To understand the need for advanced magnifying glasses, we need to discuss the limitations of simple magnifiers. The critical piece of knowledge is that a magnifying glass with perfectly made spherical surfaces does not form a perfect image. Understanding the reason for this requires an understanding of aberrations, which is an advanced topic. For now, I’ll just point out that fact.
Another important factor in our understanding is the resolution of the human eye. At the accepted minimum distance for focus, 250 mm (10″), the eye can resolve about 40 microns (0.0015″). For a magnifier, to be useful, it must give us better resolution than that. More specifically, a magnifying glass should improve our ability to resolve detail by an amount equal to the magnification. Putting numbers to this, a 2X magnifier must give us resolution of 20 microns (40 microns / 2), a 5X magnifier must give us 8 micron resolution (40 microns / 5), and a 10X magnifier must give us 4 microns (40 microns / 10).
A simple 2X magnifier (an equi-convex lens) is able to resolve 12 microns, so it does indeed help. However, a 5X simple magnifier is also limited to 12 micron resolution. Using the rule in the last paragraph, a 5X magnifier should give us 8 micron resolution. This means that a simple magnifying glass is limited to about 4X. After that, it does not help us see finer detail.
The next step up in complexity for magnifying glasses is an achromatic doublet. This type of lens is made of two different kinds of glass, crown glass and flint glass. Window glass is a type of crown glass, and fine crystal is an example of flint glass. By gluing together one piece of each type of glass, with the right curvatures, a greatly improved magnifying glass can be made. Achromatic doublets can give 0.8 micron resolution at 5X and 1.4 micron resolution at 10X, making them useful for both of these magnifications. However, there is one problem with achromatic doublets as magnifiers: they work well with the crown glass toward the eye, but not in reverse. This means that you have to be careful how you hold them for them to work properly.
The symmetry problem can be solved by making a sandwich with three pieces of glass. Either flint glass on the outside and crown in the middle or the other way around could work, but putting the flint on the outside gives much better resolution. This type of magnifier is called a Hastings or Steinheil magnifier, after the inventors. It is capable of 1.8 micron resolution at 5X and 1 micron resolution at 10X, making it useful for both of these magnifications.
Here are pictures of the two triplet magnifiers:
Both of the pictured magnifiers have a focal length of 12.5 mm (1/2″) so they give 20X magnification. These designs are taken from the Edmund Optics catalog, and unmounted lenses are available for purchase here (Steinheil) and here (Hastings) Edmund Optics Inc. kindly provides the lens prescriptions so an optical engineer can evaluate the performance. To calculate the resolution, I assumed that the eye was placed 20 mm from the lens and the pupil size was 3 mm diameter, which is a reasonable estimate for a well-lit room. For the experts reading this page I should note that the diffraction limit is about 2 microns for this setup, which is the same as the eye’s resolution at 20X.
Resolution is the main distinction between the two types of triplet magnifiers. At 10X the Hastings gives acceptable resolution over a 5 mm (1/5″) diameter while the Steinheil is acceptable out to almost 6 mm (1/4″). This is a small enough difference that either one would work well. However, the comparison changes substantially at 20X. The Hastings triplet has a spot size of about 3 microns at best, which is 50% larger than it should be to give the maximum amount of detail. Steinheil’s design exceeds the required resolution over the central 1.5mm (1/16″), so I would prefer it for high magnification. The only caveat is that the resolution of the Steinheil falls off much more quickly than the Hastings as you move away from the center of the object. For this reason, the performance of a 20X Steinheil is worse than a similar Hastings outside of the central 1.5 mm (1/16″) diameter. The bottom line is that I’d buy a Steinheil for 20X, but either is acceptable at 10X. If you need more magnification than that, you need a microscope.