Commentary on Proposition III.37¶
Back to Proposition III.37.
De Morgan observes that there is here the same defect as in I. 48, i.e. an apparent avoidance of indirect demonstration by drawing the tangent DE on the opposite side of DF from DB. The case is similar to the apparently direct proof which Campanus gave. He drew the straight line from D passing through the centre, and then (without drawing a second tangent) proved by the aid of II. 6 that the square on DF is equal to the sum of the squares on DB, BF; whence (by I. 48) the angle DBF is a right angle. But this proof uses I. 48, the very proposition to which De Morgan’s original remark relates.
The undisguised indirect proof is easy. If DB does not touch the circle, it must cut it if produced, and it follows that the square on DB must be equal to the rectangle contained by DB and a longer line: which is absurd.