/> Others, among the naturalists, benefit by the designation given to this

or that object in life's treasure-house: it is the skiff wherein they

keep afloat for a brief while. A patch of lichen on the bark of an old

tree, a blade of grass, a puny beastie: any one of these hands down a

man's name to posterity as effectively as a new comet. For all its

abuses, this manner of honouring the departed is eminently respectable.

If we would carve an epitaph of some duration, what could we find better

than a Beetle's wing-case, a Snail's shell or a Spider's web? Granite is

worth none of them. Entrusted to the hard stone, an inscription becomes

obliterated; entrusted to a Butterfly's wing, it is indestructible.

'Durand,' therefore, by all means.



But why drag in 'Clotho'? Is it the whim of a nomenclator, at a loss for

words to denote the ever-swelling tide of beasts that require

cataloguing? Not entirely. A mythological name came to his mind, one

which sounded well and which, moreover, was not out of place in

designating a spinstress. The Clotho of antiquity is the youngest of the

three Fates; she holds the distaff whence our destinies are spun, a

distaff wound with plenty of rough flocks, just a few shreds of silk and,

very rarely, a thin strand of gold.



Prettily shaped and clad, as far as a Spider can be, the Clotho of the

naturalists is, above all, a highly talented spinstress; and this is the

reason why she is called after the distaff-bearing deity of the infernal

regions. It is a pity that the analogy extends no further. The

mythological Clotho, niggardly with her silk and lavish with her coarse

flocks, spins us a harsh existence; the eight-legged Clotho uses naught

but exquisite silk. She works for herself; the other works for us, who

are hardly worth the trouble.



Would we make her acquaintance? On the rocky slopes in the oliveland,

scorched and blistered by the sun, turn over the flat stones, those of a

fair size; search, above all, the piles which the shepherds set up for a

seat whence to watch the sheep browsing amongst the lavender below. Do

not be too easily disheartened: the Clotho is rare; not every spot suits

her. If fortune smile at last upon our perseverance, we shall see,

clinging to the lower surface of the stone which we have lifted, an

edifice of a weather-beaten aspect, shaped like an over-turned cupola and

about the size of half a tangerine orange. The outside is encrusted or

hung with small shells, particles of earth and, especially, dried

insects.



The edge of the cupola is scalloped into a dozen angular lobes, the

points of which spread and are fixed to the stone. In between these

straps is the same number of spacious inverted arches. The whole

represents the Ishmaelite's camel-hair tent, but upside down. A flat

roof, stretched between the straps, closes the top of the dwelling.



Then where is the entrance? All the arches of the edge open upon the

roof; not one leads to the interior. The eye seeks in vain; there is

nothing to point to a passage between the inside and the outside. Yet

the owner of the house must go out from time to time, were it only in

search of food; on returning from her expedition, she must go in again.

How does she make her exits and her entrances? A straw will tell us the

secret.



Pass it over the threshold of the various arches. Everywhere, the

searching straw encounters resistance; everywhere, it finds the place

rigorously closed. But one of the scallops, differing in no wise from

the others in appearance, if cleverly coaxed, opens at the edge into two

lips and stands slightly ajar. This is the door, which at once shuts

again of its own elasticity. Nor is this all: the Spider, when she

returns home, often bolts herself in, that is to say, she joins and

fastens the two leaves of the door with a little silk.



The Mason Mygale is no safer in her burrow, with its lid

undistinguishable from the soil and moving on a hinge, than is the Clotho

in her tent, which is inviolable by any enemy ignorant of the device. The

Clotho, when in danger, runs quickly home; she opens the chink with a

touch of her claw, enters and disappears. The door closes of itself and

is supplied, in case of need, with a lock consisting of a few threads. No

burglar, led astray by the multiplicity of arches, one and all alike,

will ever discover how the fugitive vanished so suddenly.



While the Clotho displays a more simple ingenuity as regards her

defensive machinery, she is incomparably ahead of the Mygale in the

matter of domestic comfort. Let us open her cabin. What luxury! We are

taught how a Sybarite of old was unable to rest, owing to the presence of

a crumpled rose-leaf in his bed. The Clotho is quite as fastidious. Her

couch is more delicate than swan's-down and whiter than the fleece of the

clouds where brood the summer storms. It is the ideal blanket. Above is

a canopy or tester of equal softness. Between the two nestles the

Spider, short-legged, clad in sombre garments, with five yellow favours

on her back.



Rest in this exquisite retreat demands perfect stability, especially on

gusty days, when sharp draughts penetrate beneath the stone. This

condition is admirably fulfilled. Take a careful look at the habitation.

The arches that gird the roof with a balustrade and bear the weight of

the edifice are fixed to the slab by their extremities. Moreover, from

each point of contact, there issues a cluster of diverging threads that

creep along the stone and cling to it throughout their length, which

spreads afar. I have measured some fully nine inches long. These are so

many cables; they represent the ropes and pegs that hold the Arab's tent

in position. With such supports as these, so numerous and so

methodically arranged, the hammock cannot be torn from its bearings save

by the intervention of brutal methods with which the Spider need not

concern herself, so seldom do they occur.



Another detail attracts our attention: whereas the interior of the house

is exquisitely clean, the outside is covered with dirt, bits of earth,

chips of rotten wood, little pieces of gravel. Often there are worse

things still: the exterior of the tent becomes a charnel-house. Here,

hung up or embedded, are the dry carcasses of Opatra, Asidae and other

Tenebrionidae {39} that favour underrock shelters; segments of Iuli, {40}

bleached by the sun; shells of Pupae, {41} common among the stones; and,

lastly, Snail-shells, selected from among the smallest.



These relics are obviously, for the most part, table-leavings, broken

victuals. Unversed in the trapper's art, the Clotho courses her game and

lives upon the vagrants who wander from one stone to another. Whoso

ventures under the slab at night is strangled by the hostess; and the

dried-up carcass, instead of being flung to a distance, is hung to the

silken wall, as though the Spider wished to make a bogey-house of her

home. But this cannot be her aim. To act like the ogre who hangs his

victims from the castle battlements is the worst way to disarm suspicion

in the passers-by whom you are lying in wait to capture.



There are other reasons which increase our doubts. The shells hung up

are most often empty; but there are also some occupied by the Snail,

alive and untouched. What can the Clotho do with a _Pupa cinerea_, a

_Pupa quadridens_ and other narrow spirals wherein the animal retreats to

an inaccessible depth? The Spider is incapable of breaking the

calcareous shell or of getting at the hermit through the opening. Then

why should she collect those prizes, whose slimy flesh is probably not to

her taste? We begin to suspect a simple question of ballast and balance.

The House Spider, or _Tegenaria domestica_, prevents her web, spun in a

corner of the wall, from losing its shape at the least breath of air, by

loading it with crumbling plaster and allowing tiny fragments of mortar

to accumulate. Are we face to face with a similar process? Let us try

experiment, which is preferable to any amount of conjecture.



To rear the Clotho is not an arduous undertaking; we are not obliged to

take the heavy flagstone, on which the dwelling is built, away with us. A

very simple operation suffices. I loosen the fastenings with my pocket-

knife. The Spider has such stay-at-home ways that she very rarely makes

off. Besides, I use the utmost discretion in my rape of the house. And

so I carry away the building, together with its owner, in a paper bag.



The flat stones, which are too heavy to move and which would occupy too

much room upon my table, are replaced either by deal disks, which once

formed part of cheese-boxes, or by round pieces of cardboard. I arrange

each silken hammock under one of these by itself, fastening the angular

projections, one by one, with strips of gummed paper. The whole stands

on three short pillars and gives a very fair imitation of the underrock

shelter in the form of a small dolmen. Throughout this operation, if you

are careful to avoid shocks and jolts, the Spider remains indoors.

Finally, each apparatus is placed under a wire-gauze, bell-shaped cage,

which stands in a dish filled with sand.



We can have an answer by the next morning. If, among the cabins swung

from the ceilings of the deal or cardboard dolmens, there be one that is

all dilapidated, that was seriously knocked out of shape at the time of

removal, the Spider abandons it during the night and instals herself

elsewhere, sometimes even on the trellis-work of the wire cage.



The new tent, the work of a few hours, attains hardly the diameter of a

two-franc piece. It is built, however, on the same principles as the old

manor-house and consists of two thin sheets laid one above the other, the

upper one flat and forming a tester, the lower curved and pocket-shaped.

The texture is extremely delicate: the least trifle would deform it, to

the detriment of the available space, which is already much reduced and

only just sufficient for the recluse.



Well, what has the Spider done to keep the gossamer stretched, to steady

it and to make it retain its greatest capacity? Exactly what our static

treatises would advise her to do: she has ballasted her structure, she

has done her best to lower its centre of gravity. From the convex

surface of the pocket hang long chaplets of grains of sand strung

together with slender silken cords. To these sandy stalactites, which

form a bushy beard, are added a few heavy lumps hung separately and lower

down, at the end of a thread. The whole is a piece of ballast-work, an

apparatus for ensuring equilibrium and tension.



The present edifice, hastily constructed in the space of a night, is the

frail rough sketch of what the home will afterwards become. Successive

layers will be added to it; and the partition-wall will grow into a thick

blanket capable of partly retaining, by its own weight, the requisite

curve and capacity. The Spider now abandons the stalactites of sand,

which were used to keep the original pocket stretched, and confines

herself to dumping down on her abode any more or less heavy object,

mainly corpses of insects, because she need not look for these and finds

them ready to hand after each meal. They are weights, not trophies; they

take the place of materials that must otherwise be collected from a

distance and hoisted to the top. In this way, a breastwork is obtained

that strengthens and steadies the house. Additional equilibrium is often

supplied by tiny shells and other objects hanging a long way down.



What would happen if one robbed an old dwelling, long since completed, of

its outer covering? In case of such a disaster, would the Spider go back

to the sandy stalactites, as a ready means of restoring stability? This

is easily ascertained. In my hamlets under wire, I select a fair-sized

cabin. I strip the exterior, carefully removing any foreign body. The

silk reappears in its original whiteness. The tent looks magnificent,

but seems to me too limp.



This is also the Spider's opinion. She sets to work, next evening, to

put things right. And how? Once more with hanging strings of sand. In

a few nights, the silk bag bristles with a long, thick beard of

stalactites, a curious piece of work, excellently adapted to maintain the

web in an unvaried curve. Even so are the cables of a suspension-bridge

steadied by the weight of the superstructure.



Later, as the Spider goes on feeding, the remains of the victuals are

embedded in the wall, the sand is shaken and gradually drops away and the

home resumes its charnel-house appearance. This brings us to the same

conclusion as before: the Clotho knows her statics; by means of

additional weights, she is able to lower the centre of gravity and thus

to give her dwelling the proper equilibrium and capacity.



Now what does she do in her softly-wadded home? Nothing, that I know of.

With a full stomach, her legs luxuriously stretched over the downy

carpet, she does nothing, thinks of nothing; she listens to the sound of

earth revolving on its axis. It is not sleep, still less is it waking;

it is a middle state where naught prevails save a dreamy consciousness of

well-being. We ourselves, when comfortably in bed, enjoy, just before we

fall asleep, a few moments of bliss, the prelude to cessation of thought

and its train of worries; and those moments are among the sweetest in our

lives. The Clotho seems to know similar moments and to make the most of

them.



If I push open the door of the cabin, invariably I find the Spider lying

motionless, as though in endless meditation. It needs the teasing of a

straw to rouse her from her apathy. It needs the prick of hunger to

bring her out of doors; and, as she is extremely temperate, her

appearances outside are few and far between. During three years of

assiduous observation, in the privacy of my study, I have not once seen

her explore the domain of the wire cage by day. Not until a late hour at

night does she venture forth in quest of victuals; and it is hardly

feasible to follow her on her excursions.



Patience once enabled me to find her, at ten o'clock in the evening,

taking the air on the flat roof of her house, where she was doubtless

waiting for the game to pass. Startled by the light of my candle, the

lover of darkness at once returned indoors, refusing to reveal any of her

secrets. Only, next day, there was one more corpse hanging from the wall

of the cabin, a proof that the chase was successfully resumed after my

departure.



The Clotho, who is not only nocturnal, but also excessively shy, conceals

her habits from us; she shows us her works, those precious historical

documents, but hides her actions, especially the laying, which I estimate

approximately to take place in October. The sum total of the eggs is

divided into five or six small, flat, lentiform pockets, which, taken

together, occupy the greater part of the maternal home. These capsules

have each their own partition-wall of superb white satin, but they are so

closely soldered, both together and to the floor of the house, that it is

impossible to part them without tearing them, impossible, therefore, to

obtain them separately. The eggs in all amount to about a hundred.



The mother sits upon the heap of pockets with the same devotion as a

brooding hen. Maternity has not withered her. Although decreased in

bulk, she retains an excellent look of health; her round belly and her

well-stretched skin tell us from the first that her part is not yet

wholly played.



The hatching takes place early. November has not arrived before the

pockets contain the young: wee things clad in black, with five yellow

specks, exactly like their elders. The new-born do not leave their

respective nurseries. Packed close together, they spend the whole of the

wintry season there, while the mother, squatting on the pile of cells,

watches over the general safety, without knowing her family other than by

the gentle trepidations felt through the partitions of the tiny chambers.

The Labyrinth Spider has shown us how she maintains a permanent sitting

for two months in her guard-room, to defend, in case of need, the brood

which she will never see. The Clotho does the same during eight months,

thus earning the right to set eyes for a little while on her family

trotting around her in the main cabin and to assist at the final exodus,

the great journey undertaken at the end of a thread.



When the summer heat arrives, in June, the young ones, probably aided by

their mother, pierce the walls of their cells, leave the maternal tent,

of which they know the secret outlet well, take the air on the threshold

for a few hours and then fly away, carried to some distance by a

funicular aeroplane, the first product of their spinning-mill.



The elder Clotho remains behind, careless of this emigration which leaves

her alone. She is far from being faded indeed, she looks younger than

ever. Her fresh colour, her robust appearance suggest great length of

life, capable of producing a second family. On this subject I have but

one document, a pretty far-reaching one, however. There were a few

mothers whose actions I had the patience to watch, despite the wearisome

minutiae of the rearing and the slowness of the result. These abandoned

their dwellings after the departure of their young; and each went to

weave a new one for herself on the wire net-work of the cage.



They were rough-and-ready summaries, the work of a night. Two hangings,

one above the other, the upper one flat, the lower concave and ballasted

with stalactites of grains of sand, formed the new home, which,

strengthened daily by fresh layers, promised to become similar to the old

one. Why does the Spider desert her former mansion, which is in no way

dilapidated--far from it--and still exceedingly serviceable, as far as

one can judge? Unless I am mistaken, I think I have an inkling of the

reason.



The old cabin, comfortably wadded though it be, possesses serious

disadvantages: it is littered with the ruins of the children's nurseries.

These ruins are so close-welded to the rest of the home that my forceps

cannot extract them without difficulty; and to remove them would be an

exhausting business for the Clotho and possibly beyond her strength. It

is a case of the resistance of Gordian knots, which not even the very

spinstress who fastened them is capable of untying. The encumbering

litter, therefore, will remain.



If the Spider were to stay alone, the reduction of space, when all is

said, would hardly matter to her: she wants so little room, merely enough

to move in! Besides, when you have spent seven or eight months in the

cramping presence of those bedchambers, what can be the reason of a

sudden need for greater space? I see but one: the Spider requires a

roomy habitation, not for herself--she is satisfied with the smallest

den--but for a second family. Where is she to place the pockets of eggs,

if the ruins of the previous laying remain in the way? A new brood

requires a new home. That, no doubt, is why, feeling that her ovaries

are not yet dried up, the Spider shifts her quarters and founds a new

establishment.



The facts observed are confined to this change of dwelling. I regret

that other interests and the difficulties attendant upon a long

upbringing did not allow me to pursue the question and definitely to

settle the matter of the repeated layings and the longevity of the

Clotho, as I did in that of the Lycosa.



Before taking leave of this Spider, let us glance at a curious problem

which has already been set by the Lycosa's offspring. When carried for

seven months on the mother's back, they keep in training as agile

gymnasts without taking any nourishment. It is a familiar exercise for

them, after a fall, which frequently occurs, to scramble up a leg of

their mount and nimbly to resume their place in the saddle. They expend

energy without receiving any material sustenance.



The sons of the Clotho, the Labyrinth Spider and many others confront us

with the same riddle: they move, yet do not eat. At any period of the

nursery stage, even in the heart of winter, on the bleak days of January,

I tear the pockets of the one and the tabernacle of the other, expecting

to find the swarm of youngsters lying in a state of complete inertia,

numbed by the cold and by lack of food. Well, the result is quite

different. The instant their cells are broken open, the anchorites run

out and flee in every direction as nimbly as at the best moments of their

normal liberty. It is marvellous to see them scampering about. No brood

of Partridges, stumbled upon by a Dog, scatters more promptly.



Chicks, while still no more than tiny balls of yellow fluff, hasten up at

the mother's call and scurry towards the plate of rice. Habit has made

us indifferent to the spectacle of those pretty little animal machines,

which work so nimbly and with such precision; we pay no attention, so

simple does it all appear to us. Science examines and looks at things

differently. She says to herself:



'Nothing is made with nothing. The chick feeds itself; it consumes or

rather it assimilates and turns the food into heat, which is converted

into energy.'



Were any one to tell us of a chick which, for seven or eight months on

end, kept itself in condition for running, always fit, always brisk,

without taking the least beakful of nourishment from the day when it left

the egg, we could find no words strong enough to express our incredulity.

Now this paradox of activity maintained without the stay of food is

realized by the Clotho Spider and others.



I believe I have made it sufficiently clear that the young Lycosae take

no food as long as they remain with their mother. Strictly speaking,

doubt is just admissible, for observation is needs dumb as to what may

happen earlier or later within the mysteries of the burrow. It seems

possible that the repleted mother may there disgorge to her family a mite

of the contents of her crop. To this suggestion the Clotho undertakes to

make reply.



Like the Lycosa, she lives with her family; but the Clotho is separated

from them by the walls of the cells in which the little ones are

hermetically enclosed. In this condition, the transmission of solid

nourishment becomes impossible. Should any one entertain a theory of

nutritive humours cast up by the mother and filtering through the

partitions at which the prisoners might come and drink, the Labyrinth

Spider would at once dispel the idea. She dies a few weeks after her

young are hatched; and the children, still locked in their satin

bed-chamber for the best part of the year, are none the less active.



Can it be that they derive sustenance from the silken wrapper? Do they

eat their house? The supposition is not absurd, for we have seen the

Epeirae, before beginning a new web, swallow the ruins of the old. But

the explanation cannot be accepted, as we learn from the Lycosa, whose

family boasts no silky screen. In short, it is certain that the young,

of whatever species, take absolutely no nourishment.



Lastly, we wonder whether they may possess within themselves reserves

that come from the egg, fatty or other matters the gradual combustion of

which would be transformed into mechanical force. If the expenditure of

energy were of but short duration, a few hours or a few days, we could

gladly welcome this idea of a motor viaticum, the attribute of every

creature born into the world. The chick possesses it in a high degree:

it is steady on its legs, it moves for a little while with the sole aid

of the food wherewith the egg furnishes it; but soon, if the stomach is

not kept supplied, the centre of energy becomes extinct and the bird

dies. How would the chick fare if it were expected, for seven or eight

months without stopping, to stand on its feet, to run about, to flee in

the face of danger? Where would it stow the necessary reserves for such

an amount of work?



The little Spider, in her turn, is a minute particle of no size at all.

Where could she store enough fuel to keep up mobility during so long a

period? The imagination shrinks in dismay before the thought of an atom

endowed with inexhaustible motive oils.



We must needs, therefore, appeal to the immaterial, in particular to heat-

rays coming from the outside and converted into movement by the organism.

This is nutrition of energy reduced to its simplest expression: the

motive heat, instead of being extracted from the food, is utilized

direct, as supplied by the sun, which is the seat of all life. Inert

matter has disconcerting secrets, as witness radium; living matter has

secrets of its own, which are more wonderful still. Nothing tells us

that science will not one day turn the suspicion suggested by the Spider

into an established truth and a fundamental theory of physiology.









APPENDIX: THE GEOMETRY OF THE EPEIRA'S WEB





I find myself confronted with a subject which is not only highly

interesting, but somewhat difficult: not that the subject is obscure; but

it presupposes in the reader a certain knowledge of geometry: a strong

meat too often neglected. I am not addressing geometricians, who are

generally indifferent to questions of instinct, nor entomological

collectors, who, as such, take no interest in mathematical theorems; I

write for any one with sufficient intelligence to enjoy the lessons which

the insect teaches.



What am I to do? To suppress this chapter were to leave out the most

remarkable instance of Spider industry; to treat it as it should be

treated, that is to say, with the whole armoury of scientific formulae,

would be out of place in these modest pages. Let us take a middle

course, avoiding both abstruse truths and complete ignorance.



Let us direct our attention to the nets of the Epeirae, preferably to

those of the Silky Epeira and the Banded Epeira, so plentiful in the

autumn, in my part of the country, and so remarkable for their bulk. We

shall first observe that the radii are equally spaced; the angles formed

by each consecutive pair are of perceptibly equal value; and this in

spite of their number, which in the case of the Silky Epeira exceeds two

score. We know by what strange means the Spider attains her ends and

divides the area wherein the web is to be warped into a large number of

equal sectors, a number which is almost invariable in the work of each

species. An operation without method, governed, one might imagine, by an

irresponsible whim, results in a beautiful rose-window worthy of our

compasses.



We shall also notice that, in each sector, the various chords, the

elements of the spiral windings, are parallel to one another and

gradually draw closer together as they near the centre. With the two

radiating lines that frame them they form obtuse angles on one side and

acute angles on the other; and these angles remain constant in the same

sector, because the chords are parallel.



There is more than this: these same angles, the obtuse as well as the

acute, do not alter in value, from one sector to another, at any rate so

far as the conscientious eye can judge. Taken as a whole, therefore, the

rope-latticed edifice consists of a series of cross-bars intersecting the

several radiating lines obliquely at angles of equal value.



By this characteristic we recognize the 'logarithmic spiral.'

Geometricians give this name to the curve which intersects obliquely, at

angles of unvarying value, all the straight lines or 'radii vectores'

radiating from a centre called the 'Pole.' The Epeira's construction,

therefore, is a series of chords joining the intersections of a

logarithmic spiral with a series of radii. It would become merged in

this spiral if the number of radii were infinite, for this would reduce

the length of the rectilinear elements indefinitely and change this

polygonal line into a curve.



To suggest an explanation why this spiral has so greatly exercised the

meditations of science, let us confine ourselves for the present to a few

statements of which the reader will find the proof in any treatise on

higher geometry.



The logarithmic spiral describes an endless number of circuits around its

pole, to which it constantly draws nearer without ever being able to

reach it. This central point is indefinitely inaccessible at each

approaching turn. It is obvious that this property is beyond our sensory

scope. Even with the help of the best philosophical instruments, our

sight could not follow its interminable windings and would soon abandon

the attempt to divide the invisible. It is a volute to which the brain

conceives no limits. The trained mind, alone, more discerning than our

retina, sees clearly that which defies the perceptive faculties of the

eye.



The Epeira complies to the best of her ability with this law of the

endless volute. The spiral revolutions come closer together as they

approach the pole. At a given distance, they stop abruptly; but, at this

point, the auxiliary spiral, which is not destroyed in the central

region, takes up the thread; and we see it, not without some surprise,

draw nearer to the pole in ever-narrowing and scarcely perceptible

circles. There is not, of course, absolute mathematical accuracy, but a

very close approximation to that accuracy. The Epeira winds nearer and

nearer round her pole, so far as her equipment, which, like our own, is

defective, will allow her. One would believe her to be thoroughly versed

in the laws of the spiral.



I will continue to set forth, without explanations, some of the

properties of this curious curve. Picture a flexible thread wound round

a logarithmic spiral. If we then unwind it, keeping it taut the while,

its free extremity will describe a spiral similar at all points to the

original. The curve will merely have changed places.



Jacques Bernouilli, {42} to whom geometry owes this magnificent theorem,

had engraved on his tomb, as one of his proudest titles to fame, the

generating spiral and its double, begotten of the unwinding of the

thread. An inscription proclaimed, '_Eadem mutata resurgo_: I rise again

like unto myself.' Geometry would find it difficult to better this

splendid flight of fancy towards the great problem of the hereafter.



There is another geometrical epitaph no less famous. Cicero, when

quaestor in Sicily, searching for the tomb of Archimedes amid the thorns

and brambles that cover us with oblivion, recognized it, among the ruins,

by the geometrical figure engraved upon the stone: the cylinder

circumscribing the sphere. Archimedes, in fact, was the first to know

the approximate relation of circumference to diameter; from it he deduced

the perimeter and surface of the circle, as well as the surface and

volume of the sphere. He showed that the surface and volume of the last-

named equal two-thirds of the surface and volume of the circumscribing

cylinder. Disdaining all pompous inscription, the learned Syracusan

honoured himself with his theorem as his sole epitaph. The geometrical

figure proclaimed the individual's name as plainly as would any

alphabetical characters.



To have done with this part of our subject, here is another property of

the logarithmic spiral. Roll the curve along an indefinite straight

line. Its pole will become displaced while still keeping on one straight

line. The endless scroll leads to rectilinear progression; the

perpetually varied begets uniformity.



Now is this logarithmic spiral, with its curious properties, merely a

conception of the geometers, combining number and extent, at will, so as

to imagine a tenebrous abyss wherein to practise their analytical methods

afterwards? Is it a mere dream in the night of the intricate, an

abstract riddle flung out for our understanding to browse upon?



No, it is a reality in the service of life, a method of construction

frequently employed in animal architecture. The Mollusc, in particular,

never rolls the winding ramp of the shell without reference to the

scientific curve. The first-born of the species knew it and put it into

practice; it was as perfect in the dawn of creation as it can be to-day.



Let us study, in this connection, the Ammonites, those venerable relics

of what was once the highest expression of living things, at the time

when the solid land was taking shape from the oceanic ooze. Cut and

polished length-wise, the fossil shows a magnificent logarithmic spiral,

the general pattern of the dwelling which was a pearl palace, with

numerous chambers traversed by a siphuncular corridor.



To this day, the last representative of the Cephalopoda with partitioned

shells, the Nautilus of the Southern Seas, remains faithful to the

ancient design; it has not improved upon its distant predecessors. It

has altered the position of the siphuncle, has placed it in the centre

instead of leaving it on the back, but it still whirls its spiral

logarithmically as did the Ammonites in the earliest ages of the world's

existence.



And let us not run away with the idea that these princes of the Mollusc

tribe have a monopoly of the scientific curve. In the stagnant waters of

our grassy ditches, the flat shells, the humble Planorbes, sometimes no

bigger than a duckweed, vie with the Ammonite and the Nautilus in matters

of higher geometry. At least one of them, _Planorbis vortex_, for

example, is a marvel of logarithmic whorls.



In the long-shaped shells, the structure becomes more complex, though

remaining subject to the same fundamental laws. I have before my eyes

some species of the genus Terebra, from New Caledonia. They are

extremely tapering cones, attaining almost nine inches in length. Their

surface is smooth and quite plain, without any of the usual ornaments,

such as furrows, knots or strings of pearls. The spiral edifice is

superb, graced with its own simplicity alone. I count a score of whorls

which gradually decrease until they vanish in the delicate point. They

are edged with a fine groove.



I take a pencil and draw a rough generating line to this cone; and,

relying merely on the evidence of my eyes, which are more or less

practised in geometric measurements, I find that the spiral groove

intersects this generating line at an angle of unvarying value.



The consequence of this result is easily deduced. If projected on a

plane perpendicular to the axis of the shell, the generating lines of the

cone would become radii; and the groove which winds upwards from the base

to the apex would be converted into a plane curve which, meeting those

radii at an unvarying angle, would be neither more nor less than a

logarithmic spiral. Conversely, the groove of the shell may be

considered as the projection of this spiral on a conic surface.



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.