The Ancients used this method. (or at least James Grime said in a numberphile video) To construct the square root of a number, draw an interval of length $a+1$, and then draw a semi-circle with the interval as the diameter. The perpendicular chord from the point where a and 1 meet has a length of the square root of $a$. I understand that this is the visual display of the geometric mean of $a$ and 1, but how were the Ancient Greeks certain that this method was valid?
2026-03-31 05:41:38.1774935698
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How did the Ancient Greeks know that the circle method of finding square roots was mathematically valid? How do we know that?
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The ancients didn't actually think of it as "finding a square root of a number", so they didn't prove that.
What they did was "find a square that has the same area as this rectangle", and/or "given line segments AB and CD, find a line segment EF such that the ratio of AB to EF is the same as the ratio of EF to CD".
The latter task can be proved using Thales' theorem and some similar triangles.
The former is the capstone proposition of book II of Euclid's Elements. Euclid's proof can be seen, for example, here.
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Here's where you can find it in Euclid:
http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI13.html
The stack exchange content police tell me that links can go stale and insist I say more. So just in case the human race forgets proposition 13 of book VI of Euclid's Elements, I must explain that it tells you how to construct a geometric mean. I.e., given a line segment of length $a$ and a line segment of length $b$, it tells you how to construct a line segment of length $\sqrt{ab}$. If you take $b = a$, this tells you how to construct square roots.
That method is based in two facts:
These facts were well-known to the ancient Greek mathematicians.