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those faces likewise parallel; and the This mutual relation of the regular straight line which joins two opposite solids is very striking. We may observe angles passes through the centre of the that if lines are drawn from the centre of solid. That the opposite faces and edges the circumscribed solid to its different are parallel, is evi

angular points, these lines will be perdent from the con

A pendicular respectively to the faces of struction of the so

the inscribed solid : hence, if we cleave lid; and hence it is

or cut away the solid angles of the evident (11. and 7.)

circumscribed figure by planes perpenthat the lines XO,

dicular to these lines; and if we conX O' drawn to their

tinue the process until we arrive at the centres from the cen- A

centres of the seyeral faces, we shall tre of the solid, are in

obtain the regular solid which is inone and the same straight line: there- scribed, and which forms as it were the fore, again, because the opposite edges nucleus of the other. There are two are parallel, it may easily be shown that stages of this process, which geometers the lines XA, X A' which are drawn have marked by bestowing names upon from the centre to the opposite angles the figures which the derived solids are A, A', lie in the same straight line made to assume on arriving at them. (15. and I. 4.)

The first is when the solid angles are so

far cut away that the remaining porScholium.

tions of the faces are regular polygons,

which have twice as many sides as the Upon examining the number of the original faces. The derived solids at solid angles in each of these figures, it this stage are called the ex-tetrahedron, will appear, that the tetrahedron has four ex-cube, ex-octahedron, ex-dodecahesolid angles, which is also the number dron, and ex-icosahedron. They are of its faces; the cube eight, which is obtained from the regular solids by inthe number of faces of the octahedron; scribing in each of the faces a regular the octahedron six, which is the number of faces of the cube; the dodecahedron figure, having twice as many sides as the twenty, which is the number of faces of face, and then cutting away the small the icosahedron ; and the icosahedron pyramids which have for their vertices twelve, which is the number of faces of solid. Thus, in the face of a regular

the several solid angles of the regular the dodecahedron. Hence it is easily tetrahedron a hexagon may be inscribed inferred, that if the centres of the faces by inscribing a circle in the face, joining of a regular solid be taken, they will be the centre with the angles of the face, and the

vertices of another regular solid in- drawing tangents to the circle at the scribed in the first. In this manner a tetrahedron may be inscribed in a te- points where the circumference is cut by trahedron; an octahedron in a cube,

the joining lines: and in a similar manand a cube in an octahedron; an icosa

ner an octagon may be inscribed in a hedron in a dodecahedron, and a dode- square, and a decagon in a pentagon.

The number and character of the faces cahedron in an icosahedron.*

of any of these derived solids may be

readily obtained from the number and * With the aid of this relation it will be found, character of the faces and solid angles also, that a regular solid being given, any one of the regular solids which have a less number of faces, rived. Thus the faces of the ex-cube are

of the regular solid from which it is demay be inscribed in it by taking for the vertices certain of the vertices of the former, or else of the cen. six octagons and eight equilateral tritres of its faces, or of the middle points of its edges. Thus, in the cube AF, the vertices B, D, E, F are

angles. 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 &c., the inscribed solid HG AD PQRT is a cube ; centres of the faces E A B, EDC, FAD, FBC the hence, also, the vertices A, H, P, R are the vertices vertices of an inscribed tetrahedron.

of an inscribed tetrahedron ; and the middle points In the dodecahedron A T, the vertices H, G, A, D, of BC, UV, EL, SN, KO, MF, the vertices of an P, Q, R, T, are the vertices of an inscribed cube; for AD and GH being equal, and also, because they are In the icosahedron AG, the centres of the several parallel to BC, parallel to one another (6), the figure faces are the vertices of an inscribed dodecahedron; ADHG is a parallelogram (I. 21.); but the side A D the centres of the faces F BA, AL M, MNH, HBC, is equal to the side DH, and the diagonal A H be shown to be equal to the diagonal D G; therefore

CGD, DFE, ELK, KGN, the vertices of an inscribed

cube; the centres of the faces F BA, MNH, CGD, ADHG is a rhombus, which has its two diagonals ELK, the vertices of an inscribed tetrahedron; and equal to one another, that is, a square; and, since the middle points of the edges BC, KL, EF, HN, the saine may be shown of the other figures AD PQ, AM, DG, the vertices of an inscribed octahedron,

inscribed octahedron.

may

The second stage oceurs 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 icosado. decahedron. From the tetrahedron

PROP. 51. Prob. 16. treated in this manner we obtain the octahedron.

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 Fig. 1. 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.

A 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

Fig. 2. right-angled triangle ACB,

B having its two sides AC, CB equal, respectively, to the side and diagonal of a square, and twice the angle

B AC will be the angle of Finally, the cleavage being continued inclination required. till we arrive at the centres of the faces, 4. If a dodecahedron, the required we obtain the inscribed regular solids. inclination is that of two angles of re

B

a

B

gular pentagons which, together with XF and X G will be perpendicular to a third angle of a pentagon, form a two adjoining planes of the inscribed solid angle, and therefore may be found dodecahedron, and therefore, A XG as in Prop. 49.

Fig. 3.

being a straight line (50.Cor.3.), the angle Or thus: describe the

B

AXF will measure the inclination of those right-angled triangle A CB,

planes (17. Schol. 4.): now because XF having its sides AC, CB

is equal to X A or XG (50. Cor. 2.), to one another in the medial

the angle AFG is a right angle (I. 19. ratio, and AC the lesser of

Cor. 4.), and the angle A XF is double the two; and twice the angle

of the angle AGF (I. 6. and I. 19.); BAC will be the angle re

also, FG is the diagonal of a regular quired.

pentagon, whose side is equal to AF, and 5. If an icosahedron, the required in- therefore FG is to A F in the medial clination is that of two angles of equila. ratio (see note p. 159). Hence the con. teral triangles, which, together with the struction given for the inclination of the angle of a regular pentagon, form a solid faces of a dodecahedron. angle, and therefore may be found as in And that given for the icosahedron Prop. 49.

Fig. 4.

is similarly derived, from considering it Or thus : describe the right

as inscribed in a dodecahedron. For

if X be taken, the centre of the dodeangled triangle A CB, having its sides AC, CB to one an

cahedron LN (see the figure of Prop.

50.), XN, and XH will be perpendiother in a ratio which is the

cular to two adjoining planes of the duplicate of the medial ratio ;

inscribed icosahedron, and therefore, and twice the angle B A C will be the angle required.

LXN being a straight line (50. Cor.3.) A с

the angle L XH will measure the incliIt will be sufficient to notice briefly nation of those planes (17. Schol. 4.): the steps which lead to the foregoing now, if LK, KH, LH, be joined, the constructions.

angle LKH will be equal to the angle With regard to the tetrahedron; if a EDC of a pentagon (15.), because LK perpendicular be drawn from the centre and KH are parallel to ED and DC of an equilateral triangle to one of the respectively; therefore, the triangle sides, such perpendicular will be a LKH is similar to KOH (II. 32.), third of the whole perpendicular which and OH is to H K as H K to LH; is drawn to the same side from the angle and since O H is to H K in the medial opposite to it (see the method of in- ratio, OH or H N is to LH in a ratio scribing an equilateral triangle in a which is the duplicate of that ratio circle at III. 63.). Now, in the tetrahe- (II. def. 11.): and, because X H is equal dron the faces are equilateral triangles, to XL or X N, the angle LHN is a and the line which joins any of its solid right angle (I. 19. Cor. 4.), and L XH angles with the centre of the opposite is equal to twice LNH (I. 6. and I. 19.), face is perpendicular to that face (37. that is, the angle of inclination is equal Cor.); whence, by the aid of Prop. 4., to twice the greater acute angle of a the first construction.

right-angled triangle, whose sides are In the octahedron, the square which to one another in a ratio which is the divides the figure (see the construction duplicate of the medial ratio. in Prop. 50) bisects the angles made by Therefore, &c. the adjoining faces upon either side of it: and the line which joins the centre of this

PROP. 52. Prob. 17. square with either of the two solid angles above and below it, is equal to half The edge of any regular solid being its diagonal, while the perpendicular given, to find the radii of the inscribed drawn from the centre of the square to and circumscribed spheres. any of its sides is equal to half the side; Find AB and AC, the D whence the construction in this case. radii of the circles in

The cases of the dodecahedron and scribing and circumicosahedron admit of an easy demon- scribing a face of the stration by help of the mutual rela. given solid (III. 26.); tion of the dodecahedron and icosahe- from A draw AD perpen. dron mentioned in the last Scholium. dicular to AB or AC; at

C For, if X be the centre of the icosahe- the point B make the angle ABD equal dron AG (see the figure of Prop. 50.), to half the angle which measures the in

A

B

clination of two adjoining faces (51.); the edges of the cube, together with the let B D meet A D in D, and join C D. square of the diagonal of one of įthe Then it is evident, from the construction faces : hence, therefore, the constructions of 50. Cor. 2, that DA is the radius for the tetrahedron and cube. i of the inscribed sphere, and D C that In the octahedron, the line which of the circumscribed sphere.

joins two opposite angles is at once the Or the radius DC of the circum- diameter of the circumscribed sphere, scribed sphere may be determined in and also the diagonal of a square which the several cases, by the following con- has for its four sides four of the edges of structions; and then D A from the tri- the octahedron; hence the construction angle D AC, described with the hypo- in this case. tenuse D C and side A C.

In the dodecahedron (see the figure 1. If the given solid be a tetrahedron, of p. 160) the triangle LH N is rightdescribe the right-angled triangle ACB angled at H, and the sides LH, HN (see 51. fig. 2.), having the sides AC, have to one another a ratio which is the C B, equal respectively to the side and duplicate of the medial ratio, as was diagonal of a square; and the diame- shown in the last problem : also L N is ter of the circumscribed sphere will be the diameter of the circumscribed sphere; to the edge of the tetrahedron as AB therefore, the rule in this case is manito BC.

fest. 2. If a cube, the diameter of the cir- And, lastly, in the icosahedron (see the cumscribed sphere will be to the edge of figure of p. 158) the triangle A FG is the cube as AB to A C in the above right-angled at F, and the sides GF, FA triangle (51. fig. 2.); and that of the are to one another in the medial ratio, inscribed sphere will be equal to the as was shown in the last problem ; also edge.

A G is the diameter of the circumscribed 3. If an octahedron, the diameter of sphere: whence the construction in this the circumscribed sphere will be to the case. edge as the diagonal of a square to its Therefore, &c. side.

Cor. Every regular solid may be di4. If a dodecahedron, describe the vided into pyramids, having for their right-angled triangle ACB (see 51. bases the several faces of the solid, and fig. 4.), having its sides AC, C B to one for their common vertex the centre of the another in a ratio which is the duplicate solid; and the altitude of each of these of the medial ratio; and the diameter of pyramids will be the same, viz. the rathe circumscribed sphere will be to the dius of the inscribed sphere. By help edge as the hypotenuse A B to the lesser of this proposition, therefore, we may side A C.

find the solid content of any given regu. 5. If an icosahedron, describe the lar solid; for it will be one-third of right-angled triangle ACB (see 51. fig. the product of the above radius and 3.), having its sides A C, CB in the the convex surface of the solid (32. medial ratio; and the diameter of the Cor. 1.). circumscribed sphere will be to the edge

Scholium. as the hypotenuse A B to the lesser side

The regular solids have ceased to AC. We need not enter into the details of which was assigned to them for so long

оссиру that prominent place in science these constructions: it will be sufficient, as in the preceding problem, to point of Kepler*. A volume, replete with the

a period, from the time of Euclid to that out the considerations from which they most striking results, might indeed be are derived respectively.

written upon the subject; but as these And first, a tetrahedron may be inscribed in a cube, which shall have for have little or no concern with anything

figures,

with the exception of the cube, its four angles four of the angles of the besides themselves, such a work would cube, and for its four edges the diago- be of value to the curious only. It is nals of four faces of the cube (see note, not surprising, perhaps, if we regard p. 162); and the sphere which is cir

Euclid as the discoverer of the many cumscribed about such tetrahedron will be also circumscribed about the cube; them in the 13th, 14th, and 15th Books

elegant relations which characterize but in a cube, the line which joins two

of the Elements 1, that he should have opposite angles is the diameter of the

* See the Life of Galileo, page 27. circumscribed sphere, and the square of

+ The two last books are, however, with some pro, this line is equal to the square of one of bability, ascribed to Apollonius.

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composed his immortal treatise, as is partly curved; the plane portions being said to have been the case, with the sole two equal and parallel circles, and the object of demonstrating these relations; curved portion such, that any point being it is not, perhaps, too much to say that taken in the circumference of either at that epoch, when the properties of circle, the straight line which is drawn numbers and geometrical figures were through it parallel to the line joining investigated for their own sake, ab- their centres lies wholly in the surstractedly and without reference, as in face. the present day, to the body of mixed Such a surface may be conceived to sciences dependant upon them, the five be generated by a straight regular solids were even worthy of such line A a, which is carried a distinction. The fate of this portion round the circumference of his work (so rarely now perused) is, of a given circle ABD, however, a striking illustration of the so as to be always palasting and transcendant nature of what rallel and equal to a given is really (though humbly) useful above straight line Cc at the D! B

А the merely curious and surprising. So centre. obscure is the rank now assigned to For it is easy to perceive that the upthese once interesting and all-important per extremity of such a line will always figures, that it may be considered even lie in the circumference of a secund trifling, in the present treatise, to have circle a bd, which is of the same dimenestablished their construction, &c. at a sions with the given circle, and in a greater length than usual. We must plane parallel to it (IV. 13. Cor. 2.). refer the reader, by way of apology, to The curved surface of a cylinder is the properties above alluded to, a few of called also the convex surface; the cirwhich, capable of being verified without cles are called bases ; and the straight difficulty with the assistance which we line which joins their centres is called have afforded, are here subjoined. the axis of the cylinder.

In the tetrahedron, the radius of the 2. A cylinder is said to be right or circumscribed sphere is equal to three oblique, according as the axis is pertimes 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.

pendicular to the bases, or inclined to In the icosahedron, the distance of the them, C is a right, and C' an oblique regular pentagon, which passes through cylinder. five of the solid angles, from the centre of 3. Similar Cylinders are those whose the solid, is equal to half the radius of the axes are perpendicular, or equally incircle circumscribed about the pentagon. clined, to their respective bases, and in

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.

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the same ratio to the radii of those BOOK V.

bases. § 1. Surfaces and contents of the Right

4. A cone is a solid figure, the surCylinder and Right Cone. -- 2. Sur- face of which is partly plane, and partly face and content of the Sphere.§ 3. curved; the plane portion being a Surfaces and contents of certain por- if any point be taken in the circum

circle, and the curved portion such, that tions of the Sphere.

ference of the circle, the straight line Section 1.-Of the Right Cylinder which joins it with a certain point withand Right Cone.

out the plane of the circle, lies wholly in Def. 1. A cylinder is a solid figure, the surface. the surface of which is partly plane and A curved surface of this description

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