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absurd. for that Theon brought it into the Elements can scarce be Book Ví. doubted, as it is to be found in his Commentary upon Ptolomy's Miyaan Lurrašis, page 62. where he also gives a childish explicatiota of it, as agreeing only to such ratios as can be expressed by oumbers; 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 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 second ; and the ratio of the first to the fourthi, “ 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 so on; so in like manner if there be several “ magnitudes of the fame 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 ro. 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 ono “ 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 neceíTary that the “ intermediate ratios should be equal to one another.” Allo Mr. Edmund Scarburgh, in his English tranllation of the first six Books, page 238, 266. expressly affirms that the 5. Definition of the 6. Book, is supposititious, and that the true Definition of Compound ratio is contained in the ro. Definition of the 5. Bouk, 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 cication. Yet these, and the rest of the Moderns, do notwithstanding
Book VI. retain this 5. Def. of the 6. B. and illustrate and explain it bg long
Coinmentaries, 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 sides are E, 2 (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 wbich is the 5. Dcf. 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 abfurd that the fame Proposition Mould 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. Mhould 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 Geometer's.
But no body, as far as I know, has hitherto fhewn the true ofe 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 enuntiated and demonstrated. Now the use of Compound ratio consists whollyin 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 2 3. 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 fide of the second, so the second straight lige be made
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 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 signified, of the first to the second, and of the second tp 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 Enuntiation, but has precisely the same meaning. or yer 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 Enuntiations, the first especially, agree to the Demonstration which is now in the Greek. the Propofition may be more briefly demonstrated, as Candalla does, thus ; Let ABCD, CEFG be two equiangular parallelograms, and complete the parollelogram CDHG; then, becaufe 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 A,
H of the first AC to the second CH, and of the ratio of CH to the third CF;B
G but the parallelogram AC is to the pa
C С rallelogram CH, as the straight line BC to CG; and the parallelogram CH is to
E F 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 sides, and to this Demonftration 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 com“ pounded of their fides" is absurd. But in this Edition we have kept the Demonstration which is in the Greek text, tho' not fo short as Candalla's; because the way of finding the ratio which
Book VI. is compounded of the ratios of the sides; that is, of finding the ratis
of the parallelograms, is hewn in that, but not in Candalla's Demonitration; 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 fecond, of the second to the third, and so on to the laft, is only a name or expression by which the ratio which the first A has to the last 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 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 shewn that there are two ratios of the magnitudes from the first to the last, viz. of the first A to the lecond 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 expreísion by which the ratio of the first A to the fourth D is signified, and by which, at the same time, is shewn 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 fame with one another; and to 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 Detinitions of Duplicate and Triplicate ratio in which Euclid makes use of the word aéyetay, 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 fame word is still retained in the wrong Definition 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 Demonitration of Prop.19.B.6." the first is said to have, zer néyt7th, " to the third the Duplicate ratio, &c. which is wrong translated by Commandine and others “ has” initead of “ is said to have ;" and sometimes it is left out, as in the Demonstration of Prop. 33. of the it. Book, in which we find “ the first has, öxes, to " the third the Triplicate ratio ;” but without doubt čxet
, " has,” in this place figufics the fame as ë zer nézetay, is said to have. fo
likewise in Prop. 23. B. 6. we find this citation" but the ratio of Book VI. “ K to M is compounded, qúyxertcy, 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 expreffed by συγκαοθαι 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 and explained.
PRO P. XXIV. B. VI. It seems that some unskilful Editor has made up this Demosftration 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 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. I. and Prop. 7. B. 5. this he does not, but proceeds to Mew that the triangles and parallelograms are equiangular, and in a tedious way, by help of Prop. 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 not Euclid's. these superfluous things are now left out, and a more sinple Demonstration is given from the 4. Prop. of this Book, the fame which is in the Translation from the Arabic, by help of the 2. Prop. and Composition ; but in this the Author neglects Permutation, and does not shew the parallelograms to be equiangular, as is proper to do for the sake of beginners.
PROP. XXV. B. VI. It 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 KGH, so is the parallelogram BE to the
parallelogram EF,” nothing more should have been added but this," and the rectilineal figure ABC is equal to the parallelogram “ BE, therefore the rectilineal KGH is equal to the parallelogram “ EF,” viz. from Prop. 14, B. 5. but betwixt these two sentences he has inserted this, "wherefore, by Fermutation, as the rectilinea!