An elementary application of the principle of co-ordinates to the study of proportion, as we have here used it to illustrate the varying proportions of a bone, was in common use in the sixteenth and seventeenth centuries by artists in their study of the human form. The method is probably much more ancient, and may even be classical[654];
Fig. 364. (After Albert Dürer.)
it is fully described and put in practice by Albert Dürer in his Geometry, and especially in his Treatise on Proportion[655]. In this latter work, the manner in which the {741} human figure, features, and facial expression are all transformed and modified by slight variations in the relative magnitude of the parts is admirably and copiously illustrated (Fig. [364]).
In a tapir’s foot there is a striking difference, and yet at the same time there is an obvious underlying resemblance, between the middle toe and either of its unsymmetrical lateral neighbours. Let us take the median terminal phalanx and inscribe its outline in a net of rectangular equidistant co-ordinates (Fig. [365], a). Let us then make a similar network about axes which are no longer at right angles, but inclined to one another at an angle of about 50° (b). If into this new network we fill in, point for point, an outline precisely corresponding to our original drawing of the middle toe, we shall find that we have already represented the main features of the adjacent lateral one. We shall, however, perceive
Fig. 365.
that our new diagram looks a little too bulky on one side, the inner side, of the lateral toe. If now we substitute for our equidistant ordinates, ordinates which get gradually closer and closer together as we pass towards the median side of the toe, then we shall obtain a diagram which differs in no essential respect from an actual outline copy of the lateral toe (c). In short, the difference between the outline of the middle toe of the tapir and the next lateral toe may be almost completely expressed by saying that if the one be represented by rectangular equidistant co-ordinates, the other will be represented by oblique co-ordinates, whose axes make an angle of 50°, and in which the abscissal interspaces decrease in a certain logarithmic ratio. We treated our original complex curve or projection of the tapir’s toe as a function of the form F(x, y) = 0. The figure of the tapir’s lateral {742} toe is a precisely identical function of the form F(ex, y1) = 0, where x1 , y1 are oblique co-ordinate axes inclined to one another at an angle of 50°.
Fig. 366. (After Albert Dürer.)