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.