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Book V. like it; whereas there is not the leaft difference between the two

demonftrations, except a fingle word in the conftruction, which very probably has been owing to an unskilful Librarian. Clavius likewife gives, both the ways, but neither he nor Peletarius takes notice of the reason why one is preferable to the other.

PROP. VI. B. V.

There are two cafes of this Propofition, of which only the first and fimpleft is demonstrated in the Greek. and it is probable Theon thought it was fufficient to give this one, fince he was to make ufe of neither of them in his mutilated edition of the 4th Book; and he might as well have left out the other, as alfo the 5. Propofition for the fame reafon. the demonftration of the other cafe is now added, because both of them, as also the 5. Propofition, are neceffary to the demonftration of the 18. Prop. of this Book. the tranflation from the Arabic gives both cases briefly.

PROP. A. B. V.

This Propofition is frequently used by Geometers, and it is neceffary in the 25. Prop. of this Book, 31. of the 6. and 34. of the II. and 15. of the 12. Book. it seems to have been taken out of the Elements by Theon, because it appeared evident enough to him, and others who fubftitute the confused and indistinct idea the vulgar have of proportionals, in place of that accurate idea which is to be got from the 5. Def. of this Book. Nor can there be any doubt that Eudoxus or Euclid gave it a place in the Elements, when we fee the 7. and 9. of the fame Book demonftrated, tho' they are quite as eafy and evident as this. Alphonfus Borellus takes occafion from this Propofition to cenfure the 5. Definition of this Book very feverely, but most unjustly. in page 126. of his Euclid restored, printed at Pifa in 1658. he fays, " Nor can even this least degree "of knowledge be obtained from the forefaid property," viz. that which is contained in 5. Def. 5. "That if four magnitudes be "proportionals, the third must neceffarily be greater than the "fourth, when the first is greater than the fecond; as Clavius ac"knowledges in the 16. Prop. of the 5. Book of the Elements." But tho' Clavius makes no such acknowledgement expressly, he has given Borellus a handle to say this of him, because when Clavius in the above-cited place cenfures Commandine, and that very justly, for demonftrating this Propofition by help of the 16. of the

5. yet he himself gives no demonstration of it, but thinks it plain Book V. from the nature of Proportionals, as he writes in the end of the 14. and 16. Prop. B. 5. of his edition, and is followed by Herigon in Schol. 1. Prop. 14. B. 5. as if there was any nature of Propor tionals antecedent to that which is to be derived and understood from the Definition of them. and indeed, tho' it is very eafy to give a right demonftration of it, no body, as far as I know, has given one, except the learned Dr. Barrow, who, in anfwer to Borellus's objection, demonftrates it indirectly, but very briefly and clearly from the 5. Definition, in the 322 page of his Lect. Mathem. from which Definition it may also easily be demonftrated 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 5. Def. B. 5. and therefore is placed next to the other, for it was very ignorantly made a Corollary from the 4. Prop. of this Book. See the note on that Corollary.

PROP. C. B. V.

This is frequently made use of by Geometers, and is neceffary to the 5, and 6. Propofitions of the 10. Book. Clavius in his Notes fubjoined to the 8. Def. of Book 5. demonftrates it only in nunibers, by help of fome of the Propofitions of the 7. Book, in order to demonstrate the property contained in the 5. Definition of the 5. Book, when applied to numbers, from the property of Proportionals contained in the 20. Def. of the 7. Book. and most of the Commentators judge it difficult to prove that four magnitudes which are proportionals according to the 20. Def. of 7. B. are alfo pro portionals according to the 5. Def. of 5. Book. but this is eafily made out, as follows.

Firft, If A, B, C, D be four magnitudes, fuch that A is the fame mul-. tiple, or the fame part of B, which Cis of D; A, B, C, D are proportionals, this is demonftrated in Pro

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in this cafe likewife AB is to CD, as EF to GH.

Book V.

Let CK be a part of CD, and GL the fame part of GH; and let
AB be the fame multiple of CK, that EF is of GL. therefore by
Prop. C. of 5. Book, AB is to CK, as
EF to GL. and CD, GH are equi-

multiples of CK, GL the fecond and B
fourth; wherefore by Cor. Prop. 4.
B. 5. AB is to CD, as EF to GH.

And if four magnitudes be pro-
portionals according to the 5. Def. of
B. 5. they are alfo proportionals, ac-
cording to the 20. Def. of. B. 7.

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First, If A be to B, as C to D; then if A be any multiple or part of B, C is the fame 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 fame parts of GH. for let CK be a part of CD, and GL the fame part of GH, and let AB be a multiple of CK; EF is the fame multiple of GL. Take M thẹ fame multiple of GL that AB is of CK; therefore by Prop. C. of B. 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 Hypothefis, AB is to CD, as EF to GH; therefore M is equal to EF by Prop. 9. B. 5. and confequently EF is the fame multiple of GL that AB is of CK.

PROP. D. B. V.

This is not unfrequently used in the demonstration of other Propofitions, and is neceffary in that of Prop. 9. B. 6. it seems Theon has left it out for the reafon 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 cafes, (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 neceffarily follows that He the multiple of EB is greater than ZH the fame multiple of AE, which laft multiple, by the construction, is greater than A; whence also HQ must be greater than A. but in the second cafe, viz. that in which EB is lefs than AE, tho' ZH be greater than A, yet He may be less than the fame A; fo that

there cannot be taken a multiple of A which is the first that is Book V. greater than K, or HO, because A itself is greater than it. upon this account, the Author of this demonstration found it neceffary to change one part of the construction that was made use of in the firft cafe. but he has, without any neceffity, changed also another part of it, viz. when he orders to take N that multiple of A which is the firft that is greater than ZH; for he might have taken that multiple of A which is the firft that is greater than HO, or K, as was done in the first cafe. he likewife brings in this K into the demonftration of both cafes, without any reason, for it ferves

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the demonftration. There is also,

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a third cafe, which is not mentioned in this demonftration, viz. that in which AE in the firft, or EB in the fecond of the two other cafes, 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 cafes are at once demonftrated. and from this it is plain that Theon, or fome other unfkilful Editor has vitiated this Propofition.

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 neceffary to give another demonstration of this Propofition, because that which is in the Greek, and Latin, or other editions, is not legitimate. for the words greater, the fame or equal, lesser have a quite different meaning when applied to magnitudes and ratios, as is plain from the 5. and 7. Definitions of B. 5. by the help of these let us examine the demonftration of the 10. Prop. which proceeds thus. "Let A have to C a greater ratio, than B to C. I fay "that A is greater than B. for if it is not greater, it is either equal, " or lefs. but A cannot be equal to B, because then each of them "would have the fame ratio to C; but they have not. therefore "A is not equal to B." the force of which reasoning is this, if A

Book V. had to C, the fame 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 5. Def. of B. 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 7. Def. of B. 5. be certain equimultiples of A and B, and fome multiple of C fuch, 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 Propofition directly contradicts the preceding; wherefore A is not equal to B. the demonftration of the 10. Propofition goes on thus, "but neither is A lefs "than B, because then A would have a less ratio to C, than B has "to it. but it has not a lefs ratio, therefore A is not lefs than B," &c. here it is faid that " A would have a lefs ratio to C, than B has "to C," or, which is the fame thing, that B would have a greater ratio to C, than A to C; that is, by 7. Def. B. 5. there must be fome equimultiples of B and A, and fome multiple of C fuch, 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 lefs ratio to C, than B has to C. but this is not at all proved in the 10. Propofition; but if the 10. were once demonftrated it would immediately follow from it; but cannot without it be easily demonftrated, as he that tries to do it will find. wherefore the 10. Propofition is not fufficiently demonstrated. and it feems that he who has given the demonstration of the 10. Propofition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what was manifeft when under: stood of magnitudes, unto ratios, viz. that a magnitude cannot bẹ both greater and lefs than another. That thofe things which are equal to the fame 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 fame to the fame ratio, are the same to one another; but explicitely demonftrates this in Prop. 11. of B. 5. the demonftration we have given of the 10.

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