Page:Indian mathematics, Kaye (1915).djvu/55

degree. Of particular interest to us are the following: The area of a segment of a circle$\scriptstyle {={\frac {1}{2}}(c+a)a}$, where 'c' is the chord and 'a' the perpendicular, which actually occurs in Mahāvīra's work; in the problems dealing with the evaluation of roots, partial fractions with unit numerators are used (cf. paragraphs 5 and 7 above); the diameter of a sphere$\scriptstyle {={\sqrt[{3}]{{\frac {16}{9}}\times {\text{volume}}}}}$, which possibly accounts for Āryabhata's strange rule; the volume of the cone$\scriptstyle {=({\frac {\text{circumference}}{6}})^{2}}$ which is given by all the Indians; and the correct volume for a truncated pyramid which is reproduced by Brahmagupta and Srīdhara. One section deals with right angled triangles and gives a number of problems like the following:

"There is a bamboo 10 feet high, the upper end of which being broken reaches to the ground 3 feet from the stem. What is the height of the break?" This occurs in every Indian work after the 6th century. The problem about two travellers meeting on the hypotenuse of a right-angled triangle occurs some ten centuries later in exactly the same form in Mahāvīra's work.

The Sun-Tsū Suan-ching is an arithmetical treatise of about the first century. It indulges in big numbers and elaborate tables like those contained in Mahāvīra's work; it gives a clear explanation of square-root and it contains examples of indeterminate equations of the first degree. The example: "There are certain things whose number is unknown. Repeatedly divided by 3 the remainder is 2; by 5 the remainder is 3, and by 7 the remainder is 2. What will be the number?" re-appears in Indian works of the 7th and 9th centuries. The earliest Indian example is given by Brahmagupta and is: "What number divided by 6 has a remainder 5, and divided by 5 has a remainder of 4 and by 4 a remainder of 3, and by 3 a remainder of 2?" Mahāvīra has similar examples.

In the 3rd century the Sea Island Arithmetical Classic was written. Its distinctive problems concern the measurement