The man who has become an accomplished mathematician can use numbers much better; but if we are capable of following intelligently the intricate series of operations that he carries out on the paper before us, and can see the significance of the system of signs which he uses as an aid, we shall realize that he is only doing in more complicated ways what we have been accustomed to do almost from our childhood.
If we are interested, not so much in performing the operations, as in inquiring into what really takes place in a mind when several units are grasped together and made into a new unit,--for example, when twelve units are thought as one dozen,--the mathematician has a right to say: I leave all that to the psychologist or to the metaphysician; every one knows in a general way what is meant by a unit, and knows that units can be added and subtracted, grouped and separated; I only undertake to show how one may avoid error in doing these things.
It is with geometry as it is with arithmetic.

No man is wholly ignorant of points, lines, surfaces, and solids.

We are all aware that a short line is not a point, a narrow surface is not a line, and a thin solid is not a mere surface.

A door so thin as to have only one side would be repudiated by every man of sense as a monstrosity.

When the geometrician defines for us the point, the line, the surface, and the solid, and when he sets before us an array of axioms, or self-evident truths, we follow him with confidence because he seems to be telling us things that we can directly see to be reasonable; indeed, to be telling us things that we have always known.
The truth is that the geometrician does not introduce us to a new world at all.