Book V. like it; whereas there is not the least difference between the two demonstrations, except a single word in the construction, which very probably has been owing to an unskilful Librarian. Clavius likewise gives, both the ways, but neither he nor Peletarius takes notice of the reason why one is preferable to the other. PROP. VI. B. V. There are two cases of this Proposition, of which only the first and simplest is demonstrated in the Greek. and it is probable Theon thought it was sufficient to give this one, since he was to make use of neither of them in his mutilated edition of the 5th Book ; and he might as well have left out the other, as also the 5. Proposition for the same reason. the demonstration of the other case is now added, becaufe both of them, as also the 5. Proposition, are necessary to the demonstration of the 18. Prop. of this Book. the translation from the Arabic gives both cases briefly. PROP. A. B. V. This Proposition is frequently used by Geometers, and it is necessary in the 25. Prop. of this Book, 31. of the 6. and 34. of the 11. and 15. of the 12. Book. it seems to have been taken out of the Elements by Theon, because it appeared evident enough to him, and others who substitute the confused and indistinct idea the vulgar have of proportionals, in place of that accurate idea which is to be got from the 5. Def. of this Book. Nor can there be any doubt that Eudoxus or Euclid gave it a place in the Elements, when we Tee the 7. and 9. of the same Book demonstrated, tho' they are quite as cafy and evident as this. Alphonsus Borellus takes occasion from this Proposition to censure the 5. Definition of this Book very severely, but most unjustly. in page 126. of his Euclid restored, printed at Pisa in 1658. he fays, “ Nor can even this leåst degree “ of knowledge be obtained from the foresaid property,” viz. that which is contained in 5. Def. 5. “ That if four magnitudes be “ proportionals, the third must necessarily be greater than the “ fourth, when the first is greater than the second; as Clavius ac knowledges in the 16. Prop. of the 5. Book of the Elements." But tho' Clavius makes no such acknowledgement expressly, he has given Borellus a handle to say this of him, because when Clavius in the above-cited place censures Commandine, and that very justly, for demonstrating this Proposition by help of the 16. of the 5. yet he himself gives no demonstration of it, but thinks it plain Book v. from the nature of Proportionals, as he writes in the end of the 14. and i6. Prop. B. 5. of his edition, and is followed by Herigon in Schol. 1. Prop. 14. B. 5. as if there was any nature of Propore tionals antecedent to that which is to be derived and understood from the Definition of them. and indeed, tho' it is very ealy to give a right demonftration of it, no body, as far as I know, has given one, except the learned Dr. Barrow, who, in answer to Borellus's objection, demonstrates it indirectly, but very briefly and clearly from the 5. Definition, in the 322 page of his Lect. Mathem. from which Definition it may also easily be demonstrated directly. on which account we have placed it next to the Propositions concerning equimultiples. PROP. B. B. V. This also is easily deduced from the 5. Def. B. 5. and therefore is placed next to the other, for it was very ignorantly made a Corollary from the 4. Prop. of this Book. See the note on that Corollary. PROP. C. B. V. This is frequently made use of by Geometers, and is necessary to the 5 and 6. Propofitions of the 1o. Book. Clavius in his Notes fubjoined to the 8. Def. of Book 5. demonstrates it only in nunbers, by help of some of the Propofitions of the 7. Book, in order to demonítrate the property contained in the 5. Definition of the 5. Book, when applied to numbers, from the property of Proportionals contained in the 20. Def. of the 7. Book. and most of the Commentators judge it difficult to prove that four magnitudes, which are proportionals according to the 20. Def. of 7. B. are also pro. portionals according to the 5. Def. of s. Book, but this is easily made out, as follows.. First, If A, B, C, D be four mag F pitudes, such that A is the fame mul. B tiple, or the same part of B, which C is of D; A, B, C, D are proportionals, this is demonstrated in Pro KH position C. Secondly, If AB contain the same А A CGM parts of. CD that EF does of GH; in this case likewise AB is to CD, as EF to GH. Book V. Let CK be a part of CD, and GL the fame part of GH; and let F B H KH It ACE GM cording to the 20. Def. of. B. 7. First, If A be to B, as C to D; then if A be any multiple or any: parts of CD, EF contains the fame parts of GH. for let CK be a part of CD, and GL the same part of GH, and let AB be a multiple of CK; EF is the same multiple of GL. Take M the same multiple of GL that AB is of CK; therefore by Prop. C. of B. 5. AB is to CK, as M to GL; and CD, GH are equimultiples of CK, GL; wherefore by Cor. Prop. 4. B. 5. AB is to CD, as M to GH. and, by the Hypothesis, AB is to CD, as EF to GH; therefore M is equal to EF by Prop. 9. B. 5. and consequently EF is the same multiple of GL that AB is of CK. PROP, D. B. V. This is not unfrequently used in the demonstration of other Propositions, and is necessary in that of Prop. 9. B. 6. it seems Theon has left it out for the reason mentioned in the Notes at Prop. A. PROP. VIII. B. V. In the demonstration of this, as it is now in the Greek, there are two cases, (see the demonstration in Hervagius, or Dr. Gregory's edition) of which the first is that in which AE is less than EB; and in this, it necessarily follows that HỌ the multiple of EB is greater than ZH the same multiple of AE, which last multiple, by the cons struction, is greater than A; whence also HỘ must be greater than A. but in the second case, viz. that in which EB is less than AE, tho? ZH be greater than A, yet HỘ may be less than the fame A; so that 2 H+ A there cannot be taken a multiple of A which is the first that is Book V. greater than K, or HO, because A itself is greater than it. upon 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 cafe. but he has, without any neceflity, 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 Z ZH; for he might have taken Z that multiple of a which is the 1 first that is greater than HO, or А. K, as was done in the first case. he likewise brings in this K into the demonftration of both cafes, E HH without any reason, for it serves to no purpose but to lengthen © В. A B A the demonftration. There is also a third cafe, which is not mentioned in this demonftration, viz. that in which AE in the firft, or EB in the second of the two other cafes, is greater than Di 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. EH 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 necessary to give another demonstration of this Proposition, 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 demonstration of the 10. 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 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 diis, 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 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. 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 demona stration 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, tħan 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 should 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 1o. 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 1o. Proposition is not sufficiently demonstrated. and it seems that he who has given the demonstration of the ro. Propofition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what was manifest when undera: stood of magnitudes, unto ratios, viz. that a magnitude cannot bę both greater and less than another. That those things which are equal to the same 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 those ratios which are the same to the same ratio, are the same to one another; but explicitely demonftrates this in Prop. II, of B.5. the demonftration we have given of the 19. |