First of all, the equation that I gave is not really my equation. Second, the "+1" added to the magnification has a physical basis. In fact, it is necessary for the equation to be correct. I tried explaining it in my first answer and I gave references to more detailed derivations than I can give in an internet answer. Do you really think that your mathematical optics by intuition is correcting an equation published by Rudolf Kingslake, who maybe Kodak's best lens designer and is one of the best authors of optics books?
Originally Posted by Troy Hamon
If you had checked the references that I gave, you will find that both Kingslake and Ray use the same numerical example, going from an enlargement of 2x to 4x. Both calculate an exposure time change of x2.8. The paper dial calculator in the Kodak Darkroom Dataguide also gives an exposure change from 10 to 29 s.
What your light spread over paper area argument ignores is refocusing the enlarger lens and the consequent change in the effective f-stop.
Tn = To*(An/Ao) and Tn = To*(Dn^2/Do^2) implies a relation between "enlarger distance" (image distance, I assume -- the term wasn't precisely defined) and print area that simpy isn't correct. If a 4x5 negative is enlarged to 8x10 with a 150 mm lens, the image distance is 450 mm. Switching to a 16x20 print, the image distance becomes 750 mm. The ratio of print areas is 4. The ratio of image distances squared is 2.78. If by "enlarger distance" is meant the total distance from negative to print, the two distances are 675 mm and 937.5 mm. The square of the ratios is 1.92, which still isn't 4.
The correct equations relating image and object distance, focal length and magnfication are given near the top of the lens tutorial: http://www.photo.net/learn/optics/lensTutorial
This numerical examle (same as my previous example) shows the consistency between squaring the image distance and the m+1 squared equation, and the inconsistency with the magnification squared equation. (750 / 450 )^2 = 2.78. (4+1)^2 / (2+1)^2 = 2.78. But 4 ^2 / 2^2 = 4. This should cause worry to those who believe in both the image distance squared and magnification distance squared equations.