Book 1. a 15. J. b 4. I. C 13. F. d 14. I. a 23. I. b 13. I. E GA B F H Bifect AC in F, and draw FG perpendicular to AB; take CH in the ftraight line CD equal to AG, and on the contrary fide of AC to that on which AG is, and join FH: Therefore, in the triangles AFG, CFH the fides FA, AG are equal to FC, CH, each to each, and the angle FAG, that is, EAB is equal to the angle FCH; wherefore the angle AGF is equal to CHF, and AFG to the angle CFH: To thefe laft add the common angle AFH; therefore the two angles AFG, AFH are equal to the two angles CFH, HFA, which two laft are equal together to two right angles, therefore alfo AFG, AFH are equal to two right angles, and confequently GF and FH are in one ftraight line. And because AGF is a right angle, CHF which is equal to it is also a right angle: Therefore the ftraight lines AB, CD are at right angles to GH. PROP. 5. D If two ftraight lines AB, CD be cut by a third ACE fo as to make the interior angles BAC, ACD, on the fame fide of it, together less than two right angles; AB and CD being produced fhall meet one another towards the parts on which are the two angles which are less than two right angles. E At the point C in the ftraight line CE make the angle ECF equal to the angle EAB, and draw to AB the ftraight line CG at right angles to CF: Then, because the angles ECF, EAB are equal to one another, and that the angles ECF, FCA are together equal to two right angles, the angles EAB, FCA are equal to two right angles. MC F K But, by the hypothefis, the N D angles EAB, ACD are to L gether less than two right angles; therefore the angle A OG B FCA is greater than ACD, H and CD falls between CF and AB: And becaufe CF and CD make an angle with one another, by Prop. 3. a point may be found in either of them CD from which the perpendicular drawn to CF fhall be greater than the ftraight line CG: Let this I this point be H, and draw HK perpendicular to CF meeting Book I. PROP. XXXV. B. I. The demonftration of this Propofition is changed, because, if the method which is ufed in it was followed, there would be three cafes to be feparately demonftrated, as is done in the tranflation from the Arabic; for, in the Elements, no cafe of a Propofition that requires a different demonftration, ought to be omitted. On this account, we have chofen the method which Monf. Clairault has given, the first of any, as far as I know, in his Elements, page 21. and which afterwards Mr Simpfon gives in his page 32. But whereas Mr Simpson makes ufe of Prop. 26. B. 1. from which the equality of the two triangles does not immediately follow, because, to prove that, the 4. of B. 1. must likewife be made use of, as may be seen, in the very fame cafe in the 34. Prop. B. 1. it was thought better to make use only of the 4. of B. 1. PROP. XLV. B. I. The ftraight line KM is proved to be parallel to FL from the 33. Prop.; whereas KH is parallel to FG by conftruction, and KHM, FGL have been demonftrated to be ftraight lines. A corollary is added from Commandine, as being often used. IN PRO P. XIII. B. II. N this Propofition only acute angled triangles are mention Book II. ed, whereas it holds true of every triangle: And the demonftrations of the cafes omitted are added; Commandine and Clavius have likewife given their demonftrations of these cafes. PROP. XIV. B. II. In the demonftration of this, fome Greek editor has ignorantly inferted the words," but if not, one of the two BE, U 2 "ED 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. SEV PROP. I. B. III. EVERAL authors, efpecially 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 cannot be demonftrated any other way: Of this the prefent propofition is a very clear inftance, as no direct demonstration can be given of it: Becaufe, 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 necessary 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 because, in the demonftration, this propofition must be brought in, viz. ftraight lines from the centre of a circle to the circumference are equal, and that the point found by the conftruction cannot be affumed as the centre, for this is the thing to be demonstrated; it is manifest some other point must be affumed as the centre; and if from this affumption an abfurdity follows, as Euclid demonftrates there muft, then it is not true that the point affumed is the centre; and as any point whatever was affumed, it follows that no point, except that found by the conftruction, can be the centre, from which the neceflity of an indirect demonstration in this cafe is evident. PROP. XII 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 conftruction in the Greek text would not have fuited with this figure fo well, because the centres of the circles muft have been placed near to the cir cumferences; On which account another conftruction and de. monstration is given, which is the fame with the fecond part of that which Campanus has tranflated from the Arabic, where ⚫ where without any reafon the demonftration is divided into two Book III. parts. The converfe of the fecond part of this propofition is wanting, though in the preceding, the converfe is added, in a like cafe, both in the enunciation 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. In this we have not followed the Greek nor the Latin tranflation 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 ftraight line which is at right angles to the diameter, at its extremity; which angies 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 reafon, fufpects 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, σε κεκλάσθω δη παλιν,” are wrong tranflated by Mr Briggs and Dr Gregory "Rurfus inclinetur," for the tranflation ought to be "Rurfus infectatur," 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 firft and laft points, is inflected or broken at U 3 the Book III. the point of inflection, where the two ftraight 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 feen in the defcription 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 90th prop. of the data. 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 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 muft coincide with the fegment CFD, (fee Commandine's figure), and that it cannot fall otherwife, 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 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. 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 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. PROR |