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 Euclid 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 numbers. Let \( \frac{p}{q} \) be the logarithm of the number N (where p, q, and the other general characters which we shall use, denote integer numbers). Then \( N = 10^{\frac{p}{q}} \), whence \( N^3 = 10^{3\frac{p}{q}} = 2^p \times 5^p \). And, since the q power of N contains no prime factors but 2 and 5, N itself can contain no other. Let \( N = 2^r \times 5^s \). We have now \( 2^p \times 5^p = 2^r \times 5^s \); so that \( qr \) and \( qs \) being each equal to \( p \), we have \( \frac{p}{q} = r \), and \( N = 10^r \). Thus \( \frac{p}{q} \) 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. Yet we may conceive two such magnitudes to be related to each other 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 This affords a clew to the design of the greater part of Incommensurable Numbers, and some of the most elegant constructions. In particular, we may discover, in an attentive consideration of these formulas, the origin of those herods of irrational lines, of which he has been obliged to distinguish some by long and rather uncoth names.

The tenth book of the Elements is amongst the very finest performances of antiquity, for subtlety, clearness, and elegance.

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