88 CEITICAL NOTICES : treated in this connexion. Where the oblique lines attached to the horizontal line are directed outwards, the horizontal is over- estimated, because its ends compress the oblique lines outwards, and so compress the line itself less inwards ( ' coincidence of activities in opposite directions '), or what is the same thing, because it forms a relatively independent part of a continuous though broken line. When the oblique lines are directed in- wards the compression exerted on the oblique lines as well makes the compressive force of the ends of the original line seem greater (' coincidence of activities in similar directions '), and the line is underestimated. Section v. treats of illusions of direction, both familiar ones and also new variations of these. In the Poggendorf figure when an oblique line proceeding from a vertical seems more oblique, the illusion arises from supposing that the original movement in the vertical is deflected by a deflecting force. If we regard the oblique line as the original direction of the whole figure the vertical may in its turn appear deflected and hence ' contrast,' a distribution over both lines of the overestimated obliquity. The bending of straight lines through the proximity of curved ones, and the so- called overestimation of acute angles (i.e., small divergencies of direction) follow here. Another class of illusion is typified by the Zollner pattern where lines in one direction form a parallel series across the main line of the series as a whole. Many interesting variations of this are shown among them, convincingly, the new Miinsterberg or "Milton-Bradley" one (p. 319). Section vi. on ' Varying Limitation of Surfaces ' is the most elegant and convincing part of the whole demonstration in particular where it deals with illusions of figures which alternately bulge and contract. A chapter on tapering (Verjungung) describes the illusions of triangles and trapeziums. I find it difficult to describe the principles of these sections without the help of actual figures. But I may refer particularly to the aesthetical conclusions drawn, for instance to the contrast of Gothic and classical styles (pp. 347- 8) and the treatment of the Spitzenmotiv in the Gothic spire (c. 52). Finally we have a chapter on the illusions of solid figures contracted in the middle like an hour glass, or bulging like a vase or the base of a column. This account (faithful, I hope, though too short for those who have not read the book and superfluous for those who have) may give some idea of the way in which a single thought is made to cover a vast number of phenemona. Besides its admirable sys- tematic completeness, the book has the signal merit, in such cases as those of contrast, of attempting an actual explanation of the phenomenon instead of using the name of the phenomenon to designate a mysterious cause. The question of whether the figures in every case verify the theory is a matter for experts in this line, and I am glad therefore to refer the reader to a discussion between Prof. Heymans and Prof. Lipps contained in a criticism by the
