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THE CLOTHO SPIDER




She is named Durand's Clotho (_Clotho Durandi_, LATR.), in memory of him
who first called attention to this particular Spider. To enter on
eternity under the safe-conduct of a diminutive animal which saves us
from speedy oblivion under the mallows and rockets is no contemptible
advantage. Most men disappear without leaving an echo to repeat their
name; they lie buried in forgetfulness, the worst of graves.

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.






Previous: THE LABYRINTH SPIDER



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