E. M. NELSON ON DETERMINATION OF FOCI. 457 



d being the distance between the image and the lens, and / the 

 focus ; no confusion can then possibly arise between d for distance 

 and d for disc. 



In the figure we see that we have two similar triangles, conse- 

 quently — 



4=-^ and .?=!/. 

 d f o' 



This, then, is the method by which formula (i.) is obtained. The 

 fallacy underlying this argument is that a pinhole has no focus. 

 You are therefore introducing a term into your formula which has 

 no existence in fact ; or, to look at the question from another 

 point of view, the distance between the lens and the object (slide) 

 in an ordinary lantern is not the equivalent focal length of the pro- 

 jection lens. The object can only be at the principal focal point of 

 the lens when the image (disc) is at an infinite distance from the 

 lens, and consequently of infinite size. An excellent illustration 

 of this error may be seen in the following example : — It is required 

 to project a disc of 3 inches with a projection lens of 6 inches 

 focus, the mask being 3 inches. Here we have a projection of the 

 same size, and everyone who has had the smallest experience with 

 optical instruments knows that both the object and the image will 

 have to be at a distance from the lens equal to twice its focal 

 length ; bat what says the pinhole formula (i.) ? 



3X6 



d = -7"— = 6 mches = focus. 



o 



This is perfectly true in the case of a pinhole, as it is of no con- 

 soquenco what the distance may be, provided that the object is 

 placed at a similar distance on the other side ; but, with regard to 

 a lens, it is quite erroneous, because the distance must be twice the 

 focus, or twelve inches. The practice often adopted in elementary 

 text-books of explaining the action of lenses by the illustration of 

 a pinhole is to be regretted. It is not true that a lens is merely a 

 pinhole of enlarged aperture, as is so often stated ; a radical differ- 

 ence exists between them, viz., that a lens has two principal focal 

 points, while a pinhole has none. 



Hence, this common formula (i.) is inexact ; nevertheless, it 

 may be said that it never was intended to be absolutely accurate, 

 but that it has been generally adopted on the ground of its sim- 



