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

Chapter IV Brahmagupta’s Originality in the Khaṇḍakhādyaka Sengupta in his Indroduction to the Commentary of the Khaṇḍakhādyaka has discussed this point. We shall reproduce here some of the points mentioned by him. Brahmagupta’s Khaṇḍakhādyaka (i) Brahmagupta does not accept the system of Āryabhaṭa but has simplified it in the Khaṇḍakhādyaka proper ; and here he has given the system which he thinks to be correct. Uttar Khaṇḍa Khādyaka (ii) In the Uttara-Khaṇḍakhādyaka, he has further correc- ted some of his results, given earlier in the Khaṇḍakhādyaka proper. In the proper Khaṇḍakhādyaka, Brahmagupta assignes to the longitude of the Sun's apogee the value 80º, whereas in the Uttara text he corrects it to 77º (UKK. 4) : As the process of finding the apparent places of planets as given by Āryabhaṭa does not make them agree with observation, I shall, therefore, speak of this process. Of the Sun the apogee is at two signs and seventeen degrees (2 signs 17º=60 plus 17 degrees=77º).¹ Compare this with the value given in the Khaṇḍakhādyaka proper (I.13)² : The longitude of the Sun's apogee is 80º [KK.I,13] (The Sun's apogee is 80º or two signs plus 20 degrees) inocco means

1. न स्फुटमार्य्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये । भानुमतो मन्दोच्चं राशिद्वयमंशकाश्च सप्तदश ॥ UKK. IX 4
2. भागाशीतिरितिनोच्चं शशिनः पादोनकृत शरकृतोनाद् । भगणादि द्विस्त्रिर्दैर्वसुशुनव यम नव गुणैः सकलम् ॥ KK. I, 13

96 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA mandocca of the Sun). The value given in the Pañcasiddhān- tikā, IX.7-8) is also the same. Let us compare it with the present value. According to the astronomical constants as given in the Conn. des Temps., the longitude of the Sun's apogee in 499 A.D. (i.e. 1,400 years before 1900 A.D.) was —77° 19′19.44″ according to Conn.des Temps’ equation. —76° 40′37.22″ according to Newcomb's equation. The mean of these two values is very nearly 77⁰ as given by Brahmagupta in the Uttara text. Thus the value given by Brahmagupta is more correct than the value given by Āryabhaṭa. The Āryabhaṭīya gives the value 78⁰ which is less correct. Brahmagupta more correct than Āryabhaṭa (iii) Brahmagupta detected that Āryabhaṭa had made the Moon's apogee quicker and nodes slower, than they really are. In both the cases, Brahmagupta made rather an over-correction. We shall give the extract from Uttara-Khaṇḍakhādyaka in this connection : Multiply the ahargaṇa by 110, increase the product by 511 and divide by 30, 31; subtract the result taken as revolutions, etc., from the mean Moon; the final result is the Moon's apogee.¹ Evidently Brahmagupta assumes that the anomalistic month = 3031/110 days. This convergent to the anomalistic month was known to the author of the Vasiṣṭha Siddhānta as summarised in the Pañcasiddhāntikā² (II-2-6). According to Brahmagupta, the length of the anomalistic month = 1582236450000—4320000000 days. (BrSpSi. I.15,16,18, ───────────────────────── and 20) 577533000000—488105858 = 27.55454641 days which is for 1900 A. D. = 27.5545502 days according to Radau. = 27.554602 according to the Āryabhaṭīya.

1. द्युगणात् ख रुद्र गुणिताद् भवशरयुक्ताच्छशित्रिखाग्नि हृतात् । भगणादि फलं शोध्यं घसचन्द्राच्छशाङ्कोच्चम् ॥ (UKK. IX. 5)

UTTARAKHAṆḌA KHĀDYAKA 97 Here also Brahmagupta is more accurate. Again, the length of the sidereal period of the Moon's apogee = 1577918450000 / 488105858 days = 3232.732048 days. Āryabhaṭa's value of the same is 3231.987844 days, and the modern value is 3232.3754 days. Hence Brahmagupta's result is by 0.3566 of a day out, while Āryabhaṭa's is by 0.3876 of a day in. Further in the Uttara-Khaṇḍakhādyaka(IX.10) we have : Deduct 354½ from the ahargaṇa, divide the remainder by 6792 ; subtract the quotient that is obtained in revolutions etc. from the circle : the result is the longitude of the ascending node.¹ (IX.10) Here Brahmagupta gives the approximate period of the sidereal revolution of the Moon's node to be=6792 days. This according to his Brāhmasphuṭasiddhānta = 1577916450000 / 232311168 days = 6792.25396 days, which according to Lockyer would be 6793.39108 days and according to the Khaṇḍakhā

98 UTTARAKHAṆḌA KHĀDYAKA one-sixteenth.¹ This stanza says that in 499 A.D., Mars's aphelion point had a longitude of 127⁰ ; of Jupiter the longitude of the aphelion was 170⁰. ( KK.II. 6²) According to Newcomb's rule, the longitude of the aphelion point of Mars in 499 A. D. works out to have been = 128⁰28'12''. According to the Conn. des. Temps' rule, the same was 128⁰27'51''. Hence Brahmagupta's determination of Mars's aphelion is correct within 1⁰30' and is therefore, quite satisfactory. According to the Khaṇḍakhādyaka proper it was 110⁰, and according to the Āryabhaṭīya 118⁰. Of planets, beginning with Mars, the degrees of longitude of the apogees are respectively 11, 22, 16, 8 and 24, each multiplied by 10. (KK .II. 6) Thus the longitudes of apogees of Mars = 110 (3 signs 20⁰); of Mercury = 220⁰ (7 signs 10⁰) ; of Jupiter = 160⁰ (5 signs 10⁰); of Venus = 80⁰ (2 signs 20⁰) and of Saturn = 240⁰ (8 signs). Compare these values with those given in the Pañcasiddhāntikā XVII. 2 (the Sūrya-siddhānta). Again according to this stanza Jupiter's aphelion had a longitude of 170⁰ in 499 A.D. According to Conn. des. Temps' rule the same was 170⁰25'. Thus here too, Brahmagupta is very accurate. According to the Khaṇḍakhādyaka proper, Jupiter's aphelion had a longitude of 160⁰ (KK. II.6) and according to the Āryabhaṭīya, the value was 180⁰. Brahmagupta First to Use Second Differences All these illustrations reproduced here very well establish the point that the great Indian astronomers from Āryabhaṭa I to Brahmagupta were aware of the methods of separating the two distinct planetary inequalities, viz., that of the apsis and of conjunction in the cases of the five ‘star’ planets (PSi. Intro- duction Lii). In the Khaṇḍakhādyaka, Brahmagupta having given the “sines” and the equations of the Sun and the Moon

1. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः । सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् ।। पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ।। —UKK. IX. 11
2. मन्दांशा दशगुणिता रुद्रा द्वियमाश्व षोडशाष्टजिनाः । —KK. II. 6

BRAHMAGUPTA FIRST TO USE SECOND DIFFERENCES 99 at the interval of 15° of arc of the mean anomaly, in the Uttara Khaṇḍakhādyaka teaches, for the first time in the history of mathematics, the improved rules for interpolation by using the second difference. This very important feature I am reproducing here from the translation by Senagupta of the verse ¹ (UKK. 8) : Multiply the residual arc left after division by 900' (i.e by 15°), by half the difference of the tabular difference passed over and that to be passed over and divide by 900' (i. e. 15°) : by the result increase or decrease, as the case may be, half the sum of the same two tabular differences ; the result which, whether less or greater than the tabular difference to be passed, is the true tabular difference to be passed over. (UKK. 8) The rule given here applies to the case of all functions hitherto considered in the Khaṇḍakhādyaka, which are tabulated at the difference of 15° of arc of the argument. They are : (i) the tabular differences of the Sun's equation, (ii) the tabular differences of the Moon's equation. (iii) the tabular differences of the 'sines'. Sengupta has illustrated the rule by an example belonging to the table of sines. Illustration—To find the 'sine' of 57°. Brahmagupta's table of sines in the Khaṇḍakhādyaka is as follows : Thirty increased severally by nine, six and one; twenty- four, fifteen and five, are the tabular differences of sines at intervals of half-a-sign. For any arc, the 'sine' is the sum of the parts passed over, increased by the proportional part of the tabular difference to be passed over.² (*KK.*I.30 ; also III.6)

1. गतभोग्य खण्डकान्तरदलविकलवधात् शतैर्नवभिराप्त्या । तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम् ॥ —UKK. IX. 8
2. त्रिंशत् सन्नवरसेन्दुजिन (तिथि) विषयागृहार्द्धं चापानां । भर्द्ध्ज्या खण्डानि ज्या भुक्तैक्यं सभोग्यफलम् ॥ —KK. I. 30; also III. 6

100 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA This can be shown in the tabular form thus :

Now 57° = 3420 minutes = 900' × 3 + 720'. Thus three of the tabular differences are considered as passed over ; the last one being 31 and the one to be passed over is 24. The true tabular difference by the rule, for arc 57°, = (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 Hence the 'sine' of 57° = 39 + 36 + 31 + (720 / 900) [ (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 ] = 125.76 As worked out from the logarithm tables the same comes out to be 125.80. Again 'sine' of 57° from Brahmagupta's formula = 106 + (720 / 900) × 24 + ((31 - 24) / 2) × (720 / 900) - [(720 / 900)] × (31 - 24) / 2 = 106 + (720 / 900) × 24 + (720 / 900) { (720 / 900) - 1 } × (24 - 31) / 2 This in fact is the modern form the interpolation equation up to the term containing the second difference. Brahmagupta thus takes a decidedly improved step here and is undoubtedly the first man in the history of mathematics who has done this. One should also remember that in the case where the function is not tabula- ted at a constant interval, Brahmagupta's rule is remarkable.

BRAHMAGUPTA AND SINE RULE 101 Brahmagupta First to Introduce Sine Rule in Indian Plane Trigonometry In this connection, we shall reproduce the following verse from the Khaṇḍakhādyaka : Multiply the ‘sine’ of the (Śīghra) anomaly by the ‘sine’ of the maximum Śīghra equation and divide by the ‘sine’ of the corresponding Śīghra equation, the result is the ‘Śīghra hypotenuse’ when the (Śīghra) anomaly is half a circle, this śīghra hypotenuse is equal to the radius diminished by the ‘sine’ of the maximum equation ; when the anomaly is equal to the whole circle, the same is equal to the radius increased by the same ‘sine’ of the maximum equation.¹ Let S, E and P be the positions of the Sun, the Earth and the planet, say Mars, respecti- vely. Complete the paralle- logram SEMP ; with M as centre and MP as radius, describe a circle. This circle is the epicycle of conjunction of Mars. Produce EM to cut this circle at K. The ∠PMK = ∠S′SP (the point S′ is on [Fig. 4] ES produced ), the angle gained by the Earth over Mars since the preceding conjunction. The ∠PMK is called the śīghra anomaly or anomaly of conjunc- tion. We take EM=360, and MP=234. The ∠PEM, which is equal to ∠PES, the annual parallax of Mars, is called the śīghra equation. The ∠MPE is equal to the ∠SEP, the elongation. The ∠PMK is given, and PM and ME are also given. Hence in the triangle MPE, we have tan ½ (P−E) = (EM−MP)/(EM+MP) tan ½ PMK = 126/594 tan ½ PMK ∴ L tan ½(P−E) = log [126/594] + L tan ½ PMK We have also ½ (P+E) = ½ ∠PMK

1. केन्द्रज्याऽन्त्यफलज्या गुणिता फलजीवया हृताकर्णः । त्रिज्यास्य फलज्योना चक्रार्द्धे संयुता चक्रे ॥ KK. VI. 1.

102 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA Now log [ 126 / 594 ] = 1.3265841. The values of the ∠PMK and the ∠PEM and Brahma- gupta's values as given in the verse¹ are presented below in a tabular form : | ∠PMK = | 28° | 60° | 90° | 121° | 135° | 148° | 164° | 173° | | ∠PEM = | 10°58′ | 23°1′ | 33°1′ | 39°56′ | 40°23′ | 37°31′ | 25°32′ | 12°35 | | Brahmagupta's ∠PEM = | 11° | 23° | 33° | 40° | 40°30′ | 37°30′ | 25°30′ | 12°30′ | It will be seen that Brahmagupta gives the values of the equation within 1/8th of a degree. It seems inexplicable why such discrepancies should remain in Brahmagupta's calculations. It is probable that he wanted to state his equations to the nearest half a degree. Now we shall take up the Śīghra equations of Mars, and then revert to the Sine Rule. We have in the Khaṇḍakhā- dyaka : Mars, by the degrees of Śīghra anomaly (i.e. anomaly of conjunction) of 28 getting at the corresponding equation of 11° rises (heliacally) in the east ; by the next 32° gets 12° more of the equation ; by the next 30°, 10° more ; by the next 31°, 7°, more ; by next 14°, half a degree ; these are positive ; by the next 13°, nega- tive 3° ; by the next 16° ; negative 12° after this he is retrograde ; by the next 9°, negative 13° ; by the next 7°, negative 12½°. After this the parts of the equations occur in the reverse order¹. On the basis of this we have the following table of the Śīghra equations for Mars : | Degrees of anomaly | Equation of | Phenomena | | of conjunction | conjunction | | | 0° | 0° | Motion direct. | | 28° | +11° | Rises in the east. |

1. भौमोऽष्ट्यग्नौ रुद्रान् भुक्त्वा पूर्व्वौदितोऽदरैर्कान् । ख_णैर्दशरूपगुणैः सप्तांशा मनुभिरर्द्धांशान् ॥ धनम् 'मग्नि शशाङ्कैस्त्रिनष्ट्या भास्करात्नतो वक्री । नवभिश्च योदशनगैर्द्वादशसार्द्धान् विलोमोऽतः ॥ KK.II.8-9

BRAHMAGUPTA AND SINE RULE 103 Degrees of anomaly Equation of Phenomena of conjunction conjunction ───────────────── ─────────── ───────── 60° 11+12=+23° 90° 23+10=+33° 121° 33+ 7=+40° 135° 40+ ½=+40°30' 148° 40°30'—3°=+37° 30' 164° 37°30'—12°=+25°30' Retrograde motion begins. 173° 25°30'—13°=+12°30' 180° 12°30—12°30'=0°0' 187° —12°30' 196° —25°30' Direct motion begins. 212° —37°30' 225° —40°30' 239° —40° 270° —33° 300° —23° 332° —11° Sets in the west. 360° 0° Now we come back to our discussion on the verse VI.1. The Śīghra hypotenuse spoken of here is EP, when SP or EM is taken to be R ; when ∠PEM is a maximum, PM is its 'sine'. It would be seen from the figure that R sin PMK × PM EP = ──────────────────── Rsin PEM This may again be written as EP PM ───────────── = ────────────── sin PMK sin PEM This is equivalent to the sine rule for a triangle in plane trigonometry. Brahmagupta is here seen to be the first person to give it in Indian mathematics. This expression reminds us of the famous relationship in respect to triangle ABC : a b c ─────── = ─────────── = ─────── sin A sin B sin C

104 BRAHMASPHUṬASIDDHĀNTA'S KHAṆḌAKHĀDYAKA Brahmagupta corrects Dimensions of the Epicycle of Apsis Brahmagupta corrects the dimensions of the epicycles of apsis of the Sun and Moon by—1/42 nd part and 1/48th parts respectively. The reference may be made to the following verse in the Uttara Khaṇḍakhādyaka : The Sun's equations are to be made less by dvikṛtāṁ- śonam (1/42nd) and the Moon's equations, increased by vasuvedabhāgayutam (1/48th). Multiply the Sun's equa- tion a planet's daily motion in minutes and divide by the number of minutes of a whole circle and this is called Bhujāntara correction and applied in the same way to the planet as the equation is applied to the Sun.¹ The Sun's epicycle of apsis has the dimension 14° in the Khaṇḍakhādyaka proper. With the correction introduced here, the value becomes 14° {1 - 1/42} = 13° 40'. The correction to the Moon's equations would make the epicycle's dimension changed from 31° to 31° {1 + 1/48} = 31°38'45''. Pṛthūdaka's commentary further corrects it to 31°(1 + 1/52) = 31°35' Brahmagupta 's correction to Saturn's epicycle of apsis is—1/5th part and that to the Śīghra epicycle of Mercury 1/16th part as seen in the verse : Of Mars the apogee (the aphelion point) is to be increased by 17°, that of Jupiter by 10°; from the Śīghra of Venus 74' are to be subtracted ; Saturn's equation of apsis should be decreased by its one-fifth and the Śīghra equation of Mercury should be increased by one-sixteenth.² In the Khaṇḍakhādyaka proper (II.6), we have been given the longitudes of the apogee of planets : Mars 11°, Mercury 22°, Jupiter 160°, Venus 80° and Saturn 240°. Now with these correc- tions introduced in the Uttara Khaṇḍakhādyaka in the above

1. द्विकृतांशोनं रविफलमिन्दोर्वसुवेदभागयुतम् । अर्कफलभुक्तिघाताद् भगणकलाप्तं भुजान्तरं रविवत् ॥ UKK. IX. 9
2. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः । सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् । पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ॥ UKK.IX.11

DIMENSIONS OF THE EPICYCLE OF APSIS 105 verse, the aphelion point of Mars in 499 A.D. had a longitude of 110 plus 17=127°; of Jupiter 160 plus 10=170°. According to Newcomb's rule, the longitude of the aphelion point of Mars in 499 A.D. works out to have been 128°28'12''. The same according to Conn. des Temps rule would be 128°27'51''. Hence Brahmgupta's determination of Mars's aphelion is correct within 1°30', and may be, therefore, regarded as very satisfactory. According to the Khaṇḍakhādyaka proper this value was, as already said, 110°, while according to the Āryabhaṭīya, it was 118°. The same may be said regarding Jupiter's aphelion. Accord- ing to the Khaṇḍakhādyaka proper the value of its longitude is 160°, according to the Āryabhaṭīya it was 180°; according to Conn. des. Temps' rule, it would be in 499A. D. 170°25', and the value given by Brahmagupta in the Uttara Khaṇḍakhādyaka, it is 170°. We have thus shown by many illustrations the important corrections introduced by Brahmagupta in his Khaṇḍakhādyaka specially the Uttara part. Brahmagupta was highly original in his methods of calculations, accuracies and interpolations. He intro- duced new ideas in mathematics. He went much ahead Āryabhaṭa in many details. He so many times did not follow Āryabhaṭa in calculations. In the Khaṇḍakhādyaka proper, his treatment of parallax in the calculation of solar eclipses is different from that of Āryabhaṭa. The methods followed here are the same as propounded by him in the Brāhmasphuṭasiddhānta.¹ Senagupta is right when he says : As has already been remarked, these corrections and innovations in the Uttara Khaṇḍakhādyaka paved the way for the acceptance of his great work the Brāhmasphuṭasiddhānta as a standard work on astrono- my by the western Indian school of astronomers. The directness of the treatment of topics, and the simplicity of calculations taught in the Khaṇḍakhādyaka made it very neat handbook for the beginner. These two works of Brahmagupta were perhaps the only astronomical works in circulations in western India when the Arabs conquered Sind early in the eighth century

1. On Parallax— नाडी चतुष्कविधिना सर्वत्र समो यतस्ततः स्थूलः । मानार्थं कर्म महत् कृतमार्यभटेन लघुनि सति ॥ BrSpSi. XI. 33

106 BRHĀMASPHUṬASIDDHĀNTA'S KHAṆḌAKHĀDYAKA (712 A.D.) and the new conquerers learnt Indian astronomy and mathematics from these works as has been observed by Sachau. Alberuni who came to India early in the 11th centuty of the Christian era, learnt Indian astronomy chiefly by studying the Khaṇḍakhādyaka and the Bṛhat-Saṁhitā of Varāhamihira, and both of them with the help of commentary of Bhaṭṭotpala. —:c:— Reference P.C. Sengupta : The Khaṇḍakhādyaka, 1934.

CHAPTER VI Indian Luni– Solar Astronomy In this chapter, it is proposed to give an account of astrono- mical constants and the equations in Indian luni-solar astronomy and to present a comparative view of these quantities with the corresponding ones in Greek and modern Astronomy. This acco- unt has been reproduced from P. C. Sengupta's. Appendix I of the Khaṇḍakhādyaka It has been shown that in many cases the Indian values of these constants are more accurate than the Greek values. and in Indian lunar astronomy the equations or inequalities discovered are the most startling. Solar Astronomy In solar astronomy the length of the year was determined by Āryabhaṭa¹ from the heliacal risings of some bright star at the intervals of 365 and 366 days. (1) The year according to the Āryabhaṭīya = 1577917500 / 4320000 days = 365.2586805 days, = 365 da. 6 hrs. 12 mins. 29.64 secs. (2) The same = 1577917800 / 4320000 days = 365.25875 days. = 365 da. 6 hrs. 12 mins. 36 secs., according to the Khaṇḍakhādyaka, the Sūryasiddhānta of Varāha and the modern Sūryasiddhānta. (3) It is = 1577916450 / 4320000 days = 365.2584375 days, = 365 da. 6 hrs. 12 mins. 9 secs., according to the

1, P. C. Sengupta, "Aryabhata's Method of determining the Mean Motions of Planets," Bulletin of the Calcutta Mathematical Society. Vol. XII, No. 3.

108 INDIAN LUNI-SOLAR ASTRONOMY Brāhmasphuṭa Siddhānta of Brahmagupta. Now the mean sidereal year =365 da. 6 hrs. 9 mins. 9.3 secs. (Lockyer). The mean anomalistic year =365 da. 6 hrs. 13 mins. 49.3 secs. (Lockyer). The mean tropical year =365 da. 5 hrs. 48 mins. 46.054 secs. (Lockyer). Though we take that Indian year was designed to be the sidereal year, it approached most closely the anomalistic year ; and its excess over the sidereal year was about 3 minutes. From this consideration it appears that the Indian astronomers were justified in taking the Sun's apogee to be fixed. Against the error of +3 min. in the Indian sidereal year, we may point out that— (1) The Hipparchus-Ptolemy tropical year =365 da. 14' 48" in sexagesimal units,¹ =365 da. 5 hrs. 55 min. 12 secs., which has an error of about +6 min. (2) Meton's sidereal year = [ 365 + ¼ + 1/76 ] days² =365 da. 6 hrs. 18 min. 57 secs. which has an error of +9 min. 48 secs. nearly. (3) The Babylonian sidereal-year was 4½ min. too long.³ Thus the Indian value of it is closer to the true value. Again in 150 A.D. the longitude of the Sun's apogee according to the Conn. des Temps was =101° 13' 15". 17-6189". 03 [ (1900—150) / 100 ] =1". 63 × [ (1900—150) / 100 ]² =71° 16' 26". 37 while Ptolemy states it to be 65° 30' which was wrong by—5° 36' 27" :

1. Syntaxis, Karl Manitius's edition, Vol. I. p. 146.
2. Ibid, p. 145.
3. Encyclopaedia Britannica, History of Astronomy.
4. Syntaxis, Vol. I. p. 148. The Romaka Siddhānta of the Pañca-siddhāntikā, VIII. 2, indicates the Sun's apogee to be at longitude of 75° ; this was per- haps a correction made by Lāṭadeva to the Greek constant.

LUNAR ASTRONOMY 109 In 500 A. D (Āryabhaṭa's time) the longitude of the Sun's apogee by the same rule works out to be=77°19'19.44". Āryabhaṭa states this to be 78° in the Āryabhaṭīya, Brahmagupta in the Uttarādhyāya of the Khaṇḍakhādyaka states the same to be 77°, while the Khaṇḍakhādyaka gives it as=80°. Hence the Indian findings of the longitude of the Sun's apogee were more accurate. Again as to the Sun's equations of the centre we find that the Āryabhaṭīya states the periphery of the Sun's epicycle to be 13°30', The Khaṇḍakhādyaka gives it as 14° ; while according to the Indian form, Ptolemy's value of the same is 15°. Hence according to these writers, the Sun's equations at 90° of the mean anomaly are :— According to the Āryabhaṭīya =2° 8' 54". ” ” Khaṇḍakhādyaka=2° 1' 40". ” ” Brahmagupta=2° 7' 20". ” ” Ptolemy=2° 23' 3". The modern value =1° 55' 97". Thus the Indian equations of the Sun are in general by more correct than the Greek ones. The Indian constants in solar astronomy are thus, generally, more accurate than the Greek ones. We now turn to the Indian Lunar astronomy. Lunar Astronomy Before discussing the constants in Indian lunar astronomy it is necessary to state something as to the time when the Moon was observed by our ancient astronomers and the astronomers from Āryabhaṭa I (499 A.D. to Pṛthūdaka Svāmī (864 A.D.). The months were reckoned from the first visibility of the crescent at the time of the Mahābhārata (1400 B. C.). We have a pass- age in the Bhīṣmaparva where Vyāsa speaks of the evil omens on the eve of the Kurukṣetra war thus— चन्द्रसूर्य्यावुभौ ग्रस्तावेकामासीं त्रयोदशीम् । "That the Moon and the Sun have been both eclipsed on the 13th days of the light and dark halves of the same month." The eclipses could not take place on the 13th days of the month unless the months were reckoned from the first visibility of the crescent. This was the custom in Babylonia and it has still survived in the Mahomedan world. Even in the Pañca siddhāntikā of Varāhamihira ( 540 A. D. ), there is a special

110 INDIAN LUNI-SOLAR ASTRONOMY chapter on शशि-दर्शनम् or the fiirst visibility of the crescent. It is thus clear that the practice was to observe the Moon when very near the Sun. Again Āryabhaṭa says that 'रवीन्दुयोगात् प्रसाधितश्चेन्दुः', "the Moon was determined from her conjunctions with the Sun." The Moon was observed by him at the time of solar eclipses, or at the time of the first visibility of the crescent. Even up to the time of Pṛthūdaka, the accuracy in lunar astronomy was chiefly aimed at the time of eclipses. Thus in his commentary on the Khaṇḍakhādyaka. IV, he makes the following introductory remarks :— "All knowledge relating to (luni-solar) astronomy is desired by the wise (or cultured) specially for knowing the right instants of opposition or conjunction ; these instants are, however, not visible to the eye. Of other things such as tithis, nakṣatras and Karaṇas, as the planets, the Sun and the Moon, are not clearly observed, their beginnings and ends are not visible. Men see the agreement between calculation and observation at the times of solar and lunar eclipses. Hence the word of the astronomers is esteemed amongst men even in respect to such things as tithis, etc."¹ Thus the chief aim of the ancient Indian astronomers was to calculate the eclipses accurately and the Moon was observed chiefly at lunar or solar eclipses, though the time for observation related also to the fiinding of the first visibility of the crescent. This latter phenomena did not perhaps lead them to directly observing the Moon's position at such times by using instruments. Moon's Mean Motion The practice of observing the Moon at the time of the eclipses alone led to the determination of the synodic month with the following results :— (i) Mean synodic month according to the Āryabhaṭīya = 1577917500 / (57753336—4320000) days, =29.530582 days.

1. बाहुल्येन पर्व्वज्ञानार्थं सकलं ज्ञानमिष्यते शिष्टैः । तेषां च पर्व्वणां दर्शनं नास्ति । अन्येषामपि तिथिनक्षत्रकरणानां तस्मात् तेषां शशिभास्करयोरव्यक्तित्वात् । शशिभास्कर- ग्रहणयोर् 'ग् गणितैक्यं लोकः पश्यति । तस्मात् तिथ्यादिष्वप्यर्थेषु दैवज्ञं वाक्यं लोके आद्रियते ।

MOON'S MEAN MOTION 111 (ii) The same according to the Khaṇḍakhādyaka =29.5305874 days. (iii) The same according to the Brāhma-sphuṭa-siddhānta =29.530582 days. (iv) The same according to Ptolemy=29 da. 31′ 50″ 8″ 20″ in sexagesimal units=29.5305927 days The modern value according to Newcomb and Radaú =29.5305881 days. Hence the Khaṇḍakhādyaka mean length appears to be the closest approximation. The mean sidereal month must have been deduced from the mean synodic month and the year adopted. Hence no compari- son need be made of this element here. We will now consider the sidereal periods, the nodes and the apogee of the Moon. These are shown below :-

According to | Sid. Per. of Moon's Apogee | Sid. per. of the Ascending Node

Āryabhaṭīya | 3231,987079 da. | 6794.749511 da. Khaṇḍakhādyaka | 3231.987844 da. | 6794.750834 da. Brāhma-sphuṭa - Siddhānta | 3232.73411 da. | 6792.25396 da. Ptolemy | 3232.617656 da. | 6796.45571 da. Modern values (Lockyer) | 3232.37543 da. | 6793.39108 da.

Here also the Indian values show a closer approximation to the true values, Brahmagupta's figures representing the nearest approach. Other Constants So far the Indian values of the constants have been more accurate than the Greek ones ; but as to the inclination of the Moon's orbit the Greek value is more accurate than the Indian value. Inclination of the lunar orbit Indian value=4°30′.

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.