Chapter XXXVIII: Epilogue (3)

[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_{2}SO_{4} + 2 H_{2}O).

[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 τεχνικὴ φύσις, 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: Πλάτων δὲ καὶ ἐν τούτοις πυθαγορίζει.

[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 + √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 (θ). 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 ε^{2π sin θ cot α}, θ 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. 15_a_–_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 œufs 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_ + _mr′_ = _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.”