Obtaining a square is easy. Take two EXACTLY equal length 'rods' or
other suitable flat and true faced edge. Tie them together at one end,
and separate the other end by exactly 1.4145926536 * the length used for
the two faces. They will be precisely 90 apart.
Well yes, Jasen Betts implied that (in sed) when he said
"Pythagoras". But is that how reference squares are really
I can imagine checking a standard with Pythagoras, but you
can't just bolt three rulers together. There are gotchas.
Like, you need three perfect straight edges, and you have
to mate their reference surfaces exactly.
If you try to bolt three pieces of precision-length bar
stock to make a perfect reference square, you can imagine
the inaccuracies that would result.
The following assumes that you have a surface plate of
sufficient flatness. There are existing ways to make them starting with
three of the same size, but I won't go into them here.
So -- start with a lathe and some steel perhaps about 4"
diameter. Turn it to be a true cylinder -- measure the diameter at both
ends and several other places near the center and make sure that they
are all the same diameter (and use a toolpost grinder to get a really
nice finish). Without moving it in the chuck, turn the end flat, recess
all but about the outer 3/16", and then use the toolpost grinder to
grind that outer rim truly flat. Part off the other end where it was
held in a chuck as this is neither the right diameter. Ignoring the
parting-off part, you have a cylindrical square. Set the ground rim end
down on the surface plate, and you will have a reference which is as
square as you were able to measure with the micrometers for the diameter
of the workpiece. These are made commercially, but usually from hollow
iron castings to minimize the weight. I have a couple of these of
differing sizes and weights.
Brown & Sharpe used to offer one which had one end truly true,
and the other a precise angle off, and a series of markings chemically
etched into the cylinder to allow you to measure just how far off square
you are -- measured in steps of 0.0002" up to 0.0012". The dimensions
of this one are 6-1/4" high, 2-1/2" diameter, accuracy within 0.0001"
and 6RMS surface finish. The part number was 599-558-6, and the price
was $160.00 back in 07/1972, when the catalog which I have in hand was
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Such an exercise is detailed in one of Guy Lautard's Bedside Readers.
Both how to generate a reference square and how to check one. Neither
one involved tubs of water(except for hardening) or mirrors. You DO
have to have a true straight-edge first. For a rough check, with no
other squares needed or available, you can take your straight straight-
edge, put it on a piece of paper, park the square on top and draw a
line the length of the perpendicular blade. Flip the square and
repeat. The difference out of parallel is twice the offset of the end
of the blade.
Also detailed was lapping a square and how to make a cylindrical
I think the caliper jaws would be more useful for parallelism, and
perpendicular was determined by comparison with a toolmakers square.
First, make a surface plate or three.
Then scrape three rectangular bars into straightedges against the
Then choose an end of each of the three straightedges to be the
Scrape the three ends into conformity with one another while the
straightedge faces are also on the same surface plate. The process is
very much like the process by which one generates the surface plates.
I think this and related methods were originally invented by opticians
to make prisms, the difference being that they ground things together
rather than spotting and scraping.
I thought in SED that some variation of rubbing three cubes'
faces against each other on a surface plate might do the trick,
but I didn't press it.
After some head-scratching, I think I get your method.
The bottom surfaces of bars I-III are first scraped flat. Then,
sliding on a surface plate, faces a-c are rubbed against each
other in rotation, making their profiles square with their
bottom surfaces and the surface plate. (Fig. 1)
====== (view in Courier or other fixed font)
===== unfinished edge
| | | |
| I. a.| |b. II. |
| | | |
| III. c.|
Okay, so a, b, and c form right-angles to their straight-edged
Now, if just rubbed together by hand, faces a-c are possibly
rounded profiles, right?
===== straight edge
I. / )
) <~~~ a. (viewed from above)
I'll have to ponder this a bit further to figure how to make all
faces flat and square to each other.
Super accuracy from simple tools--elegant.
Once you have two opposite faces flat and parallel, make a fixture
with an indicator point directly above a similar-shaped stop. Push the
block up against the stop and read the indicator. Invert the block
you're making and repeat. The difference between the indicator
readings is twice the squareness error of the end of the block over
the distance between the point and the stop.
On Sat, 9 May 2009 08:36:14 -0700 (PDT), Jim Wilkins
It's cool that you can generally do arbitrary-precision measurements
of dimensionless properties with pretty much just patience. For
example, you can match resistors to any precision at any ratio with
simple equipment. Ditto weight and distance ratios.
Got it. So you make the first measurement like this:
(View in Courier or other fixed font)(angle exaggerated for clarity)
\_____ .-------. <~~ top is scraped flat
__/ / |
/ block |
.-. '-------------' <~~~ bottom is scraped flat and parallel
| | .----------------.
| | | |<~ surface plate
then flip the block over, & make the 2nd measurement.
A block placed into the chuck of the lathe, can have a pretty damned
flat face cut onto it, and a subsequent hand held lapping can be
accurately performed on it. That makes one face true, and finely
Then, it can be milled on the other face with a surface grinder or
mill. A vertical mill would be nice as it would achieve a better surface
quality than a horizontal mill would, and it would be kept more parallel
to the bottom face. Back over to the lathe to mount up, and it should
lap as true as the other face, which is a test of your upchucking
Now, you have two parallel faces, and if you mounted it into the lathe
very carefully, the sides will be perpendicular to those two faces.
You could then continue upchucking different faces of the cube and
facing them off with the lapping device you'll want to make for the tail
stock of the lathe. :-)
I agree... upchucking is fun.
Groan! Is that like when somebody says "ve have vays..." ?
I once worked in a place using a combination
shear and punch, and when they first instructed
me to weld jigging onto the work surface I
got a big knot in my gut.
Cutting it off with an angle grinder to reset
for the next part gave me a twisted gut too.
I quickly adapted, got quick at it and cycled through
many dozens of different parts and jigs.
I am now back to thinking it was so very wrong.
I used to hang around an optics company that had a bunch of the slow
wet grinding tables that made things flat. There is also an inherent
(no external artifact) way to make perfect cubes, but I don't recall
what it is. That shop was making the retroflector optics that one
Apollo mission left on the moon, in a grungy shop behind the River
Rondezvous bar at the foot of Carrolton Avenue, right near the levee,
in New Orleans.
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