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

IV.47 IV. THREE PROBLEMS 119
The following are the steps in the work to be done:
(i) The ‘First sine’ = 1,72,800 ÷ (sin colatitude × √144 + shadow²)
(ii) The Earth-sine (which is to be placed in two places) = sin latitude × sin declination ÷ sin co-
latitude.
(iii) sine I = (‘First sine’ ∓ Earth-sine) × 240 ÷ day-diameter
(iv) sine II = Earth-sine × 240 ÷ day-diameter.
(v) Find arc I and arc II of sin I and sin II
The desired time in nāḍīs = arc I/6 ± arc II/6. In (iii) and (v) the upper sign is to be taken for the
Sun in the six signs from Aries, i.e. for the Sun in the northern hemisphere; otherwise the lower
sign is to be taken.
It must be added here, in accordance with what was said in the same context in getting the
shadow from the time, that if the ‘First sine’ is less than the Earth-sine and therefore the Earth-sine
cannot be deducted in (iii), the ‘First sine’ is to be deducted from the Earth-sine, and the result, i.e.
sine I, is to taken as negative. Then in (v) the nāḍīs got from this, viz. arc I/6, are also negative, and
therefore deducted from arc II/6, to get the time. Here too, if the latitude of the place is not too
high, the reverse of the author’s method in the context can be used without any appreciable error,
though this has not been mentioned here by the author. This is the work to be done: Here the
Earth-sine is greater than the ‘First sine’; omit the Earth-sine and do (iii) and (v) with the ‘First sine’
alone, i.e. multiply the ‘First sine’ by 240, divide by the day-diameter, get the arc of this, and divide
by 6 and thus to get the nāḍīs after sunrise.
The following points must be noted here. In the work of computing the nādīs from the shadow,
as the exact time is not known, the exact Sun and therefrom the exact declination cannot be known,
and we have to use the declination of the Sun at sunrise or sunset. There may be a small error on
account of this. This can be avoided by repeating the work using the declination of the Sun for the
computed time. It has not been specifically mentioned by the author because this can be inferred
by the computer. Secondly, the author has given all this for places in the northern hemisphere in
the forenoon. For places in the southern hemisphere and the afternoon, changes have to be made
in the work, which have not been given by the author. It must also be noted that the ancients con-
sidered the computation of the time from the shadow or the shadow from the time as very impor-
tant because this was the best means available to them of knowing the times of births and muhūrtas.
Example 17 (a) For a place (in the northern hemisphere) sin lat. is 60′, and therefrom sin colat is 103′ 55″.
On a particular day the sin declination is 34′ 30″, (the sun being in the 6 signs from Aries) and therefore the
day-diameter is 229′ 51″. Find the time from sunrise if the shadow of the 12 digit gnomon is 13 aṅg 50
vyaṅgulus.
(i) ‘First sine’ = 1,72,800 ÷ (103′ 55″ × √13 14/15² + 144 = 1,72,800 ÷ (103′ 55″ × 18.389) =
90′ 26″.
(ii) Earth-sine = 60′ × 34′ 30″ ÷ 103′ 55″ = 19′ 55″
(iii) Sine I = (90′ 26″ − 19′ 55″) 240 ÷ 229′ 51″ = 73′ 38″
(iv) Sine II = 19′ 55″ × 240 ÷ 229′ 51″ = 20′ 48″