flat world, a world in which there was neither height nor depth, neither up nor down, neither top nor bottom; and now, hey presto! he had jerked me out of the flat world back into the real world in which thin pieces of paper can be picked up and set down on the top of other pieces. I was surprised in the same way as Alice when she saw the cat in Wonderland go away and leave its grin behind. -How long have we all dwelt in Wonderland, and watched other Mediterranean wizards working their famous spell that changes the cat without changing the grin?

And now I have to see whether that hard, bright crystal can be a fulcrum for the mind; if in the lexicon of Pure Earth-Measure there is any word by which I can measure the Testator's word. Was not "exact" one of the explanations of “idealist.”

II

The word Mathematics seems to mean assurance, or making sure; and that being so we can understand why Bacon called it the handmaid of the sciences, and why Descartes wanted to make it the maid-of-all-work.

The race of men before the Stone-cutters made sure of things by smelling them, a habit that still breathes in our word know. It was not till men got knives that they could cut things open, and change knowledge into science. In doing so, as well as in whole groups of words, they showed their growing trust in the eye. This turning towards the light has its bodily counterpart in the great brain-growth towards the eyeballs, a growth wrought partly by the chemic power of light, and thus a real bridge between chemistry and wakefulness; a growth which has helped to shape the human skull, and so made man a child of light.

But the eye only sees in two measurements; it can see length and breadth, but thickness it cannot see. The sense of sight is but a daughter of the sense of touch; the eye a magic finger reaching forth into the abyss. Again, there are two sides to making sure, and one of them is the thing that you make sure of. Sights come and go; the sun himself is a mere seasonticket-holder of the sky; your moon changes like a woman's temper; that big blue cave of stars is not half so steady as one's own cave. Only the firm' earth beneath your feet is fixed. You measure that, you go by steps, and you know where you are. It did not need the overflowing of the Nile, nor the Egyptian Delta, to make Geometry the mother of mathematics.

Already, when the first land surveyor measured the length of the ground by the length of his foot or stride, he showed himself more sure of one length than the other. But now another man has stepped the same length of ground, and made it longer. One

foot is not the same as another foot. It is time to measure what you measured by, and make assurance doubly sure.

In the end some unknown Andrónikos arose, and said, Let us forget there was ever real ground, or a real foot, or a real pair of human, or of compass, legs, and make believe that we are measuring Pure Ground with Pure Measures. And he called his science, or his language, Pure Assurance.

Now here is Idealism in its nakedness. I do not mean that it is the idealism of our Testator. I do not claim this bequest for works of a mathematical tendency. But, as it was in the mind of Plato, so this is in the general mind, the embryo of idealism, and therefore it cannot be passed by.

III

The first word in Euclid's definition is point. He makes-believe to begin with that. But he can only tell me what a point is by telling me what it is not. Every definition is a not. An outline can only be gained in battle. The simplest definition follows the yea and nay of electricity, as in the case of truth and verihood.

Accordingly Euclid tells me that his point has no size. And in doing so he shows that he expects to find size in my mind already-that I know all about size. And thus in the order of the thoughts size comes before the point. In the language of Pure Assurance, Euclid has assumed the three dimensions of space in telling me that his point has none. Which, as he would say himself, is absurd.-There is a good deal of absurdity in Euclid.

His aim is to make-believe that he is starting with the point. And so instead of working down to it fairly, he pretends to create it. It is another conjuring trick. The juggler claps down his magical dice-box over size, and when he lifts it up, hey presto! size has vanished, and the point is there instead. In a school book that is unfair. The abstraction ought to be abstracted before the schoolboy's eyes, and not popped on him from up the conjuror's sleeve, as if it were some real thing made of imponderable Matter. Which, to translate Euclid, is unheard of.

I have to fall back on my humble method of asking what the words mean. I cannot find that point means anything more than end, one end of a line, and so I must at least know what a line is. But line, in its turn, only means edge, the edge of a face; and face is only side, the side of a block. Thus the words themselves lead me back into the world of size, a world of three measurements, the real world in which I am accustomed to live. I can now, if I am asked to do so, make-believe to forget the block. and think only of the side, to forget that and think only of the edge, to forget that and think only of

the end, as I can to forget the end and think of nothing. And it is by such steps that this venerable quack ought to have brought his flats and lines and points before my mind's eye, when I was twelve years old.

We have not Euclid's handwriting before us, and we know that Mediterranean copyists sometimes take freedoms with their text. One of Euclid's copyists seems to have felt that there was something false about the point, and he has tried to mend matters by saying that a point is that which has position. I am reminded rather painfully of a certain Energy of Position which gave us some trouble a short way back. To say that a point has position is to say that it is fixed; and you cannot fix a point without having at least one other point to fix it by: and as soon as you have got two points side by side you have got a line. And so we work back by another road to the real starting-point, the point from which the science of fanciful Earth-Measure did indeed start, namely the solid earth.

Euclid has put the cart before the horse. Which is unheard of.

I am not ashamed to say that I have found more sense in a child's riddle than in all Euclid's definitions-though they are also in their way a child's riddle. It runs,- -How many sides has a round plum-pudding ? And the answer is - Two, an outside and an inside.

Because the science of Earthmeasure is the science of measuring shapes, and the child's plum-pudding is the truth about shape. It is the shape of the All-Thing as well as of the atom. It is, if I must