24. The origins for species

If we are going to measure anything, especially in biology, then we need to do so systematically. René Descartes provided a matchless and unarguable method. It has been exploited by Newton and all others ever since for analysing physical problems. Without it … there is no possibility for F = ma. With it … we can track any commodities we wish—even strictly biological ones:

It is given to but few men to renovate a whole department of human thought. Descartes was one of those few. …

The basic idea, like all the really great things in mathematics, is simple to the point of obviousness. Lay down any two intersecting lines on a plane. Without loss of generality, we may assume that the lines are at right angles to one another. … The whole plan will be laid out with respect to … the axes, which intersect in what is called the origin. …

Descartes did not revise geometry. He created it. (Bell, 1986, pp. 52–53).

Figure 23: A Three-Coordinate System
 A Three-Coordinate System

Descartes created the now-familiar three-coordinate Cartesian system we see in Figure 23. We shall use it to measure relevant biological attributes on our chosen axes and with suitable dimensions and scales of measure. They need only follow some basic principles.

Each axis has an origin. If Galileo’s cart moves perpetually away from its origin along the y-axis of Figure 23, in the manner of his thought experiment, then (a) it is similarly constrained to some very specific movements—i.e. none—on the x- and z-axes; and (b) it suffers no acceleration in any plane.

The situation for the circulating disc depicted above it is very different. It may be constrained to not move at all up or down upon the z-axis, but something is clearly causing it to circulate constantly about that same z-origin. It is therefore moving constantly in the x-y plane. Using the terms introduced by Descartes, Galileo, and Newton, that disc suffers constant acceleration in both those x- and y- directions. It is most likely subject to some vector force such as a variant of the F1F2j suggested and where F1 and F2 are components of F along the x- and y-axes respectively, with i and j being the units to measure them, again respectively. The disc is certainly ready for our planimeter. No matter what those axes might physically signify, we can easily determine both the net distance around that circulation, and its total area. Two partial derivatives of the form ∂F1 /∂x and ∂F2 /∂y will exist. It does not matter what causes that circulation, we will have its full measure.

Figure 24: Towards Biological Gradients
 Towards Biological Gradients

But also … if some force, whether a circulating one or not, accelerates Galileo’s cart so it travels through an arbitrary distance l from one location to another in Figure 24, then by the gradient theorem of the vector calculus, which is the gradient theorem of line integrals, we can directly relate the cart’s two terminal points. We compute a net gradient, , for the journey. That gradient will be a direct reflection of whatever force or potential might have made it move. Any such gradient is the sum of the three partial derivatives with respect to each separate axis or direction. The movement along each axis for our journey of length l will be l/x, l/y, and l/z. Any such movement is therefore in principle recoverable—i.e. we can get back to our original location—by the similar and converse movement indicated by that gradient. We get back to our original location or set of values by travelling specified distances in each of the x-, y- and z-directions: i.e. a little north-south, a little east-west, and a little up-down.

If, however, a force causes a movement along say the x-axis, then no matter how much we may subsequently move on the y- and z-axes, it is impossible to recover our original location without some compensatory reverse journey, somehow, and at some time, in the x-direction. No amount of movement in x- or y- is adequate. This will become very important very shortly when we try to measure the variations of biological populations … and the possible origins for species.