chord is to the second part ED of the other chord, but reciprocally as ED: EB; that is, AE: CEED: EB. Dem. Draw DB and AC. We have angle AEC= BED; angle ACE=EBD; angle CAE-EDB; therefore triangle AEC triangle DEB; therefore AE: EC-ED: EB, for they are homologous sides of similar triangles. 293. If two chords being produced cut each other, then the entire lines will be reciprocally proportional to the parts without the circle, (fig. 159.) OA:OC=OD : OB. Dem. Draw AD and BC. Angle AOD= angle COB; angle OAD= angle OCB; therefore triangle OAD triangle OCB; therefore ОА : ОС =OD: OB. The entire lines AO and CO are called secants. 294. If the secant AC meets the tangent CD of the same circle, then is the tangent CD a mean proportional between the entire line AC and the part without the circle, (fig. 160.) Dem. Draw DB and DA. Angle DCB = angle DCA. The angle CDB=CAD, (178,) therefore the triangles CDB and CAD are similar. Consequently the homologous sides are proportional; that is, 295. Ratio of the circumferences and of the surfaces of two circles. Every circle can be considered as a regular polygon with an infinite number of sides. Two circles can therefore be considered as regular polygons having an equal number of sides. Regular polygons having an equal number of sides are similar, and therefore the circles can be treated as similar figures. The circumferences and surfaces of circles will be to one another as the perimeters and surfaces of similar polygons are to one another, viz., as their homologous sides and the squares of those sides. The homologous sides of two circles are two chords subtending equal angles, by joining the extremities of which to the centre of the circle similar triangles will be formed. Therefore the two small chords will be to each other as the radii of their respective circles. Consequently the circumferences of two circles are to one another as their radii or diameters; and their surfaces as the squares of the radii or of the diameters. If the radii of two circles are to each other as 1 to 2 the circum'nces will be as 1 : 2 the surfaces as 1:4 Conversely, the diameters and radii of two circles are to one another as the square roots of the surfaces. 296. To describe a circle the surface of which shall be equivalent to the surfaces of two given circles taken together. Make a right angle; let one of the sides be equal to the diameter of one circle, and the other side be equal to the diameter of the other circle; draw the hypothenuse, and upon it as a diameter describe a circle, and this circle will be equivalent to the two given circles taken together. For all circles are similar figures, consequently the circle, whose diameter is the hypothenuse of a right-angled triangle, will be equivalent to both the circles whose diam eters are the sides which enclose the right angle taken togther. 6. SOLIDS. 297. Ratio of two cubes. The solidity of a cube is the product of three equal factors, each of which is the measure of a side of the cube; therefore the solidities of two cubes will be to each other in the ratio of these products; that is, they will be to each other as the cubes of their sides. If the side of one cube is to the side of another cube as 1:2 then their solidities will be as 1:8 Conversely, the sides of two cubes are to each other as the cube roots of their solidities. 298. Ratio of two cylinders or prisms. The solidity of a cylinder is the product of its base multiplied by its altitude. Consequently the solidities of two cylinders are to one another in the ratio of these products. If the altitudes of two cylinders be equal, the cylinders are to each other as their bases. If the bases be equal, they are to each other as their altitudes. If the solidities of two cylinders are equal, their bases are to each other reciprocally as their altitudes. 299. Ratio of the superficial and of the solid contents of two spheres. Since the surface of every sphere is equal to 4 times the surface of a great circle of that sphere, therefore the surfaces of two spheres are to each other in the ratio of 4 times the surfaces of their great circles, or in the ratio of their great circles, and consequently in the ratio of the squares of their radii, or of their diameters. From what has been said before, it follows (208,) that the solidity of a sphere whose diameter is D, is equal to the product of D X D X D X .523. The solidity of a sphere whose diameter is d is equal to the product of d d x d × .523. Consequently, the solidity of the two spheres will be to each other as DXD XD X.523: dxd xdx.523; that is, the solidities of two spheres are to each other as the cubes of the diameters of the spheres, and, consequently, as the cubes of the radii. If the diameter of one sphere is 1 and of another 2 their solidities are as 1:8 Conversely, the diameters of two spheres are to each other as the cube roots of the solidities of the same spheres. It would be a great mistake, therefore, to infer that the surface or the diameter of the sun is 1,400,000 as great as that of the earth, because the sun is 1,400,000 as great as the earth. 300. Ratio of the solidity of a sphere to the solidity of a circumscribed cube; that is, a cube the sides of which are each equal to the diameter of the sphere. It is evident that the solidity of such a cube would be greater than the solidity of the sphere, since the faces of the cube would be tangents to the sphere, and thus the sphere would be contained in the cube. What ratio will the solidity of the one body bear to that of the other? Sol. Let us call the diameter of the sphere d, then of the sphere will be to the solidity of the cube of its 301. Ratio of the solidity of a sphere to the solidity of a circumscribed cylinder; that is, a cylinder whose altitude is equal to the diameter of the sphere, and whose bases are equal to great circles of the sphere. Call the diameter of the sphere d; then will its solid314 ddd 314 ity = X ; the solidity of the cylinder: X 100 6 100 ddd; consequently the solidity of the sphere is to the 4 solidity of the circumscribed cylinder, as 314 ddd 314 ddd 100 6100 4 4:6=2:3. |