Enlargement magnified to divided by enlargement magnified
from obtains the factor you multiple the exposure by to
get the new enlargement exposure.

(1+M)^2
------- = exposure factor
(1+m)^2

M or m is the magnification ratio. The magnification ratio
is the size of the projected image divided by the size of
the negative along one dimension (along same side.) If
you're going from 15cm enlargement to a 40cm enlargement
using a 35mm negative m is 4.3 and M is 11.43. Thus

(1+11.43)^2     154.5
-----------  =  -----   = 5.5 exposure factor
(1+4.2)^2        28

Multiply the 10x15 print exposure by 5.7 to adjust for
the difference in intensity of illumination at the easel.

Dada,

You can forget all of this magnification talk based upon the size of the
print and just work from the height of the enlarger head.  If your enlarger
head for the smaller print was at 30 cm and for the larger size print has
raised the head to 45 cm, then take the ratio of the two and square it. So,
the ratio of 45/30 is 1.5 and the square of 1.5 is 2.25.  So the new
exposure is 2.25 times longer than the exposure for the smaller print.

All of this is based simply upon the fact the fixed energy output of the
lamp projected from the lens is a circle.  While the total light out is
fixed, you dilute the energy so the energy per unit area decreases.  The
area of the circle it is spread over will increase as the square of the
diameter of the circle.  Pretty simple.

PSsquare

 For enlargers with a diffusion source the formula is:

E2 = (M2 + 1)^2 / (M1 +1)2

Where:

E1 and M1 are the known exposure and magnification
E2 and M2 are the second exposure and magnification

 The reason this is not directly applicable to condenser
enlargers is that the amount of light available at the
source may vary with focal extension.

  The distance between the _lens_ and the easil at any
magnification is approximately (M+1)/f where f is the focal
length of the lens.

I have an Microsoft excel file to help with Darkroom calculations. I
use this when I need to make enlargements. I make a sample test print,
make color corrections and exposure corrections till I feel that the
pirnt is perfect and then I put in the details in the excel file on my
Handheld.
I usually put in the width of the exposure - say 8.25" and the exposure
that I felt was perfect - say 1.8 secs and then in the rows below all I
have to do is to put in the new size that I would want to print - say
30" ( That is the width parameter of the image- I can print either a
30"x40" or a 30"x10" or a small test print -whatever the exposure will
remain the same)
And I get the new exposure for the enlargement. - Of course, I always
take a small sample print of some portion of this enlarged image to
verify if the calculated exposure is correct-

The excel file also has tables for calculating values for either
overexposing or underexposing.

How this table works. ? Calculates the area and the light dissipated
for the new area....

Have tried out and it works for me.

BTW , I am new to forums and I would like to know as to how I can post
the file for you to check out and use if you find it accurate for your
needs . The file is in XLS format

Regards
Das

I use a trusty dial graph put out by Kodak, but I just noticed that it
doesn't go up to 16x20, so I checked in the Focal Encyclopedia of
Photography.  Considering that a centimeter is roughly 1/2 inch, what
you are asking is the difference in enlarger exposure time between a
5x7 and a 16x20 of the same photo.  Using Focal's table of approximate
exposure times at different magnifications, it gives an example of 12
seconds exposure time for a 5x photo, which translates to 80 seconds
for the same shot at 16x magnification.  That is roughly seven times as
long an exposure time.  The  8x10 requires roughly twice as long as the
5x7 and an 11x14 takes roughly four times as long as a 5x7.