पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

III.29 III. PAULIŚA-SIDDHĀNTA 69 with the expression, "Salutation to the Ketus." From the singular in the mantra of their invocation, Ketum kṛṇvannaketave, one may think that Ketu is referred to in the singular also. It is not so. Here the word does not mean the 'Comet' Ketu at all. The mantra itself is in praise of the Sun and Ketu here means the activity caused by the Sun in the sleeping inactive world. How then is this mantra used to invoke the Ketus? The utterance of the word Ketu here is sufficient, as the utterance of the many śa sounds in the Mantra śaṃ no devīḥ etc. (Ṛgveda 10.9.4) is sufficient to propitiate Śanaiścara. though the mantra itself refers to the water-deities, or the utterance of the word mayūra in the mantra, āmandravir intra haribhiḥ etc. (Ṛgveda 3.45.1) is sufficient to propitiate Subrahmaṇya, though the mantra refers to Indra, as also words like, Om, atha kalyāṇa etc. cause auspiciousness by their mere utterance. So, according to the Śāstras, Ketu refers only to the Dhūmaketus, and Rāhu is both the nodes, the recent application of the term Ketu to the descending node being un- warranted. Therefore, when the Dharmaśāstras enjoin the eclipses caused by Rāhu as sacred periods, they take in both the ascending and the descending nodes, and the suggestion by some that they do not take in the descending node on the score of some un-informed people calling it Ketu is wrong because what they call Ketu is really Rāhu. [चन्द्रविक्षेपः] वक्त्रादधिकश्चन्द्रो हीनः पुच्छाच्च याति भगणोदक् । हीनो वदने पुच्छेऽधिकेऽ(सु) राधाति दक्षिणतः ॥ ३० ॥ भागनवत्या राहोश्चन्द्रोऽन्तरितोऽतिमहति विक्षेपे । लिप्ताशतद्वये [ना] त्यशी(त्याऽ)नुपातोऽतोऽन्यत्र ॥ ३१ ॥ Moon's latitude 30. If the Moon lies between the Head and the Tail it is north of the ecliptic, (i.e. its latitude is north). If it lies between the Tail and the Head it is south of the ecliptic, (i.e., latitude is south). 31. The latitude is a maximum equal to 280 minutes when the Moon is 90 degrees distant from either Head or Tail. The latitude is to be found by pro- portion, at other places, using the distance in degrees from Head or Tail, whichever is nearer. Example 14. The true Moon is rā. 2-7-0. Rāhu's Head is rā. 5-3-0. Find the latitude of the Moon. The Tail is Head + rā. 6-0-0- = rā. 11-3-0. The Moon is between Tail and Head. Therefore the latitude is south. The distance of the Moon from the nearer limb, Head is rā. 5-3-0 − rā. 2-7-0 = rā. 2-26-0 − 86 degrees. For 90° the latitude is 280'. For 86° latitude is 86° × 280' ÷ 90° = 267½', South, as seen. 30a. B. चक्रादधिक b. A.B1.पृच्छाच्च. B.°पोदृक् c. A1.B1.पुंछे; C.वदनात् पुच्छाधिको d. A.B.°धिकोमुराघाति; C.°धिकोऽसुराद्याति; D.धिकोऽमकाद्याति 31b. B.°तोभिमहति c-d. A. येत्यशीतिमनुपातोन्पत्र ।; B. द्वयेत्यशीतमनुपातोन्पत्र ।; C. द्वयाधिकसप्ततिरनुपाततोन्पत्र ।; D. द्वय[मे] त्यशीतिमनपातोतोन्पत्र ।

70 PAÑCASIDDHĀNTIKĀ III.31 The rule is explained thus: The Moon moves in its own orbit, inclined to the ecliptic at an angle equal to the maximum latitude. Hindu astronomy assumes this motion to be on the ecliptic itself and gives the Moon's longitude, because there is only a maximum difference of 7'. Between the ascending and the descending nodes the orbit is north of the ecliptic and therefore the latitude measured on the great circle perpendicular to the ecliptic and passing through the Moon, is north. Between the descending node and the ascending, the orbit is south and the latitude is south. The distance between the node and the Moon, the latitude and the angle of inclination forming a spherical triangle, we have : sin latitude = sin interval × sin maximum latitude. The maximum being small and the latitude being generally less than this, sin latitude and sin maximum latitude are propor- tional to the latitude and maximum latitude, and we have the formula, lat = max. lat × sin interval. But the Paulīśa takes the latitude proportionate to the interval itself and gives the rule. We have to consider here whether the author intends that the latitude is to be found in propor- tion to the actual degrees of the interval or the sine of the interval in degrees. The triangle being spherical, the correct thing would be to use the sine. But we have reason to think that the degrees themselves are intended to be used in the proportion, for if the sine is to be used it must be men- tioned. Also, this Siddhānta uses only the proportion by degrees in other places also, where propor- tion by sines alone would be correct, as for e.g. in the solar eclipse, in correcting Rāhu and in calculating valana, i.e. transformation of direction (see VI. 2-4, 8). Therefore the original Paulīśa itself has instructed proportion by degrees, as being sufficiently accurate, which the author reiterates here. But in computing the parallax (in time) in the case of the solar eclipse the sine is used either by the Paulīśa itself, to avoid too much inaccuracy, or by the author VM to save the Siddhānta from ridicule. As for TS they say that the author intends here proportion only by sine, that being the proper thing to do. Another thing should be mentioned here. In using proportion by degrees, the maximum error will be in the neighbourhood of the nodes. With the maximum latitude 280', the latitude for 13° interval would be 40½ minutes, which would be incompatible with the formula for eclipses (vide VII. 5-6). Therefore it seems that the Siddhānta, though knowing that proportion by sine is the correct thing for latitudes, gives proportion by degrees for the sake of ease of computation. Or by taking liptāśatatrayeṇāśītim as the correct reading, which would make the maximum 380', the incompatibility can be avoided. It may be argued that the error in the latitude would be great in the neighbourhood of the maximum. But this is only erring one side while some others err on the other. The mean maximum latitude is 309'. If some like Ptolemy give 240', which is less by 70', what is wrong in taking that the Paulīśa gives 380', which is greater by the same amount? Now for the readings: In the fourth foot of verse 30, mu is corrected into su because murāt is meaningless. In the third foot of the 31st verse two syllables are wanting and nā is added for pur- poses of syntax. For the same reason, śī-tama-nupā in the fourth foot is corrected into śī-tyā-nupā. The correction of aśīti into saptati by TS here is unwarranted, because it is not known what it was in the original Paulīśa. If this is done in conformity with the Saura, why not in conformity with the Romaka, which gives 280', (see VIII. 11), and which is nearer to the Paulīśa? Now follow six verses devoted to criticising the views of the Romaka, and an astronomer by name Bhadraviṣṇu, the intention of the author being to create faith in his own work. (This was a custom in those days, vide for instance the Brāhma-Sphuṭa-Siddhānta, Dūṣaṇādhyāya and the Vaṭeśvara Siddhānta, Madhyamā-dhikāra, Chapter X). In several places of the text the readings are not clear and we cannot be sure of what exactly the author intends to say, though the gist is clear, the matter

III.31 III. PAULIŚA-SIDDHĀNTA 71 not being scientific and amenable to intelligent guess. Still, as there is much to say here too, we are dealing with these, unlike TS who have refrained from doing so. [भद्रविष्णुमते दोषः] तिथिनक्षत्रच्छे(द)प्रतिपत्तिर्यदि तथा ततः साधुः | न तथा च भद्रविष्णोस्तथाऽपि [न] विनिवर्तते लोकः || ३२ || Defect in Bhadraviṣṇu 32. If the tithis and nakṣatras as seen from observation of the sky agree with those computed according to the Śāstra, then the Śāstra is correct and fit to be accepted. It is not so in the case of Bhadraviṣṇu's work; still people do not turn away from that and follow the correct Śāstras. (This remark about the nature of people is true even today). For the sake of syntax the long dā has been shortened by us. Consistent with the idea intended, the negative particle na has been added in the fourth foot. [पादादित्यमते दोषः] न युगपदुदयो भानोरस्तमयो वाऽपि भवति सर्वत्र | कस्मिन् देशेऽस्तमये पादादित्येन (नोक्त)मिदम् || ३३ || Defect in Pādāditya 33. Sunrise or sunset is not at the same moment in all places on the earth; (so the place must be mentioned whose sunrise or sunset is taken as the epoch for finding the days and doing the computation). But Pādāditya, who has placed the epoch at sunset, has not mentioned the sunset of which place he refers to. (So his work is faulty.) There seems to be some error in the last foot and we are not sure-whether Pādāditya is a person as we have interpreted, or something else or whether the word is the same at all and not an nth incarnation of the original. [रोमकमते दोषः] मार्गा(द)पेतमेतत् काले लघुता न तावदतिदूरे | 'खविषयभूताष्टरसै'रब्दैः पश्याऽस्य विनिपातम् || ३४ || 32a. A.B.C.D. छेदा 33a. A.B. भानुः b. A.B1. प्रतिपत्तियदि; B2. प्रतिपतियदि. c. D. ॰ऽस्तमयः B. साधु d. B. पादादित्येन; D. पादाद्विनेन c. B. भद्रविष्टमो A. भुक्तिमिदं; B. भक्तिमिन्दुः; C. भुक्तमिदम्; d. A.B.C.D. om. [न] D. भुक्तं विदुः

72 PAÑCASIDDHĀNTIKĀ III.34 Defect in Romaka 34. This gaṇita work had deviated from the right path handed down by a hierarchy of good teachers and the day of its exposure is not far distant. Witness its downfall in 68,550 years! To agree with the fifth case in mārgāt we have corrected upeta into apeta. It may appear strange that the author calls 68,550 years as a not far distant date. But that depends on the outlook of people, and the Hindu mind, especially the old Hindu mind may con- sider even this as a comparatively short period. Or this verse may belong to the criticism of the Romaka following immediately, strayed to this place by the mistake of the scribe. In that case the first letter kha in khaviṣayabhūtāṣṭarasaiḥ should be corrected as sva and the expression taken to mean "6855 years, by its own measure", in which case the following is the meaning:- "This Romaka has not come down through a hierarchy of good teachers because it follows the Tropical year instead of the traditional sidereal year. It will be exposed in a period of 6855 of its own tropical years, and people will abandon it." How this will happen and how the number 6855 can be arrived at almost exactly, will be shown in explaining the next verse. (रौ)मकमहर्गणं (वा) (तद)र्कमिन्दुं च गणयतां ग्रा(ह्यम्) चैत्रस्य पौर्णमास्यां नवमी नक्षत्रमादित्यम् ॥ ३५ ॥ 35. If we adopt the days from epoch resulting from the tropical year as adopted by the Romaka and the Sun or Moon resulting therefrom, we must accept Punarvasu as the nakṣatra of the full moon of the month of Caitra, instead of the expected Hasta or Citrā, Punarvasu which is the nakṣatra of Caitra-Śukla-navamī. This connection has been strongly established in people's mind by the observance on Caitra- navamī as the birthday of Lord Rāma, hero of the Rāmāyaṇa, well known as born in the asterism Punarvasu. This is how this will happen: The months Caitra, Vaikśākha etc. are so called because the Moon at new moon in these months is in the vicinity of Citrā, Viśākhā etc. Thus, in a given month, the full moon, i.e. the Moon of the 15th tithi, is in a given nakṣatra or nearby, so that the other tithis also are 34a. A.B.C.D. मार्गादुपेत a-b. A.B1. ॰णं पादं मर्क॰; B2.3. C.D. ॰णं पादमर्क॰ b. B. न तावदति दूरो A. ॰यतां तां ग्राह्या; B. ॰यतां ग्राह्याः C. ॰यता ग्राह्या; c. A. षविषय; B. om ख D. ॰यतां ग्राह्य c. B2. पौर्णिमास्यां 35a. A.B.C.D. रोमक d. D. नवम्यां

III.35 III. PAULIŚA-SIDDHĀNTA 73 connected with particular nakṣatras. For e.g. as the Moon of the fifteenth tithi in Caitra is near Hasta or Citrā, the Moon of navamī, six days before, is near Punarvasu or Puṣya, because the tithi, the day and the nakṣatra, have approximately the same duration. In the same way, as the Moon of the 15th tithi of Śrāvaṇa is near Śravaṇa, the Moon at Śravaṇa Bahula Aṣṭamī, eight days after that, is near Rohiṇī, which is also a thing well known. Now, when the months (synodic) are 'tied' to the nakṣatras as mentioned, there will be this conformity. But the months are kept tied to the nakṣatras if the solar year is sidereal, and not tropical like that of the Romaka, i.e. if the solar year begins as in all other Siddhāntas with a fixed point on the ecliptic like the first point of Meṣa, and not the movable vernal equinox, the so called First Point of Aries, as in the Romaka or the new Indian Rashtriya Panchang. As the difference between the two points (i.e. the ayanāṃśa) increases, the above mentioned con- formity will gradually decrease and when the ayanāṃśa accumulates to 30° the full moon of the first month will fall in the Phalgunis instead of near Citrā and though called Caitra the first month will really be ‘Phālguna’. This non-conformity has already happened in the Rashtriya Panchang, with the ayanāṃśa more than twenty degrees now. If it accumulates to 6 nakṣatras, (i.e. 80°) the full moon of the first month, still called Caitra, will be in the nakṣatras Punarvasu or Puṣya, though the month will really be Pausha. Thus Punarvasu connected with the Navamī of the real Caitra will occur at the full moon of the so called Caitra of the Romaka (or the Rashtriya Panchang). The Navamī of the new Caitra, occurring 6 days before the full moon day, will occur in Apabharaṇī, 6 nakṣatras earlier. When a situation thus arises contradictory to their belief, people will realise that the sidereal year is the proper thing and discard the Romaka. Incidentally we may mention that the same confusion will arise in following the Rashtriya Panchang also. One thing we want to say here. We do not deny that the tropical year best suits civil purposes, but a luni-solar calendar based on the sidereal year also will suit our religious purposes best. Therefore they have to be kept apart. (A civil calendar based on the tropical year, we already have in the Christian Calendar we have been following, which is practically worldwide. As for the defects in it, the ‘Calendar Reform’ will take care of it, while in making this reform we have taken a step in isolating ourselves.) One thing could have prevented the confusion. If they had adopted the seasonal-month-names like Madhu, Mādhava etc. preserved for us in the Vedas (vide III. 25) instead of trying to fit in the sidereal luni-solar-calendar months, Caitra, Vaiśākha, etc., this confusion would have been avoided and they would not have simply added one more to the one hundred contradictory Panchangs already extant. To continue: We shall compute after how many years the exposure of the Romaka, as mentioned by the author, would happen. The ayanāṃśa was zero at the author's time, as we have shown on several occasions. We must calculate when it accumulates to 6 nakṣatras. The Siddhāntas of the author's time use a sidereal year of days, 365-15-31 nearly, and the tropical year of the Romaka is days 365-14-48 (See VIII. 1). Therefore, the Romaka year begins earlier by 43 vināḍis, every year, and this is equivalent to the rate of ayanāṃśa per annum, 42", nearly. If ayanāṃśa to become 42" takes one year, to become 6 nakṣatras it will take, 6 × 800 × 60 divided by 42 = 6857 years nearly. It is this that is given by the author as 6855 in the previous verse explained. कालाऽपेक्षा विधयः श्रौताः स्मार्ताश्च तदपचारेण । प्रायश्चित्ती भवति द्विजो यतोऽतोऽधिगम्येदम् ॥ ३६ ॥ 36. All the injunctions of the Vedas and Smṛtis are based on the proper time, and by not performing the rites at those times the performer, especially a twice-born,

74 PAÑCASIDDHĀNTIKĀ III.36 acquires sin which is to be expiated. Therefore, a study of this Romaka itself is to be expiated. How is that sin may accrue by not performing a rite at the time enjoined by the Śāstras. But by simply studying the Romaka, we cannot say one would also perform the rite at the improper time. The Romaka may give the time wrongly, but one may study it not for the sake of using its time, but for other purposes, as for instance to understand where it goes wrong, and expose its weakness to others and save them, for which the man who studies even deserves merit. Let us understand that these things, sin and merit, are subtle and cannot be known without a deep study of the Śāstras, and if the author steeped in the Dharma Śāstras says a thing, let us accept it. The Vedas promise svarga not only to the performer of the yajña but also to one who knows how to perform it properly. We frequently meet in the Vedas the expression ya u cainam evam veda. The Vedāṅga Jyotiṣa says that people who know astronomy know as it were the correct performance of the sacrifices themselves, yo jyotiṣaṃ veda sa veda yajñān and they go to svarga after establishing a long line of progeny in this world. Now, it stands to reason that if the mere study of a good thing gives merit, the mere study of a bad thing brings sin. It is said that even association with bad characters and sinners bring sin, as also doing sinful things even in dreams. As for the argument that the person even deserves merit for intending to keep off people from the improper times, he does deserve it and will get it. But that does not mean expiation is not called for. Contact with craftsmen may be necessary to keep the temple idols in form, but that does not mean that purificatory ceremonies need not be performed for the idols on that account. We can say this much that in these cases the expiation is light, like the utterances of the Lord's Name, like 'Kṛṣṇa Kṛṣṇa, Śiva Śiva! We should also take into consideration the spirit if the times in which these statements were made. स्फुटगणितविदिह लब्धा धर्माऽर्थयशांसि दिनकरादीनाम् । (कुकरणकारस्सत्यं सहते नरके कृताऽऽवासा: ॥ ३७ ॥) 37. The person having correct knowledge of the Sun, Moon, etc. gets Dharma, which will take care of his future world, Artha which will ensure his prosperity in this world and fame, which will perpetuate his memory. But the bad astronomer who misleads people by his writings will certainly have to go to hell and dwell there. 36a. A. विधय b. B. श्रौता स्मा० c. B1.2. प्रायश्चिती. A1. भवती c-d. A. कुकरणविदो द्विन्यो ये कथयन्त्यस्फुटं कुकरणकारः (A2. करः) B. अकरणविदो द्वित्यो ये कथयसत्यं अकरणकरः सहते नरके कृतावासाः । C. कुकरणविदोद्विन्यो ये कथयन्त्यस्फुटं कुकरणकरः सहते नूनं नरके कृतवासाः । D. कुकरणविदो द्विजा ये कथयन्त्यस्फुट [म] सत्यं [स गणितम्] । कुकरणकारसहि [ताच्च] ते क्षणं नरके कृतवासाः ॥ 37. In A and B, स्फुट etc. occurs as the second half of the verse. It is put here as the first half to suit the sense. a. D. लब्ध्वा

III.37 III. PAULIŚA-SIDDHĀNTA 75 It must be noted here that when even the person with correct knowledge gets so much, the writer will get more. It must also be noted that only the writer of bad astronomy goes to hell, not the reader, whose sin is small. In this verse there is a jumbling of words and phrases and induction into the text extraneous words intended as commentary. The words, sphuṭagaṇitavid etc. seem to be the first half of the verse because in the first foot there are twelve and in the second eighteen syllables. Therefore what comes before that is the second half. In that, there are may syllables more than the required twenty- seven. Selecting the required words alone, we have reconstructed the third and fourth foot. For the observations of K S. Shukla on 32-37 vis-a-vis NP, see his paper ‘The PS of VM (1)’, JIHS 9 (1974) 62-76. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां पौलिशसिद्धान्तो नाम तृतीयोऽध्यायः ||]¹

1. A.B. पौलिशसिद्धान्तः; C.D. इति पौलिशसिद्धान्तः || Thus ends Chapter Three entitled ‘Pauliśa-Siddhānta: Planetary Computations etc.’ in the Pañcasiddhāntikā composed by Varāhamihira

Chapter Four THREE PROBLEMS — TIME, PLACE AND DIRECTION ४. चतुर्थोऽध्यायः त्रिप्रश्नाधिकारः Introductory Problems on Time, Place and Direction, involving spherical trigonometry, are dealt with in this chapter. The first fifteen verses are devoted to the construction of a table of sines. As this kind of matter does not involve constants specific to any siddhānta and is commonly found in all siddhāntas, we cannot say which Siddhānta this belongs to, Pauliśa or Saura, the only two siddhāntas meant to be expounded in detail by the author. Probably it is the author’s own, meant for both, or taken from both. Two things point to this conclusion: In the part of the work dealing with the Saura, viz., chs. IX, X, XI, XIII, XIV, XV, XVI, and XVII, no space is given to the sine tables, though required, and to the problems dealt with here, and therefore if these are not meant for Saura, it would be imperfect though almost full. On the other hand, certain redundant and crude rules point to this chapter’s connection with the Pauliśa, as also its position in the chapter distribution in the PS text. [ज्यानयनम्] षष्टिशतत्रयपरिधेर्वर्गदशांशात् पदं स विष्कम्भः | तदिहां(शच)तुष्कं संप्रकल्प्य रा(श्य)ष्टभागज्या || १ || Table of R Sines

1. Take the circumference as measured in 360 units, square it, take the tenth part of the square, and find its square root. The result is the diameter of the circle in the units taken. We assume the diameter to be 4°, (i.e., 240′) and hereunder give the tabular sines of angles for 3° 45′ interval. The rule is: diameter = √circumference²/10. It comes to this: d = c/√10. The formula, d = c/π is well known, and the author has taken √10 as an approximation for π which is incommensurable and usually represented by the approximate values, 22/7, 355/133, 3.1416 etc. The Sūrya Siddhānta too gives √10 as the value of π in its instruction to find the circumference of the earth from its diameter (I.59.): “The earth’s diameter is 1600 yojanas. Square this, multiply by 10, and find the square root. This is the earth’s circumference.” By thus taking √10 for π, an error of about 0.0067% results, and for a circumference of 21,600′, we get the radius 3415′, instead of the well-known 3438′. But it must be mentioned here that this error does not affect the computation of the sines 1a-b. A.B. परिधे वर्ग c. A.B. तदिहांशाश्चतुष्कं (B. ॰ष्क) b. A. विष्कुम्भः d. A. संप्रकल्प्य; B. प्रकल्प्य. A.B. राश्याष्ट०

IV.2 IV. THREE PROBLEMS 77 mentioned in the succeeding verses, because it can be shown that the author derives the sines from a correct formula, (not dependent on this wrong ratio of the diameter to the circumference), based on 120' as the radius of the circle. If he had depended on the wrong value, the first tabular sine, i.e. sin 3° 45' would be 7' 54", (being the 96th part of the circumference, where the sine is indistin- guishable from the arc), and not the correct 7' 51" as given by the author. Taking the diameter as 4°, and thereby the maximum sine (i.e. the radius) as 120', is arbitrary. In general, the Siddhāntas give the maximum sine, 3438', as arrived at from taking the circumfer- ence as 360° or 21600'. The Vākyakaraṇa makes it 43°. In actual work, the sines enter only as a ratio to the maximum sine, and therefore no harm, will result by taking these different maximum sines. TS and NP have not understood the meaning of the second half of the verse, and mis-interpret aṁśacatuṣkam as quadrant. व्यासार्ध[स्य] कृतिर्ध्रुवसंज्ञिका कृतांशस्ततः स मेषस्य | ध्रुवकरणी मेषोना द्वयोस्तु राश्योः पदं ज्याः स्युः ॥ २ ॥ 2. The square of the radius, (i.e. 14,400), is called dhruva (karaṇī), (literally, ‘Fixed Irrational’). The fourth part of it, (i.e. 3600), is the karaṇī (Irrational) related to the first sign, (or 30°). Dhruvakaraṇī minus the karaṇī of Meṣa, (i.e. 14,400 − 3600 = 10,800), is the karaṇī of two signs, (or 60°). The square root of a karaṇī is the tabular sine. Being square of tabular sines given in minutes, the karaṇīs are squares of minutes, which is their peculiarity as given by the author, though this is not mentioned explicitly. The other well-known characteristic of a karaṇī, viz. irrationality, is found in all karaṇīs except 14,400 and 3600, though the author calls these also karaṇīs in a general way. In modern terminology the word sine used in connection with the angle is defined thus: [Figure: Right-angled triangle ABC with right angle at C and angle marked at B] Fig. IV. 1-a In the right angled triangle, (fig. 1-a), sine ∠ B = AC/AB, or sine ∠ A = BC/AB, i.e. as the ratio of the opposite side to the hypotenuse. In tabulating the sines, the hypotenuse is taken as unity, and the ratio expressed as a decimal fraction. 2a. A.B. कृते ध्रुव०; C.D. कृतिध्रुव० c. B1.3. ये योना; B3. येषोना b. A.B. ०ज्ञिता. A.B. कृताशाःस्ततः. A.B. सशेषस्य d. A. दयोस्तु; B. दयो सु

78 PAÑCASIDDHĀNTIKĀ IV. 2 The ancients however expressed the sines in minutes-length or, more accurately, in minutes and seconds-lengths, the maximum sine called Trijyā (meaning ‘the sine of three signs’, i.e. 90°), occur- ring separately in the work to make up the ratio. This is the way in which they conceived the sine (meaning ‘bow-string’ from its Sanskrit equivalent śiñjinī, synonymous with jyā). In Fig. 1-b. A₃ E F₃ D is the circumference of the circle, centre B. A part of the circumference like ADF, A₁D F₁, etc. is called dhanus (literally, ‘bow’) or arc. Fig. IV. 1-b The straight lines ACF, A₁C₁F₁, etc. forming the ‘bow-strings’ of the respective ‘bows’ are the jyās or full sines. But in actual practice, the halves of the full sines AC, A₁C₁, etc. above are used with the name of ‘sines’, with respect to the half-bows or arcs, AD, A₁D₁, etc. Because the arcs AD etc. are as the angles ABD etc., the sines AC etc. are spoken of with respect to the angles ABD (= ABC) etc. also. Thus, AC is the sine of ∠ ABD or arc AD, A₁C₁ is the sine of ∠ A₁BD or arc A₁D₁ and so on. It is this connection of the sine with the arc that has given it the nature of a length, which is expressed in minutes and seconds on account of the connection of the arc with the angle at the centre. It may be mentioned here that CD, C₁D, C₂D etc., appearing like the arrows on the respective bow-strings, are called śara (meaning ‘arrow’). If A₃BD is a right angle, i.e. three signs, then, obviously, A₃B is the sign of this angle, i.e. it is the sine of three signs, and therefore called trijyā. Its length is clearly half the diameter A₃BF₃, i.e. the radius, equal to 120′. Now, let the angle ABD be equal to one sign, i.e. 30°. ABD = DBF = 30°. ∴ ∠ ABF = 60°. AB = BF, being radii. ∴ ∠ BAF = ∠ BFA = 60°. Thus ABF is an equilateral triangle, and AF = AB = 120′. ∴ AC = AF/2 = 60′. Thus sine 30° = 60′. Its karaṇī is its square, viz. (60′)² = 3600, the karaṇī of Meṣa as mentioned by the text. Then, let ∠ A₂BD be equal two signs, or 60°. A₂BD = DBF₂ = 60°. ∴ C₂ is a right angle. So the karaṇī of 2 signs = A₂C₂² = A₂B² − BC₂² = A₂B² − AC², (∵ △ A₂B C₂ ≡ △ BAC), = 120² − 60² = 10,800, A₂B being the radius. This also agrees with what the text says. (The square root of 10800 minutes, i.e. 103′ 55″, is the sine of 2 signs, which agrees with the value given in the table.)

IV. THREE PROBLEMS 79 Incidentally, we shall derive the karaṇī and sine of one and a half signs, i.e. 45°, mentioned in verse 4. Let A₁BD be equal to 45°. ∠ A₁ = 45°, and ∠ C is a right angle. ∴ A₁C = C₁B. But, A₁B² = A₁C₁² + C₁B² = 2 A₁C₁². ∴ A₁C₁² = 120²/2 = 14400/2 = 7200 = the karaṇī of one and a half signs as mentioned in the text. Its root, 84'51", is sine 45°, agreeing with what is given in the tables. शेषेष्विष्टेषु धनु-र्द्विगु(णं) पदात् प्रोज्झ्य शेषगुणहीना [त्] । [व्यासस्याऽर्धार्द्धर्गं] द्विगुणकरण्यां समायोज्यम् ॥ ३ ॥ तत्पादोऽभिमता [स्याद्] ध्रुवा तदूनाऽवशेषपिण्डस्य । ध्रुवकरणीदलमध्यर्धसंज्ञमन्योऽत्र विधिरुक्तः ॥ ४ ॥ इष्टांशद्विगुणोनत्रिभज्ययोना त्रयस्य चापज्या । षष्टिगुणा सा करणी तया ध्रुवोनाऽवशेषस्य ॥ ५ ॥ 3-5. The other tabular sines, (i.e. sine 3° 45′, sin 7° 30′ etc. other than the four mentioned of the total 24) are formed successively in the following manner: Let the angle or arc for which the sine is required be θ. I. Sin²θ = ¼[sin²2θ + {120 − sin(90° − 2θ)}²] II. Sin²θ = 60 × {120′ − sin (90° − 2θ)}, where the sines are in minutes etc. Of the 24 sines, the karaṇī of the nth sine = 14400 − the karaṇī of the (24 − n)th sine. 7200 is the karaṇī of one and a half signs, i.e. 45°. Thus, as karaṇīs 8, 12, and 16 are known, those of their halves etc. and (24 − halves) etc. can be found successively. Thus all the sines from 1 to 24 can be found. Of the two formulae, the first is suited to geometrical representation, and the second to computation. Example 1. Given the 8th karaṇī (i.e. of 30°) 3600 and its sine 60′, the 16th karaṇī (i.e. of 60°) 10800, and its sine 103′ 55″.33, find the 4th and 20th karaṇīs and sines, using each of the two formulae. The desired sine is the 4th, i.e. of 4 × 3° 45′ = 15°. 2θ = 30°, 90° − 2θ = 60°. 3a. A. धनुर्द्वि. A.B.C.D. ॰गुणपदा॰ 4a. A. तपदो; B. पदो; C.D. त[स्य] पदो. b. A. पदायोज्य; B. पदायोज्य्; C.D. पदायॊग॰ A.B.C.D. ॰भिमतज्या A.C.D. गुणहीना; B. गुणाहीना b. A. तदुना. B. ॰विशेषे c. A.B. तृव्यासपादार्द्धाद्धर्गं; (B. om तु; B2. ०र्गं); d. A. र्द्धसंज्ञा; B1.2. र्द्धसंज्ञां; B3. र्द्धं संज्ञा C.D. [त्रिज्या तदर्धवर्गौ] द्वि C.D. संज्ञकोऽन्योऽत्र. A. विधिनुक्तः d. A. कारथो; B. कारयो 5a. A.C. इच्छांशद्विगुणेन (C. ॰णोन) A. समायोज्यं; B. समाप्रोज्यन्त; b. B3. त्रयंस A.B. वायज्या C. द्विगुणज्यार्धस्य संयोज्यः; D. द्विगुण [र्ध] करणी c. A. स कारणी; B. स करणी समायोज्यः d. B. षपा. A.B. ॰नामशेषस्य

80 PAÑCASIDDHĀNTIKĀ IV. 5 I. The 4th karaṇī = ¼ [sin² 30° + (120 − sin 60°)²] = ¼ [60² + (120 − 103′ 55″.33)²] = ¼ [3600 + (16′ 4″.67)²] = ¼ (3600 + 258 97/144) = ¼ (3858 97/144) = 964 385/576 ∴ the fourth sine, i.e. sine 15° = √964 385/576 = 31′ 4″. II. The 4th karaṇī = 60 (120 − sin 60°) = 60′ (120′ − 103′ 55″.33) = 60′ × 16′ 4″.67 = 964 385/576 From this, sin 15° = √964 385/576 as before = 31′ 4″. We shall prove the first formula geometrically, and derive the second from the first. Fig. IV. 2 In Fig. 2, FBD = θ, and DF is its arc of which the sine wanted is DE. DE² = the wanted karaṇī (i.e. sine² θ). DFA = 2DF. ∴ DE = ½DA. ∴ sin² θ = DE² = ¼DA² = ¼(AC² + CD²). Now, ∵ AC = sine 2 arc FD = sin 2θ, AC² = sin² 2θ; and ∵ CD² = (BD − BC)² = (BD − AG)² = [120′ − sin (90° − 2θ)]², sin²θ = ¼ (AC² + CD²) = ¼ [sin² 2θ + {120′ − sin (90° − 2θ)}²] From this, sinθ = √sin²θ. From I, we can derive II thus: ¼ [sin² 2θ + {120′ − sin (90° − 2θ)}²] = ¼ {sin² 2θ + 120² + sin² (90° − 2θ) − 2 × 120 × sin (90° − 2θ)} = ¼ {120² + 120² − 2 × 120 × sin (90° − 2θ)} (∵ sin² 2θ + sin² (90° − 2θ) = sin² 2θ + Cos² 2θ = radius²) 2 × 120′ = ──────── {120′ − sin (90° − 2θ)} 4 = 60′ {120′ − sin (90° − 2θ)} Now, of TS, Thibaut alone proves the formula, while Sud. Dvivedi uses it. The form of the first formula given by them differs from that given by us, and is as follows:- sin²θ = (½ sin²2θ) + [½ {120′ − sin (90° − 2θ)}]² Though their formula is correct, it entails more work, and to give the formula in this form they have made many unwarranted changes in the already correct readings. We have made only one correc- tion, and that grammatical, for the sake of syntax, viz. dhanurdviguṇapadāt into dhanurdviguṇam padāt, which entails the occurrence of an extra syllable, which can be explained, as before, by rules of prosody. Or, let the reading be śeṣe tviṣṭe dhanurdvi-guṇam padāt projjhya etc. Now follow six verses giving the sines, computed by the author himself.

IV. 9 IV. THREE PROBLEMS 81 शेषज्याः 'स्वरतिथयो' 'गुण-शिव-धृति'भिश्च 'विंशतिः' सहिता । 'पञ्चनरकं' 'शतार्धं त्रिसमेतं' 'षष्टि'रिति लिप्ता ॥ ६ ॥ सैकाऽजे पञ्चाशत् 'पञ्चाष्टकं' 'पञ्चवर्गवेदा' श्च । 'त्रिंशच्चतुर्भिरधिका' 'षट्पञ्चाशच्छराः' शून्यम्' ॥ ७ ॥ 6-7. The other sines are the following: In the first sign, the minutes parts are successively 7, 15, 20 + 3, 20 + 11, 20 + 18, 5 × 9, 50 + 3, and 60. The seconds, respectively, are: 50 + 1, 5 × 8, 25, 4, 30 + 4, 56, 5 and 0. Thus we have for the first sign –

82 PAÑCASIDDHĀNTIKĀ IV. 11 Adding the minutes and seconds, and 60' for the end of the first sign, the sines are:-

IV. 14 IV. THREE PROBLEMS 83 there is one second more. By this mistake the 24th, i.e. the radius, has become 120' 1", and even this obvious mistake they have failed to note. The promised sine-intervals are here given: मुनयोऽब्जे व्येकान्ते 'रसत्रयं' 'पञ्च(कौ)' 'कृता' (श्र) गवि । 'शिखि-पक्षचन्द्र-शून्याः' द्वित्रिमिथुने कला ज्या [नाम्] ॥१२ ॥ मेषे विकलार्धशतं सैकं 'व्येकेन्द्रि(ये)-श्वरं त्रिंशत् । (द्वा)विंशतित्रिवर्गः ------------------------------ ॥ १३ ॥ ---------------------------------------------------- । ------ 'खगुणकृतार्णवयमनवकसमुद्रशिखिवर्गैः ॥ १४ ॥ 12. Of the intervals the minute parts are, in the first sign, 7, 7, 7, 7, 7, 7, 7, 6; in the second sign: 6, 6, 6, 5, 5, 4, 4, 4; and in the third sign, 3, 3, 2, 2, 1, 1, 0, 0. 13-14. The seconds in the first sign are, 50 + 1, 50 − 1, 50 − 5, 50 − 11, 30, 22, 9, (and here is a break resulting in the loss of the 4th foot of the 13th verse and the first two feet with 4 mātrās of the 3rd foot of the 14th verse. From the values of the sines we can compute that the seconds in the 8th interval must be 55, which must have been given in the missing part. By examining the remaining part we can construct the meaning of this verse thus). The seconds of the intervals in the second sign are, respectively 10 × (4, 2, 0, 4, 2, 5, 3, 0)

• (0, 3, 4, 4, 2, 9, 4, 9), added each to each. TS have not been able to see that in what is left of the 14th verse, the digits in the unit places of the eight numbers giving the seconds of the intervals of the second sign are given. This is because they have made mistakes in the 14th and 16th sines resulting in an error of + 1" in the 14th interval, − 1" in the 15th and + 1" again in the 16th. This has prevented them from finding the correct values by comparison, so that they make many changes in the readings here, saying that the text is very corrupt, here. It may be seen that not a single word is wrong here. 12a. A. मुनयोज्ये; B. गुनयोज्ये to 14c. D suggests for 13d [पञ्चाशच्च विषयसंयुक्तम्] b. A. त्रयं को; B. त्रयपञ्चको;. C. त्रयं (त्रिः) शशः C. indicates the gap by dots as done by A. कृताच्चे गवि; B. कृता-गेवि; C. कृताब्धी गवि; us. D suggests for 14: D. कृताग्निगवि ख[समुद्र] गुण [द्विकृताः] .......... c. A. शिखिपकृत चन्द्रः; B. शिखिपक्ष कृतार्णव [द्वि] यमनवे [न्द्रि] यसमुद्रशिखिवर्गे[शाः] || d. C. द्वौद्विमिथुने. A-B. ज्या |; C. ज्यार्द्धे; D. ज्या [सु] 13a. A. विकलार्द्धशत° d. A. समुद्रा b. B1. सैकां; B2. सैव्यं; B1.2. वर्गे; B2. वर्गे A1.B1. °न्द्रिय स्वरं C. खगुणकृतार्णवयमनवसकसमुद्रा शिखिवर्गैः (?) || c. A1.B2. द्विविंशति 13d. A.B. Unindicated gap of 13d. upto खगुण of 14c.

84 PAÑCASIDDHĀNTIKĀ IV. 15 ‘मनुविषयतिथिरसा:’ स्युस्त्रिगुणाः पञ्चाष्टकं ‘स्वरोपेतम्’ | सप्तदशनवपञ्चकं षोडश चेति क्रमान्मिथुने || १५ || 15. The seconds of the intervals in the third sign, are, 3 × 14, 3 × 5, 3 × 15, 3 × 6, 5 × 8 + 7, 17, 9 × 5 and 16. The intervals can be got by deducting the previous sines from the succeeding ones, and com- pared with the author’s concern for correctness, they are correct. It is to be noted that the intervals of TS also come correct in the third sign, because there is a uniform error of 1″ in all sines. The intervals are as tabulated:

IV. THREE PROBLEMS 85 Fig. IV. 3 AK = sine AG (neglecting the first sign) because, EF, HJ, JK, GF, are all equal, as already men- tioned. Therefore, for an arc or angle in the second quadrant, (3 to 6 signs), deduct it from six signs and get the sine of the remainder. For that in the third quadrant, (6 to 9 signs), deduct six signs and find the sine of the remainder. For that in the fourth quadrant, deduct it from twelve signs and find the sine of the remainder. Example 2 (a). Find the sine of the arc or angle equal to 4 signs. It is in the second quadrant. Therefore, sine 4 signs = sine 6 signs − 4 signs = sine 2 signs = 103′ 55″. Example 2 (b). Find the sine of signs 7-11-15. This is in the third quadrant. Therefore sine of sign 7-11-15 = sine (7-11-15 − 6-0-0) = sine 1-11- 15 = 79′ 7″. Example 2 (c). Find the sine of signs 9-15-0. This is in the 4th quadrant. Therefore sine of signs 9-15-0 = sine (12 signs − sign 9-15-0) = sine 2-15-0 = 115′ 55″. Declination From here to the end of the chapter, problems based on the solution of spherical triangles are dealt with, being problems involving position, time and direction. As a preliminary, the declina- tion of the Sun and the Moon are required, which are given first. Stellar sphere (Bhagola) The ancient astronomers speak of three spheres, the Terrestrial sphere or Earth sphere on which we live, the Stellar sphere or the Sphere of the stars, and the Sky sphere or the Sphere of the sky. We shall describe the terrestrial sphere in connection with the Sāura chap. XIII-XV. Of the other two, we shall now describe the stellar sphere, a knowledge of which is immediately required. It is the apparent sphere on which the stars appear to be fixed, and form a frame of reference for

86 PAÑCASIDDHĀNTIKĀ IV. 15 the motion of bodies like the Sun, Moon, planets etc. Actually the stars are at widely different dis- tances from us and are moving in various directions at speeds of several miles per hour. But with all this they appear to be, and can be represented, as being fixed on a sphere, because of their enormous distances from us, so much so that we are practically viewing the same sphere as people several generations ago did. But the ancients believed that the stars were luminous bodies fixed on the under-surface of a sphere of radius only sixty times that of the orbit of the Sun (actually the earth) round the earth, with the centre of the sphere at the earth's centre. This sphere seems to be rotating about once a day, by the actual rotation of the earth, about once a day. Fig. IV. 4 On this sphere (Fig. 4) there is an important great-circle called the ecliptic (Krānti-vṛtta) (EC), marked by the twenty-seven asterisms, Aśvinī etc., on which the Sun (S) moves, (actually appears to move on account of the motion of the earth), completing a revolution once a year. The Moon and the planets move in orbits inclined to the ecliptic at small angles. Their longitudes are reckoned along the ecliptic from a fixed point called the first point of Meṣa or Aśvinī. Latitudes are measured on secondaries to the great circle, meeting at a point called the pole of the ecliptic (Kadamba). Another great circle called the Celestial Equator (AOB) cuts the ecliptic at r, called the First point of Aries or Vernal Equinox point, which, instead of being fixed, has a slow westward motion on the ecliptic. At the period of the authors of the first Siddhāntas, this point coincided with the first point of Meṣa, this being the reason why that particular point was taken by them for reckoning from. The angle between the two great circles (SrR) is called the Obliquity of the ecliptic. It was about 24° at the time of Varāhamihira, and now it is about 23° 27'. The declination (SR,) (Krānti) of a body like the Sun, is measured along the secondary passing through the body, (NPSRSP,) called the Declina- tion circle, all the secondaries meeting at the celestial poles, the poles of the stellar sphere, (SPNP), on the axis joining which the sphere apparently rotates. The declination is found by solving the spherical right angled triangle SrR. Spherical triangles were in general solved by the Hindu astronomers using the properties of plane right angled triangles formed by the sections of the sphere along the great circle arcs forming the spherical triangle (The

IV. 16 IV. THREE PROBLEMS 87 Greeks solved them by an extension of Manelau's Theorem to figures on the sphere.) We shall con- tent ourselves with giving the formulae for solution and refer to them whenever necessary by way of proof. Let ABC (fig. 5) be a spherical triangle, right angled at C, and R the radius of the sphere. I. Cos AB = Cos AC × Cos BC ÷ R II. sin BC = sin AB × sin A ÷ R III. Cos A = R × Cos AB. sin AC/sin AB. Cos AC IV. Cos A = Cos BC × sin B ÷ R V. Cos AB = R. Cos A × Cos B/(sin A × sin B) These, together with the general identities, cosθ = sin (90° – θ), sin² θ + Cos² θ = 1, will suffice for explaining the formulae occurring in the text. [Fig. IV. 5] [रविचन्द्रयोः क्रान्तिः] जीवाऽध्यर्धशतां' (शघ्नै) काषष्टि (र्दि) नेशकाष्ठा (ज्या) । चन्द्रस्य सविक्षेपस्तदपक्र(मो) राशिपादे(भ्यः) ॥ १६ ॥ Declination of the Sun and the Moon 16. The sine of Sun's declination is found by multiplying the sine of its longitude by 61 and dividing by 150. Its arc is the declination. The declination of the Moon found thus is the mean declination. Its true declination is the mean declination plus latitude. The intervals of declinations for intervals of quarter signs are given (in the next two verses, 17-18). This is the formula: sin dec. = sin sāyana long. × 61/150. From this the arc forming the declina- tion is found by using the tables, and then the true declination of the Moon, using this. It must be noted that when the longitude reckoned from r (i.e. sāyana long.) is within 6 signs, the declination is North, and when more than 6 signs it is South. In the case of the Moon, if the declination and the latitude are of the same direction they should be added to get the true declination. If they are of different directions, their difference is the true declination, its direction being that of the greater. The author has not mentioned this because it is obvious. Example 3 (a). Find the maximum declination of the Sun. Obviously, the maximum sine of the Sun's longitude (i.e. when the Sun is 3 signs or 9 signs) will give the maximum declination. It is therefore given by sin dec. = 120' × 61 ÷ 150 = 48' 48". Its arc, 24° is the maximum declination. 16a. A. जीवाध्यार्द्ध०; B. जीवा व्या-र्द्ध; C. जीवाव्यर्द्ध; D. [साङ्कुलिप्ता] . A.B. काष्टान्तः; C. काष्ठान्तः; D. जीवाव्यध्यर्ध. B. सिताशा; A.C.D. शतांशाः D. काष्टान्तः b. A. सैकाःषष्टि; B. सैका षष्ठि; C. सैका षष्टि; d. A.B.C.D. तदपक्रम. A.C. ०पादेन्यः

88 PAÑCASIDDHĀNTIKĀ IV. 17 Example 3 (b). Sāyana Sun is rāśi 4-7-30. Find its declination. Sin 4ʳ 7° 30′ = Sin (6ʳ-0-0 – 4ʳ-7-30) = sin 1ʳ-22-30 = 95′ 12″. Sin Declination = 95′ 12″ × (60 + 1) ÷ 150 = 95′ 12″ × (2/5 + 2/5 × 60) = 38′ 5″ + 38″ = 38′ 43″. Its arc, 18° 50′, is the declination. Since the sāyana Sun is within 6 rāśis, the declination is north. Example 3 (c). The sāyana Moon is 9ʳ0°0′. Its latitude is 4°N. Find its declination. The sāyana longitude being 9ʳ0° 0′, the mean declination is the maximum, south, i.e. 24°S. Its lat. is 4°N, i.e. of opposite direction. ∴ the true declination is 24° – 4° = 20° S. South because South is greater. The author uses the word kāṣṭhā to signify declination, which is uncommon. Sometimes this word itself is used to mean sine declination. The rule for sin declination is explained thus: Our siddhāntas take the maximum declination to be 24°. As the maximun declination occurs when the sāyana longitude is 3 signs, the angle between the ecliptic and the celestial equator (i.e. the obliquity of the ecliptic) also is 24°. In Fig. 5, take AB and AC as parts of the ecliptic and celestial equator. Then, A = 24° and AB is the sāyana longitude. BC is the declination wanted. By formula II under the present verse, sin dec. = long. × sin 24° ÷ 120′. But sin 24° ÷ 120′ = 48′ 48′′ ÷ 120′ = 61/150. Hence, sin dec = sin long × (60 + 1)/150, which is the given rule. Now for the direction: See Fig. 4. At r the Sun moving along the ecliptic crosses the celestial equator, and passes from South to North. As great circles bisect one another, till the longitude is 6 sings it moves north of the celestial equator, for which declinations are reckoned, and then moves south of it. Therefore for longitude 0 to 6 rāśis, the declination is North, and for longitude 6 to 12 rāśīs, it is South. As the Moon and other planets move in their own orbits inclined to the ecliptic, the declinations computed from their longitudes reckoned along the ecliptic are only approximate. To get correct declinations, their distances north or south from the ecliptic points, called their ‘latitudes’, should be combined in the proper manner as instructed. But the result by thus adding or subtracting will be only approximate, because the latitudes are directed towards the pole of the ecliptic (Kadamba), while the declinations are directed towards the Celestial Pole. The maximum error that can occur thus is about 24′. Combined with the error in latitude due to other factors like proportion by degrees of the argument of latitude (advocated by the Pauliśa) instead of the sine etc., the error will be considerable. Now, for the readings. For syntactical purposes, and getting the proper meaning, śatārmśāśsaikā has been corrected as śatāṁśaghnaikā, ṣaṣtidineśa as ṣaṣṭirdineśa, kāṣṭhānta as kāṣṭhā jyā, apakramarāśi as apakramo rāśi and pādenyaḥ as pādebhyaḥ. As for TS, they have not touched this verse and the next two, saying, in so many words, that they cannot interpret them. NP’s interpretation of the three verses has also been affected by the highly corrupt text. लिप्ताशत [क] म (शीत्या) मेषे 'त्रिख' यु (क्त) 'मिन्द्रिय' 'मनू' (न) म् । गवि ('मनु') 'भव' 'मुनि' 'रूपै' - श्र [तु] गुणैः संयुतं च शतम् ॥ १७ ॥