15th, the diameter AD (see Commandine's figure) is proved to be greater than the straight line BC, by means of another straight line MN; whereas it may be better done without it: on which accounts we have given a different demonstration, like to that which Euclid gives in the preceding 14th, and to that which Theodosius gives in prop. 6, b. 1, of his Spherics in this very affair.

PROP. XVI. B. III.

In this we have not followed the Greek nor the Latin translation literally, but have given what is plainly the meaning of this proposition, without mentioning the angle of the semicircle, or that which some call the cornicular angle, which they conceive to be made by the circumference and the straight line which is at right angles to the diameter, at its extremity; which angles have furnished matter of great debate between some of the modern geometers, and given occasion of deducing strange consequences from them, which are quite avoided by the manner in which we have expressed the proposition. And in like manner, we have given the true meaning of prop. 31, b. 3, without mentioning the angles of the greater or lesser segments: these passages Vieta, with good reason, suspects to be adulterated, in the 386th page of his Oper. Math.

PROP. XX. B. III.

The first words of the second part of this demonstration, “xexλaçûw on raw," are wrong translated by Mr. Briggs and Dr. Gregory "Rursus inclinetur;" for the translation ought to be "Rursus inflectatur;" as Commandine has it: a straight line is said to be inflected either to a straight or curve line, when a straight line is drawn to this line from a point, and from the point in which it meets it, a straight line making an angle with the former is drawn to another point, as is evident from the 90th prop. of Euclid's Data: for this the whole line betwixt the first and last points, is inflected or broken at the point of inflection, where the two straight lines meet. And in the like sense two straight lines are said to be inflected from two points to a third point, when they make an angle at this point; as may be seen in the description given by Pappus Alexandrinus of Apollonius's books de Locis planis, in the preface to his 7th book: we have made the expression fuller from the 90th prop. of the Data.

PROP. XXI. B. III.

There are two cases of this proposition, the second of which, viz. when the angles are in a segment not greater than a semicircle, is wanting in the Greek: and of this a more simple demonstration is given than that which is in Commandine, as being derived only from the first case, without the help of triangles.

PROP. XXIII. and XXIV. B. III.

In proposition 24 it is demonstrated, that the segment AEB must coincide with the segment CFD, (see Commandine's figure) and that it cannot fall otherwise, as CGD, so as to cut the other

circle in a third point G, from this, that, if it did, a circle could cut another in more points than two: but this ought to have been proved to be impossible in the 23d prop. as well as that one of the segments cannot fall within the other: this part then in left out in the 24th, and put in its proper place, the 23d proposition.

PROP. XXV. B. III.

This proposition is divided into three cases, of which two have the same construction and demonstration; therefore it is now divided only into two cases.

PROP. XXXIII. B. III.

This also in the Greek is divided into three cases, of which two, viz. one in which the given angle is acute, and the other in which it is obtuse, have exactly the same construction and demonstration; on which account, the demonstration of the last case is left out as quite superfluous, and the addition of some unskilful editor; besides, the demonstration of the case when the angle given is a right angle, is done a round about way, and is therefore changed to a more simple one, as was done by Clavius.

PROP. XXXV. B. III.

As the 25th and 33d propositions are divided into more cases, so this thirty-fifth is divided into fewer cases than are necessary. Nor can it be supposed that Euclid omitted them because they are easy; as he has given the case, which by far is the easiest of them all, viz. that in which both the straight lines pass through the centre; and in the following proposition he separately demonstrates the case in which the straight line passes through the centre, and that in which it does not pass through the centre: so that it seems Theon, or some other, has thought them too long to insert but cases that require different demonstrations, should not be left out in the Elements, as was before taken notice of: these cases are in the translation from the Arabic, and are now put into the text.

PROP. XXXVII. B. III.

At the end of this the words "in the same manner it may be demonstrated, if the centre be in AC," are left out as the addition of some ignorant editor.

DEFINITIONS OF BOOK IV.

WHEN a point is in a straight line, or any other line, this point is by the Greek geometers said area, to be upon, or in that line, and when a straight line or circle meets a circle any way, the one is said arreda, to meet the other: but when a straight line or circle meets a circle so as not to cut it, it is said sparsodai, to touch the circle: and these two terms are never promiscuously used by them: therefore in the fifth definition of book 4, the compound εφαπτηται must be read, instead of the simple απτηται; and

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 prop. 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 ruxs, 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 (a ruxs) 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 STUXE 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