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This is more of an optics measurements question than a photography question, so if you don't want it on the site, just delete it. I just couldn't think of a better match.

Do binoculars, loupes, magnifying glasses and microscope magnifications all give their magnification power as a magnification of area, or of length?

I have seen claims that 10x binoculars make the object appear 10x closer, which would cause a small object to appear 10x as tall. However, I believe I recall reading that as a marketing ploy (grrr!) the standard practice is to report magnification numbers as the factor by which areas grow. In that case a 10x magnification lens would be one that makes a small object appear 3.1 times taller.

Are the standard magnification powers reported for telescopes, binoculars, loupes, microscopes, etc... the factor by which areas increase, or the factor by which lengths increase?

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The text in the Wikipedia entry may be a bit confusing, so here's the TL;DR explanation.
A telescope is afocal, meaning it does not bring the light to an image plane. What the telescope does is to multiply the angle of each incoming light ray by the "M" magnification factor. As a result, the angular field of view (FoV) is reduced by the same factor. This means that your eye observes a reduced field of view as though it were M-times wider, thus magnifying the apparent linear size of an object in each axis. Quoting from one of the "bibles" of optics, Warren Smith's Optical Engineering,

enter image description here

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  • And thanks to your answer I have a copy of that book on the way (2nd edition or so to keep the cost WAY down). – John Robertson Mar 24 '17 at 16:06
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    Just to clarify, your quoted MP = f(objective) / f(eyepiece) is exactly what Wikipedia and all other sources say is the magnification of telescopes and binoculars (excepting prime focus uses). – WayneF Mar 24 '17 at 16:13
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The magnification number of binoculars is the focal length of the objective lens divided by the focal length of the eyepiece lens (used to magnify the objective image).
http://en.wikipedia.org/wiki/Binoculars#Optical_parameters

This is exactly the same concept as telescope "power", and is extremely common knowledge among telescope users (who use several eyepiece lenses to vary power), but it is not very useful to binocular users to know. Nevertheless, it is.

It does not compare to camera lenses which are used at prime focus (no magnifying eyepiece lens).

It is said that the binocular power number compares to the naked eye, that 7x power is 7 times larger than the eye sees, as if it were 7 times closer. I cannot derive that definition.

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    Camera lenses are also used with a magnifying eyepiece lens every time you look into an SLR with a TTL optical viewfinder. – Michael C Mar 24 '17 at 5:05
  • Not really. The viewfinder has a magnification factor, but the captured photo image is not affected.:) – WayneF Mar 24 '17 at 14:12
  • Didn't say anything at all about the captured image. The image coming out the exit pupil of the viewfinder is just as afocal as that coming out the exit pupil of a telescope or binoculars. – Michael C Mar 24 '17 at 14:15
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Magnification using telescope or binoculars: We see with an eye brain combination. Early in life, we learn gauge distances comparing object size intertwined with an involuntary perception of the “visual angle” induced by distance.

Example: A friend holds a yard stick, 10 feet distant. You sweep your eyes over the stick, end to end. Your eye sweeps a visual angle of 17°. If your friend withdraws to 20 feet, the visual angle becomes to 8 ½ °.

If we gaze through a telescope, we see a magnified view. The telescope or binoculars sports a converging lens, upfront called the “objective lens”. This lens alters angle of the arriving light rays. Rays, as they pass through the objective lens are caused bend inward (refract). The revised path traces the shape of a cone. An image of the vista is caused to form in air, behind the lens. We can view this image if we interpos a piece of white paper as a projection screen. We see this projection as an upside-down miniature image, in full color, of the vista.

We set another lens, with less power (less focal length), behind projected image. The second lens, called an “eyepiece” magnifies the projected image. When all is focused correctly, the light exits the eyepiece as parallel rays. We gaze at this image. Our eye/brain is fooled. We think we see light rays coming straight as an arrow to our eyes. We cannot sense that the rays have converged and then diverged. The result is; we see an altered visual angle that generates a magnified view.

The amount of magnification is the size we see with our unaided eyes vs. the size we see when looking through this instrument. The magnification amount (power) is the focal length of the objective lens multiplied by the focal length of the eyepiece lens.

We put some finishing touches on this instrument. We add lenses to invert the image to right-side-up. We add prisms to shorten the physical length of the instrument.

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