Nomograms and the foundations of geometry

Abstract

The basic properties of nomograms can be proved most easily by metrical methods, but later on the practical problems of nomogram calibration lead naturally to Pappus's theorem. Rearranging the material, we see that Pappus's theorem alone establishes the commutative and associative properties of additive and multiplicative nomograms. Use of Desargues' theorem alone likewise enables proofs to be given of all of these properties, except the commutativity of multiplication. The breakdown of the proof in this case is not sufficient to demonstrate the impossibility of constructing a proof by any other means; nevertheless the material presented indicates the fundamental role of the theorems of Pappus and Desargues, in a context which is much simpler than the usual one, where they are used to set up a coordinate system in the plane. In this way some of the central ideas of Hilbert's Grundlagen can be approached at school level.