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.

PROBLEM.

579.

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 shown, that every regular polyedron is capable of being inscribed in a sphere, and of being circumscribed about it.

Let AB be the side common to two adja- C cent 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

A

angles to AB, two indefinite lines CO and OE be drawn perpendicular to CD and ED, and meeting 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 (56.) be equal to the right-angled triangle ODE, and the perpendicular OC to OE. But, AB being perpendicular to the plane CDE, the plane ABC (349.) 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 (351.) CO is perpendicular to the plane ABC. For the same reason, EO is perpendicular to the plane ABE: hence the two straight lines CO, OE, 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, suppose that ABC and ABE represent any other two adjacent faces; the apothem will still continue of the same magnitude; and also the angle CDO, the half of CDE: hence the right-angled triangle CDO, and

4

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.

A

D

This being settled, the solution of our Problem presents no further 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 point 0; 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.

580. 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 simiJar 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, through the different edges, will divide the surface of the sphere into as many spherical polygons, all equal and similar, as the polyedron has faces.

NOTES

ON THE

ELEMENTS OF GEOMETRY.

NOTE I.

On some Names and Definitions.

SOME new expressions and definitions have been employed in this Work, where they seemed likely to give more accuracy and precision to geometrical language. We mean here to give some account of those changes, and to propose a few others, which might accomplish the same purpose more completely.

In the ordinary definition of the rectangular parallelogram and of the square, it is usual to say, that the angles of those figures are right; it would be more correct to say, that their angles are equal. For, to suppose that the four angles of a quadrilateral can be right, and even that the right angles are equal to each other, is to assume two propositions which require demonstration. This inconvenience, and several others of the same sort, might be avoided, if, instead of placing the definitions, according to the common practice, at the head of each Book, we were to disperse them over the course of the Book, each at the place where all it assumes is already proved.

The word parallelogram, according to its etymology, signifies parallel lines; it no more suits the figure of four sides, than it does that of six, of eight, &c. which have their opposite sides parallel. In like manner, the word parallelepipedon signifies parallel planes; it no more designates the solid with six faces, than the solid with eight, ten, &c. of which the opposite faces are parallel. The names, parallelogram and parallelepipedon, have the additional inconvenience of being very long. Perhaps, therefore, it would be advantageous to banish them altogether from geometry; and to substitute in their stead, the names rhombus and rhomboid, retaining the term lozenge, for quadrilaterals whose sides are all equal.

It might also be useful to extend the meaning of the word inclination, so as to make it synonymous with angle: both of them indicate a particular relation of two lines, or of two planes, which meet together, or would meet if produced. The inclination of two lines is nothing, when their angle is nothing; in other words, when the lines coincide, or lie parallel to each other. The inclination is greater when the angle is greater, or when two lines form together a very obtuse angle. The quality of sloping has a different meaning; a line slopes the more towards another, the more it deviates from the perpendicular to that other.

It is customary with Euclid, and various geometrical writers, to give the name equal triangles, to triangles which are equal only in surface; and of equal solids, to solids which are equal only in solidity. We have thought it more suitable to call such triangles or solids equivalent; reserving the denomination equal triangles, or solids, for such as coincide when applied to each other.

In solids, and curve surfaces, it is further necessary to distinguish two sorts of equality, which differ in some respects. Two solids, two solid angles, two spherical triangles or polygons, may be equal in all their constituent parts, and yet be incapable of coinciding when applied to each other, an observation which seems to have escaped the notice of elementary writers, as their inattention to it has vitiated certain demonstrations relying on the coincidence of figures, where no such coincidence can exist. Such are the demonstrations by which the equality of spherical triangles is sometimes imagined to be shewn, in the same manner as that of rectilineal triangles which are similarly related. A striking example of this oversight is exhibited by Robert Simson,* when this geometer impugns the demonstration of Euclid's Prop. 28. XI., yet falls himself into the error of grounding his own demonstration upon a coincidence which cannot take place. For these reasons, we have judged it necessary to assign a particular name to this kind of equality, which is not accompanied by coincidence; we have called it equality by symmetry, the figures to which it applies being called symmetrical.

Thus the terms equal figures, symmetrical figures, equivalent figures, refer to different objects, and should not be confounded in the same denomination.

In those propositions which relate to polygons, solid angles, and polyedrons, we have formerly excluded all figures that have re-entrant angles. Our reason was, that, besides the propriety of limiting an elementary work of the simplest cases, if this exclusion had not taken place, several propositions either would not have been true, or, at least, would have required some modification. We thought it better to restrict our reasoning to those lines which we have named convex, and which are such that a straight line cannot cut them in more than two points.

We have frequently employed the expression, product of two or more lines; by which is meant, the product of the numbers representing those lines, when valued according to a linear unit, assumed at will. The signification of the phrase once fixed, there can be no objection against using it. In the same manner must be understood what is meant by the product of a surface by a line, of a solid by a surface, &c. It is enough to have settled, once for all, that such products are, or ought to be, considered as products of numbers, each of the kind proper to it. Thus, the product of a surface by a solid, is nothing but the product of a number of superficial units by a number of solid units.

In ordinary language, the word angle is often employed to designate the point situated at the vertex. This expression is inaccurate. It would be

more correct and precise to use a particular name, such as that of vertices, for designating the points at the corners of a polygon or of a polyedron. The denomination vertices of a polyedron, as employed by us, is to be understood in this sense.

We have followed the common definition of similar rectilineal figures; we must observe, however, that it contains three superfluous conditions. In order to construct a polyedron, the number of whose sides is n, it is necessary first to know one side, and, next, the position of the vertices of

*See his work, entitled: Euclidis Elementorum libri sex, &c. Glasgow, 1756.