" and of the ratio of L to M." This definition therefore of Theon is quite useless and abfurd: For that Theon brought it into the elements can fcarce be doubted; as it is to be found in his commentary upon Ptolomy's Meyxan Eurais, page 62. where he alfo gives a childish explication of it, as agreeing only to fuch ratios as can be expreffed by numbers; and from this place the definition and explication have been exactly copied and prefixed to the definitions of the 6th book, as appears from Hervagius's edition: But Zambertus and Commandine, in their Latin tranflations, fubjoin the fame to thefe definitions. Neither Campanus, nor, as it feems, the Arabic manuscripts, from which he made his tranflation, have this definition. Clavius, in his observations upon it, rightly judges that the definition of compound ratio might have been made after the fame manner in which the definitions of duplicate and triplicate ratio are given, viz. "That as in feveral magni"tudes that are continual 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 fecond that is, the "ratio compounded of two or three intermediate ratios that "are equal to one another, and fo on; fo, in like manner, if "there be feveral magnitudes of the fame kind, following one "another, which are not continual proportionals, the first is "faid to have to the laft the ratio compounded of all the in"termediate ratios,-only for this realon, that these inter"mediate ratios are interpofed betwixt the two extremes, viz. "the first and laft magnitudes; even as, in the 10th definition "of the 5th book, the ratio of the first to the third was called "the duplicate ratio, merely upon account of two ratios be. "ing interpofed betwixt the extremes, that are equal to one "another: So that there is no difference betwixt this com"pounding of ratios, and the duplication or triplication of "them which are defined in the 5th book, but that in the du"plication, triplication, &c. of ratios, all the interpofed ratios "are equal to one another; whereas, in the compounding of "ratios, it is not neceffary that the intermediate ratios fhould Alfo Mr Edmund Scarburgh, "be equal to one another." in his English translation of the first fix books, page 238. 266. exprefsly affirms, that the 5th definition of the 6th book, is fuppofititious, and that the true definition of compound ratio

is

Book VL

n

Book VI. is contained in the roth definition of the 5th book, viz. thẻ W definition of duplicate ratio, or to be understood from it, to

wit, in the fame 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 illuftrate 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: Becaufe, in prop. 5. book 8. it is demonftrated that the plane number of which the fides are C, D has to the plane number of which the fides are E, Z, (fee Hergavius's or Gregory's edition), the ratio which is compounded of the ra tios of their fides; 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 of the confequents E, Z; that is, the ratio of the plane number of which the fides are C, D to the plane number of which the fides are E, Z. Wherefore the propofition which is the 5th def. of book 6. is the very fame with the 5th prop. of book 8. and therefore it ought neceffarily to be cancelled in one of these places; because it is abfurd that the fame proposition should ftand as a definition in one place of the elements, and be demonftrated in another place of them. Now, there is do doubt that prop. 5. book 8. fhould have a place in the elements, as the fame thing is demonftrated in it concerning plane numbers, which is demonftrated in prop. 23d 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 no body, as far as I know, has hitherto fhown the true ufe of compound ratio, or for what purpose it has been introduced into geometry; for every propofition in which compound ratio is made ufe of, may without it be both enunciated and demonftrated. Now the ufe of compound ratio confifts wholly in this, that by means of it, circumlocutions may be avoided, and thereby propofitions may be more briefly either enunciated or demonftrated, or both may be done; for inftance, if this 23d propofition of the fixth book were to be enunciated, without mentioning compound ratio, it might

be

be done as follows. If two parallelograms be equiangular, and Book VI. if as a fide of the firft to a fide of the fecond, fo any affumed ftraight line be made to a fecond ftraight line; and as the other fide of the first to the other fide of the fecond, fo the second ftraight 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 fame as we now have it. But the antient geometers, when they obferved this enunciation could be made fhorter, by giving a name to the ratio which the first straight line has to the laft, by which name the intermediate ratios might likewife be fignified, of the first to the fecond, and of the second to the third, and fo on, if there were more of them, they called this ratio of the first to the laft, the ratio compounded of the ratios of the first to the fecond, and of the second to the third ftraight line; that is, in the prefent example, of the ratios which are the fame with the ratios of the fides, and by this they expreffed the propofition more briefly thus: If there be two equiangular parallelograms, they have to one another the ratio which is the fame with that which is compounded of ratios that are the fame with the ratios of the fides. Which is fhorter than the preceding enunciation, but has precifely the fame meaning. Or yet fhorter thus: Equiangular parallelograms have to one another the ratio which is the fame with that which is compounded of the ratios of their fides. And these two enunciations, the first especially, agree to the demonftration which is now in the Greek. The propofition may be more briefly demonftrated, as Candalla does, thus: Let ABCD, CEFG be two equiangular parallelograms, and complete the parallelogram CDHG; then, becaufe there are three parallelograms AC, CH, CF, the firft AC (by the definition of compound ratio) has to the third CF, the ratio which is compounded of the ratio of A the first AC to the fecond CH, and of the ratio of CH to the third CF; but the parallelogram AC is to the rallelogram CH, as the ftraight line BC to CG; and the parallelogram CH is to CF, as the ftraight line CD is to CE; therefore the parallelogram AC has to CF the ratio which is compounded of ratios that are the fame with the ratios of the fides. And to this demonftration agrees the enunciation which is at present in the text, viz. Equiangular parallelograms have to one another the ratio which is compounded of

pa.

D

H

B

G

C

E

the

Book VI. the ratios of the fides: For the vulgar reading, "which is comw" pounded of their fides," is abfurd. But, in this edition, we

have kept the demonftration which is in the Greek text, though not fo fhort as Candalla's; because the way of finding the ratio which is compounded of the ratios of the fides, that is, of finding the ratio of the parallelograms, is fhewn in that, but not in Candalla's demonftration; whereby beginners may learn, in like cafes, how to find the ratio which is compounded of two or more given ratios.

From what has been faid, it may be obferved, that in any magnitudes whatever of the fame kind A, B, C, D, &c. the ratio compounded of the ratios of the first to the fecond, of the fecond to the third, and fo on to the laft, is only a name or expreffion by which the ratio which the firft A has to the laft D is fignified, and by which at the fame time the ratios of all the magnitudes A to B, B to C, C to D from the first to the laft, to one another, whether they be the fame, or be not the fame, are indicated; as in magnitudes which are continual proportionals A, B, C, D, &c. the duplicate ratio of the firft to the fecond is only a name, or expreffion by which the ratio of the first A to the third C is fignified, and by which, at the fame time, is fhown that there are two ratios of the magnitudes from the first to the laft, viz. of the firft A to the fecond B, and of the fecond B to the third or laft C, which are the fame with one another; and the triplicate ratio of the first to the fecond is a name or expreffion by which the ratio of the first A to the fourth D is fignified, and by which, at the fame time, is fhown that there are three ratios of the magnitudes from the first to the last, viz. of the firft A to the fecond B, and of B to the third C, and of C to the fourth or laft D, which are all the fame with one another; and fo in the cafe of any other multiplicate ratios. And that this is the right explication of the meaning of these ratíos is plain from the definitions of duplicate and triplicate ratio in which Euclid makes ufe of the word ytra, is faid to be, or is called; which word, he, no doubt, made ufe of alfo in the definition of compound ratio, which Theon, or fome other, has expunged from the elements; for the very fame word is ftill 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 fometimes retained, as in the demonftration of prop. 19.

book

book 6. "the firft is faid to have, ixtiraiyit, to the third the Book VI. duplicate ratio," &c. which is wrong tranflated by Commandine and others, "has" inftead of "is faid to have;" and fometimes it is left out, as in the demonftration of prop. 33. of the 11th Book, in which we find "the first has, x, to the "third the triplicate ratio;" but without doubt xe, "has," in this place fignifies the fame as x Aysa is faid to have: So likewife in prop. 23. B. 6. we find this citation," but the "ratio of K to M is compounded, ruya, of the ratio of "K to L, and the ratio of L to M," which is a fhorter way of expreffing the fame thing, which, according to the definition, ought to have been expreffed by συγκεῖσθαι λέκεται, is faid to be compounded.

From thefe remarks, together with the propofitions fubjoined to the 5th book, all that is found concerning compound ratio, either in the antient or modern geometers, may be understood and explained.

PRO P. XXIV. B. VI.

It feems that fome unskilful editor has made up this demonftration 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 compofition and permutation, demonftrated that the fides about the angle common to the two parallelograms are proportionals, he might have immediately concluded that the fides about the other equal angles were proportionals, viz. from prop. 34. B. 1. and prop. 7. Book 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 fame conclufion: From which it is plain that this ill compofed demonftration is not Euclid's: Thefe fuperfluous things are now left out, and a more fimple demonftration is given from the 4th prop. of this book, the fame which is in the tranflation from the Arabic, by help of the ad prop. and compofition; but in this the author neglects permutation, and does not show the parallelograms to be equiangular, as is proper to do for the fake of beginners.

PROP. XXV. B. VI.

It is very evident that the demonftration which Euclid had given of this propofition has been vitiated by fome unskilful hand: For, after this editor had demonftrated that " as the " rectilineal figure ABC is to the rectilineal KGH, so is the "parallelogram