join the several points where the right lines meet the planes by the lines A C, LM N, and B D. In the triangle B A D since L M and BD are at the same time in parallel planes, and in the same plane, they must be parallel, for they In the same way AC and M N are parallel. Hence by the two triangles we have AL: LB: AM: MD, CN: ND::AM: MD, cannot meet. therefore AL: LB::CN : N D. PROPOSITION XXIX. H K (55) If two right lines be not in the same plane, planes may be drawn through them which are parallel, and only two such planes can be drawn. B E Let the lines be A B and C D. Through any point A of the line A B draw A E parallel to C D, and through any point C of the line CD draw CF parallel to AB. The planes of the angles BAE and FCD are parallel (44). It is evident that no other parallel planes can be drawn through A B and C D. PROPOSITION XXX. (56) If two right lines be not in the same plane, a third right line may be drawn intersecting them at right angles, and only one such line can be drawn. Let parallel planes be drawn through the given right lines, and also let planes be drawn through each of them at right angles to those parallel planes. The intersection of these two planes will evidently intersect the given right lines at right angles, and will be the only line which can be so drawn. PROPOSITION XXXI. (57) The right line which intersects perpendicularly two right lines not in the same plane, is the shortest line which can be drawn between those two right lines. R BOOK II. Of Solid Figures which are bounded by Planes. (63) SINCE three planes are necessary to form a solid angle, it is evident that they cannot completely enclose a solid space. There will be in one direction a void which cannot be closed without one additional plane at least, Hence it appears, that less than four planes cannot enclose solid space, and therefore a solid figure cannot have less than four faces, the plane figures which enclose such a solid being called faces. The solid may also be conceived to be bounded by the right lines formed by the intersections of the planes of its faces. These are called its edges; and as every distinct pair of faces has a distinct intersection, a solid figure will have as many edges as it has distinct pairs of faces. By the principles of algebra, it follows, that if n be the 1. number of bounding planes or faces, is the number of edges. n n - Thus if the number of faces be 4, the number of edges is 6 × 5 or 15. In this if the number of faces be 5, the number of edges is number of faces be 6, the number of edges is manner we may construct a table showing the number of edges corresponding to any given number of faces: 6 7 8 9 10 28 36 45 Faces 4 5 which may easily be continued to any number. (64) Solid figures receive denominations expressive of the number of their faces; thus a figure with four faces is called a tetraedron, one with six faces an hexaedron, and so on. Generally, solids with more than six faces are called polyedrons. (65) Solids also receive denominations according to the figures and position of their faces, as in the following instances. (66) DEF.-A prism is a solid figure two of whose faces are equal and similar rectilinear figures so placed that their equal sides are respectively parallel, the other faces being parallelograms formed by right lines joining the vertices of the corresponding angles of these rectilinear figures. These figures are called the bases of the prism, and the edges formed by the right lines which are drawn connecting the vertices are called the sides of the prism. C Let ABCD and abcd be equal and similar rectilinear figures described upon parallel planes. And let A B and a b, two homologous sides, be parallel, and so placed that the vertices A and a of corresponding angles will be opposite. It will then follow that all the other homologous sides of the figures will be parallel each to each. For since AB and a b are parallel, and also the planes of the angles B A D and bad, and these angles themselves are equal, it follows that the sides A D and a d are parallel (45); and the same may be proved successively of each pair of homologous sides. A Since A B and ab are equal and parallel, the figure A B ba is a parallelogram, and in like manner it may be shown that the other faces formed by the lines joining the corresponding vertices of the bases are parallelograms. It is evident that all the sides of a prism are equal. (67) DEF.-The altitude of a prism is the perpendicular distance between its bases. (68) DEF.-A prism is said to be right or oblique, according as its sides are perpendicular or oblique to its bases. (69) DEF.-Prisms are denominated from the nature of their bases, triangular, quadrangular, pentagonal, &c. (70) DEF.-A prism whose bases are parallelograms is called a parallelopiped. A parallelopiped is therefore an hexaedron all whose faces are parallelograms, and each pair of faces which do not actually intersect are parallel. Any two parallel faces may be taken as the bases of the prism. If the bases of a parallelopiped be rectangles and its sides be perpendicular to them, all the faces will evidently be rectangles. In this case it is called a rectangular parallelopiped. (71) DEF.-If the bases of a rectangular parallelopiped be squares, and the altitude be equal to the side of the base, all its faces will be squares. Such a parallelopiped is called a cube. PROPOSITION I. (72) If the bases of two prisms be equal and similar, and two homologous sides of the bases be equally inclined to the sides of the prisms with which they form a solid angle, the several sides of each prism will be inclined to the sides of the base which they meet at angles which are respectively equal. P Let the base abcde be placed upon the base ABCDE, so that the several homologous sides shall coincide. Let the side bg be inclined to ba and bc, at the same angles as B G is inclined to BA and B C. Hence the side bq coincides with B G. Since the point a coincides with A and the lines BAF and b af are in the same plane, and the angles B A F and b af are equal, the line af must coincide with AF; and in the same manner it may be proved that the several sides of the prism whose base is a b c d e will coincide with the corresponding sides of the other prism, and therefore the angles under these sides and those of the base are respectively equal to each. PROPOSITION II. (73) If two prisms have equal bases and one pair of corresponding sides equal and similarly inclined to the sides of the bases with which they form solid angles, the prisms will be equal in every respect. For by the demonstration of (72) it appears, that the base of one may be so applied to the base of the other that the several sides of the one will respectively coincide with the sides of the other; and since these sides are equal the opposite bases must coincide, and therefore the several vertices of the one prism will coincide with those of the other, and the two solids will fill exactly the same spaces and be bounded by the same lines and planes. (74) COR.-Hence it obviously follows, that right prisms, which have equal and similar bases and equal altitudes, are equal in all respects. PROPOSITION III. (75) If two parallelopipeds have three conterminous edges in the one equal to three conterminous edges in the other and including angles which are equal each to each, the parallelopipeds are equal in all respects. For if two conterminous edges in one be equal to two in the other, the faces of which these edges are sides will be equal, and thus the proposition becomes a particular case of (73). (76) COR.-If the vertex of a solid angle of a parallelopiped be given in position, and the three edges terminated at that vertex be given in magnitude and position, the parallelopiped is given. (77) PROPOSITION IV. Every prism may be divided into as many triangular prisms as there are triangles into which its base may be resolved by diagonals drawn from the vertex of any of its angles. Since each pair of sides are equal and parallel, it follows that the diagonals of the bases which connect the extremities of the sides are equal and parallel, and the figure formed by the sides and diagonal is therefore a parallelogram. There are as many of these parallelograms, which we shall call diagonal planes, as there are different diagonals of the bases of the prism; and it is evident that the prism may be resolved into triangular prisms by diagonal planes, all of which pass through any one side and severally through the other sides, except those which are adjacent to that side which is their common intersection. This will be evident on inspecting the figure. (78) COR.-It is evident that each diagonal plane is parallel to the sides of the prism, and also that the intersection of any two such planes is parallel to the sides. PROPOSITION V. (79) The sections of a prism by parallel planes are similar and equal rectilinear figures. Let A B C D E and a b c d e be two parallel sections. Since A B and a b are the intersections of parallel planes with the same plane they are parallel, therefore Aab B is a parallelogram, and therefore A B and a bare equal. In the same manner it may be proved that BC is equal to bc, C D to cd, and so on. Since the sides of the angle ABC are parallel to those of abc, and in the same direction, the angle A B C is equal to a b c; and in like manner it may be proved that the angle B C D is equal to b c d, and so on. Hence the two sections ABCDE and abc de are equal and similar. (80) COR. 1.-Hence all sections of a prism parallel to its bases are equal and similar to its bases. (81) COR. 2.-All sections of a parallelopiped parallel to any face are parallelograms equal and similar to that face. |