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History.—There is little doubt that the earliest solutions of equations are given, in the Rhind papyrus, a hieratic document written some 2000 years before our era.The problems solved were of an arithmetical nature, assuming such forms as “a mass and its x = 19, which is a simple equation.In the second case the number of unknowns is limited to three, corresponding to the three dimensions of space; the degree is unlimited as before.It must be noticed, however, that by the introduction of appropriate hyperspaces, i.e.The simplest quadratic arose in the construction of a mean proportional (x) between two lines (a, b), or in the construction of a square equal to a given rectangle; for we have the proportion a:x = x:b; i.e. A simple cubic equation was presented in the problem of finding two mean proportionals, x, y, between two lines, one double the other. Abel proved the problem to be impossible; a solution involving elliptic functions has been given by C. This lacuna was not filled until the 17th century, when Descartes discovered the general theory which explained the nature of such solutions, in particular those wherein conics were employed, and, in addition, established the most important facts that every equation represents a geometrical locus, and conversely.We have a:x = x:y = y:2a, which gives x = ay and xy = 2a; eliminating y we obtain x = 2a, a simple cubic. Many fruitless attempts were made to solve algebraically the quintic equation until P. To represent equations containing two unknowns, x, y, he chose two axes of reference mutually perpendicular, and measured x along the horizontal axis and y along the vertical.

If each term has the sum of its indices the same, the equation is said to be homogeneous.

These definitions are exemplified in the equations:— In (1) the unknown is x, and the knowns a, b, c; the coefficients of x and x are a and 2b; the absolute term is c, and the degree is 2.

In (2) the unknowns are x and y, and the known a; the degree is 3, i.e. (3) is a homogeneous equation of the second degree in x and y.

The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7, by Various This e Book is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever.

You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this e Book or online at Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7 "Equation" to "Ethics" Author: Various Release Date: February 25, 2011 [EBook #35398] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK ENCYC.

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