no doubt the same with that of Eudoxus or Euclid, as it is immediately and directly derived from the definition of a greater ratio, viz. the 7th of the 5th.

The above mentioned proposition, 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 also greater than the same, is thus demonstrated.

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 10th prop. book 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.

D F E F

In Commandine's, Brigg's, and Gregory's translations, at the beginning of this demonstration, it is said, "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 translation 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 20th and 21st prop. of this book, and is as useful as the proposition.

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.

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. 1.

PROP. XVIII. B. V.

The demonstration of this is none of Euclid's, nor is it legi timate; 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 propositions 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 5th book. Euclid does not demonstrate it, nor does he show how to find the fourth proportional, before the 12th prop. of the 6th book: and he never assumes any thing in the demonstration of a proposition, which he had not before demonstrated at least, 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 17th prop. of which this 18th is the converse, is demonstrated by help of the 1st and 2d propositions of this book; so, in the demonstration now given of the 18th, the 5th prop. and both cases of the 6th are necessary, and these two propositions are the converses of the 1st and 2d. Now the 5th and 6th do not enter into the demonstration of any proposition in this book as we now have it: nor can they be of use in any proposition of the Elements, except in this 18th, 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 17th, as it ought to be, dif fers nothing from that of Eudoxus or Euclid for the 5th and 6th have undoubtedly been put into the 5th book for the sake of some propositions in it, as all the other propositions about equimultiples have been.

Hieronymus Saccherius, in his book named, Euclides ab omni nævo vindicatus, printed at Milan, anno 1733, in 4to, ac

knowledges this blemish in the demonstration of the 18th, and that he may remove it, and render the demonstration we now have of it legitimate, he endeavours to demonstrate the following proposition, which is in page 115 of his book, viz.

"Let A, B, C, D be four magnitudes, of which the two first are of the one kind, and also the two others either of the same kind with the two first, or of some other, the same kind with one another. I say the ratio of the third C to the fourth D, is either equal to, or greater, or less than the ratio of the first A to the second B."

And after two propositions premised as lemmas, he proceeds thus:

"Either among all the possible equimultiples of the first A, and of the third C, and at the same time, among all the possible equimultiples of the second B, and of the fourth D, there can be found some one multiple EF of the first A, and one IK of the second B, that are equal to one another; and also, in the same case, some one multiple GH of the third C equal to LM the multiple of the fourth D, or such equality is no where to be found. If the first case happen, [i. e. if Asuch equality is

to be found] it

is manifest from

E

B

I

-F

-K

-H

-M

as C to D; but if such simultaneous equality be not to be found upon both sides, it will be found either upon one side, as upon the side of A [and B;] or it will be found upon neither side; if the first happen; therefore (from Euclid's definition of greater and lesser ratio foregoing) A has to B a greater or less ratio than C to D; according as GH the multiple of the third C is less, or greater than LM the multiple of the fourth D: but if the second case happen; 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 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 same definition of Euclid) the ratio of the first A to the second B, is less than the ratio of the third C to the fourth D; or on the contrary.

S $

"Therefore the axiom [i. e. the proposition before set down] remains demonstrated," &c.

if

Not in the least; but it remains still undemonstrated: for what he says may happen, may, in innumerable cases, never happen; and therefore his demonstration does not hold: for example, 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 than a multiple of B, the mul tiple of C can be less than the multiple of D, nor when the multiple of A is less than that of B, the multiple of C can be greater than that 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 5th def. b. 5.

The same objection holds good against the demonstration which some give of the 1st prop. of the 6th book, which we have made against this of the 18th prop. because it depends upon the same 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 shows that the 5th book has been vitiated by editors who were not geometers: for the conversion of ratios does not depend upon this 19th, 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 demonstration of it, which we have put in proposition E; but he makes it a corollary from the 19th, and begins it with the words, "Hence it easily follows," though it does not at all follow from it.

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

The demonstrations of the 20th and 21st 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 22d and 23d propositions are, as they ought to be, extended to any number of

magnitudes and, in like manner may the 24th be, as is taken notice of in a corollary; and another corollary is added, as useful as the proposition, and the words "any whatever" are supplied near the end of prop. 23. which are wanting in the Greek text, and the translations from it.

In a paper writ by Philippus Naudæus, and published after his death, in the History of the Royal Academy of Sciences of Berlin, anno 1745, page 50, the 23d prop. of the 5th book is censured as being obscurely enunciated, and, because of this, prolixly demonstrated the enunciation there given is not Euclid's, but Tacquet's, as he acknowledges, which though not so well expressed, is, upon the matter, the same with that which is now in the Elements. Nor is there any thing obscure in it, though the author of the paper has set down the proportionals in a disadvantageous order, by which it appears to be obscure: but, no doubt, Euclid enunciated this 23d, as well as the 22d, so as to extend it to any number of magnitudes, which taken two and two are proportionals, and not of six only; and to this general case the enunciation which Naudæus gives, cannot be well applied.

The demonstration which is given of this 23d, 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 though his demonstration had been right, who does not see that it could not be made use of in the 5th book?

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

These propositions are annexed to the 5th book, because they are frequently made use of by both ancient and modern geometers and in many cases compound ratios cannot be brought into demonstration, without making use of them.

Whoever desires to see the doctrine of ratios delivered in this 5th book solidly 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 7th and 8th of the year 1666.

The 5th book being thus corrected, I most readily agree to what the learned Dr. Barrow says, "That there is nothing in the whole body of the Elements of a more subtile invention,

* See page 336.