On Growth and Form

by D'Arcy Wentworth Thompson

RS = a · α ⁄ √2.

Accordingly, the arc RS will be just equal to the arc MP when

θ cot θ = α ⁄ √2.

When

θ cot θ > α ⁄ √2 or MP > RS,

then division will take place as in RS.

When

θ cot θ < α ⁄ √2, or MP < RS,

then division will take place as in MP.

In the accompanying diagram (Fig. [150]), I have plotted the various magnitudes with which we are concerned, in order to exhibit the several limiting values. Here we see, in the first place, the curve marked α, which shews on the (left-hand) vertical scale the various possible magnitudes of that angle (viz. the angle {365} of arc of the whole sector which we wish to divide), and on the horizontal scale the cor­re­spon­ding values of θ, or the angle which determines