to each marking point
(F). Do this on both sides of the major axis, and produce these lines so
they cross the curved lines (G). When you ink in your ellipse do not
allow the circle pen to cross the lines (K), and you will have a
mechanical ellipse.

ELLIPSES AND OVALS.--It is not necessary to measure the centering points
(F) at certain specified distances from the intersection of the
horizontal and vertical lines. We may take any point along the major
axis, as shown, for instance, in Fig. 149. Let B be this point, taken at
random. Then describe the half circle (C). We may, also, arbitrarily,
take any point, as, for instance, D on the minor axis E, and by drawing
the diagonal lines (F) we find marks on the circle (C), which are the
meeting lines for the large curve (H), with the small curve (C). In this
case we have formed an ovate or an oval form. Experience will soon make
perfect in following out these directions.

FOCAL POINTS.--The focal point of a circle is its center, and is called
the _focus_. But an ellipse has two focal points, called _foci_,
represented by F, F in Fig. 148, and by B, B in Fig. 149.

A _produced line_ is one which extends out beyond the marking point.
Thus in Fig. 148 that part of the line K between F and G represents the
produced portion of line K.

[Illustration: _Fig. 149._]

SPIRALS.--There is no more difficult figure to make with a bow or a
circle pen than a spiral. In Fig. 150 a horizontal and a vertical line
(A, B), respectively, are drawn, and at their intersection a small
circle (C) is formed. This now provides for four centering points for
the circle pen, on the two lines (A, B). Intermediate these points
indicate a second set of marks halfway between the marks on the lines.
If you will now set the point of the compass at, say, the mark 3, and
the pencil point of the compass at D, and make a curved mark one-eighth
of the way around, say, to the radial line (E), then put the point of
the compass to 4, and extend the pencil point of the compass so it
coincides with the curved line just drawn, and then again make another
curve, one-eighth of a complete circle, and so on around the entire
circle of marking points, successively, you will produce a spiral,
which, although not absolutely accurate, is the nearest approach with a
circle pen. To make this neatly requires care and patience.

[Illustration: _Fig. 150._]

PERPENDICULAR AND VERTICAL.--A few words now as to terms. The boy is