NUNIBERS, TIIEORY OF, 3. branch of analysis which lias for its object, the exhibition and investigation of certain peculiar numerical properties; the relation of different orders of nunwers with each other; the forms of numerical divisors ; thc solution of indeterminate equations, and the determination of their possibility-, or impossibility, it) rational numbers.
The theory of numbers, as a distinct branch of ana lysis, cannot claini an origin anterior to the commence ment of the present century, although we may find traces ol it in the works ol some authors of great anti quity, as, fur example, in the seventh, eighth, and ninth books of Euclid, where the subject is treated geometri cally ; the "Arcnarius" or Archimedes, where it as sumes its proper numerical character, and in which is contained the first hint of that important relation be tween an arithmetical and geometrical series, which con stitutes the fundamental principle of our system of lo garithms.
At this time, however, the imperfect notation em ployed hy the Greeks interposed itself as an insuperable bar to the progress of this branch of science ; and even after some ideas had been formed of the more general language of algebra, we still find Diophantus contend ing with the (Ukuleles the numerical notation of his countrymen placed before him, and which prevented him from expressing, in an intelligible manner, various properties of numbers with which it is obvious he was by no means unacquainted.
From the time of this ancient writer, no step seems to have been tnade till about the beginning of the 17th century, when Radice, a French analyst of considerable reputation, undertook the translation of the work of Diophantus into Latin, retaining also the Greek text. This translation was published in 1621, interspersed with numerous notes, where the first glance is caught of our present theory of numbers. Thcse were afterwards considerably extended by Fermat, in his edition of the same work, published after his death in 1670, where we find many of the most elegant theorems in this branch of analysis ; hut they are generally left without demonstration, This omission accounts for in one of his notes, by stating, that he was himself preparing a treatise on the theory of numbers, which would contain many IICW and interesting nutnerical propositions; but, unfortunately, this work never appeared, and most of his propositions remained without demonstrations for a considerable time, although at present one only has re sisted the talents and perseverance of modern mathe maticians ; this one may be briefly enunciated as follows.
" The formula x"—!--- yn = zn is impossible, in rational numbers, for every value of ?I greater than 2." The denionsetation of this theorem was proposed a few years back as thc prize subject of the mathetnatical class by the National Institute of France, but received no satis factory solution.
During the latter part of the last century, and the be ginning of the present, the theory of numbers engaged the attention of many very celebrated algebraists, amongst whom are found Euler, Waring,and Lagrange. The former, besides what is contained iti the second volume of his " Algebra," and his " Analysis Infinito rum," has various papers on the subject in the Trans actions of the Academy of Petersburgh. What has been done by Waring will bc found in the fifth chapter of his "INIeditationes Algebraicx ;" and Lagrange, who, more than any other writer, has extended this branch by the invention of many new propositions, has several interesting papers on the subject in the Memoirs of Berlin, besides what are contained in his edition of the French Translation ol Euler's Algebra. It is, however, only during the present century that the theory of num bers has been reduced to a regular system—a task which was first petformed by Legendre, in his " Essai sur la Theorie des Nombres ;" and, nearly at the same time, Gauss published his " Disquisitiones Arithmeticx," in which latter work a new and highly interesting applica tion of this doctrine is made to thc solution of binomial equations of the form xn — 1= 0, on which depends the division of the circle into a given number of parts, as was before known by the Cotesian theorem. To these works we may also add an English treatise on thc same subject, by Mr. Barlow, of the Royal Military Academy, which contains several new theorems, and a simplification of some of those already alluded to in the above historical sketch.