where without any reason the demonstration is divided into two Book III. parts.

The converfe of the fecond part of this propofition is wanting, though in the preceding, the converfe is added, in a like cafe, both in the enunciation and demonftration; and it is now added in this. Befides, in the demonftration of the first part of this 15th, the diameter AD (fee Commandine's figure) is proved to be greater than the ftraight line BC by means of another ftraight line MN; whereas it may be better done without it: On which accounts we have given a different demonftration, like to that which Euclid gives in the preceding 14th, and to that which Theodofius 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 the propofition, without mentioning the angle of the femicircle, or that which fome 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 angies have furnished matter of great debate between fome of the modern geometers, and given occafion of deducing ftrange confequences from them, which are quite avoided by the manner in which we have expreffed the propofition. And in like manner, we have given the true meaning of prop. 31. b. 3. without mentioning the angles of the greater or leffer fegments: Thefe paffages, Vieta, with good reafon, suspects to be adulterated, in the 386th page of his Oper. Math.

PROP. XX. B. III.

The first words of the fecond part of this demonflration, < xexhærde da te," are wrong tranflated by Mr Briggs and Dr Gregory" Rurfus inclinetur;" for the tranflation ought to be "Rurfus inflectatur," as Commandine has it: A ftraight line is faid to be inflected either to a straight, or curve line, when a ftraight 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 thus the whole line betwixt the first and laft points, is inflected or broken at

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the

Book III. the point of inflection, where the two straight lines meet. And in the like sense two straight lines are faid 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 expreffion fuller from the 90th prop. of the data.

There are two cafes of this propofition, the fecond of which, viz. when the angles are in a fegment not greater than a femicircle, is wanting in the Greek: And of this a more fimple demonftration is given than that which is in Commandine, as being derived only from the firft cafe, without the help of triangles.

PROP. XXIII. and XXIV. B. III.

In propofition 24. it is demonftrated, that the fegment AEB muft coincide with the fegment CFD, (fee Commandine's figure), and that it cannot fall otherwife, 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 impoffible in the 23d prop. as well as that one of the fegments cannot fall within the other : This part then is left out in the 24th, and put in its proper place, the 23d Propofition.

PRO P. XXV. B. III.

This propofition is divided into three cafes, of which two have the fame conftruction and demonstration; therefore it is now divided only into two cafes.

PROP. XXXIII. B. III.

This alfo in the Greek is divided into three cafes, of which two, viz. one, in which the given angle is acute, and the other in which it is obtufe, have exactly the fame conftruction and demonstration; on which account, the demonftration of the last cafe is left out as quite fuperfluous, and the addition of some unskilful editor; befides the demonftration of the cafe when the angle given is a right angle, is done a round about way, and is therefore changed to a more fimple one, as was done by Clavius. PROP

PROP. XXXV. B. III.

As the 25th and 33d propofitions are divided into more cafes, fo this 35th is divided into fewer cafes than are neceffary. Nor can it be fuppofed that Euclid omitted them because they are eafy; as he has given the cafe, which by far, is the eafieft of them all, viz. that in which both the ftraight lines pass through the centre: And in the following propofition he feparately demonftrates the cafe in which the ftraight line paffes through the centre, and that in which it does not pafs through the centre: So that it feems Theon, or fome other, has thought them too long to infert: But cafes that require different demonftrations, should not be left out in the elements, as was before taken notice of: These cafes are in the tranflation from the Arabic, and are now put into the text

PRO P. XXXVII. B. III.

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

DEFINITIONS of BOOK IV.

HEN a point is in a ftraight, or any other line, this

W point is by the Greek geometers faid aird, to be

upon, or in that line, and when a ftraight line or circle meets a circle any way, the one is faid antaï to meet the other: But when a ftraight line or circle meets a circle fo as not to cut it, it is faid partida, to touch the circle; and these two terms are never promifcuoufly used by them: Therefore, in the 5th definition of B. 4. the compound pzz must be read, instead of the fimple aztata: And in the 1ft, 2d, 3d, and 6th definitions in Commandine's tranflation, "tangit," must be read inftead of "contingit:" And in the 2d and 3d definitions of Book 3. the fame change must be made: But in the Greek text of propofitions 11th, 12th, 13th, 18th, 19th, Book 3. the com. pound verb is to be put for the fimple.

PROP. IV. B. IV.

In this, as alfo in the 8th and 13th propofitions of this book, it is demonftrated indirectly, that the circle touches a ftraight line; whereas in the 17th, 33d, and 37th propofitions of book 3. the fame thing is directly demonftrated: And this way we

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have

Book III.

Book IV.

Book IV. have chofen to ufe in the propofitions of this book, as it is

shorter.

PROP. V. B. IV.

The demonstration of this has been fpoiled by fome unskilful hand For he does not demonftrate, as is neceffary, that the two straight lines which bifect the fides of the triangle at right angles, must meet one another; and, without any reafon, he divides the propofition into three cafes; whereas, one and the fame conftruction and demonftration ferves for them all, as Campanus has obferved; which useless repetitions are now left out: The Greek text alfo in the Corollary is manifeftly vitiated, where mention is made of a given angle, though there neither is, nor can be any thing in the propofition relating to a given angle.

PROP. XV. and XVI. B. IV.

In the corollary of the first of thefe, 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.

Book V.

MAN

DEF. III. B. V.

ANY 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 alfo he anfwers the objections made against it as well as the fubject would allow Ánd at the end gives his opinion upon the whole, as follows:

"I fhall only add, that the author had, perhaps, no "ther defign in making this definition, than (that he might "more fully explain and embellifh his fubject) to give a gene"ral and fummary idea of ratio to beginners, by premifing "this metaphyfical definition, to the more accurate defini"tions of ratios that are the fame to one another, or one of "which is greater, or lefs than the other: I call it a meta"physical, for it is not properly a mathematical definition, "fince nothing in mathematics depends on it, or is deduced, "nor, as I judge, can be deduced from it: And the defini"tion of analogy, which follows, viz. Analogy is the Gmi

"litude

"litude of ratios, is of the fame kind, and can ferve for no Book V. "purpose in mathematics, but only to give beginners fome "general, tho' grofs and confufed notion of analogy: But the. "whole of the doctrine of ratios, and the whole of mathema"tics, depend upon the accurate mathematical definitions which "follow this: To thefe 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 fpared without any "lofs to geometry; as we fee 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; tho' fuch a definition was as neceffary and ufe❝ful to be given in that book, as in this: But indeed there is "fcarce any need of it in either of them: Though I think that "a thing of fo 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 fome unfkilful editor.

DE F. XI. B. V.

pro

It was neceffary to add the word "continual" before " "portionals" 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 fpecies of compound ratio. But Theon has made it the 5th def. of B. 6. where he gives an abfurd and entirely useless definition of compound ratio: For this reafon we have placed another definition of it betwixt the 11th and 12th of this book, which, no doubt, Euclid gave; for he cites it exprefsly in prop. 23. B. 6. and which Clavius, Herigon, and Barrow, have likewife given, but they retain alfo Theon's, which they ought to have left out of the elements.

DE F. XIII. B. V.

This, and the reft of the definitions following, contain the explication of fome terms which are used in the 5th and following books; which, except a few, are eafily enough understood from

the