On Growth and Form

by D'Arcy Wentworth Thompson

Fig. 287.

In a very simple and beautiful theorem, Galileo shewed that, if we imagine a number of inclined planes, or gutters, sloping downwards (in a vertical plane) at various angles from a common starting-point, and if we imagine a number of balls rolling each down its own gutter under the influence of gravity (and without hindrance from friction), then, at any given instant, the locus of {563} all these moving bodies is a circle passing through the point of origin. For the acceleration along any one of the sloping paths, for instance AB (Fig. [287]), is such that

AB

= ½ g cos θ · t²
= ½ g · AB ⁄ AC · t² .

Therefore

t² = 2 ⁄ g · AC.

That is to say, all the balls reach the circumference of the circle at the same moment as the ball which drops vertically from A to C.

Where, then, as often happens, the generating curve of the shell is ap­prox­i­mate­ly a circle passing through the point of origin, we may consider the acceleration of growth along various radiants to be governed by a simple math­e­mat­i­cal law, closely akin to that simple law of acceleration which governs the movements of a falling body. And, mutatis mutandis, a similar definite law underlies the cases where the generating curve is continually elliptical, or where it assumes some more complex, but still regular and constant form.

It is easy to extend the proposition to the particular case where the lines of growth may be considered elliptical. In such a case we have x² ⁄ a² + y² ⁄ b² = 1, where a and b are the major and minor axes of the ellipse.