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

40 PAÑCASIDDHĀNTIKĀ III.4
The two rules for the true daily motion can be shown to be connected with the two rules for finding
the true motion in II.6, by deriving these from those, and the exact derivation itself is a proof of
the correctness of the rules. The rule for plus-padas (i.e. applicable to the first half-gati) is,
{1094 + 5(P − 1)} P/63. If m is the mean motion at the end of the last full gati, then the true Moon
= m + P° + {1094 + 5(P − 1)} P/63 minutes. The daily motion of the current day is got by subtracting
the true Moon at the end of the previous day from that at the end of the current day and if the pada
is P at the end of the current day, it is P − 9 at the end of the previous day. Therefore the motion for
the current day = [m + 9° + {1094 + 5(P − 1)} × P/63] − [m + (P − 9)° + {1094 + 5 (P − 9 − 1)} (P − 9)/
63]′ = (m − m) + P° − (P − 9°) + {1089 + 5P)} P/63 minutes − {1089 + 5(P − 9)} (P − 9)/63 minutes
= in minutes, 540 + 1089 P/63 + 5P²/63 − 1089 (P − 9)/63 − 5(P − 9)²/63 = 540 + 1089 × 9/63

• 5 × 9²/63 + 90 (P − 9)/63 = 540 + 1134/7 + 10 (P − 9)/7 = 540 + 162 + 10 (P − 9)/7 = 702 +
  10 (P − 9)/7, which is the rule here given.
  In the same way, taking the rule for the true Moon in the second half-gati, i.e. for minus-padas,
  we have, the daily motion = [m + rā. 6-0-4 + P° + {2414 − 5 (P′ − 1)} P′/63 minutes] − [m + rā.
  6-0-4 + (P′ − 9)° + {2414 − 5 (P′ − 9 − 1)} (P′ − 1)/63 minutes] = in minutes, 504 + {2419 − 5P′}
  P′/63 − {2419 − 5(P′ − 9)} (P′ − 9)/63 = 540 + 2419 P′/63 − 5P′²/63 − 2419 (P′ − 9)/63 + 5 (P′
  − 9)²/63 = 540 + 2419/7 − 45/7 − 10(P′ − 9)/7 = 540 + 339 1/7 − 10 (P′ − 9)/7 = 879 1/7 − 10
  (P′ − 9)/7, which is the rule for daily motion in the second half-gati, omitting the small fraction of
  minutes 1/7. (Note that in the example we actually got this 879 1/7 as the maximum).
  From the relationship between these two sets of rules shown here, we can understand that the
  rules for daily motion, though they have strayed into the chapter dealing with the Pauliśa, actually
  belongs to the Vāsiṣṭha. If they are to be used for the Pauliśa also, it is because they are interconnected
  and mixed up, as the use of the same technical terms, and the absence of the method to find the
  mean Moon, show, We can even say that the author does not intend this for the Pauliśa because
  another set of rules is given for this in III.9.
  Because the daily increase in padas is 9, the daily increase or decrease in the true motion is 10 ×
  9/7 = 12 6/7 minutes. By integrating the motions and deducting the mean motion during the days
  for which the integration is done, we can find the equation of the centre implied in the rules: For
  convenience let us take padas 9, 18, etc. and work out for plus-padas first. The total true motion in
  minutes for P/9 days, i.e. to the end of P padas is: 702 + 10 (9 − 9)/7 + 702 + 10 (18 − 9)/7 + 702
• 10 (27 − 9)/7 ..... + 702 + 10 (P − 9)/7 = 702 × P/9 + 9 × 10 {1 + 2 + 3 +..... (P − 9)/9}/7 = 702
  P/9 + 9 × 10 {½(P − 9)/9} P/(9 × 7) = 702 × P/9 + 5P (P − 9)/63 = 702 × 7P/63 + 5P²/63 − 45P/63
  = 4869 P/63 + 5P²/63.
  The mean motion per day can be found from 11.2–4 to be 790′ 35″ and for P/9 days, the mean
  motion is 790′ 35″ × P/9 = 5534 P/63 minutes. Subtracting this from the true motion found, the
  equation of the centre obtained is 4869 P/63 + 5 P²/63 − 5534 P/63 = 5P²/63 − 665 P/63 = (5P − 665) P/63.
  In the same way, we can find the equation of the centre connected with the second half-gati, i.e.
  minus padas. The total true motion = 879 1/7 × P′/9 − {5P′²/63 − 45 P′/63} = 6154 P′/63 + 45 P′/63 −
  5P′²/63. Deducting the total mean motion, the equation of the centre = (6154 + 45 − 5534) P′/63
  − 5P′²/63 = 665 P′/63 − 5P′²/63 = P(665 − 5P)/63.
  It is these two rules for the equation of the centre that we used in II.6, to derive the rules there.
  From inspection we see that P(5P − 665)/63 is negative for all values of P, as it ought to be in the first
  half-gati. P′ (665 − 5P′)/63 is positive, as it ought to be in the second half-gati for all the values of