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"litude of ratios, is of the fame kind, and can ferve for no Book V. "purpose in mathematics, but only to give beginners fome general, tho' grofs and confufed notion of analogy: But the "whole of the doctrine of ratios, and the whole of mathematics, depend upon the accurate mathematical definitions which "follow this: To thefe we ought principally to attend, as the "doctrine of ratios is more perfectly explained by them; this "third, and others like it; may be entirely fpared without any " lofs to geometry: As we fee in the 7th book of the elements, "where the proportion of numbers to one another is defined, " and treated of, yet without giving any definition of the ratio " of numbers; tho' fuch a definition was as neceffary and ufe"ful to be given in that book, as in this: But indeed there is "fcarce any need of it in either of them: Though I think that "a thing of fo general and abftracted a nature, and thereby the "more difficult to be conceived, and explained, cannot be more "commodioufly defined, than as the author has done: Upon "which account I thought fit to explain it at large, and defend "it against the captious objections of those who attack it." To this citation from Dr Barrow I have nothing to add, except that I fully believe the 3d and 8th definitions are not Euclid's, but added by fome unfkilful editor.

DE F. XI. B. V.

It was neceffary to add the word "continual" before " pro"portionals" in this definition; and thus it is cited in the 33d prop. of book 1I.

After this definition ought to have followed the definition of compound ratio, as this was the proper place for it; duplicate and triplicate ratio being fpecies of compound ratio. But Theon has made it the 5th def. of B. 6. where he gives an abfurd and entirely useless definition of compound ratio: For this reafon we have placed another definition of it betwixt the 11th and 12th of this book, which, no doubt, Euclid gave; for he cites it exprefsly in prop. 23. B. 6. and which Clavius, Herigon, and Barrow have likewife given, but they retain alfo Theon's, which they ought to have left out of the elements.

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This, and the rest of the definitions following, contain the explication of fome terms which are ufed in the 5th and following books; which, except a few, are eafily enough understood from

the

Book V.

the propofitions of this book where they are first mentioned. They feem to have been added by Theon, or some other. How ever it be, they are explained fomething more diftinctly for the fake of learners.

PROP. IV. B. V.

In the conftruction preceding the demonstration of this, the words a Tux, 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 demonftration; in the Greek, and in the Latin tranflation of Commandine, and in that of Mr Henry Briggs, which was published at London in 162c, 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 e"quimultiples K, L; and of B and D, any equimultiples "whatever (a TUXE) 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 ftand confiftent with truth: And in none of all these places, even in thofe which they corrected in their Latin tranf lation, have they cancelled the words a Tuxe in the Greek text, as they ought to have done.

The fame words & Tux are found in four places of prop. 11. of this book, in the first and laft of which, they are neceffary, but in the fecond and third, though 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 12th 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. Book 11.

COR. IV. PROP. B. V.

This corollary has been unfkilfully annexed to this propofition, and has been made inftead of the legitimate demonftration, which, without doubt, Theon, or fome other editor, has taken away, not from this, but from its proper place in

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this book: The author of it defigned to demonstrate, that if four Book V. magnitudes E, G, F, H be proportionals, they are also 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 4th prop. or its demonftration: For, when he fays, "be"cause it is demonstrated that if K be greater than M, L is greater than N," &c. This indeed is thewn in the demonftration of the 4th 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 5th 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 4th prop. which is often of ufe; and is neceffary 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 fhew the way of dividing straight lines, and far lefs other magnitudes, into any number of equal parts, until the 9th 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 conftruction to one, A which, without doubt, is Euclid's, in which nothing is required but to add a magnitude to itself a certain number of times; and this is to be found in the tranflation from the Arabic, though the enunciation of the propofition and the demonftration are there very much fpoiled. Jacobus Peletarius, who was the firft, as far as I know, who took B notice of this error, gives also 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 fharpen one's genius to invent others like it; whereas

E

G

C

F

D'

there

Book V. there is not the leaft difference between the two demonstrations, except a fingle word in the conftruction, 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 the 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, 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 alfo the 5th propofition, for the fame reafon : The demonftration of the other cafe is now added, becaufe both of them, as alfo the 5th propofition, are neceffary to the demonftration of the 18th propofition 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 25th prop. of this book, 31ft of the 6th, and 34th of the 11th and 15th of the 12th 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 5th 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 7th and 9th 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 5th definition of this book very feverely, but most unjustly: In p. 126. of his Euclid restored, printed at Pifa in 1658, he fays, "Nor can even this leaft de"gree of knowledge be obtained from the forefaid property," viz. that which is contained in 5th 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 acknowledges in the 16th prop. of the 5th book of the elements." But though Clavius makes no fuch acknowledgement exprefsly, he has given Borellus a handle to fay this of him; becaufe when Clavius, in the above-cited place, cenfures Commandine, and that very juftly, for demonftrating this propofition by help of the 16th of the 5th; yet he himself gives no demonftration of it, but thinks it plain

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from the nature of proportionals, as he writes in the end of the Book V. 14th and 16th prop. B. 5. of his edition, and is followed by Herigon in Schol. I. 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, though it is very easy to give a right demonftration 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 5th definition, in the 322d page of his Lect. Mathem. from which definition it may also be eafily demonftrated directly: On which account we have placed it next to the propofitions concerning equimultiples.

PROP. B. BOOK V.
В. ВО

This alfo is eafily deduced from the 5th def. B. 5. and therefore is placed next to the other; for it was very ignorantly made a corollary from the 4th prop. of this book. See the note on that corollary.

PRO P. C. B. V.

This is frequently made ufe of by geometers, and is neceffary to the 5th and 6th propofitions of the 10th book. Clavius, in his notes fubjoined to the 8th def. of book 5. demonstrates it only in numbers, by help of fome of the propofitions of the 7th book, in order to demonftrate the property contained in the 5th definition of the 5th book, when applied to numbers, from the property of proportionals contained in the 20th def. of the 7th book: And most of the commentators judge it difficult to prove that four magnitudes which are proportionals according to the 20th def. of 7th book, are alfo proportionals according to the 5th def. of 5th book. But this is eafily made out, as follows. First, If A, B, C, D be four magnitudes, 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 demonftrated in propofition C.

Secondly, If AB contain the fame parts of CD that EF does of GH; in this cafe likewife AB is to CD, as EF to GH.

D

H

L

A C

EGM

Let

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