Originally Posted by Nicholas Lindan
There is an f-stop print size exposure adjustment ruler on the Darkroom Automation web site.
To use it is simple:

Measure the size of the projected image and note the numbers. You don’t measure from zero but from ‘BASE’. The difference in the numbers is the number of stops to add/subtract to the exposure time.

You can also use it to measure the easel-lensboard height if this is easier. The ‘BASE’ is then the base of the yard-stick.

Adjusting the exposure is most easily done with an f-stop timer. Dials and tables for use with analog and digital and seconds timers are available on the web.

The ruler is in 0.1 stop increments because Darkroom Automation equipment works in a uniform system of decimal stops. If you have a timer that works in fractional stops you can make a ruler in any increment you wish. The equation for distance for each stop ‘f’ is:

Inches = length of ruler * 2 ^ (-f / 2)
= length of ruler * 10 ^ (-f / 6.6439)

The 6.6439 converts from base 2 to base 10, and makes calculations easier in some spreadsheet programs.

The theory:

The ruler, like everything in f-stop land, is logarithmic in base 2.

The ruler is made ‘inside out’ - you start with the f-stop and find the distance for the mark for that stop. As a result the equations use inverse functions: exponentiation and square roots rather than logarithms and squares. Subtracting the distances (logarithms) accomplishes division. Division by 2 “(-f / 2)” when calculating the position of the marks on the ruler takes the square root.

Subtracting the two numbers gives the result of:

R = log base 2 [(x ^2) / (y ^ 2)]

where x and y are the linear distances at the two magnifications.

If the calculations started with distance and calculated the stops the resulting ruler would be filled with numbers that are hard to work with: 27 inches would be marked 0.642; 7 inches would be marked 4.53. The present method puts the marks at even stop intervals.