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CORNELL UNIVERSITY LIBRARY
"^EKli^HS;)^
^daapJatten
Cornell University Library
BD581 .C77 Curves of
life:
being an account of spir
'"a
'*»;,..
Cornell University Library
The
original of this
book
is in
the Cornell University Library.
There are no known copyright
restrictions in
the United States on the use of the
text.
http://www.archive.org/details/cu31924028937179
THE CURVES OF
LIFE
La Nature
c'est le
modele variable
conlienl loits les styles.
el
infini qui
Elle ^lous enlouve mats nous ne la voyons pas.- Rodin.
THE CURVES OF LIFE BEING
AN ACCOUNT OF SPIRAL FORMATIONS AND THEIR APPLICATION TO GROWTH IN NATURE, TO SCIENCE AND TO ART; WITH
SPECIAL REFERENCE TO
THE MANUSCRIPTS OF LEONARDO DA
VINCI
BY
THEODORE ANDREA COOK Author of^^ Old
Toii}-nint\"
''''
Koiioi,"
WITH ELEJ'EN PLATES AND
etc.,
415
M.A. F.S.A. etc.
ILLUSTRATIONS
LONDON
CONSTABLE AND COMPANY LTD 10 ORANGE STREET LEICESTER SQUARE WC 1914
I)
/
^ -X.
v
-V
-^
^^
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Cv ;\^'Y
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^^
—
;
PREFACE Considerate la vostra semenza; Fatti non foste a viver come bruti. Ma per seguir virtute e conoscenza.
Dante.
When my attention was first turned to the subject of spiral formations, more ttian twenty years ago, it was in connection witli an artistic problem, rather than with a biological question, that I investigated them. As was inevitable, I found myself obliged to examine the forms of natural
life and I learnt that extraordinary and beautiful formation is to be seen throughout organic nature, from the microicopical foraminifera and from life forms even smaller still. In shells, in plants, in the ;
this
men and
bodily structures of tainly a
common
As
widely different. the
assistance
animals, the spiral formation
is
cer-
phenomena apparently broadened I had to secure
factor in a multitude of
whom
my investigations
expert
of
authorities
in
separate
divisions
of
cannot be sufficiently grateful, and I mention this in order to answer the obvious criticism that no one man would in these days be considered competent to deal with the separate branches of knowledge which my inquiry has gradually This is one reason why anyone who is interested necessitated. in the rough programme sketched in my first chapter may choose He may his own path to the last one, which sums up the whole. of plants and read of shells in Chapters III. and IV. and X. flowers in Chapters V. to IX. and in XI. of horns in Chapter XII. of the growth of anatomy and lefthandedness in the two next research, to
I
;
;
;
of patterns in the fifteenth
;
of architecture in
of the attribution of a building to
or of Albert Diirer's
mathematics
XVI. and XVII.
;
XVIII.
;
Leonardo da Vinci
in
in the nineteenth.
recitation of so varied a list of necessary subjects here to lay immediate stress upon the value of human curiosity about the world around us, upon that thirst for a rational explanation of phenomena which Comte so loftily despised, which
The mere
leads
me
Aristotle as
Such writers and Spinoza so clearly acknowledged. Kirchhoff seem to me to have gone back to Comte's
Mach and
most vulnerable position when they limit the use of science to For we do not want mere catalogues. If every description. generation of great thinkers had not thirsted for explanations also, we should never have evolved the complexity and beauty of modern science at all. Only by some such discovery of relationships can flux "
we ever
which
(as
try to deal rationally with that " perpetual more and more clearly recognised) the is
PREFACE
viii
and nature present. There is a deep-seated instinct which attracts our minds towards those " desperate feats of thinking " which have achieved the greater victories of humanity and it is by them that we must surely stand the
phenomena
of life
;
ultimate test of our survival. Modern research is gradually becoming free both from accidental There is a new spirit prejudices and from meretricious standards. at work upon constructive philosophy which has never been so urgent, so creative, or so strong; and knowledge has become the very time when its transmission world has been enormously facilitated by new methods of communication. This means that thought is becoming clearer and more critical, and that a breadth of outlook and a freedom of imagination have resulted which must impel
much more
accessible at
over the whole
about him, and the busiest to linger and produced this book but the very process just described has created its for specialisation is in these days so sternly own difficulty necessitated by the obvious benefits of the division of labour, that it becomes an increasingly complicated task to put the various results of separate investigations in their true relation with each Analysis, therefore, has been my first aim. Not till my other. last two chapters have I ventured to elaborate the synthesis which completes and justifies my catalogue of details and for this I have ventured to propose a convenient instrument which may be usefully employed in dealing with the varied multiplicity of alike the laziest to look
inquire.
Some
ripple of that widespread impulse has
;
;
;
natural phenomena.
may
be said that with very few exceptions the spiral formaphenomena of life and growth. When it is found in inorganic phenomena the logarithmic spiral is again connected with those forms of energy which are most closely comparable with the energy we describe as life and growth, such, for instance, as the mathematical definition of electrical It
tion
is
intimately connected with the
phenomena, or the
spiral nebulse of the astronomer. Newton theory of the movements of the celestial bodies in our own solar system by postulating perfect movement and by calculating from that the apparently erratic orbits of the planets. In just the same way may it not be possible to postulate perfect growth and from that to calculate and define the apparently erratic growths and forms of living things ? In each case the
arrived at
his
Higher Mathematics
will very properly be the instrument of philosophic inquiry, for the science of mathematics, useless as an end in itself, is, in the right hands, the most supple and perfect
instrument for defining the relations of things, and classifying phenomena in the manner so eagerly desired by every intelligent
PREFACE
IX
mind.
It was Sir John Leslie who first drew attention to the " organic aspect " of the logarithmic spiral (see p. 58). Canon
Moseley then applied it to the examination of certain turbine shells. Professor Goodsir sought in it the basis of some physiological law which should rule the form and growth of organism as gravitation prevails in the physical world. Mr. A. H. Church, of Oxford, ha ^ founded on it, in quite recent years, the whole theory of modern phyllotaxis and if the symbol illustrated on the cover of this book is correctly interpreted in its Appendix, it would appear that Chinese philosophy had adopted the logarithmic spiral as a symbol of growth as long ago as the twelfth century. But the exact form of logarithmic spiral most suitable to the fundamental conceptions involved had never yet been satisfactorily discovered. JThe jFormula for Growth now suggested in this book is here called the[(j) spiral, or Spiral of Pheidias, a new mathematical conception wdfkecT out from an ancient principle by Mr. Mark Barr and Mr. William Schooling and Mr. Schooling's exposition of some of its possibilities will be found in my twentieth chapter and the pages which immediately follow it. There is a very significant characteristic of the application of the spiral to organic forms that application invariably results in the discovery that nothing which is alive is ever simplv mathematical. In other words, there is in every organic object a factor which baffles mathematics, as we have hitherto developed them a factor which we can only describe as Life. The nautilus is perhaps the natural object which most closely approximates to a logarithmic spiral but it is onl}/ an approximation the nautilus is alive and, therefore, it cannot be exactly expressed by any we may in the future be able simple mathematical conception to define a given nautilus in the terms of its differences from a and it is these differences which are given logarithmic spiral life. It will be observed that from another characteristic of one long ago stated almost the same Darwin had view point of that the origin of species and the when he showed proposition were largely due to those differences from fittest survival of the to environment, which enabled adaptations type, those minute its bodily improvements to an pass on one living creature to improved descendant. One link between these theories of life and growth and a is contained or, if you wish, of art similar theory of beauty subtle differenin this same observation of minute variations and mathematically correct can Nothing that is simply tiations. ;
;
;
—
;
;
;
;
—
—
ever exhibit either the characteristics of hfe or the attractiveness It is by the subtle variations, which express his own characteristic personahty, that the artist gives his individual of beauty.
— PREFACE
X
charm
and the creations oi art rebelhous against the simple formuL-E of mathematics
to e\-erythmg
are just as as are the I
which he creates
phenomena
of organic hfe.
;
It is for this
reason that
have used the spiral formation (and especially the
or Ratio of Pheidias) as a kind of key, not merely
(j)
spiral,
to natural
phenomena, but to artistic and architectural phenomena as well, and I ha^•e therefore thought it right to add examples of art and architecture to the hundreds of specimens of natural growth v\hich are published in these pages. It may appear that the attempts of scientific analysts to formulate the laws of beauty follow too much the line of objective Some artists may see a champion of their view in Kant qualities. when he says that beauty has no quality in things in themselves, or but that it exists only in the mind which contemplates them ;
Arthur Balfour, when he analyses the qualities we call " sublime," " beautiful," " pathetic," " humorous," " melodious," and denies them any kind of existence apart from feeling. " Are they to be measured," he asks, " except by the emotions they produce ? Are they indeed anything but these very emotions illegitimately objectified ? We cannot describe the higher beauties of beautiful objects except in terms of sesthetic feeling, and ex vi termini such descriptions are subjective." But it is no escape from the inevitability of Nature and her ways (which are called objective) to talk about mental operations. What happens in the mind (though the word subjective may be dragged in Mr.
'
'
.
in,
as
it
too often
is)
.
.
has just as
much
of Nature's inexorable
qualitv as the formation of a mountain or the rusting of iron.
—
Both may be subject to laws namely, to sequences of events though we may not easily find or ever find the laws. Stated in terms as subjective as possible, beauty is a question of " Fit to us."
If the overpowering evidence of continuitv is anything (and for some men it has given more majesty to life than any religious tenet), it gives us the hint of a transcendental " fitting " among ah things. If I am pleased by a vase from King-te-Chen, qualities of me and qualities of the vase make up some fragment of a transcendental equation. This is
good
for
A
the vase. man of science, seeing a attraction of gravity, the bloom of a rose, the melting of metal, or even the rise of an emotion wishes to express true,
too,
if
I
dislike
phenomenon— the
—
not spoiling it. He would have more, not less of it. Mr. Mark Barr, to whose luminous suggestions this book has owed so much, imagines that the first man of science was he who found that Echo was no sprite. He asked for more of Echo, not it.
less.
That
is
And when he learned how much more there was than
creature's voice, he found
more enthrahing mystery
(if
another that
is
— PREFACE
XI
what
is wanted), not less. It is not the hope of science to dispel mystery. Mystery is widened every day. Struck by the evidence of continuity and ordered causation, the man of Science may sometimes become obsessed by the idea
real
of
formulation.
Reaction
against
that
prejudice
is
natural
enough but an opposite extreme is reached when men of artistic temperament resist too bitterly what they are pleased to call the " mechanistic theory." The aesthetically moved man looks with too much intolerance upon the claims of science, and his intoler;
ance is quite intelligible, because artistic creation, no matter how arduous it be, tends to encourage lazy habits of mind. Since analyses of things as they are play a less necessary part in his work, the artist is impatient of analysis. But there is another important fact. Appreciation of art depends largely on free gifts which we all possess in some measure. Artistic creations, therefore, make an immediate appeal to the many. Music soothes even the untutored savage and hence the artistic worker lives in a ready-made world of admiration, and very naturally resents any disturbance of the creative atmosphere or the receptive attitude. With the young Edgar Allan Poe, he cries out to Science ;
;
"Hast thou not dragged Diana from her car, And driven the Hamadryad from the wood To seek a shelter in some happier star ? "
No, would Mr. Barr reply, that has never been either the aim Yet for many years, it is but too true, the formulation of beauty has been the centre of a hot-headed wrangle, due, chiefly, to the diverse use of words. And even if antagonists may agree as to the meaning of certain words, impatience and prejudice too often cloud the issue. In this book I am bold enough to take it for granted that there is a or the result of scientific analysis.
desire to agree.
If
so,
the
man
principles less violently, while the
of science
man
will
of aesthetic
drive general
temperament
take the argument of science slowly in steps, over which intuitive people wish to bound impatiently. For my modest purposes, standpoint is everything and I feel sure we can reach some guiding principles. The attempt to do so has been far from limited to the scientific analyst. Dante, Durer, Goethe are fore-runners on the enchanted track. This same combination of the scientific study of Nature with the principles of art is the keynote of the manuscripts from which we can still strive to estimate the many-sided intellect of Leonardo da Vinci. I have reproduced a very large number of drawings from this source, because they illustrate my main theme and provide innumerable suggestions for the theory set forth in my will
;
PREFACE
xii
twentieth chapter, a theory which is not, I dare to beheve, without its interest for both the professional artist and the general spiral (essentially a public. It sugge:^ts the application of the formula for natural growth) to the proportions of a great picture, (f>
you prefer it, to the principles which underHe the instinctive " good taste " of a great artist. In " Criticism and Beauty (1910) Mr. Arthur Balfour, speaking of the two great divisions of the " Of highest value m the contemplative division emotions, said of highest value in the active division is is the feeling of beauty Love is governed by no abstract printhe feehng of love. ciples it obeys no universal rules. It knows no objective or, if
"
:
;
.
.
.
;
standard.
It
is
teristics
common
to all that
is
account of the characteristics .
.
.
For
we may '
Why
obstinately recalcitrant to logic.
we be impatient because we can
us, here
and now,
beautiful,
common must
it
should
give no account of the charac-
when we can
to all
suffice that,
that
is
give no
lovable
however
?
clearly
recognise the failure of critical theory to establish the reality of beauty, the failure finds a parallel in other
objective
'
and that nevertheless, with or without and love are the best and greatest possessions which we have it in our power to enjoy." The words " here and now " in the above passage refer to the Romanes Lecture delivered at Oxford in 1909. But, even for " the home of lost causes," it must be too depressing to seek con-
regions of speculation,
theoretical support, admiration
solation for failure in one direction
another.
A
by recognising
failure
in
mercilessly logical analysis of the foundations of
had produced the conviction in Mr. Balfour's mind was " absolutely hopeless to find a scale [in matters Eesthetic] which shall represent, even in the roughest approximation, the experiences of mankind." Now I shall not, of course, attempt the arrogance of announcing that so difficult a problem has here been solved. I suggest only that it has been mitigated, and mitigated by a formula which does " represent the experiences," not of " mankind " only, but of all life as we know it, by that new conception, called the ^ Progression, which explains not only the phenomena of vital growth, but also the principles which underlie both the artist's expression of the beautiful and
art criticism
that
it
our
own
appreciation of
course, provide a recipe
it.
This Ratio of Pheidias does not, of
by which any modern mathematician
can produce a rival to the masterpieces of Hellenic sculpture or to the paintings of a Turner or a Botticelh. For is no royal road to Beauty nor does it in any way diminish the >hA