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II. A reply to the objections of Mr Leslie against M. Legendre's theory of parallel lines.

III. A defence of the same theory by M. Le Baron Maurice.

Besides these additions communicated by the author, an Introduction on Proportion, by the translator, has been prefixed to the work, and several valuable trigonometrical tables have been added from the Trigonométrie of Cagnoli.

The greatest improvement, however, which has been made in this edition consists in the substitution of wooden cuts in the text, in place of the engraved plates which accompany all the editions of the original work. This improvement, which will be found of great use both to the teacher and the student, was considered indispensable in an English work.

D. B.

EDINBURGH, August 1, 1822.



THE doctrine of proportion belongs properly to Arithmetic, and ought to be explained in works which treat of that science. Its object being to point out the relations which subsist among magnitudes in general, when viewed as measured, or represented by numbers, the connexion it has with Geometry is not more immediate than with many other branches of knowledge, except indeed as Geometry affords the largest class of magnitudes capable of being so measured or represented, and thus offers the widest field for reducing it to practice. Owing, however, to our general and long-continued employment of Euclid's Elements, the fifth Book of which is devoted to proportion, our common systems of Arithmetic, and even of Algebra, pass over the subject in silence, or allude to it so slightly as to afford no adequate information. For the sake of the British student, therefore, it will be requisite to prefix a brief outline of the fundamental truths connected with this department of Mathematics; at least in so far as a knowledge of them is essential for understanding

the work which follows.

The proper mode of treating proportion has given rise to much controversy among mathematicians; chiefly originating from the difficulties which occur in the application of its theorems to that class of magnitudes denominated incommen surable, or having no common measure. Euclid evades this obstacle; but his method is cumbrous, and, to a learner, difficult of comprehension. All other methods have the disadvantage of frequently employing the principle of reductio ad absurdum, a species of reasoning, which, though perfectly conclusive, the mathematician wishes to employ as seldom as possible. The opposite advantages, however, have generally overcome this reluctance; and Euclid's method is now almost entirely abandoned in elementary treatises. On this matter, we are happily delivered from the necessity of making any selection; the author having himself provided for the application of proportion to incommensurable quantities, and demonstrated every case of this kind as it occurred, by means of the reductio ad absurdum. He has also in various parts of these Elements interspersed explanations of the sense in which geometrical mag

nitudes may be viewed, as coming under the dominion of numbers, and bearing a proportion to each other. So that our duty, on the present occasion, is reduced to little more than defining a few terms, and, in the briefest manner, exhibiting the leading truths of the subject, when referred to mere numbers, or to magnitudes capable of being completely represented by numbers.


I. One magnitude is a multiple of another, when the former contains the latter an exact number of times; a submultiple or measure, when it is contained by the latter an exact number of times.

Thus 6 is a multiple of 2; 2 and 3 are submultiples of 6. Like multiples and like submultiples, or equimultiples and equal submultiples, are such as contain the magnitudes they refer to, or as are contained by them, the same number of times. Thus 4 and 5 are like submultiples of 8 and 10; 8 and 10 are like multiples of 4 and 5.

II. Four magnitudes are proportional, if when the first and second are multiplied by two such numbers as make the products equal, the third and fourth being respectively multiplied by the same numbers, likewise make equal products.

Thus 6, 15, 8, 20, are proportional; because, multiplying the first and second by 5 and 2 respectively, so as to make 6 × 5

15 x 2, we have likewise 8x5=20 × 2 A; and generally, the magnitudes A, B, C, D, are proportional, if m and n being any two numbers such that n A=m B, we have likewise n Č =m D. The magnitude A is said to be to B as C is to D; the four together are named a proportion or analogy, and are written thus, A: B::C: D.·

Note. To make this definition complete, two things are required: first, that such a pair of numbers n and m be always discoverable as shall make n A =m B; secondly, we must prove that when one such pair of numbers m and n is discovered, and found likewise to make n Cm D, every other pair of numbers p and q, making p A = q B, will also make p C = q D.

First, With a view to the former of these conditions, we must find a common measure of A and B; the mode of doing which is explained at large in Problem 17, Book II. of these Elements. Suppose this common measure to be E, and that A =mE, B =nE: the numbers required will be n and m.


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