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,

F

by prop. C. of 5th book, AB is to B

H1

CK, as EF to GL: And CD, GH

Cor. prop. 4.
as EF to GH.

are equimultiples of CK, GL the
fecond and fourth; wherefore, by

AB is to CD,

A CEG M

And if four magnitudes be proportionals according to the 5th def. of Book 5. they are alfo 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 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; there fore 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 confequently EF is the fame multiple of GL that AB is of CK.

PROP. D. B. V.

This is not unfrequently used in the demonstration of other propofitions, and is neceffary in that of prop. 9. B. 6. 1t 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 Greeks there are two cafes, (fee the demonftration 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 HO must be greater than A: But, in the fecond case, viz. that in which EB is lefs than AE, tho' ZH be greater than A, yet HO may be lefs than the fame A; fo that there, cannot be taken a multiple of A which is the

first that is

greater

Δ

greater than K, or HO, because A itself is greater than it: Up. Book V. on this account, the author of this demonftration found it neceffary to change one part of the construction that was made ufe of in the firft cafe: But he has, without any neceffity, changed alfo another part of it, viz. when he orders to take N that multiple of A which

Z

2

I

A

A

E

H

E

is the firft that is greater than
ZH; for he might have taken
that multiple of A which is the
firft that is greater than HO, H
or K, as was done in the first
cafe: He likewise brings in
this K into the demonstration
of both cafes, without any rea-
ion; for it ferves to no pur-
pofe but to lengthen the de-
monftration. There is also a
third cafe, which is not mentioned in this demonstration, viz.
that in which AE in the first, or EB in the fecond of the two
other cafes, is greater than D; and in this any equimultiples,
as the doubles, of AE, KB are to be taken, as is done in this
edition, where all the cafes are at once demonstrated: And from
this it is plain that Theon, or fome other unfkilful editor has vi-
tiated this propofition.

© BA 0 B

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 neceffary to give another demonstration of this propofition, because that which is in the Greek and Latin, or other editions, is not legitimate: For the words greater, the fame or equal, leffer, 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 fay that A is greater "than B. For if it is not greater, it is either equal, or lefs. "But A cannot be equal to B, becaufe then each of them "would have the fame ratio to C; but they have not. There"fore A is not equal to B." The force of which reasoning is this, if A had to C, the fame ratio that B has to C, then if any equimultiples whatever of A and B be taken, and any

multiple

Book V. multiple whatever of C; if the multiple of A be greater than the multiple of C, then, by the 5th def. of Book 5. the multiple of B is alfo greater than that of C: But, from the hypothefis that A has a greater ratio to C, than B has to C, there muft, by the 7th def. of Book 5. be certain equimultiples of A and B, and fome multiple of C fuch, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the fame multiple of C: And this propofition directly contradicts the preceding; wherefore A is not equal to B. The demonftration of the 10th prop. goes on thus: “But nei"ther is A lefs than B; because then A would have a lefs ra❝tio to C, than B has to it: But it has not a lefs ratio, there"fore A is not lefs than B," &c. Here it is faid that "A "would have a lefs ratio to C, than B has to C," or, which is the fame thing, that B would have a greater ratio to C, than A to C; that is, by 7th def. Book 5. there must be some equimultiples of B and A, and fome multiple of C fuch, 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 cafe, the multiple of A is always greater than the multiple of C, whenever the multiple of B is greater than the multiple 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 fame thing, that A cannot have a lefs ratio to C, than B has to C: But this is not at all proved in the 10th propofition; but if the 10th were once demonstrated, it would immediately follow from it, but cannot without it be eafily demonftrated, as he that tries to do it will find. Wherefore the 10th propofition is not fufficiently demonftrated. And it feems that he who has given the demonftration of the 10th propofition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what is maniteft, when understood of magnitudes, unto ratios, viz. that a magnitude cannot be both greater and lefs than another. That thofe things which are equal to the fame are equal to one another, is a moft evident axiom when understood of magnitudes; yet Euclid does not make ufe of it to infer that thofe ratios which are the fame to the fame ratio, are the fame to one another; but explicitly demonftrates this in prop. 11. of Book 5. The demonftration we have given of the 10th prop. is

no

no doubt the fame with that of Eudoxus or Euclid, as it is immediately and directly derived from the definition of a greater ratio, viz. the 7. of the 5.

The above mentioned propofition, viz. If A have to Ca
greater ratio than B to C, and if of A and
B there be taken certain equimultiples, and
fome 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 fame,
is thus demonstrated.

A
Let D, E be equimultiples of A, B, and D
Fa multiple of C, fuch, that E the multiple
of B is greater than F; D the multiple of
A is alfo greater than F.

Because A has a greater ratio to C, than B to C, A is greater than B, by the 10th prop. B 5. therefore D the multiple of A is greater than E the fame multiple of B: And E is greater than F; much more therefore D is greater than F.

PRO P. XIII. B. V.

CB C
FE F

In Commandine's, Briggs's and Gregory's tranflations, at the beginning of this demonftration, it is faid, " And the multi"ple of C is greater than the multiple of D; but the multi"ple of E is not greater than the multiple of F;" which words are a literal tranflation from the Greek: But the fenfe evidently requires that it be read, "fo that the multiple of C "be greater than the multiple of D; but the multiple of E be "not greater than the multiple of F." And thus this place was reftored to the true reading in the first editions of Comman dine's Euclid, printed in 8vo at Oxford; but in the later editions, at least in that of 1747, the error of the Greek text was kept in.

There is a corollary added to Prop. 13. as it is neceffary to the 20th and 21ft Prop. of this book, and is as ufeful as the propofition.

PROP. XIV. B. V.

The two cafes of this, which are not in the Greek, are added; the demonftration of them not being exactly the fame with that of the firft cafe.

Book V.

Book V.

PROP. XVII. B. V.

The order of the words in a clause of this is changed to one more natural: As was also done in prop. 11.

PRO P. XVIII. B. V.

The demonstration of this is pone of Euclid's, nor is it legi timate; for it depends upon this hypothefis, that to any three magnitudes, two of which, at leaft, are of the fame kind, there may be a fourth proportional; which if not proved, the demonstration now in the text is of no force: But this is af fumed without any proof; nor can it, as far as I am able to difcern, be demonftrated by the propofitions preceding this; fo far is it from deferving to be reckoned an axiom, as Cla vius, after other commentators, would have it, at the end of the definitions of the 5th book. Euclid does not demonstrate it, nor does he fhew how to find the fourth proportional, be fore the 12th prop. of the 6th book: And he never affumes any thing in the demonftration of a propofition, which he had not before demonstrated; at least, he affumes nothing the existence of which is not evidently poffible; for a certain conclufion can never be deduced by the means of an uncertain propofition: Upon this account, we have given a legitimate demonstration of this propofition inftead of that in the Greek and other e ditions, which very probably Theon, at leaft fome other, has put in the place of Euclid's, because he thought it too prolix: And as the 17th prop. of which this 18th is the converfe, is demonftrated by help of the 1ft and 2d propofitions of this book, fo, in the demonstration now given of the 18th, the 5th prop. and both cafes of the 6th are neceffary, and these two propo fitions are the converfes of the 1ft and 2d. Now the 5th and 6th do not enter into the demonftration of any propofition in this book as we now have it: Nor can they be of use in any propofition of the Elements, except in this 18th, and this is a manifeft proof, that Euclid made ufe of them in his demonftration of it, and that the demonftration now given, which is exactly the converfe of that of the 17th, as it ought to be, dif fers nothing from that of Eudoxus or Euclid: For the 5th, and 6th have undoubtedly been put into the 5th book for the fake of fome propofitions in it, as all the other propofitions about equimultiples have been.

Hieronymus Saccherius, in his book named Euclides ab omni naevo vindicatus, printed at Milan ann. 1733, in 4to, acknowledges,