May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life.

—James Sylvester (1814–1897)

SOME MEASURE of my esteem for the French amateur mathematician and physicist Pierre de Fermat (1601–1665) is the great frequency with which he shows up in my essays. (No, I'm not going to list each of his appearances. You'll have to look for them yourself.) I suppose he is best known for his Last Theorem, but he has also shown up for his Principle of Least Time, both of which appear in at least one of these essays.

One of his lesser-known results which I'm pretty sure I haven't
mentioned is the following. Take any prime number *p*, and any
number *n* which is not a multiple of *p*. Fermat asserted,
without proof (which was his wont), that the remainder when
*n ^{ p}* is divided by

where *n* mod *p* (read "*n* modulo *p*")
means the remainder when *n* is divided by *p*. The great
Swiss mathematician Leonhard Euler (1707–1783), who begged off an attempt
to prove Fermat's Last Theorem, did succeed in proving this result, which
has ever since been called Fermat's Little Theorem. (Euler was the first
to *publish* this result. Apparently, Leibniz, the co-inventor
of calculus, also proved it, but left it unpublished in one of his
notebooks.)

Of what use is such a result? Why would Euler bend his considerable intellect to something that looks like little more than numerical tinkering? For over two centuries, anyone looking in would have judged that Euler had basically wasted his time. Newton had invented the calculus to solve a fairly theoretical problem, too, but it had found great use in engineering and physics. To what could Fermat's Little Theorem be applied?

Yet, as the old saying goes, all things come to him who knows how to wait. In the last third of the 20th century, important encryption schemes were developed that depended on certain properties of prime numbers, and fast computation of those properties. It turned out that Fermat's Little Theorem had direct applicability in this area. So Euler didn't waste his time after all. (Interestingly, the proof of Fermat's Last Theorem relies on certain properties of what are called elliptic curves, which also can be employed to speed up some encryption schemes. There's no other connection between the two, though.)

I mention all this because, naturally enough, I have a little idea I would like to share that, as far as I know, has no conceivable applicability in real life. I'm not so arrogant as to think that it will one day somehow become important somewhere—but one can always hope.

As I mentioned in my last essay, the magnification that you get out of an eyepiece and a telescope is the focal length of the telescope's objective divided by the focal length of the eyepiece. I also mentioned that as much as you might think that high magnification is the reason for being of a telescope, there are plenty of objects for which lower powers are desirable.

One reason for this is that some objects are simply too big to fit into the eyepiece field of view at high power. Consider the Moon, for instance. Its angular size is one-half degree, meaning that if you were to stack 180 Moons on end, they would span the right angle from the horizon to the zenith, since there are 90 degrees in a right angle.

If you use an 8 mm focal length eyepiece in a telescope whose objective has a focal length of 800 mm, you'll get a magnification of 800, divided by 8, or 100 power (usually written 100x). That means that the magnified Moon will have an apparent size of 100 times half a degree, or 50 degrees. It so happens that many eyepieces have an apparent field of view (often abbreviated AFOV) of about 50 degrees. In that case, the 50-degree magnified Moon will just barely fit into the field of view.

You can get higher magnification on the Moon if you use an eyepiece of 4 mm focal length. That yields 200x, magnifying the Moon to an apparent size of 100 degrees. However, unless the eyepiece has a distinctly different design, it will likely have an apparent field of view of 50 degrees, just like its longer brother. Now the magnified Moon is too big to fit into the field of view, by a factor of two.

In the case of the Moon, there are plenty of small details to look at, so the fact that the Moon doesn't fit into the field isn't a big liability.

That isn't necessarily the case, though, with other objects where a large part of the attraction is the way that they sit against the background of stars. For instance, there are congregations of stars, called clusters, where there are more stars in a given volume of space than in our galaxy generally. Ideally, that's just the way they'll look in the eyepiece: a central concentration of stars, surrounded by relative dearth.

So long as the cluster's magnified size is smaller than, say, 15 or 20 degrees, the cluster is set off nicely from the background stars and it is reasonably evident where the cluster begins and ends. But if you magnify it so much that it extends beyond the eyepiece's AFOV, all you see is the cluster's concentration of stars, and the contrast is mostly lost. In other words, it isn't the cluster itself you need to see—it's the contrast between it and the rest of space.

For this and other reasons, it's useful to have a range of available magnifications from which to pick and choose, depending on what it is you're observing (as well as the sky conditions, but I'll save that for a later essay).

One way to get such a range is to buy a so-called *zoom
eyepiece*. These eyepieces are generally made up of two groups of
lenses that can be moved closer together or further apart. If you
separate them, they behave the same as one eyepiece with a short focal
length; if you bring them closer together, they behave as though they
were a long focal length eyepiece. Since the distance between the two
groups can be varied smoothly (usually by means of a helical screw
design), the effective focal length of the zoom can also vary smoothly.

The zoom sounds like the ideal solution, but there's a catch. In order to form an accurate and pleasing image, the shapes of the lenses in eyepieces often has to be controlled quite tightly, in a way that depends on the spacing between them. But in a zoom, where the spacing changes, there will be some effective focal lengths for which the lens shapes work well, and others for which they don't. It's often thought, therefore, that zoom eyepieces are really a compromise solution, giving smooth variability in magnification at the expense of exquisite image quality.

To be sure, recent years have brought out a number of complicated zoom designs that deliver performance approaching that of fixed focal length eyepieces, but there's a catch there, too: they tend to be rather expensive.

The alternative, then, is to get a collection of eyepieces of differing focal lengths. Each one will only yield a single magnification in any given telescope, but if you have enough of them, you can cover a range of magnifications well enough. After all, there isn't likely to be a significant difference between a view at 80x and one at 85x. That raises the question: Which eyepieces should I get? And since for any telescope, each eyepiece determines a single magnification, the question is really, what set of magnifications should I have?

One property the magnifications ought to have, ostensibly, is that the range of magnifications should be covered evenly. That is, there shouldn't be any large gaps; conversely, there shouldn't be two or more eyepieces yielding very similar powers.

That still leaves some room for judgment. For instance, consider the following two series of magnifications:

20x, 40x, 60x, 80x, 100x, 120x, 140x, 160x, 180x, 200x, 220x, 240x

20x, 25x, 32x, 40x, 50x, 65x, 80x, 100x, 125x, 160x, 200x, 250x

Both series are evenly spaced out, but each in their own different
way. In the first series, each magnification is 20 higher than the
previous one. Such a series, in which the numbers are separated by a
constant difference, is called an *arithmetic series*.

In the second series, each magnification is approximately five-fourths
*times* as much as the previous one. (There are small rounding
errors, which for our purposes I'm going to ignore.) This kind of series,
in which successive numbers are related by a constant *ratio*,
rather than by a constant difference, is called a *geometric
series*.

At first glance, there doesn't seem to be much that would help us decide which series we should choose. Neither series has any noticeable gaps or unnecessary duplications.

We might, however, make one observation: An object 1.5 degrees across looks just as big at 20x as one 9 arcminutes does at 200x. If you do the multiplication, you'll see that both will appear magnified to an angular size of 1,800 arcminutes, or 30 degrees. In either case, you might want to increase the magnification to see some additional detail.

Suppose we chose the arithmetic series. From 20x, the next available magnification is 40x, or twice as much. But twice as much might be too far. It would, for instance, put the object's apparent size at 60 degrees, which might not fit in the eyepiece anymore. On the other hand, from 200x, the next available magnification is 220x, or only 10 percent greater. The new apparent size would just be 33 degrees; the view wouldn be changed hardly at all, and you would have to go perhaps one or two steps further before noticing much of a difference.

But suppose we chose the geometric series instead. Whether you're looking at the 1.5-degree object at 20x or the 9-arcminute object at 200x, the next available power is 25 percent higher, bumping the size of the object up to 37.5 degrees. Not so much that the object won't fit into the field of view, but not so little that there isn't any noticeable difference, and of course, the change in appearance is much the same for both objects. For this reason, it's my own opinion that using a geometric series makes more sense.

If we settle on using the powers in the geometric se