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the Arabic, where without any reason the Demonstration is divided Book III. into two parts.

PROP. XV. B. III.

The converfe of the fecond part of this Propofition is wanting, tho' in the preceeding, the converse is added, in a like cafe, both in the Enunciation and Demonftration; and it is now added in this. befides in the Demonstration of the first part of this 15th the diameter AD (fee 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 preceeding 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 angles have furnished matter of great debate between some 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 lesser segments. thefe paffages Vieta with good reafon fufpects to be adulterated, in the 386, page of his Oper. Math.

PROP. XX. B. III.

XE

The first words of the fecond part of this Demonftration, “xt"xacada di wax" are wrong tranflated by Mr. Briggs and Dr. Gregory "Rurfus inclinetur," for the tranflation ought to be "Rur"fus inflectatur" as Commandine has it. a ftraight line is faid 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 90. Prop. of Euclid's Data; for thus the whole line betwixt the first and last points, is inflected

or

Book III. or broken at the point of inflexion 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 Books de Locis planis, in the Preface to his 7. Book. we have made the expreffion fuller from the 90. 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 femicircle, is wanting in the Greek, and of this a more fimple Demonstration 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 must coincide with the fegment CFD (fee Commandine's figure) and that it cannot fall otherwise, as CGD, fo 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 23. Prop. as well as that one of the fegments cannot fall within the other. this part then is left out in the 24. 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.

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 demonftration; on which account the demonstration of the last cafe is left out as quite fuperfluous, and the addition of fome unskilful Editor; be fides 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 25. and 33. Propofitions are divided into more cafes, fo this 35. 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 eaficft of them all, viz. that in which both the straight lines pass thro' the center. and in the following Propofition he separately demonftrates the cafe in which the ftraight line passes thro' the center, and that in which it does not pafs thro' the center. fo that it seems 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.

PROP. XXXVII. B. III.

At the end of this the words " in the fame manner it may be deટ monstrated, if the center be in AC" are left out as the addition of fome ignorant Editor.

Book III.

DEFINITIONS of BOOK IV.

WHEN

THEN a point is in a straight, or any other line, this point is by the Greek Geometers faid today, to be upon, or in that line. and when a straight line or circle meets a circle any way, the one is faid tay to meet the other. but when a straight line or circle meets a circle fo as not to cut it, it is faid épázka, to touch the circle; and thefe two terms are never promifcuously used by them. therefore in the 5. Definition of B. 4. the compound ipálnтay must be read, inftead of the fimple anт. and in the 1, 2, 3. and 6. Definitions in Commandine's translation “ tangit " must be read instead of "contingit." and in the 2. and 3. Definitions of Book 3. the fame change muft be made. but in the Greek text of Propofitions 11, 12, 13, 18, 19. B. 3. the compound verb is to be put for the fimple.

PROP. IV. B. IV.

In this, as alfo in the 8. and 13. Propofitions of this Book, it is demonftrated indirectly that the circle touches a ftraight line;

whereas

Book IV.

Book IV. whereas in the 17. 33. and 37. Propofitions of Book 3. the fame thing is directly demonstrated. and this way we have chofen to use 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 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 construction 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, tho' 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 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.

Book V.

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 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.

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"I fhall only add, that the Author had, perhaps, no other defign in making this Definition, than (that he might more fully "explain and embellish his fubject) to give a general and fummary " idea of ratio to beginners, by premising this Metaphysical Defi" nition, to the more accurate Definitions of ratios that are the "fame to one another, or one of which is greater, or less than the "other. I call it a Metaphyfical, for it is not properly a Mathema"tical Definition, fince 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 fimi"litude

litude of ratios, is of the fame kind, and can ferve for no purpofe Book V. in Mathematics, but only to give beginners fome general tho' grofs " and confused notion of Analogy. but the whole of the doctrine of Ratios, and the whole of Mathematics depend upon the accurate "Mathematical Definitions which follow this. to thefe we ought

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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 7. 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 use"ful to be given in that Book, as in this. but indeed there is scarce any need of it in either of them. tho' 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 de"fined, 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 thofe who attack it." to this citation from Dr. Barrow I have nothing to add, except that I fully believe the 3. and 8. Definitions are not Euclid's, but added by fome unfkilful Editor.

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DEF. XI. B. V.

It was neceffary to add the word "continual" before" propor"tionals" in this Definition; and thus it is cited in the 33. 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 5. Def. of B. 6. where he gives an abfurd and entirely ufelefs Definition of Compound ratio. for this reafon we have placed another Definition of it betwixt the II. and 12. of this Book, which, no doubt, Euclid gave; for he cites it exprefly 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.

DEF. XIII. B. V.

This and the reft of the Definitions following, contain the ex plication of fome terms which are ufed in the 5. and following

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