HISTORY. No essential advance was made in the theory of numbers beyond the knowledge of the Greeks until the time of Vieta and Bachet (1612). The latter gave a satisfactory treat ment of indeterminate equations of the first degree in hi, Probkmes plaisants et ((elect ables (1612; 5th ed. 1884). Fermat (work, pub lished posthumously, 1670, 1679) enlarged the theory of primes and proved some of the most elegant properties of numbers. Legendre (1798), in Ids valuable Essai sue lu llkorie des nombres, epitomized all the results that had been pub lished up to his time, and contributed original and brilliant investigations, especially on the law of quadratic reciprocity. (;auss (1801) called this law the Theorenta Pundamcniale in Doctrina de liesitluis Quadratis. It relates to the following property of two odd and unequal prime mangers: Let (—) be the remainder which is 77 u—I left after dividing at 2 by a, and let (—) be the remainder left after dividing n'-r by m. These remainders are always +1 or —1. What ever the prime numbers m and n may be, we al ways obtain (7?) = in ease the numbers are not both of the form 4x + :3. But if both are of the form 4x + 3, then we have ) = . These two cases ore comprised in the formula (—)= (-1) 2 2 - • Propositions n embodying this law occupied the attention of Cauchy, Jacobi, Eisenstein, and Kummer. Up to 1890, twenty-five distinct demonstrations of the law of quadratic had been published, making use of induction and reduc tion, of the partition of the porigon (see POLY GON), of the theory of functions, and of the theory of forms.
theory of primes attracted many inves tigator,: during the nineteenth century. but the results were detailed rather than general. Tehebishelf (1850) was the first to reach any valuable conclusions in the way of ascertain ing the number of primes two given limits. Riemann 11859) also gave a well-known formula for the limit of the number oi primes not exceeding a given number.
To Kummer is due the treatment of (drat lttimbe rs, a part of the general theory of complex numbers. Tlu.y are defined as of prime numbers, and possess the property that there is always a power of these ideal numbers which gives a real number. E.g. there exist for the prime number p no rational factors such that = .1 B. A is diGrent from p and hut, in the theory of numbers formed from the twenty-third root, of unity. there are prime num bers p which satisfy the condition named above. In this ease. p is the product of two ideal hers, of which the third powers are the real munhers .\ and 13, '.o that = A The theory of congruences may be said to start with t:auss's ;fairs., Ile introduced the symbolism a 7: is (mod r), :Ind explored utmost of the field. Teltebishelf published in 1547 a work neon the in Russian, and Serret did much toward making the theory known in Franee.
The theory of forms (see Foams) has been developed by Gauss. Cauchy, Poinsot (1815), Lebesgmes (1859, 1808), and notably Ilermite. In the theory of ternary form Eiscnstein has been a leader. and to him and 11. J. S. Smith (q.v.) is also due a noteworthy advance in the theory of forms in general. Smith gave a complete classi
fication of ternary quadratic forms. and extended Gauss's researches concerning real quadratic forms to complex forms. The investigations eon (wiling the representation of numbers by the sum of 4. 5, 6, 7. S squares were advanced by Eisenstein, and the theory was completed by Smith.
The theory of irrational numbers (see IRRA TIONAL NUMBER), practically untouched since the time of Euclid, received new treatment at the hands of Weierstrass, Heine, G. Cantor, and Dedekind (1872). Al6ray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weier strass's method has been completely set forth by Pincherle (LS80), and Dedekind's has received additional prominence through the author's later work (1888), the recent indorsement by Tannery (1894), and, in .Anmerica, the recent (1901) translation of his work. Wcierstrass, Can tor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Sehnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker, and The theory of continued fractions (due to Cataldi, 1613) was brought into prominence by Lagrange and further developed by Drucken 'Miller (1837), Kunze (1857), Lemke (1870), and Gfinther ( 1872). Pontius ( 1855) first con nected the subject with determinants, to which phase M?ibius, and Giinther also emitrib uted. Dirichlet also added to the general theory. as have numerous contributors to the appliea tions of the subject.
Traaseendental numbers were first distin guished from algebraic irrationals by Kronecker. Lambert proved (1701) that r (see Clarl.F1 cannot be rational, and that e" (c being the base of hyperbolic logarithms and a being ra tional) is irrational. Legendre (179.1 ) showed that r is not the square root of a rational num ber. Lionville (1840) showed that neither e nor cs can he a root of an integral quadratic equation. But the existence of transcendental numbers was first established by Lionville (1844. 1851), the proof being subsequently displaced by G. Can tor's (1873). 'termite (1873) first proved e to be transcendental, and Lindemann (1882), starting from !Termite's conclusions, showed the same for 7r. 1,indemaan's proof was much sim plified by Weicrstrass (1885). still further by Hilbert 11893). and has finally been made ele• inentary by Hurwitz and Gordon.
llini.mtat.ken v. Lucas. des nom Tows (Paris, 18911: Smith, "1:eport on the Theory of in the Reports of the British Issoria lion (London. 1859-05): Ilisguisitiones .tritltm et iew ( Leipzig. 18011 : Legendre. Essai &lir In des nombres (Paris. 15301 : Dirich let, alter Zahlenthrork. edited by Dedekind (4th ed., Brnnswiek. 1894; Eng. trans. by Rennin, Chicago, 1901) ; Stolz, 1 or Tes anaen iibrr at/leap-jar .Irith un ( Le ip• zig. 1855-801: Mathews. Theory of Vomiters, part i. (Cambridge, 1502). See Flt%crtoN: but,t•