7*8 RENAISSANCES whereas Pythagoras' actual experience included the sensation of suddenly seeing the light, the belief that his was the first human mind that had ever seen it, and the emotional exultation of feeling himself to be an intellectual pioneer. All these non-mathematical elements in a mathe- matician's actual experience are expressed in Archimedes' exclamation qvpTjKa', yet the latter-day student of hydrostatics is deaf to a cry which, coming as it does from Archime~deV heart, never fails to thrill the heart of the historian, however faintly the sound may echo in the historian's ears across an ever widening gulf of fleeting Time. There is indeed a piquant difference between the mathematician's impersonal way of looking at a proposition and the way in which the same proposition interests the historian. When the historian is con- fronted with it, he does not ask himself, Ts this mathematically true ?' He asks himself, 'What can this mathematical proposition tell me about the personality and life of Pythagoras and about the history of his social milieu—the thoughts, feelings, aims, and characters of Pythagoras and his contemporaries, predecessors, and successors in Samos and in Croton, in the Hellenic World at large, and in the other societies that were this society's successors, predecessors, and contemporaries ? So long as the mathematical proposition associated with Pythagoras gives him some light on the answers to these historical questions, it is of no professional consequence to the historian whether the proposition happens to be mathematically true or false; for a mathematically true proposition might prove to be barren of historical information, while a mathematically false proposition might prove to be an historian's gold mine. Since everything that has been said about Mathematics in this Annex is likewise true of Natural Science, the case of Astrology will serve to illustrate our point. In Collingwood's and Toynbee's day in Western scientific circles, Astrology was in deep disgrace. It was perpetually being cited as the classical example of a pseudo-science whose falsity had been exposed and whose prestige had been exploded;1 yet, in this selfsame generation, masterly studies of astrological beliefs for historical purposes had been throwing floods of light on the Weltanschauung and Gefilhlsart of the children of the Babylonic Civilization who had invented —or discovered—Astrology in the eighth century B.c.s and the children hand the historian, even if he fails fully to apprehend this impersonal mathematical thought, may nevertheless succeed in utilizing it for his own ulterior purpose of making a psychic passage across the gulf of Time and Space that lies between him and some earlier thinker of the same thought with whom he is seeking to make contact (see pp, 729 and 730, below). 1 There was, of course, a long time-lag between the date, before the close of the seventeenth century of the Christian Era, at which Modern Western men of science ceased to believe that astrological propositions were true and the consummation of Astrology's ruin as a practical going concern in non-scientific circles in the West and elsewhere. In A.D. 1953. a glance at any issue of any organ of the popular press in Eng- land would inform an historian that at this date Astrology, so far from having been sent to Jericho, was fast occupying a spiritual vacuum created by the waning of Western Man's belief m the truth of Astrology's old rival, Christianity. A more surprising phenomenon, of which there were indications at this date, was an incipient readmission of this long-since scientifically discredited science into the Eden of scientific truth from which it had once been ignominiously expelled. The tree of knowledge on to which Astrology was being grafted this time was not its ungrateful natural daughter Astronomy, but an indulgent adoptive mother—Psychology. * See V/£ 56-57.