When I was about eight years old, I was staying in a very pretty little hotel in Somerset*. At night we used to sit up with the husband and wife proprietors and they'd tell us lots of stories. When it came to be our last evening there, the gentlemen went into a cupboard, got an old biscuit tin out, and showed me its contents. It was a veritable treasure trove of fossils and antiquities and wonderments.
“You can pick anything from this box to take home with you”, he said, after explaining what all of the pieces were. After much deliberation, though not so much as to be rude, I chose a palm-sized shard of Roman pot.
“Good choice”, the gentleman said. “Now, do you know how big the original pot would have been?” Of course I didn't, so the gentleman said “well, let's work it out!” He got a piece of paper out, and traced the arc of the shard on the paper. Then he got a compass and taught me how to reconstruct the circle from the arc. Because I was so young, the advice didn't fully take with me, and for years the only bit that I could remember is the general sketching that he did and the words “swinging an arc”.
Today, all these years later, I was reading about Euclid and I worked out the method that he must have used. The technique is a taken from the geometrical subfield of draughtsmanship known as “construction”. It's fully documented on A Circle Through 3 Points, but I worked it out myself from having stumbled across Euclid's Proposition 10 from Book 1, and finding a diagram from a page called Finding the Radius of an Arc.
When they gave me the pottery shard, they put it in a little brown envelope with the place that it was found written on the front. I remember that they grumbled about the fact that the county was now called Avon, but that's still what they wrote on the envelope. The county of Avon had replaced Somerset in 1974, but only lasted until 1996 when it was replaced once more by North Somerset as the administrative county and Somerset as the ceremonial county.
by Sean B. Palmer