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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.
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 definition 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 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 fifth; yet he himself gives no demonstration of it, but thinks it plaio
from the nature of proportionals, as he writes in the end of the 14th and 16th prop. book 5. of his edition, and is followed by Herigon in Schol. 1. prop. 14th. book 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 mag
F nitudes, such that A is the same multi- B
H ple, or the same part of B, which C is of
K parts of CD, that EF does of GH; in this case likewise AB is to CD, as EF to GH.
A СЕ G
Let CK be a part of CD, and GL the same part of GH, and let AB be the same multiple of CK,
F that EF is of GL: therefore, by prop. B C of 5th book, AB is to CK, as EF to
H GL: and CD, GH are equimultiples of CK, GL the second and fourth : wherefore by cor. prop. 4. book 5. AB
L is to CD, as EF to GH.
K And if four magnitudes be proportionals according to the 5th def. of book
E G 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 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 book 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 Ğ L 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 of
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 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 Ho may
be less than the same 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: 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;
Z for he might have taken that mul
1 tiple of a which is the first that is greater than Ho, or K, as
A done in the first case : he likewise H
A brings in this K into the demon- E
Hstration of both cases, without any reason'; for it serves to no purpose but to lengthen the demon
E stration. There is also a third
B 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 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, 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 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. Therefore 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 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, by the 5th 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 neither is A less than B ; because then I 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 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 should 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 ra. tio to C than B has to C: but this is not at all proved in the 10th proposition ; 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 proposition is not sufficiently demonstrated. And it seems that he who has given the demonstration of the 10th proposition 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 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 those ratios which are the same to the same ratio, are the same to one another ; but explicitly demonstrates this in prop. 11. of book 5. The demonstration we have given of the Toth prop. is