I say that A has not to B the ratio which a number number. For, if A has to B the ratio which a number number, A will be commensurable with B. But it is not; therefore A has not to B the ratio which a number has to a number. Therefore etc. PROPOSITION 8. If two magnitudes have not to one another the ra a number has to a number, the magnitudes will b mensurable. For let the two magnitudes A, B not have to on the ratio which a number has to a number; I say that the magnitudes A, B are incommensurable. For, if they are commensurable, A will have ratio which a number has to a number. But it has not; therefore the magnitudes A, B are incommensurabl Therefore etc. PROPOSITION 9. The squares on straight lines commensurable in l to one another the ratio which a square number has t number; and squares which have to one another which a square number has to a square number will their sides commensurable in length. But the s straight lines incommensurable in length have n another the ratio which a square number has to number; and squares which have not to one another which a square number has to a square number wi their sides commensurable in length either. PROPOSITIONS 7-9 x. 9] I say that the square on A For let A, B be commensurable in length; 29 has to the square on B the ratio which a square number has to a square number. For, since A is commensurable in length with B, therefore A has to B the ratio which a number has number. Let it have to it the ratio which C has to D. to a [x. 5] while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, for similar figures are in the duplicate ratio of their corresponding sides; [VI. 20, Por.] and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, for between two square numbers there is one mean proportional number, and the square number has to the square number the ratio duplicate of that which the side has to the side; [vIII. 11] therefore also, as the square on A is to the square on B, so is the square on C to the square on D. Next, as the square on A is to the square on B, so let the square on C be to the square on D; I say that A is commensurable in length with B. For since, as the square on A is to the square on B, so is the square on C to the square on D, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, therefore also, as A is to B, so is C to D. Therefore A has to B the ratio which the number Chas to the number D; therefore A is commensurable in length with B. Next, let A be incommensurable in length with B ; [x. 6] I say that the square on A has not to the square on B the ratio which a square number has to a square number. For, if the square on A has to the square on B the ratio Again, let the square on A not have to the squar the ratio which a square number has to a square num I say that A is incommensurable in length with B. For, if A is commensurable with B, the square o have to the square on B the ratio which a square nun to a square number. But it has not; therefore A is not commensurable in length with B. Therefore etc. PORISM. And it is manifest from what has been that straight lines commensurable in length are alwa mensurable in square also, but those commensurable are not always commensurable in length also. [LEMMA. It has been proved in the arithmetic that similar plane numbers have to one another which a square number has to a square number, and that, if two numbers have to one another the ra a square number has to a square number, they ar plane numbers. [Converse And it is manifest from these propositions that which are not similar plane numbers, that is, tho have not their sides proportional, have not to on the ratio which a square number has to a square nu For, if they have, they will be similar plane which is contrary to the hypothesis. Therefore numbers which are not similar plane have not to one another the ratio which a square nu to a square number.] A scholium to this proposition (Schol. x. No. 62) says categ the theorem proved in it was the discovery of Theaetetus. If a, b be straight lines, and where m, n are numbers, then and conversely. a: b = m:n, a2 : b2 = m2: n2; X. 9, 10] PROPOSITIONS 9, 10 31 This inference, which looks so easy when thus symbolically expressed, was by no means so easy for Euclid owing to the fact that a, b are straight lines, and m, n numbers. He has to pass from ab to a2: b2 by means of VI. 20, Por. through the duplicate ratio; the square on a is to the square on in the duplicate ratio of the corresponding sides a, b. On the other hand, m, n being numbers, it is VIII. 11 which has to be used to show that m2:n is the ratio duplicate of m : n. Then, in order to establish his result, Euclid assumes that, if two ratios are equal, the ratios which are their duplicates are also equal. This is nowhere proved in Euclid, but it is an easy inference from v. 22, as shown in my note on VI. 22. The converse has to be established in the same careful way, and Euclid assumes that ratios the duplicates of which are equal are themselves equal. This is much more troublesome to prove than the converse; for proofs I refer to the same note on vi. 22. The second part of the theorem, deduced by reductio ad absurdum from the first, requires no remark. In the Greek text there is an addition to the Porism which Heiberg brackets as superfluous and not in Euclid's manner. It consists (1) of a sort of proof, or rather explanation, of the Porism and (2) of a statement and explanation to the effect that straight lines incommensurable in length are not necessarily incommensurable in square also, and that straight lines incommensurable in square are, on the other hand, always incommensurable in length also. The Lemma gives expressions for two numbers which have to one another the ratio of a square number to a square number. Similar plane numbers are of the form pm . pn and qm . qn, or mnp2 and mng, the ratio of which is of course the ratio of p2 to q2. The converse theorem that, if two numbers have to one another the ratio of a square number to a square number, the numbers are similar plane numbers is not, as a matter of fact, proved in the arithmetical Books. It is the converse of VIII. 26 and is used in IX. 10. Heron gave it (see note on VIII. 27 above). Heiberg however gives strong reason for supposing the Lemma to be an interpolation. It has reference to the next proposition, x. 10, and, as we shall see, there are so many objections to x. 10 that it can hardly be accepted as genuine. Moreover there is no reason why, in the Lemma itself, numbers which are not similar plane numbers should be brought in as they are. [PROPOSITION 10. To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line. Let A be the assigned straight line; thus it is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A. Let two numbers B, C be set out which have not to one the square on D -for we have learnt how to do this [x. 6, Por.] B therefore the square on A is commensurable with the on D. And, since B has not to C the ratio which a square has to a square number, therefore neither has the square on A to the square o ratio which a square number has to a square number therefore A is incommensurable in length with D. Let E be taken a mean proportional between A, therefore, as A is to D, so is the square on A to th on E. But A is incommensurable in length with D; therefore the square on A is also incommensurable square on E; therefore A is incommensurable in square with E. Therefore two straight lines D, E have been commensurable, D in length only, and E in squar course in length also, with the assigned straight line It would appear as though this proposition was intended justification for the statement in x. Def. 3 that it is proved that infinite number of straight lines (a) incommensurable in leng commensurable in square only, and (b) incommensurable in squa given straight line. But in truth the proposition could well be dispensed wit positive objections to its genuineness are considerable. In the first place, it depends on the following proposition, x last step concludes that, since and a, x are incommensurable in length, therefore a2, y2 are incon But Euclid never commits the irregularity of proving a theorem a later one. Gregory sought to get over the difficulty by puttin X. 11; but of course, if the order were so inverted, the Lemma in the wrong place. Further, the expression éμáloμev yap, "for we have learnt (how is not in Euclid's manner and betrays the hand of a learner (tho |