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Harmonic Proportion and Form in Nature, Art, and Architecture
By Samuel Colman, C. Arthur Coan
Dover Publications, Inc.Copyright © 2003 Dover Publications, Inc.
All rights reserved.
"Order is Heaven's First Law."
WHAT does the term "Order" imply in this connection? It would seem to imply the just correlation of each of the parts of an object with the whole. This means that all of the details of a form in Nature are accurately united in perfect harmony, but the scheme of this unity can only be determined with precision by the use of mathematical measurements, angles, points, lines, and surfaces, produced by geometric principles. Unity is the highest element of beauty, and there can be no question but that the laws of growth in Nature are the fundamental ones which govern it. If it be one object of man's use of geometry to investigate the stars in space, the same science may be also similarly employed in an analysis of the laws of plant growth, polar force, or of any of the many other evidences of Nature's handiwork, as the underlying principles are the same in each. We must not consider that the circle, the triangle, and the square are simply forms, but elements representing the divine grammar of Nature. These geometric principles continue throughout the universe in orderly and exact methods, not only to secure beauty of proportion but the highest use as well.
"The heavens themselves, the planets and this centre,
Observe degree, priority, and place,
Insisture, course, proportion, form,
Office, and custom, in all line of order."
TROILUS AND CRESSIDA.
This influence has always been felt, not only by the most civilized races, but by untutored savages as well; man's history has been written in it for thousands of years, from the axehead of the Stone Age up to the mighty Pyramids of Gizeh, which have stood in their geometric form longer than almost any other work of art, dominating by their simple majesty the minds of passing generations.
The people who created them were among the first to consider philosophically the attributes of the circle, the triangle, and the square, recognizing that they contained the elements for the true measurement of proportional spaces. This knowledge, imparted to the Greeks by the great mind of Pythagoras, after his sojourn in Egypt, enabled their artists and architects to create that perfect proportion and ideal refinement of form which were the distinguishing traits of the art of that period and still remain unsurpassed. The Gothic architect and artist re-discovered many of the Greek secrets, causing beauty once more to bloom in statue, picture, and design, while cathedrals, the perfections of which are still the wonder and admiration of all classes of men, arose to glorify the land. Under the influence of this exact knowledge even the simplest village church of that period charms the world to-day by the perfect balance of its parts. Freemasonry was alive and active then, but its mysteries were known only to the brotherhood. So inviolate were its secrets that we can now learn little of the principles of proportion even from dissertations on the subject in Encyclopæias and other writings, beyond a few set phrases. Even in that remarkable controversy between Ruskin and the architects of his day, not one word was given on either side beyond similar trite sayings.
Moreover, the mistake which is made by many people, and it is a vital one, of questioning the value of the geometric theory of proportion, is this: that they consider geometry to be an invention of man involving his formally constructed equations only. I need hardly say, however, that the principles are not man's invention, although the outward form of Algebra, which is one of its visible interpreters, may be. The truth is that the system is simply the logical expression of one of the methods man has borrowed from Nature by means of which he can more easily investigate scientific questions. It is as much a part of her as the very air we breathe. Its use in an analysis of proportion is like the application of a solvent which must be suited to the object to be resolved, otherwise labor is in vain. In order to prove the statement that geometric correlations and the proportions of objects in Nature are one and the same thing, all that one need do, even though examples are innumerable, is to take a drop from a pool of fresh or salt water for examination under the microscope, when various beautiful creations will appear, revealing in their individual forms all five of the regular polyhedra, exactly as though drawn with mathematical instruments.
It is well recognized that light, sound, and heat are geometric in their action and continue on similar lines of "force" "to be amenable to accurate calculation by measurement or numbers, the elemental source of all harmony. By what right then have architects or artists in our time considered that they have any peculiar privilege of their own to seek to construct beauty without an accurate understanding of the same system by which the loveliness of all forms of beauty in Nature comes into existence?
The cant phrase of the day, that "great art can only result from a passionate spontaneity of feeling, and this feeling must be smothered or paralyzed by the employment of scientific methods," comes from a strange misapprehension of the truth. But when to this false statement are added the words, "Hellenic art, in particular, was the outcome of genius under the influence of impulse only," one cannot help thinking that those repeating this idea must be under the influence of blind prejudice, or suffer from a want of knowledge of form-composition or of the facts of history. If Ictinus and Phidias could arise from their graves they would be the first to repudiate and resent such language as applied to their works, for the Parthenon, even with its statues and ornaments in their ruins, is the most perfect form of art that the hand of man has produced, and an analysis of its portico alone will discover the fact that the relation of the details to the mass is one of most perfect unity, owing to the intelligent application of the law contained in the "Progression of the Square," or by the same sure method employed by Nature in her plan of developing the wonderful beauty of the crystal.
Under the influence of exact scientific study the artist will find that his impulses will become more vitally alive, while his imagination as well as intuitive feeling will be greatly strengthened and his achievement become vastly superior and of lasting credit. The sculptor or painter who from want of education or lack of thought condemns mathematics should bear this fact in mind, that the bolder, the swifter, the more impulsive the stroke of the modelling tool or the brush in the hand of a master, the more perfect, the more beautiful, and the more like Nature is the resultant curve, for the arm and hand have become, through training, a sort of mathematical instrument with an exact though complicated radius from which he cannot escape.
From a similar want of thought Beauty is now considered an element impossible to describe in any adequate and conclusive way, while many philosophers and æsthe-ticians agree with Johnson and his intimate friend Reynolds, the artist, who were among the first to declare, "that Beauty exists only in the mind, and just as this mind is prejudiced by its education, will an object be beautiful to one and not so to another. Who shall judge?" A frequently used definition of the word "Beauty" is to the effect "that whatever pleases the eye or the mind is beautiful," but this idea alone is of a kind that might be discussed until doomsday without arriving at any conclusion. The word "Beauty" has come, however, through centuries of use by educated people, to mean something far more than is usually given by lexicographers; very many feel that it is the highest manifestation of the Creator, revealed in mountain, cloud, and ocean, with the countless living things that they contain. But it is only through an accurate analysis of these various forms that a clear and distinct idea may be obtained, where no sophistry in argument can change the result. In this analysis we learn conclusively that the essence of "Beauty" is unity and where unity exists it can be clearly proven, remaining no longer a question of what this man thinks, or that, whose prejudices have blinded his faculties of observation.
It is true that one person may prefer the beauty of man, another one that of woman; I look with especial favor on the rose, while my neighbor takes more delight in the lily; both are right, as all are beautiful. That is, they are objects expressing divine unity of construction, or the perfect correlation of parts in form-compositions. This book is not devoted to the philosophy of beauty, these arguments are only advanced to suggest the idea that future investigators may find, through scientific methods applied to Beauty, a clearer means of interpreting the term than now exists, while these methods will ultimately expose the fallacy, now so generally looked upon as truth, that Science and Beauty are antagonistic. The more profoundly the mind considers the question of Science and Beauty the more deeply must man be impressed by Nature's methods, and under the influence of the resulting knowledge will the heart of a passionate artist be more deeply kindled with a fire which must result in work more perfect and more enduring; for the great law of numerical harmonic ratio remains unalterable, and a proper application of it to art will never fail to be productive of good effect, as its operation in Nature is universal, certain, and continuous. She appears never to tire in following out mechanical principles in her works, and her methods should be properly respected by man when he seeks to construct forms of beauty, while his theories of art should be duly founded on her laws, so exact and infallible.
In showing how the rules and principles of harmonic proportions apply to the arts and sciences, we must not look for blind adherence in every instance to the degree, minute, and second of every angle, and no vernier scale nor micrometer is needed to prove or disprove the propositions here set forth. The acid test of mathematical precision and calculation has been applied to all of the fundamental principles which are claimed, and the result proves itself at every step to be well within the limits of Nature's constant variations, as has been indicated from time to time in the Appendix, and if, in the "multiplicity of instances," cases of slight deviation be pointed out, I may venture the statement that the deviation will prove negligible and serve only to sustain the spirit of the rule.
As Nature loves variety, so she produces unexpected results by combining various forces, and while seemingly she deviates continually from the letter of her harmonic laws, she never does from their spirit. We must not look for an uninterrupted identity of mathematically derived rules with natural phenomena. The forces at work are so varied in their application that no one of them is uninfluenced by others. Botanically, torsions will be found occasionally to displace symmetry in a flower, seismic disturbances will interfere with geologic formations, or the solar influence may destroy the ellipse of an approaching comet, but in no case, actual or imaginary, does Nature show any tendency to anarchy. She abides by her rules, though through the veil of composite effects they are sometimes hard to trace.
What Nature permits, man may surely do, and it is doubtful if a clearer example of the necessity of applying to art Nature's compensatory rule of "give and take" could be offered, than the following.
It may safely be said, without fear of contradiction, that of all the Arts, Music is far and away the most mathematically exact, and in this I do not refer to the question of musical "tempo," which is comparatively simple, but to the rules of vibration and tonal production which govern the relationships of the members of the diatonic scale, and the harmony of composition.
Any studious musician will understand that the natural and perfectly constructed scale, true to the ear and to its mathematical proportions of vibration, contains perfect octaves, perfect dominants, mediants, and other intervals, but he will also understand that this natural and perfect scale is susceptible of no use whatever in any key except that of the tonic for which it was constructed. Its tone intervals, as established by the ear and by science, are not all equal, nor are its half tones equal. For example, the ratio of vibration between the tonic and its second (a whole tone) is not the same as between this second and the mediant (also called a whole tone), for if the perfect scale for a single key be examined, the interval from the tonic to the second will prove to be in proportion to the interval between the second and the dominant in the ratio of 51:46 or thereabouts, and not according to precise equality. Naturally, then, if the perfect scale based, for example, on "C" as a tonic, be used for the rendition of anything written in the key of "D," it will be found that the interval from this new tonic to its second cannot of course be the perfect first step required, since the interval to be utilized has already been restricted to the vibratory difference indicated for the second step in the "C" scale, to wit, we are obliged to use the smaller of the two so-called whole tones, whereas in this new key of "D" we need the larger whole tone for the first interval. The farther the process is carried, the worse will be found the result, except that every octave is a repetition of every other. The ideal and perfect scale for the tonic "C" thus becomes utterly useless for a modulation into any musically remote key.
To adapt any keyed instrument to general use, therefore, Nature's principle of compromise has been utilized in setting what is technically called an equalized "temperament," the greater, intervals, representing the greater ratio of difference of vibration, yielding to their lesser neighbors sufficient of their surplus so that all of the so-called whole tones represent equal, or nearly equal ratios of difference, the half tones being treated in the same way, thus leaving no key in its primary perfection at the expense of every other, but permitting all to be alike possible and pleasing. Here, if anywhere, we surely have an example of a universally adopted, measurable variation from mathematical perfection, under a system of compensation which produces a result both artistic and at the same time eminently practical.
From all of these facts it seems clear that, while Beauty is of many kinds and has many exemplars, her form is always controlled by Mother Nature, and if, by studious observation, man can re-learn a few of the principles employed by Nature in the creation of those forms which are universally acknowledged as Beauty, perhaps the determination of what is beautiful in art, music, architecture, and painting may be made simpler, more certain, and less subject to the whim and fancy of temperamental minds and the fluctuating personal equation.CHAPTER 2
Correlations of Numbers
THE examination of the relations of numbers to each other has been a favorite pursuit of the logicians and mathematical philosophers from time immemorial, and it would seem to be more or less apt that, before endeavoring to analyze the harmonies of substance we should first briefly consider the relations and harmonies of the machinery by which these concrete entities are measured. Let us understand our "yardstick" before we begin to cut our cloth, proceeding in an orderly manner, first to study a few of the fundamentals of numbers in the abstract, then to ponder the conspicuous harmonics of geometrical proportions as adopted by our Mother Nature, turning then from the generic to the specific, to examine her wonders and to see how constantly she applies these proportions; and finally we may learn in what manner these same harmonics of geometry—the veritable "Harmonics of Nature"—have been playing their part in architecture and the arts.
It is not, therefore, the intention, under this head, to treat of concrete numbers as adopted by Nature or man, or as governing factors in any science or art save in the pure science of numbers themselves. Other features will be fully developed in future chapters where the several divisions will be separately taken up, a branch at a time. For the present, let us see what are a few of the harmonic relations of mere quantity to quantity, or of numbers as such to each other, rather than to the things numbered.
Excerpted from Harmonic Proportion and Form in Nature, Art, and Architecture by Samuel Colman, C. Arthur Coan. Copyright © 2003 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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