tiples of M and N: then mXM will be to m×N, in the multiplying each member by m, and we have mxMxN=mx NX M: 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 tum 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 mxM : nXN :: mxP: nx Q. For, since we have hence, M : N :: P: MxQ=NxP; mxMxnx Q=n × N×m × P, by multiplying both members of the equation by mxn. But m XM and n X Q, may be regarded as the two extremes, and n x N 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. Let and let then will M : N :: P: Q For, since M: and since therefore, or M : P :: m : n; M : hence (P. 2), M: P: N±m : Q±n. PROPOSITION X. THEOREM. If any number of magnitudes are proportionals, any one antecedent will be to its consequent, as the sum of all the ante cedents to the sum of the consequents. then, MXN+M×Q+M×S=M×N+N×P+N×R, or, M×(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. M m N 914 Let M and N be any two magnitudes and and that is (P. 2), M : N :: M± : N± m N m PROPOSITION XII. THEOREM. If four magnitudes are proportional, their squares or cubes will Let Then will, also be proportional. By squaring both members, M3× Q3=N3×P2, and by cubing both members, M3× Q3=N3×P3; Cor. In a similar way it may be shown that like powers or roots of proportional magnitudes are proportionals. PROPOSITION XIII. THEOREM. If there be two sets of proportional magnitudes, the products of the corresponding terms will be proportionals. or, we shall have M× Q×R×V=N>P×S×T, therefore, MXR N×S :: PXT : M×R>Q×V=N>S×P×T 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 ▲, 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 er unciation. 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. 6. A STRAIGHT LINE is said to be inscribed in a circle, when its extremities are in the circumference. An inscribed angle is one which has its vertex in the circumference, and is included by two chords of the circle. 7. An inscribed triangle is one which has the vertices of its three angles in the circumference. And generally, a polygon is said to be inscribed in a circle, when the vertices of all its angles are in the circumference. The circumference of the circle is then said to circumscribe the polygon. 8. A SECANT is a line which meets the circumference in two points, and lies partly within, and partly without the circle. 9. A TANGENT is a line which has but one point in common with the circumference. The point where the tangent touches the circumference, is called the point of contact. 10. Two circumferences touch each other when they have but one point in common. The common point is called the point of tangency. 11. A polygon is circumscribed about a circle, when each of its sides is tangent to the circumference. In the same case, the circle is said to be inscribed in the poly gon. ооо POSTULATE. 12. Let it be granted that the circumference of a circle may be described from any centre, and with any radius. |