The Disquisitiones Arithmetiese ' of Gauss (Brunswick, 1801) were translated into French by N. PoulletsDelisle (Paris, 1807). The 'fliSorie des Nombres ' of Legendro (third edition, Perim, 1630) has the advantage of coming later than that of Gauss (which itself came after the first edition of Legendre s), and of using methods and note. tions which are more familiar to the methemetiehus Both are works of great originality : that of the German is condensed, and full of historical information ; that of the easier to follow, but like most French works, deficient in preclre historical refer( nee. It is not a little singular that the two great writers on this subject should have been the men who, independently of each other4introduced the method of LEAST SQUARES.
The beginner in algebra may obtain some command over equations of a simple character, not exceeding the second or third degree, by method which is, we believe, due to Playfair, or which, at least, is published in the collection of his works. Let the equation be, for instance, bxy + in which x, y, and z are to be whole numbers. Throw the equation into a form which admits of both sides being reducible into factors ; for instance, y(ay -Fbx)= (z- x) (z+ x).
If then z-x= v y, we have z+ x= (a y+bx): v, which equations give (a -v°) z (2v- b)z y=-- a-bv+v° a-bv+v° Assuming v at pleasure, z may be easily taken so as to make both x and y whole numbers ; and the same method will succeed in many equations.
Euclid's geometry, assuming only the use of his three celebrated postulates, enables him, a linear unit being given, to construct the length represented by any algebraical expression which involves only additions, subtractious, multiplication; divisions, extraction of the square root, or combinations of all these. But a cube or fifth root is beyond the power of the system. Again, from the theory of equations it is soon made obvious that the solution of the equation x° -1= 0, and the division of a circle into at equal parts, are one and the same problem. One solution of the preceding is x= cos + V-1. sin 0, where 0 is the nth part of four right angles. [Rom.] Euclid, in his fourth book, shows how to cut a circle into three, four, five, and fifteen equal parts ; and analysis shows that the sines and cosines of the'angles so involved can be obtained by formulm which involve no roots except the square. But except into halves; thirds, fifths, or fifteenths, or parts obtainable from these by one or more bisections, Euclid was not able to cut a circle into equal parts.
So the matter rested for about 2000 years, until Gauss, in his Dis quisitiones Arithmetiere ' (1801), not only pointed out how to extend Euclid's conclusions, but also in a manner how to account for them.
The statement of his results, even without demonstration, is instructive to the learner, and we shall give it accordingly : referring for the demonstration to the works of Gauss or Legendre, or to Murphy's Theory of Equations.' The expression a + Vb, a and b being rational, is the solution of a quadratic equation with rational coefficients. But if a and b them selves have the form c+ Vd, in which c and d have themselves the same form, and so on ; then a+ V b is the solution of a quadratic equation in which the coefficients are themselves the solutions of quadratic equations, and so on. Consequently, any equation, the root of which is capable of construction by Euclid's postulates, must be reducible to a system of quadratics ; and the converse. Now if n be a prime number, n -1 is an even number, and therefore has factors. Let its prime factors be 2, a, b, c, &e., and let them severally enter p, 7, r, a, &c., times so that n -1 = 25 aq br ....
Gauss succeeded in showing that when n is a prime number, the solu tion of the equation x"-1 =0, can be made to depend upon the solu tion of p equations of the second degree, q of the ath degree, r of the bth degree, and eo on. Consequently, whenever 2 is the only prime factor of n - I, or when n-1=2^, n being prime, or when 2P + 1 is a prime number, the solution of x*- I is reducible to that of p quadratic equations, and the division of the circle into II equal parts can be accomplished by Euclid's geometry. And further, it is easily demon strated that 25 + 1 can never be a prime number, except when p itself is a power of 2 (2" included) though 25 + 1 is not then always prime. Nor has it been shown that other divisions are impossible : Gauss's theorem merely points out cases in which the thing can be done, with out pronouncing the exclusion of others. Gauss, indeed, does assert that he can demonstrate all other cases to be impossible to be con structed by geometry, that is, reducible to quadratic equations : and the thing is highly probable. If we now construct the series 2c+1, 2' + 1, 2° + 1, &c., among which all our chances lie, we have 3, 5, 17, 257, 65537, 4294967297, &c. The first five are prime numbers : Euclid has disposed of the two first divisions; Gauss has added that a circle can be geometrically divided into 17, 257, and 65537 equal parts. But 4294967297 is not a prime number, being divisible by 641.