The drawing on the left of Figure 7.1 shows a fish (known as Argyropelecus olfersi among zoologists) over a grid with square units. The drawing on the right shows another fish (known as Sternoptyx diaphana), also over a grid whose cells are now parallelograms.

Their original purpose was to illustrate a way of comparing the forms of these two species which would be simpler and more precise than that favoured by morphologists at the beginning of the twentieth century. For the morphologist, Thompson writes (1961: 274), “when comparing one organism with another, describes the differences between them point by point, and ‘character’ by ‘character’” even though “he is from time to time constrained to admit the existence of ‘correlation’ between characters […].” Thompson is thus finding fault in the “local” nature of the morphologist's comparisons, the fact that they rely on an accumulation of details lacking a “global”, all encompassing, capacity of explanation. In contrast to this approach, Thompson suggests that a correlation in the forms of the two species above can be found of such a particular nature that it would explain all the “point by point differences”. More precisely, he maintains that the form on the right is the image of that on the left by a transformation of the plane.

A cursory glance at Figure 7.1 shows, however, that the required transformation is not an isometry: squares cannot become (non-square) parallelograms under the action of an isometry.