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

183

from the figure ÂPQ is the rising hour-angle = ÂPE +
ÊPQ = ÂPE + 90° where 90° correspond to 15 nādīs. So
we have to add ÂPE expressed in nādīs equal to the rising
time of AE. Let us find the duration of the day for the
place where s=3″ and when λ of s=44° ; the latitude of
the place will be 14° (from tables) when λ 44°, we have
found above that 42-20 Vinadis is the chara expressed in
time. Hence half the day = 15-42-20 or duration of day
31-25 ; duration of night = 28-35.
Note (1) In modern astronomy we have the formula
cos h = — tan ϕ tan δ where h is the rising hour angle.
Putting h = H + 90° where H stands for the arc EA of
fig. 21 cos (90+H) = — sin h = — tan ϕ tan δ so that
sin H = tan ϕ tan δ ie. R sin (Chara) = R tan ϕ tan δ
which accords with the formula found before. The word
chara used for EA means etymologically रविसञ्चारवशेन दिन-
प्रमाणे विकारः i e. the variation in 15 ghatis of the eqiunoctial
half-day on account of the Sun's variation in declination,
(2) If ϕ = 0, Chara = 0 so that duration of half-
day is 15 ghatis ie. on the terrestrial equator, whatever
be the Sun's declination, the length of the day will be
always 12 hours.
(3) Let δ = 0 so that chara = 0 ie. whatever be
the latitude (provided ϕ ≯ 90-δ as we shall see shortly).
the day and night will be each of 12 hrs.
(4) Let δ be negative, so that the arc connoting
chara ie. EA will be above the horizon, and consequently
the duration of half-day will be less than 6 hrs. by the
time that is taken for EA to rise. This can be seen other-
wise also as cos h = — tan ϕ tan δ = + ve so that h < 90°
which means half-day is less than 6 hrs.
(5) We could also treat the case when ϕ is negative,
but as this case is not in the purview of Hindu Astro-
nomers who had only India in their mind and as such
were concerned primarily with positive latitudes. If, how-
ever, we consider a negative latitude, when δ is + ve, cos h
will be positive and if δ is — ve, cos b will be negative.
This means that when the Sun is in northern latitudes, the
southern latitudes will have their day less than 12 hours
and when the Sun is in southern latitudes, their day will
be greater than 12 hrs.