I know that in the study of material things number, order and position are the threefold clue to exact knowledge; and that these three, in the mathematician’s hands, furnish the ‘first outlines for a sketch of the Universe’

D'Arcy Thompson

On Growth and Form

On Growth and Form is D’Arcy Thompson’s master work. Although it was not published until 1917, he had been thinking about it for many years. On 18 October 1889 he wrote to one of his students: “I have taken to Mathematics, and believe I have discovered some unsuspected wonders in regard to the Spirals of the Foraminifera!” He was soon passing on his interests in his teaching.

George Petrie, who studied under him in 1890-1, later recalled: “I shall never forget his description of the triradiate spicules of the calcareous sponges with their rays forming coequal angles of 120 degrees, because for the first time I became aware that mathematics may be applied to give precision to biological observations and thus to open up a fascinating vista of speculations.”


D’Arcy became increasingly convinced that the laws of mathematics could be used to explain the growth and form of living organisms, but was aware that this was a highly contentious idea. His assistant Doris Mackinnon later told D’Arcy’s daughter and biographer that “he had no thought of writing what was in his mind, and that he would walk up and down the Laboratory thinking his thoughts aloud and discussing his ‘heresies’ with her.” It would be 1908 before he first published on the topic – a paper in Nature on ‘The Shape of Eggs and the Causes which determine them’. In 1911 he raised the subject at the British Association meeting in Portsmouth, claiming that “the form of an object is a ‘diagram of Forces’,– in this sense, at least, that from it we can judge or deduce the forces that are acting or have acted upon it”.

Finally in 1915 his various ideas were assembled into book form and he sent it to Cambridge University Press under the title On Growth and Form. “I have tried to make it as little contentious as possible,” he wrote. “That is to say where it undoubtedly runs counter to conventional Darwinism, I do not rub this in, but leave the reader to draw the obvious moral for himself.” The “obvious moral” was that Darwin was wrong in seeing evolution purely as a slow, gradual process – D’Arcy’s Theory of Transformations, the most famous and radical chapter in the book, proposed that sudden changes could also occur, based on mathematical laws, to transform one species into another. The diagrams that he used to demonstrate this astonished readers at the time and continue to do so today.

A combination of wartime shortages and D’Arcy’s insistence on numerous last-minute changes meant that the book was not published until 1917, but it made an immediate impact within the scientific community. Said the review in Nature, “This book, at once substantial and stately, is to the credit of British Science and an achievement for its distinguished author to be proud of. It is like one of Darwin’s books, well-considered, patiently wrought-out, learned and cautious – a disclosure of the scientific spirit.” The non-specialist reviewers praised its accessibility, the Dundee Advertiser noting that “if Professor D’Arcy Thompson can be mathematical he never fails to translate his mathematics into English; and he is one of the relatively few men of science who can write in flawless English and who never grudge the effort to make every sentence balanced and good.” Since then even more generous compliments have been paid to D’Arcy’s great work – Stephen Jay Gould described it as “the greatest work of prose in twentieth-century science”. P D Medawar claimed it was “beyond comparison the finest work of literature in all the annals of science that have been recorded in the English tongue.” In 2014 On Growth and Form was named on ListMuse as the third best science book of all time.

In this elegantly written book, D’Arcy advanced his main thesis: that biological form can reflect physical and mathematical principles. For instance, the spicules of sponges adopt a number of characteristic shapes. D’Arcy argued that these were the consequence of slight differences in the “starting conditions” such as ionic concentrations and other physical parameters. Thus, the initial conditions might well reflect some aspect of natural selection, but the resulting morphology of the spicules did not.


One clear demonstration of his notions of the dynamic influence of starting conditions lies in the morphology of shells and horns. These are the permanent, non-living, three-dimensional record of a temporary, two-dimensional living state – the base of the horn, or the mantle of the shellfish. D’Arcy Thompson showed that all horn and shell morphologies could be described in simple mathematical terms readily derived from the incremental nature of growth.

Even if a morphology was plainly functional, this did not imply for D’Arcy that it was incorporated into the genome by natural selection. For instance, geometrical rules of packing determine cell arrangements. These need not be specified, but can arise spontaneously. Yet the packing arrangement may be “useful” in minimising the space occupied by the cells, by maximizing cell-cell contacts, by establishing different categories of cells (“inside” versus “outside”), and so on.

Perhaps the most famous images from ‘On Growth and Form’ are the transformations. D’Arcy showed that gross variation in form between related species could be modelled by the consistent deformation of a sheet. The consistency of the deformation is the crucial point here: it is obvious that any fish form could be made to look like any other fish form, if it were sketched on a perfectly deformable elastic sheet, and stretched in many directions at once. But D’Arcy Thompson showed that if the sheet were stretched in one particular pattern, then a new species form would be generated. This remarkable and curious observation has not been fully explained even today.