Page:The New International Encyclopædia 1st ed. v. 14.djvu/794

* NUMBER. 676 NUMBER. the Greeks until the time of Victa and Bachet (1612). The latter gave a satisfactory treat- ment of inclclorminate equations of the first degree in hi.s I'roblimes plaisants ct delect- ables (1012; 5th ed. 1S84). Fermat (works pub- lished po.sthuniously, 1U70, 1(J7'J) enlarged the theory of primes and proved sonic of the most elegant properties of numbers. Legendre (17!I8), in his valuable Essai sur la tluorie dcs nombres, epitomized all the results that had been pub- li.slied up to his time, and contributed original and brilliant investigations, especially on the law of quadratic reciprocity. (iauss (1801) called this law the Theorema Fundttmentale in Doctrina de Uesiduis Quadratis. It relates to the following property of two odd and unequal prime numbers: Let ( — ) be the remainder which is ^" ^ '^ (n, left after dividing m ^ bv », and let I — I be the remainder left after dividing m - by 711. These remainders are always +1 or — 1. What- ever the prime numbers »i and n may be, we al- ways obtain I — ) = (-1 in case the numbers are not both of the form ix -f 3. But if both are of the form ix + 3, then we have ( - I == - I . These two cases are comprised in the formula I - I =: ( — 1) 2 2 V~/" "ropositions embodying this law occupied the attention of Cauchy, Jacobi, Eisenstein, and Kummcr. Up to 18!)0, twenty-five distinct demonstrations of the law of quadratic reciprocity had been published, making use of induction and reduc- tion, of the partition of the perigon (see Poly- gon), of the llieory of function.s, and of the theory of forms. The theory of primes attracted many inves- tigators during the nineteenth century, but the results were detailed rather than general. Tchebisheif (1850) was the first to reach any valuable conclusions in the way of ascertain- ing the number of |)rinies between two given limits. Hiemann ( 185!)) also gave .a well-known fornuila for the liiiiil of the niunber (^ primes not exccciling a given number. To Kuninier is due the treatment of ideal numbers, a part of the general theory of complex numbers. They are defined as factors of prime numbers, and possess the property that there is always a power of these ideal numbers which gi'es a real number. E.g. there exist for the prime number p no rational factors such that p' = A H. where A is difTerent from p and ;>'; but, in the theory of numbers formed from the twenty-third roots of unity, there are priitie num- bers /) which satisfy the ccjnditinn named above. In this case, p is the product nf two ideal num- bers, of which the third powers are the real numl)ers A and B. so that ;i' = A-B. The theorji of eonririienees may be said to start with Oauss's Oisf/Mwi/ionc*. He introduced the synd)olism a rz h (mod r). and explored most of the field. TcliebishetT published in 18-)7 a work upon the subject in Hussian. and Serret diil much toward making the theory known in Trance. The theory of forms (see Forms) has been developed by Onuss. Cauchy, Pojnsot (1845), Lebesgues (1850. lSf.8) . and notablv Hermite. In the theory of ternary form Eisenstein has been a leader, and to him and H. J. S. Smith (q.v. ) is also due a noteworthy ailvance 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 con- cerning the representation of numbers by the siuu of 4, 5, 0, 7, 8 squares were advanced by Kisenstein, and the theory was completed by Smith. The theory of irratiotuil numbers (see Ibba- TIONAL Number), practically untouched since the time of Euclid, receiveil new treatment at the hands of Weierstrass, Heine. G. Cantor, and Dedekind (1872). Mgray had taken in IStiO 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 (1880), and Dedekind's has received additional prominence through the author's later work (1888), the recent indor.sement by Tannery (1804), and, in America, the recent ( inOl ) translation of his work. Weierstrass. Can- tor, and Heine base their theories on infinite series, hile Dedekind founds his on the idea of a cut (l^chnitt) 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 Jl^ray. The theory of continued fractions (due to Cataldi, ltil3) was brought into prominence by Lagi-angc and further developed bv Drucken- miiller (1837), Kunze (18.')7). Len'ike (1870), and Giinther (1872). Ramus (1855) first con- nected the subject with determinants, to which phase Heine, Jlobius, and (liinthcr also contrib- uted. Diriciilet also added to the general theory, as have numerous contributors to the applica- tions of the subject. Transcendental numbers were first distin- guished from algebraic irrationals by Kronecker. Lambert proved (1701) that ir (see Circle) cannot be rational, and that c° (e being the lUse of hypeil)olic logarithms and ji being ra- tional) is irrational. Legemlre (17114) sliowed that TT is not the square root of a rational num- ber. Liouville ( 1840) showed that neither <• nore' can be a root of an integral (piadratic equation. But the existence of transcendental numbers was first established by Liouville (1844. 1851). the proof being subscquentlv displaced bv G. Can- tor's (1873). Hermite" (1873) first" proved « to be transcendental, and Limlcmann (1882), starting from Hcrmite's conclusions, showed the same for v. Lindcmann's proof was uiuch sim- plified by Weierstrass (1885). still further by Hilbert (1803). and has finally been made ele- mentary by Hurwitz and Gordon. BinLiocRAPiiv. Lucas. Theorie den nombre» (Paris. 1801) ; Smith. "Report on the Theory of Xuml)ers." in the Rej)orts of the Rritinh .Issocia- lion (London. 1859-65) : Gauss. fUst/iiiifitioneii Arithmeticw (Leipzig. 1801): Legendre. Fssni sur hi thforie des nomhres (Paris. IS.'^O) : Dirich- let, ]'orlesiinqen iiber /nhlenlhrorie. edited by Dedekind (4th ed., Brunswick, 1804; Eng. trans. l)y Beuum, Chicago. 1001); Stolz. lor- lesiinfirn iiher allijemeine Arithmetil; (Leip- zig. 1885-8(1) : Mathews. Theoni of ynmhers, part i. (Cambridge. 1892). See Fb.ction ; Irea- i