it can never reach.

[Illustration: FIG. 94.]

Sec. 6. Again: if, along the horizontal line, A B, Fig. 94, we measure any
number of equal distances, A _b_, _b c_, &c., and raise perpendiculars
from the points _b_, _c_, _d_, &c., of which each perpendicular shall be
longer, by some given proportion (in this figure it is one third), than
the preceding one, the curve _x y_, traced through their extremities,
will continually change its direction, but will advance into space in
the direction of _y_ as long as we continue to measure distances along
the line A B, always inclining more and more to the nature of a straight
line, yet never becoming one, even if continued to infinity. It would,
in like manner, continue to infinity in the direction of _x_, always
approaching the line A B, yet never touching it.

Sec. 7. An infinite number of different lines, more or less violent in
curvature according to the measurements we adopt in designing them, are
included, or defined, by each of the laws just explained. But the number
of these laws themselves is also infinite. There is no limit to the
multitude of conditions which may be invented, each producing a group of
curves of a certain common nature. Some of these laws, indeed, produce
single curves, which, like the circle, can vary only in size; but, for
the most part, they vary also, like the lines we have just traced, in
the rapidity of their curvature. Among these innumerable lines, however,
there is one source of difference in character which divides them,
infinite as they are in number, into two great classes. The first class
consists of those which are limited in their course, either ending
abruptly, or returning to some point from which they set out; the second
class, of those lines whose nature is to proceed for ever into space.
Any portion of a circle, for instance, is, by the law of its being,
compelled, if it continue its course, to return to the point from which
it set out; so also any portion of the oval curve (called an ellipse),
produced by cutting a cylinder obliquely across. And if a single point
be marked on the rim of a carriage wheel, this point, as the wheel rolls
along the road, will trace a curve in the air from one part of the road
to another, which is called a cycloid, and to which the law of its
existence appoints that it shall always follow a similar course, and be
terminated by the level line on which the wheel rolls. All such curves
are of inferior bea