OPTICS. 359 



seen in the continuation of the perpendicular let fall from the object 

 itself upon the plane, and as far behind the plane as the object is before 

 it. This had been previously taught by Euclid; as had been the 

 equality of the angles of incidence and reflection. 



Ptolemy then recurs to the consideration of objects which appear in 

 different places at the same time, though simple ; and to those which, 

 though more than two, appear in one and the same place. We 

 cannot detail his speculations on these points, but must limit ourselves 

 to one only, and that relating to astronomy. " It results (says he) 

 from the preceding, that of things which are in the sky and subtend 

 equal angles, those which are nearest the zenith ought to appear less, 

 and those which are near the horizon appear greater ; because we see 

 the latter in a manner to which we are more accustomed. Elevated 

 objects are seen in a way with which we are less familiar, and with 

 difficulty of action." Thus, according to Ptolemy, the moon in the 

 zenith appears smaller; because he who looks towards the zenith, is 

 in a less natural position than when he looks at any object posited 

 horizontally. 



The author next returns to plane mirrors, and shows that in them 

 objects are not disfigured ; but that the right becomes the left, and 

 vice versa. 



In concave mirrors, objects appear concave ; in convex mirrors, they 

 appear convex. In convex mirrors objects seem diminished. Thus, 

 also, taught Euclid. 



In a concave mirror, a curve line may, according to circumstances, 

 appear either convex, concave, or rectilineal. In a convex mirror, 

 objects appear on the side on which they really are ; yet, by reason of 

 our habit of judging, the right will seem to be on the left, and the left 

 on the right. 



Ptolemy's fourth book relates principally to concave mirrors. 



He treats of the place of the image, and shows when it may be 

 upon the surface of the mirror ; when before that surface, when behind 

 the eye, when behind the mirror. 



When the image is behind the mirror, the distance of, the object 

 from the mirror is less than that of the image. 



When the image is between the eye and the mirror, the distance of 

 the object from the eye, will be greater, equal, or less, than the dis- 

 tance of the image from the mirror, according to circumstances. 



When the object is between the eye and the mirror, it appears in a 

 different place from that in which it really is ; and when we move it 

 in one direction, it will appear to move in the contrary one. 



Ptolemy next passes to the consideration of mirrors compounded of 

 a plane and a concave, or of a convex and a concave ; and explains the 

 cases in which the image is direct or inverted, augmented or dimi- 

 nished ; after which he traces the peculiarities of pyramidal mirrors 

 with circular or polygonal bases, having the eye placed in the axis of 

 the pyramid. In all this Ptolemy never determines the precise point 



