6 Th. Two straight lines, which are at right angles to the same plane are parallel to each other. Recite p. 47, 48 of b. 1; p. 2 and 5 of this; p. 28 of b. 1. 7 Th. If two straight lines be parallel, and one of them at right angles to a plane; the other is also at right angles to the same plane. Recite p. 6 of this; ax. 11 of b. 1. 8 Th. Two straight lines, which are, each of them, parallel to the same straight line, though not both in the same plane with it, are parallel to each other. Recite p. 4, 7, 6 of this. 9 Th. If two straight lines, which meet each other, be parallel to other two, though not in the same plane with the first two, the first two and the other two shall contain equal angles. Recite p. 33 and 8 of b. 1; p. 4,7, and def. 1 of this. 10 P. It is required to draw a straight line parallel to a plane. Recite p. 12, 11, 31 of b. 1; p. 8 of this. 11 Th. From the same point in a plane there cannot be two straight lines at right angles to the plane, upon the same side of it. And there can be but one perpendicular to a plane from a point above it. Recite p. 3, 6 of this. 12 Th. Planes to which the same straight line is perpendicular, are parallel to each other. Recite def. 1 and 7 of this; p. 17 of b. 1. 13 Th. If two straight lines meeting one another, be parallel to two straight lines which also meet one another, but are not in the same plane with the first two, the plane which passes through the first two is parallel to the plane which passes through the others. Recite p. 10, 8, 4, 12, and def. 1 of this ; p. 31, 29 of b. 1. 14 Th. If two parallel planes be cut by another plane, their lines of section with it are parallels. Recite Note def. 4 of b. 1. 15 Th. If two parallel planes be cut by a third plane, they have the same declination from that plane. Recite p. 14, 4, 7, and def. 4 of this; p. 29 of b. 1. 16 Th. If two straight lines be cut by parallel planes, they must be cut in the same ratio. Recite p. 14 of this; p. 2 of b. 6; p. 11 of b. 5. 17 Th. If a straight line be at right angles to a plane, every plane which passes through that line is at right angles to the first mentioned plane. Recite def. 1, 2, and p. 7 of this ; p. 28 of b. 1. 18 Th. If two planes cutting each other, be severally perpendicular to a third plane, their line of section is perpendicular to the same plane. Recite def. 2, 1, and p. 4 of this. 19 P. Two straight lines, not in the same plane, being given in position, to draw a straight line perpendicular to them both. Recite p. 10, 13 and its cor., and def. 1 of this. 20 Th. If a solid angle be contained by three plane angles, any two of those angles are greater than the third. Recite p. 23, 4, 20, 25 of b. 1. 21 Th. The plane angles which contain any solid angle are together less than four right angles. Recite p. 20 of this p. 32 of b. 1. THE COMPARISON OF SOLIDS. Definitions. 1. A solid is that which has length, breadth and thickness. 2. Similar solid figures are such as are contained by the same num ber of similar planes, similarly situated, and declining equally from each other. 3. A pyramid is a solid figure contained by planes that are constituted between a plane and a point above it, in which they meet. 4. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar and parallel to each other; and the others are parallelograms. 5. A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. 6. A cube is a solid figure contained by six equal squares. 7. A sphere is a solid figure described by the revolution of a semicircle about a stationary diameter. 8. The axis of a sphere is the stationary straight line about which the semicircle revolves. 9. The centre of a sphere is the middle point of the axis. 10. The diameter of a sphere is any straight line passing through the centre to the surface on either side. 11. A cone is a solid figure described by the revolution of a right angled triangle about one of the sides containing the right angle, which remains fixed. 12. The axis of a cone is the stationary straight line about which the triangle revolves. 13. The base of a cone is the circle described by the revolving side containing the right angle. 14. A cylinder is a solid figure described by the revolution of a rectangle about one of its sides, which is fixed. 15. The axis of a cylinder is the stationary side of the rectangle which revolves. 16. The bases of a cylinder are the two opposite circles described by the ends of the rectangle. 17. Similar cones and cylinders are those which have their axes, and the diameter of their bases proportionals. Propositions. i Th. If two solids be contained by the same number of equal and similar planes, similarly situated ; and if the declination of any two contiguous planes in the one solid be the same with the declination of the two equal and similarly situated planes in the other, the two solids are equal and similar. Recite ax. 8 of b. 1. 2 Th. If a solid be contained by six planes, two and two of which are parallel, the opposite planes are equal and similar parallelograms. Recite p. 14, 9, last article, and p. 4 of b. 1. 3 Th. If a solid parallelopiped be cut by a plane parallel to two of its opposite planes, it will be divided into two solids which will be to one another as the bases. Recite p. 36, 1, and def. 1, 6; p. 2, 1 of this; p. 15 of last art., def. 5 of b. 5. 4 Th. If a solid parallelopiped be cut by a plane passing through the diagonals of two of the opposite planes, it will be cut into two equal prisms. Recite p. 8, 14, 15 last art., p. 34 of b. 1; p. 2, 1 of this. 5 Th. Solid parallelopipeds upon the same base, and of the same altitude, the insisting straight lines of which are terminated in the same straight lines in the plane opposite to the base, are equal to one another. Recite p. 34, 38, 36 of b. 1; p. 15 last art., p. 2, 1 of this. 6 Th. Solid parallelopipeds upon the same base, and of the same altitude, the insisting straight lines of which are not terminated in the same straight lines in the plane opposite to the base, are equal to one another. Recite def. 5 of this ; p. 5 of the last article. 7 Th. Solid parallelopipeds, which are upon equal bases, and of the same altitude, are equal to one another. Recite p. 11 last art., p. 14, 35 of b. 1; p. 7, 9 of b. 5; p. 3, 5, 6 of this. 8 Th. Solid parallelopipeds which have the same altitude, are to each other as their bases. Recite p. 45 of b. 1 ; p. 7, 3, 4 of this; p. 4 of b. 5. 9 Th. Solid parallelopipeds are to each other in the ratio composed of the ratios of the areas of their bases and of their altitudes. Recite def. 5, and p. 8,7 of this; def. 10, and p. F, E of b. 5; p. 44 of b. 1; p. 12, 1 of b. 6, and cor. 2, p. 8 of this. 10 Th. Solid parallelopipeds which have their bases and altitudes reciprocally proportional, are equal ; and parallelopipeds which are equal, have their bases and alti. tudes reciprocally proportional. Recite p. 11, 9, and def. 10 of b. 5; p. 9 of this. 11 Th. Similar solid parallelopipeds are to each other in the triplicate ratio of their homologous sides. Recite def. 2 of this ; def. 1, and p. 23 of b. 6; def. 9 of b. 5. 12 Th. If two triangular pyramids, which have equal bases and altitudes, be cut by planes that are parallel to the bases, and at equal distances from them, the sections are equal to each other. Recite p. 14, 9, 16 of the last art., p. 18, 1 of b. 5; p. 22 of b. 6, 13 Th. A series of prisms of the same altitude may be described about any pyramid, such that the sum of the prisms shall exceed the pyramid by a solid less than any given solid. Recite cor. 1, p. 12, def. 4, aud cor. 1, p. 8 of this. 14 Th. Pyramids that have equal bases and altitudes are equal to one another. Recite p. 13, 12, and cor. 1, p. 8 of this. 15 Th. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and that are equal to one another. Recite p. 34 of b. 1; p. 14, and cor. 1, p. 8 of this. 16 Th. If from any point in the circumference of the base of a cylinder, a straight line be drawn perpendicular to the plane of the base ; that line will be wholly in the cylindric superficies. Recite def. 14 of this, p. 6, 14 last article. 17. Th. A cylinder and a parallelopiped having equal bases and altitudes, are equal to each other. Recite cor. 1, p. 4 Qu. Cir., p. 16, and cor. 2, p. 8 of this. 18 Th. If a cone and cylinder have the same base and the same altitude, the cone is the third part of the cylinder. Recite p. 4 Qu. Cir., p. 15 of this. 19 Th. If a hemisphere and a cone have equal bases . and altitudes, a series of cylinders may be inscribed in the hemisphere, and another series may be described about the cone, having all the same altitudes with one another, and such that their sum shall differ from the sum of the hemi. sphere and the cone, by a solid less than any given solid. Recite def. 7, 11, 14, and p. 13 of this. 20 Th. The same things being supposed as in the last proposition, the sum of all the cylinders inscribed in the hemisphere, and described about the cone, is equal to a cylinder, having the same base and altitude with the hemisphere. Recite cor. 2, p. 6 Qu. Cir. 21 Th. Every sphere is two-thirds of the circumscribing cylinder. Recite p. 18, 19, 20 of this. THE END. |