Euclidean Geometry and Physical Space

It takes a good deal of historical imagination to picture the kinds of debates that accompanied the slow process, which ultimately led to the acceptance of non-Euclidean geometries little more than a century ago. The difficulty stems mainly from our tendency to think of geometry as a branch of pure mathematics rather than as a science with deep empirical roots, the oldest natural science so to speak. For many of us, there is a natural tendency to think of geometry in idealized, Platonic terms. So to gain a sense of how late nineteenth-century authorities debated over the true geometry of physical space, it may help to remember the etymological roots of geometry: “geo” plus “metria” literally meant to measure the earth, of course. In fact, Herodotus reported that this was originally an Egyptian science; each spring the Egyptians had to re-measure the land after the Nile River flooded its banks altering the property lines. Among those engaged in this land surveying were the legendary Egyptian rope-stretchers, the “harpedonaptai” who were occasionally depicted in artwork relating to Egyptian ceremonials.