Book II. “ ED is the greater; let BE be the greater, and produce it to V" F,” as if it was of any confequence whether the greater or lefler be produced: Therefore, instead of these words, there ought to be read only, “ but if not, produce BE to F.” PRO P. I. B. III. Book III. EVERAL authors, especially among the modern mathe maticians and logicians, inveigh too severely against indi. rect or Apagogic demonftrations, and sometimes ignorantly enough; not being aware that there are some things that can not be demonstrated any other way : Of this the present proposition is a very clear instance, as no direct demonftration can be given of it: Because, besides 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 necessary that the point found by the constı uction 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 because, in the demonftration, this proposition 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 construction capnot be afsu. med as the center, for this is the thing to be demonstrated; ie is manifest some o! her point must be assumed as the center; and if from this assumption an absurdity follows, as Euclid demonstrates there must, then it is not true that the point assumed is the center; and as any point whatever was assumed, it follows that no point, except that found by the construction, can be the center, from which the necesity of an indirect demonstration in this case is evident. PRO P. XIII. B. III. As it is much easier to imagine that two circles may touch one another within in more points than one, upon the same fide, than upon oppofite fides; the figure of that case cught not to have been omitted; but the conliruction in the Greek text would not have fuited with this figure to well, becaute the centers of the circies must have been placed near to the cir. cumferences : On which account another ceptruction and demonstration is given, which is the same with the second part of that which Campanus has translated from the Arabic, where where without any reason the demonstration is divided into two Book III. parts. PRO P. XV. B. III. The converse of the second part of this proposition is want. ing, tho’in the preceding, the converse is added, in a like case, born in the enunciation and demonstration; and it is now add. ed in this. Besides, in the demonftration 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 preceding 14th, and to that which Theodotius gives in prop. 6. B. 1. of his Splerics, in this very affair. PRO P. 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 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 an. gles have furnithed matter of great debate between some of the modern geometers, and given occation of deducing strange conlequences 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 lefler segments : These passages, Vieta, with good reason, suspects to be adulterated, in the 386th page of his Oper. Math. PRO P. XX. B. III. ' The first words of the fecond part of this demonstration, κεκλάσθω δη παλιν, are wrong translated by Mr Briggs and “ Rursus inclinetur;” for the translation ought to be « Rurfus inflectatur,” as Commandine has it : A straight line is said to be ipflected 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 evi. dent from the goth prop. of Euclid's Data : for thus the whole line betwixt the firit and last points, is inflected or broken at the Dr Gregory U3 Book III. the point of inflection, 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 A. lexandrinus of Apollonius's Books de Locis planis, in the preface to his 7th book : We have made the expression fuller from the goth prop. of the data. : There are two cases of this proposition, the second 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 simple demonftration 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 demonstrated, that the segment AEB must coincide with the fegment CFD, (see 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 23d Prop. as well as that one of the segments cannot fall within the other : This part then is left out in the 24th, and put in its proper place, the 23d Propofition. This proposition is divided into three cases, of which two have the same construction and demonftration; therefore it is now divided only into two cases. 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 obtufe, have exactly the same construction and demonftration ; on which account, the demonstration of the last cafe is left out as quite fuperfluous, and the addition of fome unikilful edicor ; besides the demonftration of the case 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. PRO P. XXXV. B. III. Book III. As the 25th and 33d propofitions are divided into more cases, To this 35th is divided into fewer cases than are necessary. Nor can it be supposed that Euclid omitted them because they are easy ; as he has given the cafe, which, by far, is the easiest of them all, viz. that in which both the straight lines pass through the centre : And in the following propofition he separately demonstrates the case in which the straight line passes through the centre, and that in which it does not pass thro' the centre: So that it seems Theon, or some other, has thought them too long to infert : 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. PRO P. XXXVII. B. III. At the end of this, the words, “ in the same manner it may 4 be demonstrated, if the centre be in AC.” are left out as the addition of some ignorant editor. DEFINITIONS of BOOK IV. Book IV. WH HEN a point is in a straight, or any other line, this αστεσθαι, In this, as also in the 8th and 13th propofitions of this book, it is demonstrated indirectly, that the circle touches a straight line; whereas in the 17th, 33d, and 37th propositions of book 3. the same thing is directly demonstrated : And this way we have U4 Book IV. have chosen to use in the propofitions of this book, as it is Thorter. The demonstration of this has been spoiled by some unskilful hand : For he does not demonstrate, as is necessary, that the two straight lines which bisect the tides of the triangle at right angles, must meet one another ; and, without any reason, he divides the proposition into three cales; whereas, one and the same con. îtruction and demonstration ferves 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, though there neither is, nor can be any thing in the proposition relating to a given angle. PRO P. 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. : DE F. III. B. V. Book y. MANY of the modern mathematicians reject this definition : The very learned Dr Burow has explained it at large at the end of his chird 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 thall only add, that the author had, perhaps, no 0. " ther delign in making this definition, than that he might more fully explain and embellish his subject) to give a gene“ ral and fummary idea of ratio to beginners, by premising “ this metaphyical definition, to the more accurate defini" tions of ratios that are the same to one another, or one of ” which is greater, or less than the other : I call it a mera. “ physical, for it is not properly a mathematical definition, “ lince 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 fimi. " litude |