PREFACE. THE Theory of Numbers is a subject which has engaged the attention and exercised the talents of many celebrated mathematicians, both ancient and modern; under the first of which classes, may be reckoned Pythagoras and Aristotle, the former of whom is said to have invented our present multiplication table, or the Abacus Pythagoricus of the ancients; though what is alluded to under this designation was probably a much more extensive table than that now in common use: Pythagoras also attributed to numbers certain mystical properties, and seems first to have conceived the idea of what are now termed magic squares. Aristotle, amongst other numerical speculations, noticed the uniformity in almost all nations of dividing numbers into periods of tens, and attempted an explanation of the cause of this singular coincidence upon philosophical principles. But the earliest regular system of numbers is that given by Euclid in the 7th, 8th, 9th, and 10th books of his Elements, which, notwithstanding the embarrassing notation of the Greeks, and the inadequacy of geometry to the investigation of numerical propositions, is still very interesting, and displays, like all the other parts of the same celebrated work, that depth of thought and accuracy of demonstration for which its author is so eminently distinguished. Archimedes likewise paid particular attention to the powers and properties of numbers, as may be seen by consulting his tract entitled "Arenarius," in which some modern writers have thought they could perceive inculcated the principles of our present system of logarithms; but all that can be allowed on this head is, that the method by which he performed his multiplications and divisions bears a considerable analogy to that which we now commonly employ in the multiplication and division of powers; that is, by the addition and subtraction of their indices. Before the invention of analysis, however, no very extensive progress could be made in a subject, which required so much generality of investigation; and, accordingly, we find but little was effected in it till the time of Diophantus, whose treatise of algebra contains many interesting problems in the more abstruse parts of this science. But here, also, its author had to encounter the difficulties of a complicated notation, and a very deficient analysis, when compared with that of the present period; and, therefore, it cannot be expected that his work should contain a complete investigation of the theory of numbers. After Diophantus, the subject remained unnoticed, or at least unimproved, till Bachet, a French analyst of considerable reputation, undertook the translation of the abovementioned work into Latin, retaining also the Greek text, which was published by him in 1621, interspersed with many marginal notes of his own, and which may be considered as containing the first germ of our present theory. These were afterwards considerably extended by the celebrated 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; but they are generally left without demonstration, an omission which he accounted for by stating, in one of his notes, page 180, that he was himself preparing a treatise on the theory of numbers, which would contain many new and interesting numerical propositions; but, unfortunately, this work never appeared, and most of his theorems remained without demonstration for a considerable time. At length, the subject was again revived by Euler, Waring, and Lagrange, three of the most eminent analysts of modern times. The former, besides what is contained in the second volume of his "Elements of Algebra," and his "Analysis Infinitorum," has several papers in the Petersburg Acts, in which are given the demonstrations of many of Fermat's theorems. What has been done by Waring on this subject is comprised in chap. v. of his "Meditationes Algebraicæ;" and Lagrange, who has greatly extended the theory of numbers, by the invention of many new propositions, has several interesting papers on this head, in the Memoirs of Berlin, besides what are contained in his additions to Euler's Algebra. It is, however, but lately that this branch of analysis has been reduced into a regular system, a task that was first performed by Legendre, in his "Essai sur la Théorie des Nombres;" and nearly at the same time Gauss published his "Disquisitiones Arithmetica:" these two works eminently display the talents of their respective authors, and contain a complete development of this interesting theory. The latter, in particular, has opened |