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

18 PAÑCASIDDHĀNTIKĀ I.18
Now, in one solar year there are 12 R solar months. As there are R years in the yuga, there are
12 R solar months in the yuga. Therefore the length of a solar month in days = d/12R. The length
in days of a synodic month, already derived, = d/(r - R). Therefore in every solar month the end of
the synodic month (i.e. the new moon) occurs earlier by d/12R - d/(r - R) = d(r - 13R )/12R (r -
R). When this is equal to one synodic month and gets immersed in the solar, then one intercalary
month happens, and the time for this to happen is, in terms of solar months, d/(r - R) ÷ {d(r - 13R)/
12R(r - R)} = 12R/(r - 13R). Therefore, in the yuga containing 12R solar months the number of
intercalary months i = 12R/ {12R/(r - 13R)} = r - 13R (r - R) - 12R = m - 12R. Therefore 12R

• i = m...... (3), i.e. the solar months + the intercalary months give the synodic months.
  (iii) We shall explain the occurrence of elided days and derive their number: The length of a
  tithi is a little less than a day and so every day the tithi occurs earlier and earlier in the day, until the
  time so accumulated becomes equal to one tithi and gets immersed in the day, with the result that
  the correspondence, one tithi to one day, is broken. Such tithis are left out by reckoning and are cal-
  led 'submerged tithis' or 'elided days'. Now, as there are in the yuga d days and t tithis, the duration
  of one tithi = d/t. In one day, the tithi falls earlier by 1 - d/t day. This accumulates to one tithi in d/t
  ÷ (1 - d/t) = d/(t - d) days, which is the time for one elided day to happen. Therefore, the number
  of elided days happening in a Yuga = e = d/{d/(t - d)} = t - d. Therefore d = t - e ..... (4), i.e. deducting
  the elided days from the tithis we get the days.
  (iv) We shall explain the sidereal day and derive the number of sidereal days in the yuga. The
  time taken by the stellar sphere to move (apparently) one round, is the sidereal day. But the day,
  i.e. the civil day, is related to the apparent diurnal movement of the Sun, from sunset to sunset,
  from sunrise to sunrise, from midnight to midnight etc. As there are n sidereal days and d days in
  the yuga, in one sidereal day the Sun makes d/n revolution. Therefore in one sidereal day he lags
  behind by 1 - d/n = (n - d)/n, revolution. This lagging behind is due to the Sun's eastward motion
  in the Sky and its magnitude is the Sun's motion in terms of revolutions during a sidereal day. This
  is equal to R/n. Therefore, (n - d)/n = R/n. Therefore, (n - d) = R. Therefore n = R + d .....(5), i.e.
  adding the solar years to the days, we get the sidereal days. Thus all the rules of verse 16 have been
  explained.
  [वर्षाधिपः]
  'मुनियमयमद्वि'युक्ते द्युगणे 'शून्यद्विपञ्चयम' भक्ते ।
  प्रति (राश्य) 'खर्तुदहनै' लब्धं वर्षाणि यातानि ॥ १७ ॥
  तानि प्रपन्नसहिता'न्यग्नि'गुणा'न्यङ्घ्रि'वर्जितानि हरेत् ।
  सप्तभिरेवं शेषो वर्षाधिपतिः क्रमात् सूर्यात् ॥ १८ ॥
  Lord of the year

17. Add 2227 to the days from Epoch, divide out by 2520 and take the
    remainder. Set this in 3 places. In one place divide the remainder by 360 and
    take the quotient.
18. Add 1, multiply by 3, deduct 2 and divide out by 7. The remainder
    counted in the order Sun (Ravi), (Moon, Bhauma, Budha, Guru, Śukra and