MorrisKline. Math. for non-mathematicians. 20. Non-Euclidean Geometries and their significance. 01. Introduction.

  The most significant revolutions in this world are not political. Political revolutions hardly change the daily life of man or, if they do, exert a short-term effect which may even be reversed by subsequent revolutions. The significant upheaval are caused by new ideas; these far more effectively, powerfully, and lastingly alter the lives of civilized human beings. For example, the beliefs in the importance of the human being and in his right to life, liberty, and the pursuit of happiness have permanently and radically changed the lives and aspirations of hundreds of millions of human beings. Indeed, many political revolutions were inspired by the desire to realize these ideas. The two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century are evolution and non-Euclidean Geometry. The Theory of Evolution is generally well recognized as a prime influence, but non-Euclidean Geometry, despite its more fundamental and more far-reaching effects, seems to escape attention.

  In this chapter we shall examine the nature of non-Euclidean Geometry, its value for Science, its implications for the nature of Math., and, finally, its influence on our culture.