in the 1st, 2d, 3d, and 6th definitions in Commandine's translation, "tangit," must be read instead of " contingit;" and in the 2d and 3d definitions of book 3, the same change must be made: but in the Greek text of propositions 11th, 12th, 13th, 18th 19th, book 3, the compound verb is to be put for the simple.

PROP. IV. B. IV.

In this, as also in the 8th and 13th propositions of this book, it is demonstrated indirectly, that the circle touches a straight line: whereas in the 17th, 33d, and 37th propositions of book 3, the same thing is directly demonstrated: and this way we have chosen to use, in the propositions of this book, as it is shorter.

PROP. V. B. IV.

The demonstration of this has been spoiled by some unskilful hand: for he does not demonstrate, as is necessary, that the two straight lines which bisect the sides of the triangle at right angles must meet one another; and, without any reason, he divides the proposition into three cases; whereas, one and the same construction and demonstration serves for them all, as Campanus has observed; which useless repetitions are now left out: the Greek text also in the corollary is manifestly vitiated, where mention is made of a given angle, though there neither is, nor can be any thing in the proposition relating to a given angle.

PROP. XV. and XVI. B. IV.

In the corollary of the first of these, the words equilateral and equiangular are wanting in the Greek: and in prop. 16, instead of the circle ABCD, ought to be read the circumference ABCD: where mention is made of its containing fifteen equal parts.

DEF. III. B. V.

MANY of the modern mathematicians reject this definition: the very learned Dr. Barrow has explained it at large at the end of his third lecture of the year 1666; in which also he answers the objections made against it as well as the subject would allow and at the end gives his opinion upon the whole as follows:

"I shall only add, that the author had, perhaps, no other design in making this definition, than (that he might more fully explain and embellish his subject) to give a general and summary idea of ratio to beginners, by premising this metaphysical definition, to the more accurate definitions of ratios that are the same to one another, or one of which is greater, or less than the other; I call it a metaphysical, for it is not properly a mathematical definition, since nothing in mathematics depends on it, or is deduced, nor, as I judge, can be deduced from it; and the definition of analogy, which follows, viz. Analogy is the similitude of ratios, is of the same kind, and can serve for no purpose in mathematics, but only to give beginners some general, though gross and confused notions of analogy: but the whole of the doctrine of ratios, and the whole of mathematics, depend upon the accurate mathematical defini

tions which follow this: to these we ought principally to attend, as the doctrine of ratios is more perfectly explained by them: this third and others like it, may be entirely spared without any loss to geometry: as we see in the 7th book of the Elements, where the proportion of numbers to one another is defined, and treated of, yet without giving any definition of the ratio of numbers; though such a definition was as necessary and useful to be given in that book, as in this: but indeed there is scarce any need of it in either of them: though I think that a thing of so general and abstracted a nature, and thereby the more difficult to be conceived and explained, cannot be more commodiously defined than as the author has done; upon which account I thought fit to explain it at large, and defend it against the captious objections of those who attack it." To this citation from Dr. Barrow I have nothing to add, except that I fully believe the 3d and 8th definitions are not Euclid's, but added by some unskilful editor.

DEF. XI. B. V.

It was necessary to add the word "continual" before "proportionals" in this definition; and thus it is cited in the 33d próp. of book 11.

After this definition ought to have followed the definition of compound ratio, as this was the proper place for it; duplicate and triplicate ratio being species of compound ratio. But Theon has made it the 5th def. of book 6, where he gives an absurd and entirely useless definition of compound ratio: for this reason we have placed another definition of it betwixt the 11th and 12th of 'this book, which, no doubt, Euclid gave; for he cites it expressly in prop. 23, book 6, and which Clavius, Herigon, and Barrow have likewise given, but they retain also Theon's, which they ought to have left out of the Elements.

DEF. XIII. B. V.

This, and the rest of the definitions following, contain the explication of some terms which are used in the 5th and following books; which, except a few, are easily enough understood from the propositions of this book where they are first mentioned: they seem to have been added by Theon, or some other. However it be, they are explained something more distinctly for the sake of learners.

PROP. IV. B. V.

In the construction preceding the demonstration of this, the words a ɛTuxɛ, any whatever, are twice wanting in the Greek, as also in the Latin translations; and are now added, as being wholly necessary.

Ibid. in the demonstration; in the Greek, and in the Latin translation of Commandine, and in that of Mr. Henry Briggs, which was published at London in 1620, together with the Greek text of the first six books, which translation in this place is followed by Dr. Gregory in his edition of Euclid, there is this sentence following, viz. "and of A and C have been taken equimultiples K, L; and of

B and D, any equimultiples whatever (α ɛʊʊxɛ) M, N;" which is not true, the words "any whatever," ought to be left out: and it is strange that neither Mr. Briggs, who did right to leave out these words in one place of prop. 13, of this book, nor Dr. Gregory, who changed them into the word "some," in three places, and left them out in a fourth of that same prop. 13, did not also leave them out in this place of prop. 4, and in the second of the two places where they occur in prop 17, of this book, in neither of which they can stand consistent with truth and in none of all these places, even in those which they corrected in their Latin translation, have they cancelled the words. a ɛTUXE in the Greek text, as they ought to have done.

The same words a ruxs are found in four places of prop. 11, of this book, in the first and last of which they are necessary, but in the second and third, though 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. IV. PROP. 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 he 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 in the 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 shown 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 show the way of dividing straight lines, and far less other magnitudes, into any number of equal parts, until the 9th proposition of book 6; and he never requires any thing to be done in the construction, of which he had

A

G

E+

B

D

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 nothing is required but to add a magnitude to itself a certain number of times; and this is to be found in the translation from the Arabic, though the enunciation of the proposition and the demonstration are there very much spoiled. Jacobus Peletarius, who was the first, as far as I know, who took notice of this error, gives also the right con.struction 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.

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 Eudoxius 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 plain 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 fourth 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; 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

A

F

H

K

F

B

D

G

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 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

Α

H

K

E