² It turns out in fact that a = d = 8α ² – 8α + 1, b = 8α ² – 12α + 4, c = 8α ² – 4α.

³ In either case it is obvious that the roots must be reciprocals of each other, since the constant term in the quadratic equation is ad – bc = 1.

⁴ Cf. P. W. Bridgman, “The New Vision of Science,” Harper's Magazine, March 1929, for a popular exposition of the principle of indeterminacy.

⁵ The billiard ball system would furnish an excellent gambling game of this type. The ball could be set in motion in a given direction and bets could be laid as to whether the ball would be in a certain region on the table at the sound of a random gong.

⁶ The trajectories of these exceptional motions form a point set of Lebesgue measure zero in the phase space.

⁷ Let a_n be the average result of measuring simultaneously, say, an observable α in each of n like systems in the normalized state ψ. The notation of Dirac is here used. Our hypothetical causal theory would therefore have to yield an explanation, based on certain aesthetically satisfying physical “laws” yet to be discovered, of why lim a_n exists and is equal to φ∞ψ. Such a “causal” theory might be of a very abstract nature and the timelike independent variable in the underlying differential equations might have little or nothing to do with ordinary time. In fact it might be in accordance with the theory of relativity in which time and the spacial dimensions are treated almost on an equal footing.

⁸ Cf. “The Gompertz curve as a growth curve” by C. P. Winsor. Proc. of the Nat. Ac. of Sciences, vol. 18 (1932), pp. 1–8.

⁹ Cf. his book: “Pareto's General Sociology, A Physiologist's Interpretation” (Harvard University Press, 1935). Professor Henderson analyzes the notion of equilibrium in the social order as compared to equilibrium in a physico-chemical system.

¹⁰ “Sur le retour éternel,” Comptes Rendus, Jan. 21, 1935.