पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

xli INTRODUCTION
of these rules is found to occur in the Mahâ-siddhânta of Ârya-
bhaṭa II and the Gaṇita-tilaka of Śrîpati, whereas the other
two rules are found to occur in the Gaṇita-kaumudî of Nârâyaṇa,
evidently under the influence of Śrîdharâcârya.
In addition to the above arguments, the literary qualities
of composition of Śrîdharâcârya's works and other improve-
ments in them, which definitely establish the superiority of
Śrîdharâcârya's works over the Gaṇita-sâra-saṅgraha, may also
be brought in evidence to prove that Śrîdharâcârya was a
later writer than Mahâvîra. We have already seen how
successfully Śrîdharâcârya condensed the matter of six verses
of Mahâvîra into a single verse.
The following facts show that Śrîdharâcârya, though
posterior to Mahâvîra, was anterior to Âryabhaṭa II :
(i) Âryabhaṭa II has improved Śrîdharâcârya's formula
for the area of a segment of a circle by using ²²/₇ in
place of √10.¹
(ii) Âryabhaṭa II was aware of the formula for the surface
of the sphere, viz.,
circumference × diameter,
which was unknown to Śrîdharâcârya.²
(iii) Âryabhaṭa II³ criticises the formula used by Śrî-
dharâcârya for finding the area of a quadrilateral
with unequal altitudes, viz.,
√((s - a) (s - b) (s - c) (s - d)) ,
where a, b, c, d are the sides of the quadrilateral and
s=(a+b+c+d)/2.
There is another interesting argument to show that Śrî-
dharâcârya lived after Mahâvîra but before Âryabhaṭa II. It

¹ See MSi, xv. 93(ii)-94(i).
² See MSi, xvi. 35-36.
³ See MSi, xv. 70.