K b 2. 3. C Nor can two circles touch one another on the outside in Book III. more than one point: for, if it be possible, let the circle ACK touch the circle ABC in the points A, C, and join AC: therefore, because the two points A, C are in the circumference of the circle ACK, the straight line AC which joins them shall fall within the circle ACK: and the circle ACK is without the circle ABC; and therefore the straight line AC is without this last circle; but, because the points A, C are in the cir cumference of the circle ABC, the straight line AC must be within the same circle, which is absurd: therefore one circle cannot touch another on the outside in more than one point: and it has been shown that they cannot touch on the inside in more points than one. Therefore one circle, &c. Q. E. D. B A PROP. XIV. THEOR. EQUAL straight lines in a circle are equally distant' from the centre; and those which are equally distant from the centre are equal to one another. C a 3. 3. G Let the straight lines AB, CD, in the circle ABDC, be equal to one another: they are equally distant from the centre. Take E the centre of the circle ABDC, and from it draw EF, EG perpendiculars to AB, CD: then, because the straight line EF, passing through the centre, cuts the straight line AB, which does not pass through the centre, at right angles, it also bisects it: A wherefore AF is èqual to FB, and AB double of AF. For the same reason, CD is double of CG: and AB is equal to CD; therefore AF is equal to CG: F and because AE is equal to EC, the square of AE is equal to the square of EC; but the squares of AF, FE, are equal to the square of AE, because the angle AFE is a right angle; and, for the like reason, CD is double of CG: the like reason, the squares of EG, GC are equal to the square of EC: therefore the squares of AF, FE are equal to the squares of CG, GE, of which the square of AF is equal to B E b 47.1. Book III. the square of CG, because AF is equal to CG; therefore the remaining square of FE is equal to the remaining square of EG, and the straight line EF is therefore equal to EG: but straight lines in a circle are said to be equally distant from the centre, when the perpendiculars drawn to them from the 4. def, 3. centre are equal: therefore AB, CD are equally distant from the centre. Next, if the straight lines AB, CD be equally distant from the centre, that is, if FE be equal to EG, AB is equal to CD: for, the same construction being made, it may, as before, be demonstrated, that AB is double of AF, and CD double of CG, and that the squares of EF, FA are equal to the squares of EG, GC; of which the square of FE is equal to the square of EG, because FE is equal to EG; therefore the remaining square of AF is equal to the remaining square of CG; and the straight line AF is therefore equal to CG: and AB is dou ble of AF, and CD double of CG; wherefore AB is equal to CD. Therefore equal straight lines, &c. Q. E. D. See Note., PROP. XV. THEOR. THE diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less. wherefore also AD is greater than BC. And, because BC is nearer to the centre than FG, EH is less than EK; but, as it was demonstrated in the preceding, Book BC is double of BH, and FG double of FK, and the squares of EH, HB are equal the squares of EK, KF, of which the square b 5. de of EH is less than the square of EK, because EH is less than EK; therefore the square of BH is greater than the square of FK, and the straight line BH greater than FK; and therefore BC is greater than FG... Next, let BC be greater than FG; BC is nearer to the cen tre than FG, that is, the same construction being made, EH is less than EK: because BC is greater than FG, BH likewise is greater than KF: and the squares of BH, HE are equal to the squares of FK, KE, of which the square of BH is greater than the square of FK, because BH is greater than FK; therefore the square of EH is less than the square of EK, and the straight line EH less than EK. Wherefore the diameter, &c. Q. E. D. PROB. XVI. THEOR. 38 THE straight line drawn at right angles to the dia- See Note. meter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn between that straight line and the circumference from the extremity, so as not to cut the circle; or which is the same thing, no straight line can make so great an acute angle with the diameter at its extremity, or so small an angle with the straight line which is at right angles` to it, as not to cut the circle. Let ABC be a circle, the centre of which is D, and the diameter AB; the straight line drawn at right angles to AB from its extremity A, shall fall without the circle. For, if it do not, let it fall, if possible, within the circle, as AC, and draw DC to the point C where it meets the circumference: and because DA is equal to DC, the angle DAC is equal to the angle ACD; but DAC is a right angle, therefore ACD is a right angle, and the angles DAC, ACD are therefore equal to two right an C A B D a 5. gles; which is impossible: therefore the straight line drawn b 17. L k III. from A at right angles to BA does not fall within the circle : in the same manner, it may be demonstrated, that it does not fall upon the circumference; therefore it must fall without the circle, as AF. c 12. 1. d 19. 1. © 2. 3.' F C - E And between the straight line AE and the circumference no straight line can be drawn from the point A which does not cut the circle: for, if possible, let FA be between them, and from the point D drawe DG perpendicular to FA, and let it meet the circumference in H: and because AGD is a right angle, and EAG less than a right angle: DA is greater than DG: but DA is equal to DH; therefore DH is greater than DG, the less than the greater, which is impossible: therefore. straight line can be drawn from the point A between AE and the circumference, which does not cut the circle; or, which amounts B to the same thing, however great an acute angle a straight line makes with the diameter at the point A, or however small an angle it makes with AE, the cir no H D cumference passes between that straight line and the perpendicular AE. And this is all that is to be understood, when, in the Greek text and translations from it, the angle of the semicircle is said to be greater than any acute rectilineal angle, and the remaining angle less than any rectilineal angle.' COR. From this it is manifest, that the straight line which is drawn at right angles to the diameter of a circle from the extremity of it, touches the circle; and that it touches it only in one point, because, if it did meet the circle in two, it would fall within ite. Also it is evident that there can be but one straight line which touches the circle in the same point.” PROP. XVII. PROB. TO draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. First, Let A be a given point without the given circle BCD: it is required to draw a straight line from A which shall touch Book III. the circle. Find the centre E of the circle, and join AE; and from the a 1. 3. centre E, at the distance EA, describe the circle AFG; from the point I drawb DF at right angles to EA, and join EBF, b 11. 1. AB. AB touches the circle BCD. Because E is the centre * of the circles BCD), AFG, c to the triangle AEB, and the other angles to the other angles : c 4. 1. But, if the given point be in the circumference of the circle, PROP. XVIH. THEOR IF a straight line touch a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle. Let the straight line DE touch the circle ABC in the point C; take the centre F, and draw the straight line FC: FC is perpendicular to DE. For, if it be not, from the point F draw FBG perpendicular to DE; and because FGC is a right angle, GCF is an acute b 17. 1. ́angle; and to the greater angle the greatest side is opposite; c 19. 1. |