38 Mr. W. G. Horner on the Properties of the 



be described from the same centre C, and either interior or 

 exterior to the circle ABD. 



From A draw AP to any point in this second circle, and 

 from centre C describe a circle touching A P. From E 

 draw Es]), touching the same circle, and meeting the second 

 circle in p. Then it is manifest that ap = AP, and that the 

 object P, which to a spectator at A would be apparent in the 

 situation P, can only be seen from E when it has arrived in 

 the situation p, the aperture A having then arrived at a. 



Let an indefinite line be drawn through B and C, and join 

 p C. Put ■& = PCD the angular distance of P from the 

 point D diametrically opposite A; <$> = p CD, the like distance 

 of the projected point p ; \J/ = PAC, the angular distance of 

 P from D as apparent at A. Also let a = AC, d = EC, 

 .r = CD, y = pq, coordinates ; and u = PC = p C. Then 



.... , a sin \J/ 



it will be seen that u = - — — — j-., x = u cos <p, y = u sin ±; 



sin (5— \J/) J 



i-i-i * i ~ 1(t sin 4» „ , , 



likewise that <p = 3— y + sin — -= — . brom these general 



premises we may deduce all the particulars we have occasion 

 for. 



(I.) Wherever the observer is placed, whether near the 

 cylinder or remote from it, all the objects depicted on the 

 inner surface of the cylinder are visible to him at one and the 

 same time. This is a highly paradoxical pha?nomenon ; but 

 the explanation of it is involved in the deductions already 

 made. For, if we assume P to be coincident with A, or make 



a 



§ = -x and \J/ = \ %, we have <p = - -- + sin -,, which is equi- 

 valent to the arc TBD, contained between the tangent ET 

 and the diameter AD. Consequently the point T is the pro- 

 jected place of the point A, and TBD, the portion concave 

 toward the eye, is the projection of the whole semicircle ABD. 

 And so of the other semicircle A^D, which is projected on 

 t D. But the thickness of the material will obstruct the view 

 of a small portion adjacent to T and t. 



(2.) Wherever the eye is placed, the divisions of the pro- 

 jected picture are sensibly equal in breadth, especially toward 



- : -. , ' ■_, . _1 asinv(/ «\J/ a(d' — a 9 )\J/ 3 



the central parts, b or since sin i — = —, 5 jt-^ — 



d d 6d 6 



&c, if we neglect the second and following terms, which are 

 evidently small, we have <p = d— \(/+ —z- exceedingly near; 



