Z

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

Z

1

HA

E

to no purpose but to lengthen BA

the demonftration. There is alfo

A

H

E

OBA OB A

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.

PRO P. IX. B. V.

Of this there is given a more explicit demonftration than that which is now in the Elements.

PROP. X. B. V.

It was necessary 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, leffer 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 Ca 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 reafoning 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 alfo greater than that of C. but from the Hypothesis that A has a greater ratio to C, than B has 10 C, there must, by the 7. Def. of B. 5. be certain equimultiples of A and B, and fome 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 fame multiple of C. and this Propofition directly contradicts the preceding; wherefore A is not equal to B. the demonstration of the 10. Propofition goes on thus," but neither is A lefs than B, becaufe then A would have a lefs ratio to C, than B has to it. but it has not a lefs ratio, therefore A is not less 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 cafe 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 demonftrated 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 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 eafily demonftrated, as he that tries to do it will find. wherefore the 10. Propofition is not fufficiently demonftrated. and it seems that he who has given the demonftration 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 manifest when underftood of magnitudes, unto ratios, viz. that a magnitude cannot be 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 ufe of it to infer that thofe ratios which are the fame to the fame ratio, are the fame to one another; but explicitely demonstrates this in Pop. 11. of B. 5. the demonftration we have given of the 10.

313

Prop. is no doubt the fame with that of Eudoxus or Euclid, as it Book V. is immediately and directly derived from the Definition of a great-n er ratio, viz. the 7. of the 5.

The above-mentioned Propofition, viz. If A have to Ca greater ratio than B to C, and if of A and B there be taken certain equimultiples, and fome multiple of C, then if the multiple of B be greater than the multiple of C, the multiple of A is alfo greater than the fame, is thus demonftrated.

Let D, E be equimultiples of A, B, and F a multiple of C, fuch, that E the multiple of B is greater than F; D the multiple of A is alfo greater than F.

A CB C

DFE 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 fame 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 Tranflations, at the beginning of this demonstration, it is faid, " 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 tranflation from the Greek. but the fense evidently requires that it be read," fo 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 reftored 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 Propofition.

PRO P. XIV. B. V.

The two cafes of this which are not in the Greek are added; the demonftration of them not being exactly the fame with that of the firft cafe.

Book V.

PROP. XVII. B. V.

The order of the words in a claufe of this is changed to one more natural. as was alfo 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 Hypothefis, that to any three magnitudes, two of which, at least, are of the fame kind, there may be a fourth proportional; which if not proved, the Demonftration now in the text, is of no force. but this is affumed without any proof, nor can it, as far as I am able to difcern, be demonstrated by the Propofitions preceding this; fo far is it from deferving 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 demonftrate it, nor does he fhew how to find the fourth proportional, before the 12. Prop. of the 6. Book. and he never affumes any thing in the demonftration of a Propofition, which he had not be fore demonftrated; at leaft, he affumes nothing the existence of which is not evidently poffible; 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 Propofition instead of that in the Greek and other editions, which very probably Theon, at least fome other has put in the place of Euclid's, because he thought it too prolix. and as the 17. Prop. of which this 18. is the converfe, is demonftrated by help of the 1. and 2. Propofitions of this Book, fo in the demonstration now given of the 18th, the 5. Prop. and both cafes of the 6. are neceffary, and these two Propofitions are the converses of the 1. 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 ufe in any Propofition of the Elements, except in this 18. and this is a manifeft proof that Euclid made ufe of them in his demonftration of it, and that the demonstration now given, which is exactly the converfe 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 fake of fome Propofitions in it, as all the other Propofitions about equimultiples have been.

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

this blemish in the demonftration 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 demonftrate the following Propofition, which is in page 115 of his Book, viz.

"Let A, B, C, D be four magnitudes, of which the two first 66 are of one kind, and alfo the two others either of the fame kind "with the two first, or of fome other the fame kind with one "another. I fay the ratio of the third C to the fourth D, is ei"ther equal to, or greater, or less than the ratio of the first A to "the fecond B."

And after two Propofitions premised as Lemmas, he proceeds

thus.

"Either among all the poffible equimultiples of the first A, and "of the third C, and, at the fame time among all the poffible equi"multiples of the fecond B, and of the fourth D, there can be found "fome one multiple EF of the firft A, and one IK of the second B, "that are equal to one another; and alfo (in the fame cafe) fome "one multiple GH of the third C equal to LM the multiple of the "fourth D, or fuch equality is no where to be found. If the first "cafe happen, [i.e.

"if fuch equality A.

E

-K

H

M

"fimultaneous equality be not to be found upon both fides, it will "be found either upon one fide, as upon the fide of A [and B;] " or it will be found upon neither fide; if the first happen; there"fore (from Euclid's Definition of greater and leffer ratio fore

going) A has to B, a greater or lefs ratio than C to D; accor"ding as GH the multiple of the third C is lefs, or greater than "LM the multiple of the fourth D. but if the fecond cafe hap

σε

pen; therefore upon the one fide, as upon the fide of A the first "and B the fecond, it may happen that the multiple EF, [viz.of the "firft] may be lefs than IK the multiple of the fecond, while on "the contrary, upon the other fide, [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 fame Definition of Euclid) the