ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

112 INDIAN LUNI-SOLAR ASTRONOMY Greek value=5°0'. Modern mean value=5°8'43".427 (Brown) This discrepancy confirms the conclusion, that the observa- tion of the Moon was restricted to the time when she was near a node, either at solar or lunar eclipses, where a small error of observation magnified itself into about half a degree. We now turn to the parallaxes of the Sun and the Moon :-

MOON'S EQUATIONS 113 vation was up to the time of Brahmagupta, restricted to the time of eclipses perhaps also of syzygies. The modern form of the Moon's equations is =377' sin (nt—a)+13' sin 2(nt—a)+......... +76' sin [2(nt—θ)—(nt—a)]+40' sin 2(nt—θ)..............¹ where nt=mean longitude of the Moon, a the longitude of the perigee, θ=longitude of the Sun. Here the first two terms, viz., 377' sin (nt—a) + 13' sin 2(nt—a), are due to elliptic motion about the Earth in one focus; the term 76' sin [2(nt—θ)—(nt—a)] is known as the evec- tion. We combine a part of the first term with the evection term and the expression for the equation of centre becomes =301' sin (nt—a)+13' sin 2(nt—a)+............+152 sin (nt—θ) cos (θ—a)+40' sin 2(nt—θ). Now at syzygies and eclipses sin (nt—θ) and sin 2(nt—θ) will very nearly vanish. Hence according to modern astronomy at the syzygies and eclipses, the chief term of the Moon's equa- tion=301' sin(nt—a). This according to the Āryabhaṭīya =300' 15" sin (nt—a) ,, ,, Khaṇḍakhādyaka =296' sin (nt—a) ,, ,, Uttara Khaṇḍakhādyaka =301'.7 sin (nt—a) ,, ,, Brāhmasphuṭasiddhānta =293' 31" sin (nt—a) ,, ,, Greek astronomy =300' 15" sin (nt—a) very nearly. Hence both the Greek and the ancient Indian astronomers were very near the true value of the Moon's equation at the syzy- gies and eclipses. Godfray in his Lunar Theory, page 107, observes, "the hypothesis of an excentric, whose apse has a pro- gressive motion as conceived by Hipparchus served to calculate with considerable accuracy the circumstances of eclipses; and observations of eclipses, requiring no instruments, were then the only ones which could be made with sufficient exactness to test

1. The accurate values of the coefficients appear to be 377' 19".06, 12' 57".11, 76' 26" and 39' 30".

114 INDIAN LUNI-SOLAR ASTRONOMY the truth or fallacy of the supposition." We next consider the second inequality of the Moon. Moon's Second Inequality or Equation In ancient times it was Prolemy who first really found a second inequality of the Moon. According to Godfray (Lunar Theory, p. 107) "by dint of careful comparison of observations he (Ptolemy) found that the value of this second inequality in quadrature was always proportional to that of the first in the same place, and was additive or subtractive according as the first was so; and thus, when the first inequality was at its maximum or 5°1′, the second increased it to 7° 40′ which was the case when the apse line happened to be in syzygy at the same time." It is well known that though Ptolemy discovered the second inequality in the Moon's motion he was not able to ascertain its true nature. His corrections in this case are true when at the quadrature the Moon's apse line passes through the Sun or it is at right angles to the line joining the Earth and the Sun.¹ In the general case his construction does not lead to the elegant form of the evection term as we know it now, nor does it lead to the nice form in which it was given by later Indian astrono- mers from the time of Mañjula (or Muñjāla, 854 Śaka era = 932 A.D.). As has already been pointed out, the early Indian astrono- mers from Āryabhaṭa to Brahmagupta aimed at accuracy in lunar calculation only for the eclipses and syzygies, and did not interest themselves about the Moon's longitude at the quadra- tures. Hence this second inequality is absent in the works of these makers of Indian astronomy, as also in the Pre-Ptolemaic Greek astronomy. This points to the conclusion that in both the earlier Indian and Greek systems of astronomy, the modes of observation of the Moon were copied from an earlier system of astronomy whether Babylonian or Chaldean. Even in the Romaka Siddhānta of the Pañcasiddhāntikā, there is no mention of evection.² Thus inspite of the transmission of a vague system of Greek astronomy, Indian astronomy as developed by Ārya- bhaṭa and Brahmagupta must be regarded as independent and

1. Godfray's Lunar Theory. pp. 108-110.
2. Vide the Summary in P.C. Sengupta's paper, "Āryabhata the Father of Indian Epicyclic Astronomy." Journal of the Department of letters, vol. XVIII, Calcutta University Press.

MAÑJULA'S MOON'S EQUATION 115 original not only from this but also from other considerations. It sought to correct the constants as were obtained from the Babylonian and the Greek systems as has in some cases been shown already. Mañjula's Second Equation of the Moon (932 A.D.) We now take up in detail Mañjula's second equation of Moon. It is, however, necessary to say something about his first inequality. This is given in the form —488 sin (nt—a') ————————————————————— degrees, 96+ (488/120) cos (nt—a') where nt stands for the Moon's mean longitude, a'—that of the apogee. 488° Hence when nt—a'=90°, the equation = ———— =5°4'=304' 96 showing an excess of 4' over the modern value. It is further necessary to modify the expression for the Moon's modern form of the equation by chang- ing a to 180°+a, as in ancient Indian astronomy anomaly is measured not from the perigee but from the apogee. The modified form is =—301' sin (nt—a)+13 sin 2(nt—a)...... —152' sin (nt—θ) cos (θ—a)+40' sin 2(nt—θ)+... Mañjula's lines giving the second equation are— The (mean) daily motion of the Moon diminished by 11° and multiplied by the "cosine" of the longitude of the Sun diminished by that of the Moon's apogee is the multiplier of the "sine" and the "cosine" of the longi- tude of the Moon diminished by that of the Sun, divided severally by 1 and 5. The results taken as minutes are to be applied negatively and positively to the Moon and to her daily motion if the quantities multiplied together are of opposite signs and in the reverse order if they are of the same sign.¹

1. इन्दूच्चोनार्ककोटिघ्ना गत्यंशा विभवा विधोः । गुणो व्यर्केन्दुदोः कोट्यो रूप पंचाप्तयोः क्रमात् ॥ फले शशाङ्क--तद्गत्योर्लिप्तास्ते स्वर्णयोर्वधे । ऋणं चन्द्रे धनं भुक्तौ स्वर्णं साम्य वधेऽन्यथा ॥ —11, 12

116 INDIAN LUNI-SOLAR ASTRONOMY As to the positive or negative character of the "sine" and the 'cosine' he gives the rule:— The mean planet diminished by its ucca, the apogee, aphelion or the Śīghra, is called Kendra or mean anomaly; its "sine" from above six signs (180°) arises from half circles and are respectively positive or negative, and its "cosine" in different quadrants are respectively positive, negative, negative, and posi- tive.¹ The convention followed is that the "sine" is negative from 0° to 180° and positive from 180° to 360° of the arc and that the cosine is positive between 0° and 90°, negative between 90° and 270° and positive between 270° and 360°. We may now symbolically express Mañjula's second ineq- uality thus :- —(13° 11′ 35″—11°) × 8ᵖ 8′ cos (θ—α) × 8ᵖ 8′ sin (D—θ) where D stands for the Moon as corrected by the 1st equation; we leave out the correction to the Moon's daily motion as given in the stanzas quoted above. The moon's new equation comes out to be =—143′58″ cos (θ—α) sin (D—θ). This, it will be seen, is exactly the modern form of the evection as combined with a part of the equation of apsis shown before. The difference in the main is that Mañjula's constant is 144′, a quantity less by 8′. In form the equation is most perfect, it is far superior to Ptolemy's, it is above all praise. It is from this inequality, we trust, that Mañjula should have an abiding place in the history of astronomy. The next writer who gives the second equation is Śrīpati (1028 A.D.). Śrīpati's Second Inequality of the Moon The following stanzas from Śrīpati's Siddhānta Śekhara, it is said, were communicated to Sengupta by Pandit Babua Misra. Though they are probably not very correct still the general meaning is clear. They carry the following sense : "From the Moon's apogee subtract 90°, diminish the Sun by the remainder left; take the "sine" of the

1. ग्रहः स्वोच्चोनितः केन्द्रं षडूर्द्धादर्द्धजो भुजः । धनर्णैः पदशः कोटिर्धनर्णै धनात्मिका ॥

ŚRĪPATI'S SECOND INEQUALITY OF THE MOON 117 result; multiply it by 160' and divide by the radius; the result is called caraphala. Put it down in another place, multiply it by śara (i.e., R vers (D—α) or versed sine of the Moon's distance from the apogee) and divide by the difference between the Moon's distance (hypotenuse) and the radius; the result is called parama (cara) phala, which is to be considered positive or negative according as the hypotenuse put down in another place is less or greater than the radius. Multi- ply the "sine" of the Moon which has been diminished by the apparent Sun, by the apparent paramaphala and divide by the radius; the final result is to be called caraphala to be applied to Moon negatively or posi- tively as the Moon minus the Sun and the Sun minus the Moon's apogee (diminished by 90°) be of opposite signs; if these latter quantities be of the same signs, the [L1

118 INDIAN LUNI-SOLAR ASTRONOMY 160' R sin [θ – (a – 90°)] R vers (D – a) × R sin (D – θ) = ∓ ----------------------------------------------------------- R (H – R) × R 160' R cos(θ – a) × R sin(D – θ) R vers (D – a)¹ = ∓ ---------------------------------- × ----------------- R × R H – R This equation is a slightly modified one but practically the same in form as that of Mañjula, except that the constant here is 160', greater than his by 16'. The constant is 160' also in Candraśekhara's form as we shall see later on. We next consi- der the Moon's inequalities as given by Bhāskara II in his Bījo- panaya,* a special work on these inequalities composed in the Saka year 1074 (=1152A.D.) two years after he had composed the Siddhānta Śiromaṇi. Bhāskara II on Moon's Inequalities His preliminary statement runs thus .— 112' positive or negative representing the maximum difference. have been found by me in the daily obser- ved Moon (as calculated and as observed) at that point of the ecliptic where the arc from the kadamba (i.e., its pole) passing through the zenith cuts it.¹ Thus for observing the Moon he selected the nonagesimal as the suitable point where the uncertainty about her parallax is zero, and found ∓112' of arc to be the maximum difference bet- ween her calculated and observed places. Mallabhaṭṭa, perhaps a contemporary of Bhāskara II, ascribed this difference to a supposed Śīghrocca of the Moon. Bhāskara in stanzas 9-13, refutes the existence of the Śīghra in the case of the Moon, the substance of his argument begins (i) that it is against the teaching of the Sūrya-siddhānta and other accepted authorities, (ii) that there is no variation of the apparent angular diameter of the Moon corresponding to this alleged Śīghra, and (iii) that planets having a Śīghra have retrograde motion which is never the case with the Moon.

*There is some uncertainty, about this new fraction introduced by Śrīpati.

1. लिप्ता विधोरर्के महीमिता मे दृग्गोचराः प्रत्यहमीक्षितस्य । कदम्ब गोलगत-सूत्रपाते क्रान्तौ धनर्णत्वजुषो भमध्यात् ॥ – Bijopanaya, stanza 8

BHĀSKARA II ON INEQUALITY OF THE MOON 119 The reasons for his new equations are stated as follows:— When the Moon is situated at a quadrant ahead of the apogee and with the Sun at half a quadrant ahead of her, the maximum discrepancy (of 112′) is seen in the negative character. When the Moon is situated at three quadrants ahead of the apogee and with the Sun at half a quadrant behind her, the maximum discrepancy (of 112′) is seen in the positive character. When the eclipses of the Sun and the Moon take place at the apogee or the perigee of the Moon, the Moon as corrected by the equation of apsis is seen to be without any new correction called bīja. When the eclipses of the Sun and the Moon take place at the ends of the odd quadrants of the Moon's anomaly (measured from the apogee), the discrepancy is seen to be less by 34′. When the Moon is at the apogee, whether the Sun be ahead or behind her by half a quadrant, the discre- pancy amounts to be 34′. The same discrepancy of 34′ is observed when the Moon is at the perigee and the Sun is ahead or behind her by the same distance. Thus by analysis and synthesis, and by repeated obser- vations, this variable correction has been devised by me; let it be seriously considered by the learned.¹ Bhāskara here speaks of six cases and we consider them one after another :— The Moon's equations as modified to suit siddhāntas are given by −301′ sin (nt—α)+13′ sin 2(nt—α)...... −152′ sin (nt—θ) cos (θ—α)−40′ sin 2(nt—θ)+...... According to Bhāskara's Siddhānta - śiromaṇi. the Moon's equation of apsis = − (31° 36′ / 360°) × 3438′ sin (nt—α) = −301′ 46″. 8 sin (nt—α),

120 INDIAN LUNI-SOLAR ASTRONOMY this agrees well with the corresponding term of the modern equation. As Bhaskara takes in all the six cases, nt-a=90°, 270°, 0° or 180°, the second term of the equation of apsis vanishes. Case I. nt-a=90°, nt-θ=-45°, θ-a=135°; Here the total equation of the Moon =-301'-(76'+40')=-301'-116'. This fairly agrees with Bhāskara's observation, the difference being only of 4'. Case II. nt-a=270°, nt-θ=45°, θ-a=225°; the total equation of the Moon =301'+76'+40'=301'+116'. This also agrees with Bhāskara's observation. Case III. nt-a=0° or 180°, nt-θ=0° or 180°, θ-a=0° or 180°, the total equation=0', this also agrees with Bhāskara's observation. Case IV, nt-a=90° or 270°, nt-θ=0° or 180°, θ-a=90° or 270°,

1. तुङ्गादाद्यपदान्तस्थाद् विधोरर्के पदार्द्धतः । परमं चन्द्रवैषम्यं ऋणत्वेन समीक्ष्यते ॥ 20 ॥ तत् तृतीय पदान्तस्थात् पृष्ठगोऽर्के पदार्द्धतः । परं चन्द्रवैषम्यं धनत्वेन समीक्ष्यते ॥ 21 ॥ चन्द्रतुङ्गे च नीचे च शशाङ्कार्कग्रहो यदि । मन्दस्फुटगतश्चन्द्रो निर्धीजतुल्यमीक्ष्यते ॥ 22 ॥ ओजान्तयोर्विधोस्तुङ्गाच्छशाङ्केऽर्क ग्रहो यदि । चतुस्त्रिंशत्-कलाहीनं वैषम्यं तु समीक्ष्यते ॥ 23 ॥ अग्रतः पृष्ठतो वाऽपि रवेश्चन्द्रे पदार्द्धगे । तुङ्गतुल्ये चतुस्त्रिंशत् कलावैषम्यमीक्ष्यते ॥ 24 ॥ एवं तन्नीचतुल्येऽपि वैषम्यं तावदेव हि । एवं व्यासात् समासाच्च पौनःपुन्येन वेधनात् ॥ चरबीजमिदं क्लृप्तं मया सद्भिः समीक्ष्यताम् ॥ 25 ॥

ŚRĪPATI'S SECOND INEQUALITY OF THE MOON 121 the total equation = ∓301′. This does not agree with Bhās- kara's statement that the total equation = ∓(301′ ± 78′). Case V. nt—a=0, nt—θ=±45°, θ—a=±45°, the total equation =0′—76′+40′=—36′ or 0′+76′—40′=+36′. This fairly agrees with Bhāskara's observation. Case VI. nt—a=180°, nt—θ=±45°, θ—a=180°∓45° the total equation =0′+76′+40′=0′+116′, or 0—76+40′=0′—36′. This does not agree with Bhāskara's statement. Bhāskara then states his first system of 24 equations corres- ponding to 24 sines in a quadrant to be 6′, 13′, 21′, 27′, 33′, 39′, 45′, 51′, 56′, 61′, 65′, 68′, 70′, 72′, 74′, 75′, 75′, 76′, 76′ 77′, 77′, 78′, 78′, 78′.¹ These equations, he says—"are negatively added to the equation of apsis when that is negative and positively added to the same when that is positive"². In other words his new equa- tions are complements of the equation of apsis, the two together being represented by —301′ 46″. 8 sin (nt—a)—78′ sin (nt—a) i.e., by—379′ 46″. 8 sin (nt—a). Hence next states his second set of equations depending on θ—D, to be 6′, 9′, 13′, 17′, 22′, 24′, 27′, 30′, 32′, 33′, 34′, 34′, 34′, 33′, 31′, 29′, 26′, 24′, 20′, 16′, 11′, 8′, 3′, 0′³ and says : "These minutes are negative in the odd quadrants of the argument and are positive in other quadrants.⁴" When the value of the argument is 15°, the equation is 17′, ,, ,, " 45° " 34′, ,, ,, " 90° " 0′,

1. Bījopanaya, 26-28.
2. ˙˙˙फलं ऋणे ऋणं । धने धनं मन्दफलेन संयुतम् ॥ 28 ॥
3. Bījopanaya, 29-32.
4. एताः कला ओजपदे ऋणं स्युर्धनं तदन्यत्र भवन्ति भूयः ।

122 INDIAN LUNI-SOLAR ASTRONOMY Hence the new equation = - 34' sin 2 (θ - D), = 34' sin 2 (D - θ). Here the symbol D stands for the Moon as corrected by the ancient Indian equation of apsis and its complement as given by Bhāskara. It is readily seen that Bhāskara is the first of all the Indian astronomers to detect the equation known as "Variation" His constant, 34', is less than the modern value by about 6', and cannot be considered as a serious error. We now see that the sum-total of the Moon's equation as given dy Bhāskara = - 379' 46".8 sin (nt - a) + 34' sin 2(D - θ), the evection term being totally absent. This is a serious defect, and Bhāskara's new equations would make the Moon generally more incorrect at the syzygies and eclipses than what the ancient Indian equation of apsis would do. Perhaps late in life when he was 69 years old in 1105 of Śaka era (= 1183 A.D.) he discovered the inapplicability of his new equations at the times of eclipses and in his Karaṇa-kutu- hala he altogether omitted these new equations which he had given in his Bījopanaya. As to Bhāskara's second inequality which is really the com- plement of the equation of apsis without the evection term, it is far inferior to that of Mañjula and of Śrīpati; as we have seen their form of the second inequality combines the complement of the equation of apsis and evection in the mathematically cor- rect form. For the discovery of such a form of the equation as of these authors, very patient, careful and frequent observation must have been coupled with very careful and nice comparison of observed facts. As to "variation" it was first discovered by Abul-Wefa in 976 A.D.¹ which was quite forgotten when Tycho-Brahe re-dis- covered it in 1580 A.D. Hence Bhāskara, in 1152 A.D., re-dis- covered it in India four centuries before Tycho. Candraśekhara of Orissa on the Moon's Inequalities In connection with lunar inequalities it is nesefsary here to record what were the equations discovered or verified by M.M. Candraśekhara Simha of Orissa in the later half of the last century. He was educated in the orthodox Sanskrit fashion

1. Godfray's Lunar Theory, p. 114.

CANDRAŚEKHARA ON THE MOON'S INEQUALITIES 123 and had no acquaintance with English education. His work Siddhānta-darpaṇa was edited by Prof. Jogeschandra Ray, late of the Cuttack College, in 1899.¹ Candraśekhara in his work gives four equations of the Moon which are :- (1) The equation of apsis. (2) The Tungāntra equation or the complement of the equation of apsis in combination with evection. (3) The fortnightly equation or variation. (4) The Digamśa equation or the annual equation (i. e., 1/10 of the Sun's equation). (5) The first equation is of the form [31° 30' - 30' cos (nt - a)]3438 × sin (nt - a) = ─────────────────────────────────────────────── 360° = -300' 49".5 sin (nt - a) + 4' 46".5 sin (nt - a) cos (nt - a) = -300' 49".5(sin nt - a) + 2' 23".25 sin 2 (nt - a) It is seen that Candraśekhara wanted to correct the equa- tion of apsis to the second order of small quantities as in all the Indian authors from Brahmagupta but Candraśekhara's form is correct though his contant is wrong. (2) His second equation is of the form 160' × 3438 sin[a - (θ + 90°)] 3438 sin(D - θ) ────────────────────────────── × ─────────────── 3438 3438 Moon's appt. daily motion² × ────────────────────────── Moon's mean motion = -160' cos (θ - a) sin (D - θ) Moon's appt. daily motion × ───────────────────────── Moon's daily mean motion Here the constant is the same as that of Śrīpati discussed before. The symbol means the Moon as corrected by the equation of apsis. It is readily seen that the constant of the first term of the equation of apsis is increased by 80'. and that the constant of evection is taken at 80'. In both the cases the error is about +4'. (3) Candraśekhara's third equation or Variation 3438' sin 2(D' - θ) = ─────────────────── = 38' 12" sin 2(D' - θ),³ 90

1. Siddhānta-darpaṇa, V, 100-114.
2. Ibid, VI, 7-9
3. Siddhānta-darpaṇa VI. 11-12.

124 INDIAN LUNI-SOLAR ASTRONOMY where D' means the Moon as corrected by the 1st and the 2nd equations. Here the constant is wrong by −1' 18". (4) His fourth equation or the annual equation = ± 1/10 of the Sun's equation of apsis,¹ = ± 1/10 × (12 × 3438) / 360 sin (Sun's distance from the apogee). = ±11'27". 6 sin (Sun's distance from the apogee). The modern value of the constant is 11' 10". Tycho found it to be 4' 30". Horrocks' (1639) co-efficient was 11' 51". As Candraśekhara was aware of Bhāskara's Bījopanaya, as also of the work of Śrīpati, his merit here lies in the discovery of the annual equation, and correction to the constant of variation. Thus we have seen that so far as the luni-solar astronomy is concerned Indian astronomy is independent of Greek astronomy in respect of astronomical constants, that Indian astronomy is generally more accurate than Greek astronomy and that Indian astronomers were not mere "calculators"². There were observers who verified and corrected the old astronomical constants as they came down from Āryabhaṭa and Brahmagupta, who also found independently all the principal equations of the Moon.

1. Siddhānta darpaṇa VI. 13.
2. G.R. Kaye Hindu Astronomy, p. 60 — : o : — Reference P.C. Sengupta : The Khaṇḍakhādyaka, 1934 Karl Manitius's edition : Syntaxis. Godfray : Lunar Theory.

CHAPTER VII Greek and Hindu Methods in Spherical Astronomy Here we shall reproduce from Sengupta's paper a compa- rative account of the Greek and ancient Indian methods in Spherical Astronomy and to bring out the independence of the Indian Astronomers on this subject. The views on this subject would necessarily differ from those of many European scholars such as Colebrooke and Bentley (early 19th century) to Kaye (early 20th century). Kaye wrote as follows in the Journal of Asiatic Society of Bengal 1919, No. 3. The methods by which (the rules) were obtained are buried in obscurity. Braunmühl¹ has stated "that the Indians were the first to utilise the method of projec- tion in the Analemma of Ptolemy." It is intended to present the Hindu methods as clearly as possible and to show that Braunmühl has not done sufficient justice to the Indian astronomers. As to Kaye, we shall show that his remark quoted above is due to the fact that he had to rely mostly on the English tran- slation of the Sūryasiddhānta of Burgess, and perhaps he had no access to the works of Bhāskara II (1150 A.D.), who was the first to explain the ancient Indian methods clearly. Greek and Hindu Methods in Spherical Astronomy Of the Greek methods in Spherical Astronomy, the history begins with elementary principles only from Euclid (300 B.C.). Even in Theodosius' Sphaerica² (about 153 B. C.) "there is nothing that can be called trigonometrical." Heath again says,

1. Heath, Greek Mathematics, Vol. II, p. 291. Braunmuhl, Geschichte der Trigonometrie, pp. 38-42.
2. Heath, Greek Mathematics, Vol. II, p. 250.

126 GREEK AND HINDU SPHERICAL ASTRONOMY "the early spheric did not deal with the geometry of the sphere as such, still less did it contain anything of the nature of the spheri- cal trigonometry. (This deficiency was afterwards made good by Menelaus's Sphaerica).¹ Hence the Greek spherical trigonometry began with Menelaus (90 A.D.). His theorem in geometry is well-known—"If the sides of a plane triangle be cut by a trans- versal into six segments, the continued product of any three alternate segments, is equal to the continued product of the remaining three." From this proposition he deduced the so- called "regula sex quantitatum" or the theorem, if the sides of a spherical triangle be cut by an arc of a great circle into six segments, the continued product of the chords of the doubles of any three alternate segments is equal to the continued product of the chords of doubles of the remaining three segments." In plane geometry if the sides BC, CA, AB of a triangle be cut by any transversal at L, M, N, respectively, then we have (BL / LC) · (CM / MA) · (AN / NB) = 1. In spherics the theorem is : (Chord 2 BL / Chord 2 LC) · (Chord 2 CM / Chord 2 MA) · (Chord 2 AN / Chord 2 NB) = 1 Both these theorems are proved in Ptolemy's Syntaxis ( Karl Manitius's edi- Fig. 5 tion, Vol. I, pp. 45-51). If R be the radius of the sphere on which the spherical triangle ABC is constructed, then the chord of the arc 2 BL = 2 R sin BL. Hence Menelaus's theorem in spherics may be expres- sed as follows : (Sin BL / Sin LC) · (Sin CM / Sin MA) · (Sin AN / Sin NB) = 1 This theorem is true for any spherical triangle. If ∠B = AN = AM = 90° and L the pole of AB, then LMN is a secondary to the arc AB. There are four arcs of great circles; taking any three as forming a spherical triangle and the fourth as the transversal we readily get for the right-angled

1. A.A. Bjornbo, "Studien über Menelaos' Sphärik" in Abhandlun- gen Zur Geschichte der Mathematischen Wissenschaften for 1902, pp. 89 et seq.; also Heath, Greek Mathematics, vol. II p. 261-73.

GREEK METHODS IN SPHERICAL ASTRONOMY 127 triangle ABC, the relations :-- (i) sin a=sin b sin A (ii) sin c=tan a cot A (iii) cos b=cos a cos c (iv) tan c=tan b cos A The above are some of the Napier's rules for a right-angled spherical triangle, deducible from Menelaus's theorem¹. They are generally sufficient in the case of such triangles. In any spherical triangle, however, this theorem of Menelaus does not in any single step lead to any of the equivalents of the time- altitude or altazimuth equations in spherical astronomy. The ancient Indian methods, though none of them are so highly finished as Menelaus's theorem, yet are not less powerful in tackling the problems that arise in astronomy in connection with the apparent diurnal motion of the heavens. The Greek or Ptolemaic method presents no further points of interest except in its application. We now proceed to illustrate the ancient Indian methods and shall refer to the Ptolemaic method as occasion arises. Ancient Indian Methods in Spherical Astronomy In the Indian methods there is no general rule to follow. It is by properties of similar right-angled triangles that a fairly complete set of accurate formulae are obtained. These right- angled plane triangles are classified under the names,—'Krānti- kṣetras' (triangles of declination) and 'Akṣa-kṣetras' (triangles of latitude). We consider the following problems :— Problem :— To find the time of rising on the equator of a length l, of arc of the ecliptic measured from the first point of Aries. Let ω be the obliquity of the ecliptic and R. A. the right ascension corresponding to the longitude l, and δ the correspond- ing declination. The Indian form of the equation is :

1. Three more can be deduced similarly, namely, (v) sin c=sin b sin C (vi) sin a=tan c cot C (vii) tan a=cos C tan b.

128 GREEK AND HINDU METHODS *R sin R.A. = (R sin l × R sin ω) / (R cos δ) , where R is the radius of the sphere. Note:— If R be the radius of the circle of reference, the Indian trigonometrical functions for the arc θ, are (1) the 'sine,' (2) the 'cosine' and (3) the versed sine. They are respectively equal to R sin θ, R cos θ and R vers θ. In the adjoining figure, O is the centre of the armillary sphere, YQ, YC are quadrants of the equator and the ecliptic, respectively. P is the celestial pole, PCQ the summer solsti- tial colure. Join OY, CQ, OP and OC. Let YS be=l, YM=R.A., CQ=∠SYM=ω, SM=δ. Join OS, OM. PSM is the secondary to the equator. [Fig. 6] From C draw CK perpendicular to OQ. From S draw Sm and Sn perpendicular to OM and OY, respectively. Join MN and from M draw MN perpendicular to OY. Then the triangles Smn and CKO are similar. They are called 'Krānti-kṣetras'¹ or declination triangles,—similar right- angled triangles having one acute angle=ω. ∴ Sm : Sn = CK : OC or R sin δ : R sin l = R sin ω : R R sin δ = (R sin l × R sin ω) / R ......(I)

The Āryabhaṭīya, Gola, 25. Varāhamihira, in the Pañcasiddhāntikā (IV. 92) states it in the form 2R √((R² Sin ²l)—R² Sin ²δ) / (2R cos δ) = R Sin R.A., which is evident from figure. Brahmagupta's equation is identical with that of Āryabhaṭa, (BrSpSi. III. 15, Sūrya-siddhānta III. 40-41. Also Bhāskara II, Grahagaṇita cap. VIII, stanzas 54-55, is in agreement with Varāhamihira's forms.

TIME OF RISING 129 Greek Method In the same figure¹ let PSC be the triangle and γMQ be the transversal. Then Menelaus's theorem gives (sin PM / sin MS) × (sin Sγ / sin γC) × (sin CQ / sin QP) = 1 or (1 / sin δ) × (sin l / 1) + (sin ω / 1) = 1 or sin δ = sin l × sin ω. Indian Method Again by the Indian method from the same two similar triangles we get mn : nS = OK : OC or, mn : R sin l = R cos ω : R ∴ mn = (R sin l × R cos ω) / R Again MN : mn = OM : Om i.e., R sin R. A. : mn = R : R cos δ ∴ R sin R.A. = (R sin l × R cos ω) / (R cos δ) Greek Method Take PQM for the triangle and γSC for the transversal. Then, (sin PC / sin CQ) × (sin Qγ / sin γM) × (sin MS / sin SP) = 1 or (cos ω / sin ω) × (1 / sin R.A) × (sin δ / cos δ) = 1 or sin R.A. = tan δ cot ω, The Indian form of the equation is different from that of Ptolemy's. It is also better for the purpose of calculation. Note :—From the same two similar triangles we have On : ON = R cos δ ; R ∴ On : R cos l = (R cos R.A. × R cos δ) / R ......(3) Again, tan R.A. = mn / on = (R sin l × R cos ω) / (R × R cos l) ......(4) Again, mn : Sm = OK : KC

1. Manitius' Edition of Syntaxis, I, 51-53.

130 GREEK AND INDIAN METHODS or mn = (R sinδ × R cosω) / (R sin ω) ∴ R sin R. A. = (MN / mn) × mn = (R / R cosδ) × ((R sinδ × R cosω) / (R sin ω)) (5) Problem II :— Sidereal Time-intervals Indian Method The problem discussed above provides the method of find- ing the sidereal time-intervals in which the signs of the zodiac rise on the equator. To find the corresponding times at any latitude ϕ, it is necessary to calculate and apply what is the ascensional difference due to the elevation of the celestial pole. This ascensional difference is called 'carakāla' or the variation in the length of half the day. The 'sine' of this 'carakāla' is called 'carajyā.' If ch denotes this 'carakāla,' then.¹ R. Sin ch = (R sinϕ × R sin δ × R) / (R cos ϕ × R cos δ) Just as in the solution of the previous problem, the dec- linational triangles or 'Krānti Kṣetras' were constructed and used, so in the solution of this and other problems another set of similar triangles were conceived and constructed and were given the name 'Akṣa kṣetras.'² Let NPZH be the meri- dian (Fig. 7), NOH the north- south line passing through the observer O,P the celestial pole, OQ the trace of the equator on the meridian plane, Z the zenith. Join OZ. From Q draw QM perpendi- cular to OZ. Then the triangle QOM is an 'Akṣa- kṣetra' or a latitudinal right- angled triangle, as ∠QOM = ϕ, the latitude of the station. Another 'Akṣakṣetra' is thus conceived, in the same figure, let P, P' be the north and south celestial poles, N, the north point, AB A'B' the diurnal Fig 7

1. Āryabhaṭīya, Gola, 26; Pañca-siddhāntikā, IV, 34; Brāhmasphuṭasiddhānta. II, 57-58; Sūrya-siddhānta, II, 91; Grahagaṇita, VIII, 48-49.
2. Bhāskara, Golādhyāya ( Wilkinson and Bāpudeva Śāstri's tr. ) PP. 173-76; also, Bhāskara, Grahagaṇita, Ch. IX. 13-17.

SIDEREAL TIME-INTERVALS 131 circle of a heavenly body with declination δ, NEHW the horizon, PEP′ W the six O′ clock circle. Here AA′ the line of intersection of the diurnal circle with the horizon is called the “udayāsta-sūtra”¹ (or the thread joining the rising and setting points). SS′ the line of intersection of the diurnal circle and the six o′ clock circle, is the horizontal diameter of the diurnal circle. From S draw SK and SL perpendiculars respectively to AA′ and EW. Join KL. Now since PN=ϕ, the latitude of the station, in the small right-angled triangle KLS, the ∠ KLS is also=ϕ. ∴ SK : SL=QM : MO or SK = (SL × QM) / MO = (R sin δ × R sin ϕ) / (R cos ϕ) Now SK² is a “sine” in the small circle AB A′B′ of which the radius is R cos δ; this “sine” reduced to the equator (radius R) is the ‘sine’ of cara. ∴ R sin ch = R sin EPA = (R sin δ × R sin ϕ × R) / (R cos ϕ × cos δ) Greek Method Let³ the arc PA be produced to meet the equator at C. Take PCQ′ for the triangle and EAN for the transversal. Then we get, (sin PA / sin AC) × (sin CE / sin EQ′) × (sin Q′N / sin NP) = 1 or (cos δ / sin δ) × (sin CE / 1) × (cos ϕ / sin ϕ) = 1 ∴ sin CE = sin ch = (sin ϕ × sin δ) / (cos ϕ × cos δ) Note—The perpendicular distance between AA′ and Ew is called the ‘sine’ of the amplitude or the ‘Agrā’ which is thus calculated :— KL : LS = QO : OM ∴ ⁴R sin amplitude = ‘Agrā’ = KL = (LS × QO) / OM = (R sin δ × R) / (R cos ϕ) It is now evident that the ancient Indian method is different ────────────────────────────────────────

1. Bhāskara, Gola, VII, 39.
2. This is called by the name ‘kujyā’ or ‘kṣitijyā’. i, e.. earth-sine. Āryabhaṭa, Gola, 26, Brahmagupta, II, 57, Sūrya-siddhānta. II, 61 etc.
3. Manitius, ibid, p. 84.
4. Āryabhaṭa, Gola. 30, etc.