On ‘Circular Measure’ and the Product Forms of the Sine and Cosine

Extract

Were it not for Euclid’s Propositions iii. 26 (proved by superposition) and vi. 33, the statement that he nowhere defines the circumference of a circle or any curved line would surely be a truism. His definition of a circle is substantially only a locus definition. It may well be argued that neither does he define a straight line (or an angle) ; but in the case of a straight line he introduces sufficient ideas and axioms to make his subsequent propositions convincing, whereas he gives us nothing to bridge the interval between a straight line and a curved one.

And so, wise after the event, we may say that all the thought spent in old days over ‘squaring the circle’ had no adequate foundation, and was foredoomed to fail.

As to iii. 26 and vi. 33, they are only further examples of the truth with which we are familiar in the case of angles, that it is sometimes easier to prove that things are equal etc., than to say exactly what they are.