40 deg., " " 140 deg..
Decagon, " 36 deg., " " 144 deg..
Dodecagon, " 30 deg., " " 150 deg..

THE ELLIPSE.

An ellipse is a figure bounded by a continuous curve, whose nature will
be shown presently.

The dimensions of an ellipse are taken at its extreme length and
narrowest width, and they are designated in three ways, as by the length
and breadth, by the major and minor axis (the major axis meaning the
length, and the minor the breadth of the figure), and the conjugate and
transverse diameters, the transverse meaning the shortest, and the
conjugate the longest diameter of the figure.

In this book the terms major and minor axis will be used to designate
the dimensions.

The minor and major axes are at a right angle one to the other, and
their point of intersection is termed the axis of the ellipse.

In an ellipse there are two points situated upon the line representing
the major axis, and which are termed the foci when both are spoken of,
and a focus when one only is referred to, foci simply being the plural
of focus. These foci are equidistant from the centre of the ellipse,
which is formed as follows: Two pins are driven in on the major axis to
represent the foci A and B, Figure 75, and around these pins a loop of
fine twine is passed; a pencil point, C, is then placed in the loop and
pulled outwards, to take up the slack of the twine. The pencil is held
vertical and moved around, tracing an ellipse as shown.

[Illustration: Fig. 75.]

Now it is obvious, from this method of construction, that there will be
at every point in the pencil's path a length of twine from the final
point to each of the foci, and a length from one foci to the other, and
the length of twine in the loop remaining constant, it is demonstrated
that if in a true ellipse we take any number of points in its curve, and
for each point add together its distance to each focus, and to this add
the distance apart of the foci, the total sum obtained will be the same
for each point taken.

[Illustration: Fig. 76.]

[Illustration: Fig. 77.]

In Figures 76 and 77 are a series of ellipses marked with pins and a
piece of twine, as already described. The corresponding ellipses, as A
in both figures, were marked with the same loop, the difference in the
two forms being due to the difference in distance apart of the foci.
Again, the same loop was used for ellipses B in