PROPOSITION IX. THEOREM. If two quantities be increased or diminished by like parts of each, the results will be proportional to the quantities themselves. If both terms of the first couplet of a proportion bè increased or diminished by like parts of each; and if both terms of the second couplet be increased or diminished by any other like parts of each, the results will be in proportion. 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 :: C: D :: E: F:: G: H, &c., B (A + C + E + G + &c.) = A (B + D +F+II+ &c.) : hence, from Proposition II., A+ C+E+G+&c. : B+D+F+II+ &c. ·: A : B; which was to be proved. PROPOSITION XII. THEOREM. If two proportions be multiplied together, term by term, the the products will be proportional. Assume the two proportions, A : B C D; whence, E B = Α and, E: F:: G: H; whence, = Multiplying the equations, member by member, we have; ; 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 will be the square of the corresponding term in either of the given proportions hence, If four quantities are proportional, their squares will be 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. 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. :) 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. an 4. An ARC is any part of a circumference. 5. A CHORD is a straight line joining the extremities of 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 within an an arc and the 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 in the circumference, and whose 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. 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. This point is called, the point of contact, or the point of tangency. 13. A Polygon is circumscribed about a circle, when all of its sides are tangent to the circumference. 14. A Circle is inscribed in a polygon, when its circumference touches all of the sides of the polygon. OOD хо POSTULATE. A circumference can be described from any point as a centre, and with any radius. |