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absurd. 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 mo Meydan Lurrakis, page 62. where he also gives a childish explication of it, as agreeing only to such ratios as can be expressed 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 subjoin the fame 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 ratio might have been made after the same manner in whichi 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 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 intermediate ratios,— only for " this reason, that these intermediate ratios are interposed betwixt “ the two extremes, viz. the first and last magnitudes; even as in “ the 15. 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 inter posed betwixt the extremes, that are equal to one “ another : so 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 interposed ratios are equal to one another ; “ whereas in the compounding of ratios, it is not necesary that the “ intermediate ratios should be equal to one another.” Allo Mr. Edmund Scarburgh, in his English translation of the first fix Books, page 238, 266. expresiy affirms that the 5. Definition of the 6. Book, is supposititious, 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 same manner as Clavius has explained it in the preceeding citation. Yet these, and the rest of the Moderns, do notwithstanding

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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 demonstrated that the plane number of which the sides are C, D has to the plane number of which the fides are E, Ż (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. 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 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 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 these places; because it is absurd that the fame 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. B. 8. fhould have a place in the Elements, as the same thing is demonstrated in it concerning plane numbers, which is demonstrated 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 some other unskilful Geomcter's.

But no body, as far as I know, has hitherto fewn the true use of Compound ratio, or for what purpose it has been introduced into Geometry; for every Propofition in which Compound ratio is made use of, may without it be both enuntiated 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 enuntiated or demonstrated, or both may be done; for instance, if this 23. Proposition 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 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

to a third: the first parallelogram is to the second, as the first Book VI. straight line to the third. and the Demonstration would be exactly w the same as we now have it. but the antient Geometers, when they

observed this Enuntiation 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 fignified, 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 briefiy 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
preceeding Enuntiation, but has precisely the same meaning. or yet
Norter thus; equiangular parallelograms have to one another the
ratio which is the same with that which is compounded of the ra-
tios of their sides, and these two Enuntiations, the first especially,
agree to the Demonstration which is now in the Greek. the Propo-
sition may be more briefly demonstrated, as Candalla docs, thus ;
Let ABCD, CEFG be two equiangular parallelograms, and com-
plete the parallelogram CDHG; then, because there are three pa:
rallelograms AC, CH, CF, the first AC (by the Definition of Com.
pound ratio) has to the third CF, the
ratio which is compounded of the ratio

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A of the first AC to the second CH, and of the ratio of CH to the third CF; B

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С but the parallelogram AC is to the parallelogram CH, as the straight line

E BC to CG; and the parallelogram CH

F 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 Demonstrationi agrees the Enuntiation 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, tho' not so short as Candalla's; because the way of finding the ratio whicit

Bock vl. is compounded of the ratios of the sides; that is, of finding the w ratio of the parallelograms, is shewn in that, but not in Candalla's

Demonstration; 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 compoun led 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 wl.ich the frit A has to the last D is fignified, 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 fame, or be not the same, are indicated; as in magnitides which are continual proportionals A, B, C, D &c. the Dup'icate ratio of the firft to the second is only a name, or expresion by which the ratio of the first A to the third C is fignified, and by which, at the same time, is shewn that there are two ratios of the macnitudes from the first to the last, viz, of the first A to the le. cord B,and of the fecond D to the third or laft C, which are the faune with one another; and the Triplicate ratio of the first to the second is a name or expresion by which the ratio of the first A to the fourth D is fignifed, and by which, at the same time, is shown that ti ere are three ratios of the magnitudes from the first to the last, riz. of the first A to the second B, and of B to the third C., and of C to die fourth or last D, which are all the fame with one another; and so in the case of any o:her Multiplicare ratius. And that this is the right cxplication of the meaning of these ratios is plain from the Definitions of Duplicate and Triplicate ratio in which Eucli makes use of the word afgerou, is said to be, or is called; which word, lc no doubt made use of also in the Definition of Compound antio which Theon, or some other, has expunged from the Ele. ments; for the very fame word is still retained in the wrong Definifion of Compound ratio which is now the 5. of the 6. Book. but in the citation of these Definitions it is sometimes retained, as in the Demonfiration of Prop. 19. B. 6. “ the first is faid to have, men

aézetos, to the third the Duplicate ratio" &c. which is wrong trandated by Commandine and others " has” in ftcad of " is said to

bare;” and fometimes it is left out, as in the Demonstration of Prop. 33. of the 11. Book, in which we find “the first has, eye, " to the third the Triplic'te ratio ;” but without doubt yet," has," in this place fignifies the Same as ë xeir aéyé 704, is said to have. fo

likewise

likewise in Prop. 23. B. 6. we find this citation " but the ratio of Bock t!. " K to M is compounded, cuyxercy, of the ratio of K to L, and “ the ratio of L to M,” which is a shorter way of expresing the same thing, which, according to the Definition, ought to have been expreffed by συγκειοθα λέγεται, is faid to be compounded.

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

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PROP. XXIV. B. VI. It seems that some un/kiiful Editor has made up this Derrorstration as we now have it, out of iwo others; one of which may be made from the 2. Prop. and the other from the 4. of ils Book. for after he has from the 2. of this Book, and Composition and Permutation, demonstrated that the sides about the angle conmon 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. I. and Prop. 7. B. 5. this he does not, but proceeds to thew that the triangles and para:lelograms are equiangular, and in a tedious way, by help of Pro;. 4. of this Book, and the 2 2. of B. 5. deduces the same conclusion. from which it is plain that this ill composed Demonstration is nca Euclid's. these superfluous things are now left out, and a mes fimple Demonstration is given from the 4. Prop. of this book, the fame which is in the Translation from the Arabic, by hcp of the 2. Prop. and Composition; but in this the Author neglects Perma. tation, and does not thew the parallelograms to be equiangular, as is proper to do for the fake of beginners.

PRO P. XXV. B. VI. It is very evident that the Demonstration which Euclid had given of this Proposition, has been vitiated by some unskilful had. íur after this Editor had demonstrate 1 that " as the rectilineal Sgere «ABC is to the rectilineal KGH, fo is the purnllelogram BE to the

parallelogram EF,” nothing more should have been added but this, " and the rectilineal figure ABC is equal to the parallelogramı “ BE, therefore the rectilineal KGH is equal to the parallelograin " EF,” viz. from Prop. 14. B. 5. but betwixt these two feritenus he has inferted this, " wherefore, by Permutaticn, as the recilliacal

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