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I've been looking at Euclid's Elements, and I have a question about Book 1, Proposition 1. This is where Euclid constructs an equilateral triangle. The proof is straightforward, but I have a question about how Euclid identified the circles he uses in the proof. In the Heath edition of Euclid, which is considered the most authentic, the first circle is labeled BCD, and the second circle ACE. However, when proving the postulate, Euclid changes the order of the lettering when identifying the circles. The first circle is now referred to as CDB, and the second circle CAE. Same circles but different letter order. If we now go to Simson 1762 and subsequent editions derived from Simson (eg Todhunter), the ordering of letters is not changed.

My question is, is there any reason why the lettering is different in Heath's edition compared to those derived from Simson? Was this deliberately done by Euclid? It would seem easier for the reader to keep the same lettering when referring to the same circle.

[update] I checked billingsley (1527) the first English translation of Euclid and I see the letter order also changes. However when I checked a Latin translation of Adelard of Bath I (12th century) the order is not changed. I tried to read the original 888AD Theon, but I could not decipher the Greek. It would seem that the letter order has changed frequently over time.

rhody
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This by one Richard Fitzpatrick is "the original Greek beside a modern English translation". In the preface it says that it is the definitive Greek text as edited by J.L. Heiberg. In the definitive text it does indeed switch from BCD (ΒΓΔ) to CDB (ΓΔΒ) in the proof of theorem 1.

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Suzu Hirose
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  • So it looks like Euclid used different letter orderings to indicate the same circle. The only reason I think why he did this was to emphasize point C in the second reference to the circles. – rhody Oct 30 '22 at 18:44