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Book V.

Let CK be a part of CD, and GL the fame part of GH; and let AB be the same multiple of CK, that EF is of GL : Therefore,

by prop. C. of sth book, AB is to B

CK, as EF to GL: And CD, GH

are equimultiples of CK, GL the
second and fourth ; wherefore, by
Cor. prop. 4. Book 5. AB is to CD,

as EF to GH.
And if four magnitudes be pro-


Ë portionals 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, 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 fame 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 neceffary in that of prop. 9. B. 6. It seems
Theon has left it out for the reason mentioned in the notes at
Prop. A.

PRO P. VIII. B. v.
In the demonstration of this, as it is now in the Greeki
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 latt multiple, by the construction, is greater than
A; whence also HO must be greater than A : But, in the fecond
case, viz. that in which EB is less than AE, tho' ZH be greater
than A, yet Ho may be less than the fame A; so that there,
cannot be taken a multiple of a which is the first that is


greater than K, or Ho, because a itself is greater than it: Up. Book V. on 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 cafe : But he has, without any neceffity, changed also another part of it, viz. when he orders to take N that multiple of A which is the first that is greater than


z ZH; for he might have taken

2 that multiple of A which is the

I first that is greater than HO, HI


А or K, as was done in the first cafe : He likewise brings in

E this K into the demonstration

Hof both cases, without any rea

E fon ; for it serves to no pure pofe but to lengthen the de. ΒΔ Θ ΒΔ monstration. 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, KB 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 propofition.

PRO P. IX. B. V. Of this there is given a more explicit demonstration than that which is now in the elements.

PRO P. X. B. V. It was necessary to give another demonstration of this proposition, because that which is in the Greek and Latin, or o. trer editions, is not legitimate : For the words greater, the same or equal, lesser, have a quite different meaning when applied to magoitudes 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 roth 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 “ would have the same 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 same ratio that B has to C, then if ang equimultiples whatever of A and B be taken, and any



Book V. multiple whatever of C; if the multiple of A be greater than

the multiple of C, then, by the sth def. of Book 5. the multiple of B is also greater than that of C: But, from the hypothesis that A has a greater ratio to C, than B has to C, there must, by the 7th def. of Book 5. be certain equimultiples 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 nei. “ther is A less than B; because then A would have a less ra“ tio to C, than B has to ir: But it has not a less ratio, there.

fore 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 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 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 Thould have been proved, that, in this case, 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 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 propofition ; 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 roth proposition is not sufficiently demonstrated. And it seems that he who has given the demonstration of the soth propofition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what is manitost, when understood of magnitudes, unro ralios, viz. that a magnitude, 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 magnitudes; yet Euclid does not make use of it to infer that thoic ratios which are the fame to the same ratio, are ibe same to one another ; but explicitly demonstrates this in prop. 11. of Book 5. The demonitration we have given of the Toth prop. .is

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Do doubt the same with that of Eudox us or Euclid, as it is im. Book V. mediately and directly derived from the definition of a greater ratio, viz. the 7. of the 5.

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 equimultiples, 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 D F E F
Fa multiple of C, such, that E the multiple
of B is greater than ; 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 than B, by the roch prop. B 5. therefore D the multiple of A is greater than E the same multiple of B: And E is greater than F; much more therefore D is greater than F.


In Commandine's, Briggs's and Gregory's translations, at the beginning of this demonstration, it is said, “ 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 translation from the Greek : But the sense evidently requires that it, be read, “ so 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 restored 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 necessary to the 20th and 21st Prop. of this book, and is as useful as the propofition.


The two cases of this, which are not in the Greek, are add. ed; the demonstration of them not being exa&tly the same with that of the first case.



Book V.


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


The demonstration of this is pone of Euclid's, por is it legi. timate; for it depends upon this hypothesis, that to any three magnitudes, two of which, at least, are of the fame kind, there may be a fourth proportional; which if not proved, the demonstration now in the text of no force : But this is af. sumed without any proof; nor can it, as far as I am able to discern, be demonstrated by the propositions preceding this; so far is it from deserving to be reckoned an axiom, as Clavius, after other commentators, would have it, at the end of the definitions of the 5th book. Euclid does not demonftratę it, nor does he shew how to find the fourth proportional, be. fore the 12th prop. of the 6th book : And he never assumes ady thing in the demonstration of a propofition, which he had not besore demonstrated ; at least, he aflumes nothing the existence of which is not evidently possible; for a certain conclufion can never be deduced by the means of an uncertain proposition : Upon this account, we have given a legitimate demonstration of this proposition instead of that in the Greek and other e. ditions, which very probably Theon, at least some other, has put in the place of Euclid's, because he thought it 100 prolix : And as the 17th prop. of which this 18th is the converse, is demonftrated by help of the ist and ad propositions of this book, so, in the demonstration now given of the 18th, the 5th prop. and both cases of the 6th are necessary, and these two propo. fitions are the converses of the it and ad. Now the 5th and 6th do not enter into the demonstration of any propolition in this book as we now have it : Nor can they be of use in any proposition of the Elements, except in this 181h, and this is a manifest proof, that Euclid made use of them in his demonStration of it, and that the demonstration now given, wbich is exactly the converse of that of the 17th, as it ought to be, differs nothing from that of Eudoxus or Euclid: For the 5th, and 6th have undoubtedly been put into the 5th book for the sake of some propositions in it, as all the other propofitions about equimultiples have been.

Hieronyinus Saccherius, in bis book named Euclides ab omni naevo vindicatus, printed at Milan ann. 1733, in 4to, ac


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