w arc S B T.

[Illustration: Fig. 80.]

[Illustration: Fig. 81.]

A very near approach to the true form of a true ellipse may be drawn by
the construction given in Figure 81, in which A A and B B are centre
lines passing through the major and minor axis of the ellipse, of which
_a_ is the axis or centre, _b c_ is the major axis, and _a e_ half the
minor axis. Draw the rectangle _b f g c_, and then the diagonal line _b
e_; at a right angle to _b e_ draw line _f h_, cutting B B at _i_. With
radius _a e_ and from _a_ as a centre draw the dotted arc _e j_, giving
the point _j_ on line B B. From centre _k_, which is on the line B B and
central between _b_ and _j_, draw the semicircle _b m j_, cutting A A at
_l_. Draw the radius of the semicircle _b m j_, cutting it at _m_, and
cutting _f g_ at _n_. With the radius _m n_ mark on A A at and from _a_
as a centre the point _o_. With radius _h o_ and from centre _h_ draw
the arc _p o q_. With radius _a l_ and from _b_ and _c_ as centres, draw
arcs cutting _p o q_ at the points _p q_. Draw the lines _h p r_ and _h
q s_ and also the lines _p i t_ and _q v w_. From _h_ as a centre draw
that part of the ellipse lying between _r_ and _s_, with radius _p r_;
from _p_ as a centre draw that part of the ellipse lying between _r_ and
_t_, with radius _q s_, and from _q_ as a centre draw the ellipse from
_s_ to _w_, with radius _i t_; and from _i_ as a centre draw the ellipse
from _t_ to _b_ and with radius _v w_, and from _v_ as a centre draw the
ellipse from _w_ to _c_, and one-half of the ellipse will be drawn. It
will be seen that the whole construction has been performed to find the
centres _h_, _p_, _q_, _i_ and _v_, and that while _v_ and _i_ may be
used to carry the curve around on the other side of the ellipse, new
centres must be provided for _h_ _p_ and _q_, these new centres
corresponding in position to _h_ _p_ _q_. Divesting the drawing of all
the lines except those determining its dimensions and the centres from
which the ellipse is struck, we have in Figure 82 the same ellipse drawn
half as large. The centres _v_, _p_, _q_, _h_ correspond to the same
centres in Figure 81, while _v'_, _p'_, _q'_, _h'_ are in corresponding
positions to draw in the other half of the ellipse. The length of curve
drawn from each centre is denoted by the dotted lines radiating from
that centre; thus, from _h_ the part from _r_ to _s_ is drawn; from _h'_
that part from _r'_ to _s'_. At the ends th