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

200 PAÑCASIDDHĀNTIKĀ IX.4
Example 3. (a) Days 5,28,931. Find the Moon's apogee. (b) Find the Moon's apogee for Ujjain mean noon
prior to Epoch, and for Epoch.
(a) By (ii) the approx apogee in revs. = (5,28,531 × 900 + 22,60,356) ÷ 29,08,789
= Revs. 164-5-5-33.1
The additive seconds = 164 × 10 ÷ 297 = 6.
Adding, the exact apogee = rā. 5-5-33.2.
(b) For the said Ujjain mean noon, the apogee = (0 × 900 + 22,60,356) ÷ 29,08,789 = rā. 9-9-45.
Adding 2'.5, the mean motion of apogee in 22 1/3 nāḍikās, the apogee at Epoch = rā. 9-9-47.5.
(Note: The actual position was rā. 9-9-34. See how close this is.)
The explanation for the formula relating to the mean Moon is as follows: It was shown under I.14 that
in the Saura yuga consisting of 6,57,46,575 days there are 24,06,389 revolutions of the Moon. For
the sake of convenience, the author has first assumed that in whole numbers there are 9,00,000
revolutions in 2,45,89,506 days, intending to give a correction as a second step. Therefore we get
that in 6,57,46,575 days there are 6,57,46,575 × 9,00,000 ÷ 2,45,89,506 revolutions = rev.
24,06,389-0-10-55-27.
Thus we get 10° 55' 27'' more than what we should get, and this has to be deducted, proportion-
ately to the revolutions got. For one revolution the deduction is, 10° 55' 27'' /24,06,389 = 39,327''/
24,06,389. In the place of this fraction the author gives the approximate but simpler fraction 51''/
3121, since the error caused will be only plus 4'' in the yuga.
The manuscript reading, kharkāgni if read as khārkāgni as done by TS and NP, ( = 51''/3120) will
cause an error of minus 8'', which also is negligible but unlikely, since the author then would have
given the reduced form, 17''/1040. That is why we have read it as kvarkāgni, 3121.
The deduction of 6,70,217 is explained in the manner of the Sun's deduction: We have seen that at the
end of Śaka 427, the end of 3,606 solar years from the beginning of Kali fell 42/800 days, i.e. nā. 3-9,
after Ujjain mean midnight after Epoch. Under I.14 it was shown that according to the Saura there
are 24,06,389 revolutions of the Moon in 180,000 years. Therefore, in 3,606 years the revolutions
gone are 48,207.992966̇. At the beginning of Kali, the Moon, like the Sun, began a revolution,
according to the Saura. So, .007033̇ revolution remains to be completed now. We have seen that for
2,45,89,506 fractional parts there is one revolution. So, for .007033̇ revolution, the parts to go are
2,45,89,506 × .007033̇ = 1,72,946. These must go after the completion of the solar year to complete
the revolution. But the year ends nā. 3-9 + nā. 30 = nā. 33-9 from mean noon. In one day, there
are 9,00,000 parts, and for nā. 33-9, the parts to go are 9,00,000 × 33.15 ÷ 60 = 4,97,250. There-
fore at mean noon the parts to go for completing the revolution are 1,72,946 + 4,97,250 =
6,70,196. Since these have to go, this number is deducted from the total parts got by multiplying
the days by 9,00,000. Here, the author gives 6,70,217 arrived at by using approximate work in the
place of 6,70,196, for the difference is small, the error caused being only minus one second in the
yuga.
Now for the explanation of the rule to get the longitude of apogee: We do this using the element given
in Āryabhaṭa's Ārdharātrika system, or which is the same, in the Khaṇḍakhādyaka, since this is not
given in I.14, and the original Saura is not available. From them we learn that in the Mahāyuga of