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.