PROPOSITION XII. THEOREM. The products of the corresponding terms of two proportions are proportional. Assume the two proportions, A : B C D; whence, and E : F :: G : H; whence, Multiplying the equations, member by member, we have, BF = DH whence, AE : BF :: CG : DH; which was to be proved. Cor. 1. If the corresponding terms of two proportions are equal, each term of the resulting proportion is the square of the corresponding term in either of the given proportions: hence, If four quantities are proportional, their squares are proportional. Cor. 2. If the principle of the proposition be extended to three or more proportions, and the corresponding terms of each be supposed equal, it will follow that, like powers of proportional quantities are proportionals. BOOK III. THE CIRCLE AND THE MEASUREMENT OF ANGLES. DEFINITIONS. 1. A CIRCLE is a plane figure, bounded by a curved line, every point of which is equally distant from a point within, called the centre. The bounding line is called the circumference. Ө 2. A RADIUS is a straight line drawn from the centre to any point of the circumference. 3. A DIAMETER is a straight line drawn through the centre and terminating in the circumference. All radii of the same circle are equal. All diameters are also equal, and each is double the radius. 4. An ARC is any part of a circumference. 5. A CHORD is a straight line joining the extremities of an arc. Any chord belongs to two arcs: the smaller one is meant, unless the contrary is expressed. 6. A SEGMENT is a part of a circle included between an arc and its chord. 7. A SECTOR is a part of a circle included between an arc and the two radii drawn to its extremities. 8. An INSCRIBED ANGLE is an angle whose vertex is in the circumference, and whose sides are chords. 9. An INSCRIBED POLYGON is a polygon whose vertices are all in the circumference. The sides are chords. 10. A SECANT is a straight line which cuts the circumference in two points. 11. A TANGENT is a straight line which touches the circumference in one point only. This point is called, the point of contact, or the point of tangency. 12. Two circles are tangent to each other, when they touch each other in one point only. This point is called, the point of contact, or the point of tangency. 13. A Polygon is circumscribed about a circle, when each of its sides is tangent to the circumference. 14. A Circle is inscribed in a polygon, when its circumference touches each of the sides of the polygon. ооо о POSTULATE. A circumference can be described from any point as a rentre, and with any radius. PROPOSITION I. THEOREM. Any diameter divides the circle, and also its circumference, into two equal parts. Let AEBF be a circle, and AB any diameter: then will it divide the circle and its circumference into two equal parts. F E B For, let AFB be applied to AEB, the diameter AB remaining common; then will they coincide; otherwise there would be some points in either one or the other of the curves unequally distant from the centre; which is impossible (D. 1): hence, AB divides the circle, and also its circumference, into two equal parts; which was to be proved. PROPOSITION II. THEOREM. A diameter is greater than any other chord. Let AD be a chord, and AB a diameter through one extremity, as A: then will AB be greater than AD. Draw the radius CD. In the triangle ACD, we have AD less than the sum of AC and CD (B. I., P. VII.). But this sum is equal to AB (D. 3): hence, AB is greater than AD; which was to be proved. D B PROPOSITION III. THEOREM. A straight line can not meet a circumference in more than two points. Let AEBF be a circumference, and AB a straight line: then AB can not meet the circumference in more than two points. For, suppose that AB could meet the circumference in three points. By draw F B E ing radii to these points, we should have three equal straight lines drawn from the same point to the same straight line; which is impossible (B. I., P. XV., C. 2): hence, AB can not meet the circumference in more than two points; which was to be proved. In equal circles, equal arcs are subtended by equal chords; and conversely, equal chords subtend equal arcs. 1o. In the equal circles ADB and EGF, let the arcs AMD and ENG be equal: then are the chords AD and EG equal. Draw the diameters AB D N B E and EF. If the semicircle ADB be applied to the semicircle EGF, it will coincide with it, and the semi-circumference ADB will coincide with the semi-circumference EGF. But the part AMD is equal to the part ENG, by hypothesis: hence, the point D will fall on G; therefore, |