1. No category

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. 229258
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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D'Arcy Thompsonand theScience of Form
Stephen
JayGould
Our own studyof organicform,which we call by Goethe'sname of
Morphology,is but a portionof thatwiderScience of Form whichdeals
withthe formsassumedby matterunderall aspectsand conditions,and,
in a stillwidersense,withformswhichare theoretically
imaginable.
Growthand Form,p. 1026
Preface
JN
1945, the Public Oratorof Oxfordlauded D'ArcyThompson
as unicumdisciplinaeliberalioris
exemplar1;in 1969, the Whole
EarthCatalog called his majorwork"a paradigmclassic." Few
in theircompendium
of honors.
men can listsuchdiversedistinctions
But then,fewmenhave displayedso wide a rangeof talent. D'Arcy
WentworthThompson (I86O-I948), Professorof Natural Historyat
Dundee and St. Andrews,2translatedAristotle'sHistoria Animalium,
wrote glossariesof Greek birds and fishes,compiled statisticsfor the
The outstandingexample of a man of liberal education-on the occasion of his
i
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 biographywrittenby 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 abridgmentof 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
of Form (London, I951), for the I951 Exhibition at the Instituteof 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.
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230
NEW LITERARY HISTORY
thearticleon pycnogonids3
BoardofScotlandand contributed
Fishery
to the CambridgeNaturalHistory.4He also wrotea book of one
as wellas byengineers
thousandpages,revered
and architects
byartists
and biologists-the"paradigmclassic,"On Growthand Form (1917,
2nd edition,1942). To P. B. Medawar it is "beyond comparisonthe
in all theannalsof sciencethathavebeen reworkofliterature
finest
it is "one
cordedin theEnglishtongue."5 To G. EvelynHutchinson,
of the veryfewbookson a scientific
matterwrittenin thiscentury
whichwill,onemaybe confident,
culture."
6
lastas longas ourtoofragile
on organicform;these
In it D'ArcyThompsondisplayedhisthoughts
in othersand alwaysprofound.
are curiousin places,almostvisionary
Almostthirtyyearsafterthe secondedition,and morethan half a
afterthefirst,
century
theyhave gainednewimpactin a sciencethat
to deal withhisinsights.
onlynow has thetechnology
I. IneluctableModalityoftheVisible
"Extension,figure,number,and motion.. .," wroteJohnLocke, "may
calledreal,originalor primary
be properly
qualities,becausetheyare
whether
are
in thethingsthemselves,
or no."'7 Stephen
they perceived
Daedalus,walkingalongthe Irishsea side,reviewedthe dilemmasof
and evenperformed
thecrucialexperiment:
"Ineluctable
epistemology
of
he
the
Then
was
aware
of
visible...
.
them
bodies
before
modality
of themcoloured.How? By knocking
his sconceagainstthem....
Open youreyesnow.I will.One moment.Has all vanishedsince?...
See now. There all the timewithoutyou: and evershallbe, world
without
end."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 JamesJoyce,Ulysses (New York,
196I), p. 37.
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
231
Formpervadesthematerialworld.Aristotle
was willingto consider
formwithoutmatterforprimemoversand demiurges,
but notmatter
withoutform.9In studyingnatureever since,mostbiologistshave
agreedwithNeedhamthat" thecentralproblemofbiologyis theform
suited
To anyonewhoviewsformas an outdatedconcern,
problem."'10
the
of
world
dominated
for
few
a
taxonomists
only
surviving
by the
a
and
of
recommend
book
molecular
I
chemistry energetics
biology,"
thatstandssecondonlyto D'ArcyThompson'sas a 20thcentury
paean
the
to form: J.D. Watson'sThe DoubleHelix. Forthisbookdescribes
12
that
to
of
a
"too
not
a
be
true," molecularshape
pretty
pursuit shape
ofheredity.Watson'ssucwouldunderlieand explainthephenomena
froma concernwithform,froma methodology
cess emergeddirectly
tothesearchfor
ofmodelsinpreference
theconstruction
thatprescribed
He describesa lessonimpartedby Francis
a moresubtlechemistry.
Crick: "I soonwas taughtthat... thekeyto Linus' [Pauling]success
was hisrelianceon thesimplelawsofstructural
chemistry.
.... The essentialtrick... was to askwhichatomsliketo sitnextto each other.
toolswerea setofmoleIn placeofpenciland paper,themainworking
children."13
of
the
cularmodelssuperficially
resembling toys preschool
thatthey
not
follow
but
it
does
with
deal
must
All biologists
form,
treatit adequately.In myown fieldof evolutionary
biology,I detect
insufficient
whencomparedwith
threeapproachesthatseemespecially
ofD'ArcyThompson:
theinsights
to renderformin suchnon-morphological
I. Theoriesthatattempt
termsas motion,flow,and energy: Darwin'ssuccessand Lamarck's
on specific
failureis no simpleconsequenceof theirdiffering
positions
For
to
form.
their
reflects
also
it
Lamarck,
approaches
opposing
points;
motionand becomingwere primary;organicmatter,ever in flux,
mountedthe scale of being,impelledby "the forcewhichtendsin9 See J. Needham'sanalysisin "BiochemicalAspectsof Form and Growth,"in
L. L. Whyte(see footnote
2), pp. 77-86.
and
Order
Mass.,1968), p. 23.
Life (Cambridge,
Io J.Needham,
in
I fearthat the messageof molecularbiologyhas oftenbeen mistranslated
I
thiswayas it descended(or ascendedaccordingto one'sorientation)fromtechnical
Americanto theNew YorkTimesand intopopularconjournalsthroughScientific
is reladoes not preachthat "everything
sciousness.Justas Einsteinianrelativity
tive" (ratherthe opposite,in fact), neitherdoes molecularbiologyreplace the
or
withthe dynamicsof chemicalenergetics
conceptof formand spatialstructure
the abstractionof "information
models"; the greatestachievementof molecular
biology,afterall, was theelucidationby Watsonand Crickof thephysicalshape of
thisgeneticinformation.
moleculethattranscribes
DNA, theextraordinary
I
J.D. Watson,The DoubleHelix (New York,1968).
13 Ibid.,p. 38.
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232
NEW LITERARY HISTORY
cessantlyto complicateorganization."As Gillispie14writes: "Lato makea scienceout
was thelastattempt
marck'stheory
ofevolution
of theinstinct
... thattheworldis fluxand process,and thatscience
ofmatter,
northecategories
ofform,
is to study,nottheconfigurations
whichis ontologically
fundamenbutthemanifestations
ofthatactivity
tal as bodiesin motionand speciesof beingare not." To Darwin,
we sensethisin hisalparadoxicalas it mayseem,formwas primary;
oforchids,
mostpoeticaladmiration
fortheelaborateforms
exquisitely
to
insure
insects.
15
designed
pollination
by
Gillispiecontinues:"What
[Darwin]did was to treatthatwholerangeofnaturewhichhad been
tobecoming,
setofobjective
as a problemofbeing,an infinite
relegated
situations
back
time
....
The
Darwinian
theoryof
reaching
through
evolution
turnedtheproblemofbecoming
intoa problemofbeingand
the eventualmathematization
of thatvast area of nature
permitted
of
fromlogosin thewrappings
whichuntilDarwinhad beenprotected
16
process."
The triumphof Darwinism,however,did not assurean adequate
ofform.The analysisofbiologicalformmustemphasizethe
treatment
of a structure
to perform
functions
conceptof adaptation-thefitness
of animalsthat
to an organism.Good designis an attribute
beneficial
ofa Paleyor a Cuvier. It should,
intothemetaphysic
fitcomfortably
of course,have been just as congenialto Darwinism(as it was to
forevolution
had merely
Darwinhimself),
substituted
naturalselection
forGod as the efficient
cause of adaptation.Evolutionists
and creationists
wereequallyhappyto findgood designin nature,theformer
theeffects
ofnaturalselection,
becauseitillustrated
thelatterbecauseit
of hiscreativeinterference.
mainfested
God's plan fora worldworthy
Yet modernevolutionary
form,"to
theoryhas tendedto deemphasize
dissolve,
despiteitsgreatverbalemphasison function,
genuineadaptationintothe non-morphological
conceptsof gene-pool,genetical'fit17
In
etc."
ness,'adaptivezones,
part,thisis thelegacyof earlyevoluwho choseto avoidthe conceptof design(whichrepresented
tionists
the old biology,thoughit fitas wellwiththe 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 (Baltimore, 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 Contrivancesby 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 forthe Philosophyof Science, xv ( 1964), 39.
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
233
"the
enigmas,mostnotablyto thechargethatSpencer'scatch-phrase,
survivalof the fittest,"
is circularbecausefitness
is measuredas the
does so define
capacityforsurvival.Indeed,the moderngeneticist
ofform,
butDarwindid not. To him,fitness
was a property
"fitness";
a measureofgooddesignthatdidnotentailsurvival,
a priori.In many
anti-Darwinian
ofevolution,
thefit,in thissense,do notnecestheories
sarilysurvive(notionsof racial old-ageor orthogenesis
propellinga
coursesbeyondthepointof adaptation). Spencer's
lineageon straight
theprimary
of Darwin'ssystem
phrase,therefore,
expresses
prediction
and embodiesthetestbywhichit can be comparedwithothertheories
and accepted.
but do notattemptto ex2. Theoriesthatdeal withformdirectly
taxonomists
who
to
it:
Among
pretend engagein puredescripplain
ofthismost
thatexaltstheunsullied
tion,thereis a mystique
objectivity
freefromtheinhumbleserviceto Nature,thisdisplayof Her forms,
trusionof humanspeculationand thevanityof theory.The ironyof
forno human
of its attainment,
thisattemptlies in the impossibility
"Pure"
see
the
can
the
Ding-an-sich.
description, piece-by-piece
being
is as firmly
rootedin theoryas themost
of an organism,
compendium
to
theproblemis notonly
abstractand mathematical
form;
approach
is hidden,but also thatit is wrong.A standardspecies
thatitstheory
catalogstheorganismpartby part; thisimplies,in a way
description
becauseit is inexplicitand even uninthatis subtleand captivating
a framework
foritsseparatepartsand
tended,thatan animalis merely
thistheory
thatitscomplexity
is irreducible.In purelyheuristic
terms,
is slim;for
to reality
is sterileand we musthopethatitscorrespondence
onceit is statedwe can do littlewithformbutcatalogit in wonder.
3. Theoriesthattryto explainform,but do so incompletely:In
fortheregular
the i89o's D'ArcyThompsondevelopedan explanation
them
at cell conHe
ascribed
to
of
form spongespicules.
adsorption
the
reflects
their
tacts;
regularform,
(Fig. I) merely
simplegeometry
ofcells.MichaelFoster,
ofan aggregate
assumedundersurfacetension,
was displeased: "I confessI am
an eminent
evolutionary
physiologist,
of
work
.... Does yourresult
the
line
notverymuchattracted
wholly
by
in
value
of
the
the
spicules?If theformis constant
destroy diagnostic
This
1
about."
form
is
not
matter
how
the
does
a group-it
brought
the approachto formthat characterized
statement
early
epitomizes
evolutionary
thoughtand offendedD'Arcy Thompson. This is the
idea thatformis to be usedonlyas
ofmorphology-the
"sign"theory
are to be analyzedas
a guideto thetracingoflineages;thatstructures
not as designsformodemexistence;and thatform
signsof ancestry,
18
Quoted in Ruth D'Arcy Thompson, p. go.
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NEWLITERARY
HISTORY
234
...
FIo. i. D'Arcy Thompson's illustrationof the regular formof sponge spicules. From
his chapter ix, "On concretions,spicules and spicular skeletons."
is explainedonceitsantecedents
in ontogeny
areidentiand phylogeny
are
of recapitulation,
structures
fied. Thus,underthetheory
juvenile
notto deterstudiedto identify
theancestral
adultthattheyrepresent,
minehowtheyfunction
in theyounganimal. D'ArcyThompsondid
notgainsaytheuseofformto tracelineages;he did decrya morpholowhenit had donethis-forexitsworkcompleted
gythatconsidered
of
Aristotle
translator
as
the
understood,19
requiresthe
planation,
I9 So many heated (and empty) argumentsabout "proper" explanation would
be avoided if scientistsonly heeded Aristotle's ancient lesson on the multiplicity
of causes (and realized that differentdisciplinesidentifydifferentcauses and then,
in their limited perspective,wronglyclaim 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).
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
235
ofmany"causes." We try,forexample,to determine
the
identification
of a well-designed
to
its
final
This
function
cause.
structure, identify
remainsinsufficient
still. In an earlyarticleon theshapesof eggs,he
offinalcauses(prevention
ofrollingoutofthenest,
mentions
a variety
closepackingwithinthenesttopermit
moreeggsin a givenspace) and
to
comments:"Whatevertruththerebe in theseapparentadaptations
we
can
a
that
it
is
existingcircumstances, onlyby veryhastylogic
acceptthemas a veracausa or adequateexplanationof thefacts."20
cause as well: "In Aristotle's
We muststatetheefficient
parable,the
houseis therethatmenmaylivein it; butit is also therebecausethe
All the while,like
buildershave laid one stoneupon another....
and
mechanism
and
are
interwoven
and
woof,
warp
teleology
together,
we mustnotcleaveto theone nordespisetheother;fortheirunionis
rootedintheverynatureoftotality"
(p. 7).21
ofnatureto theidea oforganic
anchored
view
his
D'ArcyThompson
form(contraz); he triedto explainformby reducingitscomplexity
tosimpler
thatcouldbe identified
as causes(2) ; he recognized
elements
is multifaceted;
thatexplanation
thattheidentification
onlyof a purincoma
a
or
a
ancestor,
mechanism,
pose,
provides pitifully
phyletic
ofform(3) .
pleteanalysis
II. D'Arcy Thompson's Theory of Form
In the epilogueto Growthand Form,D'Arcy Thompsonstatedhis
is ...
aim: "to shewthata certainmathematical
aspectofmorphology
and
of
Growth
to
the
nayessential,
properstudy comprehension
helpful,
he wrote:"Numericalprecision
and Form"(p. o096). In thepreface,
affords
thebest,perhaps
is theverysoulofscience,and itsattainment
ofexperiand thecorrectness
ofthetruthoftheories
theonlycriterion
withhis
he invokedAuthority
ments"(p. 2). In theopeningsentence,
and
of
his
the
"Of
erudition:
day
generation,
chemistry
customary
Kantdeclaredthatitwas a science,butnotScience-eineWissenschaft,
oftruesciencelayin its
thatthecriterion
abernichtWissenschaft-for
This was an old story: forRogerBacon had
relationto mathematics.
and Leonardoda Vinci
calledmathematics
portaet clavisscientiarum,
20
"On the Shapes of Eggs, and the Causes whichDeterminethem,"Nature,
1 1.
lxxviii (1908),
All page numberscited in the text referto the 2nd edition of On Growth and
2
Form (Cambridge, 1942).
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NEWLITERARY
HISTORY
236
had said much the same ...."
(p. i-there followsan untranslated
Italianfootnote).
Yet D'ArcyThompson'smathematics
has a curiousring. We find
noneofthedifferential
thatadorn
statistics
equationsand mathematical
we read,instead,ofthe
modernworkin ecologyorpopulation
genetics;
of space,the tetrakaidekahedron,
the Miraldiangle,the
partitioning
enterequations;
logarithmic
spiralandthegoldenratio.Numbersrarely
For D'ArcyThompsonwas a Greek
rather,
theyexemplify
geometry.
with2othcentury
mathematician
materialand insights.Growthand
Formis thesynthesis
ofhistwolives: eminentclassicist
and eminent
AddresstotheClassicalAssociazoologist.As hestatedin a Presidential
tion (1929):
"Science and the Classics is my theme today; it could
hardlybe otherwise.For all I know,and do, and wellnighall I love
and carefor(outsideofhomeand friends)lieswithinone ortheother;
andthefactthatI havelovedthembothhas coloredall mylife,and enandmultiplied
largedmycuriosity
myinletstohappiness."22
Those who recognizethe classicalrootsof Growthand FormgenYet Aristotle
erallylinkthatworkto Aristotle.
represented
onlyone of
the two classicalinputsto D'ArcyThompson'sscience-and the one
oflesserimportance
forGrowthand Form. D'ArcyThompsonrevered
Aristotle
as a descriptive
naturalist:"In his exhaustive
accumulation
and treatment
offacts,hismethodis thatoftheobserver,
ofthescientificstudent,
and is in themaininductive."
he identified
23 Moreover,
as Aristotle's
whatmostcriticsconsiderthetwinstrengths
weaknesses
of Growthand Form: aesthetic
and mathematical
style24
skill.25
a classicist
had comparedArishimself,
D'ArcyThompson'sfather,
totle'steachings
"to theinvigorating
ofsea-bathing,
properties
openair,
wholesome
dietand quinine"and the"preaching"of
regularexercise,
Platoto "opiumor hachish."
and thelaterPlato
26 It is to Pythagoras
oftheTimaeusthatD'ArcyThompsonoweshisvision;forhe tried,as
did BertrandRussell,"to apprehend
thePythagorean
powerbywhich
numberholdssway above the flux."27D'Arcy Thompsonaccepted
22 Quoted in Ruth D'Arcy Thompson, p. 196.
23 "On Aristotleas a Biologist,"Nature, xci ( 913), 204.
24 Ibid., p. 2o2: "Wise and learned as Aristotlewas, he was neither artist nor
poet. His styleseldom 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 Insecabilibus, I am tempted to suspect that he sometimespassed 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.
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D'ARCY THOMPSON AND THE SCIENCE OF FORM
237
neitherPythagoras'
doctrinethat"thingsare numbers,"28
nor Plato's
the
from
bodies
visionofa realmofidealnumbers
existing
physical
apart
But
he
did
thatare but theirfleeting
and imperfect
representation.
sharetheirgeneralattitudes:thatsolutionsto the world'smysteries
shouldbe soughtin "the geometrical
29; thatsimaspectof number"
and truthwouldbe foundconsymmetry,
harmony
plicity,
regularity,
joined"; thattheworldof "science"couldbe approachedand coma book on the
prehendedby poeticvision.As he wrotein reviewing
or
or
"The
Pythagoreans:
Pythagorean, Platonic, Jewishconceptof
Numberis a hard sayingto the unpoetic,non-mystical
modemand
Westernworld; and manya way is foundto show that Plato and
meantsomething
afterall. Butto
Pythagoras
prosyand commonplace
of
someitis stillas plainas everthatNumberis theclueto thegreatest
what
whether
of
and
that
we
call
or
sound
beauty,
mysteries,
earthly
In Growthand Form,Plato
form,is but its resultant
expression."31
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
respectablescience as well. R. M. McInerny, A History of WesternPhilosophy: Vol.
I, From theBeginnings
of Philosophyto Plotinus(NotreDame, 1968), p. 45, has
shown how the Greek's very "manner of depicting numbers leads to speaking of
types of numbersprogressingin termsof 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 extrapolation from this to a notion that the "elements of number are the elements of all
Since their way of depicting numbers produced plane and solid
things. ...
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 influenceof 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 principlesof 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 usefuland excellent."
31 "The School of Pythagoras," Nature, xcviii (x9I6), 166. The very next
sentence reads: "It was in the very spirit of Pythagoreanmysticismthat that great
naturalist,Henri Fabre, wrote his great ode to number." And it was with a quotation 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
of imthe bee . . . who in his all but saecular life had tasted of the firstfruits
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NEW LITERARY HISTORY
238
and Pythagoras
dreamers"(p. 2).
arelinkedas "thosegreatphilosophic
of the heavenlyhostmustbe determined
"Not onlythe movements
and elucidatedby mathematics,
else
but whatsoever
by observations
can be expressed
by numberand definedby naturallaw. This is the
and themessageofGreekwisdomto
teachingofPlatoand Pythagoras,
mankind" (p. 1097).
Growthand Formis not a textin anachronistic
Greekbiology.It
modemtheories(Darwinismin parusesclassicalconceptsto criticize
withtheproblems
ofengineerticular),and itshowsa manas familiar
in
as
the
of
with
ing bridgedesign
tetractys Pythagoras.It expresses
the tensionsand conflicts
thatmustinherein any philosophy
comfrom
such
sources.
Of
two
are
these
tensions,
pounded
disparate
parin Growthand Form. I) His viewof Plato and
ticularly
important
and deduction)versusAristotle
generality
Pythagoras(mathematics,
and
His
Greek
commitment
to a pureand
(description induction).2)
of
form
abstract
versus
Baconian
idea
that
his
understanding
knowledge
is power,as expressed
lovefora good designbecause
bytheengineer's
itworks.
showedin hisperceptive
As Hutchinson
32D'ArcyThompson's
essay,
threethemes: an old idea to whichhe
theoryof formencompasses
broughtnew lightand grace,a theoryon the productionof form
and a mathematical
methodforrepresenting
thecauses
growth
through
in shapeamongrelatedorganisms.
ofdifferences
are well designedis amongthe oldestideas of
i. That organisms
Galileoand Borelli,33
biology.WhatD'ArcyThompsondid,following
was to conductthisobservation
awayfrompassivewonderat nature's
marvelstowardthe analytictechniquesof physics,
mathematics
and
When
is
the
this
of
form
becomes
a
science:
done, study
engineering.
can be madeandtested;actualforms
canbe comparedwith
predictions
the ideal configurations
that engineersprescribe;the superiority
of
one formoveranothercan 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 discussionis 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.
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
239
of thistheme,the pages of Growthand Form
As one illustration
thatorganicshapesconform
aboundwithexamplesdemonstrating
to
at theirscale. The pervasive
thephysical
forcesprevailing
ofsize
effect
upon formis a simpleconsequenceof physicallaws and geometric
arguments.As an animal (or any object) grows,its volumewillincreaseas thecubeofitslengthwhileitssurface,
ifit maintains
thesame
as
the
the
the
the
increases
larger animal, greater
only
shape,
square;
the ratioof itsvolumeto itssurface.Thus, a largeanimallivesin a
thatworkuponitsvolume
worldruledbyforces,
gravitation,
primarily
is
littleby gravity;the
A
smaller
animal
influenced
its
mass).
(i.e.
forcesthatactuponitssurfaceholdswaybecausethatsurfaceis so large
affects.Stillotherforces,
therandom
relativeto themassthatgravity
come
into
for
motion
the
of
Brownian
shocks
playat bacterial
example,
ofthemost
of
some
This
influence
size
dimensions.
explains
physical
4
of
can
walk
obvious
statements:
whyanyfly
up a wall,but
confusing
onlyJesuscould walk on water35;why elephantshave thickerlegs
thangazelles36;and whythegiantantsof Themreallycouldn'thave
ant
reachedtheLos Angelessewers.
37 Kirbyand Spence,extrapolating
of
a
such
and
the
size
to
calculating prodigious
powers
power elephant
lest
the
for
His
in
wisdom
God
thanked
leviathan,
weakening elephant,
oftheworld"(quotedon p. 36).
it shouldcause"theearlydesolation
D'ArcyThompsonshowedthatifGod had had anyhandin thematter,
34 The differencesbetween life in such a world and ours are brilliantlystated in
George Gamow's classic Mr. Tomkins in Wonderland (Cambridge, I94o). Wonderland is a world in which quantum and relativityeffectsoccur at our customarysizes
and speeds.
35 The downward force of gravitypulls animals fromwalls and through the surface-filmof a pond; surfacetensionprovides adhesion both to wall and pond surface.
Small animals have such a high ratio of surface to volume that the force of surface
tensionoutstipsthat 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 proportionto volume, as the
cube of length. A series of animals differingin size but not in shape will become
progressivelyweaker 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 sizeis called "allometric." Julian Huxley pioneered the quantitative study of allometric
growth in the Ig92's. See S. J. Gould, "Allometry and Size in Ontogeny and
Phylogeny,"Biological Reviews, xli (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 flightfroman elevated spot." At their size, the
giant ants could not even have taken off from their nest near Alamagordo. In
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NEW LITERARY HISTORY
240
he had workedin rathermundane fashionthroughthe 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 physicalconsequences. The chapter ends
thus (p. 77) :
Life has a rangeof magnitudenarrowindeed comparedto that with
whichphysicalsciencedeals; but it is wide enoughto includethreesuch
discrepantconditionsas thosein whicha man, an insect,and a bacillus
have theirbeingand play theirseveralroles. Man is ruledby gravitation,
findsthe surfaceof a pool a
and restson motherearth. A water-beetle
or an indispensable
matterof lifeand death,a perilousentanglement
support. In a thirdworld,wherethe bacilluslives,gravitationis forgotten,
of the liquid, the resistancedefinedby Stokes'law, the
and the viscosity
molecularshocksof the Brownianmovement,doubtlessalso the electric
and
chargesof the ionizedmedium,make up the physicalenvironment
on theorganism.The predomihave theirpotentand immediateinfluence
nantfactorsare no longerthoseof our scale; we have cometo theedge of
and whereall ourpreconceptions
a worldofwhichwe have no experience,
mustbe recast.
2. The relationshipof size and shape linksthe firstthemeto the second-that physicalforcesexert a direct and immediateinfluencein
shapingorganismsas theygrow. This is the guidingconceptof Growth
and Form; all but threeor four of its seventeenchaptersare devoted
betweenorganicformsand the shapes
to thecitationofcorrespondences
thatphysicalforcesproducein actingupon non-livingmaterialof similar 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 increased folding or invagination). There is a limit to this elaboration, lest there be
no room forinternal 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 notoriousforthe limited imagination that makes giant insectsact
as only normal ones could. The most incredible thing about the Incredible Shrinking Man is that, at three inches, he continues to operate as a gravity-pronesixfooter.
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 inappropriatecriterion
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.
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D'ARCY THOMPSON AND THE SCIENCE OF FORM
24I
Many criticshave failedto graspthisview of actionofphysicalforces;
theydismissGrowthand Form as a book filledwith curious and ingenious analogies that describesome aspects of formbut explain nothing. To D'Arcy Thompson,however,a correspondencebetweenorthat
ganic and inorganicis morethan an analogy; it is a demonstration
two objectsprobablyhave the same efficient
cause (p. Io) :
The wavesof thesea, the littlerippleson theshore,the sweepingcurve
of thesandybay betweentheheadlands,theoutlineof thehills,theshape
of the clouds,all theseare so manyriddlesof form,so manyproblemsof
and all of them the physicistcan more or less easilyread
morphology,
and adequately solve: solvingthem by referenceto their antecedent
phenomena,in the materialsystemof mechanicalforcesto which they
themas beingdue .... Nor is it otherbelong,and to whichwe interpret
wise withthe materialformsof livingthings.Cell and tissue,shell and
bone,leaf and flower,are so manyportionsof matter,and it is in obedience to thelaws of physicsthattheirparticleshave 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 infersfromthe correspondence(Fig. 2) that surfacetension shapes the simple, single-celledanimal. As a liquid cylinderis
stretched beyond the limits of stability, it breaks up into "a series of
equal and regularlyinterspacedbeads, oftenwith littlebeads regularly
interspacedbetweenthelargerones" (p. 386). D'Arcy Thompson finds
thesesame beads in the "dew-drops"of a spider'sweb and providesa
than that of
mechanical explanationfar more simple and satisfactory
earliernaturalists(p. 387) :
The same phenomenonis repeatedon a grosserscale when the web is
bespangledwithdew, and its threadsbestrungwith pearls innumerable.
formed
theseregularlyarrangedand beautifully
To the oldernaturalists,
of
wonderment.
web
were
a
source
on
the
frequent
spider's
globules
Blackwall,countingsometwentyglobulesin a tenthof an inch,calculated
web shouldcompriseabout I20,ooo globules;
thata largegarden-spider's
the net was spun and finishedin about fortyminutes,and Blackwallwas
filled with admiration of the skill and quickness with which the
theselittlebeads. And no wonder,foraccordingto
spidermanufactured
theabove estimatetheyhad to be made at therateof about50 per second.
At largersizes,gravitycomes into play. In the medusae of coelenterates, "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 certainsimpleorganicformsmight
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242
NEW LITERARY HISTORY
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 fluidmass withinanotherwhen gravity,
surfacetensionand fluid frictionplay theirpart" (p. 397). At still
largersizes,surfacetensionbecomes so negligiblethat rigidhard parts
are needed to maintainshape, lest gravitymake a world of pancakes.
In the internaltrabeculaeof vertebratebones D'Arcy Thompson found
patternsthat mirrorthe stressesimposed upon them by the body's
weight:
A greatengineer. . happened (in the year 1866) to come into his
wheretheanatomistwas contemplating
colleagueMeyer'sdissecting-room,
the sectionof a bone. The engineer,who had been busydesigninga new
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
243
FIG. 3. Comparisonbetweena fallingdropproducedby fuseloil in paraffin
(left)
and the medusoid"jellyfish"
Syncoryme
(right). Fromchapterv, "The formsof
cells."
and powerfulcrane,saw in a momentthatthe arrangement
of the bony
trabeculaewas nothingmorenor lessthana diagramof thelinesof stress,
or directions
of tensionand compression,
in theloaded structure:in short,
the bone in preciselythe mannerand dithat Naturewas strengthening
rectionin whichstrength
was required;and he is said to have criedout,
"That's mycrane" (pp. 976-977and Fig. 4 of thiswork).
From thisobservationof good design,D'Arcy Thompsonmade his usual inferenceto an efficient
cause (pp. 984-985):
thatour bone is notonlya livingbut a highWe mustalwaysremember
ly plasticstructure;the littletrabeculaeare constantly
beingformedand
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NEW LITERARYHISTORY
244
FIG.
inthebonytrabeculae
crane(left)andas reflected
inMeyer's
4. Linesofforce
of the human femur (right). From chapter xvi, "On formand mechanical
efficiency."
deformed,demolishedand formedanew. Here, foronce, it is safe to say
that 'heredity'need not and cannotbe invokedto accountfor the conand arrangement
of the trabeculae: for we can see themat
figuration
of
life
time
in
the
any
making,underthedirectactionand controlof the
forcesto whichthe systemis exposed ... Herein thenlies,so far as we
can discernit, a greatpart at least of the physicalcausationof what at
firstsightstrikesus as a purelyfunctionaladaptation: as a phenomenon,
in otherwords,whosephysicalcause is as obscureas itsfinalcause or end
is apparently
manifest.
For sheeringenuity,nothingin Growthand Form matchesD'Arcy
Thompson's famous,and likelycorrect,explanationforthe narwhal's
horn (actually a tooth). This horn may project eight or nine feet
beyond the creature'shead; "it never curvesnor bends, but grows as
straightas straightcan be" (p. 907); windingabout thisstraightaxis
is a screwof severallow-pitchedthreads (Fig. 5); in rare cases, when
two hornsare formed,the threadsrun the same way in each: theyare
situationforbilaterallysymmetrinot mirrorimages (an extraordinary
cal animals). D'Arcy Thompson then notes that screw-threadsare
made by combiningforwardand rotatorymotion.Now the propulsion
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D'ARCY THOMPSON AND THE SCIENCE OF FORM
245
threads.
5. Below: narwhalshowinghornwithstraightaxis and low-pitched
FIGo.
Above: skullof rare two-horned
narwhal;threadsrun in the same directionon
bothhorns. This interesting
is fromthe 1826 editionof the Compte
illustration
de Lac6pede'sHistoirenaturelledes cetace'es(originallypublishedin I804 as one
of the completing
volumesof Buffon'sHistoirenaturelle).Lac6pede, a colleague
of Cuvierand Lamarck,wrotethe last volumesof Buffon'smonumental
work.
ofclosely-related
at
orginating
dolphinscontainsa rotatory
component
thetail. To thiscomponent,
more
theflexible
actively
bodyresponds
thanthehornwhichis bothrigidand locatedfarfromthedrivingimis formed
becausethenarpulseofthetail. The horn'shelix,therefore,
as
it
swims
forward
rotates
aboutitsown
whal,
duringgrowth,
slowly
horn!
out
3. The lastchapterofGrowthand Formseems,at first,
curiously
ofplacein a deductive
work.In it D'ArcyThompsonimposesa netof
coordinates
seriesof
rectangular
upon variousanimalsand generates
relatedspeciesby subjectingthatnetto simpledeformations
(Fig. 6
fora set of crab carapaces). Some have seen thisas an exercisein
othersas a game,a scherzoto a bookthatnever
empiricaldescription,
receiveditsfinalmovement.
Butit is thefinaleto a coherent
work.For
his criticshave misseda centralpoint: thatD'Arcy Thompson was in-
in the deformed
terested
in the animalthatit gennet,notprimarily
erated.He sawthatnetas a diagramofforces;and justas thetrabecufemurreflected
theforcesresponsible
fortheirdepolae ofthestressed
sition,so would the deformednet depict the forcesthat could trans-
formone animalto another.Sincetheseforcesmightproducea form
thedeformed
netis no mereframework
fordescription;
it may
directly,
be a displayofefficient
causes. If "diverseand dissimilar
fishescan be
referred
as a wholeto identicalfunctions
ofverydifferent
co-ordinate
thisfactwillofitselfconstitute
a proofthatvariationhas prosystems,
ceeded on definiteand orderlylines,that a comprehensive
'law of
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NEW LITERARY HISTORY
246
\(a ) /1(
d
-mu
(f)
(ci)
of a coordinatenet
FIG.6. Carapacesof variouscrabsdisplayedas transformations
imposed upon Geryon (a).
From chapter xvii, "On the theoryof transformations."
initsintegrity,
haspervaded
thewholestructure
andthatsome
growth'
moreorlesssimple
andrecognizable
hasbeenin conofforces
system
trol"(p. 1037).
The method
oftransformed
coordinates
is D'ArcyThompson's
prorender
forcomplex
structures.
visional
mathematics
Thoughhe might
a simplephysical
a cellorevena jellyfish
law,he
reflecting
byforces
couldnotbe encompassed
so easily.
vertebrate
knewthata complex
mustbe soughtforthe
Yethe feltthatmathematical
representation
the
as
even
when
laws
well,
physical behindthatrepresentacomplex
oneform
andevenifthismeantaccepting
tioncouldnotbe specified,
thesimplification
andcausalinsight
ofmatheas givenandachieving
forms.Thus,as Hutchinson
ofrelated
maticsonlyforthegeneration
"whatmaybe calledbyanalogy
constructed
wrote,
D'ArcyThompson
a floating
ofthearchaeologists,
mathematwiththefloating
chronology
forthetimebeingto physical
unanchored
icsformorphology,
science,
onitsownlevel."
butcapableofvalidgeneralization
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
247
The studyof formmaybe descriptive
merely,or it maybecomeanalytical. We beginbydescribing
theshape of an objectin thesimplewordsof
commonspeech: we end by definingit in the preciselanguageof matheof a 'form'has a qualityof prematics... The mathematicaldefinition
it is
cisionwhichwas quitelackingin ourearlierstageof meredescription;
or
and
or
in
briefer
these
words
in
a
few
words
still
symbols,
expressed
symbolsare so pregnantwithmeaningthat thoughtitselfis economized;
we are broughtbymeansof it in touchwithGalileo'saphorism(as old as
as old perhapsas thewisdomof theEgyptians)
Plato,as old as Pythagoras,
that'theBook ofNatureis writtenin charactersof Geometry'(p. I026).
III. D'Arcy Thompson and his Critics
ofGrowthand Formarenumerous.Theirmajorobjections
The critics
tothreecategories.
canbe condensed
of
was a relicof pasttimesand a perpetrator
D'Arcy
Thompson
I.
here:
theirinadequatescience. I detectthreeseparatearguments
so pompousin itsVictorianmeter,
is so overblown,
i) His writing
cannot
thatharriedprofessionals
so ostentatious
in itsGreekallusions,
I
do
deem
of
time.
not
the
with
essentials
its
economy
requisite
grasp
comment.
of
thisbrandofphilistinism
worthy
approachbelongsto an earliernatural
ii) His non-experimental
modem
not
to
science;he nevermanipulatednaturein conhistory,
what nature
but merelydescribedand interpreted
trolledsituations,
and
its
roots
are with
criticism
is
serious
but
This
justified,
presented.
For
with
Linnaeus.
not
Aristotle,
D'ArcyThompson'sinadequacyis
39limitation
and decisive"
citedas the"fundamental
whatSambursky
the AncientGreeks
of Greekscience: "With veryfew exceptions,
exa periodof 8o00yearsmade no attemptat systematic
throughout
perimentation. . . The consequence was that inductionwas limited
maof such experimental
and collection
observation
to thesystematic
inducSuch
of
natural
the
terialas was offered
study
phenomena.
by
in termsof the conceptionof modem
tion was naturallyprimitive
with
science.""4But ifD'ArcyThompsondid notprovidehisinsights
he
And
are
not
falsified
of experiment
theverification
thereby.
they
of Greek
secondlimitation
to Sambursky's
did not,at least,subscribe
science: "It doesnotaim at theconquestand controlofnature,butis
intellectual
motivated
.... Forthisreasontechnology
curiosity
bypurely
39
S. Sambursky,The Physical World of the Greeks (New York, 1956), p. 2.
40
Ibid., pp. 2-3.
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248
NEW LITERARY HISTORY
findsno placein it; and ifsuffers
fromthelackofthatsynthesis
ofpure
of modem
knowledgeand practicalapplicationwhichis thestrength
science."41
iii) He gave scantnoticeto modemworkthatvied withhis basic
in the inpremisesand evenneglectedto updatematerialcontested
terimbetweeneditions.The latterchargeis trueonlyin part. In the
1917 edition(pp. 48-49), forexample,he speakswell of Arrhenius'
notionthattheminutest
fantastic
oforganisms
mightescapetheearth's
on
the
electrical
force
of
the
Aurora,thenceto be propelled
atmosphere
the
radiant
of
"like
Uriel
by
energy light,
glidingon a sunbeam"(p.
49 of 1917 ed.), to Jupiterin 80 days and to Alpha Centauriin 3,000
ourlifethroughout
theuniverse(or to seek
years-thusto disseminate
theoriginofoursin dispersion
fromotherworlds). In 1942,thisfinale
to "On Magnitude"was quietlydroppedin favorof the statement
quotedon page 240. Butit couldbe excisedbecauseitsimplausibility
did notthreatenany premiseof D'ArcyThompson'ssystem:it had
servedas a dubiousillustration
ofan incontestable
truth.42
merely
D'Arcy Thompson'sstand was this: he did not alter his basic
statements
whensomeonechallengedtheirapplicationto particular
cases.Afterall, a good theoryis thebestguideforseparating
correct
fromincorrect"facts"-especiallyin naturalhistorywhere"pure"
data are oftenthe passiveand subjectiveobservations
of men subtly
withunacknowledged
a prioris.Whenconfronted
withconprejudiced
evidence,D'ArcyThompsontrustedhistheoryand hisaestradictory
in science,thisis a dangerous
theticvision.
43 As a strategy
gamble,one
thatcan be recommended
onlyforthe great. The winnerbecomesa
thelosera blinddogmatist.
genius,
prophetic
The viewof D'ArcyThompsonas a relicwillbe embracedor disof thehistory
of science.In
misseddependingupon one'sconception
of inexorableprogress-rendered
the Comptianperspective
by many
as a marchto truthmediatedbythecontinuing
scientists
accumulation
to theAncients,
of facts-D'ArcyThompson'shearkening
his oppositionto modernfactsin conflict
withGreekTruths,mustbe decried.
ofsciencehaveanymessageforthepracticing
Butifhistorians
scientist,
withtheelusivequalitiesdeit is thatgreatmenfashiongreattheories
and "genius": a sense(oftenaesthetic)thatsomenoting"creativity"
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).
41
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D'ARCY THOMPSON
AND THE SCIENCE
OF FORM
249
thingis amisswithold ideas,an abilityto bringnewwaysof thinking
to an old problem,
fortherelative
and an "intuitive"
feeling
importance
and reliability
And nihilsub sole novum
of conflicting
information.
or no,newwaysare oftenforgotten
old ways.
2. D'ArcyThompsonwas a stubborn
opponentofmodernideas on
tonatural
evolution.ManyhavetakenD'ArcyThompson's
antagonism
obsolescence.In fact,
selectionas yetanothersignofhiscantankerous
buta specific
hisobjections
reflect
no generalstandagainstmodernity,
conflict
betweenhisvisionand Darwin's.Seenin thislight,hisopposior so we
tionis bothintelligible
and judicious.Yet it is also incorrect,
implication(but
judge today--andherehisvisionfailsin itsstrictest
He had threemajorobwe shallsee,in a reinterpretation).
triumphs,
4
the last two subjections to Darwinism: the firstmethodological,
stantive.
was "an endless
i) To mostDarwiniansof his time,morphology
of thingslivingand the pedigrees
searchaftertheblood-relationships
of thingsdead and gone" (p. 3). D'Arcy Thompson,as we have
alreadyseen (p. 5), decriedthislimitedview thatsoughtonlyfinal
he wrotethatwe
causesor antecedent
states.Of complexorganisms,
related
and fitted
to the
as
mustboth"lookuponthecoordinated
parts
of thewholeand as relatedto or resulting
end or function
fromthe
offorces
towhichthewhole
in theentire
system
physicalcausesinherent
it has comeintobeing"
had beenexposed,and underwhoseinfluence
(p. io2o; myemphasis). D'ArcyThompsonmusthave enjoyedthe
satireupon evoludialogueofKing Lear and the Fool as a prophetic
whowouldconsideronlyfinalcauses,and trivialonesat that:
tionists
Fool: Cansttellhowan oyster
makeshisshell?
Lear: No.
Fool: NorI neither:butI cantellwhya snailhasa house.
Lear: Why?
Fool: Why,toput'sheadin.
D'ArcyThompson
geometry,
organicformto abstract
ii) In relating
of
of
realm
notion
a
real
Platonic
the
form;thisrealm
pure
approached
ofitsformsamong
limitsfortheimperfect
wouldspecify
representation
oneinanimals.If mathematical
shapescannotbe transformed
earthly
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 contemporariesespecially 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 superintendingthe gradual fixationof favorable small mutations in natural populations.
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250
NEW LITERARY HISTORY
to the otherthroughinsensiblegradations,neithercan major discontinuitiesin organicformbe bridgedby theimperceptiblechain of intermediates that Darwinian theoryrequires. Thus, D'Arcy Thompson
held that major transitions
in evolutionoftenoccur suddenly,by some
formof "macromutation."45
An algebraiccurvehas itsfundamental
whichdefinesthefamiformula,
a helicoid
ly to whichit belongs.... We neverthinkof "transforming"
intoan ellipsoid,or a circleintoa frequency-curve.
So it is withtheforms
an invertebrate
into a vertebrate,
of animals. We cannot transform
nor
a coelenterateinto a worm,46by any simpleand legitimatedeformation.
. . . Natureproceedsfromone typeto another... and thesetypesvary
and are definedbyphysico-mathematiaccordingto theirown parameters,
chosen
cal conditionsof possibility.Cuvier's"types"maynot be perfectly
nor numerousenough,but typestheyare; and to seekforstepping-stones
across the gaps between is to seek in vain, forever (pp.
1094-1095).
iii) D'Arcy Thompson's belief in the direct molding of form by
physicalforcesled him to distrusta primaryguide to the tracingof
lineages-that degree of similarityis a rough measure of evolutionary
affinity(recency of common ancestry). It is somethingof a trade
secretthatmostof our evolutionarytreesare based not upon the direct
evidence of fossils,but upon inferencesthat equate reasonable series
of modem formswith actual affiliationin time. But if the formsof
theseseriesare fashioneddirectly,the sequence reflectsno history,but
only a gradationof physicalinfluencesobeyingtimelesslaws. Of the
single-celledForaminiferahe wrote (pp. 869-870):
While we can trace in the most completeand beautifulmannerthe
passageof one formintoanotheramongtheselittleshells... thequestion
staresus in thefacewhetherthisbe an 'evolution'whichwe have anyright
to correlatewithhistorictime. The mathematiciancan traceone conic
sectionintoanotherand 'evolve'forexample,throughinnumerable
graded
in variousforms,was the major evolutionary
challenge
45 "Macromutationism,"
of de Vries (circa I9oo) well into
to Darwinismfromthe "Mutationstheorie"
of
the 1940's and 1950's. The inviabilityof major mutations,the difficulty
and the
spreadingthemthroughentirepopulationsaftertheyarise in individuals,
that small mutationsprovideenoughgeneticvariabilityforevoludemonstration
tion have led to the virtualdemiseof thisconcept. D'Arcy Thompson'sreasons
foracceptingit wereunique.
today,we would not disputethis
46 And thoughwe rejectmacromutationism
of complexadult structures.Most
specificclaim as it relatesto transformations
of their
wouldtryto linkthesemajor groupsthroughtransformations
evolutionists
simpler,larvalforms.Thoughno one (to myknowledge)has evertriedto transchorand primitive
forman adult sea urchinto a man, the larvaeof echinoderms
datesare verysimilar.The mostpopulartheoryofvertebrate
originswouldseekto
to larval evolution
link the two groupsat thispoint. AlisterHardy has referred
as an "escapefromspecialization."
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D' ARCY THOMPSON
AND THE SCIENCE
OF FORM
25I
ellipses,the circle fromthe straightline: which tracingof continuous
stepsis a true'evolution,'thoughtimehas no parttherein.It was afterthis
fashionthatHegel, and for thatmatterAristotlehimself,was an evolutionist-towhomevolutionwas a mentalconcept,involvingorderand continuityin thoughtbut not an actual sequenceof eventsin time. Such a
conceptionof evolutionis not easyforthemodernbiologistto grasp,and
it is harderstillto 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 "audacityof imagination"thatJohnDewey foundin
"every great advance in science"; at least it mightfree the students
of these animals fromautomatic allegiance to theoriesdeveloped for
vertebrates(and notwhollycorrecteven 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 neitherevenlynor randomlyacross the range of
formpotentiallyavailable to them: certainshapes tend to occur again
and again. To a modern evolutionist,these recurrentshapes are selected because they are functionallysuperiorto rare or non-existent
ones; to D'Arcy Thompson,theyare simplythe shapes that controlling
physicalforcesproducemosteasily(p. 849):
It is hard indeed (to mymind) to see in such a case as thiswhereNaentersin, or to admitthatit has had any share
turalSelectionnecessarily
in theproductionofthesevariedconformations.
whatsoever
Unlessindeed
we use thetermNaturalSelectionin a senseso wide as to depriveit of any
and so recognizeas a sortofnaturalselection
purelybiologicalsignificance;
to differentiate
betweenthe likelyand
whatsoevernexusof causes suffices
theeasyand thehard: and leads
thescarceand thefrequent,
theunlikely,
one formof cloud,one chemicalcompound,to be
S.. one typeof crystal,
of frequentoccurrenceand anotherto be rare.
3. D'Arcy Thompson's centralidea-that formis fashionedby the
directaction of physicalforcesoperatingduringgrowth-is applicable
to very few of the cases he cites. We must firstunderstandD'Arcy
Thompson's own limitsupon his theory. He did not deny "a principle
beof heredity"(p. 1023); he did not attemptto ascribe differences
tween rhinocerosesand watermelonsto the action of physicalforces.
"My sole purpose," he wrote (p. I4-my italics), "is to correlate
with mathematicalstatementand physicallaw certainof the simpler
outwardphenomenaof organicgrowthand structureor form." Some
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NEW LITERARY HISTORY
252
of thesecorrelationsare surelycorrect,47
but others,includingthe fundamental comparisonof protozoanswith Plateau's figuresof surface
tension,48are almostas surelywrong.
This substantialcriticismwould seem to discreditGrowthand Form,
relegatingit to the domain of antiquarians ratherthan to historians
(not to mentionpracticingscientists). In fact,it providesthe impetus
for a reinterpretation
that explains the book's continuinginfluence
scientific
a
among
communitythatalmostmatches,in itsregardforpast
the
work,
journalist'smaxim: "yesterday'spaper wraps today's garbage."49
D'Arcy Thompson identifiedhundredsof correspondencesbetween
physicallaws and organic forms. If his own theorywill not explain
them,thenanothermustbe sought. And thatother,ironicallyenough,
is natural selection. D'Arcy Thompson claimed a directinfluencefor
physical forces; in fact, these forcesoperate indirectlyby specifying
the formsthat provideoptimal adaptation for animals subject to their
influence. "Equilibrium figures are common in organic nature,"
Hutchinsonwrites,"because any organismnot exhibitingthem would
have to be elaboratelyprotectedin otherways againstdeformationby
the stressesand strainsimposed by its environment."
5 Thus, D'Arcy
Thompson'ssecond theme (p. Io) becomesan aspect of his first--that
animalsare well designed.
Aristotle'sgreat student,in short, mixed up his causes. D'Arcy
Thompson's errorcan be epitomizedthis way: He viewed physical
cause of form;theyare, in fact,formalcauses or
forcesas the efficient
of
optimum shapes that determinethe direction which
blueprints
naturalselection(the trueefficient
cause) musttake to produce adaptation. If physicalforcesare not the kind of cause D'Arcy Thompson
thought,theyare causes nonethelessand no explanationof formis complete withoutreferenceto them. He was rightto correlatephysical
forceswithorganicformsand to claim thatthe correspondencewas no
mereanalogy; but he was rightforthe wrongreason. D'Arcy Thomp47 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 outline. 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 mustbe 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 engineeringlibraryat Harvard, for example, has exiled all its pre-195o
journals to a virtuallyunlit (and completelyunheated) attic.
50
Hutchinson,p. 58 i.
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THOMPSON
ANDTHE SCIENCEOF FORM
D'ARCY
253
fortheefficient
cause of good design;he
son thoughthe had a theory
us
instead
the
science
of
basis
for
a
gave
form--ananalyticapproach
to adaptation.
IV. D'Arcy Thompson and the Science of Form
The vindicationof D'Arcy Thompson'smethod.Amongbiologists,
Growthand Formwearsthealbatrossof Berlioz'sLes Troyens;forit
doomedby excessive
has been regardedas an unusablemasterpiece
of
are
Since
books
and
subjectto no such
length difficultyapplication.
thechargeofexcessive
ofstaging,
as theeconomics
constraint
lengthis
fornecessary
morea lamentof busymenthanan argument
neglect.
in application,
The chargeofdifficulty
hasbeenquitejustified
however,
until recently.D'Arcy Thompson'smathematicalanalysisof form
be considered
simultaneously
requiresthatall thepartsof an organism
thetransforma-that changeofshapebe graspedas a wholethrough
tionofcoordinates
imposedoverentirebodies.D'ArcyThompsondrew
hiscoordinate
diagramsas pictures;he did not,becausehe couldnotin
as mathematical
useful
expresany
way,abstractthe transformations
theonlypre-computer
sions.Thus,Medawar,whoattempted
quantifitermedthe techcationof transformed
coordinates,51
quite properly
unwieldy."52
nique"analytically
The mathematical
studyof growthand formhas been dominated
of one organagainstanotherorgan
bybivariateanalysis: theplotting
in theory
(or totalbodysize) duringgrowth.That thisis unsatisfactory
has neverbeen denied,foran animalgrowsas a whole,notas an abtechstractedseriesof pairs. Whythenwas the obviously
preferable
all partssimultaneously-multivariate
analysisnique of considering
of multivariate
not used? The theorems
analysisare not new; many
had been developedbeforeGrowthand Formreceiveditssecondedition. Buttheywerenotwidelyappliedbecausethesheerlaborofcalculationprecludedany practicalvalue. Withthe adventof electronic
the situationhas changed completelyand multivariate
computers,
analysishas takena place amongthemostexcitingof new approachesin
D'ArcyThompsonwas about one halfcenbiology.Unfortunately,
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 footnote
4), pp. 157-87.
52
P. B. Medawar, in Ruth D'Arcy Thompson, (see footnote 2), p. 231.
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NEW LITERARY HISTORY
254
henceitsuffered
themisfortune
ofmuchprophesy--it
couldnotbe used
initsowntime.
has comeonlynow,and it has just
D'ArcyThompson'svindication
begun. I shallcitebut two amongseveralexcellentstudiesin multivariateanalysisthatwereinspireddirectly
by Growthand Form.53In
his chapteron the logarithmic
spiral,D'ArcyThompsonpointedout
formsof molluscan
(p. 782) thatthe diverseand seemingly-complex
in commonby just a fewbasic variables-the
shellscan be generated
curve(the apertureof a snail,forexample),
shapeof thegenerating
to theaxis
itsrateofincreasein size,and itschangeofpositionrelative
of coiling.Byvarying
thesefewmeasures,
thewholerange
practically
ofmolluscanformcan be simulated.ButD'ArcyThompsoncouldnot
must
do this,forthenumberofcombinations
amongthebasicmeasures
foranymeaningful
runintothethousands
analysis.A computercan
generatetheseformsin seconds,and a tame machinecan even be
trainedto draw(and shade!) eachsimulated
shell(Fig 7). The result
.....
,,
':.
1:?.
?
? ,+:...'
"snails." The sequence from a to d is
generated by increasing the rate of whorl expansion and decreasing the rate of
translationdown the axis. From D. M. Raup, "Computer as an aid in describing
form in gastropod shells," Science, cxxxviii (1962), 150-52.
FIo. 7. Computer drawn (and shaded)
in mechanized
butan important
is no mereexercise
contribuwizardry,
of form-forit providesa matrixwithin
tionto our understanding
can be askedforthefirst
time. Whenwe
whichsomemajorquestions
for
of
a
generate completespectrum shapes, example,we quicklyfind
thatactualshellsoccupybuta smallrangeofshapespossiblein theory.
Whyare certainrealmsof shape unoccupied?David Raup has proof mebased upon theprinciples
videdsomefascinating
explanations
chanicsandfunctional
morphology.
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.
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D'ARCY THOMPSON AND THE SCIENCE OF FORM
255
P. H. A. Sneath has recentlypublishedthe firstsuccessfulquantification of D'Arcy Thompson's transformedcoordinates.55He has done
thiswith modificationsof a techniqueknownbest among geologiststrendsurfaceanalysis. In trendsurfaceanalysis,the distributionof a
featureon a map (rainfallin mm per yearforexample) is abstractedby
mathematical surfaces of increasing complexity. Sneath compares
chimpanzeeand humanskulls. As his feature,he uses thegeometricdistance between each pair of correspondingpoints (tip of the chin for
example); the undistortedcoordinatenet itselfservesas his map. The
surfacesfittedto these distancesare expressionsof the total difference
in shape betweentwo skulls. The differences
are renderedas numbers,
and thesecan be analyzed forpatternsthat expressthe simplestway to
one complexobjectintoanother.
transform
D'Arcy Thompson would have welcomed the computermore as a
perpetratorof his basic attitudesthan of his particulartechniques;for
he realizedthatmen's theoriesare subtlymoldedby the machinerythey
choose (or are constrained) to use. In particular,he feltthat the descriptivemethodof conventionaltaxonomy,thoughlauded by its practitionersas objective and atheoretic,impliesthe view that complexity
is irreducible.
For the morphologist,
when comparingone organismwith another,
describesthe differences
betweenthempointby pointand 'character'by
'character.'If he is fromtimeto timeconstrainedto admitthe existence
of 'correlation'betweencharacters.... yetall thewhilehe recognizesthis
fact of correlationsomewhatvaguely,as a phenomenondue to causes
which,exceptin rareinstances,he can hardlyhope to trace; and he falls
and talkingof evolutionas thoughit had
readilyintothehabitof thinking
on
his
own
the
of
lines
proceeded
descriptions,
pointbypointand character
character
(p. I136).
by
In D'Arcy Thompson's approach, we do not seek a different
explanation for each contrastbetweentwo organisms;by notingcorrelations,
to the fewestfactorsneeded to generate
we tryto reduce differences
them. We do this not for the intellectualsatisfactionprovided by
simplification,but because the abstracted systemof factors can be
linkedmoreeasilyto cause. In likeningthehuman stomachto a bubble
restrictedby a trammel,D'Arcy Thompson invokesthe glassblowerto
explainitsform(p. 1o50) :
startshis operationswitha tube,whichhe firstcloses
The glass-blower
at one end so as to forma hollowvesicle,withinwhichhisblastof air exerwhich
cisesa uniform
pressureon all sides;but thesphericalconformation
Grids,"Journal
Analysisof Transformation
55 P. H. A. Sneath,"Trend-Surface
of Zoology, cli (1967), 65-122.
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256
NEW LITERARY HISTORY
thisuniformexpansiveforcewould naturallytendto produceis modified
ofresistances
intoall kindsofformsbythetrammels
setup as theworkman
letsone part or anotherof his bubblebe unequallyheated or cooled. It
was OliverWendellHolmeswho firstshowedthiscuriousparallelbetween
theoperationsof theglass-blower
and thoseof Nature,whenshe starts,as
sheso oftendoes,witha simpletube.... Such a formas thatofthehuman
stomachis easilyexplainedwhenit is regardedfromthispointof view; it
is simplyan ill-blownbubble,a bubblethathas been renderedlopsidedby
a trammelor restraint
along one side,such a trammelas is producedif
letsone side of hisbubblegetcold,and suchas is actually
theglass-blower
presentin the stomachitselfin the formof a muscularband.
A science of form. Form and diversityare the two greatsubjectsof
natural history.The studyof speciationand systematicshas given us
a scienceof diversitywithinevolutionarytheory,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 factorsthatcan generate
it duringgrowth. Causes shall be sought among these factors,not in
theirresults. D'Arcy Thompson renderedthesefactorseithertheoretically as the physicalforcesmolding formor pictoriallyas the simple
coordinatenets. With the techniquesof compatternsof transformed
we
realize D'Arcy Thompson's unattainedgoal
can
simulation,
puter
of quantitativeexpressionforthesefactors.
2. When the "how" of formis explainedin (I), we mustachieve an
equally rigoroussolutionforits "why." Yet thestudyof adaptationhas
been plagued by vague, trivialand untestableproposals. We need a
criterionof relativeefficiency-away to determinewhich of two structuresis betterdesignedto performthesame function.D'Arcy Thompson
stressedthe mechanicalpropertiesof form. As we have seen (p. 286),
he confusedhis causes; but when we view physicalforcesnot as the
architectsof formbut as the blueprintsthat specifyoptimumshapes,
we have our testfor relativeefficiency:the comparisonof an actual
structurewithits optimumexpressedin an engineer'sterms.
56 I have mentioned several reasons for this in various parts of this work: previous lack of a technologyto handle the calculations of multivariate analysis (p.
287); the eclipse of functional anatomy by evolutionary theory and its use of
formonly for the tracingof lineages (p. 266) ; the attitude towards formimplicitin
the part-by-partdescriptionsof conventionaltaxonomy (p. 267). I presentthis general argumentmore fullyin the article cited in note 53.
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D'ARCY THOMPSON AND THE SCIENCE OF FORM
257
Withthesetwo analyticaltools,a scienceof formcan provideinsightsintothemajoreventsof life'shistory.As one example,the atof strikingly
tainment
similarexternalshapesby animalsof verydifferentancestry57
of convergence-isrecordedwith
-the phenomenon
in
the
fossil
record.
In
has called
fact,Sylvester-Bradley
greatfrequency
itsrecognition
"themostdistinctive
contribution
thatpaleontology
has
made to modernevolutionary
Older
naturalists
were
synthesis.""58
eitheran exampleofGod's plan or an orproneto see in convergence
forideal form.Today, no one would seek any other
ganic striving
cause thannaturalselection;yetfinalcausesoftenremainas
efficient
intractable
as ever. D'ArcyThompson,however,provideda general
of mechanicsspecify
a limitednumberof
explanation:theprinciples
for
the
solution
of
common
good designs
problemsfacedby animals.
Thesecan be determined
a priori,and usedto predicttheformsofundiscoveredorganismsperforming
specifiedfunctions.What seemed
can
be
and
even
mysterious
explained
predicted.
Our argumentindicates. . . that evolutionary
changes,occurringon
a comparatively
fewdefinite
ofphysico-mathelines,orplainalternatives,
maticalprobability,
are likelyto repeatthemselves:thatthe "higher"
forinstance,
notfromorthrough
oneanother,
mayhavesprung
protozoa,
fromthe simplerforms;or thatthe worm-type,
to take
but severally
another
example,
mayhavecomeintobeingagainandagain(p. 1095).
forthatis illustrated
It is easyto assessD'ArcyThompson's
influence;
the
of
his
use
work
in
the
technical
researchof distinby
continuing
and by thelocationof hismainideas at thecoreof
guishedscientists,
an emerging
scienceof form.It is muchmoredifficult,
however,to
measurehisgreatness;
forhe statednothing
and
in
new,
truly
novelty
the
is
criterion
used
most
scientists
status.
in
discovery
by
assigning
Growthand Form is an ingeniouscompendium
of classicalwisdom
fromthelater,butscarcely
withinsights
modern,age ofGalitempered
lean mechanics.To see it as a greatwork,we mustabandontheusual
of noveltyand admitthattheunionof previously
unconconception
nectedtruthscan be an intellectual
as
brilliant
and
as imdiscovery
the
of
as
disclosure
unknown.
And
portant
something
formerly
D'Arcy
knownin
Thompsondid notmerelyunitea seriesof truthsgenerally
theirisolatedstate;he combinedtruths
longforgotten
byhiscolleagues.
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 Zoological Congress,I
(1959), 193.
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258
NEW LITERARY HISTORY
Thereare somenotionsso abidingin man'sviewof naturethatwe
can scarcelydenytheirstatusas aspectsof realityor as wayswe must
perceivetheworld.I wouldplacehere,forexample,theidea thatanimalsare oftenexquisitely
designedto do whattheydo. Such a notion
at varioustimes(God, chance,natural
receivesdifferent
explanations
even
it
be
cast
asidefora timewhena newscientific
selection); may
fashionrashlydismisses
it with a properlydiscardedtheoryforits
Yet it returns,
forit must.And,whenit returns,
scienexplanation.59
tistsrediscover
whattheirforgotten
knew
better
perhaps
predecessors
thanthey.D'ArcyThompsonwas greatbecausehe 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-humanwisdom 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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