Editing On Growth and Form (section)

==Contents==

The contents of the chapters in the first edition are summarized below. All but Chapter 11 have the same titles in the second edition, but many are longer, as indicated by the page numbering of the start of each chapter. Bonner's abridgment shortened all the chapters, and removed some completely, again as indicated at the start of each chapter's entry below.

===1. Introductory===

(1st edition p. 1 – 2nd edition p. 1 – Bonner p. 1)
:: Thompson names the progress of [[chemistry]] towards [[Kant]]'s goal of a mathematical science able to explain reactions by molecular mechanics, and points out that zoology has been slow to look to mathematics. He agrees that zoologists rightly seek for  reasons in animals' [[adaptation]]s, and reminds readers of the related but far older philosophical search for [[teleology]], explanation by some [[Aristotle|Aristotelian]] final cause. His analysis of "growth and form" will try to show how these can be explained with ordinary [[physical laws]].

=== 2. On Magnitude ===
[[Image:Boat models by William Froude.JPG|thumb|right|Models used (by [[William Froude]]) to show that the [[Froude number|drag on a hull varies with square root of waterline length]]<ref>{{Cite book | last=Newman | first=John Nicholas | author-link=John Nicholas Newman | title=Marine hydrodynamics | url=https://archive.org/details/marinehydrodynam00newm | url-access=limited | year=1977 | publisher=[[MIT Press]] | location=Cambridge, Massachusetts  | isbn=978-0-262-14026-3 | page=[https://archive.org/details/marinehydrodynam00newm/page/n43 28]}}</ref>]]

(1st p. 16 – 2nd p. 22 – Bonner p. 15)
:: Thompson begins by showing that an [[Surface-area-to-volume ratio|animal's surface and volume (or weight) increase with the square and cube of its length]], respectively, and deducing simple rules for how bodies will change with size. He shows in a few short equations that the [[Froude number|speed of a fish or ship rises with the square root of its length]]. He then derives the slightly more complex scaling laws for birds or aircraft in flight. He shows that an organism thousands of times smaller than a bacterium is essentially impossible.

===3. The Rate of Growth ===

(1st p. 50 – 2nd p. 78 – Bonner removed)
:: Thompson points out that all changes of form are phenomena of growth. He analyses growth curves for man, noting rapid growth before birth and again in the teens; and then curves for other animals. In plants, growth is often in pulses, as in ''[[Spirogyra]]'', peaks at a specific temperature, and below that value roughly doubles every 10 degrees Celsius. [[Dendrochronology|Tree growth]] varies cyclically with season (less strongly in evergreens), preserving a record of historic climates. Tadpole tails [[Regeneration (biology)|regenerate]] rapidly at first, slowing exponentially.

===4. On the Internal Form and Structure of the Cell===

(1st p. 156 – 2nd p. 286 – Bonner removed)
::Thompson argues for the need to study cells with physical methods, as morphology alone had little explanatory value. He notes that in [[mitosis]] the dividing cells look like iron filings between the poles of a magnet, in other words like a [[force field (physics)|force field]].

===5. The Forms of Cells===
[[File:Vorticella campanula.jpg|thumb|''[[Vorticella campanula]]'' (stalked cup shaped organisms) attached to a green plant]]

(1st p. 201 – 2nd p. 346 – Bonner p. 49)
::He considers the forces such as [[surface tension]] acting on cells, and [[Plateau's laws|Plateau's experiments]] on [[soap film]]s. He illustrates the way a splash breaks into droplets and compares this to the shapes of [[Campanularia]]n zoophytes ([[Hydrozoa]]). He looks at the flask-like shapes of [[protista|single-celled organisms]] such as species of ''[[Vorticella]]'', considering teleological and physical explanations of their having [[minimal area]]s; and at the hanging drop shapes of some [[Foraminifera]] such as ''[[Lagena (foraminifera)|Lagena]]''. He argues that the cells of [[trypanosomatid|trypanosomes]] are similarly shaped by surface tension.

===6. A Note on Adsorption===

(1st p. 277 – 2nd p. 444 – Bonner removed)
::Thompson notes that surface tension in living cells is reduced by substances resembling oils and soaps; where the concentrations of these vary locally, the shapes of cells are affected. In the green alga ''[[Mesocarpus|Pleurocarpus]]'' ([[Zygnematales]]), potassium is concentrated near growing points in the cell.

===7. The Forms of Tissues, or Cell-aggregates===

(1st p. 293 – 2nd p. 465 – Bonner p. 88)
::Thompson observes that in multicellular organisms, cells influence each other's shapes with [[triangle of forces|triangles of forces]]. He analyses [[parenchyma]] and the cells in a [[embryogenesis|frog's egg]] as soap films, and considers the [[symmetry|symmetries]] bubbles meeting at points and edges. He compares the shapes of living and fossil [[corals]] such as ''[[Cyathophyllum]]'' and  ''[[Comoseris]]'', and the hexagonal structure of [[honeycomb]], to such soap bubble structures.

===8. The same (continued)===
(1st p. 346 – 2nd p. 566 – Bonner merged with previous chapter)
::Thompson considers the laws governing the shapes of cells, at least in simple cases such as the fine hairs (a cell thick) in the rhizoids of [[moss]]es. He analyses the geometry of cells in a frog's egg when it has divided into 4, 8 and even 64 cells. He shows that uniform growth can lead to unequal cell sizes, and argues that the way cells divide is driven by the shape of the dividing structure (and not vice versa).

===9. On Concretions, Spicules, and Spicular Skeletons===
[[File:Demospongiae spicule diversity.png|thumb|upright=1.2<!--size for low image-->|A selection of [[spicule (sponge)|spicules]] in the [[Demospongiae]]]]

(1st p. 411 – 2nd p. 645 – Bonner p. 132)
::Thompson considers the skeletal structures of [[diatom]]s, [[radiolarians]], foraminifera and [[sponge]]s, many of which contain hard [[spicule (sponge)|spicules]] with geometric shapes. He notes that these structures form outside living cells, so that physical forces must be involved.

===10. A Parenthetic Note on Geodetics===
(1st p. 488 – 2nd p. 741 – Bonner removed)
::Thompson applies the use of the [[geodesy|geodetic]] line, "the shortest distance between two points on the surface of a solid of revolution", to the spiral thickening of plant cell walls and other cases.

===11. The Logarithmic Spiral ['The Equiangular Spiral' in 2nd Ed.]===
[[Image:NautilusCutawayLogarithmicSpiral.jpg|thumb|Halved shell of ''[[Nautilus]]'' showing the chambers (camerae) in a [[logarithmic spiral]]]]

(1st p. 493 – 2nd p. 748 – Bonner p. 172)
::Thompson observes that there are many [[patterns in nature|spirals in nature]], from the horns of ruminants to the shells of molluscs; other spirals are found among the florets of the sunflower. He notes that the mathematics of these are similar but the biology differs. He describes the [[spiral of Archimedes]] before moving on to the [[logarithmic spiral]], which has the property of never changing its shape: it is equiangular and is continually [[self-similar]]. Shells as diverse as ''[[Haliotis]]'', ''[[Triton (gastropod)|Triton]]'', ''[[Terebra]]'' and ''[[Nautilus]]'' (illustrated with a halved shell and a [[radiograph]]) have this property; different shapes are generated by sweeping out curves (or arbitrary shapes) by rotation, and if desired also by moving downwards. Thompson analyses both living molluscs and fossils such as [[ammonite]]s.

===12. The Spiral Shells of the Foraminifera===
(1st p. 587 – 2nd p. 850 – Bonner merged with previous chapter)
::Thompson analyses diverse forms of minute spiral shells of the [[foraminifera]], many of which are logarithmic, others irregular, in a manner similar to the previous chapter.

===13. The Shapes of Horns, and of Teeth or Tusks: with A Note on Torsion===
[[File:Big Horn Sheep 2.jpg|thumb|The spiral horns of the male bighorn sheep, ''[[Ovis canadensis]]'']]
(1st p. 612 – 2nd p. 874 – Bonner p. 202)
::Thompson considers the three types of horn that occur in quadrupeds: the [[keratin]] horn of the [[rhinoceros]]; the paired horns of sheep or goats; and the bony [[antlers]] of deer.
:: In a note on torsion, Thompson mentions [[Charles Darwin]]'s treatment of climbing plants which often spiral around a support, noting that Darwin also observed that the spiralling stems were themselves twisted. Thompson disagrees with Darwin's [[teleological argument|teleological explanation]], that the twisting makes the stems stiffer in the same way as the twisting of a rope; Thompson's view is that the mechanical adhesion of the climbing stem to the support sets up a system of forces which act as a 'couple' offset from the centre of the stem, making it twist.

===14. On Leaf-arrangement, or Phyllotaxis===
<!--[[File:Cones14062008.jpg|thumb|upright|[[Phyllotaxis]] of cones of ''[[Abies koreana]]'']]-->
[[File:SunFlower Closeup Hungary.jpg|thumb|[[Phyllotaxis]] of [[sunflower]] florets]]

(1st p. 635 – 2nd p. 912 – Bonner removed)
::Thompson analyses [[phyllotaxis]], the arrangement of plant parts around an axis. He notes that such parts include leaves around a stem; fir cones made of scales; sunflower florets forming an elaborate crisscrossing [[patterns in nature|pattern]] of different spirals (parastichies). He recognises their beauty but dismisses any mystical notions; instead he remarks that

::{{quote|When the bricklayer builds a factory chimney, he lays his bricks in a certain steady, orderly way, with no thought of the spiral patterns to which this orderly sequence inevitably leads, and which spiral patterns are by no means "subjective".|Thompson, 1917, page 641}}

The numbers that result from such spiral arrangements are the [[Fibonacci sequence]] of ratios 1/2, 2/3, 3.5 ... converging on 0.61803..., the [[golden ratio]] which is

::{{quote|beloved of the circle-squarer, and of all those who seek to find, and then to penetrate, the secrets of the Great Pyramid. It is deep-set in [[Pythagoreanism|Pythagorean]] as well as in [[Euclidean geometry]].|Thompson, 1917, page 649}}

===15. On the Shapes of Eggs, and of certain other Hollow Structures===
(1st p. 652 – 2nd p. 934 – Bonner removed)
::Eggs are what Thompson calls simple solids of revolution, varying from the nearly spherical eggs of [[owl]]s through more typical ovoid eggs like chickens, to the markedly pointed eggs of cliff-nesting birds like the [[guillemot]]. He shows that the shape of the egg favours its movement along the oviduct, a gentle pressure on the trailing end sufficing to push it forwards. Similarly, [[sea urchin]] shells have teardrop shapes, such as would be taken up by a flexible bag of liquid.

===16. On Form and Mechanical Efficiency===
<!--[[File:Gray250.png|thumb|Lines of stress calculated for a human femur]]-->
[[File:Forth Rail & Road Bridge 2.jpg|thumb|300px|Thompson compared a dinosaur's spine to the [[Forth Railway Bridge]] (right).]]

(1st p. 670 – 2nd p. 958 – Bonner p. 221)
::Thompson criticizes talk of [[adaptation]] by [[animal coloration|coloration in animals]] for presumed purposes of [[crypsis]], [[warning coloration|warning]] and [[mimicry]] (referring readers to [[E. B. Poulton]]'s ''[[The Colours of Animals]]'', and more sceptically to [[Abbott Thayer]]'s ''[[Concealing-coloration in the Animal Kingdom]]''). He considers the mechanical engineering of bone to be a far more definite case. He compares the strength of bone and wood to materials such as steel and cast iron; illustrates the "cancellous" structure of the bone of the human [[femur]] with thin trabeculae which formed "nothing more nor less than a diagram of the lines of stress ... in the loaded structure", and compares the femur to the head of a building crane. He similarly compares the [[cantilever]]ed backbone of a quadruped or dinosaur to the [[girder]] structure of the [[Forth Railway Bridge]].

===17. On the Theory of Transformations, or the Comparison of Related Forms===
[[File:Durer face transforms.jpg|thumb|upright|[[Albrecht Dürer]]'s face transforms (1528) were among Thompson's inspirations]]

(1st p. 719 – 2nd p. 1026 – Bonner p. 268)
::Inspired by the work of [[Albrecht Dürer]], Thompson explores how the forms of organisms and their parts, whether leaves, the bones of the foot, human faces or the body shapes of [[copepod]]s, crabs or fish, can be explained by geometrical transformations. For example:

[[File:Transformation of Argyropelecus olfersi into Sternoptyx diaphana.jpg|thumb|upright=1.3<!--size for low image-->|Thompson illustrated the [[Transformation (geometry)|transformation]] of ''[[Argyropelecus|Argyropelecus olfersi]]'' into ''[[Sternoptyx|Sternoptyx diaphana]]'' by applying a [[shear mapping]].]]

::{{quote|Among the fishes we discover a great variety of deformations, some of them of a very simple kind, while others are more striking and more unexpected. A comparatively simple case, involving a simple shear, is illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian co-ordinates, a certain little oceanic fish known as ''Argyropelecus olfersi''. Fig. 374 represents precisely the same outline, transferred to a system of oblique co-ordinates whose axes are inclined at an angle of 70°; but this is now (as far as can be seen on the scale of the drawing) a very good figure of an allied fish, assigned to a different genus, under the name of ''Sternoptyx diaphana''. Thompson 1917, pages 748–749}}

::In similar style he transforms the shape of the carapace of the crab ''[[Geryon trispinosus|Geryon]]'' variously to that of ''[[Corystes]]'' by a simple [[shear mapping]], and to ''[[Rochinia|Scyramathia]]'', ''[[Paralomis]]'', ''[[Lupa (crab)|Lupa]]'', and ''[[Chorinus]]'' ([[Pisinae]]) by stretching the top or bottom of the grid sideways. The same process changes ''[[Crocodylus porosus|Crocodilus porosus]]'' to  ''[[Crocodylus acutus|Crocodilus americanus]]'' and  ''[[Notosuchus terrestris]]''; relates the hip-bones of fossil reptiles and birds such as ''[[Archaeopteryx]]'' and ''[[Apatornis]]''; the skulls of various fossil horses, and even the skulls of a horse and a rabbit. A human skull is stretched into those of the chimpanzee and baboon, and with "the mode of deformation .. on different lines" (page 773), of a dog.

===Epilogue===
(1st p. 778 – 2nd p. 1093 – Bonner p. 326)

::In the brief epilogue, Thompson writes that he will have succeeded "if I have been able to shew [the morphologist] that a certain mathematical aspect of morphology ... is ... complementary to his descriptive task, and helpful, nay essential, to his proper study and comprehension of Form."  More lyrically, he writes that "For 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" and quotes [[Isaiah]] 40:12 on measuring out the waters and heavens and the dust of the earth. He ends with a paragraph praising the French [[entomologist]] [[Jean-Henri Fabre]]<ref>{{cite web | last1=Russel | first1=Alfred | title=Decoding D'Arcy Thompson – Part 1 | url=http://www.uncommondescent.com/intelligent-design/decoding-darcy-thompson-part-1/ | access-date=13 November 2014}}</ref> who "being of the same blood and marrow with [[Plato]] and [[Pythagoras]], saw in Number 'la clef de voute' [the key to the vault (of the universe)] and found in it 'le comment et le pourquoi des choses' [the how and the why of things]".