Better still. Let us imagine a plane perpendicular to the aids of the shell and passing through its summit. Let us imagine, moreover, a thread wound along the spiral groove. Let us unroll the thread, holding it taut as we do so. Its extremity will not leave the plane and will describe a logarithmic spiral within it. It is, in a more complicated degree, a variant of Bernouilli's '_Eadem mutata resurgo_:' the logarithmic conic curve becomes a logarithmic plane curve.
A similar geometry is found in the other shells with elongated cones, Turritellae, Spindle-shells, Cerithia, as well as in the shells with flattened cones, Trochidae, Turbines. The spherical shells, those whirled into a volute, are no exception to this rule. All, down to the common Snail-shell, are constructed according to logarithmic laws. The famous spiral of the geometers is the general plan followed by the Mollusc rolling its stone sheath.
Where do these glairy creatures pick up this science? We are told that the Mollusc derives from the Worm. One day, the Worm, rendered frisky by the sun, emancipated itself, brandished its tail and twisted it into a corkscrew for sheer glee. There and then the plan of the future spiral shell was discovered.
This is what is taught quite seriously, in these days, as the very last word in scientific progress. It remains to be seen up to what point the explanation is acceptable. The Spider, for her part, will have none of it. Unrelated to the appendix-lacking, corkscrew-twirling Worm, she is nevertheless familiar with the logarithmic spiral. From the celebrated curve she obtains merely a sort of framework; but, elementary though this framework be, it clearly marks the ideal edifice. The Epeira works on the same principles as the Mollusc of the convoluted shell.
The Mollusc has years wherein to construct its spiral and it uses the utmost finish in the whirling process. The Epeira, to spread her net, has but an hour's sitting at the most, wherefore the speed at which she works compels her to rest content with a simpler production. She shortens the task by confining herself to a skeleton of the curve which the other describes to perfection.
The Epeira, therefore, is versed in the geometric secrets of the Ammonite and the _Nautilus pompilus_; she uses, in a simpler form, the logarithmic line dear to the Snail. What guides her? There is no appeal here to a wriggle of some kind, as in the case of the Worm that ambitiously aspires to become a Mollusc. The animal must needs carry within itself a virtual diagram of its spiral. Accident, however fruitful in surprises we may presume it to be, can never have taught it the higher geometry wherein our own intelligence at once goes astray, without a strict preliminary training.
Are we to recognize a mere effect of organic structure in the Epeira's art? We readily think of the legs, which, endowed with a very varying power of extension, might serve as compasses. More or less bent, more or less outstretched, they would mechanically determine the angle whereat the spiral shall intersect the radius; they would maintain the parallel of the chords in each sector.
Certain objections arise to affirm that, in this instance, the tool is not the sole regulator of the work. Were the arrangement of the thread determined by the length of the legs, we should find the spiral volutes separated more widely from one another in proportion to the greater length of implement in the spinstress. We see this in the Banded Epeira and the Silky Epeira. The first has longer limbs and spaces her cross- threads more liberally than does the second, whose legs are shorter.
But we must not rely too much on this rule, say others. The Angular Epeira, the Paletinted Epeira and the Cross Spider, all three more or less short-limbed, rival the Banded Epeira in the spacing of their lime- snares. The last two even dispose them with greater intervening distances.
We recognize in another respect that the organization of the animal does not imply an immutable type of work. Before beginning the sticky spiral, the Epeirae first spin an auxiliary intended to strengthen the stays.
This spiral, formed of plain, non-glutinous thread, starts from the centre and winds in rapidly-widening circles to the circumference. It is merely a temporary construction, whereof naught but the central part survives when the Spider has set its limy meshes. The second spiral, the essential part of the snare, proceeds, on the contrary, in serried coils from the circumference to the centre and is composed entirely of viscous cross-threads.
Here we have, following one after the other merely by a sudden alteration of the machine, two volutes of an entirely different order as regards direction, the number of whorls and intersection. Both of them are logarithmic spirals. I see no mechanism of the legs, be they long or short, that can account for this alteration.
Can it then be a premeditated design on the part of the Epeira? Can there be calculation, measurement of angles, gauging of the parallel by means of the eye or otherwise? I am inclined to think that there is none of all this, or at least nothing but an innate propensity, whose effects the animal is no more able to control than the flower is able to control the arrangement of its verticils. The Epeira practises higher geometry without knowing or caring. The thing works of itself and takes its impetus from an instinct imposed upon creation from the start.
The stone thrown by the hand returns to earth describing a certain curve; the dead leaf torn and wafted away by a breath of wind makes its journey from the tree to the ground with a similar curve. On neither the one side nor the other is there any action by the moving body to regulate the fall; nevertheless, the descent takes place according to a scientific trajectory, the 'parabola,' of which the section of a cone by a plane furnished the prototype to the geometer's speculations. A figure, which was at first but a tentative glimpse, becomes a reality by the fall of a pebble out of the vertical.
The same speculations take up the parabola once more, imagine it rolling on an indefinite straight line and ask what course does the focus of this curve follow. The answer comes: The focus of the parabola describes a 'catenary,' a line very simple in shape, but endowed with an algebraic symbol that has to resort to a kind of cabalistic number at variance with any sort of numeration, so much so that the unit refuses to express it, however much we subdivide the unit. It is called the number _e_. Its value is represented by the following series carried out ad infinitum:
e = 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) + 1/(1*2*3*4*5) + etc
If the reader had the patience to work out the few initial terms of this series, which has no limit, because the series of natural numerals itself has none, he would find:
e=2.7182818...
With this weird number are we now stationed within the strictly defined realm of the imagination? Not at all: the catenary appears actually every time that weight and flexibility act in concert. The name is given to the curve formed by a chain suspended by two of its points which are not placed on a vertical line. It is the shape taken by a flexible cord when held at each end and relaxed; it is the line that governs the shape of a sail bellying in the wind; it is the curve of the nanny-goat's milk- bag when she returns from filling her trailing udder. And all this answers to the number e.
What a quantity of abstruse science for a bit of string! Let us not be surprised. A pellet of shot swinging at the end of a thread, a drop of dew trickling down a straw, a splash of water rippling under the kisses of the air, a mere trifle, after all, requires a titanic scaffolding when we wish to examine it with the eye of calculation. We need the club of Hercules to crush a fly.
Our methods of mathematical investigation are certainly ingenious; we cannot too much admire the mighty brains that have invented them; but how slow and laborious they appear when compared with the smallest actualities! Will it never be given to us to probe reality in a simpler fashion? Will our intelligence be able one day to dispense with the heavy arsenal of formulae? Why not?
Here we have the abracadabric number _e_ reappearing, inscribed on a Spider's thread. Let us examine, on a misty morning, the meshwork that has been constructed during the night. Owing to their hygrometrical nature, the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chaplets of limpid gems, graceful chaplets arranged in exquisite order and following the curve of a swing. If the sun pierce the mist, the whole lights up with iridescent fires and becomes a resplendent cluster of diamonds. The number _e_ is in its glory.
Geometry, that is to say, the science of harmony in space, presides over everything. We find it in the arrangement of the scales of a fir-cone, as in the arrangement of an Epeira's limy web; we find it in the spiral of a Snail-shell, in the chaplet of a Spider's thread, as in the orbit of a planet; it is everywhere, as perfect in the world of atoms as in the world of immensities.
And this universal geometry tells us of an Universal Geometrician, whose divine compass has measured all things. I prefer that, as an explanation of the logarithmic curve of the Ammonite and the Epeira, to the Worm screwing up the tip of its tail. It may not perhaps be in accordance with latter-day teaching, but it takes a loftier flight.
FOOTNOTES
{1} A small or moderate-sized spider found among foliage.--Translator's Note.
{2} Leon Dufour (1780-1865) was an army surgeon who served with distinction in several campaigns and subsequently practised as a doctor in the Landes. He attained great eminence as a naturalist.--Translator's Note.
{3} The Tarantula is a Lycosa, or Wolf-spider. Fabre's Tarantula, the Black-bellied Tarantula, is identical with the Narbonne Lycosa, under which name the description is continued in Chapters iii. to vi., all of which were written at a considerably later date than the present chapter.--Translator's Note.
{4} Giorgio Baglivi (1669-1707), professor of anatomy and medicine at Rome.--Translator's Note.
{5} 'When our husbandmen wish to catch them, they approach their hiding- places, and play on a thin grass pipe, making a sound not unlike the humming of bees. Hearing which, the Tarantula rushes out fiercely that she may catch the flies or other insects of this kind, whose buzzing she thinks it to be; but she herself is caught by her rustic trapper.'
{6} Provencal for the bit of waste ground on which the author studies his insects in the natural state.--Translator's note.
{7} 'Thanks to the Bumble-bee.'
{8} Like the Dung-beetles.--Translator's Note.
{9} Like the Solitary Wasps.--Translator's Note.
{10} Such as the Hairy Ammophila, the Cerceris and the Languedocian Sphex, Digger-wasps described in other of the author's essays.--Translator's Note.
{11} The _desnucador_, the Argentine slaughterman whose methods of slaying cattle are detailed in the author's essay entitled, The Theory of Instinct.--Translator's Note.
{12} A family of Grasshoppers.--Translator's Note.
{13} A genus of Beetles.--Translator's Note.
{14} A species of Digger-wasp.--Translator's Note.
{15} The Cicada is the _Cigale_, an insect akin to the Grasshopper and found more particularly in the South of France.--Translator's Note.
{16} The generic title of the work from which these essays are taken is Entomological Memories, or, Studies relating to the Instinct and Habits of Insects.--Translator's Note.
{17} A species of Grasshopper.--Translator's Note.
{18} An insect akin to the Locusts and Crickets, which, when at rest, adopts an attitude resembling that of prayer. When attacking, it assumes what is known as 'the spectral attitude.' Its forelegs form a sort of saw-like or barbed harpoons. Cf. Social Life in the Insect World, by J.
H. Fabre, translated by Bernard Miall: chaps. v. to vii.-- Translator's Note.
{19} .39 inch.-- Translator's Note.
{20} These experiments are described in the author's essay on the Mason Bees entitled Fragments on Insect Psychology.--Translator's Note.
{21} A species of Wasp.--Translator's Note.
{22} In Chap. VIII. of the present volume.--Translator's Note.
{23} Jules Michelet (1798-1874), author of L'Oiseau and L'Insecte, in addition to the historical works for which he is chiefly known. As a lad, he helped his father, a printer by trade, in setting type.--Translator's Note.