Whilst reading an old textbook on mechanical vibrations, I came across one of those spine-chilling examples you find in old textbooks and which commence with words like...consider a spherical shell of radius R upon which is rolling without slipping a cylinder of radius r...and which then proceed with vast amounts of mathematics to a conclusion next to impossible to believe and almost impossible to accept that sane persons would expend time and effort attempting to reach. However, on this occasion, I nearly leapt out of my chair in the realisation that here was undoubtedly the seed of a practical solution to a practical problem, and one which the author (one Professor Seto) had never even considered the possibility of.
It happens that when grinding a telescope mirror, the Amateur Telescope Maker (ATM) will wish to know if the focal length of the beast is approaching the desired figure. Now, there are extant solutions to this
1) Use a spherometer. This generalises to build a spherometer, which as other web pages out there demonstrate, is not necessarily quick, cheap or easy.
2) Use a straight edge and feeler gauges. This often means go out and buy feeler gauges.
3) Wet the surface of the rough ground mirror, go outside and focus the image of the sun on a wall and measure the distance between mirror and wall. This usually means get a friend to help with the tape measure, and don't live in the North-East of England.
4) Use a precision-cut template of the mirror you wish to build. This assumes one has been published, or that you can make one yourself.
1 and 2 (and 4 if you have to create it yourself) presuppose a familiarity with the formula for finding the sagitta, whilst 3 exists at the mercy of the clemency of the weather. There now exists
5) Roll a marble or ball bearing across the surface of the mirror. Time the period of one oscillation across the mirror and back again (or several and divide the total time by the number of complete cycles, as tenths of a second matter, though not by very much.) Square this time and multiply by three and a half. That's the focal length in inches. This takes maybe five seconds for the measurement and another five to punch the buttons on the calculator. A stubby cylinder turns out to be a much better test object than a sphere, as minute flats on a sphere, less than a millimetre across, will divert the path of rolling from diametrical to elliptical and cause errors in timing. For a stubby cylinder, the focal length is equal to 0.083 times the oscillation period squared, the answer is then in metres rather than inches. Of course, if you are a smart@*$= and have a spreadsheet, you can produce a printed chart of focal length and just read off the focal length from the time directly. If you are a sad old git like me, you could even draw a graph by hand on graph paper. Instead of speaking of a 48-inch focal length mirror, ATMs can now refer to a 3.7 second mirror. The mathematics and full description are in this 82k pdf.