The Elements of Euclid

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 fomething more diftinctly for the fake of learners.

PROP. IV. B. V.

In the construction preceeding the demonstration of this, the words & Tux, any whatever, are twice wanting in the Greek, as alfo in the Latin tranflations; and are now added, as being wholly necessary.

Ibid. in the demonstration; in the Greck, 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 (Tuxe) M, N;” which is not true, the words " any whatever" ought to be left out. and it is frange that neither Mr. Briggs, who did right to leave out thefe 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 fland confiftent with truth. and in none of all thefe places, even in those which they corrected in their Latin translation, have they cancelled the words ä ruxe in the Greek text, as they ought to have done.

The fame words TX are found in four places of Prop. 11. of this Book, in the firft and laft of which, they are neceffary, but in the fecond and third, tho' they are true, they are quite fuperflu ous; 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 Propp. 22, 23. of this Book. but are wanting in the laft place of Prop. 23. as alfo in Prop. 25. B. 11.

This Corollary has been unfkilfully annexed to this Propofition, and has been made instead of the legitimate demonstration which

without

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, Gis 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 fays "because it is demonftrated that if K be greater than M, L is greater than N" &c. this indeed is shown in the demonstration 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 these words "because 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, inftead of this Corollary. and another Corollary is placed after the 4. Prop. which is often of ufe, and is necessary to the Demonstration of Prop. 18. of this Book.

PROP. V. B. V.

In the conftruction 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 shew the way of dividing ftraight 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 conftruction, 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 number of times. and this is to be found in the tranflation from the Arabic, tho' the enunciation of the Propofition and the demonstration are there very much fpoiled. Jacobus Peletarius who was the first, as far as I know, who took notice of this error, gives alfo 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, because it was fine, and might sharpen one's genius to invent others

Book V. like it; whereas there is not the leaft difference between the two demonftrations, 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 reafon why one is preferable to the other.

PROP. VI. B. V.

There are two cafes of this Propofition, of which only the first and fimpleft is demonftrated in the Greek. and it is probable Theon thought it was fufficient to give this one, fince 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. Propofition for the fame reafon. the demonstration of the other cafe is now added, because both of them, as alfo the 5. Propofition, are necessary to the demonftration of the 18. Prop. of this Book. the translation from the Arabic gives both cafes briefly.

PROP. A. B. V.

This Propofition is frequently used 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 feems to have been taken out of the Elements by Theon, because it appeared evident enough to him, and others who fubftitute the confufed and indiftinct 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 Propofition to cenfure the 5. Definition of this Book very feverely, but most unjustly. in page 126. of his Euclid restored printed at Pia in 165 8. he fays, "Nor can even this least degree of "of knowledge be obtained from the forefaid property," viz. that which is contained in 5. Def. 5. "That if four magnitudes be pro"portionals, the third muft neceffarily be greater than the fourth, "when the first is greater than the fecond; as Clavius acknow"ledges in the 16. Prop of the 5. Book of the Elements." But tho' Clavius makes no fuch acknowledgement exprefly, he has given Borellus a handle to fay this of him, becaufe when Clavius in the above cited place cenfures Commandine, and that very justly, for dcanonftrating this Propofition by help of the 16. of the 5. yet he

him elf

himself gives no demonstration of it, but thinks it plain from the Bock V. 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 underflood from the Definition of them. and indeed, tho' it is very easy to give a right de monstration 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 alfo be easily demonftrated directly. on which account we have placed it next to the Propofitions concerning equi multiples.

PROP. B. BOOK. 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 ufe of by Geometers, and is neccffary to the 5.and 6. Propofitions of the 10. Book. Clavius in his Notes fubjoined to the 8. Def of Book 5. demonftrates 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 alfo pro portionals according to the 5. Def. of 5. Book. but this is enafily made out, as follows.

First, If A, B, C, D be four mag- B

nitudes, fuch that A is the fame.

multiple, or the fame part of B,

which C is of D; A, B, C, D are proportionals. this is demonstrated in Propofition C.

Secondly, If AB contain the fame parts of CD that EF does of GH;

ACEG M

Book V. in this cafe likewife AB is to CD, as EF to GH.

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 equimultiples 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 B. 5. they are alfo proportionals according to the 20. Def. of B. 7.

A CEG M

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 muluple 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 GI!; 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 used in the demonftration of other Propofitions, and is neceflary in that of Prop. 9. B. 6. it seems 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 demonftration in Hervagius, or Dr. Gregory's edition) of which the first 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 HO must be greater than A. but in the fecond cafe, viz. that in which EB is lefs than AE, tho' ZH be greater than 4, yet HO may be lefs than the fame ▲; so that

there

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