How to make perpendicular surfaces without a reference surface? Usually in machine shops, you measure perpendicularity with a reference surface. I am interested in how the perpendicular reference itself is created. Literarily the first one ever.
Is there a similar method for creating perpendicular surfaces as it is for creating flat surfaces? Having 2 flat surfaces is a prerequisite for having a perpendicular surfaces and flatness can be achieved with sub micron tolerances by just grinding 3 plates together (Whitworth 3 plates method).
 A: In principle I believe the answer is yes, using a variant of the 3 plates method, but it would be a pain.
(This really is an engineering question, but the responses to the very similar How are perfectly square blocks made from scratch? don't really answer the question, so I'll give it a shot here.)
The mathematical basis for the method is that 3 angles (a,b,c) can only all be supplementary to each other (i.e. a+b=b+c=c+a=180°) if they are all right angles (a=b=c=90°).  This is because the only angle that is supplementary to itself is a right angle.
We'd need 3 blocks (A,B,C) and a Reference Flat. It is assumed that the bottoms of the 3 blocks are flat. The angles we want square is marked.  The sides that will be lapped against each other are marked with an arrow.

We now lap A against B until the marked sides match.  We do the same with A and C. Now flip B.

Lapping A against B only ensures A and B are flat and are supplementary to each other, and similarly for A and C.  Flipping B allows us to compare B with C.  If B and C are not perpendicular to the Reference Flat, they won't match.  In that case, lap B against C until they match.
Now lap C against A.

Iterate lapping and flipping and comparing until all three sides match when paired against each other, or until you you give up and buy a reference square.

Note that I have not done this myself. and an expert machinist may point out details I've left out or serious issues with my description.  For example, the bottoms of the blocks need to be firmly in contact with the Reference Flat, but we don't want them lapping because that could also affect the angle that we want perpendicular.
The material of all 3 blocks should to be identical and homogeneous to avoid uneven lapping, and we'd have to regularly check that all relevant surfaces (lapped sides, block bottoms, reference flat) remain flat. I've only shown the process in 2-dimensions; there are more angles to consider in 3-dimensions.
