[This chapter, one of the most wonderful in Fabre’s books, is included in a simplified form in this volume, on account of its interest to such younger readers as have studied geometry.]

When we look at the webs of the Garden Spiders, especially those of the Silky Spider and the Banded Spider, we notice first that the spokes or radii are equally spaced; the angles formed by each consecutive pair are of the same value; and this in spite of their number, which in the webs of the Silky Spider sometimes exceeds forty. We know in what a strange way the Spider weaves her web and divides the area of the web into a large number of equal parts or sectors, a number which is almost always the same in the work of each species of Spider. The Spider darts here and there when laying her spokes as if she had no plan, and this irresponsible way of working produces a beautiful web like the rose-window in a church, a web which no designer could have drawn better with compasses.

We shall also notice that, in each sector, the various chords, parts of the angular spiral, are parallel to one another and gradually draw closer together as they near the center. 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, as far as the eye can judge. Taken as a whole, therefore, the spiral consists of a series of cross-bars intersecting the several radiating lines obliquely at angles of equal value.

By this characteristic we recognize what geometricians have named the “logarithmic spiral.” It is famous in science. 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. We could not see such a line, the whole of it, even with our best philosophical instruments. It exists only in the imagination of scientists. But the Spider knows it, and winds her spiral in the same way, and very accurately at that.

Another property of this spiral is that if one in imagination winds a flexible thread around it, then unwinds the thread, keeping it taut the while, its free end will describe a spiral similar at all points to the original. The curve will merely have changed places. Jacques Bernouilli, the professor of mathematics who discovered this magnificent theorem, had engraved on his tomb, as one of his proudest titles to fame, the spiral and its double, made by the unwinding of the thread. Written underneath it was the sentence: Eadem mutata resurgo. “I rise again like unto myself.” It was a splendid flight of fancy which showed his belief in immortality.

Now is this logarithmic spiral, with its curious properties, merely an idea of the geometricians? Is it a mere dream, an abstract riddle?

No, it is a reality in the service of life, a method of construction often employed by animals in their architecture. The Mollusk never makes its 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.

There are perfect examples of this spiral found in the shells of fossils. To this day, the last representative of an ancient tribe, the Nautilus of the Southern Seas, remains faithful to the old design, and still whirls its spiral logarithmically, as did its ancestors in the earliest ages of the world’s existence. Even in the stagnant waters of our grassy ditches, a tiny Shellfish, no bigger than a duckweed, rolls its shell in the same manner. The common snail-shell is constructed according to logarithmic laws.

Where do these creatures pick up this science? We are told that the Mollusk is descended from the Worm. One day the Worm, rendered frisky by the sun, 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 science. But the Spider will have none of this theory. For she is not related to the Worm; and yet she is familiar with the logarithmic spiral and uses it in her web, in a simpler form. The Mollusk has years in which to build her spiral, so she makes it very perfectly. The Spider has only an hour at the most to spread her net, so she makes only a skeleton of the curve; but she knows the same line dear to the Snail. What guides her? Nothing but an inborn skill, whose effects the animal is no more able to control than the flower is able to control the arrangement of its petals and stamens. The Spider practices higher geometry without knowing or caring. The thing works of itself and takes its way from an instinct imposed upon creation at 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. The curve is known to science and is called the “parabola.”

The geometricians speculate still more about this curve; they imagine it rolling on an indefinite straight line and ask what course the focus of the curve follows. The answer comes that the focus of the parabola describes a “catenary,” a line whose algebraic symbol is so complicated that a numeral will not express it. The nearest it can get is this terrible sum:

The geometricians do not attempt to refer to it by this number; they give it a letter, e.

Is this line imaginary? Not at all; you may see the catenary frequently. 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 filled out by the wind. 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, becomes tremendously complicated 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 seem when compared with the smallest actual things! Shall we never be able to inquire into reality in a simpler fashion? Shall we be intelligent enough some day to do without all these heavy formulÆ? Why not?

Here we have the magic number e reappearing, written on a Spider’s thread. On a misty morning the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chains 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 rainbow-colored fires and becomes a dazzling cluster of diamonds. The number e is in its glory.

Geometry, that is to say, the science of harmony in space, rules over everything. We find it in the arrangement of the scales of a fir-cone, as in the arrangement of a Spider’s sticky snare; 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 a Universal Geometrician, whose divine compass has measured all things. I prefer that, as an explanation of the logarithmic curve of the Nautilus and the Garden Spiders, to the Worm screwing up the tip of its tail. It may not perhaps be in agreement with some latter-day teaching, but it takes a loftier flight.