Book V. Books; which, except a few, are easily enough understood from the Propofitions of this Book where they are first mentioned. they feem to have been added by Theon or fome other. However it be, they are explained something more distinctly for the fake of

learners.

PROP. IV. B. V.

In the construction, preceding the demonstration of this, the words ἅ ἔτυχε, any whatever, are twice wanting in the Greek, as also in the Latin tranflations; and are now added, as being wholly neceffary.

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 six Books, which tranflation in this place is followed by Dr. Gregory in his edition of Euclid, there is this fentence following, viz. " and of A and C have been taken equimultiples K, L; " and of B and D, any equimultiples whatever (ἅ ἔτυχε) Μ, Ν;" 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 fame Prop. 1 3. did not alfo 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 confiftent with truth. and in none of all these places, even in those which they corrected in their Latin tranflation, have they cancelled the words & ἔτυχε in the Greek text, as they ought to have done.

The fame words & ἔτυχε are found in four places of Prop. 1 1. of this Book, in the first and last of which, they are neceffary, but in the fecond and third, tho' they are true, they are quite fuperfluous; as they likewife 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 alfo in Prop 25. B. 11.

COR. PROP. IV. B. V.

This Corollary has been unskilfully annexed to this Proposition, and has been made instead of the legitimate demonftration which without doubt Theon, or fome 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 alfo proportionals inversely; that is, Gis to E, as H to F; which is true, but the demonftration of it does not in the least depend upon this 4. Prop. or its demonstration. for when he says "because it is demonftrated that if K be greater "than M, L is greater than N," &c. this is indeed shewn in the demonftration of the 4. Prop. but not from this that E, G, F, H are proportionals, for this last is the conclufion of the Propofition. Wherefore these words " because it is demonftrated," &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 Propofition 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.

G

In the construction which precedes the demonftration of this Propofition, it is required that EB may be the fame 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 9. Propofition of B. 6. and he never requires any thing to be done in the conftruction, 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 is required but to add a magnitude to itself a certain E number of times. and this is to be found in the tranflation from the Arabic, tho' the enunciation of the Propofition and the demonftration are there very much fpoiled. Jacobus Peletarius who was the firft, as far B as I know, who took notice of this error, gives also the right construction in his edition of Euclid, after he had given the other which he blames. he says he would not leave it out, because it was fine, and might sharpen one's genius to invent others

C
F+

DI

Book V. like it; whereas there is not the least difference between the two demonftrations, 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.

PROP. VI. B. V.

There are two cases of this Propofition, of which only the first and simplest is demonstrated in the Greek. and it is probable Theon thought it was fufficient to give this one, since he was to make ufe 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 fame reafon. the demonstration of the other cafe is now added, because both of them, as also the 5. Proposition, are neceffary to the demonstration of the 18. Prop. of this Book. the tranflation from the Arabic gives both cafes bricfly.

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 fee the 7. and 9. of the fame Book demonftrated, tho' they are quite as eafy and evident as this. Alphonfus Borellus takes occafion from this Proposition to cenfure the 5. Definition of this Book very feverely, 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 forefaid 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 exprefsly, he has given Borellus a handle to fay this of him, because when Clavius in the above-cited place cenfures Commandine, and that very justly, for demonftrating 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 demonstration 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 be easily demonstrated directly. on which account we have placed it next to the Propositions concerning equimultiples.

PROP. B. B. V.

This alfo 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 neceffary to the 5. and 6. Propofitions of the 10. Book. Clavius in his Notes subjoined to the 8. Def. of Book 5. demonftrates it only in numbers, by help of fome of the Propositions of the 7. Book, in order to demonftrate 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 alfo proportionals according to the 5. Def. of 5. Book. but this is eafily made out, as follows.

in this cafe likewise AB is to CD, as EF to GH.

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 alfo greater than that of C. but from the Hypothefis 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 fome 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 fame multiple of C. and this Propofition 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 faid that "A would have a less 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 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 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 less 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 demonstrated it would immediately follow from it; but cannot without it be easily demonftrated, as he that tries to do it will find. wherefore the 10. Propofition is not fufficiently demonftrated. and it seems that he who has given the demonstration 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 fame ratio, are the fame to one another; but explicitely demonstrates this in Prop. 11. of B. 5. the demonftration we have given of the 10.

1