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book 5, they are also proportionals according to the 20th def. of book 7.

First, if A be to B, as C to D; then if A be any multiple or part of B, C is the same multiple or part of D, by prop. D, of book 5.

Next, if AB be to CD, as EF to GH; then if AB contains any parts of CD, EF contains the same parts of GH: for let CK be a part of CD, and GL the same part of GH, and let AB be a multiple of CK; EF is the same multiple of GL; take M the same multiple of GL that AB is of CK; therefore by prop. C, of book 5, AB is to CK, as M to GL; and CD, GH are equimultiples of CK, GL: wherefore by cor. prop. 4, b. 5, AB is to CD, as M to GH. And, by the hypothesis, AB is to CD as EF to GH: therefore M is equal to EF, by prop. 9, book 5, and consequently EF is the same multiple of GL that AB is of CK.

PROP. D. B. V.

This is not unfrequently used in the demonstration of other propositions, and is necessary in that of prop. 9, b. 6. It seems Theon has left it out for the reasons mentioned in the notes of prop. A.

PROP. VIII. B. v.

In the demonstration of this, as it is now in the Greek, there are two cases, (see the demonstration in Hervagius, or Dr. Gregory's edition) of which the first is that in which AE is less than EB; and in this it necessarily follows, that Ho the multiple of EB is greater than ZH, the same multiple of AE, which last multiple, by the construction is greater than A; whence also HO must be greater than A. But in the second case, viz. that in which EB is less than AE, though ZH be greater than A, yet He may be less than the same A; so that there cannot be taken a multiple of a which is the first that is greater than K, or Ho because A itself is greater than it; upon this account the author of this demonstration found it necessary to change one part of the construction that was made use of in the first case: but he has, without any necessity, changed also another part of it, viz. when he orders to take N that multiple of A which is the first that is greater than ZH; for he might have taken Z that multiple of A which is the first I 1 that is greater than HO, or K, as was done in the first case: he vi likewise brings in this K into the demonstration of both cases, without any reason ; for it serves to no purpose but to lengthen the demonstration. There is also a third case, which is not mentioned in © B Å

BÅ this demonstration, viz. that in which AE in the first, or EB in the second of the two other cases, is greater than D; and in this any equimultiples, as the doubles, of AE, KB are to be taken, as is done in this edition, where all the cases are at once demonstrated : and

from this it is plain that Theon, or some other unskilful editor has vitiated this proposition.

PROP. IX. B. V. of this there is given a more explicit demonstration than that which is now in the Elements.

PROP. X. B. V.

It was necessary to give another demonstration of this proposition, because that which is in the Greek and Latin, or other editions, is not legitimate : for the words greater, the same, or equal, lesser, have a quite different meaning when applied to magnitudes and ratios, as is plain from the 5th and 7th definitions of book 5. By the help of these let us examine the demonstration of the 10th prop. which proceeds thus: “Let A have to C a greater ratio than B to C: I say that A is greater than B. For if it is not greater, it is either equal, or less. But A cannot be equal to B, because then each of them would have the same ratio to C: but they have not. Therefore 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 isnot equal to B. The demonstration of the 10th prop. goes 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 " 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 the 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 have 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, unto 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. 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.

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 great. er than the same, is thus demonstrated.

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

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.

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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 show how to find the fourth proportional, before the 12th prop. of the sixth 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, bas 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 first and second 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 converse 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 his book named Euclides ab omni navo vindicatus, printed at Milan, anno 1733, in 4to, acknowledges 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 si, e. if such equality is to be found it is mani. A

E- - F fest from what is before demonstrated, B

1

K that A is to B, as C to D; but if such si- C- G

- H multaneous equality be not to be found D

M 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 C1 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.

“ 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. b. 5.

The same objection holds good against the demonstrations 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 edi

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