Commentary on Book X Definitions

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DEFINITION 1.

Σύμμετρα μεγέθη λέγεται τὰ τῷ αὐτῷ μέτρῳ μετρούμενα, ἀσύμμετρα δέ, ὧν μηδὲν ἐνδέχεται κοινὸν μέτρον γενέσθαι.

DEFINITION 2.

Εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπ᾿ αὐτῶν τετράγωνα τῷ αὐτῷ χωρίῳ μετρῆται, ἀσύμμετροι δέ, ὅταν τοῖς ἀπ᾿ αὐτῶν τετραγώνοις μηδὲν ἐνδέχεται χωρίον κοινὸν μέτρον γενέσθαι.

Commensurable in square is in the Greek δυνάμει σύμμετρος. In earlier translations (e.g. Williamson’s) δυνάμει has been translated “in power,” but, as the particular power represented by δύναμις in Greek geometry is square, I have thought it best to use the latter word throughout. It will be observed that Euclid’s expression commensurable in square only (used in Def. 3 and constantly) corresponds to what Plato makes Theaetetus call a square root (δύναμις) in the sense of a surd. If $a$ is any straight line, $a$ and $a\sqrt{m}$, or $a\sqrt{m}$ and $a\sqrt{n}$ (where $m, n$ are integers or arithmetical fractions in their lowest terms, proper or improper, but not square) are commensurable in square only. Of course (as explained in the Porism to x. 10) all straight lines commensurable in length (μήκει), in Euclid’s phrase, are commensurable in square also; but not all straight lines which are commensurable in square are commensurable in length as well. On the other hand, straight lines incommensurable in square are necessarily incommensurable in length also; but not all straight lines which are incommensurable in length are incommensurable in square. In fact, straight lines which are commensurable in square only are incommensurable in length, but obviously not incommensurable in square.

DEFINITION 3.

Τούτων ὑποκειμένων δείκνυται, ὅτι τῇ προτεθείσῃ εὐθείᾳ ὑπάρχουσιν εὐθεῖαι πλήθει ἄπειροι σύμμετροί τε καὶ ἀσύμμετροι αἱ μὲν μήκει μόνον, αἱ δὲ καὶ δυνάμει. καλείσθω οὖν ἡ μὲν προτεθεῖσα εὐθεῖα ῥητή, καὶ αἱ ταύτῃ σύμμετροι εἴτε μήκει καὶ δυνάμει εἴτε δυνάμει μόνον ῥηταί, αἱ δὲ ταύτῃ ἀσύμμετροι ἄλογοι καλείσθωσαν.

The first sentence of the definition is decidedly elliptical. It should, strictly speaking, assert that “with a given straight line there are an infinite number of straight lines which are (1) commensurable either ($a$) in square only or ($b$) in square and in length also, and (2) incommensurable, either ($a$) in length only or ($b$) in length and in square also.”

The relativity of the terms rational and irrational is well brought out in this definition. We may set out any straight line and call it rational, and it is then with reference to this assumed rational straight line that others are called rational or irrational.

We should carefully note that the signification of rational in Euclid is wider than in our terminology. With him, not only is a straight line commensurable in length with a rational straight line rational, but a straight line is rational which is commensurable with a rational straight line in square only. That is, if $\rho$ is a rational straight line, not only is $\frac{m}{n}\rho$ rational, where $m, n$ are integers and $m/n$ in its lowest terms is not square, but $\sqrt{\frac{m}{n}} . \rho$ is rational also. We should in this case call $\sqrt{\frac{m}{n}} . \rho$ irrational. It would appear that Euclid’s terminology here differed as much from that of his predecessors as it does from ours. We are familiar with the phrase ἄρρητος διάμετρος τῆς πεμπάδος by which Plato (evidently after the Pythagoreans) describes the diagonal of a square on a straight line containing 5 units of length. This “inexpressible diameter of five (squared)” means $\sqrt{50}$, in contrast to the ῥητὴ διάμετρος, the “expressible diameter” of the same square, by which is meant the approximation $\sqrt{50 - 1}$, or $7$. Thus for Euclid’s predecessors $\sqrt{\frac{m}{n}} . \rho$ would apparently not have been rational but ἄρρητος, “inexpressible,” i.e. irrational.

I shall throughout my notes on this Book denote a rational straight line in Euclid’s sense by $\rho$, and by $\rho$ and $\sigma$ when two different rational straight lines are required. Wherever then I use $\rho$ or $\sigma$, it must be remembered that $\rho, \sigma$ may have either of the forms $a, \sqrt{k} . a$, where $a$ represents $a$ units of length, $a$ being either an integer or of the form $m/n$, where $m, n$ are both integers, and $k$ is an integer or of the form $m/n$ (where both $m, n$ are integers) but not square. In other words, $\rho, \sigma$ may have either of the forms $a$ or $\sqrt{A}$, where $A$ represents $A$ units of area and $A$ is integral or of the form $m/n$, where $m, n$ are both integers. It has been the habit of writers to give $a$ and $\sqrt{a}$ as the alternative forms of $\rho$, but I shall always use $\sqrt{A}$ for the second in order to keep the dimensions right, because it must be borne in mind throughout that $\rho$ is an irrational straight line.

As Euclid extends the signification of rational (ῥητός, literally expressible), so he limits the scope of the term ἄλογος (literally having no ratio) as applied to straight lines. That this limitation was started by himself may perhaps be inferred from the form of words “let straight lines incommensurable with it be called irrational.” Irrational straight lines then are with Euclid straight lines commensurable neither in length nor in square with the assumed rational straight line. $\sqrt{k} . a$ where $k$ is not square is not irrational; $\sqrt[4]{k} . a$ is irrational, and so (as we shall see later on) is $(\sqrt{k} \pm \sqrt{\lambda}) a$.

DEFINITION 4.

Καὶ τὸ μὲν ἀπὸ τῆς προτεθείσης εὐθείας τετράγωνον ῥητόν, καὶ τὰ τούτῳ σύμμετρα ῥητά, τὰ δὲ τούτῳ ἀσύμμετρα ἄλογα καλείσθω, καὶ αἱ δυνάμεναι αὐτὰ ἄλογοι, εἰ μὲν τετράγωνα εἴη, αὐταὶ αἱ πλευραί, εἰ δὲ ἕτερά τινα εὐθύγραμμα, αἱ ἴσα αὐτοῖς τετράγωνα ἀναγράφουσαι.

As applied to areas, the terms rational and irrational have, on the other hand, the same sense with Euclid as we should attach to them. According to Euclid, if $\rho$ is a rational straight line in his sense, $\rho^2$ is rational and any area commensurable with it, i.e. of the form $k\rho^2$ (where $k$ is an integer, or of the form $m/n$, where $m, n$ are integers), is rational; but any area of the form $\sqrt{k} . \rho^2$ is irrational. Euclid’s rational area thus contains $A$ units of area, where $A$ is an integer or of the form $m/n$, where $m, n$ are integers; and his irrational area is of the form $\sqrt{k} . A$. His irrational area is then connected with his irrational straight line by making the latter the square root of the former. This would give us for the irrational straight line $\sqrt[4]{k} . \sqrt{A}$, which of course includes $\sqrt[4]{k} . a$.

αἱ δυνάμεναι αὐτά are the straight lines the squares on which are equal to the areas, in accordance with the regular meaning of δύνασθαι. It is scarcely possible, in a book written in geometrical language, to translate δυναμένη as the square root (of an area) and δύνασθαι as to be the square root (of an area), although I can use the term “square root” when in my notes I am using an algebraical expression to represent an area; I shall therefore hereafter use the word “side” for δυναμένη and “to be the side of” for δύνασθαι, so that “side” will in such expressions be a short way of expressing the “side of a square equal to (an area).” In this particular passage it is not quite practicable to use the words “side of” or “straight line the square on which is equal to,” for these expressions occur just afterwards for two alternatives which the word δυναμένη covers. I have therefore exceptionally translated “the straight lines which produce them” (i.e. if squares are described upon them as sides).

αἱ ἴσα αὐτοῖς τετράγωνα ἀναγράφουσαι, literally “the (straight lines) which describe squares equal to them”: a peculiar use of the active of ἀναγράφειν, the meaning being of course “the straight lines on which are described the squares” which are equal to the rectilineal figures.