In a survey of books used for education throughout the history of Western civilization two books stand out: the Bible and Euclid’s Elements (Carl B. Boyer and Uta C. Merzbach, A History of Mathematics, 119). Poet and schoolmaster Edna St. Vincent Millay says of the Elements that “Euclid alone has looked on Beauty bare.” And Euclid earned a spot amongst Raphael’s School of Athens painting alongside Plato and Aristotle. What can account for such high praise and popularity? Is it that Euclid has laid the foundations for all mathematics? If so, why has Euclid been left behind in the modern classroom? Is there any value in a return to Euclid? What value might there be in studying Euclid today?
It may be too strong a claim to say that the Elements provide the foundation for all mathematics. Nevertheless, the basic principles or axioms of many of the branches of mathematics can, in fact, be seen in Euclid. In the classical mathematical Quadrivium of Arithmetic, Geometry, Music, and Astronomy, we see that Geometry is but one of the fundamental subjects of mathematics. Yet, in Euclid’s Elements there are applications and axioms for the other branches. For example, his earliest axioms like, “If equals be added to equals, the wholes are equal” have clear implications for the axioms (if not being identical) in Arithmetic. The proofs for relationships of ratios throughout Book X (and elsewhere) have clear implications for the science of Music which deal in harmonies and patterns. And certainly the principles of trigonometry that are laid down by Euclid have far reaching application from Astronomy to sea-faring to engineering.
Yet Euclid does not specifically set forth the axioms of those other branches. However, to the student who is attentive, the Elements does teach an important principle concerning the nature of learning and of certain disciplines. In demonstrative sciences one always begins with axioms and definitions and then begins to reason from those assumptions. They are the grounds or conditions of the reasoning that follows. In this sense they are indemonstrable. To ask for such demonstrations is to misunderstand the nature of the science. For example, Aristotle in the Metaphysics, sets forth to show that the Principle of Non-Contradiction (the foundations of Logic itself) cannot and should not be demonstrated. To attempt a proof is to misunderstand the nature of proof, for one cannot prove it without assuming it. The best Aristotle can do in this case is show that it is impossible to deny, because to deny it, one must assume it. In Geometry it would be improper for Euclid to attempt to prove that “a proportion in three terms is the least possible.” Rather, this definition functions as an assumption from which the proofs proceed.
As indicated from the example from Aristotle, Geometry is not the only science that proceeds in this fashion. The student who is attentive in his studies of the Elements should see parallels in other disciplines as well, such as the philosophical and theological sciences. Just as there are axioms of Geometry, so too are there axioms of philosophy and theology that are not subject to proof, but are the grounds from which reasoning proceeds. This may be one of the mistakes of Descartes in Epistemology: he attempted to assume nothing and prove everything. A task which is impossible, for all disciplines requires axioms. Even Moral Philosophy, of which Thomas Aquinas asserts the axiom of all action is: “good is to be done and pursued, and evil is to be avoided” (Summa Theologica, II-I, Q. 92, A. 1).
This may partially account for the staying-power of the Elements throughout history, the implicit lesson about the nature and procedure of demonstrative sciences. In addition to this, the one who studies Euclid does not just study Geometry. For the Elements is also a lesson in the Trivium of Grammar, Logic, and Rhetoric. That is, Euclid bridges the gap between the Trivium and the Quadrivium. This is also why Euclid may appeal to those persons who find mathematics difficult or intimidating. For as a modern student peruses the Elements they may be struck with how “unmathematical” it appears. There are no numbers, no Cartesian coordinate planes, no formulas. It is as much a book of literature as it is of geometry. This may account for the testimony throughout history of its elegance and beauty. For each of Euclid’s proofs begin with an assertion followed by the elegant “for if not” reductio ad absurdum and ending with pointed “the very thing which was to be shown” (Q.E.D.) or “the very thing which was to be done” (Q.E.F.). Thus, in the process of learning Geometry, the student also learns Grammar and Logic, as well as certain principles of persuasive argumentation (Rhetoric). This may also account for the popularity of the Elements in education.
Will Euclid ever be used again to the same degree as he was in the past? This seems unlikely for a number of reasons. First, there is a need for certain modern concepts in geometry like the Cartesian coordinate plane. Second, textbook companies have no incentive in publishing Euclid since the Elements is in the public domain. Third, the modern student (for a variety of reasons beyond the scope of this essay) may no longer have the capability to read Euclid as an introductory text on Geometry. Yet, for the student who struggles with mathematics, Euclid may be a way to bridge the gap between the humanities and mathematics. And maybe, these students too may come to see that: “Euclid alone has looked on Beauty bare.”