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

PROP. XV. B. III.

The converfe of the fecond part of this Propofition is wanting, tho' in the preceding, the converfe is added, in a like cafe, both in the Enuntiation 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 Propo fition, 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 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 reason fufpects to be adulterated, in the 3 86. page of his Oper. Math.

PROP. XX. B. III.

The first words of the fecond part of this Demonftration, "unciades di max" 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 straight 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 90. Prop. of Euclid's Data; for thus the whole line betwixt the first and last points, is inflected

meet.

Book III. or broken at the point of inflexion where the two straight lines And in the like fenfe two ftraight lines are faid to be inflected from two points to a third point, when they make an angle at this point; as may be feen in the defcription 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 first 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 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 di vided 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 construction and 'emonftration; 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. XXXV. B. III.

As the 25. and 33. Propofitions are divided into more cafes, fo this 35. is divided into fewer cafes than are necessary. Nor can it be fuppofed that Euclid omitted them because they are eafy; as he has given the cafe which by far is the easiest of them all, viz. that in which both the straight lines pass thro' the center. and in the following Propofition he feparately demonftrates the cafe in which the ftraight line paffes 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 demonstrations, 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 "demonstrated, if the center be in AC" are left out as the addition of fome ignorant Editor.

303

Book III.

W

DEFINITIONS of BOOK IV.

'HEN a point is in a straight, or any other line, this point is by the Greek Geometers faid äta, to be upon, or in that line. and when a straight line or circle meets a circle any way, the one is faid and to meet the other. but when a straight line or circle meets a circle fo as not to cut it, it is faid ipazladay, to touch the circle; and thefe two terms are never promifcuously used by them. therefore in the 5. Definition of B. 4. the compound ipánra must be read, instead of the fimple anтay. and in the 1, 2, 3. and 6. Definitions in Commandine's tranflation "tangit" must be read inftead of " contingit." and in the 2. and 3. Definitions of Book 3. the fame change must 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;

Book IV.

Book IV, whereas in the 17. 33. and 37. Propofitions of Book 3. the fame thing is directly demonftrated. and this way we have chofen to ufe in the Propofitions of this Book, as it is fhorter.

PROP. V. B. IV.

The Demonftration of this has been spoiled by fome unfkilfal hand. for he does not demonftrate, as is neceffary, that the two ftraight lines which bifect the fides of the triangle at right angles, muft 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 also 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 firft of thefe, the words equilateral and équiangular 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.

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 also he answers 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 other Defign in making this Definition, than (that he might more fully "explain and embellifh his fubject) to give a general and fummary "idea of ratio to beginners, by premifing this Metaphyfical Def

nition, to the more accurate Definitions of ratios that are the "fame to one another, or one of which is greater, or lefs 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

linde 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 doc"trine of Ratios, and the whole of Mathematics depend upon the

66

accurate Mathematical Definitions 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 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 ufeful to be given in that Book, as in this. but "indeed there is fcarce any need of it in either of them. tho' I "think that a thing of fo general and abstracted a nature, and there"by 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

ઠંડ

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 3. and 8. Definitions are not Euclid's, but added by fome unfkilful Editor.

DEF. XI B. V.

It was neceffary to add the word "continual" before " pro"portionals" in this Definition; and thus it is cited in the 33. Prop. of Book I I.

After this Definition ought to have followed the Definition of Compound ratio, as this was the proper place for it; Duplicate ani 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 11. and 12. of this Pook, which, no doubt, Euclid gave; for he cites it exprefsly in Prop. 23. B. 6. and which Clavius, Herigon and Barrow have like wife given, but they retain alfo Theon's, which they ought to' have left out in the Elements.

DE F. XIII. B. V.

This and the rest of the Definitions following, contain the explication of fome terms which are ufed in the 5. and following

U