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Book V. “ would have the same ratio to C; but they have not. There

“ fore A is not equal to B.” The force of which reasoning is this: If A had to C the same ratio that B has to C, then if any equimultiples whatever of A and B be taken, and any multiple whatever of C; if the multiple of A be greater than the multiple of C, then, by the 5th Def. of Book 5, the multiple of B is also greater than that of C: but, from the hypothesis that A has a greater ratio to C, than B has to C, there must, by the 7th Def. of Book 5, be certain equimultiples of A and B, and some multiple of C, such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the same multiple of C: And this proposition directly contradicts the preceding; wherefore A is not equal to B. The demonstration of the 10th Prop. goes on thus: “ But “ neither is A less than B; because then A would have a less “ ratio to C than B has to it: But it has not a less ratio, “ therefore A is not less than B,” &c. Here it is said, that 6 A would have a less ratio to C than B has to C,” or, which is the same thing, that B would have a greater ratio to C than A to C; that is, by 7th Def. Book 5, there must be some equimultiples of B and A, and some multiple of C, such that the multiple of B is greater than the multiple of C, but the multiple of A is not greater than it: And it ought to bave been proved, that this can never happen if the ratio of A to C be greater than the ratio of B to C; that is, it should have been proved, that, in this case, the multiple of A is always greater than the multiple of C, whenever the multiple of B is greater than the multiple of C; for when this is demonstrated, it will be evident that B cannot have a greater ratio to C, than A has to C, or, which is the same thing, that A cannot have a less ratio to C than B has to C; But this is not at all proved in the 10th proposition: but if the 10th were once demonstrated, it would immediately follow from it, but cannot without it be easily demonstrated, as he that tries to do it will find. Wherefore the 10th Proposition is not sufficiently demonstrated. And it seems that he who has given the demonstration of the 10th proposition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what is manifest, when understood of magnitudes, to ratios, viz. that a magnitude cannot be both greater and less than another. That those things which are equal to the same are equal to one another, is a most evident axiom when understood of magnitudes; yet Euclid does not make use of it to infer, that those ratios which are the same to the same ratio, are the same to one another; but explicitly demonstrates this in Prop. 11. of

Book 5. The demonstration we have given of the 10th Prop. Book V. is 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 Book.

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 demonstrated. À CB C

Let D, E be equimultiples of A, B, and F a multiple of C, such, that E the D F E F 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. 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 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.

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 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 shew how to find the fourth proportional, before the 12th Prop. of the 6th Book: And be 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, differs 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 this book named “ Euclides ab "omni nævo vindicatus,” printed at Milan ann. 1733, in 4to, acknowledges this blemish in the demonstration of the 18th;

66 if such equa

A

and that he may remove it, and render the demonstration we Book V. now have of it legitimate, he endeavours to demonstrate the following proposition, which is in page 115 of this 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 first two, or of some other the same 66 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 6 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 66 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.

E

F
“ lity is to be
found) it is
6 manifest from

B
I-

K " what is be

G.

H н
66 fore demon-
“ strated, that

D
-L

M
“ A is to B, as
6 C to D; but
66 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 less ratio foregoing) A has to B a greater or less ratio
66 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 hap-
“ pen that the multiple EF (viz, of the first] may be less than
“ IK the multiple of the second, while, on the contrary, upon
6 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
66 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,

C

Book V.

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

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, 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 than a multiple of B, the multiple 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. Book 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 which 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,

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