M N Let M and N be any two magnitudes, and and parts of each: then will m m M: N :: M+MN+N each is equal to M.N+N.M m tities are proportional (Prop. II.). Let then will and N For, it is obvious that M× (N±. =N× (M±M) m For, or, and Let If four quantities are proportional, their squares or cubes will also be proportional. then will For since and MxQ=Nx P, since M: N:: P: Q M2 × Q2=N2 × P2, therefore, M2 : No : : P2 : Q2 and M3 N3 P3: Q3 or m therefore, PROPOSITION XII. THEOREM. m MNP Q, M2 : N2 : : P2 : Q2 Cor. In the same way it may be shown that like powers or roots of proportional quantities are proportionals. Consequently, the four quan If there be two sets of proportional quantities, the products of the corresponding terms will be proportionø PROPOSITION XIII. THEOREM. MxQ=NxP be like by squaring both members, by cubing both members; MN: P: Q MXR NXS :: P×T: Q×V : since RXV SXT, we shall have 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.* F E 2. Every straight line, CA, CE, CD, drawn from the centre to the circumference, is called a radius or semidiameter; 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. An inscribed triangle is one which, like BAC, has its three angular points in the circumference. And, generally, an inscribed figure is one, of which all the angles have their vertices in the circumference. The circle is then said to circumscribe such a figure. 7. A secant is a line which meets the circumference in two points, and lies partly within and partly without the circle. AB is a secant. 8. A tangent is a line which has but one point in common with the circumference. CD is a tangent. The point M, where the tangent touches the circumference, is called the point of contact. In like manner, two circumferences touch each other when they have but one point in common. 9. A polygon is circumscribed about a circle, when all its sides are tangents to the circumference: in the same case, the circle is said to be inscribed in the polygon. PROPOSITION I. THEOREM. B Every diameter divides the circle and its circumference into two equal parts. Let AEDF be a circle, and AB a diameter. Now, if the figure AEB be applied to AFB, their common base AB retaining its position, the curve line AEB must fall exactly on the A curve line AFB, otherwise there would, in the one or the other, be points unequally distant from the centre, which is contrary to the definition of a circle. PROPOSITION II. THEOREM. Every chord is less than the diameter. Draw the radii Let AD be any chord. CA, CD, to its extremities. have AD AC+CD (Book I. Prop. VII.*); A or AD<AB. Cor. Hence the greatest line which can be inscribed in a circle is its diameter. PROPOSITION III. THEOREM. A straight line cannot meet the circumference of a circle in more than two points. For, if it could meet it in three, those three points would be equally distant from the centre; and hence, there would be three equal straight lines drawn from the same point to the same straight line, which is impossible (Book Ï. Prop. XV. Cor. 2.). PROPOSITION IV. THEOREM. In the same circle, or in equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs. Note. When reference is made from one proposition to another, in the same Book, the number of the proposition referred to is alone given; but when the proposition is found in a different Book, the number of the Book is also given. If the radii AC, EO, are equal, and also the arcs AMD, ENG; then the chord AD will be equal to the chord EG. M N BE K For, since the diameters AB, EF, are equal, the semicircle AMDB may be applied exactly to the semicircle ENGF, and the curve line AMDB will coincide entirely with the curve line ENGF. But the part AMD is equal to the part ENG, by hypothesis; hence, the point D will fall on G; therefore, the chord AD is equal to the chord EG. Conversely, supposing again the radii AC, EO, to be equal, if the chord AD is equal to the chord EG, the arcs AMD, ENG will also be equal. PROPOSITION V. THEOREM. F For, if the radii CD, OG, be drawn, the triangles ACD, EOG, will have all their sides equal, each to each, namely, AC=EO, CD=OG, and AD=EG; hence the triangles are themselves equal; and, consequently, the angle ACD is equal EOG (Book 1. Prop. X.). Now, placing the semicircle ADB on its equal EGF, since the angles ACD, EOG, are equal, it is plain that the radius CD will fall on the radius OG, and the point D on the point G; therefore the arc AMD is equal to the arc ENG Let the arc AH be greater than the arc AD; then will the chord AH be greater than the chord AD. In the same circle, or in equal circles, a greater arc is subtended by a greater chord, and conversely, the greater chord subtende the greater arc. D H A For, draw the radii CD, CH. The two sides AC, CH, of the triangle ACH are equal to the two AC, CD, of the triangle ACD, and the angle ACH is greater than ACD; hence, the third side AH is greater than the third side AD (Book I. Prop. IX.); therefore the chord, which subtends the greater arc, is the greater. Conversely, if the chord AH is greater than AD, it will follow, on comparing the same triangles, that the angle ACH is K B |