s is greater than the multiple of D; but the multiple of E is not

Prop. is no doubt the same with that of Eudoxus or Euclid, as it Book V. is immediately and dire&ly derived from the Definition of a greater ratio, viz. the 7. of the 5.

The above-mentioned Proposition, viz. If A have to C a 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 also greater than the same, is thus demon- A CBC strated.

D F E F Let D, E be equimultiples of A, B, and Fa multiple of C, such, that E the multiple of B is greater than F; D the multiple of A is also greater than 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 same multiple of B. and E is greater than F; much more therefore D is greater than F.

PROP. XIII. B. V. In Commandine's, Briggs's and Gregory's Translations, at the beginning of this demonstration, it is faid, “ And the multiple of C

“greater than the multiple of F,” which words are a literal tranflation from the Greek, but the sense 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 restored to the true reading in the first editions of Commandine's Euclid printed in 8vo. at Oxford; but in the later editions, at least in that of 1747, the error of the Greek text was kept in.

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

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PROP. XIV. B. V.. The two cases of this which are not in the Greek are added ; the demonstration of them'not being exactly the same with that of the first case.

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 Hypothesis, that to any three magnitudes, , two of which, at least, are of the same kind, there may be a fourth proportional; which if not proved, the Demonstration now in the text, is of no force. but this is assumed without any proof, nor can it, as far as I am able to discern, be demonstrated by the Propofitions preceding this; so far is it from deserving to be reckoned an Axiom, as Clavius, after other Commentators, would have it, at the end of the Definitions of the


Book. Euclid does not demonstrate it, nor does he shew how to find the fourth proportional, before the 12. Prop. of the 6. Book. and he never assumes any thing in the demonftration of a Propofition, which he had not before demonstrated; at leaft, he assumes nothing the existence of which is not evidently possible; for a certain conclusion can never . be deduced by the means of an uncertain Proposition. upon this account we have given a legitimate Demonstration of this Proposition instead of that in the Greek and other editions, which very probably Theon, at least some 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 i. and 2. Propositions of this Book, so in the demonstration now given of the 18th, the 5. Prop. and both cases of the 6. are neceffary, and these two Propositions are the converses of the I. and 2. Now the 5. and 6. do not enter into the demonstration of any Proposition in this Book as we now have it, nor can they be of use in


Proposition of the Elements, except in this 18. and this is a manifest proof that Euclid made use of them in his demonstration of it, and that the demonstration 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 sake 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 demonstrate 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 same kind “ with the two firft, or of fome other the same 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 first A (6 to the fecond B.”

And after two Propofitions premifed as Lemmas, he proceeds chus.

« Either among all the possible equimultiples of the first A, and # of the third C, and, at the same 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) fome “ 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 such equality Ar E

F « is to be found, « it is manifest from B

I. K 56 what is before de« monstrated, that C


H “ A is to B, as CD


M « to D. but if such « fimultaneous equality be not to be found upon both sides, it will “ be found either upon one side, as upon the fide of A [and B;] « or it will be found upon neither side; if the first happen; there« fore (from Euclid's Definition of greater and lesser ratio fore« going) A has to B, a greater or less ratio than C to D; accora “ ding as GH the multiple of the third C is less, or greater than “ LM the multiple of the fourth D. but if the second cafe hap“pen; therefore upon the one side, as upon the side of A the first “ and B the second, it may happen that the multiple EF, (viz. of the “ first] may be less than IK the multiple of the second, while on “ the contrary, upon the other side, (viz. of Cand 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

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Book V. « ratio of the first A to the second B, is less than the ratio of

o the third C to the fourth D; or on the contrary.

“ Therefore the Axiom, [i. e. the Proposition before set down,] « remains demonstrated," &c.

Not in the leaft; but it remains still undemonstrated: for what he says may happen, may in innumerable cafes never happen, and therefore his demonftration does not hold. for example, if A be the side and B the diameter of a square; and C the side and D the diameter of another square; there can in no case 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 less 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 so if the multiple of A be greater, &c. than that of B, so must that of C be greater &c. than that of D, by 5. Def. B. 5.

The same objection holds good against the Demonstration which some give of the 1. Prop. of the 6. Book, which we have made against this of the 18. Proposition, because it depends upon the fame insufficient foundation with the other,

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

PROP. XX, XXI, XXII, XXIII, XXIV. B. V. The Demonstrations of the 20. and 21. Propositions are shorter than those Euclid gives of easier Propositions, 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' useful as the Propofition, and the words “ any whatever" are supplied 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 History of the Royal Academy of Sciences of Berlin, Ann. 1745. page 50. the 23. Prop. of the 5. Book, is censured as being obscurely enuntiated, and, because of this, prolixly demonstrated. the Enuntiation there given is not Euclid's but Tacquets, as he acknowledges, which, tho' not fo well exprefled, is, upon the matter; the same with that which is now in the Elements. Nor is there any thing obscure in it, tho' the Author of the paper has fet down the proportionals in a disadvantageous order, by which it appears to be obscure. but no doubt Euclid enuntiated this 23. as well as the 22. so 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 case the Enuntiation which Naudaeus gives, cannot be well anplied.

The Demonstration 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 said, that in this case straight lines are to be taken which are proportional to the figures, the Demonstration 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 use of in the 5. Book ?

PROP. F, G, H, K. B. V. Thefe Propositions are annexed to the 5. Book, because they are frequently made use of by both antient and modern Geometers, and in many cases Compound ratios cannot be brought into Demonstrations, without making use of them.

Whoever desires to see 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 most readily agree to what the learned Dr. Barrow says *, " That there is nothing in the whole * Page 336.

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