Book I. 3. 15. I. b. 4. I. E that on which AG is, and join FH. therefore, in the triangles AFG, GA B c. 13. 1. gether to two right angles, therefore сн alfo AFG, AFH are equal to two right Ꭰ d. 14. 1. angles, and confequently d GF and FH are in one straight line. and because AGF is a right angle, CHF which is equal to it is also a right angle. therefore the straight lines AB, CD are at right angles to GH a. 23. I. b. 13. I. PROP. 5. 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 shall meet one another towards the parts on which are the two angles which are lefs than two right angles. 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 E gles, therefore the angle FCA A OG B is greater than ACD, and CD falls between CF and AB. and because 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 point be H, and draw HK perpendicular to CF meeting AB in L. and because AB, CF contain equal angles with AC on the same side of it, by Prop. 4. AB and CF are at right angles to the straight line MNO which bifects AC in N and is perpendicular to CF. therefore, by Cor. Prop. 2. CG and KL which are at right angles to CF are equal to one another. and HK is greater than CG, and therefore is greater than KL, and confequently the point H is in KL produced. Wherefore the straight line CDH drawn betwixt the points C, H which are on contrary fides of AL, muft neceffarily cut the ftraight line AB. PROP. XXXV. B. I. The Demonstration of this Propofition is changed, because if the method which is used in it was followed, there would be three cafes to be separately demonstrated, as is done in the translation from the Arabic; for in the Elements no cafe of a Propofition that requires a different demonstration 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 use 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 likewise 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 straight 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 demonstrated to be straight lines. a Corollary is added from Commandine, as being often used, IN PROP. XIII. B. II. N this Propofition only acute angled triangles are mentioned, whereas it holds true of every triangle, and the Demonstrations of the cafes omitted, are added; Commandine and Clavius have likewife given their Demonstrations 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, ED is the Book I. Book II. Book II. «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 instead of these words, there ought to be read only, "but if not, produce BE to F." Book III. PROP. I. B. III. EVERAL Authors, efpecially among the modern Mathemati SEV cians and Logicians, inveigh too feverely against indirect, or Apagogic Demonstrations, and sometimes ignorantly enough; not being aware that there are some things that cannot be demonstrated any other way. of this the present Propofition is a very clear inftance, as no direct demonstration can be given of it. because, 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 Demonftration can be deduced. wherefore it is neceffary that the point found by the construction of the Problem be proved to be the center of the circle, by the help of this Definition, and some of the preceding propofitions. and because in the Demonstration, this Propofition 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 conftruction cannot be affumed as the center, for this is the thing to be demonftrated; it is manifest some other point must be affumed as the center; and if from this affumption an abfurdity follows, as Euclid demonstrates there muft; then it is not true that the point affumed is the center; and as any point whatever was affumed, it follows that no point, except that found by the construction can be the center. from which the neceffity of an indirect Demonftration in this cafe is evident. *PROP. 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 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 centers of the circles must have been placed near to the circumferences. on which account another construction and demonstration is given which is the fame with the second part of that which Campanus has tranflated from the Arabic, where without any reason the Demonstration is Book III. divided into two parts. PROP. XV. B. III. The converse of the second part of this Propofition is wanting, tho' in the preceding, the converfe is added, in a like case, both in the Enuntiation 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 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 Theodofius gives, in Prop. 6. B. 1. of his Spherics, in this very affair. it 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 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 segments. these paffages Vieta with good reason fufpects to be adulterated, in the 386. page of his Oper. Math. PROP. XX. B. III. The first words of the fecond part of this Demonstration, “nenλdow ♪ù má" are wrong tranflated by Mr. Briggs and Dr. Gregory "Rurfus inclinetur," for the translation ought to be "Rurfus inflectatur" as Commandine has it. a straight 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 Book III. or broken at the point of inflexion where the two ftraight 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 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 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 segment AEB must 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 23. Prop. as well as that one of the segments 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 demonstration; therefore it is now divided only into two cafes. 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 obtuse, have exactly the fame conftruction and demonstration; on which account the demonstration of the laft cafe is left out as quite fuperfluous, and the addition of fome unskilful Editor; befides 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. |