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

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will be the same on both the occasions. Then the spherical
triangles PZS₁ and PZS₂ where P is the celestial pole, Z the
Zenith and S₁ and S₂ are the positions of the Sun on the
two occasion, are congruent their three sides being respe-
ctively equal, provided we take PS₁ = PS₂ ie. 90 — δ₁ =
90 — δ₂ ie. δ₁ = δ₂ on both the occasions. When the
two triangles are thus congruent, PẐS₁ = PẐS₂ ie. S₁ and
S₂ are equidistant from the plane of the Prime-Vertical.
Hence the extremities of the shadows will be equidistant
from the East-West line ; or this may be seen in another
way ; S₁ S₂ will be perpendicular to the meridian plane
and as such parallel to the plane of the Prime-Vertical.
The correction mentioned in the verse is known as
the Agrāntara correction which was originally given by
Chaturvedā chārya and then accepted by Sripati. Why
it is called Agrāntara is because it is a change in what
are called Karnavrittāgras of the two occasions where we
shall see in due course that the formula for Karnavrittāgra
is (K sin δ) / (cos φ) where K is hypotense of the gnomonic triangle
formed by the gnomon and its shadow S at any place and
time. This correction is a very minute correction and
as a matter of fact could be ignored. But the fact that the
correction was cognized and correctly formulated testifies
to the knowledge of the sphere which the above
acharyas had. Assuming the formula of the Karnāgra
here (it will be proved by us later in this chapter) if
δ₁ and δ₂ be the declinations on the two occasions
respectively the Karnāgras will be (K sin δ₁) / (cos φ) and (K sin δ₂) / (cos φ) ,
K being equal on the two occasions because the shadows
are equal. Hence the correction being the difference of
the Agrās, it is [K (sin δ₁ ~ sin δ₂)] / (cos φ) as stated by Bhās-
kara,