. 36! or 11*,,10°,. 45/,, 24".—B. D.] XIII. 35.] Siddhdnta-s’iromani. 249 or cquinoctial shadow is 5 digits, tell me thon, my clevor friend, if quick in working questions of latitudinal triangles and capable of abstracting your attention, what are the separate amounts of each quantity ? 34. First assume the sine of declination to be equal to 12 times the shadow paLaBHi: and then find the amounts of the remain- ing quantitics upon this supposition. Then those on tlio sup- position mado, multiplied severally by the given sum and divided by their sum on tho supposition made, will respectively make manifest the actual amounts of those quantities the sum of which is given.* 35. If you have a knowledge of mathematical questions involving the doctrine of the sphere, tell me what will be the several amounts of sines of amplitude, declination and the एए (where tho PALABHA is 5 digits) when their sum 18 2000.f | Ruts. Question. # Solution. Hero ratanuk = 5 digits ००, Suppor the sine of declination = 5 X% 12 = 60: aud thon say. IfranapudA;: akslakaeya : : sine of dccln.: SAMA 8 ANKU 13 X 60 ° ण 5 : 13 =: : 60: sama 8/4 ब्र <. == 156} 5 166 X 13. Gnomon: AKSHAKABNA : : SAMA 8’ ANKU : TADDIIRITI = = 160, 12 60 x 5 न 12 : 741, 7“ : : sine of declan. : KUSYA = = 25, 00 % 13 0४९ and 12 : akswakakya: : sino of decln.: sino of amplitude =- 2 °. If the eum: sine of decln. supposed : : given sum : sine of decln. required. or 475 : 9600 : 1200 If 475 : 166 :: 9500 ; 8120 sama 8’aNKU required. and so on 8380 TADDICKITI 600 KUJYA 1300 eine of amplitudo. Aner. L. W. †+ Solution. Toro also rananita = 56, then suppose sine of declination as bofore = 60, and .°, sine of amplitude = 65, and KUJYA == 26; tho sum = 150, 250 Translation of the (XIII. 36. 36. But dropping for a moment thoso questions of tho SIDDHANTAS involying a knowledgo of the doctrine of the sphere, tell mo, my learned friend, why in finding the point of the ecliptic rising above the horizon at any given time, (that is tho LAaNa or horoscope of that time,) you first calculate tho Sun’s apparent or true place for that time, 1. e. the Sun’s instantaneous place : and further tell me, when the Sun’s sdvana day, 1. e. terrestrial day, consists of 60 sidereal auafix4s and 10 74148, the LAGNA calculated for a whole terrestrial day should bo in advanco of the Sun’s instantaneous place, and the Laana calculated for the time equal to the terrestrial day minus 10 21.48 should be equal to the Sun’s instantaneous place. 37. Are the aHafixds used in finding the LaGNa, GHATIKAs of sidereal or common sfvana time? If they are sfvana anaTixis, then tell me why are the hours taken by tho several signs of tho ocliptic in rising, i. o. tho rfs‘yupaYa which so sidereal, subtracted from them, being of a different denomina- tion? If on the other hand you say they are sidereal, then I ask why, in calculating the Laana for a period equal toa whole sf{vana day 1. 6. 60 sidereal GHatrKas and 10 74148, the LAGNA does not correspond with, but is somewhat in advance of, the Sun’s instantaneous place; and then why tho Sun’s instantaneous place is used in finding tho Laana or horoscopo.* - 88. Given the length of the shadow of gnomon at 10 aaffs after sun-rise equal to 9 digits at a place where the pataBné in 5 digits, tell me what is the longitude of the Sun, if you are au fait in solving questions involving a knowledge of tho sphere.+ Question 6. Question. Then say as before as 160 : 60 +:: 2000 : 800 sine of declination, as 160 3 65 :: 2000 : 866% sine of amplitude, as 150 : 25 :: 2000 : 8838} xuvzgxya.—L. भ. a oh to these questions see the note on the 27th verse of the 7th । Peel? 2) ५ t (For solving this question, it is necossary to define somo lines drawn in tho Armillary sphore and show some of their relations. XIII. 39.] Siddhdnta-s'tromani. 251 39. ‘Toll mo, O Astronomer, what is tho एतत at that place where the gnomon’s shadow fall- ing due west is equal to the gnomon’s Question, Tet BODE be meridian of the given place, 0 A ए the diameter of the Horizon, ए the Zenith, P and Q tne north and south poles, B A D the diameter of the Prime Vertical, ए AG that of the Equinootial, P A Q that of the six. o'clock line, Hf L that of one of the diurnal circles, s the Sun's projected place in it and f4, sm, 7 + perpendiculars toO E, Then 1 ¥ or E P = tho latitude of the place, A f= tho sino of the Sun’s declination, A g = 4०१९4 or the siuo of amplitude, Sg == xusya’, (1४ is called onanasya’ or sine of the ascensional difference when reduced to the radius of a great circle) - म = Kata’, (It is called s6TRa when reduced to the radius of > great circle.) ० ¢ = 1sHTa uRITI. (It is called TappHRITI whon ॐ 18 at ¢, HRITI when 8 is at H and xusya when ¢ is at /.) The 18nfA HRITI reduced to the radius of > great circle 1s called IsntTa ~ तकर ^^) but s coincides with JT, it is cnallod antya’ only. It is evident from tho figure above described that (1) 18क^ HRITI = KALa’ + Kusya’, (2) 1sntTa ANTYA’ = B6TRA + CIABAJYA’, (3) 1 एत == एर ९५८ or cosino of declination & कम ^ (4) AnTYa’ = radius + 0141१4२ २८८. 110० tho positive or negative sign is to be taken according as the San १8 in the northern or southern hemisphere. ए 2 252 Translation of the (XIII. 39. height when the Sun is in the middle of tho sign Loo, i. ©. when his longitude is 4 signs and 15 degrees.* ` Now at a given hour of the day, the IsHta HRITI and others can be found as follows. Half the length of the day diminished by the time from noon (or the Nata Ka‘La properly so cullud) is the UNNaTA KaLA (or olevated time). Subtract from or add to tho UNNaTA Ka’LA the asconsional dilfurence according as the Sun is in the northern or southern hemisphere: reduce the remainder to degrees: the sine of tho degrees is 804४८, Tho 867४4 multiplied by the cosine of declination and divided by the radius gives the KaLa’, Then from the above formule wo can easily find the 1snta HRITI and others. Now to find the answer to the present question. Square the length of the Gnomonic shadow and add it to the squaro of tho Gnomon or 144: and square-root of the sum is callod the hypothenuse of tho shadow. Fron this hypothonuso find tho Maua‘s’anxu or tho sine of the Sun's altitude by the following proportion. As the hypothenuse of the shadow : Gnomon or 12 :: Radius : The Mawas’anxv or the sine of the Sun’s altitude. Then by similar latitudinal triangles, as the Gnomon of 12 digits : 41८84 KABNA found from givon PALADICA’ 2: MAHAS'ANKO : उशाक+ MRITI (866 vorecs from 45 to 49 of the 7th Chapter). 1०५८५ tho givon UNNaTa KA’LA to dogrvos nid assume the sino of tho degroos as isHTaNTY small circle on the supposition that the Sun has the declination now found. As Raul: ९ 0 : : cosino of declination: N B. Now find also B A by the same proportion. Then N B + BA=N 1" 8 new value of TADDHRITI. If N: gaveO B:: HN’: 0 B’ corrected value © B. Hence a corrected longitude of the Sun. Tho operation to be repeated till rightness is found. 2nd.—To fiud the declination from the naTa Ka’La or timo from noon = pin 30°, Jat a = tho sino of waTA १4114 ६ {° ~~ कन = sina’, and x = tho sino of declination ६ R* — x" = ००8० of declination. The sérRa reduced to value of diurnal circle will give KaLa’ Tho proportion 18. As R: sGrea:: cos of declination: Kaba’, but I do not know what cos of declination is but only its square. I must therefore make this proportion in squares (B* — at) (R* — <) As R®: sdrcna®:: 006१ of declination: Kava) ‘= —— 7. ES Now by similar latitudinal triangles As 1४ ) * : PALABILA’ ) 9: : Kana ) ॐ ; sine*® of declination PALADUA ) * __ ¬ ° 25 (R* -- a*) (R* — रग) e 2 [| 9 = canis Sai commen KALA a. खातकं ea ee sine” of declination 7 . : x 144 {२१ = 2 254 Translation of the (XIII. 41. 41. Ina 1४66 of which tho latitude is unknown ond on a day which is unknown, the Sun was observed, on entering the primo vor- tical, to give a shadow of 16 digits from a gnomon (12 digits long) at 8 anatiKAs after sun-rise. If you will tcll mo tho declination of the Sun, and the patasaf I will hold you to be expert without an equal in the great expanse of the questions on directions space and time.* 42. O Astronomer, tell me, if you have a thorough know- 16406 of the latitudinal figures, tho PALABHA and the longitude of the Sun Question. Quostion. Now R? —a? = 25 (R® — ०२) = 25 >९ 8804883 = 221622075 and 144 R* = 144 (3438) * = 1702067536 221622075 (R* — 2°) 1702057536 1702087586 R? — a? = ——— —— 2* = 7j o* nearly 221622076 8R 262°=8 R*: & ~~ —~ = 1363828 26 and 2 = a/ 1363828 = 1167’ = sino of 19°... 51’ Hence the Sun’s place may be found.—L. W. ® To find the sino of altitude or aawa’ BANKU (16)? + (12)* = (20)* ... hypothenuse of the shadow = 20. Then say As 20: 12 : : 8488’ : 2062’.. 48” = the mana’ 8’aNKU. Now suppose the sine of UNwaTa Ka’La or 8 QHaTiKa’s to bo tho TADDHRITI Then by similar triangles | 2655’ X 12 2062’ .. 48”: 2666“ : : 12: aksHa KaBya = ——-—— 20624 From this find the PaLaBHa’. To find declination says As aKSHA KARNA: PALABHA’:: 2062’... 48” : sine of declination. From this find the cosine of declination, the Kusya, the ascensional difference, &o. The UNNaTA KALA diminished by the asconsional differonce gives the time from 6 o'clock : tho sino of this timo will be tho sérra and henco tho Kaba : thence (Kugya’ buing addod) the +^ काश्य : and thonce the aksita KaRNa and doclination. The न to be repeated till the crror of tho original assump- tion vanishes.—L. W. शा. 43.] Siddhdnta-s'iromant. 255 at that place, where (at 9 certain time) the KuJYA is equal to 245 and the TADDHRITI is equal to 3125.* 43. Given the sum of the 3 following quantities, viz. of the sines of declination, and of the alti- tude of the Sun (when in the prime vertical) and of the rappHRITI decreased by the amount of the KUJYA cqual to 6720, and given also the sum of the kusyA, tho sinos of amplitude and doclination (at the samo time) oqual to 1960. Iwill hold him, who can tell me the longitude of the Sun and also 11५4 एप्त { from the given sums, to be a bright instructor of astronomers, enlightening them as the Sun makes the buds of the lotus to expand _by his genial heat.t Question. ® Aner. Let ॐ == the PALABIA é 2940 then say. Asw:12:: 245: sine of declination = ——. @ Now 0०4 the TADDHRITI minus एरर, 2940 85280 Asa: 12: : —— : TADDHRITI — KUJYa = -——. @ | But + णाभि — xusya = 3126 — 245 = 2880. 36280 36280 49 ० 2880 = and >° = =— . + 2880 4 ० ‰& == { == 23 PALADUA. To find declination sey As 3 : 12: ; 245: 840 sine of declination. Hence the longitude of the Sun may be discovered as before.—L. W. ¢ Lhis question admite of a ready solution in consequence of its peculiarities, The sine of declination SAMA 84.14 ए > = 6720 and TADDIRITI — KUSYA aro all throo respectively perpondiculars in the threo latitudinal trianglos, And the kusya the sine of amplitude} = 1960 and the TADDURITI — KUJYA are bases in the same 8 triangles. Ifence we may take the suin of the 3 porpendiculars and also the eum of the three bases and uso them to find the PaALabua, As the sum of the } sum of the 3 bases Gnomon PALABHA 3 perpendiculars } in the same triangles 1960 > 12 6720 ४ 1960 :: 12 : ~ <== 3}. 6720 Now tho xusya, sine of amplitude and sino of declination are the throo sides of a latitudinal triangle. Those throe I may compare with the three Gnomon, PALABILA aud AKSHA KakWA to find the value of any one. 256 . Translation of the (XITT. 44. 44. Given the sum of tho sine of declination, sino of the Sun’s altitude in prime vertical and the TADDHRITI minus एरर equal to 1440’, and given also the sum of the sine of amplitude, the sino of tho Sun’s altitude in primo vortical and tho rappugirt equal to 1800. I will hold him, who having observed the given sums.* 45, Given the equinoctial shadow equal to 9. What longi- । tude must the Sun have in that lati- Question. Question. : + mir tudo to givo an ascensional differonco of three anatis? I will hold you to be the best of astrono- mers if you will answer me this question.t’ 46. Hitherto it has been usual to find the length of the Sun’s midday shadow, of the shadow tion. 1100 > of the Sun whon in the pmmo vorti- But the ^ इश + KaRya must be first found to complete the sum of thoso threo. ^^ KARNA = J (2 44+ (‡)* == 0८० = ध: yA ~ 2) > 4 [स 2 Gnomon = 13 24.1.74 = 3h = 28 sum of the 8 sides of a latitudinal triangle. . AK8Ha KaARNA = 12} Now if 28: 12: : 1960: 840 the sine of declination, Hence the place of the Sun as before.—L. W. * This question is similar to the preceding. . In the firat sum we have the sun of three perpendiculars in threo differont latitudinal Triangles. In the second we have tho sum of tho throo hypotho- nuses of those same throe Triangles. Flonce wo may suy. , sum 8 por. sum of 8 corresponding hy. Gnomon aksiwaKanNa As 1440 : 1800 2; 12; 16 _ Now from axsua Karwa to find PALABHA PALABHA = 4/(16)*—(12)* = a/ 81 = 9. Now sine of amplitude, sine of the Sun’s altitnde in tho Primo Vertical, and tho TaDDURITI aro tho three sides of a lutitudinal.—L. W. ¶ Let «= sine of the Sun’s declination. thon 12; 9:: @: Kosy4 = 4 a, Again ./K*—z* = cosino of declination. Then as R: cos of declination : : sine of usoensional differce. : xusvf Bine of asceusl. diffve. or Of ५ ८2५२ इ 4 = sine of 3 GuaTis = ain 18° = 1062’, cosin of decln. K cHaRasyA ० ~~ ————— = एण्ड R a/ 1५*-- >> >€ 1002 or <== ‡ ॐ 11111. ww bo found the sine of tho Sun’s docln. and thonco his longi- fude.— L. W. XIII. 46.] Siddhduta-s‘iromani. 257 cal, and when in an intermediate circle (1. ९. wheu he has an azimuth of 45°) by three different modes of calculation: now he who will by a single calculation tell me tho length of those three shadows and of the shadows at any intermediate points at the wish of the querist, shall be held to be a very Sun on the Karth to expand the lotus-intellects of learned astronomers.* # [Tere tho problom is this :—Given tho Sun’s declination or amplitude, वअ shadow of tho place and the Sun’s azimuth, to find the Sun’s shadow. For solving this problem BuisxarfonAMrya has stated two different Rules in the GanitApuydya. Of them, we now shew hore the second. “Multiply the square of the Radius by the square of the equinoctial shadow, and the square of the cosine of the azimuth by 144. The sum of the products divided by the difference between the squares of the cosine of tho azimuth and the sine of the amplitude, is called the PrarHaMa (first) and: tho continued product of the Radius, cquinoctial shadow and the sine of the amplitude divided by the (same) difference is called the anya (second). Take the square- root of t he square of the anya added tothe PRaTHAMA: this root decreased or increasod by tho aNYA according as the Sun is in the northern or southorn hemisphere gives the hypothenuse of the shadow (of the Sun) whon the Sun is in any given direction of the compass.” ‘But when the cosine of the azimuth is less than the sine of the amplitude, take tho square-root of the square of the anyA diminished by the Pratuama: the anya decreased and increased (separately) by the equare-root (just found) ives tho two values of the hypothenuse (of the Sun's shadow) when the Suu is in the northern hemisphere.” This rule is proved algobraically thus. Let a = the sine of amplitude, A = the sine of azimuth, e == the Equinoctial shadow, and ॐ == tho hy (1 of the shadow when the Sun is in any given प1166. tion of the compass. Then say 12 R 98 ॐ : 12:: RB: the mani s'anxv or the sine of the Sun’s altitude = —— ` "क 717 Rr? R 1 aud .°. tho sine of tho Sun’s zonith distance =/ R?— ( ~) << - VJ —144, ७ ® | . 12R eR Now, a8 12 : € == ~ : s’ANKUTALA = —-. x x ० Bkav or tho sine of an 876 of a circle of position contained between the e Sun and the Prime Vertical = a@ कूः —-: (see Ch. VII. ए, 41) here the sign— @ or ~+ is used according as the Sun ie in the northern or southern homisphere. Then say R eR as — A/ 2*—144 :ay—::R:A: (~ (. 1 RA eR ७०6 =— A/ #*—144 ~~ (“ + ~-) R 3 a x 258 T'ranslution of tho (XIII. 47. 47. He who, knowing both the azimuth and the longitude of the Sun, observes one shadow of the gnomon at any time, or he who know- ing the azimuth observes two shadows and can find the PaLa- BHA, I shall conceive him to be a very Garupa in destroying conceited snakes of astronomers. {On this Bafsxardcufeya has given an example in the Gani- 747 १८१८ as follows. “Given the hypothenuse of the shadow (at any hour of the day) equal to 30 digits and tho south BHUJA* equal to 3 digits: givon also Question. Rramplo, or A a/ 2144 =an+eR; A‘ c*—14%4 A* = a? 2*§ कु 91२०००४ + e* R'; (A°—a*) ॐ + 2 Reaw=e' R® + 144 A?; Rea ० R? + 144 4.3 ॐ +3 च or 2° + 2 aNYa > = PRATHAMA (1) = छग त 2 ANYA ॐ + aANYA? = PRATHAMA + ANYA* and .*. ० = A/PRATHAMA त ANYA? क ५४४५. But when A < aand the Sun is in the northern hemisphere, the equation (1) will be 2*—2 anya 2 = — PraTuama, end then ॐ = anya + 4/anya*—firet: i. ९. the value of the hypothenuse of the shadow will be of two kinds here. Hence the Rule. | . Brasxkanacuarya was the first Hindu who has given a general rule for finding the Sun’s shadow whatever be the azimuth; and he was the first who has shewn that in certain cases the solution gives two different results,—B. D.] * (On a levelled plane draw east and west and south and north lines and on their intersecting point, place Gnomon of 12 digits: the distance between the ond of tho shadow of that Gnomon and tho cast and west lino is called tho BUUJA. . Jt isto be known here that the value of the great Buusa (as stated in 41st verse of the 7th Ob.) being reduced to the hypothenuse of the shadow becomes equal to the BHUJa (above found). Or as the Radius : the great एर + १ १ tho Fy thonuse of tho shadow ‡ tho roc [५७०५ एर or tho ५1919160 of tho ond of the shadow from tho east and woet line. This reduced BHusa is called north or south according as the end of the shadow falls north or south of the east and west line. It ia very olear from this that the reduced BHUJa will be the cosine of the azimuth in a small circle described by the radius equal to the shudow. Or as the shadow : the reduced BHUJA : 3 radius of a great circle : the cosine of the azimuth. This 18 the method by which all Hindus roughly determine the azimuth of the Sun from the savsa of his gnomonic shadow.—B. D XIII. £9.] Siddhdnta-s'tromayi. 259 the hypothenuse equal to 15 digits, and the north फा equal to 1 digit, to find the paLaBHs. Or, given the declination equal to 846 and only one hypothenuse and its corresponding BHUJA at the time, to find the 2414 एषठ ८.72 |] 48. First of all multiply one प्र ०३5 of the shadow by the hypothenuse of the other, and the se- cond BHUJA by the hypothenuse of the first: then tako the difference of these two bitusas thus multi- plied, if they are both north or if both south, and their sum if of different denominations, and divide the difference or the sum by the difference of the two hypothenuses ; it will be the PALABHA.* 49. Low should he who, like a man just drawn up from tho bottom of a well, is utterly ignorant of the paLaBHA, the place of the Sun, the points of the compass, the number of the years elapsed from RULE. Question. * The rule mentioned here for finding the PataBita/ when thotwo shadows and their rospective छा एर 48 are given, is proved thus, th, == tho firat hypothonues of the shadow, b, = its corresponding ए एरक h the second hypothonuee, and = ©$ = its corresponding BHUJA, Thien 12 ए Ash,: 12 : : R: —— =the first mamta s’anxu ; a 12 7 and in the same mannor —— = the second MAUA B’ANKU ; R and also ns hy: 7: : R —— = the first great nitvsa , h ४, & and .°. —— = the socond great एर + , h ४, ४ A, ९५ 1 + ne oe hy Then the raLabna’ = —— (७९० Ch. XI. ४. 32) 2h 197 he A, ¢, 09 = 690 h,—A Henco the Rulo.—L. W. x 2 200 Translation of the [XIIL. 50. the commencement of the yuua, the month, the णका or lunar day and the day of week, being asked by others to tell quickly the points of the compass, the place of the Sun, &c., give a correct answer? He, however, who can do so, has my humble reverence, and what astronomers will not acknowledge lim worthy of admiration ?* 50. He, who can know merely with the staff in his hand, the height and distance of a bamboo of which he has observed the root and top, knows the uso of that instruimont of instruments—Cenius (the pufyantRa) : and tell me what is there that ho cannot find out ! 51. There is a high famous bamboo, the lower part of which, being concealed by houses, &c. Question. Question. ee was invisiblo: tho ground, however, was perfectly level. If you, my friend, remaining on this same spot, by observing the top, will tell moe the distance and its height, 1 acknowledge you shall have the title of being the most skil- ful of observers, and expert in the use of the best of instru- ments, DH{YANTRA. 52. Having seen only the top of a bamboo reflected in water, whether the bamboo be near or at a distance, visible or invisible, if you, remaining on this same spot, will toll ino tho distanco and height of the bamboo, I will hold you, though appearing on the Earth as a plain mortal, to have attributes of superhuman knowledge.t 53. Given the places of the Sun and the Moon increased by tho amount of tho procession of tho oquinox, 1. 0. thoir longitudes, equal to four and two signs (respectively) and the place of the Moon decreased by the place of the ascending node equal to 8 signs, tell me whether the Sun and the Moon have the same declination (either both south or one north Question. # This refers to the 84th verse of the Ch. XI.—U. W. + [Answers to these questions will be found in the 11th Oh, —B. 7.1 XIII. 59.] Siddhdnta-s’tromant. 261 and ono south), if you have a perfect acquaintance with the Daufvempuwa Tantra. 54. Ifthe placo of the Moon with the amount of tho pro- cession of the equinox be equal to 100 degrees, and the place of the Sun increased by the same amount to 80 degrees, and the place of the Moon diminished by that of the ascending node cqual to 200 degrees, tell me whether the Sun and the Moon have the samo dcclination, if you havo a porfect acquain- tanco with the Dufveippuipa Tantra. | 55. If you understand the subject of the ९(7+ i. e. the equality of the declinations (of the Sun and the Moon), tell me the reason why there is in reality an impossibility of the p{ta when there is its possibility (in the opinion of Lauta), and why there is a possibility when there is an impossibility of it (according to the same author). | 56. If the places of the Sun and the Moon with the amount of the precession of the equinox be equal to $ signs plus and minus 1 degree (i. €. 28. 29° and 858. 1° respectively) and the place of the Moon decreased by that of the ascending nodo equal to 115. 28°, tell me whether the Sun and the Moon have the same declination, if you perfectly know the subject. 57. (In the DufvrippuHa Tantra), it is stated that the rfra ts {0 come in some places when it has already taken place (in reality), and also it has happened where its to come. It is a strange thing in this work when the possibility and impos- ibility of the ९474 aro also reversely mentioned. Tell me, O you best of astronomers, all this after considering it well.* 58. I (एप (81424), born in the year of 1036 of the S’A11- Date of the Author’s birth VAHANA era, have composed this Srp- and his work. DUANTA-8’IROMANI, whon I was 86 years old. 59. Le who has a penetrating genius like the sharp point 0 of & large DARBHA straw, 18 qualified to compose a good work in mathe- # [Answers to these questions will be found in the last Ohapter of the Ganrra- payaya.— B, D.] 262 Translation of the Siddhdnta-s'iromant. [XIII. 60. matics: excuse, therefore, my impudence, O leatned astrono- mers, (in composing this work for which I am not qualified). 60. I, having lifted my folded hands to my forehead, bey the old and young astronomers (who live at this time) to excuse me for having refuted the (erroneous) rules prescribod by my predecessors; because, those who fix their belief in the rules of the predecessors will not know what is the truth, unless I refute the rules when I am going to state astronomical truths. 61. The learned Manes wara, the head of all astronomers, the most good humoured man, the store of all sciences, skilful in the discussion of acts connected with law and religion, and a BRAH- MANA descended from S‘Anpitya (a MUNI), flourished in a city, thickly inhabited by learned and dull persons, virtuous men of all sorts, and men competent in the three VEDAs, and situated near the mountain SAHYA. 62. His son, tho poet and intolligont Butskara, mado this clear composition of the SrppH4nra by the favour of the lotus- like feet of his father; this SippuAnta is the guidance for ignorant persons, propagator of delight to the learned astro- nomers, full of easy and elegant style and good proofs, easily comprehensible by the learned, and remover of mistaken ideas. 63. I have repeated here some questions, which I havo stated before, for persons who wish to study only this Pras NA- DHYAYA, . 64. The genius of the person who studies these questions becomes unentangled, and flourishes like a creeping plant watered at its root by tho consideration of tho questions and answers, by getting hundreds of leaves of clear proofs, shoot- ing from the Spheric as from a bulbous root. Author's birth-place, &o. End of the 18th and last Chapter of the Gondpuyfya of the SIDDHANTA-S IROMANL. APPENDIX. ON THE CONSTRUCTION OF THE CANON OF SINKS. 1. As tho Astronomer can acquire the rank of an Acuarya in the science only by a thorough knowledge of the mode of constructing the canon of sines, Badsxara therefore now pro- ceeds to treat upon this (interesting and manifold) subject in the hope of giving pleasure to accomplished astronomers. 2 and 3. Draw a circlo with o radius oqual to any number of digits: mark on it the four points of the compass and 360°. Now by dividing 90° by the number of sines (you wish to draw in a quadrant), you will get the arc of the first sine. This arc, when multiplied by 2, 3 &c., will successively be the arcs of other sines. Now set off the first arc on the circumference on both sides of one of the points of the compass and join the extremities of these arcs by a transverse straight line, the half of which should be known the sine of the first arc: All the other sines are thus to be known. 4, Or, now, I proceed to state those very sines by mathe- matical precision with exactness. The square-root of the dif- feronco botween tho squares of tho radius and the sino is cosine. 5. Deduct the sine of an arc from the radius the remainder will be the vorsed sine of the complement of that arc, and the cosino of an arc deducted from the radius will give tho verscd sine of that arc. ‘The vorsed sine has been compared to the 26. Appendix. arrow between the bow and the bow-string: but here 1४ has received the name of versed-sine. 6. The half of the radius is the sine of 30°: the cosine of 80° will then bo the sine of 60°. ‘The square-root of half squaro of radius will bo the sino of 450. 7. Deduct the square-root of five times the fourth power of radius from five times the square of radius and divide the remainder by 8: the square-root of the quotient will be the sine of 36,. Or ५/० १8 Send *x 5 eS Rare Ai = sine 36°.* 8 8. Or the radius multiplied by 5878 and divided by 10000 will give the sine of 36°, (where the radius = 3438.) ‘The cosine of this is the sino of 54०. 9. Deduct the radius from the square-root of the product of ® (This is proved thus. Let # = sine 18°; and .°. R — @ = covors 18° or vers 72°. | B56 Then (3 3 == 91०७ Y°: (see the 10th verse.) or VSR R— 2) = sine 86° ; ^^ but a = श > (see the Sth verse) .*. sine 86° = J tian yew} = 321 9. 7.1 R x 5878 + The Rulo in 8th vorse viz.. ——--—— scoms to bo the same as above and 10000 to be deduced from it ; 5R*— SS KY __ 6 ~ WS for fae = KR. —— ५^इ = 2.287411 &o, and .*. & ~ ./§ = 2.762589 whieh divided by 8 = .845323 ॐ 5878 7 ०, sine 36° = B ./ 345323 = R K .5878 = ——_-——.—L. v 10000 Appendix. 265 tho squaro of radius and fivo and divido tho remainder by 4: the quotient thus found will give the oxact sine of 18°.* 10. र्भा the root of tho sum of the squares of the sine and versed sine of any arc, is the sine of half that arc. Or, the sine of half that arc is the square-root of half the product of the radius and the versed sine. 11. From the sine of any arc thus found, the sine of half the arc may bo found (and so on with tho half of this last). In like manner from the complement of any arc may be ascer- tained the sine of half the complement (and from that again the sine of half of the last arc). Thus the former Astronomers prescribed 1 mode for deter- mining tho other sincs (from a given one), but I proceed now to givo a mode dillerent from that stated by them. 12. Deduct and add the product of radius and sino of णाएर+ from and to the square of radius and oxtract the square- roots of tho halves of tho results (thus found), these roots will respectively give the sines of the half of 90° decreased and increased by the pirusa. In liko manner, the sinos of half of 90° decreased and in- creased by the Kofi can be found from assuming the cosino for the sino of ए... 13. Take the sines of Bausas of two arcs and find their difforence, then find also the difference of their cosines, square * (This ia proved thus. Let © bo centro of tho cirelo ABE and ~ O = 36°, then AB = 2 sin 18°, and — # (CAB, CBA) each of thom == 2 (~. raw AD bisecting tho <~ CAB, then AB, AD, CD will bo cqual to cach other. Now 1९४ 2 = ain 18”, then by simi- lay triangles CB: AB = AB: BD or 1९; % = 2: 1२--23; ,*, 422% ~~ 1४* — 2 Re which gives a5 a/5 ९ B.D] 4 ~ 266 Appendiz. these differences, add these squares, extract their square-root and halve it. This half will be the sine of half the difference of the sines.* Thus sines can be determined by several ways. 14. The square-root of half the squaro of the difference of the sine and the cosine of the Bausa of an arc is cqual to the sine of half the difference of the bHUJA and its complement.+ | 1 will now give some rules for constructing sines without having recourse to the extraction of roots. | 15. Divide the square of the sine of the एप्रठठ+ by the half radius. The difference between the quotient thus found and the radius is equal to the sine of tho differonce botween tho # This rulo is obvious, for ac = dilfee. of 8168 bd & af and cb = 010५8. of cosines bg & ah and ८०} + bc ab ) ab = chord of difference of arcs ab — = sine of half that difference. 2 L. W. t Lot bc = sino of any aro and bg = 1८8 | cosino, Draw the sine ad = cosine 3g. ॥ sine will be equal to 8 Srey sc rea १ „९ + ८* = ०29 ; but as af? = fu? ab af? b ० of? = — and —- = —., । 3 . 2 4 uf % a ५० Jt = = W.]) Appendiz. 267 degrees of एाएर+ and its complement.* In this way several sines may be found here [As these several rules suffice for finding ouly tho sines of arcs differing by 3 degrees from each other and not the sines of the intermediate arcs, the author therefore now pro- ceeds to detail the mode of finding the intermediate sines, that is tho sine of every degree of the quadrant. This mode, therofuro, 18 called PRATIBITAGAJYAKA-VIDII. | 16. Deduct from the sine of BHUJA its उक part and divide Rules for finding the sine the ten-fold sine of Kofi by 578. ५ every degree from 1° to 17. The sum of these two results will give the following sine (i. 0., the sino of BituJA ono dogrco more than original एम and the difference between the same results will give the preceding sine, i. e., the sine of BHUJA one degree less than original BuusA). Here the first sine, 1. €. the sine of 1°, will be 60 and the sines of the remaining arcs may be successively found. 18. The rule, however, supposes that the radius = 3438. Thus tho 87168 of 90° of tho quadrant may be found. ` Multiply the cosine by 100 and divide the product by 1529. Rules for finding tho 24 19. And subtract the 73, part of sines vis., of 8°2, 7°, 11°, tho sine from it. ‘Che sum of these will a be the following sine (i. e., the sine of arc of 3°? degrees moro than original aro): and the differ- ® Let ab bo any are, and ac = ab, ad then ad = its complement, ed = their differenco, and be = 2 ab. ४ | ५, -------- ८ Now / BX १९१ ०८ => gin — or ain ab, 2 2 R ५ vers be a or —— = sin? ad, 2 sin? ad or vers dc == ; sin? ab thon R — vers be or sin cd = ~ द —r. W. 208 Appendix. ence of them will be preceding sine (i. e., the sino of arc इग degrees less than original arc). 20. But the first sine (or the sine of 8०) is hero equal to 2244 (and not to 225 as it is usually stated to be). By this 11116 24 sines may be successively found.* 21 and 22. Ifthe sines of any two arcs of a quadrant be Rules for finding the sines multiplied by their cosines reciprocally eee difference of any (that is the sine of the first arc by the cosine of the 2d and tho sino of the 2d by tho cosine of tho first arc) and tho two products divided by radius, then tho quotients will, when added to- gether, be the sine of the sum of the two arcs, and the differ- ence of these quotients will be the sine of their difference.t This excellent rule called Jya-Bafvand lias been prescribed for ascertaining the other sines. 23. This rule is of two sorts, tho first of which is called saMAsa-bit{vanf (i. 0., tho rulo for finding tho sino of sum of two arcs) and the second anTara-Buivand (i. ©., the rule to find the sine of difference of arcs). [If it be desired to reduce the sines to the value of any other radius than that above given of 3438.] Find the first sine by the aid of the above-mentioned rule pratipHaasyaKi- VIDHI. 24 and 25. And then reduce it to the value of any now radius by applying the proportion. After that apply the ग्र एप्त4ए^ प्र rule through the aid of the first sine and the cosine thus found, for as many sines as are roquired. ‘The sines will thus be successively climinated to the valuo of any now radius. The rule given in my [arf or Litdvari is not sufficiently accurate (for nice calculations) I have not therefore repeated here that rough rule. # (These rules given in the verses from 16 to 20 are easily deduced from the rules given in tho vorses 21 and 22.—B. D.] ¢ BudskandonAuya hus given those rules in his work without any domon- stration.—3%, D.] INDEX. Age, birth, &c. of tho Author, pago 261. Mandaphalé, 109. Armillary Sphore 151, 210. Mandochcha, 109. Astronomical Instruments, 209. Mouth, 129. Atmosphere, 127. Moon, Eolipses of, 176. Celestial latitude, 200. Phalaka- Yantra, 213. Clepaydrn, 211. 12111808 of tho Moon, 206. Chakra, 212 Planots, 128, 185. Canon of Siucs, 263. | Day of Brahma, 163. Questions, 231. Day of the Pitris, 163. Days and nights, 161. Rising and setting of the heavenly bo- Doluges, 126. dies, 196. — Drikkarma, 110. ————— signs, 164, Driyautro, or gonius instrument, 221. Seasons, 228. Farth, 112, Soven Winds. 127. 11111118 diamelor, 122. Bighrochcha,109. Eclipses, 176. Signs, rising of tho, 164. Jpicycles, 144. Sphere, 107. Equation of tho centro, 141, 144. Sun, Eclipses of, 176. 1011078 of Lalla, &c. 169, 165, 205. Swayanvaha Yantra, or solf-revolving instrument, 227. Gnomon, 212. Syphon, 227. Horoscope, 166, 211. Time, 160. Kalpa, 108. i Kendra, 109. Winda, 137, Lagna, 166, 211. Year, 129. Longitudos, 212 Yugas, 110. ॐ Digitized by G oogle This book is a preservation photocopy It was produced on Hammermill Laser Print natural white a 60 # book weight acid-free archival paper which meets the requirements of ANSI/NISO Z39.48-1992 (permanence of paper) Preservation photocopying and binding by Acme Bookbinding अ 1996