If we begin by drawing a net of rectangular equidistant co-ordinates (about the axes x and y), we may alter or deform this {729} network in various ways, several of which are very simple indeed. Thus (1) we may alter the dimensions of our system, extending it along one or other axis, and so converting each little square into a corresponding and directly proportionate oblong (Fig. [353]). It follows that any figure which we may have inscribed in the
| Fig. 352. | Fig. 353. |
| Fig. 354. | Fig. 355. |
original net, and which we transfer to the new, will thereby be deformed in strict proportion to the deformation of the entire configuration, being still defined by corresponding points in the network and being throughout in conformity with the original figure. For instance, a circle inscribed in the original “Cartesian” net will now, after extension in the y-direction, be found elongated {730} into an ellipse. In elementary mathematical language, for the original x and y we have substituted x1 and cy1 , and the equation to our original circle, x2 + y2 = a2 , becomes that of the ellipse, x12 + c2 y12 = a2 .
If I draw the cannon-bone of an ox (Fig. [354], A), for instance, within a system of rectangular co-ordinates, and then transfer the same drawing, point for point, to a system in which for the x of the original diagram we substitute x′ = 2x ⁄ 3, we obtain a drawing (B) which is a very close approximation to the cannon-bone of the sheep. In other words, the main (and perhaps the only) difference between the two bones is simply that that of the sheep is elongated, along the vertical axis, as compared with that of the ox in the relation of 3 ⁄ 2. And similarly, the long slender cannon-bone of the giraffe (C) is referable to the same identical type, subject to a reduction of breadth, or increase of length, corresponding to x″ = x ⁄ 3.
(2) The second type is that where extension is not equal or uniform at all distances from the origin: but grows greater or less, as, for instance, when we stretch a tapering elastic band. In such cases, as I have represented it in Fig. [355], the ordinate increases logarithmically, and for y we substitute εy . It is obvious that this logarithmic extension may involve both abscissae and ordinates, x becoming εx , while y becomes εy . The circle in our original figure is now deformed into some such shape as that of Fig. [356]. This method of deformation is a common one, and will often be of use to us in our comparison of organic forms.
(3) Our third type is the “simple shear,” where the rectangular co-ordinates become “oblique,” their axes being inclined to one another at a certain angle ω. Our original rectangle now becomes such a figure as that of Fig. [357]. The system may now be described in terms of the oblique axes X, Y; or may be directly referred to new rectangular co-ordinates ξ, η by the simple transposition x = ξ − η cot ω, y = η cosec ω.
(4) Yet another important class of deformations may be represented by the use of radial co-ordinates, in which one set of lines are represented as radiating from a point or “focus,” while the other set are transformed into circular arcs cutting the radii orthogonally. These radial co-ordinates are especially applicable {731} to cases where there exists (either within or without the figure) some part which is supposed to suffer no deformation; a simple illustration is afforded by the diagrams which illustrate the flexure of a beam (Fig. [358]). In biology these co-ordinates will