Page:A History of Mathematics (1893).djvu/113

 by the formula $$\scriptstyle{\sqrt{a+\sqrt{b}}=\sqrt{a+\sqrt{a^2-b}\over 2}+\sqrt{a-\sqrt{a^2-b}\over 2}}$$ the square root of the sum of rational and irrational numbers could be found. The Hindoos never discerned the dividing line between numbers and magnitudes, set up by the Greeks, which, though the product of a scientific spirit, greatly retarded the progress of mathematics. They passed from magnitudes to numbers and from numbers to magnitudes without anticipating that gap which to a sharply discriminating mind exists between the continuous and discontinuous. Yet by doing so the Indians greatly aided the general progress of mathematics. "Indeed, if one understands by algebra the application of arithmetical operations to complex magnitudes of all sorts, whether rational or irrational numbers or space-magnitudes, then the learned Brahmins of Hindostan are the real inventors of algebra."[7]

Let us now examine more closely the Indian algebra. In extracting the square and cube roots they used the formulas $$\scriptstyle{(a+b)^2=a^2+2ab+b^2}$$ and $$\scriptstyle{(a+b)^3=a^3+3a^2b+3ab^2+b^3}$$. In this connection Aryabhatta speaks of dividing a number into periods of two and three digits. From this we infer that the principle of position and the zero in the numeral notation were already known to him. In figuring with zeros, a statement of Bhaskara is interesting. A fraction whose denominator is zero, says he, admits of no alteration, though much be added or subtracted. Indeed, in the same way, no change takes place in the infinite and immutable Deity when worlds are destroyed or created, even though numerous orders of beings be taken up or brought forth. Though in this he apparently evinces clear mathematical notions, yet in other places he makes a complete failure in figuring with fractions of zero denominator.

In the Hindoo solutions of determinate equations, Cantor