Isn't there a better way to pick reading glasses?

OK, this one is not a question a 3-year old would ask, but more like a 40 year old with (until then) 20/20 vision would ask. It is a question that cropped up in my life very recently for which some simple physics comes VERY handy.

Most people develop presbyopia somewhere in their late thirties to early forties. The eye loses its flexibility and the ability to adjust the lens for near vision. Seeing far is still OK, as that is the eye's relaxed position. That is when one has to go shopping for reading glasses. Reading glasses, like prescription lenses, are rated by their power, which is just 1 over the focal length of the lenses.

I found lots of websites with reading charts to help me pick the power of my reading glasses. I found these charts completely useless. The issue with them is that I can still focus, it just takes effort. It is only recently that I realized that the chronic fatigue and headaches I have been living with for a few months now were likely partly due to my declining eyesight. It's hard for me to tell which line of small characters on these charts I can no longer read. I can read all of them (especially when it is an easily recognizable sentence), it just takes increasing effort. How much effort is too much effort? I found that very subjective and could not decide.

When faced with a problem, a physicist goes back to the underlying principles. In this case, the thin lens equation:

f is the focal length of the lens and 1/f is the power of the lens (in diopters). Stronger lenses are more curved, have shorter focal lengths, and higher power. s is the distance between the lens and the object we are looking at, and s' is the distance between the lens and the image formed. Distances must be measured in meters. Translated to our problem:

The left hand side is the power of the lens, what we are trying to determine. s is the distance between our eyes and whatever it is that we are looking at: the book lying on the desk, the computer screen, a project in our hands. s' is a little more abstract, but translated to our problem, it is the distance at which we wish the thing we are looking at were. It is the distance that is comfortable for us. The book held at fully stretched arms length for instance.

In the case of reading glasses, s' must be entered as a negative number.

To find the power lenses that you need, do the following:
1) Determine s'. That is the toughest part. The goal is to find the minimum distance of comfortable focus. That is the s' we are looking for. To find it, stand at increasing distance from an object that has fine details on it. Start at reading distance. Relax your eye completely. You will see two objects. Slowly bring the object into focus until there is just one object. Go slow. Resist the temptation to bring the object completely into focus if it doesn't do that without extra effort. You will have to tune into your newly discovered lack of ability to focus and practice. If you do it right, you will discover with dismay that at reading distance, the object is all blurred. Progressively move away from the object until you can bring it into a single object and in sharp focus without strenuous effort.

That's your minimum distance of comfortable focus. Translate that distance to meters (or measure it with a metric ruler), put a negative sign in front of it, and stick it into the thin lens equation.

2) Determine s. s is I think pretty standard for everybody. You may need different power glasses for different purposes. For instance, if you are using your glasses to look at a computer screen on your desk at work, it's probably something like 0.50 meters away from your face (measure it). On the other hand, if you are working on crafts or reading a book, you are holding your work closer to 0.30 meters away from your face.

In my case, for instance, I have discovered that my presbyopia is more serious than I would have thought for something I was not noticing or at least misinterpreting as tiredness. My minimum distance of comfortable focus is around 1.5 meters. That is to say, holding things at arms length don't make it much more comfortable to read. In the thin lens equation, my 1/s' is -0.67. If your near point is greater than 5 meters (that is you can't find a point where it starts to be comfortable to focus), use 0 for 1/s'. It means your minimum distance of comfortable focus has moved to "infinity" and 1/infinity is zero (in physicist's math).

So for a computer screen fairly far from my face, the power I need is

Such glasses would be rated "+1.33" at the store. Ready-made reading glasses go in increments of 0.25, so I am somewhere between a +1.25 and a +1.50. I went with +1.25. The slightly lower power can be compensated for by moving my computer screen back some.

For close work, the power I need is

That is, glasses rated as close to "+2.33" as possible.

When you shop for reading glasses, you will notice that the maximum power is +4.00. This corresponds to someone who has a near point at infinity (the worse it can get), trying to look at work 0.25 meter (25 cm) from their face (the closest you would normally be looking at things). In many states, you can't buy reading glasses over the counter when they are much stronger than 2.75 diopters.

To summarize, the first term on the right hand side of the thin lens equation depends on what you are trying to do, while the second term depends on the quality of your eyesight. While determining s' is still subjective (that's the nature of the problem), I still prefer this method. Determining s' is a traceable source of uncertainty, while the reading chart is just complete black magic. I did find a use for the reading chart at the store, but not the way they suggested using it. I stood a reading distance away from the chart and tried the "comfort focus" method detailed above while wearing different strengths reading glasses. The ones that were the right ones brought the chart into sharp focus with the minimum of effort.