Comparison of Related Forms

We are apt to think of mathematical definitions as too strict and rigid for common use, but their rigour is combined with all but endless freedom.  The precise definition of an ellipse introduces us to all the ellipses in the world; the definition of a ‘conic section’ enlarges our concept, and a ‘curve of higher order’ all the more extends our range of freedom.  By means of these large limitations, by this controlled and regulated freedom, we reach through mathematical analysis to mathematical synthesis.  We discover homologies or identities which were not obvious before, and which our descriptions obscured rather than revealed: as for instance, when we learn that, however we hold our chain, or however we fire our bullet, the contour of the one or the path of the other is always mathematically homologous.

Once more, and this is the greatest gain of all, we pass quickly and easily from the mathematical concept of form in its statical aspect to form in its dynamical relations: we rise from the conception of form to an understanding of the forces which gave rise to it; and in the representation of form and in the comparison of kindred forms, we see in the one case a diagram of forces in equilibrium, and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other…

Every natural phenomenon, however simple, is really composite, and every visible action and effect is a summation of countless subordinate actions.  Here mathematics shows her peculiar power, to combine and to generalize… Growth and Form are throughout of this composite nature; therefore the laws of mathematics are bound to underlie them, and her methods to be peculiarly fitted to interpret them… We must learn from the mathematician to eliminate and to discard; to keep the type in mind and leave the single case, with all its accidents, alone; and to find in this sacrifice of what matters little and conservation of what matters much one of the peculiar excellences of the method of mathematics.

In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalyzed and undefined.  This process of comparison, of recognizing in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original ‘type’ or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician.  This method is the Method of Co-ordinates, on which is based the Theory of Transformations…

A system of transformation

Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull.  We may now treat this complicated figure, in general terms, as a function of x, y. 

If we submit our rectangular system to deformation on simple and recognized lines, altering, for instance, the direction of the axes, the ratio of x/y, or substituting for x and y some more complicated expressions, then we obtain a new system of co-ordinates, whose deformation from the original type the inscribed figure will precisely follow.  In other words, we obtain a new figure which represents the old figure under a more or less homologous strain, and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates x and y.

The problem is closely akin to that of the cartographer who transfers identical data to one projection or another; and whose object is to secure (if it be possible) a complete correspondence, in each small unit of area, between the one representation and the other.  The morphologist will not only seek to draw his organic forms in a new and artificial projection; but, in the converse aspect of the problem, he will enquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other.  This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation… For it is a maxim in physics that an effect ought not to be ascribed to the joint operation of many causes if few are adequate to the production of it…

It is clear, I think, that we may account for many ordinary biological processes of development or transformation of form by the existence of trammels or lines of constraint, which limit and determine the action of the expansive forces of growth that would otherwise be uniform and symmetrical.  This case has a close parallel in the operations of the glass-blower, to which we have already, more than once, referred in passing.  The glass-blower starts his operation with a tube, which he first closes at one end so as to form a hollow vesicle, within which his blast of air exercises a uniform pressure on all sides; but the spherical conformation which this uniform expansive force would naturally tend to produce is modified into all kinds of forms by the trammels or resistances set up as the workman lets one part or another of his bubble be unequally heated or cooled.  It was Oliver Wendell Holmes who first showed this curious parallel between the operations of the glass-blower and those of Nature, when she starts, as she so often does, with a simple tube…