“There was no meaning in the stuff - it didn't hang together. It had been the same all through grade school. Arithmetic - algebra - they werent right. His mind rejected them. He’d trained himself to memorize stuff - take it on faith—parrot it out when the teachers asked for it, but this geometry was the worst of the lot. It was twisted - distorted - wrong.”

"Mr. Andrews, he said thoughtfully; this book—it’s ‘plain’ geometry. Isn’t there any fancy geometry?" Lying in his bed, snug up under the slope of the roof, with the moonlight making ranges of hills across the bed-quilt, Tommy wondered about it. Why shouldn’t there be a fancy geometry as well as the plain kind? Why shouldn’t there be a kind of geometry that didn’t refuse to fit together when he tried to reason it out? Why wasn’t there another kind of geometry— his kind of geometry?”

“In school Euclid was the law, but out of school, when the chores were done and he could go up to his room or climb up to the open ledges back of town, the other, fancier brand of geometry gradually took shape in his mind. Once it was started it came easily to him. It was not too much different from the "plain” variety of the little brown book, but what differences there were seemed to be fundamental, and he began to see why Euclid seemed alien and unreasonable. He had all of the simpler theorems in his head before he wrote any of it down, and then he did it in the hieratic script of Keer, which he had never before put on paper. It somehow seemed to fit the odd, non-Euclidian structure which he was building. It was soon clear to him that this was the geometry of Keer itself that he was setting down —not alone of the misted moorland kingdom with its five towns, but of its world —its entire universe. When he had the rudiments of the geometry' down on paper he went back to the algebra, and saw where it was that he had gone astray, and where the altered laws should lead It all fell together simply and beautifully.”