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Prop. is no doubt the fame with that of Eudoxus or Euclid, as it Book V. is immediately and directly derived from the Definition of a greater

ratio, viz. the 7. of the 5.

i

The above-mentioned Propofition, viz. If A have to Ca greater ratio than B to C, and if of A and B there be taken certain equimultiples, and some multiple of C, then if the multiple of B be greater than the multiple of C, the multiple of A is alfo greater than the fame, is thus demonftrated.

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Let D, E be equimultiples of A, B, and Fa multiple of C, fuch, that E the multiple of B is greater than F; D the multiple of A is also greater than F.

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D FE F

Because A has a greater ratio to C, than B to C, A is greater than B, by the 10. Prop. B. 5. therefore D the multiple of A is greater than E the fame multiple of B. and E is greater than E; much more therefore D is greater than F.

PROP. XIII. B. V.

In Commandine's, Briggs's and Gregory's Tranflations, at the beginning of this demonftration, it is faid, " And the multiple of C "is greater than the multiple of D; but the multiple of E is not "greater than the multiple of F," which words are a literal tranflation from the Greek. but the fenfe evidently requires that it be read, “so that the multiple of C be greater than the multiple of D; "but the multiple of E be not greater than the multiple of F." and thus this place was reftored to the true reading in the first editions of Commandine's Euclid printed in 8vo. at Oxford; but in the later editions, at leaft in that of 1747, the error of the Greek text was kept in.

There is a Corollary added to Prop. 13. as it is neceffary to the 20. and 21. Prop. of this Book, and is as useful as the Propofition.

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The two cafes of this which are not in the Greek are added; the demonstration of them not being exactly the fame with that of the firft cafe.

Book V.

PROP. XVII. B. V.

The order of the words in a clause of this is changed to one more natural. as was also done in Prop. 11.

PROP. XVIII. B. V.

The demonstration of this is none of Euclid's, nor is it legitimate. for it depends upon this Hypothefis, that to any three magnitudes, two of which, at least, are of the fame kind, there may be a fourth proportional; which if not proved, the Demonstration now in the text, is of no force. but this is affumed without any proof, nor can it, as far as I am able to difcern, be demonftrated by the Propofitions preceding this; fo far is it from deferving to be reckoned an Axiom, as Clavius, after other Commentators, would have it, at the end of the Definitions of the 5. Book. Euclid does not demonstrate it, nor does he fhew how to find the fourth proportional, before the 12. Prop. of the 6. Book. and he never affumes any thing in the demonftration of a Propofition, which he had not before demonftrated; at least, he affumes nothing the existence of which is not evidently poffible; for a certain conclufion can never-. be deduced by the means of an uncertain Propofition. upon this account we have given a legitimate Demonstration of this Propofition instead of that in the Greek and other editions, which very probably Theon, at least fome other has put in the place of Euclid's, because he thought it too prolix. and as the 17. Prop. of which this 18. is the converse, is demonstrated by help of the 1. and 2. Propofitions of this Book, fo in the demonftration now given of the 18th, the 5. Prop. and both cafes of the 6. are neceffary, and these two Propofitions are the converfes of the I. and 2. Now the 5. and 6. do not enter into the demonstration of any Propofition in this Book as we now have it, nor can they be of use in any Propofition of the Elements, except in this 18. and this is a manifest proof that Euclid made use of them in his demonftration of it, and that the demonftration now given, which is exactly the converse of that of the 17. as it ought to be, differs nothing from that of Eudoxus or Euclid. for the 5. and 6. have undoubtedly been put into the 5. Book for the fake of fome Propofitions in it, as all the other Propofitions about equimultiples have been.

Hieronymus Saccherius in his Book named Euclides ab omni naevo vindicatus, printed at Milan Ann. 1733 in 4to, acknowledges

this blemish in the demonstration of the 18. and that he may re- Book V. move it, and render the demonstration we now have of it legitimate, he endeavours to demonftrate the following Propofition, which is in page 115 of his Book, viz.

“Let A, B, C, D be four magnitudes, of which the two first ❝are of one kind, and also the two others either of the fame kind "with the two first, or of fome other the fame kind with one "another. I fay the ratio of the third C to the fourth D, is " either equal to, or greater, or lefs than the ratio of the firft A "to the fecond B."

And after two Propofitions premifed as Lemmas, he proceeds

thus.

"Either among all the poffible equimultiples of the firft A, and "of the third C, and, at the fame time among all the poffible equi"multiples of the second B, and of the fourth D, there can be found "fome one multiple EF of the first A, and one IK of the fecond B, "that are equal to one another; and also (in the fame cafe) some "one multiple GH of the third C equal to LM the multiple of the "fourth D. or fuch equality is no where to be found. If the first "cafe happen, [i.e.

"if fuch equality A

E

"is to be found,]

"it is manifeft from B

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"what is before de

C

G

-H

"monftrated, that

"A is to B, as CD.

L.

M

"to D. but if fuch
"fimultaneous equality be not to be found upon both fides, it will
"be found either upon one fide, as upon the fide of A [and B;]
"or it will be found upon neither fide; if the first happen; there-
"fore (from Euclid's Definition of greater and leffer ratio fore-
"going) A has to B, a greater or lefs ratio than C to D; accor-
"ding as GH the multiple of the third C is lefs, or greater than
"LM the multiple of the fourth D. but if the second cafe hap-
pen; therefore upon the one fide, as upon the fide of A the first
" and B the second, it may happen that the multiple EF, [viz. of the
"firft] may be less than IK the multiple of the fecond, while on
"the contrary, upon the other fide, [viz. of C and D] the multiple
"GH [of the third C] is greater than the other multiple LM [of
❝ the fourth D.] and then (from the fame Definition of Euclid) the

Book V." ratio of the first A to the second B, is lefs than the ratio of ❝ the third C to the fourth D; or on the contrary.

"Therefore the Axiom, [i. e. the Propofition before fet down,] "remains demonftrated," &c.

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Not in the leaft; but it remains ftill undemonftrated: for what he fays may happen, may in innumerable cafes never happen, and therefore his demonstration does not hold. for example, if A be the fide and B the diameter of a square; and C the fide and D the diameter of another fquare; there can in no cafe be any multiple of A equal to any of B; nor any one of C equal to one of D, as is well known; and yet it can never happen that when any multiple of A is greater or less than a multiple of B, the multiple of C can, upon the contrary, be lefs or greater than the multiple of D, viz. taking equimultiples of A and C, and equimultiples of B and D. for A, B, C, D are proportionals, and fo if the multiple of A be greater, &c. than that of B, fo muft that of C be greater &c. than that of D, by 5. Def. B. 5.

The fame objection holds good against the Demonftration which fome give of the 1. Prop. of the 6. Book, which we have made against this of the 18. Propofition, because it depends upon the fame infufficient foundation with the other.

PROP. XIX. B. V.

A Corollary is added to this, which is as frequently used as the Propofition itself. the Corollary which is fubjoined to it in the Greek, plainly fhews that the 5. Book has been vitiated by Editors who were not Geometers. for the converfion of ratios does not depend upon this 19. and the Demonstration which several of the Commentators on Euclid give of Converfion, is not legitimate, as Clavius has rightly observed, who has given a good Demonftration of it which we have put in Propofition E; but he makes it a Corollary from the 19. and begins it with the words, "Hence "it eafily follows," tho' it does not at all follow from it,

PROP. XX, XXI, XXII, XXIII, XXIV. B. V.

The Demonftrations of the 20. and 21. Propofitions are fhorter than thofe Euclid gives of cafier Propofitions, either in the preceding, or following Books. wherefore it was proper to make them more explicit. and the 22. and 23. Propofitions are, as they ought to be, extended to any number of magnitudes. and in like manner

may the 24. be, as is taken notice of in a Corollary; and another Book V. Corollary is added, as ufeful as the Propofition. and the words "any whatever" are fupplied near the end of Prop. 23. which are wanting in the Greek text, and the tranflations from it.

In a paper writ by Philippus Naudaeus, and published, after his death, in the Hiftory of the Royal Academy of Sciences of Berlin, Ann. 1745. page 50. the 23. Prop. of the 5. Book, is cenfured as being obfcurely enuntiated, and, because of this, prolixly demonftrated. the Enuntiation there given is not Euclid's but Tacquet's, as he acknowledges, which, tho' not fo well expreffed, is, upon the matter, the fame with that which is now in the Elements. Nor is there any thing obfcure in it, tho' the Author of the paper has fet down the proportionals in a disadvantageous order, by which it appears to be obfcure. but no doubt Euclid enuntiated this 23. as well as the 22. fo as to extend it to any number of magnitudes, which taken two and two, are proportionals, and not of fix only; and to this general cafe the Enuntiation which Naudaeus gives, cannot be well applied.

The Demonftration which is given of this 23. in that paper, is quite wrong; because if the proportional magnitudes be plane or solid figures, there can no rectangle (which he improperly calls a Product) be conceived to be made by any two of them. and if it should be faid, that in this cafe ftraight lines are to be taken which are proportional to the figures, the Demonftration would this way become much longer than Euclid's. but even tho' his Demonstration had been right, who does not see that it could not be made ufe of in the 5. Book?

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PROP. F, G, H, K. B. V.

Thefe Propofitions are annexed to the 5. Book, because they are frequently made ufe of by both antient and modern Geometers. and in many cafes Compound ratios cannot be brought into Demonstrations, without making ufe of them.

Whoever defires to fee the doctrine of Ratios delivered in this 5. Book, folidly defended, and the arguments brought against it by And. Tacquet, Alph. Borellus and others, fully refuted, may read Dr. Barrow's Mathematical Lectures, viz. the 7. and 8. of the year 1665.

The 5. Book being thus corrected, I moft readily agree to what the learned Dr. Barrow fays*, "That there is nothing in the whole *Page 336.

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