Conversely, Two tetraedrons are similar, ist, when their homologous faces are similar ; 2d, when their homologous triedrals are equal. First.—When the homologous faces are similar, the homologous edges are proportional, and therefore the tetraedrons similar. Secondly.—When the homologous triedrals are equal, the plane angles which form them are respectively equal, and therefore the homologous faces have the angles of the one equal to the angles of the other, and are therefore similar. Hence, the tetraedrons are similar. Cor. 1. Two similar tetraedrons have their six homologous diedrals equal each to each, and conversely. Cor. 2. Every section, abcde, parallel to the base of a pyramid, S-ABCDE, determines another pyramid, S-abcde, similar to the first. For the planes SEC, SEB, divide the two pyramids into tetraedrons, S-ABE and S-abe, S-BCE and S-bce, S-CDE and S-cde, similar each to each, because their faces are similar. These tetraedrons are also similarly situated. Hence, by definition, the two pyramids are similar. E B PROPOSITION XXX. THEOREM. In two similar polyedrons : 1. The homologous faces are similar, each to each, and their inclinations are the same. 2. The homologous solid angles are equal. First. The two polyedrons being composed of the same number of tetraedrons, similar each to each and similarly situated, their surfaces are also composed of the same number of triangles, similar each to each and similarly grouped. Moreover, the inclination of two adjacent triangles of the first surface is equal to the inclination of the two homologous triangles of the second; for these inclinations are either the homologous diedrals of two similar tetraedrons, or they are the sums of a like number of homologous diedrals : whence it results that two similar polyedrons are contained by the same number of faces, similar each to each, and equally inclined to each other. Secondly.—The homologous solid angles are equal; for all their plane angles and diedrals are equal each to each and similarly grouped. CoR.-The edges, the diagonals, and in general all the homologous lines of two similar polyedrons, are proportional. PROPOSITION XXXI. THEOREM. Similar tetraedrons are to each other as the cubes of their homologous edges. D We can always place the two tetraedrons so that they shall have a triedral, S, in common (Prop. XXIX, Cor. 2.). Then, since the bases, ABC, DEF, are similar, we have, ABC : DEF :: AB? : DE?. (1) And since the angles SAB and SDE are equal, as also SBC and SEF, the plane DEF is parallel to the plane ABC. Therefore, SH : Sh :: SA : SD : : AB : DE, or, SH : Sh :: AB : DE. (2) E A с H 1 B Multiplying the proportions (1) and (2), term by term, we have, ABC x LSH : DEF x {Sh :: AB® : DE. That is, S-ABC : S-DEF :: ABS : DE». PROPOSITION XXXII. THEOREM. In two similar polyedrons : 1. The surfaces are to each other as the squares of their homologous edges. 2. The volumes are to each other as the cubes of these edges. First. — The areas of similar polygons being proportional to the squares of their homologous sides, the homologous faces of the two polyedrons form a series of equal ratios; hence, the sums of these faces, that is, the surfaces of the two polyedrons, are to each other as the squares of these same sides or edges. Secondly. --Similar tetraedrons being proportional to the cubes of their homologous edges, the tetraedrons of which the two polyedrons are composed form a series of equal ratios; hence the sums of the antecedents and of the consequents, that is, the volumes of the two polyedrons, are to each other as the cubes of these same edges. Cor. —Two similar pyramids are to each other as the cubes of their homologous edges, as also two similar prisms. EXERCISES ON BOOK VI. THEOREMS. 1. Show that two tetraedrons are equal, First.—When they have an equal diedral contained by two plane faces equal each to each, and similarly situated. Second. —When they have an equal face adjacent to three diedrals, equal each to each, and similarly situated. Third. —When they have three faces equal each to each, and similarly situated. Fourth. When they have one edge equal, and the plane angles of three faces equal, and similarly situated. Fifth.--When they have an edge and five diedrals equal each to each, and similarly situated. 2. Two triangular prisms are equal when they have their lateral faces equal and arranged in the same manner. 3. The volume of a triangular prism is equal to the product of one of its lateral faces by half the distance from this face to the opposite edge. 4. Every plane which contains the line joining the middle points of two opposite faces of a parallelopipedon will divide that parallelopipedon into two equal parts. 5. Show that the formula a is verified by the geometrical construction when a and b are two parts of a given line. 6. Two tetraedrons which have a solid angle equal, are to each other as the product of the three edges which meet in the vertices of these equal solid angles. 7. The plane bisector of a diedral angle of a tetraedron divides the opposite edge into two parts proportional to the faces adjacent to this diedral. 8. If a plane be drawn, containing one edge of a tetraedron and the middle point of the opposite edge, it will divide the tetraedron into two equivalent tetraedrons. 9. Every plane which passes through the middle points of two opposite edges of a tetraedron, divides this body into two equivalent parts. 10. The six planes drawn perpendicular to the six edges of a tetraedron at their middle points, meet in a common point which is equidistant from the four vertices of the tetraedron. II. The six plane bisectors of the diedral angles of a tetraedron meet in a common point equidistant from the four faces of the tetraedron. 12. The four straight lines which join the vertices of a tetraedron with the intersections of the medians of the opposite faces meet in a common point which divides each line of junction in the ratio of 3 to 1. (That is, the part of the line towards the vertex will be to the part towards the face as 3 to 1.) a 13. The four perpendiculars erected to the faces of a tetraedron at the centres of their circumscribed circles meet in a common point. 14. The three straight lines which join the middle points of the opposite edges of a tetraedron meet in common point which bisects these lines. 15. The different points of intersection mentioned in Exercises 9 to 14 inclusive, all become one and the same point when the tetraedron is regular. 16. When a tetraedron has one of its triedrals trirectangular, then the square of the area of the face opposite to this right triedral is equal to the sum of the squares of the other three faces. 17. The distance of the centre of a parallelopipedon from any plane is equal to one-eighth of the sum of the distances of its eight vertices from the same plane. 18. If a point is at a constant distance from the centre of a parallelopipedon, the sum of the squares of its distances from the vertices is constant. 19. The altitude of a regular tetraedron is equal to the sum of the perpendiculars let fall from any point taken in the interior of the tetraedron on its four faces. 20. First.—The volume of a truncated triangular prism is equivalent to three pyramids, having the base of the prism for a common base, and the three points in which the three edges pierce the inclined cutting plane as vertices. Second. It is also equivalent to its base multiplied by one-third of the distance of this base to the point of intersection of the medians of the inclined upper base. Third.-It is also equal to its right section multiplied by one-third of the sum of its three edges. 21. The volume of a truncated parallelopipedon is equal to the product of its right section by the arithmetical mean of its four lateral edges. 22. If planes be drawn through the vertices of a tetraedron parallel to the opposite faces, the tetraedron formed by these planes is not similar to the first. |