反演法Inversion

92° Inversion. The Hindus were gifted arithmeticians and made significant contributions to algebra. Many of the arithmetic problems were solved by the method of inversion, where one works backward from a given piece of information. Consider, for example, the following problem given during the sixth century by the elder Aryabhata: "Beautiful maiden with beaming eyes, tell me, as thou understandst the right method of inversion, which is the number which multiplied by 3, then increased by 3/4 of the product, then divided by 7, diminished by 1/3 of the quotient, multiplied by itself, diminished by 52, by the extraction of the square root, addition of 8, and division by 10 gives the number 2?" By the method of inversion we start with the number 2 and work backward. Thus $[(2)(10) - 8]^2 + 52 = 196$, $\sqrt{196} = 14$, $(14)(3/2)(7)(4/7)/3 = 28$, the answer. Note that where the problem instructed us to divide by 10 we multiply by 10, where we were told to add 8 we subtract 8, where we were told to extract a square root we take the square, and so forth. It is the replacement of each operation by its inverse that accounts for the name inversion. It is, of course, just what we would do if we were to solve the problem by modern methods. Thus, if we let $x$ represent the sought number, we have

To solve this we multiply both sides by 10, then subtract 8 from each side, then square both sides, and so forth.