But these sciences are far less highly developed, far less exact and systematic, that is to say, far less scientific, at present, than is either physics or chemistry. However, the application of statistical methods promises good results, and there are not wanting generalisations already arrived at which are expressible mathematically;Weber's Law in psychology, and the law concerning the arrangement of the leaves about the stems of plants in biology, may be instanced as cases in point."[1]
[1] Quoted from a lecture by the present writer on "The Law of Correspondences Mathematically Considered," delivered before The Theological and Philosophical Society on 26th April 1912, and published in _Morning Light_, vol. xxxv (1912), p.
434 _et seq_.
The Pythagorean doctrine of the Cosmos, in its most reasonable form, however, is confronted with one great difficulty which it seems incapable of overcoming, namely, that of continuity. Modern science, with its atomic theories of matter and electricity, does, indeed, show us that the apparent continuity of material things is spurious, that all material things consist of discrete particles, and are hence measurable in numerical terms.
But modern science is also obliged to postulate an ether behind these atoms, an ether which is wholly continuous, and hence transcends the domain of number.[1] It is true that, in quite recent times, a certain school of thought has argued that the ether is also atomic in constitution--that all things, indeed, have a grained structure, even forces being made up of a large number of quantums or indivisible units of force.
But this view has not gained general acceptance, and it seems to necessitate the postulation of an ether beyond the ether, filling the interspaces between its atoms, to obviate the difficulty of conceiving of action at a distance.
[1] Cf. chap. iii., "On Nature as the Embodiment of Number,"of my _A Mathematical Theory of Spirit_, to which reference has already been made.
According to BERGSON, life--the reality that can only be lived, not understood--is absolutely continuous (_i.e_. not amenable to numerical treatment). It is because life is absolutely continuous that we cannot, he says, understand it; for reason acts discontinuously, grasping only, so to speak, a cinematographic view of life, made up of an immense number of instantaneous glimpses.
All that passes between the glimpses is lost, and so the true whole, reason can never synthesise from that which it possesses.
On the other hand, one might also argue--extending, in a way, the teaching of the physical sciences of the period between the postulation of DALTON'S atomic theory and the discovery of the significance of the ether of space--that reality is essentially discontinuous, our idea that it is continuous being a mere illusion arising from the coarseness of our senses.
That might provide a complete vindication of the Pythagorean view;but a better vindication, if not of that theory, at any rate of PYTHAGORAS' philosophical attitude, is forthcoming, I think, in the fact that modern mathematics has transcended the shackles of number, and has enlarged her kingdom, so as to include quantities other than numerical. PYTHAGORAS, had he been born in these latter centuries, would surely have rejoiced in this, enlargement, whereby the continuous as well as the discontinuous is brought, if not under the rule of number, under the rule of mathematics indeed.
PYTHAGORAS' foremost achievement in mathematics I have already mentioned.
Another notable piece of work in the same department was the discovery of a method of constructing a parallelogram having a side equal to a given line, an angle equal to a given angle, and its area equal to that of a given triangle. PYTHAGORAS is said to have celebrated this discovery by the sacrifice of a whole ox. The problem appears in the first book of EUCLID'S _Elements of Geometry_ as proposition 44.
In fact, many of the propositions of EUCLID'S first, second, fourth, and sixth books were worked out by PYTHAGORAS and the Pythagoreans;but, curiously enough, they seem greatly to have neglected the geometry of the circle.
The symmetrical solids were regarded by PYTHAGORAS, and by the Greek thinkers after him, as of the greatest importance.
To be perfectly symmetrical or regular, a solid must have an equal number of faces meeting at each of its angles, and these faces must be equal regular polygons, _i.e_. figures whose sides and angles are all equal. PYTHAGORAS, perhaps, may be credited with the great discovery that there are only five such solids.
These are as follows:--
The Tetrahedron, having four equilateral triangles as faces.
The Cube, having six squares as faces.
The Octahedron, having eight equilateral triangles as faces.
The Dodecahedron, having twelve regular pentagons (or five-sided figures) as faces.
The Icosahedron, having twenty equilateral triangles as faces.[1]
[1] If the reader will copy figs. 4 to 8 on cardboard or stiff paper, bend each along the dotted lines so as to form a solid, fastening together the free edges with gummed paper, he will be in possession of models of the five solids in question.
Now, the Greeks believed the world to be composed of four elements--earth, air, fire, water,--and to the Greek mind the conclusion was inevitable[2a] that the shapes of the particles of the elements were those of the regular solids.
Earth-particles were cubical, the cube being the regular solid possessed of greatest stability; fire-particles were tetrahedral, the tetrahedron being the simplest and, hence, lightest solid.
Water-particles were icosahedral for exactly the reverse reason, whilst air-particles, as intermediate between the two latter, were octahedral. The dodecahedron was, to these ancient mathematicians, the most mysterious of the solids:
