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the 14th and 16th Prop. B. 5, of his edition, and is followed Book v. by Herigon in Schol. i, Prop. 14, B. 5, as if there was any nature of proportionals antecedent to that which is to be derived and understood from the definition of them; and indeed though it is very easy to give a right demonstration of it, nobody, as far as I know, has given one, except the learned Dr Barrow, who, in answer to Borelluss objection, demonstrates it indirectly, but very briefly and clearly, from the 5th definition in the 322d page of his Lect. Mathem., from which definition it may also be easily demonstrated directly : On which account we have placed it next to the propositions concerning equimultiples.

PROP. B. B. V.

This also is easily deduced from the fifth def. B. 5, and therefore is placed next to the other; for it was very ignorantly made a corollary from the 4th Prop. of this Book. See the note on that corollary.

PROP. C. B. V.

This is frequently made use of by geometers, and is necessary to the 5th and 6th Propositions of the 10th Book. Clavius, in his notes subjoined to the 8th def. of Book 5, demonstrates it only in numbers, by help of some of the propositions of the 7th book : in order to demonstrate the property contained in the 5th definition of the 5th book, when applied to numbers, from the property of proportionals contained in the 20th def. of the 7th book : And most of the commentators judge it difficult to prove that four magnitudes which are proportionals according to the 20th def. of 7th book, are also proportionals according to the 5th def. of 5th book. But this is easily made out as follows:

First, if A, B, C, D, be four magnitudes, such that A is the same mul

E tiple, or the same part of B, which B C is of D; A, B, C, D, are propor

H

D tionals: This is demonstrated in Proposition C.

Secondly, if A B contain the same K! L parts of CD that EF does of GH; in this case likewise AB is to CD, as EF to GH.

A CE G

Book V.

Let CK be a part of CD, and GL the same part of GH,
and let AB be the same multiple of
CK, that EF is of GL: Therefore,

E
by Prop. C. of 5th Book, AB is to B
CK, as EF to GL: And CD, GH

H

D are equimultiples of CK, GL the second and fourth; wherefore, by Cor. Prop. 4, B. 5, AB is to CD, as EF KH Lt to GH.

And if four magnitudes be proportionals according to the 5th def. of

A CE G 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 B. 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 B. 5, A B 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 reason mentioned in the notes at 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 A E, which last multiple, by the construction, is greater than A, whence also Ho must be greater than : But in the second case, viz. that in which EB is less than A E, though ZH be

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greater than 4, yet Ho may be less than the same A; so that Book V. 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

Z

Z than ZH; for he might have

1

2 taken that multiple of A, which is the first that is

A greater than Ho or K, as was done in the first case : He likewise brings in this E H K into the demonstration of

E both cases, without any reason; for it serves no purpose BAOB

© В А but to lengthen the demonstration. There is also a third case which is not mentioned in this demonstration, viz. that in which A E 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, EB 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, less, 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 6 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

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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, tben 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 6 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 tban 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 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, 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 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. А С ВС

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 X 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.

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