PROP. XXXI. B. VI. In the demonstration of this, the inversion of proportionals is twice neglected, and is now added, that the conclusion may be legitimately made by help of the 24th prop. of B. 5. as Clavius had done. PROP. XXXII. B. VI. The enunciation of the preceding 26th prop. is not gneral enough; because not only two similar parallelograms that have an angle common to both, are about the same diameter but likewise two similar parallelograms that have vertically oppsite angles, have their diameters in the same straight line: but here seems to have been another, and that a direct demonstratio of these cases, to which this 32d proposition was needful: anche 32d may be otherwise and something more briefly demonstred as follows: 339 Book VI. PROP. XXXII. B. VI. If two triangles which have two sides of the one, &c. Draw CK parallel a to FH, and let it meet GF produced in K: because AG, KC are each of them parallel to FH, they are parallel b to one another, and therefore the alternate angles AGF, FKC are A E B G K Ꭰ H a 31. 1, b 30. 1. C equal and AG is to GF, as (FH to HC, that is) CK to KF;: 34. 1. wherefore the triangles AGF, CKF are equiangulard, and the d 6 6 angle AFG equal to the angle CFK: but GFK is a straight line, therefore AF and FC are in a straight line. The 26th prop. is demonstrated from the 32d, as follows: First, Let the parallelograms ABCD, AEFG have the angle 14. 1. Book VI. standing retain this 5th def. of the 6th book, and illustrate and explain it by long commentaries, when they ought rather to have taken it quite away from the Elements. For, by comparing def. 5, book 6, with prop. 5, book 8, it will clearly appear that this definition has been put into the Elements in place of the right one, which has been taken out of them because, in prop. 5, book 8, it is demonstrated that the plane number of which the sides are C, D has to the plane number of which the sides are E, Z (see Hergavius' or Gregory's edition), the ratio which is compounded of the ratios of their sides; that is, of the ratios of C to E, and D to Z: and, by def. 5, book 6, and the explication given of it by all the commentators, the ratio which is compounded of the ra tios of C to E, and D to Z, is the ratio of the product made by the multiplication of the antecedents C, D to the product by the consequents E, Z, that is, the ratio of the plane number f which the sides are C, D to the plane number of which e sides are E, Z. Wherefore the proposition which is the 5th d of book 6, is the very same with the 5th prop. of book 8, an therefore it ought necessarily to be cancelled in one of these plaes; because it is absurd that the same proposition should stant as a definition in one place of the Elements, and be demonstrated in another place of them. Now, there is no doubt that prop. 5, hook 8, should have a place in the Elements, as the same thing is demonstrated in it concerning plane numbers, which is demonstrated in prop. 23, book 6, of equiangu parallelograms; wherefore def. 5, book 6, ought not to be in the Elements. And from this it is evident that this definition is not Euid's, but Theon's, or some other unskilful geometer's. But nobody, as far as I know, has hitherto shown the true use of compoundatio, or for what purpose it has been introduced into geomen, for every proposition in which compound ratio is made usef, may without it be both enunciated and demonstrated. Now the use of compound ratio consists wholly in this, that by means of it, circumlocutions may be avoided, and thereby propositions may be more briefly either enunciated or demonstrated, or both may be done: for instance, if this 23d proposition of the sixth boo be enunciated, without mentioning compound ratio, it night be done as follows. If two parallelograms be equiangular, and if as a side of the first to a side of the second, so any assumed straight line be made to a second straight line; and as the other side of the first to the other side of the second, so the second straight line be made to a third. The first parallelogram is to the second, as the first straight line to the third. And the lar were to demonstration would be exactly the same as we now have it. Book VI But the ancient geometers, when they observed this enunciation could be made shorter, by giving a name to the ratio which the first straight line has to the last, by which name the intermediate ratios might likewise be signified, of the first to the second, and of the second to the third, and so on, if there were more of them, they called this ratio of the first to the last the ratio compounded of the ratios of the first to the second, and of the second to the third straight line: that is, in the present example, of the ratios which are the same with the ratios of the sides, and by this they expressed the proposition more briefly thus: if there be two equiangular parallelograms, they have to one another, the ratio which is the same with that which is compounded of ratios that are the same with the ratios of the sides. Which is shorter than the preceding enunciation, but has precisely the same meaning. Or yet shorter thus: equiangular parallelograms have to one another the ratio which is the same with that which is compounded of the ratios of their sides. And these two enunciations, the first especially, agree to the demonstration which is now in the Greek. The proposition may be more briefly demonstrated, as Candalla does, thus: let ABCD, CEFG be two equiangular parallelograms, and complete the parallelogram CDHG; then, because there are three parallelograms AC, CH, CF, the first AC (by the definition of compound ratio) has to the third CF, the ratio H which is compounded of the ratio of the first AC to the second CH, and of the ratio of CH to the third CF; but the parallelogram AC is to the pa- B rallelogram CH, as the straight line BC to CG; and the parallelogram CH is to CF, as the straight line A D E F CD, is to CE; therefore the parallelogram AC has to CF the ratio which is compounded of ratios that are the same with the ratios of the sides. And to this demonstration agrees the enunciation which is at present in the text, viz. Equiangular parallelograms have to one another the ratio which is compounded of the ratios of the sides: for the vulgar reading," which is compounded of their sides," is absurd. But, in this edition, we have kept the demonstration which is in the Greek text, though not so short as Candalla's; because the way of finding the ratio which is compounded of the ratios of the sides, that is, of finding the ratio of the parallelograms, is shown in that, but not in Candalla's demonstration; whereby beginners may learn, in like Book VI. Produce EF, GF, to H, K, and join FA, FC: then because the parallelograms ABCD, AEFG are similar, DA is to AB, as GA to AE: where a Cor. 19. fore the remainder DG is a to the A F G B K 32. 6. F; herefore AF, FC are in the same straight line b. D H C Nxt, Let the parallelograms KFHC, GFEA, which are similar. andimilarly placed, have their angles KFH, GFE vertically op pose; their diameters AF, FC are in the same straight line. ecause AG, GF are parallel to FH, HC; and that AG is GF, as FH to HC; therefore AF, FC are in the same stright line b. PROP. XXXIII. B. VI. The words "because they are at the centre," are left out, as ae addition of some unskilful hand. In the Greek, as also in the Latin translation, the words & etuxe, 'any whatever," are left out in the demonstration of both parts of the proposition, and are now added as quite necessary; and in the demonstration of the second part, where the triangle BGC is proved to be equal to CGK, the illative particle aga in the Greek ext ought to be omitted. The second part of the proposition is an addition of Theon's, as ne tells us in his commentary on Ptolemy's Mayáλn Zortážis, p. 50. PROP. B. C. D. B. VI. These three propositions are added, because they are frequently made use of by geometers. DEF. IX. and XI. B. XI, THE similitude of plane figures is defined from the equality of their angles, and the proportionality of the sides about the equal angles; for from the proportionality of the sides only, or only from the equality of the angles, the similitude of the figures does not follow, except in the case when the figures are triangles: the similar position of the sides which contain the figures, to one another, depending partly upon each of these: and, for the same reason, those are similar solid figures which have all their solid angles equal, each to each, and are contained by the same number of similar plane figures: for there are some solid figures contained by similar plane figures, of the same number, and even of the same magnitude, that are neither similar nor equal, as shall be demonstrated after the notes on the 10th definition: upon this account it was necessary to amend the definition of similar solid figures, and to place the definition of a solid angle before it and from this and the 10th definition, it is sufficiently plain how much the Elements have been spoiled by unskilful editors. DEF. X. B. XI. Book XI. Since the meaning of the word " equal" is known and established before it comes to be used in this definition; therefore the proposition which is the 10th definition of this book, is a theorem, the truth or falsehood of which ought to be demonstrated, not assumed; so that Theon, or some other editor, has ignorantly turned a theorem which ought to be demonstrated into this 10th definition: that figures are similar, ought to be proved from the definition of similar figures; that they are equal, ought to be demonstrated from the axiom," Magnitudes that wholly coincide, are equal "to one another;" or from prop. A. of book 5, or the 9th prop. or the 14th of the same book, from one of which the equality of all kinds of figures must ultimately be deduced. In the preceding books, Euclid has given no definition of equal figures, and it is certain he did not give this: for what is |