It is no wonder then, that the Elements of Geometry is the most imitated text of all time. In one field of study after another, thinkers have wanted their systems of thought to become public truths in the same way as Euclidean geometry. Descartes' Meditations is one such effort. He attempts to assume nothing that is not self-evident, and to proceed rigorously to conclusions which he hopes will be universally acceptable. Thus, he is not even willing to assume that he himself exists, but rather concludes it from the observation that he thinks. This was self-evident to Descartes, because the opposite would be absurd: How could he think about the possibility that he was not thinking?
The high point of Catholic theological thought, the Summa Theologica of St. Thomas Aquinas, also imitates the style of Euclid, as does the Ethics of Spinoza, where we read “I shall consider human actions and appetites just as if I were considering lines, planes, or solids.”
The influence of the Euclidean form was amplified by the astounding success of Newton. His laws of motion may have been based on experiment and observation rather than pure reason or intuition, but once these laws were granted his reasoning proceeded to conclusions seemingly far removed from his assumptions. The same rules applied to falling rocks and planets moving across the sky. It is hard for us to recapture the shock of this discovery to Newton’s contemporaries. For more than a thousand years the Lord’s Prayer’s affirmation “Thy will be done on Earth as it is in Heaven” had been more than just a metaphor. The regularities of the planets were seen as a direct expression of the will of God. The learned man who read and believed the works of Newton must have felt an awe of the invisible order every bit as strong as the ancient Egyptians felt when they saw the miracles wrought by their priestly surveyors and architects.
With Spinoza’s claim that “the clear and distinct ideas of the mind are as true as those of God,” it became possible for human reason to oppose the claims of both tradition and divine revelation. Jefferson is claiming this authority when he writes "We hold these truths to be self-evident …" The rights of each individual, he was claiming, were not just his personal beliefs; they were universal truths that each person could validate by using his own Cartesian intuition.
By the middle of the 18th-century, the Euclidean form of reason with its definition/assumption/theorem structure reigned supreme in intellectual circles. It was the only rigorous way to think; it gave access to the highest form of truth; it forced all reasonable people to fall into line; and there was no limit to how far it could go.
Like the mythic Tower of Babel, the rational worldview of the Enlightenment was a structure that would reach to Heaven and make men as gods. In the myth, God deals with this possibility by splitting the language of the builders into many languages. Early in the 19th century something similar happened: geometry split into many geometries.
Return to the outline of The Unreasonable Influence of Geometry