P H TG ; tilineal angle ; it is required to ciescribe upon the given straight line AB a segment of a circle containing an angle equal to the angle C. First, let the angle at C be a right angle, and bisect (10. 1.) AB in F, and from the centre F, at the distance FB, describe the semicircle AHB; :herefore the angle AHB in a semicircle, is (31. 3.) equal to the right angle at C. But, if the angle C be not a right angle, at the point A, in the straight line AB, make (23. 1.) the angle BAD equal to the angle C, and from the point A draw (11. 1.) AE at right angles to AD; bisect (10. 1.) AB in F, and from F draw (11. 1.) FG at right angles to AB, and join GB : And because AF is equal to FB, and FG common to the triangles AFG, BFG, the two sides AF, FG are equal to the two BF, and the angle AFG is equal to the angle BFG: therefore the base AG is equal (4. 1.) to the base GB : and the circle described from the centre G, at the distance GA, shall pass through the point B ; let this be the circie AHB : And because from the point A the extremity of the diameter AE, AD is drawn at right angles to AE, therefore AD (Cor. 16. 3.) touches the circle ; and because AB drawn from the point of contact A cuts the circle, the angle DAB is equal to the angle in the alternate segment AHB : (32. 3.) But the angle DAB is equal to the angle C, Therefore also the angle C is equal to the angle in the D segment AHB. Wherefore upon the given straight line AB the segment AHB of a circle is described, which contains an angle equal to the given angle at C. Which was to be done. Proposition XXXIV. Problem. D a To cut off a segment from a given circle which shall contain an angle equal to a given rectilineal angle. Let ABC be the given circle, and D the given rectilineal angle ; it is required to cut off a segment from the circle ABC that shall contain an angle equal to the given angle D. Drav (17.3.) the straight line EF touching the circle ABC in the point B, and at the point B, in the straight line BF, make (23. 1.) the angle FBC equal to the angle D: de Therefore because the straight line EF touches the circle ABC, and BC is drawn from the point of contact B, the angle FBC is equal (23.3.) to the angle in the alternate segment BAC of the circle : But the angle FBC is equal to the angle D; therefore the angle in the segment BAC is equal to the angle D; Wherefore the segment BAC is cut off from the given circle ABC, containing an angle equal to the given angle D. Which was to be done. : Proposition XXXV. Theorem. It two straight lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. Let the two straight lines AC, BD, within the circle ABCD, cut one an her in the point E: the rectangle contained by AE, EC is equal to the rectangle contained by BE, ED. If AC, BD pass each of them through the centre, so that E is the centre ; it is evident, that AE, EC, BE, ED, being all equal, the rectangle AE, EC is likewise equal to the rectangle BE, ED. But let one of them BD pass through the centre, and cut the other AC which does not pass through the centre at right angles, in the point E : Then, if BD be bisected in F, F is the centre of the circle ABCD; join AF: And because BÔ, which passes through the centre, cuts the straight line AC, which does not pass through the centre, at right angles in E, AE, EC are equal (3. 3.) to one another : And because the straight line BD is cut into two equal parts in the point F, and into two unequal in the point E, the rectangle BE, ED, together with the square of EF, is equal (5. 2.) to the square of FB ; that is, to the square of FA; but the squares of AE, EF are equal (47. 1.) to the square of FA ; therefore the rectangle BE, FD, together with the square of EF, is equal to the squares of AE, EF: Take away the common square of EF, and the remaining rectangle BE, ED is equal to the remaining square of AE; that is, to the rectangle AE, EC. Next, let BD, which passes through the centre, cut the other AC, which does not pass through the centre in E, but not at right angles : Then, as before, if BD be bisected in F, F is the centre of the circle. Join AF, and from F draw (12. 1.) FG perpendicular to AC; therefore AG is equal (3.3.) to GC; wherefore the rectangle AE, EC, together with the square of EG, is equal (5. 2.) to the square of AG: To each of these equals add the square of GF; therefore the rectangle AE, EC, together with the squares of EG, GF, is equal to the squares of AG, GF: But the squares of EG, GF are equal (47. 1.) to the square of EF ; and the squares of AG, GF are equal to the square of AF: Therefore the rectangle AE, EC, together with the square of EF, is equal to the square of AF; that is to the square of FB : But the square of FB is equal (5. 2.) to the rectangle BE, ED, together with the square of EF; therefore the rectangle AE, EC, together with the square of EF is equal to the rectangle BE, ED, together with the square of EF: Take away the common square of EF, and the remaining rectangle AE, EC, is there. fore equal to the remaining rectangle BE, ED. Lastly, let neither of the straight lines AC, BD pass through the con H : B tre : Take the centre F, and through E the intersection of the straight lines AC, DB draw the diameter GEFH: And because the rectangle AE, EC is equal, as bas been shewn to the rectangle GE, EH; and, for the same reason, the rectangle BE, ED is equal to the same rectangle GE, EH; therefore the rectangle AE, EC is equal to the rectangle BE, ED Where. fore, if two straight lines, &c. Q. E. D. Proposition XXXVI. Theorem. If from any point without a circle two straight lines be drawn, ono of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it. Let D be any point without the circle ABC, and DCA, DB two straight lives drawn from it, of which DCA cuts the circle, and DB touches the same : The rectangle AD, DC is equal to the square of DB. Either DCA passes through the centre, or it does not ; first, let it pass through the centre E, and join EB : therefore the angle EBD is a right (18. 3.) angle : And because the straight line AC is bisected in E, and produced to the point D the rectangle AD, DC, together with the square o EC, is equal (6. 2.) to the square of ED, and CE i equal to EB : Therefore the rectangle AD, DC, to gether with the square of EB, is equal to the square of ED: But the square of ED is equal (47. 2.) to B. E the squares of EB, BD, because EBD is a right angle : Therefore the rectangle AD, DC, together with the square of EB, is equal to the squares of EB, BD: Take away the coinmon square of EB : therefore the remaining rectangle AD, DC is equal to the square of the tangent DB. But if DCA does not pass through the centre of the circle ABC: take (1.3.) the centre E, and draw EF perpendicular (12. 1.) to AC, and join EB, EC, ED: And because the straight line EF, which passe. through the centre, cuts the straight line AC which does not pass through the centre, at right angles, it shall likewise bisect (3. 3.) it; therefore AF is equal to FC · And because the straight line AC is bisected in F, and produced to D, the rectangle AD, DC, to gether with the squarc of FC is equal (6. 2.) to the square of FD: To each of these equals add the square of FE; therefore the rectangle AD, DC, together with the squares of CF, FE, is equal to the squares of DF, FE : But the square of ED is equal (47. 1.) to the squares of DF, FE, because EFD is a right angle : and the square of EC is equal to the squares of CF, FE; therefore the rectangle AD, DC, together with the square of EC, is equal to the square of ED, and CE is equal : B to EB ; therefore the rectargle AD, DC, together with the square of EB is equal to the square of ED : But the squares of EB, BD are equal to the square (47. 1.) of ED, because EBD is a right angle ; therefore the rectangle AD, DC, together with the square of EB, is equal to the squares of EB, BD: Take away the common square of EB ; therefore the remaining rectangle AD, DC is equal to the square of DB. Wherefore, if from any point, &c. Q. E. D. Cor. If from any point without a circle, there be drawn iwo 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 of the straight I'ne AD which touches the circle. : Proposition XXXVII. Theorern. 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 of 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 lives DCA and DB be drawn, of which DCA cuts the circle, and DB meets it; if the rectangle AD, DC be equal to the square of DB ; DB touches the circle. Draw (17.3.) the straight line DE, touching the circle ABC, find its centre F, and join FE, FB, FD; then FED is a right (18.3) angle : And because DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal (36. 3.) to the square of DE : But the rectangle AD, DC is, by hypothesis, equal to the square of DB : Therefore the square of DE is equal to the square of DB ; and the straight line DE equal to the straight line DB. And FE is equal to PB, wherefore DE, EF are equal to DB, BF; and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF, is equal (8. 1.) to the angle DBF; BA but DEF is a right angle, therefore also DBF is a right angle: And FB, if produced, is a diameter, and the straight line which is drawn at right angles to a diameter, from the extremity of it touches (16. 3.) the circle: There fore DB touches the circle ABC. Wherefore, if from a point, &c. Q. E. D. D C I. scribed about a circle, when cach A RECTILINBAL figure is said to side of the circumscribed figure be inscribed in another rectilineal fin touches the circumference of the gure, when all the angles of the in- circle. scribed figure are upon the sides of v. the figure in which it is inscribed, Inlike manner, a circle each upon each. is said to be inscribII. ed in a rectilineal fi. In like manner, a figure gure, when the ciris said to be described cumference of the about another figure, circle touches each when all the sides of side of the figure. the circumscribed fin VI. gure pass through the angular A circle is said to be described about points of the figure about which a rectilineal figure, it is described, each through each. when the circumIII. ference of the circle A rectilineal figure is passes through all said to be inscribed the angular points in a circle, when all of the figure about the angles of the in which it is described. scribed figure are VII. upon the circumfer A straight line is said to be placed ence of the circle. in a circle, when the extremities IV. of it are in the circumference of A rectilineal figure is said to be de. the circle. PROP. I. PROB. In a given circle to place a straight line, equal to a given straight line not greater than the diameter of the circle. Let ABC be the given circle, and BC is greater D the given straight line, not greater than D; make. than the diameter of the circle. CE equal(3.1.) Draw BC the diameter of the circle to D, aud from ABC; then, if BC is equal to D, the the centre C, thing required is done; for in the at the distance circle ABC a straight line BC is CE, describe placed equal to D: But, if it is not, the circle AEF, and join CA: There |