THE PEECEPTION OF SPACE. (IV.) 535

equivalencies about equivalencies, mind, of different retinal forms and sizes, not forms and sizes themselves. Super- position is the way in which the eye-movements accomplish this result. An object traces the line AB on a peripheral tract of the retina. Quickly we move the eye so that the same object traces the line ab on a central tract. Forthwith, to our mind, AB and ab are judged equivalent. But, as Helmholtz admits, the equivalence-judgment is independent of the way we may feel the form and length of the several retinal pictures themselves : "The retina is like a pair of compasses, whose points we apply in succession to the ends of several lines to see whether they agree or not in length. All we need know meanwhile about the compasses is that the distance of their points remains unchanged. What that distance is, and what is the shape of the compasses, is a matter of no account " (Pliysiol. Optik. p. 547). Measurement implies a stuff to measure. Retinal sensa- tions give the stuff; objective things form the yard-stick; motion does the measuring operation; which can, of course, be well performed only where it is possible to make the same object fall on many retinal tracts. This is practically impossible where the tracts make a wide angle with each other. But there are certain directions in the field of view, certain retinal lines, along which it is particularly easy to make the image of an object slide. The object then be- comes a "ruler" for these lines, as Helmholtz puts it, 1 making them seem straight throughout if the object looked straight to us in that part of them at which it was most distinctly seen. But all this need of superposition shows how devoid of exact space-import the feelings of movement are per se. As we compare the space-value of two retinal tracts by super- posing them successively upon the same objective line, so 1 " We can with a short ruler draw a line as long as we please on a plane surface by first drawing one as long as the ruler permits, and then sliding the ruler somewhat along the drawn line and drawing again, &c. If the ruler is exactly straight we get in this way a straight line. If it is somewhat curved we get a circle. Now, instead of the sliding ruler we use in the field of sight the central spot of distinctest vision impressed with a linear sensation of sight, which at times may be intensified till it becomes an after-image. We follow, in looking, the direction of this line, and in so doing we slide the line along itself and get a prolongation of its length. On a plane surface we can carry on this procedure on any sort of a straight or curved ruler, but in the field of vision there is for each direction and movement of the eye only one sort of line which it is possible for us to slide along in its own direction continually." These are what Helmholtz calls the "circles of direction" of the visual field lines which he has studied with his usual care. Cp. Pliysiol. Optik, p. 548 ff.