making an angle with the former, is drawn to another point, as is evident from the 90th prop. of Euclid's Data: for thus 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 the 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, is 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 S5th 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 anteobal, to be upon or in that line; and when a straight line or circle meets a circle any way, the one is said á ATEOBAI, to meet the other: but when a straight line or circle meets a circle so as not to cut it, it is said toánteo , to touch the circle; and these two terms are never promiscuously used by them : therefore, in the 5th definition of B. 4. the compound épámtntai must be read, instead of the simple amtntai: 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 the 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 be ginners, by premising this metaphysical definition, “ to the more accurate definition 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 s 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 simili66 tude of ratios, is of the same kind, and can serve for “no purpose in mathematics, but only to give begin“ners some general, though gross and confused, notion “ of analogy: but the whole of the doctrine of ratios, " and the whole of mathematics, depend upon the ac" curate mathematical definitions which follow this: “ to these we ought principally to attend, as the doc66 trine 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 66 book of the Elements, where the proportion of num“bers to one another is defined, and treated of, yet 66 without giving any definition of the ratio of numbers; 6 though such a definition was as necessary and useful “ to be given in that book as in this: but indeed there s is scarce any need of it in either of them : though I " think that a thing of so general and abstracted a na“ ture, and thereby the more difficult to be conceived “ and explained, cannot be more commodiously de-66 fined than as the author has done: upon which ac“ count I thought fit to explain it at large, and defend s it against the captious objections of those who attack 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 B. 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 ! 66 it.” it expressly in Prop. 23. B. 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 & ÉTUXE, 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 in 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 equi“ multiples K, L; and of B and D, any equimultiples “ whatever (à Étuxe) 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 & ÉTUXE in the Greek text, as they ought to have done. The same words à štuxe 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 |