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Book VI. 20th Prop. of this book; and it is extended to five fided figures, 'by which it may be seen how to extend it to figures of any num. ber of fides.

PROP. XXIII. B. VI.

Nothing is ufually reckoned more difficult in the elements of geometry by learners, than the doctrine of compound ratio, which Theon has rendered abfurd and ungeometrical, by fubftituring the 5th definition of the 6th book in place of the right definition, which without doubt Eudoxus or Euclid gave, in its proper place, after the definition of triplicate ratio, &c, in the 5th book. Thcon's definition is this; a ratio is

faid to be compounded of ratios όταν αι των λογων πηλικότητες εφ' εαυτας πολλαπλασιασθείσαι ποιωσι τινα : Which Commandine thus tranflates, "quando rationum quantitates inter fe multi"plicatæ aliquam efficiunt rationem;" that is, when the quantities of the ratios being multiplied by one another make a certain ratio. Dr Wallis tranflates the word nxornteS "rationum exponentes," the exponents of the ratios: And Dr Gregory renders the laft words of the definition by "illius

facit quantitatem," makes the quantity of that ratio: But in whatever fenfe the "quantities" or "exponents of the ra"tios," and their "multiplication" be taken, the definition will be ungeometrical and uleless: For there can be no multiplication but by a number: now the quantity or exponent of a ratio (according as Eutocius in his Comment, on prop. 4. book 2. of Arch. de Sph. et Cyl. and the moderns explain that term) is the number which multiplied into the confequent term of a ratio produces the antecedent, or, which is the fame thing, the number which arifes by dividing the antecedent by the con fequent; but there are many ratios fuch, that no number can arife from the division of the antecedent by the confequent; ex. gr. the ratio which the diameter of a fquare has to the fide of it; and the ratio which the circumference of a circle has to its diameter, and fuch like. Befides, there is not the leaft mention made of this definition in the writings of Eu clid, Archimedes, Apollonius, or other ancients, tho' they fre quently make ufe of compound ratio: And in this 23d prop. of the 6th book, where compound ratio is first mentioned, there is not one word which can relate to this definition, tho' here, if in any place, it was neceffary to be brought in; but the right definition is expressly cited in thefe words: "But the " ratio of K to M is compounded of the ratio of K to L,

"and

"and of the ratio of L to M," This definition therefore of Book VI. 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 Meyaan Zurtažis, 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 manufcripts, from which he made his tranflation, 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 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 several magnitudes of the fame kind, following one "another, which are not continual proportionals, the firft is "faid to have to the laft the ratio compounded of all the in "termediate ratios, only for this reafon, that these inter"mediate ratios are interpofed betwixt the two extremes, viz. ❝ the first and last 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 "be equal to one another." Alfo Mr Edmund Scarburgh, in his English tranflation of the firft 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 VI. 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 fame manner as Clavius has explained it in the preceeding citation. Yet these, and the rest of the moderns, do notwithstanding retain this 5th def. of the 6th book, and illu strate and explain it by long commentaries, when they ought ra. ther 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: Because, in prop. 5. book 8. it is demonstrated 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 de monstrated in another place of them. Now there is no 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. 23. book 6. of equiangu lar 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 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 purpose it has been introduced into geometry; for every propofition in which compound ratio is made use of, may without it be both enun-' ciated and demonftrated. Now the ufe of compound ratio confifts wholly in this, that by means of it, circumlocutions may be avoided, and thereby propofitious 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 first to a fide of the fecond, fo any affumed' heftraight line be made to a fecond ftraight line; and as the o

ther fide of the first to the other fide of the fecond, fo the fedi coad ftraight line be made to a third. The first parallelogram is to the fecond, 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 shorter, by giving a name to the ratio which the first straight line has to the laft, by which name the intermediate ratios might likewise 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 first 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 propo. fition 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 firft efpecially, agree to the demonftration which is now in the Greek. The propofition may be more briefly demoostrated, 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 firft AC (by the definition of compound ratio) has to the third CF, the ratio which is compounded of the ratio of A

B

D

H

G

F

the firft AC to the fecond CH, and of
the ratio of CH to the third CF; but
the parallelogram AC is to the pa-
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 enun-
ciation which is at prefent in the text, viz. Equiangular paralle
lograms have to one another the ratio which is compounded of

the

Book VI. the ratios of the fides: For the vulgar reading, "which is com pounded of their fides," is abfurd. But, in this edition, we have kept the demonftration which is in the Greek text, tho' 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 observed, 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 second, of the fecond to the third, and fo on to the last, is only a name or expreffion by which the ratio which the first A has to the laft 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 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 fhewn that there are two ratios of the magnitudes from the first to the last, 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 fhewn that there are three ratios of the magnitudes from the first to the last, 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 these ratios is plain from the definitions of duplicate and triplicate ratio in which Euclid makes ufe of the word xéytral, is faid to be, or is called; which word, he no doubt made ufe 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 ic is fometimes retained, as in the demonftration of prop. 19. book

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