Commentary on Proposition III.2¶
Back to Proposition III.2.
The reductio ad absurdum form of proof is not really necessary in this case, and it has the additional disadvantage that it requires the destruction of two hypotheses, namely that the chord is (1) outside, (2) on the circle. To prove the proposition directly, we have only to show that, if $E$ be any point on the straight line AB between A and B, DE is less than the radius of the circle. This may be done by the method shown above, under I. 24, for proving what is assumed in that proposition, namely that, in the figure of the proposition, F falls below EG if DE is not greater than DF. The assumption amounts to the following proposition, which De Morgan would make to precede I. 24: “Every straight line drawn from the vertex of a triangle to the base is less than the greater of the two sides, or than either if they be equal.” The case here is that in which the two sides are equal; and, since the angle DAB is equal to the angle DBA, while the exterior angle DEA is greater than the interior and opposite angle DBA, it follows that the angle DEA is greater than the angle DAE, whence DE must be less than DA or DB.
Camerer points out that we may add to this proposition the further statement that all points on AB produced in either direction are outside the circle. This follows from the proposition (also proved by means of the theorems that the exterior angle of a triangle is greater than either of the interior and opposite angles and that the greater angle is subtended by the greater side) which De Morgan proposes to introduce after I. 21, namely, “The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line, and of others that which is nearer to the perpendicular is less than the more remote, and the converse; also not more than two equal straight lines can be drawn from the point to the line, one on each side of the perpendicular.”
The fact that not more than two equal straight lines can be drawn from a given point to a given straight line not passing through it is proved by Proclus on I. 16 (see the note to that proposition) and can alternatively be proved by means of I. 7, as shown above in the note on I. 12. It follows that
a proposition which De Morgan would introduce here after III. 2. The proof given does not apply to a straight line passing through the centre; but that such a line only cuts the circle in two points is self-evident.