this sort of thing not only as a hallucination but also as evidence of an utterly nonmathematical mind. In a certain measure, geometry may be treated algebraically and algebra geometrically, that is, the eye may be switched off or it may be allowed to govern. We take the first alternative, the Greeks the second. Archimedes, in his beautiful management of spirals, touches upon certain general facts that are also fundamentals in Leibniz’s method of the definite integral; but his processes, for all their superficial appearance of modernity, are subordinated to stereometric principles; in like case, an Indian mathematician would naturally have found some trigonometrical formulation.[72] ----- Footnote 70: A beginning is now being made with the application of non-Euclidean geometries to astronomy. The hypothesis of curved space, closed but without limits, filled by the system of fixed stars on a radius of about 470,000,000 earth-distances, would lead to the hypothesis of a counter-image of the sun which to us appears as a star of medium brilliancy. (See translator’s footnote, p. 332.) Footnote 71: That only one parallel to a given straight line is possible through a given point—a proposition that is incapable of proof. Footnote 72: It is impossible to say, with certainty, how much of the Indian mathematics that we possess is old, i.e., before Buddha. ----- XIII From this fundamental opposition of Classical and Western numbers there arises an equally radical difference in the relationship of element to element in each of these number-worlds. The nexus of _magnitudes_ is called _proportion_, that of _relations_ is comprised in the notion of _function_. The significance of these two words is not confined to mathematics proper; they are of high importance also in the allied arts of sculpture and music. Quite apart from the rôle of proportion in ordering the parts of the _individual_ statue, the typically Classical artforms of the statue, the relief, and the fresco, admit _enlargements and reductions of scale—words that in music have no meaning at all_—as we see in the art of the gems, in which the subjects are essentially reductions from life-sized originals. In the domain of Function, on the contrary, it is the idea of _transformation of groups_ that is of decisive importance, and the musician will readily agree that similar ideas play an essential part in modern composition-theory. I need only allude to one of the most elegant orchestral forms of the 18th Century, the _Tema con Variazioni_. All proportion assumes the constancy, all transformation the variability of the constituents. Compare, for instance, the congruence theorems of Euclid, the proof of which depends in fact on the assumed ratio 1 : 1, with the modern deduction of the same by means of angular functions. XIV