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S. Book V. Books; which, except a few, are eafily 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 fomething more diftinctly for the fake of learners.

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PROP. IV. B. V.

In the conftruction, preceding the demonftration of this, the words a Tux, any whatever, are twice wanting in the Greek, as alfo in the Latin tranflations; and are now added, as being wholly neceffary.

Ibid. in the demonftration; 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 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 (a ruxe) M, N;" which is not true. the words " any whatever" ought to be left out. and it is ftrange 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 "fome" in three places, and left them out in a fourth of that fame 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 confiftent with truth. and in none of all thefe places, even in those which they corrected in their Latin tranflation, have they cancelled the words a ruxe in the Greek text, as they ought to have done.

The fame words & Tuxt are found in four places of Prop. 1 1. of this Book, in the first and laft 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 fecond 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. PRO P. IV. B. V.

This Corollary has been unfkilfully annexed to this Propofition, and has been made inftead of the legitimate demonstration 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 defigned to demonftrate that if four magnitudes E, G, F, H be proportionals, they are alfo proportionals inverfely; that is, G is 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 fays "because it is demonftrated that if K be greater "than M, L is greater than N," &c. this is indeed fhewn in the demonftration of the 4. Prop. but not from this that E, G, F, H are proportionals, for this laft is the conclufion of the Propofition. Wherefore thefe words "becaufe it is demonftrated," &c. are wholly foreign to his defign. 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 ufe, and is neceffary to the Demonftration of Prop. 18. of this Book.

PROP. V. B. V.

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 conftruction is not Euclid's. for he does not fhew the way of dividing straight lines, and far lefs 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 contruction, of which he had not before given the method of doing. for this reafon we have changed the conftruction to one which without doubt is Euclid's, in which nothing is required but to add a magnitude to itfelf a certain Enumber 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 alfo

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the right conftruction in his edition of Euclid, after he had given the other which he blames. he fays he would not leave it out, becaufe it was fine, and might sharpen one's genius to invent others

Book V. like it; whereas there is not the least difference between the two demonstrations, except a fingle word in the construction, which very probably has been owing to an unfkilful 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 cafes of this Propofition, of which only the firft and fimpleft is demonstrated in the Greek. and it is probable Theon thought it was fufficient to give this one, fince 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. Propofition for the fame reafon. the demonftration of the other cafe is now added, because both of them, as alfo the 5. Propofition, are neceffary to the demonftration of the 18. Prop. of this Book. the tranflation from the Arabic gives both cafes briefly.

PROP. A. B. V.

This Propofition is frequently ufed by Geometers, and it is neceffary 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 fubftitute 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 easy and evident as this. Alphonfus Borellus takes occafion from this Propofition to cenfure the 5. Definition of this Book very feverely, but most unjustly. in page 126. of his Euclid restored printed at Pifa in 1658. he fays, " 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 muft neceffarily be greater than the "fourth, when the firft is greater than the fecond; as Clavius ac"knowledges in the 16. Prop. of the 5. Book of the Elements." But tho' Clavius makes no fuch acknowledgement exprefsly, he has given Borellus a handle to say this of him, because when Clavius in the above-cited place cenfures Commandine, and that very justly, for demonftrating this Propofition 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 eafy to give a right demonstration of it, no body, as far as I know, has given one, except the learned Dr. Barrow, who, in anfwer to Borellus's objection, demonftrates 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 demonftrated directly. on which account we have placed it next to the Propofitions concerning equimultiples.

PROP. B. B. V.

This alfo is eafily 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 fubjoined to the 8. Def. of Book 5. demonstrates it only in numbers, by help of fome of the Propofitions 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 also proportionals according to the 5. Def. of 5. Book. but this is eafily made out, as follows.

First, If A, B, C, D be four mag-, nitudes, fuch that A is the fame mul-B tiple, or the fame part of B, which C is of D; A, B, C, D are proportionals. this is demonftrated in Propofition C.

Secondly, If AB contain the fame

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parts of CD that EF does of GH; A CE GM

in this cafe likewife 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 fame multiple of CK, that EF is of GL. therefore by Prop. C. of 5. Book, AB is to CK, as

EF to GL. and CD, GH are equi-B

multiples of CK, GL the fecond and
fourth; wherefore by Cor. Prop. 4.

B. 5. AB is to CD, as EF to GH.

And if four magnitudes be proportionals according to the 5. Def. of

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B. 5. they are alfo proportionals, ac- ACE 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 fame multiple of GL. Take M the fame 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 Hypothefis, AB is to CD, as EF to GH; therefore M is equal to EF by Prop. 9. B. 5. and confequently EF is the fame multiple of GL that AB is of CK.

PROP. D. B. V.

This is not unfrequently ufed in the demonftration of other Propofitions, and is neceffary in that of Prop. 9. B. 6. it feems Theon has left it out for the reafon mentioned in the Notes at Prop. A.

PROP. VIII. B. V.

In the demonftration of this, as it is now in the Greek, there are two cafes, (fee the demonstration in Hervagius, or Dr. Gregory's edition) of which the firft is that in which AE is lefs than EB; and in this, it neceffarily follows that HO the multiple of EB is greater than ZH the fame multiple of AE, which laft multiple, by the conftruction, is greater than A; whence alfo H must be greater than A. but in the fecond cafe, viz. that in which EB is lefs than AE, tho' ZH be greater than ▲, yet HO may be less than the fame A; fo that

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