Popular Science Monthly/Volume 9/June 1876/Mathematics in Evolution

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WHILE we know that only Infinite Intelligence could reduce the entire phenomena of the universe to mathematical expression, it affords an observer constant surprise to find primitive laws of order and number recur again and again amid the infinite variety of Nature.

The spectroscope would seem to indicate that the elements of our present chemistry are really very complex structures, yet we find them, when grouped in all sorts of proportions as molecules, capable of crystallizing in forms of perfect geometrical symmetry, often of much simplicity. In botany, where the factors both chemically and mechanically are extremely various, we find simple laws obeyed in the disposition of leaves, flowers, and parts of flowers; a remarkable instance of which occurs in the growth of leaves on spirally-leaved plants. In the first order of them, a leaf is found in 1/2 the circumference of the stem, and throughout the series the arcs occupied by a leaf are respectively 1/3, 2/5, 3/8, 5/13, 8/21, and 13/34, of a circle, the numerator and denominator of each fraction being those of the two next preceding added together.

In the highest plane of Nature, that of animal forms, the conditions fulfilled are too complex to permit any formulation of lines and angles, but natural history in its first chapters gives us the habitations of the nautilus and other organisms low in the scale of life, which in their beautiful volutes and spirals embody simple geometry. So also does the architecture of our common insects, the bee, wasp, and spider, which, wonderful as it is, must remain less so than the work of the microscopic coral zoöphytes, which, while severally living and building where it is easiest, yet unconsciously coöperate through successive generations to complete a structure of comparatively vast proportions and much symmetrical unity.

These few examples, which might be multiplied indefinitely, may serve as bases for the opinion that complex wholes, acting in many cases like simple ones, may be more easily reducible to mathematical treatment than might at first view be supposed, from the number and variety of ultimate factors concerned in any given problem. Nature would seem to act by but few first principles, which she constantly repeats in her various fields, and which, combined in different ways, yield all her infinite manifestations. The scientific progress of our times is marked by the continual absorption of diverse laws into higher and more general ones; thus the forms of force that used to be thought distinct entities are now proved to be interchangeable, and therefore essentially the same. A minor instance of a like kind occurs in the recent investigation of wave-motion. The old notion was that the particles in water-waves moved up and down in straight lines, but the fact has been demonstrated that they roll in circles having a diameter equal to the amplitude of the wave; this holds of all wave-motion, including light, so that the movements of the planets, as they turn on their axes and circle round the sun, are conveyed to our sight by an ethereal motion of precisely the same kind.

Although mathematical studies find ample illustration in Nature, an exaggerated love of symmetry may be induced by them, causing an enthusiast to pass legitimate bounds in an effort to over-simplify intricate problems; thus Kepler attempted to harmonize the orbits of the five planets with the boundaries of the five regular solids successively contained in each other. Such a vagary, however, could be pardoned in the author of the three immortal laws of astronomy.

In the present stage of knowledge so few of the sciences are exact, that any application of mathematics to the vast and complex processes of evolution is only allowable when the laws considered would be so powerful, did they work in an open field, that, though veiled by many weaker ones, they remain distinctly discernible in the salient features of Nature.

A valid application of this kind is made by Mr. Darwin in his theory of natural selection, where he states the tendency of organisms to multiply according to the law of geometrical progression—a tendency which he shows counts throughout the mazy conflict of forces affecting organic life. The purpose of this paper is to trace some effects of other such laws, in their theoretical simplicity so extremely potent, that their results persist through all practical qualifications, and so, when shown to account for observed facts, may serve as tenable ground for inference and deduction.

In evolution heterogeneity is a constant measure of progress, hence the laws stating the variety of effects producible from given elements have a direct interest and value. These are the laws of combination and permutation. Combinations, mathematically, are groups where the presence and not the position of an element counts for difference—thus B C A and B A C are the same combination but different permutations. As additions are made to the elements, combinations increase in geometrical progression with 2 as constant factor. Thus 2 elements yield 4; 3, 8; 4, 16; until, when we reach 63, the number of elements in chemistry, we find more than nine quintillions of combinations to be possible. This law tends to hold only in cases where the particular position of an element in a group is indifferent, as in the superimposition of colors in light; as in the simple molecules of chemistry, where, for instance, the result is the same, whether H2 unites with O, or O with H2; and as in all merely mechanical mixing of ingredients in manufacture, as pottery, gunpowder, and so on. Such cases are less common in Nature and art than those in which definite positions are points of difference, as we find in the atomic grouping of compound molecules, where the phenomena of isomerism appear; in the order of successive sounds, whether in language or music; and as in the various series in which muscular and nervous forces coordinate in animal movements. In all such cases the multiplication of effects tends to follow a law of even greater increase than that of geometrical progression—namely, the law of permutations.

If A B C be elements given, their permutations in groups of 3 are 6 (3 x 2 x 1), in groups of 2, 6 more, and adding 3 for the elements taken singly, 15 is obtained as the number of permutations of all kinds. The addition of a new element increases them to 64 (15 x 4 + 4), and so on in a ratio increasing with every additional element, until we find that 10 produce 9,856,900 permutations, and but 1,024 combinations.

These abstract laws are paralleled by the multiplied results which follow in the wake of any important invention or discovery. Forty years ago the main arts of representation were five in number—sculpture, painting, printing, engraving, and lithography. The art of photography, introduced by Daguerre in 1839, and since so beautifully developed, is continually increasing derivative arts. It is applicable to every other main art, and may become an element in new permutative groups of them. It has already given aid to the sculptor, the painter, and the engraver, and in the heliotype and woodbury type exhibits relations with lithography and printing; besides, it has added to human power in many other ways, has made the stereoscope available, bringing the natural beauties and artistic treasures of distant lands vividly near; it has aided astronomy in fixing views of transits and eclipses of brief duration, and in mapping the sun and moon; the physiologist has used it to preserve the evanescent exhibitions of dissection; and in observatories it accurately marks the minute movements of delicate apparatus. It limns the interiors of pyramids, catacombs, caves, and mines, giving incidental help to archaeology and geology; and, in regions inaccessible to man, pictures the depths of the sea. It serves in war—and might in peace—to aid the topographer in mapping plans of city and country; in times of siege it has reduced correspondence to microscopic limits for carriage in the only possible way—by birds; and from year to year this wonderful art continues to be applied in new and valuable uses.

The illustration it affords of the manner in which human resources are multiplied by the accession of a new discovery might be repeated, were all the applications and results of the steam-engine, locomotive, or telegraph, traced in their numerous ramifications. So far from these mighty achievements exhausting the conquests possible to man, they are merely centres of new circles of power from which he may successively penetrate into the ever-boundless regions of the unknown.

The late Mr. Mill, at a period of great depression in his early life, found relief in the charms of music, and strangely enough dreaded an exhaustion of it, just as many other people who have not the excuse of morbid ailment think that all the greatest possible discoveries have been made, and that all the finest things in prose and verse have been said. Such notions are denied by the laws which have been stated, as exemplified not only in the diversity and might of modern achievement, but also in the deep relations between the elements of natural action divulged by their very multiplication of effects; the generalizations of this age have never been equaled in scope and force—the persistence of force and the theory of evolution.

As sciences advance, their essential unity becomes more and more evident; methods that at first view would seem utterly unconnected are being constantly found to have a secret and helpful family tie. The comparative value of various types of bridges has been investigated by submitting glass models duly weighted to polarized light, which shows at once the distributions of strain and pressure. A common magnetic needle has been successfully employed in finding weak places in iron and steel axles by its unequal deflection at such points, due to internal heterogeneity in the mass examined. At Paris recently an underground pneumatic tube became obstructed at an unknown point; excavation was correctly guided by the adoption of an acoustic principle; a loud sound was made at the tube's entrance, and the time occupied before the reflected wave returned was carefully noted, from which was inferred the distance traversed by it to and from the obstacle. Many instruments at first made for purely philosophical study have been drafted into the world's practical uses. Applications of the rheostat and Wheatstone's bridge serve to locate the oft-recurring breaks in ocean-cables and telegraph-lines, and have very lately yielded the marvellous duplex and quadruplex telegraphs. The spectroscope, originally directed to the heavens, has now found uses on earth of great value; it detects adulteration, marks defectiveness in drainage, and points out impurities in water-supplies.[1]

So that, in the tree of knowledge, as the branches grow in all directions, their offshoots come to touch at innumerable points.

The multiplication of effects may be traced not only in physics, chemistry, and cognate sciences, but also in the chapters of natural history and the facts of human life. The organized faculties of an animal which are distinctly different may be considered—of course, with proper qualification—as elements which may be grouped permutatively in the various actions directed to aid maintenance or promote safety; although, in the case of any particular variety of a species, a vast discrepancy must exist between the theoretical results of the mathematical law and the number of different groupings really made, yet, if the discrepancy is tolerably constant in degree in any two successive cases, the relations between two such cases may be stated by the law with an approximation to truth. Thus if a variety of quadrupeds with, say, four distinct and presumedly averaged powers be taken, at first sight it would seem but one-third better off in the struggle for existence than another variety with three several powers; yet the one may have an advantage over the other as great as four to one, for the variety of actions possible to the former may cover a field four times as great as the others. This aids us in understanding why variations in useful rather than those in useless directions tend strongly to persist. They do so because of the immense exaltation of power that comes with the development of any new faculty, any new means of securing a livelihood or escaping danger; and so great is this exaltation that even minor degrees of development have an appreciable value and tend to become permanent and to increase.

The effects of the laws under consideration also help to make clear why transition periods in organic Nature have been brief as revealed in their infrequent traces in such geological records as we possess.

When new circumstances have demanded the acquisition of new powers, or rather the development of dormant ones, the odds have been overwhelming against such individuals of a race as have been inelastic in the required direction, so that in a comparatively short period all that lived knew the new lesson.

A further corollary which harmonizes with observed facts is that, as species progress, an ever-increasing width of gap would separate kind from kind, and the highest individual of a kind from the next below it. The lowest organisms, monera, have no definite shape; polyps, some grades above them, conform very tolerably to a certain outline; and so on in the scale of life an increasing individuality keeps pace with an increasing divergency, until man and the tree mark the two great summits of Nature in her animal and vegetal forms.

Many able students of the theory of evolution stop short at the chasm which divides the human climax from the allied primates, and hesitate to believe that there can be a common origin for apes and the race which has produced a Beethoven and a Raphael; but a consideration of the laws which have been stated, and which are closely borne out by observation, would lead us to expect just what we find, namely, in the processes of development intermediate links would drop out after comparatively brief existence between planes of life increasingly separated, so that the last difference of power and intelligence would be the greatest of all.

And, furthermore, the same laws make intelligible the vast gulfs we find fixed between our intellectual giants and the rest of mankind, so that they form a small solitary band above us all, leaving a mere understanding of their mighty works the test of our highest powers. A single English dramatist and a single English mathematician have probably equaled in scope and excellence of original work, in their several fields, all the like labors of their countrymen put together.

Two other mathematical laws, abstractedly of great power and generality, may be noticed in the many phases of evolution, namely, those treating of the relations between areas and solids of the same form, varying in size. In like plane figures, boundaries increase directly as like dimensions and areas, as the square; in similar solids, surfaces increase as the square and contents, as the cube of like dimensions. These laws state in an abstract way the economy of aggregation, whether domestic, industrial, social, or political. The farmer profits by them when he takes down costly fences in enlarging his domain; the ship-builder avails himself of them when he models his monster craft which shall carry the cargo of half a dozen small vessels at half the expense; the Broadway architect embodies them in his lofty designs, rivaling in a business structure the height of a common church-steeple, putting two ordinary buildings on one lot of ground.

From the time when animals first noticed that two together were stronger than two singly, the gregarious instinct has been assisted in taking a firm hold on many species from its usefulness in attack and defense; where it is not exhibited, exceptional circumstances prevent: for instance, a spider would have nothing to gain by going into partnership, for it preys on flies much weaker than itself, and no company of spiders, however large, could do battle with a swallow, or a housemaid armed with a broom.

Speaking in a general way, such savage tribes of men as have had the strongest social feeling, and the largest mutual confidence, have, other things equal, had an advantage over less coherent neighbors, and so on, until now modern history deals with national groups fewer than ever before, and becoming fewer still.

In commerce, also, the largest banks, mercantile firms, and factories, grow continually larger by virtue of the less expense attending the management of extensive groups. The costly competition of many small manufactories and merchants is passing away before the more economical methods of a few strong concerns. Coöperation in labor, and in the supply of a community with goods, has succeeded to an encouraging extent in Europe, and in some degree on this continent.

In domestic life, also, the burden of sustaining the usual isolated homes is beginning to be thought grievous and unnecessary. The constant repetition of the same details on a small scale, in cooking, warming, and attendance, is evidently subject to a large discount in cost, and increase of comfort, when a number of families combine to have a single kitchen, heating-furnace, and corps of servants. Many solutions of this problem have been attempted with various success; large houses rented in fiats, copied from European models, adorn some of the chief streets of New York and Boston, and hotels on all sorts of systems are to be found in our principal cities, numbering among their patrons thousands of families. It may be reasonably expected that in the near future some plan will be arrived at, and widely accepted, combining the benefits of individual homes with the advantages of association; but, for this result, an improvement in our present crudeness of social feelings must take place. Great is the premium placed on the growth of mutual harmony and confidence, yet how slow that growth is!

A process analogous to aggregation is that of concentration, which marks many of the forms of progress. When a force operates against a lesser one of constant amount, concentration multiplies its efficiency.

If a common furnace's heat is 3,500°, and a temperature of say 3,000° is required to melt iron, then but 500° of 3,500° are available for that purpose; but, when the same quantity of heat is presented at 4,200°, 1,200° of 4,200° may be utilized, an efficiency twice as great as the former. Hence the value of such an invention as the hot-blast, increasing the intensity of flame: the inert and diluting nitrogen is mingled with the oxygen of common air by the feeble force of diffusion; if they could be cheaply separated, it would mightily enhance the value of coal. Steam-engines, as now constructed, rarely yield in work more than a tenth the equivalent of heat applied; the chief waste is in the exhaust-steam, which, although in immense quantity, is of too low a temperature to raise more steam. Any feasible plan of concentration is all that is wanted to make the steam-engine more powerful; its duty has already been nearly doubled by the use of much higher pressures than Watt employed or sanctioned. A pebble on a sea-beach may have been exposed to the sun for ages without perceptible effect, but the focusing of a lens may reduce it to the liquid state in a few moments with no more solar beams than might have otherwise idly fallen upon it in an hour. This same principle also obtains in the operations of trade and business: the expenses of a railroad, steamship, or hotel, are pretty constant, and a certain amount of patronage pays them; beyond this point profits rapidly accumulate, and below it so do losses; small fluctuations produce large results in the balance-sheet.

Successive increments of difference in degree may gradually merge and become exalted into a difference in kind. A number of pendulums might, if unresisted, vibrate in an arc forever, but, if on one of them the movements of the others are suitably concentrated, its arc will gradually increase in amplitude until it becomes a circle.

This principle of concentration appears in organic Nature in the physiological division of labor, and in the adaptation of every organism to some particular environment which may be to it its field and kingdom. Analogy would lead us to suppose that the different duties of the brain are performed by special parts. So directly profitable has the division of labor been found in manufacturing industry, that in many cases it has been pushed to an injurious extreme, for a man is stunted in development when all his powers of mind and body but one remain unexercised. Specialists in art and science discover that their highest excellence can only come with a comprehension of wide principles and study in many various fields.

So far from concentration being invariably useful, diffusion may be a process incident to progress. A lump is soonest leavened by leaven distributed throughout it, crystallization proceeds more swiftly from separate nuclei than from a concrete mass. Analogously, the best, wisest, and most talented men of a people exert a larger influence when scattered through it than if gathered into an over-centralized capital, where they radiate chiefly on each other.

In the laws which have been considered thus briefly, it has appeared that their tendencies are continually progressive; that, while the capital of evolution is being increased, so also is the rate of compound interest by which it accumulates. It is now fitting that some of the causes should be noticed which reduce these tendencies from their theoretical power to the moderate activity we find them really presenting.

A minor and unfavorable sort of natural selection is that made by animals not carnivorous when they have a choice of food; they take the best to be had, and leave the rest to propagate its kind. This residue may be very bad indeed, when the total supply is scanty; in crowded pastures the grazing herds only permit the worst parts of the clover to come to seed, and squirrels always first eat the best nuts stored in their hiding-places, and any surplus that might germinate and grow is commonly of a very poor kind. The acquisition of new powers by an animal is usually accompanied by a gradual and injurious loss of its original ones; neither the omnivorous hog nor the higher primates can number readiness in swimming among their resources, although their inferior ancestry doubtless could. The introduction of machinery is steadily causing us to lose the deftness and dexterity of the old, unaided handicrafts, yet never so much as now were knack and skill of value, for they are indispensable to the designer and inventor in their work. A highly-cultivated citizen of New York, when he penetrates the wilds of the far West, must have an Indian to guide him through prairie and forest, for the red-man's perceptions of the phenomena around him remain keen and almost intuitive.

Modern arts vastly outnumber ancient ones, yet do not include them all; antiquity possessed many, either lost by neglect or by being secret with individuals and perishing with them, or perhaps in the extirpation of small, highly-gifted communities by overwhelming barbarous hordes.

The vast preponderance of mediocrity over exalted talent has always limited the influence of intellectual greatness, and at times even perverted it to confirm the low standard of a community's intelligence instead of raising it. A key in metaphor is always something unlocking or unfolding the hidden—this refers to but half the business of a key—it is also used to bind, lock up, and secrete. History furnishes many examples of an unusual might of mind permitted, by the lack of appreciation for its best work, not only to leave it undone, but induced to acquire power by mystifying difficulties instead of resolving them, and so to retard progress by an exertion of the very capacity that might assist it.

The individuals of a community rise pretty much together, and the voice of circumstances is not so loudly "Be your best," as "Be fit." The limit to the practical value of greatness becomes plain if we imagine Kepler, while making a scientific journey, to be suddenly surrounded by hostile Sioux. We can believe that the world may not know some of its greatest sons, for greatness is known only when allied with the talents of publicity and the circumstances of appreciation.

Truths and suggestions beyond the comprehension of hearers have doubtless often been uttered in vain. Our guides in the path of knowledge must keep within easy distance if they are to be useful. Huyghens, the great Dutch philosopher, clearly propounded the wave-theory of light, but it remained unnoticed in his times, to be rediscovered a century afterward, when the minds of scientific men had been prepared to receive it.

Then, again, the very intensity of appreciation bestowed upon genius may be hurtful, in the diversion of men of some original power from the development of themselves into the army of mere repeaters, imitators, and quoters. Besides, when the leaders of thought and investigation have erred, as at times they inevitably must, the mistaken opinion from the weight of a great name becomes a clog and a barrier. Newton's emission-theory of light delayed the true explanation through many weary years; and zoology is still suffering from the belief in catastrophes entertained in the mighty brain of Cuvier. And, further, physiologically, the antagonism of growth and reproduction has left the chiefs of men either childless, as Kant, or continued in a puny race, as Cromwell. Talent is hereditary, but genius scarcely.

Progress is also thwarted by the sub-evolution of evil. In human societies, as mutual trust and confidence advance, they are liable to be rudely checked from time to time as the rewards of the liar and thief temptingly increase. The very perfection of mechanical appliances is used by the burglar and counterfeiter, and only a high degree of educated ingenuity and a world-wide mercantile good faith could have made such a fiend as Thomassen possible. The invention of new machinery, the manufacture of new chemicals, the extensions of mining, and the commingling of increased travel, in their accidents and sometimes in their baneful results in common pursuit, render the tasks of physician and surgeon more difficult than ever before. The complications of modern life are so great and varied, that the moral laws do not possess the direct and simple force they had of old; in the surge and vortex of to-day it takes a keen intellect to separate right from wrong, and many err because their consciences are not reinforced by education for the new exigencies.

Evolution is underlaid, as is all change, by the greater law of the persistence of force, ever holding the even balance through all complexity, maintaining throughout all a just compensation. Every new faculty and enjoyment is earned by its equivalent of work, trouble, or ill; with every addition to power comes an addition to wants, to labor, and the possibilities of pain. As the stores of the mind increase so also do ideals craving satisfaction become higher and wider: ever "on the isthmus of a middle state," man is at once a record of the past and a prophecy of the future; limited by his inheritance to definite acquirement, he yet aspires, by nascent impulses, for such better things as only his posterity can ever possess.

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  1. A proposition in pure mathematics may receive elucidation and extension by an illustration taken from optics. In Newton's "Principia," book i., section xii., prop. 70, he proves, in a manner very difficult to follow, that a corpuscle placed within a hollow sphere, if attracted as the square of the distance by all the points in the concave surface, will remain unmoved wherever placed, as the sums of attraction always balance.

    This may be made clear not only of a spherical surface, but the closed interior of any surface whatever, provided it has no reëntrant angles, as a pyramid or an obliquely-truncated cone.


    For, imagine the corpuscle to be luminous and to be bisected by any plane extended so as to cut the containing hollow surface into two parts, it is evident that equal amounts of light are radiated by each half of the corpuscle on each of the two parts of the surface containing it. Now, these rays diminish in intensity as the square of the distance, and so reciprocally correspond with a force emanating according to the same law from the surface and affecting the corpuscle. Hence, the area of the surface of any hollow body, having no reëntrant angles, varies as the square of its average distance from any point within it.