abfurd. for that Theon brought it into the Elements can scarce be Book VI. doubted, as it is to be found in his Commentary upon Ptolomy's Μεγάλη Συντάξις, Meydan Zurrasic, page 62. where he also 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 6. Book, as appears from Hervagius's edition. but Zambertus and Commandine in their Latin Translations fubjoin the fame to thefe Definitions; Neither Campanus, nor, as it feems, the Arabic manuscripts from which he made his Translation, have this Definition. Clavius in his Obfervations 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 several magnitudes that are continual proportionals, Eu"clid named the ratio of the first to the third, the Duplicate ratio "of the first to the fecond; and the ratio of the first to the fourth, "the Triplicate ratio of the firft to the fecond; that is, the ratio

66

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 last "the ratio compounded of all the intermediate ratios,only for "this reafon, that thefe intermediate ratios are interpofed betwixt "the two extremes, viz. the first and last magnitudes; even as in "the 1. Definition of the 5. Book, the ratio of the first to the "third was called the Duplicate ratio, merely upon account of two " ratios being interpofed betwixt the extremes, that are equal to one "another: fo that there is no difference betwixt this compounding "of ratios, and the duplication or triplication of them which are “defined in the 5. Book, but that in the duplication, 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 be equal to one another." Alfo Mr. Edmund Scarburgh, in his English translation of the firft fix Books, page 238, 266. exprefly affirms that the 5. Definition of the 6. Book, is fuppofititious, and that the true Definition of Compound ratio is contained in the 10. Definition of the 5. Book, viz. the Definition of Duplicate ratio, or to be understood from it, to' wit, in the fame manner as Clavius has explained it in the preceeding citation. Yet thefe, and the reft of the Moderns, do notwithstanding

Book VI. retain this 5. Def. of the 6. B. 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. B. 6. with Prop. 5. B. 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. B. 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 Hervagius's or Gregory's Edition) the ratio which is compounded of the ratios of their fides; that is, of the ratios of C to E, and D to Z. and by Def. 5. B. 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 5. Def. of B. 6. is the very fame with the 5. Prop. of B. 8. and therefore it ought necessarily to be cancelled in one of thefe places; because it is abfurd that the fame Propofition fhould ftand as a Definition in one place of the Elements, and be demonftrated in another place of them. Now there is no doubt that Prop. 5. B. 8. fhould have a place in the Elements, as the fame thing is demonftrated in it concerning plane numbers, which is demonftrated in Prop. 23. B. 6. of equiangular parallelograms; wherefore Def. 5. B. 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 fome other unfkilful Geometer's.

But no body, as far as I know, has hitherto fhewn the true ufe of Compound ratio, or for what purpofe it has been introduced into Geometry; for every Propofition in which Compound ratio is made ufe of, may without it be both enuntiated 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 enuntiated or demonftrated, or both may be done; for inftance, if this 23. Propofition of the 6. Book were to be enuntiated, without mentioning Compound ratio, it might be done as follows; If two parallelograms be equiangular, and if as a fide of the firft to a fide of the fecond, fo any affumed straight line be made to a fecond straight line; and as the other fide of the first to the other fide of the fecond, fo the fecond straight line be made

to

to a third: the firft parallelogram is to the fecond, as the firft Book VI. ftraight line to the third. and the Demonftration would be exactly the fame as we now have it. but the antient Geometers, when they obferved this Enuntiation could be made fhorter, by giving a name to the ratio which the firft ftraight line has to the laft, by which name the intermediate ratios might likewife be fignified, of the firft to the fecond, and of the fecond 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 firft to the fecond, and of the fecond 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 preceeding Enuntiation, 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 thefe two Enuntiations, the firft 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 coinplete the parallelogram CDHG; then, because there are three pa rallelograms AC, CH, CF, the firft AC (by the Definition of Com pound ratio) has to the third CF, the ratio which is compounded of the ratio of the first AC to the fecond CH, and of the ratio of CH to the third CF; B but the parallelogram AC is to the parallelogram CH, as the ftraight line BC to CG; and the parallelogram CH

A

D

H

G

C

E

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 fame with the ratios of the fides. and to this Demonftration agrees the Enuntiation which is at prefent in the text, viz. equiangular parallelograms have to one another the ratio which is com pounded of the ratios of the fides. for the vulgar reading " which "is compounded of their fides" is abfurd. But in this Edition we i have kept the Demonftration which is in the Greek text, tho' not fo short as Candalla's; becaufe the way of finding the ratio which

Bock VI. 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 magniAndes whatever of the fame kind A, B, C, D &c. the ratio compounded of the ratios of the firft 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 firft A to the third C is fignified, and by which, at the fame time, is fhewn that there are two ratios of the magnitudes from the first to the laft, viz. of the firft A to the fe cond 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 firft A to the fourth D is fignifed, and by which, at the fame time, is fhewn that there are three ratios of the magnitudes from the firft to the laft, viz. of the first 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 thefe ratios is plain from the Definitions of Duplicate and Triplicate ratio in which Euclid makes ufc of the word ayer, 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 5. of the 6. Book. but in the citation of the fe Definitions it is fometimes retained, as in the Demonflration of Prop. 19. B. 6. "the firft is faid to have, exer "Ta, to the third the 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 11. Book, in which we find "the firft has, ye, "to the third the Triplicate ratio;" but without doubt xe," has." in this place fignifies the same as xer xéye704, is faid to have. fo

r

likewife

likewife in Prop. 23. B. 6. we find this citation "but the ratio of Bock !. "K to M is compounded, ovyxera, 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 ovynesday λéyera, is faid to be compounded.

From these Remarks, together with the Propofitions subjoined to the 5. Book, all that is found concerning Compound ratio cither in the antient or modern Geometers may be understood and explained.

PROP. XXIV. B. VI.

It feems that fome unfkilful Editor has made up this Demonftration as we now have it, out of two others; one of which may be made from the 2. Prop. and the other from the 4. of this Book. for after he has from the 2. 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. B. 5. this he does not, but proceeds to fhew that the triangles and parallelograms are equiangular, and in a tedious way, by help of Fro ̧ . 4. of this Book, and the 22. of B. 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 mea fimple Demonftration is given from the 4. Prop. of this Book, the fame which is in the Tranflation from the Arabic, by help of the 2. Prop. and Compofition; but in this the Author neglects Permu. tation, and does not fhew the parallelograms to be equiangalar," as is proper to do for the fake of beginners.

PRO P. XXV. B. VI.

It is very evident that the Demonftration which Euclid had given of this Propofition, has been vitiated by fome unfkilful hand. for after this Editor had demonftrated that " as the rectilineal figure "ABC is to the rectilineal KGH, fo is the parallelogram BE to the "parallelogram EF," nothing more fhould have been added but this, "and the rectilineal figure ABC is equal to the parallelogram BE, therefore the rectilineal KGH is equal to the parallelograin "EF," viz. from Prop. 14. B. 5. but betwixt thefe two fentences he has inferted this," wherefore, by Permutation, as the reincal