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

PROP. XV. B. III. The converse of the second part of this Proposition is wanting, tho' in the preceeding, the converse is added, in a like case, both in the Enunciation and Demonstration; and it is now added in this. besides in the Demonstration of the first part of this 15th the diameter AD (see 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 Theodosius 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 semicircle, or that which some 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 occasion of deducing strange consequences from them, which are quite avoided by the manner in which we have expressed the Proposition. 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, these passages Vieta with good reason suspects to be adulterated, in the 386, page of his Oper. Math.

PROP. XX. B. III. The first words of the second part of this Demonstration, “xe“xadal w dni nánar" are wrong translated by Mr. Briggs and Dr. Gregory “Rursus inclinetur,” for the translation ought to be “Rur“ sus inflectatur” as Commandine has it. a straight line is said 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 go. Prop. of Euclid's Data; for thus the whole line betwixt the first and last points, is inflected

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Book III. or broken at the point of inflexion 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 Books de Locis planis, in the Preface to his 7. Book. We have made the expression fuller from the 90. 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 fimple 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 demonftrated that the segment AEB must coincide with the segment CFD (fee 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 23. Prop. as well as that one of the segments can: not fall within the other. this part then is left out in the 24. 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 die vided 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 fame 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; be. fides he 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,



Book III. PROP. XXXV. B. III. As the 25. and 33. Propofitions are divided into more cafes, so this 35. is divided into fewer cases than are necessary. Nor can it be supposed that Euclid omitted thern 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 thro' the center. and in the following Proposition he separately demonstrates the case in which the straight line passes thro' the center, and that in which it does not pass thro' the center. fo 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 dede monstrated, if the center be in AC” are left out as the addition of fome ignorant Editor.


Book IV.


THEN a point is in a straight, or any other line, this point is

by the Greek Geometers faid atleasay, to be upon, or in that line. and when a straight line or circle meets a circle any way, the one is said ärl3cy to meet the other. but when a straight line or circle meets a circle so as not to cut it, it is said épdaltatay, to touch the circle; and these two terms are never promiscuously used by them. therefore in the 5. Definition of B. 4. the compound ipatintoy must be read, instead of the simple axlntcy. 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 same change must be made. but in the Greek text of Propositions 11, 12, 13, 18, 19. B. 3. the compound verb is to be put for the simple.

PROP. IV, B. IV. In this, as also in the 8. and 13. Propositions of this Book, it is demonstrated indirectly that the circle touches a straight line ;


Book IV. whereas in the 17. 33. and 37. Propositions of Book 3. the fame

thing is directly demonstrated, and this way we have chosen to use in the Propofitions of this Book, as it is shorter.

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PROP. V. B. IV. The Demonstration of this has been spoiled by some anskilful hand, for he does not demonstrate, as is necessary, that the two straight lines which bifect 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, tho' there neither is, nor can be any thing in the Proposition relating to a given angle.

PROP. xv. and XVÌ. 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.



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 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 de. sign 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 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 Metaphysical, for it is not properly a Mathema" tical Definition, since 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 simi

“ litude

litude of ratios, is of the same kind, and can serve for no purpose Book v. “ in Mathematics, but only to give beginners some general tho' gross m “ and confused notion of Analogy. but the whole of the doctrine of « Ratios, and the whole of Mathematics depend upon the accurate i 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, “ spared without any loss 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' such a Definition was as necessary 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 so

general and abstracted a nature, and thereby the more difficult to “ be concived, 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 capti"ous objections of those who attack it.” to this citation fronz Dr. Barrow I have nothing to add, except that I fully believe the 3. and 8. Definitions are not Euclid's, but added by some unikilful Editor.

DEF. XI. B. V. It was necessary 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 species of Compound ratio. But Theon has made it the 5. 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 ul. 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 likewise given, but they retain alfo 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 ihe 5. and following


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