PROPOSITION V. THEOREM. If four magnitudes are in proportion, they will be in proportion when taken inversely. If M : N :: P: Q, then will N: M:: Q : P. For, from the given proportion, we have MXQ=NXP, or, N×P=M× Q. PROPOSITION VI. THEOREM. Now, N and P may be made the extremes, and M and Q the means of a proportion (P. 2): hence N: M :: Q: P. If we have M: N:P : Q, we shall also have M±N : M :: P±Q: P. For, from the first proportion, we have If four magnitudes are in proportion, they will be in proportion by composition or division. MxQ=NxP, or NxP=MxQ Add each of the members of the last equation to, and subtract it from MXP, and we shall have, MXP±NX P=M×P±M×xQ; or PROPOSITION VII. THEOREM. (M±N)×P=(P±Q)×M. But MN and P, may be considered the two extremes, and PQ and M, the two means of a proportion (P. 2): hence, (M±N): M :: (P±Q) : P. Equimultiples of any two magnitudes, have the same ratio as the magnitudes themselves. Let M and N be any two magnitudes, and m any num. ber whatever; then will mXM, and mXN, be equal mul tiples of M and N: then mXM will be to m×N, in the ratio of M to N. For, MXN=NXM: multiplying each member by m, and we have mxMxN=mxNXM: then (P. 2), mxM: mxN:: M: N. PROPOSITION VIII. THEOREM. Of four proportional magnitudes, if there be taken any equimul tiples of the two antecedents, and any equimultiples of the two consequents, such equimultiples will be proportional. Let M, N, P, Q, be four magnitudes in proportion; and let m and n be any numbers whatever, then will m x P : nxQ. mxM : nx N For, since we have hence, m× M× n× Q=n× N×m×P, by multiplying both members of the equation by m×n. But m XM and n X Q, may be regarded as the two extremes, and n XN and mXP, as the means of a proportion; hence, mxM : nxN :: mxP : nx Q. PROPOSITION IX. THEOREM. Of four proportional magnitudes, if the two consequents be either augmented or diminished by magnitudes which have the same ratio as the antecedents, the resulting magnitudes and the antecedents will be proportional. :: M: N :: P: Q, MxQ=NxP; Let and let then will M.: N :: P: Q, M: P :: m : n; M: P :: N±m : Q±n. M : N P : :: Q, M×Q=N×P. MxQ±Mxn=N×P±P×m, or Mx(Q±n)=Px(N±m): hence (P. 2), M: P: N±m : Q±n. For, since and since therefore, PROPOSITION X. THEOREM. If any number of magnitudes are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of the consequents. P Q R S, &c. and, M: N :: add to each Let M : N :: Then since, MN:: PQ, we have M×Q=N×P, then, M×N+M×Q+M×S=M×N+N×P+N×R, or, MX(N+Q+S)=N×(M+P+R); therefore (P. 2), M: N :: M+P+R: N+Q+S. PROPOSITION XI. THEOREM. If two magnitudes be each increased or diminished by like parts of each, the resulting magnitudes will have the same ratio as the magnitudes themselves. Let M and N be any two magnitudes and like parts of each. We have add to both, or subt. or, MXN MXN MXN MXN m and we have (A. 2), M×N± = Let Then will, m MXN m M M(N± ~~) = N(M± 12), M that is (P. 2), M: N :: M± : N± m N M m m and member by member MXN =MXN± m PROPOSITION XII. THEOREM. If four magnitudes are proportional, their squares or cubes will also be proportional. · M : N: P: Q, By squaring both members, MxQ=N°×P3, and by cubing both members, M3× Q3=N3×P3; 3 therefore, and Let M3 : N3 p3 Cor. In a similar way it may be shown that like powers or roots of proportional magnitudes are proportionals. and then will For, since 2 M2 N2 :: p? If there be two sets of proportional magnitudes, the products of the corresponding terms will be proportionals. therefore, PROPOSITION XIII. THEOREM. N :: P: Q, [, Q3. : M : R: MXR : and we shall have M×Q×R×V=N×P×S×T, or, NXS :: PXT : MxQ=NxP, RX V=SXT, MX RX QX V=NXSXPXT; MXR NXS :: PXT : QX V. QX V. PROPOSITION XIV. THEOREM. If any number of magnitudes are continued proportionals; then, the ratio of the first to the third will be duplicate of the common ratio; and the ratio of the first to the fourth will be triplicate of the common ratio; and so on. For, let A be the first term, and m the common ratio: the proportional magnitudes will then be represented by A, m1×A, m3×A, m3×A, m1×A, &c.: Now, the ratio of the first to any one of the following terms exactly corresponds with the enunciation. BOOK III. THE CIRCLE, AND THE MEASUREMENT OF ANGLES. DEFINITIONS. 1. The CIRCUMFERENCE OF A CIRCLE is a curve line, all the points of which are equally distant from a point within, called the centre. The circle is the portion of the plane terminated by the circumference. 2. Every straight line, drawn from the centre to the circumference, is called a radius, or, semidiameter. Every line which 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 also equal, and each double the radius. 3. Any part of the circumference is called an arc. A straight line joining the extremities of an arc, and not passing through the centre, is called a chord, or subtense of the arc.* 4. A SEGMENT is the part of a circle included between an arc and its chord. 5. A SECTOR is the part of the circle included between an arc, and the two radii drawn to the extremities of the arc. * In all cases, the same chord belongs to two arcs, and consequently, also to two segments: but the smaller one is always meant, unless the contrary is expressed. |