THE DIVINE GEOMETRY
By Bro. H.L. Haywood, Iowa
Proverbs 8:27, "he set a compass upon the face of the deep."
AFTER Euclid had shown Ptolemy his treatise on geometry the king inquired, somewhat wistfully, "Cannot the problems be made easier?" to which the geometer replied, "There is no royal road to geometry." True enough, but geometry itself is a royal road, and one that will lead us to Divine things if we will but follow it, as I now ask you to do.
It is difficult, if not impossible, for us to retrace our steps into the ancient day when men had not yet learned the orderliness of nature. Before the calendar was discovered or clocks invented the navigator steered his ship by the landmarks on the coast, and the farmer planted his crops by chance, for it was not known that the seasons repeat their regular ritual or that the heavens are ruled by order. "They saw things come and saw them go, but whence or whither they could not know." Everything changed or passed away and all things seemed to be in an eternal flux. In the midst of that everlasting stream of circumstance, that wildering maze of vicissitude, the early people felt helpless, if not mocked, for it always seemed that Nature was making sport of them. Even Renan, so far removed from them in time, recognized the pathos of this, for he said that "Nothing is so painful as the universal flow of things," while Tennyson set the mood to his music of accustomed sweetness:
The hills are shadows, and they flow
From form to form, and nothing stands;
They melt like mist, the solid lands,
Like clouds they shape themselves, and go.
If the mutability of all things was so oppressive to the recent thinkers, having at their hand science's unveiling of the lucid order of the universe, how much more painful must it have seemed to human minds before science came! "We are strangers before thee," they cried in their prayers, "and sojourners, as all our fathers were: our days on the earth are as a shadow, and there is no abiding."
Little wonder that the discovery of the North Star, one fixed body among all the others that moved perpetually, was an event of such importance that the simple folk worshipped it as a god and hung its symbol above the altars of their temples ! Little wonder that Heraclitus, the first thinker to state the fact with the thoroughness and system of philosophy, was called "The Weeping Philosopher !" Where there is no stability the mind hangs in the air and grows weary like a land bird at sea that finds no solid ground for its feet.
It was for this reason that the discovery of numbers, and especially of geometry, which is the application of numbers to form, was hailed as a visitation from on high. This discovery was not made in a day but came so gradually that men could hardly discern the lifting of the changing mists. And it was after this wise it came, if we have rightly pieced together the fragments of the story. The Egyptians lived along the Nile, their fields lying adjacent to its current in order to profit from the rich deposits of its overflow. But this very flood itself, source as it was of all fertility, gave rise to great difficulties, for the rising waters obliterated all landmarks each season and thus caused confusion among the owners of the fields. It was in their efforts to discover some method of fixing their boundaries that the Egyptians learned how to trace out the regular motions of the heavens, the periodicity of the seasons, and the properties of numbers. How much the race is indebted to those sun-browned workers in the fluviatile valley nobody can compute !
Inasmuch as numbers had won them order from chaos of their first impressions these early peoples exalted mathematics to the level of divinity, seeing in it, and rightly we may believe, a revelation, an uncovering, of the Creative Mind. Triangles and squares were engraved on their monuments and hung in their temples. The numbers three, five and seven were held especially sacred for in them were many qualities not possessed by other numerals. The cult of numbers arose at last and men formed secret societies for studying and teaching the properties of geometry.
It was among these secret societies that there came at a later day Pythagoras, one of the noblest of all thinkers, and the first to raise mathematics to the level of an exact science. From his hidden schools in Greece he taught his initiates the mystery of arithmetic, calling God "the Great Geometrician" and telling his pupils that "All things are in numbers; crystals are solid geometry."
Plato, also, the most opulent thinker of antiquity, found in geometry a revelation of the Infinite Mind, looking upon it as the very essence of religion, the knowledge of God. "What does Deity do all the while?" one of his pupils asked him. "God is always geometrizing," was the reply. "Geometry must ever tend to draw the soul towards truth." Over the portal of his school he inscribed the legend: "Let no one who is ignorant of geometry enter my doors."
What science is to all modern thinking the one science of mathematics, "the sacred mathematics," was to early thinking; and those first teachers felt it a sacred duty to transmit so valuable a knowledge to their descendants. Therefore was it that, three hundred years before Christ, Euclid wrote the treatise in which he embodied all that was known of the science at that time. Indeed, the work of Euclid is still the standard treatise on the subject, being used as the basis of every textbook in our schools. Better methods for proving the problems have been worked out, and new propositions have been discovered, but the fundamentals stand like adamant, and always will stand.
After the breakup of the ancient world and the general inundation of culture under the Barbarian Invasion, geometry was lost. For hundreds of years the people of Europe wandered among the mazes of chance and caprice, as primitive men had done before them. Then at last along came Simon Grynaeus, a contemporary of Luther, who rediscovered Euclid and gave his science to the new peoples. How much this influenced the Reformation no historian has yet undertaken to estimate but it is certain that it had far reaching consequences and paved the way for modern science, which is itself a superstructure built on mathematics.
If the earlier peoples were overjoyed to make their few discoveries of the hidden but fixed order of Nature how delighted they would now be to learn that all the endeavors of science have only served to make more clear and more universal the reign of number and form throughout the universe. For through a prophetic inspiration of the geometers we have had uncurtained to us a spectacle of mathematical order throughout the universe which is as revealing as it is beautiful.
Matter itself, immobile as it may appear to the eye, is in reality a composite of atoms that move through the mazes of an everlasting dance, every evolution of which seems timed to some exact pattern. Even the chemical elements, which so long baffled the system makers, were proved by Newlands to lie in a regular order of periodicity strangely grouped around the number seven. Order is the first law of the elements. Crystallization is a solid geometry. If one observes ice crystals forming across a window pane he will see them grouping themselves together into symmetrical forms, intricate, involved, beautiful, as if some unseen artist were at work depicting a scene from an arctic fairyland.
Even when life gathers matter up about itself into its organisms the same rhythm is preserved. Vitality is free and flowing, often apparently erratic, and moving by the law of its own, yet it will always be found at last to keep step with the geometrical motions of the world. If one would expect the eternal harmony absent from any field surely it would be in that little known realm which the insects inhabit; yet John Henri Fabre was so impressed by the reign of numbers among these insignificant creatures that he was moved to write this magnificent paragraph:
"He will admire as much as we do geometry the eternal balancer of space. There is a severe beauty, belonging to the domain of reason, the same in every world, the same under every sun, whether the suns be single or many, white or red, blue or yellow. This universal beauty is order. Everything is done by weight and measure, a great statement whose truth breaks upon us all the more vividly as we probe more deeply into the mystery of things. Is this order, upon which the equilibrium of the universe is based, the predestined result of a blind mechanism? Does it enter into the plans of an eternal Geometer, as Plato had it? Is it the ideal of a supreme lover of beauty, which would explain everything? Why all this regularity in the curve of the petals of a flower, why all this elegance in the chasings on a beetle's wing-cases? Is that infinite grace, even in the tiniest details compatible with the brutality of uncontrolled forces? One might as well attribute the artist's exquisite medallion to the steamhammer which makes the slag sweat in the melting !"
The "regularity in the curve of the petals of the flower" has attracted the attention of others as well as Fabre. Maeterlinck, who learned so much from the veteran French naturalist, made a prolonged study of the Mind that is at work in plants with what result anyone can read in a book of lovely pages, "The Intelligence of the Flowers." Why are leaves set around the stem in such mathematical regularity ? Why do flowers seem to love numbers, as the trilium is partial to three, and the rose to five ? Surely it must be because there is that in them which responds to the universal order. Like Plato's deity they are always geometrizing.
An animal is a plant that has taken to moving about, and just because it is so often apparently ungoverned in its movements, we lose sight of the regular laws which rule among animals as much as among plants and minerals. But those laws are there as many a scientist has proved. In the Mid-nineteenth Century days, before the evolution theory was so well understood, men fell to theorizing as if the universe had happened into existence through chance. Life itself was defined as the result of a "fortuitous concourse of atoms." The absurdity of this "thinking"--it was really an abdication of thought--was never more clearly revealed than by the Duke of Argyll, whose work on "The Reign of Law" is almost classical. The learned Duke took the wing of a common bird and showed that the mechanism of flight is so unimaginably complicated, so perfect, and solves so many mathematical problems, many of them beyond the ken of a Lord Kelvin, that it tasks our credulity too much to be asked to believe that this exquisite machinery could possibly have come through "chance." In a more recent time, Sir Oliver Lodge has made the same use of the human eye, an organ so intricate and nice in its adjustments and functions, that a Swiss watch is simple by comparison.
What is true of the things we find on the earth holds good in equal measure of the great bodies that sail round us through the sky. The astronomer's charts are strangely like a page of Euclid. He has found that order is the first law of the heavens as it is of Heaven. The wildest comet, careening irresponsibly through space, moves in an orbit as rigidly fixed as the passing of the hands about the clock. Surely it must be that an Infinite Mind has set His compasses upon the face of the deeps of space, else how explain the periodicity, the regularity, of the sidereal universe, the movement of any one body of which may be predicted for thousands of years in advance!
This law of geometric harmony holds as true among the arts of man as in those realms which are the art of God. Every building is geometric demonstration. As we may read in the pages of a learned student of this: "The language (geometry) spoke in the sloping wall and massive pillar and flat roof of Egypt, or in the mighty piles of Chaldea, or in the Corinthian grace, or in Roman boldness; the heart was that of the geometrician who spoke as he dreamed, in anger, in epic, in poetry of stone and graceful curve--who planned by the plumb and the square, by the secret of the arch and the balance of accurate measure."
Even painting, when lightly understood, conforms to the ancient patterns, being based on the principle described by one of its most magisterial exponents: "All nature is modelled either like a cone, a sphere or a cylinder. Painting is a colored mathematics of things." As for music, that is geometry that has taken to wings, its freedom evermore being inbound in law. It is the child of rhythm which is the purest manifestation of the law of numbers. From of old it has been dreamed that the morning stars sang together, that the rafters and beams of creation were laid deep in melody, that the spheres make music as they move, that all "deep things are song." Of this truth every musician is the priest as every poet is its apostle. As Dryden sings:
"From Harmony, from heavenly Harmony
This universal frame began;
When Nature underneath a heap
Of jarring atoms lay And could not heave her head,
The tuneful voice was heard from high,
Arise, ye more than dead!
Then cold, and hot, and moist, and dry
In order to their stations leap,
And music's power obey.
From Harmony, from heavenly Harmony
This universal frame began:
From harmony to harmony
Through all the compass of the notes it ran,
The diapason closing full in Man."
Yes, in Man, truly, for order holds in the soul as much as in the heavens where the astronomer thinks God's thoughts after Him. Character is no chance product but builds according to laws as immutable and as ascertainable as any to be found in the builder's art. For the freedom of the soul is not capriciousness, least of all lawlessness, but voluntary co-operation with the fixed rules of the spirit. He who will build according to that principle will erect a character as stable as that house which the wise architect builded on the rock. Glorious will be the day when men learn the geometry of the heart and square their actions to the fixed rules of moral life.
The significance of this geometry of the cosmos for our faith has been know ever since men discovered it. At bottom there ale but two philosophies: that which holds that this universe is a heap of dirt governed by chance; and that which finds in it a reasoned reign of order resting in an Infinite Mind. As between dirt and deity a man may make his choice, but surely the thinker who sees everywhere the beautiful sweep of order will not for a moment believe that this mighty music could have come to us out of the falling atoms of chance. One might as well throw a handful of type into the air and expect them to write a poem in their fall !
Twenty-five centuries ago Socrates labored to show the little atheist, Aristodemus, that as a statue by Polytectetus could not possibly have emerged from the quarries through mere chance, so is it impossible to believe that the cosmos, infinitely greater in complexity as well as in beauty, could ever have come into existence through mere fortuitousness. In the same wise, Franklin, who may typify the modern thinker, exposed the fallacy of an atheist astronomer friend of his. The astronomer was showing him an orrery, which is a working model of the solar system, when Franklin said, "It is strange that such a thing could build itself by chance." "Chance !" exclaimed the astronomer, "I made that myself. How could so complicated a device have come by chance?" "Then," said the philosopher, turning upon him, "how can you believe that the solar system itself, of which this is a mere model, could have come by chance ?"
Surely, when we have our minds with us, it must be apparent that the everywhere present order of things is the revelation of a Divine Orderer! Where there is so much intelligence there must be an Intelligence! Where there is so much harmony there must stand near a great Musician! The poetry of earth is the song of an Infinite Poet! The beauty of all creation is the outshining, the splendor of an Eternal Artist !
Long ago a psalmist cried, "Whither shall I flee from Thy presence?" We cannot flee from His presence. While we dig in the dirt He is there, present in the dance of the atoms that compose the soil: while we walk through the snow He draws His pictures about us in the traceries of the crystals: the bird that wings above us is His angel, making hieroglyphics in the air: the very tides move along the circle which His compasses draw upon the deep. Everywhere He is. We live imbedded in His mind. To escape from Him is as impossible as to climb out of the atmosphere !
Where there is so much order all must be ordered. King Alphonso of Castile, looking out over the general muddle of affairs into which Spain had fallen, doubted that a Mind ruled all. "If God had called me to His councils," he sighed, "things would have been in better order." In these days when it seems that the bottom has gone out of the world and chaos has come again, we may fall into the mood of the old king. But let us despair not. The plain is there; we have lost the perspective, or the key. It is said that the frescoes on the ceiling of St. Peter's look like an inartistic jumble to the man who climbs close to them; but from a station three hundred feet below they spring up into a majestic beauty. They are wrought on too large a plan for a close view. We humans, with our near-sightedness, our myopic eyes, are standing too close to the program of creation; it may appear all jumble to us now. Let us wait with patience. Some morning, soon or late, will find us on a mountain of vision where we can see things as they are and watch the Divine Geometer draw His circles across the deep.
- Source: The Builder June 1918