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: B: A+ C+E:B+D+ F. :: For, from the given proportion, we have AX DBX C, and AX FBX E. By adding A X B to the sum of the corresponding sides of these equations, we have AX BAXD+A×F=AXB+BXC+BXE. Therefore, A × (B+ D +F) = B × (A + C + E). Hence, by Prop. II., A: B:: A+ C+E:B+D+ F. PROPOSITION XII. THEOREM. of 148. If four magnitudes are in proportion, the sum 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+B: A-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 AB: A:: C-D: C. Hence, from these two proportions, by Prop. X. Cor. 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 AXE:BX F:: C × G: DX H. 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 EX HFX G. Multiplying together the corresponding members of these equations, we have AX DX EX HBX CXFX G. Hence, by Prop. II., AXE:BX F:: CX G: DX 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 АХС: Multiplying each side of this equation by A gives A2 X CA X B2. Hence, by Prop. II., A: C: A2 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: B : : CÀ : Da. For, from the given proportion, we have Raising both members of this equation to the nth power, we have and extracting the nth root of each member, we have Hence, by Prop. II., the last two equations give 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. All the radii of a circle are equal; all the diameters are also equal, and each is double the radius. 156. An ARC of a circle is any part of the circumference; as the part AD, AE, or EGF. 157. The CHORD of an arc is the straight line joining its extremities; thus EF is the chord of the arc EGF. D A C E G B F 158. The SEGMENT of a circle is the part of a circle included between an arc and its chord; as the surface included between the arc A EGF and the chord EF. 159. The SECTOR of a circle is the part of a circle included between an D B C E F G arc, and the two radii drawn to the extremities of the arc; as the surface included between the arc AD, and the two radii CA, CD. 161. A TANGENT to a circle is a straight line which, how far so ever produced, meets the circumference in but one point; as the line CD. The point of meeting is called the POINT OF CONTACT; as the point M. 162. Two circumferences TOUCH each other, when they have a point of contact without cutting one another; thus two circumferences touch each other at the point A, and two at the point B. 163. A STRAIGHT LINE is INSCRIBED in a circle when its ex C A B B tremities are in the circumference; as the line A B, or B C. A 164. An INSCRIBED ANGLE is one which has its vertex in the circumference, and is formed by two chords; as the angle ABC. |