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 such as that of a flat fish, looking upwards, need only turn as much as 48° 36′ from the vertical, to see distinctly every object above the surface of the water. Such vision must, however, be rather confused as to the relative position of objects near the surface, or far from the place where the eye is, as the rays by which they are made visible must be crowded near the circumference of the cone, the half angle of which is this limit of the angles of incidence.

58.Let $$QR,$$ (Fig. 56.) represent a ray of light incident at $$R$$ on the plane surface $$MN$$ of a refracting medium $$B$$ of a different power from $$A,$$ that in which $$QR$$ lies; $$RS$$ the refracted ray, $$XRY$$ the perpendicular. Let $$OP$$ be a second surface parallel to $$MN, $$ separating $$B$$ from another medium $$A,$$ or another part of the same, as if $$B$$ were glass, and $$A$$ the air on both sides of it.

Let $$ST$$ be the course of the ray on passing out of $$B$$ into $$A; \ ZSV$$ the perpendicular.

Then since the difference of media at $$R$$ and $$S$$ is the same, and since the angles $$YRS, \ ZSR,$$ which we may call the angles of incidence, are equal, the angles of refraction $$QRX,$$ and $$VST$$ must likewise be equal, so that $$QR$$ and $$ST$$ must be parallel.

59.We may here recall to the reader’s attention a remark that was made at the end of the introductory observations, namely, that it is indifferent in which direction the light is moving along a ray or system of rays connected by reflexion or refraction. In fact, the subject is purely geometrical, and all that we have to consider is the angles made by certain lines with certain other lines according to certain laws.

60.Suppose now (Fig. 57.) that the ray passes from $$S$$ not immediately into the medium $$A,$$ but into another medium $$C,$$ of a power different from either of the former, and then into the medium $$A,$$ the surfaces being all parallel as before. In this case experiment shows that the first and last rays are parallel as before, so that the actions at $$S$$ and $$T$$ produce, when combined, the same effect as the single refraction at $$S$$ in the former instance.

61.It appears then that whether a ray passes from one medium into another, immediately, or through any number of intermediate