Michael W. Grenke’s “Another Euclid”

Euclid presents the reader with proofs about eyes and what they see.  He gives us proofs about the height of trees and the depth of ditches.  Some of the parts of the proofs are sunlight and shadows and mirrors and chariot wheels.  Euclid offers comments about natural beings that move and grow.  He offers comments that are phenomenological.  He even offers comments about the workings of the human mind.  But this Euclid of whom I am now speaking is not the writer and thinker that many of you may think you know.  The Euclid of whom I am speaking is the author of the work known as the Optics.  This Euclid can be found just a few inches farther along the bookshelf from the Euclid of the Elements.   The tradition that passes both works down to us tells us that the author of both works is the same.  Yet, the Euclid of the Optics does not seem to think the same things that I have heard many students of the Euclid of the Elements claim about that author.  One of the most striking and persistent claims I have heard is that in the Elements Euclid is not making any claims about the “real” world.  The Euclid of the Optics does not seem to think straight lines or other geometrical figures possessing similar degrees of perfection are not part of the “real” world.  Rather this Euclid seems to think our power of perception to be inadequate to the perfections of the beings that are in principle perceptible.  This Euclid does not seem to present mathematics as a bunch of made up things, rather he presents human beings as beings that make up some part of their experiences in order to compensate for deficiencies of our powers of perception in some cases and in order to avoid the unsettling consequences of purer or more honest perceiving on the other.