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PROP. VI. THEOR. TRIANGULAR pyramids are similar, if two faces in one of them be similar to two faces in the other, each to each, and their inclinations equal.
Let ABC, abc be the bases, and D, d the vertices of two triangular pyramids, in which ABC, DBC are respectively similar to abc, dbc, and the inclination of ABC, DBC equal to that of abc, dbc : the pyramids are similar.
To demonstrate this, it is sufficient to show that the triangles ABD, ACD are similar to abd, acd, for then the solid angles (XI. B.) will be equal, each to each, and (XI. def. 8.) the pyra. mids similar. Now the angles ABD, abd are equal. For, if they be equal, since the other plane angles at B and b are equal, the inclinations of ABC, DBC, and of abc, dbc, are (XI. A.) equal. But if ABD be not equal to abd, conceive DBC to revolve about BC till ABD become i
b equal to abd: then (XI. A.) the inclination of ABC, DBC would still be equal to that of abc, dbc, which is absurd; since the inclinations of these planes, at first (hyp.) equal, are rendered unequal by the motion of DBC: therefore ABD, abd are equal. Then (hyp.) DB : BC:: db : bc, and BC: BA :: bc : ba; whence, ex æquo, DB : BA :: db: and therefore (VI. 6.) the triangles ABD, abd are equiangular, and consequently similar : and it would be proved in the same manner, that ACD, acd are similar. Therefore (XI. def. 8.) the pyramids are similar.
PROP. VII. THEOR. TRIANGULAR pyramids are similar, if three faces of one of them be respectively similar to three faces of the other.
In the triangular pyramids ABCD, abcd (see the preceding figure) let the faces ABC, ABD, DBC be similar to abc, abd, dbc, each to each : the pyramids are similar.
For (VI. def. 1.) AD: DB :: ad : db, and DB: DC :: db: dc; whence, ex æquo, AD: DC : : ad : dc. Also DC : CB : : dc : cb, and CB: CA :: cb : ca ; whence, ex æquo, DC : CA :: dc : ca; and therefore (VI. 5.) the triangles ADC, adc are equiangular, and (XI. B. and def. 8.) the pyramids are similar.
PROP. VIII. THEOR. SIMILAR polyhedrons may be divided into the same
number of triangular pyramids, similar, each to each, and similarly situated.
Let ABCDEFG and abcdefg be similar polyhedrons, having the solid angles equal which are marked with the corresponding large and small letters : they may be divided into the same number of similar triangular pyramids similarly situated.
The surfaces of the polygons may be divided (APP. V. 5.) into the same number of similar triangles, similarly situated : then planes passing through any two corresponding solid angles, A, a, and through the sides of all these triangles, except those forming
the solid angles, A, a, will divide the polyhedrons into triangular pyramids, similar to one another, and similarly situated.
The pyramids thus formed have each one solid angle at the common vertex A or a : and these solid angles may be of three classes ; lst, those which have two of their faces coinciding with faces of one of the polyhedrons ; 2d, those which have only one face coinciding; and 3d, those which lie wholly within the solid angle A or a. Now those of the first kind in one of the polyhedrons, are similar to the corresponding ones in the other, by the seventh proposition of this book; and those of the second kind by the sixth. From the polyhedrons take two of these similar pyramids, and the remaining bodies will be similar ; as the boundaries common to them and the pyramids are (APP. V. 6. or 7.) similar triangles ; and their other boundaries are similar, being faces of the proposed polyhedrons. Also the solid angles of the remaining bodies are equal, as some of them are angles of the primitive polyhedrons, and the rest are either trihedral angles which are contained by equal plane angles, or may be divided into such. From these remaining bodies other similar triangular pyramids may be taken in a similar manner; and the process may be continued till only two similar triangular pyramids remain ; and thus the polyhedrons are resolved into the same number of similar triangular pyramids.
PROP. IX. THEOR.
SIMILAR polyhedrons are to one another in the triplicate ratio of their homologous sides.
For, by the preceding proposition, they may be divided into
the same number of similar triangular pyramids : and these pyramids (XII. 8.) are to one another in the triplicate ratio of their homologous sides. But because the polyhedrons are similar, their homologous sides and diagonals are all in the same ratio ; and, ex æquo, ratios which are triplicate of equal ratios are equal. Hence, as any of the pyramids in one of the polyhedrons, to the similar one in the other, so is any other in the first to the similar one in the other; and therefore (V. 12.) the one polyhedron is to the other, as any pyramid in the first, to the similar one in the other ; that is, in the triplicate ratio of their homologous sides or edges.
Cor. l. Since cubes are similar bodies, they have to one another the triplicate ratio of that which their sides have; and therefore similar polyhedrons are to one another as the cubes of their homologous sides.
Cor. 2. If four straight lines be continual proportionals, the first is to the fourth as any polyhedron described on the first, is to a similar polyhedron, similarly described on the second.
PROP. X. THEOR.
A section of a sphere by a plane is a circle.
Since the radii of the sphere are all equal, each of them being equal to the radius of the describing semicircle, it is plain that if the section pass through the centre, it is a circle of the same radius as the sphere. But if the plane do not pass through the centre, draw (XI. 11.) a perpendicular to it from the centre, and draw any number of radii of the sphere to the intersection of its surface with the plane. These radii, which are equal, are the hypotenuses of right-angled triangles, which have the perpendicular from the centre as a common leg ; and therefore (I. 47. cor. 5.) their other legs are all equal: wherefore the section of the sphere by the plane is a circle, the centre of which is the point in which the perpendicular cuts the plane.
Schol. 1. All the sections through the centre are equal to one another, and are greater than the others. The former are therefore called great circles, the latter small or less circles.
Schol. 2. A straight line drawn through the centre of a circle of the sphere perpendicular to its plane, is a diameter of the sphere. The extremities of this diameter are called the poles of the circle. It is plain (I. 47. cor. 5.) that chords drawn in the sphere from either pole of a circle to the circumference, are all equal; and therefore (III. 28.) that arcs of great circles between the pole and circumference are likewise equal.
PROP. XI. PROB.
To find the diameter of a given sphere. Let A be any point in the surface of the given sphere, and take any three points B, C, D at equal distances from A.. Describe the triangle bed having bc equal to the distance or chord BC, cd equal to CD, and bd to BD. Find e the centre of the circle described about bcd, and join be: draw aef perpendicular to be, and make ba equal to BA; draw of perpendicular to ba, and af is
d equal to the diameter of the sphere. Conceive a circle to be described
¿ through BCD, and E to be its centre; that circle will evidently be the section of the sphere by a plane through B, C, D; and it will be equal to the circle described about bcd. Conceive the diameter AEF to be drawn, and BA, BE, BF to be joined. Then, in the right-angled triangles ABE, abe, the sides AB, BE are respectively equal to ab, be, and therefore (I. 47. cor. 5.) the angles A, a are equal. Again, in the right-angled triangles ABF, abf, the angles A, a are equal, and also the sides AB, ab: hence (I. 26.) the sides AF, af are equal ; that is, af is equal to the diameter of the sphere.