take away the common square on EB; therefore the remaining rectangle AD, DC is equal to the square on the tangent DB. (ax. 3.) Next, if DCA does not pass through the center of the circle ABC. D B F E A Take E the center of the circle, (III. 1.) draw EF perpendicular to AC, (1. 12.) and join EB, EC, ED. Because the straight line EF, which passes through the center, cuts the straight line AC, which does not pass through the center, at right angles; it also bisects AC, (III. 3.) therefore AF is equal to FC; and because the straight line AC is bisected in F, and produced to D, the rectangle AD, DC, together with the square on FC, is equal to the square on FD: (11. 6.) to each of these equals add the square on FE; therefore the rectangle AD, DC, together with the squares on CF, FE, is equal to the squares on DF, FE: (1. àx. 2.) but the square on ED is equal to the squares on DF, FE, (1. 47.) because EFD is a right angle; and for the same reason, the square on EC' is equal to the squares on CF, FE; therefore the rectangle AD, DC, together with the square on EC, is equal to the square on ED: (ax. 1.) but CE is equal to EB; therefore the rectangle AD, DC, together with the square on EB, is equal to the square on ED: but the squares on EB, BD, are equal to the square on ED, (1.47.) because EBD is a right angle: therefore the rectangle Al), DC, together with the square on EB, is equal to the squares on EB, BD; take away the common square on EB; and the remaining rectangle AD, DC is equal to the square on DB. (1. ax. 3.) Wherefore, if from any point, &c. Q. E. D. COR. If from any point without a circle, there be drawn two straight lines cutting it, as AB, AC, the rectangles contained by the whole lines and the parts of them without the circle, are equal to one another, viz. the rectangle BA, AE, to the rectangle CA, AF: for each of them is equal to the square on the straight line AD, which touches the circle. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on the line which meets it, the line which meets, shall touch the circle. Let any point D be taken without the circle ABC, and from it let two straight lines DCA and DB be drawn, of which DCA cuts the circle in the points C, A, and DB meets it in the point B. If the rectangle AD, DC be equal to the square on DB; D E B A Draw the straight line DE, touching the circle ABC, in the point E; (u. 17.) find F, the center of the circle, (II. 1.) and join FE, FB, FD. Then FED is a right angle: (II. 18.) and because DE touches the circle ABC, and DCA cuts it, therefore the rectangle AD, DC'is equal to the square on DE: (111. 36.) but the rectangle AD, DC, is, by hypothesis, equal to the square on DB: therefore the square on DE is equal to the square on DB; (1. ax. 1.) and the straight line DE equal to the straight line DB: and FE is equal to FB; (1. def. 15.) wherefore DE, EF are equal to DB, BF, each to each; and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF is equal to the angle DBF: (1. 8.) but DEF was shewn to be a right angle; therefore also DBF is a right angle: (1. ax. 1.) and BF, if produced, is a diameter; and the straight line which is drawn at right angles to a diameter, from the extremity of it, touches the circle; (III. 16. Cor.) therefore DB touches the circle ABC. Wherefore, if from a point, &c. Q.E.D IN the Third Book of the Elements are demonstrated the most elementary properties of the circle, assuming all the properties of figures demonstrated in the First and Second Books. It may be worthy of remark, that the word circle will be found sometimes taken to mean the surface included within the circumference, and sometimes the circumference itself. Euclid has employed the word (TEPpepsia) periphery, both for the whole, and for a part of the circumference of a circle. If the word circumference were restricted to mean the whole circumference, and the word arc to mean a part of it, ambiguity might be avoided when speaking of the circumference of a circle, where only a part of it is the subject under consideration. A circle is said to be given in position, when the position of its center is known, and in magnitude, when its radius is known. Def. I. And it may be added, or of which the circumferences are equal. And conversely: if two circles be equal, their diameters and radii are equal; as also their circumferences. Def. 1. states the criterion of equal circles. Simson calls it a theorem ; and Euclid seems to have considered it as one of those theorems, or axioms, which might be admitted as a basis for reasoning on the equality of circles. Def. II. There seems to be tacitly assumed in this definition, that a straight line, when it meets a circle and does not touch it, must necessarily, when produced, cut the circle. A straight line which touches a circle, is called a tangent to the circle; and a straight line which cuts a circle is called a secant. Def. IV. The distance of a straight line from the center of a circle is the distance of a point from a straight line, which has been already explained in note to Prop. XI. page 53. Def. vI. X. An arc of a circle is any portion of the circumference ; and a chord is the straight line joining the extremities of an arc. Every chord except a diameter divides a circle into two unequal segments, one greater than, and the other less than a semicircle. And in the same manner, two radii drawn from the center to the circumference, divide the circle into two unequal sectors, which become equal when the two radii are in the same straight line. As Euclid, however, does not notice re-entering angles, a sector of the circle seems necessarily restricted to the figure which is less than a semicircle. A quadrant is a sector whose radii are perpendicular to one another, and which contains a fourth part of the circle. Def. VII. No use is made of this definition in the Elements. Def. XI. The definition of similar segments of circles as employed in the Third Book is restricted to such segments as are also equal. Props. XXIII. and XXIV. are the only two instances, in which reference is made to similar segments of circles. Prop. I. "Lines drawn in a circle," always mean in Euclid, such lines only as are terminated at their extremities by the circumference. If the point G be in the diameter CE, but not coinciding with the point F, the demonstration given in the text does not hold good. At the same time, it is obvious that G cannot be the centre of the circle, because GC is not equal to GE. Indirect demonstrations are more frequently employed in the Third Book than in the First Book of the Elements. Of the demonstrations of the forty-eight propositions of the First Book, nine are indirect: but of the thirty-seven of the Third Book, no less than fifteen are indirect demonstrations. The indirect is, in general, less readily appreciated by the learner, than the direct form of demonstration. The indirect form, however, is equally satisfactory, as it excludes every assumed hypothesis as false, except that which is made in the enunciation of the proposition. It may be here remarked that Euclid employs three methods of demonstrating converse propositions. First, by indirect demonstrations as in Euc. 1. 6: III. 1, &c. Secondly, by shewing that neither side of a possible alternative can be true, and thence inferring the truth of the proposition, as in Euc. 1. 19, 25. Thirdly, by means of a construction, thereby avoiding the indirect mode of demonstration, as in Euc. 1. 47. III. 37. Prop. II. In this proposition, the circumference of a circle is proved to be essentially different from a straight line, by shewing that every straight line joining any two points in the arc falls entirely within the circle, and can neither coincide with any part of the circumference, nor meet it except in the two assumed points. It excludes the idea of the circumference of a circle being flexible, or capable under any circumstances, of admitting the possibility of the line falling outside the circle. If the line could fall partly within and partly without the circle, the circumference of the circle would intersect the line at some point between its extremities, and any part without the circle has been shewn to be impossible, and the part within the circle is in accordance with the enunciation of the Proposition. If the line could fall upon the circumference and coincide with it, it would follow that a straight line coincides with a curved line. From this proposition follows the corollary, that "a straight line cannot cut the circumference of a circle in more points than two.' Commandine's direct demonstration of Prop. II. depends on the fol lowing axiom, "If a point be taken nearer to the center of a circle than the circumference, that point falls within the circle." Take any point E in AB, and join DA, DE, DB. (fig. Euc. III. 2.) Then because DA is equal to DB in the triangle DAB; therefore the angle DAB is equal to the angle DBA; (1. 5.) but since the side AE of the triangle DAE is produced to B, therefore the exterior angle DEB is greater than the interior and opposite angle DAE; (1. 16.) but the angle DAE is equal to the angle DBE, therefore the angle DEB is greater than the angle DBE. And in every triangle, the greater side is subtended by the greater angle; therefore the side DB is greater than the side DE; but DB from the center meets the circumference of the circle, therefore DE does not meet it. Wherefore the point E falls within the circle: and E is any point in the straight line AB: therefore the straight line AB falls within the circle. Prop. VII. and Prop. vIII. exhibit the same property; in the former, the point is taken in the diameter, and in the latter, in the diameter produced. PROP. VIII. An arc of a circle is said to be convex or concave with respect to a point, according as the straight lines drawn from the point meet the outside or inside of the circular arc: and the two points found in the circumference of a circle by two straight lines drawn from a given point to touch the circle, divide the circumference into two portions, one of which is convex and the other concave, with respect to the given point. Prop. IX. This appears to follow as a Corollary from Euc. III. 7. Prop. xI. and Prop. XII. In the enunciation it is not asserted that the contact of two circles is confined to a single point. The meaning appears to be, that supposing two circles to touch each other in any point, the straight line which joins their centers being produced, shall pass through that point in which the circles touch each other. In Prop. XIII. it is proved that a circle cannot touch another in more points than one, by assuming two points of contact, and proving that this is impossible. Prop. XIII. The following is Euclid's demonstration of the case, in which one circle touches another on the inside. If possible, let the circle EBF touch the circle ABC on the inside, in more points than in one point, namely in the points B, D. (fig. Euc. III. 13.) Let P be the center of the circle ABC, and Q the center of EBF. Join P, Q; then PQ produced shall pass through the points of contact B, D. For since P is the center of the circle ABC, PB is equal to PD, but PB is greater than QD, much more then is QB greater than QD. Again, since the point Q is the center of the circle EBF, QB is equal to QD; but QB has been shewn to be greater than QD, which is impossible. One circle therefore cannot touch anotheron the inside in more points than in one point. Prop. xvi. may be demonstrated directly by assuming the following axiom; "If a point be taken further from the center of a circle than the circumference, that point falls without the circle." If one circle touch another, either internally or externally, the two circles can have, at the point of contact, only one common tangent. Prop. XVII. When the given point is without the circumference of the given circle, it is obvious that two equal tangents may be drawn from the given point to touch the circle, as may be seen from the diagram to Prop. VIII. The best practical method of drawing a tangent to a circle from a given point without the circumference, is the following: join the given point and the center of the circle, upon this line describe a semicircle cutting the given circle, then the line drawn from the given point to the intersection will be the tangent required. Circles are called concentric circles when they have the same center. Prop. XVIII. appears to be nothing more than the converse to Prop. XVI., because a tangent to any point of a circumference of a circle is a straight line at right angles at the extremity of the diameter which meets the circumference in that point. Prop. xx. This proposition is proved by Euclid only in the case in which the angle at the circumference is less than a right angle, and the demonstration is free from objection. If, however, the angle at the circumference be a right angle, the angle at the center disappears, by the two straight lines from the center to the extremities of the arc becoming one straight line. And, if the angle at the circumference be an obtuse angle, the angle formed by the two lines from the center, does not stand on the same arc, but upon the arc which the assumed arc wants of the whole circumference. If Euclid's definition of an angle he strictly observed, Prop. xx. is geometrically true, only when the angle at the center is less than two |