[640] “There can be no doubt that Fraas is correct in regarding this type (Procetus) as an annectant form between the Zeuglodonts and the Creodonta, but, although the origin of the Zeuglodonts is thus made clear, it still seems to be by no means so certain as that author believes, that they may not themselves be the ancestral forms of the Odontoceti”; Andrews, Tertiary Vertebrata of the Fayum, 1906, p. 235.
[641] Reprinted, with some changes and additions, from a paper in the Trans. Roy. Soc. Edin. L, pp. 857–95, 1915.
[642] M. Bergson repudiates, with peculiar confidence, the application of mathematics to biology. Cf. Creative Evolution, p. 21, “Calculation touches, at most, certain phenomena of organic destruction. Organic creation, on the contrary, the evolutionary phenomena which properly constitute life, we cannot in any way subject to a mathematical treatment.”
[643] In this there lies a certain justification for a saying of Minot’s, of the greater part of which, nevertheless, I am heartily inclined to disapprove. “We biologists,” he says, “cannot deplore too frequently or too emphatically the great mathematical delusion by which men often of great if limited ability have been misled into becoming advocates of an erroneous conception of accuracy. The delusion is that no science is accurate until its results can be expressed mathematically. The error comes from the assumption that mathematics can express complex relations. Unfortunately mathematics have a very limited scope, and are based upon a few extremely rudimentary experiences, which we make as very little children and of which no adult has any recollection. The fact that from this basis men of genius have evolved wonderful methods of dealing with numerical relations should not blind us to another fact, namely, that the observational basis of mathematics is, psychologically speaking, very minute compared with the observational basis of even a single minor branch of biology .... While therefore here and there the mathematical methods may aid us, we need a kind and degree of accuracy of which mathematics is absolutely incapable .... With human minds constituted as they actually are, we cannot anticipate that there will ever be a mathematical expression for any organ or even a single cell, although formulae will continue to be useful for dealing now and then with isolated details...” (op. cit., p. 19, 1911). It were easy to discuss and criticise these sweeping assertions, which perhaps had their origin and parentage in an obiter dictum of Huxley’s, to the effect that “Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation” (cit. Cajori, Hist of Elem. Mathematics, p. 283). But Gauss called mathematics “a science of the eye”; and Sylvester assures us that “most, if not all, of the great ideas of modern mathematics have had their origin in observation” (Brit. Ass. Address, 1869, and Laws of Verse, p. 120, 1870).
[644] Historia Animalium I, 1.
[645] Cf. supra, p. 714.
[646] Cf. Osborn, H. F., On the Origin of Single Characters, as observed in fossil and living Animals and Plants, Amer. Nat. XLIX, pp. 193–239, 1915 (and other papers); ibid. p. 194, “Each individual is composed of a vast number of somewhat similar new or old characters, each character has its independent and separate history, each character is in a certain stage of evolution, each character is correlated with the other characters of the individual .... The real problem has always been that of the origin and development of characters. Since the Origin of Species appeared, the terms variation and variability have always referred to single characters; if a species is said to be variable, we mean that a considerable number of the single characters or groups of characters of which it is composed are variable,” etc.
[647] Cf. Sorby, Quart. Journ. Geol. Soc. (Proc.), 1879, p. 88.
[648] Cf. D’Orbigny, Alc., Cours élém. de Paléontologie, etc., I, pp. 144–148, 1849; see also Sharpe, Daniel, On Slaty Cleavage, Q.J.G.S. III, p. 74, 1847.
[649] Thus Ammonites erugatus, when compressed, has been described as A. planorbis: cf. Blake, J. F., Phil. Mag. (5), VI, p. 260, 1878. Wettstein has shewn that several species of the fish-genus Lepidopus have been based on specimens artificially deformed in various ways: Ueber die Fischfauna des Tertiären Glarnerschiefers, Abh. Schw. Palaeont. Gesellsch. XIII, 1886 (see especially pp. 23–38, pl. I). The whole subject, interesting as it is, has been little studied: both Blake and Wettstein deal with it mathematically.