INCOMMENSURABLE, in Geometry, is a term applied to homogeneous magnitudes which have no common measure, or whereof one cannot be denoted as either multiple, aliquot part, or aliquot parts, of the other, or whose ratio cannot be represented by numbers.

The great Σror>∕ε∕ωnj5 of the ancients has not expressly called attention to this negative relation of magnitudes earlier than in the tenth book of his Elements ; but he has kept it steadily in view in the preceding parts of the work. Hence he has two distinct treatises of proportion; the one of proportion in magnitudes, the other of proportion in numbers.

In the second proposition of the tenth book it is shown, that if from the greater of two magnitudes we take the less, or the highest multiple of the less which it contains, then take from the less the remainder, or the highest multiple of the remainder which is contained in it, and so on continually; whenever this process becomes interminable, the magnitudes have no common measure.

The simplest instance of this interminable process to which we can refer, is in the case of a straight line and the greater segment of the same divided in extreme and mean ratio. For, by proposition 5, book xiii. it appears, that when the greater segment is taken from the whole, the remainder (that is, the less segment) has exactly the same relation to this greater segment which the greater has to the whole, and so on for ever.

If we begin a similar process with the diagonal and side of a square, at the end of every two operations the two lines with which we have to proceed have the same relative magnitude as the two with which we began; and thus we should never come to an end. If, therefore, the side of a square be one foot, we cannot possibly express the diagonal in feet or parts of a foot.

In fact, although, in ultimate practice, every quantity with which the mathematician has to deal is represented by numbers, whole or fractional, the cases where this representation is not metaphysically accurate are far more numerous than where it is perfect.

Take, for instance, the vulgar logarithms of the natural p numbers. Let^ be the logarithm of the number N (where p, q, and the other general characters which we shall use, p denote integer numbers). Then N = 10 y , whence N ¢ > = 10 p = 2 p × 5 p . And, since the q power of N contains no prime factors but 2 and 5, N itself can contain no other. Let N = 2 r × 5 i . We have now 2 qr × 5 Çf = 2 p × 5 P ; so that qr and qs being each equal to p, we have ĩ- = r, and N = 10 r . Thus^ is necessarily integer, consequently not one logarithm of the series can be properly a fraction; and those which are integer succeed only at intervals, of which each is ten times as great as the preceding.

We have said that the impossibility of reducing the relation of concrete magnitudes to that of numbers, in an infinity of cases, has caused Euclid to form two distinct treatises of proportion. And it is easy to see by what considerations he has passed from the simpler to the more complex, though this last has priority in the order of the Elements.

Two numbers are called proportional to two other, or “the first is said to have to the second the same ratio which the third has to the fourth, when the first is the same multiple (aliquot) part or parts of the second which the third is of the fourth.” But we have seen that there may be magnitudes of the same kind, whereof one is neither multiple, part, nor parts of another; in other words, that have no common measure, no numerical ratio. ìet we may conceive two such magnitudes to be related to each other κατα Ω∙^λ∕zorjjra, exactly like other two.

If D and S be the diagonal and side of a square; and Δ and Σ the diagonal and side of another square; and if S and Σ be divided into the same number of equal parts, however great the number and small the parts, we may [7:12:259]conceive and easily prove, that whenever D is greater than m of the parts of S, but less than m + 1, Δ is also greater than m of the parts of S, but less than m + 1. So that, amongst incommensurable magnitudes, the first mi<dit be said to have the same ratio to the second which the third has to the fourth, when, “according as the first is greater or less than any multiple, part, or parts whatsoever of the second, the third is greater or less than the same multiple, part, or parts of the fourth.” And both commensurable and incommensurable magnitudes might be brought under the following definition: “The first of four magnitudes has the same ratio to the second which the third has to the fourth, when, according as the first is greater than any multiple, part, or parts whatsoever of the second, equal to it, or less, the third is greater than the same multiple, part, or parts of the fourth, equal to it, or less.”

7Z . .

Thus, if — B signify n of the magnitudes, of which B contains rn, we might say, A has the same ratio to B which C has to D, if, according as A is greater than —B,

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equal to it, or less, C is greater than —D, equal to it, or less;anand η being not particular numbers, but any whatsoever. But if A = —B, we have mA = nB; and m thence, according as A is greater than - B, equal to it, or less, we have mA greater than nB, equal to it, or less; and similarly, rnC greater than nD, equal to it, or less. Thus we are brought to Euclid’s definition (book v. def. 5), Ev τlji αυτψ λογω μεγεάη asysτα∕ eivai, tfξωτ( ∣ v ffgog δsursξοt ×ai τρίτον ∙~ξος τετaξτοv , orav τά του nfi>τoυ xal τξiτoυ ίσάχις vολλa- τλάΰια των του δίυτίζου καί ητάξτου ισάκις πολλαπλασίων, καό’ ίπωονουν πολλαπλασιασμόν, sκaτiξoυ άμα ελλείπη, r ∣ άμα ίσα η, γ άμα i ∣ πεξεχτ1 ληφόίντα κατάλληλα-

On the foundation of this definition he has constructed, in his sixth book, what appears to us a far more elegant treatise than any by which modern writers of elements have endeavoured to supersede it. That he has been able to accomplish it logically, without even mentioning the existence of incommensurable magnitudes, is the best possible evidence of its perfection.

His tenth book has often been called a treatise of incommensurables, or of surds. Not that Euclid has any mode of expressing abstract surds; but he treats of lince and areas whose relation could not be signified arithmetically without surds. The extent of his doctrines on this subject, in more than a hundred propositions, only embraces relations expressed by square roots, and their union with integral or fractional numbers, including repeated extractions of square roots, so as to produce biquadratic roots, &c. in infinitum.

It is manifest that he knew what is expressed in the algebraic formula √A + B = √ {⅜(A + √A 2 — B 2 )} + √ {⅜ (A — √A 2 — B 2 ) }; or that, as the same may be expressed implicitly, “the square root of A + B is the sum of the square roots of the numbers whose sum is A and product ⅜B 2 ;” which gives this geometrical proposition: “The straight line equal in power to the rectangle under a given line, and the sum of two unequal lines, is composed of two straight lines, which are severally equal m power to the rectangles under the given line and those segments of the greater of the unequal lines whose rectangle is a fourth of the square of the less.” He discusses with equal fulness the propositions contained in the kindred formula √AB = √U(A + √A 2 — B 2 )} — ^{⅜(A — √A 2 — B 2 )}∙

This affords a clew to the design of the greater part of the book, and some of the most elegant constructions. In particular, we may discover, in an attentive consideration of these formulae, the origin of those hexads of irrational lines, of which he has been obliged to distinguish some by long and rather uncouth names.

The tenth book of the Elements is amongst the very finest performances of antiquity, for subtilty, clearness, and elegance. (o. o. o.)

Incommensurable Numbers are such as have no common divisor that will divide them both equally.