THE BUILDER JUNE, 1918

THE DIVINE GEOMETRY
BY BRO. H.L. HAYWOOD, IOWA

Text: 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.

WHERE THE RAINBOW NEVER FADES

It can not be that the earth is man's only abiding-place. It can
not be that our life is a mere bubble cast up by eternity to float
a moment on its waves and then sink into nothingness. Else why is
it that the glorious aspirations which leap like angels from the
temple of our hearts are forever wandering unsatisfied ?

Why is it that all the stars that hold their festival around the
midnight throne are set above the grasp of our limited faculties,
forever mocking us with their unapproachable glory ?

And, finally, why is it that bright forms of human beauty presented
to our view are taken from us, leaving the thousand streams of our
affections to flow back in Alpine torrents upon our hearts?

There is a realm where the rainbow never fades; where the stars
will be spread out before us like islands that slumber in the
ocean; and where the beautiful beings which now pass before us like
shadows will stay in our presence forever.
--George D. Prentice.
