« AnteriorContinuar »
those faces likewise parallel; and the straight line which joms two opposite angles passes through the centre of the solid. That the opposite faces and edges are parallel, is evident from the construction of the solid; and hence it is evident (11. and 7.) that the lines XO, X O' drawn to their centres from the cen- A tre of the solid, are in
one and the same straight line: therefore, again, because the opposite edges are parallel, it may easily be shown that the lines XA, XA' which are drawn from the centre to the opposite angles A, A', lie in the same straight line (15. and I. 4.)
Upon examining the number of the solid angles in each of these figures, it will appear, that the tetrahedron has four solid angles, which is also the number of its faces; the cube eight, which is the number of faces of the octahedron;
the octahedron six, which is the number of faces of the cube; the dodecahedron twenty, which is the number of faces of the icosahedron; and the icosahedron twelve, which is the number of faces of the dodecahedron. Hence it is easily inferred, that if the centres of the faces of a regular solid be taken, they will be the vertices of another regular solid inscribed in the first. In this manner a tetrahedron may be inscribed in a tetrahedron; an octahedron in a cube, and a cube in an octahedron; an icosahedron in a dodecahedron, and a dodecahedron in an icosahedron.*
With the aid of this relation it will be found, also, that a regular solid being given, any one of the regular solids which have a less number of faces, may be inscribed in it by taking for the vertices certain of the vertices of the former, or else of the centres of its faces, or of the middle points of its edges.
Thus, in the cube AF, the vertices B, D, E, F are the vertices also of an inscribed tetrahedron.
In the octahedron EF, the centres of the several faces are the vertices of an inscribed cube; and the centres of the faces E A B, EDC, FAD, FBC the vertices of an inscribed tetrahedron.
In the dodecahedron AT, the vertices H, G, A, D, P, Q, R, T, are the vertices of an inscribed cube; for AD and GH being equal, and also, because they are parallel to BC, parallel to one another (6), the figure ADHG is a parallelogram (I. 21.); but the side A D is equal to the side DH, and the diagonal A H may be shown to be equal to the diagonal DG; therefore ADHG is a rhombus, which has its two diagonals equal to one another, that is, a square; and, since the same may be shown of the other figures ADPQ,
This mutual relation of the regular solids is very striking. We may observe that if lines are drawn from the centre of the circumscribed solid to its different angular points, these lines will be perpendicular respectively to the faces of the inscribed solid: hence, if we cleave or cut away the solid angles of the circumscribed figure by planes perpendicular to these lines; and if we continue the process until we arrive at the centres of the several faces, we shall obtain the regular solid which is inscribed, and which forms as it were the nucleus of the other. There are two stages of this process, which geometers have marked by bestowing names upon the figures which the derived solids are made to assume on arriving at them. The first is when the solid angles are so far cut away that the remaining portions of the faces are regular polygons, which have twice as many sides as the original faces. The derived solids at this stage are called the ex-tetrahedron, ex-cube, ex-octahedron, ex-dodecahedron, and ex-icosahedron. They are scribing in each of the faces a regular obtained from the regular solids by infigure, having twice as many sides as the face, and then cutting away the small Pyramids which have for their vertices solid. Thus, in the face of a regular the several solid angles of the regular tetrahedron a hexagon may be inscribed by inscribing a circle in the face, joining the centre with the angles of the face, and drawing tangents to the circle at the points where the circumference is cut by the joining lines: and in a similar manner an octagon may be inscribed in a square, and a decagon in a pentagon. The number and character of the faces of any of these derived solids may be readily obtained from the number and character of the faces and solid angles rived. Thus the faces of the ex-cube are of the regular solid from which it is desix octagons and eight equilateral triangles.
&c., the inscribed solid HG ADPQRT is a cube; hence, also, the vertices A, H, P, R are the vertices of an inscribed tetrahedron; and the middle points of BC, UV, EL, SN, KO, MF, the vertices of an inscribed octahedron.
In the icosahedron AG, the centres of the several faces are the vertices of an inscribed dodecahedron ; the centres of the faces FBA, ALM, MNH, HBC, CGD, DFE, ELK, KGN, the vertices of an inscribed cube; the centres of the faces F BA, MNH, CGD, ELK, the vertices of an inscribed tetrahedron; and the middle points of the edges BC, KL, EF, HN, AM, DG, the vertices of an inscribed octahedron,
The second stage occurs when, the solid angles being still further cut away, the planes of cleavage meet at the middle points of the edges, thus reducing the original faces to regular polygons which have the same number of sides with the faces, and are inscribed in them by joining the middle points of the edges. In fact, if the edges of a regular solid be bisected, and the points of bisection joined, there will be inscribed in each of its faces a figure similar to that face, that is, an equilateral triangle, if the face be an equilateral triangle; a square, if a square; and a pentagon, if a pentagon. Here the forms of the derived solids apprise us at once of the mutual relations of their originals; the two derived from the cube and octahedron being precisely similar, as are likewise those derived from the dodecahedron and icosahedron; from which circumstance the new figures with which we are thus presented have received the names of the cuboctahedron and the icosadodecahedron. From the tetrahedron treated in this manner we obtain the octahedron.
Finally, the cleavage being continued till we arrive at the centres of the faces, we obtain the inscribed regular solids.
PROP. 51. Prob. 16.
To find the inclination of two adjoining planes of a given regular solid.
1. If the given solid be a regular tetrahedron, the required inclination is that of two angles of equilateral triangles, which, together with a third, form a solid angle, and therefore may be found by the construction given in Prop. 49.
Or thus: describe the rightangled triangle ACB, having the hypotenuse A B equal to three times the side AC; and the angle BAC will be the angle of inclination required.
2. If the solid be a cube, the angle of inclination will be a right angle (17. Cor.)
3. If an octahedron, the required inclination is that of two angles of equilateral triangles, which, together with the angle of a square, form a solid angle, and may be found as in Prop. 49.
Or thus describe the right-angled triangle ACB, having its two sides A C, CB equal, respectively, to the side and diagonal of a square, and twice the angle BA C will be the angle of A inclination required.
4. If a dodecahedron, the required inclination is that of two angles of re
It will be sufficient to notice briefly the steps which lead to the foregoing constructions.
With regard to the tetrahedron; if a perpendicular be drawn from the centre of an equilateral triangle to one of the sides, such perpendicular will be a third of the whole perpendicular which is drawn to the same side from the angle opposite to it (see the method of inscribing an equilateral triangle in a circle at III. 63.). Now, in the tetrahedron the faces are equilateral triangles, and the line which joins any of its solid angles with the centre of the opposite face is perpendicular to that face (37. Cor.); whence, by the aid of Prop. 4., the first construction.
In the octahedron, the square which divides the figure (see the construction in Prop. 50) bisects the angles made by the adjoining faces upon either side of it: and the line which joins the centre of this square with either of the two solid angles above and below it, is equal to half its diagonal, while the perpendicular drawn from the centre of the square to any of its sides is equal to half the side; whence the construction in this case.
The cases of the dodecahedron and icosahedron admit of an easy demonstration by help of the mutual relation of the dodecahedron and icosahedron mentioned in the last Scholium. For, if X be the centre of the icosahedron AG (see the figure of Prop. 50.),
XF and XG will be perpendicular to two adjoining planes of the inscribed dodecahedron, and therefore, AXG being a straight line (50. Cor. 3.), the angle AXF will measure the inclination of those planes (17. Schol. 4.): now because X F is equal to XA or XG (50. Cor. 2.), the angle AFG is a right angle (I. 19. Cor. 4.), and the angle AX F is double of the angle AGF (I. 6. and I. 19.); also, FG is the diagonal of a regular pentagon, whose side is equal to AF, and therefore FG is to AF in the medial ratio (see note p. 159). Hence the construction given for the inclination of the faces of a dodecahedron.
And that given for the icosahedron is similarly derived, from considering it as inscribed in a dodecahedron. For if X be taken, the centre of the dodecahedron LN (see the figure of Prop. 50.), XN, and XH will be perpendicular to two adjoining planes of the inscribed icosahedron, and therefore, LXN being a straight line (50. Cor. 3.) the angle LX H will measure the inclination of those planes (17. Schol. 4.) : now, if L K, KH, LH, be joined, the angle LKH will be equal to the angle EDC of a pentagon (15.), because LK and KH are parallel to ED and DC respectively; therefore, the triangle LKH is similar to KOH (II. 32.), and OH is to HK as HK to LH; and since OH is to HK in the medial ratio, OH or H N is to LH in a ratio which is the duplicate of that ratio (II. def. 11.): and, because X H is equal to XL or XN, the angle LHN is a right angle (I. 19. Cor. 4.), and LX H is equal to twice LNH (I. 6. and I. 19.), that is, the angle of inclination is equal to twice the greater acute angle of a right-angled triangle, whose sides are to one another in a ratio which is the duplicate of the medial ratio. Therefore, &c.
clination of two adjoining faces (51.); let B D meet AD in D, and join CD. Then it is evident, from the construction of 50. Cor. 2, that DA is the radius of the inscribed sphere, and DC that of the circumscribed sphere.
Or the radius DC of the circumscribed sphere may be determined in the several cases, by the following constructions; and then D A from the triangle DAC, described with the hypotenuse D C and side A C.
1. If the given solid be a tetrahedron, describe the right-angled triangle ACB (see 51. fig. 2.), having the sides A C, CB, equal respectively to the side and diagonal of a square; and the diameter of the circumscribed sphere will be to the edge of the tetrahedron as A B to B C.
2. If a cube, the diameter of the circumscribed sphere will be to the edge of the cube as AB to AC in the above triangle (51. fig. 2.); and that of the inscribed sphere will be equal to the edge.
3. If an octahedron, the diameter of the circumscribed sphere will be to the edge as the diagonal of a square to its side.
4. If a dodecahedron, describe the right-angled triangle ACB (see 51. fig. 4.), having its sides A C, C B to one another in a ratio which is the duplicate of the medial ratio; and the diameter of the circumscribed sphere will be to the edge as the hypotenuse A B to the lesser side A C.
5. If an icosahedron, describe the right-angled triangle A CB (see 51. fig. 3.), having its sides AC, CB in the medial ratio; and the diameter of the circumscribed sphere will be to the edge as the hypotenuse A B to the lesser side A C.
We need not enter into the details of these constructions: it will be sufficient,
as in the preceding problem, to point out the considerations from which they are derived respectively.
And first, a tetrahedron may be inscribed in a cube, which shall have for its four angles four of the angles of the cube, and for its four edges the diagonals of four faces of the cube (see note, p. 162); and the sphere which is cir
the edges of the cube, together with the square of the diagonal of one of the faces: hence, therefore, the constructions for the tetrahedron and cube. į
In the octahedron, the line which joins two opposite angles is at once the diameter of the circumscribed sphere, and also the diagonal of a square which has for its four sides four of the edges of the octahedron; hence the construction in this case.
In the dodecahedron (see the figure of p. 160) the triangle LHN is rightangled at H, and the sides LH, HN have to one another a ratio which is the duplicate of the medial ratio, as was shown in the last problem: also LN is the diameter of the circumscribed sphere; therefore, the rule in this case is manifest.
And, lastly, in the icosahedron (see the figure of p. 158) the triangle AFG is right-angled at F, and the sides GF, FA are to one another in the medial ratio, as was shown in the last problem; also A G is the diameter of the circumscribed sphere: whence the construction in this
Cor. Every regular solid may be divided into pyramids, having for their bases the several faces of the solid, and for their common vertex the centre of the solid; and the altitude of each of these pyramids will be the same, viz. the radius of the inscribed sphere. By help of this proposition, therefore, we may find the solid content of any given regular solid; for it will be one-third of the product of the above radius and the convex surface of the solid (32. Cor. 1.).
The regular solids have ceased to which was assigned to them for so long оссиру that prominent place in science a period, from the time of Euclid to that of Kepler". A volume, replete with the most striking results, might indeed be written upon the subject; but as these have little or no concern with anything figures, with the exception of the cube, besides themselves, such a work would be of value to the curious only. It is not surprising, perhaps, if we regard Euclid as the discoverer of the many them in the 13th, 14th, and 15th Books elegant relations which characterize of the Elements, that he should have *See the Life of Galileo, page 27.
cumscribed about such tetrahedron will
The two last books are, however, with some pro,
composed his immortal treatise, as is said to have been the case, with the sole object of demonstrating these relations; it is not, perhaps, too much to say that at that epoch, when the properties of numbers and geometrical figures were investigated for their own sake, abstractedly and without reference, as in the present day, to the body of mixed sciences dependant upon them, the five regular solids were even worthy of such a distinction. The fate of this portion of his work (so rarely now perused) is, however, a striking illustration of the lasting and transcendant nature of what is really (though humbly) useful above the merely curious and surprising. So obscure is the rank now assigned to these once interesting and all-important figures, that it may be considered even trifling, in the present treatise, to have established their construction, &c. at a greater length than usual. We must refer the reader, by way of apology, to the properties above alluded to, a few of which, capable of being verified without difficulty with the assistance which we have afforded, are here subjoined.
In the tetrahedron, the radius of the circumscribed sphere is equal to three times the radius of the inscribed sphere: in the cube (as we have seen) the two radii are to one another in the subduplicate of this ratio.
In the cube, the radii of the inscribed and circumscribed spheres have to one another the same ratio as in the octahedron: and the same is true of the dodecahedron and icosahedron.
In the icosahedron, the distance of the regular pentagon, which passes through five of the solid angles, from the centre of the solid, is equal to half the radius of the circle circumscribed about the pentagon.
These few may serve as a sample of the rest, which are occupied with the mutual inscription and circumscription of these figures, and the proportions of their surfaces and contents when inscribed in one and the same sphere.
§ 1. Surfaces and contents of the Right Cylinder and Right Cone.-§ 2. Surface and content of the Sphere.-§ 3. Surfaces and contents of certain portions of the Sphere.
SECTION 1.-Of the Right Cylinder and Right Cone.
Def. 1. A cylinder is a solid figure, the surface of which is partly plane and
Such a surface may be conceived to be generated by a straight line A a, which is carried round the circumference of a given circle ABD, so as to be always parallel and equal to a given straight line Cc at the D centre.
For it is easy to perceive that the upper extremity of such a line will always lie in the circumference of a second circle a bd, which is of the same dimensions with the given circle, and in a plane parallel to it (IV. 13. Cor. 2.).
The curved surface of a cylinder is called also the convex surface; the circles are called bases; and the straight line which joins their centres is called the axis of the cylinder.
2. A cylinder is said to be right or oblique, according as the axis is per
pendicular to the bases, or inclined to them. C is a right, and C' an oblique cylinder.
3. Similar Cylinders are those whose axes are perpendicular, or equally inclined, to their respective bases, and in
the same ratio to the radii of those bases.
4. A cone is a solid figure, the surface of which is partly plane, and partly curved; the plane portion being a if any point be taken in the circumcircle, and the curved portion such, that ference of the circle, the straight line which joins it with a certain point without the plane of the circle, lies wholly in the surface.
A curved surface of this description