they are true, they are quite superfluous; as they likewise are in the second 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 also in Prop. 25, Book 11.

CoR. PROP. IV. B. V. This corollary has been unskilfully annexed to this proposition, and has been made instead of the legitimate demonstration, which, without doubt, Theon, or some other editor, has taken away, not from this, but from its proper place in this book : the author of it designed to demonstrate, that if four magnitudes E, G, F, H, be proportionals, they are also proportionals inversely; that is, G is to E, as H to F; which is true; but the demonstration of it does not inthe 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 shewn 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 demonstrated,” &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 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 shew the way of dividing straight lines, and far less 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 of which he had not before given the method of doing : for this reason,

we have changed the construction to one, which, without doubt, is Euclid's, in which

A nothing is required but to add a magnitude G to itself a certain number of times; and this

E is to be found in the translation from the

С Arabic, though the enunciation of the pro

F position and the demonstration are there very much spoiled. Jacobus Peletarius, who was the first, as far as I know, who B В DI 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 likewise gives both the ways; but neither he nor Peletarius takes notice of the reason why the one is preferable to the other.


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 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 obtained from the aforesaid property," viz. that which is contained in 5th def. 5. « That, if “ four magnitudes be proportionals, the third must ne“ cessarily 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 acknowledgement 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 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 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.

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 multiple, or the same part of B, which C is of D:

F A, B, C, D, are proportionals : B

H This is demonstrated in proposi

D tion C. Secondly, if AB contain the

KI same parts of CD that EF does of

LL GH; in this case likewise AB is to CD, as EF to GH.

À Č E Ġ 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, by Prop. C. of the 5th Book, AB is to 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 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, by Prop. D. of Book 5.

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 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; there- M fore 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 reasons mentioned in the notes at Prop. A.


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 HỘ 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 HR must be greater than A: but in the second case, viz. that in which EB is less than AE, though ZH be greater than 4, yet HR may be less than the same 4; 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: 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
than ZH; for he might
have taken that multi-


ple of A which is the

first that is greater
than HO, or K, as was


done in the first case:

he likewise brings this
K into the demon-

В А В. Д
stration of both cases,
without any reason; for it serves to no purpose but to
lengthen the demonstration. 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 A; and in this any equimul-


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