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that the object producing them is nearer than the point of sight. We are now prepared to explain.

In the diagram (Fig. 83) P P is the experimental plane, R and I represent a portion of the retinæ of the two eyes, of which n n' are the nodal points. The eyes are supposed by convergence to be fixed on the points b b', which impressing corresponding points--viz., central

Fig. 83.

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spot b b'will be united and seen single at point of sight, B. At the same time the points a a' and c c' of the plane wonld also combine by geometric construction and be seen at A and C, as a phantom plane parallel to the experimental plane. Such would be the case by geometric construction and by the law of direction. And such would be the case also by the law of corresponding points if the retinæ were planes parallel to the experi

mental planes. But the retinæ are concaves at right angles to the lines of sight b n b, b'n'b'. It is evident, therefore, that, taking the retinal points b' b (central spots) as corresponding points, a a and c c are not exactly corresponding points. They are nearer together than corresponding points, and therefore the objects which produced them will seem farther off than the point of sight. Therefore in the phantom surface the point A on one side and on the other will seem farther off than the middle point, B-i.e., the plane will be arched from side to side, as shown by tlie dotted line.

The law of direction would make the phantom a plane, but the law of corresponding points would make it curved. Therefore, when these two laws are in conflict the law of corresponding points prevails. We will see other proofs of this hereafter (page 293).

It is seen that the convexity of the phantom in the last experiment is due to the extreme obliquity in opposite directions of the visual lines to the experimental plane, and this can not be brought about except by extreme convergence. But Prof. Le Conte Stevens * has made the ingenious suggestion that the same obliquity to the two visual lines and in opposite directions may be easily effected, and the same result in the phantom produced by dividing the experi. mental plane along the middle and bending the two halves in opposite directions. This method has the great advantage of allowing combination beyond the plane of the object also. But although the result is similar, viz., a curved phantom, yet the phenomena and the explanation are different, as we now proceed to show.

* Philosophical Magazine, May, 1882, p. 314.

Fig. 84.

Experiment 4.-In the diagram Fig. 84 a e a' e' are the regularly figured experimental planes inclined to each about 90°. R and L are the positions of the two eyes. If now the right eye be directed on the middle point, c, of the right plane, and the left on the middle point, c', of the left plane, these will combine and be seen single at C. At the same time, by geometric construction, all the other figures of the two planes will be seen as A, B, D, E, showing a strongly convexly curved phantom, A, B, C, D, E. If, on the contrary, the figures cc' be combined by convergence, a concave surface is developed by geometric construction. It is evident, however, as already said, that the explanation in this case is at least partly different, for the curvature of the phantom is brought out by geometric construction alone, although it is probably increased by the property of corresponding points. Experiment 5. — We


R are indebted to Prof. Stevens * for the discovery and explanation of another very striking and beautiful phenomenon.

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* Philosophical Magazine, May, 1882, p. 316.



Let a similar series of concentric circles be drawn on the two halves of a stereoscopic card, partly cut through along the middle line, so that the card may be bent either way at any angle. I have placed the two series of concentric circles on opposite pages of the

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book so that the experiment may be made by placing the pages at any angle with one another (Fig. 85). Now if, lying flat, the circles be combined by convergence with the naked eye, or without convergence with the stereoscope, the combined phantom is sensibly flat. But if the card be bent along the middle line toward the observer (book partly closed) so as to make an angle of 90° or less with one another, and then the circles be combined by convergence, the phantom becomes a most beautiful elliptic concave or elliptic saucer. By varying the angle between the two planes, it is seen that both the ellipticity and the concavity increases as the angle is less. For best effect the line of intersec

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tion of the planes (back of the book) must be at right angles to the visual plane, so that the lines C, D, E, F coincide perfectly, and also the planes must be equally inclined to the median plane of sight. We will call this the normal position of the plane.

If, next, the planes be inclined the other way—i. e., away from the observer, by bending the book backward—then by combination the phantom becomes a

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