both figures, as also
for C and D. From these figures we perceive that--

1st. With a given width or distance apart of foci, the larger the
dimensions are the nearer the form of the figure will approach to that
of a circle.

2d. The nearer the foci are together in an ellipse, having any given
dimensions, the nearer the form of the figure will approach that of a
circle.

3d. That the proportion of length to width in an ellipse is determined
by the distance apart of the foci.

4th. That the area enclosed within an ellipse of a given circumference
is greater in proportion as the distance apart of the foci is
diminished; and,

5th. That an ellipse may be given any required proportion of width to
length by locating the foci at the requisite distance apart.

The form of a true ellipse may be very nearly approached by means of the
arcs of circles, if the centres from which those arcs are struck are
located in the most desirable positions for the form of ellipse to be
drawn.

[Illustration: Fig. 78.]

Thus in Figure 78 are three ellipses whose forms were pencilled in by
means of pins and a loop of twine, as already described, but which were
inked in by finding four arcs of circles of a radius that would most
closely approach the pencilled line; _a b_ are the foci of all three
ellipses A, B, and C; the centre for the end curves of _a_ are at _c_
and _d_, and those for its side arcs are at _e_ and _f_. For B the end
centres are at _g_ and _h_, and the side centres at _i_ and _j_. For C
the end centres are at _k_, _l_, and the side centres at _m_ and _n_.
It will be noted that, first, all the centres for the end curves fall on
the line of the length or major axis, while all those for the sides fall
on the line of width or the minor axis; and, second, that as the
dimensions of the ellipses increase, the centres for the arcs fall
nearer to the axis of the ellipse. Now in proportion as a greater number
of arcs of circles are employed to form the figure, the nearer it will
approach the form of a true ellipse; but in practice it is not usual to
employ more than eight, while it is obvious that not less than four can
be used. When four are used they will always fall somewhere on the lines
on the major and minor axis; but if eight are used, two will fall on the
line of the major axis, two on the line of the minor axis, and the
remaining four elsewhere.

[Illustration: Fig. 79.]

In Figure 79 is a construction wherein four a