42 MATHEMATICS. 
this circumference. This rectangle must, in fact, pe constructed upon the 
elements first mentioned. The semi-circumference, in fig. 53, is to be 
divided into 8 equal parts, as shown at a, b, c, d, &c.: these must be so 
small that, without material error, the arcs may be considered as straight: 
lines. These eight parts are to be transferred to the rectangle at a’, 0’, &e., 
and the perpendiculars a’a’*, b’b*,——— 771°, drawn, which will all lie in 
the convex surface of the cylinder. From the points a, b, c, ———i, in the 
horizontal projection of the cylinder, lines are to be drawn until they intersect 
the oblique section of the cylinder at a’, b—-—7’. From the points 
a’, b'.—— —— 1, draw parallels to the base, zy,-until they intersect their 
corresponding lines; the points a’, b’, c*, ———7’, will thus be obtained, 
which may be connected by a curve. This will be the development of half 
the ellipse of which the line FG represents the vertical projection. The 
perpendiculars aa’, bb’, — — — 1’, also intersect the projections CB and BE 
of the semi-ellipses of the lower cylinder sections ; accordingly, here, as in 
the upper ellipse, the corresponding points may be connected by parallels to 
the base, xy, and points of the curve obtained on the lines a’*a*, b*b*,— — —7v’. 
Fig. 54 represents rather more than half the development of the cylinder. 
Fig. 55 exhibits‘ the horizontal and vertical projection of a right cone, 
intersected in the three conic sections, and projected according to the rules 
given for figs. 27, 28, 29. The convex surface of this cone is now to be 
found, and upon it the developments of the three conic sections, described. 
The convex surface of a right cone is‘a circular sector, whose radius equals 
the slant height of the cone, and whose arc equals the circumference of the 
base. If, then, from any point, with a radius equal to the slant height of the 
cone, an are, zy’ ( fig. 56), be described, this sector, with its two radii, will 
determine the convex surface of the cone, provided. that the proper length 
of the arc has been obtained. This may be done by dividing the circle 
whose projection is zy’( fig. 55), just as was done in the case of fig. 53, and 
transferring the arcs of division. ‘The sum of these arcs, which will be few 
or many as the result is to be less or more accurate, will determine the 
extent of the circumference of the base. Lines drawn from the individual 
points of division to the centre, in fig. 56, will represent so many lines of 
.the convex surface of the cone. Their projections in elevation ( fig. 55) 
will be obtained by transferring the parts from the plan, z’y', of the base, to 
its vertical projection, zy, by means of perpendiculars. Lines must then be 
drawn from_ the points of intersection thus obtained, to the apex. 
These will intersect the vertical projections of the conic sections. 
From the vertex, in fig: 56, lay off on the middle line the distance from 
the apex of the cone (fig. 55), to the point G’; G’ will then be a point in 
the development of the ellipse. The distance from the apex of the cone 
( fig. 55) to the first intersection of the ellipse by the projection of the sides 
of the cone in jig. 55, laid off on both sides of the point G’ in fig. 56, 
gives two new points in the development: of the ellipse; and the same is 
to be done with respect to the remaining lateral lines of fig. 55. Connecting 
these points in fig..56, will give the development of the ellipse. In like 
manner the curve B’A*C? is found, as the projection of the hyperbola The 
42 
