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The Foucault test is one of many different tests designed to test the focal length, smoothness, and accuracy of the surface figure of a mirror (most of the other tests are just elaborations of this basic set-up). A good description is given in Texereau's classic How to Make a Telescope, copyright Interscience Publishers, Inc. 1957. Newer editions are available, but the basic design has not changed.
Front view
Side view
Top view
Railing view
Mirror test stand
The screw at front adjusts the front-to-back tilt. Sideways adjustment is done by nudging the front or back of the rolling platform that the stand sits on. The mirror has to reflect light from the slit back to the knife edge.

Using the Foucault tester was by far my (JM) favorite part of making the mirrors.  The idea that a contraption made out of- wood, basic hardware, scrap plastic, razor blades, rubber bands and a car's taillight bulb- could measure a mirror's surface to an accuracy of about 2 millionths of an inch- is nothing short of astounding to me.

Our tester was built sometime in the past by some unknown excellent woodworker, but needed some minor mechanical refurbishing, such as replacing rusty screws, adding bottle caps as knobs, and installing rubber bands.  Then we were ready for the first test...
The larger shiny object holds a light which shines through a very narrow slit between two razor blades (seen taped at the top).
The smaller shiny object is a support for another razor blade which partially intercepts the light after it has bounced back off the mirror. This third blade is what gives this and related tests the nickname "knife edge test".
The vertical shiny object is a thick-walled aluminum tube which acts as a railing that the upper portion of the tester slides forward and back on to adjust for focal length (also seen in front view above).  The large knob in back (also seen below) is graduated and turns a threaded rod which pushes the upper part.  This gives a fine-scale adjustment and reading of the distance from the mirror.
Here the upper platform has been tilted up a bit to show the tube-railing at the far side. The platform-angle screw is at top, lifted up. The screw normally slides on the lower (dark brown) block when the upper section moves forward or back. The block is polished wood, but was later covered with hard smooth plastic since the wood was dented from earlier use.
The upper wood block (aircraft-grade plywood) pivots on the upper right screw to move the intercepting razor blade (the "knife edge"). The rubber band maintains tension while the upper left screw adjusts the position so the blade is just at the interception point (no knob attached yet, but soda bottle caps worked well!). The center left bolt allows coarse adjustment. The lower center screw (also seen below) adjusts the tilt of the upper platform so the intercepting blade is parallel to the slit. The shiny part at upper left is the pointer for the large graduated knob.