Geometry and faith; a supplement to the ninth Bridgewater treatise

Front matter

OMAS HILL D.D.

UUSB LIBRARY

GEOMETRY AND FAITH.

Chap. I. Symmetry In Space I

II. SYMMETRY IN TIME 10

III. NUMBER 15

IV. THE CALCULUS 20

V. APPLIED MATHEMATICS . . . . .24

VI. MOTION 27

VII. MUSCULAR ACTION 32

VIII. GEOMETRICAL INSTINCTS .... 37

IX. MOTION ETERNAL IN DURATION ... 43

X. MOTION OMNIPRESENT IN SPACE . . 46

XI. THE SPHERE OF HUMAN INFLUENCE . . 50

XII. MAGNITUDE 54

XIII. CHANCE AND AVERAGE 64

XIV. PHYLLOTAXIS 1\

XV. NUMBER AND PROPORTION .... 90

XVI. THE DEVELOPMENT OF FORMS . 102

Preface

" THE present work lays no claim to originality." A humble gleaner in the fields of Ma thesis and Theology, I offer only a few of the common fruits, well known to those who have more thoroughly surveyed the boundaries of these two domains. And I have ventured to connect them with the name of the " Bridgewater Treatises," not because I consider myself worthy to appear in company with their writers, but simply that I may thus the more earnestly express my admiration for the treatise of Babbage.

WALTHAM, MASS., 1849.

Preface To The Second Edition

I have rewritten the greater part of this work, and altered so much the expression and illustration of my thought, that I might have given it a new title, but for the affection which twenty -five years' familiarity has bred for the old one.

PORTLAND, ME., 1874.

GEOMETRY AND FAITH.

I. Symmetry In Space

THE universe, actual, possible and impossible, is composed of four elements, spirit, matter, space, and time, which are by no alchemy transmutable into each other. Many alchemists continue, even in this closing half of the nineteenth century, to make the attempt, and some even flatter themselves that they are succeeding ; but the sturdy reply of human consciousness is, that the four elements are diverse and not transmutable ; or, if any transmutation is possible, it must be confined to this, that matter may, in some manner, be an effect of spirit. But to us, finite spirits, nothing more is granted than the re-arrangement, the partial control of matter, not its creation. Matter, as we know it, is distinguished by its being the recipient and dispenser offeree ; which force, so far as we know it, is from spirit alone. This obedience of matter to spirit gives justification to our suspicion that it is the creation of spirit.

Space and time are without parts, and are indivisible except by a mental act. This division is suggested to us by manifest motion in matter. Force shows itself in matter by moving it ; that motion calls our attention to the space and time, within which the motion is taking place ; and we divide mentally this space and time, first from the remainder of the boundless contiguities, secondly into smaller parts. Thus geometry and algebra are generated, the sciences which deal respectively with space and time, those pure entities, the relation of which to the Infinite Spirit we cannot comprehend, but which we become familiar with in the finite portion embraced in our experience, in the universe and its history.

In geometry, the mind imposes upon indivisible space arbitrary boundaries of division, according to arbitrarily selected laws or conditions. These boundaries are of three kinds, surfaces, lines and points. The point is a zero of magnitude in space, but nevertheless is not nothing ; which is nowhere, while the point is somewhere. This contradiction in terms, that a point should have no extension, and yet have a position, is one of those instances, in which geometry abounds, in which the mind is compelled, by the necessity of direct vision, to admit each of two truths, which are to' logic mutual contradictories. The mathematician modifies the law of non-contradiction by confining it to propositions concerning finite quantities.

A lower form of a zero -of magnitude in space is the line, which is extended, at each point, only in two opposite directions ; and the lowest form is the surface ; for which there can, at each point, be drawn a line, such that the surface extends, in every direction, only perpendicular to that line. Geometers define these lower forms of zeroes, or boundaries in space, by the further self-contradiction of imagining the movement of a point ; a double contradiction, since space is itself incapable of motion, much more a zero of magnitude.

A geometrical line is defined as the path of a point, moving according to certain conditions, which always limit its motion, in each of its positions, to one of two opposite directions. Or, it may be defined as a continuous series of all the points which fulfill certain conditions, among which must be the condition that each point is contiguous only to two others, one on the opposite side to the other. So also a surface may be defined as the space in which a point moves, when, in each position which it assumes, a straight line may be drawn through it, and its motion be permitted, in any direction at right angles to that line, and in no other. Or, the surface may be defined as a series of points, through any one of which a straight line may be drawn, such that all the contiguous points lie in a direction at right angles to that line. To either of these definitions of a surface, we must add, in order to make a geometrical surface, some other conditions which the points must fulfill.

When the geometer has selected these conditions and would investigate the form which the points, so conditioned, would enclose, he is not contented with the mere act of reason ; he endeavors to bring imagination to his aid ; to make a sensible image of the form. If he has been blind from his birth, he imagines his fingers feeling out the form ; otherwise he embodies it visibly, as in a drawing, or in a model. If he would convey a knowledge of it to others, he calls matter to his aid, and forces atoms of chalk, black lead, wood or thread, to fulfill approximately the conditions which his geometric law imposes upon the series of points. This drawing, or model, is an expression of his idea, an enunciation of his law. A geometrical figure, whether upon the blackboard, or the printed page, or in a block of wood, or a set of stretched threads, is incontrovertible evidence that a geometer has been expressing, by this means, a geometrical thought.

The laws which please the geometer most highly are those which give us symmetrical figures, figures in which part answers to part ; either on opposite sides of one line or one surface, or about more than one line or surface. This taste is not peculiar to the geometer ; symmetry pleases the most savage, as it does the civilized man; and men whose whole ability lies in other directions, as well as the mathematician. A striking proof of the universality of this taste was shown in the sudden and universal popularity attained by the kaleidoscope. In a few years that toy of Brewster found its way to every parlor, and the heart of every child, ay, and every man in Christendom. Yet its sole magic consists in the symmetry which it imparts to a few fragments of irregular form. But that magic is sufficient to enchant all who come within its sway. We have never found any one uninterested in an extempore kaleidoscope, made by throwing open the piano, and placing brightly colored articles at one end of the folding lid.

All regularity of form is as truly an expression of thought as a geometrical diagram can be. The particles of matter take the form in obedience to a force which is acting according to an intellectual law, imposing conditions on its exercise. It does not alter the reality of this ultimate dependence of symmetry upon thought, simply to introduce a chain of secondary canses, between the original thinking and the final expression of the thought.

Many of the geometer's a priori laws were, indeed, first suggested by the forms of nature. Natural symmetry leads us to investigate, first, the mathematical law which it embodies ; then, the mechanical law which embodies it. Thus all the benefits which have come to our race from the pursuit and discovery and use of the keys of physical science, have been bestowed upon us through these suggestions of geometrical thoughts in the outward creation.

But in the pursuit of mathematical knowledge men began, at an early age, to invent and investigate a priori laws, laws of which they had not received any suggestion from nature. And the intellectual origin of the forms of nature was made still more manifest when these a priori laws, of man's invention, Avere, in many cases, afterwards discovered to have been truly embodied in the universe from the beginning; as, for example, Plato's conic sections in the forms and orbits of the heavenly bodies, and Euclid's division in extreme and mean ratio.

This division in the extreme and mean ratio was invented by the early geometers, without any known suggestion. It is evident that this division might be illustrated in a great variety of ways. A whole must be divided into two parts, such that the first shall bear the same relation to the second that the second does to the whole. No matter what the whole is, a division of it approximately in this manner would be an expression of the idea of extreme and mean ratio. If the whole were a quantity (distance, angle, surface, volume, value, time, velocity, &c.), and the relation were that of magnitude, the whole would be to the smaller part as unity is to half the difference between three and the square root of five. If, on the other hand, the whole were a work of art of any kind, or a system of thought, the relation would not be one of mere magnitude ; and the division would be a work of more ingenuity. But, whatever the whole, or the relation, the proper division would be an expression of the idea.

Now we have, in nature, at least three embodiments of the law of extreme and mean ratio, two of which are very striking. The botanists find that two successive leaves, counting upward on the stem, stand at an angle with each other that is either one-half, one-third, two-fifths, three-eighths of the whole circle ; or some higher approximation to this peculiar proportion. The seed vessels and buds on a spike of broad-leaved plantain afford one of the most instructive examples. They are usually set on a high approximation, so that the order is not apparent. Take a piece of the spike, an inch or so in length, between your hands, and gently twisting reduces it to three ; while a slight twist in the opposite direction brings out five rows, which a harder twist reduces to two.

The efficient cause of this arrangement we do not know. It has been ingeniously suggested that it might be produced by a simple law of the genesis of cells. Let us suppose that each cell emits a new cell at regularly recurring intervals of time, and that the new cell begins to generate cells at the expiration of two intervals after its birth. A cell developing on a plane, under this law, would produce its cells in the phyllotactic order of the leaves, in the terminal rosette of a plant. But it is difficult to see how this hypothesis can be made to include and explain the whole phenomena of the arrangement.

The final causes, although the devout mind always recognizes the impossibility of man's attaining a certainty concerning all the final causes of the phenomenon, are more obvious. It has been shown that this division of the circle insures in the only perfect way to each leaf its chance at zenith light, its best chance at air ; in short, that this phyllotactic law distributes the leaves most evenly about the stem.

In the solar system, if we divide the periodic time of each planet by that of the planet next farthest from the sun, we shall have, beginning with the quotient of Uranus' year divided by that of Neptune and ending with the quotient of Mercury's year divided by that of Venus, a series of fractions agreeing very closely with the approximations of the" phyllotactic law. The problem was similar. The planets would not have remained in proper subjection to the sun had they been allowed to group themselves too frequently in one rebellious line, hanging upon the golden chain of his attraction, dragging him and themselves from their proper orbits. They must be kept evenly distributed about the sun; and since they are moving, the times of their revolution, their angular velocities, must be divided by the same law as that which divides the stationary angles of the leaves.

We have then in the plants a geometrical or angular illustration, and in the planets an algebraical or temporal illustration, of the mathematical idea of extreme and mean rati6. The inference seems irresistible, — these two illustrations, which cannot be imagined as having any causal or genetic connection, owe their intellectual relation to having sprung from One Mind.

This is a striking illustration, but the same inference may be drawn from every form in nature, — planet, crystal, plant, and animals. All natural forms conform more or less closely to geometrical ideals ; sufficiently near to suggest these ideals to men fitted to receive the suggestion ; sufficiently near to show that the whole of nature may, in one sense, be regarded as a series of drawings and models, by which to teach the mathematics to students in the school of life.

The final causes may never, however, be considered as wholly known. The perfection of the Divine workmanship is shown in the adaptation of each object in nature to a great variety of ends. The geometrical laws, on which the world is buijt, are adapted to all the wants and all the needs of every creature. Our human needs are innumerably various, and nature finds means to satisfy them all. Our intellect craves symmetry, and through symmetry is first led to the perception of geometric law. But we love the symmetry before we perceive the law. The sense of beauty i? satisfied, even in externals, most perfectly, and fills us with most pleasure, in things that the understanding fails to analyze and define. Much has been written concerning an analysis of the beauty of outline ; one great painter thinking it consists in flexure, others assigning it to a spiral, or a helix, or an ellipse ; while Darwin refers it to early association, while yet a suckling, with the form of the mother's breast. I venture with diffidence to give my own opinion, that the perception of beauty in outline is the unconscious perception of geometric law, — just as the perception of harmony has been demonstrated to be the unconscious perception of arithmetical ratios in time, or algebraic law. The beauty of outline, I would say of external form, independently of expression, is in proportion to the simplicity of the geometric law, and to the variety of the outline which embodies it. Nor is it essential to the highest enjoyment of beauty that the conformity to geometric ideals should be perfect, any more than it is essential to the highest music to have the harmony perfect. On the contrary, the higher degrees of beauty are apt to be found in forms that suggest, rather than embody, the ideal ; and especially in figures potentially, but not actually, symmetrical. The monotony, which might result from unbroken regularity of form, is avoided, and a new grace is given, for example, to the higher animals, by their temporary disguise of symmetry, in their varied positions and movements. In the sea shells, the same end is attained by the spiral form, which so many of them take ; in which there is not an actual symmetry, but only a law of symmetry, the perfect development of which would require an infinite number of convolutions.

In the forms of vegetative life, there is the widest departure from actual symmetry, and yet a constant suggestion of its laws.

The phyllotactic law secures to the tree a general regularity, and equal growth upon every side ; and yet, by complication of detail, combined with occasional failure or destruction of buds, secures an endless variety of graceful forms, in each species. May we not then name beauty as another final cause, another end secured by the adoption of the division in extreme and mean ratio ? The approximations are beautiful to us, and the pleasure given to us was foreseen when the law was adopted. May it not also have been felt ; and may not the forms of flowers be but approximations toward the expression of an infinite beauty, hidden, from all finite sense, in the incommensurable ratio of that surd ? That the external symmetry of animals may have beauty as its final cause, is rendered probable from the lack of symmetry in the viscera, which are hidden from sight.

Whatever be our speculations upon such points, this at least is manifest, that the sense and the presence of beauty are kindly adapted to each other hi the world. Even shapeless matter declares its Creator's power; the perfect symmetry of crystalline forms, the potential symmetry of all the organic worlds, show forth His wisdom and His love.

II. Symmetry In Time

TIME has but one dimension, and is divisible only into before and after. In the zero of now, the future is becoming the past ; and this suggests the division of the future, and of the past, by the insertion of imagined presents, zero boundaries, dividing time into periods ; these imagined future nows becoming actual, as we successively reach them, and those past having been actual, as we passed them. As time flows only forward, the imagination runs backward into the past with the greatest difficulty ; indeed I am not certain whether it is possible for the imagination to run back ; when I attempt to do so, I find that I leap backward, by longer or shorter leaps, but never run continuously in imagination through time, except forward, from the moment to which I have leaped.

By symmetry in time, therefore, we do not mean a similar arrangement of intervals before and after a certain moment. This has occasionally been attempted in per rede et retro chanting ; but it is a transference of geometrical symmetry to time, where it is out of place, and tasteless. Symmetry in time is the arrangement of two or more similar series of intervals, to follow the same, or successive movements. When the set of similar intervals follows the same moment, it constitutes keeping of time ; when successive moments, rhythm; unless the intervals are very short, when rhythm becomes tone, or color, and keeping time becomes harmony.

SYMMETRY IN TIME. 11

The passion for harmony and rhythm is an essential element in human nature. It is a passion which varies greatly in intensity in different persons, but it is never wholly absent. Savage nations ha/e some rudiments of music and of song. The naked Fuegian, when the stormy winds of his inhospitable straits pause for a while in their wild uproar, chants his songs or hymns in a rude measure and melody. The dullest ear for harmony has an ear for rhythm sufficient to perceive the difference between prose and verse. It is when the intervals of time in rhythm become so short, as to be 'separately imperceptible, that the rhythm is called simply tone ; and harmony is the simultaneous movement of tones. Man finds pleasure in all forms of symmetry in time, whether the parts are perceptible, or imperceptibly short ; and the world has been made in exquisite adaptation to this taste of man.

Space has three dimensions, time but one. Yet, in some respects, time is richer in its contents for man, than is space. The beauty of forms in space, is almost equalled by the beauty of color, and color arises simply from symmetry of times ; it is a kind of tone. Color, indeed, is more expressive, more directly productive of pleasure to the eye, than form. The latter appeals more to the intellect, and is more directly expressive of intellectual ideas ; the former appeals more to the heart, and gives a sweeter pledge of the Divine Love. But beside color, symmetry in time gives us music in all its vast variety of forms and expressions. In music there is a beauty as distinctly intellectual as that of geometrical figures, and a power of expression which geometric form attains, scarcely even in the human figure and face. Nor can we omit to mention heat, which although not giving direct pleasure to the mind, and the heart, as beauty, color, and music do, is still essential to the life of the body and to its comfort Many chemical changes are also produced by minute symmetric motions.

A minute symmetrical division in space produces no sensation for us, except as it may lead to symmetrical motion. Thus the minute symmetry of the particles of a solid in a clear liquid solution, is revealed to us only through the motion of light, and the changes which that motion experiences in going through the solution. In like manner the temporal symmetry of motion in the ray of light coming from the stars, brings us more information than its mere property of making visible can give ; for, on cross-examination by the spectroscope, it confesses the chemical secrets, and the degree of heat in the star at the time of its leaving, ages ago ; and also the direction of the star's motion.

Man's organization, and his surroundings, are adapted to his love of music. His voice is capable of being regulated to musical tones of various pitch. The metals are sufficiently elastic to render them sonorous ; and, as the foundation of all, the air itself, by its elasticity, becomes the vehicle of sound and the instrument of music.

Two gases, intermingled, remain to a certain extent independent of each other ; and, inasmuch as sound travels in each gas with a velocity proportioned to the density and elasticity of that gas, there will be, from a single source of sound, two sounds propagated in the mixture, with different velocities, interfering with each other, and destroying the pure tone of a musical sound. Now the atmosphere is a mixture of heavy oxygen, with lighter nitrogen. The elasticities are, however, so nearly adjusted to the densities, that sounds travel in either gas with nearly the same velocity, so that the air sounds in an organ pipe, as if -one gas. Had sound travelled in these two gases at rates differing as much,

SYMMETRY IN TIME. 13

as the rate in them differs from that in most of the gases known to us, the use of wind instruments of music would have been impossible ; probably all music, even the tones of the human voice, would, in that case, have been discordant to an ear at any considerable distance from the source of sound. With the intense and elevating character of the pleasure derived, first from the tones of human speech, from the melody of birds, and other natural music, and secondly from the art of music, in our minds, we cannot but be grateful for this adaptation of the mingled atmosphere to the needs of man, in his higher nature.

All undulatory motion produces a symmetrical division of time. The beauty of color, like that of tone, arises from an implicit perception of rhythm. The harmony of tints in the landscape, like that of the sounds in a strain of music, arises from the harmony of times in which the vibrations of the mediums occur. The pleasure, in either case, arises from an implicit, or unconscious, perception of keeping time. Heat also has its colors, or tones, as is known to all who have noticed that the sun's heat passes freely through glass ; which is impervious to the heat of a fire. What other advantages to man may hereafter be discovered, in this coloration of heat, time alone can show ; but when we consider to what an extent, through the providence of God, glass is employed, it seems not irreverent to own our gratitude to Him that this substance reflects back the warmth of our apartments, and keeps it within, but allows the heat of the sun to pass through from without.

Besides these hidden proofs of creative foresight, and beneficence, in the concealed, or minute symmetry of time, we shall find open and abundant proofs in manifest rhythmic movements. In the play of alternating muscles, we perceive an adaptation of the physical frame to the intellectual taste. In walking, for example, there are few persons who do not feel the increase of pleasure and of power gained by keeping step. A single drum-tap, regulating the tramp of a large body of men, has sometimes an effect almost equal to that of music. The rhythm of verse, and of music, delights many who are comparatively insensible to both melody and harmony properly so called. Who that reflects upon the genius of Bach, of Handel, of Haydn, and Beethoven, and considers the effect which music, such as theirs, has upon the world, can doubt the kindness of that superintending Power, who kindled the fire in their hearts, and through that in ours ; who also adapted the air, and the various materials, for man, by which he pours out his musical conceptions ? Who that reflects upon the genius of a Homer, and a Shakespeare, and remembers to how many millions their verse has given delight and instruction, can doubt that the same Beneficence gave the poet his power, and men the heart to be touched with poetry ?

III. Number

NUMBER Is not, like space and time, matter and spirit, an integral part of the Universe, nor is it a necessary attribute of either of these. Space and time are without parts or limits, and are, in themselves, so diverse that they would not suggest even the idea of duality. Thus also amorphous matter suggests no number. Number is an impress of thought, it is a pure creation of Spirit ; and its constant suggestion in the forms and periods of nature, is a clear demonstration that nature is the work of an Intellect which controls both space and time in thought. The human intellect early learns number from the text-book of outward nature ; and delights in tracing, further than nature goes, the laws of number.

From the great usefulness of this earliest abstract science, and from the fascination of its pursuit, arithmetic has, in modern schools, been allowed to usurp the place of geometry ; and the pupil has been taught to reason upon abstract numbers before he has learned to conceive clearly imaginary forms. From the same fascinating power, number has sometimes, in the minds of great men, like Pythagoras, been allowed to occupy a disproportionate share of attention ; as though number included all proportion and beauty. Even the Hebrews, with all their clearer light of truth, appear to attach a mysterious power to number.

There is a power in number. When our human thought attempts the survey of space and time, and would subdue these realms to obedience under our intellect, we find ourselves compelled, before we can attain any precision in our forms, to introduce number. The reason can deal, to some extent, with continuous quantity, moving under continuous law, and not in the proportion of numbers. But the imagination cannot take a step with any clearness, much less can the hand build with any satisfaction, without referring quantities to a unit of quantity, to which the ratio shall be that of two numbers to " each other. And of course our finite intellects handle with most ease the smaller numbers ; so that these become to us the most important ; and there is not a number under ten which has not some strong associations with it in the human mind, which give it a kind of sanctity. These mystic charms cluster especially around the odd numbers three, five, seven, and nine ; which seem to have an individuality ; the first-named three being primes, while four is but two twos ; and six, two threes ; and these charms were felt in the earliest ages of human history.

But nature also loves these numbers ; and they are illustrated, even to the untaught mind, by many phenomena ; organic beings possess a unity, which is absolute ; the sexes, of both plants and animals, give us duality ; the powers within, and those above, suggest the threefold division ; the points of the compass, the limbs of mammals, give us the number four ; the fingers of the hand, five, and so on. And the increasing knowledge of the physical world, in our nineteenth century, brings us increasing proof that God, who planned heaven and earth, was acquainted with numbers ; made all things in number, weight, and measure ; and adopted the smaller numbers, either out of preference for them, or in condescension to the minds of his children, whom he has placed here for their preparatory education.

NUMBER. 17

Chemistry is a science of this century, and it teaches us that, from the beginning, the numbers two and three have been dominant powers in the Universe. Simple unites with simple to form a couple, a compound. This couple rarely takes a third element to form a triplet. The 'couplets and triplets unite again in compound couplets, and thus the innumerable variety of substances is built up under the simplest possible combinations of number. Follow these substances through all their various modes of motion and action ; in their weights, in their attractions, their gaseous condition, their volume, their specific heat, their color at a high temperature ; and they are found still to be bound together by simple laws of arithmetical proportion.