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

60 The Dinādya may be also obtained from the beginning of the Kali, which begins with Friday. The Dhruvas calculated for the commencement of the solar year are to be added to the planetary positions at the beginning of the Kali, in the event of computing the Ahargaṇa and thereby the planetary positions from the beginning of the Kaliyuga for the sake of convenience. Comm. The Dinādya at the commencement of Kali is Zero, since the number of civil days during the length of time equal to a Kaliyuga is integral and equal to 157791645 according to Brahmagupta and Bhaskara who follows him. The Suryasiddhānta, it may be noted here, gives the number of days in a Kaliyuga as not integral but equal to 157791782.8. Hence the Dinādya computed from the beginning of the Kali is to be increased by .8 to obtain its value according to Suryasiddhānta. The plane- tary positions at the commencement of Kali were given by Bhaskara already. Verse 12. The number of what are called Kshepadinās to obtain the Ahargaṇa. Hereafter Bhaskara is going to obtain the planetary positions for any day during the current solar year having obtained the Dhruvakas or the planetary positions for the beginning of the Solar year. In that behalf the Ahargaṇa or the collection of days which have elapsed from the commencement of the Solar year is to be found. This Ahargaṇa is obtained by subtracting the number of Ksha- yāhās from the number of tithis that have elapsed. In finding these Kshayāhās, we have to take note that there is a little remnant of Kshayāhās at the beginning of the Solar year which is also to be taken into account while computing the number of Kshayāhās during the course of the year. In other words, the number of Kshayāhās that are going to be computed during the course of the year, for the elapsed part of the year will be in default of the

61 actual number if we ignore the accrued fraction of Ksha- yāhās at the commencement of the Solar year. To make amends for that default we have to add some number to the numerator of the improper fraction which is going to give us the number of the elapsed Kshayāhās during the course of the year. The formula that is going to be used to obtain the number of Kshayāhās during the elapsed tithis is x/64 (1 + 1/702). This formula arises out of the fact that there are 55739 Kshayāhas during 3562220 tithis, so that for 64 tithis the number of Kshayāhās is equal to (64 × 55739) / 3562220 = 3567296 / 3562220 = 1 + 5076 / 3562220 = 1 + 1 / (3562220 / 5076) = 1 + 1/702. Let the fraction of Kshayāha at the commencement of the solar year, to be taken into account be x ghatis i.e. x/60 of a tithis (for Kshayāhās are are computed out of tithis) i.e. (x tithis) / 60. Let y be the number of tithis elapsed after the commencement of the solar year. Then to compute the Kshayāhās that ensue after the commencement of the solar year upto the day concerned during the course of the year, the formula to be used is y/64 (1 + 1/702). To this we have to add (x tithis) / 60 as the balance of Kshāyāha at the commencement of the solar year to be taken into acccount (x tithis) / 60 = (x × 64) / (64 × 60) = (x × 64/60) / 60 Now, in computing the number of tithis which have elapsed after the commencement of the Solar year, we subtract Śuddhi from the number of tithis that have elapsed after the beginning of the luni-Solar year. But

62 in this Śuddhi we have subtracted the fractional part of the Kshayāhas for a different purpose so that in sub- tracting the Suddhi, we have increased the tithis by the fractional part of the Ksbayāhās. This increase there- fore should be nullified, which means that the quantity to be added is (x × 64/60) / 64 − x/60 = (x × 63/60) / 64 = (21/20) (x/64) = x (1 + 1/20) / 64 Hence the Kshepadinas or the tithis to be added to the number of tithis which have elapsed from the commence- ment of the Solar year, are x (1 + 1/20). This means that the ghaṭis x which form the fractional part of the Kshayāhas are to be increased by one twentieth part of themselves and are to be viewed as tithis and not ghaṭis as mentioned. Verse 12 (contd

next Sun-rise, T₃ the ending moment of the tithi preceding the Sun-rise R₄ upto which point the Ahargaṇa has to be found from the commencement of the Solar year. Thus the Ahargaṇa to be found is SR₄ = ST₃ + T₃ R₄. Here T₃ R₄ is the Kshayāghatis, at the Sun-rise concerned. SR₃ is a fraction of a day that is to be there in the Ahar- gaṇa SR₄ we are seeking. While subtracting the Śuddhi T₁ S diminished by the Kshayāghatis T₂ R₃ from the number of elapsed integral number of tithis from the commencement of the luni-Solar year namely T₁ T₃ we have T₁ T₃ — (T₁ S — T₂ R₃) = T₁ T₃ — (T₁ T₂ — SR₃) = T₂ T₃ + SR₃. Thus instead of SR₃ + R₃ R₄ we have by the above procedure SR₃ + T₂ T₃. Though both T₂ T₃ and R₃ R₄ are integral numbers and should be the same if the interval is small, they may differ by an integral number, if the interval happens to be long. Bhaskara says that this difference of an integral number will be rectified by the Kshepadinas found under verse No. 12, for, from these Kshepadinas, the Kshayāhas are found and subtracted from the tithis. It will be noted that the difference of an integral number between T₂ T₃ and R₃ R₄ is no other than Kshayāhas. Or, this may be seen in an other way. We are to find SR₄ = T₁ R₄ — T₁ S = T₁ T₃ + T₃ R₄ — T₁ S = T₁ T₃ — (T₁ S — T₃ R₄) = No. of elapsed tithis — (Suddhi — Kshayaghatis at the day concerned). But instead of the Kshayaghatis at the day concerned Bhaskara prescribes the subtraction of the Kshayāghatis at the end of the Solar year ie instead of subtracting T₃ R₄ it is prescribed to sub- tract T₂ R₃. This difference, Bhaskara says is made up by taking into account the Kshepadinas pertaining to the Kshayāhas. Verse 14. In case the Ahargaṇa is required for a day preceeding the commencement of the Solar year, then the elapsed tithis are less than the Tithis of the Suddhi; so subtraction is not feasible. In this case, it is prescribed

64 to take the elapsed tithis from the previous Chaitra and the Suddhi of the previous year. Also in this case the Dhruvakas pertain to the commencement of the previous Solar year. Comm. Easy. Verse 15. To obtain the position of the Sun. The number of the days in the Ahargaṇa is to be diminished by 1/60 part of itself to obtain the number of degrees, and fractions thereof; then the Ahargaṇa multi- plied by three and divided by 22 gives the minutes and fractions thereof. Adding the two results we get the position of the Sun. Comm. The mean daily motion of the Sun is 0-59-8-10-21. Here 59' = 1° - 1' = (1 - 1/60)°; for x days x (1 - 1/60)° = x° - x°/60 as mentioned. The remain- ing part namely 0-0-8-10-21 = 9807'/72000; Converting this into a continued fraction it is equal to 1/(7 +) 1/(2 +) 1/(1 +) 1/(12 +) ... of which a good convergent is 3'/22. Hence for x days 3x/22 as stated in the verse. Verse 16. To obtain the position of the Moon. The number of elapsed integral tithis multiplied by 12 and added to the Sun's position in degrees, gives the Moon's position in degrees at the ending moment of the tithi preceeding the day at the Sun-rise of which the planetary positions are sought. To find the position at the Sun-rise required, ten times the Kshaya-dina-Sesha increased by 1/7 th of itself gives the number of minutes to be added to the position got above.

65 Comm. The first part is clear, because for every tithi, there will be an increase of elongation of 12°. To get the fractional part of the tithi in between the ending moment of the tithi and the subsequent Sun-rise, the interval which is Sāvana ie expressed in the unit of civil day has to be converted into luni-Solar units. The rule of three is ‘ If for 63 Sāvana days there are 64 tithis, what will it be for the above interval ?’ Here the approximate ratio of 64/63 is used because the Ahargaṇa which is Laghu is small ie is less than 365. The Kshaya-dina-Sesha which was got under verses 12, 13, has a divisor of 64, and is of the form y/64 . Hence the quantity to be added is (y × 64)/(64 × 63) = y/63 of a tithi = y/63 × 12 × 60 minutes of arc = 8/7 × 10 y = 10 y (1 + 1/7) minutes as given in the verse. This has to be added to 12t°, got before where t is the number of elapsed tithis. Verse 17. Computation of Mars. The mean daily motion of Mars is 0-31-26-28-7=0-30

• 0-0-90 minus 0-0-3-31-53. If x be the number of days x × 0-30' = x°/2 ; x × 0-0-90" = x/2 × 3' ; thus the rule prescribed is “ Half of the number of days gives the degrees ; half the number of days multiplied by 3 gives the number of minutes ” from this we have to subtract x × (0-0-3-31-53). The quantity within the brackets is approximately 1/17 of a minute for 1/17' = 0-0-3"-31"'-46"'. Since the Ahargaṇa is small the precision required is there. Thus the rule is x°/2 + x/2 × 3' + x'/17 + the Dhruvaka at the beginning of the Solar year. Verse 18. Computation of the S'īghroccha of Mercury. The mean daily motion of the Budha-S'īghra is 4°-5'-32"-18"'-28"". 9

66 Rule. If the Ahargaṇa be x days then the Śīghra will be 4x° + (4x × 3') / 130 + Dhruvaka. Proof. For x days, the mean motion is x × 4° + x × (0-5′-32″-18‴-28⁗). The second part is taken to be 4x × 3° / 130 ie x × 12° / 130 = x × 6/65° for 6/65 of a degree = 0-5′-32″-18‴-"28"". The fraction 6/65 can be seen to be a convergent of the remainder for 5′-32″- 18‴-28⁗ = 299077° / 3240000 = 1/(10 +) 1/(1 +) 1/(4 +) 1/(1 +) 1/9968 of which the penultimate convergent is 6/65. Verse 19. The Ahargaṇa divided by 12 and by 71 gives respectively the positive degrees and negative minutes to be added to the Dhruvaka of Jupiter. Comm. If x be the Ahargaṇa x° / 12 − x' / 71 + Dhruvaka = Guru. The mean daily motion of Jupiter is 0-4-59-9.9 = 0-5′ minus 0-0-0-50-51. Evidently 0-5′ × x = x° / 12. The remainder 0-0-0-50-51 = 61′ / 4320 of which the continued fraction is 1/(70 +) 1/(1 +) 1/4. A very approximate convergent is 1/71 as given. Verse 19 contd. To obtain the position of the Śīghra of venus. The Ahargaṇa multiplied by 10 and divided by 6 and 155 respectively gives degrees positive and negative to be added to the Dhruvaka of Venus to give his position. Comm. The formula is 10 x / 6 − 10 x / 155. The mean daily motion of the Śīghra of Venus is 1°-36′-7″-44‴-35⁗ =

67 1°-40' minus 0-3'052"-15"' approximately. For x days x ✕ 1 2/3° = 5/3 x° = 10 x° / 6 which gives the first part. Also 3'-52"- 15"' = 929° / 14400 = 1 / (15+) 1 / (1+) 1 / (1+) 1 / 464 of which a very ap- proximate convergent is 2/31 = 10/155 ie for x days 10x° / 155. Hence the position is given by 10 x° / 6 - 10 x° / 155 + Dhruvaka. Verses 20. The position of Saturn is given by 2x' / 5 + 2x" / 5 + Dhruvaka = Saturn's position. Comm. The mean daily motion of Saturn is 0-2-0-22-51. For x days 2x' + x ✕ 0"-22"'-51"". The latter part is 137" / 360 approximately = 1 / (2+) 1 / (1+) 1 / (1+) 1 / (1+) 1 / 2 of which a near covergent 2/5 so that for x days, we have 2x' + 2x" / 5

• Dhruvaka position of Saturn. Verses 20 contd. To find the position of the apogee of Moon. The Ahargaṇa divided successively by 10 and 88 and added gives the degrees to be added to the Dhruvaka of the Apogee of the Moon to obtain its position. Comm. The mean daily motion of the apogee of the Moon is 0-6-40-53-56. At the rate of 6' per day, in x days the number of degrees covered is x / 10 ; the remainder 0-0-40-53-56 = 4601° / 405000 = 1 / (88+) 1 / 41 of which a very approximate convergent is 1/88. Hence the position is given

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