Book v. Books; which, except a few, are easily enough understood from the Propositions of this Book where they are first mentioned. they seem to have been added by Theon or some other. However it be, they are explained fomething more distinctly for the sake of learners. PROP. IV. B. V. In the construction, preceding the demonstration of this, the words a ČTuxt, any whatever, are twice wanting in the Greek, as also in the Latin translations; and are now added, as being wholly necessary. Ibid. in the demonstration; in the Greek, and in the Latin tranflation of Commandine, and in that of Mr. Henry Briggs, which was published at London in 1620, together with the Greek text of the first fix Books, which translation in this place is followed by Dr. Gregory in his edition of Euclid, there is this sentence following, viz." and of A and C have been taken equimultiples K, L; “ and of B and D, any equimultiples whatever (ő éruxe) M, N;" which is not true. the words “any whatever” ought to be left out. and it is strange that neither Mr Briggs, who did right to leave out these words in one place of Prop. 13. of this Book, nor Dr. Gregory who changed them into the word “ some" in three places, and left them out in a fourth of that same Prop. 13. did not also leave them out in this place of Prop. 4. and in the fecond of the two places where they occur in Prop. 17. of this Book, in neither of which they can stand consistent with truth. and in none of all these places, even in those which they corrected in their Latin tranflation, have they cancelled the words ä ituye in the Greek text, as they ought to have done. The same words á é Tuxe are found in four places of Prop. II. of this Book, in the first and last of which, they are necessary, but in the second and third, tho' they are true, they are quite superfluous; as they likewise are in the second of the two places in which they are found in the 12. Prop. and in the like places of Prop. 22, 23. of this Book. but are wanting in the last place of Prop. 23. as also in Prop 25. B. 11. COR. PRO P. IV. B. V. This Corollary has been unskilfully annexed to this Proposition, and has been made instead of the legitimate demonstration which without doubt Theon, or some other Editor has taken away, not Book v. from this, but from its proper place in this Book. the Author of it designed to demonstrate that if four magnitudes E, G, F, H be proportionals, they are also proportionals inversely; that is, G is to E, as H to F; which is true, but the demonstration of it does not in the least depend upon this 4. Prop. or its demonstration. for when he says “ because it is demonstrated that if K be greater “ than M, L is greater than N,” &c. this is indeed shewn in the demonstration of the 4. Prop. but not from this that E, G, F, H áre proportionals, for this last is the conclusion of the Proposition. Wherefore these words “ because it is demonstrated,” &c. are wholly foreign to his design. and he should have proved that if K be greater than M, L is greater than N, from this, that E, G, F, H are proportionals, and from the 5. Def. of this Book, which he has not; but is done in Proposition B, which we have given, in its proper place, instead of this Corollary, and another Corollary is placed after the 4. Prop. which is often of use, and is neceffary to the Demonstration of Prop. 18. of this Book. PROP. V. B. V. In the construction which precedes the demonstration of this Proposition, it is required that EB may be the same multiple of CG, that AE is of CF; that is, that EB be divided into as many equal parts, as there are parts in AE equal to CF. from which it is evident that this construction is not Euclid's. for he does not fhew the way of dividing straight lines, and far less other magnitudes, into any number of equal parts, until the y. Proposition of B.6. and he never requires any thing to be done in the contruction, of which he had not before given the method of doing. for this reason we have changed the construction to A one which without doubt is Euclid's, in which nothing GO is required but to add a magnitude to itself a certain E number of times. and this is to be found in the tran C flation from the Arabic, tho' the enunciation of the F Proposition and the demonstration are there very much spoiled. Jacobus Peletarius who was the first, as far B DI as I know, who took notice of this error, gives allo the right construction in his edition of Euclid, after he had given the other which he blames. he fays he would not leave it out, because it was fine, and might sharpen one's genius to invent others Book v. like ir; 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 likewife gives both the ways, but neither he nor Peletarius takes notice of the reason why one is preferable to the other. PRO P. 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 Theoa 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, because both of them, as also the 5. Proposition, are necefsary to the demonstration of the 18. Prop. of this Book. the translation from the Arabic gives both cafes brichiy. 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 ri. and 15. of the 1 2. 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 see the 7. and 9. of the fame Book demonftrated, tho' they are quite as easy 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 says, “ Nor can even this least 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 Borelius a handle to say this of him, because when Clavius in the above-cited place censures Commandine, and that very jastly, 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 16. 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 Proportionals antecedent to that which is to be derived and understood from the Definition of them. and indeed, tho' it is very easy to give a right deinonstration 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 3 2 2 page of his Lect. Mathem. from which Definition it may also be easily demonstrated directly. on which account we have placed it next to the Propositions concerning equimultiples. PRO P. 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. PRO P. C. B. V. This is frequently made use of by Geometers, and is necessary to the 5. and 6. Propositions of the ro. Book. Clavius in his Notes fubjoined to the 8. Def. of Book 5. demonstrates it only in numbers, by help of some of the Propositions of the 7. Book, in order to demonstrate 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 Coinmentators judge it difficult to prove that four magnitudes which are proportionals according to the 20. Def. of 7. B. are also proportionals according to the 5. Def. of 5. Book. but his is easily made out, as follows. First, If A, B, C, D be four mag T nitudes, such that A is the same mul-B tiple, or the same part of B, which H DI C is of D; A, B, C, D are proportionals. this is demonstrated in Pro KH position c. Secondly, If AB contain the same parts of CD that EF does of GH; A CE GM 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 AB be the same multiple of CK, that EF is of GL. therefore by F H DI And if four magnitudes be pro KH L , ac- A C F GM cording to the 20. Def. of B. 7. Firft, If A be to B, as C to D; then if A be any multiple or part of B, C is the fame multiple or part of D, by Prop. D. of B. 5: Next, If AB be to CD, as EF to GH; then if AB contains any parts of CD, EF contains the fame parts of GH. for let CK be a part of CD, and GL the fame 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. PRO P. 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. 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 Ho the multiple of EB is greater than ZH the same multiple of AE, which last multiple, by the conItruction, is greater than A; whence also HA must be greater than a. but in the fecond cate, viz. that in which EB is less than AE, tho' ZH be greater than 4, yet H may be less than the fame A; fo that |