352 THE FORMS OF TISSUES [ch. 



which we cannot fully master, the same guiding principle is at 

 the root of the matter. 



The bisection of a solid (or the subdivision of its volume in 

 other definite proportions) soon leads us into a geometry which, 

 if not necessarily difficult, is apt to be unfamiliar; but in such 

 problems we can go a long way, and often far enough for our 

 particular purpose, if we merely consider the plane geometry of 

 a side or section of our figure. For instance, in the case of the 

 cube which we have been just considering, and in the case of the 

 plane and cyHndrical partitions by which it has been divided, it 

 is obvious that, since these two partitions extend symmetrically 

 from top to bottom of our cube, that we need only consider (so 

 far as they are concerned) the manner in which they subdivide 

 the base of the cube. The whole problem of the solid, up to a 

 certain point, is contained in our plane diagram of Fig. 138. And 

 when our particular sohd is a solid of revolution, then it is obvious 

 that a study of its plane of symmetry (that is to say any plane 

 passing through its axis of rotation) gives us the solution of the 

 whole problem. The right cone is a case in point, for here the 

 investigation of its modes of symmetrical subdivision is completely 

 met by an examination of the isosceles triangle which constitutes 

 its plane of symmetry. 



The bisection of an isosceles triangle by a line which shall 

 be the shortest possible is a very easy problem. Let ABC be 

 such a triangle of which A is the apex ; it may be shewn that, 

 for its shortest line of bisection, we are limited to three cases : 

 viz. to a vertical line AD, bisecting the angle at A and the side 

 BC ; to a transverse line parallel to the base BC ; or to an oblique 

 line parallel to AB or to AC. The respective magnitudes, or 

 lengths, of these partition lines follow at once from the magnitudes 

 of the angles of our triangle. For we know, to begin with, since 

 the areas of similar figures vary as the squares of their linear 

 dimensions, that, in order to bisect the area, a line parallel to one 

 side of our triangle must always have a length equal to l/v2 

 of that side. If then, we take our base, BC, in all cases of 

 a length = 2, the transverse partition drawn parallel to it will 

 always have a length equal to 2/-\/2, or = y/'I. The vertical 



