In the beginning of this book I said that its scope and treatment were of so prefatory a kind that of other preface it had no need; and now, for the same reason, with no formal and elaborate conclusion do I bring it to a close. The fact that I set little store by certain postulates (often deemed to be fundamental) of our present-day biology the reader will have discovered and I have not endeavoured to conceal. But it is not for the sake of polemical argument that I have written, and the doctrines which I do not subscribe to I have only spoken of by the way. My task is finished if I have been able to shew that a certain math­e­mat­i­cal aspect of morphology, to which as yet the morphologist gives little heed, is interwoven with his problems, complementary to his descriptive task, and helpful, nay essential, to his proper study and comprehension of Form. Hic artem remumque repono.

And while I have sought to shew the naturalist how a few math­e­mat­i­cal concepts and dynamical principles may help and guide him, I have tried to shew the mathematician a field for his labour,—a field which few have entered and no man has explored. Here may be found homely problems, such as often tax the highest skill of the mathematician, and reward his ingenuity all the more for their trivial associations and outward semblance of simplicity.

That I am no skilled mathematician I have had little need to confess, but something of the use and beauty of mathematics I think I am able to understand. I know that in the study of material things, number, order and position are the threefold clue to exact knowledge; that these three, in the mathematician’s hands, furnish the “first outlines for a sketch of the Universe”; that by square and circle we are helped, like Emile Verhaeren’s carpenter, to conceive “Les lois indubitables et fÉcondes Qui sont la rÈgle et la clartÉ du monde.”

For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural {779} Philosophy are embodied in the concept of math­e­mat­i­cal beauty. A greater than Verhaeren had this in mind when he told of “the golden compasses, prepared In God’s eternal store.” A greater than Milton had magnified the theme and glorified Him “who sitteth upon the circle of the earth,” saying: He measureth the waters in the hollow of his hand, he meteth out the heavens with his span, he comprehendeth the dust of the earth in a measure.

Moreover the perfection of math­e­mat­i­cal beauty is such (as Maclaurin learned of the bee), that whatsoever is most beautiful and regular is also found to be most useful and excellent.

The living and the dead, things animate and inanimate, we dwellers in the world and this world wherein we dwell,—p??ta ?a ?? t? ?????s??e?a,—are bound alike by physical and math­e­mat­i­cal law. “Conterminous with space and coeval with time is the kingdom of Mathematics; within this range her dominion is supreme; otherwise than according to her order nothing can exist, and nothing takes place in contradiction to her laws.” So said, some forty years ago, a certain mathematician; and Philolaus the Pythagorean had said much the same.

But with no less love and insight has the science of Form and Number been appraised in our own day and generation by a very great Naturalist indeed:—by that old man eloquent, that wise student and pupil of the ant and the bee, who died but yesterday, and who in his all but saecular life tasted of the firstfruits of immortality; who curiously conjoined the wisdom of antiquity with the learning of to-day; whose ProvenÇal verse seems set to Dorian music; in whose plainest words is a sound as of bees’ industrious murmur; and who, being of the same blood and marrow with Plato and Pythagoras, saw in Number “la clef de la voÛte,” and found in it “le comment et le pourquoi des choses.”