GEOMETRIC Loci. 1. Find the locus of all the points at a given distance from a given plane. 2. Find the locus of all the points, any one of which is equally distant from two given points. 3. Find the locus of all the points in space, any one of which is equally distant from two given straight lines which lie in the same plane. 4. Find the locus of all the points, any one of which is equidistant from two given planes. 5. Find the locus of all the points, any one of which is equidistant from three given points. 6. Find the locus of all the points, any one of which is at equal distances from three given straight lines situated in the same plane. 7. Find the locus of all the points, any one of which is equally distant from three given planes. 8. Find the locus of all the points, any one of which is equally distant from the three edges of a given triedral. 9. Find the locus of the points, any one of which is equidistant from two given points, and also equidistant from two given straight lines which lie in the same plane. 10. Find the locus of the points, any one of which is equidistant from two given points and also from two given planes. 11. Find the locus of the points, any one of which is equidistant from two given straight lines in the same plane, and also from two given planes. NOTE.-In problems 9, 10, and 11, we determine a locus by the intersection of two loci, as in a plane we solve a determinate problem by the intersection of two loci. 12. Find the locus of two points, the difference of the squares of the distances of each one of which from two given points is constant. 13. Find the locus of the points in space, any one of which is equally distant from all points of the circumference of a circle. 14. Find the locus of all the points in a given plane, which are at a given distance from a given point, A, without the plane. 15. Find the locus of the points in a plane such that the sum of the squares of each one of them from two given points, A and B, without the plane, is constant. 16. Find the locus of the points in a given plane, the difference of the squares of the distances of each one of which from two given points, A and B, without the plane, is constant. 17. Find the locus of the feet of the perpendiculars drawn from a given point, A, without a given plane, to the different straight lines drawn through a given point, B, in the plane. PROBLEMS. 1. Through a given point draw a straight line parallel to two given planes. 2. Through a given point drawn a plane parallel to two given straight lines. 3. Through a given point draw a plane perpendicular to two given planes. 4. Through a given point draw a straight line which shall meet two given straight lines not situated in the same plane. 5. Draw a line parallel to a given straight line which shall meet two given straight lines not situated in the same plane. 6. Find a point equidistant from four given points not in the same plane. Discuss the problem if the four points are all in the same plane. 7. Find upon a straight line a point such that the difference of the squares of its distances from two given points is a given square. 8. Through a given straight line draw a plane which shall be parallel to a given straight line. 9. In a given plane and through a given point in this plane, draw a straight line perpendicular to a straight line in space. 10. Given a plane, P, and two points, A and B, situated on the same side of the plane, find a point, M, in P, such that the sum of the distances AM, BM shall be the least possible. 11. Given a plane, P, and a triangle, ABC, find the point of the plane equidistant from the three points A, B, C. 12. Cut a quadriedral angle by a plane, so as to make the section a parallelogram. BOOK VI. POLYEDRONS. DEFINITIONS I. 1. The name solid polyedron, or simply polyedron, is given to every solid terminated by planes or plane faces. (These planes will themselves be terminated, it is evident, by straight lines.) The polyedron which has four faces, is named a tetraedron ; that which has six, a hexaedron ; that which has eight, an octaedron; that which has twelve, a dodecaedron ; that which has twenty, an icosaedron, and so on. The tetraedron is the simplest of all polyedrons; because at least three planes are required to form a solid angle, and these three planes leave an opening which requires at least a fourth plane to close it. 2. The common intersection of two adjacent faces of a polyedron is called the side or edge of the polyedron. 3. A regular polyedron is one whose faces are all equal regular polygons, and all whose solid angles are equal to each other. There are five such polyedrons. (See Appendix to Book VI.) 4. The prism is a solid bounded by several parallelograms, terminated at both ends by equal and parallel polygons. To construct this solid let ABCDE be any polygon ; then if in a plane parallel to ABC, the lines FG, GH, HI, etc., be drawn, equal and parallel to the sides AB, BC, CD, etc., thus forming the polygon FGHIK, equal to ABCDE; if, in the next place, the vertices of the angles in one plane be joined with the homologous vertices in the other,' by the straight lines AF, BG, CH, etc., the faces ABGF, BCHG, etc., will be parallelograms, and the solid ABCDEFGHIK, thus formed, will be a prism. 5. The equal and parallel polygons ABCDE, FGHIK are called the bases of the prism; the parallelograms, taken together, constitute K I E B с the lateral or convex surface of the prism ; the equal straight lines AF, BG, CH, etc., are called the sides of the prism. 6. The altitude of a prism is the distance between its two bases, or the perpendicular let fall from a point in the upper base on the plane of the lower base. 7. A prism is right when the sides, AF, BG, etc., are perpendicular to the planes of the bases; and then each of them is equal to the altitude of the prism, and the faces are rectangles. In any other case the prism is oblique, and the altitude less than the side. 8. A prism is triangular, quadrangular, pentagonal, hexagonal, etc., according as its base is a triangle, a quadrilateral, a pentagon, a hexagon, etc. 9. The prism whose base is a parallelogram has all its faces parallelograms. It is called a parallelopipedon. Parallelopipedons are right or oblique. The faces of the right parallelopipedon are rectangles. E H G F B 10. Among right parallelopipedons we distinguish the E rectangular parallelopipedon, whose bases are rectangles as well as its faces. 11. Among rectangular parallelopipedons we distinguish the regular hexaedron or cube, bounded by six equal squares. The lengths of three edges, AB, AD, AE, of a rectangular parallelopipedon, which meet in the same vertex, are called the dimensions of the parallelopipedon. The dimensions of a cube are equal. 12. A pyramid is the solid bounded by several triangular planes, proceeding from the same point, S, and terminating in different sides of the same polygon, ABCDE. The polygon ABCDE is called the base of the pyramid ; the point S its vertex, and the triangles ASB, BSC, etc., taken together, form the convex or lateral surface of the pyramid. B T D 13. The altitude of a pyramid is the perpendicular let fall from the vertex upon the plane of the base (produced, if necessary). 14. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc. Among these we note, especially, the triangular pyramid or tetraedron (mentioned in Def. 1.), a figure with four triangular faces, four vertices, and six edges. 15. A pyramid is regular when its base is a regular polygon, and when, at the same time, a perpendicular let fall from the vertex on the plane of the base passes through the centre of the base; this line is called the axis of the pyramid. 16. The frustum of a pyramid or truncated pyramid, is the part of the pyramid which remains when any part towards the vertex is cut off by a plane parallel to the base. Thus ABCDE, abcde, is a frustum of a pyramid, S, ABCDE. B 17. The diagonal of a polyedron is the straight line which joins the vertices of two solid angles which are not adjacent. 18. By the vertices of a polyedron, we mean the points situated at the vertices of its different solid angles. NOTE.—The only polyedrons we intend at present to treat of, are polyedrons with salient angles, or convex polyedrons. They are such that their surface cannot be intersected by a straight line in more than two points. In polyedrons of this kind, the plane of any face, when produced, can in no case cut the solid; the polyedron, therefore, cannot be in part above the plane of any face and part below it. It must be wholly on the same side of this plane. 19. By the volume or solidity of a polyedron, we mean its magnitude or extent. PROPOSITION I. THEOREM. Two polyedrons having the same number of vertices, and these vertices being the same points, will coincide. For, suppose one polyedron to be already constructed ; if a second is to be formed, having the same vertices, and in the same number, the planes of the latter cannot all pass through the same points with |