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

xviii INTRODUCTION
Pâtīgaṇita. It gives the important rules and examples of the
Pâtīgaṇita with slight modifications, alterations and additions
here and there, and provides a short course of arithmetic and
mensuration. From the study of the latter part of this work,
which does not have its counterpart in our incomplete manu-
script of the Pâtīgaṇita, we find that the following two approxi-
mate formulæ are Śrîdharâcârya’s own contributions to the
subject :
(1) Area of the segment of a circle
= (√10 / 3) × ((chord + arrow) / 2) × arrow .¹
This formula occurs for the first time in this work and is
better than the following formula given by Mahâvîra² :
(√10 × chord × arrow) / 4
It is noteworthy that Mahâvîra has called this formula
accurate (sûkṣma).
(2) Volume of a sphere = ((diameter)³ / 2) (1 + 1/18)
= 4 (radius)³ (1+1/18)
= (4·22 . . )r³, where r = radius.³
The formulæ given by the earlier writers are as follows :
(i) Âryabhaṭa I’s formula⁴ : (πr²)³ᐟ² = (5·56....)r³.
(ii) Formula⁵ quoted by Bhâskara I : (9/2)r³ = 4·5r³.
(iii) Mahâvîra’s accurate formula⁶ : 9/2 × 9/10 r³ = 4·05r³.
Śrîdharâcârya’s formula is the best, as it more closely
approximates to the modern formula :
(4/3)πr³ = (4·188....)r³.
¹ Triś, p. 35, Rule 47.
² See GSS, vii. 70½.
³ See Triś, p. 39, Rule 56.
⁴ See Â, ii. 7.
⁵ See Bhâskara I’s comm. on Â, ii. 7.
⁶ GSS, viii. 28½.