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four magnitudes E, G, F, H, be proportionals, they are also Book V. 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 4th 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 indeed is shown in the demonstration of the 4th Prop., but not from this, that E, G, F, H, are proportionals; for this last is the conclusion of the proposition. Wherefore these words, “ because it is demon“ strated,” &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 5th def. of this book, which he has not; but this is done in Proposition B, which we have given in its proper place, instead of this corollary; and another corollary is placed after the 4th Prop., which is often of use; and is necessary 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 show the way of dividing straight lines, and far less of other magnitudes, into any number of equal parts, until the 9th Proposition of B. 6.; and he never requires any thing to be done in the construction in which he had not before given the method of doing: For this reason, we have changed the construction to one, A which, without doubt, is Euclid's, in which no

G thing is required but to add a magnitude to itself a certain number of times; and this is to be found Et

C in the translation from the Arabic, though the enunciation of the proposition and the demonstra- F tion are there very much spoiled. Jacobus Peletarius, who was the first, as far as I know, who Bl D 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 like it: 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

Book v. likewise gives both the ways; but neither he nor Peletarius

takes notice of the reason why the one is preferable to the other.

PROP. 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 Theon 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 5th Proposition, for the same reason. The demonstration of the other case is now added, because both of them, as also the 5th Proposition, are necessary to the demonstration of the 18th Proposition of this Book. The translation from the Arabic gives both cases briefly.

PROP. A. B. V.

This Proposition is frequently used by geometers, and it is necessary in the 25th Prop. of this Book, 31st of the 6th, and 34th of the 11th, and 15th of the 12th 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 5th def. of this Book. Nor can there be any doubt that Eudoxus or Euclid give it a place in the Elements, when we see the 7th and 9th of the same book demonstrated, though they are quite as easy and evident as this. Alphonsus Borellus takes occasion from this proposition to censure the 5th definition of this Book very severely, but most unjustly. In p. 126 of his Euclid Restored, printed at Pisa in 1658, he says, “ Nor can even this least de

gree of knowledge be obtained from the foresaid property, viz. that which is contained in 5th 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 acknowledges in the 16th Prop. of the “ 5th Book of the Elements.” But though Clavius makes no such acknowledgment expressly, he has given Borellus a handle to say this of him; because when Clavius, in the above-cited place, censures Commandine, and that very justly, for demonstrating this proposition by help of the 16th of the 5th ; yet he himself gives no demonstration of it, but thinks it plain from the nature of proportionals, as he writes in the end of the 14th and 16th Prop. B. 5, of his edition, and is followed Book V. 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 demonstration of it, nobody, 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 5th definition in the 322d 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 also is easily deduced from the fifth 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.

PROP. C. B. V.

This is frequently made use of by geometers, and is necessary to the 5th and 6th Propositions of the 10th Book. Clavius, in his notes subjoined to the 8th def. of Book 5, demonstrates it only in numbers, by help of some of the propositions of the 7th book : in order to demonstrate 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 also proportionals according to the 5th def. of 5th book. But this is easily made out as follows:

First, if A, B, C, D, be four magnitudes, such that A is the same mul

E tiple, or the same part of B, which B C is of D; A, B, C, D, are propor

H

D tionals: This is demonstrated in Proposition C.

Secondly, if AB contain the same KH L parts of CD that EF does of GH; in this case likewise AB is to CD, as EF to GH.

А СЕ G

Book v. Let CK be a part of CD, and GL the same part of GH,

and let AB be the same multiple of
CK, that EF is of GL: Therefore,

E
by Prop. C. of 5th Book, AB is to B
CK, as EF to GL: And CD, GH

H

D
are equimultiples of CK, GL the se-
cond and fourth; wherefore, by Cor.
Prop. 4, B. 5, AB is to CD, as EF KL

L
to GH.

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

A A CEG Book 5, they are also proportionals according to the 20th def. of Book 7.

First, if A be to B, as C to D; then if A be any multiple or part of B, C is the same 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 same parts of GH: For let CK be a part of CD, and GL the same 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, Book 5, and consequently EF is the same multiple of GL that AB is of CK.

PROP. 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.

PROP. VIII. B. V.

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 construction, is greater than 4, whence also Ho must be greater than A: But in the second case, viz. that in which EB is less than AE, though ZH be

1

greater than 4, yet Ho may

be less than the same A ; so that Book v. there cannot be taken a multiple of 4, which is the first that is greater than K or Ho, because A itself is greater than it: Upon this account, the author of this demonstration found it necessary to change one part of the construction that was made use of in the first case : But he has, without any necessity, changed also another part of it, viz. when he orders to take N that multiple of a which is the first that is greater

Z

Z than ZH; for he might have taken that multiple of 1,

1

2 which is the first that is

H greater than Ho or K, as

А.

A
was done in the first case :
He likewise brings in this E HI
K into the demonstration of

E
both cases, without any rea-
son; for it serves no purpose © BA B A
but to lengthen the demon-
stration. There is also a
third case which is not mentioned in this demonstration, viz.
that in which AE in the first, or EB in the second of the two
other cases, is greater than D; and in this any equimultiples,
as the doubles of AE, EB are to be taken, as is done in this
edition, where all the cases are at once demonstrated: And
from this it is plain that Theon, or some other unskilful editor,
has vitiated this proposition.

PROP. IX. B. V.

Of this there is given a more explicit demonstration than that which is now in the Elements.

PROP. X. B. V.

* It was necessary to give another demonstration of this proposition, because that which is in the Greek and Latin, or other editions, is not legitimate: For the words greater, the same, or equal, less, have a quite different meaning when applied to magnitudes and ratios, as is plain from the 5th and 7th definitions of Book 5. By the help of these let us examine the demonstration of the 10th Prop. which proceeds thus: “ Let A “ have to C a greater ratio than B to C: I say that A is greater “ than B; for if it is not greater, it is either equal or less. “ But A cannot be equal to B, because then each of them

U

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