to trace, if we can, their deadly effects both in the course of human history and in the present status of our human world. The task of disengaging the two monstrous misconceptions from the tangled skein of inherited beliefs and framing them in words, I have already repeatedly performed. Let us keep the results in mind. Here they are in their nakedness: (1) Human beings—men, women, and children—are animals (and so they are natural): (2) human beings are neither natural nor _super_natural, neither wholly animal nor wholly “divine,” but are _both_ natural and _super_natural _at once_—a sort of mysterious hybrid compound of brute and gods. The second part of our task—which is the reader’s task as much as mine—is not so easy; and the reason is evident. It is this: The false creeds in question—the fatal misconceptions they involve—are so _familiar_ to us—they have been so long and so deeply imbedded in our thought and speech and ways of life—we have been so thoroughly _bred_ in them by home and school and church and state—that we _habitually_ and _unconsciously_ take them for granted and have to be virtually _stung_ into an awareness of the fact that we do actually hold them and that they do actually reign to-day throughout the world and have so reigned from time immemorial. We have, therefore, to shake ourselves awake, to _prick_ ourselves into a realization of the truth. I assume that the reader is at once hard-headed, rational, I mean, and interested in the welfare of mankind. If he is not, he will not be a “reader” of this book. He, therefore, knows that the third task—the task of detecting and exposing the fundamental error of the misconceptions in question—is a task of the utmost importance. What is that error? It is, I have said, an error in logic. But logical errors are not all alike—they are of many kinds. What is the “kind” of _this_ one? It is the kind that consists in what mathematicians call “confusion of types,” or “mixing of dimensions.” The answer can not be made too clear nor too emphatic, for its importance in the criticism of _all_ our thinking is great beyond measure. There are millions of examples that help to make the matter clear. I will again employ the simplest of them—one so simple that a child can understand it. It is a mathematical example, as it ought to be, for the whole question of logical types, or dimensions, is a mathematical one. I beg the reader not to shy at, or run away from, the mere word mathematical, for, although most of us have but little mathematical _knowledge_, we all of us have the mathematical _spirit_, for else we should not be human—we are all of us mathematicians _at heart_. Let us, then, proceed confidently and at once to our simple example. Here is a _surface_, say a _plane_ surface. It has length and breadth—and so it has, we say, _two_ dimensions; next consider a _solid_, say a _cube_. It has length, breadth and thickness—and so _it_ has, we say, _three_ dimensions.