H 2 I A A E H E+ BA OBA greater than K, or He, because A itself is greater than it: Up- PROP. 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 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, leffer, have a quite different meaning when applied to magnitudes and ratios, as is plain from the 5th and 7th definitions of Book 5. By the help of these let us examine the demonstration of the 10th prop. 'which proceeds thus: "Let A "have to C a greater ratio, than B to C: I say 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. There"fore A is not equal to B." The force of which reasoning is this, if A 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 Book V. Book V. multiple whatever of C; if the multiple of A be greater than the multiple of C, then, by the 5th def. of Book 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 7th def. of Book 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 preceding; wherefore A is not equal to B. The demonftration of the 10th prop. goes on thus: "But nei"ther is A lefs than B; because then A would have a less ra"tio to C, than B has to it: But it has not a lefs ratio, there"fore 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 7th def. Book 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 10th propofition; but if the 10th were once demonftrated, it would immediately follow from it, but cannot without it be easily demonstrated, as he that tries to do it will find. Wherefore the 10th propofition is not fufficiently demonftrated. And it seems that he who has given the demonftration of the 10th propofition as we now have it, inftead of that which Eudoxus or Euclid had given, 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 understood 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 explicitly demonftrates this in prop. 11. of Book 5. The demonftration we have given of the toth prop. is no no doubt the fame with that of Eudoxus or Euclid, as it is im- Book V. mediately 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 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 Fa multiple of C, fuch, 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 10th 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. A CBC PROP. XIII. B. V. In Commandine's, Briggs's, and Gregory's tranflations, at the beginning of this demonftration, it is faid, " And the multi"ple of C is greater than the multiple of D; but the multi"ple 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 neceffary to the 20th and 21ft Prop. of this book, and is as ufeful as the propofition. PROP. 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 first cafe. Book V. PROP. XVII. B. V. The order of the words in a clause of this is changed to one more natural: As was alfo done in prop. 11, 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 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 5th book. Euclid does not demonftrate it, nor does he fhew how to find the fourth proportional, before the 12th prop. of the 6th book: And he never affumes any thing in the demonftration of a propofition, which he had not before 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 demonftration of this propofition inftead of that in the Greek and other editions, which very probably Theon, at leaft fome other, has put in the place of Euclid's, because he thought it too prolix: And as the 17th prop. of which this 18th is the converse, is demonftrated by help of the ft and 2d propofitions of this book, fo, in the demonftration now given of the 18th, the 5th prop. and both cafes of the 6th are neceffary, and these two propostions are the converses of the 1ft and 2d. Now the 5th and 6th 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 18th, and this is a manifeft proof, that Euclid made ufe of them in his demonftration of it, and that the demonftration now given, which is exactly the converfe of that of the 17th, as it ought to be, dif. fers nothing from that of Eudoxus or Euclid: For the 5th and 6th have undoubtedly been put into the 5th 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, ac knowledges knowledges this blemish in the demonftration of the 18th, and Book V. that he may remove 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 are of the one kind, and alfo the two others either of the "fame kind with the two first, or of some other the same "kind with one another. I fay 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 fecond B." 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 equimultiples 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 first cafe happen, A. "[i. e. if fuch "equality is to E -F "be found] it is "A is to B, 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; 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 happen; therefore upon the one fide, as upon the fide "of A the firft and B the fecond, it may happen that the "multiple EF, [viz. of the firft] may be less than IK the "multiple of the fecond, while, on the contrary, upon the o"ther 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 ra X 2 "tio |