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Updated: May 17, 2025
Some geometers who are not philosophers have rejected final causes, but real philosophers admit them; a catechist proclaims God to the children, and Newton demonstrates Him to the learned. If there are atheists, whom must one blame, if not the mercenary tyrants of souls, who, making us revolt against their knaveries, force a few weak minds to deny the God whom these monsters dishonour.
From the time of the Ptolemies the scientific spirit of the Alexandrian school declined; for though such mathematicians as Theodosius, whose work on Spherical Geometry was greatly valued by the Arab geometers; and Pappus, whose mathematical collections, in eight books, still for the most part remain; and Theon, doubly celebrated for his geometrical attainments, and as being the father of the unfortunate Hypatia, A.D. 415, lived in the next three centuries, they were not men like their great predecessors.
I have ever remained firm in my original resolution to suppose no other principle than that of which I have recently availed myself in demonstrating the existence of God and of the soul, and to accept as true nothing that did not appear to me more clear and certain than the demonstrations of the geometers had formerly appeared; and yet I venture to state that not only have I found means to satisfy myself in a short time on all the principal difficulties which are usually treated of in philosophy, but I have also observed certain laws established in nature by God in such a manner, and of which he has impressed on our minds such notions, that after we have reflected sufficiently upon these, we cannot doubt that they are accurately observed in all that exists or takes place in the world and farther, by considering the concatenation of these laws, it appears to me that I have discovered many truths more useful and more important than all I had before learned, or even had expected to learn.
Such diagrammatic presentments, while of course primarily for the purpose of clear expression and comparison, are also frequently suggestive by "inspection," as geometers say of relations not previously noticed.
Modern mathematicians may not have the patience to go through his investigations, since the conclusions he arrived at may now be reached by shorter methods, but the great conclusions of the old geometers were only reached by prodigious mathematical power.
He was gaining, too, from some source, what the ancient geometers would themselves have claimed as partly the product of their study: the plain fact and its plain consequences were not only clear in calm hours of thought, but remained present to him, felt and instinctive, through seasons of confusion, passion, and dismay.
The intuitions from which mathematical deduction starts are highly generic notions drawn from observation. The lines and angles of geometers are ideals, and their ideal context is entirely independent of what may be their context in the world; but they are found in the world, and their ideals are suggested by very common sensations.
As an instance of what is here referred to, we may cite the extraordinary notion which, under the designation of a discovery, first brought Kepler into fame. Geometers had long known that there were five, but no more than five, regular solid figures. There is, for instance, the cube with six sides, which is, of course, the most familiar of these solids.
Theological dicta were, to the thinkers of those days, that which the axioms and definitions of Euclid are to the geometers of these. The business of the philosophers of the middle ages was to deduce from the data furnished by the theologians, conclusions in accordance with ecclesiastical decrees.
Its pole will become displaced while still keeping on one straight line. The endless scroll leads to rectilinear progression; the perpetually varied begets uniformity. Now is this logarithmic spiral, with its curious properties, merely a conception of the geometers, combining number and extent, at will, so as to imagine a tenebrous abyss wherein to practise their analytical methods afterwards?
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