Commentary on Proposition III.9¶
Back to Proposition III.9.
The result of this proposition is quoted by Aristotle, Meteorologica III. 3, 373 a 13—16 (cf. note on I. 8).
9 is, as De Morgan remarks, a logical equivalent of part of III. 7, where it is proved that every non-central point is not a point from which three equal straight lines can be drawn to the circle. Thus III. 7 says that every not-$A$ is not-$B$, and III. 9 states the equivalent fact that every $B$ is $A$. Mr H. M. Taylor does in effect make a logical inference of the theorem that, If from a point three equal straight lines can be drawn to a circle, that point is the centre, by making it a corollary to his proposition which includes the part of III. 7 referred to. Euclid does not allow himself these logical inferences, as we shall have occasion to observe elsewhere also.
Of the two proofs of this proposition given in earlier texts of Euclid, August and Heiberg regard that translated above as genuine, relegating the other, which Simson gave alone, to a place in an Appendix. Camerer remarks that the genuine proof should also have contemplated the case in which one or other of the straight lines AB, BC passes through D. This would however have been a departure from Euclid’s manner of taking the most obscure case for proof and leaving others to the reader.
The other proof, that selected by Simson, is as follows: “For let a point D be taken within the circle ABC, and from D let more than two equal straight lines, namely AD, DB, DC, fall on the circle ABC;
I say that the point D so taken is the centre of the circle ABC.
For suppose it is not; but, if possible, let it be E, and let DE be joined and carried through to the points F, G.
Therefore FG is a diameter of the circle ABC.
Since, then, on the diameter FG of the circle ABC a point has been taken which is not the centre of the circle, namely D,
But the latter are also equal: which is impossible.
Similarly we can prove that neither is any other point except D;
On this Todhunter correctly points out that the point $E$ might be supposed to fall within the angle ADC. It cannot then be shown that DC is greater than DB and DB than DA, but only that either DC or DA is less than DB; this however is sufficient for establishing the proposition.