The Success of Euclid’s ‘Elements’

Scuola_di_atene_07In 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.

200px-EuclidStatueOxfordYet 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.”

An Apologia for the Study of Logic

In the never ending world of education reform, from “No Child Left Behind” to “Common Core Standards,” we are continually told of the need for “critical thinking,” reading, and writing skills—along with technical skills for future employment. A survey of the reforms and initiatives put into law and practice, however, all have a similar defect: a failure to teach Logic. When, exactly, Logic was dropped from the curriculum, I do not know, but it’s reintroduction does not seem to be a goal of any reformers of public education.

metalogicon250One of the most cogent and eloquent defenses of the teaching of Logic comes from the 12th century thinker John of Salisbury. In his book, The Metalogicon, John argues persuasively that a study and knowledge of Logic is necessary for myriad reasons. The title, while admittedly daunting, means simply “on behalf of Logic,” and in the book John sets forth to refute those thinkers of his own time (who he refers to collectively as Cornificius) who were adversaries of the teaching of Logic (or the trivium more generally). Cornificius, says John, is the “ignorant and malevolent foes of studies pertaining to eloquence, attacks not merely one, or even a few persons, but all civilization and political organization” (11-12). Bold words indeed, for as John sees it, to oppose the teaching of Logic, is to oppose civilization. John’s apologia includes more than just a defense of Logic, undeniably it is a robust defense of the whole of liberal arts education, but I will restrict my discussion here to the focus on Logic.

What exactly is “Logic” as a field of study? For John, Logic has a twofold meaning: “the science of verbal expression and reasoning” (32). That is, Logic (in the narrow sense) covers the rules of rational thinking and (in the broad sense) knowledge and skill of how to express reason with speech—or, as John puts it, “all instruction relative to words” (32). This broader sense, Augustine referred to as, “the science of argumentation” (80). Thus, John suggests that the traditional trivium of Grammar, Logic, and Rhetoric, is what he has in mind by “Logic.”

This seems to go beyond the traditional definition of Logic which was restrictive to the art and science of reasoning, with Rhetoric taking up the ability to express ideas eloquently and winsomely and Grammar the science of words. The reason John extends the use of Logic to encompass all of the art of argumentation seems to be due to the nature of Logic itself as the hinge to both proper Grammar and effective Rhetoric. Grammar, affirms John, “is the science of speaking and writing correctly—the starting point of all liberal studies” (37). Rhetoric is the art of expresses those words eloquently, or as John puts it, “Rhetoric, where persuasion is in order, supplies the silvery luster of its resplendent eloquence” (67). In order to show why Logic is the “linchpin,” so to speak, of the verbal arts, it’s nature and purpose must first be explored.

The kind of Logic, in the narrower sense, John has in mind is that formalized by Aristotle. Aristotelian Logic, certainly at the time of John, was the only game in town. Aristotle, being its one and only founder, dominated Logic studies and John did not depart from this tradition.

So, why study Aristotelian Logic? John gives several reasons. First, logic provides the groundwork or rules which give birth to Prudence. Says John, “Of all things most desirable is wisdom, whose fruit consists in the love of what is good and the practice of virtue” (74). Could we considered anyone wise who reasons illogically? In fact, is that not a true oxymoron, to “reason illogically”? Logic provides the tools for the mind to operate and enable it to judge (if not to act) wisely. In order to act Prudently, one’s mind must operate along the rules of logic, which guide the mind to the proper course of action. This, of course, is not a perfect road map, but without it, one could only follow the proper course by accident. With all the roads that we would take, who could navigate without the ability to read the map?

In an age which is tempted to worship science, it is a wonder that training in Logic is not mandatory, given that the presumed object of science is truth and logic is the mind’s aid to discover truth. As John says, “Prudence consists entirely in insight into the truth, together with a certain skill in investigating the latter; whereas justice embraces the truth and fortitude defends it, while temperance moderates the activities of the aforesaid virtues” (74). That is, at the center of the Cardinal Virtues is Truth and Logic provides the means to attain the Truth.

Given all of this, we can now see why Logic is the linchpin of the verbal arts, and why John calls the whole of the trivium “Logic.” Logic is what connects the grammar of the word with the eloquence of expression. Without Logic, Rhetoric becomes Sophistry. Logic aids in judging propositions, it is what guides the mind in the discovery of Truth. If the mind is not aimed at Truth, Rhetoric is merely aimed at power, overcoming one’s opponent. As John puts it, Rhetoric “unenlightened by reason, is rash and blind” (10). The uniting of the trivium John explains poetically:

If we may resort to a fable, antiquity considered that Prudence, the sister of Truth, was not sterile, but bore a wonderful daughter [Philology], whom she committed to the chaste embrace of Mercury [Eloquence]. In other words, Prudence, the sister of Truth, arranged that [her daughter], the Love of [Logical] Reasoning and Knowledge, would acquire fertility and luster from Eloquence. Such is the union of Philology and Mercury. (78-79)

It may be that the modern educator’s failure to teach and instruct in the art and science of Logic is an implicit rejection of Truth. For if there is no Truth, Logic is irrelevant. So too is reason, knowledge, and science. If, however, Truth is deemed possible, to ignore the study of Logic is to handicap the mind. It is to lead the student in the study of truth but not to give the student the tools to discover it. A recovery of the study of Logic, therefore, is one of the truly necessary areas of “reform” for modern education, even if it is not on the agenda of any modern school boards or legislators.


John of Salisbury, The Metalogicon: A Twelfth-Century Defense of the Verbal and Logical Arts of the Trivium, translated by Daniel D. McGarry, Philadelphia: Paul Dry Books, 2009.

Hilaire Belloc’s “Cautionary Tales for Children”

Today in my Rhetoric class I read some selections from Hilaire Belloc’s Cautionary Tales for Children for my students.  We are preparing for our annual Fine Arts Festival, and I was modelling poetry recitation (though I did not have the pieces memorized for which my students scolded me). Several of the students snorted with delight, while several stared at me in horror as I read these pieces. “Cautionary tales” is a special genre of poetry in which children who do bad things suffer awful consequences, yet in an absurdly light manner in order to teach a lesson. Here is one of my favorites:



Who ran away from his Nurse, and was eaten by a Lion.

There was a Boy whose name was Jim;
His Friends were very good to him.
They gave him Tea, and Cakes, and Jam,
And slices of delicious Ham,
And Chocolate with pink inside,
And little Tricycles to ride,
And read him Stories through and through,
And even took him to the Zoo—
But there it was the dreadful Fate
Befell him, which I now relate.

You know—at least you ought to know.
For I have often told you so—
That Children never are allowed
To leave their Nurses in a Crowd;
Now this was Jim’s especial Foible,
He ran away when he was able,
And on this inauspicious day
He slipped his hand and ran away!
He hadn’t gone a yard when—Bang!
With open Jaws, a Lion sprang,
And hungrily began to eat
The Boy: beginning at his feet.

Now just imagine how it feels
When first your toes and then your heels,
And then by gradual degrees,
Your shins and ankles, calves and knees,
Are slowly eaten, bit by bit.
No wonder Jim detested it!

No wonder that he shouted “Hi!”
The Honest Keeper heard his cry,
Though very fat he almost ran
To help the little gentleman.
“Ponto!” he ordered as he came
(For Ponto was the Lion’s name),
“Ponto!” he cried, with angry Frown.
“Let go, Sir! Down, Sir! Put it down!”

The Lion made a sudden Stop,
He let the Dainty Morsel drop,
And slunk reluctant to his Cage,
Snarling with Disappointed Rage
But when he bent him over Jim,
The Honest Keeper’s Eyes were dim.
The Lion having reached his Head,
The Miserable Boy was dead!

When Nurse informed his Parents, they
Were more Concerned than I can say:—
His Mother, as She dried her eyes,
Said, “Well—it gives me no surprise,
He would not do as he was told!”
His Father, who was self-controlled,
Bade all the children round attend
To James’ miserable end,
And always keep a-hold of Nurse
For fear of finding something worse.