Page:Popular Science Monthly Volume 8.djvu/458

442 But when we examine his argument we find that he has made three unproved assumptions—namely: 1. That a thing cannot at the same time be and not be; 2. That if equals be added to equals, the wholes are equal; 3. That things which are equal to the same are equal to one another. It so happens that each of these propositions which he has assumed to be true is, if true, much more important than the proposition which he has proved. Let us point out these three assumptions to our bright student, and then resume our catechism.

Q. Could you possibly prove this proposition in geometry if any one of those three assumed propositions were not granted?

A. No.

Q. Then, if we deny these assumptions, can you prove them?

A. No; but can you deny them?

No, we cannot deny them, and cannot prove them; but we believe them, and therefore have granted them to you for argument, and know your proposition of the two right angles to be true, because you have proved it.

Now, here is the proposition which Euclid selected as the simplest of all demonstrable theorems of geometry, in the demonstration of which the logical understanding of a student cannot take the first step without the aid of faith.

From the student let us go to the master. We go to such a teacher as Euclid, and in the beginning he requires us to believe three propositions, without which there can be no geometry, but which have never been proved, and, in the nature of things, it would seem never could be proved—namely, that space is infinite in extent, that space is infinitely divisible, and that space is infinitely continuous. And we believe them, and use that faith as knowledge, and no more distrust it than we do the results of our logical understandings, and are obliged to admit that geometry lays its broad foundations on our faith.

Now, geometry is the science which treats of forms in their relations in space. The value of such a science for intellectual culture and practical life must be indescribably important, as might be shown in a million of instances. No form can exist without boundaries, no boundaries without lines, no line without points. The beginning of geometric knowledge, then, lies in knowing what a "point" is, the existence of forms depending, it is said, upon the motion of points. The first utterance of geometry, therefore, must be a definition of a point. And here it is: "A point is that which has no parts, or which has no magnitude." At the threshold of this science we meet with a mystery. "A point is"—then, it has existence—"is" what? In fact, in form, in substance, it is nothing. A logical definition requires that the genus and differentia shall be given. What is the genus of a "point?" Position, of course. Its differentia is plainly seen. It is distinguished from every thing else in this, that every thing else is