My approach to understanding Euclid, both mathematically and philosophically, is fully presented in Chapter 1. To understand Euclid’s method as abstract measurement is to understand both Euclid’s logical structure and, most critically, the way that Euclid’s geometry relates to the world.
Chapter 3 continues the discussion of Euclid’s fifth postulate. It further examines the context of the parallel postulate and explains its application to Euclid’s theory of geometric area and geometric proportion.
The 19th through the early 20th century, brought the transformation of mathematics from the science of quantity, with a Euclidean geometric base, to a formal system built on ontologically meaningless axioms. This transformation was driven by a philosophical perspective. But it transpired on a stage set by the development of non-Euclidean geometry and the later application of non-Euclidean geometry to Einstein’s relativity.
But these mathematical and scientific developments in no way impact my thesis that mathematics is about the world. It is not only true that the fifth postulate is independent of the other four, but, in light of its measurement implications, this independence should be expected. And since all measurements require physical means, one should not expect to understand geometric relationships established by light rays (general relativity) without reference to the nature of light rays. Yet the perceptual level remains the base and relativistic corrections consist in relating such relationships to the Euclidean geometry of the perceptual level. So the first part of chapter 3 examines the context of the parallel postulate.
The measurement of area and the application of geometric proportion to trigonometric calculation is based, inescapably on the parallel postulate. In modern treatments, this relationship is hidden, even if it’s hidden in plain sight. In Euclid, this dependence is front and center and Euclid’s treatment provides a level of understanding and a perspective on area and proportion that is missing from modern treatments. Understanding area and proportion the way that Euclid did illuminates the geometry and provides an ideal theater for watching Euclid’s method in action. Elucidating these applications completes Chapter 3.