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there cannot be taken a multiple of A which is the first that is Bcok V. greater than K, or Ho, because A itself is greater than it. upon m this account, the Author of this demonstration found it neceffary 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


2 taken that multiple of a which

I is the first that is greater than HO, or K, as was done in the



A first cafe. he likewise brings in

E this K into the demonstration of

HI both cases, without any reason,

E for it serves to no purpose but

Θ Β Δ Θ B to lengthen the demonstration. There is also a third case, which is not mentioned in this demonstra

а tion, 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, 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 necesary 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 same or equal, lejer 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 demonstration of the 10. Prop. which proceeds thus. “Let A have to C a greater ratio, then B to C. I Ay “ that A is greater than B. for if it is not greater, it is cither 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




Bock V. haŭ 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 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. Dcf. of B. 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 muliple of C. and this Proposition directly contra. dicts the preceeding; wherefore A is not equal to B. the demonfration of the 1o. Froposition goes on thus,“ but neither is A les

$u " than E, because then A would have a lels ratio to C, than B hus

a “ to it. but it has not a kís ratio, therefore A is not less than 5" &c. here it is said that" A would hare : less ratio to C, than B has

to C,” or, which is the same thing, iliat B wouid have a greater ratio to C, than A to C; that is, by 7. Dcf. B. 5. there must be fome cquimultiples of B and A, and some mul iple 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 cae the multiple of A is always greater than the multipic of C, wherever 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 fame thing, that A cannot have a less ratio to C, than B has to C. but this is not at all prored in the ro. Propofition; but if the 10. were once demonftated it would immediately follow from it; but cannot without it be cally dumonftratcd, as he that trics to do it will find. wherefore thie io. Propofition is not fufficiently demonftrated. and it seems that he who has given the demonftration of the 10. Proposition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what is manifest when underftooi of magnitudes, unto ratius, viz. that a magnitude cannot be both greater and less than another. That those things which are cqual to the fame are equal to one another, is a most evident Axiom whun understood of magnitades, yet Euclid does not make use of it to infur that those ratios which are the same to the fame ratio, are the line to one another; but explicitely demonstrates this in Prop. 1.5. the demonstraticn we have given of the 10. Prop. is

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no doubt the same with that of Eudoxus or Euclid, as it is imme- Book v. diately and directly derived from the Definition of a greater ratio, viz. the 7. of the 5.

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 demon

A CB C frated.

D F E F Let D, E be equimultiples of A, B, and F a 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 10. 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.

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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 20. and 21. 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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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.

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 Propofiti-
ons preceeding 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 5. Book. Euclid does not de-
monstrate it, nor does he thew how to find the fourth proportio-
nal, before the 12. Prop. of the 6. Book. and he never assumes any
thing in the demonstration of a Proposition, which he had not be-
fore demonstrated ; at least, he assumes nothing the existence of
which is not evidently poflible; for a certain conclufion can never
be deduced by the means of an uncertain Propofition. upon this
account we have given a legitimate Demonstration of this Propo-
sition 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 hc thought it too prolix. and as the 17. Prop. of which
this 18. is the converse, is demonstrated by help of the 1. and
2. Propositions of this Book, so in the demonstration now given of
the 18th, the 5. Prop. and both cases of the 6. are necessary, and
these two Propositions are the converses of the I. and 2. Now the
5. and 6. 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 Pro-
position of the Elements, except in this 18. 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 con-
verse of that of the 17. as it ought to be, differs nothing from that
of Eudoxus or Euclid. for the 5. and 6. have undoubtedly been
put into the 5. Book for the sake of fome Propositions in it, as all
the other Propositions about equimultiples have been.

Hieronymus Saccherius in his Book named Euclides ab omni
naevo vindicatus, printed at Milan Ann, 1733, in 4to, acknowledges

are of

this blemilh in the demonstration of the 18. and that he may re- Book V. move 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

one kind, and also the two others either of the fame kind so with the two first, or of some other the same kind with one ano“ ther. 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 66 fecond B.”

And after two Propositions premised as Lemmas, he proceeds thus.

“ Either among all the possible cquimultiples of the firit A, and “ of the third C, and, at the fame time among all the possible equis multiples of the fecond B, and of the fourth D, there can be found * some one maltiple EF of the firft A, and one IK of the second B, “ that are equal to one another; and also (in the fame 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 " cafe happen,

A " [i.e. if such e

E -F “ quality is to be B" found, ) it is


-K “ manifest from C


-H 66 what is before - demonstrated , D


M “ that A is to B,

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 lefser 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

rs ratio

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