book, as appears from Hervagius's edition: But Zambertus Book VI. and Commandine, in their Latin translations, subjoin the same to these definitions. Neither Campanus, nor, as it seems, the Arabic manuscripts from which he made his translation, have this definition. Clavius, in his observations upon it, rightly judges that the definition of compound ratios might have been made after the same manner in which the definitions of duplicate and triplicate ratio are given, viz. "That as in several magnitudes that are conti"nual proportionals, Euclid named the ratio of the first to "the third, the duplicate ratio of the first to the second; and the ratio of the first to the fourth, the triplicate ratio "of the first to the second; that is, the ratio compounded "of two or three intermediate ratios that are equal to one "another, and so on; so, in like manner, if there be several magnitudes of the same kind, following one another, "which are not continual proportionals, the first is said to "have to the last the ratio compounded of all the interme"diate ratios,only for this reason, that these interme "diate ratios are interposed betwixt the two extremes, viz. "the first and last magnitudes; even as, in the 10th defi"nition of the 5th book, the ratio of the first to the third "was called the duplicate ratio, merely upon account of "two ratios being interposed betwixt the extremes, that are દ equal to one another: so that there is no difference be"twixt the compounding of ratios, and the duplication or "triplication of them which are defined in the 5th book, "but that in the duplication, triplication, &c. of ratios, all "the interposed ratios are equal to one another; whereas, "in the compounding of ratios, it is not necessary that the "intermediate ratios should be equal to one another." Also Mr. Edmund Scarburgh, in his English translation of the first six books, pages 238, 266, expressly affirms, that the 5th definition of the 6th book is supposititious, and that the true definition of compound ratio is contained in the 10th definition of the 5th book, viz. the definition of duplicate ratio, or to be understood from it, to wit, in the same manner as Clavius has explained it in the preceding citation. Yet these, and the rest of the moderns, do notwithstanding 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 Ele

ments.

For, by comparing def. 5, book 6, with prop. 5, book 8, it will clearly appear that this definition has been put into Y

BOOK VI. 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,

(see Hervagius's 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 ratios 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 of which the sides are C, D, to the plane number of which the sides are E, Z. Wherefore the proposition which is the 5th def. of book 6, is the very same with the 5th prop. of book 8, and therefore it ought necessarily to be cancelled in one of these places; because it is absurd that the same proposition should stand 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, book 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 equiangular parallelograms; wherefore def. 5, book 6, ought not to be in the Elements. And from this it is evident that this definition is not Euclid's, but Theon's, or some other unskilful geometer's.

But nobody, as far as I know, has hitherto shown the true use of compound ratio, or for what purpose it has been introduced into geometry; for every proposition in which compound ratio is made use of, 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 6th book were to be enunciated, without mentioning compound ratio, it might 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 demonstration would be exactly the same as we now have it. But the ancient geometers, when they observed this enuncia

tion could be made shorter, by giving a name to the ratio Book VI. 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 first to the last, the ratio compounded of the ratio 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 which is compounded of the ratio of the first AC to the second

D

H

is to CF, as the straight line 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 ratio of the sides, that is, of finding the ratio of parallelograms, is shown in that, but not in Candalla's de

BOOK VI. monstration; whereby beginners may learn, in like cases, how to find the ratio which is compounded of two or more given ratios.

From what has been said, it may be observed, that in any magnitudes whatever of the same kind, A, B, C, D, &c. the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last, is only a name or expression by which the ratio which the first A has to the last D is signified, and by which at the same time the ratios of all the magnitudes A to B, B to C, C to D, from the first to the last, to one' another, whether they be the same, or be not the same, are indicated; as in magnitudes which are continual proportionals A, B, C, D, &c. the duplicate ratio of the first to the second is only a name, or expression by which the ratio of the first A to the third C is signified, and by which, at the same time, is shown, that there are two ratios of the magnitudes from the first to the last, viz. of the first A to the second B, and of the second B to the third or last C, which are the same with one another; and the triplicate ratio of the first to the second is a name or expression by which the ratio of the first A to the fourth D is signified, and by which, at the same time, is shown, that there are three ratios of the magnitudes from the first to the last, viz, of the first A to the second B, and of B to the third C, and of C to the fourth or last D, which are all the same with one another; and so in the case of any other multiplicate ratios. And that this is the right explication of the meaning of these ratios is plain from the definitions of duplicate and triplicate ratio, in which Euclid makes use of the word λɛyɛtal, is said to be, or is called; which word he, no doubt, made use of also in the definition of compound ratio, which Theon, or some other, has expunged from the Elements; for the very same word is still retained in the wrong definition of compound ratio, which is now the 5th of the 6th book: But in the citation of these definitions it is sometimes retained, as in the demonstration of prop. 19, book 6, "the "first is said to have, έxɛiv λsyɛtal, to the third the dupli"cate ratio," &c. which is wrong translated by Commandine and others, "has" instead of " is said to have," and sometimes it is left out, as in the demonstration of prop. 33, of the 11th book, in which we find, "the first has, exel, "to the third the triplicate ratio;" but without doubt xa, "has," in this place signifies the same as xev AsYETZ, is said to have: so likewise in Prop. 23, B. 6, we find this

citation, "but the ratio of K to M is compounded, σvyxsirai, Book VI. "of the ratio of K to L, and the ratio of L to M," which is a shorter way of expressing the same thing, which, according to the definition, ought to have been expressed by συγκείσθαι λέγεται, is said to be compounded.

From these remarks, together with the propositions, subjoined to the 5th book, all that is found concerning compound ratio, either in the ancient or modern geometers, may be understood and explained.

PROP. XXIV. B. VI.

Ir seems that some unskiful editor has made up this demonstration as we now have it, out of two others; one of which may be made from the 2d prop. and the other from the 4th of this book. For after he has, from the 2d of this book, and composition and permutation, demonstrated, that the sides about the angle common to the two parallelograms, are proportionals, he might have immediately concluded, that the sides about the other equal angles were proportionals, viz. from Prop. 34, B. 1. and Prop. 7, B. 5. This he does not, but proceeds to show, that the triangles and parallelograms are equiangular: and in a tedious way, by help of Prop. 4. of this book, and the 22d of book 5, deduces the same conclusion: From which it is plain, that this ill-composed demonstration is not Euclid's: These superfluous things are now left out, and a more simple demonstration is given from the 4th prop. of this book, the same which is in the translation from the Arabic, by help of the 2d prop. and composition; but in this the author neglects permutation, and does not show the parallelograms to be equiangular, as it is proper to do for the sake of beginners.

"

PROP. XXV. B. VI.

Ir is very evident, that the demonstration which Euclid had given of this proposition has been vitiated by some unskilful hand: For, after this editor had demonstrated, that "as the rectilineal figure ABC is to the rectilineal figure "KGH, so is the parallelogram BE to the parallelogram EF;" nothing more should have been added but this," and the