# Summary

Chapter 1: Introduction
The author sets up the basic premise behind the title On Growth and Form in this introductory chapter. Thompson describes the framework for the text, stating that the study of science is not alone dependent on mathematics, nor is it simply to be viewed as unexplainable, divinely created phenomena. His criticism is that both the scientist and the naturalist, among others, attempt to explain the natural world with a limited focus. The intent of his book is to foster a more diverse approach to the study of the concepts of growth and form from a mathematical perspective.

Kant declared that the criterion of true science is in its relation to mathematics. Adds Thompson, ‘‘numerical precision is the very soul of science.’’ Thompson is very careful to point out that he has no interest in reducing the wonders and mystery of the living body to a mechanism (to mathematical formula). He remains an individual who in all creatures is impressed by the beauty manifested in adaptation, that is, the flower for the bee, the berry for the bird. However, he maintains that inquiry into the way in which both living things and physical phenomena grow and take on a specific shape should be approached in the spirit of both scientific theory and mystery. Thompson describes his objective in writing the work as follows:

We want to see how . . . the forms of living things, of the parts of living things, can be explained by physical considerations and to realize that in general no organic forms exist save such as are in conformity with physical and mathematical laws.

More simply put, growth deserves to be studied in relation to form. Both growth and form are necessarily related by mathematical principles, hence the need for a mechanical approach to morphology, or the study of growth and form.

Chapter 2: Magnitude
The form of an object is defined, says Thompson, when we know its magnitude and direction related to the further concept or dimension of time. Growth in length and growth in volume are both parts of one process or phenomenon. Understanding the correlation between length and weight enables us at any time to translate one magnitude to the other by means of mathematical formula.

From a philosophical perspective, the author claims that, concerning magnitude, there is ‘‘no absolute scale of size in the universe, for it is boundless towards the great and also boundless towards the small.’’ Magnitude presents itself conceptually or as an idea in any combination of ways, based on contrasts such as big and small or near and far, for example.

Chapter 3: The Forms of Cells
Surface tension is due to molecular force, arising from the action of one molecule upon another. In the case of a liquid, the molecules of a surface layer are being constantly attracted into the interior by those molecules more deeply situated within the liquid, explains Thompson. He continues, stating, ‘‘the surface shrinks as molecules keep quitting it for the interior, and this surface shrinkage exhibits itself as surface tension. The cell forms explained in the chapter are those forms that a fluid surface can assume under the mere influence of surface tension.

This surface tension accounts, for example, for the spherical form of a raindrop. It is smaller organisms or small cellular elements of larger organisms whose forms will be governed by surface tension. Forms of other larger organisms are dictated by entirely unrelated, non-molecular forces. The author points out, for example, that the surface of a larger body of water is level because it is dictated by...

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gravity, but the surface of water within a narrow tube is curved due to molecular attraction (as in the case of a raindrop).

Thompson discusses various surface tension forms and their surfaces of revolution (surfaces symmetrical about an axis), using soap bubbles to illustrate what happens to create various surface tension forms. For example, if a soap bubble is caught by the end of a pipe and the other side of that same bubble is caught by another pipe, the slow pulling apart of both pipes will cause the bubble to take on a cylindrical form. Eventually, the bubble will break. The fragility of the bubble illustrates Thompson’s next point—that such surfaces realize complete equilibrium within particular dimensional limits or only demonstrate a certain amount of stability, with the exception of the plane or the sphere, whose perfect symmetry allows for perfect stability.

Chapter 4: Forms of Tissues or Cell Aggregates
The movement of the text has been to discuss the nature of solitary cells and then to discuss the action between two cells. This chapter discusses the impact of cell aggregates, or clusters of cells, and the impact of such groupings upon form.

Thompson begins his discussion by restating a general principle underlying the theory of surface tension, namely, that in the case of fluid to fluid or fluid to gas contact, ‘‘a portion of the total energy of the system is proportional to the area of the surface of contact.’’ The author goes on to say that total energy is also ‘‘proportional to a coefficient that is specific for each particular pair of substances and is constant for these,’’ with the exception of changes inspired by temperature or electrical charge. In other words, there are numerical measures (coeffi- cients) that dictate the total energy of a system. Equilibrium, or the state in which minimum potential energy is realized in a system, will be achieved by a reduction in surface contact.

In the case of three bodies, the solution to such a problem becomes a bit more complex. In the example of a substance floating on water, there are three surfaces involved in the process of equilibrium, namely, the water, the air above it, and the substance itself. The condition of equilibrium will be reached by ‘‘contracting those surfaces whose specific energy happens to be large and extending those where it is small.’’ This contraction will result in production of a drop and extension to a spreading film. Turpentine will contract into a drop when it comes in contact with water, as opposed to olive oil, which will instead spread out and form a film on the water’s surface. In the context of either of the previously mentioned examples, a pull for equilibrium on a water surface exists as the result of tensions existing in the other two surfaces of contact. Thompson instructs us to imagine a single particle. If a drop is pulled out by another water particle, without finding yet another providing a counter-pull, ‘‘it will be drawn upon by three different forces, whose directions lie in the three surfaceplanes and whose magnitudes are proportional to the specific tensions characteristic of the three ‘interfacial,’ or smoothly coordinating, surfaces.’’

The chapter then moves from a discussion on the interactions of particles as a function of surface tensions to the forms aggregates of cells tend to take. Thompson elaborates on the tendency not only of cells, ‘‘but of any bodies of uniform size and originally circular outline, close-packed in a plane’’ to adapt the hexagonal pattern, each cell or body being in contact with six others surrounding it ‘‘under widely varying circumstances.’’ Such patterns are evidenced in the bee’s cell as it is formed to create the honeycomb, a circumstance Thompson elaborates on in much detail.

Chapter 5: On Spicules and Spicular Skeletons
The deposition of inorganic material in the human body ‘‘begins by the appearance of small, isolated particles, crystalline or non-crystalline, whose form has little relation or none to the structure of the organism.’’ This deposition then culminates in the formation of spicules, needle-like structures or parts supporting the tissues of certain invertebrates, as well as in the more complex skeletons of the vertebrate animals.

Thompson warns of the complications in studying such forms. In speaking of what he identifies to be ‘‘two distinct problems,’’ Thompson states that what form an inorganic material may take involves two scenarios or situations. ‘‘The form of a spicule or other skeletal element may depend solely on its chemical nature,’’ or ‘‘the inorganic material may be laid down in conformity with the shapes assumed by cells, tissues or organs moulded to the living organism.’’ The author is suggesting, in this case, that points of attachment, or the way bones conform to other tissues, determine their form. The problem between the two scenarios or situations influencing these bony forms arises from the fact that ‘‘there may well be intermediate stages in which both phenomena are simultaneously at work, the molecular forces playing their part in conjunction with other forces inherent in the system.’’

To clarify the problem of the study of such forms, Thompson turns to the problem of studying crystallization in the presence of colloids, or crystals. Such large crystals of salt may be found in a root of rhubarb or the leaf stalk of a begonia. These crystals, extremely numerous in their varieties and form, are ‘‘crystalline forms proper to the salt itself, and belong to two systems, cubic and monoclinic.’’ And in either one, ‘‘according to the amount of crystallisation, this salt is known to crystallise.’’

In contrast, calcium carbonate occurring in plant cells ‘‘appears in the form of fine rounded granules, whose inherent crystalline structure is only revealed under polarised light.’’ Thompson adds that there is a variety of forms a skeleton of carbonate lime takes in accordance with an endless variety of animals. In addition, in death, such skeletal constructs change form due to precipitation of the dead tissues surrounding them, contributing to the final crystallization of such forms. In some cases, an organism can continue to exist without the benefit of materials it may employ to make spicules or its shell.

With these problems in mind, Thompson continues with a careful consideration of the forms of various spicules and other skeletal configurations, focusing on the symmetry of crystallization in the sponge as well as the snow crystal. Thompson also considers the complex patterns arising from crystallized forms, as well as the problem of studying skeletal arrangements arising from fluid crystallization.

Chapter 6: The Equiangular Spiral
‘‘A spiral is a curve, which, starting from a point of origin, continually diminishes in curvature as it recedes from that point; or, in other words, whose radius of curvature continually increases,’’ states Thompson. Basically, as the spiral spins outward, it expands outward as well. Thompson sees the presence of spirals everywhere, in the outline of a leaf, in a lock of hair, or in the coil of an elephant’s trunk. He adds that, although they express themselves in a mathematically similar fashion, they are fundamentally, from a biological standpoint, different. Some spirals formed by natural phenomena, such as the spiral shape a snake makes in movement, are momentary, the result of ‘‘certain muscular forces acting upon a structure of a definite, and normally an essentially different, form.’’ In other words, according to Thompson, these spirals take on more of an attitude or a position than a form, with little or no relation to the phenomenon of growth.

In studying spiral forms, Thompson mentions ‘‘several mathematical curves whose form and development may be so conceived, the two most important (1) the equable spiral, or spiral of Archimedes, and (2) the equiangular or logarithmic spiral.’’ The spiral of Archimedes takes a form resembling a rope being coiled in such a manner that each looping of the spiral coil is identical to the one preceding it. A snail shell, in contrast, spirals out, increasing as the shell increases. Each whorl, or circling out, signifies a geometrical progression as the radius increases. Spirals of both plant and shell then are expounded upon in the chapter’s remaining pages, in keeping with both mathematical curves.

Chapter 7: The Shapes of Horns and Teeth or Tusks
Mathematical study of the forms that horns take is set aside due to the lack of symmetry of horns. Thompson proceeds from this point to consider the morphology of three types of horns—the rhinoceros horn; the horns of the sheep, goats, and antelope; and the solid bony horns, or ‘‘antlers,’’ characteristic of deer.

The rhinoceros horn is ‘‘physiologically equivalent to a mass of consolidated hairs.’’ Thompson states that ‘‘like ordinary hair, it consists of nonliving or formed material, continually added to by the living tissue at its base.’’ The shape of the horn is elliptical, but some horns may be nearly circular. The horn curves into the form of a logarithmic spiral of a small angle. If the horn occupies a median position on the head, there is no tendency for the horn to twist, and the horn is said to develop into a plane logarithmic spiral. Two median horns are not identical in size, one being smaller than the other and ‘‘the rate of growth diminishes as we pass backwards,’’ or move from the first horn ‘‘up the rhino’s head towards the tail end of the animal.’’ Both horns are essentially the same shape, one (the shorter of the two) with less curvature due to its length.

Thompson describes paired horns, those of a sheep or goat, as being hollow-horned, growing under similar yet sometimes varied conditions from those of the rhinoceros. The horn consists of ‘‘a bony core with a covering of skin.’’ The inner layer is supplied with blood vessels, while the outer layer is a fibrous material, ‘‘chemically and morphologically akin to a mass of cemented or consolidated hairs, which constitutes the sheath of the horn.’’ A ‘‘zone of active growth,’’ or ring at the base of the horn, keeps adding to this sheath, ‘‘ring by ring,’’ what Thompson refers to as the ‘‘generating curve’’ of the horn. The two horns, unlike the rhinoceros, are not symmetrical with the animal but are symmetrical with each other, and they form spirals, one being the mirror image of the other. The growth of the horny sheath is described as being periodic rather than continuous (growth in spurts), and this frequency of growth is reflected in the rings of the sheep’s horns, which supposedly tell the age of the animal.

In some cases, the horn does not grow in a continuous spiral, its shape changing as growth proceeds. The reason, says Thompson, is that ‘‘the bony horn-cores about which the bony sheath is shaped and moulded, neither grow continuously nor even remain of constant size after attaining their full growth.’’ The heavy growth of the horns does cause their bony core to bend down due to their weight, and the growth of the horn is guided in a new direction. In older animals, the core is weakened or absorbed, and the horny sheath continues to grow but in a flattened curve different from its original spiral. But later on, says Thompson, the core stiffens and causes the path of the horn to become straighter in course.

‘‘Nail, claw, beak and tooth,’’ states Thompson, all follow the growth pattern of a logarithmic spiral, and the rest of the chapter is dediO cated to these forms as they express themselves morphologically. Chapter 8: On Form and Mechanical Efficiency
The form bones take is the subject of Thompson’s next morphological investigation. Tension and compression are first explained to shape a discussion on bone strength. The author says that ‘‘in all structures raised by engineers,’’ there is a need ‘‘for strength of two kinds, strength to resist compression or crushing, and strength to resist tension or pulling asunder.’’ Whereas some structures, like a wire rope, realize power in tensile stress (resistance by pulling), others, like a column, are loaded to support a downward pressure. Each has a separate function yet can function effectively, complementing one another as part of a system. Thompson refers to the architecture of a suspension bridge to illustrate this point. In this structure, the ropes and wires of the fabric are subject to tensile strain only, built throughout a series of ropes and wires, while the piers at either end of the bridge support the weight of the entire structure. There is an easilydrawn parallel aptly made to the structure of living organisms:

ligament and membrane, muscle and tendon, run between bone and bone; and the beauty and strength of the mechanical construction lie not in one part or in another, but in the harmonics concatenation which all parts, soft and hard, rigid and flexible, tension bearing and pressure-bearing, make up together.

It is the form the matter adopts that determines its strength, no matter how hollow or solid. The long bone of a bird’s wing, for example, is a tubular structure that, although carrying very little weight, is forced to endure ‘‘powerful bending moments.’’ Because this bone is a hollow rather than solid bone, it reacts to such force, (i.e., the wind, as it is flying) without snapping, the hollow construction providing some flexibility. It is this hollow construction that creates a stiffness or rigidity necessary to support the bird’s wing in flight without snapping. Thin, cylindrical tubes, by the very nature of their shape, also tend not to buckle, as in the joints of a crab’s leg, says the author.

Essential to the study of the way bones form to act as supporting structures is an understanding of how stress and compression influence two specific forms. The first situation involves a bending beam. If the stress is applied to a beam, it will be greatest in the middle, and the beam will naturally snap midway, as would a walking stick pressed down against the ground. In reaction to such a strong load, it would be prudent to construct a walking stick or a beam with thick walls in the middle, thinning off gradually at the ends. The resulting structure creates bending moments all along the structure, thus diverting the possibility of what Thompson calls a ‘‘danger point,’’ as in the midsection of a walking stick, that would cause the structure to give.

The second point Thompson considers vital to a discussion on form and mechanical efficiency is best explained in the construction of a beam supported by a bracket. The beam attached to a wall by way of a bracket might react a bit differently to an applied load. The point at which the beam is mounted is subject to some pressure, that is, where the lower surface or part of the beam and the wall meet, directed horizontally against the wall. The beam is also subject to pressure or force from an immediate load bearing down upon it. There is an intersection of pressure, called compression lines, crossing as a result of the load, reflecting both the beam’s reaction to a specific load and the horizontal load at the base of the beam and bracket. Accordingly, it is tension lines functioning together to carry a load.

Based on the principles of load-bearing and strength, Thompson works throughout the remainder of the chapter to explain the variety in shape of various bones and their structures and how these forms interact to support the structure of an organism.

Chapter 9: On the Theory of Transformations, or the Comparison of Related Forms
Recalling the progression of the text in a sweeping fashion, Thompson recounts his thesis and his mathematical approach to the study of growth and form. He points out that the movement of the text in its description of morphology moves from more common forms of speech to precise mathematical language. The result of this movement is a more desirable definition of morphology, one based on economy, on a few words or symbols ‘‘pregnant with meaning.’’

To Thompson, such strict or rigid language is not limiting but instead opens up myriad possibilities. ‘‘The precise definition of an ellipse,’’ he states, ‘‘introduces us to all of the ellipses in the world.’’ In other words, the ellipse expresses itself in countless numbers of natural forms. Every natural phenomenon is a composite and every visible action and effect is the result of countless subordiO nate actions. In other words, more elementary mathematical principles lead to any number of related conclusions. Though there are many forms that are indescribable by mathematical terms, such as a fish or the vertebrate skull, ‘‘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,’’ can all be described mathematically.

Although a figure may be left undefined by such principle, the process of comparison, of recognizing in one form a deformation or alteration in the other, does lie within the province of mathematics, states Thompson, and is called the ‘‘Method of Coordinates.’’ He defines this method as ‘‘a way of translating the form of a curve (as well as the position of a point) into numbers and into words.’’ The study of a statistical curve, for example, is accomplished using the method of coordinates, as is the series of numbers used to plot such a curve. In a related scenario or exercise, the outline of a fish could be plotted in a net of rectangular coordinates and then translated into a table of numbers, as could the outlines of other natural forms. The balance of the chapter is dedicated to such endeavors.

Chapter 10: Epilogue
Summing up the efforts of his work, Thompson reiterates his intentions in writing the work: ‘‘I have sought to show the naturalist how a few mathematical concepts and dynamical principles may help and guide him.’’ He adds, ‘‘I have tried to show the mathematician a field for his labour—a field which few have entered and no man has explored.’’ He finishes the work by praising the importance of morphology, stating during his argument that ‘‘the harmony of the world 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.’’