It is easy to find a very flat surface, "perfect" straight edge, or squares for ensuring things are perpendicular. The machines that make these need to be flat/orthogonal to do so, but how were the first squares and straight edges that were fairly accurate made?

"Fairly accurate" is subjective, and I don't know exactly how to define it, but in my mind I am imagining measurements made in a time before the age of heavy machinery and modern manufacturing.

  • $\begingroup$ Don't trivialise the problem of finding a flat surface ... the machines that make them rely on already having them! Lapping and honing are crude (and rely on the flatness of the lap!) - there's alaborious process due to Maudslay, of marking and hand scraping groups of three plates to achieve truly flat surfaces. Two plates give the danger of making one convex and one concave surface ... a 3 way comparison eliminates that. $\endgroup$ Apr 14, 2017 at 12:53
  • $\begingroup$ See en.wikipedia.org/wiki/Surface_plate and an extension to right angles here practicalmachinist.com/vb/… $\endgroup$ Apr 14, 2017 at 12:53
  • $\begingroup$ A 3/4/5 cubit square would do it. $\endgroup$
    – Paul Uszak
    Apr 14, 2017 at 18:28
  • $\begingroup$ Very true, it is then affected by how accurate the ratio is I guess. $\endgroup$
    – samhh1
    Apr 16, 2017 at 17:26

2 Answers 2


Perpendicularity can be achieved by using any item of constant length.

Knotted cord was used by the Greeks, Romans, and Ancient Egyptians. Sticks work well too.

The simplest way to establish perpendicularity (and one I frequently use in fabrication) is to measure diagonals and confirm they are equal. Even though I own a large steel square, this method has much less error on projects that are larger than the square. As joojaa pointed out in the comments, this only works if opposing sides are also equal as shown by the congruency symbols on the diagram below.

perpendicularity via equal diagonals

A 3:4:5 triangle also has a 90 degree corner, and can be more convenient that constructing a whole rectangle if it doesn't already exist in the design.

3 4 5 triangle

The wiki "construction of the perpendicular" section and the links below have the compass method and a few others I have not used.

perpendicularity via compass

Instructables, tricks to find square
Math Educators SE 3:4:5 Triangle Question
MathIsFun 3:4:5 Triangle

  • 1
    $\begingroup$ I would like to note that (due to students just yesterday doing this) measuring diagonals only does not guarantee that the thing is square if your sides arent equally long also. $\endgroup$
    – joojaa
    Apr 14, 2017 at 5:35
  • $\begingroup$ True, the symbols on that diagram show it better than i explained it. I will add that note. $\endgroup$
    – ericnutsch
    Apr 14, 2017 at 5:48

It is fairly easy to construct a pretty accurate right angle using a divider to bisect a straight line with the limiting factors on accuracy being the straightness of your lines and the consistency of the arc length that your dividers can scribe (the accuracy of the arc length doesn't matter as long as it is consistent).

You can also use the angle between a horizontal line (achieved with a spirit level) and a vertical line (from a pendulum bob) to give you a right angle.

For flat surfaces and straight lines there are various methods a taunt wire being the simplest way to achieve a straight line. For flatness lapping can achieve better flatness than the tools used in the process. Similarly any liquid will form surface parallel to the surface of the earth under gravity which is actually pretty flat for most practical purposes, for example float glass is generally pretty flat and parallel just as a product of the manufacturing process.

A key thing here is that when you are talking about geometric tolerances eg flatness, straightness, parallelism, perpendicularity ect there are often various tricks you can exploit to get decent accuracy with quite simple tools with a bit of creativity.

another useful example is honing which essentially uses free floating abrasive blocks to 'average out' irregularities in a surface and can produce flatness/roundness much better than the tolerances of the tool itself.

Also don't discount the importance of the skill of individual craftsmen, especially in the contest of pre and proto industrial technology, medieval stone masons are quiet a good example of this.

  • $\begingroup$ The pendulum and level is so simple and genius $\endgroup$
    – ebrohman
    Apr 14, 2017 at 1:27

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