Now, every civilized man who is not mentally deficient can perform the fundamental operations of arithmetic. He can add and subtract, multiply and divide. In other words, he can use _numbers_. 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. He merely gives us a fuller and a more exact account than was before within our reach of the space relations which obtain in the world of external objects, a world we already know pretty well. Suppose that we say to him: You have spent many years in dividing up space and in scrutinizing the relations that are to be discovered in that realm; now tell us, what is space? Is it real? Is it a thing, or a quality of a thing, or merely a relation between things? And how can any man think space, when the ideas through which he must think it are supposed to be themselves non-extended? The space itself is not supposed to be in the mind; how can a collection of non-extended ideas give any inkling of what is meant by extension? Would any teacher of mathematics dream of discussing these questions with his class before proceeding to the proof of his propositions? It is generally admitted that, if such questions are to be answered at all, it is not with the aid of geometrical reasonings that they will be answered.