Three different forms of the antagonism of force and counter-force are recognised: (1) that of contraction, or compression, or limitation and expansion (Begrenzung and Ausdehnung); (2) that of gravity and vertical expansion; (3) that of identity and change of direction. The mass of the book falls into four sections, in which these three antagonisms are exhibited, though the division does not closely follow this enumeration. Under various heads an enormous number of illusions are described and arranged (some familiar, many new and of very ingenious devising), the mere collection and arrangement of which is itself a work of value, while their demonstration from a single principle is a wonderfully educative piece of systematic thinking, even if the reader is not yet convinced that the last word has been said. It is impossible to follow the argument closely. It will be best to outline the method of each section and give a few illustrations which can be described without diagrams.
Section iii. Expansion and Contraction (or Limitation). In horizontal or lateral directions, the figure is conceived as compressed (like a spring, I suppose) and then expanding against the pressure. The compressing force being primary, the figure, e.g., a line, is underestimated, the boundaries are pushed inwards. In general a compressed form is underestimated. So a shilling looks smaller than its mould in wax because the limiting circumference acts in different directions in the two cases. A striking illusion (p. 75) is produced by a vertical row of points which have other 'points placed beside them horizontally, in one group to the right and in the next to the left, and so on: the points in the vertical row are deflected out of the vertical towards their companion points. The underestimation may be reduced by diminishing the compressing effect of the limits in degree, e.g., including the points of a point-distance in circles, which makes them independent. According to this the straight line should look smaller than the corresponding distance, but this is counteracted by the greater solidity of the line, which gives it more expansiveness, or elasticity, while, at the same time, it expands only in one direction, whereas the mere point-distance is free to expand at right angles to its direction. There are secondary effects of limits on one another which I must pass over (c. xviii). With forms whose trend is vertical, the primary force is either gravity or the upward force according as the figure is viewed from above downwards, or from below upwards. In the first case the lower limit drops, in the second the higher limit rises. In both cases the vertical direction is overestimated (as against an equal horizontal length). This explains the familiar illusion by which of two oblique lengths in one straight line the upper seems to be continued above the lower.
Besides degrees of limitation, an interesting and important use is made of the notion of stages of limitation, which may be briefly explained. We may have a series of figures in which the com-
