INCOMMENSURABLE

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

The great Στοιχεῖον of the ancients has not expressly
called attention to this negative relation of magnitudes ear-

lier 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 contin-
ually; 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 remain-
der (that is, the less segment) has exactly the same rela-
tion 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 rela-
tive 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 re-
presentation is not metaphysically accurate are far more
numerous than where it is perfect.

Take, for instance, the vulgar logarithms of the natural
numbers. Let $\backslash frac\{p\}\{q\}$ be the logarithm of the number $N$ (where
$p,\; q$, and the other general characters which we shall use,
denote integer numbers). Then $N\; =\; 10^\{\backslash frac\{p\}\{q\}\}$, whence $N^q\; =$
$10^p\; =\; 2^p\; \backslash times\; 5^p$. And, since the $q$ power of $N$ con-
tains no prime factors but 2 and 5, $N$ itself can contain no
other. Let $N\; =\; 2^r\; \backslash times\; 5^s$. We have now $2^r\; \backslash times\; 5^s\; =\; 2^p\; \backslash times$
$5^p$; so that $qr$ and $qs$ being each equal to $p$, we have
$\backslash frac\{p\}\{q\}\; =\; r$, and $N\; =\; 10^r$. Thus $\backslash frac\{p\}\{q\}$ is necessarily integer, con-
sequently 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 pre-
ceding.

We have said that the impossibility of reducing the re-
lation 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 nei-
ther 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 κατὰ πρᾶγμα, exactly like other two.

If $D$ and $S$ be the diagonal and side of a square; and
$\backslash Delta$ and $\backslash Sigma$ the diagonal and side of another square; and if
$S$ and $\backslash Sigma$ be divided into the same number of equal parts,
however great the number and small the parts, we may

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 might 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."

Thus, if $\backslash frac\{n\}\{m\}B$ signify n of the magnitudes, of which B contains m, we might say, A has the same ratio to B which C has to D, if, according as A is greater than $\backslash frac\{n\}\{m\}B$, equal to it, or less, C is greater than $\backslash frac\{n\}\{m\}D$, equal to it, or less; m and n being not particular numbers, but any whatsoever. But if $A\; =\; \backslash frac\{n\}\{m\}B$, we have $mA\; =\; nB$; and

thence, according as A is greater than $\backslash frac\{n\}\{m\}B$, equal to it, or less, we have $mA$ greater than $nB$, equal to it, or less; and similarly, $mC$ greater than $nD$, equal to it, or less. Thus we are brought to Euclid's definition (book v. def. 5), Εἰ τὸ αὐτὸ λόγῳ μεγίστη λέγεται ἴσαι, πρώτον πρὸς δεύτερον καὶ τῆν πρὸς τῆν ἕταρον ἵσται τὰ τὸ πρῶτον καὶ τῆν ἵσαις πολλὰ πλάτων τὸ τὸ δεύτερον καὶ τὸ δεύτερον ἵσαις πολλὰ πλάτων, καὶ ἴσους πολλὰ πλάτων, ἵσους ἢ ἄμα ἰσλῶν, ἢ ἄμα ἵσῶν, ἢ ἄμα ἰσλῶν καὶ ἰσλῶν.

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 lines 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 $\backslash sqrt\{A+B\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{2\}(A\; +\; \backslash sqrt\{A^2\; -\; B^2\})\}\; +\; \backslash sqrt\{\backslash frac\{1\}\{2\}(A\; -\; \backslash sqrt\{A^2\; -\; B^2\})\}$; 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 $\backslash frac\{1\}\{2\}B^2$;" 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 in 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 $\backslash sqrt\{A-B\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{2\}(A\; +\; \backslash sqrt\{A^2\; -\; B^2\})\}\; -\; \backslash sqrt\{\backslash frac\{1\}\{2\}(A\; -\; \backslash sqrt\{A^2\; -\; B^2\})\}$.

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 formulæ, 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 subtlety, clearness, and elegance. (o. o. o.)

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