Commentary on Proposition III.36¶
Back to Proposition III.36.
Cf. note on the preceding proposition. Observe that, whereas it would be natural with us to prove first that, if A is an external point, and two straight lines AEB, AFC cut the circle in E, B and F, C respectively, the rectangle BA, AE is equal to the rectangle CA, AF, and thence that, the tangent from A being a straight line like AEB in its limiting position when E and B coincide, either rectangle is equal to the square on the tangent (cf. Mr H. M. Taylor, p. 253), Euclid and the Greek geometers generally did not allow themselves to infer the truth of a proposition in a limiting case directly from the general case including it, but preferred a separate proof of the limiting case (cf. Apollonius of Perga, p. 40, 139—140). This accounts for the form of III. 36.