DESCRIPTIVE GEOMETRY. 4] 
it is only the circumferences of the surfaces of intersection that require to 
be attended to here. These surfaces of intersection must, in all cases, be 
curves: they are obtained by dividing the circumference of the cylinder 
into any number of equal parts, and, through these, drawing parallels to the 
sides of the cylinder. The figure represents only a few of these parallels. 
Where these lines intersect the perpendicular diameter of the sphere, in the 
vertical view, planes of intersection are to be passed through the sphere. 
These can be very readily transferred to the horizontal view, where they 
appear as circles. The points where the parallels to the circumference of 
the cylinder intersect the corresponding circular sections, are points of the 
surface of intersection, which may then be readily described. On account 
of the small scale of our figure, only a few points have been determined ; 
the rest are readily formed in the same manner. A concluding example of 
the intersection of bodies is presented in fig. 35. Here, an oblique cone 
penetrates an oblique cylinder, in such a manner that part of the cone 
passes through the cylinder. To develope the intersection, the method em- 
ployed in reference to fig. 33 must be again brought into play, with this 
difference only, that the lines drawn from the points of division of the base 
to the cone must not be parallel to the lateral edges, but converging to the 
apex of the cone. 
b. The Reticulations of Bodies, and the Unfolding or Development of 
Surfaces. 
By the reticulation of a body is meant the continuous description of its 
inclosing surfaces in one plane. This is easiest in bodies which are inclosed 
entirely by plane surfaces, as is the case in the so-called regular bodies. It 
is only in this case that the reticulation of a body can exhibit a perfectly 
true picture of its surface. Plate 4, fig. 49, is the reticulation of a tetrahe- 
dron, formed by four equal equilateral triangles; fig. 50, that of a cube, or 
hexahedron, formed by six equal squares; fig. 51, that of a dodecahedron, 
formed by twelve equal regular polygons; fig. 52, that of an icosahedron. 
The figure is not quite complete, as in addition to the fourteen equal equi- _ 
lateral triangles, six more must be added, viz. one in the upper row, next to 
11, three in the middle row, next to 7, and two in the lower row, next to 14. 
In conclusion, we will present one or two examples, in which not the 
entire reticulation, but merely the convex surface, will be referred to. The 
bottoms, so to speak, are very easily constructed. 
Fig. 53 represents, at A, the horizontal intersection of a cylinder by the 
plane CD, this latter being itself visible in fig. 54. Let portions be cut off 
obliquely from the lower part of the cylinder, by the lines BC and BE, and 
an oblique portion from the upper part by the lines FG. Suppose, now, that 
half the convex surface of this remnant of the cylinder is to be ascertained. 
If the cylinder had not been mutilated in this manner, its development 
would be a rectangle, the altitude being equal to the height, and the base 
equal to the circumferenee of the cylinder, or, as in our illustration, to half 
41 
