there cannot be taken a multiple of ▲ which is the first 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 neceffary to change one part of the conftruction that was made ufe of in the first cafe. but he has, without any neceffity, changed alfo 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 taken that multiple of ▲ which is the first that is greater than HO, or K, as was done in the firft cafe. he likewife brings in this K into the demonftration of both cafes, without any reafon, for it ferves to no purpose but Z 2 I A A E H E H to lengthen the demonftration, BA OB There is alfo 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. Of this there is given a more explicit demonftration than that which is now in the Elements. PROP. X. B. V. It was neceffary to give another demonftration 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, lefer 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 Bock 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 Hypothefis that A has a greater ratio to C, than B has to C, there muft, 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 fame multiple of C. and this Propofition directly contradicts the preceeding; wherefore A is not equal to B. the demonfiration of the 10. Fropofition goes on thus," but neither is A les "than E, because 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 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 fhould 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 demonfurated it would immediately follow from it; but cannot without it be cafily demonftrated, as he that trics to do it will find. wherefore the 10. Propofition is not fufficiently demonftrated. and it feems that he who has given the demonftration of the 10. Propofition as we now have it, inflead of that which Eudoxus or Euclid had gi ven, has been deceived in applying what is manifeft 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 moft evident Axiom when underflood of magnitudes, yet Euclid does not make use of it to infer that thofe ratios which are the fame to the fame ratio, are the fame to one another; but explicitely demonftrates this in Prop. 11. of B. 5. the demonftration we have given of the 10. Prop. is no no doubt the fame 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 Propofition, 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 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 demonfrated. 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. 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 A CB C D FE FE F 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 demonftration, it is faid, "And the multiple of C is greater than the multiple of D; but the multiple of E is not 66 66 greater than the multiple of F," which words are a literal tranflation from the Greek. but the fenfe 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 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 neceffary to the 20. and 21. Prop. of this Book, and is as useful as the Propofition. PROP. XIV. B. V. The two cafes of this which are not in the Greek are added; the demonstration of them not being exactly the fame with that of the firft cafe. PROP. 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. PRO P. XVIII. B. V. The demonftration 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 demonftrated by the Propofitions preceeding 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 before demonstrated; at leaft, he affumes 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 Demonftration of this Propofition inftead 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 converfe, is demonftrated by help of the 1. and 2. Propofitions of this Book, fo in the demonftration now given of the 18th, the 5. Prop. and both cafes of the 6. are neceffary, 'and these two Propofitions are the converfes of the 1. and 2. Now the 5. and 6. do not enter into the demonftration of any Propofition in this Book as we now have it, nor can they be of use in any Propofition of the Elements, except in this 18. and this is a manifeft proof that Euclid made ufe of them in his demonstration of it, and that the demonftration 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 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 this blemish in the demonftration of the 18. and that he may re- Book V. move it, and render the demonftration we now have of it legitimate, he endeavours to demonftrate the following Propofition, which is in page 115 of his Book, viz. 66 are "Let A, B, C, D be four magnitudes, of which the two first of one kind, and also the two others either of the fame kind "with the two firft, or of fome other the fame kind with one ano"ther. I fay the ratio of the third C to the fourth D, is either equal to, or greater, or lefs than the ratio of the first A to the "fecond B." 66 And after two Propofitions premifed 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 first A, and one IK of the fecond 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 firft "cafe happen, A "[i.e. if fuch e E as C to D. but if fuch 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; therefore (from Euclid's Definition of greater "and leffer ratio foregoing) A has to B, a greater or lefs ratio than "C to D; according 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 happen; therefore upon the one fide, as upon the fide 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 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 |