silence, as she had known he would. “What of the other

[1] Poggendorf's Annelen der Physik, Vol. 110, p. 500; 114, 587; 117, 477.

The same lines looked at directly or backwards seem, in Fig. 1, convex, in Fig. 2 concave.

Still more significant is the illusion in Fig. 3, in which the convexity is very clear. The length, etc., of the lines makes no difference in the illusion.

On the other hand, in Fig. 4 the diagonals must be definitely thicker than the parallel horizontal lines, if those are to appear not parallel. That the inclination is what destroys the appearance of parallels is shown by the simple case given in Fig. 5, where the distance from A to B is as great as from B to C, and yet where the first seems definitely smaller than the second.

Still more deceptive is Fig. 6 where the first line with the angle inclined inwards seems incomparably smaller than the second with the angle inclined outwards.

All who have described this remarkable subject have attempted to explain it. The possession of such an explanation might put

us in a position to account for a large number of practical difficulties. But certain as the facts are, we are still far from their _*why_ and _*how_. We may believe that the phenomenon shown in Figs. 1 and 2 appears when the boundaries of a field come straight up to a street with parallel sides, with the result that at the point of meeting the street seems to be bent in. Probably we have observed this frequently without being aware of it, and have laid no particular stress on it, first of all, because it was really unimportant, and secondly, because we thought that the street was really not straight at that point.

In a like manner we may have seen the effect of angles as shown in Figs. 5 and 6 on streets where houses or house-fronts were built cornerwise. Then the line between the corners seemed longer or shorter, and as we had no reason for seeking an accurate judgment

we paid no attention to its status. We simply should have made a false estimate of length if we had been required to judge it. It is also likely that we may have supposed an actual or suppository line on the side of the gables of a house enclosed by angles of the gables, to be short,--but until now the knowledge of this supposition has had no practical value. Nevertheless, the significance of these illusions should not be underestimated. They mean most of all the fact that we really can be much deceived, even to the degree of swearing to the size of a simple thing and yet being quite innocently mistaken. This possibility shows, moreover, that the certainty of our judgment according to sensible standards is inadequate and we have no way of determining how great this inadequacy is. We have already indicated that we know only the examples cited by Zllner, Delboeuf and others. It is probable that they were hit upon by accident and that similar ones can not be discovered empirically or intentionally. Hence, it may be assumed that such illusions occur in great number and even in large dimensions. For example, it is known that Thompson discovered his familiar ``optical circle illusion'' (six circles arranged in a circle, another in the middle. Each possesses bent radii which turn individually if the whole drawing is itself turned in a circle) by the accident of having seen the geometrical ornament drawn by a pupil. Whoever deals with such optical illusions may see very remarkable ones in almost every sample of ladies' clothes, particularly percale, and also in types of carpets and furniture. And these are too complicated to be described. In the course of time another collection of such illusions will be discovered and an explanation of them will be forthcoming, and then it may be possible to determine how our knowledge of their existence can be turned to practical use.