History of Hindu Mathematics: A Source Book v2.c17.s18.42

v2.c17.s18.42

**Mahâvîra's Definitions.** A triangle or a quadrilateral whose sides, altitudes and other dimensions can be expressed in terms of *rational* numbers is called *janya* (meaning generated, formed or that which is generated or formed) by Mahâvîra.² Numbers which are employed in forming a particular figure are called its *bîja-saṁkhyâ* (element-numbers) or simply *bîja* (element or seed). For instance, Mahâvîra has said: "Forming O friend! the generated figure from the *bîja* 2, 3,"¹ "forming another from half the base of the figure (rectangle) from the *bîja* 2, 3,"² etc. Thus, according to Mahâvîra, "forming a rectangle from the *bîja* *m*, *n*" means taking a rectangle with the upright, base and diagonal as m² - n², 2mn, m² + n² respectively. It is noteworthy that Mahâvîra's mode of expression in this respect very closely resembles that of Diophantus who also says, "Forming now a right-angled triangle from 7, 4," meaning "taking a right-angled triangle with sides 7² - 4², 2.7.⁴, 7² + 4² or 33, 56, 65."³ It should also be noted that Mahâvîra never speaks of "right-angled triangle." What Diophantus called "forming a right-angled triangle from *m*, *n*," Mahâvîra calls "forming a longish quadrilateral or rectangle from *m*, *n*."