PROPOSITION IX.-THEOREM. 143. Equimultiples of two magnitudes have the same ratio as the magnitudes themselves. Let A and B be two magnitudes, and m X A and m X B their equimultiples, then will m× A: mx B:: A: B. AX B=BxA; For Multiplying each side of this equation by any number, m, we have mXAX Bmx B XA; therefore (m x A) x B = (m × B) × A. Hence, by Prop. II., mXA: m X B:: A: B. PROPOSITION X.-THEOREM. 144. Magnitudes which are proportional to the same proportionals, will be proportional to each other. : Let A B E: F, and C: D:: E: F; then will For, by the given proportions, we have A E с E = and B F' D F Therefore, it is evident (Art. 34, Ax. 1), A C B D = = Hence A: B::C: D. 145. Cor. 1. If two proportions have an antecedent and its consequent the same in both, the remaining terms will be in proportion. 146. Cor. 2. Therefore, by alternation (Prop. VI.), if two proportions have the two antecedents or the two co sequents the same in both, the remaining terms will be in proportion. PROPOSITION XI.-THEOREM. 147. If any number of magnitudes are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let ABC:D::E:F; then will A: BA+C+E: B+ D + F. By adding AX B to the sum of the corresponding sides of these equations, we have AX BAXD+A×F=AXB+B× C+ B × E. Therefore, AX (B+D+F) BX (A + C + E). Hence, by Prop. II., = AX FBX E. 148. If four magnitudes are in proportion, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. Let A B C D; then will : : A+BA-B::C+D: C-D. For, from the given proportion, by Prop. VII., we have A+BA::C+D: C; and from the given proportion, by Prop. VIII., we have ABA:: C-D: C. Hence, from these two proportions, by Prop. X. Còr. 2, we have A+BA-B::C+D: C-D. PROPOSITION XIII.-THEOREM. 149. If there be two sets of proportional magnitudes, the products of the corresponding terms will be proportionals. : Let A B C D, and E: F:: G: H; then will For, from the first of the given proportions, by Prop. I., we have AX DBX C; and from the second of the given proportions, by Prop. I., we have Hence, by Prop. II., EX HFX G. Multiplying together the corresponding members of these equations, we have AX DX EX H = BX CXFX G. AXE:BX F:: CxG: D X H. PROPOSITION XIV.-THEOREM. 150. If three magnitudes are proportionals, the first will be to the third as the square of the first is to the square of the second. Let A B B: C; then will A: C:: A2: B2. For, from the given proportion, by Prop. III., we have AXC B2. = Multiplying each side of this equation by A gives Hence, by Prop. II., A: CA: B2. PROPOSITION XV.-THEOREM. 151. If four magnitudes are proportionals, their like powers and roots will also be proportional. Let A B C : D; then will A" : B" :: C" : D", and A: BA:: C: Da. For, from the given proportion, we have A C B Ꭰ and Raising both members of this equation to the nth power, we have = A" C" B D" B = and extracting the nth root of each member, we have A CA DA Hence, by Prop. II., the last two equations give A: B" :: C" : D", AA: BA :: CA: DA, = BOOK III. THE CIRCLE, AND THE MEASURE OF ANGLES. DEFINITIONS. 152. A CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure AD BE. A D E 153. The CIRCUMFERENCE or PERIPHERY of a circle is its entire bounding line; or it is a curved line, all points of which are equally distant from a point within called the centre. 154. A RADIUS of a circle is any straight line drawn from the centre to the circumference; as the line CA, CD, or CB. 155. A DIAMETER of a circle is any straight line drawn through the centre, and terminating in both directions in the circumference; as the line AB. 156. An ARC of a circle is any part of the circumference; as the part AD, AE, or EGF. B All the radii of a circle are equal; all the diameters are also equal, and each is double the radius. Ꭰ A 157. The CHORD of an arc is the straight line joining its extremities; E thus EF is the chord of the arc EGF. |