59 PROPOSITION XI. THEOREM. In any continued proportion, the sum of the antecedents is to the sum of the consequents, as any antecedent to its corresponding consequent. From the definition of a continued proportion (D. 3), A : B hence, :: C: D :: E : F :: G: H, &c.; hence, from Proposition II., = A (B+D+F+ H+ &c.): A+ C+E+G+ &c. : B+D+F+H+ &c. :: A: B; which was to be proved. PROPOSITION XII. THEOREM. The products of the corresponding terms of two proportions are proportional. Multiplying the equations, member by member, we have, 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, bound ed by a The bounding line is called the circumference. 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. are also equal, and each is double the radius. 4. An ARC is any part of a circumference. All diameters 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 7. A SECTOR is a part of a circle included between an arc and the two radii drawn to its extremities. GEOMETRY. 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. POSTULATE. A circumference can be described from any point as a centre, and with any radius. 63 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 A B E 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. In the tri Draw the radius CD. angle ACD, we have AD less than the sum of AC and CD (B. I., P. VII.). this sum is equal to AB (D. 3): hence, AB is greater than AD; which was to be proved. But C |