ples will be enough to put the reader in
the way of pursuing the subject for himself.

[Illustration: FIG. 91.]

Take any number of lines, _a b_, _b c_, _c d_, &c., Fig. 91, bearing any
fixed proportion to each other. In this figure, _b c_ is one third
longer than _a b_, and _c d_ than _b c_; and so on. Arrange them in
succession, keeping the inclination, or angle, which each makes with the
preceding one always the same. Then a curve drawn through the
extremities of the lines will be a beautiful curve; for it is governed
by consistent laws; every part of it is connected by those laws with
every other, yet every part is different from every other; and the mode
of its construction implies the possibility of its continuance to
infinity; it would never return upon itself though prolonged for ever.
These characters must be possessed by every perfectly beautiful curve.

If we make the difference between the component or measuring lines less,
as in Fig. 92, in which each line is longer than the preceding one only
by a fifth, the curve will be more contracted and less beautiful. If we
enlarge the difference, as in Fig. 93, in which each line is double the
preceding one, the curve will suggest a more rapid proceeding into
infinite space, and will be more beautiful. Of two curves, the same in
other respects, that which suggests the quickest attainment of infinity
is always the most beautiful.

[Illustration: FIG. 92.]

[Illustration: FIG. 93.]

Sec. 5. These three curves being all governed by the same general law, with
a difference only in dimensions of lines, together with all the other
curves so constructible, varied as they may be infinitely, either by
changing the lengths of line, or the inclination of the lines to each
other, are considered by mathematicians only as one curve, having this
peculiar character about it, different from that of most other infinite
lines, that any portion of it is a magnified repetition of the preceding
portion; that is to say, the portion between _e_ and _g_ is precisely
what that between _c_ and _e_ would look, if seen through a lens which
magnified somewhat more than twice. There is therefore a peculiar
equanimity and harmony about the look of lines of this kind, differing,
I think, from the expression of any others except the circle. Beyond the
point _a_ the curve may be imagined to continue to an infinite degree of
smallness, always circling nearer and nearer to a point, which, however,