This article is written by Prof. S. D. Agashe. I greatly enjoyed it when I was a master student.
Euclidean geometry and the axiomatic method
Euclid’s Elements constitutes the earliest extant substantial presentation of a body of material in the axiomatico-deductive form . Through it the subject of geometry got permanently associated with axiomatico-deductive formulation which was then viewed as a method, so much so that the expression ‘more geometrico’ (the geometric way) became synonymous with axiomatico-deductive formulation. Thus arose the general belief, especially in methodological quarters, that Euclid’s Elements and, in particular, Euclid’s geometry were merely instances of the application of a previously thought out/discovered/known method, and, thus, that the axiomatico-deductive method existed prior to the axiomatico-deductive formulation of geometry. Using Euclid’s Elements as my principal evidence, I want to suggest that the true state of affairs is the other way round. The axiomatico-deductive formulation of geometry emerged out of a successful attempt- most probably by some of Euclid’s predecessors – to solve some geometrical problems. Once this was done, it was seen by these geometers and also, of course, by Euclid as an instrument of open-ended discovery. Only, then, could the germs of a method be seen in it. My view of the genesis of the axiomatic method emboldens me to suggest further that in general a method, which is something consciously conceived, arises as the result of reﬂection on an activity that is already being pursued ‘intuitively’. Again, once the method is consciously con- ceived, it can engender new activity being pursued consciously in accordance with the method, i.e. methodically.