Commentary on Proposition X.19¶
Back to Proposition X.19.
There is a difficulty in the text of the enunciation of this proposition. The Greek runs τὸ ὑπὸ ῥητῶν μήκει συμμέτρων κατά τινα τῶν προειρημένων τρόπων εὐθειῶν περιεχόμενον ὀρθογώνιον ῥητόν ἐστιν, where the rectangle is said to be contained by “rational straight lines commensurable in length in any of the aforesaid ways.” Now straight lines can only be commensurable in length in one way, the degrees of commensurability being commensurability in length and commensurability in square only. But a straight line may be rational in two ways in relation to a given rational straight line, since it may be either commensurable in length, or commensurable in square only, with the latter. Hence Billingsley takes κατά τινα τῶν προειρημένων τρόπων with ῥητῶν, translating “straight lines commensurable in length and rational in any of the aforesaid ways,” and this agrees with the expression in the next proposition “a straight line once more rational in any of the aforesaid ways”; but the order of words in the Greek seems to be fatal to this way of translating the passage.
The best solution of the difficulty seems to be to reject the words “in any of the aforesaid ways” altogether. They have reference to the Lemma which immediately precedes and which is itself open to the gravest suspicion. It is very prolix, and cannot be called necessary; it appears moreover in connexion with an addition clearly spurious and therefore relegated by Heiberg to the Appendix. The addition does not even pretend to be Euclid’s, for it begins with the words “for he calls rational straight lines those….” Hence we should no doubt relegate the Lemma itself to the Appendix. August does so and leaves out the suspected words in the enunciation, as I have done.
Exactly the same arguments apply to the Lemma added (without the heading “Lemma”) to X. 23 and the same words “in any of the aforesaid ways” used with “medial straight lines commensurable in length” in the enunciation of X. 24. The said Lemma must stand or fall with that now in question, since it refers to it in terms: “And in the same way as was explained in the case of rationals….”
Hence I have bracketed the Lemma added to X. 23 and left out the objectionable words in the enunciation of X. 24.
If $\rho$ be one of the given rational straight lines (rational of course in the sense of X. Def. 3), the other can be denoted by $k\rho$, where $k$ is, as usual, of the form $m/n$ (where $m, n$ are integers). Thus the rectangle is $k\rho^2$, which is obviously rational since it is commensurable with $\rho^2$. [X. Def. 4.]
A rational rectangle may have any of the forms $ab$, $ka^2$, $kA$ or $A$, where $a, b$ are commensurable with the unit of length, and $A$ with the unit of area.
Since Euclid is not able to use $k\rho$ as a symbol for a straight line commensurable in length with $\rho$, he has to put his proof in a form corresponding to
$$\rho^2 : k\rho^2 = \rho : k\rho,$$
whence, $\rho, k\rho$ being commensurable, $\rho^2, k\rho^2$ are so also. [X. 11]