Of the Sphere. This polyedron may be considered as formed of pyramids, each having for its vertex the centre of the sphere, and for its base one of the faces of the polyedron. Now, the solidity of each pyramid, will be equal to one third of the product of its base by its altitude (Th. xvii). But if we suppose the faces of the polyedron u be continu ally diminished, and consequently, the number of the pyramids to be constantly increased, the polyedron will finally become the sphere, and the bases of all the pyramids will become the surface of the sphere. When this takes place, the solidities of the pyramids will still be equal to one third the product of the bases by the common altitude, which will then be equal to the radius of the sphere. Hence, the solidity of a sphere is equal to one third of the product of the surface by the radius. The surface of a sphere is equal to the convex surface of the circumscribing cylinder; and the solidity of the sphere is two thirds the solidity of the circumscribing cylinder. Let MPNQ be a great circle of the sphere; ABCD the circumscribing square if the semicircle PMQ, and the half square PADQ, are at the same time made to revolve about the diameter PQ, the semicircle will describe the sphere, while the half square will describe the cylinder circumscribed about that sphere. D M N The altitude AD, of the cylinder, is equal to the diameter Of the Sphere. PQ; the base of the cylinder is equal to the great circle, since its diameter AB is equal MN; hence, the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter (Th. ii). This meas D M N B ure is the same as that of the surface of the sphere (Th. xxiii): hence, the surface of the sphere is equal to the convex surface of the circumscribing cylinder. In the next place, since the base of the circumscribing cylinder is equal to a great circle, and its altitude to the diameter, the solidity of the cylinder will be equal to a great circle multiplied by a diameter (Th. xiv. Cor). But the solidity of the sphere is equal to its surface multiplied by a third of its radius; and since the surface is equal to four great circies (Th. xxiii. Cor.), the solidity is equal to four great circles multiplied by a third of the radius; in other words, to one great circle multiplied by four-thirds of the radius, or by two-thirds of the diameter; hence, the sphere is two-thirds of the circumscribir g cylinder. Appendix. 4. The dodecaedron, is a solid bounded by twelve equal 5. The icosaedron, is a solid, bounded by twenty equa squilateral triangles. 6. The regular solids may easily be made of pastebcard. Draw the figures of the regular solids accurately on pasteboard, and then cut through the bounding lines: this will give figures of pasteboard similar to the diagrams. Then, cut the other lines half through the pasteboard, after which, turn up the parts, and glue them together, and you will form the bodies which have been described. ELEMENTS OF TRIGONOMETRY. INTRODUCTION. SECTION I. OF LOGARITHMS. 1. The logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number, in order to produce the first number. This fixed number is called the base of the system, and may be any number except 1: in the common system 10 is assumed as the base. 2. If we form those powers of 10, which are denoted by entire exponents, we shall have From the above table, it is plain, that 0, 1, 2, 3, 4, &c., are respectively the logarithms of 1, 10, 100, 1000, 10000, &c.; we also see that the logarithm of any number between 1 and 10 is greater than 0 and less than 1: thus Log 2 0.301030 |