It is true, mathematicians pretend they give an exact definition of a
right line, when they say, it is the shortest way betwixt two points. But
in the first place I observe, that this is more properly the discovery of
one of the properties of a right line, than a just deflation of it. For I
ask any one, if upon mention of a right line he thinks not immediately on
such a particular appearance, and if it is not by accident only that he
considers this property? A right line can be comprehended alone; but this
definition is unintelligible without a comparison with other lines, which
we conceive to be 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