164 On Spandrel Groins. 
set the compasses on 7, and revolve the lines-bb, &c. to come 
in contact with the cone: these lines then draw parallel to the 
base of the cone; and where they intersect the inclined lines on 
the conic surface will be the undulating or serpentine line; the 
spiral one will now be seen to a demonstration by the diagram 
fig. OQ above, which diagram is formed by the revolutions 
round the respective points brought up from the cone below 
Figs. DEFG show the conic sections by a very simple method, 
which is new*, 72,7, for example, shows the cone. Now sup- 
pose the line ¢¢ to he the section-line for the hyperbola, (a 
section perpendicular to the base of the cone,) take the distance 
fu at the base, and set it to h,, on each side of the line A; 
next set the compasses on the cone at O, and revolve round the 
dotted lines ec; then take the measures ¢7 7, and set them on 
each side the line 2: these lines draw perpendicular to 64, and 
produce the parailel lines ¢p, &c. then at the intersections of 
these perpendicular lines with the horizontal ones will be points 
through which to describe the hyperbola. For the parabola 
(a section cut parallel to the side of the cone) take the distances 
123 at fig. D, and set them up the line 123 fig. F; these 
lines draw parallel to dd; then from the cone D take the 
measures ec,eéc, and set them on each side the line & at fig. F, 
which lines draw perpendicular to dd, and the intersections with 
the parallel lines will give the points through which to describe 
the parabola. 
For the ellipsis (a section cut obliquely through the cone), find 
the ceutre of the section-line marked 1,2, at fig. D, and from 
the apex of the cone draw a dotted lineto the base through 1; 
then with the centre 2 revolve round the dotted line tox. The 
dotted line a is also produced at right angles to the dotted line 
2, #2, proceeding from the apex of the cone; next draw the line 
1,2 at right angles to the section line 1, 2, and revolve the dotted 
line ia@ round to m; then will the line 1,2 be the minor, and 
the line 7,2 the major diameters of the ellipsis; to describe 
which, take the two diameters and place them at fig. G, then 
form the figure as there shown; this is plain to inspection. 
III. On Spandrel Groins. 
I have also sent for insertion in your next number, a draw- 
ing of the new and more effectual method of building groined 
* The sections of the Scalene Cunel have shown in my Treatise on the 
Practical Principles of Perspective, 
arches 
