How Did They Calculate the Square-Root of 3 & Things Like That Before They Had Calculators, or Even Slide Rules?

Published: October 20, 2014

When I'm not teaching math at CWI, I am working on violins. I repair and make violins.

Given my educational background (physics & math), I have an interest in the conceptual design of the violin. As far as we know, the violin was invented in northern Italy in the 1500s.

Of course it had predecessors, but it appears to be unique to the area, and more importantly (for math people) to have grown out of the same design concepts that were used to design buildings constructed during the Renaissance. Think of the great Dome in Florence, Italy, as a model of what they could do without electronic calculators.

To make it even worse, they did not even have a standard measuring system; it usually varied from town to town. Nor did they have a very good way of even transmitting any information in a way we are used to. They couldnʼt say: “Make it 14 inches long.” A 20th century carpenterʼs tape measure is actually a sophisticated device, made of highly engineered materials that are resistant to changes of temperature and humidity.

So how did they do it? They did know about Euclid and Pythagoras. They knew how to draw straight lines and circles. They thought of measurements as proportions -- ratios or fractions, in modern terms. Small-integer ratios were used to develop structure and form, such as 1/2, 3/5, and so on. Rational numbers, we call them in the math department.

Suppose, however, you were building something that required you to make a box with sides that were in the proportion of the square-root of 3 to 1. I know, that sounds weird to us, but it did come up from time to time. If you were into bad puns, you could even say it sounds irrational.

Hereʼs one way to do it.

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