PROPOSITION IV. THEOREM. If there be four proportional quantities, and four other proportional quantities, having the antecedents the same in both, the consequents will be proportional. Q S hence N=R; or N:Q::R: S. Cor. If there be two sets of proportionals, having an antecedent and consequent of the first, equal to an antecedent and consequent of the second, the remaining terms will be proportional. PROPOSITION V. THEOREM. If four quantities be in proportion, they will be in proportion when taken inversely. For, from the first proportion we have MxQ=Nx P, or NxP=MxQ. But the products Nx P and MQ are the products of the extremes and means of the four quantities N, M, Q, P, and these products being equal, N:M::Q:P (Prop. II.). PROPOSITION VI. THEOREM. If four quantities are in proportion, they will be in proportion by composition, or division. Let, as before, M, N, P, Q, be the numerical representatives of the four quantities, so that M:N::P.Q; then will For, from the first proportion, we have MxQ=Nx P, or Nx P=MxQ; Add each of the members of the last equation to, or subtract it from M.P, and we shall have, M.P+N.P-M.P+M.Q; or (M±N) × P=(P±Q) × M. But M±N and P, may be considered the two extremes, and P+Q and M, the two means of a proportion: hence, M±N:M:: P+Q: P. PROPOSITION VII. THEOREM. Equimultiples of any two quantities, have the same ratio as the quantities themselves. Let M and N be any two quantities, and m any integral number; then will m. M: m. N::M:N. For m. MxN=m. Nx M, since the quantities in each member are the same; therefore, the quantities are proportional (Prop. II.); or m. M: m. N::M: N. PROPOSITION VIII. THEOREM. Of four proportional quantities, if there be taken any equimultiples of the two antecedents, and any equimultiples of the two consequents, the four resulting quantities will be proportional. Let M, N, P, Q, be the numerical representatives of four quantities in proportion; and let m and n be any numbers whatever, then will m. M: n. N:: m. P: n. Q. For, since M:N::P: Q, we have MxQ=NxP; hence, m. Mxn. Q=n. Nxm. P, by multiplying both members of the equation by mxn. But m. M and n. Q, may be regarded as the two extremes, and n. N and m. P, as the means of a proportion; hence, m. M: n. N::m. P: n. Q. PROPOSITION IX. THEOREM. Of four proportional quantities, if the two consequents be either augmented or diminished by quantities which have the same ratio as the antecedents, the resulting quantities and the antecedents will be proportional. If any number of quantities are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of the consequents. Let For, since And since M: NP: Q : : R : S, &c. then will and we have, M.N+M.Q+M.S=M.N+N.P+N.R or therefore, MN: M+P+R: N+Q+S. PROPOSITION XI. THEOREM. If two magnitudes be each increased or diminished by like parts of each, the resulting quantities will have the same ratio as the magnitudes themselves. Let M and N be any two magnitudes, and and parts of each: then will M N be like m m MN: M± :N± Mx For, it is obvious that M× (N±) =N× (M± m tities are proportional (Prop. II.). Consequently, the four M N since m m quan PROPOSITION XII. THEOREM. If four quantities are proportional, their squares or cubes will M2x Q2 N2x P2, by squaring both members, therefore, M2 : N2 : : P2 : Q2 and M3 N3: P3: Q3 by cubing both members; Cor. In the same way it may be shown that like powers or roots of proportional quantities are proportionals. PROPOSITION XIII. THEOREM. If there be two sets of proportional quantities, the products of the corresponding terms will be proportional. BOOK III. THE CIRCLE, AND THE MEASUREMENT OF ANGLES. Definitions. 1. The circumference of a circle is a curved line, all the points of which are equally distant from a point within, called the centre. The circle is the space terminated by A this curved line.* 2. Every straight line, CA, CE, CD, drawn from the centre to the circumference, is called a radius or semidiam F H eter; every line which, like AB, passes through the centre, and is terminated on both sides by the circumference, is called a diameter. From the definition of a circle, it follows that all the radii are equal; that all the diameters are equal also, and each double of the radius. 3. A portion of the circumference, such as FHG, is called an arc. The chord, or subtense of an arc, is the straight line FG, which joins its two extremities.† 4. A segment is the surface or portion of a circle, included between an arc and its chord. 5. A sector is the part of the circle included between an arc DE, and the two radii CD, CE, drawn to the extremities of the arc. 6. A straight line is said to be inscribed in a circle, when its extremities are in the circumference, as AB. An inscribed angle is one which, like BAC, has its vertex in the circumference, and is formed by two chords. B *Note. In common language, the circle is sometimes confounded with its circumference: but the correct expression may always be easily recurred to if we bear in mind that the circle is a surface which has length and breadth, while the circumference is but a line. Note. In all cases, the same chord FG belongs to two arcs, FGH, FEG, and consequently also to two segments: but the smaller one is always meant, unless the contrary is expressed. |