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Someone told me that reading glasses (a priori with a magnifying glass effect only) improve one's eyesight of objects lying in the long range distance. I am really sceptic about it since everything is blurry if I look at objects through those glasses! Is it possible (for this particular person at least)?

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I'm not really sure what you are getting at here. I, for one, and near sighted, which means I cannot see things clearly in the distance. For me, this is at about 10 meters or so and I cant clearly read something as large as a scoreboard. So for me, glasses do vastly improve my eyesight for objects lying in the long range. Is the question can this be done in the general (normal 20/20) case? I know there is a limit to what you can clearly see based on the properties of light and the distance between your eyes, so that would be the limit, I'm willing to bet that is nearly reachable. – CJB Mar 25 '11 at 18:31
@CJB -- I think the key word here is reading glasses, which are converging lenses, unlike a nearsighted person's eyeglasses, which are diverging lenses. That's what wok means by the comment about "magnifying glass effect." – Ted Bunn Mar 25 '11 at 18:34
Ahhh, decent chance I read over that without even registering it. Then this question makes alot more sense, and is alot more clear – CJB Mar 25 '11 at 18:49
"Reading glasses" are those, when Granny says: My eyes are quite ok, but my arms are not long enough any more. – Georg Mar 25 '11 at 19:24
up vote 3 down vote accepted

I don't think this is possible. Suppose that your unaided eye can focus on objects over some range of distanced from $d_1$ (closest) to $d_2$ (furthest). Putting a converging lens in front of your eye reduces both $d_1$ and $d_2$. That is, it shifts the range of distances over which you can focus closer to you, not further.

Here's the geometric-optics proof of this. Varying the distance at which you can focus is equivalent to varying the effective focal length of your eye. If $D$ is the diameter of your eye, and $d$ is the distance of the object you're focusing on, then the focal length is given by $$ {1\over f}={1\over d}+{1\over D}. $$ When you put on your reading glasses, the effective $f$ is decreased, or to put it another way, the effective power of the lens, $1/f$, is increased. (To be specific, $1/f_{\rm new}=1/f_{\rm old}+1/f_{\rm lens}$.) So the left side of the equation becomes larger. $D$ doesn't change, so $1/d$ must become larger. So $d$, the distance to which you focus, becomes less.

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This makes sense: I have looked at the benefits of a converging lens on a random page about vision of Wikipedia, and reducing the focal distance makes objects more focused in some cases. However, does it still work if we consider $d=+\infty$ for long range distance? – Wok Mar 25 '11 at 18:56
If you can already focus to $d=\infty$, then you may or may not still be able to focus to $d=\infty$ after you put on reading glasses. That is, they may not hurt your ability to focus to long distances, but they can't help -- because your distance vision is already perfect. – Ted Bunn Mar 25 '11 at 18:59
True. Thanks! So it is a mystery, the mind is sometimes not rational! – Wok Mar 25 '11 at 19:02
@Ted Bunn, there might be one exeption, when ones eyes are extremely far-sighted, that much, that the shortest focus is farther away than infinity. In such a case, the convex lens would help to get the focus down to infinity or less. I don't know, whether such extreme far-sightedness really occurs. – Georg Mar 25 '11 at 19:38
@Georg -- Certainly true in principle. Such a person would be someone who couldn't see anything in focus, at any distance, with the unaided eye. I don't know if such people exist. If @wok's interlocutor was such a person, that'd be important information! – Ted Bunn Mar 25 '11 at 20:02

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