Recently I re-worked my way through all forty-eight propositions of Book I of Euclid’s Elements. In the book, Euclid offers a captivating introduction to classical geometry, which straddles the world of perfect abstraction on the one hand, yet also Euclidean geometry relies upon certain principles of the physical world around us. For example, in some propositions we experience a kind of imagined motion, or even a gravitational pull that governs Euclid’s visual demonstrations. Euclid serves as a guide as we learn about lines, circles, angles, triangles, and rectilineal figures (all of which rely upon the unproven acceptance of Postulate #5, the Parallel Postulate).
As I went through the book this time I was struck by the unfolding plot of the Elements. Insofar as the Elements has a narrative, the plot of Book I begins by introducing us to equilateral triangles and it concludes by proving the equality of all right angles in equilateral figures (triangles and squares). Throughout the book we glean a sense of equality (equilateral triangles and rectilineal figures) while other shapes like circles are merely used to create triangles and rectilineal figures (in other words circles are subordinate to other shapes). Right angles are knowable and equal everywhere, whereas obtuse or acute angles may have varying degrees of distinction -the word mathematics comes from the Greek for ‘the knowable things’ or ‘the art of the knowable things.’ In Book I of Euclid’s Elements, the knowable things share equal attributes.
In Euclid, the concept of spatial area also begins to appear from Proposition #4 onward (“parallellogramic area”). We imagine a sense of transposition that may possible (i.e. that equal triangles may be placed directly on top of one another). What purpose does the concept of spatial area serve for Euclid?
Euclid’s Elements concludes each proposition with the Latin short-script of either Q.E.D. or Q.E.F. Q.E.D. stands for quod erat demonstrandum (or “what was to be shown”) or Q.E.F. or Quod erat faciendum (or “which had to be done”). Euclid used the Greek original of Quod Erat Faciendum (Q.E.F.) to conclude certain propositions that were demonstrations of figures rather than proofs of theorems. For example, Euclid’s first proposition of Book I shows how to construct an equilateral triangle, given one side, and it is concluded with Q.E.F. however most propositions are concluded with Q.E.D.
Lastly, there are by my count 11 reductio ad absurdum proofs (in Latin “reduction to absurdity”) in Book I of the Elements. The first reductio occurs in Proposition #6 (one proposition contains two reductios, as well). The reductios add to the force of the previous propositions by adding a hypothetical converse which, when proven, becomes impossible (“atopon” or literally ‘placeless’ or ‘impossible’ or ‘strange’ or even ‘foreign’). The reductios show us what is truly ‘impossible’ or ‘foreign’ in classical mathematics.
The following images capture my notes on completing the forty-eight propositions of Book I of Euclid’s Elements:
For this reading I used the wonderful translation of Euclid’s Elements by Thomas L. Heath from Green Lion Press. Mr. Heath was a Cambridge scholar who translated Euclid directly from the original Greek in the early 20th century.