Cor. If the antecedents of one proportion are equal to the antecedents of another proportion, the consequents are pro portional. If and then will A: B::C: D, A:E::C: F; B:D::E: F. For, by alternation (Prop. III.), the first proportion be If four quantities are proportional, they are also proportional when taken inversely. If four quantities are proportional, they are also proportion. al by composition. (A+B) ×C=A× (C+D). Therefore, by Prop. II., A+B: A:: C+D: C. PROPOSITION VII. THEOREM. If four quantities are proportional, they are also proportion al by division. Equimultiples of two quantities have the same ratio as the quantities themselves. Let A and B be any two quantities, and mA, mB their equimultiples; then will AB:: mA : mB. mxAxB=mxAxB, AxmB=BxmA. For or, Therefore, by Prop. II., A:B::mA : mB. PROPOSITION IX. THEOREM. If any number of quantities are proportional, any one ante cedent is to its consequent, as the sum of all the antecedents. is to the sum of all the consequents. and we have or, AxB+AxD+AxF=AxB+BxC+BxE; Therefore, by Prop. II., A: B:: A+C+E : B+D+F. PROPOSITION X. THEOREM. If four quantities are proportional, their squares or cubes are also proportional. or, multiplying each of these equals by itself (Axiom 1), we have and multiplying these last equals by AxD-BXC, we have A'XD3=B'XC. Therefore, by Prop. II., and A: B2: C: D', A3 : B3 : : C3 : D3. PROPOSITION XI. THEOREM. If there are two sets of proportional quantities, the producis of the corresponding terms are proportional. Let and then will For, since by Prop. I., And, since by Prop. I., or, A:B::C:D, E:F::G: H; AXE: BxF:: CxG: DXH. A:B::C: D, AxD-BXC. E: F::G: H, ExH=F×G. Multiplying together these equal quantities, we have AXDXEXH=B×C×F×G; (A × E) × (D×H)=(B×F)×(C×G); If three quantities are proportional, the first is to the third, as the square of the first to the square of the second. But, by Prop. VIII., AxB: BXC::A:C; hence, by Prop. IV., A: C:: A' : B'. BOOK III. THE CIRCLE, AND THE MEASURE OF ANGLES. Definitions. 1. A circle is a plane figure bounded by a line, every point of which is equally distant from a point within, called the center. This bounding line is called the circumference of the circle. 2. A radius of a circle is a straight line drawn from the center to the circumference. A diameter of a circle is a straight line passing through the center, and terminated both ways by the circumference. Cor. All the radii of a circle are equal; all the diameters are equal also, and each double of the radius. 3. An arc of a circle is any part of the circumference. The chord of an arc is the straight line which joins its two extremities. 4. A segment of a circle is the figure included between an arc and its chord. 5. A sector of a circle is the figure included between an arc, and the two radii drawn to the extremities of the arc. 6. A straight line is said to be inscribed in a circle, when its extremities are on the circumference. An inscribed angle is one whose sides are inscribed. 7. A polygon is said to be inscribed in a circle, when all its sides are inscribed. The circle is then said to be described about the polygon. 8. A secant is a line which cuts the circumference, and lies partly within and partly without the circle. 9. A straight line is said to touch a circle, when it meets the circumference, and, being produced, does not cut it. Such a line is called a tangent, and the point in which it meets the circumference, is called the point of contact. |