The mathematician has reached the highest rung on the ladder of human thought. But it is the same ladder which we have all of us been always ascending, alike from the infancy of the individual and the infancy of the race. Molière’s Jourdain had been speaking prose for more than forty years without knowing it. Mankind has been thinking poetry throughout its long career and remained equally ignorant. Footnote 26: It would not appear that the pioneers of the Mathematical Renaissance of the twentieth century are inclined to imitate Descartes in this matter. Einstein would certainly not, and many apostles of physical science to-day (see, e.g., Professor Smithells, _From a Modern University: Some Aims and Aspirations of Science_) insist on the æsthetic, imaginative, and other “art” qualities of science. Footnote 27: C. Singer. “What is Science?” _British Medical Journal_, 25th June, 1921. Singer refuses the name of “science” in the strict sense to fields of completely organised knowledge which have ceased growing, like human anatomy (though, of course, the anatomist still remains a man of science by working outwards into adjoining related fields), preferring to term any such field of completed knowledge a _discipline_. This seems convenient and I should like to regard it as sound. It is not, however, compatible with the old doctrine of Mill and Colvin and Ray Lankester, for it excludes from the field of science exactly what they regarded as most typically science, and some one might possibly ask whether in other departments, like Hellenic sculpture or Sung pottery, a completed art ceases to be art. Footnote 28: It has often been pointed out that the imaginative application of science—artistic ideas like that of the steam locomotive, the flying-machine heavier than air, the telegraph, the telephone, and many others—were even at the moment of their being achieved, elaborately shown to be “impossible” by men who had been too hastily hoisted up to positions of “scientific” eminence. Footnote 29: J. B. Baillie, _Studies in Human Nature_ (1921), p. 221. This point has become familiar ever since F. A. Lange published his almost epoch-marking work, _The History of Materialism_, which has made so deep an impress on many modern thinkers from Nietzsche to Vaihinger; it is indeed a book which can never be forgotten (I speak from experience) by any one who read it in youth. Footnote 30: G. Wallas, _The Great Society_, p. 107. Footnote 31: Gomperz, _Greek Thinkers_, vol. I, chap. III, where will be found an attractive account of Pythagoras’ career and position. Footnote 32: