AN ELEMENTARY INVESTIGATION OF THE THEORY OF NUMBERS, WITH ITS APPLICATION TO THE INDETERMINATE AND DIOPHANTINE ANALYSIS, THE ANALYTICAL AND GEOMETRICAL DIVISION OF THE CIRCLE, AND SEVERAL OTHER CURIOUS ALGEBRAICAL AND ARITHMETICAL PROBLEMS, BY PETER BARLOW, OF THE ROYAL MILITARY ACADEMY. LONDON: PRINTED FOR J. JOHNSON AND CO., ST. PAUL'S CHURCH-YARD. 1 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. A 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 BRITISH |