Mahāvīra

Mahāvīra or Mahāvīrācārya was a 9th-century Jain mathematician born in Mysore, in Southern India. He was born in the year 815 AD. Gaṇitasārasan̄graha was authored by Mahāvīra. He was in the royal court of king Amoghavarṣa of Rāṣṭrakūṭa  dynasty.

Gaṇitasārasan̄graha has the following  chapters


 * 1) Saṃjñādhikāraḥ (Terminology)
 * 2) Parikarmavyavahāraḥ (Arithmetical operations)
 * 3) Kalāsavarṇavyavahāraḥ (Fractions)
 * 4) Prakīrṇakavyavahāraḥ (Miscellaneous problems)
 * 5) Trairāśikavyavahāraḥ (Rule of three)
 * 6) Miśrakavyavahāraḥ (Mixed problems)
 * 7) Kṣetragaṇitavyavahāraḥ (Measurement of Areas)
 * 8) Khātavyavahāraḥ (calculations regarding excavations)
 * 9) Chāyāvyavahāraḥ (Calculations relating to shadows)


 * Mahāvīrācārya has praised mathematics in Gaṇitasārasan̄graha
 * लौकिके वैदिके वापि तथा सामयिकेऽपि यः।
 * व्यापारस्तत्र सर्वत्र संख्यानमुपयुज्यते॥
 * Meaning : Where there is business in worldly, Vedic and contemporary, only numbers are used everywhere.
 * It was Mahāvīra who first treats the series in Geometric progressions and gives almost all the formulae required therein.
 * गुणसङ्कलितान्त्यधनं विगतैकपदस्य गुणधनं भवति ।
 * तद्गुणगुणं मुखोनं व्येकोत्तर भाजितं सारम् ॥
 * अन्त्यधन - the value of the last term. गुण - common ratio.
 * The verse states that $$S_n =\frac{ar^{n-1} X \ { r-a } }{r-1}

$$
 * $$=\frac{a(r^n-1)}{r-1}

$$ where a is the first term and r is the common ratio and Sn is the sum to n terms.
 * Mahāvīra's work differs from that of others in respect of the definitions he introduced of various figures. He has given the definitions of a triangle-equilateral, isosceles and scalene-a square, a rectangle, isosceles trapezium, trapezium with three sides equal, a quadrilateral, a circle, a semicircle, an ellipse, a hollow hemisphere and the lune. It is true that the results he derived regarding the area of an ellipse and the length of the curve of an ellipse are not accurate; but as a pioneer in this line his place stands high. Almost all the properties of the cyclic quadrilateral derived by Brahmagupta, have been more lucidly explained by him.