PROPOSITION A.

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 gave 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 degree of knowledge be ob"tained from the aforesaid property," viz. that which is contained in 5th Def. 5. "That, if four magnitudes be propor"tionals, the third must necessarily be greater than the "fourth, when the first is greater than the second: as Cla"vius 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 by Herigon in Schol. 1. Prop. 4. 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 fas 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 Def. 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.

PROPOSITION B.

This also is easily 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.

PROPOSITION C.

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 Def. 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 multiple, or the same part of B, which C is of D: A, B, C, D are proportionals: This is demonstrated in proposition C.

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

B

F

D

H

K

A CEG

• C. 5.

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*, AB is to CK, as EF to GL: and CD, GH are equimultiples of CK, GL, the second and fourth; wherefore*, AB is to CD, as EF Cor. 4. 5. to GH.

And if four magnitudes be proportionals according to the 5th Def. of 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. Next, if AB be to CD, as EF to GH: then if AB contain any parts of CD, EF contains the same parts of GH: for let CK be

• D. 5.

* C. 5.

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*, AB is to CK, as M to GL: and CD, GH, are Cor. 4. 5. equimultiples of CK, GL; wherefore*, 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, and consequently EF is the same multiple of GL that AB is of

• 9.5.

CK.

*

M

PROPOSITION D.

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 reasons mentioned in the notes at Prop. A.

PROPOSITION VIII.

Ꮓ

1

2

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 He the multiple of EB is greater than ZH the same multiple of AE, which last multiple, by the construction, is greater than A; 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 greater than A, yet HC may be less than the same A; so that there cannot be taken a multiple of ▲ which is the first that is greater than K or HO, because ▲ 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 ▲ which is the first that is greater than ZH; for he might have taken that multiple of A which is the first that is greater than HO, or K, as was done in the first case: he likewise brings in this K into the demonstration of both cases, without any reason; for it serves to No purpose but to lengthen the demonstration. There is also a

H

H

BA

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 ▲ : 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.

PROPOSITION IX.

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

PROPOSITION X.

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, lesser, 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 be not greater, it is either equal "or less. But A cannot be equal to B, because then each of "them would have the same ratio to C; but they have not. "Therefore A is not equal to B." The force of which reason

ing is this: if A had to C the same 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 * the multiple of B is also greater than 5 Def. 5. that of C; but from the hypothesis, that A has a greater ratio

to C, than B has to C, there must be certain equimultiples +7 Def. 5. of A and B, and some 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 same multiple of C, and this proposition directly contradicts the preceding; wherefore A is not equal to B. The demonstration of the 10th Prop. 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 said, that "A would have a less ratio to C than B has to C", or, which is the same thing, that B would have a greater ratio to C than

⚫ 7 Def. 5.

† 10. 5.

:

A to C ; that is, there must be some equimultiples of B and
A, and some 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 case, the multiple of A is always greater than the multi-
ple of C, whenever the multiple of B is greater than the multi-
ple of C; for when this is demonstrated, it will be evident
that B cannot have a greater ratio to C, than A has to C, or,
which is the same thing, that A cannot have a less ratio to C
than B has to C. But this is not at all proved in the 10th
Prop. but if the 10th were once demonstrated, it would
immediately follow from it, but cannot without it be easily
demonstrated, as he that tries to do it will find. Wherefore
the 10th Prop. is not sufficiently demonstrated. And it seems
that he who has given the demonstration of the 10th Prop.
as we now have it instead of that which Eudoxus or Euclid
had given, has been deceived in applying what is manifest,
when understood of magnitudes, unto ratios, viz. that a mag-
nitude cannot be both greater and less than another. That
those things which are equal to the same are equal to one
another, is a most evident axiom when understood of mag-
nitudes; yet Euclid does not make use of it to infer, that
those ratios, which are the same to the same
ratio, are the same to one another, but ex-
plicitly demonstrates this in Prop. 11. of Book

5.

The demonstration we have given of the 10th Prop. is no doubt the same with that of Eudoxus or Euclid, as it is immediately and directly derived from the definition of a greater ratio, viz. the 7th of the 5th.

The above-mentioned proposition, viz. If A have to C a greater ratio than B to C; and if of A and B there be taken certain equi

multiples, and some multiple of C; then if the multiple of B be greater than the multiple of C, the multiple of A is also greater than the same, is thus demonstrated.

Let D, E be equimultiples of A, B, and F a multiple of C, such, that E the multiple of B is greater than F; D the multiple of A is also greater than F.

Because A has a greater ratio to C, than B to C, A† is greater