Section 2 - Concerning the extensive usefulness peculiar to this mode of proof in particular

Summary

FIRST REFLECTION: IN WHICH THE EXISTENCE OF GOD IS INFERRED A POSTERIORI FROM THE UNITY PERCEIVED IN THE ESSENCES OF THINGS

1. The unity in the manifold? of the essences of things is demonstrated by appeal to the properties of space

The necessary determinations of space afford the geometer a pleasure which is far from ordinary. They do so because of the certainty of their conviction, the exactitude of their execution, and the extensiveness of their application. The whole range of human knowledge has nothing to show which equals it, far less anything which exceeds it. But, for the present, I wish to examine this same object from an entirely different point of view. Looking at it with a philosophical eye, I come to notice that order and harmony, along with such necessary determinations, prevail throughout space, and that concord and unity prevail throughout its immense manifold. Let us suppose, for example, that I wish to produce a bounded space by moving a straight line around a fixed point. I have no difficulty at all in grasping that the result is a circle, the circumference of which is at all points equidistant from the aforementioned fixed point. But I have no reason at all to suppose that such a simple construction should conceal something highly complex which is itself subject, in virtue of that very construction, to major rules of order. And yet I discover that all the straight lines which intersect each other inside a circle at any given point, when they are extended to its circumference, are always divided in geometrical proportion. Likewise, I discover that all the straight lines which extend from a given point outside a circle so as to intersect its circumference are always divided into parts which are related to each other in inverse proportion to their wholes.