Book II. "ED is the greater; let BE be the greater, and produce it to "F," as if it was of any confequence whether the greater or leffer be produced: Therefore, inftead of these words, there ought to be read only," but if not, produce BE to F."

Book III.

SEVE

PRO P. I. B. III.

EVERAL authors, especially among the modern mathematicians and logicians, inveigh too feverely against indirect or Apagogic demonftrations, and fometimes ignorantly enough; not being aware that there are fome things that can. not be demonftrated any other way: Of this the prefent propofition is a very clear inftance, as no direct demonftration can be given of it: Because, befides the definition of a circle, there is no principle or property relating to a circle antecedent to this problem, from which either a direct or indirect demonstration can be deduced: Wherefore it is neceffary that the point found by the conftruction of the problem be proved to be the centre of the circle, by the help of this definition, and fome of the preceding propofitions: And becaufe, in the demonftration, this propofition must be brought in, viz, straight lines from the center of a circle to the circumference are equal, and that the point found by the conftruction cannot be affumed as the center, for this is the thing to be demonstrated; it is manifeft fome other point must be affumed as the center; and if from this affumption an abfurdity follows, as Euclid demonftrates these muft, then it is not true that the point affumed is the center; and as any point whatever was affumed, it follows that no point, except that found by the conftruction, can be the center, from which the neceflity of an indirect demonstration in this cafe is evident.

PRO P. XIII. B. III.

As it is much eafier to imagine that two circles may touch one another within in more points than one, upon the fame fide, than upon oppofite fides; the figure of that cafe ought not to have been omitted; but the contruction in the Greek text would not have fuited with this figure fo well, becaufe the centers of the circles must have been placed near to the cir cumferences: On which account another conftruction and demonftration is given, which is the fame with the fecond part of that which Campanus has tranflated from the Arabic, where

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

PROP. XV. B. III.

The converse of the fecond part of this propofition is wanting, tho' in the preceding, the converfe is added, in a like cafe, both in the enunciation and demonstration; 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 demonstration, 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 tranf. lation 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 angles have furnished matter of great debate between fome of the modern geometers, and given occation 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: These paffages, Vieta, with good reason, fufpects to be adulterated, in the 386th page of his Oper. Math.

PROP. XX. B. III.

The first words of the fecond part of this demonstration, 4 κεκλασθω δη παλιν, 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 ftraight 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 fense 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 goth prop. of the data.

PROP. XXI. B. III.

There are two cafes of this propofition, the fecond of which, viz. when the angles are in a fegment not greater than a semicircle, 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 fi gure), 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 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.

PROP. XXV. B. III.

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

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 fome unfkilful 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

Γ

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 supposed 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 separately demonftrates the cafe in which the ftraight line paffes through the centre, and that in which it does not pafs thro' the centre: So that it feems Theon, or fome other, has thought them too long to infert: But cafes that require different demonftrations, fhould 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,

PROP. XXXVII. B. III.

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

DEFINITIONS of BOOK IV.

WHE

HEN a point is in a ftraight, or any other line, this point is by the Greek geometers faid area, to be upon, or in that line, and when a ftraight line or circle meets a circle any way, the one is faid a77edα to meet the other: But when a ftraight line or circle meets a circle so as not to cut it, it is faid padá, to touch the circle; and these two terms are never promifcuously used by them: Therefore, in the 5th definition of B. 4. the compound paятnтαι must be read, instead of the fimple aтитα: And in the 1st, 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, B. 3. the compound verb is to be put for the fimple.

In this, as alfo in the 8th and 13th propofitions of this book, it is demonftrated indirectly, that the circle touches a straight line; whereas in the 17th, 33d, and 37th propositions 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 use in the propofitions of this book, as it is horter.

PRO P. V. B. IV.

The demonstration of this has been spoiled by fome unskilful hand: For he does not demonstrate, as is neceffary, that the two ftraight lines which bifect the fides of the triangle at right angles, must meet one another; and, without any reason, he divides the propofition into three cafes; whereas, one and the fame conftruction and demonftration ferves for them all, as Campanus has obferved; which ufeless 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.

PRO P. 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. inftead of the circle ABCD, ought to be read the circumference ABCD: Where mention is made of its containing fifteen equal parts.

Book V.

DE F. 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 alfo he anfwers the objections made against it as well as the fubject would allow And at the end gives his opinion upon the whole, as follows:

"I fhall only add, that the author had, perhaps, no o"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 metaphysical 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" phyfical, for it is not properly a mathematical definition, "fince nothing in mathematics depends on it, or is deduced,

nor, as 1 judge, can be deduced from it: And the defini tion of analogy, which follows, viz. Analogy is the fimi

"litude