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When I see through someone's spectacles, things seem very small to me? Why does it happen? Do people wearing spectacles see things smaller than we do?

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That would depend on what type of lenses they have. For me (farsighted) everything looks closer and bigger. I cannot judge distances without my glasses and if I want to make my living room seem really big I can just take off my glasses.

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If you are short sighted then the correcting spectacle lens are diverging (overall concave) and they form a smaller image of the object on the retina than when looking at the object direct.

If you are long sighted then the correcting spectacle lens are converging (overall convex) and they form a larger image of the object on the retina than when looking at the object direct.

To test what type of spectacle lens you have just put the lens above a book and see what happens to the size of the front.
If the print is smaller than the spectacle lens is overall diverging and is correcting for short sight and if the print is larger than the spectacle lens is overall converging (and acting like a magnifying glass) and is correcting for long sight.

enter image description here

The correction for my wife's converging reading lens at the top is much less than the correction for my varifocal (note the change in magnification from top to bottom) overall diverging lenses.

There is more detail about defects of the eye and correction at this website.

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