सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

68 by x°/10 + x°/88 + Dhruvaka as given by the verse. Verse 21. To obtain the position of the lunar Node Rahu. The Ahargaṇa multiplied by 30 and divided by 566 gives the number of degrees to be added to the Dhruvaka to give the position of Rahu. Comm. If the number of the sidereal revolutions of Rahu in a Kalpa multiplied by twelve divide the number of days in a Kalpa we have very approximately 566, which means that Rahu traverses a Rasi very nearly in 566 days. Hence dividing the Ahargaṇa by 566, and multiplying by 30, we have the degrees covered, which being added to the Dhruvaka gives the position of Rahu. Verses 22, 23, 24. Alternative method of obtaining the planetary positions. The Ahargaṇa multiplied by 100000, and divided successively by 101461, 151787, 190833, 24436, 1203400, 62416, 2990000, 898000, 1886800 gives the respective posi- tions of theplanets beginning from the Sun and those of the apogee and Node of the Moon; in the case of the Moon, however, the result is to be multiplied by 20. The results in degrees added to the Dhruvakas give their posi- tions. Comm. The degrees covered by the planets etc in D days are (R × 360 × D°) / M where R is the number of side- real revolutions in a Kalpa, M the number of mean solar days in a Kalpa, and D the Ahargaṇa. Now (R × 360 × D) / M = (R × 360 × D × 100000) / (M × 100000) = (D × 100000) / ((M × 100000) / (R × 360)) . Then (M × 100000) / (R × 360) is found for every planet. In the case

69 of the Moon, however a multiplier 20 also is used in addi- tion to 100000, because he has such a quicker motion. Verses 25, 26. To obtain the mean daily motion of the planets. The number of minutes of arc moved by a planet per day gives the mean daily motion of that planet. Though, however, the spatial velocity of each planet is the same per day, in angle, the velocities differ, (on account of the varied distances) and so we perceive slowness or fastness in the movement of the planets. Comm. Easy. The assumption of equal daily spatial velocities for all the planets has been explained before. The daily angular velocity of a planet in minutes of arc is R × 360 × 60' ─────────────── where R is the number of its sidereal re- M volutions and M the number of mean solar days in a Kalpa. Verse 27. The reason for unequal angular velocity of the planets. Since the planetary orbits are all construed as com- prising of 360° alone, a minute of arc of a smaller orbit has a smaller spatial distance which will be covered more rapidly, the velocity being constant, whereas of a longer orbit, a minute of arc means a longer distance which will be covered in a longer time at the same spatial velocity, which means that the planet appears to be slow in motion. Thus the Moon, the Mercury, Venus, Sun, Mars, Jupiter, Saturn being placed at longer distances from the earth in ascending order their orbits are longer in ascend- ing order so that they are slower in angular motion in that order. Here ends the section called Pratyabda Suddhi in the chapter Madhyādhikāra.

THE SECTION KNOWN AS ADHIMĀSĀDI- NIRNAYA IN Ch. I Verse 1. A special feature of the computation of Ahargaṇa. If the Ahargaṇa is to be increased or decreased by unity to adjust it with the week-day, the tithis also are to be increased or decreased by unity to be adjusted with the week-day. Then in that context of increasing or decreasing the Ahargaṇa by a day the Adhimāsa-Sesha is to be increased or decreased by the number of Adhimāsās of a Kalpa and the Avama-Sesha is to be inoreased or decreased by the Avamās or Kshayāhās of a Kalpa. Comm. After having computed the Ahargaṇa as per the directions given under verses 1-3 of the section of Grahānayana, we have to test its correctness on the basis of the week-day taking it that the Kalpa began with Sunday. In other words, since the number of the elapsed week-days accord with the number of Sun rises, the week- day on the day on which the Ahargaṇa is computed should accord with the Ahargaṇa ie dividing the Ahargaṇa by seven, the remainder must give us the week-day of the day in question. But it so happens that we may have to add or subtract one from the Ahargaṇa arrived at to adjust the Ahargaṇa with the week-day. Why does this happen ? The reason is as follows. When we find the Ahargaṇa for a particular number of elapsed tithis, we are unwit- tingly taking the number of elapsed true tithis in the place of the average tithis. The fact that we are given in the table of constants, the number of mean units of time in a Kalpa, for example mean solar days, mean tithis etc, and the fact that computation proceeds only on the basis of these mean units, means that we have taken into account only the number of mean tithis elapsed. Hence this is to be corrected by us. If we are computing the Ahargaṇa

71 for, say, the pratipat of phālguna, we automatically take that the elapsed true tithis from the beginning of the luni-Solar year is 11 × 30 = 330 tithis. But it may so happen that had we taken only the number of mean tithis elapsed, either 329 tithis or 331 tithis might have elapsed and not 330 as construed which therefore works an error amounting to unity in the Ahargaṇa arrived at. At the same time the error does not exceed unity, for, during the course of every lunation, the longer tithis are almost compensated by tithis of shorter duration in the same lunation. The variance in the length of a tithi being wrought by the variable motion of the Moon, which again depends on the distance of the Moon from his apogee, is rounded off in every lunation, as the Moon completes a circle with respect to the apogee in what is called an anomalistic month. Thus it is that the Ahargaṇa is to be rectified on the basis of the week day. When this is done and the Ahargaṇa happens to be increased or decreased by unity, automatically it goes without saying, that the tithis are also increased or decreased by one. In this context there is a still more deeper significance as detailed below. The Adhimāsās are computed from the following formula viz. (A × a) / S = I + (F / S) × 30 (1) where A = Ahargaṇa, a = adhi- masās in a Kalpa, S = solar days in a Kalpa, I = Integral quotient obtained by division and (F / S) × 30 the Adhimasa- Sesha-tithis. If now A is to be increased or decreased by 1 ((A ± 1) a) / S = (Aa / S) ± (a / s) = I + (F / S) × 30 ± a/S from (1) = I + (F × 30 ± a) / S ; hence the adhimāsa-sesha namely F × 30 is increased or decreased by a ie the number or adhimasās. Similarly the Avama-Sesha or the Kshayāha-Sesha is to be increased or decreased by the Kshayāhas in a Kalpa.

72 The particular mention of the increase or decrease in the Adhimāsa-Sesha or the Avama-Sesha is necessitated in the context of computing the positions of the Sun or Moon, given the Adhimāsa-Sesha and Avama-Sesha as mentioned under verses 6, 7 in the section of grahānayanā. Verse 2. Pertaining to the smaller Ahargaṇa compu- ted from the beginning of the current Solar year, called Laghu-Ahargaṇa. In the case of computing the Ahargaṇa from the beginning of the Solar year also, when the Ahargaṇa is to be increased or decreased by unity, the tithis are to be increased or decreased by unity. The Avama-Sesha here is to be increased or decreased not by the Kshayāhās of the Kalpa but only by unity because we have used the formula 1/64 of tithis to get the Kshayāhās. If, an Adhika- māsa happens to occur during the course of the current year, the tithis, 30 in number of this Adhikamāsa must be also taken into account to obtain the Ahargaṇa. Comm. Easy. Verse 3, 4. The Ahargaṇa is to be computed (larger Ahargaṇa) after taking into account an Adhimāsa which has conspicuously occured but which is not obtained by computation or by rejecting an Adhikamāsa which has not occured but which is obtained by calculation. The Adhimāsa-Sesha is to be increased or decreased by the Adhimāsas of the Kalpa; the elapsed months from the beginning of the luni-Solar year are to be increased or decreased by unity and then the positions of the Sun and the Moon are to be computed from such an Adhimāsa-Sesha and such an Avama-Sesha. Comm. Already explained. Verse 5. A point to be noted with respect to the Suddhi,

73 In the case of obtaining the Suddhi, if an Adhikamāsa, which did not actually occur, is obtained by calculation, then the Suddhi is to be increased by 30, so that the Ahargaṇa is not affected by the un-occuring Adhikamāsa. Comm. The computation of the Adhikamāsas or inter- calary months proceeds under the consideration of mean lengths. So, it is likely that an Adhikamāsa may occur un-warranted by calculation or may not occur in spite of its being shown by calculation. Further, an Adhikamāsa may be delayed in occurence by the fact that though the luni-Solar reckoning has gained over the solar by one mean lunation, the lunation at that point may still con- tain a Samkrānti, the preceding particular lunar month being smaller in length than the mean. Thus the conven- tion made with respect to the occurence of an adhikamāsa, namely that the lunation which does not carry a Samkrānti is to be construed as an adhikamāsa, may also delay the occurence of the Adhikamāsa, though shown in calculation. Similarly an Adhikamāsa may be preponed though not warranted by computation by the same logic. Verse 6. The criteria of an Adhikamāsa and a Kshayamāsa. A lunation which does not carry a Samkrānti is an Adhikamāsa; whereas a lunation which carries two Sam- krāntis is to be taken as a Kshayamāsa. The Kshaya- māsa, occurs only in the course of the three lunar months named Kārtica, Mārgasīrsha and pausha and not during any other lunar month; when a Kshayamāsa occurs, then during the course of that year there will be two Adhikamāsās occuring on either side of the Kshaya- māsa. Comm. The institution of intercalation has been explained to some extent under verse 10 of the Bhagaṇādh- yāya. We shall see some more particulars of intercalation. 10

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1. Bhaskara has given that 1593300000 Adhika- māsās occur in a Kalpa of 4320000000 Solar years, which means that 15933 Adhikamāsās occur in 43200 Solar years ie. 5311 Adhikamāsas in 14400 solar months. Converting 14400 / 5311 into a continued fraction, we have 2 + 1/(1+) 1/(2+) 1/(2+) 1/(6+) 1/(1+) 1/(1+) 1/(7+) 1/(3+) 1/2. The successive convergents are 2/1 3/1 8/3 19/7 122/45 141/52. Let us see what these convergents signify. (a) The convergent 19/7 means that on an average there are 7 Adhikamāsās per 19 years. This ratio was adopted in the Romaka Siddhānta of Panchasiddhāntika. It means that 19 × 12 = 228 Solar months are equal to 235 lunations. The Metonic cycle described in modern astronomy is based upon this equivalence. The recurrence of Moon's phases in 19 years ie correspondence of the Moon's phases or tithis with the dates of the English year and Meton's formula are based on this equivalence. Recurrence of phase means recurrence of the relative positions of the Sun and the Moon, which again means recurrence of the Suddhi, for Suddhi is no other than the interval between the New Moon and Saṁkrānti. In the next verse Bhāskara says that a Kshayamāsa recurs after a lapse of either 19 years or 122 years or 141 years. The numerators of the last three convergents are 19, 122 and 141 which are the numbers of Solar years that effect recurrence of the same Suddhi and as is going to be mentioned shortly a Suddhi of 21 tithis is likely to bring in a Kshayamāsa. Hence a Kshayamāsa recurs either in 19 years or 121 years or 141 years.
2. Mention of the occurence of a Kshayamāsa was made by Sripati first and not by the preceding astro- nomers. However, mention of it is there in the Vedic

76 literature and it is not clear whether observance of this Kshayamāsa was defunct for some centuries in between. 3. We shall now proceed to see how there occur two Adhikamāsas on either side of a Kshayamāsa. A simple argument is as follows. The convergents cited above namely 19/7 or 122/45 or 141/52 signify that either 7 or 45 or 52 Adhikamāsās are to occur in the course of 19 or 122 or 141 Solar years normally. But when the S'uddhi happens to be 21 days at the beginning of the Solar year, it so happens that the S'uddhi goes on increasing for the first five months because the Sun is in his apogee when his longitude is 78°, and his motion being slow for three months when he is on either side of his apogee, the Moon gains over him in shorter intervals of time and as a consequence the lunations are of shorter duration. This means that the S'uddhi goes on increasing during those months and rapidly increases from its value 21 at the beginning to 30 by about Bhādrapada month. Under these circumstances, a Samkramaṇa occurs generally just before the beginning of Bhādrapada. The next Samkra- maṇa happens just a little after the lapse of Bhādrapada, so that the month of Bhādrapada goes without a Samkra- maṇa and as a consequence, it becomes an Adhikamāsa. Thus far it is alright that an Adhikamāsa has occured as per the meaning of the convergents. But when the Bhādrapada thus becomes an Adhikamāsa, the subsequent months from Kārtica to Mārgasira, being of longer duration than the corresponding Solar months, the Sun having a quicker motion on either side of his perigee, there is every likelihood of a Solar month being contained between two conjunctions or New Moon days. In other words two Samkramaṇas occur either in Kārtica or Mārgasira or Pausha. This means that as per the convention for the occurence of a Khayamāsa, one of the aforesaid lunations must become a Kshayamāsa. Thus the Adhikamāsa which

76 is due to occur during the course of the year, though it has occured has been lost. So, to make amends, another Adhikamāsa is to occur as is warranted by the convergents cited above. It might be asked what if two Adhikamāsas occur and why a Kshayamāsa be instituted at all. The reason is not that a religious convention warrants it but because the wedding of the luni-Solar year to the Solar year has to be made on a particular principle. Nor- mally, so long as a Samkramaṇa goes on occuring during the course of a lunation, the two systems of reckoning may be seen to be running parallel. But if a particular lunar month does not contain a Samkramaṇa it is to be taken as a warning that the luni-Solar reckoning has overtaken the Solar by one lunation. This lunar month has to be curtailed to make the two kinds of reckoning to proceed side by side. This convention naturally raised the ques- tion as to how to deal with a lunation which contains two Saṁkrāntis. The Solar month their has to be deleted to make the two systems run concurrently. This deletion of a Solar month is achieved not by declaring the particular Solar month as an Adhikamāsa but what is virtually the same two lunar months are deemed to lapse during the course of that solar month. This kind convention helped the occurrence of one Adhikamāsa alone as scheduled be- cause one of the two Adhikamāsas has been nullified by the convention of a Kshayamāsa. Why the proposition that a Kshayamāsa generally occurs when the Śuddhi at the beginning of the Solar year happens to be 21 tithis is quite evident because in such a case generally Bhādrapada becomes an Adhikamāsa, which again entails the occurence of two Samkramaṇas during the course of one of the three lunations beginning with Kārtica, which happen to be longer than the corresponding Solar months. In other words a Kshayamāsa is expected to occur only when Bhādrapada happens to be an Adhika- māsa, and this in turns happens only when the Śuddhi happens to be 21 at the beginning of the Solar year. Adhi-

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kamāsās do occur when the Suddhi happens to be more than 21, but in this case one of the months prior to Bhādra- pada happens to be an Adhikamāsa and then by the time Kārtica is reached, the Samkramaṇa occurs not at the beginning of Kārtica but a little later, which means that a second Samkramaṇa could not occur before the lapse of that lunation. This therefore will not be a Kshaya month.

The Suddhi which is defined as the number of tithis in between a New-Moon and a Samkramaṇa, is clearly the in- terval which has been gained by the luni-Solar system over the Solar. This S'uddhi increases at the average rate of 11 days, 3 ghatis, 52 palas and 30 Vipalas per an year as we have seen already. For the occurrence of an Adhika- māsa during the course of an year the Suddhi at the begin- ning of the year must be such that it accrues to a lunation during the course of the year. The apogee of the Sun being almost stationary having a very slow motion the lengths of the Solar months do not vary appreciably for a good number of years. Taking that the Sun's apogee is roughly at 80° longitude (78° according to Hindu Astronomy) the Sun's motion is less than his average from the moment when he has 350° longitude upto the moment when he has 170°, so that the Moon gains rapidly over him during this period. In other words the lunations during this period are of shorter duration i.e. the luni-Solar months of chaitra upto Sravana will be shorter in length so that S'uddhi increases rapidly from March upto August. Thereafter it will attain a stationary value just for a little time and then decreases for the next six months upto February. The increase, however, far exceeds the decrease and the balance of in- crease per year is as mentioned above is 11-3-52-30.

Bhāskara mentions in the course of the commentary that (1) the average length of a lunation is 29 days, 31 ghatis and 50 palas; (2) the average length of a Solar month is 30-26-17 ; (3) when the daily motion of the Sun happens to be 61', then the Solar month will have a length

of 29-30 and as such falls short of a lunation and (4) that the minimum length of a Solar month is 29-20-40 only. Then he gives a general and broad explanation as to how a Kshayamāsa occurs on the following lines. Let us call NS as the interval between a New Moon and the subsequent Saṁkrānti. In other words it is the Suddhi. Suppose Bhādrapada goes without a Saṁkrānti thus becoming an Adhikamāsa. Then in Āswayuja NS will have a small value, because the Saṁkrānti S which should have occured at the general rate of one per a lunation, has not occured before the lapse of Bhādrapada and being belated a little must have occured close on the heels of the New Moon of Bhādrapada. Let this interval NS have a particular value say x. Thereafter the Solar months grow gradually smaller i.e. the Saṁkrāntis occur earlier ; so in Kārtica the value of NS decreases. It might decrease to such an extent that the next Saṁkrānti might occur before the next New Moon. Suppose this does not happen in Kārtica ; then in Marga- sira the value of this NS is still smaller and there is a greater likelihood of another Saṁkrānti occuring during this Mārgasira. Then the value of NS being the smallest in the month of Pausha, another Saṁkrānti is bound to take place during this month at the latest. In other words S now occurs before the next N i.e. the Saṁkrānti occurs just a little before the next New Moon. Let now this SN be a small quantity say y; or what is the same the next NS or the Suddhi will be nearly a lunation. There after the Solar months begin to gain in length i.e. S occurs later and later. This means NS gains in length. Being nearly of a length equal to a lunation, and now gradually increasing it is bound to exceed a lunation shortly thereafter. This again means that S occurs later than the next New Moon, which is to say that one more lunation goes without a Saṁkrānti. Thus it is that another Adhikamāsa occurs shortly after the occurence of a Kshayamāsa. Hence in the course of one year there occur two Adhikamāsās and one Kshayamāsa which means again that the balance of an

79 Adhikamāsa is there in that year. In other words, the occurence of a Kshayamāsa during the course of an year as per a stipulated convention does not preclude the average occurence of an Adhikamāsa which is normally due after an average lapse of 32½ Solar months. That the occurence of a Kshayamāsa entails the occurrence of two Adhikamāsās on either side may be also seen in another way. Twelve lunations put together have a length of 354⅓ days approxi- mately during the course of which only eleven Saṁkrāntis could occur in case the initial Śuddhi is 21 days, for, sub- tracting this 21 from 354⅓, there remain 333⅓ days only which could contain only ten Solar months and not eleven. Ten Solar months are contained if there be only eleven Saṁkrāntis. Out of these eleven, two are consumed in a single lunation which happens to be a Kshayamāsa. Hence the remaining eleven lunations have to contain only nine Saṁkrāntis which means that two lunations have to go without Saṁkrāntis. In other words there are to be two Adhikamāsās during that year. Further both these Adhi- kamāsās could not occur on one side of the Kshayamāsa for the following reason. Since a Kshayamāsa could occur only during the course of Kārtica or Mārgasirsha or Pansha which alone are longer than the corresponding solar months, in the event of both the Adhikamāsās taking their place on one side of these months, they are to take place in nine lunations which are on one side of the three months beginning with Kārtica. This means that only seven Saṁkrāntis are to occur during the course of 9 × 29½ days = 265½ days. Even seven Solar months fall short of this period so that it is impossible that only seven Saṁkrāntis should occur in this period. Hence, the two Adhikamāsās which are to occur during the year containing a Kshaya- māsa have to take their place on either side of the Kshaya- māsa. Bhāskara mentions that when the daily motion of the Sun equals 61', the length of the Solar month will be 1800/61 = 29–30 approximately. He also states that the minimum

80 length of a Solar month will be 29-20-40. This will be so when the average daily motion during the course of a month is 61-20-30. Verse 7. A mention of the years past and future that had and will have Kshayamāsās. A Kshayamāsa occured in the Saka year 974 and will occur in the years 1115, 1256, 1378. Thus generally it occurs once in 141 years or even 19 years. Comm. We have seen before that according to the convergents given above a Kshayamāsa generally occurs in 19 years or 122 years or 141 years. In the course of the commentary Bhāskara says "19 years before or after" so that 141-19=122 years was also meant by him. He gives how in the course of 141 years or 19 years the Suddhi at the be- ginning of the year happens to be approximately 21 days. In this behalf, he invokes his formulation of the Adhika- māsās in the verse 6 under Pratyabdasuddhi. As per that formula the number of Adhikamāsās in 19 years will be 7-13-37-30 taken by Bhāskara as 7-13-40. In other words the Suddhi increases by 13 ghatis and 38 palas from what it was 19 years ago; thus practically the value of the Suddhi recurs in periods of 19 years which means that if a Kshayamāsa occurred this year, there is a likelihood of its recurrence after nineteen years. Of course the recurrence would not be certain because there is an excess of 13 ghatis Suddhi which might preclude its occurrence. Similarly in 122 years and 141 years the numbers of Adhikamāsās would be respectively 45 minus 7-15 ghatis and 52 plus 6 ghatis. 22 palas and 30 Vipalas, the latter being taken by Bhaskara to be 52 Adhikamāsās plus 6 ghatis and 20 palas. Thus Kshayamāsas are more and still more likely to recur in intervals of 122 years and 141 years respectively, the Suddhis at the beginning of the year almost recurring and assuming the original value of 21 days for a Kshayamāsa to occur.

81 Bhāskara adds that a Kshayamāsa occured in 974 Saka i.e. 62 years before his birth which knowledge he must have derived from hearsay or even by calculation. He then predicted its recurrence in the Saka years 1115, 1256 and 1378 which were at intervals of 141 years, 141 years and 122 years respectively. In the year 974 Saka i.e. 4153 Kali year, applying the above formula Dwidhābdāḥ the number of Adhikamāsās elapsed from the beginning of the Kali were 1531-21-12-52-30 i.e. the S'uddhi at the begin- ning of that year was 21-12-52-30. This naturally entailed the occurence of a Kshayamāsa. Thereafter during the years 1115, 1256 and 1378, the S'uddhis in the beginning of the years should be as per the above analysis 21-12-52-30 plus 68-22-30 i.e. 21-19-15; 21-25-37-30 and 21-25-37-30 and 21-18-23 respectively so that Bhāskara could forecast the occurrence of the Kshayamāsās in those years also. In this context it is worth-noting that Gaṇeśa notified in a verse that Kshayamāsās would recur in the years 1462, 1603, 1744, 1885, 2167, 2232, 2373, 2392, 2524, 2533, 2655, 2674, 2796, 2815 according to Sūrya Siddhānta and accord- ing to Āryabhaṭīya during the years 1482, 1793, 1904, 2129, 2186, 2251. These years also may be verified by computing the S'uddhis as said before and also with respect to Ārya- bhaṭīya according to which the number of Adhimāsas during a yuga differs by a little and as such effects a differ- ence in the sequence of the Kshayamāsa years. Verse 8. Tell me, how, when and in the course of how many years do two Adhikamāsas occur as mentioned by the Rishis? Questioned accordingly by an expert in questioning, if a mathematician could know the answer, I would reckon him as no other than Bhāskara (either the Sun-god or he himself i.e. Bhāskarāchārya) who could make the lotus-buds of mathematicians blossom. Comm. Probably taking up this challenge alone Gaṇeśa answered the question and gave the years cited above which should bring in a recurrence of Kshayamāsās. 11

82 N.B. It does not suffice merely to compute the S'uddhis alone in the beginning of the years but a rigorous compu- tation necessitates the calculation of the moments of New Moons and Samkrāntis also during the particular years as well, in as much as, the motion of the Moon also comes in- to the picture and it differs from month to month on account of the rapid motion of his apogee. Here ends the section named Adhimāsādi-nirṇaya.

THE SECTION BHŪ-PARIDHI-MĀNĀDIKA- CIRCUMFERENCE OF THE EARTH Verse 1. The circumference of the earth's globe is 4967 yojanas ; its diameter 1581. A yojana is equal to (d × 360) / (C × δϕ) where δϕ is the difference in the latitudes of two places on the same terrestrial meridian in degrees, C the circumference of the earth's globe given above and d the distance between the two places. Comm. The second half of the verse gives the method of computing the circumference of the earth's globe, which is mathematically correct ; for, by the rule of three "If by a difference of 90° in latitude we have ¼ C, what shall we have for 1° difference in latitude ? " The answer is C / 360° Again "If by distance of d between two places on the same meridian, we have a difference of δϕ° in latitude, what shall we have for 1° difference in latitude ? ' The answer is d / δϕ. Both the answers must be the same ; so equating them, we have C / 360 = d / δϕ = number of yojanas = x say. Hence one yojana = x / x = (d / δϕ) ÷ (C / 360) = (d × 360) / (C × δϕ) N.B. Here it must be noted that a yojana's length is derived from the number of yojanas contained in the circumference as reported in the Āgama. In the course of the commentary under this verse, Bhāskara explains why he had recourse to this kind of definition, which is based upon āgama and as such does not contain a proof. He says that in as much as the defi-

84 nition of a yojana was given basing ultimately on the units of Angulas and yavas (grains of paddy) differently by differ- ent authorities, and in as much as the circumference of the earth’s globe was more or less unanimously accepted, he has chosen to define a yojana on the accepted measure of the circumference. In some other place Bhāskara says that a yojana is equal to 4 Krōsās, where the word Krōsa etymologically means that distance through which the topmost voice of a healthy person could be heard by another healthy person having good audition. In this context it may be recalled how Śrīpati defined a yojana under verses 69, 70 Madhyamādhyāya of his Sid- dhānta Śekhara. “The minute speck of dust that is seen flying in the light of the Sun’s rays entering a house through the windows, is called a paramāṇu (not an atom as is being translated now). Eight such paramāṇus equal one Reṇu. Eight Reṇus equal the breadth of the end of a hair known as Vālāgra or Kaccha-mukha or yūka. Eight Vālāgras equal one Likshā; Eight Likshās equal one Yava; eight Yavas equal one Angulā; twelve Angulās make what is called a Vitasti (i.e. the expanded length of a palm of the tallest person). Two Vitastis equal one Hāsta. Four has- tās make one Chāpa. Two thousand Chāpas make one Krōsa and four Krōsās are reported to be equal to a yojana by Astronomers” Verse 2. Rectification of the circumference .ie. find- ing the length of the circumference of the earth parallel to the terrestrial equator and passing through the locality in question. Also the definition of the primary meridian. The equatorial circumference of the earth multiplied by cos ϕ and divided by R or multiplied by 12 and divided by the hypotenuse of the right angled triangle formed by the gnomon and the equinoctial midday shadow thereof. (Hereafter called equinoctial hypotenuse) gives the circum- ference of the earth parallel to the equator and passing

85 through the locality (hereafter called the rectified circum- ference). Also the primary terrestrial meridian is that longitudinal line passing through the places (1) Lanka (2) Ujjain (3) Kurukshetra and (4) the north pole. Fig. 4 Comm. (Ref. fig. 4) Let ABC be the terrestrial Equa- tor, and QST be a circle parallel to this terrestrial equator and passing through the locality S. This QST is called the rectified circumference of the earth at the place S. The terrestrial equator is defined as follows. It is known that the earth rotates about herself about the axis P₁ P₂ where P₁ P₂ is called the polar axis or Dhruvayashti. In other words the entire heavens appear to revolve round the earth in such a way that any star will appear to be revolving in a circle called its diurnal circle which is parallel to the circle ABC. The points P₁ and P₂ are called the north and

86 south pole respectively. These two points evidently do not move, though the earth is herself rotating in the clockwise direction. If O be the centre of the earth, OP₁ produced meets the skies at a point called the north celes- tial pole and OP₂ produced meets the skies in the point called the south pole. It so happens that the north celes- tial pole is very near a star which is called the pole star. This star is known as the Dhruva-Tāra in as much as it does not appear to move at all while all the heavens (i.e. all the stars of the sky) appear to be revolving round the earth rising and setting as seen at any place. (The word Dhruva means fixed). Aryabhaṭāchārya mentioned in so many words that it is the earth that really rotates and so the stars which are themselves fixed appear to be going round the earth in circles parallel to the circle ABC. अनुलोमगतिः नौस्थः पश्यत्यचलं विलोमगं यद्वत्। अचलानि भानि तद्वत्समपश्चिमगानि लङ्कायाम्॥ (Explained under verse 7 Bhagaṇādhyāya). The celestial Equator is the great circle which is the circle of intersection of a plane perpendicular to the polar axis and passing through the earth's centre with the celestial sphere (celestial sphere is the sphere-like surface which shape the Sky takes and on which the stars and the planets appear to be studded.) Similarly the terrestrial equator ABC is the circle of intersection of the earth's globe with the same plane. Thus the terrestrial and the celestial equators are concentric coplanar circles with the earth's centre as the common centre. A great circle of a sphere is a circle whose plane passes through its centre. Thus ABC is a great circle on the earth's surface, because its plane passes through the earth's centre. Similarly the celestial equator is a great circle of the skies. The circle QST is called a small circle, just as the diurnal circle traced by any star in its diurnal rotation is also a small circle parallel to the celestial equa- tor. Thus small circles, an infinity of them can be drawn parallel to the terrestrial equator ABC and they will be in decreasing dimension as we proceed towards the pole. Hence the Sphutaparidhi or the rectified circumference

87 QST at the locality S is a small circle whose circumference is smaller than that of ABC. The problem is now to find the length of this circle QST. Evidently QST / ABC = 2 π r / 2 π R = r / R where r and R are respectively the radii of the two circles. But r / R = Cos ϕ from the triangle OSO', where Ô'₁SO = ŜOA = latitude of the place S. ∴ QST = ABC × Cos ϕ = (ABC × H Cos ϕ) / R I where H Cos ϕ is the Hindu cosine of ϕ known as lambajyā and stands for OS₁ = O'₁S = r. This lambajyā is also called Dyujya if the small circle is the diurnal circle of a Star. The diurnal circle of a Star is called Dyujya-Vritta and its radius is called Dyujyā-Equation I proves the first statement of the Fig. 5 verse. In fig. 5 Gn N is called the fundamental gnomonic triangle where Gn is the vertical gnomon pointing to the Zenith Z of the celestial sphere and is considered to be of 12 units (Angulas as they are called), GN the midday- shadow of the gnomon cast on an equinoctial day when the