On Growth and Form

NOTES:

1 These sayings of Kant and of Du Bois, and others like to them, have been the text of many discourses: see, for instance, Stallo’s Concepts, p. 21, 1882; HÖber, Biol. Centralbl. XIX, p. 284, 1890, etc. Cf. also Jellett, Rep. Brit. Ass. 1874, p. 1.

2 “Quum enim mundi universi fabrica sit perfectissima, atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum quin omnes mundi effectus ex causis finalibus, ope methodi maximorum et minimorum, aeque feliciter determinari queant atque ex ipsis causis efficientibus.” Methodus inveniendi, etc. 1744 (cit. Mach, Science of Mechanics, 1902, p. 455).

3 Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les causes finales..., c’est de lÀ qu’il faut tout dÉduire en Physique.”

4 Cf. p. 162. “La force vitale dirige des phÉnomÈnes qu’elle ne produit pas: les agents physiques produisent des phÉnomÈnes qu’ils ne dirigent pas.”

5 It is now and then conceded with reluctance. Thus Enriques, a learned and philosophic naturalist, writing “della economia di sostanza nelle osse cave” (Arch. f. Entw. Mech. XX, 1906), says “una certa impronta di teleologismo quÀ e lÀ È rimasta, mio malgrado, in questo scritto.”

6 Cf. Cleland, On Terminal Forms of Life, J. Anat. and Phys. XVIII, 1884.

7 Conklin, Embryology of Crepidula, Journ. of Morphol. XIII, p. 203, 1897; Lillie, F. R., Adaptation in Cleavage, Woods Holl Biol. Lectures, pp. 43–67, 1899.

8 I am inclined to trace back Driesch’s teaching of Entelechy to no less a person than Melanchthon. When Bacon (de Augm. IV, 3) states with disapproval that the soul “has been regarded rather as a function than as a substance,” R. L. Ellis points out that he is referring to Melanchthon’s exposition of the Aristotelian doctrine. For Melanchthon, whose view of the peripatetic philosophy had long great influence in the Protestant Universities, affirmed that, according to the true view of Aristotle’s opinion, the soul is not a substance, but an ??te???e?a, or function. He defined it as d??a?? quaedam ciens actiones—a description all but identical with that of Claude Bernard’s “force vitale.”

9 Ray Lankester, Encycl. Brit. (9th ed.), art. “Zoology,” p. 806, 1888.

10 Alfred Russel Wallace, especially in his later years, relied upon a direct but somewhat crude teleology. Cf. his World of Life, a Manifestation of Creative Power, Directive Mind and Ultimate Purpose, 1910.

11 Janet, Les Causes Finales, 1876, p. 350.

12 The phrase is Leibniz’s, in his ThÉodicÉe.

13 Cf. (int. al.) Bosanquet, The Meaning of Teleology, Proc. Brit. Acad. 1905–6, pp. 235–245. Cf. also Leibniz (Discours de MÉtaphysique; Lettres inÉdites, ed. de Careil, 1857, p. 354; cit. Janet, p. 643), “L’un et l’autre est bon, l’un et l’autre peut Être utile ... et les auteurs qui suivent ces routes diffÉrentes ne devraient point se maltraiter: et seq.”

14 The reader will understand that I speak, not of the “severe and diligent inquiry” of variation or of “fortuity,” but merely of the easy assumption that these phenomena are a sufficient basis on which to rest, with the all-powerful help of natural selection, a theory of definite and progressive evolution.

15 Revue Philosophique. XXXIII, 1892.

16 This general principle was clearly grasped by Dr George Rainey (a learned physician of St Bartholomew’s) many years ago, and expressed in such words as the following: “......it is illogical to suppose that in the case of vital organisms a distinct force exists to produce results perfectly within the reach of physical agencies, especially as in many instances no end could be attained were that the case, but that of opposing one force by another capable of effecting exactly the same purpose.” (On Artificial Calculi, Q.J.M.S. (Trans. Microsc. Soc.), VI, p. 49, 1858.) Cf. also Helmholtz, infra cit., p. 9.

17 Whereby he incurred the reproach of Socrates, in the Phaedo.

18 In a famous lecture (Conservation of Forces applied to Organic Nature, Proc. Roy. Instit., April 12, 1861), Helmholtz laid it down, as “the fundamental principle of physiology,” that “There may be other agents acting in the living body than those agents which act in the inorganic world; but those forces, as far as they cause chemical and mechanical influence in the body, must be quite of the same character as inorganic forces: in this at least, that their effects must be ruled by necessity, and must always be the same when acting in the same conditions; and so there cannot exist any arbitrary choice in the direction of their actions.” It would follow from this, that, like the other “physical” forces, they must be subject to mathematical analysis and deduction. Cf. also Dr T. Young’s Croonian Lecture On the Heart and Arteries, Phil. Trans. 1809, p. 1; Coll. Works, I, 511.

19 Ektropismus, oder die physikalische Theorie des Lebens, Leipzig, 1910.

20 Wilde Lecture, Nature, March 12, 1908; ibid. Sept. 6, 1900, p. 485; Aether and Matter, p. 288. Cf. also Lord Kelvin, Fortnightly Review, 1892, p. 313.

21 Joly, The Abundance of Life, Proc. Roy. Dublin Soc. VII, 1890; and in Scientific Essays, etc. 1915, p. 60 et seq.

22 Papillon, Histoire de la philosophie moderne, I, p. 300.

23 With the special and important properties of colloidal matter we are, for the time being, not concerned.

24 Cf. Hans Przibram, Anwendung elementarer Mathematik auf Biologische Probleme (in Roux’s VortrÄge, Heft III), Leipzig, 1908, p. 10.

25 The subject is treated from an engineering point of view by Prof. James Thomson, Comparisons of Similar Structures as to Elasticity, Strength, and Stability, Trans. Inst. Engineers, Scotland, 1876 (Collected Papers, 1912, pp. 361–372), and by Prof. A. Barr, ibid. 1899; see also Rayleigh, Nature, April 22, 1915.

26 Cf. Spencer, The Form of the Earth, etc., Phil. Mag. XXX, pp. 194–6, 1847; also Principles of Biology, pt. II, ch. I, 1864 (p. 123, etc.).

27 George Louis Lesage (1724–1803), well known as the author of one of the few attempts to explain gravitation. (Cf. Leray, Constitution de la MatiÈre, 1869; Kelvin, Proc. R. S. E. VII, p. 577, 1872, etc.; Clerk Maxwell, Phil. Trans. vol. 157, p. 50, 1867; art. “Atom,” Encycl. Brit. 1875, p. 46.)

28 Cf. Pierre PrÉvost, Notices de la vie et des Écrits de Lesage, 1805; quoted by Janet, Causes Finales, app. III.

29 Discorsi e Dimostrazioni matematiche, intorno À due nuove scienze, attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli Elzevirii, MDCXXXVIII. Opere, ed. Favaro, VIII, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 1914, p. 130, etc. See Nature, June 17, 1915.

30 So Werner remarked that Michael Angelo and Bramanti could not have built of gypsum at Paris on the scale they built of travertin in Rome.

31 Sir G. Greenhill, Determination of the greatest height to which a Tree of given proportions can grow, Cambr. Phil. Soc. Pr. IV, p. 65, 1881, and Chree, ibid. VII, 1892. Cf. Poynting and Thomson’s Properties of Matter, 1907, p 99.

32 In like manner the wheat-straw bends over under the weight of the loaded ear, and the tip of the cat’s tail bends over when held upright,—not because they “possess flexibility,” but because they outstrip the dimensions within which stable equilibrium is possible in a vertical position. The kitten’s tail, on the other hand, stands up spiky and straight.

33 Modern Painters.

34 The stem of the giant bamboo may attain a height of 60 metres, while not more than about 40 cm. in diameter near its base, which dimensions are not very far short of the theoretical limits (A. J. Ewart, Phil. Trans. vol. 198, p. 71, 1906).

35 Trans. Zool. Soc. IV, 1850, p. 27.

36 It would seem to be a common if not a general rule that marine organisms, zoophytes, molluscs, etc., tend to be larger than the corresponding and closely related forms living in fresh water. While the phenomenon may have various causes, it has been attributed (among others) to the simple fact that the forces of growth are less antagonised by gravity in the denser medium (cf. Houssay, La Forme et la Vie, 1900, p. 815). The effect of gravity on outward form is illustrated, for instance, by the contrast between the uniformly upward branching of a sea-weed and the drooping curves of a shrub or tree.

37 The analogy is not a very strict one. We are not taking account, for instance, of a proportionate increase in thickness of the boiler-plates.

38 Let L be the length, S the (wetted) surface, T the tonnage, D the displacement (or volume) of a ship; and let it cross the Atlantic at a speed V. Then, in comparing two ships, similarly constructed but of different magnitudes, we know that L =V?², S =L?² =V?⁴, D =T =L?³ =V?⁶; also R (resistance) =S·V?² =V?⁶; H (horse-power) =R·V =V?⁷; and the coal (C) necessary for the voyage =H/V =V?⁶. That is to say, in ordinary engineering language, to increase the speed across the Atlantic by 1 per cent. the ship’s length must be increased 2 per cent., her tonnage or displacement 6 per cent., her coal-consumpt also 6 per cent., her horse-power, and therefore her boiler-capacity, 7 per cent. Her bunkers, accordingly, keep pace with the enlargement of the ship, but her boilers tend to increase out of proportion to the space available.

39 This is the result arrived at by Helmholtz, Ueber ein Theorem geometrisch Ähnliche Bewegungen flÜssiger KÖrper betreffend, nebst Anwendung auf das Problem Luftballons zu lenken, Monatsber. Akad. Berlin, 1873, pp. 501–14. It was criticised and challenged (somewhat rashly) by K. MÜllenhof, Die GrÖsse der FlugflÄchen, etc., PflÜger’s Archiv, XXXV, p. 407, XXXVI, p. 548, 1885.

40 Cf. also Chabrier, Vol des Insectes, MÉm. Mus. Hist. Nat. Paris, VI–VIII, 1820–22.

41 Aerial Flight, vol. II (Aerodonetics), 1908, p. 150.

42 By Lanchester, op. cit. p. 131.

43 Cf. L’empire de l’air; ornithologie appliquÉe À l’aviation. 1881.

44 De Motu Animalium, I, prop. cciv, ed. 1685, p. 243.

45 HarlÉ, On Atmospheric Pressure in past Geological Ages, Bull. Geol. Soc. Fr. XI, pp. 118–121; or Cosmos, p. 30, July 8, 1911.

46 Introduction to Entomology, 1826, II, p. 190. K. and S., like many less learned authors, are fond of popular illustrations of the “wonders of Nature,” to the neglect of dynamical principles. They suggest, for instance, that if the white ant were as big as a man, its tunnels would be “magnificent cylinders of more than three hundred feet in diameter”; and that if a certain noisy Brazilian insect were as big as a man, its voice would be heard all the world over: “so that Stentor becomes a mute when compared with these insects!” It is an easy consequence of anthropomorphism, and hence a common characteristic of fairy-tales, to neglect the principle of dynamical, while dwelling on the aspect of geometrical, similarity.

47 I.e. the available energy of muscle, in ft.-lbs. per lb. of muscle, is the same for all animals: a postulate which requires considerable qualification when we are comparing very different kinds of muscle, such as the insect’s and the mammal’s.

48 Prop. clxxvii. Animalia minora et minus ponderosa majores saltus efficiunt respectu sui corporis, si caetera fuerint paria.

49 See also (int. al.), John Bernoulli, de Motu Musculorum, Basil., 1694; Chabry, MÉcanisme du Saut, J. de l’Anat. et de la Physiol. XIX, 1883; Sur la longueur des membres des animaux sauteurs, ibid. XXI, p. 356, 1885; Le Hello, De l’action des organes locomoteurs, etc., ibid. XXIX, p. 65–93, 1893, etc.

50 Recherches sur la force absolue des muscles des InvertÉbrÉs, Bull. Acad. E. de Belgique (3), VI, VII, 1883–84; see also ibid. (2), XX, 1865, XXII, 1866; Ann. Mag. N. H. XVII, p. 139, 1866, XIX, p. 95, 1867. The subject was also well treated by Straus-DÜrckheim, in his ConsidÉrations gÉnÉrales sur l’anatomie comparÉe des animaux articulÉs, 1828.

51 The fact that the limb tends to swing in pendulum-time was first observed by the brothers Weber (Mechanik der menschl. Gehwerkzeuge, GÖttingen, 1836). Some later writers have criticised the statement (e.g. Fischer, Die Kinematik des Beinschwingens etc., Abh. math. phys. Kl. k. SÄchs. Ges. XXV–XXVIII, 1899–1903), but for all that, with proper qualifications, it remains substantially true.

52 Quoted in Mr John Bishop’s interesting article in Todd’s Cyclopaedia, III, p. 443.

53 There is probably also another factor involved here: for in bending, and therefore shortening, the leg we bring its centre of gravity nearer to the pivot, that is to say, to the joint, and so the muscle tends to move it the more quickly.

54 Proc. Psychical Soc. XII, pp. 338–355, 1897.

55 For various calculations of the increase of surface due to histological and anatomical subdivision, see E. Babak, Ueber die OberflÄchenentwickelung bei Organismen, Biol. Centralbl. XXX, pp. 225–239, 257–267, 1910. In connection with the physical theory of surface-energy, Wolfgang Ostwald has introduced the conception of specific surface, that is to say the ratio of surface to volume, or S/V. In a cube, V =l?³, and S =6l?²; therefore S/V =6/l. Therefore if the side l measure 6 cm., the ratio S/V =1, and such a cube may be taken as our standard, or unit of specific surface. A human blood-corpuscle has, accordingly, a specific surface of somewhere about 14,000 or 15,000. It is found in physical chemistry that surface energy becomes an important factor when the specific surface reaches a value of 10,000 or thereby.

56 Though the entire egg is not increasing in mass, this is not to say that its living protoplasm is not increasing all the while at the expense of the reserve material.

57 Cf. Tait, Proc. R.S.E. V, 1866, and VI, 1868.

58 Physiolog. Notizen (9), p. 425, 1895. Cf. StrasbÜrger, Ueber die WirkungssphÄre der Kerne und die ZellgrÖsse, Histolog. Beitr. (5), pp. 95–129, 1893; J. J. Gerassimow, Ueber die GrÖsse des Zellkernes, Beih. Bot. Centralbl. XVIII, 1905; also G. Levi and T. Terni, Le variazioni dell’ indice plasmatico-nucleare durante l’intercinesi, Arch. Ital. di Anat. X, p. 545, 1911.

59 Arch. f. Entw. Mech. IV, 1898, pp. 75, 247.

60 Conklin, E. G., Cell-size and nuclear-size, J. Exp. Zool. XII. pp. 1–98, 1912.

61 Thus the fibres of the crystalline lens are of the same size in large and small dogs; Rabl, Z. f. w. Z. LXVII, 1899. Cf. (int. al.) Pearson, On the Size of the Blood-corpuscles in Rana, Biometrika, VI, p. 403, 1909. Dr Thomas Young caught sight of the phenomenon, early in last century: “The solid particles of the blood do not by any means vary in magnitude in the same ratio with the bulk of the animal,” Natural Philosophy, ed. 1845, p. 466; and Leeuwenhoek and Stephen Hales were aware of it a hundred years before. But in this case, though the blood-corpuscles show no relation of magnitude to the size of the animal, they do seem to have some relation to its activity. At least the corpuscles in the sluggish Amphibia are much the largest known to us, while the smallest are found among the deer and other agile and speedy mammals. (Cf. Gulliver, P.Z.S. 1875, p. 474, etc.) This apparent correlation may have its bearing on modern views of the surface-condensation or adsorption of oxygen in the blood-corpuscles, a process which would be greatly facilitated and intensified by the increase of surface due to their minuteness.

62 Cf. P. Enriques, La forma come funzione della grandezza: Ricerche sui gangli nervosi degli Invertebrati, Arch. f. Entw. Mech. XXV, p. 655, 1907–8.

63 While the difference in cell-volume is vastly less than that between the volumes, and very much less also than that between the surfaces, of the respective animals, yet there is a certain difference; and this it has been attempted to correlate with the need for each cell in the many-celled ganglion of the larger animal to possess a more complex “exchange-system” of branches, for intercommunication with its more numerous neighbours. Another explanation is based on the fact that, while such cells as continue to divide throughout life tend to uniformity of size in all mammals, those which do not do so, and in particular the ganglion cells, continue to grow, and their size becomes, therefore, a function of the duration of life. Cf. G. Levi, Studii sulla grandezza delle cellule, Arch. Ital. di Anat. e di Embryolog. V, p. 291, 1906.

64 Boveri. Zellen-studien, V. Ueber die AbhÄngigkeit der KerngrÖsse und Zellenzahl der Seeigellarven von der Chromosomenzahl der Ausgangszellen. Jena, 1905.

65 Recent important researches suggest that such ultra-minute “filter-passers” are the true cause of certain acute maladies commonly ascribed to the presence of much larger organisms; cf. Hort, Lakin and Benians, The true infective Agent in Cerebrospinal Fever, etc., J. Roy. Army Med. Corps, Feb. 1910.

66 Zur Erkenntniss der Kolloide, 1905, p. 122; where there will be found an interesting discussion of various molecular and other minute magnitudes.

67 Encyclopaedia Britannica, 9th edit., vol. III, p. 42, 1875.

68 Sur la limite de petitesse des organismes, Bull. Soc. R. des Sc. mÉd. et nat. de Bruxelles, Jan. 1903; Rec. d’oeuvres (Physiol. gÉnÉrale), p. 325.

69 Cf. A. Fischer, Vorlesungen Über Bakterien, 1897, p. 50.

70 F. Hofmeister, quoted in Cohnheim’s Chemie der EiweisskÖrper, 1900, p. 18.

71 McKendrick arrived at a still lower estimate, of about 1250 proteid molecules in the minutest organisms. Brit. Ass. Rep. 1901, p. 808.

72 Cf. Perrin, Les Atomes, 1914, p. 74.

73 Cf. Tait, On Compression of Air in small Bubbles, Proc. R. S. E. V, 1865.

74 Phil. Mag. XLVIII, 1899; Collected Papers, IV, p. 430.

75 Carpenter, The Microscope, edit. 1862, p. 185.

76 The modern literature on the Brownian Movement is very large, owing to the value which the phenomenon is shewn to have in determining the size of the atom. For a fuller, but still elementary account, see J. Cox, Beyond the Atom, 1913, pp. 118–128; and see, further, Perrin, Les Atomes, pp. 119–189.

77 Cf. R. Gans, Wie fallen StÄbe und Scheiben in einer reibenden FlÜssigkeit? MÜnchener Bericht, 1911, p. 191; K. Przibram, Ueber die Brown’sche Bewegung nicht kugelfÖrmiger Teilchen, Wiener Ber. 1912, p. 2339.

78 Ueber die ungeordnete Bewegung niederer Thiere, PflÜger’s Archiv, CLIII, p. 401, 1913.

79 Sometimes we find one and the same diagram suffice, whether the intervals of time be great or small; and we then invoke “Wolff’s Law,” and assert that the life-history of the individual repeats, or recapitulates, the history of the race.

80 Our subject is one of Bacon’s “Instances of the Course,” or studies wherein we “measure Nature by periods of Time.” In Bacon’s Catalogue of Particular Histories, one of the odd hundred histories or investigations which he foreshadowed is precisely that which we are engaged on, viz. a “History of the Growth and Increase of the Body, in the whole and in its parts.”

81 Cf. Aristotle, Phys. vi, 5, 235 a 11, ?pe? ??? ?pasa ????s?? ?? ?????, ?t?. Bacon emphasised, in like manner, the fact that “all motion or natural action is performed in time: some more quickly, some more slowly, but all in periods determined and fixed in the nature of things. Even those actions which seem to be performed suddenly, and (as we say) in the twinkling of an eye, are found to admit of degree in respect of duration.” Nov. Org. XLVI.

82 Cf. (e.g.) Elem. Physiol. ed. 1766, VIII, p. 114, “Ducimur autem ad evolutionem potissimum, quando a perfecto animale retrorsum progredimur, et incrementorum atque mutationum seriem relegimus. Ita inveniemus perfectum illud animal fuisse imperfectius, alterius figurae et fabricae, et denique rude et informe: et tamen idem semper animal sub iis diversis phasibus fuisse, quae absque ullo saltu perpetuos parvosque per gradus cohaereant.”

83 BeitrÄge zur Entwickelungsgeschichte des HÜhnchens im Ei, p. 40, 1817. Roux ascribes the same views also to Von Baer and to R. H. Lotze (Allg. Physiologie, p. 353, 1851).

84 Roux, Die Entwickelungsmechanik, p. 99, 1905.

85 Op. cit. p. 302, “Magnum hoc naturae instrumentum, etiam in corpore animato evolvendo potenter operatur; etc.”

86 Ibid. p. 306. “Subtiliora ista, et aliquantum hypothesi mista, tamen magnum mihi videntur speciem veri habere.”

87 Cf. His, On the Principles of Animal Morphology, Proc. R. S. E. XV, 1888, p. 294: “My own attempts to introduce some elementary mechanical or physiological conceptions into embryology have not generally been agreed to by morphologists. To one it seemed ridiculous to speak of the elasticity of the germinal layers; another thought that, by such considerations, we ‘put the cart before the horse’: and one more recent author states, that we have better things to do in embryology than to discuss tensions of germinal layers and similar questions, since all explanations must of necessity be of a phylogenetic nature. This opposition to the application of the fundamental principles of science to embryological questions would scarcely be intelligible had it not a dogmatic background. No other explanation of living forms is allowed than heredity, and any which is founded on another basis must be rejected ....... To think that heredity will build organic beings without mechanical means is a piece of unscientific mysticism.”

88 Hertwig, O., Zeit und Streitfragen der Biologie, II. 1897.

89 Cf. Roux, Gesammelte Abhandlungen, II, p. 31, 1895.

90 Treatise on Comparative Embryology, I, p. 4, 1881.

91 Cf. Fick, Anal. Anzeiger, XXV, p. 190, 1904.

92 1st ed. p. 444; 6th ed. p. 390. The student should not fail to consult the passage in question; for there is always a risk of misunderstanding or misinterpretation when one attempts to epitomise Darwin’s carefully condensed arguments.

93 “In omni rerum naturalium historia utile est mensuras definiri et numeros,” Haller, Elem. Physiol. II, p. 258, 1760. Cf. Hales, Vegetable Staticks, Introduction.

94 Brussels, 1871. Cf. the same author’s Physique sociale, 1835, and Lettres sur la thÉorie des probabilitÉs, 1846. See also, for the general subject, Boyd, R., Tables of weights of the Human Body, etc. Phil. Trans. vol. CLI, 1861; Roberts, C., Manual of Anthropometry, 1878; Daffner, F., Das Wachsthum des Menschen (2nd ed.), 1902, etc.

95 Dr Johnson was not far wrong in saying that “life declines from thirty-five”; though the Autocrat of the Breakfast-table, like Cicero, declares that “the furnace is in full blast for ten years longer.”

96 Joly, The Abundance of Life, 1915 (1890), p. 86.

97 “Lou pes, mÈstre de tout [Le poids, maÎtre de tout], mÈstre sÈnso vergougno, Que te tirasso en bas de sa brutalo pougno,” J. H. Fabre, Oubreto prouvenÇalo, p. 61.

98 The continuity of the phenomenon of growth, and the natural passage from the phase of increase to that of decrease or decay, are admirably discussed by Enriques, in “La morte,” Riv. di Scienza, 1907, and in “Wachsthum und seine analytische Darstellung,” Biol. Centralbl. June, 1909. Haller (Elem. VII, p. 68) recognised decrementum as a phase of growth, not less important (theoretically) than incrementum: “tristis, sed copiosa, haec est materies.”

99 Cf. (int. al.), Friedenthal, H., Das Wachstum des KÖrpergewichtes ... in verschiedenen LebensÄltern, Zeit. f. allg. Physiol. IX, pp. 487–514, 1909.

100 As Haller observed it to do in the chick (Elem. VIII, p. 294): “Hoc iterum incrementum miro ordine ita distribuitur, ut in principio incubationis maximum est: inde perpetuo minuatur.”

101 There is a famous passage in Lucretius (v. 883) where he compares the course of life, or rate of growth, in the horse and his boyish master: Principio circum tribus actis impiger annis Floret equus, puer hautquaquam, etc.

102 Minot, C. S., Senescence and Rejuvenation, Journ. of Physiol. XII, pp. 97–153, 1891; The Problem of Age, Growth and Death, Pop. Science Monthly (June–Dec.), 1907.

103 Quoted in Vierordt’s Anatomische ... Daten und Tabellen, 1906. p. 13.

104 Unsere KÖrperform, Leipzig, 1874.

105 No such point of inflection appears in the curve of weight according to C. M. Jackson’s data (On the Prenatal Growth of the Human Body, etc., Amer. Journ. of Anat. IX, 1009, pp. 126, 156), nor in those quoted by him from Ahlfeld, Fehling and others. But it is plain that the very rapid increase of the monthly weights, approximately in the ratio of the cubes of the corresponding lengths, would tend to conceal any such breach of continuity, unless it happened to be very marked indeed. Moreover in the case of Jackson’s data (and probably also in the others) the actual age of the embryos was not determined, but was estimated from their lengths. The following is Jackson’s estimate of average weights at intervals of a lunar month:

Months	0	1	2	3	4	5	6	7	8	9	10
Wt in gms.	·0	·04	3	36	120	330	600	1000	1500	2200	3200

106 G. Kraus (after Wallich-Martius), Ann. du Jardin bot. de Buitenzorg, XII, 1, 1894, p. 210. Cf. W. Ostwald, Zeitliche Eigenschaften, etc. p. 56.

107 Cf. Chodat, R., et Monnier, A., Sur la courbe de croissance des vÉgÉtaux, Bull. Herb. Boissier (2), V, pp. 615, 616, 1905.

108 Cf. Fr. Boas, Growth of Toronto Children, Rep. of U.S. Comm. of Education, 1896–7, pp. 1541–1599, 1898; Boas and Clark Wissler, Statistics of Growth, Education Rep. 1904, pp. 25–132, 1906; H. P. Bowditch, Rep. Mass. State Board of Health, 1877; K. Pearson, On the Magnitude of certain coefficients of Correlation in Man, Pr. R. S. LXVI, 1900.

109 l.c. p. 42, and other papers there quoted.

110 See, for an admirable rÉsumÉ of facts, Wolfgang Ostwald, Ueber die Zeitliche Eigenschaften der EntwickelungsvorgÄnge (71 pp.), Leipzig, 1908 (Roux’s VortrÄge, Heft V): to which work I am much indebted. A long list of observations on the growth-rate of various animals is also given by H. Przibram, Exp. Zoologie, 1913, pt. IV (VitalitÄt), pp. 85–87.

111 Cf. St Loup, Vitesse de croissance chez les Souris, Bull. Soc. Zool. Fr. XVIII, 242, 1893; Robertson, Arch. f. Entwickelungsmech. XXV, p. 587, 1908; Donaldson. Boas Memorial Volume, New York, 1906.

112 Luciani e Lo Monaco, Arch. Ital. de Biologie, XXVII, p. 340, 1897.

113 Schaper, Arch. f. Entwickelungsmech. XIV, p. 356, 1902. Cf. Barfurth, Versuche Über die Verwandlung der Froschlarven, Arch. f. mikr. Anat. XXIX, 1887.

114 Joh. Schmidt, Contributions to the Life-history of the Eel, Rapports du Conseil Intern. pour l’exploration de la Mer, vol. V, pp. 137–274, Copenhague, 1906.

115 That the metamorphoses of an insect are but phases in a process of growth, was firstly clearly recognised by Swammerdam, Biblia Naturae, 1737, pp. 6, 579 etc.

116 From Bose, J. C., Plant Response, London, 1906, p. 417.

117 This phenomenon, of incrementum inequale, as opposed to incrementum in universum, was most carefully studied by Haller: “Incrementum inequale multis modis fit, ut aliae partes corporis aliis celerius increscant. Diximus hepar minus fieri, majorem pulmonem, minimum thymum, etc.” (Elem. VIII (2), p. 34).

118 See (inter alia) Fischel, A., VariabilitÄt und Wachsthum des embryonalen KÖrpers, Morphol. Jahrb. XXIV, pp. 369–404, 1896. Oppel, Vergleichung des Entwickelungsgrades der Organe zu verschiedenen Entwickelungszeiten bei Wirbelthieren, Jena, 1891. Faucon, A., PesÉes et Mensurations foetales À diffÉrents Âges de la grossesse. (ThÈse.) Paris, 1897. Loisel, G., Croissance comparÉe en poids et en longueur des foetus mÂle et femelle dans l’espÈce humaine, C. R. Soc. de Biologie, Paris, 1903. Jackson, C. M., Pre-natal growth of the human body and the relative growth of the various organs and parts, Am. J. of Anat. IX, 1909; Post-natal growth and variability of the body and of the various organs in the albino rat, ibid. XV, 1913.

119 l.c. p. 1542.

120 Variation and Correlation in Brain-weight, Biometrika, IV, pp. 13–104, 1905.

121 Die SÄugethiere, p. 117.

122 Amer. J. of Anatomy, VIII, pp. 319–353, 1908. Donaldson (Journ. Comp. Neur. and Psychol. XVIII, pp. 345–392, 1908) also gives a logarithmic formula for brain-weight (y) as compared with body-weight (x), which in the case of the white rat is y =·554-·569 log(x-8·7), and the agreement is very close. But the formula is admittedly empirical and as Raymond Pearl says (Amer. Nat. 1909, p. 303), “no ulterior biological significance is to be attached to it.”

123 Biometrika, IV, pp. 13–104, 1904.

124 Donaldson, H. H., A Comparison of the White Rat with Man in respect to the Growth of the entire Body, Boas Memorial Vol., New York, 1906, pp. 5–26.

125 Besides many papers quoted by Dubois on the growth and weight of the brain, and numerous papers in Biometrika, see also the following: Ziehen, Th., Das Gehirn: MassverhÄltnisse, in Bardeleben’s Handb. der Anat. des Menschen, IV, pp. 353–386, 1899. Spitzka, E. A., Brain-weight of Animals with special reference to the Weight of the Brain in the Macaque Monkey, J. Comp. Neurol. XIII, pp. 9–17, 1903. Warneke, P., Mitteilung neuer Gehirn und KÖrpergewichtsbestimmungen bei SÄugern, nebst Zusammenstellung der gesammten bisher beobachteten absoluten und relativen Gehirngewichte bei den verschiedenen Species, J. f. Psychol. u. Neurol. XIII, pp. 355–403, 1909. Donaldson, H. H., On the regular seasonal Changes in the relative Weight of the Central Nervous System of the Leopard Frog, Journ. of Morph. XXII, pp. 663–694, 1911.

126 Cf. Jenkinson, Growth, Variability and Correlation in Young Trout, Biometrika, VIII, pp. 444–455, 1912.

127 Cf. chap. xvii, p. 739.

128 “...I marked in the same manner as the Vine, young Honeysuckle shoots, etc....; and I found in them all a gradual scale of unequal extensions, those parts extending most which were tenderest,” Vegetable Staticks, Exp. cxxiii.

129 From Sachs, Textbook of Botany, 1882, p. 820.

130 Variation and Differentiation in Ceratophyllum, Carnegie Inst. Publications, No. 58, Washington, 1907.

131 Cf. LÄmmel, Ueber periodische Variationen in Organismen, Biol. Centralbl. XXII, pp. 368–376, 1903.

132 Herein lies the easy answer to a contention frequently raised by Bergson, and to which he ascribes great importance, that “a mere variation of size is one thing, and a change of form is another.” Thus he considers “a change in the form of leaves” to constitute “a profound morphological difference.” Creative Evolution, p. 71.

133 I do not say that the assumption that these two groups of earwigs were of different ages is altogether an easy one; for of course, even in an insect whose metamorphosis is so simple as the earwig’s, consisting only in the acquisition of wings or wing-cases, we usually take it for granted that growth proceeds no more after the final stage, or “adult form” is attained, and further that this adult form is attained at an approximately constant age, and constant magnitude. But even if we are not permitted to think that the earwig may have grown, or moulted, after once the elytra were produced, it seems to me far from impossible, and far from unlikely, that prior to the appearance of the elytra one more stage of growth, or one more moult took place in some cases than in others: for the number of moults is known to be variable in many species of Orthoptera. Unfortunately Bateson tells us nothing about the sizes or total lengths of his earwigs; but his figures suggest that it was bigger earwigs that had the longer tails; and that the rate of growth of the tails had had a certain definite ratio to that of the bodies, but not necessarily a simple ratio of equality.

134 Jackson, C. M., J. of Exp. Zool. XIX, 1915, p. 99; cf. also Hans Aron, Unters. Über die BeeinflÜssung der Wachstum durch die ErnÄhrung, Berl. klin. Wochenbl. LI, pp. 972–977, 1913, etc.

135 The temperature limitations of life, and to some extent of growth, are summarised for a large number of species by Davenport, Exper. Morphology, cc. viii, xviii, and by Hans Przibram, Exp. Zoologie, IV, c. v.

136 RÉaumur: L’art de faire Éclore et Élever en toute saison des oiseaux domestiques, foit par le moyen de la chaleur du fumier, Paris, 1749.

137 Cf. (int. al.) de Vries, H., MatÉriaux pour la connaissance de l’influence de la tempÉrature sur les plantes, Arch. NÉerl. V, 385–401, 1870. KÖppen, WÄrme und Pflanzenwachstum, Bull. Soc. Imp. Nat. Moscou. XLIII, pp. 41–110, 1870.

138 Blackman, F. F., Ann. of Botany, XIX, p. 281, 1905.

139 For various instances of a “temperature coefficient” in physiological processes, see Kanitz, Zeitschr. f. Elektrochemie, 1907, p. 707; Biol. Centralbl. XXVII, p. 11, 1907; Hertzog, R. O., Temperatureinfluss auf die Entwicklungsgeschwindigkeit der Organismen, Zeitschr. f. Elektrochemie, XI, p 820, 1905; Krogh, Quantitative Relation between Temperature and Standard Metabolism, Int. Zeitschr. f. physik.-chem. Biologie, I, p. 491, 1914; PÜtter, A., Ueber Temperaturkoefficienten, Zeitschr. f. allgem. Physiol. XVI, p. 574, 1914. Also Cohen, Physical Chemistry for Physicians and Biologists (English edition), 1903; Pike, F. H., and Scott. E. L., The Regulation of the Physico-chemical Condition of the Organism, American Naturalist, Jan. 1915, and various papers quoted therein.

140 Cf. Errera, L., L’Optimum, 1896 (Rec. d’Oeuvres, Physiol. gÉnÉrale, pp. 338–368, 1910); Sachs, Physiologie d. Pflanzen, 1882, p. 233; Pfeffer, Pflanzenphysiologie, ii, p. 78, 1904; and cf. Jost, Ueber die Reactionsgeschwindigkeit im Organismus, Biol. Centralbl. XXVI, pp. 225–244, 1906.

141 After KÖppen, Bull. Soc. Nat. Moscou, XLIII, pp. 41–110, 1871.

142 Botany, p. 387.

143 Leitch, I., Some Experiments on the Influence of Temperature on the Rate of Growth in Pisum sativum, Ann. of Botany, XXX, pp. 25–46, 1916. (Cf. especially Table III, p. 45.)

144 Blackman, F. F., Presidential Address in Botany, Brit. Ass. Dublin, 1908.

145 Rec. de l’Inst. Bot. de Bruxelles, VI, 1906.

146 Hertwig, O., Einfluss der Temperatur auf die Entwicklung von Rana fusca und R. esculenta, Arch. f. mikrosk. Anat. LI, p. 319, 1898. Cf. also Bialaszewicz, K., BeitrÄge z. Kenntniss d. WachsthumsvorgÄnge bei Amphibienembryonen, Bull. Acad. Sci. de Cracovie, p. 783, 1908; Abstr. in Arch. f. Entwicklungsmech. XXVIII, p. 160, 1909.

147 Der Grad der Beschleunigung tierischer Entwickelung durch erhÖhte Temperatur, A. f. Entw. Mech. XX. p. 130, 1905. More recently, Bialaszewicz has determined the coefficient for the rate of segmentation in Rana as being 2·4 per 10° C.

148 Das Wachstum des Menschen, p. 329, 1902.

149 The diurnal periodicity is beautifully shewn in the case of the Hop by Joh. Schmidt (C. R. du Laboratoire de Carlsberg, X, pp. 235–248, Copenhague, 1913).

150 Trans. Botan. Soc. Edinburgh, XVIII, 1891, p. 456.

151 I had not received, when this was written, Mr Douglass’s paper, On a method of estimating Rainfall by the Growth of Trees, Bull. Amer. Geograph. Soc. XLVI, pp. 321–335, 1914. Mr Douglass does not fail to notice the long period here described; but he lays more stress on the occurrence of shorter cycles (of 11, 21 and 33 years), well known to meteorologists. Mr Douglass is inclined (and I think rightly) to correlate the variations in growth directly with fluctuations in rainfall, that is to say with alternate periods of moisture and aridity; but he points out that the temperature curves (and also the sunspot curves) are markedly similar.

152 It may well be that the effect is not due to light after all; but to increased absorption of heat by the soil, as a result of the long hours of exposure to the sun.

153 On growth in relation to light, see Davenport, Exp. Morphology, II, ch. xvii. In some cases (as in the roots of Peas), exposure to light seems to have no effect on growth; in other cases, as in diatoms (according to Whipple’s experiments, quoted by Davenport, II, p. 423), the effect of light on growth or multiplication is well-marked, measurable, and apparently capable of expression by a logarithmic formula. The discrepancy would seem to arise from the fact that, while light-energy always tends to be absorbed by the chlorophyll of the plant, converted into chemical energy, and stored in the shape of starch or other reserve materials, the actual rate of growth depends on the rate at which these reserves are drawn on: and this is another matter, in which light-energy is no longer directly concerned.

154 Cf. for instance, NÄgeli’s classical account of the effect of change of habitat on Alpine and other plants: Sitzungsber. Baier. Akad. Wiss. 1865, pp. 228–284.

155 Cf. Blackman, F. F., Presidential Address in Botany, Brit. Ass. Dublin, 1908. The fact was first enunciated by Baudrimont and St Ange, Recherches sur le dÉveloppement du foetus, MÉm. Acad. Sci. XI, p. 469, 1851.

156 Cf. Loeb, Untersuchungen zur physiol. Morphologie der Thiere, 1892; also Experiments on Cleavage, J. of Morph. VII, p. 253, 1892; Zusammenstellung der Ergebnisse einiger Arbeiten Über die Dynamik des thierischen Wachsthum, Arch. f. Entw. Mech. XV, 1902–3, p. 669; Davenport, On the RÔle of Water in Growth, Boston Soc. N. H. 1897; Ida H. Hyde, Am. J. of Physiol. XII, 1905, p. 241, etc.

157 PflÜger’s Archiv, LV, 1893.

158 BeitrÄge zur Kenntniss der WachstumsvorgÄnge bei Amphibienembryonen, Bull. Acad. Sci. de Cracovie, 1908, p. 783; cf. Arch. f. Entw. Mech. XXVIII, p. 160, 1909; XXXIV, p. 489, 1912.

159 Fehling, H., Arch. fÜr Gynaekologie, XI, 1877; cf. Morgan, Experimental Zoology, p. 240, 1907.

160 HÖber, R., Bedeutung der Theorie der LÖsungen fÜr Physiologie und Medizin, Biol. Centralbl. XIX, 1899; cf. pp. 272–274.

161 Schmankewitsch has made other interesting observations on change of size and form, after some generations, in relation to change of density; e.g. in the flagellate infusorian Anisonema acinus, BÜtschli (Z. f. w. Z. XXIX, p. 429, 1877).

162 These “Fezzan-worms,” when first described, were supposed to be “insects’ eggs”; cf. Humboldt, Personal Narrative, VI, i, 8, note; Kirby and Spence, Letter X.

163 Cf. Introd. À l’Étude de la mÉdecine expÉrimentale, 1885, p. 110.

164 Cf. Abonyi, Z. f. w. Z. CXIV, p. 134, 1915. But FrÉdÉricq has shewn that the amount of NaCl in the blood of Crustacea (Carcinus moenas) varies, and all but corresponds, with the density of the water in which the creature has been kept (Arch. de Zool. Exp. et GÉn. (2), III, p. xxxv, 1885); and other results of FrÉdÉricq’s, and various data given or quoted by Bottazzi (Osmotischer Druck und elektrische LeitungsfÄhigkeit der FlÜssigkeiten der Organismen, in Asher-Spiro’s Ergebn. d. Physiologie, VII, pp. 160–402, 1908) suggest that the case of the brine-shrimps must be looked upon as an extreme or exceptional one.

165 Cf. Schmankewitsch, Z. f. w. Zool. XXV, 1875, XXIX, 1877, etc.; transl. in appendix to Packard’s Monogr. of N. American Phyllopoda, 1883, pp. 466–514; Daday de DeÉs, Ann. Sci. Nat. (Zool.), (9), XI, 1910; Samter und Heymons, Abh. d. K. pr. Akad. Wiss. 1902; Bateson, Mat. for the Study of Variation, 1894, pp. 96–101; Anikin, Mitth. Kais. Univ. Tomsk, XIV: Zool. Centralbl. VI, pp. 756–760, 1908; Abonyi, Z. f. w. Z. CXIV, pp. 96–168, 1915 (with copious bibliography), etc.

166 According to the empirical canon of physiology, that (as FrÉdÉricq expresses it) “L’Être vivant est agencÉ de telle maniÈre que chaque influence perturbatrice provoque d’elle-mÊme la mise en activitÉ de l’appareil compensateur qui doit neutraliser et rÉparer le dommage.”

167 Such phenomena come precisely under the head of what Bacon called Instances of Magic: “By which I mean those wherein the material or efficient cause is scanty and small as compared with the work or effect produced; so that even when they are common, they seem like miracles, some at first sight, others even after attentive consideration. These magical effects are brought about in three ways ... [of which one is] by excitation or invitation in another body, as in the magnet which excites numberless needles without losing any of its virtue, or in yeast and such-like.” Nov. Org., cap. li.

168 Monnier, A., Les matiÈres minÉrales, et la loi d’accroissement des VÉgÉtaux, Publ. de l’Inst. de Bot. de l’Univ. de GenÈve (7), III, 1905. Cf. Robertson, On the Normal Rate of Growth of an Individual, and its Biochemical Significance, Arch. f. Entw. Mech. XXV, pp. 581–614, XXVI, pp. 108–118, 1908; Wolfgang Ostwald, Die zeitlichen Eigenschaften der EntwickelungsvorgÄnge, 1908; Hatai, S., Interpretation of Growth-curves from a Dynamical Standpoint, Anat. Record, V, p. 373, 1911.

169 Biochem. Zeitschr. II, 1906, p. 34.

170 Even a crystal may be said, in a sense, to display “autocatalysis”: for the bigger its surface becomes, the more rapidly does the mass go on increasing.

171 Cf. Loeb, The Stimulation of Growth, Science, May 14, 1915.

172 B. coli-communis, according to Buchner, tends to double in 22 minutes; in 24 hours, therefore, a single individual would be multiplied by something like 10?²⁸; Sitzungsber. MÜnchen. Ges. Morphol. u. Physiol. III, pp. 65–71, 1888. Cf. Marshall Ward, Biology of Bacillus ramosus, etc. Pr. R. S. LVIII, 265–468, 1895. The comparatively large infusorian Stylonichia, according to Maupas, would multiply in a month by 10?⁴³.

173 Cf. Enriques, Wachsthum und seine analytisehe Darstellung, Biol. Centralbl. 1909, p. 337.

174 Cf. (int. al.) Mellor, Chemical Statics and Dynamics, 1904, p. 291.

175 Cf. Robertson, l.c.

176 See, for a brief resumÉ of this subject, Morgan’s Experimental Zoology, chap. xvi.

177 Amer. J. of Physiol., X, 1904.

178 C.R. CXXI, CXXII, 1895–96.

179 Cf. Loeb, Science, May 14, 1915.

180 Cf. Baumann u. Roos, Vorkommen von Iod im ThierkÖrper, Zeitschr. fÜr Physiol. Chem. XXI, XXII, 1895, 6.

181 Le NÉo-Vitalisme, Rev. Scientifique, Mars 1911, p. 22 (of reprint).

182 La vie et la mort, p. 43, 1902.

183 Cf. Dendy, Evolutionary Biology, 1912, p. 408; Brit. Ass. Report (Portsmouth), 1911, p. 278.

184 Lucret. v, 877. “Lucretius nowhere seems to recognise the possibility of improvement or change of species by ‘natural selection’; the animals remain as they were at the first, except that the weaker and more useless kinds have been crushed out. Hence he stands in marked contrast with modern evolutionists.” Kelsey’s note, ad loc.

185 Even after we have so narrowed the scope and sphere of natural selection, it is still hard to understand; for the causes of extinction are often wellnigh as hard to comprehend as are those of the origin of species. If we assert (as has been lightly done) that Smilodon perished owing to its gigantic tusks, that Teleosaurus was handicapped by its exaggerated snout, or Stegosaurus weighed down by its intolerable load of armour, we may be reminded of other kindred forms to show that similar conditions did not necessarily lead to extermination, or that rapid extinction ensued apart from any such visible or apparent disadvantages. Cf. Lucas, F. A., On Momentum in Variation, Amer. Nat. xli, p. 46, 1907.

186 See Professor T. H. Morgan’s Regeneration (316 pp.), 1901 for a full account and copious bibliography. The early experiments on regeneration, by Vallisneri, RÉaumur, Bonnet, Trembley, Baster, and others, are epitomised by Haller, Elem. Physiologiae, VIII, p. 156 seq.

187 Journ. Experim. Zool. VII, p. 397, 1909.

188 Op. cit. p. 406, Exp. IV.

189 The experiments of Loeb on the growth of Tubularia in various saline solutions, referred to on p. 125, might as well or better have been referred to under the heading of regeneration, as they were performed on cut pieces of the zoophyte. (Cf. Morgan, op. cit. p. 35.)

190 Powers of the Creator, I, p. 7, 1851. See also Rare and Remarkable Animals, II, pp. 17–19, 90, 1847.

191 Lillie, F. R., The smallest Parts of Stentor capable of Regeneration, Journ. of Morphology, XII, p. 239, 1897.

192 Boveri, EntwicklungsfÄhigkeit kernloser Seeigeleier, etc., Arch. f. Entw. Mech. II, 1895. See also Morgan, Studies of the partial larvae of Sphaerechinus, ibid. 1895; J. Loeb, On the Limits of Divisibility of Living Matter, Biol. Lectures, 1894, etc.

193 Cf. Przibram, H., Scheerenumkehr bei dekapoden Crustaceen, Arch. f. Entw. Mech. XIX, 181–247, 1905; XXV, 266–344, 1907. Emmel, ibid. XXII, 542, 1906; Regeneration of lost parts in Lobster, Rep. Comm. Inland Fisheries, Rhode Island, XXXV, XXXVI, 1905–6; Science (n.s.), XXVI, 83–87, 1907. Zeleny, Compensatory Regulation, J. Exp. Zool. II, 1–102, 347–369, 1905; etc.

194 Lobsters are occasionally found with two symmetrical claws: which are then usually serrated, sometimes (but very rarely) both blunt-toothed. Cf. Calman, P.Z.S. 1906, pp. 633, 634, and reff.

195 Wilson, E. B., Reversal of Symmetry in Alpheus heterochelis, Biol. Bull. IV, p. 197, 1903.

196 J. Exp. Zool. VII, p. 457, 1909.

197 Biologica, III, p. 161, June. 1913.

198 Anatomical and Pathological Observations, p. 3, 1845; Anatomical Memoirs, II, p. 392, 1868.

199 Giard, A., L’oeuf et les dÉbuts de l’Évolution, Bull. Sci. du Nord de la Fr. VIII, pp. 252–258, 1876.

200 EntwickelungsvorgÄnge der Eizelle, 1876; Investigations on Microscopic Foams and Protoplasm, p. 1, 1894.

201 Journ. of Morphology, I, p. 229, 1887.

202 While it has been very common to look upon the phenomena of mitosis as sufficiently explained by the results towards which they seem to lead, we may find here and there a strong protest against this mode of interpretation. The following is a case in point: “On a tentÉ d’Établir dans la mitose dite primitive plusieurs catÉgories, plusieurs types de mitose. On a choisi le plus souvent comme base de ces systÈmes des concepts abstraits et tÉlÉologiques: rÉpartition plus ou moins exacte de la chromatine entre les deux noyaux-fils suivant qu’il y a ou non des chromosomes (Dangeard), distribution particuliÈre et signification dualiste des substances nuclÉaires (substance kinÉtique et substance gÉnÉrative ou hÉrÉditaire, Hartmann et ses ÉlÈves), etc. Pour moi tous ces essais sont À rejeter catÉgoriquement À cause de leur caractÈre finaliste; de plus, ils sont construits sur des concepts non dÉmontrÉs, et qui parfois reprÉsentent des gÉnÉralisations absolument erronÉes.” A. Alexeieff, Archiv fÜr Protistenkunde, XIX, p. 344, 1913.

203 This is the old philosophic axiom writ large: Ignorato motu, ignoratur natura; which again is but an adaptation of Aristotle’s phrase, ? ???? t?? ????se??, as equivalent to the “Efficient Cause.” FitzGerald holds that “all explanation consists in a description of underlying motions”; Scientific Writings, 1902, p. 385.

204 As when NÄgeli concluded that the organism is, in a certain sense, “vorgebildet”; Beitr. zur wiss. Botanik, II, 1860. Cf. E. B. Wilson, The Cell, etc., p. 302.

205 “La matiÈre arrangÉe par une sagesse divine doit Être essentiellement organisÉe partout ... il y a machine dans les parties de la machine Naturelle À l’infini.” Sur le principe de la Vie, p. 431 (Erdmann). This is the very converse of the doctrine of the Atomists, who could not conceive a condition “ubi dimidiae partis pars semper habebit Dimidiam partem, nec res praefiniet ulla.”

206 Cf. an interesting passage from the Elements (I, p. 445, Molesworth’s edit.), quoted by Owen, Hunterian Lectures on the Invertebrates, 2nd ed. pp. 40, 41, 1855.

207 “Wir mÜssen deshalb den lebenden Zellen, abgesehen von der Molekularstructur der organischen Verbindungen welche sie enthÄlt, noch eine andere und in anderer Weise complicirte Structur zuschreiben, und diese es ist welche wir mit dem Namen Organisation bezeichnen,” BrÜcke, Die Elementarorganismen, Wiener Sitzungsber. XLIV, 1861, p. 386; quoted by Wilson, The Cell, etc. p. 289. Cf. also Hardy, Journ. of Physiol. XXIV, 1899, p. 159.

208 Precisely as in the Lucretian concursus, motus, ordo, positura, figurae, whereby bodies mutato ordine mutant naturam.

209 Otto Warburg, BeitrÄge zur Physiologie der Zelle, insbesondere Über die Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro’s Ergebnisse der Physiologie, XIV, pp. 253–337, 1914 (see p. 315). (Cf. Bayliss, General Physiology, 1915, p. 590).

210 Hardy, W. B., On some Problems of Living Matter (Guthrie Lecture), Tr. Physical Soc. London, xxviii, p. 99–118, 1916.

211 As a matter of fact both phrases occur, side by side, in Graham’s classical paper on “Liquid Diffusion applied to Analysis,” Phil. Trans. CLI, p. 184, 1861; Chem. and Phys. Researches (ed. Angus Smith), 1876, p. 554.

212 L. Rhumbler, Mechanische ErklÄrung der Aehnlichkeit zwischen Magnetischen Kraftliniensystemen und Zelltheilungsfiguren, Arch. f. Entw. Mech. XV, p. 482, 1903.

213 Gallardo, A., Essai d’interpretation des figures caryocinÉtiques, Anales del Museo de Buenos-Aires (2), II, 1896; La division de la cellule, phenomÈne bipolaire de caractÈre electro-colloidal, Arch. f. Entw. Mech. XXVIII, 1909, etc.

214 Arch. f. Entw. Mech. III, IV, 1896–97.

215 On various theories of the mechanism of mitosis, see (e.g.) Wilson, The Cell in Development, etc., pp. 100–114; Meves, Zelltheilung, in Merkel u. Bonnet’s Ergebnisse der Anatomie, etc., VII, VIII, 1897–8; Ida H. Hyde, Amer. Journ. of Physiol. XII, pp. 241–275, 1905; and especially Prenant, A., Theories et interprÉtations physiques de la mitose, J. de l’Anat. et Physiol. XLVI, pp. 511–578, 1910.

216 Hartog, M., Une force nouvelle: le mitokinÉtisme, C.R. 11 Juli, 1910; Mitokinetism in the Mitotic Spindle and in the Polyasters, Arch. f. Entw. Mech. XXVII, pp. 141–145, 1909; cf. ibid. XL, pp. 33–64, 1914. Cf. also Hartog’s papers in Proc. R. S. (B), LXXVI, 1905; Science Progress (n. s.), I, 1907; Riv. di Scienza, II, 1908; C. R. Assoc. fr. pour l’Avancem. des Sc. 1914, etc.

217 The configurations, as obtained by the usual experimental methods, were of course known long before Faraday’s day, and constituted the “convergent and divergent magnetic curves” of eighteenth century mathematicians. As Leslie said, in 1821, they were “regarded with wonder by a certain class of dreaming philosophers, who did not hesitate to consider them as the actual traces of an invisible fluid, perpetually circulating between the poles of the magnet.” Faraday’s great advance was to interpret them as indications of stress in a medium,—of tension or attraction along the lines, and of repulsion transverse to the lines, of the diagram.

218 Cf. also the curious phenomenon in a dividing egg described as “spinning” by Mrs G. F. Andrews, J. of Morph. XII, pp. 367–389, 1897.

219 Whitman, J. of Morph. II, p. 40, 1889.

220 “Souvent il n’y a qu’une sÉparation physique entre le cytoplasme et le suc nuclÉaire, comme entre deux liquides immiscibles, etc.;” Alexeieff, Sur la mitose dite “primitive,” Arch. f. Protistenk. XXIX, p. 357, 1913.

221 The appearance of “vacuolation” is a result of endosmosis or the diffusion of a less dense fluid into the denser plasma of the cell. Caeteris paribus, it is less apparent in marine organisms than in those of freshwater, and in many or most marine Ciliates and even Rhizopods a contractile vacuole has not been observed (BÜtschli, in Bronn’s Protozoa, p. 1414); it is also absent, and probably for the same reason, in parasitic Protozoa, such as the Gregarines and the Entamoebae. Rossbach shewed that the contractile vacuole of ordinary freshwater Ciliates was very greatly diminished in a 5 per cent. solution of NaCl, and all but disappeared in a 1 per cent. solution of sugar (Arb. z. z. Inst. WÜrzburg, 1872, cf. Massart, Arch. de Biol. LX, p. 515, 1889). Actinophrys sol, when gradually acclimatised to sea-water, loses its vacuoles, and vice versa (Gruber, Biol. Centralbl. IX, p. 22, 1889); and the same is true of Amoeba (Zuelzer, Arch. f. Entw. Mech. 1910, p. 632). The gradual enlargement of the contractile vacuole is precisely analogous to the change of size of a bubble until the gases on either side of the film are equally diffused, as described long ago by Draper (Phil. Mag. (n. s.), XI, p. 559, 1837). Rhumbler has shewn that contractile or pulsating vacuoles may be well imitated in chloroform-drops, suspended in water in which various substances are dissolved (Arch. f. Entw. Mech. VII, 1898, p. 103). The pressure within the contractile vacuole, always greater than without, diminishes with its size, being inversely proportional to its radius; and when it lies near the surface of the cell, as in a Heliozoon, it bursts as soon as it reaches a thinness which its viscosity or molecular cohesion no longer permits it to maintain.

222 Cf. p. 660.

223 The elongated or curved “macronucleus” of an Infusorian is to be looked upon as a single mass of chromatin, rather than as an aggregation of particles in a fluid drop, as in the case described. It has a shape of its own, in which ordinary surface-tension plays a very subordinate part.

224 ThÉorie physico-chimique de la Vie, p. 73, 1910; Mechanism of Life, p. 56, 1911.

225 Whence the name “mitosis” (Greek ?t??, a thread), applied first by Flemming to the whole phenomenon. Kollmann (Biol. Centralbl. II, p. 107, 1882) called it divisio per fila, or divisio laqueis implicata. Many of the earlier students, such as Van Beneden (Rech. sur la maturation de l’oeuf, Arch. de Biol. IV, 1883), and Hermann (Zur Lehre v. d. Entstehung d. karyokinetischen Spindel, Arch. f. mikrosk. Anat. XXXVII, 1891) thought they recognised actual muscular threads, drawing the nuclear material asunder towards the respective foci or poles; and some such view was long maintained by other writers, Boveri, Heidenhain, Flemming, R. Hertwig, and many more. In fact, the existence of contractile threads, or the ascription to the spindle rather than to the poles or centrosomes of the active forces concerned in nuclear division, formed the main tenet of all those who declined to go beyond the “contractile properties of protoplasm” for an explanation of the phenomenon. (Cf. also J. W. Jenkinson, Q. J. M. S. XLVIII, p. 471, 1904.)

226 Cf. BÜtschli, O., Ueber die kÜnstliche Nachahmung der karyokinetischen Figur, Verh. Med. Nat. Ver. Heidelberg, V, pp. 28–41 (1892), 1897.

227 Arrhenius, in describing a typical colloid precipitate, does so in terms that are very closely applicable to the ordinary microscopic appearance of the protoplasm of the cell. The precipitate consists, he says, “en un rÉseau d’une substance solide contenant peu d’eau, dans les mailles duquel est inclus un fluide contenant un peu de colloide dans beaucoup d’eau ... Evidemment cette structure se forme À cause de la petite diffÉrence de poids spÉcifique des deux phases, et de la consistance gluante des particules sÉparÉes, qui s’attachent en forme de rÉseau.” Rev. Scientifique, Feb. 1911.

228 F. Schwartz, in Cohn’s Beitr. z. Biologie der Pflanzen, V, p. 1, 1887.

229 Fischer, Anat. Anzeiger, IX, p. 678, 1894, X, p. 769, 1895.

230 See, in particular, W. B. Hardy, On the structure of Cell Protoplasm, Journ. of Physiol. XXIV, pp. 158–207, 1889; also HÖber, Physikalische Chemie der Zelle und der Gewebe, 1902. Cf. (int. al.) Flemming, Zellsubstanz, Kern und Zelltheilung 1882, p. 51, etc.

231 My description and diagrams (Figs 42–51) are based on those of Professor E. B. Wilson.

232 If the word permeability be deemed too directly suggestive of the phenomena of magnetism we may replace it by the more general term of specific inductive capacity. This would cover the particular case, which is by no means an improbable one, of our phenomena being due to a “surface charge” borne by the nucleus itself and also by the chromosomes: this surface charge being in turn the result of a difference in inductive capacity between the body or particle and its surrounding medium. (Cf. footnote, p. 187.)

233 On the effect of electrical influences in altering the surface-tensions of the colloid particles, see Bredig, Anorganische Fermente, pp. 15, 16, 1901.

234 The Cell, etc. p. 66.

235 Lillie, R. S., Amer. J. of Physiol. VIII, p. 282, 1903.

236 We have not taken account in the above paragraphs of the obvious fact that the supposed symmetrical field of force is distorted by the presence in it of the more or less permeable bodies; nor is it necessary for us to do so, for to that distorted field the above argument continues to apply, word for word.

237 M. Foster, Lectures on the History of Physiology, 1901, p. 62.

238 Op. cit. pp. 110 and 91.

239 Lamb, A. B., A new Explanation of the Mechanism of Mitosis, Journ. Exp. Zool. V, pp. 27–33, 1908.

240 Amer. J. of Physiol. VIII, pp. 273–283, 1903 (vide supra, p. 181); cf. ibid. XV, pp. 46–84, 1905. Cf. also Biological Bulletin, IV, p. 175. 1903.

241 In like manner Hardy has shewn that colloid particles migrate with the negative stream if the reaction of the surrounding fluid be alkaline, and vice versa. The whole subject is much wider than these brief allusions suggest, and is essentially part of Quincke’s theory of Electrical Diffusion or Endosmosis: according to which the particles and the fluid in which they float (or the fluid and the capillary walls through which it flows) each carry a charge, there being a discontinuity of potential at the surface of contact, and hence a field of force leading to powerful tangential or shearing stresses, communicating to the particles a velocity which varies with the density per unit area of the surface charge. See W. B. Hardy’s paper on Coagulation by Electricity, Journ. of Physiol. XXIV, p. 288–304, 1899, also Hardy and H. W. Harvey, Surface Electric Charges of Living Cells, Proc. R. S. LXXXIV (B), pp. 217–226, 1911, and papers quoted therein. Cf. also E. N. Harvey’s observations on the convection of unicellular organisms in an electric field (Studies on the Permeability of Cells, Journ. of Exper. Zool. X, pp. 508–556, 1911).

242 On Differences in Electrical Potential in Developing Eggs, Amer. Journ. of Physiol. XII, pp. 241–275, 1905. This paper contains an excellent summary of various physical theories of the segmentation of the cell.

243 Gray has recently demonstrated a temporary increase of electrical conductivity in sea-urchin eggs during the process of fertilisation (The Electrical Conductivity of fertilised and unfertilised Eggs, Journ. Mar. Biol. Assoc. X, pp. 50–59, 1913).

244 Schewiakoff, Ueber die karyokinetische Kerntheilung der Euglypha alveolata, Morph. Jahrb. XIII, pp. 193–258, 1888 (see p. 216).

245 Coe, W. R., Maturation and Fertilization of the Egg of Cerebratulus, Zool. JahrbÜcher (Anat. Abth.), XII, pp. 425–476, 1899.

246 Thus, for example, Farmer and Digby (On Dimensions of Chromosomes considered in relation to Phylogeny, Phil. Trans. (B), CCV, pp. 1–23, 1914) have been at pains to shew, in confutation of Meek (ibid. CCIII, pp. 1–74, 1912), that the width of the chromosomes cannot be correlated with the order of phylogeny.

247 Cf. also Arch. f. Entw. Mech. X, p. 52, 1900.

248 Cf. Loeb, Am. J. of Physiol. VI, p. 32, 1902; Erlanger, Biol. Centralbl. XVII, pp. 152, 339, 1897; Conklin, Biol. Lectures, Woods Holl, p. 69, etc. 1898–9.

249 Robertson, T. B., Note on the Chemical Mechanics of Cell Division, Arch. f. Entw. Mech. XXVII, p. 29, 1909, XXXV, p. 692. 1913. Cf. R. S. Lillie, J. Exp. Zool. XXI, pp. 369–402, 1916.

250 Cf. D’Arsonval, Arch. de Physiol. p. 460, 1889; Ida H. Hyde, op. cit. p. 242.

251 Cf. Plateau’s remarks (Statique des liquides, II, p. 154) on the tendency towards equilibrium, rather than actual equilibrium, in many of his systems of soap-films.

252 But under artificial conditions, “polyspermy” may take place, e.g. under the action of dilute poisons, or of an abnormally high temperature, these being all, doubtless, conditions under which the surface-tension is diminished.

253 Fol, H., Recherches sur la fÉcondation, 1879. Roux, W., BeitrÄge zur Entwickelungsmechanik des Embryo, Arch. f. Mikr. Anat. XIX, 1887. Whitman, C. O., OÖkinesis, Journ. of Morph. I, 1887.

254 Wilson. The Cell, p. 77.

255 Eight and twelve are by much the commonest numbers, six and sixteen coming next in order. If we may judge by the list given by E. B. Wilson (The Cell, p. 206), over 80% of the observed cases lie between 6 and 16, and nearly 60% between 8 and 12.

256 Theory of Cells, p. 191.

257 The Cell in Development, etc. p. 59; cf. pp. 388, 413.

258 E.g. BrÜcke, Elementarorganismen, p. 387: “Wir mÜssen in der Zelle einen kleinen Thierleib sehen, und dÜrfen die Analogien, welche zwischen ihr und den kleinsten Thierformen existiren, niemals aus den Augen lassen.”

259 Whitman, C. O., The Inadequacy of the Cell-theory, Journ. of Morphol. VIII, pp. 639–658, 1893; Sedgwick, A., On the Inadequacy of the Cellular Theory of Development, Q.J.M.S. XXXVII, pp. 87–101, 1895, XXXVIII, pp. 331–337, 1896. Cf. Bourne, G. C., A Criticism of the Cell-theory; being an answer to Mr Sedgwick’s article, etc., ibid. XXXVIII, pp. 137–174, 1896.

260 Cf. Hertwig, O., Die Zelle und die Gewebe, 1893, p. 1; “Die Zellen, in welche der Anatom die pflanzlichen und thierischen Organismen zerlegt, sind die TrÄger der Lebensfunktionen; sie sind, wie Virchow sich ausgedrÜckt hat, die ‘Lebenseinheiten.’ Von diesem Gesichtspunkt aus betrachtet, erscheint der Gesammtlebensprocess eines zusammengesetzten Organismus nichts Anderes zu sein als das hÖchst verwickelte Resultat der einzelnen Lebensprocesse seiner zahlreichen, verschieden functionirenden Zellen.”

261 Journ. of Morph. VIII, p. 653, 1893.

262 Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. VIII, pp. 272, 313, 333, 1883.

263 Journ. of Morph. II, p. 49, 1889.

264 Phil. Trans. CLI, p. 183, 1861; Researches, ed. Angus Smith, 1877, p. 553.

265 Cf. Kelvin, On the Molecular Tactics of a Crystal, The Boyle Lecture, Oxford, 1893, Baltimore Lectures, 1904, pp. 612–642. Here Kelvin was mainly following Bravais’s (and Frankenheim’s) theory of “space-lattices,” but he had been largely anticipated by the crystallographers. For an account of the development of the subject in modern crystallography, by Sohncke, von Fedorow, SchÖnfliess, Barlow and others, see Tutton’s Crystallography, chap. ix, pp. 118–134, 1911.

266 In a homogeneous crystalline arrangement, symmetry compels a locus of one property to be a plane or set of planes; the locus in this case being that of least surface potential energy.

267 This is what Graham called the water of gelatination, on the analogy of water of crystallisation; Chem. and Phys. Researches, p. 597.

268 Here, in a non-crystalline or random arrangement of particles, symmetry ensures that the potential energy shall be the same per unit area of all surfaces; and it follows from geometrical considerations that the total surface energy will be least if the surface be spherical.

269 Lehmann, O., FlÜssige Krystalle, sowie PlasticitÄt von Krystallen im allgemeinen, etc., 264 pp. 39 pll., Leipsig, 1904. For a semi-popular, illustrated account, see Tutton’s Crystals (Int. Sci. Series), 1911.

270 As Graham said of an allied phenomenon (the so-called blood-crystals of Funke), it “illustrates the maxim that in nature there are no abrupt transitions, and that distinctions of class are never absolute.”

271 Cf. Przibram, H., Kristall-analogien zur Entwickelungsmechanik der Organismen, Arch. f. Entw. Mech. XXII, p. 207, 1906 (with copious bibliography); Lehmann, Scheinbar lebende Kristalle und Myelinformen, ibid. XXVI, p. 483, 1908.

272 The idea of a “surface-tension” in liquids was first enunciated by Segner, De figuris superficierum fluidarum, in Comment. Soc. Roy. GÖttingen, 1751, p. 301. Hooke, in the Micrographia (1665, Obs. VIII, etc.), had called attention to the globular or spherical form of the little morsels of steel struck off by a flint, and had shewn how to make a powder of such spherical grains, by heating fine filings to melting point. “This Phaenomenon” he said “proceeds from a propriety which belongs to all kinds of fluid Bodies more or less, and is caused by the Incongruity of the Ambient and included Fluid, which so acts and modulates each other, that they acquire, as neer as is possible, a spherical or globular form....”

273 Science of Mechanics, 1902, p. 395; see also Mach’s article Ueber die physikalische Bedeutung der Gesetze der Symmetrie, Lotos, XXI, pp. 139–147, 1871.

274 Similarly, Sir David Brewster and others made powerful lenses by simply dropping small drops of Canada balsam, castor oil, or other strongly refractive liquids, on to a glass plate: On New Philosophical Instruments (Description of a new Fluid Microscope), Edinburgh, 1813, p. 413.

275 BeitrÄge z. Physiologie d. Protoplasma, PflÜger’s Archiv, II, p. 307, 1869.

276 Poggend. Annalen, XCIV, pp. 447–459, 1855. Cf. Strethill Wright, Phil. Mag. Feb. 1860.

277 Haycraft and Carlier pointed out (Proc. R.S.E. XV, pp. 220–224, 1888) that the amoeboid movements of a white blood-corpuscle are only manifested when the corpuscle is in contact with some solid substance: while floating freely in the plasma or serum of the blood, these corpuscles are spherical, that is to say they are at rest and in equilibrium. The same fact has recently been recorded anew by Ledingham (On Phagocytosis from an adsorptive point of view, Journ. of Hygiene, XII, p. 324, 1912). On the emission of pseudopodia as brought about by changes in surface tension, see also (int. al.) Jensen, Ueber den Geotropismus niederer Organismen, PflÜger’s Archiv, LIII, 1893. Jensen remarks that in Orbitolites, the pseudopodia issuing through the pores of the shell first float freely, then as they grow longer bend over till they touch the ground, whereupon they begin to display amoeboid and streaming motions. Verworn indicates (Allg. Physiol. 1895, p. 429), and Davenport says (Experim. Morphology, II, p. 376) that “this persistent clinging to the substratum is a ‘thigmotropic’ reaction, and one which belongs clearly to the category of ‘response.’ ” (Cf. PÜtter, Thigmotaxis bei Protisten, A. f. Physiol. 1900, Suppl. p. 247.) But it is not clear to my mind that to account for this simple phenomenon we need invoke other factors than gravity and surface-action.

278 Cf. Pauli, Allgemeine physikalische Chemie d. Zellen u. Gewebe, in Asher-Spiro’s Ergebnisse der Physiologie, 1912; Przibram, VitalitÄt, 1913, p. 6.

279 The surface-tension theory of protoplasmic movement has been denied by many. Cf. (e.g.), Jennings, H. S., Contributions to the Study of the Behaviour of the Lower Organisms, Carnegie Inst. 1904, pp. 130–230; Dellinger, O. P., Locomotion of Amoebae, etc. Journ. Exp. Zool. III, pp. 337–357, 1906; also various papers by Max Heidenhain, in Anatom. Hefte (Merkel und Bonnet), etc.

280 These various movements of a liquid surface, and other still more striking movements such as those of a piece of camphor floating on water, were at one time ascribed by certain physicists to a peculiar force, sui generis, the force Épipolique of Dutrochet: until van der Mensbrugghe shewed that differences of surface tension were enough to account for this whole series of phenomena (Sur la tension superficielle des liquides considÉrÉe au point de vue de certains mouvements observÉs À leur surface, MÉm. Cour. Acad. de Belgique, XXXIV, 1869; cf. Plateau, p. 283).

281 Cf. infra, p. 306.

282 Cf. p. 32.

283 Or, more strictly speaking, unless its thickness be less than twice the range of the molecular forces.

284 It follows that the tension, depending only on the surface-conditions, is independent of the thickness of the film.

285 This simple but immensely important formula is due to Laplace (MÉcanique CÉleste, Bk. x. suppl. ThÉorie de l’action capillaire, 1806).

286 Sur la surface de rÉvolution dont la courbure moyenne est constante, Journ. de M. Liouville, VI, p. 309, 1841.

287 See Liquid Drops and Globules, 1914, p. 11. Robert Boyle used turpentine in much the same way. For other methods see Plateau, op. cit. p. 154.

288 Felix Plateau recommends the use of a weighted thread, or plumb-line, drawn up out of a jar of water or oil; Phil. Mag. XXXIV, p. 246, 1867.

289 Cf. Boys, C. V., On Quartz Fibres, Nature, July 11, 1889; Warburton, C., The Spinning Apparatus of Geometric Spiders, Q.J.M.S. XXXI, pp. 29–39, 1890.

290 J. Blackwall, Spiders of Great Britain (Ray Society), 1859, p. 10; Trans. Linn. Soc. XVI, p. 477, 1833.

291 The intermediate spherules appear, with great regularity and beauty, whenever a liquid jet breaks up into drops; see the instantaneous photographs in Poynting and Thomson’s Properties of Matter, pp. 151, 152, (ed. 1907).

292 KÜhne, Untersuchungen Über das Protoplasma, 1864, p. 75, etc.

293 A Study of Splashes, 1908, p. 38, etc.; Segmentation of a Liquid Annulus, Proc. Roy. Soc. XXX, pp. 49–60, 1880.

294 Cf. ibid. pp. 17, 77. The same phenomenon is beautifully and continuously evident when a strong jet of water from a tap impinges on a curved surface and then shoots off it.

295 See a Study of Splashes, p. 54.

296 A case which we have not specially considered, but which may be found to deserve consideration in biology, is that of a cell or drop suspended in a liquid of varying density, for instance in the upper layers of a fluid (e.g. sea-water) at whose surface condensation is going on, so as to produce a steady density-gradient. In this case the normally spherical drop will be flattened into an oval form, with its maximum surface-curvature lying at the level where the densities of the drop and the surrounding liquid are just equal. The sectional outline of the drop has been shewn to be not a true oval or ellipse, but a somewhat complicated quartic curve. (Rice, Phil. Mag. Jan. 1915.)

297 Indeed any non-isotropic stiffness, even though T remained uniform, would simulate, and be indistinguishable from, a condition of non-stiffness and non-isotropic T.

298 A non-symmetry of T and T?' might also be capable of explanation as a result of “liquid crystallisation.” This hypothesis is referred to, in connection with the blood-corpuscles, on p. 272.

299 The case of the snow-crystals is a particularly interesting one; for their “distribution” is in some ways analogous to what we find, for instance, among our microscopic skeletons of Radiolarians. That is to say, we may one day meet with myriads of some one particular form or species only, and another day with myriads of another; while at another time and place we may find species intermingled in inexhaustible variety. (Cf. e.g. J. Glaisher, Ill. London News, Feb. 17, 1855; Q.J.M.S. III, pp. 179–185, 1855).

300 Cf. Bergson, Creative Evolution, p. 107: “Certain Foraminifera have not varied since the Silurian epoch. Unmoved witnesses of the innumerable revolutions that have upheaved our planet, the Lingulae are today what they were at the remotest times of the palaeozoic era.”

301 Ray Lankester, A.M.N.H. (4), XI, p. 321, 1873.

302 Leidy, Parasites of the Termites, J. Nat. Sci., Philadelphia, VIII, pp. 425–447, 1874–81; cf. Saville Kent’s Infusoria, II, p. 551.

303 Op. cit. p. 79.

304 Brady, Challenger Monograph, pl. XX, p. 233.

305 That the Foraminifera not only can but do hang from the surface of the water is confirmed by the following apt quotation which I owe to Mr E. Heron-Allen: “Quand on place, comme il a ÉtÉ dit, le dÉpÔt provenant du lavage des fucus dans un flacon que l’on remplit de nouvelle eau, on voit au bout d’une heure environ les animaux [Gromia dujardinii] se mettre en mouvement et commencer À grimper. Six heures aprÈs ils tapissent l’extÉrieur du flacon, de sorte que les plus ÉlevÉs sont À trente-six ou quarante-deux millimetres du fond; le lendemain beaucoup d’entre eux, aprÈs avoir atteint le niveau du liquide, ont continuÉ À ramper À sa surface, en se laissant pendre au-dessous comme certains mollusques gastÉropodes.” (Dujardin, F., Observations nouvelles sur les prÉtendus cÉphalopodes microscopiques, Ann. des Sci. Nat. (2), III, p. 312, 1835.)

306 Cf. Boas, Spolia Atlantica, 1886, pl. 6.

307 This cellular pattern would seem to be related to the “cohesion figures” described by Tomlinson in various surface-films (Phil. Mag. 1861 to 1870); to the “tesselated structure” in liquids described by Professor James Thomson in 1882 (Collected Papers, p. 136); and to the tourbillons cellulaires of Prof. H. BÉnard (Ann. de Chimie (7), XXIII, pp. 62–144, 1901, (8), XXIV, pp. 563–566, 1911), Rev. gÉnÉr. des Sci. XI, p. 1268, 1900; cf. also E. H. Weber. (Poggend. Ann. XCIV, p. 452, 1855, etc.). The phenomenon is of great interest and various appearances have been referred to it, in biology, geology, metallurgy and even astronomy: for the flocculent clouds in the solar photosphere shew an analogous configuration. (See letters by Kerr Grant, Larmor, Wager and others, in Nature, April 16 to June 11, 1914.) In many instances, marked by strict symmetry or regularity, it is very possible that the interference of waves or ripples may play its part in the phenomenon. But in the majority of cases, it is fairly certain that localised centres of action, or of diminished tension, are present, such as might be provided by dust-particles in the case of Darling’s experiment (cf. infra, p. 590).

308 Ueber physikalischen Eigenschaften dÜnner, fester Lamellen, S.B. Berlin. Akad. 1888, pp. 789, 790.

309 Certain palaeontologists (e.g. Haeusler and Spandel) have maintained that in each family or genus the plain smooth-shelled forms are the primitive and ancient ones, and that the ribbed and otherwise ornamented shells make their appearance at later dates in the course of a definite evolution (cf. Rhumbler, Foraminiferen der Plankton-Expedition, 1911, i, p. 21). If this were true it would be of fundamental importance: but this book of mine would not deserve to be written.

310 A Study of Splashes, p. 116.

311 See Silliman’s Journal, II, p. 179, 1820; and cf. Plateau, op. cit. II, pp. 134, 461.

312 The presence or absence of the contractile vacuole or vacuoles is one of the chief distinctions, in systematic zoology, between the Heliozoa and the Radiolaria. As we have seen on p. 165 (footnote), it is probably no more than a physical consequence of the different conditions of existence in fresh water and in salt.

313 Cf. Doflein, Lehrbuch der Protozoenkunde, 1911, p. 422.

314 Cf. Minchin, Introduction to the Study of the Protozoa, 1914 p. 293, Fig. 127.

315 Cf. C. A. Kofoid and Olive Swezy, On Trichomonad Flagellates, etc. Pr. Amer. Acad. of Arts and Sci. LI, pp. 289–378, 1915.

316 D. L. Mackinnon, Herpetomonads from the Alimentary Tract of certain Dungflies, Parasitology, III, p. 268, 1910.

317 Proc. Roy. Soc. XII, pp. 251–257, 1862–3.

318 Cf. (int. al.) Lehmann, Ueber scheinbar lebende Kristalle und Myelinformen, Arch. f. Entw. Mech. XXVI, p. 483, 1908; Ann. d. Physik, XLIV, p. 969, 1914.

319 Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the Red Blood Corpuscle, Biochem. Journal, III, p. 55, 1908.

320 For an attempt to explain the form of a blood-corpuscle by surface-tension alone, see Rice, Phil. Mag. Nov. 1914; but cf. Shorter, ibid. Jan. 1915.

321 Koltzoff, N. K., Studien Über die Gestalt der Zelle, Arch. f. mikrosk. Anat. LXVII, pp. 364–571, 1905; Biol. Centralbl. XXIII, pp. 680–696, 1903, XXVI, pp. 854–863, 1906; Arch. f. Zellforschung, II, pp. 1–65, 1908, VII, pp. 344–423, 1911; Anat. Anzeiger, XLI, pp. 183–206, 1912.

322 Cf. supra, p. 129.

323 As Bethe points out (Zellgestalt, Plateausche FlÜssigkeitstigur und Neurofibrille, Anat. Anz. XL. p. 209, 1911), the spiral fibres of which Koltzoff speaks must lie in the surface, and not within the substance, of the cell whose conformation is affected by them.

324 See for a further but still elementary account, Michaelis, Dynamics of Surfaces, 1914, p. 22 seq.; Macallum, OberflÄchenspannung und Lebenserscheinungen, in Asher-Spiro’s Ergebnisse der Physiologie, XI, pp. 598–658, 1911; see also W. W. Taylor’s Chemistry of Colloids, 1915, p. 221 seq., Wolfgang Ostwald, Grundriss der Kolloidchemie, 1909, and other text-books of physical chemistry; and Bayliss’s Principles of General Physiology, pp. 54–73, 1915.

325 The first instance of what we now call an adsorptive phenomenon was observed in soap-bubbles. Leidenfrost, in 1756, was aware that the outer layer of the bubble was covered by an “oily” layer. A hundred years later DuprÉ shewed that in a soap-solution the soap tends to concentrate at the surface, so that the surface-tension of a very weak solution is very little different from that of a strong one (ThÉorie mÉcanique de la chaleur, 1869, p. 376; cf. Plateau, II, p. 100).

326 This identical phenomenon was the basis of Quincke’s theory of amoeboid movement (Ueber periodische Ausbreitung von FlÜssigkeitsoberflÄchen, etc., SB. Berlin. Akad. 1888, pp. 791–806; cf. PflÜger’s Archiv, 1879, p. 136).

327 J. Willard Gibbs, Equilibrium of Heterogeneous Substances, Tr. Conn. Acad. III, pp. 380–400, 1876, also in Collected Papers, I, pp. 185–218, London, 1906; J. J. Thomson, Applications of Dynamics to Physics and Chemistry, 1888 (Surface tension of solutions), p. 190. See also (int. al.) the various papers by C. M. Lewis, Phil. Mag. (6), XV, p. 499, 1908, XVII, p. 466, 1909, Zeitschr. f. physik. Chemie, LXX, p. 129, 1910; Milner, Phil. Mag. (6), XIII, p. 96, 1907, etc.

328 G. F. FitzGerald, On the Theory of Muscular Contraction, Brit. Ass. Rep. 1878; also in Scientific Writings, ed. Larmor, 1902, pp. 34, 75. A. d’Arsonval, Relations entre l’ÉlectricitÉ animale et la tension superficielle, C. R. CVI, p. 1740. 1888; cf. A. Imbert, Le mÉcanisme de la contraction musculaire, dÉduit de la considÉration des forces de tension superficielle, Arch. de Phys. (5), IX, pp. 289–301, 1897.

329 Ueber die Natur der Bindung der Gase im Blut und in seinen Bestandtheilen, Kolloid. Zeitschr. II, pp. 264–272, 294–301, 1908; cf. Loewy, Dissociationsspannung des Oxyhaemoglobin im Blut, Arch. f. Anat. und Physiol. 1904, p. 231.

330 We may trace the first steps in the study of this phenomenon to Melsens, who found that thin films of white of egg become firm and insoluble (Sur les modifications apportÉes À l’albumine ... par l’action purement mÉcanique, C. R. Acad. Sci. XXXIII, p. 247, 1851); and Harting made similar observations about the same time. Ramsden has investigated the same subject, and also the more general phenomenon of the formation of albuminoid and fatty membranes by adsorption: cf. Koagulierung der EiweisskÖrper auf mechanischer Wege, Arch. f. Anat. u. Phys. (Phys. Abth.) 1894, p. 517; Abscheidung fester KÖrper in OberflÄchenschichten Z. f. phys. Chem. XLVII, p. 341, 1902; Proc. R. S. LXXII, p. 156, 1904. For a general review of the whole subject see H. Zangger, Ueber Membranen und Membranfunktionen, in Asher-Spiro’s Ergebnisse der Physiologie, VII, pp. 99–160, 1908.

331 Cf. Taylor, Chemistry of Colloids, p. 252.

332 StrasbÜrger, Ueber Cytoplasmastrukturen, etc. Jahrb. f. wiss. Bot. XXX, 1897; R. A. Harper, Kerntheilung und freie Zellbildung im Ascus, ibid.; cf. Wilson, The Cell in Development, etc. pp. 53–55.

333 Cf. A. Gurwitsch, Morphologie und Biologie der Zelle, 1904, pp. 169–185; Meves, Die Chondriosomen als TrÄger erblicher Anlagen, Arch. f. mikrosk. Anat. 1908, p. 72; J. O. W. Barratt, Changes in Chondriosomes, etc. Q.J.M.S. LVIII, pp. 553–566, 1913, etc.; A. Mathews, Changes in Structure of the Pancreas Cell, etc., J. of Morph. XV (Suppl.), pp. 171–222, 1899.

334 The question whether chromosomes, chondriosomes or chromidia be the true vehicles or transmitters of “heredity” is not without its analogy to the older problem of whether the pineal gland or the pituitary body were the actual seat and domicile of the soul.

335 Cf. C. C. Dobell, Chromidia and the Binuclearity Hypotheses; a review and a criticism, Q.J.M.S. LIII, 279–326, 1909; Prenant, A., Les Mitochondries et l’Ergastoplasme, Journ. de l’Anat. et de la Physiol. XLVI, pp. 217–285, 1910 (both with copious bibliography).

336 Traube in particular has maintained that in differences of surface-tension we have the origin of the active force productive of osmotic currents, and that herein we find an explanation, or an approach to an explanation, of many phenomena which were formerly deemed peculiarly “vital” in their character. “Die Differenz der OberflÄchenspannungen oder der OberflÄchendruck eine Kraft darstellt, welche als treibende Kraft der Osmose, an die Stelle des nicht mit dem OberflÄchendruck identischen osmotischen Druckes, zu setzen ist, etc.” (OberflÄchendruck und seine Bedeutung im Organismus, PflÜger’s Archiv, CV, p. 559, 1904.) Cf. also Hardy (Pr. Phys. Soc. XXVIII, p. 116, 1916), “If the surface film of a colloid membrane separating two masses of fluid were to change in such a way as to lower the potential of the water in it, water would enter the region from both sides at once. But if the change of state were to be propagated as a wave of change, starting at one face and dying out at the other face, water would be carried along from one side of the membrane to the other. A succession of such waves would maintain a flow of fluid.”

337 On the Distribution of Potassium in animal and vegetable Cells; Journ. of Physiol. XXXII, p. 95, 1905.

338 The reader will recognise that there is a fundamental difference, and contrast, between such experiments as these of Professor Macallum’s and the ordinary staining processes of the histologist. The latter are (as a general rule) purely empirical, while the former endeavour to reveal the true microchemistry of the cell. “On peut dire que la microchimie n’est encore qu’À la pÉriode d’essai, et que l’avenir de l’histologie et spÉcialement de la cytologie est tout entier dans la microchimie” (Prenant, A., MÉthodes et rÉsultats de la Microchimie, Journ. de l’Anat. et de la Physiol. XLVI, pp. 343–404, 1910).

339 Cf. Macallum, Presidential Address, Section I, Brit. Ass. Rep. (Sheffield), 1910, p. 744.

340 In accordance with a simple corollary to the Gibbs-Thomson law.

341 It can easily be proved (by equating the increase of energy stored in an increased surface to the work done in increasing that surface), that the tension measured per unit breadth, T?_ab, is equal to the energy per unit area, E?_ab.

342 The presence of this little liquid “bourrelet,” drawn from the material of which the partition-walls themselves are composed, is obviously tending to a reduction of the internal surface-area. And it may be that it is as well, or better, accounted for on this ground than on Plateau’s assumption that it represents a “surface of continuity.”

343 A similar “bourrelet” is admirably seen at the line of junction between a floating bubble and the liquid on which it floats; in which case it constitutes a “masse annulaire,” whose mathematical properties and relation to the form of the nearly hemispherical bubble, have been investigated by van der Mensbrugghe (cf. Plateau, op. cit., p. 386). The form of the superficial vacuoles in Actinophrys or Actinosphaerium involves an identical problem.

344 In an actual calculation we must of course always take account of the tensions on both sides of each film or membrane.

345 Hofmeister, Pringsheim’s Jahrb. III, p. 272, 1863; Hdb. d. physiol. Bot. I, 1867, p. 129.

346 Sachs, Ueber die Anordnung der Zellen in jÜngsten Pflanzentheilen, Verh. phys. med. Ges. WÜrzburg, XI, pp. 219–242, 1877; Ueber Zellenanordnung und Wachsthum, ibid. XII, 1878; Ueber die durch Wachsthum bedingte Verschiebung kleinster Theilchen in trajectorischen Curven, Monatsber. k. Akad. Wiss. Berlin, 1880; Physiology of Plants, chap. xxvii, pp. 431–459, Oxford, 1887.

347 Schwendener, Ueber den Bau und das Wachsthum des Flechtenthallus, Naturf. Ges. ZÜrich, Febr. 1860, pp. 272–296.

348 Reinke, Lehrbuch der Botanik, 1880, p. 519.

349 Cf. Leitgeb, Unters. Über die Lebermoose, II, p. 4, Graz, 1881.

350 Rauber, Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. VIII, pp. 279, 334, 1882.

351 C. R. Acad. Sc. XXXIII, p. 247, 1851; Ann. de chimie et de phys. (3), XXXIII, p. 170, 1851; Bull. R. Acad. Belg. XXIV, p. 531, 1857.

352 Klebs, Biolog. Centralbl. VII, pp. 193–201, 1887.

353 L. Errera, Sur une condition fondamentale d’Équilibre des cellules vivantes, C. R., CIII, p. 822, 1886; Bull. Soc. Belge de Microscopie, XIII, Oct. 1886; Recueil d’oeuvres (Physiologie gÉnÉrale), 1910, pp. 201–205.

354 L. Chabry, Embryologie des Ascidiens, J. Anat. et Physiol. XXIII, p. 266, 1887.

355 Robert, Embryologie des Troques, Arch. de Zool. exp. et gÉn. (3), X, 1892.

356 “Dass der Furchungsmodus etwas fÜr das ZukÜnftige unwesentliches ist,” Z. f. w. Z. LV, 1893, p. 37. With this statement compare, or contrast, that of Conklin, quoted on p. 4; cf. also pp. 157, 348 (footnotes).

357 de Wildeman, Etudes sur l’attache des cloisons cellulaires, MÉm. Couronn. de l’Acad. R. de Belgique, LIII, 84 pp., 1893–4.

358 It was so termed by Conklin in 1897, in his paper on Crepidula (J. of Morph. XIII, 1897). It is the Querfurche of Rabl (Morph. Jahrb. V, 1879); the Polarfurche of O. Hertwig (Jen. Zeitschr. XIV, 1880); the Brechungslinie of Rauber (Neue Grundlage zur K. der Zelle, M. Jb. VIII, 1882). It is carefully discussed by Robert, DÉv. des Troques, Arch. de Zool. Exp. et GÉn. (3), X, 1892, p. 307 seq.

359 Thus Wilson (J. of Morph. VIII, 1895) declared that in Amphioxus the polar furrow was occasionally absent, and Driesch took occasion to criticise and to throw doubt upon the statement (Arch. f. Entw. Mech. I, 1895, p. 418).

360 Precisely the same remark was made long ago by Driesch: “Das so oft sehematisch gezeichnete Vierzellenstadium mit zwei sich in zwei Punkten scheidende Medianen kann man wohl getrost aus der Reihe des Existierenden streichen,” Entw. mech. Studien, Z. f. w. Z. LIII, p. 166, 1892. Cf. also his Math. mechanische Bedeutung morphologischer Probleme der Biologie, Jena, 59 pp. 1891.

361 Compare, however, p. 299.

362 Ricreatione dell’ occhio e della mente, nell’ Osservatione delle Chiocciole, Roma, 1681.

363 Cf. some of J. H. Vincent’s photographs of ripples, in Phil. Mag. 1897–1899; or those of F. R. Watson, in Phys. Review, 1897, 1901, 1916. The appearance will depend on the rate of the wave, and in turn on the surface-tension; with a low tension one would probably see only a moving “jabble.” FitzGerald thought diatom-patterns might be due to electromagnetic vibrations (Works, p. 503, 1902).

364 Cushman, J. A. and Henderson, W. P., Amer. Nat. XL, pp. 797–802, 1906.

365 This does not merely neglect the broken ones but all whose centres lie between this circle and a hexagon inscribed in it.

366 For more detailed calculations see a paper by “H.M.” [? H. Munro], in Q. J. M. S. VI, p. 83, 1858.

367 Cf. Hartog, The Dual Force of the Dividing Cell, Science Progress (n.s.), I, Oct. 1907, and other papers. Also Baltzer, Ueber mehrpolige Mitosen bei Seeigeleiern, Inaug. Diss. 1908.

368 Observations sur les Abeilles, MÉm. Acad. Sc. Paris, 1712, p. 299.

369 As explained by Leslie Ellis, in his essay “On the Form of Bees’ Cells,” in Mathematical and other Writings, 1853, p. 353; cf. O. Terquem, Nouv. Ann. Math. 1856, p. 178.

370 Phil. Trans. XLII, 1743, pp. 565–571.

371 MÉm. de l’Acad. de Berlin, 1781.

372 Cf. Gregory, Examples, p. 106, Wood’s Homes without Hands, 1865, p. 428, Mach, Science of Mechanics, 1902, p. 453, etc., etc.

373 Origin of Species, ch. VIII (6th ed., p. 221). The cells of various bees, humble-bees and social wasps have been described and mathematically investigated by K. MÜllenhoff, PflÜger’s Archiv XXXII, p. 589, 1883; but his many interesting results are too complex to epitomise. For figures of various nests and combs see (e.g.) von BÜttel-Reepen, Biol. Centralbl. XXXIII, pp. 4, 89, 129, 183, 1903.

374 Darwin had a somewhat similar idea, though he allowed more play to the bee’s instinct or conscious intention. Thus, when he noticed certain half-completed cell-walls to be concave on one side and convex on the other, but to become perfectly flat when restored for a short time to the hive, he says: “It was absolutely impossible, from the extreme thinness of the little plate, that they could have effected this by gnawing away the convex side; and I suspect that the bees in such cases stand on opposite sides and push and bend the ductile and warm wax (which as I have tried is easily done) into its proper intermediate plane, and thus flatten it.”

375 Since writing the above, I see that MÜllenhoff gives the same explanation, and declares that the waxen wall is actually a FlÜssigkeitshÄutchen, or liquid film.

376 Bonnet criticised Buffon’s explanation, on the ground that his description was incomplete; for Buffon took no account of the Maraldi pyramids.

377 Buffon, Histoire Naturelle, IV, p. 99. Among many other papers on the Bee’s cell, see Barclay, Mem. Wernerian Soc. II, p. 259 (1812), 1818; Sharpe, Phil. Mag. IV, 1828, pp. 19–21; L. Lalanne, Ann. Sci. Nat. (2) Zool. XIII, pp. 358–374, 1840; Haughton, Ann. Mag. Nat. Hist. (3), XI, pp. 415–429, 1863; A. R. Wallace, ibid. XII, p. 303, 1863; Jeffries Wyman. Pr. Amer. Acad. of Arts and Sc. VII, pp. 68–83, 1868; Chauncey Wright, ibid. IV, p. 432, 1860.

378 Sir W. Thomson, On the Division of Space with Minimum Partitional Area, Phil. Mag. (5), XXIV, pp. 503–514, Dec. 1887; cf. Baltimore Lectures, 1904, p. 615.

379 Also discovered independently by Sir David Brewster, Trans. R.S.E. XXIV, p. 505, 1867, XXV, p. 115, 1869.

380 Von Fedorow had already described (in Russian) the same figure, under the name of cubo-octahedron, or hepta-parallelohedron, limited however to the case where all the faces are plane. This figure, together with the cube, the hexagonal prism, the rhombic dodecahedron and the “elongated dodecahedron,” constituted the five plane-faced, parallel-sided figures by which space is capable of being completely filled and symmetrically partitioned; the series so forming the foundation of Von Fedorow’s theory of crystalline structure. The elongated dodecahedron is, essentially, the figure of the bee’s cell.

381 F. R. Lillie, Embryology of the Unionidae, Journ. of Morphology, X, p. 12, 1895.

382 E. B. Wilson, The Cell-lineage of Nereis, Journ. of Morphology, VI, p. 452, 1892.

383 It is highly probable, and we may reasonably assume, that the two little triangles do not actually meet at an apical point, but merge into one another by a twist, or minute surface of complex curvature, so as not to contravene the normal conditions of equilibrium.

384 Professor Peddie has given me this interesting and important result, but the mathematical reasoning is too lengthy to be set forth here.

385 Cf. Rhumbler, Arch. f. Entw. Mech. XIV, p. 401, 1902; Assheton, ibid. XXXI, pp. 46–78, 1910.

386 M. Robert (l. c. p. 305) has compiled a long list of cases among the molluscs and the worms, where the initial segmentation of the egg proceeds by equal or unequal division. The two cases are about equally numerous. But like many other writers, he would ascribe this equality or inequality rather to a provision for the future than to a direct effect of immediate physical causation: “Il semble assez probable, comme on l’a dit souvent, que la plus grande taille d’un blastomÈre est liÉe À l’importance et au dÉveloppement prÉcoce des parties du corps qui doivent en naÎtre: il y aurait lÀ une sorte de reflet des stades postÉrieures du dÉveloppement sur les premiÈres phÉnomÈnes, ce que M. Ray Lankester appelle precocious segregation. Il faut avouer pourtant qu’on est parfois assez embarrassÉ pour assigner une cause À pareilles diffÉrences.”

387 The principle is well illustrated in an experiment of Sir David Brewster’s (Trans. R.S.E. XXV, p. 111, 1869). A soap-film is drawn over the rim of a wine-glass, and then covered by a watch-glass. The film is inclined or shaken till it becomes attached to the glass covering, and it then immediately changes place, leaving its transverse position to take up that of a spherical segment extending from one side of the wine-glass to its cover, and so enclosing the same volume of air as formerly but with a great economy of surface, precisely as in the case of our spherical partition cutting off one corner of a cube.

388 Cf. Wildeman, Attache des Cloisons, etc., pls. 1, 2.

389 Nova Acta K. Leop. Akad. XI, 1, pl. IV.

390 Cf. Protoplasmamechanik, p. 229: “Insofern liegen also die VerhÄltnisse hier wesentlich anders als bei der Zertheilung hohler KÖrperformen durch flÜssige Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der kleinsten FlÄchen geforderte Lage und KrÜmmung annimmt, so werden wir den Grund dafÜr in andrer Weise abzuleiten haben.”

391 There is, I think, some ambiguity or disagreement among botanists as to the use of this latter term: the sense in which I am using it, viz. for any partition which meets the outer or peripheral wall at right angles (the strictly radial partition being for the present excluded), is, however, clear.

392 Cit. Plateau, Statique des Liquides, i, p. 358.

393 Even in a Protozoon (Euglena viridis), when kept alive under artificial compression, Ryder found a process of cell-division to occur which he compares to the segmenting blastoderm of a fish’s egg, and which corresponds in its essential features with that here described. Contrib. Zool. Lab. Univ. Pennsylvania, I, pp. 37–50, 1893.

394 This, like many similar figures, is manifestly drawn under the influence of Sachs’s theoretical views, or assumptions, regarding orthogonal trajectories, coaxial circles, confocal ellipses, etc.

395 Such preconceptions as Rauber entertained were all in a direction likely to lead him away from such phenomena as he has faithfully depicted. Rauber had no idea whatsoever of the principles by which we are guided in this discussion, nor does he introduce at all the analogy of surface-tension, or any other purely physical concept. But he was deeply under the influence of Sachs’s rule of rectangular intersection; and he was accordingly disposed to look upon the configuration represented above in Fig. 168, 6, as the most typical or most primitive.

396 Cf. Rauber, Neue Grundlage z. K. der Zelle, Morph. Jahrb. VIII, 1883, pp. 273, 274:

“Ich betone noch, dass unter meinen Figuren diejenige gar nicht enthalten ist, welche zum Typus der Batrachierfurchung gehÖrig am meisten bekannt ist .... Es haben so ausgezeichnete Beobachter sie als vorhanden beschrieben, dass es mir nicht einfallen kann, sie Überhaupt nicht anzuerkennen.”

397 Roux’s experiments were performed with drops of paraffin suspended in dilute alcohol, to which a little calcium acetate was added to form a soapy pellicle over the drops and prevent them from reuniting with one another.

398 Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., Trans. R. S. E. XXVI, p. 9, 1870.

399 See Greville, K. R., Monograph of the Genus Asterolampra, Q.J.M.S. VIII, (Trans.), pp. 102–124, 1860; cf. IBID. (n.s.), II, pp. 41–55, 1862.

400 The same is true of the insect’s wing; but in this case I do not hazard a conjectural explanation.

401 Ann. Mag. N. H. (2), III, p. 126, 1849.

402 Phil. Trans. CLVII, pp. 643–656, 1867.

403 Sachs, Pflanzenphysiologie (Vorlesung XXIV), 1882; cf. Rauber, Neue Grundlage zur Kenntniss der Zelle, Morphol. Jahrb. VIII, p. 303 seq., 1883; E. B. Wilson, Cell-lineage of Nereis, Journ. of Morphology, VI, p. 448, 1892, etc.

404 In the following account I follow closely on the lines laid down by Berthold; Protoplasmamechanik, cap. vii. Many botanical phenomena identical and similar to those here dealt with, are elaborately discussed by Sachs in his Physiology of Plants (chap. xxvii, pp. 431–459, Oxford, 1887); and in his earlier papers, Ueber die Anordnung der Zellen in jÜngsten Pflanzentheilen, and Ueber Zellenanordnung und Wachsthum (Arb. d. botan. Inst. WÜrzburg, 1878, 1879). But Sachs’s treatment differs entirely from that which I adopt and advocate here: his explanations being based on his “law” of rectangular succession, and involving complicated systems of confocal conics, with their orthogonally intersecting ellipses and hyperbolas.

405 Cf. p. 369.

406 There is much information regarding the chemical composition and mineralogical structure of shells and other organic products in H. C. Sorby’s Presidential Address to the Geological Society (Proc. Geol. Soc. 1879, pp. 56–93); but Sorby failed to recognise that association with “organic” matter, or with colloid matter whether living or dead, introduced a new series of purely physical phenomena.

407 Vesque, Ann. des Sc. Nat. (Bot.) (5), XIX, p. 310, 1874.

408 Cf. KÖlliker, Icones Histiologicae, 1864, pp. 119, etc.

409 In an interesting paper by Irvine and Sims Woodhead on the “Secretion of Carbonate of Lime by Animals” (Proc. R. S. E. XVI, 1889, p. 351) it is asserted that “lime salts, of whatever form, are deposited only in vitally inactive tissue.”

410 The tube of Teredo shews no trace of organic matter, but consists of irregular prismatic crystals: the whole structure “being identical with that of small veins of calcite, such as are seen in thin sections of rocks” (Sorby, Proc. Geol. Soc. 1879, p. 58). This, then, would seem to be a somewhat exceptional case of a shell laid down completely outside of the animal’s external layer of organic or colloid substance.

411 C. R. Soc. Biol. Paris (9), I, pp. 17–20, 1889; C. R. Ac. Sc. CVIII, pp. 196–8, 1889.

412 Cf. Heron-Allen, Phil. Trans. (B), vol. CCVI, p. 262, 1915.

413 See Leduc, Mechanism of Life (1911), ch. X, for copious references to other works on the artificial production of “organic” forms.

414 Lectures on the Molecular Asymmetry of Natural Organic Compounds, Chemical Soc. of Paris, 1860, and also in Ostwald’s Klassiker d. ex. Wiss. No. 28, and in Alembic Club Reprints, No. 14, Edinburgh, 1897; cf. Richardson, G. M., Foundations of Stereochemistry, N. Y. 1901.

415 Japp, Stereometry and Vitalism, Brit. Ass. Rep. (Bristol), p. 813, 1898; cf. also a voluminous discussion in Nature, 1898–9.

416 They represent the general theorem of which particular cases are found, for instance, in the asymmetry of the ferments (or enzymes) which act upon asymmetrical bodies, the one fitting the other, according to Emil Fischer’s well-known phrase, as lock and key. Cf. his Bedeutung der Stereochemie fÜr die Physiologie, Z. f. physiol. Chemie, V, p. 60, 1899, and various papers in the Ber. d. d. chem. Ges. from 1894.

417 In accordance with Emil Fischer’s conception of “asymmetric synthesis,” it is now held to be more likely that the process is synthetic than analytic: more likely, that is to say, that the plant builds up from the first one asymmetric body to the exclusion of the other, than that it “selects” or “picks out” (as Japp supposed) the right-handed or the left-handed molecules from an original, optically inactive, mixture of the two; cf. A. McKenzie, Studies in Asymmetric Synthesis, Journ. Chem. Soc. (Trans.), LXXXV, p. 1249, 1904.

418 See for a fuller discussion, Hans Przibram, VitalitÄt, 1913, Kap. iv, Stoffwechsel (Assimilation und Katalyse).

419 Cf. Cotton, Ann. de Chim. et de Phys. (7), VIII, pp. 347–432 (cf. p. 373), 1896.

420 Byk, A., Zur Frage der Spaltbarkeit von Razemverbindungen durch Zirkularpolarisiertes Licht, ein Beitrag zur primÄren Entstehung optisch-activer Substanzen, Zeitsch. f. physikal. Chemie, XLIX, p. 641, 1904. It must be admitted that further positive evidence on these lines is still awanting.

421 Cf. (int. al.) Emil Fischer, Untersuchungen Über AminosÄuren, Proteine, etc. Berlin, 1906.

422 Japp, l. c. p. 828.

423 Rainey, G., On the Elementary Formation of the Skeletons of Animals, and other Hard Structures formed in connection with Living Tissue, Brit. For. Med. Ch. Rev. XX, pp. 451–476, 1857; published separately with additions, 8vo. London, 1858. For other papers by Rainey on kindred subjects see Q. J. M. S. VI (Tr. Microsc. Soc.), pp. 41–50, 1858, VII, pp. 212–225, 1859, VIII, pp. 1–10, 1860, I (n. s.), pp. 23–32, 1861. Cf. also Ord, W. M., On Molecular Coalescence, and on the influence exercised by Colloids upon the Forms of Inorganic Matter, Q. J. M. S. XII, pp. 219–239, 1872; and also the early but still interesting observations of Mr Charles Hatchett, Chemical Experiments on Zoophytes; with some observations on the component parts of Membrane, Phil. Trans. 1800. pp. 327–402.

424 Cf. Quincke, Ueber unsichtbare FlÜssigkeitsschichten, Ann. der Physik, 1902.

425 See for instance other excellent illustrations in Carpenter’s article “Shell,” in Todd’s CyclopÆdia, vol. IV. pp. 550–571, 1847–49. According to Carpenter, the shells of the mollusca (and also of the crustacea) are “essentially composed of cells, consolidated by a deposit of carbonate of lime in their interior.” That is to say, Carpenter supposed that the spherulites, or calcospherites of Harting, were, to begin with, just so many living protoplasmic cells. Soon afterwards however, Huxley pointed out that the mode of formation, while at first sight “irresistibly suggesting a cellular structure, ... is in reality nothing of the kind,” but “is simply the result of the concretionary manner in which the calcareous matter is deposited”; ibid. art. “Tegumentary Organs,” vol. V, p. 487, 1859. Quekett (Lectures on Histology, vol. II, p. 393, 1854, and Q. J. M. S. XI, pp. 95–104, 1863) supported Carpenter; but Williamson (Histological Features in the Shells of the Crustacea, Q. J. M. S. VIII, pp. 35–47, 1860) amply confirmed Huxley’s view, which in the end Carpenter himself adopted (The Microscope, 1862, p. 604). A like controversy arose later in regard to corals. Mrs Gordon (M. M. Ogilvie) asserted that the coral was built up “of successive layers of calcified cells, which hang together at first by their cell-walls, and ultimately, as crystalline changes continue, form the individual laminae of the skeletal structures” (Phil. Trans. CLXXXVII, p. 102, 1896): whereas v. Koch had figured the coral as formed out of a mass of “Kalkconcremente” or “crystalline spheroids,” laid down outside the ectoderm, and precisely similar both in their early rounded and later polygonal stages (though von Koch was not aware of the fact) to the calcospherites of Harting (Entw. d. Kalkskelettes von Asteroides, Mitth. Zool. St. Neapel, III, pp. 284–290, pl. XX, 1882). Lastly Duerden shewed that external to, and apparently secreted by the ectoderm lies a homogeneous organic matrix or membrane, “in which the minute calcareous crystals forming the skeleton are laid down” (The Coral Siderastraea radians, etc., Carnegie Inst. Washington, 1904, p. 34). Cf. also M. M. Ogilvie-Gordon, Q. J. M. S. XLIX, p. 203, 1905, etc.

426 Cf. ClaparÈde, Z. f. w. Z. XIX, p. 604, 1869.

427 Spicules extremely like those of the Alcyonaria occur also in a few sponges; cf. (e.g.), Vaughan Jennings, Journ. Linn. Soc. XXIII, p. 531, pl. 13, fig. 8, 1891.

428 Mem. Manchester Lit. and Phil. Soc. LX, p. 11, 1916.

429 Mummery, J. H., On Calcification in Enamel and Dentine, Phil. Trans. CCV (B), pp. 95–111, 1914.

430 The artificial concretion represented in Fig. 202 is identical in appearance with the concretions found in the kidney of Nautilus, as figured by Willey (Zoological Results, p. lxxvi, Fig. 2, 1902).

431 Cf. Taylor’s Chemistry of Colloids, p. 18, etc., 1915.

432 This rule, undreamed of by Errera, supports and justifies the cardinal assumption (of which we have had so much to say in discussing the forms of cells and tissues) that the incipient cell-wall behaves as, and indeed actually is, a liquid film (cf. p. 306).

433 Cf. p. 254.

434 Cf. Harting, op. cit., pp. 22, 50: “J’avais cru d’abord que ces couches concentriques Étaient produites par l’alternance de la chaleur ou de la lumiÈre, pendant le jour et la nuit. Mais l’expÉrience, expressÉment instituÉe pour examiner cette question, y a rÉpondu nÉgativement.”

435 Liesegang, R. E., Ueber die Schichtungen bei Diffusionen, Leipzig, 1907, and other earlier papers.

436 Cf. Taylor’s Chemistry of Colloids, pp. 146–148, 1915.

437 Cf. S. C. Bradford, The Liesegang Phenomenon and Concretionary Structure in Rocks, Nature, XCVII, p. 80, 1916; cf. Sci. Progress, X, p. 369, 1916.

438 Cf. Faraday, On Ice of Irregular Fusibility, Phil. Trans., 1858, p. 228; Researches in Chemistry, etc., 1859, p. 374; Tyndall, Forms of Water, p. 178, 1872; Tomlinson, C., On some effects of small Quantities of Foreign Matter on Crystallisation, Phil. Mag. (5) XXXI, p. 393, 1891, and other papers.

439 A Study in Crystallisation, J. of Soc. of Chem. Industry, XXV, p. 143, 1906.

440 Ueber Zonenbildung in kolloidalen Medien, Jena, 1913.

441 Verh. d. d. Zool. Gesellsch. p. 179, 1912.

442 Descent of Man, II, pp. 132–153, 1871.

443 As a matter of fact, the phenomena associated with the development of an “ocellus” are or may be of great complexity, inasmuch as they involve not only a graded distribution of pigment, but also, in “optical” coloration, a symmetrical distribution of structure or form. The subject therefore deserves very careful discussion, such as Bateson gives to it (Variation, chap. xii). This, by the way, is one of the very rare cases in which Bateson appears inclined to suggest a purely physical explanation of an organic phenomenon: “The suggestion is strong that the whole series of rings (in Morpho) may have been formed by some one central disturbance, somewhat as a series of concentric waves may be formed by the splash of a stone thrown into a pool, etc.”

444 Cf. also Sir D. Brewster, On optical properties of Mother of Pearl, Phil. Trans. 1814, p. 397.

445 Biedermann, W., Ueber die Bedeutung von Kristallisationsprozessen der Skelette wirbelloser Thiere, namentlich der Molluskenschalen, Z. f. allg. Physiol. I, p. 154, 1902; Ueber Bau und Entstehung der Molluskenschale, Jen. Zeitschr. XXXVI, pp. 1–164, 1902. Cf. also Steinmann, Ueber Schale und Kalksteinbildungen, Ber. Naturf. Ges. Freiburg i. Br IV, 1889; Liesegang, Naturw. Wochenschr. p. 641, 1910.

446 Cf. BÜtschli, Ueber die Herstellung kÜnstlicher StÄrkekÖrner oder von SphÄrokrystallen der StÄrke, Verh. nat. med. Ver. Heidelberg, V, pp. 457–472, 1896.

447 Untersuchungen Über die StÄrkekÖrner, Jena, 1905.

448 Cf. Winge, Meddel. fra Komm. for HavundersÖgelse (Fiskeri), IV, p. 20, Copenhagen, 1915.

449 The anhydrite is sulphate of lime (CaSO?₄); the polyhalite is a triple sulphate of lime, magnesia and potash (2 CaSO?₄.MgSO?₄.K?₂SO?₄ +2 H?₂O).

450 Cf. van’t Hoff, Physical Chemistry in the Service of the Sciences, p. 99 seq. Chicago, 1903.

451 SphÄrocrystalle von Kalkoxalat bei Kakteen, Ber. d. d. Bot. Gesellsch. p. 178, 1885.

452 Pauli, W. u. Samec, M., Ueber LÖslichkeitsbeeinflÜssung von Elektrolyten durch EiweisskÖrper, Biochem. Zeitschr. XVII, p. 235, 1910. Some of these results were known much earlier; cf. Fokker in PflÜger’s Archiv, VII, p. 274, 1873; also Irvine and Sims Woodhead, op. cit. p. 347.

453 Which, in 1000 parts of ash, contains about 840 parts of phosphate and 76 parts of calcium carbonate.

454 Cf. Dreyer, Fr., Die Principien der GerÜstbildung bei Rhizopoden, Spongien und Echinodermen, Jen. Zeitschr. XXVI, pp. 204–468, 1892.

455 In an anomalous and very remarkable Australian sponge, just described by Professor Dendy (Nature, May 18, 1916, p. 253) under the name of Collosclerophora, the spicules are “gelatinous,” consisting of a gel of colloid silica with a high percentage of water. It is not stated whether an organic colloid is present together with the silica. These gelatinous spicules arise as exudations on the outer surface of cells, and come to lie in intercellular spaces or vesicles.

456 Lister, in Willey’s Zoological Results, pt IV, p. 459, 1900.

457 The peculiar spicules of Astrosclera are now said to consist of spherules, or calcospherites, of aragonite, spores of a certain red seaweed forming the nuclei, or starting-points, of the concretions (R. Kirkpatrick, Proc. R. S. LXXXIV (B), p. 579, 1911).

458 See for instance the plates in ThÉel’s Monograph of the Challenger Holothuroidea; also Sollas’s Tetractinellida, p. lxi.

459 For very numerous illustrations of the triradiate and quadriradiate spicules of the calcareous sponges, see (int. al.), papers by Dendy (Q. J. M. S. XXXV, 1893), Minchin (P. Z. S. 1904), Jenkin (P. Z. S. 1908), etc.

460 Cf. again BÉnard’s Tourbillons cellulaires, Ann. de Chimie, 1901, p. 84.

461 LÉger, Stolc and others, in Doflein’s Lehrbuch d. Protozoenkunde, 1911, p. 912.

462 See, for instance, the figures of the segmenting egg of Synapta (after Selenka), in Korschelt and Heider’s Vergleichende Entwicklungsgeschichte (Allgem. Th., 3?^te Lief.), p. 19, 1909. On the spiral type of segmentation as a secondary derivative, due to mechanical causes, of the “radial” type of segmentation, see E. B. Wilson, Cell-lineage of Nereis, Journ. of Morphology, VI, p. 450, 1892.

463 Korschelt and Heider, p. 16.

464 Chall. Rep. Hexactinellida, pls. xvi, liii, lxxvi, lxxxviii.

465 “Hierbei nahm der kohlensaure Kalk eine halb-krystallinische Beschaffenheit an, und gestaltete sich unter Aufnahme von Krystallwasser und in Verbindung mit einer geringen QuantitÄt von organischer Substanz zu jenen individuellen, festen KÖrpern, welche durch die natÜrliche ZÜchtung als Spicula zur Skeletbildung benÜtzt, und spÄterhin durch die Wechselwirkung von Anpassung und Vererbung im Kampfe ums Dasein auf das VielfÄltigste umgebildet und differenziert wurden.” Die KalkschwÄmme, I, p. 377, 1872; cf. also pp. 482, 483.

466 Op. cit. p. 483. “Die geordnete, oft so sehr regelmÄssige und zierliche Zusammensetzung des Skeletsystems ist zum grÖssten Theile unmittelbares Product der WasserstrÖmung; die characteristische Lagerung der Spicula ist von der constanten Richtung des Wasserstroms hervorgebracht; zum kleinsten Theile ist sie die Folge von Anpassungen an untergeordnete Äussere Existenzbedingungen.”

467 Materials for a Monograph of the Ascones, Q. J. M. S. XL. pp. 469–587, 1898.

468 Haeckel, in his Challenger Monograph, p. clxxxviii (1887) estimated the number of known forms at 4314 species, included in 739 genera. Of these, 3508 species were described for the first time in that work.

469 Cf. Gamble, Radiolaria (Lankester’s Treatise on Zoology), vol. I, p. 131, 1909. Cf. also papers by V. HÄcker, in Jen. Zeitschr. XXXIX, p. 581, 1905, Z. f. wiss. Zool. LXXXIII, p. 336, 1905, Arch. f. Protistenkunde, IX, p. 139, 1907, etc.

470 BÜtschli, Ueber die chemische Natur der Skeletsubstanz der Acantharia, Zool. Anz. XXX, p. 784, 1906.

471 For figures of these crystals see Brandt, F. u. Fl. d. Golfes von Neapel, XIII, Radiolaria, 1885, pl. v. Cf. J. MÜller, Ueber die Thalassicollen, etc. Abh. K. Akad. Wiss. Berlin, 1858.

472 Celestine, or celestite, is SrSO?₄ with some BaO replacing SrO.

473 With the colloid chemists, we may adopt (as Rhumbler has done) the terms spumoid or emulsoid to denote an agglomeration of fluid-filled vesicles, restricting the name froth to such vesicles when filled with air or some other gas.

474 Cf. Koltzoff, Zur Frage der Zellgestalt, Anat. Anzeiger, XLI, p. 190, 1912.

475 MÉm. de l’Acad. des Sci., St. PÉtersbourg, XII, Nr. 10, 1902.

476 The manner in which the minute spicules of Raphidiophrys arrange themselves round the bases of the pseudopodial rays is a similar phenomenon.

477 Rhumbler, Physikalische Analyse von Lebenserscheinungen der Zelle, Arch. f. Entw. Mech. VII, p. 103, 1898.

478 The whole phenomenon is described by biologists as a “surprising exhibition of constructive and selective activity,” and is ascribed, in varying phraseology, to intelligence, skill, purpose, psychical activity, or “microscopic mentality”: that is to say, to Galen’s te????? f?s??, or “artistic creativeness” (cf. Brock’s Galen, 1916, p. xxix). Cf. Carpenter, Mental Physiology, 1874, p. 41; Norman, Architectural achievements of Little Masons, etc., Ann. Mag. Nat. Hist. (5), I, p. 284, 1878; Heron-Allen, Contributions ... to the Study of the Foraminifera, Phil. Trans. (B), CCVI, pp. 227–279, 1915; Theory and Phenomena of Purpose and Intelligence exhibited by the Protozoa, as illustrated by selection and behaviour in the Foraminifera, Journ. R. Microscop. Soc. pp. 547–557, 1915; ibid., pp. 137–140, 1916. Prof. J. A. Thomson (New Statesman, Oct. 23, 1915) describes a certain little foraminifer, whose protoplasmic body is overlaid by a crust of sponge-spicules, as “a psycho-physical individuality whose experiments in self-expression include a masterly treatment of sponge-spicules, and illustrate that organic skill which came before the dawn of Art.” Sir Ray Lankester finds it “not difficult to conceive of the existence of a mechanism in the protoplasm of the Protozoa which selects and rejects building-material, and determines the shapes of the structures built, comparable to that mechanism which is assumed to exist in the nervous system of insects and other animals which ‘automatically’ go through wonderfully elaborate series of complicated actions.” And he agrees with “Darwin and others [who] have attributed the building up of these inherited mechanisms to the age-long action of Natural Selection, and the survival of those individuals possessing qualities or ‘tricks’ of life-saving value,” J. R. Microsc. Soc. April, 1916, p. 136.

479 Rhumbler, Das Protoplasma als physikalisches System, Jena, p. 591, 1914; also in Arch. f. Entwickelungsmech. VII, pp. 279–335, 1898.

480 Verworn, Psycho-physiologische Protisten-Studien, Jena, 1889 (219 pp.).

481 Leidy, J., Fresh-water Rhizopods of N. America, 1879, p. 262, pl. xli, figs. 11, 12.

482 Carnoy, Biologie Cellulaire, p. 244, fig. 108; cf. Dreyer, op. cit. 1892, fig. 185.

483 In all these latter cases we recognise a relation to, or extension of, the principle of Plateau’s bourrelet, or van der Mensbrugghe’s masse annulaire, of which we have already spoken (p. 297).

484 Apart from the fact that the apex of each pyramid is interrupted, or truncated, by the presence of the little central cell, it is also possible that the solid angles are not precisely equivalent to those of Maraldi’s pyramids, owing to the fact that there is a certain amount of distortion, or axial asymmetry, in the Nassellarian system. In other words (to judge from Haeckel’s figures), the tetrahedral symmetry in Nassellaria is not absolutely regular, but has a main axis about which three of the trihedral pyramids are symmetrical, the fourth having its solid angle somewhat diminished.

485 Cf. Faraday’s beautiful experiments, On the Moving Groups of Particles found on Vibrating Elastic Surfaces, etc., Phil. Trans. 1831, p. 299; Researches in Chem. and Phys. 1859, pp. 314–358.

486 We need not go so far as to suppose that the external layer of cells wholly lacked the power of secreting a skeleton. In many of the Nassellariae figured by Haeckel (for there are many variant forms or species besides that represented here), the skeleton of the partition-walls is very slightly and scantily developed. In such a case, if we imagine its few and scanty strands to be broken away, the central tetrahedral figure would be set free, and would have all the appearance of a complete and independent structure.

487 The “bourrelet” is not only, as Plateau expresses it, a “surface of continuity,” but we also recognise that it tends (so far as material is available for its production) to further lessen the free surface-area. On its relation to vapour-pressure and to the stability of foam, see FitzGerald’s interesting note in Nature, Feb. 1, 1894 (Works, p. 309).

488 Of the many thousand figures in the hundred and forty plates of this beautifully illustrated book, there is scarcely one which does not depict, now patently, now in pregnant suggestion, some subtle and elegant geometrical configuration.

489 They were known (of course) long before Plato: ???t?? d? ?a? ?? t??t??? p??a?????e?.

490 If the equation of any plane face of a crystal be written in the form h x+k y+l z =1, then h, k, l are the indices of which we are speaking. They are the reciprocals of the parameters, or reciprocals of the distances from the origin at which the plane meets the several axes. In the case of the regular or pentagonal dodecahedron these indices are 2, 1+v?5, 0. Kepler described as follows, briefly but adequately, the common characteristics of the dodecahedron and icosahedron: “Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quinquangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli structura perfici non potest sine proportione illa, quam hodierni geometrae divinam appellant” (De nive sexangula (1611), Opera, ed. Frisch, VII, p. 723). Here Kepler was dealing, somewhat after the manner of Sir Thomas Browne, with the mysteries of the quincunx, and also of the hexagon; and was seeking for an explanation of the mysterious or even mystical beauty of the 5-petalled or 3-petalled flower,—pulchritudinis aut proprietatis figurae, quae animam harum plantarum characterisavit.

491 Cf. Tutton, Crystallography, p. 932, 1911.

492 However, we can often recognise, in a small artery for instance, that the so-called “circular” fibres tend to take a slightly oblique, or spiral, course.

493 The spiral fibres, or a large portion of them, constitute what Searle called “the rope of the heart” (Todd’s Cyclopaedia, II, p. 621, 1836). The “twisted sinews of the heart” were known to early anatomists, and have been frequently and elaborately studied: for instance, by Gerdy (Bull. Fac. Med. Paris, 1820, pp. 40–148), and by Pettigrew (Phil. Trans. 1864), and of late by J. B. Macallum (Johns Hopkins Hospital Report, IX, 1900) and by Franklin P. Mall (Amer. J. of Anat. XI, 1911).

494 Cf. BÜtschli, “Protozoa,” in Bronn’s Thierreich, II, p. 848, III, p. 1785, etc., 1883–87; Jennings, Amer. Nat. XXXV, p. 369, 1901; PÜtter, Thigmotaxie bei Protisten, Arch. f. Anat. u. Phys. (Phys. Abth. Suppl.), pp. 243–302, 1900.

495 A great number of spiral forms, both organic and artificial, are described and beautifully illustrated in Sir T. A. Cook’s Curves of Life, 1914, and Spirals in Nature and Art, 1903.

496 Cf. Vines, The History of the Scorpioid Cyme, Journ. of Botany (n.s.), X, pp. 3–9, 1881.

497 Leslie’s Geometry of Curved Lines, p. 417, 1821. This is practically identical with Archimedes’ own definition (ed. Torelli, p. 219); cf. Cantor, Geschichte der Mathematik, I, p. 262, 1880.

498 See an interesting paper by Whitworth, W. A., “The Equiangular Spiral, its chief properties proved geometrically,” in the Messenger of Mathematics (1), I, p. 5, 1862.

499 I am well aware that the debt of Greek science to Egypt and the East is vigorously denied by many scholars, some of whom go so far as to believe that the Egyptians never had any science, save only some “rough rules of thumb for measuring fields and pyramids” (Burnet’s Greek Philosophy, 1914, p. 5).

500 Euclid (II, def. 2).

501 Cf. Treutlein, Z. f. Math. u. Phys. (Hist. litt. Abth.), XXVIII, p. 209, 1883.

502 This is the so-called Dreifachgleichschenkelige Dreieck; cf. Naber, op. infra cit. The ratio 1:0·618 is again not hard to find in this construction.

503 See, on the mathematical history of the Gnomon, Heath’s Euclid, I, passim, 1908; Zeuthen, TheorÈme de Pythagore, GenÈve, 1904; also a curious and interesting book, Das Theorem des Pythagoras, by Dr. H. A. Naber, Haarlem, 1908.

504 For many beautiful geometrical constructions based on the molluscan shell, see Colman, S. and Coan, C. A., Nature’s Harmonic Unity (ch. ix, Conchology), New York, 1912.

505 The Rev. H. Moseley, On the Geometrical Forms of Turbinated and Discoid Shells, Phil. Trans. pp. 351–370. 1838.

506 It will be observed that here Moseley, speaking as a mathematician and considering the linear spiral, speaks of whorls when he means the linear boundaries, or lines traced by the revolving radius vector; while the conchologist usually applies the term whorl to the whole space between the two boundaries. As conchologists, therefore, we call the breadth of a whorl what Moseley looked upon as the distance between two consecutive whorls. But this latter nomenclature Moseley himself often uses.

507 In the case of Turbo, and all other “turbinate” shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a “gauche” spiral, such that the radius vector no longer revolves in a plane perpendicular to the axis of the system, but is inclined to that axis at some constant angle (?). The figure still preserves its continued similarity, and may with strict accuracy be called a logarithmic spiral in space. It is evident that its envelope will be a right circular cone; and indeed it is commonly spoken of as a logarithmic spiral wrapped upon a cone, its pole coinciding with the apex of the cone. It follows that the distances of successive whorls of the spiral measured on the same straight line passing through the apex of the cone, are in geometrical progression, and conversely just as in the former case. But the ratio between any two consecutive interspaces (i.e. R?₃-R?₂/R?₂-R?₁) is now equal to e?^2psin?cota, ? being the semi-angle of the enveloping cone. (Cf. Moseley, Phil. Mag. XXI, p. 300, 1842.)

508 As the successive increments evidently constitute similar figures, similarly related to the pole (P), it follows that their linear dimensions are to one another as the radii vectores drawn to similar points in them: for instance as P P?₁, P P?₂, which (in Fig. 264, 1) are radii vectores drawn to the points where they meet the common boundary.

509 The equation to the surface of a turbinate shell is discussed by Moseley (Phil. Trans. tom. cit. p. 370), both in terms of polar coordinates and of the rectangular coordinates x, y, z. A more elegant representation can be given in vector notation, by the method of quaternions.

510 J. C. M. Reinecke, Maris protogaei Nautilos, etc., Coburg, 1818. Leopold von Buch, Ueber die Ammoniten in den Älteren Gebirgsschichten, Abh. Berlin. Akad., Phys. Kl. pp. 135–158, 1830; Ann. Sc. Nat. XXVIII, pp. 5–43, 1833; cf. Elie de Beaumont, Sur l’enroulement des Ammonites, Soc. Philom., Pr. verb. pp. 45–48, 1841.

511 Biblia Naturae sive Historia Insectorum, Leydae, 1737, p. 152.

512 Alcide D’Orbigny, Bull. de la soc. gÉol. Fr. XIII, p. 200, 1842; Cours ÉlÉm. de PalÉontologie, II, p. 5, 1851. A somewhat similar instrument was described by BoubÉe. in Bull. soc. gÉol. I, p. 232, 1831. Naumann’s Conchyliometer (Poggend. Ann. LIV, p. 544, 1845) was an application of the screw-micrometer; it was provided also with a rotating stage, for angular measurement. It was adapted for the Study of a discoid or ammonitoid shell, while D’Orbigny’s instrument was meant for the study of a turbinate shell.

513 It is obvious that the ratios of opposite whorls, or of radii 180° apart, are represented by the square roots of these values; and the ratios of whorls or radii 90° apart, by the square roots of these again.

514 For the correction to be applied in the case of the helicoid, or “turbinate” shells, see p. 557.

515 On the Measurement of the Curves formed by Cephalopods and other Mollusks. Phil. Mag. (5), VI, pp. 241–263, 1878.

516 For an example of this method, see Blake, l.c. p. 251.

517 Naumann, C. F., Ueber die Spiralen von Conchylien, Abh. k. sÄchs. Ges. pp. 153–196, 1846; Ueber die cyclocentrische Conchospirale u. Über das Windungsgesetz von Planorbis corneus, ibid. I, pp. 171–195, 1849; Spirale von Nautilus u. Ammonites galeatus, Ber. k. sÄchs. Ges. II, p. 26, 1848; Spirale von Amm. Ramsaueri, ibid. XVI, p. 21, 1864; see also Poggendorff’s Annalen, L, p. 223, 1840; LI, p. 245, 1841; LIV, p. 541, 1845, etc.

518 Sandberger, G., Spiralen des Ammonites Amaltheus, A. Gaytani, und Goniatites intumescens, Zeitschr. d. d. Geol. Gesellsch. X, pp. 446–449, 1858.

519 Grabau, A. H., Ueber die Naumannsche Conchospirale, etc. Inauguraldiss. Leipzig, 1872; Die Spiralen von Conchylien, etc. Programm, Nr. 502, Leipzig, 1882.

520 It has been pointed out to me that it does not follow at once and obviously that, because the interspace AB is a mean proportional between the breadths of the adjacent whorls, therefore the whole distance OB is a mean proportional between OA and OC. This is a corollary which requires to be proved; but the proof is easy.

521 A beautiful construction: stupendum Naturae artificium, Linnaeus.

522 English edition, p. 537, 1900. The chapter is revised by Prof. Alpheus Hyatt, to whom the nomenclature is largely due. For a more copious terminology, see Hyatt, Phylogeny of an Acquired Characteristic, p. 422 seq., 1894.

523 This latter conclusion is adopted by Willey, Zoological Results, p. 747, 1902.

524 See Moseley, op. cit. pp. 361 seq.

525 In Nautilus, the “hood” has somewhat different dimensions in the two sexes, and these differences are impressed upon the shell, that is to say upon its “generating curve.” The latter constitutes a somewhat broader ellipse in the male than in the female. But this difference is not to be detected in the young; in other words, the form of the generating curve perceptibly alters with advancing age. Somewhat similar differences in the shells of Ammonites were long ago suspected, by D’Orbigny, to be due to sexual differences. (Cf. Willey, Natural Science, VI, p. 411, 1895; Zoological Results, p. 742, 1902.)

526 Macalister, Alex., Observations on the Mode of Growth of Discoid and Turbinated Shells, P. R. S. XVIII, pp. 529–532, 1870.

527 See figures in Arnold Lang’s Comparative Anatomy (English translation), II, p. 161, 1902.

528 Kappers, C. U. A., Die Bildung kÜnstlicher Molluskenschalen, Zeitschr. f. allg. Physiol. VII, p. 166, 1908.

529 We need not assume a close relationship, nor indeed any more than such a one as permits us to compare the shell of a Nautilus with that of a Gastropod.

530 Cf. Owen, “These shells [Nautilus and Ammonites] are revolutely spiral or coiled over the back of the animal, not involute like Spirula”: Palaeontology, 1861, p. 97; cf. Mem. on the Pearly Nautilus, 1832; also P.Z.S. 1878, p. 955.

531 The case of Terebratula or of Gryphaea would be closely analogous, if the smaller valve were less closely connected and co-articulated with the larger.

532 “It has been suggested, and I think in some quarters adopted as a dogma, that the formation of successive septa [in Nautilus] is correlated with the recurrence of reproductive periods. This is not the case, since, according to my observations, propagation only takes place after the last septum is formed;” Willey, Zoological Results, p. 746, 1902.

533 Cf. Woodward, Henry, On the Structure of Camerated Shells, Pop. Sci. Rev. XI, pp. 113–120, 1872.

534 See Willey, Contributions to the Natural History of the Pearly Nautilus, Zoological Results, etc. p. 749, 1902. Cf. also Bather, Shell-growth in Cephalopoda, Ann. Mag. N. H. (6), I, pp 298–310, 1888; ibid. pp. 421–427, and other papers by Blake, Riefstahl, etc. quoted therein.

535 It was this that led James Bernoulli, in imitation of Archimedes, to have the logarithmic spiral graven on his tomb, with the pious motto, Eadem mutata resurgam. On Goodsir’s grave the same symbol is reinscribed.

536 The “lobes” and “saddles” which arise in this manner, and on whose arrangement the modern classification of the nautiloid and ammonitoid shells largely depends, were first recognised and named by Leopold von Buch, Ann. Sci. Nat. XXVII, XXVIII, 1829.

537 Blake has remarked upon the fact (op. cit. p. 248) that in some Cyrtocerata we may have a curved shell in which the ornaments approximately run at a constant angular distance from the pole, while the septa approximate to a radial direction; and that “thus one law of growth is illustrated by the inside, and another by the outside.” In this there is nothing at which we need wonder. It is merely a case where the generating curve is set very obliquely to the axis of the shell; but where the septa, which have no necessary relation to the mouth of the shell, take their places, as usual, at a certain definite angle to the walls of the tube. This relation of the septa to the walls of the tube arises after the tube itself is fully formed, and the obliquity of growth of the open end of the tube has no relation to the matter.

538 Cf. pp. 255, 463, etc.

539 In a few cases, according to Awerinzew and Rhumbler, where the chambers are added on in concentric series, as in Orbitolites, we have the crystalline structure arranged radially in the radial walls but tangentially in the concentric ones: whereby we tend to obtain, on a minute scale, a system of orthogonal trajectories, comparable to that which we shall presently study in connection with the structure of bone. Cf. S. Awerinzew, Kalkschale der Rhizopoden, Z. f. w. Z. LXXIV, pp. 478–490, 1903.

540 Rhumbler, L., Die Doppelschalen von Orbitolites und anderer Foraminiferen, etc., Arch. f. Protistenkunde, I, pp. 193–296, 1902; and other papers. Also Die Foraminiferen der Planktonexpedition, I, 1911, pp. 50–56.

541 BÉnard, H, Les tourbillons cellulaires, Ann. de Chimie (8), XXIV, 1901. Cf. also the pattern of cilia on an Infusorian, as figured by BÜtschli in Bronn’s Protozoa, III, p. 1281, 1887.

542 A similar hexagonal pattern is obtained by the mutual repulsion of floating magnets in Mr R. W. Wood’s experiments, Phil. Mag. XLVI, pp. 162–164, 1898.

543 Cf. D’Orbigny, Alc., Tableau mÉthodique de la classe des CÉphalopodes, Ann. des Sci. Nat. (1), VII, pp. 245–315, 1826; Dujardin. FÉlix, Observations nouvelles sur les prÉtendus CÉphalopodes microscopiques, ibid. (2), III, pp. 108, 109, 312–315, 1835; Recherches sur les organismes infÉrieurs, ibid. IV, pp. 343–377, 1835, etc.

544 It is obvious that the actual outline of a foraminiferal, just as of a molluscan shell, may depart widely from a logarithmic spiral. When we say here, for short, that the shell is a logarithmic spiral, we merely mean that it is essentially related to one: that it can be inscribed in such a spiral, or that corresponding points (such, for instance, as the centres of gravity of successive chambers, or the extremities of successive septa) wall always be found to lie upon such a spiral.

545 von MÖller, V., Die spiral-gewundenen Foraminifera des russischen Kohlenkalks, MÉm. de l’Acad. Imp. Sci., St PÉtersbourg (7), XXV, 1878.

546 As von MÖller is careful to explain, Naumann’s formula for the “cyclocentric conchospiral” is appropriate to this and other spiral Foraminifera, since we have in all these cases a central or initial chamber, approximately spherical, about which the logarithmic spiral is coiled (cf. Fig. 309). In species where the central chamber is especially large, Naumann’s formula is all the more advantageous. But it is plain that it is only required when we are dealing with diameters, or with radii; so long as we are merely comparing the breadths of successive whorls, the two formulae come to the same thing.

547 Van Iterson, G., Mathem. u. mikrosk.-anat. Studien Über Blattstellungen, nebst Betrachtungen Über den Schalenbau der Miliolinen, 331 pp., Jena, 1907.

548 Hans Przibram asserts that the linear ratio of successive chambers tends in many Foraminifera to approximate to 1·26, which =??2; in other words, that the volumes of successive chambers tend to double. This Przibram would bring into relation with another law, viz. that insects and other arthropods tend to moult, or to metamorphose, just when they double their weights, or increase their linear dimensions in the ratio of 1:??2. (Die Kammerprogression der Foraminiferen als Parallele zur HÄutungsprogression der Mantiden, Arch. f. Entw. Mech. XXXIV p. 680, 1813.) Neither rule seems to me to be well grounded.

549 Cf. Schacko, G., Ueber Globigerina-Einschluss bei Orbulina, Wiegmann’s Archiv, XLIX, p. 428, 1883; Brady, Chall. Rep., p. 607, 1884.

550 Cf. Brady, H. B., Challenger Rep., Foraminifera, 1884, p. 203, pl. XIII.

551 Brady, op. cit., p. 206; Batsch, one of the earliest writers on Foraminifera, had already noticed that this whole series of ear-shaped and crozier-shaped shells was filled in by gradational forms; Conchylien des Seesandes, 1791, p. 4, pl. VI, fig. 15a–f. See also, in particular, Dreyer, Peneroplis; eine Studie zur biologischen Morphologie und zur Speciesfrage, Leipzig, 1898; also Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen, TÜbinger zool. Arbeiten, III, p. 35, 1899.

552 Doflein, Protozoenkunde, 1911, p. 263; “Was diese Art veranlÄsst in dieser Weise gelegentlich zu varÜren, ist vorlÄufig noch ganz rÄthselhaft.”

553 In the case of Globigerina, some fourteen species (out of a very much larger number of described forms) were allowed by Brady (in 1884) to be distinct; and this list has been, I believe, rather added to than diminished. But these so-called species depend for the most part on slight differences of degree, differences in the angle of the spiral, in the ratio of magnitude of the segments, or in their area of contact one with another. Moreover with the exception of one or two “dwarf” forms, said to be limited to Arctic and Antarctic waters, there is no principle of geographical distribution to be discerned amongst them. A species found fossil in New Britain turns up in the North Atlantic: a species described from the West Indies is rediscovered at the ice-barrier of the Antarctic.

554 Dreyer, F., Principien der GerÜstbildung bei Rhizopoden, etc., Jen. Zeitschr. XXVI, pp. 204–468, 1892.

555 A difficulty arises in the case of forms (like Peneroplis) where the young shell appears to be more complex than the old, the first formed portion being closely coiled while the later additions become straight and simple: “die biformen Arten verhalten sich, kurz gesagt. gerade umgekehrt als man nach dem biogenetischen Grundgesetz erwarten sollte,” Rhumbler, op. cit., p. 33 etc.

556 “Das Festigkeitsprinzip als Movens der Weiterentwicklung ist zu interessant und fÜr die Aufstellung meines Systems zu wichtig um die Frage unerÖrtert zu lassen, warum diese BevorzÜgung der Festigkeit stattgefunden hat. Meiner Ansicht nach lautet die Antwort auf diese Frage einfach, weil die Foraminiferen meistens unter VerhÄltnissen leben, die ihre Schalen in hohem Grade der Gefahr des Zerbrechens aussetzen; es muss also eine fortwahrende Auslese des Festeren stattfinden,” Rhumbler, op. cit., p. 22.

557 “Die Foraminiferen kiesige oder grobsandige Gebiete des Meeresbodens nicht lieben, u.s.w.”: where the last two words have no particular meaning, save only that (as M. Aurelius says) “of things that use to be, we say commonly that they love to be.”

558 In regard to the Foraminifera, “die Palaeontologie lÄsst uns leider an Anfang der Stammesgeschichte fast gÄnzlich im Stiche,” Rhumbler, op. cit., p. 14.

559 The evolutionist theory, as Bergson puts it, “consists above all in establishing relations of ideal kinship, and in maintaining that wherever there is this relation of, so to speak, logical affiliation between forms, there is also a relation of chronological succession between the species in which these forms are materialised”: Creative Evolution, 1911, p. 26. Cf. supra, p. 251.

560 In the case of the ram’s horn, the assumption that the rings are annual is probably justified. In cattle they are much less conspicuous, but are sometimes well-marked in the cow; and in Sweden they are then called “calf-rings,” from a belief that they record the number of offspring. That is to say, the growth of the horn is supposed to be retarded during gestation, and to be accelerated after parturition, when superfluous nourishment seeks a new outlet. (Cf. LÖnnberg, P.Z.S., p. 689, 1900.)

561 Cf. Sir V. Brooke, On the Large Sheep of the Thian Shan, P.Z.S., p. 511, 1875.

562 Cf. LÖnnberg, E., On the Structure of the Musk Ox, P.Z.S., pp. 686–718, 1900.

563 St Venant, De la torsion des prismes, avec des considÉrations sur leur flexion, etc., MÉm. des Savants Étrangers, Paris, XIV, pp. 233–560, 1856.

564 This is not difficult to do, with considerable accuracy, if the clay be kept well wetted, or semi-fluid, and the smoothing be done with a large wet brush.

565 The curves are well shewn in most of Sir V. Brooke’s figures of the various species of Argali, in the paper quoted on p. 614.

566 Climbing Plants, 1865 (2nd edit. 1875); Power of Movement in Plants, 1880.

567 Palm, Ueber das Winden der Pflanzen, 1827; von Mohl, Bau und Winden der Ranken, etc., 1827; Dutrochet, Mouvements rÉvolutifs spontanÉs, C.R. 1843, etc.

568 Cf. (e.g.) Lepeschkin, Zur Kenntnis des Mechanismus der Variationsbewegungen, Ber. d. d. Bot. Gesellsch. XXVI A, pp. 724–735, 1908; also A. TrÖndle, Der Einfluss des Lichtes auf die PermeabilitÄt des Plasmahaut, Jahrb. wiss. Bot. XLVIII, pp. 171–282, 1910.

569 For an elaborate study of antlers, see RÖrig, A., Arch. f. Entw. Mech. X, pp. 525–644, 1900, XI, pp. 65–148, 225–309, 1901; Hoffmann, C., Zur Morphologie der rezenten Hirschen, 75 pp., 23 pls., 1901: also Sir Victor Brooke, On the Classification of the Cervidae, P.Z.S., pp. 883–928, 1878. For a discussion of the development of horns and antlers, see Gadow, H., P.Z.S., pp. 206–222, 1902, and works quoted therein.

570 Cf. Rhumbler, L., Ueber die AbhÄngigkeit des Geweihwachstums der Hirsche, speziell des Edelhirsches, vom Verlauf der BlutgefÄsse im Kolbengeweih, Zeitschr. f. Forst. und Jagdwesen, 1911, pp. 295–314.

571 The fact that in one very small deer, the little South American Coassus, the antler is reduced to a simple short spike, does not preclude the general distinction which I have drawn. In Coassus we have the beginnings of an antler, which has not yet manifested its tendency to expand; and in the many allied species of the American genus Cariacus, we find the expansion manifested in various simple modes of ramification or bifurcation. (Cf. Sir V. Brooke, Classification of the Cervidae, p. 897.)

572 Cf. also the immense range of variation in elks’ horns, as described by LÖnnberg, P.Z.S. II, pp. 352–360, 1902.

573 Besides papers referred to below, and many others quoted in Sach’s Botany and elsewhere, the following are important: Braun, Alex., Vergl. Untersuchung Über die Ordnung der Schuppen an den Tannenzapfen, etc., Verh. Car. Leop. Akad. XV, pp. 199–401, 1831; Dr C. Schimper’s VortrÄge Über die MÖglichkeit eines wissenschaftlichen VerstÄndnisses der Blattstellung, etc., Flora, XVIII, pp. 145–191, 737–756, 1835; Schimper, C. F., Geometrische Anordnung der um eine Axe peripherische Blattgebilde, Verhandl. Schweiz. Ges., pp. 113–117, 1836; Bravais, L. and A., Essai sur la disposition des feuilles curvisÉriÉes, Ann. Sci. Nat. (2), VII, pp. 42–110, 1837; Sur la disposition symmÉtrique des inflorescences, ibid., pp. 193–221, 291–348, VIII, pp. 11–42, 1838; Sur la disposition gÉnÉrale des feuilles rectisÉriÉes, ibid. XII, pp. 5–41, 65–77, 1839; Zeising, NormalverhÄltniss der chemischen und morphologischen Proportionen, Leipzig, 1856; Naumann, C. F., Ueber den Quincunx als Gesetz der Blattstellung bei Sigillaria, etc., Neues Jahrb. f. Miner. 1842, pp. 410–417; Lestiboudois, T., Phyllotaxie anatomique, Paris, 1848; Henslow, G., Phyllotaxis, London, 1871; Wiesner, Bemerkungen Über rationale und irrationale Divergenzen, Flora, LVIII, pp. 113–115, 139–143, 1875; Airy, H., On Leaf Arrangement, Proc. R. S. XXI, p. 176, 1873; Schwendener, S., Mechanische Theorie der Blattstellungen, Leipzig, 1878; Delpino, F., Causa meccanica della filotassi quincunciale, Genova, 1880; de Candolle, C., Étude de Phyllotaxie, GenÈve, 1881.

574 Allgemeine Morphologie der GewÄchse, p. 442, etc. 1868.

575 Relation of Phyllotaxis to Mechanical Laws, Oxford, 1901–1903; cf. Ann. of Botany, XV, p. 481, 1901.

576 “The proposition is that the genetic spiral is a logarithmic spiral, homologous with the line of current-flow in a spiral vortex; and that in such a system the action of orthogonal forces will be mapped out by other orthogonally intersecting logarithmic spirals—the ‘parastichies’ ”; Church, op. cit. I, p. 42.

577 Mr Church’s whole theory, if it be not based upon, is interwoven with, Sachs’s theory of the orthogonal intersection of cell-walls, and the elaborate theories of the symmetry of a growing point or apical cell which are connected therewith. According to Mr Church, “the law of the orthogonal intersection of cell-walls at a growing apex may be taken as generally accepted” (p. 32); but I have taken a very different view of Sachs’s law, in the eighth chapter of the present book. With regard to his own and Sachs’s hypotheses, Mr Church makes the following curious remark (p. 42): “Nor are the hypotheses here put forward more imaginative than that of the paraboloid apex of Sachs which remains incapable of proof, or his construction for the apical cell of Pteris which does not satisfy the evidence of his own drawings.”

578 Amer. Naturalist, VII, p. 449, 1873.

579 This celebrated series, which appears in the continued fraction (1/1)+(1/(1+)) etc. and is closely connected with the Sectio aurea or Golden Mean, is commonly called the Fibonacci series, after a very learned twelfth century arithmetician (known also as Leonardo of Pisa), who has some claims to be considered the introducer of Arabic numerals into christian Europe. It is called Lami’s series by some, after Father Bernard Lami, a contemporary of Newton’s, and one of the co-discoverers of the parallelogram of forces. It was well-known to Kepler, who, in his paper De nive sexangula (cf. supra, p. 480), discussed it in connection with the form of the dodecahedron and icosahedron, and with the ternary or quinary symmetry of the flower. (Cf. Ludwig, F., Kepler Über das Vorkommen der Fibonaccireihe im Pflanzenreich, Bot. Centralbl. LXVIII, p. 7, 1896). Professor William Allman, Professor of Botany in Dublin (father of the historian of Greek geometry), speculating on the same facts, put forward the curious suggestion that the cellular tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra. and that of the monocotyledons or endogens of icosahedra (On the mathematical connexion between the parts of Vegetables: abstract of a Memoir read before the Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, OrganogÉnie vÉgÉtale, I, p. 534).

580 Proc. Roy. Soc. Edin. VII, p. 391, 1872.

581 The necessary existence of these recurring spirals is also proved, in a somewhat different way, by Leslie Ellis, On the Theory of Vegetable Spirals, in Mathematical and other Writings, 1853, pp. 358–372.

582 Proc. Roy. Soc. Edin. VII, p. 397, 1872; Trans. Roy. Soc. Edin. XXVI, p. 505, 1870–71.

583 A common form of pail-shaped waste-paper basket, with wide rhomboidal meshes of cane, is well-nigh as good a model as is required.

584 Deutsche Vierteljahrsschrift, p. 261, 1868.

585 Memoirs of Amer. Acad. IX, p. 389.

586 De avibus circa aquas Danubii vagantibus et de ipsarum Nidis (Vol. V of the Danubius Pannonico-mysicus), Hagae Com., 1726.

587 Sir Thomas Browne had a collection of eggs at Norwich, according to Evelyn, in 1671.

588 Cf. Lapierre, in Buffon’s Histoire Naturelle, ed. Sonnini, 1800.

589 Eier der VÖgel Deutschlands, 1818–28 (cit. des Murs, p. 36).

590 TraitÉ d’Oologie, 1860.

591 Lafresnaye, F. de, Comparaison des oeufs des Oiseaux avec leurs squelettes, comme seul moven de reconnaÎtre la cause de leurs diffÉrentes formes, Rev. Zool., 1845, pp. 180–187, 239–244.

592 Cf. Des Murs, p. 67: “Elle devait encore penser au moment oÙ ce germe aurait besoin de l’espace nÉcessaire À son accroissement, À ce moment oÙ ... il devra remplir exactement l’intervalle circonscrit par sa fragile prison, etc.”

593 Thienemann, F. A. L., Syst. Darstellung der Fortpflanzung der VÖgel Europas. Leipzig, 1825–38.

594 Cf. Newton’s Dictionary of Birds, 1893, p. 191; Szielasko, Gestalt der Vogeleier, J. f. Ornith. LIII, pp. 273–297, 1905.

595 Jacob Steiner suggested a Cartesian oval, r+m r?' =c, as a general formula for all eggs (cf. Fechner, Ber. sÄchs. Ges., 1849, p. 57); but this formula (which fails in such a case as the guillemot), is purely empirical, and has no mechanical foundation.

596 GÜnther, F. C., Sammlung von Nestern und Eyern verschiedener VÖgel, NÜrnb. 1772. Cf. also Raymond Pearl, Morphogenetic Activity of the Oviduct, J. Exp. Zool. VI, pp. 339–359, 1909.

597 The following account is in part reprinted from Nature, June 4, 1908.

598 In so far as our explanation involves a shaping or moulding of the egg by the uterus or “oviduct” (an agency supplemented by the proper tensions of the egg), it is curious to note that this is very much the same as that old view of Telesius regarding the formation of the embryo (De rerum natura, VI, cc. 4 and 10), which he had inherited from Galen, and of which Bacon speaks (Nov. Org. cap. 50; cf. Ellis’s note). Bacon expressly remarks that “Telesius should have been able to shew the like formation in the shells of eggs.” This old theory of embryonic modelling survives only in our usage of the term “matrix” for a “mould.”

599 Journal of Tropical Medicine, 15th June, 1911. I leave this paragraph as it was written, though it is now once more asserted that the terminal and lateral-spined eggs belong to separate and distinct species of Bilharzia (Leiper, Brit. Med. Journ., 18th March, 1916, p. 411).

600 Cf. Bashforth and Adams, Theoretical Forms of Drops, etc., Cambridge, 1883.

601 Woods, R. H., On a Physical Theorem applied to tense Membranes, Journ. of Anat. and Phys. XXVI, pp. 362–371, 1892. A similar investigation of the tensions in the uterine wall, and of the varying thickness of its muscles, was attempted by Haughton in his Animal Mechanics, pp. 151–158, 1873.

602 This corresponds with a determination of the normal pressures (in systole) by Krohl, as being in the ratio of 1:6·8.

603 Cf. Schwalbe, G., Ueber Wechselbeziehungen und ihr Einfluss auf die Gestaltung des Arteriensystem, Jen. Zeitschr. XII, p. 267, 1878, Roux, Ueber die Verzweigungen der BlutgefÄssen des Menschen, ibid. XII, p. 205, 1878; Ueber die Bedeutung der Ablenkung des ArterienstÄmmen bei der Astaufgabe, ibid. XIII, p. 301, 1879; Hess, Walter, Eine mechanisch bedingte GesetzmÄssigkeit im Bau des BlutgefÄsssystems, A. f. Entw. Mech. XVI, p. 632, 1903; Thoma, R., Ueber die Histogenese und Histomechanik des BlutgefÄsssystems, 1893.

604 Essays, etc., edited by Owen, I, p. 134, 1861.

605 On the Functions of the Heart and Arteries, Phil. Trans. 1809, pp. 1–31, cf. 1808, pp. 164–186; Collected Works, I, pp. 511–534, 1855. The same lesson is conveyed by all such work as that of Volkmann, E. H. Weber and Poiseuille. Cf. Stephen Hales’ Statical Essays, II, Introduction: “Especially considering that they [i.e. animal Bodies] are in a manner framed of one continued Maze of innumerable Canals, in which Fluids are incessantly circulating, some with great Force and Rapidity, others with very different Degrees of rebated Velocity: Hence, etc.”

606 “Sizes” is Owen’s editorial emendation, which seems amply justified.

607 For a more elaborate classification, into colours cryptic, procryptic, anticryptic, apatetic, epigamic, sematic, episematic, aposematic, etc., see Poulton’s Colours of Animals (Int. Scientific Series, LXVIII), 1890; cf. also Meldola, R., Variable Protective Colouring in Insects, P.Z.S. 1873, pp. 153–162, etc.

608 Dendy, Evolutionary Biology, p. 336, 1912.

609 Delight in beauty is one of the pleasures of the imagination; there is no limit to its indulgence, and no end to the results which we may ascribe to its exercise. But as for the particular “standard of beauty” which the bird (for instance) admires and selects (as Darwin says in the Origin, p. 70, edit. 1884), we are very much in the dark, and we run the risk of arguing in a circle: for wellnigh all we can safely say is what Addison says (in the 412th Spectator)—that each different species “is most affected with the beauties of its own kind .... Hinc merula in nigro se oblectat nigra marito; ... hinc noctua tetram Canitiem alarum et glaucos miratur ocellos.”

610 Cf. Bridge, T. W., Cambridge Natural History (Fishes), VII, p. 173, 1904; also Frisch, K. v., Ueber farbige Anpassung bei Fische, Zool. Jahrb. (Abt. Allg. Zool.), XXXII, pp. 171–230, 1914.

611 Nature, L, p. 572; LI, pp. 33, 57, 533, 1894–95.

612 They are “wonderfully fitted for ‘vanishment’ against the flushed, rich-coloured skies of early morning and evening .... their chief feeding-times”; and “look like a real sunset or dawn, repeated on the opposite side of the heavens,—either east or west as the case may be”: Thayer, Concealing-coloration in the Animal Kingdom, New York, 1909, pp. 154–155. This hypothesis, like the rest, is not free from difficulty. Twilight is apt to be short in the homes of the flamingo: and moreover, Mr Abel Chapman, who watched them on the Guadalquivir, tells us that they feed by day.

613 Principal Galloway, Philosophy of Religion, p. 344, 1914.

614 Cf. Professor Flint, in his Preface to Affleck’s translation of Janet’s Causes finales: “We are, no doubt, still a long way from a mechanical theory of organic growth, but it may be said to be the quaesitum of modern science, and no one can say that it is a chimaera.”

615 Cf. Sir Donald MacAlister, How a Bone is Built, Engl. Ill. Mag. 1884.

616 Professor Claxton Fidler, On Bridge Construction, p. 22 (4th ed.), 1909; cf. (int. al.) Love’s Elasticity, p. 20 (Historical Introduction), 2nd ed., 1906.

617 In preparing or “macerating” a skeleton, the naturalist nowadays carries on the process till nothing is left but the whitened bones. But the old anatomists, whose object was not the study of “comparative” morphology but the wider theme of comparative physiology, were wont to macerate by easy stages; and in many of their most instructive preparations, the ligaments were intentionally left in connection with the bones, and as part of the “skeleton.”

618 In a few anatomical diagrams, for instance in some of the drawings in Schmaltz’s Atlas der Anatomie des Pferdes, we may see the system of “ties” diagrammatically inserted in the figure of the skeleton. Cf. Gregory, On the principles of Quadrupedal Locomotion, Ann. N. Y. Acad. of Sciences, XXII, p. 289, 1912.

619 Galileo, Dialogues concerning Two New Sciences (1638), Crew and Salvio’s translation, New York, 1914, p. 150; Opere, ed. Favaro, VIII, p. 186. Cf. Borelli, De Motu Animalium, I, prop. CLXXX, 1685. Cf. also Camper, P., La structure des os dans les oiseaux, Opp. III, p. 459, ed. 1803; Rauber, A., Galileo Über Knochenformen, Morphol. Jahrb. VII, pp. 327, 328, 1881; Paolo Enriques, Della economia di sostanza nelle osse cave, Arch. f. Ent. Mech. XX, pp. 427–465, 1906.

620 Das mechanische Prinzip. im anatomischen Bau der Monocotylen, Leipzig, 1874.

621 For further botanical illustrations, see (int. al.) Hegler, Einfluss der Zugkraften auf die Festigkeit und die Ausbildung mechanischer Gewebe in Pflanzen, SB. sÄchs. Ges. d. Wiss. p. 638, 1891; Kny, L., Einfluss von Zug und Druck auf die Richtung der Scheidewande in sich teilenden Pflanzenzellen, Ber. d. bot. Gesellsch. XIV, 1896; Sachs, Mechanomorphose und Phylogenie, Flora, LXXVIII, 1894; cf. also PflÜger, Einwirkung der Schwerkraft, etc., Über die Richtung der Zelltheilung, Archiv, XXXIV, 1884.

622 Among other works on the mechanical construction of bone see: Bourgery, TraitÉ de l’anatomie (I. OstÉologie), 1832 (with admirable illustrations of trabecular structure); Fick, L., Die Ursachen der Knochenformen, GÖttingen, 1857; Meyer, H., Die Architektur der Spongiosa, Archiv f. Anat. und Physiol. XLVII, pp. 615–628, 1867; Statik u. Mechanik des menschlichen KnochengerÜstes, Leipzig, 1873; Wolff, J., Die innere Architektur der Knochen, Arch. f. Anat, und Phys. L, 1870; Das Gesetz der Transformation bei Knochen, 1892; von Ebner, V., Der feinere Bau der Knochensubstanz, Wiener Bericht, LXXII, 1875; Rauber, Anton, ElastizitÄt und Festigkeit der Knochen, Leipzig, 1876; O. Meserer, Elast, u. Festigk. d. menschlichen Knochen, Stuttgart, 1880; MacAlister, Sir Donald, How a Bone is Built, English Illustr. Mag. pp. 640–649, 1884; Rasumowsky, Architektonik des Fussskelets, Int. Monatsschr. f. Anat. p. 197, 1889; Zschokke, Weitere Unters. Über das VerhÄltniss der Knochenbildung zur Statik und Mechanik des Vertebratenskelets, ZÜrich, 1892; Roux, W., Ges. Abhandlungen Über Entwicklungsmechanik der Organismen, Bd. I, Funktionelle Anpassung, Leipzig, 1895; Triepel, H., Die Stossfestigkeit der Knochen, Arch. f. Anat. u. Phys. 1900; Gebhardt, Funktionell wichtige Anordnungsweisen der feineren und grÖberen Bauelemente des Wirbelthierknochens, etc., Arch. f. Entw. Mech. 1900–1910; Kirchner. A., Architektur der Metatarsalien, A. f. E. M. XXIV, 1907; Triepel, Herm., Die trajectorielle Structuren (in Einf. in die Physikalische Anatomie, 1908); Dixon, A. F., Architecture of the Cancellous Tissue forming the Upper End of the Femur, Journ. of Anat. and Phys. (3) XLIV, pp. 223–230, 1910.

623 SÉdillot, De l’influence des fonctions sur la structure et la forme des organes; C. R. LIX, p. 539, 1864; cf. LX, p. 97, 1865, LXVIII. p. 1444. 1869.

624 E.g. (1) the head, nodding backwards and forwards on a fulcrum, represented by the atlas vertebra, lying between the weight and the power; (2) the foot, raising on tip-toe the weight of the body against the fulcrum of the ground, where the weight is between the fulcrum and the power, the latter being represented by the tendo Achillis; (3) the arm, lifting a weight in the hand, with the power (i.e. the biceps muscle) between the fulcrum and the weight. (The second case, by the way, has been much disputed; cf. Haycraft in SchÄfer’s Textbook of Physiology, p. 251, 1900.)

625 Our problem is analogous to Dr Thomas Young’s problem of the best disposition of the timbers in a wooden ship (Phil. Trans. 1814, p. 303). He was not long of finding that the forces which may act upon the fabric are very numerous and very variable, and that the best mode of resisting them, or best structural arrangement for ultimate strength, becomes an immensely complicated problem.

626 In like manner, Clerk Maxwell could not help employing the term “skeleton” in defining the mathematical conception of a “frame,” constituted by points and their interconnecting lines: in studying the equilibrium of which, we consider its different points as mutually acting on each other with forces whose directions are those of the lines joining each pair of points. Hence (says Maxwell), “in order to exhibit the mechanical action of the frame in the most elementary manner, we may draw it as a skeleton, in which the different points are joined by straight lines, and we may indicate by numbers attached to these lines the tensions or compressions in the corresponding pieces of the frame” (Trans. R. S. E. XXVI, p. 1, 1870). It follows that the diagram so constructed represents a “diagram of forces,” in this limited sense that it is geometrical as regards the position and direction of the forces, but arithmetical as regards their magnitude. It is to just such a diagram that the animal’s skeleton tends to approximate.

627 When the jockey crouches over the neck of his race-horse, and when Tod Sloan introduced the “American seat,” the object in both cases is to relieve the hind-legs of weight, and so leave them free for the work of propulsion. Nevertheless, we must not exaggerate the share taken by the hind-limbs in this latter duty; cf. Stillman, The Horse in Motion, p. 69, 1882.

628 This and the following diagrams are borrowed and adapted from Professor Fidler’s Bridge Construction.

629 The method of constructing reciprocal diagrams, in which one should represent the outlines of a frame, and the other the system of forces necessary to keep it in equilibrium, was first indicated in Culmann’s Graphische Statik; it was greatly developed soon afterwards by Macquorn Rankine (Phil. Mag. Feb. 1864, and Applied Mechanics, passim), to whom is mainly due the general application of the principle to engineering practice.

630 Dialogues concerning Two New Sciences (1638): Crew and Salvio’s translation, p. 140 seq.

631 The form and direction of the vertebral spines have been frequently and elaborately described; cf. (e.g.) Gottlieb, H., Die Anticlinie der WirbelsÄule der SÄugethiere, Morphol. Jahrb. LXIX, pp. 179–220, 1915, and many works quoted therein. According to Morita, Ueber die Ursachen der Richtung und Gestalt der thoracalen DornfortsÄtze der SÄugethierwirbelsÄule (ibi cit. p. 201), various changes take place in the direction or inclination of these processes in rabbits, after section of the interspinous ligaments and muscles. These changes seem to be very much what we should expect, on simple mechanical grounds. See also Fischer, O., Theoretische Grundlagen fÜr eine Mechanik der lebenden KÖrper, Leipzig, pp. 3, 372, 1906.

632 I owe the first four of these determinations to the kindness of Dr Chalmers Mitchell, who had them made for me at the Zoological Society’s Gardens; while the great Clydesdale carthorse was weighed for me by a friend in Dundee.

633 This pose of Diplodocus, and of other Sauropodous reptiles, has been much discussed. Cf. (int. al.) Abel, O., Abh. k. k. zool. bot. Ges. Wien, V. 1909–10 (60 pp.); Tornier, SB. Ges. Naturf. Fr. Berlin, pp. 193–209, 1909; Hay, O. P., Amer. Nat. Oct. 1908; Tr. Wash. Acad. Sci. XLII, pp. 1–25, 1910; Holland, Amer. Nat. May, 1910, pp. 259–283; Matthew, ibid. pp. 547–560; Gilmore, C. W. (Restoration of Stegosaurus). Pr. U.S. Nat. Museum, 1915.

634 The form of the cantilever is much less typical in the small flying birds, where the strength of the pelvic region is insured in another way, with which we need not here stop to deal.

635 The motto was Macquorn Rankine’s.

636 John Hunter was seldom wrong; but I cannot believe that he was right when he said (Scientific Works, ed. Owen, I, p. 371), “The bones, in a mechanical view, appear to be the first that are to be considered. We can study their shape, connexions, number, uses, etc., without considering any other part of the body.”

637 Origin of Species, 6th ed. p. 118.

638 Amer. Naturalist, April, 1915, p. 198, etc. Cf. infra, p. 727.

639 Driesch sees in “Entelechy” that something which differentiates the whole from the sum of its parts in the case of the organism: “The organism, we know, is a system the single constituents of which are inorganic in themselves; only the whole constituted by them in their typical order or arrangement owes its specificity to ‘Entelechy’ ” (Gifford Lectures, p. 229, 1908): and I think it could be shewn that many other philosophers have said precisely the same thing. So far as the argument goes, I fail to see how this Entelechy is shewn to be peculiarly or specifically related to the living organism. The conception that the whole is always something very different from its parts is a very ancient doctrine. The reader will perhaps remember how, in another vein, the theme is treated by Martinus Scriblerus: “In every Jack there is a meat-roasting Quality, which neither resides in the fly, nor in the weight, nor in any particular wheel of the Jack, but is the result of the whole composition; etc., etc.”

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.

650 Cf. Sir Thomas Browne, in The Garden of Cyrus: “But why ofttimes one side of the leaf is unequall unto the other, as in Hazell and Oaks, why on either side the master vein the lesser and derivative channels stand not directly opposite, nor at equall angles, respectively unto the adverse side, but those of one side do often exceed the other, as the Wallnut and many more, deserves another enquiry.”

651 Where gourds are common, the glass-blower is still apt to take them for a prototype, as the prehistoric potter also did. For instance, a tall, annulated Florence oil-flask is an exact but no longer a conscious imitation of a gourd which has been converted into a bottle in the manner described.

652 Cf. Elsie Venner, chap. ii.

653 This significance is particularly remarkable in connection with the development of speed, for the metacarpal region is the seat of very important leverage in the propulsion of the body. In the Museum of the Royal College of Surgeons in Edinburgh, there stand side by side the skeleton of an immense carthorse (celebrated for having drawn all the stones of the Bell Rock Lighthouse to the shore), and a beautiful skeleton of a racehorse, which (though the fact is disputed) there is good reason to believe is the actual skeleton of Eclipse. When I was a boy my grandfather used to point out to me that the cannon-bone of the little racer is not only relatively, but actually, longer than that of the great Clydesdale.

654 Cf. Vitruvius, III, 1.

655 Les quatres livres d’Albert DÜrer de la proportion des parties et pourtraicts des corps humains, Arnheim, 1613, folio (and earlier editions). Cf. also Lavater, Essays on Physiognomy, III, p. 271, 1799.

656 It was these very drawings of DÜrer’s that gave to Peter Camper his notion of the “facial angle.” Camper’s method of comparison was the very same as ours, save that he only drew the axes, without filling in the network, of his coordinate system; he saw clearly the essential fact, that the skull varies as a whole, and that the “facial angle” is the index to a general deformation. “The great object was to shew that natural differences might be reduced to rules, of which the direction of the facial line forms the norma or canon; and that these directions and inclinations are always accompanied by correspondent form, size and position of the other parts of the cranium,” etc.; from Dr T. Cogan’s preface to Camper’s work On the Connexion between the Science of Anatomy and the Arts of Drawing, Painting and Sculpture (1768?), quoted in Dr R. Hamilton’s Memoir of Camper, in Lives of Eminent Naturalists (Nat. Libr.), Edin. 1840.

657 The co-ordinate system of Fig. 382 is somewhat different from that which I drew and published in my former paper. It is not unlikely that further investigation will further simplify the comparison, and shew it to involve a still more symmetrical system.

658 Dinosaurs of North America, pl. LXXXI, etc. 1896.

659 Mem. Amer. Mus. of Nat. Hist. I, III, 1898.

660 These and also other coordinate diagrams will be found in Mr G. Heilmann’s book Fuglenes Afstamning, 398 pp., Copenhagen, 1916; see especially pp. 368–380.

661 Cf. W. B. Scott (Amer. Journ. of Science, XLVIII, pp. 335–374, 1894), “We find that any mammalian series at all complete, such as that of the horses, is remarkably continuous, and that the progress of discovery is steadily filling up what few gaps remain. So closely do successive stages follow upon one another that it is sometimes extremely difficult to arrange them all in order, and to distinguish clearly those members which belong in the main line of descent, and those which represent incipient branches. Some phylogenies actually suffer from an embarrassment of riches.”

662 Cf. Dwight, T., The Range of Variation of the Human Scapula, Amer. Nat. XXI, pp. 627–638, 1887. Cf. also Turner, Challenger Rep. XLVII, on Human Skeletons, p. 86, 1886: “I gather both from my own measurements, and those of other observers, that the range of variation in the relative length and breadth of the scapula is very considerable in the same race, so that it needs a large number of bones to enable one to obtain an accurate idea of the mean of the race.”

663 There is a paper on the mathematical study of organic forms and organic processes by the learned and celebrated Gustav Theodor Fechner, which I have only lately read, but which would have been of no little use and help to our argument had I known it before. (Ueber die mathematische Behandlung organischer Gestalten und Processe, Berichte d. k. sÄchs. Gesellsch., Math.-phys. Cl., Leipzig, 1849, pp. 50–64.) Fechner’s treatment is more purely mathematical and less physical in its scope and bearing than ours, and his paper is but a short one; but the conclusions to which he is led differ little from our own. Let me quote a single sentence which, together with its context, runs precisely on the lines of the discussion with which this chapter of ours began. “So ist also die mathematische Bestimmbarkeit im Gebiete des Organischen ganz eben so gut vorhanden als in dem des Unorganischen, und in letzterem eben solchen oder Äquivalenten BeschrÄnkungen unterworfen als in ersterem; und nur sofern die unorganischen Formen und das unorganische Geschehen sich einer einfacheren Gesetzlichkeit mehr nÄhern als die organischen, kann die Approximation im unorganischen Gebiet leichter und weiter getrieben werden als im organischen. Dies wÄre der ganze, sonach rein relative, Unterschied.” Here in a nutshell, in words written some seventy years ago, is the gist of the whole matter.

An interesting little book of Schiaparelli’s (which I ought to have known long ago)—Forme organiche naturali e forme geometriche pure, Milano, Hoepli, 1898—has likewise come into my hands too late for discussion.

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