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form a solid angle being given, the angle which two of these plane angles form with each other may be determined.
In the tetraedron. Each solid angle is formed of three angles of equilateral triangles; therefore seek, by the Problem referred to, the angle which two of these planes contain between them: it will be the inclination of two adjacent faces of the tetraedron.
In the hexaedron. The angle contained by two adjacent faces is a right angle.
In the octaedron. Form a solid angle with two angles of equilateral triangles and a right angle: the inclination of the two planes, in which the triangular angles lie, will be that of two adjacent faces of the octaedron.
In the dodecaedron. Every solid angle is formed with three angles of regular pentagons: the inclination of the planes of two of these angles will be that of two adjacent faces of the dodecaedron.
In the icosaedron. Form a solid angle with two angles of equilateral triangles and one of a regular pentagon; the inclination of the two planes, in which the triangular angles lie, will be that of two adjacent faces of the icosaedron.
PROPOSITION IV. PROBLEM.
The side of a regular polyedron being given, to find the radii of the spheres, inscribed in this polyedron and circumscribing it.
It must first be shewn, that every regular polyedron is capable of being inscribed in a sphere, and of circumscribing it.
Let AB be the side common to two adjacent faces; C and E the centres of those faces; CD, ED the perpendiculars let fall from these centres upon the common side AB, and therefore terminating in D the middle point of that side. The two perpendiculars CD, DE make with each other an angle which is known, being the inclination of two adjacent faces, and determinable by the last Problem. Now, if in the plane CDE, at right angles to AB, two indefinite lines CO and ŎE be drawn perpendicular to CD and ED, and meet
ing each other in O; this point O will be the centre of the inscribed and of the circumscribed sphere, the radius of the first being OC, that of the second OA.
For, since the apothems CD, DE are equal, and the hypotenuse DO is common, the right-angled triangle CDO must (18. I.) be equal to the right-angled triangle ŎDE, and the perpendicular OC to ÔE. But, AB being perpendicular to the plane CDE, the plane ABC (17. V.) is perpendicular to CDE, or CDE to ABC; likewise CO, in the plane CDE is perpendicular to CD, the common intersection of the planes CDE, ABC; hence (18. V.) CO is perpendicular to the plane ABC. For the same reason, EO is perpendicular to the plane ABE: hence the two straight lines CO, EO, drawn perpendicular to the planes of two adjacent faces, through the centres of those faces, will meet in the same point O, and be equal to each other. Now, sup
pose that ABC and ABE represent any other two adjacent faces; the apothem CD will still continue of the same magnitude; and also the angle CDO, the half of CDE: hence the right-angled triangle CDO, and its side CO will be equal in all the faces of the polyedron; hence, if from the point O as a centre with the radius OC, a sphere be described, it will touch all the faces of the polyedron at their centres, the planes ABC, ABE, &c. being each perpendicular to a radius at its extremity: hence the sphere will be inscribed in the polyedron, or the polyedron circumscribed about the sphere.
Again, join OA, OB: by reason of CA=CB, the two oblique lines OA, OB, lying equally remote from the perpendicular, will be equal ; so also will any other two lines drawn from the centre O to the extremities of any one side hence all those lines will be equal; hence, if from the point O as a centre, with the radius OA, a spherical surface be described, it will pass through the vertices of all the solid angles of the polyedron; hence the sphere will be circumscribed about the polyedron, or the polyedron inscribed in the sphere.
This being settled, the solution of our Problem presents no farther difficulty, and may be effected thus:
One face of the polyedron being given, describe that face; and let CD be its apothem. Find, by the last Problem, the inclination of two adjacent faces of the polyedron, and make the angle CDE equal to this inclination; take DE-CD; draw CO and EO perpendicular to CD and ED, respectively these two perpendiculars will meet in a point O; and CO will be the radius of the sphere inscribed in the polyedron.
On the prolongation of DC, take CA
equal to a radius of the circle, which circumscribes a face of the polyedron; AO will be the radius of the sphere circumscribed about this same polyedron.
For, the right-angled triangles CDO, CAO, in the present diagram, are equal to the triangles of the same name in the preceding diagram: and thus, while CD and CA are the radii of the inscribed and the circumscribed circles belonging to any one face of the polyedron, OC and OA are the radii of the inscribed and the circumscribed spheres which belong to the polyedron itself.
Scholium. From the foregoing Propositions, several consequences may be deduced.
1. Any regular polyedron may be divided into as many regular pyramids as the polyedron has faces; the common vertex of these pyramids will be the centre of the polyedron; and at the same time, that of the inscribed and of the circumscribed sphere.
2. The solidity of a regular polyedron is equal to its surface multiplied by a third part of the radius of the inscribed sphere.
3. Two regular polyedrons of the same name are two similar solids, and their homologous dimensions are proportional; hence the radii
of the inscribed or of the circumscribed spheres are to each other as the sides of the polyedrons.
4. If a regular polyedron is inscribed in a sphere, the planes drawn from the centre, along the different edges, will divide the surface of the sphere into as many spherical polygons, all equal and similar as the polyedron has faces.
THE THREE ROUND BODIES.
I. A cylinder is the solid produced by the revolution of a rectangle ABCD, conceived to turn about the immoveable side AB.
In this movement, the sides AD, BC, continuing always perpendicular to AB, describe E equal circular planes DHP, CGQ, which are called the bases of the cylinder, the side CD at the same time describing the convex surface.
The immoveable line AB is called the axis of the cylinder.
Every section KLM, made in the cylinder, at right angles to the axis, is a circle equal to either of the bases; for, whilst the rectangle ABCD turns about AB, the line KI, perpendicular to AB, describes a circular plane, equal to the base, and this plane is nothing else than the section made perpendicular to the axis at the point I. Every section PQGH, made along the axis, is a rectangle double of the generating rectangle ABCD.
II. A cone is the solid produced by the revolution of a right-angled triangle SAB, conceived to turn about the immoveable side SA.
In this movement, the side AB describes a circular plane BDCE, named the base of the cone; the hypotenuse SB describes its convex surface.
The point S is named the vertex of the cone, SA the axis or the altitude, and SB the side or the apothem.
Every section HKFI, formed at right angles to the axis, is a circle; every section SDE, formed along the axis, is an isosceles triangle double of the generating triangle SAB.
III. If from the cone SCDB, the cone SFKH be cut off by a section parallel to the base, the remaining solid CBHF is called a truncated cone, or the frustum of a cone.
We may conceive it to be described by the revolution of a trapezium ABHG, whose angles A and C are right, about the side AG. The immoveable line AG is called the axis or altitude of the frustum, the circles BDC, HFK, are its bases, and BH is its side.
IV. Two cylinders, or two cones, are similar, when their axes are to each other as the diameters of their bases.
V. If in the circle ACD, which forms the base of a cylinder, a polygon ABCDE is inscribed, a right prism, constructed on this base ABCDE, and equal in altitude to the cylinder, is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism.
The edges AF, BG, CH, &c. of the prism, being perpendicular to the plane of the base, are evidently included in the convex surface of the cylinder; hence the prism A and the cylinder touch one another along these edges.
VI. In like manner, if ABCD is a polygon, circumscribed about the base of a cylinder, a right prism, constructed on this base ABCD, and equal in altitude to the cylinder, is said to be circumscribed about the cylinder, or the cylinder to be inscribed in the prism.
Let M, N, &c. be the points of contact in the sides AB, BC, &c.; and through the points M, N, &c. let MX, NY, &c. be drawn perpendicular to the plane of the base those perpendiculars will evidently lie both in the surface of the cylinder, and
in that of the circumscribed prism; hence they will be their lines of contact.
Note. The Cylinder, the Cone, and the Sphere, are the three round bodies treated of in the Elements of Geometry.
PRELIMINARY LEMMAS CONCERNING SURFACES.
I. A plane surface OABDC is less than any other surface PABCD, terminated by the same contour ABCD.
This proposition is almost evident enough to be ranked in the class of axioms; for the plane may be regarded among surfaces, as being what the straight line is among lines; the straight line is the shortest distance between two given points; and so also, it may
easily be conceived, is the plane the least of all the surfaces having the same perimeter. Yet, since it is advisable to reduce the number of axioms as far as possible, we have subjoined a demonstration, which will remove all doubt concerning this truth.
A surface being extended in length and breadth, one surface cannot be imagined to be greater than another, unless the dimensions of the first, in some direction, exceed those of the second and if it should happen that the dimensions of one surface were, in all directions, less than the dimensions of another, the first surface would evidently be the less of the two. Now, in whatever direction we stretch the plane BPD to cut the plane surface along BD, and the other surface along BPD, the straight line BD will always be less than BPD; hence the plane surface OABCD is less than the surface PABCD, which envelopes it.
II. Every convex surface OABCD, is less than any other surface enveloping it, and resting on the same contour ABCD. We shall here repeat, that, by convex surface, we understand a surface which cannot be cut by a straight line in more than two points: a straight line, however, may in some cases be capable of applying itself A exactly in a certain direction to a convex surface; examples of this are to be seen in the surfaces of the cone and the cylinder. We may farther observe, that the name convex surface, is not limited to curve surfaces alone; it includes polyedral surfaces, or surfaces composed of several planes, and likewise surfaces partly curve and partly polyedral.
This being settled, if the surface OABCD, is not less than all those which envelope it, there must be among the latter a