e more extended. In
common life it is established as a maxim, that the straightest way is
always the shortest; which would be as absurd as to say, the shortest
way is always the shortest, if our idea of a right line was not
different from that of the shortest way betwixt two points.

Secondly, I repeat what I have already established, that we have no
precise idea of equality and inequality, shorter and longer, more than
of a right line or a curve; and consequently that the one can never
afford us a perfect standard for the other. An exact idea can never be
built on such as are loose and undetermined.

The idea of a plain surface is as little susceptible of a precise
standard as that of a right line; nor have we any other means of
distinguishing such a surface, than its general appearance. It is in
vain, that mathematicians represent a plain surface as produced by the
flowing of a right line. It will immediately be objected, that our idea
of a surface is as independent of this method of forming a surface, as
our idea of an ellipse is of that of a cone; that the idea of a right
line is no more precise than that of a plain surface; that a right line
may flow irregularly, and by that means form a figure quite different
from a plane; and that therefore we must suppose it to flow along two
right lines, parallel to each other, and on the same plane; which is a
description, that explains a thing by itself, and returns in a circle.

It appears, then, that the ideas which are most essential to geometry,
viz. those of equality and inequality, of a right line and a plain
surface, are far from being exact and determinate, according to our
common method of conceiving them. Not only we are incapable of telling,
if the case be in any degree doubtful, when such particular figures are
equal; when such a line is a right one, and such a surface a plain one;
but we can form no idea of that proportion, or of these figures, which
is firm and invariable. Our appeal is still to the weak and fallible
judgment, which we make from the appearance of the objects, and correct
by a compass or common measure; and if we join the supposition of
any farther correction, it is of such-a-one as is either useless or
imaginary. In vain should we have recourse to the common topic, and
employ the supposition of a deity, whose omnipotence may enable him to
form a perfect geometrical figure, and describe a right line without any
curve or inflexion. As the ultima