Stephen Jay Gould Source: New Literary History, Vol

D'Arcy Thompson and the Science of Form Author(s): Stephen Jay Gould Source: New Literary History, Vol. 2, No. 2, Form and Its Alternatives (Winter, 1971), pp. 229- 258 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/468601 Accessed: 22-04-2015 15:47 UTC

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This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'Arcy Thompsonand the Science of Form

StephenJay Gould

Our own studyof organicform, which we call by Goethe's name of Morphology,is but a portionof that wider Science of Form whichdeals withthe formsassumed by matterunder all aspectsand conditions,and, in a stillwider sense, with forms which are theoreticallyimaginable. Growthand Form,p. 1026

Preface

JN1945, the Public Oratorof Oxfordlauded D'Arcy Thompson as unicumdisciplinae liberalioris exemplar1; in 1969, the Whole Earth Catalog called his major work"a paradigmclassic." Few men can listsuch diversedistinctions in theircompendium of honors. But then,few men have displayedso wide a rangeof talent. D'Arcy WentworthThompson (I86O-I948), Professorof Natural Historyat Dundee and St. Andrews,2translated Aristotle's Historia Animalium, wrote glossariesof Greek birds and fishes,compiled statisticsfor the i The outstandingexample of a man of liberal education-on the occasion of his admission to the honorarydegree of Civil Law. Quoted in Ruth D'Arcy Thompson, D'Arcy Wentworth Thompson: The Scholar-Naturalist (London, 1958), P. 238. 2 Of D'Arcy Thompson's life and personalityI shall say little in the limited space available here. See the biographywritten by his daughter (cited in footnote I) and the best three of many articles written to celebrate Growth and Form: J. T. Bonner, editor's introductionto abridgment of D'Arcy Thompson, On Growth and Form (Cambridge, 1966), pp. vii-xiv; G. Evelyn Hutchinson, "In Memoriam, D'Arcy WentworthThompson," American Scientist, xxxvi (1948), 577-606; P. B. Medawar, "D'Arcy Thompson and Growth and Form," postscriptto Ruth D'Arcy Thompson's biography of her father,pp. 219-33, and reprintedin The Art of the Soluble (London, 1967), pp. 21-35. As a furthersource of informationand tribute to the syntheticattraction of D'Arcy Thompson's ideas on form,see the books that arose from two recent art exhibitionson organic form: L. L. Whyte, ed., Aspects at the Institute of of Form (London, I951), for the I951 Exhibition Contemporary Art, London, designed as a tribute to D'Arcy Thompson; and P. C. Ritterbush, The Art of Organic Forms (Washington, 1968), for the 1968 Exhibition at the Museum of Natural Historyin Washington,D. C.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 230 NEW LITERARY HISTORY FisheryBoard of Scotlandand contributedthe article on pycnogonids3 to the CambridgeNatural History.4He also wrotea book of one thousandpages, revered by artistsand architectsas wellas byengineers and biologists-the"paradigm classic," On Growthand Form (1917, 2nd edition,1942). To P. B. Medawar it is "beyond comparisonthe finestwork of literaturein all the annalsof sciencethat have been re- cordedin theEnglish tongue." 5 To G. EvelynHutchinson, it is "one of the veryfew books on a scientificmatter written in this century whichwill, one may be confident,last as longas ourtoo fragile culture." 6 In it D'ArcyThompson displayed his thoughts on organicform; these are curiousin places,almost visionary in othersand alwaysprofound. Almostthirty years after the secondedition, and morethan half a centuryafter the first, they have gainednew impactin a sciencethat onlynow has thetechnology to deal withhis insights.

I. IneluctableModality of the Visible

"Extension,figure, number, and motion.. .," wroteJohn Locke, "may be properlycalled real, original or primaryqualities, because they are in thethings themselves, whether they are perceivedor no."'7 Stephen Daedalus,walking along the Irishsea side,reviewed the dilemmasof epistemologyand evenperformed the crucial experiment: "Ineluctable modalityof thevisible... . Then he was awareof thembodies before of themcoloured. How? By knockinghis sconceagainst them.... Open youreyes now. I will.One moment.Has all vanishedsince? ... See now. There all the timewithout you: and evershall be, world withoutend." 8

3 A small group of marine arthropods that would, I am sure, be considered obscure by all but the veryfew who love (and study) them. 4 D'Arcy Thompson's complete bibliography to I945 will be found in: G. H. Bushnell, "A list of the published writingsof D'Arcy WentworthThompson," in W. E. Le Gros Clark and P. B. Medawar, eds., Essays on Growth and Form presented to D'Arcy Wentworth Thompson (Oxford, 1945), PP. 386-400. The list contains 279 items. This volume is a series of essays presented to D'Arcy Thompson on the occasion of his 6oth year as an active professor. 5 Medawar (see footnote2), p. 232. 6 Hutchinson (see footnote2), p. 579. 7 John Locke, "Some Farther Considerations Concerning our Simple Ideas of Sensation," from Book I of An Essay Concerning Human Understanding (1690). 8 James Joyce,Ulysses (New York, 196I), p. 37.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 231 Formpervades the material world. Aristotlewas willingto consider formwithout matter for prime movers and demiurges,but notmatter withoutform.9 In studyingnature ever since,most biologists have agreedwith Needham that " thecentral problem of biology is theform problem."'10To anyonewho views form as an outdatedconcern, suited onlyfor the fewsurviving taxonomists of a worlddominated by the chemistryand energeticsof molecularbiology," I recommenda book thatstands second only to D'ArcyThompson's as a 20thcentury paean to form: J.D. Watson'sThe DoubleHelix. For thisbook describes the pursuitof a shape"too prettynot to be true,"12 a molecularshape that wouldunderlie and explainthe phenomena of heredity.Watson's success emergeddirectly from a concernwith form, from a methodology thatprescribed the construction ofmodels in preferenceto thesearch for a moresubtle chemistry. He describesa lessonimparted by Francis Crick: "I soonwas taughtthat ... thekey to Linus' [Pauling]success was hisreliance on thesimple laws ofstructural chemistry. .... The es- sentialtrick ... was to ask whichatoms like to sitnext to each other. In placeof pencil and paper,the main working tools were a setof mole- cularmodels superficially resembling the toys of preschoolchildren."13 All biologistsmust deal withform, but it does not followthat they treatit adequately.In myown fieldof evolutionarybiology, I detect threeapproaches that seem especially insufficient when compared with theinsights of D'Arcy Thompson: I. Theoriesthat attempt to renderform in suchnon-morphological termsas motion,flow, and energy: Darwin'ssuccess and Lamarck's failureis no simpleconsequence of theirdiffering positions on specific points;it alsoreflects their opposing approaches to form.For Lamarck, motionand becomingwere primary;organic matter, ever in flux, mountedthe scale of being,impelled by "the forcewhich tends in-

9 See J. Needham'sanalysis in "BiochemicalAspects of Form and Growth,"in L. L. Whyte(see footnote2), pp. 77-86. Io J.Needham, Order and Life (Cambridge,Mass., 1968), p. 23. I I fearthat the messageof molecularbiology has oftenbeen mistranslatedin thisway as it descended(or ascendedaccording to one's orientation)from technical journalsthrough Scientific American to theNew York Timesand intopopular con- sciousness.Just as Einsteinianrelativity does not preach that "everythingis relative" (ratherthe opposite,in fact), neitherdoes molecularbiology replace the conceptof formand spatial structurewith the dynamicsof chemicalenergetics or the abstractionof "informationmodels"; the greatestachievement of molecular biology,after all, was the elucidationby Watsonand Crickof thephysical shape of DNA, theextraordinary molecule that transcribes this genetic information. I J.D. Watson,The Double Helix (New York,1968). 13 Ibid., p. 38.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 232 NEW LITERARY HISTORY cessantlyto complicateorganization." As Gillispie14writes: "La- marck'stheory of evolutionwas thelast attempt to makea scienceout of theinstinct ... thatthe world is fluxand process,and thatscience is to study,not the configurations ofmatter, nor the categories of form, butthe manifestations ofthat activity which is ontologicallyfundamen- tal as bodiesin motionand speciesof beingare not." To Darwin, paradoxicalas it mayseem, form was primary;we sensethis in his al- mostpoetical admiration for the elaborate forms of orchids,exquisitely designedto insurepollination by insects. 15 Gillispiecontinues: "What [Darwin]did was to treatthat whole range of naturewhich had been relegatedto becoming,as a problemof being, an infiniteset of objective situationsreaching back throughtime .... The Darwiniantheory of evolutionturned the problem of becoming into a problemof being and permittedthe eventualmathematization of thatvast area of nature whichuntil Darwin had beenprotected from logos in thewrappings of process."16 The triumphof Darwinism,however, did not assurean adequate treatmentof form. The analysisof biological form must emphasize the conceptof adaptation-thefitness of a structureto performfunctions beneficialto an organism.Good designis an attributeof animalsthat fitcomfortably into the metaphysic of a Paleyor a Cuvier. It should, of course,have been just as congenialto Darwinism(as it was to Darwinhimself), for evolution had merelysubstituted natural selection forGod as the efficientcause of adaptation.Evolutionists and crea- tionistswere equally happy to findgood designin nature,the former becauseit illustrated the effects of natural selection, the latter because it mainfestedGod's plan fora worldworthy of his creativeinterference. Yet modernevolutionary theory has tendedto deemphasizeform, "to dissolve,despite its great verbal emphasis on function,genuine adapta- tioninto the non-morphologicalconcepts of gene-pool,genetical 'fitness,'adaptive zones, etc." 17 In part,this is thelegacy of earlyevolu- tionistswho choseto avoid the conceptof design(which represented the old biology,though it fitas well withthe new) and to use form onlyas a clue to the tracingof lineages.It has led to somecurious

14 C. C. Gillispie, "Lamarck and Darwin in the History of Science," in B. Glass, O. Temkin and W. L. Strauss, Jr., eds., Forerunnersof Darwin: 1745-1859 (Balti- more, 1959), PP. 268-69. 15 See Origin of Species (6th ed.): account of Criiger's experimentsin chapter 6. Darwin also devoted an entire book to the subject: On the Various Con- trivancesby which Britishand Foreign Orchids are Fertilized by Insects (1862). 16 Gillispie, "Lamarck and Darwin," p. 291. I7 M. J. S. Rudwick, "The Inference of Function from Structure in Fossils," BritishYournal for the Philosophy of Science, xv ( 1964), 39.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 233 enigmas,most notably to thecharge that Spencer's catch-phrase, "the survivalof the fittest,"is circularbecause fitnessis measuredas the capacityfor survival.Indeed, the moderngeneticist does so define "fitness";but Darwin did not. To him,fitness was a propertyof form, a measureof good design that did notentail survival, a priori.In many anti-Darwiniantheories of evolution, the fit, in thissense, do notneces- sarilysurvive (notions of racial old-ageor orthogenesispropelling a lineageon straightcourses beyond the point of adaptation). Spencer's phrase,therefore, expresses the primary prediction of Darwin'ssystem and embodiesthe test by which it can be comparedwith other theories and accepted. 2. Theoriesthat deal withform directly but do not attemptto ex- plainit: Amongtaxonomists who pretendto engagein puredescrip- tion,there is a mystiquethat exalts the unsullied objectivity of this most humbleservice to Nature,this display of Her forms,free from the in- trusionof humanspeculation and the vanityof theory.The ironyof thisattempt lies in the impossibilityof its attainment,for no human beingcan see theDing-an-sich. "Pure" description,the piece-by-piece compendiumof an organism,is as firmlyrooted in theoryas themost abstractand mathematicalapproach to form;the problem is notonly thatits theory is hidden,but also thatit is wrong.A standardspecies descriptioncatalogs the organismpart by part; thisimplies, in a way thatis subtleand captivatingbecause it is inexplicitand even unin- tended,that an animalis merelya frameworkfor its separate parts and thatits complexity is irreducible.In purelyheuristic terms, this theory is sterileand we musthope that its correspondence to reality is slim;for onceit is statedwe can do littlewith form but catalogit in wonder. 3. Theoriesthat try to explainform, but do so incompletely:In the i89o's D'ArcyThompson developed an explanationfor the regular formof spongespicules. He ascribedthem to adsorptionat cell con- tacts;their simple geometry (Fig. I) merelyreflects the regular form, assumedunder surface tension, of an aggregateof cells. Michael Foster, an eminentevolutionary physiologist, was displeased: "I confessI am notvery much attracted by the line of work .... Does yourresult wholly destroythe diagnostic value of thespicules? If theform is constantin a group-it does notmatter how theform is broughtabout." 1 This statementepitomizes the approachto formthat characterizedearly evolutionarythought and offendedD'Arcy Thompson. This is the "sign"theory of morphology-theidea thatform is to be used onlyas a guideto thetracing of lineages;that structures are to be analyzedas signsof ancestry,not as designsfor modem existence; and thatform

18 Quoted in Ruth D'Arcy Thompson, p. go.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 234 NEWLITERARY HISTORY

...

FIo. i. D'Arcy Thompson's illustrationof the regular formof sponge spicules. From his chapter ix, "On concretions,spicules and spicular skeletons."

is explainedonce its antecedents in ontogenyand phylogenyare identified. Thus,under the theory of recapitulation,juvenile structures are studiedto identifythe ancestral adult that they represent, not to deter- minehow theyfunction in theyoung animal. D'ArcyThompson did notgainsay the use ofform to tracelineages; he did decrya morpholo- gythat considered its work completed when it had donethis-for explanation,as the translatorof Aristotleunderstood,19 requires the

I9 So many heated (and empty) arguments about "proper" explanation would be avoided if scientistsonly heeded Aristotle's ancient lesson on the multiplicity of causes (and realized that differentdisciplines identifydifferent causes and then, in their limited perspective, wrongly claim a full explanation). In what D'Arcy Thompson called Aristotle's"parable" of the house (next paragraph of main text), we must consider the various factors,in the absence of which a particular house could not have been built: the stones that compose it (material cause), the mason who laid them (efficientcause), the blueprintthat he followed (formal cause), and the purpose for which the house was built (final cause).

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 235 identificationofmany "causes." We try,for example, to determinethe functionof a well-designedstructure, to identifyits finalcause. This remainsinsufficient still. In an earlyarticle on theshapes of eggs,he mentionsa varietyof final causes (prevention of rolling out of the nest, closepacking within the nest to permitmore eggs in a givenspace) and comments:"Whatever truth there be in theseapparent adaptations to existingcircumstances, it is onlyby a veryhasty logic that we can acceptthem as a vera causa or adequateexplanation of the facts."20 We muststate the efficient cause as well: "In Aristotle'sparable, the houseis therethat men may live in it; but it is also therebecause the buildershave laid one stoneupon another.... All the while,like warpand woof,mechanism and teleologyare interwoventogether, and we mustnot cleave to theone nordespise the other; for their union is rootedin thevery nature of totality" (p. 7).21 D'ArcyThompson anchored his view of nature to theidea oforganic form(contra z); he triedto explainform by reducingits complexity tosimpler elements that could be identifiedas causes(2) ; he recognized thatexplanation is multifaceted;that the identification only of a purpose,a mechanism,or a phyleticancestor, provides a pitifullyincom- pleteanalysis of form (3) .

II. D'Arcy Thompson's Theory of Form

In the epilogueto Growthand Form,D'Arcy Thompsonstated his aim: "to shewthat a certainmathematical aspect of morphology is ... helpful,nay essential, to theproper study and comprehensionofGrowth and Form" (p. o096). In thepreface, he wrote:"Numerical precision is thevery soul of science,and its attainmentaffords the best, perhaps theonly criterion of the truth of theoriesand thecorrectness of experi- ments"(p. 2). In theopening sentence, he invokedAuthority with his customaryerudition: "Of the chemistryof his day and generation, Kant declaredthat it was a science,but not Science-eine Wissenschaft, abernicht Wissenschaft-for that the criterion of true science lay in its relationto mathematics.This was an old story: forRoger Bacon had calledmathematics porta et clavisscientiarum, and Leonardoda Vinci

20 "On the Shapes of Eggs, and the Causes whichDetermine them," Nature, lxxviii (1908), 1 1. 2 All page numberscited in the text referto the 2nd edition of On Growth and Form (Cambridge, 1942).

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 236 NEWLITERARY HISTORY had said much the same ...." (p. i-there followsan untranslated Italianfootnote). Yet D'ArcyThompson's mathematics has a curiousring. We find noneof the differential equations and mathematicalstatistics that adorn modernwork in ecologyor populationgenetics; we read,instead, of the partitioningof space,the tetrakaidekahedron,the Miraldiangle, the logarithmicspiral and thegolden ratio. Numbers rarely enter equations; rather,they exemplify geometry. For D'ArcyThompson was a Greek mathematicianwith 2oth century material and insights.Growth and Formis thesynthesis of his twolives: eminentclassicist and eminent zoologist.As he statedin a PresidentialAddress to theClassical Associa- tion (1929): "Science and the Classics is my theme today; it could hardlybe otherwise.For all I know,and do, and well nighall I love and carefor (outside of home and friends)lies within one or theother; and thefact that I haveloved them both has coloredall mylife, and en- largedmy curiosity and multipliedmy inlets to happiness."22 Those who recognizethe classicalroots of Growthand Formgen- erallylink that work to Aristotle.Yet Aristotlerepresented only one of the two classicalinputs to D'ArcyThompson's science-and the one oflesser importance for Growth and Form. D'ArcyThompson revered Aristotleas a descriptivenaturalist: "In his exhaustiveaccumulation and treatmentof facts, his method is thatof the observer, of the scien- tificstudent, and is in themain inductive." 23 Moreover,he identified as Aristotle'sweaknesses what most critics consider the twinstrengths of Growthand Form: aestheticstyle24 and mathematicalskill.25 D'ArcyThompson's father, a classicisthimself, had comparedAris- totle'steachings "to theinvigorating properties of sea-bathing, open air, regularexercise, wholesome diet and quinine"and the "preaching"of Platoto "opiumor hachish."26 It is to Pythagorasand thelater Plato ofthe Timaeus that D'Arcy Thompson owes his vision; for he tried,as did BertrandRussell, "to apprehendthe Pythagorean power by which numberholds sway above the flux."27D'Arcy Thompsonaccepted

22 Quoted in Ruth D'Arcy Thompson, p. 196. "On Aristotleas a xci 23 Biologist,"Nature, ( 913), 204. 24 Ibid., p. 2o2: "Wise and learned as Aristotlewas, he was neither artist nor poet. His style seldom rises ... above its level, didactic plane." 25 Ibid., p. 20o: "I have my doubts as to his mathematics. In spite of certain formidible passages in the 'Ethics,' in spite even of his treatise, De Lineis In- secabilibus, I am tempted to suspect that he sometimes passed shylybeneath the superscriptionover Plato's door." According to legend, the inscription over the door of the Academy read: "Let no one enter here who is ignorant of geometry." 26 Quoted in Hutchinson (see footnote2), p. 578. 27 Bertrand Russell, "What I have Lived For," in The Autobiographyof Bertrand Russell, Vol. I, z872-z914 (Boston, 1967), pp. 3-4.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 237 neitherPythagoras' doctrine that "things are numbers,"28nor Plato's visionof a realmof ideal numbers existing apart from the physical bodies that are but theirfleeting and imperfectrepresentation. But he did sharetheir general attitudes: that solutions to the world'smysteries shouldbe soughtin "the geometricalaspect of number"29; thatsim- plicity,regularity, symmetry, harmony and truthwould be foundcon- joined"; thatthe world of "science"could be approachedand com- prehendedby poeticvision. As he wrotein reviewinga book on the Pythagoreans:"The Pythagorean,or Platonic,or Jewishconcept of Numberis a hard sayingto the unpoetic,non-mystical modem and Westernworld; and manya way is foundto show that Plato and Pythagorasmeant something prosy and commonplaceafter all. But to someit is stillas plainas everthat Number is theclue to thegreatest of earthlymysteries, and thatwhat we call beauty,whether of soundor form,is but its resultantexpression."31 In Growthand Form,Plato

28 In the Metaphysics, Aristotlewrote of the Pythagoreans: "They assumed that the elementsof numberswere the elementsof all things,and that the whole heavens were harmonyand number." This view seems so arcane in today's context that we tend to see in it the most abstract of mysteries. In an Ionian setting,it makes for respectable science as well. R. M. McInerny, A History of WesternPhilosophy: Vol. I, From theBeginnings of Philosophyto Plotinus(Notre Dame, 1968), p. 45, has shown how the Greek's very "manner of depicting numbers leads to speaking of types of numbers progressingin terms of dimensions." The lonians had no written numerals; the Pythagoreans, in fact, represented numbers with sets of pebbles arranged in geometric form. Bodies were then generated from points (pebbles) representingunits: the number I produced a point, 2 a line, 3 a triangle,and 4 a pyramid; moreover, the Pythagoreans had no notion of a continuum and viewed lines as series of discrete points (- units, = numbers). It is no distant extrapola- tion from this to a notion that the "elements of number are the elements of all things. ... Since their way of depicting numbers produced plane and solid figures,bodies and even physical bodies, were taken by the Pythagoreans to be composed of units and, consequently, these bodies are numbers" (McInerny, p. 47). D'Arcy Thompson, of course denied that numbersare the material cause of physical objects; he did, however, view number as a formal cause: numbers generate the symmetricaland regular shapes that organic formsassume under the in- fluenceof physical forces. 29 D'Arcy Thompson's words in speaking of Pythagorean mathematics: "The Hellenic Element in the Development of Science," Nature, cxxiii (1929), 732. 30o Discussing the honeycomb in Growth and Form (p. 529), D'Arcy Thompson speaks of "the two principles of simplicityand mathematical beauty as ... sure and sufficientguides." And later, in the epilogue (p. 1o97): "the perfectionof mathematical beauty is such . . . that whatsoever is most beautiful and regular is also found to be most useful and excellent." 31 "The School of Pythagoras," Nature, xcviii (x9I6), 166. The very next sentence reads: "It was in the very spirit of Pythagoreanmysticism that that great naturalist,Henri Fabre, wrote his great ode to number." And it was with a quota- tion fromthat "great ode" that D'Arcy Thompson chose to end Growth and Form, calling Fabre "that old man eloquent, that wise student and pupil of the ant and the bee . . . who in his all but saecular life had tasted of the firstfruitsof im-

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 238 NEW LITERARY HISTORY and Pythagorasare linked as "thosegreat philosophic dreamers" (p. 2). "Not onlythe movementsof the heavenlyhost must be determined by observationsand elucidatedby mathematics,but whatsoeverelse can be expressedby numberand definedby naturallaw. This is the teachingof Plato and Pythagoras,and themessage of Greek wisdom to mankind" (p. 1097). Growthand Formis not a textin anachronisticGreek biology. It usesclassical concepts to criticizemodem theories (Darwinism in particular),and it showsa man as familiarwith the problems of engineering in bridgedesign as withthe tetractysof Pythagoras.It expresses the tensionsand conflictsthat mustinhere in any philosophycom- poundedfrom such disparate sources. Of thesetensions, two are par- ticularlyimportant in Growthand Form. I) His view of Plato and Pythagoras(mathematics, generality and deduction)versus Aristotle (descriptionand induction).2) His Greekcommitment to a pureand abstractunderstanding ofform versus his Baconian idea thatknowledge is power,as expressedby the engineer's love fora good designbecause itworks. As Hutchinsonshowed in hisperceptive essay, 32 D'Arcy Thompson's theoryof formencompasses three themes: an old idea to whichhe broughtnew lightand grace,a theoryon the productionof form throughgrowth and a mathematicalmethod for representing the causes ofdifferences in shape among related organisms. i. That organismsare well designedis amongthe oldestideas of biology.What D'Arcy Thompson did, following Galileo and Borelli,33 was to conductthis observation away from passive wonder at nature's marvelstoward the analytictechniques of physics,mathematics and engineering.When this is done,the study of formbecomes a science: predictionscan be madeand tested;actual forms can be comparedwith the ideal configurationsthat engineersprescribe; the superiorityof one formover another can be assessedand evenmeasured. mortality... in whose plainest words is a sound as of bees' industriousmurmur; and who, being of the same blood and marrow with Plato and Pythagoras, saw in Number le comment et le pourquoi des choses, and found in it la clef de voiite de l'Univers" (p. og97). I thinkwe may include D'Arcy Thompson among the "some" to whom "it is still as plain as ever. ... 32 Hutchinson (see footnote2). 33 Galileo's major discussion is in the "Dialogue of the Second Day" in Dialogues Concerning Two New Sciences (1638), trans.H. Crew and A. de Salvio (New York, 1914). Here he displays the mechanical consequences of size increase, demonstrates the impossibilityof giants and explains why large animals must have relatively thickerlegs than smaller relatives. Borelli's treatise,De Motu Animalium, was published in 1685.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 239 As one illustrationof thistheme, the pages of Growthand Form aboundwith examples demonstrating that organic shapes conform to thephysical forces prevailing at theirscale. The pervasiveeffect of size upon formis a simpleconsequence of physicallaws and geometric arguments.As an animal (or any object) grows,its volumewill in- creaseas thecube of its length while its surface, if it maintainsthe same shape,increases only as thesquare; thelarger the animal,the greater the ratioof itsvolume to itssurface. Thus, a largeanimal lives in a worldruled by forces, primarily gravitation, that work upon its volume (i.e. its mass). A smalleranimal is influencedlittle by gravity;the forcesthat act upon its surface hold sway because that surface is so large relativeto themass that gravity affects. Still other forces, the random shocksof the Brownian motion for example, come into play at bacterial dimensions.4 This physicalinfluence of size explains some of the most confusingof obviousstatements: why any fly can walkup a wall,but onlyJesus could walk on water35;why elephantshave thickerlegs thangazelles36; and whythe giantants of Them reallycouldn't have reachedthe Los Angelessewers. 37 Kirbyand Spence,extrapolating ant powerto elephantsize and calculatingthe prodigious powers of such a leviathan,thanked God in His wisdomfor weakening the elephant, lest it shouldcause "theearly desolation of theworld" (quoted on p. 36). D'ArcyThompson showed that if God had had anyhand in thematter,

34 The differencesbetween life in such a world and ours are brilliantlystated in George Gamow's classic Mr. Tomkins in Wonderland (Cambridge, I94o). Wonder- land is a world in which quantum and relativityeffects occur at our customarysizes and speeds. 35 The downward force of gravitypulls animals from walls and through the surface-filmof a pond; surface tensionprovides adhesion both to wall and pond surface. Small animals have such a high ratio of surface to volume that the force of surface tensionoutstips that of gravity. 36 The strengthof a leg bone is a functionof its cross sectional area; it will increase as the square of length if shape remains constant with increase in size. Yet, the weight that the legs must support increases in proportion to volume, as the cube of length. A series of animals differingin size but not in shape will become progressively weaker at larger sizes. In nature, large animals overcome this potential problem by having relativelythick legs with a large cross-sectionalarea; in addition, legs tend to thicken disproportionatelyduring growth. This type of growth-in which shape changes to meet the physical demands of increase in size- is called "allometric." Julian Huxley pioneered the quantitative study of allometric and Size in and growth in the Ig92's. See S. J. Gould, "Allometry Ontogeny xli Phylogeny,"Biological Reviews, (1966), 587-64o. 37 The ability to stay aloft depends upon wing surface area, but body weight increases as the cube of length. This is the classic problem of "wing loading" in aeronautics; though Hollywood knows little of it, Pliny understood the issue when he wrote in the Historia Naturalis that "the heavier birds can flyonly after taking a run, or else by commencing their flightfrom an elevated spot." At their size, the giant ants could not even have taken off from their nest near Alamagordo. In

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 240 NEW LITERARY HISTORY he had workedin rathermundane fashionthrough the physicallaws of size.38 Following the general introduction,D'Arcy Thompson began his workwith the eloquent chapter "On Magnitude"-a testimonyto the significanceof size and its physical consequences. The chapter ends thus (p. 77) : Life has a range of magnitudenarrow indeed comparedto that with whichphysical science deals; but it is wide enoughto includethree such discrepantconditions as thosein which a man, an insect,and a bacillus have theirbeing and play theirseveral roles. Man is ruledby gravitation, and restson motherearth. A water-beetlefinds the surfaceof a pool a matterof lifeand death,a perilousentanglement or an indispensablesup- port. In a thirdworld, where the bacilluslives, gravitation is forgotten, and the viscosityof the liquid, the resistancedefined by Stokes'law, the molecularshocks of the Brownianmovement, doubtless also the electric chargesof the ionizedmedium, make up the physicalenvironment and have theirpotent and immediateinfluence on theorganism. The predomi- nantfactors are no longerthose of our scale; we have come to the edge of a worldof whichwe have no experience,and whereall our preconceptions mustbe recast. 2. The relationshipof size and shape linksthe firsttheme to the second-that physical forcesexert a direct and immediateinfluence in shapingorganisms as theygrow. This is the guidingconcept of Growth and Form; all but threeor four of its seventeenchapters are devoted to the citationof correspondencesbetween organic forms and the shapes that physicalforces produce in actingupon non-livingmaterial of simi- lar size, density,viscosity, and rigidity.From these correspondences, D'Arcy Thompson inferredthat organic formshad been fashionedin the same way--by the directaction of physicalforces. fact, they couldn't even have breathed; the tracheae (respiratoryorgans) of insects are invaginations of the external body surface; they must increase in constant proportion to the body weight that they service. For surface to grow as fast as weight, shape must change: the surface must become more elaborate (by in- creased folding or invagination). There is a limit to this elaboration, lest there be no room for internal organs. Large animals have avoided this dilemma by evolving separate internal organs (lungs) to harbor a vast amount of surface for respiration. Horror movies are notoriousfor the limited imagination that makes giant insectsact as only normal ones could. The most incredible thing about the Incredible Shrink- ing Man is that, at three inches, he continues to operate as a gravity-pronesix- footer. 38 Strengthis primarilya functionof the cross-sectionalareas of muscles. Since elephants have smaller areas relative to their weight than small animals, they cannot match the featsof insectswhen these are measured by the inappropriate criterion of weight liftedvs. body weight. That insects can drag an object many times their own weight is no testimonyto any superior design or prodigious will, but merely a functionof theirsmall size.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 24I Many criticshave failedto graspthis view of actionof physicalforces; they dismissGrowth and Form as a book filledwith curious and in- genious analogies that describesome aspects of formbut explain nothing. To D'Arcy Thompson, however,a correspondencebetween organic and inorganicis morethan an analogy; it is a demonstrationthat two objects probablyhave the same efficientcause (p. Io) : The waves of the sea, the littleripples on the shore,the sweepingcurve of thesandy bay betweenthe headlands,the outlineof the hills,the shape of the clouds,all theseare so manyriddles of form,so manyproblems of morphology,and all of them the physicistcan more or less easily read and adequately solve: solving them by referenceto their antecedent phenomena,in the materialsystem of mechanicalforces to which they belong,and to whichwe interpretthem as beingdue .... Nor is it otherwise with the materialforms of livingthings. Cell and tissue,shell and bone, leaf and flower,are so manyportions of matter,and it is in obedi- ence to the laws of physicsthat their particles have been moved,moulded and conformed. D'Arcy Thompson's analysisbegins at small sizes, in the realm of surface forces. He compares protozoans with Plateau's surfaces of revolution (surfaces of minimal area radially symmetricalabout an axis), and infersfrom the correspondence(Fig. 2) that surface tension shapes the simple, single-celledanimal. As a liquid cylinderis stretched beyond the limits of stability, it breaks up into "a series of equal and regularlyinterspaced beads, oftenwith littlebeads regularly interspacedbetween the largerones" (p. 386). D'Arcy Thompson finds thesesame beads in the "dew-drops" of a spider'sweb and providesa mechanical explanationfar more simple and satisfactorythan that of earliernaturalists (p. 387) : The same phenomenonis repeatedon a grosserscale when the web is bespangledwith dew, and its threadsbestrung with pearls innumerable. To the older naturalists,these regularly arranged and beautifullyformed globules on the spider's web were a frequentsource of wonderment. Blackwall,counting some twenty globules in a tenthof an inch,calculated thata large garden-spider'sweb shouldcomprise about I20,ooo globules; the net was spun and finishedin about fortyminutes, and Blackwallwas filled with admiration of the skill and quickness with which the spidermanufactured these little beads. And no wonder,for according to theabove estimatethey had to be made at the rateof about 50 per second. At largersizes, gravity comes into play. In the medusae of coelente- rates, "jellyfish"and their relatives,D'Arcy Thompson was "able to discover various actual phases of the splash or drop [Fig 3] .... It is hard indeed to say how much or littleall these analogies imply. But theyindicate, at the veryleast, how certainsimple organic forms might

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FIG. 2. Comparison between one of Plateau's surfaces of revolution (left-unduloid with positiveand negative curvature) and "the flagellate'monad' Distigma proteus." From chapter v, "The formsof cells." be naturallyassumed by one fluid mass withinanother when gravity, surface tension and fluid frictionplay their part" (p. 397). At still largersizes, surface tension becomes so negligiblethat rigid hard parts are needed to maintainshape, lest gravitymake a world of pancakes. In the internaltrabeculae of vertebratebones D'Arcy Thompson found patternsthat mirrorthe stressesimposed upon them by the body's weight: A great engineer. . happened (in the year 1866) to come into his colleagueMeyer's dissecting-room, where the anatomist was contemplating the sectionof a bone. The engineer,who had been busydesigning a new

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FIG. 3. Comparisonbetween a fallingdrop producedby fuseloil in paraffin(left) and the medusoid"jellyfish" Syncoryme (right). From chapterv, "The formsof cells." and powerfulcrane, saw in a momentthat the arrangementof the bony trabeculaewas nothingmore nor less thana diagramof thelines of stress, or directionsof tensionand compression,in the loaded structure:in short, that Nature was strengtheningthe bone in preciselythe mannerand di- rectionin whichstrength was required;and he is said to have criedout, "That's mycrane" (pp. 976-977and Fig. 4 of thiswork). From thisobservation of good design,D'Arcy Thompson made his usual inferenceto an efficientcause (pp. 984-985): We mustalways remember that our bone is not onlya livingbut a high- ly plasticstructure; the littletrabeculae are constantlybeing formed and

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FIG. 4. Linesof force in Meyer's crane (left) and as reflectedin thebony trabeculae of the human femur (right). From chapter xvi, "On formand mechanical efficiency."

deformed,demolished and formedanew. Here, foronce, it is safe to say that 'heredity'need not and cannot be invokedto account for the con- figurationand arrangementof the trabeculae: for we can see them at any timeof lifein the making,under the directaction and controlof the forcesto which the systemis exposed ... Herein then lies, so far as we can discernit, a greatpart at least of the physicalcausation of what at firstsight strikes us as a purelyfunctional adaptation: as a phenomenon, in otherwords, whose physical cause is as obscureas itsfinal cause or end is apparentlymanifest.

For sheer ingenuity,nothing in Growthand Form matchesD'Arcy Thompson's famous,and likelycorrect, explanation for the narwhal's horn (actually a tooth). This horn may project eight or nine feet beyond the creature'shead; "it never curves nor bends, but grows as straightas straightcan be" (p. 907); windingabout this straightaxis is a screw of severallow-pitched threads (Fig. 5); in rare cases, when two hornsare formed,the threadsrun the same way in each: theyare not mirrorimages (an extraordinarysituation for bilaterally symmetri- cal animals). D'Arcy Thompson then notes that screw-threadsare made by combiningforward and rotatorymotion. Now the propulsion

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FIGo.5. Below: narwhalshowing horn with straight axis and low-pitchedthreads. Above: skullof rare two-hornednarwhal; threads run in the same directionon both horns. This interestingillustration is fromthe 1826 editionof the Compte de Lac6pede's Histoirenaturelle des cetace'es(originally published in I804 as one of the completingvolumes of Buffon'sHistoire naturelle). Lac6pede, a colleague of Cuvierand Lamarck,wrote the last volumesof Buffon'smonumental work. of closely-relateddolphins contains a rotatorycomponent orginating at thetail. To thiscomponent, the flexiblebody responds more actively thanthe horn which is bothrigid and locatedfar from the driving im- pulseof the tail. The horn'shelix, therefore, is formed because the narwhal,as it swimsforward during growth, slowly rotates about its own horn! 3. The lastchapter of Growthand Formseems, at first,curiously out ofplace in a deductivework. In it D'ArcyThompson imposes a netof rectangularcoordinates upon variousanimals and generatesseries of relatedspecies by subjectingthat net to simpledeformations (Fig. 6 fora set of crab carapaces). Some have seen thisas an exercisein empiricaldescription, others as a game,a scherzoto a bookthat never receivedits final movement. But it is thefinale to a coherentwork. For his criticshave misseda centralpoint: that D'Arcy Thompson was in- terestedin the deformednet, not primarilyin the animalthat it generated.He saw thatnet as a diagramof forces; and justas thetrabecu- lae of thestressed femur reflected the forces responsible for their depo- sition,so would the deformednet depict the forcesthat could transformone animalto another.Since these forces might produce a form directly,the deformed net is no mereframework for description; it may be a displayof efficientcauses. If "diverseand dissimilarfishes can be referredas a wholeto identicalfunctions of verydifferent co-ordinate systems,this fact will of itself constitute a proof that variation has pro- ceeded on definiteand orderlylines, that a comprehensive'law of

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\(a ) /1( d

-mu

(f)

(ci)

FIG.6. Carapaces of variouscrabs displayed as transformationsof a coordinatenet imposed upon Geryon (a). From chapter xvii, "On the theoryof transformations." growth'has pervaded the whole structure inits integrity, and that some moreor less simple and recognizablesystem of forces has been in con- trol"(p. 1037). The methodof transformed coordinates is D'Arcy Thompson's pro- visionalmathematics for complex structures. Though he mightrender a cellor evena jellyfishby forces reflecting a simple physical law, he knewthat a complexvertebrate could not be encompassedso easily. Yet he feltthat mathematical representation must be soughtfor the complexas well,even when the physical laws behind that representa- tioncould not be specified,and evenif this meant accepting one form as givenand achievingthe simplification and causalinsight of mathe- maticsonly for the generation of relatedforms. Thus, as Hutchinson wrote,D'Arcy Thompson constructed "what may be calledby analogy withthe floating chronology ofthe archaeologists, a floating mathemat- icsfor morphology, unanchored for the time being to physicalscience, butcapable of valid generalization onits own level."

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 247 The studyof formmay be descriptivemerely, or it may becomeanalyti- cal. We beginby describingthe shape of an object in the simplewords of commonspeech: we end by definingit in the preciselanguage of mathematics... The mathematicaldefinition of a 'form'has a qualityof pre- cisionwhich was quite lackingin our earlierstage of meredescription; it is expressedin a few wordsor in stillbriefer symbols, and thesewords or symbolsare so pregnantwith meaning that thoughtitself is economized; we are broughtby means of it in touchwith Galileo's aphorism(as old as Plato,as old as Pythagoras,as old perhapsas thewisdom of theEgyptians) that'the Book of Natureis writtenin charactersof Geometry'(p. I026).

III. D'Arcy Thompson and his Critics

The criticsof Growth and Formare numerous.Their major objections can be condensedto threecategories. I. D'ArcyThompson was a relicof pasttimes and a perpetratorof theirinadequate science. I detectthree separate arguments here: i) His writingis so overblown,so pompousin itsVictorian meter, so ostentatiousin its Greekallusions, that harried professionals cannot graspits essentials with the requisite economy of time. I do not deem thisbrand of philistinism worthy of comment. ii) His non-experimentalapproach belongs to an earliernatural history,not to modemscience; he nevermanipulated nature in con- trolledsituations, but merelydescribed and interpretedwhat nature presented.This criticismis seriousand justified,but its rootsare with Aristotle,not withLinnaeus. For D'Arcy Thompson'sinadequacy is whatSambursky cited as the "fundamentaland decisive"39 limitation of Greekscience: "With veryfew exceptions,the AncientGreeks throughouta periodof 8o00years made no attemptat systematicex- perimentation. . . The consequence was that induction was limited to thesystematic observation and collectionof such experimentalma- terialas was offeredby thestudy of naturalphenomena. Such induction was naturallyprimitive in termsof the conceptionof modem science.""4But if D'ArcyThompson did notprovide his insights with the verificationof experimentthey are not falsifiedthereby. And he did not,at least,subscribe to Sambursky'ssecond limitation of Greek science: "It doesnot aim at theconquest and controlof nature,but is motivatedby purely intellectual curiosity .... Forthis reason technology

39 S. Sambursky,The Physical World of the Greeks (New York, 1956), p. 2. 40 Ibid., pp. 2-3.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 248 NEW LITERARY HISTORY findsno placein it; and ifsuffers from the lack of that synthesis of pure knowledgeand practicalapplication which is thestrength of modem science."41 iii) He gave scantnotice to modemwork that vied withhis basic premisesand even neglectedto updatematerial contested in the in- terimbetween editions. The lattercharge is trueonly in part. In the 1917 edition(pp. 48-49), forexample, he speakswell of Arrhenius' fantasticnotion that the minutest of organisms might escape the earth's atmosphereon theelectrical force of the Aurora, thence to be propelled by the radiantenergy of light,"like Uriel gliding on a sunbeam"(p. 49 of 1917 ed.), to Jupiterin 80 days and to Alpha Centauriin 3,000 years-thusto disseminateour lifethroughout the universe (or to seek theorigin of ours in dispersionfrom other worlds). In 1942,this finale to "On Magnitude"was quietlydropped in favorof the statement quotedon page 240. But it couldbe excisedbecause its implausibility did not threatenany premiseof D'ArcyThompson's system: it had merelyserved as a dubiousillustration of an incontestabletruth.42 D'Arcy Thompson'sstand was this: he did not alter his basic statementswhen someonechallenged their application to particular cases.After all, a good theoryis thebest guide for separating correct fromincorrect "facts"-especially in naturalhistory where "pure" data are oftenthe passiveand subjectiveobservations of men subtly prejudicedwith unacknowledged a prioris. When confronted with con- tradictoryevidence, D'Arcy Thompson trusted his theory and his aes- theticvision. 43 As a strategyin science,this is a dangerousgamble, one thatcan be recommendedonly for the great. The winnerbecomes a propheticgenius, the loser a blinddogmatist. The viewof D'ArcyThompson as a relicwill be embracedor dis- misseddepending upon one's conceptionof thehistory of science. In the Comptianperspective of inexorableprogress-rendered by many scientistsas a marchto truthmediated by the continuing accumulation of facts-D'ArcyThompson's hearkening to theAncients, his opposi- tionto modernfacts in conflictwith Greek Truths, must be decried. Butif historians of science have any message for the practicing scientist, it is thatgreat men fashion great theories with the elusive qualities de- noting"creativity" and "genius": a sense(often aesthetic) that some-

41 Ibid., p. 3. 42 That the smallest organisms are subject to forces (here electricityand the "radiant pressure"of light) that do not affectus. 43 For an account of the positive role of aesthetic convictionsin major scientific discoveries see: E. A. Burtt, The Metaphysical Foundations of Modern Science (New York, 1932).

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 249 thingis amisswith old ideas,an abilityto bringnew ways of thinking to an old problem,and an "intuitive"feeling for the relative importance and reliabilityof conflictinginformation. And nihilsub sole novum or no, newways are oftenforgotten old ways. 2. D'ArcyThompson was a stubbornopponent of modernideas on evolution.Many have taken D'Arcy Thompson's antagonism to natural selectionas yetanother sign of his cantankerousobsolescence. In fact, hisobjections reflect no generalstand against modernity, but a specific conflictbetween his vision and Darwin's. Seen in thislight, his opposi- tionis bothintelligible and judicious.Yet it is also incorrect,or so we judge today--andhere his visionfails in its strictestimplication (but triumphs,we shallsee, in a reinterpretation).He had threemajor ob- 4 jections to Darwinism: the firstmethodological, the last two sub- stantive. i) To mostDarwinians of his time,morphology was "an endless searchafter the blood-relationshipsof things living and the pedigrees of thingsdead and gone" (p. 3). D'Arcy Thompson,as we have alreadyseen (p. 5), decriedthis limited view thatsought only final causesor antecedentstates. Of complexorganisms, he wrotethat we mustboth "look upon the coordinated parts as relatedand fittedto the end or functionof the wholeand as relatedto or resultingfrom the physicalcauses inherent in theentire system of forces to which the whole had beenexposed, and underwhose influence it has comeinto being" (p. io2o; myemphasis). D'ArcyThompson must have enjoyedthe dialogueof King Lear and the Fool as a propheticsatire upon evolu- tionistswho wouldconsider only final causes, and trivialones at that: Fool: Cansttell how an oystermakes his shell? Lear: No. Fool: NorI neither:but I cantell why a snailhas a house. Lear: Why? Fool: Why,to put's head in. ii) In relatingorganic form to abstractgeometry, D'Arcy Thompson approachedthe Platonic notion of a realrealm of pure form; this realm wouldspecify limits for the imperfect representation ofits forms among earthlyanimals. If mathematicalshapes cannot be transformedone in-

44 I speak here only of the objections that arose from his personal theory of form. He also shared many of the common doubts of his contemporaries- especially their reluctance to grant natural selection a creative role: for if selection were only the headsman for the unfit,what created the fit; and if this creation be by "random" mutation, how can exquisite adaptation arise from "chance." The achievements of population genetics in the 193o's resolved these dilemmas by showing that very small selective pressurescould be effectivein super- intending the gradual fixationof favorable small mutations in natural populations.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 250 NEW LITERARY HISTORY to the other throughinsensible gradations, neither can major discon- tinuitiesin organicform be bridgedby the imperceptiblechain of inter- mediates that Darwinian theoryrequires. Thus, D'Arcy Thompson held that major transitionsin evolutionoften occur suddenly,by some formof "macromutation."45

An algebraiccurve has itsfundamental formula, which defines the fami- ly to whichit belongs.... We neverthink of "transforming"a helicoid intoan ellipsoid,or a circleinto a frequency-curve.So it is withthe forms of animals. We cannot transforman invertebrateinto a vertebrate,nor a coelenterateinto a worm,46by any simpleand legitimatedeformation. . . . Nature proceedsfrom one typeto another... and thesetypes vary accordingto theirown parameters,and are definedby physico-mathematical conditionsof possibility.Cuvier's "types" may not be perfectlychosen nor numerousenough, but typesthey are; and to seek forstepping-stones across the between is to seek in forever gaps vain, (pp. 1094-1095). iii) D'Arcy Thompson's belief in the direct molding of form by physical forcesled him to distrusta primaryguide to the tracing of lineages-that degree of similarityis a rough measure of evolutionary affinity(recency of common ancestry). It is somethingof a trade secretthat mostof our evolutionarytrees are based not upon the direct evidence of fossils,but upon inferencesthat equate reasonable series of modem formswith actual affiliationin time. But if the formsof these seriesare fashioneddirectly, the sequence reflectsno history,but only a gradationof physicalinfluences obeying timeless laws. Of the single-celledForaminifera he wrote (pp. 869-870): While we can trace in the most completeand beautifulmanner the passageof one forminto another among these little shells ... thequestion staresus in theface whether this be an 'evolution'which we have anyright to correlatewith historictime. The mathematiciancan trace one conic sectioninto another and 'evolve'for example, through innumerable graded

45 "Macromutationism,"in variousforms, was the major evolutionarychallenge to Darwinismfrom the "Mutationstheorie"of de Vries (circa I9oo) well into the 1940's and 1950's. The inviabilityof major mutations,the difficultyof spreadingthem through entire populations after they arise in individuals,and the demonstrationthat small mutationsprovide enough genetic variability for evolution have led to the virtualdemise of this concept. D'Arcy Thompson'sreasons foraccepting it wereunique. 46 And thoughwe reject macromutationismtoday, we would not disputethis specificclaim as it relatesto transformationsof complexadult structures.Most evolutionistswould tryto linkthese major groupsthrough transformations of their simpler,larval forms.Though no one (to my knowledge)has ever triedto transforman adult sea urchinto a man, the larvaeof echinodermsand primitivechor- datesare verysimilar. The mostpopular theory of vertebrateorigins would seek to link the two groupsat thispoint. AlisterHardy has referredto larval evolution as an "escapefrom specialization."

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D' ARCY THOMPSON AND THE SCIENCE OF FORM 25I ellipses,the circle fromthe straightline: which tracingof continuous stepsis a true'evolution,' though time has no parttherein. It was afterthis fashionthat Hegel, and for that matterAristotle himself, was an evolutionist-towhom evolution was a mentalconcept, involving order and con- tinuityin thoughtbut not an actual sequence of eventsin time. Such a conceptionof evolutionis not easy forthe modernbiologist to grasp,and it is harderstill to appreciate.

For these simple forms,subject by their small size to surficialand molecular forces,D'Arcy Thompson may well be right. At least his suggestionhas the "audacity of imagination"that John Dewey foundin "every great advance in science"; at least it might free the students of these animals from automatic allegiance to theoriesdeveloped for vertebrates(and not whollycorrect even forthem). However,when he applies it to large and complex forms,I begin to doubt its propriety while continuingto admire its sheer ingenuity. Molluscan shells,for example, are spread neitherevenly nor randomlyacross the range of formpotentially available to them: certainshapes tend to occur again and again. To a modern evolutionist,these recurrentshapes are se- lected because they are functionallysuperior to rare or non-existent ones; to D'Arcy Thompson,they are simplythe shapes that controlling physicalforces produce most easily (p. 849):

It is hard indeed (to mymind) to see in such a case as thiswhere Na- turalSelection necessarily enters in, or to admitthat it has had any share whatsoeverin theproduction of thesevaried conformations. Unless indeed we use the termNatural Selection in a senseso wide as to depriveit of any purelybiological significance; and so recognizeas a sortof naturalselection whatsoevernexus of causes sufficesto differentiatebetween the likelyand theunlikely, the scarce and thefrequent, the easy and the hard: and leads S.. one typeof crystal,one formof cloud,one chemicalcompound, to be of frequentoccurrence and anotherto be rare.

3. D'Arcy Thompson's centralidea-that formis fashionedby the directaction of physicalforces operating during growth-is applicable to very few of the cases he cites. We must firstunderstand D'Arcy Thompson's own limitsupon his theory. He did not deny "a principle of heredity"(p. 1023); he did not attemptto ascribe differencesbetween rhinocerosesand watermelonsto the action of physicalforces. "My sole purpose," he wrote (p. I4-my italics), "is to correlate with mathematicalstatement and physicallaw certain of the simpler outwardphenomena of organic growthand structureor form." Some

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 252 NEW LITERARY HISTORY of thesecorrelations are surelycorrect,47 but others,including the fun- damental comparisonof protozoans with Plateau's figuresof surface tension,48are almostas surelywrong. This substantialcriticism would seem to discreditGrowth and Form, relegatingit to the domain of antiquarians ratherthan to historians (not to mentionpracticing scientists). In fact,it providesthe impetus for a reinterpretationthat explains the book's continuinginfluence among a scientificcommunity that almostmatches, in itsregard for past work, the journalist'smaxim: "yesterday'spaper wraps today's gar- bage."49 D'Arcy Thompson identifiedhundreds of correspondencesbetween physicallaws and organic forms. If his own theorywill not explain them,then anothermust be sought. And that other,ironically enough, is natural selection. D'Arcy Thompson claimed a directinfluence for physical forces; in fact, these forcesoperate indirectlyby specifying the formsthat provide optimal adaptation for animals subject to their influence. "Equilibrium figures are common in organic nature," Hutchinsonwrites, "because any organismnot exhibitingthem would have to be elaboratelyprotected in otherways against deformationby the stressesand strainsimposed by its environment."5 Thus, D'Arcy Thompson's second theme (p. Io) becomes an aspect of his first--that animals are well designed. Aristotle'sgreat student, in short, mixed up his causes. D'Arcy Thompson's error can be epitomizedthis way: He viewed physical forcesas the efficientcause of form;they are, in fact,formal causes or blueprintsof optimum shapes that determinethe direction which natural selection (the true efficientcause) must take to produce adaptation. If physicalforces are not the kind of cause D'Arcy Thompson thought,they are causes nonethelessand no explanationof formis complete without referenceto them. He was rightto correlatephysical forceswith organicforms and to claim that the correspondencewas no mere analogy; but he was rightfor the wrongreason. D'Arcy Thomp-

47 Certain extinct corals (cited on p. 513), for example. When the individuals of a colony are crowded together,each assumes the hexagonal form that laws of closest packing require. When uncrowded, the individuals remain circular in out- line. Since thereare no genetic differences(circular and hexagonal can occur within the same colony, always correlated to the extent of crowding), direct shaping by the pressuresof contact must be the efficientcause. 48 Surface-tensionsof cells have been measured directlyand they are too low to shape the cell. See J. T. Bonner, (note 2), p. 49. 49 The engineering library at Harvard, for example, has exiled all its pre-195o journals to a virtuallyunlit (and completelyunheated) attic. 50 Hutchinson,p. 58 i.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCYTHOMPSON AND THE SCIENCEOF FORM 253 son thoughthe had a theoryfor the efficient cause of good design;he gave us insteadthe basis for a scienceof form--ananalytic approach to adaptation.

IV. D'Arcy Thompson and the Science of Form

The vindicationof D'Arcy Thompson'smethod. Amongbiologists, Growthand Formwears the albatrossof Berlioz'sLes Troyens;for it has been regardedas an unusablemasterpiece doomed by excessive lengthand difficultyof application.Since books are subjectto no such constraintas theeconomics of staging, the charge of excessivelength is morea lamentof busymen thanan argumentfor necessary neglect. The chargeof difficulty in application, however, has beenquite justified until recently.D'Arcy Thompson'smathematical analysis of form requiresthat all theparts of an organismbe consideredsimultaneously -that changeof shape be graspedas a wholethrough the transforma- tionof coordinates imposed over entire bodies. D'Arcy Thompson drew hiscoordinate diagrams as pictures;he did not,because he couldnot in any usefulway, abstract the transformationsas mathematical expres- sions.Thus, Medawar, who attempted the only pre-computer quantifi- cationof transformedcoordinates,51 quite properlytermed the tech- nique"analytically unwieldy."52 The mathematicalstudy of growthand formhas been dominated by bivariateanalysis: theplotting of one organagainst another organ (or totalbody size) duringgrowth. That thisis unsatisfactoryin theory has neverbeen denied,for an animalgrows as a whole,not as an ab- stractedseries of pairs. Whythen was the obviouslypreferable tech- nique of consideringall partssimultaneously-multivariate analysis- not used? The theoremsof multivariateanalysis are not new; many had been developedbefore Growth and Formreceived its second edition. Butthey were not widely applied because the sheer labor of calcu- lationprecluded any practicalvalue. With the adventof electronic computers,the situationhas changed completelyand multivariate analysishas takena place among the mostexciting of new approachesin biology.Unfortunately, D'Arcy Thompsonwas about one half cen- turytoo early. His approach to formwas multivariatein conception;

51 P. B. Medawar, "Size, Shape and Age," in W. E. Le Gros Clark and P. B. Medawar (see footnote4), pp. 157-87. 52 P. B. Medawar, in Ruth D'Arcy Thompson, (see footnote 2), p. 231.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 254 NEW LITERARY HISTORY henceit suffered the misfortune ofmuch prophesy--it could not be used in itsown time. D'ArcyThompson's vindication has comeonly now, and it has just begun. I shallcite but two amongseveral excellent studies in multivariateanalysis that were inspired directly by Growthand Form.53In his chapteron the logarithmicspiral, D'Arcy Thompson pointed out (p. 782) thatthe diverseand seemingly-complexforms of molluscan shellscan be generatedin commonby just a fewbasic variables-the shape of thegenerating curve (the apertureof a snail,for example), itsrate of increase in size,and itschange of position relative to theaxis of coiling.By varyingthese few measures, practically the whole range ofmolluscan form can be simulated.But D'ArcyThompson could not do this,for the number of combinations among the basic measures must runinto the thousands for any meaningfulanalysis. A computercan generatethese forms in seconds,and a tame machinecan even be trainedto draw (and shade!) each simulatedshell (Fig 7). The result

..... ? ,+:...' ? ,, ':. 1:?.

FIo. 7. Computer drawn (and shaded) "snails." The sequence from a to d is generated by increasing the rate of whorl expansion and decreasing the rate of translation down the axis. From D. M. Raup, "Computer as an aid in describing form in gastropod shells," Science, cxxxviii (1962), 150-52. is no mereexercise in mechanizedwizardry, but an importantcontribu- tionto our understandingof form-forit providesa matrixwithin whichsome major questions can be askedfor the first time. Whenwe generatea completespectrum of shapes,for example, we quicklyfind thatactual shells occupy but a smallrange of shapes possible in theory. Whyare certainrealms of shape unoccupied?David Raup has pro- videdsome fascinating explanations based upon theprinciples of me- chanicsand functionalmorphology. 54

53 A compendium of other examples will be found in my article "Evolutionary Paleontology and the Science of Form," Earth-Science Reviews, vi (1970), 77-I19. 54 D. M. Raup and A. Michelson, "Theoretical Morphology of the Coiled Shell," Science, cxlvii (1965), I294-95. D. M. Raup, "Geometric Analysisof Shell Coiling: General Problems,"Journal of Paleontology,xl (1966), 1 I78-90.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 255 P. H. A. Sneath has recentlypublished the firstsuccessful quantifica- tion of D'Arcy Thompson's transformedcoordinates.55 He has done this with modificationsof a techniqueknown best among geologists- trend surfaceanalysis. In trendsurface analysis, the distributionof a featureon a map (rainfallin mm per yearfor example) is abstractedby mathematical surfaces of increasing complexity. Sneath compares chimpanzeeand human skulls. As his feature,he uses the geometricdis- tance between each pair of correspondingpoints (tip of the chin for example); the undistortedcoordinate net itselfserves as his map. The surfacesfitted to these distancesare expressionsof the total difference in shape betweentwo skulls. The differencesare renderedas numbers, and thesecan be analyzed forpatterns that expressthe simplestway to transformone complexobject intoanother. D'Arcy Thompson would have welcomed the computermore as a perpetratorof his basic attitudesthan of his particulartechniques; for he realizedthat men's theoriesare subtlymolded by the machinerythey choose (or are constrained) to use. In particular,he feltthat the de- scriptivemethod of conventionaltaxonomy, though lauded by its prac- titionersas objective and atheoretic,implies the view that complexity is irreducible.

For the morphologist,when comparingone organismwith another, describesthe differencesbetween them point by pointand 'character'by 'character.'If he is fromtime to timeconstrained to admit the existence of 'correlation'between characters.... yetall thewhile he recognizesthis fact of correlationsomewhat vaguely, as a phenomenondue to causes which,except in rareinstances, he can hardlyhope to trace; and he falls readilyinto the habit of thinkingand talkingof evolutionas thoughit had proceededon thelines of his own descriptions, point by point and character bycharacter (p. I136). In D'Arcy Thompson's approach, we do not seek a differentexplana- tion for each contrastbetween two organisms;by notingcorrelations, we tryto reduce differencesto the fewestfactors needed to generate them. We do this not for the intellectualsatisfaction provided by simplification,but because the abstracted systemof factors can be linkedmore easilyto cause. In likeningthe human stomachto a bubble restrictedby a trammel,D'Arcy Thompson invokesthe glassblowerto itsform : explain (p. 1o50) The glass-blowerstarts his operationswith a tube,which he firstcloses at one end so as to forma hollowvesicle, within which his blast of air exer- cisesa uniformpressure on all sides;but thespherical conformation which

55 P. H. A. Sneath,"Trend-Surface Analysis of TransformationGrids," Journal of Zoology, cli (1967), 65-122.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 256 NEW LITERARY HISTORY thisuniform expansive force would naturallytend to produceis modified intoall kindsof formsby the trammels of resistancesset up as theworkman lets one part or anotherof his bubble be unequallyheated or cooled. It was OliverWendell Holmes who firstshowed this curious parallel between theoperations of the glass-blowerand thoseof Nature,when she starts,as sheso oftendoes, with a simpletube .... Such a formas thatof the human stomachis easilyexplained when it is regardedfrom this point of view; it is simplyan ill-blownbubble, a bubblethat has been renderedlopsided by a trammelor restraintalong one side, such a trammelas is producedif the glass-blowerlets one side of hisbubble get cold, and such as is actually presentin the stomachitself in the formof a muscularband.

A science of form. Form and diversityare the two greatsubjects of natural history.The studyof speciation and systematicshas given us a science of diversitywithin evolutionary theory, but we have lacked a science of form.56I believe that a science of formis now arisingand that D'Arcy Thompson will be regarded as its godfatherbecause he anticipated and developed the two principlesupon which it will be based: i. Adult form,in all its complexity,shall not serve as a primary datum. It mustbe reducedto a smallerset of factorsthat can generate it during growth. Causes shall be sought among these factors,not in theirresults. D'Arcy Thompson renderedthese factorseither theoreti- cally as the physicalforces molding formor pictoriallyas the simple patternsof transformedcoordinate nets. With the techniquesof computer simulation,we can realize D'Arcy Thompson's unattained goal of quantitativeexpression for these factors. 2. When the "how" of formis explainedin (I), we mustachieve an equally rigoroussolution for its "why." Yet the studyof adaptationhas been plagued by vague, trivialand untestableproposals. We need a criterionof relativeefficiency-a way to determinewhich of two struc- turesis betterdesigned to performthe same function.D'Arcy Thompson stressedthe mechanical propertiesof form. As we have seen (p. 286), he confusedhis causes; but when we view physicalforces not as the architectsof formbut as the blueprintsthat specifyoptimum shapes, we have our test for relativeefficiency: the comparisonof an actual structurewith its optimumexpressed in an engineer'sterms.

56 I have mentioned several reasons for this in various parts of this work: pre- vious lack of a technology to handle the calculations of multivariate analysis (p. 287); the eclipse of functional anatomy by evolutionary theory and its use of formonly for the tracing of lineages (p. 266) ; the attitude towards formimplicit in the part-by-partdescriptions of conventionaltaxonomy (p. 267). I presentthis general argumentmore fullyin the article cited in note 53.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions D'ARCY THOMPSON AND THE SCIENCE OF FORM 257 Withthese two analyticaltools, a scienceof formcan providein- sightsinto the majorevents of life'shistory. As one example,the at- tainmentof strikinglysimilar external shapes by animalsof verydif- ferentancestry57 -the phenomenonof convergence-isrecorded with greatfrequency in thefossil record. In fact,Sylvester-Bradley has called itsrecognition "the most distinctive contribution that paleontology has made to modernevolutionary synthesis.""58 Older naturalistswere proneto see in convergenceeither an exampleof God's plan or an organic strivingfor ideal form. Today, no one would seek any other efficientcause thannatural selection; yet final causes often remain as intractableas ever. D'ArcyThompson, however, provided a general explanation:the principlesof mechanicsspecify a limitednumber of good designsfor the solutionof commonproblems faced by animals. Thesecan be determineda priori, and usedto predictthe forms of un- discoveredorganisms performing specified functions. What seemed mysteriouscan be explainedand evenpredicted. Our argumentindicates . . . that evolutionarychanges, occurring on a comparativelyfew definite lines, or plainalternatives, ofphysico-mathe- maticalprobability, are likelyto repeatthemselves: that the "higher" protozoa,for instance, may have sprung not from or throughone another, but severallyfrom the simplerforms; or thatthe worm-type,to take anotherexample, may have come into being again and again(p. 1095). It is easyto assessD'Arcy Thompson's influence; for that is illustrated by the continuinguse of his workin the technicalresearch of distin- guishedscientists, and by thelocation of his mainideas at thecore of an emergingscience of form.It is muchmore difficult, however, to measurehis greatness;for he statednothing truly new, and noveltyin discoveryis the criterionused by mostscientists in assigningstatus. Growthand Form is an ingeniouscompendium of classicalwisdom temperedwith insights from the later, but scarcely modern, age ofGali- lean mechanics.To see it as a greatwork, we mustabandon the usual conceptionof noveltyand admitthat the unionof previouslyuncon- nectedtruths can be an intellectualdiscovery as brilliantand as im- portantas thedisclosure of something formerly unknown. And D'Arcy Thompsondid not merelyunite a seriesof truthsgenerally known in theirisolated state; he combinedtruths long forgotten by his colleagues.

57 Standard examples include the attainment of cup-coral shapes by a Permian brachiopod and a Jurassic clam and numerous correspondencesbetween Australian marsupials and placentals of the major continents (mole and marsupial "mole" for example). 58 P. C. Sylvester-Bradley,"Iterative Evolution in Fossil Oysters," Proceedings, International I Zoological Congress, (1959), 193.

This content downloaded from 134.84.3.112 on Wed, 22 Apr 2015 15:47:09 UTC All use subject to JSTOR Terms and Conditions 258 NEW LITERARY HISTORY Thereare somenotions so abidingin man'sview of naturethat we can scarcelydeny their status as aspectsof realityor as wayswe must perceivethe world. I wouldplace here, for example, the idea thatani- malsare oftenexquisitely designed to do whatthey do. Such a notion receivesdifferent explanations at varioustimes (God, chance,natural selection);it mayeven be cast aside fora timewhen a new scientific fashionrashly dismisses it with a properlydiscarded theory for its explanation.59Yet it returns,for it must.And, when it returns,scien- tistsrediscover what their forgotten predecessors knew perhaps better thanthey. D'Arcy Thompson was greatbecause he had notforgotten.

MUSEUM OF COMPARATIVE ZOOLOGY, HARVARD UNIVERSITY

59 It is still unfashionable, in biological circles, to use such words as "design," "purpose," or "teleology." Since final cause is so indispensable a concept in the elucidation of adaptation, and since natural selection can produce a well-designed structurewithout any conscious interventionof God's super-human wisdom or the sub-human intelligenceof the animal in question, one would think that these terms would again be admitted into orthodoxy. Evidently, however, in our choice of words, we are still fightingthe battle with theologians that we won in deeds almost a centuryago.

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