8 ५.7 4 । प 4, ॐ TRANSLATION SURYA SIDDHANTA PUNDIT BAPU’ DEVA SASTRI, SIDDHANTA SIROMANI BY THE LATE LANCELOT WILKINSON, ESQ, C. 8. REVISED BY PUNDIT BAPU’ DEVA SASTRI, FROM THE SANSKRIT. CALCUTTA : PRINTED BY C. B. LEWIS, AT THE BAPTIST MISSION PRESS. ee 1861. ) £ (१ ५ ee ॥ F NEW YORK PUBLIC LIBRA) be ee es ~" १ ध pay | क & ५ > ’ funy &. Ss. FS Oh ASTCR, Ersoy ay TH DEN | = + ` iD 1608 5. GENERAL INDEX. Additive months, - sea 9८ wee Airs, Different kinds of, _... ० + Amplitude, The sine of, and the udayasta-sutra, ... Apogee, ea we as ००७ Apology, The Author’s, ४ ६ see Apsis, The reason for the invention of the higher, Seis. - Arc of sines, How to find the, ... x Armillary sphere, On the construction of the, 87,88, 106, 151, 209 —— Uses of the, ,,, ६ aus Ascensional difference and its place, ~ of the sun, ah sea —— The sine of the, Asterisms, The Bhogas of the, ४ ies Asterism, Rohini, crossing the cart of a planet Asterisms or principal stars of the Yoga-Taérds,... Astronomical Instruments, On the constuction of the, Astronomy, Origin of, ००५ ००३ ००० ००७ The science of. ०९१ ४ ००, ————-- Originally, taught by the sun, es Atmosphere, Extension of, ० * Aurigae, The longitude and latitude of the star Projdpati, Author’s birth-place, &c., ६ Autumn, The sératkéla or season of early, Avaha or atmosphere, The, Azimuth or vertical circle, ... Bhaskara, Brahma, Length of a day and night of , Period of his life, ४ ae Brahmagupta, Praise of, ... ae ae ee . 9 INDEX, Brahmanda, Dimensions of the, ... ie ००१ Canon of 81768, The construction of the, ae ० acs Circle, ५४ ‘ wel 209, ——-— To find the longitude of planets by the,... se — Centre of, ‘ay ave ee इ ee Circles, diurnal, ... us jee aes see -— of the 12 signs, 99६ oe ००७ ae Clepsydra, Kapala Yantra or, ses see ०० 91, Compass, the points of. Day, Determination of the place where the, becomes of 60 Ghati- kas, ; re Sus re 0: ume Terrestrial ‘dé ४8 विप we 9) 6, Subtractive, ia Sas. , tees ——— of the week begins, To find the instant when, ... ००९ To find the lunar, at a given time, see bee: has —— Period of ०७ ०० aie ose va —— of the gods, ४ és ak To find the Ruler of the present, ... a —— Subtractive, called Avama, the reason of, .., aa hes of the pitris, Length of the, ००७ --- of Brahma, The explanation of the, ००१ -- ४०१ night, Definition of the artificial, and the day and night of the pitris, ००९ . ०० and night, Determination of place and time of perpetual, ... Days and nights, Cause of increase and decrease in the length of, from the time the planetary motions commenced to the present midnight me + ००७ ००७ ००७ -- Sidereal and terrestrial, their lengths, aes es Lunar, be ७०५ aes ves ४ Degree, measure of, ... : ००७ Degrees of latitude are produced from the distance in Yojanas and vice versa, ... ४४ ees ee ree Deluges or dissolutions, _... oe ४ - Fourfold, ४ at ‘ie eee abs Demons, The day of the, _... ००१ ees ‘ae. wee The night of the, ००५ Ere चक The year of the, ooe eee eee Cee eee INDEX. Earth, Description of the, ००७ ०* Refutation of the supposition that the earth has suc- cessive supporters, _... ०० ९» ——-— Refutation of the objection as to how the earth has its own inherent power, eee sas eos ००५ Attraction of the, ... sae see 3४. - wie ——— Bauddhas, opinion of the,... ००७ भ ००७ -~——— Jainas, opinion of the soe ००९ ge. eee ——v- Refutation of the opinion of the Bauddhas, regarding the, ——-— Refutation of the opinion of the Jainas, regarding the,.., ——— Refutation of the supposition that it is level,... —- Reason of the false appearance of plane form of the, ——— Proof of the correctness of alleged circumference of the, ——-— Confirmation of the alleged circumference of the, ——-— Questions regarding the, ४ Ses ide, 19) —- Superficial area of the, .., ७६४ ००५ ——-— Middle line of the, ... ४ ६६ ste —— circumference of the, ... ६० ve ue its diameter and circumference, tee sues “as diameter of the shadow of, at the moon, ... ॥ Eclipse, Given the quantity of the eclipsed part to find its cor- responding time, ove ००७ To find the Valanas used in the projection of an, To find the Angulas or digits contained in the moon’s latitude,diameter,eclipsed part, &c., at a given time during an, 46 of the sun, Les ie = 48, 11], of the moon, ... eve ४2 41, ——— the science of, very secret, ... soe ००५ To find the magnitude of an, at ००* To ascertain the occurrence of a total, partial or no, To find the half duration of the, and that of the total darkness, To find the time of the phases of an, To find the quantity of the eclipsed part at a given time during the first half of an, .. To find the quantity of the eclipsed part at a given time during the latter half of an, ves 176 26. 56 42 ab. 43 26. 4 ४ ab. 4 INDEX. Page Eclipse, To mark the latitudes found at the beginning and end of an, ... es ves sis ,०„ 98 To find the magnitude of an, ‘ 54 The limit of the magnitude of the eclipsed portion which | is invisible in the solar or lunar, bs ५ wee 26, To find the path of the coverer in an, ... eee: wae 100 To find the direction of the beginning of total darkness by the projection of an, ... ‘ ० 59 To find the direction of the end of the total darkness 1b. The cause of the directions of the beginning and end of a solar, ‘ae २० 176 The cause of the directions of the beginning and end of the lunar, iss : ४ we AE The determination of the coverer in the, of the sun and moon $ ००९ see, oun 178 Eclipses, What covers the sun and the moon in them, we 42 — Projection of solar, 52 The directions of the beginning and end of the lunar and solar ४६ sas ves ०० 2९. To find the probable times of the occurrences of, ... 41 Kcliptic, the,... ` ane oes aes .. 88, 89, 153 ~ Variation of the, ... ० 184 To find the sine of the zenith distance of the culminat- ing point of the, ... ट ve ०४; = 48 — Four common points of the, ., ध 92 To find the Horoscope or the point of the, just rising at a given time from sunrise on ww. «39 —— The Madhya Lagna or the culminating point of the,... 89 Epicycle, Construction of diagram to illustrate the theory 9... 144 Construction of the mixed diagrams of the excentric and, २ ध ‘a ee ..„ cee 146 Epicycles of the sun and moon, ... we COAT Equation, The reason for assuming the manda-spashta planet as a mean in finding the 2 ends, __... sae 142 of centre, The principle on which the rule for finding the amount of is based, a ses + 241 Equator, The four cities placed at the, 9५७ we eee 80 INDEX. Equator, There is no equinoctial shadow at the, —————— Too find the rising periods of the signs of the ecliptic at the, on ४९ = ००१ ००० Equinoxes, the precession of the, ... ००७ ०५५ 29, Equinoctial, the, ... Bais i aes ००, 152, — how to make an, ... es ब — shadow, from the latitude to find the, ‘oe 6 — shadow to find the co-latitude and latitude, Ganges, Source of the, as ae tase Geographical, Anomaly, as curious fact rehearsed, .., Goladhyaya, sie ००५ eee Globe, equatorial, circumference of the, ... ४ इ Gnomon, the, १७ इ is (& 91, 209, — vertical, “es es ees Gnomonic shadow, ae as 9 व, Gods, the Mana of the, sii aes > ive —— the day of the, ००९ ०९ ine een a0 —— the night of the, ७ see aes ००७ —— the year of the, ‘ee ००७ ०० ००७ ००७ Grammar, in praise of, nee * Heavenly bodies, Rules for finding the times of the rising and setting of 76, ... ०९० sa sve Himalaya mountain, The, ... sie ००० ००० ०७ Horizon, the, ... ‘es ae eee Horoscope, the, ... és bes ee Hypothenuse, Given the shadow to find its, ——-— to find the, and the equation of the centre, —— explanation of the reason of omission of, in the manda process, = ००, ssi ००७ Instrument, the self-revolving Spheric, ०० — for measuring time, ०००. ००९ ००९ — ————— a self-revolving or Swayanvaha yantra, ००९ —— the praise of Dhiyantra or genius, ००५ Instruments, astronomical, ... se ie ००७ Jambudvipa, position of Mountains in the, ००० Juga, The number of days in a, abe ००० Jupiter, years of, see ००९ see oe ००९ 6 INDEX. Jupiter, The node of, wile es ee ०० ०७, ——— revolutions of, ... 8 ६४ ‘es aes Jupiter’s apogee, ... oes ee sos ००१ Kalpa, The length of, ... ०० ००७ a Krita yuga, solar years elapsed from the time when the planeta- ry motions commenced to the end of the last ०० tee Lagna, etymology of the word, ... ae wie Lalla, the error of, ००९ --- The wrongness of the Rule given by, ———- an error of, exposed, ... eee see wee eee ———- another gross error of, ... ‘es wet ove - cause of error in, and others stated, cae bogs slaps Latitude of a place, to find from the gnomon’s shadow, ee — rectified is ००१ ००९ 6 0 — celestial ६ Ses Ne 201, Latitudes, determination of, in which different signs are always above and below the horizon, ... | eae ‘ae Sas Lokas, arrangement of the seven, vee as ००, Longitude of the sun, how to find, ., my १2. -xas - of a place, how to find, vee ५ bee Lunar Mana, the, ae sus ह ०७५ — Mana, use of the, १०५ see ०१९ vee Mathematics, in praise of, ... ००९ ०४२ | Mathematical calculations, two kinds of, ... ००० Matters, Cosmographical, ... ००७ ०० ०७ eee Mars, 2nd equation of,... ee ००९ ००७ ——- epicycles of, we see a ia. 4 - nodes of, jee ४95 ies 0 ies न= revolution of, ध इ ae seer “ae Mars’ apogee, an ees ००७ ide Manu, the length of, __... sai see - Mercury, node of, oe see ४26 ००७ $ — revolution of, he (44 ae we Mercury’s apogee, ie ‘eis ve sie wee Meridian, The, how to determine, =, ४ aie: pee. line, $ : ६६ sis hat Meru, why due north of all places, ... one see INDEX. Meru, place of, sis ४ aes 79, ——- in [lavrita, position of the mountain, ... Wei: ewe Minute, measure 00 ... ies see ase ie Month, Savana, ... as sia ba bee: ४8 solar, oes see 4 ०० lunar, ... iss ous nis vos 2, 6, ‘period of, ००९ or ००५ one as ~~ intercalary, vee + ००५ to find the ruler of the present terrestrial, ... ००* length of lunar or lunation, ... ००७ ५७ the reason of additive called 4 00104888; = ०, see Months, seasons and year, ... vee ००७ ००* Moon, node of 116, = ०, eee ००५ ses ies — eclipse of the, ४ Mi os tee — on the phases of the, and the position of the moon’s cusps, diameter of the, see ies peer 68s ———— the colour of the eclipsed portion of the, ... ०* ——-— rules for finding the time at which the declination of the sun become equal to that of the,... ५२ baa? eae -——-— to find the true places of the, ss ०७० ` ०७ ——-— to find the time of daily setting of the, cee ०० -——-— to find the time of daily rising of the, vos —— — to find the phases of the, ies ००७ dee She ——-— cause of the phases of the,... इ ५ 111, to find the true diurnal motions of the, ... ४४, She -———— revolution of the, ewe aes 28 ००५ ~--- epicycles of the, ०७७ ४ ००० aes Case Moon’s parallax, aa wai ००७ eee * Moon’s apogee, ... ene ००१ ००५ ००७ ००७ Motion of planets, se क Gs Ses ‘i heliocentric, nee vee इ ——— different kinds of, ॐ eee avs इन ————~ decreasing retrograde, ००९ ०० ००५ increasing retrograde, = ,,, ००७ ०* cr direct, een ०७९ oes 9०५ eee eo ——— mean, (न stationary, an 8 INDEX. Night, determination of the place where the, becomes of 60 Ghatikas, eis Node, ... ४ ies : Nonagesimal, to find the sine and cosine of the zenith dis- tance of the, ... i 48 was dee. es Ocean, situation of the great, ... ध sae ००९ Orbits of Planets, es ans ies. “ees Parallax in longitude and that in latitude, ००५ 48, -- not being necessary in lunar eclipses, ००७ ————-- what is the cause of, and why it is calculated from the radius ofthe earth, ... a vos Age. ede Parallel sphere, $ श seis a ove and Right spheres, ०९५ ne ००७ Paravaha,... ००७ re aa: ००७ Parivaha, dis ee ००७ eh: ~ Persons, praise of,ingenious, ... ie ००५ ००५ Pitris, day and night of, ... ००० ** ०७० ०० Planet, to find the conjunction of a, with a star, ... ahs rectified mean place of a, —«.. ००७ ०७१ to find the motion of a minor, ar ats " Ss —— lst equation of, ... ee see and star, to know whether the time of conjunction is past or future, ... eee to find the mean place of a, at a given time, ... ०० to find the dimensions of the rectified periphery of the epicycles, ००० vee ००० . ००७ to find the time at which a, rises or sets heliacally, Reason of correction which is required to find the true, from the mean place of a, i Planets, to find the mean places of, .... ae sss. 20, to find apogees and nodes of, re we ——— an easy method for finding the mean places of, ... determination of the dimensions of the orbits of the, and their daily motion in Yojanas, : of their daily motions in minutes or angular motions, —— to find the radius of the diurnal circle of, ... ie ————— to find the ascensional difference of, ... bai Page 137 INDEX. 9 Page Planets, cause of the motions ef, ., ees a . 13 apsis of, ... श ase ६ See: ake 0 ——— observation of,... 9 as ४ ,.. 59 the fight and association of, ... see ved eter “OO which is conquered in the fight, ... sie ,०, 6. which is the conqueror, és es saa. Bea 00: rules for finding the true places of, ध se. 28 ——— deflection of, ००७ ०७ se we. aa at ——- attraction of, ... ४ ५ es ०० 14 ————— on the conjunction of the, with the stars, Sie. sae. (60 order of the orbits of the stars and, ies toe. 19 = and stars, on the heliacal rising and setting of the, ... 65 to find the length of a day of the, see ee. number of risings of, in 6 -—————— rules for finding the mean places of, = - ... १९ 1 motion eastward of, 2 द seks 46 to find the longitude of ‘es ses ww. 212 an illustration of the motions of, =... 128 -—— the minor 5, Why they require both the Ist and 2nd equations to their true places, i ww. 147 ———= how the lst and 2nd equations are to be applied .., 20 to find the true place of, sis बः eee «=D which set heliacally in the western horizon and rise =` heliacally in the eastern horizon, ७ ००* - 66 = why their mean and true motions coincide, wee vee 149 manner of observing the retrogression, &c. of, ww. (749 on the principles of the Rules for finding the mean places of, ५5 127 on the principles on which the Rules for finding the true places of the, are grounded ans rae © 19) the cause of variation of apparent size of, the ५18९8 of, 143 —— their conjunction with the sun, Hee == vee 56 on the conjunction of, ... ae coe rn kinds of conjunction of, isi „ 0, to find whether the time of conjunction is past or future, 70. ——— to find the time of conjunction from a given time, .., 26. conjunction the correction called the aksha-drik-karma, 57 10 INDEX. Planets, to find the distance of two, in the same circle of posi- tion, the apparent diameters of the, in minutes, Poetry, sweets of, Points of the compass, Pole, North, of the Earth, Poles, the inhabitants of the two, Projection of Kclipses, Quadrant Questions, Pras‘n4dhy4ya containing useful Miscellaneous, ... Radius of the diurnal circle, Rainy season, .. Retrogression of planets, Samviha, ne Sandhi, height of, Saturn, revolutions of, ba Saturn’s apogee, ... node, ... ‘a Seas and Dvipas, positions of the, Seasons, months and year description of the, ... Seconds, measure of, sii . 209, Shadow, determination at noon, of the direction of the gnomonic, Sphere, oblique, ... Sidereal month, (0 day and night, revolution, Mana, the, € Signs, positions where same are always invisible, Sine of amplitude, es Rules for finding the, of every degree from 1° to 90°, 15 way of refutation, of using the versed 190, Sines, versed sisi can. than Rules for finding the 24, viz. 3° ३, 7° 3, 11° 3, 15°, &.... ~~ Rules for finding the, of sum and difference of any two ares, Semi-circles, Page INDEX. Sines, on the canon of, Solar Mana, use of the, Sphere, oblique, ——— parallel ge ४२ wee ^ र Spheric, in praise of the advantages of, ००७ Spring, Star, the longitudes and latitudes of the Agastya, Mrigavyddha, Agni and Brahmahridaya र Stars, the position of the polar, soe to find longitudes of the principal of the Asterisms or, ... Summer season, the Grishma or, Sun, revolutions of the, ... es : — 1168 for finding the time at which the declination of the sun and moon become equal, —— to find the true places of the, --- to find the true diurnal motion of the, —— epicycles of the, a —— duration of the eclipse of the, —— Eclipse of, vide Eclipse. —— Rules for finding the time at which the declination of the moon become equal, with that of the, 5 —— and Moon, to find the time of their true declination —— question about the revolution of the, -- - diameters of the Moon and the, ... —— Apogee of the, eee ene —— Revolutions of the—in a year are less than the revolu- tions of stars by one, ... a —— the reason for finding the exact place of the, in order to find Lagna, : —-— declination and longitude of, how to find —-— the prime vertical of the, —-— mean place of the, to find, —-— zenith distance of the, at noon, ... —-— shadow of the, and its hypothenuse, ... —-— amplitude and the sine of amplitude of the, reduced — altitude, — altitude, zenith distance, &c, at given time from noon see eee eee Page 136 121 105 228 12 | INDEX. Sundial, a new ` ००, tas = ००७ how to use a, See as ies aes Supreme Being, the excellence of the, Suvaha, ४ इ “ive Syphon, description of a Syzygy, to reduce the places of the sun, the moon and her ascending node as given at midnight to the instant of the, Terrestrial Mana, its use, ४; ध ३५६ Terrestrial and Lunar days in a yuga, shadow equinoctial, Time, rules for resolving the questions on, , kinds of, ——, measurable (Murta), ... ——, immeasurable (Amurta), ... , number of kinds of, Triangles arising from latitude, ... Tropic, Terrestrial, oe क Sas a Udvaha, ६ sae ene Universe, the, sas eet oe aes sae. hes Unmandala or six o’clock line,... Venus, resolution of, se se ies See „9; ~, node of, ... sisi sae ——-, Apogee of, vee ००७ Virginis, of the stars Apamvatsa and Apa or, ... Sec lees Water, Observations in, ais ar Te ian Winter, hemanta or early, ... ba ००७ es es Winter S'isira or close of, Year, to find the ruler of the present terrestrial , solar, ... wie ~~, the season and months of the, ... im: (४: ` 9 ——, two halves of a tropical, ... sue , Length of the solar, ... ee oe Yuga, length of the great Say ००९ ध Yugas, length of the four small asi see Yuga, number of months and days in a subtractive and additive, 6 INDEX OF SANSKRITA Abhijit, Acharya, ... Adhara-Kaksha, Adhara- Vrittas, Adhimésa, 6, 130, 8, 131 129 Adhimésa-sesha, 131, 182 240, 241 ... 61, 88, 63, Aditya ~ ies Adhyatma, ... a Agastya 63, 68 Agn "63, 68, 119 Agra, 172, 111, 174, 251, 171 Agradi, । Agradi-Khanda, 174, Agragra-K handa, 174 Ahargana, 8, 131, 237, 239 232, 109, 236, 8, 234 235, 236, 29, 20, 130, 133, 240, 108, ९३ 4101६678, __... 2, Ahor4tra-Vrittas, ६8 Aindra Aksha, 188, 201, 196, 58 57, 191, 46 Aksha-Drikkarma, 110, 201, 69, ... die Akshaja, __... is Akshajy4, ००. ss Aksha-Karna, 252, 253, 254, 255, 256, 243, 217, 249, 256, 215, 173 Aksha-Kshetras, Aksha-Valana, 199, 194, 46, 198, 195, 192, 188, 197, 47, 190, 189, 187, 194, 185, 189, Aksha-Vritta, vs Alakananda, ... 244, 220, 198, 200, TERMS. 0.1६ १88१६, 130, Angulas, = ०० ००९ Amurta, wee ee Aniruddha, ... 77, Aiikusa, 227, Ansuvi-Marda, fee Antara-bhavana, es Antyajas,... “isa Antya, .. 201, 36,37, Anuradha, ... 68, 94, Anvala, Sep 123, Anya, ६ 257, Apa, श 65, Apém-vatsa, ... 65, Apasavya,... ०५० 4.108.811 6234188} ies Apséras, ee wee Ardra, ese 246, 68, Arupa, ies eH Aryabhatta, ... 122, Ashadha,_... 8, 94, 4816811६, ००, 75, 65, 94, Aspashta, 202, Aspashta-sara, Asta, 198, 197, 201 Astalagna, 20i, 198, 197 ` 89, ae sis Astamana, ... wee Asu, ४६४ wie Asuddha, ०. 167, 39, As’uras, 84, 82, 79, 85, 81, 116, 81, 95, 80, 83, Asus, 197, 198, 194, 168, 215, 161, 173, 188, 129 Aswina, 129, 8, 94, 68, 64 75 Atisighra, ... Ativakra, ... 15, Atyantika, ... see Ansa, ves re Avama, ००, 130, 131, e 2 INDEX OF SANSKRITA TERMS. | ` iP Ayana, 190, 191, 29, 196 203 Ayana-Drikkarma,110,202 Ayana-Valana, 198, 200 202, 197, 199, 185, 187 47, 195, 46, 191, 189 188, 192, 193 Bahu, 257, 172, 194, 71, 72 Baisékha, ... Balva, ses Bapudeva, Bauddha, Bava, Bh4-Bhoga, ... Bhédra, sis Bhadrapada, .. Bhadraswa, ... Bhadr4swa- Varsha, 80, 117 5068019 153, 156, 159 160 : ee0 113, 24, 25, 8 64, 94, 82 Bhaganas, ... 5 Bhagana-sesha, 239, Bhajya, ee 241, Bhanji, ‘ve ae Bharani, _«.. 65, 94, Bharata, ies 85, Bhérata-varsha, 120, 80, 117, 120 ses Bhasandhi , 3148119, 268, 223, 114 221, 218 Bhaskarécharya, 138, 136, 126, 124, 149, 142, 125 148, 119, 122, 108, 37, 182, 179, 240, 169, 158, 241, 258, 242, 247, 268, 233, 257, 209, 215, 205, 203, 201, 188, Bhavana, ; ००७ Bheda, sui ies Bhoga, we. 24, 25, 62, Bhogya, 167, 39, 168, Bhogya-khanda, 9 01108 $ ६7888 Bhuja, 185, 16, 30, 18, 31 27, 44, 28, 29, 27, 85, 17 22, 176, 173, 194, 172 245, 266, 267, 265, 258 222, 225, 217, 174, 224, © 68 258, 269, 219, 209, 215, 220, 208, 195, 194, 207, 175, 225, 219, 224, 223 219 Bhuja-phala, 148, 146, 19 145 : Bhujantara, ... Bhukta, oe Bhumi, ig Bhur, aus ee Bhurloka, _... wee Bhuvaloka, ... see Bhuvanakos’/a, ae Bhuvar, Bowris Brahmé, 4, 79, 78, 112, 127 189, 117, 95, 125, 118 91, 110, 116, 88, 178, 164, aoe eas Br&hmana, ... 9 Brahma-Maina, 150, 20 39, 40 129, Brahménda,... 112, 126, Brahmandee, ini Brahmagupta, 208, 201, 148, 122, 202, 209, =... Brahma-hyidaya, 69, 68, 6 iy Brahma-siddhaénta 285, Brahma-siddhanta-vit, ... Brischika, _... see Buddhitattwa, Chaitra, 131, 129, 228, 8 232 94 eee श Chaitraratha, sive Chakra, ans Ses Chakshu, _... * Chfépa ‘ Chara, 109, 215, 184, 183 Charajya 256, 161 Chara-kalas, ... sae Chara-khanda, 169, 16], 264, 165, 160, 1, Chatushpéda, ae Chaturaénana, eos Chaturyuga, ०००, Chheda, र 87, 49 Chhedaka, ... 176 Chitra, des 94, 68 Dainandjna, .., INDEX OF SANSKRITA Daityas, 110, 115, 162, 122, Dakshinéyana, 93, Darbha, 261, Danavas,_.... ध Danujas, Desantara, 11, 134, 109, 183, Dhanishtha, 61, 68, 65, 69 94 Dhanu, avs Dhara Dhivriddhida, Dhivriddhida, tantra, 128 237, 205, 236 aes Dhriti, Dhiyantra, Dhruvas Dhruva-yashti, Digvalanaja - Digvalanaja, tryasra, Diks, Divya me Dreshidnas, ... Dridha, Driggati, Driggola, = ,,. Drigjy4 Drigyé, see Drikkarma, 66, 58, 196, 110, 204, 205, 208, 197 200 129 245 221, 260, 91, 153, 85, 171, 37 Drikkarama-vasana, 196 Drikshepa, 50, 49, 183 Drikshepa-vritta, 152 Driksutra, ot Drifiimandala, (४४ 0 कक्षा) ०, 110 Dwipas, 116, 80 Dyujya, 110, 159, 189, 197 200, 251, 20], 24 os Dynjy4 194, 123, Gabhastimat Gamya, aie Gandanta, =... sah Gandhamédana, Gandharva, .., _ Ganes’a, ree Ganitadhyaéya, 208, 258 257, 144, 261, 247, 145, 125, 148, 158, 126, 156, 25, Page 112 163 150 126 112 12 TERMS. Garaja, vee Garuda, Gata, ids Gatiphala, ... Gatis : Ghati, 211, 209, 164, 167 211, 256, 216, 214, 130 159, 168, 218, 168, 184 256 Ghatika, 2, 128, 254, 25 11, 184, 24, 49, 93, 250 254, 210, 48, 57, 43, 49, 68, 76, 25, 218, 130, 128, 83, 1, 40, 129, 44, 12, 45, 49, 74, ... 25, Golabandh Golédhydya, 105, 262, 37 Gomedaka, ... Grahana,... ses Grahayuti, sas Grishma, 229, Guhyakas, ... Yaa: Hara, ade 242, Harivarsha, ... see Hasta, । 94, 65, Hemanta, ,,, 93, Hemakuta, ... ‘as Himflaya, ... ००९ Hiranmaya, ... oes Hiranya-garbha, ०० Hriti .,. 201, 252, Tlavrita, 4६ Ilavrita-varsha, eis Indra, ..„ 80, 119, 184, कि वि Ishta, dee ee Ishta-hriti, ... 251, Ishtantya, ... 251, Jaina aoe 114, Jambu धी sete Jambudwipa, 116 Jambunad{, ... sae Janaloka,... ४ Jina-vritta, ... Jyé-bhavana, J yautishopanishat Tyeshthd, 241, 242, 129, 8 75, 68 Kachnér see 253 4 INDEX OF SANSKRITA TERMS. Kadamba, 229, 190, 118, Kadamba-bhrama-vritta, 192, ६ es Kakshavritta 137 Kala, 254, 252, 253, 251 133 श 16147823, ,,, 66, 67 Kali, nat 110, 3 Kalpa, 235, 238, 242, 240, 130, 233, 234, 238, 4, 157, 29, 95, 7, 4, 86, 118, 131, 132, 130, 127, 108, Kalpa-adhimasas Kalpa-bhaganas, 239, Kalpa-saura, ... 243, Kama-deva, ... os Kanya, ses 8, Kanishtha, ... 241, K 4pala-yantra, os Karma, 132 Karana, ga 93, 26 Karna, Karan, 34 Karka, ००9 8 Karkyadi, 142, 141, Karky4di-kendra 142, Kartika, 9, 4, 129, 8, Kaseru, ae : Katahas,__... ane Kaulava, ; Kausa Kendra, 21, 144, 159, 146 158, 18, 19, 150, 142 148, 16, 145, 14], 109, 140, 20, 109, 16, _... Kendra-gati, 146, Ketumala 84 Ketumala varsha, 80, 119 Khagola, 151, 152, 153, 160 Khakaksha, ... Khanda, क 175 11021088, .., 117, 217 Khandakas, ... 215 K hecharagola, i Khila, ea 240, Kinnaravarsha, sii Kinstughna, .., ०० Kokilas, iets ate Kona-saiiku,... 171, 34, Page 191 191 139 215 68 108 2 35 Kona-vrittas, 152, Koti, 185, 16, 142, 70, 185, 141, 27, 265, 222, 223, 267, 208, 209, 144, 207, 202, 208, 18, 175, 174, 44, 172, 173, 194, 224, 45, 225, 176, 44, 17, 71, Koti-phala, ... 19, 145 Krama Kranti-pata, 157, 154, 153 Krantijya, Krauncha, ... eis Krishna, 3५8 ‘ee Krishna-paksha, sas Krita, 108, 3, Krita-yuga, ... 4, 8, 1, 10, Krittiké, ... 65, 68, Kshepa, 241, 157, Kshepa-vritta, 157, 155 Kshepa-patas, Kujya, 110, 23, 174, 175, 176, 160, 215, 255, 254 256, 252, 249, 25], 248, Kulachalas, ... wate Kumarika, ... < Kumbha, _... 8 Kuru . 84,1457, Kurukshetra 134, Kuru-varsha, Has Kusakasa,... 1 Kuta, 95 - Kuttaka, 242, 236, Lagna, 250, 167, 200, 210, 89, 197, 166, 211, 20], 167 ee ‘ Lakshmf, ,.. Lalla, 108, 128, 149, 205, 188, 169, 205 Lambana, 111, 183, 182 181 : me Lam bana-kalas, Lamba-rekha, 217, 218, Lank&, 89, 109, 80, 115, 9, 134, 10, 133, 182, 82, 11, 20, 243, 184, 118, 120, 117, 108, 8 ००५ Lankodayas, .., ave Lilavati, 136, Loka, ००९ Page 117 38 173 268 116, 126, 120 INDEX OF SANSKRITA TERMS, Lokaloka, श Madhya, ... 12, Madhya-gati 134, 1, Madhya-gati-vasana, =... Madhyajya, 48, 49, Madhya-lagna, "gee Madhyama, .., ००० Madhyarekhaé, =. soe Madhya-safiku vee Magha, 94, 129, 8, 68, 65, 212 ०० Mahan, ४ Mahéhrada, ... Maharloka, ... ad Mahésafiku, 257, 252, 254, Mahattattwa, na Mahayugas, ... Mahendra, ... Maheswara, ... Makara, Malatt, Mallika, me Malaya, 120, Malyavan, ... Mana, 94, 96, 95, 93, 91, 92 Manasa ae Manda, 17, 142, 147, 21 15, 20, 14, 154 sa Mandakendra, Mandaphala, 109, 19, 150 137 $ १ Manda-prativritta, 142, Mandspashta, ae Mardardha, 44, 43, nas Mandira, 118 Manda-spashta, 157, 138 154, 155, 21, 137, 142 143, इ Mandatara, ... 14 Mandochcha, 13, 14, 10, 16 109, 146, 143, 137, 16 109 vei Manu, 108, 7, 3, Manus, Seg 7, Manwantara, Margasirsha,,... 129, 94, Masha, 77, 112, Page 126 89 209 127 216 166 89 134 171 Maya-Asura, Meru, 119, 79, 162, 121 117, 120, 119, 81, 134 118, 88, 163, 84, 169, 85, 117, 114, 80, oe Mesha, we 181,.129, Mina, ००७ 8, Misra, ies ae Mithuna, _... 8, Mithuna-sankranti, a Mriga, ६ 68, 64 Mrigédi, ०० 140,141 Mrigasirsha, ... Mrigavyadha 63 Mula 9 94, 65 Muni क ee Munjala, pis Murta, sas 2, Nadana, re ee Nadi-valaya, .. 209, Naga, . 25, 120, Nairrita, 118 Nakshatra, 246, 75, 24 Nakshatra Ahoratra, ... Nakshatra masa, ५ Nakshatra-vritta, न Narak Nata, 187, 195, 189, 190 Nataghatijya, Natajya, Natakarma, ... Nati, 111, 182, 188, 181 Nati-kala, ... 183, Navansas, __... ee Nichochcha, ... Nichochcha-vritta, 138, ६८१ es Nila, Ses aes Nishadha, ष Paksha, sins ves Pala, 4 ,९४ sie Palas, 129, 214, 215, 25, Palabha, 251, 249, 248, 245, 244, 243, 250, 256, 246, 254, 252, 253, 258, 220, 222, 255, 259, 219, 215, 161, 30, 178, 81, es Palétmakas, ... Para, 143, 215 112 6 INDEX OF SANSKRITA TERMS. Pariyatra, Parama-lambana, sae Parvata, ६४ Pata, 159, 261, 154, 158 155, 13, 153, 75, 72, 73 Patadhikara, ... 75, Pata-kala, 75, Patas, ००, 159, 158, Patala, a 116, Patdéla-bhumis, ‘a Pati, as 268, Pattika, 215, 214, 218, Paurapika, 114, 126, Pausha, : 8, 129, Phala, 34, 35, 138, 144 Phalaka, 214, 209 Phalaka-yantra, 216, Phalguna, .. 94, 129, Pippala, Pitris, 110, 163, 76, 110, 85 94, 162 Pitrya, nee ००७ Préchyapara,... ००* Prajapati, ... 65, Prajapatya, ... 95, Prajapatya-ména, ००७ Prakasa, ae cae Prakritika, ... ; Prakriti, ey Pralaya, Prana, Pranas, 238, 40, 37, 39, 68 66, 2, 70, 38, 24 ies Prasnadhyaya, Sie Prathama, ... 257, Pratibhagajyaka-vidhi, 267, Prativritta, 137, Pravaha, 14, 72, 13, 150, Pravipalas, ... Punarvasu, ... 68, 94, Puranas, . 96, 178, Purnimé, ° ‘a Purusha, arte see Purva Purva-Bhadrapada, 94, 64 Purva-phaélguni, 68 Purvashadhaé, 68, 94, 63, Pushkara, ,,, es Pushpavati, ... Page Pushya, ,,, 212, 68, 65, Rahu, : 178, Ramyaka, Rasi, a Rasis, eas Rasi-vritta, ... Rasyudaya, ... Revati, 212, 5, 3, 65, 75 94, Rig-veda,... oo Rikshaka, ... र Rohini, ००, 68, 64, 65 Romaka, 9६ Romakapattana, 118, 117 115 Rujya, see ie Sahya, 120, S/aka, os S’akuni, we S/alivéhana, ... 108, 84101819 S4ma, 193, 252, 255, 14, 15 8411088, Samasa, 53, Samasa-bhavana, ode Samagama, .. Samasanka, 111, 171, 253 175, 174, 249 Sambhu-Horaprakasa, Samputa, ... Sama-kala, ... vo Samvatsaras,.., 92, Sama-veda, ... a Samslishta, ... 9 Sanhitikas, ... ae Sanhita, ०० Sanaischara, ... Sandhis, ... 8, 108, 7, Sandhya, ,,. 108, Sandhyanga, ... 108 द 8, .. Sankranti, 130, 93, 8, 92 Sankrantis-karka, Safikarshaga, Safiku, 36, 37, 28, 171, 49 172, 173, 170, 175, 176 174, 252, 220, 219, 255, 176, 245, 257, 172, 28, 172, 35, Safikutala, 29, 194, 4 94 13 117 INDEX OF SANSKRITA TERMS. Sanmya, ००७ “0 Sara, 197, 200, 192, 202, Sarat, wee wee Saratkéla, ,,, ४ Saraswati, ... Saura, 129, 131, 182, 243 235, 282, 242 8४४9119, 2, 170, 250, 130 131, 129, 95, 6, 170, 169 171, व ee Savana-ghatis, ००७ Savita, Ge Sashadbha, ... ges Sasi, sas ४ aes Satatéré, ... a Satatdraka, ... 212, Satyaloka, ... see Sesha, dea See Setha, = Shadasiti-mukhas, Shadvarga, ... one Siddh Siddhapura, 115, 120, 118 $2 : sisi Siddha-puri Siddhas, 79, 118, Siddhénta, 262, 175, 165, 105, 170, 110, 156, 187 96, 185, 182, 166, 250 207, 180 Siddhantis, ... Siddhanta-Siromani, 10, 109, 87, 14, 16, 149 143 Sighrochchas, 21, 158, 138 Sighra-karna 154 Sighra-kendra, 150, 159 109 Sighra-phala, 109, 19, 19 158 Sighra-prativritta, 138, 157, Sighratara or Atisighra, ... Sihgra-dhivriddhida, Sinha, ००७ 81101011 see Sisira, 262, Sighra, 147, 142, 18, 19, 15, Sighrochcha, 138, 158, 146, 236, 159, 5, 6, 22, 13, 129, 126, 93, Page 120 184 98 230 107 180 8118, sae 5४ Siva 113 Slokas, 8, 13, 81, 61, 29 92, 97, 75, 79, 92, 28, 86 42, 81, 57, 94, 19, 148 Soma-Siddhanta, Somamandala, Spashta, 142, 188, 47, 155 46, 159, 158, 154, 182 205, 197, aes Spashta-paridhi Spashta-patas, 156 ‘ Spashta-séra, Spashta-valana, 191 190 wee Spashta-valana-sutra, .. Sphata, 71, 11, 286, 14] 142,41, ... Sphuta-gati, ... Sphuta-kaksha, Sphuta-koti, ... Sphuta-lambana-lipta, =, Srévana, 8, 63, 61, 68, 65 129, 94 Sridharécharya, ७०४ 311168ए810, = -.. ००५ Sringonnati-vasana, 209, Sripati, wae 188 Sthityardha, ००७ Suchi, wae = Sugandha, ... 119 Sukla, so 1:7, Sukla-paksha, soe Sukti 3५ ००० Sumeru, cee bee Suparswa, ... S‘urya, S‘urya-siddhanta, 209, 157 96, 21, 26, 97, 75, 12 Sutra, 254, 252, 220, 251 Sutaka, we Swarloka, Swati, 8 Swayanvéha., ... sae Swayanvaha-yantra, ०५० 8५61४ Taddhriti, 256, 249, 255 85, 154, 153, 197, 185; 146, 68, 69, 8 INDEX OF SANSKRITA TERMS, 248, 254, 253, 255, 252 256, 175 Taddhriti-Kujya, Taitila, se ८४ Tala ००९ Tamraparna Tantra, 237, 261, 108, 286 Tapoloka,... ‘es Tatkélika, sei Timi, . 26, 71, 27, Tithis, 181, 132, 25, 24, 93, 260, 25, 24, 184, Tithi-kshaya, Treta = Tretéyuga, ... Trijya, Trinsanoas, ... Triprasna, ... Triprasnadhyayas, Triprasna-vasana, Trita, Trysra, Tula, Tuladi, Turiya, Turti ००९ Uchch Uchcha-rekhaé, 189, 144 141, 145, ... 108, 40, Udaya, 48, 198, 200, Udaya-lagna,... 98 Udayasta, Udaydsta-sutra, 172, 219 172, 220, ... Udayantara, 133, 132 Udi 17] Ujjayin{, 184, 248, 11, 253 15 Ullekha, Unmandala, 175, 152, 161 165, 162, 164, 174 Unnata, Upavritta, ... Upavritta-trijya, Uttara 64 Uttara-Bhadrapada, 94, 64 68 Uttara-phaélguni, 64, 68 Uttardshaédha, 68, 64, 61,94 252, 254, 46, Page 174 174 25 . 227 120 205 ` 120 12 Uttara-yana, ... 93 Vadavanala, ... 116, 122, Vaibhraja, ... ७ Vaidhrita, a Vais/Akha, 94, Vaivaswata, ... 4, Vayu bes Vakra 15 Valana, 205, 2038, 194, 197 195, 46, 203, 196, 192 193, 187, 190, 191, 189, 184, 46, 47, 58, 51, 54, 188, 45, 185, 186, 46, 47 52 Valanas, 205, 45, 196, 188 Valana sutra, Vanija, ००७ Varga-prakriti Varshas, 5, 71, 93 Varuna ००, 118, 120 Vasana, 176 Vasana-Bhashya, 142, 179 233 Vasanta, Vasishtha, Vaskara, Vasudeva, = ०० Vata, wee eee Vedas, 178, 107, 77, 79 78, 262, ... Vedavadana, ... Vidy4dhara, ... Vigraha, Vijaya, ००० Vikala, es Vikalas, ४ Vikshepa, ०० 14 Vikshepa-kendra, 156, 155 154, 158 Vikshepa-vritta, Vimandala, ... 156, Vindhya,... ०० Vipalas, ‘es Vipula, ove Visakhé, . “68, 94 Vishkambhas, 119 Vishpu-padi, ... 53 191 203 25 241 117 119, 134 136 93 139 107 INDEX OF SANSKRITA TERMS. 9 Page | Page , Vishnu, ne 118, 119 Yantrddhydya, 228, 209 - Vrischika, ... ... 129 Yashti, 217, 215, 201, 202, Vrishabha, ... 8, 129 160, 209, 220,219, ... 218 Vrishabhasankranti, .» 180 Yashti-chinha, .. 218 Vritta, ee ०० 147 Yoga, ०० 25, 75, 24, 90 Vyasa, sie .. 119 Yoga-téré,... 65,62, 64 Vyatipata, ... 72, 75 Yojanas, 244, 183, 83, 11, Vyomakaksha, eee . 19 126, 87, 86, 49, 125, 84, Vyulkrama, ... 167, 39 134, 122, 207, 127, 86, Yajamas, a. ०० 150 121, 41 Yajur-veda, ... „=, 78 Yuga, 260, 286, 108, 7, Yama, aes 118, 119 236, 6, 5,9, 3, 29, 180, Yama-koti, 120, 243, 80, 109, 110, 237,8,2,110,4, 108 118, 117, ... .. 115 Yuganghri, ... 110, 108 Yamyottara-vritta, ,०, 152 Yujanas, ,,, 115 Yantra, „०, 216,227, 91 १५ ५ 4a 1 = क (श ४ न ro = oe = = ~ a * . . ५ ५ + ध ५ ५ ध ४ + ' red r ye TRANSLATION OF THE SURYA SIDDHANTA. Digitized by Google CONTENTS. Page CuHaptER I.—Called ^ एप ^-^ which treats of the Rules for finding the mean places of the planets Cuapter II.—Called Spuuta-eati which treats of the Rules for finding the true places of the planets, Cuapter II[.—Called the Tripras‘Na, which treats of the Rules for resolving the questions on time, the position of places, and directions ४९ Cuaprer IV.—On the Eclipses of the Moon, sis CuapteR V.—On the Eclipses of the Sun, ४ धि (प ^ एए VI.—On the projection of Solar and Lunar Eclipses, घ ^ ए VII.—On the conjunction of the planets, Cuapter VIII.—On the conjunction of the planets with the stars oe “ Cuarter IX.—On the heliacal rising and setting of the planets and stars CuarTer X.—On the phases of the Moon and the position of the Moon’s cusps,__... or ४ (^ एए XI.—Called PATADHIKARA, which treats of the Rules for finding the time at which the declination of the Sun and Moon become equal aes ase CuapterR XII.—On Cosmographical matters CuHapTerR XIII.—On the construction of the armillary sphere and other astronomical instruments, CuapterR XIV.—On kinds of time, Postscript by the Translator, ५८० ग ५०० ख + क 9 9900 ^ BELLA 1 18 26 41 48 52 56 61 65 69 72 76 87 91 96 TRANSLATION OF SU'RYA-SIDDHANTA. CHAPTER I. Called Mapuya-aati which treats of the Rules for finding the mean places of the planets. 1. Salutation to that Supreme Be- ing which is of inconceivable and imperceptible form,void of properties (of all created things), the external source of wisdom and happiness, and the supporter of the whole world in the shapes (of Braum4, Visunvu and Siva.) 2&3. Some time before the end of the Krira एए6५, a great Demon named Maya, being desirous of obtaining the sound, secret, excellent, sacred and complete knowledge of Astronomy, which is the best of the six sciences subordinate to the ४ 74, practised the most difficult penance, the worship of the Sun. 4. The self-delightful Sun, being gratified at such (difficult) penance of Maya, bestowed on him the knowledge of the science of Astronomy which he was inquiring after. The illustrious Sun said. 5. (O Maya,) I am informed of your intention (of attaining the knowledge of the science of Astronomy) and pleased with Invocation. Introductory. your penance. I, therefore will grant you the great knowledge of Astronomy which treats of time. 6. (But since) nobody can bear my light and Ihave no time — to teach you (the science,) this man who partakes of my nature will impart to you the whole of the science. B १ Translation of the 7. The God Sun, having thus spoken to, and ordered the man born from himself (to teach Maya), disappeared. That man spoke to Maya, who stood bending and folding his hands close to his forehead, in the following manner. 8. (O Maya), hear attentively the excellent knowledge (of the science of Astronomy) which the Sun himself formerly taught to the great saints in each of the Yuaas. 9. Iteach you the same ancient science, which the Sun himself formerly taught. (But) the difference (between the present and the ancient works) is caused only by time, on account of the revolution of the Yuaas. 10. Time is of two kinds ; the first (is continuous and endless which) de- stroys all animate and inanimate things (which is also the cause of creation and preservation), the second is that which can be known. This (latter kind of time) is also of two kinds; the one is called Mérra (measurable) and the other is Amtrra (immeasurable, by reason of bulkiness and smallness respec- tively). Kinds of time. 11. The time called Mtrra, begins with Priya (a portion of time which contains four seconds,) and the time called AmUrra begins with (ष्णा (8 very small portion of time which is the ड ठ) part of asecond.) The time which contains six 174 48 18 called @ Paua, and that which contains sixty Patas is called a GHATIKA, Pala and Ghatiké. 12. The time, which contains sixty Guatikfs is called a NAksHatTra AHO- RATRA (a sidereal day and night) and a Ndxsuatra MAsa (a sidereal month) consists of thirty NAxsaarra AHORATRAS. Thirty SXvana (terrestrial) days (a terrestrial day being reckoned from sun-rise to sun-rise) make a SAvana month 13. Thirty lunar days make a lunar क न month, and a solar month is the time and the Divine Day. : which the Sun requires to move from Day and Month. Surya-Siddhanta. 3 one sion* of the Zodiac to the next. A solar year consists of twelve solar months ; and this is called a day of the Gods. 14, An ^ 04774 (day and night) 1 7 4 of of the Gods and that of the Demons are mutually the reverse of each other, (viz. a day of the Gods is the night of the Demons; and con- versly, a night of the Gods is the day of the Demons). Sixty 6 प् ०६८7१५8, multiplied by six, make a year of the Gods and Demons. 15 & 16. The time containing twelve व length of 9 great thousand years of the Gods is called a Cuaturyuaa (the aggregate of the four yuaas, Keita, एण, Dwdrara and Kati). These four yugas including their SanpnyMt and San- DHYANS A contain 4,320,000 solar years. The numbers of years included in these four small ई ०6५५8 are proportional to the numbers of the legs of DHarmat (virtue personified). 17. The tenth part of 4,320,000 amet त OF the four the number of years in a great ruaa, multiplied by 4, 3, 2, 1 respectively make up the years of each of the four rucas, KrytTa and others, the years of each yuaa include their own sixth part, which is collectively the number of years of SanpHyf and SanpHy4ns’a, (the periods at the commencement and expiration of each yuaa). The length of a period 18. (According to the technicality of ed Manv and that of its the time called Mérra,) 71 great yuaas न (containing 306,720,000 solar years) constitute a Manwantara (a period from the beginning of a # It is to be observed here that the signs Aries, Taurus, &c., are reckoned from the star Rrevati (¢ Piscium,) and a solar year corresponds to a sidereal year. B.D + These two words will be explained in the sequel, B.D. ‡ It is stated that Dharma stands with four legs in the Keyra, with three legs in the Trt, with two legs in the DwApaka and with one leg in the Karr. Therefore the number of the years of the Kuita, 18774, Dwdpara, and Katt are proportional to 4, 3, 2 and 1 respectively. 8. D. B 2 4 Translation of the Manu to its end) and at the end of it, 1,728,000 the whole number of the (solar) years of the Krita, is called its SanpH ; and it is the time when a universal deluge happens. 19. Fourteen such Manus with their Sanpgis (as mentioned before), constitute a Kapa, at the beginning of which is the fifteenth Sanpu1 which contains as many years as a Krfra does. 20. Thus a thousand of the great yucas make a Kapa, a period which destroys the whole world. It is a day of the God Braum(, and his night is equal to his day. 21. And the age of Brau con- sists of a hundred years—according to the enumeration of day and night (mentioned in the preceding 87074}. One half of his age has elapsed, and this present Kaxra is the first in the remaining half of his age. 22. Out of this present Kapa six Manus with their Sanpats, and twenty-seven yuaas of the seventh Manu called Vaivas- waTa have passed away. 23. Of the twenty-eighth great ए ०६५, the Kryta # ए५ has passed away. Let (a calculator,) reckoning the time from the end of the KrjTa compute the number of years passed. 24, 47,400 years of the Gods have elapsed in the creation of the God Braum(, of animate and inanimate things, of the planets, stars, Gods, Demons, &c 25. Now the planets (such as the Sun) being on their orbits, go very rapidly and continually with the stars towards the west and hang down (from their places towards east) at an equal distance, (i. e. they describe equal spaces daily towards the east,)* as if overpowered by the stars (by reason of their very rapid motion caused by the air called PRAVABA.) The length of a Kapa. The lengths of a day and night of the God BranmMA The period of his life and that of his passed age. How the planets move eastward. The Hindu Astronomers suppose that all the planets move in their orbits with the same velocity D Surya-Siddhdnta. 5 26. Therefore, the motions of the planets appear towards the east, and their daily motions determined by their revolutions (by applying the rule of proportion to them) are unequal to each other, ’in consequence of the circumferences of their orbits ; and by this unequal motion, they pass the signs (of the Zodiac.) 27. The planet which moves rapid- ly, requires a short time, to pass the signs (of the Zodiac,) and the planet that moves slowly, passes the signs (of the Zodiac) in a long time. Buracana means that revolution through the signs (of the Zodiac which a planet makes by passing round) up to the end of the true place of the star called एर (¢ Piscium, from which end they set out.) Bhagana or a sidereal re- volution. 28. Sixty Vixa.as (seconds) make a Kalé (a minute) and sixty minutes constitute an Ans’A (a degree.) A 487 (a sign) consists of thirty degrees and just twelve 14878 (signs) make a Bhagana (revolution.) The circular measures. 29. In a great yuaa each of the ae aa ae व oT Veta planets, the Sun, Mercury, Venus and and the S’ighrochcha of the S/faHrocucHa (i. e. the farthest a peer = a +. point from the centre of the Earth in the orbit of each of the planets) of Mars, Saturn and Jupiter moving towards the east make 4,320,000 revolutions (about the Harth). 80. There are 57,753,336 revolu- tions of the Moon and 2,296,832 reyo- lutions of the planet Mars. 31. There are 17,937,060 revolu- Of Mercury's Sighrochcha {008 of the S'faHRocucua of the planet and Jupiter. Mercury* and 364,220 revolutions of Of Moon and Mars. the planet Jupiter. * The revolutions of the Sighrochchas of Mercury and Venus correspond to their revolutions about the Sun. B. D. 6 Translation of the 32. There are 7,022,376 revolu- tions of the S’faHrocucua of the planet Venus* and 146,568 revolutions of Of Venus’s S’ighrochcha and of Saturn. the planet Saturn. 33. In a great yuaa, there are a. Moon’s Apogee and 488903 revolutions of the Moon’s Manpocucaa (apogee,) and the number of the retrograde revolutions of the Moon’s ascending node is 232,238. 34. There are 1,582,237,828 sidere- 0 ee al revolutions in a great yuaa (a sidereal 0 pale revolution is the time from one rising ofa star to the next at the equator and it is a sidereal day as mentioned in the twelfth S’toxa.) These sidereal revolutions diminished by each planet’s own revolutions (before mentioned) are its own risings in a great yuGA. 85. The number of Lunar months ay 1 gta hoa is equal to the difference between the Hi अ additive months revolutions of the Moon and those of the Sun; and the remainder of the Lunar Months lessened by the Solar months is the number of AvuimAsas (additive months.) 36. Ifthe Savana (terrestrial) days coe न oe be subtracted from the Lunar days, yoga and the definition of the remainder constitute the days a terrestrial day. called the Tirsi.ksHayA (subtractive days.) There the Savana days are those in which a Sivana day or terrestrialt day is equal to the time from sun-rise to sun-rise (at the equator). 37. There are 1,577,917,828 terres- Pah of terrestrialand Iunar trig) days and 1,608,000,080 lunar days in a great YUGA. # The revolutions of the Sighrochchas of Mercury and Venus correspond to their revolutions about the Sun. B. D. + A terrestrial day is that which the English call a solar day. 2. D. Surya-Siddhanta. 7 38. (In a great yuaa) there are No. additi the and a ॥ rear Sabie clive day ०५ 1,598,336 additive months and 25,082,252 subtractive days. 89. There are 51,840,000 Solar No. of Solar months in a j yuaa and the way to know months ina great yuca, and the ter- the No. of terrestrial days, = trial days are the sidereal days diminished by the Sun’s revolutions. 40. The revolutions of the planets, the additive months, the subtractive days, the sidereal days, the lunar days and the terrestrial days (mentioned above) separately multiplied by 1000 make the revolutions, the additive months &c., in a Katpa, (because a Katpa consists of 1000 great ए ०५५8.) 41 & 42. In a Kapa, there are 0: the 387 revolutions of the Sun’s Apogee (about the Earth), 204 of Mars’ apogee, 368 of Mercury’s apogee, 900 of Jupiter’s apogee, 535 of Venus’ apogee and 39 of Saturn’s apogee. Now we proceed to mention the retrograde revolutions of the Nodes (of the planets Mars, &c.) 43 & 44. There are 214, 488, 174, 903, 662 revolutions of the Nodes of the planets Mars, Mercury, Jupiter, Venus and Saturn respectively. We have already mentioned the revolu- tions of the apogee and node of the Moon. 45, 46 & 47. Collect together the The number of the solar years of the six Manus, with their six years elapsed from the time ch ies ; when the planetary motions SANDHIS, and the SanDHI which lies in commenced, to the end of oe, it Shia last Krita-vUGA. the beginning of the Kapa, those of twenty-seven great yuaas of the pre- sent Manu named Varvaswata and those of the Krita yuaa ; and subtract from the sum, the said number of years of the Gods, reduced to solar years, required (by the God Brahmé) in the creation of the universe, (before the commencement of the planetary montions,) and the remainder 1,958,720,000 is the number of solar years before the end of the Kryra yuaa. 8 Translation of the To find the Auarcana or 48: 170 1,953,720,000 the number the No. of terrestrial days of elapsed years, add the number of from the time the planetary motions commenced to the years elapsed (from the end of the last 1. Krfta yuaa to the present year;) reduce the sum to months (by multiplying it by 12 ;) to the result add the number of lunar months from the beginning of the light half of the CuaiTra* (of the current year to the present lunar month.) 49, Write down the result separately; multiply it by the number of additive months (in a yuaa) and divide the product by the number of solar months (in a yuaa) ; the quotient, (without the remainder,) will be the elapsed additive months. Add the quotient (without the remainder) to the said result, reduce the sum to days (by multiplying it by thirty) and increase it by the number of (lunar) days (passed of the present lunar month). 50 and 51. Write down the amount in two places; (in one place,) multiply it by the number of subtractive days (in a yu@a) ; divide the product by the number of lunar days (in a yuea) and the quotient (without the remainder) will be the number of elapsed subtractive days. Take the number of these days from the amount (which is written in the other place) and the remainder will be the number of elapsed terrestrial days (from the time, when the planetary motions commenced) to the pre- sent midnight at LanKA.t * That lunar month which ends, when the Sun is in Mzswa (stellar Aries) the first sign of the Zodiac, is called CHatrra, and that which terminates when the Sun is in VaisHaBpua (Taurus) the second sign of the Zodiac, is called Vais‘AKHA and so on. Thus the lunar months corresponding to the twelve signs Mesha (Aries,) VrisHaBHA (Taurus,) MirHuna (GEMINI,) Karxa (Cancer,) अक्र्त (Leo,) KanyA (Virgo,) TuxA (Libra,) Vris’cuixa (Scorpio,) Duanu (Sagittarius,) Makara (Capricornus,) KumByHa (Aquarius) and Mina (Pisces, ) are CHAITRA, Vals‘AkHA, JYESHTHA, AsHADHA, S’rA4vana, BHADRAPADA, As’wina, KArtika, MAre@asirnsHa, Pausua, MAGua and PHALGUNA. If two lunar months terminate when the Sun is only in one sign of the Zodiac, the second of these is called ADHIMA8A (an additive or intercalary month.) The 30th part of a lunar month is called Tithi (a lunar day.) ए. D. + The proof of the process for finding the AHARGANA stated in the S’LoKas from 48th to 51st will be clearly understood from the following statement. In order to find the AHaRr@ana, let the number of the Solar years elapsed be multiplied by 12; and the product is the number of elapsed solar months to the last mean MrsHa SANKRANTI (i. e. the time when the mean Sun enters the first stellar sign of the Zodiac called stellar Aries ;) to this letthe number of passed Surya-Siddhanta. 9 From the number of these elapsed days, the Rulers of the present day month and year can be known (by reckoning the order of them) from the Sun. Divide the number of elapsed ter- restrial days by 7, and reckoning the remainder from the sun-day, the Ruler of the present day will be found. a Te 52. Divide the number of elapsed present terrestrial monthand terrestrial days by the number of days ue in a month and by that in a year (i. 6. by 30 and 360) multiply the quotients (rejecting the remainders) by 2 and 3 respectively, and increase the products by 1. Divide the results by 7, and reckoning (the order of the Rulers) from the Sun, the remainders will give the Rulers of the present (terrestrial) month and year respectively. To find the Ruler of the present day. lunar months (प्र 41724, &c., considering them as solar, be added: the sum is the elapsed solar months up to the time when the Sun enters the stellar sign of the Zodiac corresponding to the present lunar month. ‘To make these solar months lunar, let the elapsed additive months be determined by proportion in the following manner. As the number of solar months in a yuGa : the number of additive months in that period : : the number of solar months just found : the number of additive months elapsed. If these additive months with their remainder be added to the solar months elapsed, the sum will be the number of lunar months to the end of the solar month; but we require it to the end of the last lunar month. And as the remainder of the additive months lies between the end of the lunar month and that of its corresponding solar month, let the whole number of additive months, without the remainder, be added to the solar months elapsed; and the sum is the number of the lunar months elapsed to the end of the last lunar month. This number of lunar months elapsed, multiplied by 30 and increased by the number of the passed lunar days of the present lunar month, is the number of lunar days elapsed. To make these lunar days terrestrial, the elapsed subtrac- tive days should be determined by proportion as follows. As the number of lunar days in a yuGA : the number of subtractive days in that period :: the number of lunar days just found : the number of subtractive days elapsed. If these subtractive days be subtracted with their remainder from the lunar | days, the difference will be the number of terrestrial days elapsed to the end of the last lunar day; but it is required to the preset mid-night. As the remainder of the terrestrial days lies between the end of the lunar day and the mid-night, the whole number of the subtractive days, (without the remainder) should be subtracted from the lunar days elapsed, and the difference is, of course, the number of terrestrial days elapsed from the time, when planetary motions commenced, to the present mid-night at Land. B. D. ¢ 10 ` Translation of the To find the menn places of 53. Multip ly the number of the planets at a given mid- elapsed terrestrial days by the number night at LankA, - + 5 of a planet’s revolutions (in a Katpa) ; divide the product by the number of terrestrial days (in a Katra) ; and the quotient will be the elapsed revolutions, signs, degrees &c. of the planet. Thus the mean place of each of the planets can be found. To find the places of the 54. In the ‘same way, the mean S'ighrochchas, apogees and places of the S feHRocHCHA and Man- nodes of the planets. न DOCHCHA (apogee) whose direct revo- lutions (in a Kapa) are mentioned before, and those of the nodes of the planets can be found. But the places of the nodes, thus found, must be subtracted from twelve signs, because their motions are contrary to the order of the signs. 55. Multiply the number of elapsed revolutions of Jupiter by 12; to the product add the number of the signs from the stellar Aries to that occupied by Jupiter; divide the amount by 60, and reckoning the remainder from Visaya,* you will find the present SAMVATSARA. To find the present Sam- VATSABA, | An easy method for find- 56. These processes are mentioned 0 mean places of the (from 45th S’Loxa to 54th) indetail, but, for convenience’ sake, let (an astrono- mer) computing the elapsed terrestrial days from the beginning of the Trerf uaa, find easily the mean places of the planets. 07. At the end of this Krfra yuca the mean places of all the planets, except their nodes and apogees, coincide with each other in the first point of stellar Aries. 58. (At the same instant) the place of the Moon’s apogee= nine signs, her ascending node=six signs, and the places of the other slow moving apogees and nodes, whose revolutions are mentioned before, are. not without degrees (i. e. they contain some signs and also degrees). * Astrologers reckon 60 SamvatTsaras, Visaya &., which answer successively to the periods required by mean Jupiter to move from one sign to the next. B. D. Stirya-Siddhanta. 11 _ The lengths of the Earth’s 59. The diameter of the Karth is diameter and its circumfer- 1600 Yosanas. Multiply the square ence di of the diameter by 10, the square-root of the product will be the circumference of the Earth. ee eee 60. The Earth’s circumference mul- ence of the Karth, and tiplied by the sine of co-latitude (of = a ~ correction 1 the given place) and divided by the radius is the SpHuta or rectified cir- cumiference (i. €. the parallel of latitude) at that place Multiply the daily motion (in minutes) by the distance of the given place from the Middle Line of the Earth, and divide the product by the rectified circumference of the Earth. 61. Subtract the quotient in minutes from the place of the planet (which is found at the mid-night of LanKA, as mentioned in 81.0६4 53,) if the given place be east of the Middle Line, but if it be west, add the quotient to it, and (you will get) the planet’s place at (the mid-night of) the given place. | 62. (The cities named) Roufraka, Ussayini, Kuruxsuetra &c. are at the ee Lanx4 and the north pole of the Harth, (this line tsCalled the Middle Line of the Harth.) 63, 64 and 65. At the given place 9 (1 if the Moon’s total darkness (in her eclipse) begins or ends after the instant when it begins or ends at the Middle Line of the Earth, then the given place is east of the Middle Line, (but if it begins or ends) before the instant (when it begins or ends at the Middle Line, then) the given place is west of the Middle Line. Find the difference in GuaTixds between the times (of the beginnings or ends of the Moon’s total darkness at the given place and the mid-night, which difference 18 called the Drs’4n- TARA GHAfIKAs.) ¦ Middle Line of the Earth * Des’AnTARA is tle correction necessary to be applied to the place of a planet in consequence of the longitude of a place, reckoned from the Middle Line of the Earth or the Meridian of LanxA. B. D 12 Translation of the Multiply the rectified circumference of the Earth by this difference and divide the product by 60. The quotient will be the east or west distance (in Yosanas) of the given place from the Middle Line. Apply the minutes, found by this distance, to the places of the planets (as directed before in S’Loxas 60 and 61). 66. A day of the week begins at the Des’{ntara Guatixds after or be- fore the mid-night at the given place according as it is east or west of the Middle Line. 67. (If you want to know the place of a planet at a given time after or before a given mid-night,) multiply the daily motion of the planet by the given time in GaatTiKAs, divide the product by 60, and add or subtract the quotient, in minutes, to or from the place of the planet found at the given (mid-night,) and you have the place of the planet at the given time after or before the given mid-night. The place of the planet, thus found, is called its (वह (+ or instantaneous place 68. The Moon’s deflection to the north and south from the end of the declination of her corresponding point at the Ecliptic is caused by her node. The measure of her greatest deflection is equal to the ,1,th part of the minutes in a circle. 69. The measures of the greatest deflections of Jupiter and Mars caused by their nodes are respectively ई and § of that of the Moon, and that of Mercury, Venus and Saturn is ई of | the Moon’s greatest deflection. 70. Thus the mean greatest latitudes of the Moon, Mars, Mercury, Jupiter, Venus and Saturn are declared to be 270, 90, 120, 60, 120 and 120 minutes respectively. To find the instant when a day of the week begins To find the mean place of a planet at a given time End of the Ist chapter of Sdrya-stippHdnta called Mapayé. art (which treats of the Rules for finding the mean places of the planets.) . Surya-Siddhanta. 13 CHAPTER ILI. Called Spuvuqa-cati which treats of the Rules for finding the true places of the planets. Cause of the planetary mo- 1, The Deities, invisible (to hu- Hons: man sight), named S’fcnrocucna, ManpocucHa (Apogees) and Pata (Nodes,) consisting of (continuous and endless) time, being situated at the ecliptic, produce the motions of the planets. 2. The Deities, (8 प ६०८८४ and Manpocucaa) attract the planets (from their uniform course) fastened by the reins of winds borne by the Deities towards themselves to the east or the west, with their right or left hands according as they are to their right or left.*. 3. (Besides this) a (great) wind called Pravaua carries the planets (westward) which are also attracted towards their apogees. Thus the planets being attracted (at once) to the east and west get the various motions. 4. The Deity called Ucucna (apogee) draws the planet to the east or west (from its uniform progress) according as the Deity is east or west of the planet at a distance less than six signs. 5. As many degrees &c., as the planets, being attracted by their apogees, move to the east or the west, so many are called additive or subtractive (to or from their mean places). 6. In the same way, the Deity node named RAxu by its power deflects the planet, such as the Moon, to the north or to the south from (the end of) the declination (of its corresponding * The place of a planet rectified by the Ist or 2nd cquation is nearer to its higher apsis (ManpocucHa or S’I@HROCHCHA) in its orbit, than the planet’s unrectified place. The cause of this is that the Deities have hands furnished with reins of winds ana by them they attract the planet towards themselves. This will expluin the meaning of the 2nd S’toxa. B.D. 14 Translation of the point at the ecliptic). This deflection is called ViksHepa (celestial latitude). 7. The Deity node draws the planet to the north or to the south (from the echptic) according as the node is west or east of the planet at a distance less than six signs. 8. (But im respect of Mercury and Venus) when their Paras (or nodes) are in the same direction at the same distance (as mentioned in the preceding Stoxa) from their S’fauRocucuas, they deflect in the same manner (as mentioned before) by the attractions of their S’feHRrocHcnas. 9. The attraction of the Sun (by its apogee) is very small by reason of the bulkiness of its body, but that of the Moon is greater than that of the Sun, on account of the smallness of the Moon’s body. | 10. As the bodies of the (five) minor planets, Mars, &c. are very small, they are attracted by the Deities S faurocucHa and Manpocucua very violently. | 11. And for this reason, the additive or subtractive equation of the minor planets caused by their movement (which is pro- duced by the attraction by their Ucucuas) is very great. Thus, the minor planets, being attracted by their S’fgnrocucHa and Manpocucwa and carried by the wind Pravana, move in the heavens. 12. (And therefore) the motion of the planets is of eight kinds, 1. e. I. _Vaxek (decreasing retrograde motion). II. AtivaKeé (increasing retrograde motion). : III. Vrxa (stationary). IV. Manpé (increasing direct motion less than the mean motion). V. Manpararé (decreasing direct motion less than the mean motion). VI. Samx (mean motion). VII. Sfauratarg or Atis’faHrx (increasing direct motion greater than the mean motion). Kinds of motion. Surya-Sid dhdanta. 15 VIII. Sfaurx (decreasing direct motion greater than the mean motion). 13. Of these kinds, the five motions Atis faHRa, 8 {0 प्र ५, Manna, Manpatar&A and SaMA are direct and the two motions Vaxkr& and ATIVAKRA are retrograde. 14. (Now) I explain carefully the Rules for finding the true places (of the planets) in such a manner that the places found by the Rules coincide with those, determined by observation, of the planets which move constantly with various motions. The Rule for finding the 15. The eighth part of the number sines for every 3० in a 006 fas ९ ६ व ग 06400 of minutes contained in a sign (i. e. Radius—3438 1800) is the first sine. Divide the first sine by itself, subtract the quotient from that sine, and add the remainder to that sine: the sum will be the second sine. 16. In the same manner, divide successively the sines (found) by the first sine ; subtract (the sum of) the quotients from the divisor and add the remainder to the sine last found and the sum will be the next sine.* Thus you will get twenty- * This method is proved thus. Let sing A—sin. O=d,; sin. 2 A—sin. A—=d,; sin. 3 A—sin. 2 A = d, ; &८ = &e sin.» A—sin. (sn—1) A = dn; sin (n--1) A—sin. » A = dy +1 Then ७।।,५८५ व, -- ८ = 2vers A.sin A=>R; d,—d, = 2 vers A.sin2A~R; d,—d, = 2versA.sin3A~>R &५. = ke dy—dy + , = 2 vera A. sinnA R we have by addition = 2 A iS ~~ A ~ sin. 2 A + ...... + sin. » A) or, 111 (sin A ~ sin.2A...... + sin. # A) ०» sin. (# ~+ 1) A= sin.n A + 9०. A —s- (sin. A == 811. 2 A...... -+ sin. # A.) Here, A = 8° 45’ : ie : 0042822 = — which is roughly given the text = : in the text = 5 16 Translation of the four sines (in a quadrant of a circle whose radius is 3438), These are as follows. 17 to 22. 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438. । Subtract these sines separately from the Radius 84868 in the inverse order, the remainders will be the versed smes (for every 3#°), The sines. 23 to 27. There are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1815, 1528, 1719, 1918, 2128, 2333, 2548, 2767, 2989, 3213, 3438, versed sines (in a quadrant). 28. The sine of the (mean) greatest declination, (of each of the planets)—1307 (the sine of 24°). The Rule for finding the Multiply the sine (of the longitude of ee । 1 aa a planet) by the said sine 1307 ; divide - the product by the radius 3438; find the arc whose sine is equal to the quotient. This arc is the (mean*) declination (of the planet required). 29. Subtract the place of the planet from those of the Man- pocHcHAt and SiaHRocucHa: and the remainderst are the Kenpras. From the Kenpra determine the quadrant (in which the Kendra ends,) and the sines of the Buusa and 1९011 (of the Krnpra). 80. The sine of the Buuya (of the arc which terminates) in an odd quadrant (1. e. 1st and 8rd,) is the sine of that part of The versed sines. * The mean declination of a planet is the declination of its corresponding point in the ecliptic: but the Sun’s mean declination is the same as his true _ declination. B.D. + Manpocacua is equivalent to the higher apsis. The Sun’s and Moon’s Manpocucuas (higher apsides) are the saine as their apogees while the other planets’ ManpocucuHas are equivalent to their aphelions. ए. D. ‡ The first remainder is called the first KzgNpRa which corresponds with the anomaly, and the second, the second KENDRA which is equivalent to the com- mutation added to or subtracted from 180° as the second KENDRA is greater or less than 180°. 3B. D. § The Buvsa of any given arc is that arc, less than 90°, the sine of which is equal to the sine of that given arc; and the Kori of any arc is the complement of the Buvsa of that arc. ए. D Surya-Siddhanta. 17 the given arc which falls in the quadrant where it terminates, but the sine of the Koti (of that arc) is the sine of that arc which it wants to complete the quadrant where the given arc ends; and the sine of the Bauya (of the arc) which ends in an even quadrant (i. €. 2nd and 4th) is the sine of that arc which it wants to complete the quadrant where the given arc ends ; but the sine of the Koti (of that arc) is the sine of that part of the given arc which falls in that quadrant where it terminates. To find the sine of the 31. (Reduce the given degrees &c., piven degrees: to minutes.) Divide the minutes by 225: and the sine (in S’LoKas 17—22) corresponding to the quotient is called the cata (the past) sine, (and the next sine is called the gamya to be past sine): multiply (the remainder in the said division) by the difference between the 6414 and GAMYA sine and divide the product by 225. 32. Add the quotient to the sine past: (the sum will be the sine required). 018 18 the Rule for finding the right sines (of the given degrees &c.) Inthe same way, the versed sines (of the given degrees &c.) can be found. Given the sine to find its 33. Subtract the (next less) sine anne (from the given sine) ; multiply the re- mainder by 225 and divide the product by the difference (between the next less and greater sines) : add the quotient to the product of 225, and that number (which corresponds to the next less sine) ; the sum will be (the number of minutes con- tained in) the arc (required). Dimensions of the lst 34. There are fourteen degrees (of epicycles of the Sun and t न ri t Moon in degrees of the de- he concentric) in the periphery of the ferent or concentric. MANDA or first epicycle of the Sun, and thirty-two degrees (in the periphery of the 1st epicycle) of the Moon, when these epicycles are described at the end of an even quadrant (of the concentric or on the Line of the Apsides.) But when they are described at the end of an odd quadrant (of the concentric, ‘or on the diameter of the concentric per- pendicular to the Line of the Apsides) the degrees in both are D 18 Translation of the diminished by twenty minutes; (then the degrees in the pe- riphery of the Sun’s epicycle=138° 40 and in that of the Moon’s==31° 40’.) ` Dimensions of the Ist epi- 85. There are 75, 30, 33, 12 and (क apie in 49, (degrees of the concentric in the peripheries of the first epicycles of Mars, Mercury, Jupiter, Venus and Saturn respectively) at the end of an even quadrant (of the concentric, but) at the end of an odd quadrant, there are 72, 28, 82, 11, 48 (degrees of tho concentric.) | Dimensions of the 2nd 86. There are 285, 138, 70, 262 (0 and 389 (degrees of the concentric) in the peripheries of the S‘iaHra or second epicycles of Mars &c., at the end of an even quadrant (of the concentric). 87. At the end ofan odd quadrant (of the concentric,) there are 232, 182, 72, 260, 40 degrees of the concentric in the -peri- pheries of the second epicycles of Mars &c. Given the Krxpra of a 38. Take the difference between न ( the peripheries of epicycles of a planet ry of the epicycle. at the ends of an even and an odd quadrant ; multiply it by the sine of the Buusa (of the given Kenpra of the planet,) and divide the product by the radius. Add or subtract the quotient to or from the periphery which is at the end of an even quadrant according as it is less or greater than that which is at the end of an odd quadrant: the result will be the Spuuta or rectified periphery (of the epicycle of the planet.) (ican ane: Feb oe. aya 89. Multiply the sines of the Buv- Kenpra of a planet, tofind ya and Kori (of the given Ist and 2nd the Ist or 2nd BHUJA-PHA- . ta and Kort-p#ata andthe KENDRA of a planet) by the rectified 19 equation of the planet. = herinhery (of the 18४ and 2nd epicycle of the planet), and divide the products by the degrees in a circle or 360° (the quotients are called the Ist or 2nd Bausa- ` PHALA and [ई णा -एप् ^+ respectively). Find the arc whose sine is equal to the 1st Buusa-paaLta: the number of the minutes Stirya-Siddhanta. 19 contained in this arc is tho MANDA-PHALA* (or the Ist equation of the planet.) To find the 2nd equation 40. Find the 2nd Koti-paata (from of the minor planets Mars a planet’s 2nd Kenpra as mentioned = before :) it is to be added to the radius when the Kendra is less than 3 signs or greater than 9 signs, but when the Kenpra is greater than 3 signs and less than 9, (then the 2nd'Korfi-pHa.a) 18 to be subtracted (from the radius). 41. Add the square of the result (just found) to that of the sine of the 2nd Buusa-pHata: the square root of the sum is the 8 {0७ प् ^-^ एप्त + or 2nd hypothenuse.t Find the (2nd) Buusa-pHata of the planet (as mentioned in sLOKA 39th ;) multiply it by the radius and divide the product by the 2nd hypothenuse (above found). 42. Find the arc whose sine is equal to the quotient (just found) ; the number of the minutes contained in the arc is called the S’/fanra-PHaLat (or 2nd equation of the planet.) : The 2nd equation of Mars &c. is employed in the first and fourth operations (which will be explained in the sequel). ‹ pe tind Chertrusyplacse ot 48. (In order to find the true places the Sun, the Moon and other of the Sun and Moon,) a single ope- planets ration called manpa (or operation of finding the first equation,) is to be employed (that is to say, when you want to find the true places of the Sun and Moon find their first equation and apply it, as will be mentioned in 45th S'Loxa, to their mean places: thus you have the true places of the Sun and Moon) But in respect of Mars &c. 1st S’fanRa operation (or operation of finding the 2nd equation,) 2nd Manpa operation, 38rd ManpA operation, and 4th S’fanRa operation, are to be employed successively. * Manpa-PHALa is the same as the equation of the centre of a planet. B.D + The S‘ianRa-KaRya or 2nd hypothenuse is equivalent to the distance (in minutes) of the planet from the Earth’s centre. B.D ‡ SiauRa-PHALa or 2nd equation is equivalent to the annual parallax of the superior planets ; and the elongation of the inferior planets. 3B. D. D 2 20 Translation of the 44, Find the second equation (from the mean place of a planet :) apply the half of it to the mean place, and (to the result) apply the half of the first equation (found from that result ; from the amount) find the 1st equation again, and apply the whole of it to the mean place of the planet and (to that rectified mean place)* apply the whole of the 2nd equation (found from the rectified mean place: thus you will find the true place of the planet). 6 Gated! aaa 45. In the S’fonra and Manna equations of the planets are operations, the (second or first) equa- to be applied. = ; . tion of a planet in minutes is to be additive when the (second or first) Kenpra (of the planet) 18 less than 6 signs; but when it is greater than 6 signs, the (2nd or 181) equation is to be subtractive. The BuousknraRaft correc- 46. Multiply the diurnal motion Rann: of a planet by the number of minutes contained in the first equation of the Sun, and divide the pro- duct by the number of minutes contained in a circle or 21600’ : add or subtract the quotient, in minutes, according as the Sun’s equation is additive or subtractive, to or from the place of the planet (which is found from the Amaraana at the mean mid-night at Lanx{, the result will be the place of the planet at the true mid-night at Lanxé.) 47. Subtract the diurnal motion of the Apogee of the Moon from her mean diurnal motion ; (the remainder will be the Moon’s motion from her apogee;) from this remainder find the Ist equation of her motion (by the rule which will be explained further on). This equation is to be subtractive or additive to her mean motion (for finding the true motion of the moon). | * The rectified mean place of a planet is called its Manpa spHuta place. The Manpa-spuHuta places of Mars, Jupiter and Saturn correspond with their heliocentric places. B.D + The BuusANTanRa correction is to be applied to the place of a planet found from the AnaRr@ana for finding the place of the planet at the true mid-night at LanKA, arising from that portion of the equation of time which is due to the unequal motion of the Sun in the Ecliptic. B.D. Stirya-Siddhanta. 21 Find the true diurnal 48. In the manna operation, find ee 01 the (first) equation of a planet’s diurnal ta motions of the others. motion from the motion itself, in the same way in which the planet’s first equation is found. (Take the difference between the GATA and GAMYA sines which have been found in finding the sine of the first KENDRA of the planet) ; by the difference between the sines (cata and GAMYA) multiply the (planet’s mean) motion (from its apogee) and divide the product by 225 49, The quotient multiplied by the (rectified) periphery of the first epicycle of the planet and divided by 360° (becomes the first equation of the planet’s motion) in minutes. Add this equation (to the mean diurnal motion of the planet) when the first Kenpra is greater than 3 signs and less than 9; but when the first Kenpra is greater than 9 signs or less than 3, subtract the equation of the motion from it: (thus you have the true diurnal motions of the Sun and Moon, and the MANDA- sPHUTA motions of the others which are equivalent to their heliocentric motions.) To find the true: diurnal 50. Subtract the MANDA-SPHUTA di- म urnal motion of a (minor) planet from its 8 faHROcHCHa’s diurnal motion, and multiply the remainder by the difference between the radius* and the 2nd hypothenuse found in the 4th operation for finding the 2nd equation. 51. Divide the product by the (said) 2nd hypothenuse, add the quotient (to the MANDA-sPHUTA motion of the planet) when the 2nd hypothenuse is greater than the radius ;* but when it is less than the radius subtract the quotient (from the MANDA-SPHUTA motion, the result will be the true motion of the planet). (But in the latter case), if the quotient be greater (than the MANDA-sPHUTA motion,) subtract (the MANDA-SPHUTA motion from the quotient) ; the remainder will be the retro- grade motion of the planet. * Notes on 50 and 51. Some commentators of the SGeyva SIDDUANTA under- stand by the term radius the cosine of the 2nd equation found in the 4th opera tion 22 Translation of the The cause of the retrogres- 52, When a planct is at a great प distance (more than 3 signs) from its S‘fanrocucna and (therefore) its body is attracted by the loose reins (borne by the S‘fasrocucaa,) to its left or right, then the planet’s motion becomes retrograde. Wien The silencer hee 53 and 54. The planets Mars, and (0 they others (i. 6. Mars, Mercury, Jupiter, Venus and Saturn) get the retrograde motion about the same time when the degrees of (their 2nd) Kewpras, found in the 4th operation, are equal to 164, 144, 130, 163 and 115 (respectively) : and when the degrees of (their 2nd) KEnDRAS are equal to the remainders (196, 216, 230, 197 and 245,) found by subtracting the (said) numbers (164, 144, 130, 163 and 115,) from 360° (separately,) the planets leave their retrogression. 55. Venus and Mars (leave their retrogression about the same time) when (their 2nd Kenpra) is equal to 7 signs, on account of the greatness (of the rectified dimension) of their 2nd epicycle: so Jupiter and Mercury (leave their retrogression) when (their 2nd Kenpra)=8 signs, and Saturn leaves its retro- gression when (its 2nd KenpRra)= 9 signs. To find the latitude of a 56. Add or subtract the 2nd equa- planet: tions of Mars, Saturn and Jupiter (found in the 4th operation) to or from their nodes according as the 2nd equations applied to the (rectified mean) places of the planets: but in respect of Mercury and Venus add or sub- tract their 180 equations (found in the 3rd operation, to or from their nodes) according as their ] st equations are subtractive or additive respectively (the results are the rectified nodes). 57. (For the argument of latitude of each of the planetst Mars, Jupiter and Saturn) take its rectified node from its true place: but for (the argument of latitude of) Mercury or Venus take its rectified node from its S’fanrocucHa ; find the sine (of t+ Notes on 56 and 57. It is evident that the argument of latitude of each of the planets, found here, equals the heliocentric place of the planet diminished by the place of ita node, B.D. Surya-Siddhanta. . 23 the argument of latitude of a planet); multiply it by the (greatest) latitude of the planet (mentioned in S’LoKa 70th of Ist Chapter) and divide the product by the 2nd hypothenuse found in the 4th operation; but in respect of the Moon divide it by the radius: the quotient will be the latitude (of the planet). To find the true declina- 58. The (mean) declination (of a honobe planet planet or the declination found by computation from its corresponding point in the ecliptic) in- creased or diminished by its latitude, according as they are both of the same or different denominations, becomes the true (declination of the planet). But the Sun’s (true declination) is (the same as) his mean declination. | To find the length of a 59. Multiply the diurnal motion planet's day. (in minutes) of a planet by the number of Prkyas which the sign, in which the planet is, takes in its rising (at a given place;) divide the product by 1800’ (the number of minutes which each sign of the ecliptic contains in itself,) add the quotient, in Priyas, to the number of the Prinas contained in a (sidereal) day: the sum will be the number of Prdyas contained in the day and night of that planet (at the given place). Given the declination, ४७ 60. Find the right. and versed find the radius of the diur- gines of the declination (of a planet) : भ | take the versed sine (just found) from the radius, the remainder will be the radius of the diurnal circle south or north of the equinoctial. (This radius is called Dyvusy4). To find the ascensional 61. Multiply the sine of declination १९. (above found) by the length (in digits) of the equinoctial shadow,* divide the product by 12, the quotient 18 the Kusya:¢ The इ एरर multiplied by the radius * The equinoctial shadow is the shadow of a vertical gnomon of 12 digits when the Sun is in the equinoctial at the mid-day at a given place. B.D. ¢ KusvyA is the sine of that arc of a diurnal circle which is intercepted between the Horizon and the six o’clock line. B,D. 24 Translation of the and divided by the [रए (above found) becomes the sine of the ascensional difference. The arc of that sine (in minutes) is the ascensional difference in PRANAS. To find the lengths of the 62. Add and subtract the ascen- day and night of a planet sional difference to and from the fourth Meenas part of the length of the day and night of the planet (as found in 87.014 59) separately, the results will be lengths of the half day and half night respectively of the planet when its declination is north. = 63. But when the planet’s declination 1s south, the reverse of this takes place (i. e. the results, just found, will be the lengths of the half night and half day of the planet respec- tively). (In both cases,) twice the results are the lengths of the day and night (respectively). In the same way, the lengths of the day and night of any fixed star can be determined from its declination which is to be found by adding or subtracting its latitude to or from the declination (of its corresponding point in the ecliptic). The Boga of a NaxsHa- 64. The Bua-snoaa (or the space of See Soo UTIEEE a NaksHaTRA or an Asterism) contains 800° minutes, and the Buoga of a (प्ता (or the space which the Moon describes from the Sun in tithi or lunar day) contains 720’ minutes. To find the क पश 1106 Place of a planet, reduced to in which a planet is at a minutes, divided by the Buasyoga or + 800’, gives the number of those Nax- sHaTRA or Asterisms (counted from As‘winf which are passed by the planet: and the remainder is that portion of the present NaxksuHatRA which is passed by the planet.) (This remainder divided) by the diurnal motion (of the planet) gives the quotient in the days, aHaTiK4s, &c. which the planet has taken to pass that portion of the present NaksHaTRA. To find the Yoaa* at a 65. The sum of the places of the ^. Sun and Moon (found ata given time,) * Yoaa is a period of time in which the sum of the places of the Sun and Moon increases by 13° 20’ or 800^ B. D. Surya-Siddhanta. 29. reduced to minutes, is to be divided by the एप ^ -एप् ००५५ (or 800.) The quotient is the number of the elapsed Yoaas (counted from VisHKaroBHa): (The remainder is called the cata of the present Yoca, and the Buaa-sxHoaa (or 800’) dimi- nished by the 6474 is called the camya of that २०७८५.) The 6474 and aamya of the present २०५५ multiplied by 60 and divided by the sum of the diurnal motions (of the Sun and Moon) become the numbers of the past and to be past GHATIKAS (respectively of the present Yoaa at the given time.) To find the lunar day at a 66. ‘Take the place of the Sun from given time. that of the Moon (found at a given time) ; divide the remainder, reduced to minutes, by the Buoea (of a TITHI or 720’; the quotient is the number of the elapsed tithis or lunar days.) (The remainder is the gata of the present TITHI, and the Broaa of a TITHI diminished by the cata is the GaAmya of the present TITHI.) The cata and camya of the present TITHI, multiplied by 60 and divided by the difference between the diurnal motions (of the Sun and Moon) become the numbers of the past and to be past aHaTIKAs (respectively of the present वप्रा at the given time). 67. The four invariable Karayas called S’axuni, Naca, CHatusHpaDA and KinstucHNna (always appropriate to themselves succes- sively the halves of the T1THI1s,) from the latter half of the fourteenth क्षपा of the dark half (of a lunar month to the first half of the first कत्रा of the hight half of the next lunar month inclusive). Invariable Karnanas. 68. And the seven variable Kara- was, Bava* &c. afterwards succeed each other regularly, through eight repetitions in a (lunar) month. Variable KARANAS. * 1. Bava. 2, BAtva. 3, ^ एए. 4, (404. 5. Garagva. 6. Vayt- za. 7, BuHapDRA. B.D. E 26 Translation of the 69. It 13 to be known that all the Karanas answer succes- sively to half of a TITH1. (O Maya,) thus I have explained to you the Rules for finding the true places of the heavenly bodies, the Sun &c. End of the 2nd Chapter of the SGrra-SippHanta. CHAPTER III. Called the Trrpras’Na, which treats of the Rules for resolving the questions on Time, the position of places, and directions. ; 7 1. On the surface of astone levelled To determine the meridian : and east and west lines and with water or on the levelled floor of the points of the Horizon. = ल्भा work, describe a circle with a radius of a certain number of digits. 2and3. Place the vertical Gnomon of 12 digits at its centre and mark the two points where the shadow (of the Gnomon) before and after noon meets the circumference of the circle: these two points are called the west and the east points (respectively). Then, draw a line through the णा formed between the * To draw a line perpendicular to and bisecting the line joining two given points, it is usual to describe two arcs from the two given points 83 centres with @ common radius, intersecting each other in two points : the line passing through the intersecting points is the line required. In this construction, the space contained by the intersecting arcs is called TIMI “a fish,” on account of its form. It is evident that the line drawn through the TIm1 formed between two given points, must be perpendicular to and bisect the line which joins the given points. B.D. Strya-Siddhénta. 27 (said) east and west points, and it will be the north and south line or the Meridian Line. 4, And thus, draw a line through the फा formed between the north and south points of the Meridian Line; this line will be the east and west line. In the same manner, determine the intermediate directions through the Trmis formed between the points of the determined directions (east, south &c.). 0: 5. (In order to find the direction dow and its Bavsa, to find of a given shadow of the Gnomon at a 4 given time, describe a circle in the plane of the Horizon with a radius whose length is equal to that of the given shadow and at its circumference determine the points of the Horizon, the Meridian and east and west lines as mentioned before:) Then describe a square about the said circle through the lines drawn from the centre (of the circle to the points of the Horizon, in such a manner that the square shall touch the circle at the cardinal points) and in this circle (towards the western or eastern part of it according as the given time is before or after noon), draw a line (as & sine,) equal to Buusa* (of the given shadow and perpendicular to the east and west line towards the north or south according as the Buvsa is north or south. To the end of this perpendicular, draw a line from the centre). This (line) will denote the direction of the given shadow (at the given time). 6. The line representing the Prime Vertical, the six o’clock line or the equinoctial, passes through the east and west points of the Horizon. * The distance (in digits) of the end of the shadow of the Gnomon (which is placed at the intersecting point of the Meridian and east and west line) is called the Buusa (of the shadow north or south according as the end of the shadow is north or south of the east and west line: and the distance of the end of the shadow from the Meridian Line is called the Kort (of the shadow) east or west according as the end of the shadow is east or west of the Meridian Line. B. D. E 2 28 Translation of the To find the sine of ampli- 7. (In the said circle)* from the tude reduced to the hypo- ४ thenuse of the given shadow. east and west line (to its north) at a distance equal to the equinoctial shadow, draw another line parallel to the former; the distance between the end of the # Note on the 7th S’LoKa. Let Z GN H be the ९- plane of the Meridian of the given place of north lati- tude; and in that plane let @ A H be the diame- ter of the Horizon, Z the ् zenith, Pand Q the north and south poles, E A F the diameter of the Equinoc- | NAc ५, . tial, P A Q that ofthe six o’clock line, Z A N that of © the Prime vertical, C a D that of one of the diurnal circle in which the Sun is supposed to revolve at the given day ands the projec- @ tion of the Sun’s place; and let $ ©, 5 b be the perpendi- culars to Z N, © H respec- ¢ tively. Then, A a= AGRA or 6 the sine of the Sun’s am- N plitude ; 8 ® = Shyxv or the sine of the Sun’s altitude; esor Ab = Buvsa or the sine of the distance of the Sun from the Prime Vertical measured on a great circle passing through the Sun and at right angles to the Prime Vertical. a b = S’ANKUTALA or the distance of the perpendicular drawn from the Sun’s place to the horizorital plane, from the line (called the UpayAsTa-s6TRA in Sanskrit) in which the plane of the Horizon intersects that of the diurnal circle: and it is evident from the figure that ^ ८ = ८6 ¬+ 4 ९: [वा or Agri = S‘anxuTata + Buvsa: in this the upper or lower sign must evidently be used according as the Sun is north or south of the Prime Vertical. Now if these Agri, S’anxuraza and Buvsa which are in terms of the radius of a great circle, be reduced to the hypothenuse of the gnomonic shadow at the given time, it is clear that the reduced Buusa will be equal to the distance of the end of the shadow from the east and west line, but the reduced SANKUTALA will equal the Equinoctial shadow. It is showed thus: | । let R = the radius of a great circle : ¢ = the hypothenuse of the shadow ; 12R then, 2:R=12:806,..86=——; h Now, in the triangle s ab °." Z as b = the latitude of the given place; ८2 thesine of latitude the Equinoctial shadow ॐ © the cos. of latitude 12 Surya-Siddhanta. 29 given shadow and the latter line is equal to the sine of ampli- tude (reduced to the hypothenuse of the given shadow). Given the shadow to find 8. The square-root of the sum of ua hypetienness the squares (of the lengths) of the Gnomon and the given shadow is called the hypothenuse of the shadow: from the square of the hypothenuse subtract the square of the Gnomon; the square-root of the remainder will be equal to the shadow; and the length of the Gnomon is to be known (from the shadow) by the inverse calculation. The precession of the 9. The circle of Asterisms lbrates Pine 600 times in a great Yuca (that is to say, all the Asterisms, at first, move westward 27°. Then returning from that limit they reach their former places. Then from those places they move eastward the same number of degrees ; and returning thence come again to their own places. Thus they complete one libration or revolution, as it is called. In this way the number of revolutions in a Yuaa is 600 which answers to 600,000 in a Katpa). Multiplying the AnarGana (or the number of elapsed days) by the said revolutions and dividing by the number of terres- trial days in a Kapa; the quotient is the elapsed revolutions, signs, degrees, &c. 10. (Rejecting the revolutions), find the Buusa of the rest (1. €. signs, degrees &c.as mentioned in S‘Loxa 30th of the 2nd Chapter). The Buusa (just found) multiplied by 3 and divided by 10* gives the degrees &c. called the Ayana (this is the same with the amountt of the precession of the equinoxes). h the Equinoctial shadow or ab ce = h ab € — or the reduced S’ankuraLa = the Equinoctial shadow ; R e ॐ 12 ०० The reduced sine of amplitude == the Equinoctial shadow + the reduced Buvusa; this explains the 7th 8/0 4. ए. 1). * 27°: 90°. B.D. ¶† This is the distance of the Stellar Aries from the vernalequinoxr. ए, 1, 30 Translation of the Add or subtract the amount of the precession of the equi- noxes (according as the asterisms are moving eastward or westward at the given time) to or from the place of a planet: from the result (which is equivalent to the longitude of the planet) find the declination, the shadow of the Gnomon, the the ascensional difference &c. This motion of the asterisms (or the precession of the equi- noxes) will be verified by the actual observation of the Sun when he is at the equinoctial or the solstitial points. 11. According as the Sun’s true place found by computa- tion (as stated in the 2nd Chapter) is less or greater than that which is found by observation (i. e. the longitude of the Sun), the circle of asterisms is to the east or west (from its original place) as many degrees as these are in the difference (between the Sun’s true place and the longitude). 12. At a given place, when the Sun comes to the equinoctial, the sha- dow (of the Gnomon of 12 digits) cast on the Meridian Line at noon is called the ^^ एप्त or the equinoctial shadow (for that place). The equinoctial shadow. ग्‌ । tink 1 Given the equinoctial sha- 18. he Radius multiplied DY he dow, to find the co-latitude Gnomon (or 12) and the equinoctial latitude. ar म shadow (separately) and divided by the equinoctial hypothenuse* gives the -cosine and sine of the latitude (respectively). The arcs of these sines are the co-lati- tude and the latitude which are always southt (at the given place from whose zenith the equinoctial circle is inclined to the south). Given the Gnomon’s sha- 14and 15. The Buousa of the sha- dow at noon and Sun’s de- t i + clination, to find the latitude dow of the Gnomon at noon is the of the place. same as the shadow itself. Multiply * The equinoctial hypothenuse is the hypothenuse of the equinoctial shadow found by taking the square-root of the sum of the squares of the equinoctial shadow and the Gnomon (or 12). B.D. + The south latitudes of Sanskrit correspond to the north latitudes of the Europeans. ए. D ~ Surya-Siddhanta. 31 the Radius by that Bausa and divide the product by the hypo- thenuse of the said shadow; the quotient will be the sine of the zenith distance: the zenith distance, found from that sine in minutes, is north or south according as the Bavsa is south or north respectively (at a given place). Find the sum of the zenith distance and the Sun’s declination in minutes when they are both of the same name, but when they are of contrary names, find the difference between them. This sum or differ- ence is the latitude in minutes (at the given place). To find the equinoctial 16. Find the sine of the latitude, sia COM १०११. (just found) ; take the square of that sine from that of the Radius; the square root of the remainder is the cosine of the latitude. The sine of the latitude multiplied by 12 and divided by the cosine of the latitude gives the PaLaBHA& or the equinoctial shadow. Given the latitude of the 17. Find the difference between the tape 1 degrees of the latitude (at a given place) declination and longitude. === and those of the Sun’s zenith distance at noon when they are both of the same name, but when they are of contrary names find the sum of them; the result will be the Sun’s declination: multiply its sine by the Radius. 18. Divide the product by the sine of the Sun’s greatest declination (or 1397) : find the arc (in signs &c. whose sine 18 equal to the quotient, just found) ; this arc will be the longi- tude of the Sun when he is in the first quarter of the Kcliptic : but when he is in the second or third quarter, subtract or add the signs &c. (contained in the arc) from or to 6 signs; (the remainder or the sum will be the longitude of the Sun). 19. And when the Sun is in the fourth quarter of the Kcliptic subtract the signs &c. (which compose the arc) from 12 signs; the remainder will be the true longitude of the Sun at noon. (To the longitude, just found, apply the amount of the pre- cession of the equinoxes inversely for the Sun’s trve place.) 32 Translation of the To-find the Sun's: wean (In order to find the mean place of Place from his true place. the Sun from his true place above found,) find the 1st equation from the true place of the Sun and apply it inversely to the place repeatedly, the result is the mean place of the Sun (that is, assume the true place as his mean place, find the Sun’s first equation from it and add this equation to the true place if the equation be subtractive, but if it be additive, subtract it from the true place; the result will be somewhat nearer to the exact mean place of the Sun at the given noon ; assuming this result as the Sun’s mean place apply the said mode of calculation, and repeat the process until you get the exact mean place of the Sun). Given the latitude of the 20. Find the sum of the latitude of cee the declination of 4 given place and the Sun’s declination distance at noon. when they are of the same name, but when they are of contrary names find the difference between them ; the result will be the zenith distance of the Sun (at noon). Find the sine and cosine of the (found) zenith dis- tance. Given the Sun’s zenith 21. The sine (just found) and the distance at noon, to find the Radius multiplied by the length of the shadow and its hypothenuse. (1 digits (or by 12) an Fie oe vided by the cosine (above found) give the shadow of the Gnomon and its hypothenuse (respectively) at noon. Given the Sun’s declina- 22. The sine of the Sun’s declina- tion and shadow, to find his ti ultipli + ti amplitude and the sine of tion multiplied by the equinoctial hy- amplitude reduced. pothenuse and divided by 12 gives the sine of the Sun’s amplitude. This sine multiplied by the hypothenuse of the Gnomonic shadow at noon, and divided by the Radius, becomes the sine of amplitude reduced to the shadow’s hypothenuse. Given the equinoctial sha- 23. To this reduced sine of the 1 0 Ue ine Of Sun’s amplitude add the equinoctial ०८. shadow ; the sum will be the north Buvsa (of the shadow at the given time) when the Sun is Sdrya-Siddhanta. 33 in the southern hemisphere, but when he is in the northern hemisphere, take the reduced sine of amplitude from the equinoctial shadow, and the remainder will be the north Buvsa. 24. In the latter case, when the reduced sine of amplitude is greater than the equinoctial shadow, subtract this shadow from the reduced sine; the remainder will be the south Buusa between east and west line and the end of the shadow at the given time. Every day the Buusa at noon equals the Gno- mon’s shadow at that time. —§ - 25. Multiply the cosine of the lati- Given the latitude and the Sun’s declination, to find the hypothenuse of the sha- dow at the time when the Sun reaches the Prime Verti tude by the Equinoctial shadow or the sine of the latitude by 12; the product (which is the same in both cases) a divided by the sine of the Sun’s de- clination gives the hypothenuse of the gnomonic shadow at the time when the Sun reaches the prime vertical.* 26. When the (Sun’s) north declin- ation is less than the latitude, the hypothenuse of the shadow at noon multiplied by the equinoc- tial shadow and divided by the reduced sine of amplitude at noon, gives the (same) hypothenuset (which is found in the preceding S'LoKA). Otherwise. This is shown thus Let 2 = latitude of the place e = the equinoctial shadow d = the sine of the Sun’s declination when the Sun reaches the prime . altitude : verti 2 . + x == the hypothenuse of the shadow Then, sin ए: द ==: p; and R:p= 2: 12; 12 80९ € 008 द ० ® == = (because cos 7: sin? == 12: ए and .. ©, 008 ए = d d ` 1292). + This is proved thus. Let 4 = the hypothenuse of the shadow at noon ; छ = the sine of amplitude reduced to that hyp. aR "| ——— = the sine of amplitude in terms of the radius. ¥ 84 Translation of the 27. The sine of the declination (of the Sun) multiplied by the radius and divided by the cosine of the latitude becomes the sine of amplitude. Multiply this sine by the hypothenuse of the shadow at a given time and divide the product by the radius: the quantity obtained 78 the sine of amplitude in digits (reduced to the hypothenuse of the shadow at the given time). — 28 and 29. Subtract the square of Given the equinoctial sha- , ]; dow and the Sun’s ampli- the sine of amplitude from the half of tude, to find his altitude + a t ° alii when situated in the vertical he square of the radius; multiply circle of which the azimuth the remainder by 144; divide the distance is 45°, product by the half of the square of the gnomon (i. e. 72) increased by the square of the equinoc- tial shadow. Let the name of the result be the Karanf. Let the calculator write down this number (for future reference). 30. Multiply twelve times the equinoctial shadow by the sine of amplitude and divide the product by the former divisor (i. €. 72 added to the square of the equinoctial shadow). Let the result be called the PHata. Add the square of the Karanf to the PHaa and take the square-root of the sum. 81 and 32. The square-root, (just found), diminished or increased by the PHata according as the Sun is south or north of the equinoctial, becomes the Kona-s‘anku* or the sine of ak Then —— : p (the sine of the Sun’s altitude when he is at the prime vertical) = cos ¢ : sin ¢ = € (equinoctial shadow) : 12; 12 ० = 3 | he and ,*. 2 : R= 12: % (the hypothenuse of the Sun’s shadow when he reaches the prime vertical) : 12 1 he he ० % == --- == 12 ‰%€ == —— ; supposing the Sun’s declination to 2 12 aR @ undergo no change during the day. # This is demonstrated thus. Let e = the equinoctial shadow, छ = the sine of amplitude, k = the Karanf, J = the PHata, and 2 = the Kowa-s/anxvu. Surya-Siddhanta. 35 altitude of the Sun when situated at an intermediate vertical (intersecting the Horizon at the N. E. and 3. W. or N. W. and 9. 1. points). Ifthe sun be south of the prime vertical, then the Kowna-s’anxv will be south-east or south-west, but if he be north of it, then it will be north-east or north-west. The square-root of the difference between the square of the Kownas’anxu and that of the radius, is called the दग or the sine of the zenith distance. 33. Multiply the (said) sine of the zenith distance and the radius by 12 and divide the products by the Kowna-s‘ANKU (above found); the quotients will be the shadow (of the gno- mon) and its hypothenuse (respectively, when the Sun will come on an intermediate vertical) at the proper place and time. e Then, 12 : e = > : —— 2 =S’auxurTata (as shown in the note on 7th S’LoKA) ; 12 and since it is manifest from the same note that the S’aukuTata applied with the sine of amplitude by addition or subtraction according as the Sun is south or north of the equinoctial, becomes BuvJa (i. 6, the sine of the Sun’s distance from the prime vertical), ९ ° ~~ ॐ -+~ ¢ ~ Baus; 12 but when the Sun is N. E., N. W., 8. E., or 8. W., it is equidistant from the prime vertical and the meridian. Therefore the hypothenuse of a right-angled triangle, of which one side is the Buuga and the other equal to it, is the sine of the zenith distance. e + ae , hyp.)? = 2 (—— = + ¢ ): = — a? + — > 4 2 २. 12 72 3 Now, since the square of the sine of the zenith distance added to the square of the sine of the altitude is equal to the square of the radius, e* ae “Ob ets —2e2+2a79=R’'; । 72 3 or (© + 72) 7? + 24 ० ९ > == 72 R?— 144 ८* ; 24 a6 72 ° - 144 ८ 144(4 R*? -- a’) ° ¢ ~+ -------- ॐ = = = ~> ४ e* ~ 72 €* ~ 72 e? ~~ 72 Now, in the foregoing equation it will be observed that the value of the side containing the known quantities is what has been already spoken of under the name of Karanf, and that the half of the co-efficient of is what has been already spoken of under the name of PHALA, ०» et 2fr—hk, which gives ॐ == 4.7 + ए Ef B.D. F 2 36 Translation of the The latitude of the place 84. Add or subtract the sine of and the Sun’s declination the ascensional difference to or from being given, to find the Sun’s adi din त) altitude, Zenith distance &, the radius according as the Sun is in abgiven time irom.no0n. the northern or southern hemisphere. The result is called the पकर, From the ^ प र subtract the versed sine of the time from noon (reduced to degrees) ; Multi- ply the remainder by the cosine of the declination. 85 and 86. Divide the product, (thus found), by the radius ; the quotient is called the cHHEDA; the CHHEDA multiplied by the cosine of latitude and divided by the radius becomes the S‘anxku* or the sine of the Sun’s altitude (at the given time). Subtract the square of the S'anxu from that of the radius; the square root of the remainder 18 DRIG-JYA or the sine of the zenith distance (at the given time). (From the s’anxu and the DR}G-JYA) find the shadow and its hypothenuse as mentioned before (in S'toKa 32). Givens ties eaombaioal Multiply the radius by the given 9 shadow (of the gnomon) and divide the "product by the shadow’s hypothenuse. * This will be manifest thus. Let / = latitude of the place north of the equator ; @ = the Sun’s declination ; @ = the ascensional difference, ट = the time from noon in degrees, and ~ = the Sun’s altitude. Then we have the equation which is very common: coe ¢. cos 4, cos d + 7. sin 2, sind. R?2 tan J. tan क —) cos ¢, cos @ sin 2 == ; (cos ६ + — eee Geet — 1 R (cos £ + sin a) cos ¢ cos व R Z (R © sina — vers £) 608 क ०08 व r ॐ RB R It is to be observed here, that when the latitude of the place is north, the sin @ becomes plus or minus according as the declination is north or south. B. D. or = Surya-Siddhanta. 37 87. The quotient is the pria-sy4 ; the square-root of the square of the radius diminished by that of the pric-syf (just found), is the 8’ANKU: multiply it by the radius and divide the product by the cosine of latitude (of the place). 88 and 39. The result (thus found) is the cauEDA ; multiply the cHHEDA by the radius; divide the product by the cosine of the declination. Subtract the quotient from the Antyf and take the remainder. From the versed sines (given in s’LOKAS 23—27 of the second chapter) find the arc whose versed sine equals the remainder: The minutes contained in the arc are the Pr&nas in the time before or after noon.* Given the latitude of the Multiply the cosine of latitude by oe ब्‌ स ९९ the given reduced sine of amplitude Sun’s declination and longi- and divide the product by the given en shadow’s hypothenuse (at a given time), 40. The quotient, (thus found), is the sine of the Sun’s declination ; multiply it by the radius and divide the product by the sine of the greatest declination ; find the arc in signs, degrees, &c.; from this arc and that quarter of the ecliptic in which the Sun is situated at the given time the Sun’s longitude can be determined (as mentioned before in 9 LoKas 18 and 19 of this Chapter). To draw 9 line in which ५1. 00 any day place a vertical the Gnomonic shadow’s end gnomon on an horizontal plane ; mark ave the end of the shadow at three dif- ferent times on the plane, and describe a circle passing through these points. Then the end of the shadow of that gnomon will revolve in the circumference of this circle through that day.t 1 * This Rule is the converse of the preceding one. B. D. + This Rule is refuted by BoAskandcHArya in his GoLapHyAya, and he is right, because the end of the gnomonical shadow revolves in an hyperbola in the places between the arctic and antarctic circles. B.D. 38 Translation of the To find the right ascen- 42. (In order to find the right sions of the first three signs ascensions of the ends of the three ५ first signs of the ecliptic i. €. Aries, Taurus and Gemini, find the declinations of the said ends) then multiply the sines of one, two, and three sines by the cosine of the greatest declination of the Sun separately, and divide the products by the cosines of the declinations (above found), respectively: The quotients will be the sines of the arcs; find the arcs in minutes. (These arcs will be the right ascensions of the ends of the three first signs of the ecliptic). To find the rising periods 43. The number of minutes con- of those signsatthe Equator. tained in the first right ascension, (above found), is the number of Pr4nas which Aries takes in its rising at Lanxa (or the equator) ; then take the first right ascension from the second and the second from the third; the remainders in minutes will denote the numbers of 28.448 in which Taurus and Gemini rise at the equator. (The numbers of the Pranas, thus found, contained in the rising periods of Aries, Taurus and Gemini at the equator are) 1670, 1795 and 1935 (respectively). To :Had'the tising periods (In order to find the rising periods of those signs at a given of the first three signs of the ecliptic न at a given place of N. L., find at first the ascensional differences of the ends of the said signs at that place and subtract the first ascensional difference from the second and the second from the third. The first ascen- sional difference and these remainders are severally called the Cuarakuanpas of the said signs for the given place). Sub- tract the CoarakHanpas (of the first three signs) for the given place from their rising periods at the equator: the remainders will be the rising periods in Priyas of the said signs at the given place. To find the rising periods 44, The rising periods of the first ०८ three signs of the ecliptic at the Equator successively increased by their CHARAKHANDAS give in Surya-Siddhanta. 89 a contrary order the लश periods of the following three signs (i. e. Cancer, Leo and Vergo). The rising periods of the first 6 signs, thus found, answer in an inverse order to those of the latter six Libra, &c. for the given place. : 45. From the Sun’s longitude as- the eter ee ecliptic Gust certained at the given time, find the msing at agiven time from Broxra and Buoaya times in PRANAS, (in the following manner. Find the sign in which the Sun is and find the Buuxta degrees or the degrees which the Sun has passed and the Buocya degrees or those which he has to pass). Multiply the numbers of the Buvuxra and Broeya degrees (separately) by the rising period of the said sign (at the given place) and divide the products by 30. (The first quotient is the Bauxta time in Pranas in which the Sun has passed the Buuxra degrees, and the latter is the BHoara time in Prdyas in which he has to pass the Buoaya degrees.) 46 and 47. From the given time in Priyas (at the end of which the Horoscope is to be found) subtract the ए ०५१ ^ time in Pranas and the rising periods of the next signs (to that in which the Sun is, as long 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 con- sequently called the as uppHaA sign or the sign incapable of being subtracted, and its rising period the as’UDDHA rising). Multiply the remainder thus found by 30 and divide the product by the as uppHA rising period: add the quotient, in degrees, to the preceding signs (to the as’UDpHA sign) reckoned from Aries: (and to the sum apply the amount of the preces- sion of the equinoxes by subtraction or addition according as it will be additive or subtractive): the result, (thus found), will be the place of the Horoscope* at the eastern horizon. If the time at the end of which the Horoscope is to be found, * Thus there are two processes for finding the Horoscope, one when the given time is after sun-rise and the other when it is before sun-rise, and which are consequently called Krama or direct and VYULKRAMA or indirect processes respectively. B.D. 40 Translation of the be given before sun-rise, then take the Buukta time (above found) and the rising periods of the preceding signs, to that which is occupied by the Sun) in a contrary order from the given time; multiply the remainder by 80 and divide the product by the as’uDDHA rising period ; subtract the quotient, in degrees, from the signs (reckoned from Aries to the as UD- DHA sion inclusive) ; the remainder (inversely applied with the amount of the precession of the equinoxes) will be the place of the Horoscope at the eastern horizon. To find the culminating 48. From the time, in GHATIEAS, point oe the Ecliptic at the from noon, before or after, the Sun’s given time from noon. ; । place found at the given time, and the rising periods of the signs ascertained for the equator, find the arc, in signs, degrees, &c. (intercepted between the Sun and the meridian at the given place) subtract or add the signs &c. (just found) from or to the Sun’s place (according.as the given time is before or after noon); the result will be the place of the culminating point of the ecliptic (at the given time). Given the place of the 49. (Of the given place of the च ease Horoscope and that of the Sun), find sun-rise. the Buocya time in 284 प 48, of the less and the Buuxta time, in Prayas of the greater, add toge- ther these times and the rising periods of the intermediate signs (between those which are occupied by the Sun and the Horoscope) ; and you will find the time (from sun-rise at the end of which the given place of the Horoscope is just rising in the eastern horizon). 50. When the given place of the Horoscope is less than that of the Sun, the time (above found) will be before sun-rise, but when it 18 greater, the time will be after sun-rise. And when the given place of the Horoscope is greater than that of the Sum increased by 6 signs, the time found (as men- tioned before) from the place of the Horoscope and that of the Sun added to 6 signs will be after sun-set. End of the third Chapter called the Tripras'Na. Strya-Siddhanta. Al CHAPTER IV. On the Eclipses of the Mvon. _ 1. The diameter of the Sun’s orb The diameters of the Sun and Moon in yosanas and is 6,500 yoyanas and that of the their rectification. Moon’s is 480 १0१५१५8. 2 811 3. The diameters of the Sun and Moon multiphed by their true diurnal 11101078 and divided by (their) mean diurnal motions become the spuuta or rectified diameters. Pips 0 The rectified diameter of the Sun at the Moon and their 09 multiplied by his revolutions (in a व ^^) and divided by the Moon’s revolutions (in that cycle), or multiplied by the periphery of the Moon’s orbit and divided by that of the Sun, becomes the diameter of the Sun at the Moon’s orbit. The diameter of the Sun at the Moon’s orbit and the Moon’s rectified diameter divided by 15, give the numbers of minutes contained in the diameters (of the discs of the Sun and the Moon respectively) ha athe ae neha 4and5. Multiply the true diurnal bee ares shadow at the motion of the Moon by the Earth’s diameter (or 1,600) and divide the product by her mean diurnal motion ; the quantity obtained is called the Stcuf. Multiply the difference between the Earth’ diameter and the rectified diameter of the Sun by the mean diameter of the Moon (or 480) and divide the product by that of the Sun (or 6,500) ; subtract the quotient from the Stcuf the remainder will be the diameter (in yosanas) of the Earth’s shadow (at the moon); reduce it to minutes as mentioned before (i. €, by dividing it by 15). To find the probable times 6. The Earth’s shadow (alway s) of the occurrences of eclipses. remains at the distance of 6 signs from the Sun. When the place of the Moon’s ascending node equals the place of the shadow or that of the Sun, there will be an G 42 Translation of the eclipse (lunar or solar). Or when that node is beyond or within the place of the shadow or that of the Sun, by some degrees, the same thing will take place. 7. The places of the Sun and the Moon found at the time of the new moon are equal (to each other) in signs, (degrees) | &c. and at the instant of the full moon they are at the distance of 6 signs from each other. 1, 8. (Find the changes of the places of the Sun, the Moon and her the Sun, the Moon and her ascending ascending node as giyen at ह ६ se mid-night to the instant of node in the instant from midnight to the eyzyey. the instant of the syzygy as men- tioned in 6 L0Ka 67th of Ist Chapter). To the places of the Sun and the Moon (as found at the midnight) apply by sub- traction or addition their changes according as the instant of the syzygy is before or after midnight: the results are called the sama-KALa places of the Sun and the Moon: But increase the place of the node (at midnight) by its change, if the instant of the syzygy be before midnight, or diminish it if it be after midnight, What covers the Sun and 9. The Moon being like a cloud in the Moon in their eclipses. == 8 lower sphere covers the Sun (in a solar eclipse); but in a lunar one the Moon moving eastward enters the Earth’s shadow and (therefore) the shadow obscures her disc. To find the magnitude of 10. Take the Moon’s latitude (at Bn eclipse: the time of syzygy) from half the sum of the diameter of that which is to be covered and that of the coverer (in a lunar or solar eclipse) ; the remainder is the greatest quantity of the eclipsed part of the disc. 7 ll. If this quantity should be of a total, partial or no greater than the diameter of the disc si which is to be eclipsed, the eclipse will be a total one, otherwise it will be partial. But if the Moon’s latitude be greater than half the sum (mentioned in the preceding 31.014) there cannot be an eclipse, Surya-Siddhanta. 43 To find the half duration 12. Find the halves separately of of the eclipse and that of the the sum and difference of the diameter + of that which is to be covered and that which is the coverer. Subtract the square of the (Moon’s) latitude (as found at the time of the syzygy) severally from the squares of the half sum and the half difference and take the square-roots of the results. 18. These roots multiplied by 60 and divided by the diurnal motion of the Moon from the Sun give the StairyarDHA the half duration of the Eclipse and marpdrpaa the half dura- tion of the total darkness in aHaTix4s (respectively). To find the exact Srarry- 14 and 15. Thediurnal motions (of ARDHa and MARDARDHA, = tho Sun, the Moon and her ascending node) multiplied by the SruiryarpHa (above found) in Ga#ati- ४648 and divided by 60 give their changes in minutes. Then to find the first exact StuityaRDHA, subtract the changes of the Sun and the Moon from their places and add the node’s change to its place; from these applied places find the Moon’s latitude and the Sruiryarpga. This Sta#ityarpHa will be somewhat nearer the exact one, from this find the changes and apply the same mode of calculation (as mentioned before) and repeat the process until you get the same STHITYARDHA in every repeti- tion. This Srmirvarpaa will be the exact first SruityarDHa. But to find the latter SrniryazpHa add the changes of the Sun and Moon to their places and subtract the node’s change from its place; from these applied places find the Moon’s latitude and the StuiryarDHa again and repeat the same process until the exact latter Srairyarpaa be found. In the same manner determine the first and second exact MaARDARDHAS by repeated calculations. To find the times of the 16. . the beginning of the eclipse, draw a line equal to half the difference between the diameters (of the coveror and.that which is to be covered) in such a manner that its end fall on the coveror’s path. About the end of that line describe a circle with an extent equal to the semi-diameter of the coveror. Then you will find the direction of the beginning of total darkness where the third circle touches internally the circle above described. To find the direction of 22. 19 the same way draw the said the end of the total dark- line towards the end of the eclipse and भ describe a circle as above. Then you will find the direction of the end of total darkness just as mentioned before. The colour of the eclipsed 23. When the eclipsed portion of portion: of Ene noon. the Moon’s disc is less than the half, it appears of a smoky colour, when it is greater than the half, it appears of a black colour: and when the Moon’s eclipsed portion is greater than 3ths of the whole it appears of a dusky copper hue, and in a total eclipse it appears of a tawny hue. 56 Translation of the 24, (0 Maya) this science, secret even to the Gods, is not to be given to any body, but to the well-examined pupil who has attended one whole year. This science is very secret. End of the sixth Chapter. CHAPTER VII. On the conjunction of the planets. 1. The conjunction of the five mi- nor planets 18 considered their fight or association with each other (according to their light and position as will be explained afterwards) : but their conjunction with the Moon, is considered their association with her and with the Sun is their astamana disappearance. ग 4 2. The conjunction of two planets, of coujunction is past or both moving eastward, is past when future. : + the place of the quick moving planet is beyond that of the slow-moving one, otherwise (i. e. when the place of the quick-moving planet is within that of the slow-moving) their conjunction is future. But when both the planets have retrograde motions, the reverse of this takes Kinds of conjunction. place. 3. When, of the two planets, (only) one is moving east- ward and its place is beyond that of the other (which move to the west) their conjunction is past: but when the place of the retrograde is beyond that of the other ©. 6. the east- moving) the conjunction is future. To find the time of con- (When you want to know the exact junction from a given time, time of conjunction of two planets, find their true places at any given time near the time of con- junction :) (then) multiply the difference in minutes between Surya-Siddhanta. 57 the places (above found) by the diurnal motions of the planets in minutes (separately), 4. And divide the two products by the difference between the diurnal motions, when the motions of the planets are both direct or both retrograde ; but when of the planets one is retrograde, divide the two products (above found) by the sum of the diurnal motions: (the results are the changes of the planets.) 5. From the places of these two planets (found at the given time) subtract their changes when the conjunction is past, but when it is future add the changes to the places. (This rule applies when the planets move eastward,) but when they retro- grade, the reverse of this takes place. When one of the two planets is retrograde, add or subtract its change to or from its place (according as the conjunction is past or future). 6. Thus the places of the planets on the ecliptic applied with their changes become equal (to each other): divide the difference between the places of the planets (found at the given time) by the divisor which is taken before in finding their changes, the quotient will be the interval in days, Guatixas &c (between the given time and the time of conjunction) 7. Having found the lengths of the day and night of the ‘places of the planets (found at the time of conjunction) and their latitudes in minutes, (determine for that time), the time* from noon (i. e. from the time when the planet’s place comes to the meridian) and that from rising or setting of the place of each of the two planets with the horoscope (at that time according as the planet’s place is east or west of the meridian of the place). The correction called the 8. Multiply the latitude of the पय planet by the equinoctial shadow and divide the product by 12 ; the quantity obtained being multi- plied by the time in Guatixas from noon of the planet’s place * The time can be found by the Rule mentioned in 31.04 49th of the 3rd Chapter. B.D. घ 58 Translation of the and divided by half the length of the day of the planet’s place (as found before), gives the correction called the AKksH#a एषृ KARMA. | 9. Subtract the correction from the planet’s place when it is east of the meridian, and add when it is west: this holds when the latitude of the planet is north, but when it is south add the correction to the planet’s place when it is east of the meridian and subtract when it is west. च place and find the declination from the sum. Then the number of miuutes contained in the planet’s latitude multiplied by the number of degrees contained in the declination (above found) gives the correction in seconds (called the AYANA DRJKKARMA). Add or subtract this correction (to or from the place of the planet) according as the declination (above found) and the planet’s latitude are of the same name or of different names. The use of the DRIKKAR- 11, In finding the times of con- ma in finding the conjuno- junctions of the stars and planets and tions &e. व व those of rising and setting of the planets and in finding the phases of the Moon, this DRyJKKARMA correction must be applied (to the place of the planet) at first. To find the distance of 12. (Thus apply the two portions two planete in the same cir- of the DRIKKARMA correction above cle of position. found, to the equal places of the two planets as found in 6th s‘Loxa of this Chapter, and from these places applied, find the apparent time of conjunction by the Rule as mentioned in the sLoKas 2nd to 6th: and repeat the operation until you get the time at which the places of the two * DrjKKaRMA 28 the correction requisite to be applied to the place of a pla- net, for finding the point of the ecliptic on the cirele of position which passes through the planet. This correction is to be applied to the place of the planet by means of its two portions, one called the AYaNA DRIKKaRMA and the other the AKSHA DRyKKaRMA. The place of a planet with the AYANA DRE)KKARMA applied, gives the point of the ecliptic on the hour circle which passes through the planet: and thie corrected placeof the planet again, with the AksHAa DRIK- KARMA Applied, gives the point of the ecliptic on the circle of position whieh passes through the planet. B.D. Stirya-Siddhanta. 59 planets with the two portions of the pRyKKAEMA applied become equal to each other. This time is the exact apparent time of conjunction of those two planets.) Find again the places of the planets (at the time of their exact apparent time) and their latitudes from them: then find the difference between the lati- tudes when they are of the same name and the sum when they are of different names; the result will be the north and south distance (between those two planets at that time). The apparent diameters of 18. The diameters of Mars, Sa- 0 minutes: turn, Mercury, Jupiter and Venus re- duced to the Moon’s orbit are 30, 374, 45, 524 & 60 (yojanas respectively). 14, These diameters multiplied by 2 and the radius and divided by the sum of the radius and the hypothenuse found in the fourth operation (as mentioned in the 2nd Chapter) become their rectified diameters. Divide these rectified diame- ters by 15, the quotients are the minutes contained in the ap- parent diameters of the planets. 15. On the levelled floor (place a gnomon & ) mark the shadow (found at any assigned time from the bottom of the gnomon) to the opposite side of the planet: then show the planet in the mirror placed at the end of the shadow (just marked): the planet will be seen in the direction passing through the end of the shadow and the reflected end of the gnomon. 16. (When, at the time of conjunction of two planets, they will be above the horizon) erect two styles, five cubits long, one cubit buried in the ground, in the north and south line, at the distance equal to that of the*two planets (as found in the 12th stoKa of this Chapter, (reduced to digits by the Rule as mentioned in s LOKA 26th of the 4th Chapter). 17. Mark the shadows from the bottoms of the styles (as mentioned in 8 LOKA 15th) and draw lines from the ends of the shadows to those of the styles: then the astronomer may _ show the planets in the lines (above drawn). H 2 Observation of the planets, 60. Translation of the 18. (Thus) the planets will be seen in the heaven at the ends of the styles. The fight and association In the conjunction of any two minor of the planets. planets, there is their fight called the Uxtexua (paring) when their discs only touch each other: but when the discs cross each other, the fight is called the Buepa (breaking) 19. When in the conjunction, the rays of the two planets mix with each other, it is their fight, called the Ans एण- MARDA (the mixture of the rays). When in the conjunction of the two planets, their distance (found in 81.0६ ^ 12th) is less than one degree, it is their fight called the apasavya (the contrary) if one of the two planets be smaller; (otherwise the fight is not distinct). 20. (In the conjunction) when the distance of the planets is greater than one degree, it is their association, if the discs of the planets are both large and bright; (otherwise the association is indistinct). Which planet is conquer- In the fight called APASAVYA that ed inthe Hgts planet is conquered which is obscure, small and gloomy. 21. And that planet is overcome which is rough, dis- coloured or south (of the other). And that is the conqueror of which the disc is the brighter and larger, whether it be north or south (of the other). 22. If (in the conjunction) the planets both be very near to each other and bright, then their fight is called the samaama: If both the planets be small or overpowered, then the fight is called the KUya or vicrana (respectively). 23, (In the fight of Venus with any other minor planet,) Venus is usually the conqueror whether she be north or south (of the other). Which is the conqueror. Kinds of fight. Surya-Siddhanta. 61 Find the time of conjunction of the moon with any of the minor planets in the same way as mentioned before. 24. This (1, €. the association and fight of the planets) is (only) imaginary, intended to foretel the good and evil fortune people, since the planets being distant from each other move in their own (separate) orbits. End of the Seventh Chapter called the Geanaruti or the planetary conjunctions. CHAPTER VIII. On the conjunction of the planets with the Stars. To find the longitudes of 1. 1 declare the number of the the principal stars of the minutes contained in the BHoaas* of न | (श्रा) the Asterisms (As’winf, Buaranf, &c. except the UrrardsHxpHa, Asuiyit, S’ravana and Dua- * Dividing the number of minutes contained in the longitude of the principal star of an Asterism by 800 and dividing the remainder by 10, the quotient obtained 18 here called the Booga of the AstERism. B. D. Note on V 2 to 9. For convenience’ sake the longitudes of the principal stars of the four Asterisms UrrarMsHApHA, ABHIJIT, S’RAVANA and DHANISHTHA only are given and the 8980648 of the others from which the longitudes of the remaining principal stars can easily be found by the rule mentioned in lst S’LoKa, are given. The longitudes and latitudes of the stars mentioned here are the apparent ones. The apparent longitude of a star is the distance from the origin of the Ecliptic to the intersecting point of this circle and the circle of declination passing through the star: and the apparent latitude of a star is the sum or difference of its true declination and the declination of the intersecting point of the Ecliptic and the circle of latitude passing through the star, according 88 the said declinations are of different names or of the same name. The following table will exhibit the names of the Asterisms and of their principal stars as supposed to be meant, their apparent longitudes as will be found from their 2380648, and their apparent latitudes. 62 Translation of the NIsHTHA). Multiply the 38064 of each Asterism by 10 and to the product add the spaces of the antecedent Asterisms (each of which contains 800 minutes as mentioned in S‘Loxa 64th of the second Chapter), the sum is the longitude (of the principal star of the asterism). The Buoaas of the Aste- 2. (The number of minutes in the स Buoga of the Asterism called A’swinf is) 48, (of Boaranf) 40, (of [इ पणा) 65, (of Routyf) 57, (of Mryaa) 58, (of Arprd) 4, (of Punarvasv) 78, (of PusHya) 76, and (of As‘tesHf) 14. 3. (The Bxoga, 70 minutes, of Maaui is) 54, (of Pérvi- pHdtauni) 64, (of Urrard-padiaunf) 50, (of Hasta) 60, (of HITRA) 40, (of SwArf) 74, (of #*18 ^ एप) 78, (and of Anugé- DHA) 64 0 Yooa-TARAs or prin- Apparent longi- Apparent latitudes cipal stars. udes, 8 ° , © Ac’ winf, a Arietis, 0 8 0 10 N. Bharani, Musca, 0 20 90 12 N. Krittika, x Tauri, Pleiades, 1 7 30 5 N. Rohinf, © Tauri, Aldeharan, 1 19 30 5 N. Mriga, A Orionis, 2 8 10 “8. Ardra, a Orionis, 2 7 20 9 S. Punarvasu, 8 Geminorum, $ 38 6 N. Pushya, 8 Cancri, $ 16 0 N. As’lesha, © 1 and 2 Cancri, 8 19 7 8. Maghé, a Leonis, Regulus, 4 9 0 पि. Purva-phalgunf 8 Leonis, 4 24 12 N. Uttara-phalgunf, 6 Leonis, 5 56 13 ON. Hasta, y or 3 Corvi, 5 20 11 3. (01 a Virginis, Epica, 6 0 2 8. Swati, «@ Bootis; Arcturus, 6 19 ॐ ON. Vi/sakha, a or x Libra, 7 #8 1 30" 8. Anuradha, 8 Scorpionis, 7 14 $ 8. J yeshtha, a Scorpionis, Antares, 7 19 4 8. Mil y Scorpionis, 8 1 9 8. Purvashadha, 3 Sagittarii, 8 14 6 980“ 8. Uttardshadha, 7 Sagittarii, 8 20 6 8. Abhijit a Lyri, 8 26 40’ 60 N. 87४९ 818, a Aquile, 9 10 80 N. Dhanishthé, @ Delphini, 9 20 36 कि. tatarak4 A Aquarii, 10 20 0० 30 8. Purvébhadrapada, a Pegasi, 10 26 24 दि. Uttarabhadrapadaé, «a Andromedo, 11 3 26 N. Revati, ¢ Piscium, 11 29 5 OON. . B. D. Surya-Siddhanta. 63 4. (The Baoaa, in minutes, of Jyrsupui is) 14, (of Mota) 6, and (of एर ए६8प्र (प्र) 4. The principal star of Urrari- sHADHA is in the middle of the space of Pbrvdsuipna (i. e. the longitude of the principal star of UrrarasHipaa is 8 signs and 20 degrees). The principal star of ^ छपरा is at the end of the space of PérvAsnanua (i. ©. the longitude of the principal star of Asarsr is 8 signs, 26 degrees and 40 minutes) and (the principal star of) S’ravaya is situated at the end (of the space) of Urrardsufpui (i. €. the longitude of the principal star of S'Ravana is 9 signs and 10 degrees). 5. The principal star of DuanisaTHX is at the junction of the third and fourth quarters of the space of S’ravaya (i. €. the longitude of the principal star of DuanisuTHf is 9 signs and 20 degrees). (The Broaa, in minutes, of 847474६4 is) 80 (of POrvasydprapapa) 36, (and of Urrardsyaprapapa) 22, 6 to 9. (The Boga of Revarf is) 79. ‘The latitudes of the principal stars of the Asterisms As‘winf, &c. from the ends of their mean declinations are 10° N 9: 190 N., 5° N., $° 8., 10* 8., 9° 8., 6° N., 0°, 7° S., 0°., 12° N., 13° N., 11° 8. 2°S., 87° N., 192 8., 3° 9. 4° 8. 9° 8., 5% 8,, 5° 9. 60° N., 80° N., 86° N., 2° S., 24° N., 26°N., and 0° re- spectively, The longitudes and Jati- 10, 11 and 12. The star AGASTYA Mec (or Canopus) is at the end of the sign BranMannypara, Gemini at a distance of 80° south (from its corresponding point in the ecliptic, i. e. the longitude of 4 9487५ is 90° and its latitude is 80° S.) and the star Mrfaa- ` vyxpuHa or the Hunter (which is evidently Sirius) is situated in the 20th degree of the sign Gemini (i. e. its longitude is 2 signs and 20 degrees) and its latitude from the end of its mean declination (from its corresponding point in the ecliptic,) to the south is 40°, The stars called Acni (or 6 Tauri) and BrawManripaya (or Capella) are in the 22nd degree of the sign Taurus (i. e. the 64 Translation of the longitude of both of them is 1 sign and 22°. The latitudes of these two stars are 8° and 30° N. respectively. Having framed a spherical instrument examine each of the (said) apparent latitudes and longitudes. Crossing the cart of Ro- 13. That planet will cross the cart EINE | (of the Asterism) Rournyf (i. e. the place of Rohini which is figured as a cart) which is placed at the 17th degree of the sign Taurus and of which the south latitude is greater than 2°. To find the conjunction of 14. (When you want to know the ह. time of conjunction of a planet with a star) find the lengths of the day and night of the star as you found those of a planet (in the preceding chapter): and apply the AKsHA-DRIKKARMA (only) to the longitude of the star as men- tioned before ; then proceed just in the same way as in finding them in planetary conjunctions : and find the days (past or future from the given time to that of conjunction of the planet with the star) from the diurnal motion of the planet (only). To know whether the time 19. (At 8 given time), when the of conjunction is past or longitude of the planet (with the two future, : ट $ portions of the DryxKarMa applied) is less than that of the star (with the AKSHA-DRJKKARMA applied) the conjunction is future: and when the longitude of the pla- net is greater than that of the star, the conjunction is past: (this holds when the planet is direct) (but) when it is retrograde the conjunction is contrariwise (i. e. when the longitude of the planet is less or greater than that of the star the conjunction is past or future). Yooa-rkeis or principal 16. The north star of (each of the stare pene eter uane: Asterisms) PUrvApHALGuNI, Uttara- PHALGUNI, एष BHADRAPADA, UTTARAX BHADRAPADA PURvA- SHADHA, UTTARASHADHA, VISAKHA, As’WINI and Marfea is called its YOGA-TARA or the principal star. Surya-Siddhdnta, — 65 17. € अहम which is.near to and west of the north-western star of the Asterism Hasta is its $ 064 7.{744 ; and the western star of the Asterism Daanisa7ua is its YoGA-TARA. 18. The middle star of (each of the Asterisms) JYEsHTHA, S‘RavANA, ANURADBA, and Pusuya is its Yoca-TaRrs: and the southern star of each of the Asterisms Baaranf, KrirrixkA, Maaui, and Revarf is its Yooa-TARA. 19. The eastern star of each of the Asterisms Routnf, Pu- NARVASU, Muza, and As'tEsHf is its Yoaa-Thrf and of the remaining Asterisms that is the Yoaa-tdRf which is the brightest (in each Asterism) | The longitude and lati- 20. The star Prasxpati (Aurige) tude of the star PrasdraTl. i, $ doprees to the east of the star BHRAHMA-HRIDAYA. Its longitude is 1 sign and 27° and the latitude is 38° N Of the etars Apém-vates 21. The star Apfm-vatsa (b 1. 2. भ 8) is situated in the Asterism Currrd five degrees north (of its principal star) (i. e. the longitude of 4 2८4१५784 is equal to that of the principal star of Cuirrd or 180°; and its latitude is 8° N.). (And in the same Asterism) the star Apa (Virginis), somewhat larger than APAM-VATSA, 18 north of it at a distance of 6° (i. e. the longitude of Ara is 180° and the latitude 9° N.) End of the eighth Chapter on the conjunction of the planets with the stars ष मि ५ CHAPTER IX. On the heliacal rising and setting of the planets and stars. 1. I now explain the heliacal rising and setting of the bodies (the moon and other planets and stars) which have little light and (consequently) disappear on account of the brilliancy of the sun (when he approaches them). I 66 Translation of the The planets which set 2. Jupiter, Mars and Saturn set aria At helically - heliacally in the western horizon when in the eastern horizon. their places are beyond that of the sun: and they rise heliacally in the eastern horizon when their places are within that of the sun: and the same thing takes place with respect to Venus and Mercury when they have retrograde motion. T ; : Tid planats- Which’ tise ia 3. The moon, whose motion 18 the eastern horizon and set quicker than that of the sun, and im the western horizon. Mercury and Venus when they have quicker motion, set heliacally in the eastern horizon when their places are within the place of the sun: and rise heliacally in the western horizon when their places are beyond it. To find the time at which 4, (When you want to determine a planet rises or sets helia- the time of the heliacal rising or set- cally. ; : ting of a planet), find (at any given day near to that time) the true places of the sun and the planet at the sun’s setting, when the planet’s heliacal rising or setting is in the western horizon; (but) when it is in the eastern horizon, determine the places at the rising of the sun : then apply the DRIKKARMA Correction to the planet’s place (as mentioned in the seventh Chapter). &. (When the planet’s heliacal rising or setting is in the eastern horizon) find the time in prdyas, from the places (just found) of the san and the planet (by the rule mentioned in S‘toxa 49th Chapter III.): (It will be the time from the planet’s rising to the rising of the sun). But when the heliacal rising or setting of the planet is in the western horizon, find the time, in pRANAS, from the places of the sun and the planet with 6 signs added: (It will be the time from the setting of the planet to that of the sun). The time, in prdvnas, (thus found) divided by 60 gives the KALans‘as, the degrees of time (1. €. the time turned into degrees at the given rising or setting of the sun.) Surya-Siddhanta. 67 6. (The degrees of time at which before the sun’s rising or after the sun’s setting a heavenly body rises or sets heliacally, are called the [64.48.48 of that body). Thus the 1९414848 of Jupiter are 11, of Saturn 15 and of Mars 17. (i. €. when the degrees of time found by the rule mentioned in S'toxa 5th are 11, 15 or 17 of Jupiter, Saturn or Mars respectively, the planet will rise or set heliacally). 7. Venus sets heliacally in the western horizon and rises in the eastern horizon by its 8 degrees (of time) on account of the greatness of its disc (when it has retrograde motion, but when it has direct motion) and hence its disc becomes small, it sets heliacally in the eastern horizon and rises in the western horizon by 10 degrees (of time). 8. Thus Mercury rises or sets heliacally at the distance of 12 degrees of time from the sun, when it becomes retrograde ; but when it is moving quick it rises or sets heliacally at the distance of 14 degrees. 9. When (ata given time) the 1 (14848 (found from the places of the planets by the rule mentioned in 5th S’Loka) are greater than the planet’s own KALANs‘as (just mentioned), the planets become visible ; (but) when less, the planets having their discs involved in the rays of the sun, become invisible on the earth. 10. Find the difference, in minutes, between the KALANS‘a8 (i. €. Kxians‘as found from the place of the planet at the given time, and those which are the planet’s own as mentioned before) : and divide it by the difference between the diurnal motions* of the sun and the planet ; the quantity obtained is the interval in days, (ghatikas) &c., between the given time and that of the planet’s heliacal rising or setting. (‘This holds when the planet is direct ; but) when it is retrograde, take the sum of the diurnal motions of the sun and the planet for the difference of the diurnal motions. * Here motions should first be turned into time (as directed in S’LoKA 1 1111 to make the dividend and divisor similar D 12 68 ` | Translation of the 11. The diurnal motions of the’ sun and the planet multi- plied by the numbers of Pr4yas contained in the rising periods of the signs occupied by the sun and the planet, and divided by 1,800, become the motions in time. From these motions (turned into time) find the time past or future in days, GHATI- xs &c., from the given time to the time of heliacal rising or setting of the planet. 12. The stars Swfrf (Arcturus), Acastya (Canopus) Maiaa- vydpua (Sirius), Caitrx (Spica), Jyesa#pui (Antares), Punar- vasu (8 Geminorum), Abhijit (« Lyre) and BranManrfpaya (Capella) rise or set heliacally by 18 degrees of time. 18. The stars Hasta (8 Corvi), S’ravana (a Aquile) Ptrva- pudiaunf (8 Leonis), Urrarx-pHAteunf (8 Leonis), DHANISHTHA (2 Delphini), Rouryf (a Tauri), Maca (Regulus), Vis AKHA (a Libree) and As’winf (a Arietis) rise (or set) heliacally by 14 degrees of time 14. The stars Kgirrixa (x Tauri, Pleiades), AnurapHs (8 Scorpionis), Mt a (v Scorpionis), As’LEsHA (a 1 and 2 Cancri), & 4 (a Orionis) P6gyAsHapHA (6 Sagittarii) and UrrardsHa- DHA (7 Sagellari) rise (or set) by 15 degrees of time, 15. The stars Boaranf (Musca), Pusnya (8 Cancri) and 11104 (A Orionis), on account of their smallness, rise or set heliacally by 21 degrees of time; and the others [1. €. S’ata- TABAKS (A Aquarii), POrvA-BHADRAPADA (a Pegasi), Urrara- BHADRAPADA (० Andromedz), Revati (€ Piscium), Aani (8 Tauri), Prasfpati (8 Aurige), ApAmvatsa (b 1. 2. 8.) and Apa (8 Virginis)] rise and set by 17 degrees of time. 16. The 41.848 (of 8 planet and those which are found at a given time from the place of the planet) multiplied by 1,800 and divided by the rising period of the sign which is occupied by the planet, give the degrees of the ecliptic. (Then in 93107६4 10th) take the degrees of the ecliptic for their corresponding degrees of time and from them find the time of heliacal rising or setting of the planet, Stirya-Siddhdnta. 69 17. The said stars rise heliacally in the eastern horizon and set heliacally in the western. Apply the AksHA-DRIKKARMA to their longitudes and (through them) find the days past or future from the given time to the time of heliacal rising or setting of the stars from the diurnal motion of the sun only (by the rule mentioned in 10th S’toxa). 18. Thestars Apaisit (a Lyre), BRanMA-HRIDAYA (Capella), कव (Arcturus), S’ravana (a Aquile), DuanisutHf (a Del- phini) and Urrard-suaprapapd (a Andromede) never disappear owing to the sun’s light on account of the greatness of their north latitudes (1. e. these stars having great north latitudes never set heliacally) in the northern hemisphere. End of the ninth Chapter on the heliacal rising and setting of the planets and stars. CHAPTER X. On the phases of the Moon and the position of the Moon’s cusps. 1. Find the time also at which the Moon will rise or set heliacally in the same way as mentioned before. She becomes visible in the western horizon and invisible in the eastern horizon by 12 degrees of time. To find the time of daily 2. Find the true places of the Sun setting of the Moon. and the Moon (at Sun-set of that day of the light half of a lunar month at which you want to know the time of daily setting of the Moon) and apply the two por- tions of the pgikKARMA to the moon’s place) ; from those places, with 6 signs added, find the time in praNas (just in the same way) as mentioned before (in 5th S’Loxa of the preceding. Chapter). At these prawas after the sun-set, the Moon will set (on that day). : 70 Translation of the ~ To find the time of daily 3. (But when you want to know mieing.of the Moon, the time of the Moon’s daily rising on a day of the dark half of a lunar month) find the true places of the Sun and the Moon (at sun-set) and add 6 signs to the Sun’s place (and apply the two portions of the DRIKKARMA to the Moon’s place) ; from these places (i. €, from the Sun’s place with 6 signs added and from the Moon’s place with the DRIKKARMA applied) find the time in pra&Nas (in the same way as mentioned before in 5th S’toxa of the preceding Chapter). At this time in prayas after sun-set the Moon will rise (on that day) To find the phases of the 4. (When you want to know the Moon. phase of the moon on a day of the first quarter of a lunar month, find the true declinations of the Sun and the Moon at sun-set or sun-rise of that day) find the difference of the sines of the declinations (just found), when they are of the same name, otherwise find the sum: to this result (the difference or the sum) give the name of the same direction south or north at which the Moon is from the Sun. 5. Multiply the result by the hypothenuse of the gnomonic shadow of the Moon (at the same time as can be found by the rule mentioned in the third Chapter): find the difference between the product and twelve times the equinoctial shadow if the result be north (but) if it be south find the sum of them. 6. The amount (thus found) divided by the sine of co-lati- tude of the place, gives the Bénu or base (of a right angled triangle) : this is of the same name of which the amount is: and the sine of the altitude of the Moon is the Kofi (or perpen- dicular of the triangle). The square-root of the sum of the squares of the BAsu and Kofi is the hypothenuse (of the triangle). | 7. Subtract the Sun’s place from that of the Moon. The minutes contained in the remainder divided by 900 give the illuminated part of the Moon: This part multiplied by the Surya-Siddhanta. 71 Moon’s disc (in minutes) and divided by 12 becomes the Spuv- va or rectified illuminated part. 8. (Ona board or levelled floor) having marked a point repre- senting the Sun, draw from that point a line equal to the Banu (above found) in the same direction in which the Baut 18, and from the end of the Baut a line (perpendicular to it) equal to the Kort (as above found) to the west, and draw the hypothenuse between the end of the Kofi and the point (denoting the Sun). 9. About the point where the Koti and the hypothenuse meet, describe the disc of the Moon (found at the given time). In this disc suppose the directions (east, west d&c.,) through the line of the hypothenuse (i. e. in the disc suppose the east where the line of the hypothenuse cuts the disc, the west where the same line produced intersects it, and the north and south where a line passing through the centre of the dise and being perpendicular to the line of the hypothenuse cuts the disc). 10. Take a part of the hypothenuse within the disc from the (latter) intersection of the disc and the hypothenuse equal to the (rectified) illuminated part: and between the end of that part and the north and south points of the disc describe two TIMIS. 11. From the intersecting point of the two lines, drawn through the Timis, describe the arc which will pass through the three points (the end of the illuminated part and the north and south points of the disc). The disc thus cut by the are will represent the form of the Moon as it will be seen on the evening of the given day. 12. Marking the directions in the disc through the Kop (above drawn), show the horn elevated ai the end of the trans- verse line ; this figure will represent the phase of the Moon. 13. In the dark half of the lunar month subtract the place of the Sun with 6 signs added to it, from the Moon’s place, and from the remainder find the dark part of the Moon (in the same way as you found the illuminated part in the 7th S’Loxa) : 72 Translation of the (and in the diagram) change the direction of the 84 ए and show the dark portion of the Moon in the west. End of the tenth Chapter called S’ringonnati which treats of the phases of the moon. CHAPTER XI. Called PATADHIKARA which treats of the Rules for finding the time at which the declinations of the Sun and Moun become equal. 1. It 18 called Varpurfta when the Sun and Moon are in the same AYANA (i. ©. when they are both in the ascending or descending signs), the sum of their longitudes equal to 12 signs (nearly) and their declinations equal. 2. It is called VratiraxTa when the Moon and the Sun are in different Avyawas, the sum of their longitudes equal to 6 signs (nearly) and their declinations equal. 8. The Fire (named Pata) which arises from the mixture of the rays of the sun and the moon in equal quantities, being burnt by the air called Pravana produces evil to mankind. 4. Since the (said) Pata frequently destroys people at the time (when the declinations of the Sun and Moon become equal) it is called एर 4702476. It is also called Vamurfta. 5, This ८74 18 of black colour and hard body, red eyed and gorbellied, destroyer of all people and horrible: it happens frequently. ` ` VaIDHRITA. VyArTfpAta. त 8 6. When the sum of the places of true declinations of the Sun the Sun and Moon, applied with the and Moon become equal. d : । egrees of the precession of the equi- noxes as found by observation, is 12 or 6 signs find their declinations. Surya-Siddhanta. 73 7. Now, if the Moon’s mean declination (i. e. the declina- tion of her corresponding point in the ecliptic) with her latitude applied (i. e. her true declination) be greater than that of the Sun, when the Moon is in an odd (18४ or 3rd) quarter of the ecliptic, the 2474 (or the instant when the declinations of the Sun and Moon become equal) is past. 8. And (if the Moon’s declination be) less, (the Pata is future. But when the place of the Moon is in an even i.e. 2nd or 4th) quarter (of the ecliptic) the reverse of this takes place (i. e. if the Moon’s true declination be greater than that of the Sun the Pata is future, and if less the Pata 18 past). When the Moon’s (mean) declination is subtracted from her latitude (for her true declination change the name of the Moon’s quarter. 9. Multiply the sines of the declinations (as found .in the 6th 87.014) by the radius and divide the products by the sine of the greatest declination (i. e. 24°): take the arcs whose 81068 are equal to the quotients, and add the difference or half the difference of the arcs to the Moon’s place when the Pata is future. (This result which is just applied to the Moon’s place is called the moon’s change). 10. But when the Pata is past, subtract the Moon’s change from her place. The Moon’s change multiplied by the true daily motion of the Sun and divided by that of the Moon gives the Sun’s change: apply it to the Sun’s place as in the case of the Moon. 11. Find the change of the Moon’s ascending node in the same way (i. ©, multiply the Moon’s change by the daily motion of the node and divide the product by the Moon’s true daily motion) : apply this change inversely to the node’s place. Find the declinations of the Sun and the Moon again (from their places with their changes applied) and apply the same process (mentioned in the preceding S:Loxas) repeatedly until ‘you get their declinations equal. K 74 Translation of the To find when a 47 is 12. The Pata is that instant at past or to be past. which the declinations (of the Sun and the Moon) become equal. Now, according as the Moon’s true place found at the Pata by applying the Moon’s change (as mentioned before) is less or greater than that found at mid- night (of that day), the Pata is before or after (the mid-night.) To find the true sime of 13. The difference, in minutes, Srp Reta: between the Moon’s true places found at the Pira and the mid-night, multiplied by 60 and divided by the true daily motion of the Moon gives the @HATIKAS between the Pata and the mid-night. (Then you will get the time of the Para by adding or subtracting the GHaTIKAs, just found, to or from the mid-night according as the Pata is past or future). | Fo find half the duration 14. (Find the semi-diameters, in phthe PATE Ae: minutes, of the Sun and the Moon by the Rule mentioned in the 4th Chapter.) The sum of the semi-diameters of the Sun and the Moon multiplied by 60 and divided by the Moon’s true daily motion from the Sun gives half the duration of the PAra-KAta.* To find the beginning, 15. The true time of the Pata middle and endof the PAta. (found in the 13th S’toxa) is called the middle of the P4ra: This time diminished by half the duration of the Pdta, just found, gives the beginning of the Pita and increased by half the duration gives the end of the Pda, 16. The interval between the beginning and end of a Pita is horrible; being in the form of burning fire, all rites are prohibited during its continuance. 17. As long as the distance of any point of the sun’s disc (from the equi- noctial) is equal to that of any point of the Moon’s disc, the Form of the PAtTAa-KALA. # The Pa’Ta-Ka’LA, or duration of the Pa’ra, is the time during which the declination of any point of the Sun’s disc and that of any point in the Moon's are equal.—B. D- Surya-Siddhanta. 75 474 -ए 41. lasts and destroys the (happy results of) all rites (performed during that time). 18. People get very great religious merits from such (virtuous) acts as bathing, alms-giving, prayers, funeral cere- monies, religious obligations, burnt offerings, &c. (performed in the 2.74 -ए (1.4), as well as from the knowledge of that time. 19. When the (mean) declinations of the Sun and the Moon become equal, near the equinoctial points, the Pfra of the two kinds (i. €. Vyatfrdta and Vaipurfta) happens twice: contrari- wise (i. e. when the mean declinations become equal near the solstitial points, and the true declination of the Moon is less than that of the Sun) no PAta happens. 20. There becomes a third 247 called (also) Vyatfpkra* when the minutes, contained in the sum of the places of the Moon and the Sun, divided by the Buassoaa (or 800) give a quotient which terminates in 17 (1. 6. which is more than 16 aud less than 17). Third 2474. 21. The last quarters of the Nax- SHATRASt ASLESHA, JYESHTHA and Revatf are called the Baasanpui (or junctions of Naxswarras) and the first quarter of each of their following ones (i. e. Maaua, 21. and As‘winf) is called the GANDANTA. 22. During the three frightful Vyatfexs, GanpANTas and Baasanpuls (just mentioned), all (joyful) acts are prohibited. 23. (0 Maya,) thus far have I told you the excellent, virtu- ous, useful secret and great knowledge of Astronomy, what more do you want to hear ? End of the 11th Chapter called Pérapuixara. End of the First Part of the SGRrya-sIDDHANTA. Ganpa’NTA and BHASANDHI. * This is the ४0०७८ or the period of time in which the sum of the places of the Sun and the Moon increases by 800’. This Yooa is the 17th reckoned from VISHKAMBHA. See 65th S’Loxa of the second Coapter.—B. D. t+ These are the periods 9th, 18th and 27th from As’winf: they are found from the Moon’s place by the Rule mentioned in the 64th S’LoKa of the 2nd CuapTEeR.—B. D. Kk 2 76 Translation of the CHAPTER XII. On CosmogPaphical Matters. 1. Now, Maya-asvRa joining the palms of his hands, saluted (his teacher) the man who partakes of the Sun’s nature, and worshipping him with his best respects asked this :— 2. (Tell me, O my) omnipotent (master,) What is the magnitude of the Earth? what is its form? what supports it ? how is it divided ? and how are the seven PATALA-BHUMIS8 or lower regions situated Question about the Earth. in it ? Question about the sun’s 3. How does the Sun cause day Peso and night ? How does he, enlighten- ing (all) the worlds, circumvolve the Earth ? 4, Why are the day and night of of the (Gods) and Asuras mutually the reverse of each other (i. e. why is it day to the Gods when it is night to the Asuras and vice versa) : and how is it that the (said) day and night is equal to the time in which the Sun completes one revolution ? 5. By what reason does the day and night of the Pirris con- sists of a lunar month and that of man consists of 60 aHafiK phases and cusps of,... as ‘és ... 69 Planets, on finding their mean places, ... 3 ~ dk ५ 5 » true places, ... es .. 18 » revolutions of, ०, es ve im ` 9 » cause of their motion, vee sa .. 13 » conjunctions of, ... ase iy 56, 61 » order of, ye ees ०० 29 +» dimensions of, their orbits and daily motions, ,,, 86 Position, questions on, __.... wee ०७७ ०० 26 Precession of the equinoxes, ive ०० .. 29 Signs of the ecliptic (or zodiac), right ascension of, ... 88 Sun, longitude, declination, &c., of, ... si . Ol » eclipses, .. es sine 48, 52 Sun and Moon, when declinations are equal oe . 7 Time, kinds of, “es श ons 4, 91 + questions on, ae ei ०० .. 26 Yugas, ००९ aes sae + ०० 38 (1 म as. «> ; ) ८ ~ ५“ ae tA . TRANSLATION OF THE SIDDHANTA SIROMANI. CONTENTS. CrapteR I.—In praise of the advantages of the study of the Spheric, rer ee re CHaPtEeR II.—Questions on the general view of the Sphere, ... CuaPTER IJI.—Called Bhuvana-kos/a or Cosmography, ......... Cuaprer IV.—Called Madhya-gati-vasané; on the principles of the Rules for finding the mean places of the Planets,... CuapteR V.—On the principles on which the Rules for finding the true places of the Planets are grounded Cuarter VI.—Called Golabandha; on the construction of an Armillary Sphere, wee 7 (^^ ९777 VII.—Called Tripras’na-vasana ; on the principles of the Rules resolving the questions on time, space, and directions, ate ध 4 24 Cuapter VIII.—Called Grahana-vasana ; in explanation of the cause of Kclipses of the Sun and Moon,............... CuapteR IX.—Called Drikkarama-vasané; on the principles of the Rules for finding the times of the rising and setting of the heavenly bodies, अ CuapteR X.—Called S/ringonnati-vasané; in explanation of the cause of the Phases of the Moon, .............0.cecceeee CuaprerR XJ.—Called Yantradhydya; on the use of astronomical instruments, (न ees 2.3 (^ 278 XII.—Description of the Seasons, 1 (^ ९787 XIJI.—Containing useful questions called Pras’né4- dhyaya,..,... 13 Page 105 107 112 127 160 Jos TRANSLATION OF THE GOLADHYAYA OF THE SIDDHANTA-S TROMANI. CHAPTER I In praise of the advantages of the study of the Spheric Salutation to Gayezsa ! 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. ~ [nvocation. 2. Inasmuch as no calculator can hope to acquire in the assemblage of the learned a distinguished reputation as 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 temove the doubts which may arise in his own mind, I therefore proceed to treat of the sphere, in such a manner as to make the reasons of all my calculations manifest. On inspecting the Globe they become clear and manifest as if submitted to the eye, and are as completely at command, as the wild apple (4uwlé) held in the palm of the hand. Object of the work. B 106 Translation of the [I. 3. = | 3. As a feast with abundance of of the Seberio, on rgnoran’? all things but without clarified butter, and as a kingdom without a king, and an assemblage without eloquent speakers have little to recom- mend them ; so the Astronomer who has no knowledge of the spheric, commands no 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 he, 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 is said, by ppnere: the wise, to be a representation of the celestial sphere, for the purpose of ascertaining the proofs of the positions of the Earth, the stars, and the planets: 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 cana man, ignorant of mathematics, comprehend the doctrine of the sphere &c. ? Who is likely to under- 7. Mathematical calculations are 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. II. 3.] Sidhanta-siromani. 107 8. He who has acquired a perfect knowledge of Grammar, which has been termed VeEpAvapDANA i. e. the mouth of the Vrepas and domi- cile of Saraswati, may acquire a knowledge of every other sclence—nay of the Vrepas 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 otherson 9. O learned man; if you intend ee न hae ay 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 18 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 ० 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 16 falls 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 ee १ oe place of a planet found out from well taining planets’ true places calculated AHARGANA (or enumeration aud their causes. of mean terrestrial days, elapsed from B 2 108 Translation of the [11. 3. the commencement of the Kaupa)* by applying the rule of pro- * (A Katpa is that portion of time, which intervenes between one conjunction of all the planets at the Horizon of LanxA (that place at the terrestrial equator, where the longitude is 76° E., reckoned from Greenwich) at the first point of Aries, and a subsequent similar conjunction. A Kapa consists of 14 Manus and their 15 sANDHIS; each MANU lying between 2 sanDHIs. Each MaNU eontains 71 yu@as; each yuGa is divided into 4 yvuagAnG@uris viz., Krivra, Teeta’, Dwipapa and Katt, the length of each of these is as the numbers 4,3,2and 1. The beginning and end of each yu@a’NGuRIs being each one 12th part of it are respectively called its saspHYA and SanpHya’Nsa. The number of sidereal years contained in each yuGa’NGHRI, &c. are shewn below ; ॐ ००७ ०००००७० ०७०७००९ ०७७ ०००७७०० ०७७ ७००७००७ ०००७ ००९७००७ ००७७ ७०००००५ ०००००४ 432,000, न COC COO CEE ००७००७० Od ०७७१०७० ७७००७०० ०७७०००७ ००००७७००००००००७ 864,000, TA ०७० ०७9 -@०9 ०७७ ००७ EET ०७००७००० LOH SEH EES HOH SEH HOH HOE HOE HEE TEE O88 1,296,000 Knir,...... ००० ७७०००७० ००७ ७००००००१ ०७०७००७ ००० ००७९००७ ७०७० ००७०००७ ००७००० 1.728.000, Yuaa, oetcceveee ०००७७०७५ 4,320,000, १1 >€ YUGA = MANU, | ,,१११११११११११०१०००१०११००००००००,,०,०, 4,294,080,000, 0.01 id vawar cues viens esuintstakeuecdaus ves 306,720,000, 15 Manv sanpuis each egual to a हा YUGANGHRI, 25,920,000, Of the present Kapa 6 Manus with their 7 sawpu1s, 27 yc@as and their three YuGa’NGHRIi. ९. 7८, Teeta, and Dwa’para, and 3179 sidereal years of the fourth yua@a’NeHRI of the 28th Yuaa of the 7th manu, that is to say, 1,972,947,179 sidereal years have elapsed from the beginning of the present Kapa to the commencement of the Sa’t1wa’HANa era. Now we can easily find out the number of ior that have elapsed from the beginning of the present KaLPa 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 fouud, their number in a Kaira or Yua@a can be calculated by the rule of proportion. By this method ancient Astronomers found out the number of lunar and ter- restrial days in a Kaupa as 0 below. 1,602,999,000,000 (synodic) lunar days } . and 1,577,916,450,000 1 days म { in a Kapa. With the foregoing results and a knowledge of the number of sidereal years contained in a KaLpa as well of those that have passed, we caa find out the number of mean terrestrial days from the beginning of a Kaupa to any given day. This number is called AHarG@ana and the method of finding it is given in GanitTADHyAya by BuHa’sKaRa’CHA’RYA. By the daily mean motions of the planets, ascertained by astronomical observa- tions, the numbers of their revolutions in 8 [६ ^ 24 are known and are given in works on Astronomy. To find the place of a planet by the number of its revolutions, the number of days contained in a Kapa and the ^ प्^ ४6५ ए toa given day, the following pro- portion is used. As the terrestrial days in a Katpa, : the number of revolutions of a planet in a Kaupa : the AHARGANA: : the number of revolutions and signs &c. of the planet in the AHARGANA. By leaving out the number of revolutions, contained in the result found, the remaining signs &¢. 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 GayitipHyAya. 28687 ^ 8.६९ ५/४ ५ has stated the revolutions in a Katpa, but he has here mentioned the revolutions in a YUGA on account of his constant study of the 8S’IsHya-DHivRiDDHIDA-TANTRA, a Trea- ni Astronomy by Latza who has stated in it the revolutions in a Yuaa.— II. 4.] Sidhanta-siromani. 109 portion to the revolutions in the Yuaa* &c. is not a 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 Desantaza, UDAYANTARA, एषण, andCuara corrections?+ What is the ManpocucHat (slow or 18४ Apogee) and S/feHrocucHa§ (quick or 2nd Apogee)? What is the node ? 4. Whatis the Krnpra|| and that which arises from it (i. e. the sine, cosine, &c. of it)? What is the ManpapHata|| (the first. equation) and S'f@HRaPHALAg (the 2nd equation) which depend on the sine of the Kenpra? Why does the place of a planet become true, when the Manpapua.a or S'faHRAPHALA = * [It may be proper to give notes explaining concisely the technical terms occurring in these questions, which have no corresponding terms in English, in order that the English Astronomer may at once apprehend these questions with- out waiting for the explanation of them which the Author gives in the sequel.— + [To find the place of a planet at the time ofsun-rise at a given place, the several important corrections, i. e. the Upaya’NTARA, BaUJa’NTARA, Des'ANTARA, and (प ^ ए.^ are to be applied to the mean place of the planet found out from the Anareana by the fact of the mean place being found from the छ ^+ 204 त + for the time when a fictitious body, which is supposed to move uniformly in the Equi- noctial, and to perform a complete revolution in the same time as the Sun, reaches the horizon of Lanxa’. We now proceed to explain the corrections. The Upaya’nTaRa and BuuJa’NTaRa corrections are to be applied to the mean place of a planet found from the AnarGana for finding the place of the planet at the. true time when the Sun comes to the horizon of Lanxa’ arising from those 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 Dgs’a’NTaRA and CHABA corrections are to be applied to the mean place of a planet applied with the Upaya’yTaga and BHUJA’NTARA corrections, for finding the place of the planet at the time of sun rise at a given place. The Dxs’a/nTaRa correction due to the longitude of the place reckoned from the meridian of Lanxa’ and the CHara correction to the ascentional differ- ence. B. D.] { [ManpocucHa is equivalent to the higher Apsis. The Sun’s and Moon’s Manpocucuas (higher Apsides) are the same as their Apogees, while the other planets’ ManpDocHCHas are equivalent to their Aphelions. B. D.]} § [S‘V’aurocHona is that point of the orbit of each of the primary planets (1. 6. Mars, Mercury, Jupiter, Venus and Saturn) which is furthest from the Earth. B. D. | 1 is of two kinds, one called MANDA-KENDRA corresponds with the anomaly and the other called 8’I’@HRa-KENDRa is equivalent to the commutation added to or subtracted from 180° as the SIGRA-KENDRa« is greater or less than 180° 8. D.] q [Manpa-PHALA is the same as the equation of the centre of a planet and §’GHRA-PHALA is equivalent to the annual parallax of the superior planet ; and the elongation of the inferior planets. B. D.] 110 Translation of the [II. 5. are (at one time) added to and (at another) subtracted from it ? What is the twofold correction called Drixxarma* 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 9. Tell me, O you acute astrono- length of the day and night. er why, when the Sun is on the 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 ? 1 6. How is it that the day and length of the day and night night of the Gods and their enemies oF the Gods Dartyas,PITBIS Oarryas correspond in length with the solar years? How is it that the night and day of the Pirris 18 equal in length to a (synodic) lunar month, and how is it that the day and night of 22 प 4 18 2000 ए ०५48 in length ? (00 the 7. Why, O Astronomer, is it that periods of risings of the the 12 signs of the Zodiac which are signs of the Zodiac. $ + 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 as to the places of 8. Shew me, O learned one, the the DrusvA, the Kusya,&. laces of the Dyusyf (the radius of the diurnal circle), the [इ एर (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 * (DaikkarMa 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 planet reaches it. This correction is to be applied to the place of a planet by means of its two portions, one called the AyYana-DRIKKARMA and the other the AksHa-DRIK- KakMA. The place of a planet with the AYANA-DRIKKARMA applied, gives the point of the ecliptic on the six o’clock line when the planet arrives at it: and this corrected place of the planet, again with the AKsHA-DRIKKARMA applied, gives the point of the ecliptic on the horizon when the planet comes to it. B. D.} + The Krita, Teeta/, Dwa’para and Kati are usually called Yuaas: but the four together form only one Yuaa, according to the SippHa’nTa system, each of these four being held to be individually but a Yuea’Neusr, L. क, IT. 10.] Sidhanta-siromani. 111 of a small circle), and show me also the places of the declination, Sama-s4nxu,* Aara (the sine of amplitude), latitude and co-latitude &c. in-this Armillary sphere as these places are in the heavens. ८ If the middle of a lunar Eclipse tain differences in the times takes place at the end of the Tirni 12 of solar andlunar (gt the full moon), why does not the middle of the solar Eclipse take place in like 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 Eclipse ?+ Questions regarding the 9. What, 0 most intelligent one, + is the Lampanat and what is the Natit? why is the Lamsana applied to the Tirui 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 the 10. Ah! why, after being full, does Biases? on, be atoon. 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, does she thenceforth, by successive augmentation of her pure * (Sama-sa/nxkv is the sine of the Sun’s altitude when it comes to the prime vertical. B. D.] + [An Eclipse of the Moon is caused by her entering into the Earth’s shadow and as the place of the Earth’s shadow and that of the Moon is the same at the full moon, the conjunction of the Earth’s shadow and the Moon must happen at the same time ; and an Eclipse of the Sun is caused by the interposi- tion of the Moon between the Earth and the Sun, and the conjunction of the Sun and Moon in like manner must happen at the new moon, as then the place of the Sun and Moon is the same. As this is the case with the eclipses of both of them (i. e. both the Sun and Moon) the querist asks, “Ifthe middle of a lunar eclipse &c.” It is scarcely necessary to add that the assumption that the middle of a lunar eclipse takes place exactly at the full moon, is only approximately correct. B. D.] ‡ [The Lampana is equivalent to the Moon’s parallax in longitude from the Sun reduced into time by means of the Moon’s motion from the Sun: and the Nati is the same as the Moon’s parallax in latitude from the Sun. B, D.] 112 Translation of the (IIT. 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. aie: eenslinnee:. fe? is 1. TheSupreme Being Para Brau- Supreme Being. mA the first principle, excels eternally. From the soul (PurusHA) and nature (Praxriti,) when excited by the first principle, arose the first Great Intelligence called the Manartattwa or Buppuitatrwa: from it sprung self-con- sciousness (AHANKARA :) from it were produced the Ether, Air, Fire, Water, and Earth ; and by the combination of these was made the universe BraumMAnpa, in the centre of which is the Earth: and from Braumd Cuaturdnana, residing on the sur- face of the Earth, sprung all animate and inanimate things. 2. This Globe of the Earth 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 Dairtyas. Description of the Earth. * This verse has a double meaning, all the native writers, however grave the subject, being much addicted to conceits. 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 after having met with a Brahman skilled in the ए 7748, and by having recourse to him, thenceforth becomes distinguished for his eminent good conduct by gradual augmentation of his illustriousness. L. W ITT. 6.] Sidhanta-s'iromani. 118 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 tion that the Earth has suc- any material substance or living crea- cessive supporters. ; 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 one of the forms of the eight-fold divinity i. €. of S’rva. Refutation of the objeo- 5. As heatis an inherent property Lean bas ee coger: of the Sun and of Fire, as cold of the Moon, fimdity 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. The* property of attraction.is inherent in the Earth. By this property tie Earth attacts any unsupported heavy thing towardsit: The thing appears to be falling [but it is in a state of being drawn to the Earth]. The etherial expanse being equally outspread all around, where can the Earth fall ? Opinion of the Bavp- 7. Observing the revolution of the DHAS. constellations, the BauppHas thought that the Earth had no support, and as no heavy body is seen stationary in the air, they asserted that the earth} goes eternal- ly downwards in space. 8. The Jarmnas and others main- Opinion of the JaInas. : tain that there are two Suns and two * It is manifest from this that neither can the Earth by any means fall downwards, nor the men situated at the distances of a fourth part of the circum- ference from us or in the opposite hemisphere. ए. D.] + [He who resides on the Earth, is not conscious of the motion of it down- wards in space, as a man sitting on a moving ship does not perceive its motion, B. D. ¢ 114. Translation of the [III. 9. Moons, and also two sets of constellations, which rise in con- stant alternation. To them I give this appropriate answer. Refutation of the opinion 9 Observing अ. ^ do, O Bavp- of the Bauppwas. DHA, that every heavy body projected into the air, comes back again to, and overtakes the Harth, how then can you idly maintain that the Earth is falling down in space? [If true, the Earth being the heavier body, would, he imagines*—perpetually gain on the higher projectile and never allow its overtaking it. ] Refutation of the opinion 10. But what shall I say to thy ^ 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 ? Refutation of the supposi- 11. Ifthis blessed Earth were level, tion that the Earth is level. 1-6 8 plane mirror, then why is not the sun, revolving above at a distance from the Earth, visible to men as well as to the Gods? (on the ^ एदद्वा ^ hypothesis, that itis 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 between us and the Sun? And Merv being admitted (by the Pauranixas) to lie to the North, how comes it to pass that the Sun rises (for half the year) to the South ? iis sob Chace oe: 13. As the one-hundredth part of pearance of the plane form the circumference of a circle is (scarce- oer 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 BHASKARA’S own notion :—but even on the more correct principle, that all bodies fall with equal rapidity, the argument holds good. ए. D.] IIT. 18.] Sidhanta-s‘tromant. 115 Proof ofthe correctness of 14. गं the correct dimensions of alleged circumference of the the circumference of the Earth have न been stated may be proved by: the simple Rule of proportion in this mode: (ascertain the differ- ence in Yusanas between two towns in 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 YusANAs, what will the whole circumference of 360 degrees give ? To confirm the same cir- 15. As itis ascertained by calcula- cumference of the Harth, = ‘tion that the city of Ussayinf is situated at a distance from the equator equal to the one-sixteenth part of the whole circumference: this distance, therefore, multiplied by 16 will be the measure of the LEarth’s cir- cumference. What reason then is there in attributing (as the PaurdyixKas 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 consistent with this (estimate of the extent of the) 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. Lanxf is situated in the middle of the Karth: Yama- KOT! is situated to the East of LANKA, and RoMAKAPATTANA to the west. The city of SrppHapura 1168 underneath [^^ प्रह द. SUMERU is situated to the North (under the North Pole,) and Vapavdnata to the South of LanxA (under the south Pole) : 18. These six places are situated at a distance of one-fourth part of the Earth’s circumference each from its adjoining one. So those who have a knowledge of Geography maintain. At Mert reside the Gods and the ७7770048, whilst at VADAVANALA are situated all the hells and the Darryas. | ५ ४ 116 Translation of the [III. 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 night angles to himself. 20. Those who are placed at the distance of half the Earth’s circumference from each other are mutually antipodes, as a 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 the same ease as we do here in our position. Positiona: 1 21. Most learned astronomers have and Beas. stated that JamB0pwfra embraces the whole northern hemisphere lying to the north of the salt sea: and that the other six Dwfpas and the (seven) Seas viz. those of salt, milk, &c. are all situated in the southern hemis- 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 Laxsumf, and where the Omni- present VAsupeva, to whose Lotus-feet Braumi 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 24741. 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 Sippwas enjoy themselves with the pleasing persons of beautiful females resembling the finest gold in purity. 25. The 94 ^, 9 AtMata, Kavs’a, Kefuncua, Gomepa&ka, and III. 81.] ` Sidhdnta-s'iromani. 117 Pusuxara Dwfpas are situated [in the intervals of the above mentioned seas] in regular alternation: each Dwipa lying, it is said, between two of these seas. Positions of the Moun- 26. To the North of LanxA lies ५ a : त the Himdtaya mountain, and beyond caused by the mountains. that the HermaxttTa mountain and beyond that again the NisHaDHA mountain. These three Mountains stretch from sea to sea. In like manner to the north of SippHapura lie in succession the S’RINGAVAN Suxia and Nita mountains. To the valleys lying between these mountains the wise have given the name of VarsHas. 27. This valley which we inhabit is called the BuxRata- VARSHA; to the North of it lies the KiINNARAVARSHA, and beyond it again the HarivarsHa, and know that the north of SippHapura in like manner are situated the Kuru, Hiran- mAYA and Ramyaxka VARSHAS. 28. To the north of Yamaxort! lies the Matyavd4n mountain, and to the north of Romaxapatrana the GANDHAMADANA mountain. These two mountains are terminated by the Niza and NisHADHA mountains, and the space between these two is called the ILAvgita VARSHA. 29. The country lying between the Ma tyavdn mountain and the sea, is called the BaaprAs wa-vaRsHa by the learned ; and geographers have denominated the country between the GANDHAMADANA and the sea, the KETUMALA-VARSHA. 30. The ItAvgira-varsHa, which is bounded by the Nisnapya, वि, GANDHAMADANA and MatyavAN 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 31. In the middle of the It4vrira eee ee VarsHa stands the mountain Merv, which is composed of gold and of precious stones, the abode of the immortal Gods. Expounders of the Purfnas have further described this Merv to be the pericarp of the earth- lotus whence Brauma had his birth, 118 Translation of the (III. 32. 32. The four mountains Manpara, SucanpHa, Virus and SupARS Wa serve as buttresses to support this Merv, and upon these four hills grow severally the KapamBa, Jampd, Vata and [17414 trees which are as banners on those four hills. 33. From the clear juice which flows from the fruit of the JaMBO springs the jamBO-napf; from contact with this juice earth becomes gold: and it is from this fact that gold is called sAmBUNADA : [this juice is of so exquisite a flavour that] the multitude of the immortal Gods and Srppuas, turning with distaste from nectar, delight to quaff this delicious beverage. 84. And it is well known that upon those four hills [the buttresses of Meru] are four gardens, (1s?) CHAITRARATHA of varied brilliancy [sacred to Kusrra], (2nd) Nanpana which is the delight of the Apsaras, (37d) the Dariti which gives refreshment to the Gods, and (4th) the resplendent VAIBHRAJA. 85. And in these gardens are beautified four reservoirs, viz. the Aruna, the Mdnasa, the ManAurapa and the S’weta- JALA, 111 due order: and these are the lakes in the waters of which the celestial spirits, when fatigued with their dalliance with the fair Goddesses, love to disport themselves. 36. Merv divided itself into three peaks, upon which are situated the three cities sacred to VisHnu, 214 ४५ and S‘1va [denominated VaixuntHa, Braumapura, and Kaiiasa], and beneath them are the eight cities sacred to Inpra, Aani, YAMA, Narrgita, Varuna, VAyu, S’asf, and Isa, [i. e. the regents of the eight Dixs or directions,* viz., the east sacred to * [As the point where the equator cuts the horizon is the east, the sun therefore rises due east at time of the equinoxes but.on this ground, we cannot determine the direction at Mrru [the north pole] because there the equator coincides with the horizon and consequently the sun moves at MERU under the horizon the whole day of the equinox. Yet the ancient astronomers maintained that the direction in which the yamMaxkorTi hes from MERU is the east, because, according to their opinion, the inhabitants of Mrru saw the sun rising towards the YAMAKOTI at the beginning of the KALPA. In the same manner, the direction in which LanKA lies from mount Mrgu is south, that in which RomaKapPaTTaNa lies, is west, and the direction in whieh SippHa- III. 40.] Sidhanta-siromani. 119 Inpra, the south-east sacred to ^+ अक्रा, the south sacred to Yama, the south-west sacred to Narrrita, the west sacred to Varuna, the north-west sacred to Vayu, the north sacred to S’as’f and the north-east sacred to 18. .] 37. The sacred Ganges, springing from the Foot of VisHnv, falls upon mount Meru, and thence separating itself into four streams descends through the heavens down upon the four VIsHKAM- ‘BHAS or buttress hills, and thus falis into the four reservoirs [above described]. 38. [Of the four streams above mentioned], the first called Sitx, went to BHAaDRAS WA-VARSHA, the second, called ALAKANANDA, to BuApata-vaRsHA, the third, called CHAKsuHv, to Kerum4La-varsHA, and the fourth, called ए ^ 084 to Urtara (एषण [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 be seen, if seen, touched or bathed in, if her waters be tasted, if her name be uttered, or brought to mind, and her virtues be celebrated she purifies in many ways thousands of sinful men [from their 81118 | 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 Naraxa [the infernal regions], secure an abode in the happy regions of Heaven. Some peculiarity. PuRA lies from Merv is north. The buttresses of Merv, Manpara, SUGANDHA, &c. are situated in the east, south &c. from एए respectively. B. D Note on verses from 21 to 43 :—BHaskaRa’OHa’RY< superficial area.—B.D.] 126 Translation of the [IIT. 66. Supreme Being, and after their death, as they attain the state from which there is no return, the wise men therefore denomi- nate this state the ATYANTIKA 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 planets and the Loxas which, it is said, are arranged one above the other, are all included in what has been denominated the BrauMAnDaA (universe). The universe. Dimensions of the Bran ©/* Some astronomers have assert- 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 BraumMdnpa. Some Paurdnikas say that this is the length of the circumference of the LoxALoxa PaRvaTa .* * Vide verses 67,68,69, Bua’sKaRa’cHa BRYA does 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 the BraHMa NpA—the centre of light whose boundary is supposed fixed: but if the Sun moves then the Hindoo BrauMma’nNpA must be supposed to be constantly changing its Boundaries. Subbuji Bapd had not failed to use this argument in favour of the Newtonian system in his S’IROMANI PRaxka’s’A, vide pages 55, 66. Bua’skaRa’CHa’RYA however denies that he can father the opinion that this is the length of the circumference limiting the BRanMa’NpA and thus saves him- self from a difficulty. L. W. | [Mr. Wilkinson has thus shown the objection which Subbaje Bapd made to the assumption of the Sun’s motion, but I think that the objection is not a judicious one. Because had the length of the circumference of the BRaHMA’NDA been changed on account of the alteration of the boundary of the Sun’s light with him, or had any sort of motion of the stars been assumed, as would have been granted if the earth is supposed to be fixed, then, the inconvenience would have occurred ; but this is not the case. In fact, as we cannot fix any boundary of the light which issued from the sun, the stated length of the circumference of the BkanMANDA is an imaginary one. For this reason, BHASKARACHARYA does not admit this stated length of the circumference of the BRanMa'NDA. He stated in his GaniTa’DHYAYa’ in the commentary on the verse 68th of this Chapter that ^^ those only, who have a perfect knowledge of the BraHMA’NDA as they have of an a’NvaLa’ fruit held in their palm, can say that this length of the circumference of the BkanMa’Npa is the true one;” that is, as it is not in man’s power to fix any limit of the BkanMma’NpA, the said limit is unreasonable. Therefore no objection can be possibly made to the system that the Sun moves, by assuming such an imaginary limit of the BranmAnpa which is little less impossible than the existence of the heavenly lotus.—B. D.] IV. 3.] Sidhanta-s’iromani. 127 68. Those, however, who have had a most perfect mastery of the clear doctrine of the sphere, have declared that this is the 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 Braumdnpa or not: [of that I have no sure knowledge] but it is my opinion that each planet traverses a distance correspond- ing to this number of Yosanas in the course of a KAtpa or a day of BrauMd and that it has been called the KuaxaxsHA by the ancients. End of third Chapter called the Bauvana-Kos’a or cosmo- graphy. CHAPTER IV. CaLLED MADHYA-GATI-VASANA, On the principles of the Rules for finding the mean places of the Planets. Places of the several 1. The seven [grand] winds have mance: thus been named : viz.— 1st. The Avaha or atmosphere. 2०१. The Pravaha beyond it. 8rd. The Udvaha. 4th. The Samvaha. 5th. The Suvaha. 6th. The Parivaha. 7th. The Pardvaha. 2. The atmosphere extends to the height of 12 Yousanas from the Earth: within this limit are the clouds, lightning, &c. The Pravaha wind which is above the atmosphere moves con- stantly to the westward with uniform motion. 8. As this sphere of the universe includes the fixed stars and planets, it therefore being impelled by the Pravaha wind, is carried round with the stars and planets in a constant revolution. ` 128 | Translation of the [IV. 4. An illustration of the 4. The Planets moving eastward Zao De Ob ete planes in the Heavens with a slow motion, appear as if fixed on account of the rapid motion of the sphere of the Heavens to the west, as insects moving reversely on a whirling potter’s wheel appear to be stationary [by reason of their comparatively slow motion]. Sidereal and terrestrial 5. I?fastar and the Sun rise simulta- Raye ang: ther lerig the. neously [on any day], the star will rise again (on the following morning) in 60 sidereal auatiKAs : 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 the 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. The time thus found added to the 60 sidereal auatixis forms a true terrestrial day or natural day. The length of this day is variable, as it depends on the Sun’s daily motion and on the time [which different signs of the ecliptic take] in rising, [in different latitudes: both of which are variable elements] .* * (Had the Sun moving with uniform motion on the equinoctial, the each minute of which rises in each asv, the number of asus equal to the number of the minutes of the Sun’s daily motion, being added to the 60 sidereal @HaTIKAs, would have invariably made the exact length of the true terrestrial day as Lata and others say. But this is not the case, because the Sun moves with 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 daily motion take in rising and then add this time to 60 sidereal @maTiKa’s. For this reason, the terres- trial day determined by Lata and others is not a true but it is a mean. The difference between the oblique ascension at the beginning of any given day, and that at the end of it or at the beginning of the next day, is the time which the minutes of the Sun’s motion at the day above alluded to take iu rising, but as this cannot be easily determined, the ancient Astronomers having determined the periods which the signs of the ecliptic take in rising at a given place, find the time which any portion of a given sign of the ecliptic takes in rising, by the following proportion. If 30° or 1800’ of a sign: take number of the asus (which any given sign of the ecliptic takes) in rising at a given place : : what time will any portion of the sign above alluded to take in rising ? The calculation which is shown in the 5th verse depends on this proportion.— 8. 7.] IV. 10.] Sidhanta-s’tromani. 129 Revolutions of the Sun fa 7. A sidereal day consists invari- a year are less than the ably of 60 sidereal GHATIKAS: a mean १ sMvana 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 minutes]. Thus the number of terrestrial days in a year is less by one than the number of revolutions made by the fixed stars. 8. The length ofthe (solar) year is 869 days, 15 Guafixis, 30 paras, 224 VIPALAS reckoned in Buti sXvVANA or terrestrial days: The ण of this 18 called a gsaura (solar) month, viz. 30 days, 26 GHATIKAS, 17 pauas, 31 VIPALAS, 524 pRAvipALAS. ‘Thirty sXvana or terrestrial days make a sAvANA month.* Length of luner month 9. The time in which the Moon न [after being in conjunction with the Sun] completing a revolution with the difference between the daily motion and that of the Sun, again overtakes the Sun, (which moves at a slower rate) is called a Lunar month. It is 29 days, 31 GuaqtiKAs, 50 PALAS in length. Whe: Seagon:- ok addikive 10. An ApurmAsa or additive month months called ADHIMA’SA8. = which is lunar, occurs in the duration of 324 saura (solar) months found by dividing the lunar month by the difference between this and the savzA month. From Length of solar year. ॐ (Here 8 solar year consists of 365 days, 15 GHatrkAs, 30 241,48, 22% VIPALAS, i. 6. 365 d. 6 4, 12m. 9 5, and in Sdrya-sippHa’NTA the length of the year is 365 d. 16 ¢. 31 p. 31. 4 ४. 1. ०. 365 d. 6 #. 12 m. 36. 56 s.—B. D.] (t That lunar month which ends, when the Sun is in Mgsma stellar Aries is called CHAITRA and that which terminates when the Sun is in vraisHaBHA stellar Taurus, is called Vats akHa and soon. Thus, the lunar months corresponding to the 12 stellar signs एप्त + (Aries) VaisHaBHA (Taurus) Mirauna (Gemini) Karka (Cancer), Stnna (Leo), Kawnya’ (Virgo), Tuxa’ (Libra), Vais’cHIKkA (Scorpio), Daanu (Sagittarius), Makara (Capricornus), Kumpya (Aquarius) and Mina (Pisces), are CHaITRA, VAIs’a’RKHA, JYESHTHA, A’sHA’DHA S’Ra’vVANA, Bua’pRaPapa, A’s’wina, Ka’etrka, Ma’paas’frsua, PausHa, Ma’aua, and Poatauna. If two lunar months terminate when the Sun is only in one stellar sign, the second of these is called 4 एप्रा ४५/8८ an additive month. Tho 30th part of a lunar month is called Tithi (a lunar day).—B. D.] E 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 shorter in length than a ‘mean sAuRA month, the lunar months are therefore more in number than the SAURAin a KALPA. The difference between the number of lunar and savuRA months in a KALpa is Called by astronomers the number of ApHIMAsas in that period. The reason of subtractive 12. An avama or subtractive day 0 ae which is sXVANA occurs in 64,), TITHIS (lunar days) found by dividing 30 by the difference between the lunar and sfvANA month. From this, the number of avamas in a yuaa may be found by proportion.t 13.t Ifthe Aparm4sas are found from saura days or months, then the result found is in the lunar months, [as for instance in finding the Anaryana. If in the saura days of 8 KALPA: are * [After the commencement of a yuaa, a lunar month terminates at the end of AMava’sya’ (new moon) anda saURA month at the mean VRISHABHA- BANKRA’NTI (i. €. when the mean Sun enters the second stellar sign) which takes place with 54 g. 27 p. 31 ४, 624 p. after the new moon. Afterwards a second lunar month ends at the 2nd new moon after which the MITHUNA-SANKRA’NTI takes place with twice the Ghatis. &c. above mentioned. Thus the following San- KRA/NTIS KARKA &c. take place with thrice four times &c. those Guatis, &c. In this manner, when the SankRa’NTI thus going forward, again takes place at new moon, the number of the passed lunar months exceeds that of the sauRa by one. This one month is called an additive month: and the savRA months which an additive month requires for its happening can be found by the proportion as follows. As 54 ghatis, 27 p. &c. the difference between a lunar and 8 saura : One saura month : : 29, 31, 50 the number terrestrial day &c. in a lunar month : 82, 15, 31, &c. the number of saura months, days, &c.—B. D.] + [At the beginning of a KatPa or 8 yu@a, the terrestrial and lunar days begun simultaneously, but the lunar day being less than the terrestrial day, terminated before the end of the terrestrial day, i. ©. before the next sun-rise. The interval between the end of the lunar day and the next sunrise, is called AvaMa-s’Esua the remainder of the subtractive day. This remainder increases every day, therefore, when it is 60 Guatikds (24 hours), this constitutes a AvaMa day or subtractive day. The lunar days in which a subtractive day occurs, are found by the following proportion. If 0 ०.28 4. 10 2. the difference between the lengths of terrestrial and of a lunar month. : 1 lunar month or 80 tithis x: a whole terrestrial day: 64-,tithis nearly.—B. D.] + The objects of these two verses seems not to be more than to assert that the fourth term of a proportion is of the same denomination as the 2nd.—L. W. IV. 16.] Sidhénta-s’tromant. 131 80 many ApHimAsas :: then in given number of solar days; how many Apaimdsas?] If the Apximdsas are found from lunar days or months, then the result is in 84 एष months, and the remainder is of the like denomination. | 14. [In lke manner] the avamas or subtractive days if found from lunar days, are in sfvana time: if found from sAVANA time they are lunar and the remainder is so likewise. 15. Why, O Astronomer, in find- ing the AHaARGANA do you add saURA months to the lunar months Cuaitra &c. [which may have elapsed from the commencement of the current year]: and tell me also why the [fractional] remainders of ADHIMAsAs and Avama days are rejected: for you know that to give a true result in using the rule of proportion, remainders should be taken into account ? A question. Beeson: OF omitting tothe 16.* As the lunar month ends at clude the Apuimdsa s'zsHA the change of the Moon and the in finding the AHARGANA. , SUARA month terminates when the Sun enters a stellar sign, the accumulating portion of an ADHIMASA always lies after each new Moon and before the Sun enters the sign. * [The meaning of these 4 verses will be well understood by a knowledge of the rule for finding the AHARGANA, we therefore show the rule here. In order to find the AHABGANA (elap3ed terrestrial days from the commence- ment of the Kapa to the required time) astronomers multiply the number of SAURA years expired from the beginning of the Kaupa by 12, and thus they get the number of 8८^ ए84 months till the last 88 ^ SanxRranri (that is, the time when the Sun enters the Ist sign of the Zodiac called Aries.) To these months they add then the passed lunar months CHairra &c., considering them as SAURA. These SAURA months become, up to the time when the Sun enters the sign of the Zodiac corresponding to the required lunar month. They multiply then the num- ber of these months by 30 and add to this product the number of the passed TITHIS (lunar days) of the required month considering them as 84 ए 7.4 days. The number of sauBa days thus found becomes greater than that of those till the end of the required TITHI by the ADHIMASA s’ETHA. To make these SAURA days lunar, they determine the elapsed additive months by the proportion in the following manner As the number of sauRna days in 8 Kapa : the number of additive months in that period : : the number of sauRA days just found ; the number of additive nonthsa elapsed E 2 132 Translation of the [IV. 17. 17. Now the number of TITHIS (lunardays) elapsed since the change of the Moon and supposed as if saura, is added to the number of sauRA days [found in finding the Anarcaya] : but as this number exceeds the proper amount by the quantity of the Apuimasa-s ESHA therefore the ApHIMAS-SESHA 18 omit- ted [to be added]. _ 18. [In the same manner] there is always a portion of a Avama-s‘esHa between the time of sun-rise and the end of the [preceding] TITHI. By omitting to subtract it, the AHARGANA is found at the time of sun-rise: if it were not omitted, the Auarcana would represent the time of the end of the व्यता [which is not required but that of the sun-rise]. R 0 19, 20, 21 and 22. As the true, eason © called the UparAwraza terrestrial day is of variable length, the ब ^ प ^ १७८ प्र has been found in mean terrestrial days: the places of the planets found by this AHARGANA when rectified by the amount of the correction called the UpayAnrara whether additive or subtractive will be found to be at the time of sun-rise at Lanxd.* The ancient If these additive months with their remainder be added to the 8484 days above found, the sum will be the number of lunar days to the end of the sauRA days, but we require it to the end of the required TITHI. Andas the remainder of the additive months lies between the end of the TITHI and that of its corre- sponding 8AURA days, they therefore add the whole number of ApHI-MAsa8 just found to that of the sauRa days omitting the remainder to find the lunar days to the end of the required श्क्षा, Moreover, to make these lunar days terrestrial, they determine AVaMa subtractive days by the proportion such as follows. As the number of lunar days in a Kapa : the number of subtractive days in that period १ ६ the number of lunar days just found : the number of Avama elapsed with their remainder. If these Avamas be subtracted with their remainder from the lunar days, the difference will be the number of the Avama days elapsed to the end of the requir- ed TITHTI; but it is required at the time of sun-rise. And as the remainder of the subtractive days lies between the end of the TITHI and the sun-rise, they therefore subtract the AVamas above found from the number of lunar days omitting their remainder 1. e. AVAMA-8’ESHA. Thus the AHARGANA itself becomes at the sun- rise.—B. D.] * [If the Sun been moving on the equinoctial with an equal motion, the terrestrial day would have been of an invariable length and consequently the Sun would have reached the horizon at LanKA at the end of the AHARGANA which is an enumeration of the days of invariable length that is of the mean terrestrial days. But the Sun moves on the ecliptic whose equal parts do not IV. 28. Sidhdnta-s'iromani. 133 Astronomers have not thus rectified the places of the planets by this correction, as itis of a variable and small amount. The difference between the number of asus of the right ascension of the mean Sun [found at the end of the AHARGANA |' and 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.+ The result [thus found] m 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 Kats or minutes [of the mean longitude of the Sun], otherwise the result is to be added to the places of the 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 called the Dzs‘a’nTaRA, = Which are found being rectified by this UDAYANTARA correction at the time of sun-rise at Lanka may be found, being applied with the Drsanrara 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 Lanxa’ at the end of the Anaraana. Therefore the places of the planets determined by the mean AHARGANA, will not be at the sun-rise at Lanka’, Hence 8 correction 18 necessary to be applied to the places of the planets. This correction called UpayAntTara has been first invented by BHAsHaRACHARYA who consequently abuses them who say that the places of the planets determined by the mean AHARGANA become at the time of the sun-rise at LanxA.—B. D.] * The difference between the mean and true AHARGANAS is that part of the equation of time which is due to the obliquity by the ecliptic.—L. W, ¶ [This calculation is nothing else than the following simple proportion If the number of Asus in a nycthemeron : daily motion of the planet :: the difference between the true and mean AHARGANAS give.—B. D.] 134 Translation of the “LTV. 24. is north and south. This north and south correction is called CHARA, 24, The line which passes from Lanxi, Ussayinf, Kurv- KSHETRA and other places to Meru (or the North Pole of the EKarth) has been denominated the MapHyarEeKkui 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 be- tween the time of sun-rise at the mid-line and that at a given place, is subtractive or additive to the places of the planets, as the given place be east or west of the mid-line [in order 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 Mrru or North Pole of the Earth, at the distance in Yosanas reckon- ed from Merv to given place and produced from co-latitude of the place [as mentioned in the verse 50th, Chapter 11. ] is called rectified circumference of the Earth (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 the radius. End of 4th Chapter called Mapuya-Gati Vasana. * This amount of correction is determined in the following manner. The yosanas between the midline and the given place, in the parallel of latitude at that place, which is denominated SPasHTA-PARIDHI ara called, Ders’a’NTARA YOJANAS of that place. Then by the proportion. As the number of Yosanas in the SPASHTA-PARIDHI: 60 GHATIKAS: : DEsa’N- TARA YOJANAS: the difference between the time of sun-rise at midline and that at a given place. ‘This difference called DEs’a’NTARA GHATIKA’S is the longitude in time east or west from Lanxa’. Again | As 60 GHaTiKa’s: daily motion of the planet: : DESANTA’RA GHATIKA’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 yosanas in the SPASHTA-PARIDHI: daily motion of the planet: : Des’ANTABA yYoJANAS: the same amount of the correction above found.— B. D.] V. 2.] Sidhanta-s'tromant. 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 sines of BHUJA and KOTI,* as a piece of cloth by upright and transverse threads. Before describing the spheric, I shall first explain the canon of sines. On the canon of sines. 2. Take any radius, and suppose it the hypothenuse (of a right-angled triangle), The sine of BHuUJA is the base, and the sine of Koy1 is the square root of the difference of the squares of the radius and the base. The sines of degrees of BHUJA and KoTI subtracted separately from the radius will be the versed sines of एण्या and BHUJA (respectively). [* The BHusa of any given arc is that arc, less than 90°, the sine of which is equal to the sine of that given arc, (the consideration of the positiveness and negation of the sine is here neglected). For this reason, the BHUJA of that arc which terminates in the odd quadrants i. €. the 1st and 3rd is that part of the given arc which falls in the quadrant where it terminates, and the BuoJa of the arc which ends in the even quadrants, i.e. in the 2nd and 4th, is ari arc which is wanted to complete the quadrant where the given aro is ended. The Kor! of any arc is the complement of the BHUJA of that arc. Let the 4 quadrants of a circle A BCD be successively A ए, B ¢, C D and D A, then the BHuUsas of the arcs AP,, AB 729, ACP,, ADP, will be A P,,C 29, C P,, A P, and the complements of these BHUJAS are the arcs B P,, 8 P,, DP,, D P, respectively. — B. D.] B 136 Translation of the [V. 3. 3. The versed sine is like the arrow intersecting the bow and the string, or the arc and the sine.* The square root of half the square of the radius is the sine of an arc of 45°. The co-sine of an arc of 45° is of the same length as the sine of that arc. * These methods are grounded upon the following principles, written by Bua’sKARA’CHARYA, in the commentary VasANa’-BHA’SHYA. (1) Let the are A ए = 90° and ^ © == 4 45० -. AD (=-= $ A 2) =—sin. 45°; and let OAorOB=the radius (R) then A B? == 0 4 ` ~ 0 B? = 20 A? = 2 RB? ~ A 8 = ^^2 पः and AD == } ^ B= ^° 2 or sin, 45° = Je ° 2 (2) It is evident and stated also in the Lita’vatt, that the side of a regular hexagon is equal to the radius of its circumscribing circle (i.e. ch. 60° = R). Hence, 811. 30° = ॐ R. (3) Let A B be the half ofa given arc A P, whose sine P M and versed sine A Mare given. Then | 0 3 AP=A/P ४ + ^ M? and} 4 2 == ^ पि = 81. + B » sin, A B=} / PM? + + M (4) The proof of the last method by Algebra cos = R — versed sine . cos? = R?— 2R.0+ v? subtracting both sides fromR’, € र R?7—cos? = 2 ए . ० —v? or 5171. == 2 ए . ० -- ४ adding v* to both sides sin? — ४*-- 2K. % and वरै (हा. 4 ४२) =4R.v0 extracting the square root, + /sint + ०* = 4८3 8. but by the preceding method 3 हि, 810. ~ v* = the sine of half the given arc ; ०० Sin. ‡ arc = J ‡ ४ , ०.-8. D. | V. 7.4] Sidhdntu-s'iromani. 137 4. Half the radius is the sine of an arc of 30°: The co-sine of an arc of 30° is the sine of an arc of 60°. Half the root of the 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 ired to find the tr : । . fan ee Gan le. of the constellation of the Zodiac coin- planet. cides with the centre of the Earth: * [Whien, 24 sines are to be determined in a quadrant of a circle, the 3 sines, 1. €. 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. e. 66°. Then by means of these three sines, the rest can be found by the method for finding the sine of half an arc, as follows. From the 8th sine, the 4th and the co-sine of the 4th 1. e., the 20th sine, can be determined. Again, from the 4th, the 2nd and 22nd, and from the 2nd, the lst and 23rd, can be found. In like manner, the 10th 14th, 5th, 19th, 7th, 17th, 11th, and 13th, can also be found from the 8th sine. From the 12th again, the 6th, 18th, 3rd, 2181, 9th and 15th can be determined, and the radius is 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.] {+ Bua’skaRa’CHA’RYA maintains that the Earth isin the centre of the Universe, and the Sun, Moon and the five minor planets, Mars, Mercury, &c. 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 called PRATIVRITTA, or excentric circle, and a circle of the same size which is supposed to have the same centre with that of the Harth, is called KAKSHA VRITTA or concentric circle. In the circle, the planet appears to revolve with unequal motion, though it revolves in the excentric with equal motion. The place where tlie planet revolving in the excentric appears in the concentric is its true place and to find this, astronomers apply a correction called MANDA- PHALA (Ist equation of the centre) to the mean place of the planet. A mean planet thus corrected is called MANDA-SPASHTA, the circle in which it revolves MANDA-PRATIVRITTA (1st excentric) and its farthest point from the centre of the concentric, MANDOCHCH (lst higher Apsis). As the mean places of the Sun and Moon when corrected by lst equation become true at the centre of the Earth, this correction alone is sufficient for them. But the five minor planets, Mars, Mercury, &c. when corrected by the 1st equation are not true at the 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 ॥ abONe tect: running due east and west, let the teacher draw a diagram illustrative of the fact for the satis- faction of his pupils. A verse to encourage those 9, But this science 18 of divine who may be disposed to de- < ५ lino fact t : spond in consequence of the origin, reveaung facts no cognizable difficulties of the science. by the senses. Springing from the the concentric circle as second excentric of these five planets, take another circle of the same size and of the same centre with the Earth as concentric, and in order to find the place where the planet revolving in the 2nd excentric appears, in this concentric, they apply a correction called s’{GHRA-PHALA, or 2nd equation of the centre, to the mean place corrected by the lst equation. The MANDA-SPASHTA planet, when corrected by the 2nd equation is called 8’PasHTA, or true planet, the 2ndexcentric, 8’ f{@HRA-PRATIVRITTA, and its farthest point from the centre of the Earth, s’Ig¢Hrocuca the 2nd higher 4 0818. If a man wishes to draw a diagram of the arrangement of the planets accord- ing to what we have briefly stated here, he should first describe the excentric circle, and through this excentric the concentric, and then he may determine the place of the ManDa-sPasHTa planet in the concentric thus described. Again, having assumed the concentric as 2nd excentric and described the concentric 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 excentric, find the corrected mean place of the planet in the concentric. After this, having described the 2nd excentric through the same concentric, they find the true place in the concentric, through the 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 excentric circle, and in the latter, the excentric is drawn through the concentric, but this can easily be understood that both of these modes are equivalent and produce the same result. In order to find the lst and 2nd equations through a different theory, astro- nomers assume that the centre of a small circle called NfCHOCHCHA-VRITTA or epicycle, revolves in the concentric circle with the mean motion of the planet and the planet revolves in the epicycle with a reverse motion equal to the mean motion. BHa’sKaRa’cHa’RYA, himself will show in the sequel that the motion of the planet is the same in both these theories of excentrics and epicycles. It is to be observed here that, in the case of the planets Mars, Jupiter and Saturn, the motion in the excentric is in fact their proper revolution, in their orbits, and the revolution of their s’‘I@HROCHCHA, or quick apogee, corresponds to a revolution of the Sun. But in the case of the planets Mercury and Venus, the revolution in the excentric is performed in the same time with the Sun, and the revolutions of their sI@HROCHCHAS are in fact their proper revolutions in their orbits,—B. D.] V. 12.) Sidhdnta-s‘iromant. 189 supreme Brauma himself it was brought down to the एधा by VasisHTHaA 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 1s 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. 0 ee ae aD 10. In the first place then, de- gram to illustrate the ex- scribe a circle with the compass opened centric theory. : to the length of the radius (3438). This is called the KAKsHAvVRITTA, or concentric circle; at the centre of the circle draw a small sphere of the Harth 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) diameter passing through the centre of the Earth and the higher apsis (which is called ucHCHA-REKHA, the line of the apsides) and 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 Harth’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 parallax PARAMA-LAMBANA Of all the planets amounts to a quantity equal to चष of their daily motion.—L. W. 7 2 140 Translation of the [V. 138. 13 and 14.* Where the ucHcHA-REKHA perpendicular dia- meter (when produced) cuts the excentric circle, that is the * Fig. 1. “4 । / \ | Cc [ In fig. 1st let 9 be the centre of the concentric circle A B © D, ¥ the place of the stellar Aries, A that of the higher apsis, and M that of the mean planet in it: then E A will be the UcHCHA-REKHA (the line of the apsides). Again let E O be the excentricity and H F LG the excentric which has O for its centre; then H, ¥ P, will be the places of the higher apsis, the stellar Aries and the planet respectively in it. Hence H P will be the KENDRA; P K the sine of the KENDRA; P I the co-sine of the KENDRA. The KENDRA which is more than 9 signs and less than 3 is called MRIGADI (i. e. that which terminates in the six signs beginning with Capricornus) and that which is above 3 and less than 9 is called KaRKyaDI (i. e. that which ends in the six signs beginning with Cancer). Thus (Fig. 1) that which terminates in@ H Fis MRiGaDI KENDRA, and that which ends in F L § is Karxya’p1.—B. D.] V. 15.] Sidhdnta-s'iromant. 141 place of the higher 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 18) 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-REKHA is the sine of sHUJA of the KENDRA. 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 principle on which 15. As the distance between the the rule for finding the . t t t ~ 9 amount of equation of centre 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 18 MRIGADI and KARKYADI (respectively).t * The word KENnpRa or centre is evidently derived from the Greek word xevrpoy and means the true centre of the planet.—L. W. + (In (Fig. 1) P K is the spHuta Kori and P E the Karna (the hypothenuse) which cuts the concentric at T. Hence the point T will be the apparent place of the planet and T M the equation of the centre. This equation can be determined as follows. Draw M » perpendicular to E व, it will be the sine of the equation and the triangle P M » will be similar to the triangle P E K. „ PE:EK=PM:Mn; PM.EK —— = sine of the equation $ PE EO.EK — — eae = PE hence M a ==: , for PM =IK=EO Now, let ‰ = KENDRA, a = the distance between the centres of the two circles excentric and concentric, x = sine of the equation, and A = hypothenuse : then the SPHUTA KOTI <== cos. ¢ + a, according as the KENDRA is MRIGADI or KARKYADI, and $ = sin.’ & & (cos. ‰ ‡ a)? hence by substitution @ , 811. ¢ @ , 811. ¢ h ` +न k + (cos. ¢ & 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 sPHuUTA KOTI (i. 6. 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 एप्ए7+ [of the KENDRA] is the BHUJA (the base) and the square-root of the sum of the squares of the sPHUTA KoTI 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 be observed in that point of the 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, the equation is additive.* | The reason for assuming 18. The mean planet moves in its the MAND-SPASHTA planet @8 sranDA-PRATIVRITTA (first excentric) ; amean in finding the 2nd equation. the MANDA-SPASHTA planet (i. e. 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. ‰ 18 equal to a in the KARKYADI KENDRA, then / will be equal to sin. &, otherwise & will always be greater than sin. ¢ and consequently x will be less than a. Hence, when 4 is equal to sin &, x will then be greatest and equal to a, i.e. the equation of the centre will be greatest 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 excentric ia marked at the distance equal to the excentricity from the centre of the concentric (as stated in the V 12th.)—B. D.] * (Thus, the mean planet, corrected by the Ist equation, becomes MANDA-SPASHTA and this process is called the MaNnDA process. After this, the MANDA-SPASHTA when rectified by the sI‘GHRA PHALA, or 2nd equation, is the sPasHTA planet, and this 2nd process is termed the s’IGHRA process. Both of these processes, MANDA-SPASHTA and SPASHTA are reckoned in the VIMAN- DALA or the orbit of the planet as hinted at by BHaskKaRacHaRYA in the commentary called VasaNA-BHASHYA in the sequel. These places are assumed for the ecliptic also without applying any correction to them, because the correction required is very small.—B. 12. ] ४. 23.) Sidhdnta-s'tromani. 143 is therefore here assumed as the mean planet in the second process (i. e. in finding the second equation).* ग. invene 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 been inserted by former Astronomers. 20. That point of the excentric which is most distant from the Earth has been denominated the higher apsis (or ucHcHA) : that point 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 a 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 The cause of variation of 90818 at its least distance, therefore its apparent size of planet's disc. disc appears small and large accord- ingly. In like manner, its disc appears small and large 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 Prativritta Buaner of the diagram of the excentric. I shall now proceed to explain the same proof in a different manner by the diagram of a NfcHOCcHCHA-VRITTA (epicycle). * (For this reason, having assumed the MANDA-SPASHTA planet for the mean, Which MANDA-8PASHTA can be determined in the concentric by describing the excentric circle &c. through the mean planet and ManpocHcHA, make the place of the stellar Aries from the MaNDa-SPASHTA place in the reverse order of the signs and then determine the place of the sIGHrocuHcHa in the order of the signs. Through the places of the stellar Aries and s‘I@HRocuCHA describe the 2nd excentric circle &c. in the way mentioned before, and then find the place of the true planet in the concentric.—B. J). ] 141 Translation of the [V. 24. ( । 24. Taking the mean place of the onstruction of Diagram : . to illustrate the theory of planet in the concentric as the centre, epicycle. with a radius equal to the excentricity of the planet, draw a circle. This is called nficHocHcHa एण or epicycle. Then draw a line from the centre of the Earth passing through the mean place of the planet [to the circumference of the epicycle]. 25. That place in the epicycle most distant from the centre of the arth, cut by the line [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 EKarth’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-REKHA]. 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 360 degrees towards the 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 (perpendicular) line from the planet to the UCHCHA-REKHA 18 the sine of the BHUJA of the KENDRA: and from the planet on the transverse line is the cosine [of the KENDRA].* (See note neat page.) To find the hypothenuse 28 and 29. The BHUJA PHALA and and the equation of centre. = roy pHaLA of the KENDRA which are found [in the GanitApHyAya] are sine and cosine in the epicycle. As the KoTI 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 प्र (the base) and the xarya hypothenuse (to complete V. 27.) Sidhdnta-s‘iromani. 145 the right-angled triangle) is the line intercepted between the centre of the Earth and the planet. The equation of the centre is here the arc [of the concentric] intercepted between # Note on verses from 24 to 27. © [In fig. 2, let ABC D be the concentric, % the place of the stellar Aries, E the centre of the Earth, M the mean place of the planet in the concentric, k f ८ g, the Epicycle, 4 the place of the higher apsis in it, E & the UCHCHA-REKHA ithe place of the lower apsis, P that of the planet, 4 P the KENDRA, P & the sine of the KENDRA and P $ the cosine of it The sine and co-sine of the. KENDRA in the excentric, reduced to their dimensions in the epicycle in parts of the radius of the concentric, are named BHUJA-PHALA and ROTI-PHALA respectively in the GayitApHyAya. That is As the radius or 360° of the concentric : the sine and cosine of the KENDRA in the excentric : : excentricity or the periphery of the epicycle : BHUJA-PHALA and KOTI-PHALA respectively Therefore the BHUJA-PHALA and KOTI-PHALA must be equal tothe sine and cosine of the KENDRA in the epicycle.—B. D. ] G 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 18 higher apsis, in the excentric diagrams of the excentric 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 st/eHrocucHa, 2nd higher apsis, it moves in antecedentia or reversely, as it 18 thrown backwards. 31. When the epicycle however is used, the reverse of this takes place, the planet ‘moving in antecedentia from its Ist 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 astronomers to ascertain the amount of equation. * In (Fig 2) E & 18 the spHuta-Kot1, P E the hypothenuse, T the apparent place of the planet in the concentric and 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 >, the KENDRA in the excentric by &, the excentricity by a, and the hypothenuse by 4: then ¢ R: 811 == ५ : Pk the एष्एर+ एप. asin ¢ ० the BHUJA-PHALA = 3 Now, the triangles E T % and E P & are similar to each other „ EP: PK=ET: Ta or h :Pk=R :2 PkEXR = f= ————— |} 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 2 ¢ = 3 R ०० by substitution asink @ asink Pig ge, "gee h found ity by the theory of the excentric in the note on the verses 15, 16 and 17.—B. D. $ the sine of the equation as- V. 37.) Sidhanta-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 the 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 lst epicycle is in the concentric, Explains why the 5 minor let the planet therefore move in the planets requiré both the ei th it ra Ist and 2nd equations to 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’GHRA NicHocHCcHA, veitta or of the 2nd epicycle: In the second or s‘IcHRA epicycle is found the true place of the planet. 30. The first process, or process of finding the Ist equa- tion, is used in the first place, in order to ascertain the position of the centre of the siaHra 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 these 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 1 the 1st process, because the difference sion of hypothenuse inthe (in the two modes of computation) MANDA process. te i _ 18 inconsiderable, but others maintain that since in this process the periphery of the first epicycle . beng multiplied by the hypothenuse and divided by the radius becomes true, and that, if the hypothenuse then be used, the result is the same as it was before, therefore the hypothenuse is ५ 148 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.* 38. 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 NataKarma is declared (by astronomers). The proof of this is the same as in finding the parallax.t Reason of NavTaKaRMA, * [The BHUJA-PHALA, determined by means of the sine of the first KENDRA of the planet (i.e. by multiplying it by the periphery of the Ist epicycle and dividing it by 3600) has been taken for the sine of the 18४ equatiqn of the centre : and what we have shown in the note on the V. 28 and 29, that the BHUJa- PHALA, when multiplied by the radius and divided by the hypothenuse, becomes the sine of the equation may be understood only for finding the 2nd equation of the five minor planets and not for determining the 1st equation. Some say that the omission of the hypothenuse in the lst process has no other ground but the very inconsiderable difference of the result. But BRAHMA- GUPTA maintains that the periphery of the lst epicycle, varies: according to the hypothenuse ; that is, their ratio ia always the same, and the periphery of the Ist epicycle, mentioned in the GanITADHYAYA, 18 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 RK: lst periphery = the hypothenuse: the true periphery PXh ० the true periphery = + and consequently the BHUJA-PHALA in Px sink the true epicycle = —; 360° PXh sink R ० the sine of the 1st equation — ——— % —— y — and abridging == R 860० fh P .sin & which is equal to the BHUJa-PHALA, Hence the hypothenuse is not 360° used in the 1st process. BraHMaGuPta’s opinion is much approved of by BHa’skKaRa’CHA’RYA.—B. D.] + But this is not the case, because the NATAKARM which 2 घ ^ (87424708 ^ RYA has stated in the GaniTADHYAYa 1188 no connection with the fact stated in this SLOKa and therefore many say that this 81106 does not belong to the text.—B. D.] Er ge = ea, | ॥ wh eee ' ast, re ८.4५ ^ ~ ५ ~ = » PN. So nat 2 ६ lg 0 अ र area oe ५ ५ -^ 4-~ 4 ~. "५८ ५५८५१ Y 7 ASTCT. LENOX ८ ~) (त +©“. = a ee ig om ee, V. 40.) Sidhdnta-s’tromani. 149 39. The mean motion of a planet is also its true motion Bea laine where the-anaad when the planet reaches that point in and true motions of all the the excentric cut by the transverse =o 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. [Lata 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 &९. of Planets. J Jained before, and having changed the marks of the places of the planet and its s‘iaHrocucHa 2nd higher apsis with their daily motions, an astronomer may quickly show the retrogressions, &c.+ * The ancient astonomers Latxa, S’rrpati &. say that the true motion of a planet equals to its mean motion when it reaches the point of intersection of the concentric and excentric. But BHa’skKaRA’CHARYA denying this, says, that when the planet reaches the point when the transverse axis of the concentric cuts the excentric and when the amount of equation is a maximum, the true motion of a planet becomes equal to its mean motion. For, suppose, 79 22) Ps, &e., are the mean places of a planet found on successive days at sun-rise when the planet proceeded from its higher or lower apsis and ९1, ég, ¢5, &c. are the amounts 1 then p तै ¢,, Do, त es, Ps + ९३) &c. will be the true places of the planct, ~ Po—p, + (¢,—¢,), Ps—Pa + (८१--८३), PsP ॐ (¢,—e,), &०. will be the true motions of the planet on successive days. Now, as the difference between the true and mean motions is called the GaTIPHALA, by cancelling therefore, Pa—P; 2353-7 9) &e. the parts of the true motions which are equal to the mean motion, the remaining parts e,—e,, ¢,—e, &c. will evidently be the GATIPHALAS that is the differences between two successive amounts of equation are the GATIPHALAS. Thus, it is plain that the GaTrPHaLa entirely depends upon the amount of equation, but as the amount of equation increases, so the GATIPHALA is decreased and therefore when it is a maximum, the GaTIPHALA will indifintely be decreased i. 6. will be equal to nothing. Now as the amount of equation becomes a maximum in that place where the transverse diameter of the con- centric circle cuts the excentric, (see 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 each other. Having thus shown a proof of his own assertion, BHASKARA'CHA’RYA says that what the ancient astronomers stated, that the true and mean motions of 8 planet are eqnal to each other when the planet comes in the intersection point of the concentric and 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 the 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. Circle: itis 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 SpHura-KaksHa or cor- being multiplied by the s‘iagHRA-KARNA १० (or 2nd hypothenuse), and divided by the radius (3438) is spHuTa-KaxsH< (corrected orbit). The planet is (that moment) being carried [round the earth] by the ९६५१५६५ wind, and moves at a distance equal to half the diameter of the spHuTA-KaxKsui from the earth’s centre. 43. When the sun’s MANDA-PHALA i. e. the equation of the Reason of Busa'ntaza centre is subtractive, the apparent or veer real time of sun-rise takes place before the time of mean sun-rise: when the equation of the centre is 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 explained by diagrams, a matter of no difficulty whatever : but men of weak and blunt understanding find this subject as heavy and immovable as the high mountain} that has been shorn of its wings by the thunderbolt of Indra. End of Chapter V. on the principles on which the rules for finding the true places of the planets are grounded. It is to be observed here that when the planet comes to the places a, a &c. in the dotted line, it is then at its higher apsis, when it comes to the places c, c and ९, it is at its lower, and when it comes to ©, © &c. it appears, stationary : and when it is moving in the upper arc da ©, its motion being direct appears quicker, and when in the lower arc ¢ ¢ 6, its retrograde motion is seen.— B. 12. | # [These asus are equivalent to that part of the equation of time, which is due to the unequal motion of the sun on the ecliptic.—B. D.] + Mountains are said by Hindu theologians to have originally had wings. VI. 3.] Sidhdanta-s’iromant. 151 CHAPTER VI. Called GoLaBANDHA, on the construction of an Armillary Sphere. 1. Leta mathematician, who is as skilful im 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 degrees in the circle. 2. In the first place, let him mark a straight and cylindn- cal DHRUVA-YASHTI, or polar axis, of any excellent wood he pleases: then let him place loosely in the middle of it a small sphere to represent the earth [80 that the 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 DRIGGOLA unconnected with each other, and fastened to the hollow cylinders [in which the axis is to be inserted] .* [Description in detail of the fact above alluded to.] 8. Fix vertically the four circles and another circle called horizon transversely in the middle of The prime vertical, th meridian and the konavkit- them, so that one of those vertical ae circles called SAMAMANDALA, prime vertical, may pass through the east and west points of the horizon, the other called yAMyorraRa-veftrTa, meridian the * The sphere of the fixed stars which is mentioned here is called the BHAGOLA starry sphere. This BHAGOLA is assumed for all the planets, instead of fixing 8. separate sphere for each planet. This sphere consists of the circles ecliptic, equinoctial, diurnal circles, &c. which are moveable. For this reason, this sphere is to be firmly fixed to the polar axis, so that it may move freely by moving the axis. Beyond this sphere, the kHaGoLa celestial sphere which consists of the prime vertical, meridian, horizon, &c. 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 KHaGOLA and BHAGOLA are mixed together. For this reason the latter is called DRia@Gota the double sphere. And as the spherical fingers are well seen by mixing together the two spheres KHAGOLA 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- vrittas the N. ए. and 8. W. and N. W. and 8. E. points. 4. Then fix a circle passing through the points of the The UNMANDaLa or six horizon intersected by the prime verti- o'clock line. 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 ‘ncrease and decrease in the length of the days and nights.* 5. The equinoctial (called nipf-vaLaya), 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 a distance south from the zenith equal to the latitude, and at a distance north of the nadir also equal to the latitude of the place [for which the sphere 18 constructed]. 6. Let the azimuth or vertical circle be next attached 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 them]. It should be capable of being placed so as to cover the planet, wherever it may happen to be. 7. Only one azimuth circle may be used for all the planets ; 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 DRIK- SHEPA-VRITTA. The equinoctial. Azimuth or vertical circle. * The circle of declination or the hour circle passing through the east and west points of the horizon is called 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 six 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, the UNMANDALA. The prime vertical or the SAMA-MANDALA of the Sanskrit cannot, with propriety, be called the six o’clock line; because it is only twice a Ir that it is six o’clock when the sun is at this circle, the prime vertical.— VI. 11.] Sidhdnta-s'iromant. 153 8. Let two hollow cylinders project beyond the two poles north and south of the KHAGOLA ce- lestial sphere, and on these cylinders let the skilful astronomer place the 66014 double sphere as follows. 9. When the system of the KHaGoLA, celestial sphere, 18 mixed with the ecliptic, and all the other circles forming the BHAGOLA (which will be presently shown) it is then called DRIGGOLA, double sphere. As in this the figures formed by the circles of the two spheres KHAGOLA and BHAGOLA are seen, it 18 therefore called pRiacoLa double sphere.* THE BHAGOLA. 10. Let two circles be firmly fixed on the axis of the poles answering to the meridian and horizon (of the KHAGOLA) ; they are called the ApHARA-vRITTAs, or circles of support: Let the equinoctial circle also be fixed on them marked with 60 ghatis like the prime vertical (of the KHAGOLA). 11. Make the ecliptic (of the same size) and mark it with 12 signs; in this the Sun moves: and also in it revolves the Harth’s shadow at a distance of 6 signs from the Sun. The KRAnNTI-PATA or vernal equinox, moves in it contrary to the order of the signs: The spasHta-pPdtas [of the other planets] have a like motion: the places of these should be marked in 1४. The Driaaora. The Ecliptic. * See the note on 2 Verse. + [The Sun revolves in the ecliptic, but the planets, Moon, Mars, &c. do not revolve in that circle, and the planes of their orbits are inclined to that of the ecliptic. Of the two points where the planetary orbit cuts the plane of the ecliptic, that in which the planet in its revolution rises to the north of the ecliptic is called its Pa’TA or ascending node (it is usually called the mean 2.4.74.) and that which is at the distance of six signs from the former is called its SASHADBHA 2.4.74 or descending node. The Pa’ta of the Moon lies in its concentric, because the plane of its orbit passes through the centre of the concentric, i. e. through the centre of the Earth; but the pa’Tas of the other planets are in their second excentric, because the planes of their orbits pass through the centres of their 2nd excentrics, which centres lie in the plane of the ecliptic. When the planet is at any other .place than its nodes, the distance between it and the plane of the ecliptic is called its north or south latitude as the planet is north or south of the ecliptic. When the planet is at the distance of 3 signs forward or backward from its 24/74) it is then at the greatest distance north or south from the ecliptic : This distance is its greatest latitude. Thus, H 154 Trauslation of the [VI. 12. 12. Let the ecliptic be fixed on the equinoctial in the point of vernal equinox KRAnqfI-PATA and in a point (autumnal equinox) 6 signs from that: it should be so placed that the 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 equinoctial. 18. Divide a circle called kKsHEPA-VRITTA representing the | orbit of a planet into 12 signs and mark in it the places of the spasHta- 24748, rectified nodes, 98 ` has been before prescribed [for the ecliptic]. Then this circle should be so placed in connection with the ecliptic as it has been placed in connection with the equinoctial. 14. The ecliptic and the KsHEPA-vRITTA should be so placed that the latter may intersect the former at the [rectified] ascending and descending nodes, and pass through points distant 3 signs from the ascending node east and west at a 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 planets being multiplied by the radius and divided by the siGHRA-KARNA Pianet’s orbit. the latitude of the planet begins from its pa’ta and becomes extreme at the distance of 3 signe from it, therefore, in order to find the latitude, it is necessary to know the distance between the planet and its 24/74. This distance is equal to the sum of the places of the planet and its pa’ra, because all Pa’Tas move in antecedentia from the stellar aries. This sum is called the VIK8HEPA-KENDRA or the argument of latitude of the planet. As the pa’ta of the Moon lies in her concentric, and in this circle is her true place, the sum of these two is her VIKSHEPA-KENDRA, but the 2444 of any other planet, Mars, &c. lies in its 2ud excentric and its MANDA-SPasHTA place (which 18 equivalent to its heliocentric place) is in that circle, therefore its VIK8HEPA-KENDRA is found by adding the place of its Pa’TA to its MANDA-8PASHTA place. The spasHta-Pa'Ta of the planet is that which being added to the true place of the planet, equals its VIKSHEPA-KENDRA for this reason, it is found by reversely applying the 2nd equation to its mean Pa’Ta. As ¦ ० SPASHTA 2474 न true place of the planet, == VIKSHEPA-KENDRA, == place of the MANDA SPA8HTA planet न= mean Pa’TA, = 2, of the m. s. p. ¬ 2nd equation न m. २. त 2nd equation, == true place of the planet +- mean 24/74 + 2nd equation, ० SPASHTA PATA = mean Pa’Ta यूः 2nd equation. The place of this sPasHTa Pa’TA is to be reversely marked in the ecliptic from the stellar aries.— B. D | VI. 15.] Sidhanta-s'tromani. 155 second hypothenuse becomes spasHfa, rectified. The KswEpa- VRITTA, or circles representing the orbits of the six planets, should be made separately. The Moon and the rest revolve in their own orbits.* * [As the pa’ta of the Moon and her true place lie in her concentric, the sun of these two, which is called her VIKSHEPA-KENDRA or the argument of latitude, must be measured in the same circle, and her latitude, there 0 found through her VIKSHEPA-KENDRA, will be as seen from the centre of her concentric i. e. from the centre of the Earth. But the pa’va of any other planet and its MANDA- SPASHTA place (which is its heliocentric place) lie in its 2nd excentric, theretore its latitude, determined by means of its VIKSHEPA-KENDRA, which is equal to the sum of its MANDA-SPasHTA place and 24/74 and measured in the same circle, will be such as seen 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 the ecliptic, N O that of the 2nd excentric, N ८ the node and P the planet. ee, Suppose O E and P p (parts of N Pp great circles) to be drawn from O and P perpendicularly to the plane of the ecliptic: then 0 © will be the 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 centre of the Earth, will see the planet distant from the ecliptic. This latitude, therefore, is called a mean latitude which can be found as follows, 810 पि 0 : 800: : snNP: sin P ¢, or R.sin Pp = sin OE. sinN 2, consequently, in order to determine P g, it is necessary to know previously O E, the greatest latitude and N P, the distance of the place of the planet from the node, which distance is evidently equal to the VIKSHEPA-KENDRA that is, to the sum of the MANDA-sPasHta 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 be the centre of the earth, O that of the 2nd excentric, P the MANDA sPAsHTA place of the planet init: then E P will be the 2nd hypothenuse which is supposed to cut the concentric at A: then A will be the true place of the planet in the concentric. Again let P 4 be a circle with the centre O, whose plane is perpendicular to the eclip- tic plane and A © another circle with the centre E whose place is also perpendicular to the same piane, then Pg will be the mean latitude of the planet and A © will be the true. Let P pandAa lines be perpendicularly drawn to the plane of the ecliptic, these lines will also be at right angles to the line Ep: then P p will be the sine of the mean latitude P g and A a that of the true lati- tude Ad. Now by the similar triangles E P p and E A a, EP:Pp::EA:Aa; EA.Pp ae Be ee os EP ` 156 Translation of the [VI. 16. 16. The declination is an arc of a great meridian circle: cutting the equinoctial at right angles, and continued till it touch the ecliptic. R X sine of the mean latitude or the sine of the true latitude = —————_——- —_____—_____-———- h sinOE. sin N P R R sinOE. श) पि the sine of the true latitude == —— x h R sinOE . sinN P Declination and latitude. but, the sine of the mean latitude = .. by substitution h As the latitude of the planet is of a smaller amount, the arc of a latitude it, therefore taken in the SIDDHANTAS instead of the sine of the latitude, OE.sinNP Hence, the true latitude = A $ that is, the sine of the argument of latitude multiplied by the greatest ee and divided by the 2nd hypothenuse is equal to the true latitude of the 811९6. ह Now in the एष 46016, 8 circle should be so fixed to the ecliptic, that the former may intersect the latter at the spasHTa-PATA and the point six signs from it, and whose extreme north and south distance from the ecliptic may be such that the distance between the circle 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 spasHTa- श Pata, N P the VIKSHEPA-KEN- DRA, P p the true latitude, E E 0 the true extreme latitude: च ee क then sin No: sin E O:: sin NP: sin P p sinNO.sinPp “sin EO = —— 5 sin N P R.Pp sin NP or EO= L.sin NP but if L be taken for the mean extreme latitude the P p = L.sn NP 2.1 ae [क ००७ E O == K sin N P h h This is the mean extreme latitude stated in the GanitapHYAyYA multiplied “by the radius and divided by the 2nd hypothenuse equals the true or rectified extreme latitude.— ए, D.] VI. 21.] Sidhanta-s'tromani. 157 celestial latitude is in like manner an arc of a great circle (which passes through the ecliptic poles) intercepted between the ecliptic and the KSHEPA-VRITTA. The corrected declination [of any of the small planets and Moon] is the distance of the planet from the equinoctial in a circle of declination. 17. The point of intersection of the equinoctial and ecliptic circles is the KRANTI-PATa or inter- secting point for declination. The retrograde* revolutions of that point in a Kapa amount to 30,000 according to the author of the Strya-sipDHANTA. 18. The motion of the solstitial pomts spoken of by Mun- JXLA and others is the same with this motion of the equinox: according to these authors its revolutions are 199,669 in a Kapa. 19. The place of the KRAntTI-pATa, or the amount of the precession of the equinox determined through the revolutions of the krA4nTI-PATA must be added to the place of a planet ; and the declination then ascertained. The ascensional differ- ence 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 the KsHEpa-PATAs are also contrary to the order of the signs, hence to find their latitudes, the places of the KSHEPA-PATAS 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 the s’IGHRA-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. Precession of the equinox. * The motion of the KRANTI-PATA is in a contrary direction to that of the order of the signs.—L. W. 158 Translation of the - (VI. 22. 22. Or the amount of the latitude may be found from the SPASHTA planet added to the node which the 8 iGHRA-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 circle, 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’faHra- KENDRAS to the revolutions of their nodes which have been stated [in the GayrrApuy(yal: 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 s'‘{aHRA-KENDRaS [to get the exact places of the nodes].+ 24. To find the xenpra [of any of the planets] the place of the planet is subtracted from the s’faHrocucHa: then take ॐ [366 the nodes on V. 11, and V. 13, 14, 15. —B. D.] ¶ [In all the original astronomical works, the sum of the 24/74 and s’f{@HROCH- ¢+ of Mercury and Venus, is assumed for their VIKSHEPA-KENDRA, and through this, their latitude is determined. But the latitude thus found would be at the place of their s‘f@HROcHCHA and not at their own place, because their places are different from those of their s’faHrocHcHas. To remove this difficulty, Bua’skra’cHa’Rya writes, ‘The exact revolutions &.” But the difficulty arises in the supposition that, the earth is stationary in the centre of the universe and all the planets revolve round her, because we are then bound to grant that the mean places of Mercury and Venus are equal to that of the Sun, and hence their places will be different from those of their s’f¢HROCHCHaS. 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’fa@HrocucHas of Mercury and Venus are their own heliocentric places, and consequently the sum of the places of their 8’f@HROCHCHAS and 24748 will be equal to the sum of their own places and those of their Pa’Tas, that is to their VIKSHEPAKENDEA. For this reason, their latitude found through this, will be at their own places. Now, it is a curious fact that, the revolutions of the patas of Mercury and Venus, stated in the original works, are such as ought to be mentioned when it is supposed that the Sun is in the middle of the universe and the planets revolve round him, and not when the Earth is supposed to be stationary in the centre of the universe. From this fact, we can infer that the original Authors of the As- tronomical works knew that all the planets together with the Earth revolve round the Sun, and consequently they stated the smaller amounts of the revolutions of the Pa’tas of the Mercury and Venus. When this is the case, why is it supposed that all the planets revolve round the Earth, because the ५ can more easily be understood by this supposition than by the other.— VI. 27.) Sidhdanta-s'tromani. 159 the KENDRA with the pAta added [to get the exact amount of the 2474 or node] and let the place of the planet be added thereto, [we thus get the VIKSHEPA-KENDRA or the argument of the latitude of Mercury or Venus]. Therefore from the s’faHrocHcHas of these two planets with the 24178 added, their latitudes are directed by the ancient astronomers to be found.* | 25 and 26. The patas or nodes of these two planets added to the s/igHRocucHRAS from which the true places of the planets have been subtracted, become spasuya 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 the nodes of the (superior) planets, and also mark in it the mean place of the nodes of Mercury and Venus added to their respective 8 f4HRA-KENDRAS.+ 27. Next the AHORATRA-VRITTAS or diurnal circles, must be Diurnal circles calleqd made on both sides of the equinoctial AHOB ATEA-VRITTAS. [and parallel to it] at every or any degree of declination that may be required:—and they must all be marked with 60 GHatis: The radius of the diurnal circle [on which the Sun may move on any day] is called DYuJYA. * (Let, ¢ = s’fanrocucua or the place of 2d higher apsis ¢ = the s’i@HRa-KENDRA p = the place of the planet 2 = Pa'Ta or the place of the ascending node and NV, = the exact 2474 then ¢ = h—p; andh=—=k + n=h.—p+n ^.“ VIKSHEPA KENDRA or argument of latitude of Mercury or Venus + ,2 == ¢ -- 2 ¬+ 2. ~+ p=h + n—B. D [See the note on verses 13, 14 and 15 :- 8. 2). ] 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, three diurnal circles should be attached in the order of the signs: these again will answer for the three following signs. The BHacoLa has thus been described. This is to be known also as the KHECHARA-GOLA, the sphere of a planet. 30. Or in the plane of the ecliptic bind the orbits of Saturn and of the other planets with cross diameters 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. 81. Having thus secured the BHAGOLA on the axis or YASHTI, 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 reference to the KHAGOLA 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. Called Trrpras’Na-vdsank on the Principles of the Rules for resolving the questions on time, space, and directions. The aacensional difference 1. The time called cHARA-KHUNDA and its place. or ascensional difference is found by that arc of a diurnal circle intercepted between the horizon aud the six o’clock line. The sine of that arc is called the KuJYA in the diurnal circle: but, when reduced to relative VII. 5.) Siddhdnta-s'iromani. 161 value in a great circle, it is called cHaRAJyA or sine of as- censional difference.* 2. The 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 place and the equatorial region under the same meridian] is the ascensional difference. 3. When the sun is in the nor- Determination of the : 4 ॥ is t question when the cuara thern hemisphere, it mses at any correction is additive and rt t 1 Oe 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 isin the southern hemisphere the reverse of this takes place, as the part of the UNMANDALA in that hemisphere lies below the horizon. The halves of the sphere north and south of the equinoctial are called the northern and southern hemispheres. _ Cause of increase and decrease 9. [And it is m consequence of in length of days and nights. = this ascensional difference that] the days are longer and the nights shorter (than they are on the # [The times found by the arcs intercepted between the horizon and the six o’clock line, of the three diurnal circles attached at the end of the first 3 signsi, €, Aries, Taurus and Gemini are called the CHARA-KA‘LAS or the ascensional differences of these signs, and the differences of these CHARA-Ka'LAS are called the CHABRA-KHANDAS of those three signs. As, where the PALABHA is 5 digits or the latitude 18 nearly 224° north, the as- censional differences of the 3 first signs are 297, 541 and 642 asus, and the dif- ferences of those 1. €. 297, 244 and 101 are the CHARA-KHANDAS of those signs. Theee are again the OHARA-KHANDAS of the following three signs inversely 1. o. 101, 244 and 297 asus. Thus the CHaka-KHaNpas of the first six sigus answer for the following six sigus.—B. D.]| I 162 Translution of the [ VII. 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 the horizon. 6. But atthe equator the days and nights are always of the same length, as there is no UNMANDALA there except the horizon [on the distance between which, the variation in the length of days and nights depends]. A circumstance of: peculiar curiosity, however, occurs im those places having a latitude greater than 66° N. viz. than the complement of the Sun’s greatest declination Determination of place and 7. Whenever the northern declina— nai of perpetual day and प्ता of the Sun exceeds the comple- ment of the latitude, then there will be perpetual day for such time as that excess continued ; and when 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 MERU, 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 Darryas [at the south pole]. For, the northern and southern poles are situated respectively in their zeniths. 9. The Celestial Beings on MERU behold the Sun whilst he is in the northern hemisphere, always revolving above the horizon from left to right: but Dairyas 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. सका 10. Thus it is day whilst the Sun Definition of the artificial . .., . hi A ts day and night and the day 18 visible, and night whilst he is in- + visible. As the determination of VII. 15.] Siddhdnta-s'iromani. 163 night and day is made in regard to men residing on the sur- face of the Earth, so also is that of the prrris or deceased ancestors who dwell on the upper part of the Moon. 11. Asfor the doctrine of astro- व ह nee a logers, that it was day with the Gods professors or 8a'NHITIKAS. at MERU whilst the Sun was in the ए. TARAYANA (or moving from the winter to the summer solstice) and night whilst the 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 prrris reside on the upper oss Ope cay err part of the Moon and fancy the foun- tain of nectar to be beneath themselves. They behold the Sun on the day of our am4vdsyf or new Moon in their zenith. That therefore is the time of their midday. 14. They (i.e. the prrRis) cannot see the Sun when he 18 opposite the lower part of the Moon: it is therefore, midnight with the Pirris on the day of the एष्टा or full Moon. The Sun rises to them in the middle of the KRIgHNA 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 the context. 19. As Braumé being at an im- 1 of aday mense distance from the Earth, always 8668 the Sun till the time of the pra- LAYA or general deluge, and sleeps for the same time, therefore I 2 164 Translation of the [VIT. 16. the day and night of BrauMa are together of 2000 mMaHayuGas in length. a 16. As the portion of the ecliptic वि न taken by ech which is more oblique than the other, rizon. rises and sets in a shorter time and that which is more upright takes a longer time in rismg and setting, hence the times of msing 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 even at the equator are 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 Cancer or de- scending signs incline whilst they rise to the northerly direc- 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 cHARAKHANDA of that sign. 19. Hach quarter of the ecliptic rises in 15 auatis 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 30 GHaTis in every or any latitude. 20. The three sins from the commencement of Aries to the end of Gemini, 1. e. the first quarter of the ecliptic, pass the UNMANDALA in 15 eHatis; but the horizon [of a place in north latitude] is below the unmManpaLa, they therefore pre- viously pass it in time less than 15 auatis by the CHARAKHANDAS. 21. The three signs from the end of Virgo to the end of Sagittarius, 1. e. the 3rd quarter of the ecliptic, pass the UNMAN- DALA in 15 @HaTIs; but they pass the horizon of a place VIL. 23.) Siddhdnta-s'tromani. 165 afterwards which is above the uNMANDALA [in north latitude] in 15 eHatTis 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 1. e. the 4th quarter of the ecliptic, pass the horizon in the time equal to the remainder of 30 gHafis diminished by the time which the first or third quarter takes to pass the horizon respectively. For this reason, the times which the signs contained in the Ist and 4th quar- ters 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 CHARAKHANDAS of the signs from and adding them to the 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 latitudes will be seen from the following memo. according to the SippHaNTa. ६0. 2 = हव 0. © + ~o ar ८ cam +S a = == 04 ~ o 14 ८ nm ae we > ~ 2.2 |> a S55 (2. ~~, (८2० m “3 = 3% ~ चर) OO ष्ट ८ asa |8 <£ 2:28, 2 £ lass .:83-23:: € va = GO ~ "८ हि ^ +> => ASUS. ASUS. | ASUS. TT t t क 2070 | — 297 | 1978. |), ome Sand fhe tee था (००००००००. 1798 | — 244 | 1549 | ( + 8 take ०९6 Tite ४ Gemini, ,,,,,,०,५,,०,५,०,.| 1987 | - 101 | 1836 १." - de | than at the equator. GROG 14, 2987 + 101 2038 |) "1 ७ ,०००| L799 + 244 | 2037 || ‘These 6 signs take a क + ००. ,०५१००* | 2070 + 297 | 1967 + longer time to rise in TRE 3५... 1670 +- 297 1967 | north latitude than at SOFIE, icy: cosas १० ०४.०७ 1793 =+ 244, 2037 । the equator, Sagittarius,..............) 1937 न 101 | 2038 |) Capricorn, ,,,,,,.१०,,९१५,| 1937 — 101 1836 AQUuaATIUS, = ,,,,,०,०,,००,..| 1798 — 244 |. 1549 1218668 ०,००००,०* ७५११५१५१ १ 1670 ae 297 1373 L. W. LS 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 mses 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 Etymology of the word the eastern horizon is called the LAGNA eee or horoscope. This is expressed in signs, degrees, &c. reckoned from the first point of stellar Aries. That point which is on the western horizon is called the ASTA-LAGNA or setting horoscope. The point of the ecliptic on the meridian is called the MADHYA-LAGNA or middle horoscope (culminating point of the ecliptic).* * [When the place of the horoscope is to be determined at a given time it is necessary at first to ascertain the height and longitude of the nonagesimal point from the right ascension of mid-heaven, and then by adding 3 signs to the longitude of the nonagesimal point, the place of the horoscope is found: but as this way for finding the place of the horoscope is very tedious, it has been determined otherwise in the SippHa’nTas. As, from the periods of risings of the 12 signs of the eeliptic 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 ecliptic corresponding to the given time from sun-rise through the longitude of the Sun then determined and the given time. The portion of the ecliptic which can be thus found is evidently that portion of the ecliptic intercepted between the place of the Sun and the horizon. Therefore by adding this portion to the place of the Sun, the place of the horoscope is found. Upon this principle, the following common rule which is given in the SrppHayNTas 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 VIL. 27. Siddhdnta-s'iromani. 167 27. If when you want to find the Lacna, the given GHATIS are SAVANA-GHATIS, then they will be- The reason for finding the ४ : 5 exact place of the Sun at come sidereal by finding the Sun’s the time of question in order jinctant < t os and ee, instantaneous place 1. ©. 16 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 BHUKTANS aS and BHOGYANS’aS respectively. Now the time which the Sun requires to pass the BHOGYANS’‘AS is called the BHOGYA time, and is found by the following proportion. If 300 : the period of rising of the sign in which the Sun is : : BHOGYANS aS : BHOGYA time. In the same manner, the BHUKTA time can also be found through the BHUKTANSAS, Now from the time at the end of which the horoscope is to be found, and which 18 called the IsHTa or given time, subtract the BHOGYA time just found, and from the remainder subtract the periods of risings of the next successive signs to that in which the Sun is as long as youcan. ‘Then at last you will find the sign, the rising period of which being greater than the remainder you will mot be able to subtract, and which is consequently called the 48.077 4 sign, or the sign incapable of being subtracted, and its rising period, as‘UDDHA rising. From this it is evident that the as’upDHA sign is of course on the horizon at the given time. The degrees of the as’‘UDDHa sign which are above the horizon and therefore called the BHUKTa or passed degrees, are found as follows. If the rising period of the as’UDDHA sign : 800 : : the remainder of the given time : the passed degrees of the as’UDDHA sign. Add to these passed degrees thus found, the preceding signs reckoned from the 1st point of Aries, and from 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 BHUKTA, or passed time of the sign in which the Sun 18, in the way above shown, and subtract it and the rising periods of the pre- ceding signs from the given time. After this find the degrees of the as'UDpDHA sign corresponding to {116 remainder of the given time which will evidently be the BHOGYA degrees of the horoscope by proportion as shown above, and subtract the sum of the BHoGyYa degrees of the horoscope, the signs the rising periods of which are subtracted and the BHUKTA 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 horoscope is to be found, is after sun-rise, and the other when that time is given before sun-rise, and which are consequently called KRaMa, or direct, and VYUTKEAMA or undirect processes respectively. It 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 making the sum of the BHoeya time of the sign in which the Sun is and the BHUKTA time of the horoscope and by adding to this sum the rising periods of intermediate signs.—B, D.] 168 Translation of the (VII. 27. of rising of the signs which are sidereal must be subtracted from these GHaTIs (of the question) reduced to a like deno- mination. When the hours of the question are already sidereal, there is no necessity for finding the sun’s real place for that time.* * [If 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. the place determined for the hour given is used to ascertain the BHOGYA time, the given time is reckoned from sun-rise and the BHOGYA degrees of the sign in which the sun is, rise gradually above the horizon after sun-rise. Hence the BHOGYA degrees of the sign of the Sun’s longitude, determined at the time of sun-rise, should be taken to find the place of the Horoscope, otherwise the place of the Horoscope will be greater than the real one. As for example, take the time from sun-rise, at the end of which the Horoscope is to be found, equal to 60 sidereal GuaTIs and 44 asus when the Sun is in the vernal equinox at a place where the PaLaBua is 6 digits or the latitude is 22°} nearly, and ascertain the place of the Horoscope through the instantaneous place of the sun. Then, the place of the Horoscope thus found will be greater than the place of the Sun found at the time of next sun-rise, but this ought to be equal to it, and you will not be able to make this equal to the place of the Sun determined at the time of next sun-rise, unless you determine this through the place of the sun ascertained at sun-rise, and not through the Sun’s instantaneous place. Hence it appears wrong to ascertain the place of the Horoscope through the Sun’s instantaneous place. But the answer to this is as follows, The अ 48 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 GRATIS, but the GHATIS contained in the 876 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, @HaTIs. Thus it is plain from this that if the Sun’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 evidently be the side- real time and consequently the place of the Horoscope determined through this will be right. But if the instantaneous place of the Sun be given, the time given must be the sAvaNa time, because let the instantaneous place of the 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 sAVANA, will evidently be the sidereal time. Hence the fact as stated in tha verse 27th is right. Therefore if the Sun’s instantaneous place and the place of the Horoscope be given, the time found through these will be the sAvana time, but if the place of the Horoscope and that of the Sun determined at the time of sun-rise be given, the time ascertained through these will be the sidereal time. And if you wish to find the sAVANA time through the place of the Horoscope and that of the Sun determined at the time of the sun-rise assumed the sidereal time just found as a rough sAVANA time and determined through this the instantaneous place of the Sun by the following proportion. If 60 GuaTis : Sun’s daily motion : : these rough sAVANA GHATIS ; the Sun’s motion relating to this time; and add then this result to the placo of the Sun found at the time of sun-rise. The sum thus found will be the instantaneous [01966 of the Sun nearly. Find the time again through this VII. 32.] Siddhdnta-s' iromani. 169 28. In those countries having a north latitude of 69° 20’ the eae signs sagittarius and capricornus are Determination of latitudes a व ace in which different signs are ever visible: and the signs gemini ates pore 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 are never seen, and the four signs taurus, gemini, cancer, and leo, always appear revolving above the horizon. ` 80. On that far-famed hill of gold Meru 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. : 31. Latta 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 is 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 once on all occasions, as the times of their rising on the equator are equal to the asus of their cHARA-KHANDAS: but this is not the fact. 82. Latta has also stated in his work on the sphere that where the north latitude is 66° 30, 1420 gross error of sagittarius and capricornus are not 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 Latia exposed. instantaneous place of the Sun, and through this time ascertain the instantaneous place of the Sun. Thus you will get at last the exact sAvaNa 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 sAvana time of any planet or any planetary. time from the planet’s rising to the hour given by the repetition of the aforesaid process.—B. D.] # [BHASKABACHARYA means here that Latta mentioning the degrees of lati- tudes, has committed a grand mistake in omitting 3 degrees, because 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 the LAMBANSA or complement of the latitude. Or the zenith distance and altitude of the Sun at mid-day when on the equi- noctial give the latitude and its complement. 34. 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 axsHA-KARNA, by reason of the lati- tude, is called cHHEDAKA and not s’ANKU because it is upright.* 35. In order to find the shadow of the Moon, the एणा (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) 18 not sdvana. The labour of the astronomer that does not thoroughly understand mathematics as well the doctrine of the stated in his work that sagittarius and capricornus ore 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 versea 28 and 29.—B. D.] * (When the Sun is above the Horizon, the shadow caused by a gnomon 12 digits, high, is called the Sun’s shadow according to the s'IpDHANTA languages and having at first determined the sine of the Sun’s altitude and that of it eomplement through his UDITA time, astronomers ascertained this by the 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 planets, 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, like that of the Sun and Moon, as to make their shadow visible, yet it is necessary to determine the shadow of any heavenly 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 SAVANA time, because it is necessary to determine the degrees of altitude of a planet to know its shadow, and the degreea can be determined through the time contained in that arc of the diurnal circle intercepted between the planet and horizon. But the time containcd in this are cannot be other than the sAVANA time.—B. D.) VII. 39.] Siddhdnta-s tromani. 171 sphere, in writing a book of instruction on the science 18 utter- ly futile and useless.* 36. The degrees of altitude are found in the DRINMANDALA or vertical circle, being the degrees of ५ 0 of SANKU glovation 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 ‘DRIGJYA is the sine of the zenith distance. 87. When the Sun in his ascent arrives at the prime verti- cal, the s‘anku found at the moment 18 SE AMA SIANET, FONA- the SAMA-S‘ANKU : the s’ANKUS found at the moments of his passing the KONA- VRITTA and the meridian are respectively termed the Kona- §’ANKU and MADHYA-S/ANKU. | 38. One-half of the vertical circle in which a planet is . observed should be visible, but only Reason of the correction . ४ of parallax to the sine of alti- one-half less the portion opposite the noe radius of the Earth is visible to observ- ers on the surface of the Harth. Therefore कष part of the 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 T १ १ ‘ie ee and prime vertical and the planet’s diurnal circle in the east or west 1. e. between * (In order to determine the Moon’s shadow at a given time at full moon, some astronomers find her up1Ta time i. €. the time elapsed from her rising to the hour given by the repeated calculation, through her instantaneous place and the place of the horoscope determined at the given hour. But they greatly err in this, because the time thus found will not be the s’avana time and consequently they cannot use this in finding the Moon’s shadow. Their way for finding the UDITA time by the repeated calculation would be right, then only if the given place of the Moon would be such as found at the time of her rising and not her instantaneous place. Because her upiTa time found through her instantaneous place becomes s’AVANA at once without having a recourse to the repeated calcu- lation, as it is shown in the note on the verse 27 of this Chapter.—B D.] K 2 172 Translation of the [VII. 40. the east or west point of the horizon, and the point of the horizon at which the planet rises or sets. The line connecting the points of the extremities of the east and west 46. is called the uDaYAsTa-sUTRA, the line of rising and setting. 40. The s’anxu-TaLa or base of the s’anku stretches dur- ing the day to the south of the upyAsTa-suTRA; because the diurnal circle have during the day a southern inclination (in nor- thern latitude) above the horizon. But, below the horizon at night, the base lies to the north of the upDAYAsTA-sUTRA as then the diurnal circles incline to the north. The s'anxu- TaLa’s place has thus been rightly defined. 41, The s’AnKu-TALA lies to the south of the extreme point of AGRA when that aGRA is north and when the 4७764 is south, the s’ANKU-TALA lies still to the south of it. The difference and sum of the sine of amplitude and s’aNku-TaLa has been denominated the BAHU or BHUJA; itis the sine of the degrees lying between the prime vertical and the planet on the plane of the horizon. 42. [Taking this Bfuv as one side of a right-angled triangle.] The sine of the zenith distance being the hypothe- nuse then the third side or the kof! 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 are created by reason of the Sun’s varying declination : and shall then proceed to explain briefly also the latitudinal triangles or those created by different latitudes. [The former are called KRANTI-KSHETRAS and the latter AKSHA-KSHETRAS. | * Vide accompanying dia- gram. a being place of the Sun: d its place of rising in the horizon : d h the upaYa’sTa-stTRa d f the aara’: @ © the 8’ aNKU-TALA: then ag 18 the Ba’HU and the triangle @ 2 ¢ isthe one here represented to.—L. W. £ ध ¢ 1 निमी णि) | VIT. 45.] Siddhdnta-siromani. 173 43. In the Ist triangle of declination. lst. The sine of declination == BHUJA or base, the radius of diurnal circle cor- | । == Kori or per- responding with the declination pendicular, above given and radius of large circle = hypothenuse. 2nd. Or in a right sphere. The sine of I, 2 or 3 signs = hypothenuse : The declination of 1, 2 or 3 signs insix| __ ‘in == BHUJAS. o’clock line 44, Sines of arcs of diurnal circles cor- responding with the dination | = KOTIS. above given ‘These 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 i.e. the mght ascensions of those signs or the LANKODAYAS, that 1s the 2nd will be found when the Ist is subtracted from two found conjointly, and the 3rd will be found when the sum of the 1st and 2nd is subtracted from three found con- jointly. 45. In the right-angled triangle formed by the sanxu Triangles arise from lati: OF gnomon when the Sun is on the tude. equinoctial.* 18४. The s’anxu of 12 digits = the KOTI. The pataBué or the shadow of s‘ANKU or gnomon | } = the BHUJA and the AKSHA-KARNA =the Karya or hypothenuse or 2nd. The sine of latitude == BHUJA. The sine of co-latitude = KOTI and radius = hypothenuse This triangle is found in the plane of the meridian. [* The right angle triangles stated in the five verses from 45 to 49, are clearly seen by fastening some diametrial threads within the armillary sphere. As 174 Translation of the [VII. 46. 46. Or the sine of declination reckoned on the UNMANDALA from theeast and west Bey य. a Kusy4, the sine of ascensional difference in the diurnal circle of the given day } = BHUJA. Let Z © N Hi be the meridian of the given place, & A H the diameter of the horizon, Z the Zenith, P and Q the north and south poles, E A F the diameter of the equinoctial, P A Q that of the six o’clock line, C f D that of one of the diurnal circles, and E B, f% the perpendiculars to G@ H. Then it is clear from this that 2 E or H P = the latitude, A B = the sine of it, B = the co-sine of it, J = the declination of a planet revolving in the diurnal circle whose diameter is C D, g = the a@ra or the sine of amplitude, . g = the xusya’, € = the 8aMA-8A'NEU or the sine of the planet’s altitude when it reaches the prime vertical. € g = the TADDAHRITI, ef = the TADDHRITI—KUJYA, ¢ = the UNMANDALA 8’ANKU or the sine of the planet’s altitude when it reaches the six o’clock line, A ¢ = the aGga’DI-KHANDA or thie lst portion of the sine of amplitude, aud & g = the aGra’GRa-KHaNDA or the 2nd portion of the sine of amplitude ; E A and .. A St A VII. 48.] Siddhdnta-s’iromani. 175 The sine of amplitude in the horizon == hypothenuse This is a well known triangle. 47. Or the sama s’ANKU in the prime ver- : == KOTI tical being } The sine of amplitude == BHUJA The TADDHRITI in the diurnal circle = hypothenuse Or Taking the sine of declination == BHUJA and the SAMA-8 ANKU = hypothenuse TADDHRITI minus KUJYA == KOTI. 48. The UNMANDALA S ANKU being == BHUJA The sine of declination will then be = hypothenuse And AGRADI KHANDA or lst portion of ms Se 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 ए, Agf,Aeg,Aef, Afh and 4.7 ¢ aud the first triangle you will get by dividing the three sides of the E B triangle A E B by —— and for the last see the note on the verse 49. 12 It is clear from the above described diagram that all of these triangles are similar to each other and consequently they can be known by means of propor- tion if any of thera be known. The srppHAnTISs, having thus produced several triangles similar to these original by fastening the threads within the armillary sphere, find answers of the several questions of the spherical trigonometry. Some problems of the spherical trigonometry can be solved with greater facility by this SippHaNnTA 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. The way for finding the answer of this problem according to the SIDDHANTA is as follows. Draw Ce L A 2, (See the proceeding diagram) then © c e will be a lati- tudinal triangle. Now, let a = © © the sine of zenith distance, ¢ = Ac, the co-sine of Z c, ¢ = A €, the SAMA-8 ANKU, and zx = the latitude. Then Ce = ५८० + (b—0)% a? +- (6—c)*, and Ce:Ce:: AE: AB, or ^; + (b—c)*: ८ : : rad: sin 2; a >€ Rad ० 8111 ¢ = /a? + (b—c)*.—B, 7. ] 176 Translation of the [ ४11. 49. Or Making the UNMANDALA SANKU == KOTI the AGRAGRA-KHANDA or 2nd portion of त । | | == BHUJYA sine of amplitude is the Kusya then becomes = hypothenuse 49.* The s’anxu being = KOT and the 8 ANKU-TALA = BHUJA Then the CHHEDAKA or HRITI = hypothenuse Those who have a clear knowledge of the spherics having thus immediately formed thousands of triangles should explain 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. Cuapter VIII. Called GRAHANA VASANA. In explanation of the cause of eclipses of the Sun and Moon. 1. The Moon, moving like a cloud in a lower sphere, 7 overtakes the Sun [by reason of its Hons as Deginning and quicker motion and obscures its shin- of the solar eclipse. ; ६ : . ing disk by its own dark body :] hence it arises that the western side of the Sun’s disk is first obscured, and that the eastern side is the last part relieved from the 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 Ist of the 47th verse only in this respect that the base of the triangle in the 47th verse is equal to the sine of the whole ampli- tude while the base found when the Sun is not in the prime vertical, will alwaya be more or less than the sine of amplitude and is therefore generally called SANKUTALA.—L. W. VITI. 6.] Siddhdnta-s'tromoni. 177 2. At the change of the Moon it often so happens that an Tike eaudesob he parallax observer placed at the centre of the 1 and that in Harth, would find the Sun when far from the zenith, obscured by the intervening body of the Moon, whilst another observer on the surface of the Earth will not at the same time find him to be so obscured, as the Moon will appear to him [on the higher elevation] to be depressed from the line of vision extending from his eye tothe 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. 3. When the Sun and Moon are in opposition, the Earth’s The reason of the correc. शतक्त envelopes the Moon in dark- tion of parallax not being ness. As the Moon is actually enve- ^ 1 loped in darkness, its eclipse 18 equally seen by every one on the Harth’s surface [above whose horizon it may be at the time]: and as the Harth’s shadow and the Moon which enters it, are at the same distance from the Earth, there is therefore no call for the correction of the parallax in a lunar eclipse. 4. Asthe Moon moving eastward enters the dark sha- The cause of the direc. GOW Of the Karth : therefore its eastern tions of the beginning and side is first of all involved in obscurity, न and its western is the last portion of its disc which emerges from darkness as 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 made by the Sun from the Earth is therefore of a conical form terminating in a sharp point. It extends to a distance considerably beyond that of the Moon’s orbit. 6. The length of the Earth’s shadow, and its breadth at the part traversed by the Moon, may be easily found by propor- tion. | L 178 Translation of the (VIII. 7. In the lunar eclipse the Earth’s shadow is northwards or southwards of the Moon when its latitude is south or north. Hence the latitude of the Moon is here to be supposed inverse (i. ©. it is to be marked reversly in the projection to find the centre of the Earth’s shadow from the Moon.) 7. As the horns of the Moon, when it is half obscured form ० ohihe. MOD obtuse angles: and the duration coverer in the eclipse of the of a lunar eclipse is also very great, a 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 inferred that the dimensions of the body causing the obscuration in a solar eclipse are smaller than and different from the body causing an eclipse of the Moon.* 9. Those learned astronomers, who, being too exclusively devoted to the doctrine of the sphere, believe and maintain that Riuv 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 Sanur, the Vrepas and Purdnas. 10. All discrepancy, however, between the assertions above referred to and the sacred scriptures may be reconciled by understanding that it is the dark Ra&au which entermg the Harth’s shadow obscures the Moon, and which again entering the Moon (in a solar eclipse) obscures the Sun by the power conferred upon it by the favour of Branma. * [Had the Sun’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 Sun, forming acute angles, if its coverer had been the 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.) Siddhanta-s’iromoni. 179 11. As the spectator is elevated above the centre of the rt it $ t h ree What is the cause of paral. © h by half its diameter, he the re lax, and why it is calculated fore sees the Moon depressed from its ius of the Earth. । च bere eee place [as found by a calculation made for the centre of the Harth]. 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, the sphere of the earth © reduced to any convenient scale, and onstruction of diagram to illustrate the cause of the orbits of the Moon and Sun at 193. १ proportionate distances: next draw a transverse diameter and also a perpendicular diameter to both orbits.* 18, 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, let E be the ¢ centre of the earth; A a spectator on her surface; C D, ¥F © the vertical circles passing through the Moon M, and the Sun 8; D, G@ the points of the horizon cut by the vertical circles C D, F G; and C, the zenith in the Moon’s sphere, and F in that of the Sun. Now, let EMS be a line drawn from the centre of the Earth to the Sun in which the Moon lies always at the time of conjunction,and A 9 the vision line drawn from the spectator A to the Sun. The distance at which the Moon appears depressed from the vision line in the vertical circle is her parallax from the Sun. When the Sun reaches the zenith F, it is evident that the Moon also will then be at C and the vision line, and the line drawn from the centre of the Earth will be coincident. Hence there is no parallax in the zenith. Thus the parallax of the Moon from the Sun in the vertical circle is here shown by means of a diagram which becomes equal to the difference between the parallaxes of the Sun and Moon separately found in the vertical circle as stated by Bua’sKABA’OHA'BYA in the chapter on eclipses in the commentary Va/SANA’BHA’- sHya and the theories and methods are also given by him on the parallaxes of the Sun and Moon. This parallax in the vertical circle which arises from the zenith distance of the planet is called the common parallax or the parallax in altitude. | | 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 sines and arcs,| let the learned astronomer show the manner in which Fig. 2. As in Fig. 2, let A be a specta- A tor on the earth’s surface; Z the zenith ; and 2 9 the vertical circle passing through the planet 8 : Let a circle Z’ m r be described with centre A and radius E 8 which cuts the lines A Zand A 8 produced in the points Z’ = andr: Let a line s m be drawn pa- rallel to E Z, then the arc Z’ m will 9 be equal to the arc 2 8. Now the planet 8 seen from E has a zenith dis- tance 2 8 and from A, a zenith distance Z’ r greater than 2 8 or Z’ m by the arc mr, hence the apparent place r of the planet is depressed by mr in the vertical circle. This arc mr 18 there- fore the common parallax of the planet, which can be found as follows. A Draw mn perpendicular to A r and # 0 ४0 4 2 84 1९४ 2 = 7 8 ०८ ^ ^“; h=EAormS; p= ‘i r the paral- true zenith distance of the planet ; and ,, @ ~ p = Z’r the apparent zenith distance of the planet, Then m » = sin p and r o = sin (d + p). Now by similar triangles A r 0, 8 mn. Ar: ro=Sm: ma, or $ sin (द + p) =A: शा; & X sin (d + 2) RB e Hence, it 18 evident from this that when the sin (द 4- 2) = Rord +p = 90°, then the parallax will be greatest and if it he denoted by P, sin P X sin (द sin P = ¢ and 1 ltateh +?) ००७ sin p — R .. Now, the parallax is generally so small that no sensible error is introduced by ynaking sin p = p and sin P = P; P X sin (d +p) । ह. न्ष रल - र R Again, for the reason just mentioned sin क is assumed for sin (द + 2) in the SIDDHANTAS, P. sin d १ 2 = R that is, the common parallax af a planet is found by multiplying the greatest parallax by the sine of the zenith distance and dividing the product by the radius.— 2. D.] VIII. 20.] Siddhanta-s'iromont. 181 the parallax arises. [For this purpose] let him draw one line passing the centre of the earth to the Sun’s disc: and another which 18 called the pgixsdrra or line 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 line drawn from the centre of the earth to be in exactly the same place and to have the same longitude: but when the Moon is observed from the surface of the Harth in the pRixsttra 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 line drawn from the Harth’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 Harth, and also the orbits of the Sun and Moon at proportionate distances from the Earth, and also the diameter transverse and perpendicular, &c. | 18. The orbits now drawn, must be considered as DRIKSHE- PA-VRITTAS 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 DRIKSHEPA 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 Narti-xatfs, i. e. the minutes of the parallax in latitude of the Moon from the Sun. 20. The difference north and south between the two orbits 1, e. the measure of their mutual inclination, is the same in every part of the orbit as it is in the nonagesimal point, hence this difference called nati is ascertained through the DRIKSHE- pa or the sine of the zenith distance of the nonagesimal.* [* When the planet is depressed in the vertical 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 east and west difference between the Sun and Moon in a vertical circle.* distance from its orbit caused by this depression is called NaTI or the parallax in latitude. Fig. 3. As, in Fig. 3, let Z be the zenith; N the nona- . gesimal; ZN P its vertical circle; N s r the ecliptic ; P its pole; Z s ६ the vertical circle passing through the true place 8 and the depress- ed or apparent place ¢ of theSun; P ¢ r a secondary to the ecliptic passing through the apparent place ¢ of the Sun; then s r is the SPASHTA LAMBANA or the parallax in longitude and ¢r the NaTI or the parallax in latitude which can be found in the foliowing manner according to the sIpDHANTAs. Let ZN be the zenith distance of the nona- gesimal and ZS that of the Sun; then by the triangles ZN S,isr sin ZS: sin ZN <= 8118६: 8107४, 8115८ X sin ZN *. sin rt = ———— ॐ sin ZS Now, s ¢ is taken for sin s ¢, and r ¢ for sin r ¢, on account of their being very small st > sin ZN sin ZS but according to the sIDDHANTAS P.sinZ8 (see the preceding note).. (1) P . 810 ZN Serr st= ~~ ort = R that is, the Natr is found by multiplying the sine of the latitude of the nona- gesimal by the greatest parallax and dividing the product by the radius. _ It is clear from this that the north and south distance frem the Sun depressed in the vertical circle to the ecliptic wherever he may be in it, becomes equal to the common parallax at the nonagesimal, and hence the NAT! is to be determined from the zenith distance of the nonagesimal. For this reason, by subtracting the nati of the Sun from that of the Moon, which are separately found in the way above mentioned, the parallax in latitude of the Moon from the Sun is found : and this becomes equal to the difference between the mean parallaxes of the Sun and Moon at the nonagesimal. The same fact is shown by BuAskarsoninya through the diagrams stated in the verses 12th &c. Af 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 the parallax in longitude and that in latitude of the Moon is here determined from her corresponding place in the ecliptic, on account of the difference being very small.—B. D. | ; , # [According to the technicality of the Siddhantas, the distance taken in any circle from any point in it, is called the east and west distance of the point, and VITI. 27.] Siddhanta-s’iromoni. 183 22. For this reason, the difference is two-fold, beimg partly east and west, and partly north and south. And the ecliptic is here east and west, and the circle secondary 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. The difference east and west has been denominated LAMBANA or parallax in longitude, whilst that runnimg 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 munutes 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 ` distance of the minutes of the extreme or horizontal parallax and dividing the product by the radius. Thus the nati will be found from the pRixsHEpa or the sine of the nonagesimal zenith distance.t 26. The extreme or horizontal parallax of the Moon from the Sun amounts to कृषः part of the difference of the Sun’s and Moon’s daily motion. For त्न part of the yosanas, the distance of which any planet traverses per diem (according to the sID- DHANTAS) 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 the distance taken in the secondary to that circle from the same point, is called the north and south distance of that point.--B. D.] * [See Fig. 3, in which by assuming the triungle r ॐ ¢ as a plane right-angled triangle, r ¢ = base, ॐ ¢ = hypothenuse and s r = perpendicular, and therefore sr=,/st'—r t*,—B, D.] f [This is clear from the equations (1) and (2) shown in the preceding large note.—B. D.] 184. Translation of the [VIII. 28. motions of the Sun and Moon will be converted into GHATIS [1. €. 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 is thrown backward (by the parallax). 28. And ifthe 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 TITHI or the hour of ecliptic conjunction, and to be added when the Moon is to the west [of the nonagesimal]. 29. The latitude of the Moon is north and south distance between the Sun and Moon, and the nati also is north and south. Hence the sara or latitude applied with the nati 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 VaLana or variation (of the ecliptic). It is evident from this, that the variation is equivalent to the arc which is the measure of the angle formed by the ecliptic and the secondary to the circle of position at the planet’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 @HaTIS — what will : : given LAMBANA—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 the parallax = acceleration or delay of con- diff. in degreea of Sun’s and Moon’s motions junction arising from parallax.—L. W. VIII. 29.] Siddhanta-s'iromont. 185 measure of the 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-vaLANa (latitudinal variation) and the dyaNa-vALANA (solstitial variation). The AKSHA-VALANA is the arc which is the measure of the angle formed by the circle of position, and the circle of declination at the place of the planet in the ecliptic, 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- ence of these two arcs, the arc which is the measure of the angle formed by the circle of position and the circle of latitude is ascertained, and hence it is sometimes called the s'PasHTa- VALANA or rectified variation. | Now, according to the phraseology of the Sippudntas, the point at a distance of 90° forward from any place im any circle is the east point of that place, and the point at an equal distance backwards from it is the west point. And, the night hand point, 90° distant from that place, in the secondary to the former circle, is the south point, and the left hand point, is the north point. According to this language, the deviation of the east point of the place of the planet in the ecliptic, 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 prime vertical at a distance of 90° forward from the place of the planet, and hence the deviation of the east point in the ecliptic from the east 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- tions 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 [ VIII. 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 vautana. This direction being known, the exact directions of the beginning, 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 vALANaA will exactly be understood by seeing the follow- ing diagram Let E P C be the ecliptic, P the place of the planet in it, A h B the equinoctial, V the vernal equinox, D h F the prime vertical, ॥ the poirt of intersection of the prime vertical and VITTI. 29.] Siddhanta-s' iromoni. 187 the equinoctial, hence h the east or west point of the horizon and D h equivalent to the nata which is found in the V. 36. Again, lete Pe,a 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 ¢ whichis the measure of Z ? P f = the 48६ - VALANA : the are 0c scsscsevssivindsiwiewnsie Z, ¢-P b= the AYANAs VALANA : and the arc fie ,,,.१०१..--.१०१११,०००० ^ ¢ Pf=the spasnta- VALANA. Or according to the phraseology of the SippHAntas E the east point of P in the ecliptic ; Pe poster 9 the equinoctial ; 00 nieces the prime vertical ; hence, the distance from D to A or arc D A or fb = the Axsna- VALANA : 9 A to E 0 © A E or ¢ ८ = the AYANA-VALANA: and ......00. Dto E or arc D KE or fc = the sPasHTA-VALANA or rectified variation. These arcs can be found as follows Let, ( = 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 == XKSHA-VALANA, and Z = rectified VALANA. Then, in the spherical triangle A V BK, sn HAV:snAVE=>smEV:sinA HB, or cos ¢ ; sine =cosl ; sin a, ४ 2 188 Translation of the [ VIII. 30. sine.ecosl. “. sin wor sine of the AYANA-VALANA = (A) cos ¢ see V. 32, 33, 34. This vaLANA is called north or south as the point E be north or south to the point A. And, in the triangle A ॥ D. sn DAh:snAhD=smnDh:snDA; here, sin DAh= sin EA V = 008 ५, sin A ॥ D = sin L, and sin Dh= sinn, cos d: sin L = 87 2! : sin y, sin L. sin n sin y or sine of the AKSHA-VALANA = (B) cos d See V. 37. The AksHA-VALANA is Called north or south as the poimt A be north or south to the point D. And the rectified vatana D E=DA-+ AE, when the point A lies between the points D and H, but if the point A be beyond them, the rectified vaLana will be equal to the difference between the AxsHa and AyANA-vALANA. This also is called north or south as the point E be north or south to the point D. The ancient astronomers Latua, S’rfpatt &c. used the co-versed sin / instead of cos / and the radius for the cos d in (A) and the versed sin n in the place of sin n and radius for the cos d in (B) and hence, the vaLanas, found by them are wrong. BuAsKkarAcHArRyA therefore, in order to convince the people of the said mistake made by Latta, S’rfpati, &c. in finding the vaLaNnas refuted them in several ways in the subsequent parts of this chapter.—B. D.] 30. In either the 18 Libra or the 1st Aries in the equi- noctial poimt of intersection of the equinoctial and ecliptic, the north and south lines of the two circles i. e. their secondaries are different AYANA-VALANA. VITI. 36.] Siddhdnta-s'ivomoni. 189 and are at a distance* of the extreme declination (of tho Sun 1. €. 24°) from each other. 31. Hence, the AYANA-VALANA 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. 32, 33 and 34. And the north and south lines being there coincident, it follows as a matter of course that the east of those two circles will be the same. Hence at the 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 the planet by the sine of 24°, and dividing the product by the pyusyaA or the co-sine of the declination of the planet. ‘This AYANA-VALANA 18 called north or south as the planet be in the ascending or descending signs respectively. Thus in like manner at the point of intersection of the prime vertical and equinoctial, the six o’clock line is the north and south line of the equinoctial, whilst the hcrizon (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). 30. 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 the same. Hence at midday there is no AKSHA-VALANA. 86. For any intervening spot, the AKSHA-VALANA 18 to be found from the sine of tha 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 the half length of day. AKSHA-VALANA. * [By the distance of any two great circles is here meant an arc intercepted between them, of a great circle through the poles of which they pass.—B. D. | + [Here the nara is the arc of the prime vertical intercepted between the - zenith and the secondary circle to it passing through the place of the planet.— B. D. ] 190 Translation of the (VIII. 37. 37. Then the sine of the nata degrees multiplied by the sine of latitude, and divided by the co-sine of the declination of the 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. e. 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 AKSHA-VALANAS (as they happen to be of the same or different denominations) is for that time at its maximum. 39. But at a point of the ecliptic distant from the point of intersection three signs either forward or backward, there is no SPASHTA-VALANA : for, at those points the north and the south EPASHTA-VALANA. lines of the two circles are coincident. 40. However, were you to attempt to shew by the use of the versed sine, that there was then no sPASHTA-VALANA at those points, you could not succeed. The calculation must be worked by the right sine. I repeat this to impress the rule more strongly on your mind. 41. As all the circles of declination meet at the poles; it Another way of refutation 18 therefore evident that the north of using the versed sine. and south line perpendicular to the east and west line in the plane of the equinoctial, will fall in the poles. 42. But all the circles of celestial latitude meet in the pole of the ecliptic-called the kapamBa, 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-siromoni. 191 line which is perpendicular to the east and west line in the plane of the ecliptic. To illustrate this, a circle should be attached to the sphere, taking the equinoctial pole for a centre, and 24° for radius. This circle is called the KADAMBA-BHRAMA-VRITTA or the circle in which the KADAMBA revolves (round the pole). The sines in this circle correspond with the sines of the declination. All the secondary circles to the prime vertical meet in the point of intersection of the meridian and horizon, and this point of intersection 18 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 AksHA VALANA is the distance between the two circles just described passing through the SAMA and equinoctial pole. 2. The AYANA-vVALANA 18 the distance between the circles passing through the ecliptic and equinoctial poles. 3. The SPASHTA-VALANA 18 the distance between the circles 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 a sphere, ing the SPasHTA-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 the point cut by the prime vertical. The distance of the point cut by the equinoctial from that cut by the ecliptic is the AyANA—and the distance between the points cut by the ecliptic and prime vertical the spasuTa- VALANA. 192 Translation of the [VIIT. 50. 50. In this case the plane of the ecliptic 1s always east and west—celestial latitude forming its north and south line. Those therefore who (like s‘rfpatt or Lata) would add the s‘ara celestial latitude to find the vaLana, labour under a grievous delusion. 51. The 1st of Capricorn and the ecliptic pole reach the meridian at the same time (in any latitude): so also with regard to the Ist Cancer. Hence at the solstitial points there 18 NO AYANA-VALANA. 52. As the lst Capricorn revolves in the sphere, so the ecliptic pole revolves in its own small circle (called the Ka- DAMBA-BHRAMA-VRITTA round the pole). 53 and 54. When the Ist of Aquarius or the Ist of Pisces comes to the meridian, the distance in the form of a sine in the KADAMBA-BHRAMA-VRITTA, between the ecliptic pole and the meridian is the AYANA-vALANA. This VALANA corresponds with the krantisyf or the sine of declination found from the degrees corresponding to the time elapsed from the 1st Capri- cornus leaving the meridian. 55. As the versed sine is like the sagitta and the sine is the half chord (therefore the versed sine of the distance of the ecliptic pole from the meridian will not express the proper quantity of vaLana as has been asserted by Latia &c.: but the right sine of that distance does so precisely). The Ayana- VALANA will be found from the declination of the longitude of the Sun added with three signs or 90°. 56. Those people who have directed that the versed sine of the declination of that point three signs in advance of the Sun should be used, have thereby vitiated the 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 does not shew the error of their authors is much to be blamed. Hence I am acquitted of blame having thus clearly exposed the errors of my predecessors. ४111, 64.] Siddhanta-s'tromont. 193 58. The inapplicability of the versed sine may be further Another way of refutation, ilustrated as follows. Make the eclip- of using the versed sine. tic pole the centre and draw the circle called the Jrna-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 the equinoctial poles, when it comes to the end of the sign of Gemini. : 60. By whatever 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 s1va-vgitta. The sine of those degrees will be there found to correspond exactly with and increase as does the sine of the declination. 61. And this sine is the Ayana-vaLana: This vALANA is the 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. 6. the cosine of the declination. 62. But as the value of the result found is required in terms of the radius, it is consequently to be converted into those terms. As the JINA-vRITTA 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 circle from the equinoctial pole will be exactly equal to that of the planet from the zenith in the prime vertical. The sine of the planet’s zenith distance in the prime vertical, will, when reduced to the value of the radius of AKSHA-VRITTA represent the AKSHA-VALANA. N 194 Translation of the [ VIII. 65. 65. As in the AYANA-vaLawa 80 also in this € AKSHAJYA’ UPAVEITTA-TRIJYA’ Here our author makes use of the diurnal circle and uravgirra in term of the equator and prime vertical, whose portions determine the VataNa. The smaller circles being parallel to the larger, the object sought is equally attained, —L. W. == sine of AKSHA-VALANA. VIII, 74.] Siddhanta-sivomoni. 195 68. Or the 4xsHA-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 AKSHA-VALANA, 69. Place the disc of the Sun at the point at which the diurnal circle intersects the 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 vatana 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 BHoagyA-KHANDA of the BHUsA of the Sun’s longitude and divide by 225. 71. Then multiply this result by sine of 24° and divide by the radius : the quotient is the difference of the two sine of declination. This again multiplied by the radius and divided . by the radius of Sun’s disc will give the value in terms of the 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) therefore cancel both. There will then remain rule, multiply the Sun’s 8०७१५ 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 sine of declination as above alleged, (and not by the versed sine). Abandon therefore, O foolish men, your erroneous 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 (TX. 1. the proportion of the एरर 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 the Sun and Moon CHAPTER IX. Called DRIKKARAMA-VA8ANK on the principles of the Rules for jinding the times of the rising and setting of the heavenly bodies 1. A planet is not found on the horizon at the time at Oj ect of the correction Which its corresponding point im the psi is 0 ecliptic (or that point of the ecliptic plied to the place of the having the same longitude) reaches planet, for finding the point । . wots of the ecliptic onthehorizon the horizon, inasmuch as it is elevated when the planet reaches it. 90७ or depressed below the horizon, by the operation of its latitude. A correction called pRIK- KARAMA to find the exact time of rising and setting of a planet, is therefore necessary. 2. When the planet’s corresponding point in the ecliptic reaches the horizon, the latitude then does not coincide with the horizon, but with the circle of latitude. The elevation of the latitude above and depression of it below the horizon, 18 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, e. the AyANA and the AKSHAJA or AxsHA. The detail and mode of performing these two sorts of the correction are now clearly unfolded 3. When the two vaLanas are north and the planet’ ४ 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. DrikKARMA. IX. 9.] Stddhanta-s tromoni. 197 4. When the two kinds of vatana are south, then the reverse of this takes place ; the reverse of this also takes place when the planet’s corresponding point is in the western hori- zon. | [And the difference in the 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 latitude give ? 5. And if cosine of the latitude of the evn} ‡ AKSHA-VALANA place : : what will spAsHTA s ARA give ? Multiply the two results thus found by these two propor- tions, by the radius and divide the products by the pyusyA 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 the direct process (as shown in the note on the verse 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 the eastern horizon at the same time with the planet. When the planet’s corresponding point is in the western horizon, the LaaNna horoscope found then by the rule converse of that above given, by means of the place of the planet added with 6 signs, 18 its Asta LAGNA setting horoscope or the point of the ecliptic which is on the eastern horizon when the planet comes to the western horizon. 8and9. For the fixed stars whose latitudes are very considerable the resulted time of the DRIKKARMA is found in a 198 Translation of the [ TX. 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. e. 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 are of the different or of the same denominations respectively, are the asus of depression or elevation depending on the AKsHA DRIKKARMA. (Find also the time depending on the Ayana- DRIKKARMA): and from the sum of or the 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).* * Let A D BC be the meridian; © E 7 the horizon, A the zenith; E the east point of the horizon; F E अ the equinoctial; K the north pole; L the south ; P the planet $ p its corresponding point in the ecliptic; H P 2 J the secondary to the ecliptic passing through the planet P, and hence p P the latitude. Let ? g the diurnal circle passing through the planet P and hence 2 R the rectified latitude. Now, when the corresponding place of the planet is in the horizon, it is then evident from the accompanying figure, that the planet is elevated above or depressed below the horizon by its latitude p P and as it is very difficult to find the elevation or depression at once, it is therefore ascertained by means of its two parts, the one of which is from the horizon to the circle of declination, i. e. @ ४० R. This partial elevation or depression takes place by the planet’s rectified latitude p R. And the other part of the elevation or depression 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 parts, the exact elevation of the planet above the horizon or the depression below it, can be determined. When the terrestrial latitude, of the given place 18 north and the planet’s corresponding place in the ecliptic is in the eastern horizon, the A’/KSHA-VALANA is then north and the circ 9 of declination is elevated above the horizon to the north. For this reason, when the a’KsHA- VaLaNa is north, the planet will be elevated above the eastern horizon if its latitude be north, and if it be south, the planet will be depressed below the horizon. But the reverse of this takes place when the a/KSHA-VALANA is south which occurs on account of the south latitude of the given place, 1, ९. when the A’KSHA-VALANA is south, the circle of declination is depressed below the horizon to the north and hence the planet is depressed below it, if its latitude be 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 Ayana-VALANA becomes then north, and the north pole of the ecliptic is elevated above the circle of declination passing through the planet. Hence, when the a’YaNa-VALANA is 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, when the a’YANA-VALANA is south, i. e. the planet is depressed below or elevated above the circle of declination, as its latitude is north or south. Because when the a’YANA-VALANA is south IX. 9.1]. Siddhénta-s'iromoni. 199 the north pole of the ecliptic lies below the circle of declination and the south above it. Again, when the planet is in the western horizon, the circle of declination passing through the place of the planet in the ecliptic lies to the north above the horizon, but the aKsHA-VALANA, becomes south and hence the reverse takes place of what is said about the elevation or depression when the planet is in the eastern horizon. But as to the AyaNa-vaLaNa, 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 planet 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 its latitude is north, and the depression when it is south. Thus 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 [1X. 10. 10. The [AspasHta] S‘aRa or true latitude [of the planet] To find the value of ce. Multiplied by the Dyusya& or cosine lestial latitude in terms of a f inati ° i: व of declination. to wis. declination of the point of the eclip = it Att be added to or tic, three signs in advance of the t lina ti din e di + व planet’s corresponding point and di- See the 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 BR, and the time of elevation of the planet from Rto P is denoted either by the angle P KR or by the equinoctial arc P’ p’. Out of these two times Q’ p’ and P’ ¢) we show at first how to find P’ p’. In the triangle 2 2 R, P p = the latitude of the planet, ८ ए p R = the A’YANA-VALANA and <= P R 2 = —, and R: snPpR=sin Pp: snRP; or if radius : sin of A’YANA-VALANA == the sine of latitude : sin ‰ ९. Again, by the similar triangles K P R and K P’ p’ sin K P: sin R P = asin K P’: sin ए“ p’, here, sin K P = cosine of declination and K ˆ = R, R >< sin RP as sin Pp ¢ p’ — cos of declination Now, the time p’ Q’ is found as follows. In the triangle p R Q, p R = the spasuta-s’aRkA which can be found by the rule given in the ए. 10 of this chapter, Z BR p Q = AKSHA-VALANA and <~ RQ p = co-latitude of place nearly and ..sinp QR: sin Ry Q::sinp B: sin BQ or, if cosine of latitude, : sine of AKSHA-VALANA, == SPASHTA-8/ARA : sin RQ; again, by the triangles K @ R, K Q’ p’, sin K Q: sin Q R = sin £ Q’: sin p’ Q’; here, sin K Q = cosine of declination and sine K Q’ = R, RX snQR ee sin 2 Q’ न= cos of declination. If both of these times thus found, be of the elevation or both of the depres- sion, the planet will be 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 elevation and the other for its depression, the planet will be elevated above 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’IDDHANTAS. That point of the ecliptic which is on the eastern horizon when the planet reaches it, is called the UDAYA LAGNA rising horoscope of the planet. As it is necessary to know this UDAYa La@Na for finding the time of the planet’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 when the planet will come to the eastern horizon, its corresponding place in the IX. 11.] Siddhanta-s'iromoni. 201 vided by the radius becomes [nearly] the spasuta or rectified latitude, [i. €. 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. The celestial latitude is not reduced by BranMacupta 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 it by the same time when the planet will come to it. Therefore, the horoscope found by the indirect process through the resulted time ; will be the rising horoscope 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 La@Na 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 the indirect process, through the regulted time and this will be the asta LAGNA setting horoscope. But if the planet be depressed below the western horizon, its corresponding place added with six signs will then be elevated above the eastern horizon by the resulted time and hence the horoscope found by the direct process will then be the 4874 LaGNa setting horoscope. Now the time p’ Q’ which is determined aboye through the triangle p R Q, is not the exact one, because, in that triangle the angle p Q R is 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 considerable error is caused in the time p’ Q’ thus found, if the latitude be of a planet, as it is always small. As to the star whose latitude is considerable, the time p’ Q’ thus found cannot be the exact time. The exact time can be found as follows. See the preceding figure and in that take R for a star and p the intersecting point of the ecliptic, and the circle of declination passing through the star R then 2 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 be equal to p’ Q’i. €. E 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 case will be equal to the sum of the two ascensional differences.—B. 7. ] * This rule is admitted by BuAskanAcHARya to be incorrect; but the error being small, is neglected. Instead of using the pyugyA, the yasutT1 should have been adopted. 0 202 Translation of the [1-९. 12. _ and other early astronomers to its value Omission of the Iast , . : mentioned correction or re- 10 declination: and the reason of this duction of Celestial latitude os + _ 3 to its value in declination, = नपाता, seems to have been its (क and 6108.11716€88 of amount. And also it is the uncorrected latitude which 18 used in finding the half duration of the eclipses and in their projections &c. 12. As the constellations are fixed, their latitudes as given in the books of these early astronomers are the sPASHAT- s/ARAS, 1. ©. the reduced values of thé latitudes so as to render them fit to be added to or subtracted from the declination ; and the puruvas or longitude of these constellations are given, after being corrected by the AYANA DRIKKARMA 80 as to suit those corrected latitudes that 1s, the star will appear to rise at the equator at the same time with longitude found by the correction. Let ad be equinoctial and P the equinoctial pole, d b = Ecliptie, b ॐ = Celestial latitude, | ¢ c= Celestial latitude reduced to its value in declination 18 KOTI, 8 c= BuUsa being arc of diurnal circle ¢ s g $ c= ¢ 6 portion of diurnal circle of the planet’s longitude at 0. The triangle sc b or s ‰ 8 18 assumed to be a 7716- VALANAJA TRYABRA. The angle $ b c = AyaNa-vaLana or the angle of the inclination of ड which goes to ecliptic pole with ® ९ which goes to equinoctial ole. : Hence this triangle s ® © is called DIG-VALANAJA mryska, the angle s dc varying with the Ayana-varana. If} were at the Ist Cancer, then the north line a 6 ¢ which goes to the pole would go also to the ecliptic pole. Hence the ASPASHTA 84१4, and sPAsHTA 8’ARA of a star of 90° of latitude being both represented by 6 ¢ would be the same. ‘To the longitude of a star being 270°, its ASPASHTA and SPASHTA SARA would be the same.—-L. W. [The rule stated in this verse is founded upon the following principle. Assuming the triangle s 2 ० as a plane right-angled triangle and the angle s bc, as the declination of the point of the ecliptic three signs in advance of the planet’s corresponding place, because this declination is nearly equal to the AYANa-VALANA, we have, sinsecb:cossbc=bs:be; or B : YASHTI or nearly the cosine of the declination of the planet’s place 9०0 ~ == Celestial latitude : rectified latitude. —B. D.] IX. 18.] Siddhdnta-s'iromoni. 203 18. Those astronomers, who have mentioned that celestial Mae latitude is an arc of a circle of de- BHA’SKARA CHARYA ex- , ॥ poses the incorrect theory clination, are stupid. Were the ce- Of certain of his predecessors.“ [81419016 nothing: more: than: an by quoting their own prac- 5 tice which is irreconcilable arc of a 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 the six o’clock line at the time 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-SUTRA or on the line denoting the 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 xKorTi,i. e. perpendicular to the ecliptic and thus found the half duration of the eclipse? If the latitude were of this nature, it would never be ascertained by the proportion (which is used in finding it). 16. A certain astronomer has (first) erroneously stated the Censure of the astrono- DRIKKARMA and VALANA by the versed नक त sues sine. This course has been followed KARMA and VALANA. by others who followed him like blind men following each other in succession: [without seeing their way]. 17. Braumacupta’s rule, however, is wholly unexceptionable, but it has been misinterpreted by his followers. My observations cannot be said to be presumptuous, but if they are alleged to be so, I have only to request able mathematicians to weigh them Praise of BRAHMAGUPTA. with candour. 18. The DRIKKARMA and VALANA found by the former astro- o 2 204 Translation of the [1.९. 19. nomers through the versed sine are erroneous: And J 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 181 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 respectively, 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 the ecliptic [which is on the horizon], as the celestial latitude is perpen- dicular to the ecliptic.* 21. In this case the resulted times of the ष्मा ^ ए + 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 the versed sine—hence let the right sine (as prescribed) be always used for the DRIKKARMA. * [It is evident that the longitude of this point is equal to the arc through which it is found, and as the point of the ecliptic 3 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 90० the declination of this point will be north. This declination equals to the latitude in question. For R X sin latitude The sine of the latitude of the point = (by the as- sin 24° sumption) sin 24° >€ sin longitude of the point — ——_——————,, but this = sin de- Radius sin latitude = clination. The declination of that point or nonagesimal equal to the latitude of the place. And hence, 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.] IX. 26. | Siddhanta-s'iromoni. 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 BHAsKARACHARYA 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 point 90° in advance of the Sun’s place in the horizon. If you, 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 vatana found in the Dufvrip- DHIDA TANTRA by Lata and in the other works to be correct. [To this Buisxaracudrya adds a further most important and curious illustration : | 24. In the place where the latitude is 66° N. when the Sun at the time of his rising is in 18४ Aries, 1st 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 SPASHTA VALANA will be equal to the radius by means of the versed sine ! [In the same manner 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 inLatza 10 their own quick intelligence, by म 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'rinGonnati-vAsand 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 the imhabit- 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 the 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 the 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 the earth to the Moon. Then the Sun illumines half of the visible side of the X. 5.] Siddhdnta-s'tromoni. 2८१7 Moon. वा 18 when the Moon is 85°. . 45’ from the Sun east or west, it will appear half full to us.* 4, The 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 sphere, its horns assume a pointed or cusped appearance (varying in acuteness according to its distance from the Sun). 5. (To illustrate the subject, a diagram should be drawn Diagram for illustrating 98 follows). Let the distance north and the subject. south between the Sun and Moon re- present the BHusa, the 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 is KoTI at the extremity 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 the direction of the hypothenuse. * This is thus illustrated. Let @ repre- sent the Karth, © ¢ d the orbit of the Sun, € f = do. of the Moon. Then it is ob- vious that half of the side of the Moon visible to us will be illuminated when the Sun is at cand not at व, when the Sun 18 at द it will illumine more than half of the Moon’s disc; ¢ ८ is less than a quadrant by the arce d, the sine of which (= aeoreg 11) 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 us follows :— In the triangle a ec right angled at. e, ae — 51566 yosanas, ac =< 689377 a gy @ ००५५8 according to the SippHa’NTas, ae 51566 ; Then, cos e ८ ० == —— न == . 0748 = cos 85° .. 43 ac „५, arc bc = 85°.. 48’ nearly.-- 2. D.] 208 Translation of the [X. 5. [For instance Let 8 be the Sun and m the Moon, then a 8 = sausa, am == KoTI,i S = hypothenuse. Then / g a line drawn at right angles to extremity of hypotenuse will represent line of direction of the enlightened horns and the angle ॥ m d opposite to BHUJA will be equal to < ¢ mc = the amount of angle by which the northern cusp 18 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 cusp of the Moon is elevated: if south the southern cusp. lL. ४४. | [Mr. Wilkinson has extracted the following two verses from the GANITADHYAYA. 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, then it will appear to be bisected by the meridian and the eastern half will be enlightened. But according to Braumacupta this would not occur, for he has declared that the kori will be cqual to radius in this case whereas it 18 obviously “nil,” and it is the BHUJA which is’ equal to radius when there is no north and south difference XI, 2.] Siddhanta-siromoni. 209 between the Sun and Moon then the xotr would be equal to the hypothenuse or radius and the sxvsa would be “ nil.” TI.* And the Moon’s horns are of equal altitude when there is nO BHUJA, whilst they become perpendicular when there is no koTi. That the xkoti and BHvsA 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? BranMacupta 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 memtioned in the Chapters on Mapuyaaati &c. fearing to lengthen my work; but the talented astronomer should understand the principles of all the subjects in completion, because this is the result to be obtained by a complete know- ledge of the spheric. End of Chapter X. called S’rrinconnati-vasani. a i ee CHAPTER XI. Called YaANTRADHYAYA, on the use of astronomical instruments. 1. As minute portions of time elapsed from sun-rise cannot be ascertained without instruments, I shall therefore briefly detail a fow instruments which are of established use for this purpose. 2. The Armillary sphere, NApf-vaLaya (the equinoctial), the yasHTi or staff, the gnomon, the प्रकषण or clepsydra, the circle, the semi-circle, the quadrant, and the pHaLaxKa: but of all instruments, it is ९ INcENUITY”’ which is the best. | Object of the Chapter. ॐ BuisxarhowArya is here very severe on BRAHMAGUPTA who of all his pre- decessors is evidently his favorite, but truth seemed to require this condemn- ation. He at the same here does justice to ARYA-BHATTA and the author of the SGrya-sippHa’NTa. They both justly concur in saying there is no KoTI in this case.—L. W. P 210 Translation of the [XI. 3. © 3 and 4. (This instrument is to be made as before de- scribed, placing the Buagota starry sphere, which consists of the ecliptic, diurnal circles, the Moon’s path, and the circles of declination &c. within the kaacoLa celestial sphere, 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 BHAGOLA) intersected by the horizon, viz. east pot. 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. The Laena or horoscope will then be found in that point of the ecliptic which is cut by the horizon. Take a wooden circle and divide its outer 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 ares as shall correspond with their periods of rising in the place. of observation (the twelve periods are to be thus marked on either side, which are to be again each subdivided into two प 0748 (or hours), three DRESH- KANAS, into NAVANS’As or ninths of 38°. . 20’ each, twelfths of 2°. . 10’ and into rrins’ans’4s or thirtieths. These are called the sHAPVARGA or six classes). These signs, however, must be inscribed in the inverse order of the signs, that is 1st 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 longitude in signs, degrees, &c. for the sun-rise of the given day (by calculation) and find the same degree in the circle. Mark there the Sun’s place, turn the Use of Armillary Sphere. The Na’DI-VaLaYa. XI. 8.] Siddhanta-siromoni. 211 circle round the axis, so that the shadow of the axis will fall on the mark of the Sun’s place at sun-rise and then fix the circle. Now as the Sun rises, the shadow of the axis will advance from the mark made for the point of sun-rise to the nadir and will indicate the hour from sun-rise, and also the Laana (horos- cope) : the number of hours will be seen between the point of sun-rise and the shadow: and the Lana will be found on the shadow itself. [While the Sun goes from east to west the shadow travels from west to east and hence the signs with their periods of rising must be reversed in order—the arc from W to Laena represents the hour arc: and the Laena 18 at the word Lacna 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 GHatI made of copper like the lower half of a water- pot, should have a 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 1s placed. Divide 60 auatis of day and night by the quotient P 2 The GuHarI 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 @HaTI will be of one GHaTIKA; if 24 times it will be of one hour or 24 GHaTIKAs.) 9. For a gnomon take a cylindrical piece of ivory, and let it be turned on a lathe, taking care that the circumference 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 elsewhere explained). 10. The circle should be marked with 360° on its outer circumference, and should be 808- | pended by a string or chain moveable _ on the circumference. The horizon or Harth 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 the @nomon. The CHAKRA or circle. zenith. 11. Through its centre put a thin axis: and placing the 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 @ rough calculation of the time, viz. multiply the 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 longitudes of Of the following fixed stars appear to planets by the circle. touch the circumference, viz. Ma- GHA (a Leonis, Regulus), Pusuya (8 Cancri), Revarf ({ Piscium) and 8 474747.^६4 (or A Aquarii). [These stars are on the ecliptic and having no latitude, are to be preferred.] Or, that any star (out of the (प्राणा ora 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. | XT. 17.] Siddhdnta-siromont. 213 14 and 15. Then look from the bottom of the 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 the 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 cHaPa 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, 1 have therefore laboured 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 Sphere. 17. I Brsxara now proceed to describe this excellent instrument, which is calculated to remove always the darkness of igno- rance, which is moreover the delight of clever astronomers ` 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 is distinguished by having a circle in its centre. I proceed to describe this instrument 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.* Semi-circle and quadrant. PHALAKA-YANTRA. Addresses to the Sun. * This verse is another instance of the double entendre, in which even the 214 Translation of the [ XI. .18. 18. Let a clever 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-REKHA. 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 are called sines. 20. At that point of the perpendicular imtersected by the 30th sine at the 380th digit, a small hole 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 80 digits: the circle will then cut the 60th sine), 60 digits forming the diameter. Now mark the circumference. of this circle with 60 GHatis and 360 degrees, each degree being subdivided into 10 paLas | 22. Let a thin २८ or index arm with a hole 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 patfixs be of half digit. Let the pattiK4 be so suspended by the pin above mentioned, that one side may coincide with -~- ~ the tamBA-REKHA. The accompanying figure +--+} will represent the form of the एवष. The rough ascensional difference in 24148 determined by the KHANDPAKAS or 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 double meaning of course does not admit of translation.— L The sines of ascensional difference for each sign of the ecliptic were found by the following proportions = ऋ ~ = XT. 26.] Siddhdnta-siromont. 215. . 23, The numbers 4, 11, 17, 18, 18, 5 multiplied severally by the AKSHA-KARNA and divided by 12, will be the KHANDAKAS or portions at the given place ; each of these being for each. 15 degrees (of Husa 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 together as many portions as correspond to the Buusa of the Sun’s longitude above found, divide by 60 and add the quo- tient to aKsHa-KARNA. Now multiply the result by 10 and divide by 4 (or multiply by 24). The quotient is here called the YasHTI in digits and the number of digits thus found is to be marked off on the arm of the एवष counting from its hole penetrated 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 in the circumference marked out by the 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 the Sun’s place, from the point at the end of the yasuHqI set off carefully above or below (parallel to the LAMBA-REKHA) on the instrument, the sine of the ascensional difference above found, setting it off above if the Sun be in the northern 1. If cosine of latitude : sine of lat. or as 12: Pacapua’ sign or 2 or 3 signs, give. : kusya’ of 1, 2 or 3 signs. 9, If cosine of declination : this result : : what will radius ;: sine of ascen- sional difference in KALAs. The arc of this will give ascensional difference. This is the plain rule: but Bua’skara cua’rya had recourse to another short rule by which the ascensional differences for 1, 2 and 3 signs, for the place in which the PALABHA’ was 1 digit, were 10, 8, 84 24148. These three multiplied by PaLaBHa’ would give the ascensional differences with tolerable accuracy for a place of any latitude not having a greater PaLaBHA’ than 8 digits. Now take these three PALATMAKAS 10, 8, 34 and multiplied by six, then the 24148 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 30 digits say, 60 > 30 As 3438: 10 x 6 = 60’: : 30 digits : aa => quantity of cHara forl 3438 i : what will sine of declination of I sign in this instrument, but instead of multiplying the 10 by 6 >€ 30 or 180 and dividing by 3438, the author taking 180 = ;'5 part of 3438, divides at once by 19.—L. W. 216 Translation of the [XT. 26. hemisphere, and below if it be in the southern hemisphere. The distance from the point where the sine which meeting the end of the sine of the ascensional difference thus set off, cuts the circle, to the lowest part of the circle will represent the GHATIS to or after midday.* * In the accompanying diagram of the PHALAKA YANTRA, 0 is the centre of the circle abc and the line o » passing through ० is called MADHYAJYA’ or middle sine. If the shadow of the pin touches the circumference in 8 when the instrument is held in the vertical circle passing through the Sun, 8 ¢ will then be the zenith distance of the Sun, From this the time to or after midday can be found in the following manner. Let a = altitude of the Sun, d = declination, A = ascensional difference, i = north latitude of the place, 2 = degrees in time to or after midday. Then, we have the equation which is common in the astronomical works, R*. sing = 1 . 80. 810 4 cos 2 = ------ 3 cos 1. cos ध R ०. sina tan /.tand + -------~ ; cos ¢ , 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 = sin A or sine of ascensional difference. R' , sin ८ ०, 008 } == — cos , 008 ध + sin, A. XI. 27.) Siddhdnta-s'iromani. 217 27. Set off the time from midday on the instrument To find the place of the counting from the LAMBA-REKHA ; from shadow of axis from time. the end of the sine of this time, set off the sine of ascensional difference in a line parallel to the _ Now, cos J a R= 12: ‰ 1. 6, AKSHAKARNA (See Chapter VII. v. 45.) h or — wun — 9 cos ९ 12 h R. sina ०० &08 ® == -- . ~ sin A 12 cos द sin a h R? =. = sin A, when y = -——. + which is called 12 cosd YASHTI and can be found as follows. ४ 12 ४ == ~ ° = ङ्ध 12 cos व 12 12 cosd R h 12 versed -) = . --- § 1 12 12 cos व When the savsa of the Sun’s longitude is 15, 30, 45, 60, 75, 90, the value of 12 versed d cos d these values are 4, 11, 17, 18, 13, 5 which are written in the text. Multiply these differences by 4 or the aKsHAKARNA, divide the products by 12 and the quotients thus found are called the kKHanpas for the given place. By assuming the sHuga of the Sun’s longitude as an argument, find the result through the KHANDAS and take r for this result, is 4,15, 32, 50, 63, 68 sixtieths respectively. The differences of r h r Then —— = --( ht+— 60 12 60 ध R # and hence, y == ~~ § ^ —) 12 60 e But in this instrument BR = 30 10 # ० youu fh + —) which exactly coincide, with the rule given in the 4 60 text for determining the YasHTI. The value of the yasHtr will certainly be more than 30, because the value of the AKSHAKABNA or ¢ is more than 12. _ Now, (see the diagram) suppose m is the end of the YAsHTI in the PATTIKA or index o m which touches the circle in 8, then, in the triangle ० m n R:om=sinmon: mn; or R: y=sine mn; y X sina ० 2४ # == ————— ; 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 be 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 paTTIKA or index is now to be placed till the yasHTI-cHINHA or point of yasuTI 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 equal to radius of a great circle, mark east 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 declination i. e. with a radius of diurnal circle, and mark this circle with 60 aHatis. Now take the YASHTI, equal to the radius (of the great circle) 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 YASHTI 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 auatixds from sun- rise, and if after midday an arc of the time to sun-set.* The Yasurt or Staff. that is, the sine of the ascensional difference is subtracted from or added to mn the distance between the end of the YasHTI and the 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 have, ९08 2 == % # कृ- 8171 4 == # 2 कूः mr, == # 7०८६ == cos c ४, ० p=ct—B. 7.] * [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 arc of the diurnal circle passing through the Sun, intercepted between the horizon and the Sun. For this reason, the arc subtended by the distance in question in this interior circle described with a radius of the diurnal circle which is equal to the cosine of the declination, will denote the time after sun-rise or to sun-set.—B. D.]} . 5 ete XI. 88.] Siddhdnta-s'iromani. 219 81. The perpendicular let fall from the point of the yasuTr To find the PaLABHA with 18 the 8 ANKU or sine of altitude: the the एक, place between the 84 प्रष्टए and centre is equivalent to pRIGYA 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^ उपयु will rest at sun-rise and sun-set, and the east and west line the PRACHYAPARA, 82 and 83. The distance between the s’anxu 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 yasHT1: observe the s’anxus of the two times and the BHusas. | Add the two suusas, 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/anxus, the result will be the pataBHi.* The difference between the east and west line and the root of 8 ANKU is called BHUJA. * [Let O be the east or west point of the horizon O a, Z the zenith, as S the diurnal circle on which 8 and $ are the Sun’s two places at different times and S mand ॐ n the 8’aNKus or the sines of altitudes of the Sun, then O m, on will be the BHUJAs, n m or ऽ p the difference between the BHUJ4S and 8 p the difference between the s’ANKUS, Q 2 220 Translation of the [ XI. 34. If the s’anxu be observed three different times by the To find patapya’, declin- YASHTI, then the time, declination &c. ध may be found (by simply observing observations by the Yasurt, of three s’ANKUS. the Sun) ए 84. First of all find three s‘anxus: draw a line from the top of the first to the top of the last ; from the top of the second 8’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. + The observer first directs a ® his staff to व, the root of. the tree: The staff XI. 46.] Siddhdnta-s‘iromani. 223 116 digits and एण्या = 87 digits. Then tell me the height of bamboo and the distance of it. As, 68 x 144 ———._ = 576 digits or 24 cubits distance to bamboo ;. 17 576 Xx 87 and = 432 height of tree above observer’s eye, 116 68 add the eye’s height, 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 length), let him observe again the top of the same object whilst sitting. 46. Then divide the two xotis 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.* d h ६ € is furnished at either end with drop lines ah, bk:bk—ah=be=sin of ८ bac. Then say ह aang eee ¢, e then observes the top of object and finds which is easy, 88 f © has been found. —L. W. | १८ > * BHASK ABA founds this rule on the following 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 the BHuJA, withthe first staff, to be 4 cubits or 96 digits: he then sits down and finds with another staff the Buusa to be 90 digits. In both cases the Kofi 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 water : but in this case the height of observer’s eye 1s to be sub- Question. Example. Observation in water. 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 KoTIs: and the two distances from the observer to the Let 2 = base, distance to bamboo. Then say x x if96:1::a2:——: then —— ~+ 72 = height of bamboo. 96 96 x x By second observation 90: 1::2:——, then —— -++ 24 = height of bam- 90 90 boo. x ‘2 x x 62 Then 72 ~+ —— = 24 ~+ ——= ; ~ ~ —— = 48, ०" - = 48 96 90 90 96 8640 ० © == 69,120 digits == 2880 cubits. 2 2 That is ——~ — —— = 72 — 24 90 96 72 — 24 or t= that is difference of observer’s height—difference of two एग toss divided by their respective BuuUsa’s.—L. W. = i ~ XI. 49.] Siddhdnta-siromani. , 225 places in the water where the top of the object is reflected, the BHUSAS. 49. 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 tell me the distance and height of bamboo, I will hold you, though appearing on Harth 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 xot1 = 3 digits and BHUJA = 4 digits. Then sitting down he makes a second observation and finds the sHuUJa = 11 digits and koti = 8 digits. His eye’s height standing = 3 cubits or 72 digits, and sitting == 1 cubit or 24 digits. Tell me height of bamboo and its distance.* Question. Example, * Let df= fc = height of bamboo = h ¢ then © a or y = height of bamboo and man’s height together. Let ¢ c = breadth of water = 2 then by first observation R वि 911 य 4 226 Translution of the [ XI. 49. A man standing up sees the shadow of a bamboo in the water—the point of the water at which Example. p ais 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.* 4y 4:3::e2:yor37=4 yore = — 3 by 2nd observation 11:8: : 2: 9 — 48 digits 11 y - 528 or 8 r= 11 y — 528 or ge = —— 8 y 11 y — 528 thas z = —— andr = -- 3 8 4४ 11 y — 528 33 y — 1584 —_—= — or4y => ———__ ~ 3 or 32 y = 33 y — 1584, or y = 1584 .°. 1584 ~ 72 = 1512 digits = 63 cubits = height of bamboo. 2nd part. To find width of water or x 4y 1584 > 4 ® ~~ स = ————. = 219 digits = 88 cubits.—L, W. 3 * Let ce = 96 digits © व = 33 ८ ८ == 172 bc = 24 let > == distance from observer to bamboo. Nowce:ac=jh:ja 72 2 3 or 96: 72 = 2 : &# = ~ == - 96 4 3४ Then a — 3 = height of bamboo Aganed:be::jh:jb 24 z or 88 : 24::2:y— 48 = —— 33 82 11 8 then सत — 1 == height of bamboo XT. 55.) Siddhanta-siromani. 227 50 and 51. Make a wheel of light wood and in its circum- १ ference put hollow spokes all having ment or SWAYANVAHA YAN- bores of the same diameter, and let eae them be placed at equal distances from each other; and letthem 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 an 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 TMua 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 come out, and then cork up the orifice left open for filling the wheel. The wheel will then revolve of itself, 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 end into a reservoir of water, let the other end remain suspended outside. Now uncork both ends. The water of the reservoir will be wholly sucked up and fall outside. 55. Now attach to the rim of the before described self- revolving whee] a number of water-pots, and place the wheel and these pots like the water-wheel so that the water from the lower end of the tube flowing into them on one side shall set the wheel in motion, impelled 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 82 32 3 8 ट — -- 1 ~~ — - 3 ०८ 2 == ~ - - <= ~ 11 4 4, 11 44 ०, @ == 44 ॐ 2 = 88 3 3 X 88 Then # = - = —— == 3 x 22 = 66, height of bamboo. 4 4 228 Translation of the ` [-1. 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 machine on account of its being dependant, because that which mani- fests an ingenious and not a rustic contrivance is said to be 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. Kind of Chapter XI. called Yanrrapuy(ya. CHAPTER 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 (branches) rais- 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 cHarTRA, 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) MALAT] (Jasminum grandiflorum) which appears all black soiled and without leaf or flower (at this season), and appears to beckon her forlorn sister to leave the grove and garden with her Spring. XII. 7.] Siddhdnta-siromant. 229 tender budding arms, agitated by the sweet breezes from the fragrant groves of the hill of Mazaya. 3. In the summer (which follows), the lovers of pleasure The auisnma or mid-sum- and their sweethearts quitting their Ree ernee 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 the glare of the fierce heat, and to disport themselves in the (well shaded) waters of its Bowris (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 mAuatf and the turbid state of every passing torrent proclaims the season of the rains and of all-powerful love to have arrived. Why, therefore, do you not have compassion on my miserable 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 he 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 and 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, several letters however are to be read differently. —L. Ww. 230 Translation of the (XII. 8. 8. The mountain burning with remorse at the guilt of The sXRatKa’La or season having received the forbidden em- of early autumn. braces of his own pusHPAvAT{ 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 fields all bedecked with the pearl-like dew, and teeming with joyous herds of plump kine, rejoice (at the grateful sight). 10. When the s/Is'ira season sets in what unspeakable S’'1s’‘rra or close of win- beauty and what sweet and endless ५ variety of red and purple does not the ‘ KacHNAR’ grove unceasingly present, when its leaf is in full bloom, and its bright glories are all expanded. 11. The rays of the Sun fall midday on the earth, hence in this 818 IRA season, they avail not utterly to driveaway the cold : HeEManvTa or early winter. नै नै नै of 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 write something calculated to please the fancy of men of literary taste. 13. Whereis the man, whose heart is not captivated by the ever sweet notes of accomplished poets, whilst they dis- course on every subject with refinement and taste? or whose heart 1s not enchanted by the blooming budding beauties of the handsome willing fair one, whilst she prattles sweetly 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 the genuine poets? or who, whilst he listens to the soft notes of the water-swans on Sweets of poetry. XIII. 2.] Siddhanta-siromant. 231 the 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 themselves in well rounded periods abounding in displays of a playful taste. End of Chapter XII. CHAPTER XIII. Containing useful questions called PRas/NADHYAYA. 1. Inasmuch as a mathematician generally fails to acquire Object of the Chapter and distinction in an assemblage of learned its praise. men, unless well practised in answer- ing questions, I shall therefore propose a few for the enter- tainment of men of ingenuity, who delight in solving all descriptions of problems. At the bare proposition of the questions, he, who fancies in his idle conceit, that he has attained the pinnacle of perfection, is often utterly discon- certed and appalled, and finds his smiling cheeks deserted of their 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, 1. 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 a few. 232 Translation of the (XIII. 3. 8. All arithmetic is nothing but the rule of proportion : Praise of ingenious एन ®nd Algebra is but another name for ^ 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 denominated PAtf canita, which has been composed in many ways by the wisest of former mathema- ticians, 18 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 essential properties. But Algebra is wholly and simply a talent and facility of invention, because the faculties of inventive genius are infinite. 6. Why, O astronomer, in finding the anarcaya, do you add saura months to the lunar months cHaITRA &c. (which may have elapsed from the commencement of the current year): and tell me also why the (fractional) remainders of ADHIMASAS 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 1st. 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 are the revolutions of the planet whose place when added to or subtracted from the remainder will give the place of Saturn ? Question 2nd. XIII. 9.] Siddhénta-siromani. 983 8 and 9. In order to work this proposition in the first place proceed with the whole numbers RULE. : of revolutions of the several planets 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 1s to be subtracted, then add the number of terrestrial days in a KALPA, or if the remainder exceed the number of terrestrial days in the Kapa, then reduce it into the remainder by dividing it by the number of days in the KALPa.* | * एप^/8 ^^ CHA BRYA himself has given the following example in his com- mentary vA/SANA-BHA SHYA Suppose Moon to have 4 revolutions in a KALPA of 60 days DUNS i662 ce uel 0 web eee esac ci Sees wees Sees Mars, ee ® @ ® ड 8 .. # @ ® @ @ @ en @ ® eevee @@ > क 200080688468 ® ८११1 0 Saturn, .... 9 reer Then 4 ॐ 1 + 3 ‰ 12 ~ 5 %€ 6 == 70 814 7 x, 3 = 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 — 2 = 9 ०४ 11 - 9 == 2 == but 11 ~+ 7 == 9 orp =9 — 11 = 60 + 9 - 11 = 58 It thus appears that the unknown planet has 2 or 58 revolutions in the KALPA Now let us see if this holds true on the 23rd day of this KALPA: revolutions signs for Moon, if60: 4 :: 28: 6 .. 12० ४18 > 1=6.. 12° Sun, 60:3 :: 23: 1 .. 24 thisx12=9.. 18, Moers, 60:5 :: 23:11 .. O thsx 66. 90 signs 10... 0 subtracted Saturn, 60: 9 23: 6 .. 12 Jupiter, 60: 7 23: 8 .. 6 thisX 3= 0O.. 18 from forp, 60: 2 23: 9 .. 6 this sub. from 2.. 18 remainder 9.. 6 corresponding with Satarn, 5 .. 12 23-4. Translation of the (XIII. 10. 10. The algebraical learned, who knowing the sum of the 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 the student who is puffed up with a conceit of his knowledge of the exact pulverizer called sam’sLIsHTA united, as the lion triumphs over the poor. Question 3rd. trembling deer he tears to pieces in play. 11. For the solution of this question, you must multiply the given number of additive months, RULE. ; ‘ : subtractive days and their remainders, by 863374491684 and divide by one less than the number of lunar days in a KALPA 1. €. by 1602998999999, the remainder will be the number of lunar days elapsed from the beginning of the kaupa. From these lunar days the terrestrial days may be readily found.* 8 @ ‘orif, 60:58 : : 23:2: 24, Then2.. 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 18 added to the 9. The reason for adding 60 is that this number 18 always be denominator of the fractional remainder in finding the place of the planets ; for the proposition. If days of KaLPA: revolutions :: given days give: here the daya of KALPA are assumed to be 60 hence 60 is added. —L. फ. * [When the additive months and subtractive days and their remainders are given to find the AHARGANA. : Let ¢ = 1602999000000 the number of lunar days in a KALPA. € = 159300000 the number of additive months in a KALPA. @ == 25082550000 the number of subtractive days in a KALPA. A = additive months elapsed. A’ = their remainder. B = subtractive days elapsed. B’ = their remainder, a = the given sum of the elapsed additive months, subtractive daye and their remainders, and > == lunar days elapsed ; K then say =. B’ 4 XITY. 12.] Siddhanta-siromani. 239 12. Given the sum of the elapsed additive months, sub- tractive days and their remainders, equal (according to BRAHMAGUPTA’S system) to 648426000171; to find the anarcaya. He who shall answer my question shall be dubbed a ^ BRAHMA-sID- DHANTA-VIT” 1. e. shall be held to have a thorough knowledge of the BRAHMA-SIDDHANTA.* Example. A’ +- B’ A’ + ए“ ० 8 ए : छ ~+ द : : = : A+B + ——— ory + १ we (€ ~+ @) 2 == ८ + A’ + ए; or (८ +d) ० - 1४ = ^“ + B’ y=A + ०० by addition, (e+ °) ४ - ( - 1) # = ^+ + 2 ~+ A't B’ =a by substitution, 26675850000 « — 1602998999999 y =a now let. 26675850000 «’ — 1602998999999 »’ = 1 then we shall have by the process of indeterminate problems = 863374491684 Again, let m =e + danda=/— 1, then mx—ny =a; (1) and ma —ny == 1 भ amz’—any =a, and mnt—mnt = 0 se (ax’—nt)—n(ay’—mt)=a which is similar to (1) ; : ० 2 == ५ 2 -- १ ६ = 863374491684 a — (1 — 1) ६. -Hence the rule in the text.—B. D.] * Solution. The given sum == 648426000171 and ४ he lunar days in a KALPA == 1602999600000 ४ 648426000171 > 863374491684 ee ~~~ == 349941932336 1602998999999 and 10300 remainder : 10300 these are lunar days elapsed. To reduce them to their equivalent in terrestrial days says 16] subtractive If lunar daysin| , Number of a _, Lunar days a- ( | days and remain- 9 KALPA ˆ tractive days ` ˆ bove found ` der amounting 267426000000. , From 10300 Lunar days subtract 161 Subtractive days remainder 10139 Terrestrial days or AHARGANA. Now to find additive months elapsed. If-lunar aa „ additive months | _lunar days] . 10 additive months and in a KALPA ˆ of KALPA ` * 10300 ` remn, 381000000000. 10 additive months = 300 lunar days 10300 — 300 = 100,00 sauRa days elapsed Hence 27 years 9 months and 10 days elapsed from the commencement of KALPA.—L. ण, 3 2 236 Translation of the [> 111. 18. 18 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‘faHrocucHas of Mercury and Venus and of Saturn according to the pHfvRIDDHIDA, including the remainder of subtractive days in finding the anarGana, abraded (reduced into remainder by division) by the number of terrestrial days (in 8 yuca). He who, well-skilled in the management of SPHUTA KUPTAKA (exact pulverizer), shall tell me the places of the planets and the anarcaya from the abraded sum just mentioned, shall be held to be like the lion which longs to make its 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 Yuaa, on being divided by 4, leaves a remainder, then the question is not to be solved. It is then called a KHiLa or an ^< impos- sible’ question. If, on dividing by 4, no remainder remain, then multiply the quotient by 293627203, and divide the product by 394479375. The number remaining will give the AHARGANA. Ifthe day of the week does not correspond with that of the question, then add this ^ प ^ 6५ प्न to the divisor (394479375) until the desired day of the week be found.* Question 4th. RULE. * [According to the DH{fVRIDDHIDA TANTRA of LaLua the terrestrial days in a YUGA = 1577917500 and the sum of all the 86 remainders for one day = 118407188600968 : this abraded by the terrestrial days in a YUGA == 259400968. Let 2 = AHARGANA then say As 1 : 259400968 : : # : 259400968 x x This abraded by 1577917500 the terrestrial days in a yuGa will be equal to 1491227500 the given abraded sum of the 36 remainders, now let y == the quotient got in abrading 259400968 2 by 1577917500, then 259400968 2 — 1577917500 y == 1491227500. It is evident from this that as the coefficients of z 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. Honce, dividing the both sides of the above question by 4, 64850242 2 — 894479375 y = 372806875 : ००००००००. (A) and let 64850242 x’ प 394479375 y — Lee 99 ०० 00 08 00 ७७०७०००७ (B) XIII. 17.) Siddhanta-siromani. 237 16. Tell me, my friend, what is the aAHARGANA when on a Thursday, Monday or Tuesday, the 30 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'faHrocucHas of Mercury and Venus) together with the remainder of the subtractive days according to the pufvriD- DHIDA, give, when abraded by the number of terrestrial days in a YuGA, a remainder of 1491227500.* 17. The place of the Moon is of such an amount, Question 5th. that The minutes 2 the minutes — seconds + 3 = degrees the degrees ० == signs. Example. -+ 10 = the seconds ध == 293627208 by the processes of indeterminate problems. Now let a = 64850242, 6 = 394479375, and c = 372806875 we have the equations (A) and (B) in the forms axz—by=e and az—by=1 : & == ८ 2“ — bt (866 the preceding note) == 293627203 c — 394479375 ८ as stated in the text.—B. D.] # Solution. The given sum of the 36 remainders in a ruga = 1491227500 according to the DH{VRIDDHIDA TANTRA ,, 1491227500 ~ 4 = 372806875 372808875 x 293627203 and .°. = 297495471 and remainder 10000 i. e, 394479375 AHARGANA. == 1428 ~ 4 remainder, i. e. 10000 = anaraaya on a Tuesday, for the yuGa commenced on Friday This would be the aHaRGANA on a Tuesday. To find the aHARGANA on Monday, it would be necessary to add the reduced terrestrial days in a yu@a to this 10000, till the remainder when divided by Tw 10000 -++ 894479375 X 2 788968750 -~---------~ ~~ —— == 112909821 — 3 remainder = 7 | Monda 10000 + 3944793875 X 3 1183448125 and —_——- <= प ~~ = 169064017 — 6 remainder or = ण ॥ Thursday.—L. W. 238 Translation of the [XIII. 18. And the signs, degrees, minutes and seconds together equal to 130. On the supposition that the sum of these four quantities is of this amount on a Monday then tell me, if you are expert 17 rules of Arithmetic and Algebra, when it will be of the same amount on a Friday.* 18. Reduce the signs, degrees and minutes to seconds, adding the seconds, then reducing the terrestrial days and the planet’s re- volutions in a KALPA to their lowest terms, multiply the seconds of the planet (such as the Moon) by the terrestrial days (reduced) and divide by the number of seconds in 12 signs: then omitting the remainder, take the quotient and add 1 to it, the sum will be the remainder of the BHAGANAS revolutions.t RULE. # Let z = minutes ~+ 20 then == seconds x ~+ 20 % ~ ——— ++ 3 = degrees. x and ¢ ~ ०० & = 58 minutes. 58 ~+ 22 2 58 — 39 ~+ 3 = 22 degrees. 22 — = 11 signs. 2 = 39 seconds, Hence the Moon’s place = 11s. .. 220 .. 58’ .. 39”. + The mean place of the Moon = 11s, .. 220 .. 58’ .. 39“ = 1270719” The number of seconds in 12 signs = 1296000. Terrestrial days in a KALPA = canes | These divided by a 1650000 become DRI- Revolutions of Moon == 57753300000 DHA or reduced. 35002. XIII. 20.] Siddhanta-siromani. 239 19. The remainder before omitted subtracted from the divisor will give the remainder of seconds: if that remainder of the seconds is greater than the terrestrial days in a 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 seconds the aHaRGANA may be found (by the KUPTAKA pulverizer as given in the LfLAvSti and Bfya-aanita) Or, 20. That number is the number of anarcana by which the reduced number of revolutions multiplied, diminished by the remainder of the revolutions and divided by the reduced number of terrestrial days in the Kapa, will bear no remainder. The reduced number of terrestrial days in a KALPA 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 rule was As the Terrestrial Revolutions in a Given days or + daysinakaLpa. | ` KALPA. he: 1 | : Revolutions. If any remainder existed, it, when multiplied by the number of seconds in 12 signs and divided by KaLPA, terrestrial days gave the Moon’s mean place in seconds. We now wish to find the BHAGANA-S’ESHA or the remainder of revo- lutions, from the Moon’s given place in seconds: we must therefore reverse the operation Moon’s place in seconds X KALPa terrestrial days ०" ~~~ -——_ = BHAGANA-S ESHA. seconds in 12 signs The terrestrial days, however, to be used, must to be reduced to the lowes} terms to which it, in conjunction with the KALPA-BHAGANA4Ss or revolutions in a KALPA can be reduced: the lowest terms as above stated were of the terrestrial days == 956313, of the Moon’s KaLPA-BHAGANAS == 35002. 1270719 >€ 656313 1215205099047 ——_—— ------- = ———- == 937658 quotient — remainder 1296000 1296000 331047, 937658 quotient 1 adding one gives 937659 for the BHAGANA-8 ESHA. The reason for adding one is, that we have got a remainder of 331047, which we 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 — 331047 = 964953. This remainder 964953 being greater than the terrestrial days reduced to lowest terms, viz. 956313, the question does not admit of being solyed.—L. W. 240 Translation of the [XIIT. 21. 21. Ifthe Moon’s pHacaya-s’/EsHA or the remainder after finding the complete revolutions admits of being divided by 1650000, without leaving any remainder, the question may then be solved: the reduced BHAGANA-8 ESHA on being multiplied by 886834 and divided by 951363, then the remainder will give the 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-rise, equal to 0 signs, 5 degrees, 86 minutes and 19 seconds. 23. When will the square of the apHimAsa-s/EsHA remainder of the additive months, multiplied by 10 and the product increased by one, be a square : or when will the square of the aADHIMASA-8 ESHA decreased by one and the remainder divided by 10 be a square? The man who shall tell me at what period of the Katpra this Impossible question. Question 6th. * [To find the aHaRGaNA from the Moon’s BHAGAN 4-8 ESHA. Let R = BHAGANA-S8’ESHA, T = 1577916450000 terrestrial days in a KALPA, M = 57753300000 the Moon’s revolutions in a KALPA, 2 == AHARGANA. R R Then, as T: M :: x: revolutions + — ory ~+ —: T T a Mr—Ty=R: 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 > — 956313 y = R’ or M’r —T’ y= RB Now let ^“ —T’ & = 1 : or 35002 2’ — 956313 y’ = 1 : hence we have x’ = 886834. ; and ® = R’ x’ — T ८ (see the note on the verse 11th) = 886834 R' — 956313 ¢. Hence the rule in the text And, as the reduced BHAGANAS’ ESHA == 937659 (see the preceding note) hence 937659 X 886834 == 831547881606 This divided by 956313 will give as quotient 869555 (i.e. ¢) leaving a re- mainder of 257151 which should be the ^+ घ ^ ८७८ ^ , but as the BHAGANAS’ESHA 1. 6. 937659 does not admit of being divided by 1650000 (the numbers by which the terrestrial days were reduced) it ought to have been KHILA or insoluble question: but BHAskaRACHAnya here still stated this number to be the true AHARGANA,—B. D.j XIII. 24.] Siddhdnta-s’iromani. 241 will take place—will be humbly saluted even by the wise, who generally speaking, gaze about in utter amazement and confu- sion at such questions, like the bee that wanders in the bound- less expanse of heaven without place of rest. 24. (In working questions of KUFTAKA pulverizer, the aug- Remark on the preceding ment must be reduced by the same question. number by which the pudsya dividend and Haga divisor are reduced to their lowest terms, and when the augment is not reducible by the same number as the BHAJYA and HARA, the question is always insoluble.) But here, in working questions of KuTTaKA, those acquainted with the subject should know that the given augment is not to be reduced, i. e. it belongs to the reduced Budsya and Hara, otherwise in some places the desired answer will not be obtained, or in others the question will be impossible.* ° * [The questions in the 23rd verse are the questions of the VARGA-PRAKRITI or the affected square, i. 6. questions of indeterminate problems of the second degree. Ist question. Let a = the aDHIMASA-8'ESHA : then by question 10 2? + 1 = y?. In such questions the coefficient of x is called PRaxkRITI, the value of x Ka- NISHTHA, that of the augment KsH=uPa and that of ¢ गरड वप्र. Now assume y = max +1, then 10 ४: + 1 = (max + 1), == # 2? + 2 + 1; 2 10 — m? Hence the rule given by BHASKARACHARYA in his Algebra Ch. ए.) verse VI., for finding the KANISHTHA where the KsHEPA is 1, is “ Multiply any assumed number by 2 and divide by the difference between the square of the number and the PRAKRITI, the quotient will be the KANISHTHA where the KSHEPA is 1.” 2m 2x3 Now assume m == 3, then > = ———— = —6: 10— mm 10—9 and .* Y= 4^10 ० + 1= /361 = 19: ADHIMAIA-S ESHA == 6. From two sets, whether identical or otherwise, of KANISHTHA, JYESHTHA and KSHEPA belonging to the same PRAKRITI, all others can be derived such as follows. Let @ == PRAERITI, and १५२२4 ० + the two sets of KANISHTHA, JYESHTHA and KSHEPA, then 2 ) &2 ) a ति Dae we have Gai 4 ०3 = 9; ack + ba ==: b, == +: == 4 x? 9 9 = ४६ — ५ ८६ : T 242 Translation of the (XTIT. 25. 25. Tell me, O you competent in the spheric, considermg it frequently in your mind for awhile, what is the latitude of the city (A) which is situated at a distance of 90° from एग राढ, and bears Question. and .°. b, xb, = (४१ — a 22) (४६ —a 23), == ५१४६-० > ys —ariys + ०22; ^ ०2४९ + ०2 yt +b, bo =+ ys + ० 32: adding -+-* @ 2} 29 &\ Yq to both sides a 2१५६-० 42४, ४० + Ore yi 61 09 ~--५१9६ +° ०, 23994 a* x} x3. or @ (८, Yo + 2291) ° + 5, bg = ($, Yo £92, 23) >: thus we get a new set of KaNIFTHA, JYESHTHA and KSHEPA: i.e. new KANISHTHA 2, #9 £259; 3 new JYESHTHA == Y, ¥g ५ 1 2; and new KSHEPA — ०, ¢: Hence the Rule called BHAVANA given by BHASKARACHARYA in his Algebra Ch VI. verses III. & 1V. Now in the present question 2) == 6, ४, == 19 84९, == 1, and also 22 = 6,4, = 19 andd,=1: ०, new KANISHTHA — 6 >< 721 + 228 x 19 - 4326 +4 4332 == 8658 ; new JYESHTHA = 721 K 19 + 10 X 6 >< 228 = 13699 ~+ 13680 = 17379 ; and new KSHREPA = 1 ॐ 1 =}. Thus 2 = 8658 &c., according to the Bhavana aasumed. The second question is z?—1 == $° 9 10 or ॐ = 10 # +1, Here then we have an equation similar to the former one, but z? is now be in the place of y* and y? in the place of =, ae zx will be=— 19, or = 721 &e. Now given ADHIMASA-SESHA 88 found by the first case == 6. The proportion by which this remainder was got, was if KALPA SAUKA days : KALPA-ADHIMABAS : : 2 or elapsed SAURA days 6 : # + - F KALPA SAURA days ० KALPA-ADHIMASAS € 2 == KALPA SAURA days X y + 6 KALPA-ADHIMASAS > ॐ — 6 or y=. KALPA SAURA days From this we get a new question: ^ What are the integer values of 2 and y in this equation ?” which question is one of the questions of KUTTAKa and in which the coefficient of the unknown quantity in the numerator is called BHAJYA or dividend, the denominator HaRa or divisor and the augment KSHEPA. It is clear that in this equation, if the augment be not divisible by the same number as the dividend and divisor, the values of z 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 taken, reduced to their lowest terms, otherwise the question will be insoluble. As in the present question, if the dividend KaLPA-aDHIMasaS and the divisor KALPA 84 ए४८ days be taken not reduced to their lowest terms, 1, e, not divided by XIII. 25.] Siddhdnta-s'iromani. 243 due east from that city (ussayinf) ? What is the latitude 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. EK. from (B): and of the place (D) which is situated at a distance of 90° from (C) and bears S. W. from (C) ?* the number 300000, the question will be an impossible one, because the augment 6 is not divisible by the same number. For this reason the dividend and divisor must be taken here reduced to their lowest terms. 1593300000 —— = 5311; and Hence, dividend = reduced KAaLPA-ADHIMASAS = 1555200000000 divisor = reduced KALPA 84724 days == ———~-—~—— = 5184000. 300000 6311 « — 6 4 ० By substitution, 9 = - 6 5184000 which gives ॐ = 826746 the elapsed saURA days or 2276 years 6 months and 6 days.—B. D.] * Let a = the azimuth degrees, == the distance in degrees between the two cities, p = PALABHA’ at the given city, ¢ = aKSHA-KARNA, and x = the latitude of the other city. 810 d KX cos a cos d Xp 12 Then sn a= ( = ----) > = ad 12 k Now in the 18६ question, a = 90°, d = 90°, p = 5 digits, the paLaBHa’ at UssaYinf, ००१ ‰ == ^ /19? + 5? = 13 8438 ॐ 0 OX5 12 न (न Pd 8438 12 13 = (0 + 0) X 4#=—0 . = == 0 = latitude of (A) or of र ^ प्राण (2). In the second question, a = 90°, द = 90°, p = 0 digits at २५४५ ०४, and .*. ¢ = 12 34388 >< 0 ०0 12 ०० 81 = ( ) x — 3438 12 12 = (0+ 0) ‰$ = 0 ॐ == 0 Latitude of city (B) or Lanka. (3). In the 3rd question, 6 = 45°, द = 90°, p = 0 at LANKA and ¢ = 12: 3488 K 2431 0 9 0 12 „० memo ——-— +--—— 8438 12 12 = (2431 + 0) > 1 = 2481: T 2 244 Translation of the [XIIT. 26. 26 and 27. Convert the distance of yosanas (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 the azimuth of the other city and paLaBHA 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 the reverse 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 axksHaxKarya will give the sine of the latitude sought.* RULE. . aw == 45° Latitude of city (C). (4). In the 4th question, a= 45°, d= 90°, p= 12 at C and ..k= 12 42 ; 3488 K 3475 / 2 0 % 12 12 ee sin 2 = eee eee A eee x = 3 3438 12 ) 124. 2 3488 1 8438 । <~ 8 ~~ 2 ^~0 = = ~ 5 2 Vv )* 2 ` ॐ == 30° Latitude of D.—L. W. * (Let Z be the Zenith of the given city bearing a north latitude, Z HN © the Meridian, G A H the 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 pass- ing through 8. Then the arc 2 9 (which is equal to the distance in degrees between the two cities) willG be the Zenith distance of 8 ; the - arc H G, the arc containing the given azimuth degrees, and 84 which is equal to the declination of the point S, the latitude of the other city which can be found as follows. Let ० =H g the given azimath degrees, d= ZS the distance in degrees between the two cities, p = PALABHA, k = AKSHA-KARN 4 XIII. 28.] Siddhénta-s'tromani. 245 28. Tell. me quickly, O Astronomer, what is the latitude of a place (A) which is distant ई of the earth’s circumference from the city of एप 474 and bears 90° due east from 10 7 What also is the latitude of a place distant 60° from puri, but bearing 45° N. HE. from it? What also is the latitude of a place distant 60° from 7 प्र 484 and bears 9, HK. from it? What also are the latitudes of three places 120° from DHARA and bearing respectively due east, N. E., and 9. E. from it ?* Question. and «= Sh the declination of the point § i. e. the latitude of the other city Then say, As sine Zg: sine Ag:: sine Z 8: the BHuJA 1. €. the sine of distance from S to the Prime Vertical. or R:cosa::sind: BHUJA cos a sin d ’. BHUJA == - ° R And by similar latitudinal triangles, 12 : 2 : : cos d: 8’ANKUTALA, pKecosd ०, SANKUTALA = —————. 12 Now when the other city is north of east of the given city, it is evident that the BHUJA will be north and consequently the sine of amplitude == BHUJA ~+ 8’ANKUTALA but when the other city is south, the BHUJA also will be south and then, the sine of amplitude = BHUJA ^~ S’ANKUTALA, cosaXsind poosd रकया ॥ R 12 or the sine of amplitude = And by latitudinal triangles k:12:: sine of amplitude : sine of declination i. © sin @ ९08 @ ॐ 8111 @ p x cos -) — विक 12 + 12 X sine of amplitude ( R 12 ००, 811) ८ = म emnteeia a k ॐ k hence the rule in the text If the distance in degrees between the two cities be more than 90°, the point 8 will then lie below the Horizon, and consequently the direction of the BHUJA will be changed. Therefore the reverse of the signs + will take place in that case.—B D.] pee sc. R 12 k (1.) In the first question, a = 90°, d= 60°, p = 5 digits the paLaBHa of DHARA and .. ¢ == 13 2977 ॐ 01719 ॐ 5 12 ४ == ( ) >~ 3438 12 1719x 5 12 8595 9 = x - = - = 662 —: 12 13 13 13 811) @ ॐ 008 छ ९08 ८ >< 12 * Here also sin > = + ———*) 246 Translation of the [XITI. 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 NAKSHATRA (i. ©. having the longitude 2 signs 13° 20") rises in the north-east point.* Question. 2 = 11°..15’..1” Latitude of city due east from DHARA. (2). Inthe 2nd equation, a = 45°, d= 60°, p=5&«. K=18: 2977 K 2481 1719 >< -)} 12 8 —— ॥ >€ -—— >< 3438 12 19399109 1913 = 0 —— ‡ in 2 $ 13 ` 7449 7 ०, 2 = 49० . 18... 24” Latitude of city bearing 45° N. E. from pHaRa. (3). In the 8rd question, a = 45°, d = 60°, p = 5 and ¢ = 18. 2977 < 2481 17195 12 १५ ५०= ( eer» 9 ) न 13 9549289 7070 = ——- = 1281 —. 7449 7449 ० ॐ == 219... 54/..34” Latitude of city bearing the S. E. from DHaBa. (4). To find latitude of place 120° from pHaRa and due east. Here, sin d = sin 120° = sin 60° = 2977, cos d= cos 1200 = — sin 30° = — 1719 cos a=0,p=5 ४104 ¢ 13 : 2977 0 1719 + 8 12 sin = ( +——) x=; 3438 12 13 9 = 662 —: 13 ० ॐ or latitude == 11°..15’..1% | The latitudes of the places 120° bearing N. E. & 8, E., will be the same as the latitudes of those places distant 60° and bearing S. E& N. ४. Hence the latitudes are 21° . 54. . 84“ and 49° 18’ 24’.—L. W. * Ansr. Sun’s amplitude = sine of 45° = 2431’, the sine of longitude of middle uf anpra = sine of 2 signs 13° 20’ = sin 73° 20" == 3292’..6” 40’ and the sine of the Sun’s greatest declination = sin 24° = 1397’. Then say: As Rad: sin 24० : : sin (78° 20’): sine of declination, and as sine of amplitude : sine of declination : : Rad : cos of latitude, + sine of amplitude : sin 24० : : sin (73° 20’) : cos of latitude. sin 24° >< sin (73०., 20/) 1897! x (3292’.. 6“. . 40’”’) °, cos of latitude = - ~~~ - ~ = - ~ -~-- sine of amplitude 2431/ = 1891’ 50’ 48” = sine of 33° 23’ 37” : whence latitude will be 56° 36’ 28’ .*. sine of latitude = 2870’ 13. . Then say: As cos of latitude: sine of latitude: : Gnomon : equinoctial shadow 1891’..51” : 2870' 13” : : 12 12 >< (2870'.. 13”) 13 *. equinoctial shadow = = 18 — digits.—L. W. 60 1891’. 61“ XIII, 32.] Siddhdnta-siromani. 247 30. Tell me the several latitudes in which the Sun remains above the horizon for one, two, three, tion. | ee four, five and six months before he sets again.* 81. 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 declination, the sine of decli- nation, and the sine of the Sun’s longitude: equal to 5000 is (the radius is 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 337, the quotient is to be substracted from 910678. Take the square-root of the remainder. That root must then be subtracted from the apyAa above found: the remainder will be the declination, when the radius is equal to 3438. From the declination the Sun’s longitude may be found.t | Question. RULE. * Ansr, When the Sun has northern declination he remains above the horizon for one month in 67° N. 1, two months in 69° three months 73° four months 78° five months 84° six months 90° These are roughly wrought: for BaoasKaRACHARYa«’s rule for finding these Latitudes see the TRIPRAS’NADHYAYAS of the @OLADHYAYA and also the GANITA- DHYaya.—L. W + [Let a = the given sum p = the sine of the Sun’s extreme declination az = the sine of the Sun’s declination Then the cosine of declination will be ९ 1६*--४* and the sine of the Sun’s Rx longitude == ——: P Re ० by question R?—a? + 7+ — =a: / अ or 2 / Ra? + (R + p) c= ap, md p 4८ ap— Rta: R? 0 — p? x = a? p®? — 2ap (BR + 2) x + (R? + 2 Rp + >) 2’; (R? + 2 Rp + 2p’) x? —2ap (BR + 2) > = — (a? — RB?) p 248 Translution of the (XIII. 33. 33. Given the sum of the sines of the declination and of the altitude of the Sun when in the prime vertical ; the TADDHRITI, the kusyf and sine of amplitude equal to 9500, at a place where the paLaBHA Question. 2a p (R + p) (० — ए) 2 or ८ ~ ----“ = — —_—_-—_——_ ;sy R* + 2Rp + 2p? R? + 2Rp + 2 p? 2ap (ॐ + 2) a” p? (R + 2) > completing the square, 2? — ————- -————_ « + ——_—— pa एः 12.88 4 22 (RP? 4+2R p+ 2.29) * ८०2 (R +p)? (a? — R?) p — ete aoe Ome चवि e (R?4+2Rp+2p?)*? 7२ + 2 8 + 22 R‘ p* + 2 R* 28 + 2 R* p* — ८ p (° + 2 2 + 2 p’)* R? p? a? ५ ` उ+ 20128 (RF IRD + 28० ` ap (BR ~+ 2) J a? p tc ee en र ° + 2 5.2 + 22 R?42Rp42p? (R?’+2Rp+2p%)? ap (R + 2) ane or > == ——— + ,/ Dt a । RT ee Tee 73 + 224 28 (R?4+2Rp+2p*)* Now here R = 8438 and 2 = 1397, ap(R-+ p) ap (R + 2) ५ >< 1897 > 4835 6734495 a ` ` ° + 2 28 + 28* (B+ 2) 2 + 22 (4835)? ~+ (1397)? 25328834 4 — ८ nearly = ADYA; 15 ५० + 3808777688881 a? 2 a® --- ----- = — = — nearly; (® + 8 ‰ # ~ 222) 2 641549831799556 337 R? p 3967713928996 and ----- ———— = 910729, in place of this the Au- 7 + 27 + 22 25328884 thor has taken the number 910678 but of these, the positive value is excluded by the nature of the case, because the sine of declination is always less than 1397. Hence the Rule in the text. Solution. The given sum = 6000, 5000 x 4 ० ADYA == ————_ = 1333’ 20” and 52, a? = 148367! 57" 9.“ 15 ^. sine of declination = 1333’ 20’/ — 4910678 — 1488671 677 9” = 1888. 20” — 873/ 6/ 13/4 = 460’ 13” 474 : from which we have the longitude of he Sun = 08, 19. .. 14/ 36” or 6. . 10". . 45/ 24” or 6 19¢ 14/. . 36/ or 119, 10. . 45/,. 24”.—B. D.] XIII. 35.] Siddhdnta-s’iromant. 249 or equinoctial shadow is 5 digits, tell me then, my clever 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 paLaBHdé: and then find the amounts of the remain- ing quantities upon this supposition. Then these on the sup- position made, multiplied severally by the given sum and divided by their sum on the 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 एयर (where the PALABHA is 5 digits) when their sum is 2000.+ RULE. Question. # Solution. Here panaBah = 5 digits ० Suppose the sine of declination = 5 K 12 = 60: and then say. 1 241. 84 : aKSHAKABNA: : sine of decln.: SAMA 8'ANKU 13 > 60 - or 5 : 18 : : 60 : 847 8’ANKU —- = 156) 5 156 > 13 Gnomon: AKSHAKAERNA : : SAMA 8’ANKU : TADDHRITI = —————- = 169 , 12 60 x 5 ° 12 : PALABHA’ : : sine of वन्न. : KUJYA = = 25, 12 60 24 13 and 12: aAKSsHAKARYA:: sine of decln.: sine of amplitude ==. =65. 12 .. If the sum: sine of decln. supposed : : given sum: sine of decln, required. or 475 : 60 : : 9500 : 1200. If 475 : 156 :: 9500 : 3120 8५74 8’ANKU required. and so on 3380 TADDHRITI 500 KUJYA 1300 sine of amplitude. ; Ansr, L. W. + Solution. Here also paLABHA = 5, | then suppose sine of declination as before == 60, and .°, sine of amplitude = 65 , and एए द्व -29. the sum = 150, 250 Translation of the — [र 111. 86. 36. But dropping for a moment those questions of the SIDDHANTAS involving a knowledge of the doctrine of the sphere, tell me, my learned friend, why in finding the point of the ecliptic rising above the horizon at any given time, (that is the LAGNA or horoscope of that time,) you first calculate the Sun’s apparent or true place for that time, 1. e. the Sun’s instantaneous place: and further tell me, when the Sun’s savana day, 1. e. terrestrial day, consists of 60 sidereal auatikd4s and 10 pauas, the LAGNA calculated for a whole terrestrial day should be in advance of the Sun’s instantaneous place, and the Lacna calculated for the time equal to the terrestrial day minus 10 Pauas should be equal to the Sun’s instantaneous place. 87. Are the auaqtikds used in finding the LAGNA, GHATIKAS of sidereal or common sMvana time? If they are sivANa cuaTik4s, then tell me why are the hours taken by the several signs of the ecliptic in rising, 1. €. the Rds‘yuDAYA which are sidereal, subtracted from them, being of a different denomina- tion? If on the other hand you say they are sidereal, then 1 ask why, in calculating the taana for a period equal toa whole sMvana day i. e. 60 sidereal cHaTIKAS and 10 ९.1.48, the LAGNA does not correspond with, but is somewhat in advance of, the Sun’s instantaneous place; and then why the Sun’s instantaneous place is used in finding the Laana or horoscope.* 88. Given the length of the shadow of gnomon at 10 aarfs after sun-rise equal to 9 digits at a place where the PaLaBHA in 5 digits; tell me what is the longitude of the Sun, if you are au fait in solving questions involving a knowledge of the sphere.f Questions, Question. Then say as before as 150 : 60 :: 2000 : 800 sine of declination, as 150 : 65 :: 2000 : 866% sine of amplitude, as 150 : 25 :: 2000 : 3334 xusya.—L, W. ध * [For answers to these questions see the note on the 27th verse of the 7th h.—B. D. + [For solving this question, it is necessary to define some lines drawn in the Armillury sphere and shew some of their relations. XITI. 39.) Siddhdnta-s’iromani. 251 39. Tell me, O Astronomer, what is the PALABHA at that place where the gnomon’s shadow fall- uestion, : ‘ = ing due west is equal to the gnomon’s Let BODE be meridian of the given place, C A E the diameter of tha Horizon, B the Zenith, P and Q tne north and south poles, B A D the diameter of the Prime Vertical, ए A G that of the Equinoctial, 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, H » perpendiculars toC E, Then B F or E P = the latitude of the place, A f = thie sine of the Sun’s declination, A ¢ == aGRa or the sine of amplitude, Sg = kusya’. (It 18 called cpapasya’ or sine of the ascensional difference when reduced to the radius of a great circle.) J s= kata’, (It is called sérka when reduced to the radius of a great circle.) $9 = ISHTA HRITI. (It 18 called TADDHRITI when 5 is at e, HRITI when 8 is at H and kuJya when s is at 7.) The 18HTA HRITI reduced to the radius of a great circle 18 called IsHta ANTYa’, but s coincides with H, it is called antya’ only. It is evident from the figure above described that (1) ISHTA HRITI = Kaa’ + Kusyas’, (2) isHTa aNTYA’ = SOTRA + CHARAJYA’, (3) HRITI = DyvuJyYa’ or cosine of declination ॐ एवाग्र ^ ^ (4) Anrya’ = radius + cHaRasya’. Here the positive or negative sign is to be taken according as the Sun is in the northern or southern hemisphere. vu 2 252 Translation of the [XITT. 39. height when the Sun is in the middle of the sign Leo, i. e. 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 culled) is the UNNATA KALA (or elevated time). Subtract from or add to the UNNATA ए ^^. the ascensional difference according as the Sun is in the northern or southern hemisphere: reduce the remainder to degrees: the sine of the degrees is 8८784. -The sGTRA multiplied by the cosine of declination and divided by the radius gives the Kata’. Then from the above formule we can easily find the IsHTa HRITI and others, Now to find the answer to the present question. Square the length of the Gnomonic shadow and add it to the square of the Gnomon or 144: and square-root of the sum is called the hypothenuse of the shadow. From this hypothenuse find the Mana’s’ANKU or the sine of the Sun’s altitude by the following proportion. As the hypothenuse of the shadow : Gnomon or 12 :: Radius : The Manas’anxv or the sine of the Sun’s altitude. Then by similar latitudinal triangles, as the Gnomon of 12 digits 9 : AKSHA KARNA found from given PALABHA’ 2: MAHAS‘ANKU : ISHTA HRITI (see verses from 45 to 49 of the 7th Chapter). Reduce the given UNNATA Ka’LA to degrees and assume the sine of the degrees as IsHPFANTYA (for this will always be very near to the IsHTa’NTya). Then cosine of declination = IsHTA HRITI ˆ ` Radius ISHTA’NTYA From this the cosine of declination will nearly be found, and thence the declination and ascensional difference can also be found. From the ascensional difference, just found, find the IsHTa’Ntya’ of two kinds, one when the Sun is supposed to be in the northern hemisphere and the other when the Sun is supposed to be in the southern hemisphere. Of these two IsHTanrTya’s that is nearly true which is nearer to the rough 1sHTa’NTY4’ first assumed (i. e. the sine of the UNNaTA Ka’LA). From this new IsHTa’NTyYa’ find again the declination and repeat the process until the roughness of declination vanishes. From the declination, last found, the longitude of the Sun can be found.—B. D.] * The hypothenuse of the shadow is first to be found, ‘Then say As hypothenuse of the shadow : Gnomon :: Rad : the MaHa’ 8’ANKU or the sine of the Sun’s altitude. Here we shall find sine of 45°. This is the sama 8’ANKU. It is 2431’ signs Sine of declination of the Sun when in 4 .. 150 == 987’ 48” .. 2487: } ° — 9877 .. 48”) *== (TADDHRITI — एएरर^ ^): or 5909761 — 975749 .. 9 == 4934011 51. „^ TADDHRITI — KusYA’ = 4/4934011 .. 51 = 2221“ .. 15” Here we have 3 sides of the latitudinal triangle consisting 84244 8’ANKU, declination and TADDHEITI — KuJya’. Hence we may find the latitude. Then by similar latitudinal triangles As TADDHRITI — KUJYA’ 2221’. 15’ : sine of declination 987’ .. 48” :: Gnomon 19 : PALABHA’ 5} digits.—L. W. ९ 111. 40.) Siddhdnta-s‘iromani. 203 40, When the Sun enters the prime vertical of a person at ussayInf either at 5 auatis after sun-rise or 5 GHaTIs before or after midday, what are his declinations? If you will answer me this ब will hold you to be the sharp anxus'a (goad) for the guidance of the intoxicated elephants, the proud astronomers.* Question. * First of all assume H N the rap- DHRITI = sine of the given elevated time that is=sin 30 From this find the s’aNkKU or the sine of altitude by similar triangles, । If akKsHa KaRya or hypothenuse of equinoctial shadow. : Gnomon 12 : : TADDHRITI 12 KX TADDHRITI : SAMA 8S ANEU == ——- = 13 ON From 0 पि, to find 0 B the sine of declination say PaLaBHa’ KON ध 88 AKSHA KARBNA: PALABHA’:: ON: 0 B = ————-——— = sine of de- 13 clination. From O B we may now find the longitude of the Sun and O D the ascen- sional difference: Now deduct this ascensional difference from the sine of elevated time converted into degrees. Hence 0 7 - 0 7 = © 0. Now reduce © 0 to terms of a small circle on the supposition that the Sun has the declination now found. As Rad: © © : ; cosine of declination: N B. Now find also B A by the same proportion. Then N 8 + B A =N H’a new value of TADDHRITI. IfH N: gaveO B:: HN’: 0 B’ corrected value of O B. Hence a corrected longitude of the Sun. The operation to be repeated till rightness is found. 2nd.—To find the declination from the NaTA Ka’LA or time from noon = 8111 30°. Let a = the sine of Nata KA’LA: R? — a? = 80774», and x = the sine of declination : R? — 2? = cos? of declination. The sGTRa reduced to value of diurnal circle will give KaLa’ The proportion is, As R: 80784 : : cos of declination : Kaba’, but I do not know what cos of declination is but only ite square. I must therefore make this proportion in squares (R? — a*) (R? — 2%) As R?: 87749 : ; cos? of declination: Kata) = —— ह Now by similar latitudinal triangles As }2) 7: PanapHa’) 7:! Kava ) *? sine® of declination PALABHA ) ` 25 Pl — a’) (R? — >>) ae Bg ha en ene १ ~~ ---------- ०० Since? of declination = | 8 x KALA | 144 R? = 2 254 Translation of the [XTII. 41. 41. Ina place of which the latitude is unknown and on a day which is unknown, the Sun was observed, on entering the prime ver- tical, to give 9 shadow of 16 digits from a gnomon (12 digits long) at 8 cHatik4s after sun-rise. If you will tell me the declination of the Sun, and the patasgf 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- ledge of the latitudinal figures, the PALABHA and the longitude of the Sun Question. Question. Now R? —a? = 8864883 25 (R? — a?) = 25 x 8864883 = 221622075 and 144 R? = 144 (3438)? = 1702057536 221622075 (R? — 2) bd मीर oS 2? 1702057536 1702057536 R? — 2? = ——— — 2? = 73 2* nearly 221622075 3 26 and 2 = 4/1363828 = 1167’ = sine of 19°... 61’ Hence the Sun’s place may be found.—L. W. * To find the sine of altitude or Maa’ 8८ प्राता (16)? + (12)? = (20)* ,,, hypothenuse of the shadow = 20, Then say As 20: 12:: 8438’ : 2062’.. 48” == the ऋ ^ प्र ^ ˆ 8’ANKU. Now suppose the sine of UNNATA Ka’La or 8 GHATIKa’S to be the TADDHRITI == 2655. Then by similar triangles 2655’ X 12 2062’ .. 48“ : 2655’: : 12: axsHa KARNA == ———--—— 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, &९. The UNNATA KALA diminished by the ascensional difference gives the time from 6 o’clock : the sine of this time will be the 8८784 and hence the Kata : thence (Kusya’ being added) the TADDHRITI : and thence the AK8SHA KARNA and declination. The ope eration to be repeated till the error of the original assump- tion vanishes.—L. W. 111. 43. ] Siddhdnta-s'iromani. 299 at that place, where (at a certain time) the KusYA 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 TrappHRITI decreased by the amount of the KUJYA equal to 6720, and given also the sum of the xusyx, the sines of amplitude and declination (at the same time) equal to 1960. Iwill hold him, who can tell me the longitude of the Sun and also paLaBHf 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. # Ansr. Let x = the PALABHA 2940 then say. -#8 2 : 19 : : 245: sine of declination == ——. x Now find the TADDHRITI minus KUIJYA’. 2940 35280 As ॐ : 12: : —— : TADDHRITI ~ KUJYA == -——., x 29 But TADDHRITI — KUJYA = 3125 — 245 == 2880. 35280 35280 49 ,, 2880 = ——- and 22 = ——-- = — x? 2880 4; ० ©== {= 34 PALABHA. To find declination say As 34 : 12:: 245: 840 sine of declination. Hence the longitude of the Sun may be discovered as before.—L. W. + This question admits of a ready solution in consequence of its peculiarities, The sine of declination SAMA anf = 6720 and TADDHRITI — KUJYA are all three respectively perpendiculars in the three latitudinal triangles, And the KUJYa the sine of amplitude > == 1960 and the TADDHRITI — KUJYA are bases in the same 3 triangles. Hence we may take the sum of the 3 perpendiculars and also the sum of the three bases and use them to find the PaLaBHA. As the sum of the 2 sum of the 3 bases Gnomon PALABHA 3 1 in the same triangles 1960 x 12 6720 : 1960 :: 12 ; ———-—= 3}. 6720 Now the KusYA, sine of amplitude and sine of declination are the three sides’ of a latitudinal triangle. These three I may compare with the threo Gnomon, PALABHA and AKSHA KARNA to find the value of any one. 256 | Translation of the [शाा. 44. 44, Given the sum of the sine of declination, sine of the Sun’s altitude in prime vertical and the TADDHRITI minus KusyYA equal to 1440’, and given also the sum of the sine of amplitude, the sine of the Sun’s altitude in prime vertical and the TADDHRITI 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- tude to give an ascensional difference of three auatis? 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 of the Sun when in the prime verti- Question. Question. Question, But the aKsHa KaRNA must be first found to complete the sum of those three. त 625 __ 2 ASHA ४५१५ = 4/ (12)? + > = +^ => Gnomon = 12 | PALABILA = 34 == 28 sum of the 8 sides of a latitudinal triangle. AKSHA KARNA == 12} Now if 28 : 12: : 1960: 840 the sine of declination. Hence the place of the Sun as before.—L. भ. This question is similar to the preceding In the first sum we have the sum of three perpendiculars in three different latitudinal Triangles. In the second we have the sum of the three hypothe- nuses of those same three Triangles. Hence we may say sum 3 per. sum of 3 corresponding hy. Gnomon axkSHAKABNA As 1440 : 1800 १६ 12. 4 15 Now from aKsHa KARNA to find PALABHA PALABHA = 4/(15)*—(12)* = / 81 =9 Now sine of amplitude, sine of the Sun’s altitude in the Prime Vertical, and the TADDHRITI are the three sides of a latitudinal_—_L. W + Let 2 = sine of the Sun’s declination. then 12: 9:: >: KusYA=42. Again 4/R*—2* = cosine of declination. Then as R:; cos of declination : : sine of ascensional differce. : KUIYA Sine of ascensl. १1८५९. or CHARAJYA = sine of 3 G@HATIS = 8111 18° == 1062’. cosin of decln. K CHABAJYA | oo. = Kor R a/ B—a* % 1062 or == ‡ R Hence may be found the sine of the Sun’s decln. and thence his longi- tude.—L. W 111. 46.] Siddhduta-s'tromani. 257 cal, and when in an intermediate circle (i. e. when he has ‘an azimuth of 45°) by three different modes of calculation: now he who will by a single calculation tell me the length of these -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 Earth to expand the lotus-intellects of learned astronomers.* * (Here the problem is this:—Given the Sun’s declination or amplitude, + shadow of the place and the Sun’s azimuth, to find the Sun's shadow. For solving this problem Buadsxkardcufgya has stated two different Rules in the GaniTADHYAya. Of them, we now shew here ४116 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 the azimuth and the sine of the amplitude, is called the PRaTHAMA (first) and the continued product of the Radius, equinoctial shadow and the sine of the amplitude divided by the (same) difference is called the anya (second). Take the square- root of the square of the anya added tothe PRATHAMA: this root decreased or increased by the anya according as the Sun is in the northern or southern hemisphere gives the hypothenuse of the shadow (of the Sun) when 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 the square-root of the square of the anya diminished by the PRaTHAMa: the ANyA decreased and increased (separately) by the square-root (just found) gives the two values of the hypothenuse (of the Sun’s shadow) when the Sun is in the northern hemisphere.” This rule is proved algebraically thus, Let a = the sine of amplitude, A = the sine of azimuth, e == the Equinoctial shadow, and «=the hypothenuse of the shadow when the Sun is in any given direc- tion of the compass. Then say 12R 88 % : 12 ; : R: the MawA s‘aNKU or the sine of the Sun’s altitude = —— x 2R\? » ,____ and .°. the sine of the Sun’s zenith distance == ८ ४- (न ) =— 4, ८*-- 144. xv 12 R eR Now, as 12 : ८ == - : S‘ANKUTALA = —-. x x ० BAut or the sine of an arc of 8 cirele of position contained between the eR Sun and the Prime Vertical = ८ कुः —-: (see Ch. VII, V. 41) here the sign— x or + is used according as the Sun 18 in the northern or southern hemisphere. Then say | é as — A/a*—144 : ० त —-::R A x xz RA eR ° — fem = (लक) प @ 2.58 Translation of the (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 2616 BHA, I shall conceive him to be a very Garupa in destroying conceited snakes of astronomers. [On this Bufsxardcudrya has given an example in the Gayt- 740 प्र ८१ as follows. : ‘‘ Given the hypothenuse of the shadow (at any hoar of the day) equal to 30 digits and the south BHUJA* equal to 8 digits: given also Question. Example. or A A/a*—144 == ० > eR; A‘ 2*—144 ^ ° == ८० 2 F2Reaxt e? R'; (A?—a*) a? + 2 २९०2 = 9 R? + 144 4०; Rea e? R? + 144 A* 2 # 2 c= ; A*—a?* A2— a? or 2? + 2 ANYA 2 = PRATHAMA (1) ^ ऋ 2 ANYA 2 =+ ANYA? = PRATHAMA + ANYA? and .. 2 == «/ PRATHAMA - ANYA? 3 ARYA. But when A < aand the Sun is in the northern hemisphere, the equation (1) will be 2*—2 anya 2 = — PRATHAMA, and then 2 == anya + ^ aANYa*—tirst : i. e. the value of the hypothenuse of the shadow will be of two kinds here, Hence the Rule. BHASKARACHARYA 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.]} ॐ (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 end of the shadow of that Gnomon and the east and west line is called the BHUJA. It is to be known here that the value of the great BuUJA (as stated in 41st. verse of the 7th Ch.) being reduced to the hypothenuse of the shadow becomes equal to the BHUJa (above found). Or as the Radius : the great BHUJA :: the hypothenuse of the shadow : the reduced BHUJa or the distance of the end of the shadow from the east and west line. This reduced BHUJA 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 clear from this that the reduced BHUJA will be the cosine of the azimuth in a small circle described by the radius equal to the shadow. Or as the shadow : the reduced BHUJA : : radius of a great circle : the cosine of the azimuth. This is the method by which all Hindus roughly determine the azimuth of the Sun from the Buvsa of his gnomonic shadow.—B. D.] XIII. 49.] Stddhdnta-a'tromani. 259 the hypothenuse equal to 15 digits, and the north Bausa equal to 1 digit, to find the patasndé. Or, given the declination equal to 846 and only one hypothenuse and its corresponding BHUsJA at the time, to find the 2 1.^ एप {.7 |] 48. First of all multiply one इष एर+ of the shadow by the hypothenuse of the other, and the se- cond BHusaA by the hypothenuse of the first: then take the difference of these two BHuJas 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. How should he who, like a man just drawn up from the 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 PaLanHa/ when the two shadows and their respective BHUJas are given, is proved thus, Let 4, = the first hypothenuee of the shadow, b, = its corresponding BHUJA, h, = the second hypothenuse, and 6, = its corresponding BHUJA, Then 12R As h,: 12:: R: —— =the first MAWA 8’ANKU ; h 128 and in the same manner = the second MAHA 8’ANKU ; 2, % and also ash, : 2६ : : R: == the first great BHUJA, hy b, R and .*, —— = the second great BHUJ: , h KR 8, ——— + ———we ae A, hg Then the PALABHA’ = ~ (see Ch. XI. V. 32) 12 127 he hy 2१ he + 09 #1 h,—Ah Hence the Rule.—L. W. x 2 260 Translation of the [XIII. 50. the commencement of the yuca, the month, the TITHI 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 him 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 use of that instrument of instruments—Genius (the pafyanTra): and tell me what is there that he cannot find out ! 51. There is a high famous bamboo, the lower part of which, being concealed by houses, &c. Question. Question. 2 । was invisible: the ground, however, was perfectly level. If you, my friend, remaining on this same spot, by observing the top, will tell me 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 8 a distance, visible or invisible, if you, remaining on this same spot, will tell me the distance 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 the amount of the precession of the equinox, 1. €. their 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, telt me whether the Sun and the Moon have the same declination (either both south or one north Question. * This refers to the 34th verse of the Ch. XI.—L. W + [Answers to these questions will be fourd in the 11४11 Oh. —B. D.] XIII. 59.) Siddhdnta-s’iromant. 261 and one south), if you have a perfect acquaintance with the DufvRippHIDA TANTRA. 54. Ifthe place of the Moon with the amount of the pre- 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 equal to 200 degrees, tell me whether the Sun and the Moon have the same declination, if you have a perfect acquain- tance with the DufvraippHipa Tantra. | | 55. If you understand the subject of the pdta i. €. the equality of the declinations (of the Sun and the Moon), tell me the reason why there is in reality an impossibility of the 2474 when there 18 its possibility (in the opinion of Latta), and why there is a possibility when there is an impossibility of it (according to the same author) 56. Ifthe places of the Sun and the Moon with the amount of the precession of the equinox be equal to 3 signs plus and minus 1 degree (i. €. 2s. 29° and 3s. 1° respectively) and the place of the Moon decreased by that of the ascending node equal to 115. 28°, tell me whether the Sun and the Moon have the same declination, if you perfectly know the subject. 97, (In the Dafvrippuipa Tantra), it is stated that the p{Ta as to come in some places when it has already taken place (in reality), and also it has happened where itis to come. It is a strange thing in this work when the possibility and impos- ibility of the pPATa are also reversely mentioned. Tell me, O you best of astronomers, all this after considering it well.* 98. ` I (Budsxara), born in the year of 1086 of the S’A.I- Date of the Author’s birth VAHANA era, have composed this S1p- and his work DHANTA-8/IROMANI, when I was 86 years old 59. He who has a penetrating genius like the sharp point of alarge paRBHA straw, is qualified Author’s apology. ; to compose a good work in mathe- * [Answers to these questions will be found in the last Chapter of the GanitTa- pDayaya.— B, D. ” 262 Translation of the Siddhanta-s’iromani. [XIII. 60. matics : excuse, therefore, my impudence, O learned astrono- mers, {in composing this work for which I am not qualified). 60. I, having lifted my folded hands to my forehead, beg the old and young astronomers (who live at this time) to excuse me for having refuted the (erroneous) rules prescribed 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 # 8748, and situated near the mountain Sanya. 62. His son, the poet and intelligent ए घ 48 ह ५१, made this clear composition of the SippHAnta by the favour of the lotus- like feet of his father; this SippHAnra 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 have stated before, for persons who wish to study only this PRras'Né- DHYAYA. 64, The genius of the person who studies these questions becomes unentangled, and flourishes like a creeping plant watered at its root by the consideration of the 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, &c. End of the 13th and last Chapter of the Goudpuydya of the SIDDHANTA-S IROMANI. APPENDIX. ON THE CONSTRUCTION OF THE CANON OF SINES. 1. Asthe Astronomer can acquire the rank of an Acwarya 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 8. Draw acircle with a radius equal 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 iathe- matical precision with exactness. The square-root of the dif- ference between the squares of the radius and the sine is cosine. 5. Deduct the sine of an arc from the radius the remainder will be the versed sine of the complement of that arc, and the cosine of an arc deducted from the radius will give the versed sine of that arc. The versed sine has been compared to the 264 Appendix. ~~ arrow between the bow and the bow-string: but here it has received the name of versed-sine. 6. The half of the radius is the sine of 30°: the cosine of 30° will then be the sine of 60°. The square-root of half square of radius will be the sine 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 J rad ˆ > 8 — ofrad #x 5 = 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 sine of 54°.f 9. Deduct the radius from the square-root of the product of ° ® (This is proved thus. Let © = 81116 18°; 8०... R — ¢ = covers 18° or vers 72°. Then ^^ Rx sore = 81116 4°: (see the 10४0 verse.) or SERA 9) = sine 36°; +, sine 36° = JS ok CES (0 5 - 94 =f er 7.] . R >< 5878 + The Rule in 8th verse viz.. -—--——~ seems to be the same as above and 10000 to be deduced from it ; | 5 R?— +^5 R* _, (5 —V/5 ; ०. 1 ५७ ह = 2.237411 &०. ९१,“ 5 — ,/§ = 2.762589 which divided by 8 = .345323 R ॐ 5878 : ,*, sine 36° = R ,/ 345323 = R KX .5878 = ——_-——.—-L. | ^ 10000 Appendix. 265 the square of radius and five and divide the remainder by 4: the quotient thus found will give the exact sine of 18°.* 10. Half the root of the 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 be found (and so on with the 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 a mode for deter- mining the other sines (from a given one), but I proceed now to give a mode different from that stated by them. 12. Deduct and add the product of radius and sine of BHUJA from and to the square of radius and extract the square- roots of the halves of the results (thus found), these roots will respectively give the sines of the half of 90° decreased and increased by the BHUJA. In like manner, the sines of half of 90° decreased and in- creased by the Kofi can be found from assuming the cosine for the sine of BHUJA. 13. Take the sines of Bausas of two arcs and find their difference, then find also the difference of their cosines, square * (This ia proved thus. Let C be centre of the circle ABE and —. © = 36°, then AB == 2 sin 18°, and „~ s (CAB, CBA) each of them = 2 ¢^. Draw AD bisecting the <~ CAB, then AB, AD, CD will be equal to each other. Now let x = sin 18°, then by simi- lar triangles CB: A3 = AB: BD or १; 2 = 2: ८ - 2; ,- 4 2 = R? — 2 Re which gives = र न =. 70. ] ~~ 266 Appendia. 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 square of the difference of the sine and the cosine of the BauJa of an arc is equal to the sine of half the difference of the BHUJA and its complement.t+ I 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 suusa by the half radius. The difference between the quotient thus found and the radius is equal to the sine of the difference between the # This rule is obvious, for ac = diffce. of sines bd & af and cb = diffce. of cosines bg & ah and ac |" + bc )* == abd ) ab = chord of difference of arcs . ab — = 8116 of half that difference. 2 L. W. , % Let ९८ = sine of any arc and 3 cosine. ‘i oa I < Draw the sine वद == cosine bg, then ah its sine will be equal to dc and af = f ^ af? + fb" = ab*: but as af? = ए" ab af* Appendix. 967 degrees of BHuUJA and its complement.* In this way several sines may be found here. [As these several rules suffice for finding only the 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 the sine of every degree of the quadrant. This mode, therefore, is called PRaTIBHAGAJYAKA-VIDHI. | 16. Deduct from the sine of BHusa its उदन part and divide Rules for finding the sine the ten-fold sine of एण्या by 573. of every degree from 1° to 17. The sum of these two results a will give the following sine (i. e., the sine of BHUJA one degree more than original BHUJA and the difference between the same results will give the preceding sine, i. e., the sine of BHUJA one degree less than original ष्च ए). Here the first sine, i. e., 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 the sines of 90° of the quadrant may be found. Multiply the cosine by 100 and divide the product by 1529. Rules for finding the 24 19. And subtract the 71, part of a १५ of 8०, 7, 1173, the sine from it. The sum of these will —- be the following sine (i. e., the sine of arc of 3°} degrees more than original arc): and the differ- * Let ad be any arc, and ac = ab, d then ad = its complement, cd = their difference, and bc == 2 ab. (कवन क be Now RX vers 0८ — gin — or sin ab, 2 2 . R ॐ vers be > or —— == sin? ad, 2 sin® ad or vers be = ; 81109 ab —L. W. R then R — vers dc or sin cd = R ~ 268 Appendin.. ence of them will be preceding sine (i. e., the sine of arc 3°% degrees less than original arc). 20. But the first sine (or the sine of 3,%) is here equal to 2248 (and not to 225 as it is usually stated to be). By this rule 24 sines may be successively found.* 21 and 22. Ifthe sines of any two arcs of a quadrant be Riles for finding the eines multiplied by their cosines reciprocally (५ difference ofany (that is the sine of the first arc by the cosine of the 2d and the sine of the 2d by the cosine of the first arc) and the two products divided by radius, then the 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.+ This excellent rule called Jya-pHivand has been prescribed for ascertaining the other sines. 23. This rule is of two sorts, the first of which is. called saAMASA-BHAVANA (i. €. the rule for finding the sine of sum of two arcs) and the second ANTARA-BHAVANA (i. 6.9 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 praTipHdaasyaK4- VIDHI. 24 and 25. And then reduce it to the value of any new radius by applying the proportion. After that apply the गर BHMvANA rule through the aid of the first sine and the cosine thus found, for as many sines as are required. The sines will thus be successively eliminated to the value of any new radius. The rule given in my Parf or Lidvarti 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 the verses 21 and 22.—B. D.] + BuAskarACHARY,a has given these rules in his work without any demons stration.—B. D.] INDEX. Age, birth, &c. of the Author, page 261. Armillary Sphere 151, 210. Astronomical Instruments, 209. Atmosphere, 127. Celestial latitude, 200. Clepsydra, 211. Chakra, 212. Canon of Sines, 263. Day of Brahma, 163. Day of the Pitris, 163. Days and nights, 161. Deluges, 125. Drikkarma, 110. Driyantra, or genius instrument, 221. Earth, 112. Earth’s diameter, 122. Kclipses, 176. Epicycles, 144. Equation of the centre, 141, 144. Errors of Lalla, &c, 169, 165, 205. Gnomon, 212. Horoscope, 166, 211. Kalpa, 108. Kendra, 109. Lagna, 166, 211. Longitudes, 212. [000 Mandaphala, 109. Mandochcha, 109. Month, 129. Moon, Eclipses of, 176, Phalaka- Yantra, 213. Phases of the Moon, 206. Planets, 128, 135. Questions, 231. Rising and setting of the heavenly bo- dies, 196. —— —— signs, 164. Seasons, 228. Seven Winds, 127. Sighrochcha,109. Signs, rising of the, 164. Sphere, 107. Sun, Eclipses of, 176. Swayanvaha Yantra, or self-revolving instrument, 227. | Syphon, 227. Time, 160. Winds, 127. Year, 129. Yugas, 110, < ध ८८. lh a eee ध ( उणा णण BIBLIOTHECA INDICA; bd ~ (द | | COLLECTION OF ORIENTAL WORKS | | PUBLISHED UNDER THE SUPERINTENDENCE OF THE | | | ASIATIC SOCIETY OF BENGAL. | 4 | ex | New Surtzs. 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