there cannot be taken a multiple of A which is the firft that is Book v. Z Z 2 1 H+ А. A E! H ET О В А С В А PRO P. IX. B. V. PRO P. X. B. V. U 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 10 C, there muli, by the 7. Def. 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 multiple of C. and this Proposition directly contradicts the preceding; wherefore A is not equal to B. the demonstration of the 10. Proposition goes on 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 7. Def. B. 5. there must be fome 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 fhould 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 10. Proposition ; but if the 10. 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 10. Proposition is not sufficiently demonstrated. 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 was 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 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 same ratio, are the same to one another ; but explicitely demonstrates this in 2:07. 11. of B. 5. the demonstration we have given of the 10. Prop. is no dcubt the same with that of Eudoxus or Euclid, as it Book V. is immediately and directly derived from the Definition of a greater rario, 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 muluple of C, the multiple of A is also greater than the same, is thus demon- À CB C strated. D F E F Let ), E be equimultiples of A, B, and Fa 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. thercfore 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. 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; s 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 Propofition. PRO P. 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 firft cafe. Book v. PRO P. 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. PRO P. 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 Propofitions 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 5. Book. Euclid does not demonstrate it, nor does he few how to find the fourth proportional, before the 12. Prop. of the 6. 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 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 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 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 use in any Proposition of the Elements, except in this 18. and this is a manifeft proof that Euclid made use of them in his demonftration of it, and that the demonstration now given, which is exactly the converse 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 some Propositions in it, as all the other Propositions about equimultiples have been. Hieronymus Saccherius in his Book named Euclides ab omni paevo vindicatus, printed at Milan Ann. 1733 in 4to, acknowledges are this blemish in the demonstration of the 18. and that he may re- Book v. move it, and render the deinonstration we now have of it legitimate, he endeavours to demonstrate the following Proposition, which is in page 15 of his Book, viz. “ Let A, B, C, D be four magnitudes, of which the two first of 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 ei“ther 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 equi“ multiples of the second B, and of the fourth D, there can be found “ fome 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 « cafe happen, [i.e. “ if such equality A E“ is to be found,] “it is manifeft from B I K « what is before de G -H “monstrated, that “ A is to B, as CD -L M “ to D. but if such “ fimultaneous 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; there“ fore (from Euclid's Definition of greater and lesser ratio fore going) A has to B, a greater or less ratio than C to D; accor ding as GH the multiple of the third C is less, or greater than « LM the multiple of the fourth D. but if the second cafe hap pen; 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 es the fourth D.) and then (from the fame Definition of Euclid) the |