Douglass,
You are trying to change the definition of "magnification." Magnification is a linear, dimensionless quantity according to any book on optics you care the check. In no definition I am aware of is the multiplication of length and width used in calculating the magnifying power of an optical system.
You have provided a useful example of the inverse square relationship, but it is not how magnification is defined in science.
Starting with Wiki on Magnification:
Photography: The image recorded by a photographic film or image sensor is always a real image and is usually inverted. When measuring the height of an inverted image using the cartesian sign convention (where the x-axis is the optical axis) the value for hi will be negative, and as a result M will also be negative. However, the traditional sign convention used in photography is "real is positive, virtual is negative".[1] Therefore in photography: Object height and distance are always real and positive. When the focal length is positive the image's height, distance and magnification are real and positive. Only if the focal length is negative, the image's height, distance and magnification are virtual and negative. Therefore the photographic magnification formulae are traditionally presented as:
M = {di \do} = {hi \ho} = {f \do-f} = {di-f \f}
The symbols are presented in this diagram:
http://upload.wikimedia.org/wikipedia/en/thumb/e/ee/Basic_optic_geometry.png/220px-Basic_optic_geometry.pngMoving to Basu's
Dictionary of Pure and Applied Physics:
The magnification of an optical system indicates the effectiveness of enlarging or reducing an image. There are several kinds of magnification:
lateral magnification of an image,
axial magnification of an image or
magnification of the magnifying power of an optical instrument. It is important which magnification should be considered for use to treat optical magnification. The term
magnification is sometimes used simply to mean
lateral magnification of a lens without qualification.
Basu continues with the formulae to calculate each type of magnification. In no instance is the height and width of the magnified object multiplied in the calculation.
I agree with everything you say, but you are NOT defining magnification of an optical system. Enlarge is not a term defined in Basu's Dictionary nor does it give a true indication of the performance of an optical system. If I tell someone my 1600x microscope enlarged 2.56 million times, no one is going to believe it unless it's an electron microscope.
I expect most of us think lateral magnification and not the multiplication of the lateral mag of the height and width of the image formed by an optical system. It's the same working the other direction on the focal length range. If a 50mm lens is "normal", a 500mm will magnify the image 10 times when compared to the normal lens. Yes, the tile you used will cover a 100x greater area, but that ain't magnification.
Whew...