relative proportions in length and breadth of the leaves themselves; or, more strictly speaking, on the ratio of the diagonals of the rhomboidal figure by which each leaf-area is circumscribed. When, as in the fir-cone, the scales by mutual compression conform to these rhomboidal outlines, their inclined edges at once guide the eye in the direction of some one particular spiral; and we shall not fail to notice that in such cases the usual {649} result is to give us arrangements corresponding to the middle diagrams in Fig. [327], which are the configurations in which the quadrilateral outlines approach most nearly to a rectangular form, and give us accordingly the least possible ratio (under the given conditions) of sectional boundary-wall to surface area.
The manner in which one system of spirals may be caused to slide, so to speak, into another, has been ingeniously demonstrated by Schwendener on a mechanical model, consisting essentially of a framework which can be opened or closed to correspond with one after another of the above series of diagrams[583].
The determination of the precise angle of divergence of two consecutive leaves of the generating spiral does not enter into the above general investigation (though Tait gives, in the same paper, a method by which it may be easily determined); and the very fact that it does not so enter shews it to be essentially unimportant. The determination of so-called “orthostichies,” or precisely vertical successions of leaves, is also unimportant. We have no means, other than observation, of determining that one leaf is vertically above another, and spiral series such as we have been dealing with will appear, whether such orthostichies exist, whether they be near or remote, or whether the angle of divergence be such that no precise vertical superposition ever occurs. And lastly, the fact that the successional numbers, expressed as fractions, 1 ⁄ 2, 2 ⁄ 3, 3 ⁄ 5, represent a convergent series, whose final term is equal to 0·61803..., the sectio aurea or “golden mean” of unity, is seen to be a mathematical coincidence, devoid of biological significance; it is but a particular case of Lagrange’s theorem that the roots of every numerical equation of the second degree can be expressed by a periodic continued fraction. The same number has a multitude of curious arithmetical properties. It is the final term of all similar series to that with which we have been dealing, such for instance as 1 ⁄ 3, 3 ⁄ 4, 4 ⁄ 7, etc., or 1 ⁄ 4, 4 ⁄ 5, 5 ⁄ 9, etc. It is a number beloved of the circle-squarer, and of all those who seek to find, and then to penetrate, the secrets of the Great Pyramid. It is deep-set in Pythagorean as well as in Euclidean geometry. It enters (as the chord of an angle of 36°), {650} into the thrice-isosceles triangle of which we have spoken on p. [511]; it is a number which becomes (by the addition of unity) its own reciprocal; its properties never end. To Kepler (as Naber tells us) it was a symbol of Creation, or Generation. Its recent application to biology and art-criticism by Sir Theodore Cook and others is not new. Naber’s book, already quoted, is full of it. Zeising, in 1854, found in it the key to all morphology, and the same writer, later on[584], declared it to dominate both architecture and music. But indeed, to use Sir Thomas Browne’s words (though it was of another number that he spoke): “To enlarge this contemplation into all the mysteries and secrets accommodable unto this number, were inexcusable Pythagorisme.”
If this number has any serious claim at all to enter into the biological question of phyllotaxis, this must depend on the fact, first emphasized by Chauncey Wright[585], that, if the successive leaves of the fundamental spiral be placed at the particular azimuth which divides the circle in this “sectio aurea,” then no two leaves will ever be superposed; and thus we are said to have “the most thorough and rapid distribution of the leaves round the stem, each new or higher leaf falling over the angular space between the two older ones which are nearest in direction, so as to divide it in the same ratio (K), in which the first two or any two successive ones divide the circumference. Now 5 ⁄ 8 and all successive fractions differ inappreciably from K.” To this view there are many simple objections. In the first place, even 5 ⁄ 8, or ·625, is but a moderately close approximation to the “golden mean”; in the second place the arrangements by which a better approximation is got, such as 8 ⁄ 13, 13 ⁄ 21, and the very close approximations such as 34 ⁄ 55, 55 ⁄ 89, 89 ⁄ 144, etc., are comparatively rare, while the much less close approximations of 3 ⁄ 5 or 2 ⁄ 3, or even 1 ⁄ 2, are extremely common. Again, the general type of argument such as that which asserts that the plant is “aiming at” something which we may call an “ideal angle” is one that cannot commend itself to a plain student of physical science: nor is the hypothesis rendered more acceptably when Sir T. Cook qualifies it by telling us that “all that a plant can do {651} is to vary, to make blind shots at constructions, or to ‘mutate’ as it is now termed; and the most suitable of these constructions will in the long run be isolated by the action of Natural Selection.” Finally, and this is the most concrete objection of all, the supposed isolation of the leaves, or their most complete “distribution to the action of the surrounding atmosphere” is manifestly very little affected by any conditions which are confined to the angle of azimuth. If we could imagine a case in which all the leaves of the stem, or all the scales of a fir-cone, were crushed down to one and the same level, into a simple ring or whorl of leaves, then indeed they would have their most equable distribution under the condition of the “ideal angle,” that is to say of the “golden mean.” But if it be (so to speak) Nature’s object to set them further apart than they actually are, to give them freer exposure to the air than they actually have, then it is surely manifest that the simple way to do so is to elongate the axis, and to set the leaves further apart, lengthways on the stem. This has at once a far more potent effect than any nice manipulation of the “angle of divergence.” For it is obvious that in F(ϕ · sin θ) we have a greater range of variation by altering θ than by altering ϕ. We come then, without more ado, to the conclusion that the “Fibonacci series,” and its supposed usefulness, and the hypothesis of its introduction into plant-structure through natural selection, are all matters which deserve no place in the plain study of botanical phenomena. As Sachs shrewdly recognised years ago, all such speculations as these hark back to a school of mystical idealism.
CHAPTER XV ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW STRUCTURES
The eggs of birds and all other hard-shelled eggs, such as those of the tortoise and the crocodile, are simple solids of revolution; but they differ greatly in form, according to the configuration of the plane curve by the revolution of which the egg is, in a mathematical sense, generated. Some few eggs, such as those of the owl, the penguin, or the tortoise, are spherical or very nearly so; a few more, such as the grebe’s, the cormorant’s or the pelican’s, are approximately ellipsoidal, with symmetrical or nearly symmetrical ends, and somewhat similar are the so-called “cylindrical” eggs of the megapodes and the sand-grouse; the great majority, like the hen’s egg, are ovoid, a little blunter at one end than the other; and some, by an exaggeration of this lack of antero-posterior symmetry, are blunt at one end but characteristically pointed at the other, as is the case with the eggs of the guillemot and puffin, the sandpiper, plover and curlew. It is an obvious but by no means negligible fact that the egg, while often pointed, is never flattened or discoidal; it is a prolate, but never an oblate, spheroid.
The careful study and collection of birds’ eggs would seem to have begun with the Count de Marsigli[586], the same celebrated naturalist who first studied the “flowers” of the coral, and who wrote the Histoire physique de la mer; and the specific form, as well as the colour and other attributes of the egg have been again and again discussed, and not least by the many dilettanti naturalists of the eighteenth century who soon followed in Marsigli’s footsteps[587]. {653}
We need do no more than mention Aristotle’s belief, doubtless old in his time, that the more pointed egg produces the male chicken, and the blunter egg the hen; though this theory survived into modern times[588] and perhaps still lingers on. Several naturalists, such as Günther (1772) and Bühle (1818), have taken the trouble to disprove it by experiment. A more modern and more generally accepted explanation has been that the form of the egg is in direct relation to that of the bird which has to be hatched within—a view that would seem to have been first set forth by Naumann and Bühle, in their great treatise on eggs[589], and adopted by Des Murs[590] and many other well-known writers.
In a treatise by de Lafresnaye[591], an elaborate comparison is made between the skeleton and the egg of the various birds, to shew, for instance, how those birds with a deep-keeled sternum laid rounded eggs, which alone could accommodate the form of the young. According to this view, that “Nature had foreseen[592]” the form adapted to and necessary for the growing embryo, it was easy to correlate the owl with its spherical egg, the diver with its elliptical one, and in like manner the round egg of the tortoise and the elongated one of the crocodile with the shape of the creatures which had afterwards to be hatched therein. A few writers, such as Thienemann[593], looked at the same facts the other way, and asserted that the form of the egg was determined by that of the bird by which it was laid, and in whose body it had been conformed.
In more recent times, other theories, based upon the principles of Natural Selection, have been current and very generally accepted, to account for these diversities of form. The pointed, conical egg of the guillemot is generally supposed to be an adaptation, {654} advantageous to the species in the circumstances under which the egg is laid; the pointed egg is less apt than a spherical one to roll off the narrow ledge of rock on which this bird is said to lay its solitary egg, and the more pointed the egg, so much the fitter and likelier is it to survive. The fact that the plover or the sandpiper, breeding in very different situations, lay eggs that are also conical, elicits another explanation, to the effect that here the conical form permits the many large eggs to be packed closely under the mother bird[594]. Whatever truth there be in these apparent adaptations to existing circumstances, it is only by a very hasty logic that we can accept them as a vera causa, or adequate explanation of the facts; and it is obvious that, in the bird’s egg, we have an admirable case for the direct investigation of the mechanical or physical significance of its form[595].