A big part of what I do with the microscope comes directly out of books. I have no formal training with the microscope so books dealing with it usually contain something that I was totally unaware of, or just provide a solid footing. Occasionally they clear up misconceptions, which happed just this morning actually.
Microscope objectives are designated in many ways, but chief among them is by their magnification. Unfortunately for the beginner, there is more than one way to describe the magnification of an objective. For quite a long time objectives (and oculars) were described and marketed primarily by their focal length. They might also be described by the times they magnify an image, (as in 10x, 40x, 100x) or by the diameters of magnification they provide; a way that until recently I thought synonymous with the times designation.
The focal length of an objective (if you’ll forgive a generalization) is really just the distance from an object at which the objective is in primary focus. There are a few factors that can alter the distance for practical reasons, but in essence an objective with a focal length of 4mm needs to be 4mm from the object it is focused upon, an 8mm at 8mm, a 1/12in at one twelfth of an inch and so on.
An objective might also be described by the number of times it magnifies an image, the today ubiquitous “X” marked on nearly anything with a lens. The square root of the number of times magnified, provides the number of diameters of magnification. So that it may be said an objective that magnifies twenty five times, provides five diameters of magnification. Who knew?
I can say that I have seen articles and forums where the term “diameters” is used when speaking of “times,” I’ve even done it myself. Let us now turn to an old authority, Alfred Cheatham Stokes, in his book Microscopical Praxis:
A lens of any kind magnifying ten diameters is said to magnify one hundred times, or ten diameters in each direction, “times” representing the square of the “diameters,” and the diameters the square-root of the times.
It’s certainly odd that this does not appear to be more widely known, or Stokes more widely read. Microscopical Praxis is out of print and has been for years, so long in fact that it is in the public domain and one can find it nearly anywhere that handles or reprints public domain books. Here’s a link to it at the Internet Archive.