History of Hindu Mathematics: A Source Book v2.c17.s1.85

v2.c17.s1.85

**Power.** The oldest Hindu terms for the power of a quantity, known or unknown, are found in the *Uttarâdhyayana-sûtra* (c. 300 B.C. or earlier).² In it the second power is called *varga* (square), the third power *ghana* (cube), the fourth power *varga-varga* (square-square), the sixth power *ghana-varga* (cube-square), and the twelfth power *ghana-varga-varga* (cube-square-square), using the multiplicative instead of the additive principle. In this work we do not find any method for indicating odd powers higher than the third. In later times, the fifth power is called *varga-ghana-ghâta* (product of cube and square, *ghâta* = product), the seventh power *varga-varga-ghana-ghâta* (product of square-square and cube) and so on. Brahmagupta's system of expressing powers higher than the fourth is scientifically better. He calls the fifth power *pañca-gata* (literally, raised to the fifth), the sixth power *ṣaḍ-gata* (raised to the sixth); similarly the term for any power is coined by adding the suffix *gata* to the name of the number indicating that power.³ Bhâskara II has sometimes followed it consistently for the powers one and upwards.⁴ In the *Anuyogadvâra-sûtra*⁵, a work written before the commencement of the Christian Era, we find certain interesting terms for higher powers, integral as well as fractional, particularly successive squares (*varga*) and square-roots (*varga-mûla*). According to it the term *prathama-varga* (first square) of a quantity, say *a*, means a²; *dvitîyavarga* (second square) = (a²)² = a⁴; *tṛtîya-varga* (third square) = ((a²)²)² = a⁸; and so on. In general,