The Influence of Classic Greek Thought

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On Growth and Form is an amazing testimony to the power of nature and its impact upon the organism, both organic and inorganic. Many scientists marvel at D’Arcy Thompson’s discourse, his consideration of morphology in the light of mathematics, and his use of illustrations. Thompson’s contributions as a classicist are perhaps most significant in respect to the overall text of On Growth and Form. Looking back to the work of classical Greek scholars and comparing their work to Thompson’s is of paramount importance in understanding the work as a whole. Such a comparison reveals that the very foundation of Thompson’s work is based upon classical convention—not surprisingly, as Thompson was very familiar with the classics due to the influence of his father and perhaps time spent working on his own education.

Greek literature is but a reflection of thoughts and ideas at the root of Western civilization. The Greeks were linguistic masters, and they enjoyed the challenge of a conversation, having a deep appreciation for clean, economical language to exO press their ideas. This time in history is also considered a time of great intellectualism, represented by numerous scholars of various disciplines, from theatre to philosophy to science to writing. The concept of ‘‘know thyself,’’ or the desire for self-awareness, was intensely Greek; for this reason, the Greeks were also considered to be practical, perceptive individuals. All of these qualities seem to resonate in Thompson’s work. Thompson indulges in an intellectualism akin to the Greek tradition. He takes great pains to look at any given topic from various perspectives. All of the topics are also supported by evidence from scholars, scientists, and the like. Thompson, to illuminate a particular point for the reader, draws on the views of these men. A deep consideration of the topic of morphology is apparent in Thompson’s use of a vast array of resources. Further, it is apparent that the author respects and defers to the expertise of other scholars to enhance his own areas of weakness:

That I am no skilled mathematician I have had little need to confess. I am advanced in these enquiries no farther than the threshhold; but something of the use and beauty of mathematics I think I am able to understand.

This quote reveals a certain humility in Thompson. As educated a man as he was, as well-rounded a man as he was, he was also open to considering his inadequacies, admitting that perhaps his abilities were not as polished or advanced as he might have liked them to be. Such candor in a text working at such a high-level of intellectual engagement is a surprise, yet this insistence on self-awareness and honest self-assessment does fall within the realm of Greek ideals.

Greek influences abound in the construction of Thompson’s text, particularly those of Aristotle to whose authority Thompson refers to support the most fundamental and important premise behind his book, that is, his thesis concerning the relevance of mathematical application to the study of morphology. Thompson, in the introductory chapter of the text, draws on a parable of Aristotle’s to further defend the need for employing mathematics in the study of morphology. He explains that, ‘‘in Aristotle’s parable, the house is there that men may live in it; but it is also there because the builders have laid one stone upon another.’’ The words of Aristotle inspire reverence, and in Thompson’s choice of words, he seems to equate his parable with the celestial, or higher life or intelligence. In fact, Aristotle was one of the greatest philosophical minds in the history of the world. During his lifetime,...

(This entire section contains 1463 words.)

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Aristotle studied biology and developed investigative techniques that would become a useful contribution to the study of science. From such ideological roots, Thompson has built his morphological studies.

Comparing ‘‘Nichomachean Ethics,’’ considered to be one of Aristotle’s most popular and influential works, to Thompson’s work reveals the influence of the ancient philosopher’s writings on Thompson. Aristotole’s discussion in ‘‘Ethics’’ deals primarily with the positive side of life. The work is divided into ten books, the first of which will be compared to Thompson’s works. Chapter 1, book 1 opens with the following statement: ‘‘Every art and every scientific inquiry, and similarly, every action and purpose, may be said to aim at some good.’’ From this premise forward, the work is organized hierarchically. Each of these actions or purposes is addressed in subsequent chapters of the work. Politics, to which Aristotle assigns great importance, is the focus of chapter 2. The aim of politics is primarily to create optimal conditions for its citizenry. In chapters 3 and 4, Aristotle’s methodology is established as he engages in the study of politics and ethics. Here, he contrasts the imprecision of scientific inquiry into the study of politics and ethics with that of the study of mathematics, to demonstrate the uncertainty of such study. In chapter 6, Plato’s notion of universal good is introduced to further his discussion. Aristotle believes Plato’s concept to be impractical, viewing good as a subject defined rather by needs, circumstance, and the individual. Aristotle then considers the effect of the dead upon the living in chapter 11, and chapter 12 is an inquiry into the nature of virtue.

On Growth and Form is constructed in a similar manner, although, whereas Thompson’s volume is divided into ten chapters, Aristotle’s is divided into ten books. All of the chapters in On Growth and Form relate to one main topic of discussion: the study of growth and form. Like Aristotle’s opening, chapter 1 of Thompson’s work opens with the following: ‘‘for that the criterion of true science lay in relation to mathematics.’’ He also works from this premise in a hierarchical fashion, working from the essence of form to its greater manifestations. Thompson communicates this movement clearly in chapter 4, stating, ‘‘we pass from the solitary cell to cells in contact with one another—to what we may call in the first instance ‘cell aggregates,’ through which we shall be led ultimately to the study of complex tissues.’’ Considering Aristotle’s admission in chapters 3 and 4 that the study of politics and ethics, in contrast to the study of mathematics, is rendered imprecise, Thompson has much to share regarding his own experiences as a scientist. In chapter 7, he points out the lack of symmetry as an obstacle to his mode of scientific investigation, stating ‘‘let us dispense altogether in this case with mathematics; and be content with a very simple account of the configuration of the horn.’’ Thomp- son is also actively drawing on the theories, discoveries, and observations of countless scholars, but unlike Aristotle in his refutation of Plato, Thompson’s use of such references serves to support or assert his agenda. Curiously, too, he supports, like his mentor, the impact of death on the shape and form of skeletal structure. In chapter 5, he speaks on the impact of precipitates in bone formation, stating, ‘‘for the actual precipitation takes place, as a rule, not in actively living, but in dead or at least inactive tissue.’’ In other words, deposits of organic matter become evident and influence the forms of bones as they separate from the dead tissues that produce them to cause skeletal changes in form.

In comparison to Thompson’s work, however, it is important to note that Aristotle’s work is divided into several books that follow book 1, discussed above. It is also important to make note of Aristotle’s concluding statements in book 10, in which he says, ‘‘It is not enough to know what virtue is, we must strive to have and use it, and try whatever ways we may to become good.’’ Similarly, Thompson has this to share, in conclusion, about his discussion on morphology: ‘‘for the harmony of form is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.’’ Aristotle and Thompson, each in his own right, end their theses, solidifying their arguments in clearly identifiable closing statements.

Like Aristotle and other ancient philosophers, Thompson expressed an appreciation for the literature that supports or describes the scientific, the philosophical, surrounding morphology and the drama that played out in the processes of growth and form. His elation and wonder at such developments are evident in a quote Thompson incorporates in his On Growth and Form from Milton’s Paradise Lost. He recounts the quote thus: ‘‘He hath measured the waters in the hollow of his hand, and meted out heaven with a span, and comprehended the dust of the earth in a measure.’’

Source: Laura Kryhoski, Critical Essay on On Growth and Form, in Nonfiction Classics for Students, The Gale Group, 2002.

A True Naturalist

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On Growth and Form, D’arcy Wentworth Thompson’s classic 1000 and plus page ‘‘essay’’ is not the final word on natural history. It is not a definitive statement of Darwinism, or even anti-Darwinism. It is not required reading for students of evolution. Thompson did not leave a school of inquiry behind him or found a point of view that other zoologists and naturalists could follow and expand upon. And yet, it continues to be read, both by scientists and non-scientists, all over the world.

But what he did do, and the reason why On Growth and Form continues to be read, is to understand nature in a unique way; a way, moreover, which was a departure at the time and has become increasingly rare in the decades since. On Growth and Form is not a book likely to be written today by a respected scientist; nor does it seem likely that a scientist would gain much respect from the writing of it. It is not based on experimentation, and does not suggest directions for further experiment. Although written by a learned zoologist, it hardly engages at all with scholarly literature. Instead, it is written in a style rich with classical, literary, and philosophical allusions, which no one writing for either a purely scientific or a general audience would presume to publish. Yet it continues to be republished, and its readers are willing to go the extra step—to look up Thompson’s allusions, to stick with his difficult passages, to make allowances they would not make for an ordinary scientist.

The reason is that Thompson is not just a scientist. He is a philosopher, an enthusiast, and an evangelist for nature. What gives a reader pleasure from On Growth and Form is its brilliant insight and its beautiful and expansive prose. It is one of the great works of imagination in our language. For those who are interested in such things, it works as an essential corrective to the main line of evolutionary thought.

Perhaps looking at the book in its historical context is the best way to begin to understand and appreciate On Growth and Form. Over fifty years had passed since the world of natural science was revolutionized by Charles Darwin’s theories about evolution and natural selection. Throughout human history, scientists and philosophers had tried to explain how the world’s living creatures, including ourselves, had come into being. Most were ready to credit God as being the first cause; it was the intermediate mechanisms that were so mysterious. Darwin’s theories, so ably, soberly, and logically propounded in The Origin of Species, provided a natural explanation that essentially solved the problem to the satisfaction of most scientists. According to Darwin, random mutation, spread out over geological time, was the mechanism by which species were formed. Individuals with favorable mutations (a little more speed say, or better vision) would tend to reproduce more often in the long run, thus passing these favorable mutations on to their offspring, who would do so in turn. In this way, new species were developed, many of which derived from a common ancestor. Darwin’s natural world was a family tree, with larger and more complex branches extending from the earliest, simplest forms.

Men of Thompson’s age, therefore, who belonged to the first generation of evolutionary zoologists, naturally tended to concentrate on the action of environments on living things. The emphasis continued for half a century, during which time the grand vision of Darwinism shrank down to a merely technical scholarship. ‘‘British zoology,’’ writes P. D. Medawar in D’arcy Wenworth Thompson: The Scholar-Naturalist, ‘‘after fifty years, was still almost wholly occupied with problems of phylogeny and comparative anatomy, that is, with the apportioning out of evolutionary principles and the unraveling of relationships of descent.’’ Thompson was an evolutionary zoologist, but this overconcentration on anatomy made him uneasy. Why study adaptations only for what they say about evolution? Why not consider the mechanisms by which these adoptions happen? Like many naturalists since, Thompson appreciated the explanatory power and beauty of Darwinian theory but was impatient with those of his colleagues who settled for a pat understanding of it. Surely there was more to explore? For Thompson, Darwin was the beginning of understanding, not the end.

It was here that Thompson made his great contribution to biology. Like most sciences, biology had begun to be highly professionalized by the late nineteenth century, a fact that helped to account for the airless, insular atmosphere that frustrated Thompson. Thompson belonged to an older tradition, scientists who were learned in the classics, in literature, and other sciences such as mathematics and physics. It was these last fields that Thompson brought to bear on zoology, in On Growth and Form.

Thompson is primarily interested in natural forms as they are grow within the matrix of the whole physical world—not just the world of predators and prey. In his second chapter, ‘‘On Magnitude,’’ he brings these ideas forward in the most approachable way: how greater size makes whales faster, or why getting wet is so much harder on a fly than it is on a man. As the book develops, he looks more and more closely at the physical laws defining the shape of things. Not every form, Thompson says, has an adaptive benefit; some things look the way they do because of the stuff from which they are made. The spicules of sponges, Thompson says, are a product of crystallization and of adsorption and diffusion. Likewise, his analysis of spiral forms like the chambered nautilus is a tour-de-force based almost entirely on classical geometry.

But On Growth and Form is more than just a book advancing a particular theory of organic development. If that is all it were, it would be no more than a footnote today. Few scientists since World War II have neglected physical environment as a factor in evolution. What is more interesting in the long run is the organic wholeness of Thompson’s approach. Just as Thompson was trained as a mathematician and a classicist as well as a zoologist, his point of view transcends the tunnel-vision of scholarly discourse. Too often, specialists become preoccupied with arcane points of controversy, and develop a kind of technical jargon whose main purpose is to keep the untrained out. Thompson’s mind, in contrast, was supremely inclusive. At first, it might not seem so: Stephen Jay Gould notes, in his introduction to the 1994 edition of On Growth and Form, that ‘‘few people today have his literary and linguistic background; even fewer will grasp the classical allusions (not to mention the untranslated lines of Greek and Latin) that are not mere adornments but intrinsic parts of the text.’’ Thompson does not put these difficult allusions in as a way of showing off, or (as disgruntled readers today might suggest) of practicing an intellectual elitism. His intellect was naturally discursive, unifying, connecting—he could not comfortably keep to one subject, with one vocabulary. Add to this the fact that educated people of his generation could reasonably have been expected to understand most or all of his allusions, and one sees that far from looking to keep readers out, he was trying to bring them in, using all of his background as a humanist to bridge the gap between science and the world outside science.

Wanting to bridge this gap between science and those outside it was part of his complaint with the comparative anatomists, but it also explains something very central to his method of thinking, and to the achievement of On Growth and Form. The book’s most famous and influential chapter, ‘‘On the Theory of Transformations, or the Comparison of Related Forms,’’ is an eloquent demonstration of the folly of looking narrowly at nature. By placing an even grid over natural forms—a prehistoric fish, a skull, a leaf—and distorting that grid mathematically, Thompson powerfully unlocks one of the inner keys of evolution, showing how all parts of a body change over time. Thompson looks at how the totality of a creature’s form changes, where naturalists of his time had been preoccupied with teeth getting sharper, or how a particular quirk of skeletal structure might help an organism survive.

It was Thompson’s great gift to look at morphological change holistically.That gift grew naturally out of his wide-ranging mind. It was because he was a classicist and a mathematician and a humanist as well as a zoologist that he was able to see the larger picture and to inspire later students of living forms to do likewise.

No doubt, some of Thompson’s conclusions, and many of his arguments, have been weakened or refuted entirely by time. It is generally admitted that he has no modern disciples, strictly speaking, but as Medawar notes,

To many of D’Arcy’s contemporaries it must have seemed strange and even perverse that he should have combined a physico-mathematical analysis of Nature with, at all times, a most intense consciousness of its wonder and beauty . . . [but] it is by such diffused and widely pervasive effects as these that we must measure the influence of ‘On Growth and Form’ upon biological science.

It was as a visionary, a ‘‘natural philosopher,’’ and a poet that this nineteenth-century scientist still continues to influence science, even in the twenty- first century.

Source: Josh Ozersky, Critical Essay on On Growth and Form, in Nonfiction Classics for Students, The Gale Group, 2002.

Thompson's Interdisciplinary Approachy to the Theory of Evolution

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When the Scottish classicist, mathematician, and naturalist D’Arcy Thompson wrote On Growth and Form in 1917, Darwin’s theory of evolution was not only relatively new, it was at the height of intellectual fashion, certainly within the scientific community if not with the religious community. Thompson’s highly original thesis on morphology—the branch of biology that examines the forms and structures of living organisms—dared to question the principle of an ever-upward, genetically directed, ascending spiral toward perfection that Darwin had put forth as the notion of natural selection in his seminal work, The Origin of Species, in 1859. Where Darwin draws frequent analogies between Nature and a cow-breeder culling his herd, Thompson sees the physicist’s underlying harmony of cause and effect; he sees energy interacting with matter in mathematical terms.

If Thompson’s work was regarded as controversial in its own time—because it first of all questioned, and then largely ignored, this prevailing assumption of an inevitable progression toward a supposed ideal—today his work seems nothing less than heretical. Or refreshing, depending on one’s point of view. Today, Thompson’s audacity is all the more striking, particularly as his challenge to Darwin comes from the scientific, and not the creationist, quarter. Thompson, while not denying the divine mystery as the absolute final cause, also does not deny the phenomenon of natural selection. He simply points out that this process of elimination of the unfit, of unsuccessful forms, need not imply teleology—that is, a purposeful development or progression toward a final end. To dispute Darwin’s emphasis on ever-improving adaptation through inherited traits, was then and remains now, quite a radical break with the prevailing thinking about nature.

But are not all the changes over time in organic forms, after all, improvements in structure, better suited to meeting challenging changes in the environment? The answer is, actually, no, not necessarily. In fact, while most adjustments in form were at one point in time advantageous to the organism, many of these features have become obsolete. These ‘‘leftovers’’ are simply lingering on past their functional purpose and have not been ‘‘cleaned up’’ after their usefulness has expired. There are also many failures in mutation, many ‘‘wrong turns’’ down blind alleys.

It is Thompson’s contention that this philosophical interpretation of progress becomes a dogma. This thought then blinds the naturalist to the more immediate causes for ‘‘fortuitous variations’’ in form: the interplay of physical forces with geometric structures. Thompson maintains that one does not need to look quite so far afield from the core duality of force and form to learn more about changes in form, whether in terms of the growth of an individual organism or part of an organism, or whether one is looking at evolutionary variations within a species over time. Feeding the doctrine of teleology—the assumption of steady movement toward an end or purpose—starves the scientist of the empirical observation of the direct mechanical causes of adaptation and changes in form. The fundamental mechanics of growth and change in form tend to be overlooked. They can be explored, quantified, and described in mathematical detail, furthering the understanding of morphology.

Thompson’s Introduction to On Growth and Form begins with these words:

Of the chemistry of his day and generation, Kant declared that it was a science, but not Science, for that the criterion of true science lay in its relation to mathematics. . . . A hundred years after Kant, Du Bois Raymond declared that the chemistry of the future must deal with molecular mechanics by the methods and in the strict language of mathematics, as the astronomy of Newton and Laplace dealt with the stars in their courses. . . . [On the mathematical definition of form]: We are brought by means of it in touch with Galileo’s aphorism (as old as Plato, as old as Pythagoras, as old perhaps as the wisdom of the Egyptians), that the book of nature is written in characters of Geometry.

Thompson sees the application of physics and mathematics to the field of zoology as no less an occasion for awe in the face of natural phenomena than the prevailing teleological assumptions. He feels his approach enhances an even deeper appreciation of the intricacies of the ultimate mysteries of life. Thompson expresses his love of nature and wonder at its mysteries in such poetic terms that this has been called as much a work of literature as a work of science. It could also be called an extended meditation on the mathematical patterns inherent in nature everywhere, from the inorganic crystalline structures of snowflakes to the spirals in a nautilus shell to the curves of a living and growing ram’s horn.

While some may recoil from the ‘‘mechanization’’ of the living body of nature, who can resist the engineer’s delight in the design of a bird’s wing? But Thompson can hardly be accused of reductionism when he muses on ‘‘the shape of a snail-shell, the twist of a horn, the outline of a leaf, the texture of a bone, the fabric of a skeleton, the stream-lines of fish or bird, the fairy lace-work of an insect’s wing.’’ These are no less beautiful under closer examination, and he intends no slight when he points out that ‘‘no organic forms exist save such as are in conformity with physical and mathematical laws’’ and demonstrates that all these wonders can be described mathematically. For Thompson, ‘‘Numerical precision is the very soul of science.’’

He elaborates:

To seek, not for ends but for antecedents is the way of the physicist, who finds ‘causes’ in what he has learned to recognize as fundamental properties, or unchanging laws, of matter and energy. In Aristotle’s parable, the house is there that men may live in it; but, the house is also there because the builders have laid one stone upon another. It is as a mechanism, or a mechanical construction, that the physicist looks upon the world; and Democritus, first of physicists and one of the greatest of the Greeks, chose to refer all natural phenomena to mechanism and set the final cause aside.

Throughout On Growth and Form, D’Arcy Thompson’s unique perspective as a triple scholar is called into play. His fully professional background within each discipline does more than just add to the others—there is a blending between them. They have been ‘‘chemically combined,’’ to create a new alloy. His grounding in classics and mathematics assures that he has read (even translated) not only Plato and Aristotle, but Archimedes, Euclid, Pythagoras, et al. He brings the full depth of his decidedly Greek sensibility and insight to bear on his astute observations of nature. Thompson was not only a modern Renaissance man, he was an embodiment of one of his favorite principles, synergy: ‘‘The interaction of two or more agents or forces so that their combined effect is greater than the sum of their individual effects.’’ This is exactly the cumulative effect of his arguments. They combine to form a perspective as multifaceted as one of Thompson’s beloved dodecahedrons (a 12-sided geometric figure).

Each chapter takes a small question of form and examines it from one angle and then another, and then another, in much the same way that Bach introduces a simple melodic phrase, which is then amplified and extended and interpolated by various instruments. Various points invite reciprocal counterpoints. A certain tension escalates in the call and response. Then there is an explosion, as the individual elements come together and develop their own dance, and a full fugue emerges.

Thompson’s overarching theme is that one need not anthropomorphize a celestial engineer working out the immediate advantages of hollowboned birds’ wings with a slide rule and protractor. The laws of nature, divinely inspired, do that naturally. The solid-boned birds, whose more rigid, heavier wings weighed them down or which snapped in the wind, were eliminated from the gene pool, allowing the hollow-boned specimens to thrive and multiply. Thin, cylindrical tubes, by the very nature of their shape, tend not to buckle under their own weight or under outside loads or pressures, such as wind. Thompson celebrates the success of this model in a detailed mathematical contemplation of the interplay of forces and stresses in the air pressure working ‘‘against’’ the wing, and the structural resistance and advantages in the actual construction of the wing. He illustrates his points with ancient and recently discovered ratios and formulas in such a way as to suggest that the aerodynamic circumstances themselves, interacting with the developing organism, gave rise to the wing’s design. Similarly, the strength of a tree is directly related to the windpressure it must withstand.

Thompson gives an hilarious and informative example of this subtle principle in his examination of honeycombs. There was much impassioned discussion at his time of writing concerning the marvel of the bees’ ingenious economy in employing their famous hexagonal shape. There was some debate as to whether it was the bees’ own intelligence that devised this architectural plan, or whether it was divine instruction guiding their instincts. But everyone agreed, including the mathematicians, that the hexagonal shape afforded the absolute most effi- cient use of space within the hive, and the most economical use of wax in the construction of the cells. There was much specious comparison of ‘‘ideal angles’’ (110 degrees and 72 degrees), with impossible ‘‘measurements’’ of the bees’ roughly hewn angles, but they did come close enough to warrant general amazement.

However, Thompson observes that along the outside edges of the hives, the cells are round, and not hexagonal at all. In other words, when the cells of the hive are not adjacent to other cells, they do not assume the shape of a hexagon. He further observes that as the bees are working, they are not creating flat edges to the sides, but rather, simple rounded out hollows or cylinders. The tubular cell walls only become flat, by default, as neighboring bees scrape down ‘‘common’’ walls to a final degree of thinness, each working to maximize the interior area of his cell, pushing up against the adjacent cells of his neighbors.

Thompson drives his point home with the observation that when soap bubbles, for that matter, are amassed together, their spherical shape tends to distort and collapse into flat edges where the bubbles’ walls meet. He concludes his discussion of this subject with the simple fact of geometry that a circle can be surrounded by exactly six circles of similar size; hence, any natural circular or spherical cluster flattens into a hexagonal honeycomb shape, be it soap bubbles or living cells or honeycombs—provided the material with which they are constructed is pliable enough. It is a necessity of physical law, geometric, mathematical law, that they assume this shape. In short, honeycomb cells are hexagons because they have to be.

Actually, flat circles become hexagons, but spheres become dodecahedrons, which are solid. One can observe the same ‘‘packing’’ phenomenon in crates of oranges or the compressed, faceted flesh of pomegranate seeds. This phenomenon has far more to do with intrinsic physical dynamics such as surface tension and matter interacting with forces (such as mutual pressure and weight), than with the mysterious motives and causes, immediate or final, behind insect behavior. This is but one example of why it is important for naturalists to take physics and mathematics into account.

This example of crates of oranges and pomegranate seeds is a very clear and simple illustration of what Thompson means when he asserts that the form of an object is a ‘diagram of forces.’ The object or organism’s shape is like a footprint in the sand, tracing where and how various forces have acted, or are acting, upon matter. The organism’s own process of growth is one such force. That genetics enters the dialogue between force and matter, such that fetal development anticipates environmental stresses, does not detract from Thompson’s argument that force determines form. That is simply nature’s prerogative at work to accelerate the process of ‘‘adaptation,’’ or, response to a force.

This view also need take nothing whatsoever from one’s wonder at the mystery of creation, or the notion of a design or purpose to it. It may not bring one any closer to knowing what that purpose might be, but it certainly doesn’t preclude the question. If anything, one is privileged to glimpse the implicate order at an even deeper level of the overall design. This view only enhances the appreciation of Nature’s creativity:

The harmony of the world is made manifest in Form and Number, and the heart and soul of all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

Thompson is fond of quoting Kant, who says, ‘‘it is nature herself, and not the mathematician, who brings mathematics into natural philosophy.’’

Matter as Diagram of Forces is the introduction of a theme that Thompson builds into a brilliant fugue in his chapter on ratio in logarithmic spirals. What more Greek a concept is there than ratio? And this theme interweaves, warp to woof, with his earlier theme of geometric necessitas

. Thompson seems to delight in presenting ratios throughout the text. He begins with the simple. The ratio between the thickness of a stalk of grass to the stress of its own weight is a square; but, its length to its weight? Cube. This he readily correlates to bone mass and body weight in animals. Gradually he builds to one of the most elegant, and prevalent, ratios known to exist throughout nature. Sometimes called, after Euclid, the Divine or Golden Section, it is also known as the Fibonacci series. Expressed as a ratio, it is roughly 3:5. Expressed mathematically it is a recursive sequence: 1, 1, 2, 3, 5, 8, 13, 21; each value is the sum of the previous two.

Expressed geometrically, this ratio takes the shape of the spiral that makes a nautilus shell a nautilus shell. Each chamber within the nautilus shell equals its two predecessors in size. One sees exactly this type of growth throughout not only the nonliving tissues of seashells and animal horns, but with an added corkscrew twist, in the intervals between leaves on a stem; even between the planets in the Milky Way. Wherever growth occurs in this sequential, additive manner it is always in this 3:5 ratio. Wherever this ratio occurs, this spiral occurs. These spirals exist, like the honeycomb hexagons, not because of genetic programming or mutation or instinct, but because of mathematical necessity.

It is perhaps here that one can best see the signature of nature, signed with a flourish.

Source: Marjorie Partch, Critical Essay on On Growth and Form, in Nonfiction Classics for Students, The Gale Group, 2002.

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