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.”

Of Idle Tales and Stupefying Joy


Resurrection Icon

Virgil Nemoianu claims that “‘Christian humanism’ is rooted primarily in the Gospels of Luke and John.” Nowhere is this more evident than the 24th chapter of Luke. In this remarkable chapter, Jesus’ tomb is discovered empty, Jesus appears to two of his followers on the road to Emmaus, he has a meal with his disciples, and he ascends into Heaven—all in 53 verses. Yet, in spite of these tremendously important events, dare we say, the most important events of Jesus’ life, we are given little detail. The reader may be struck by the lack of specifics concerning these events, and we can only speculate as to why Luke did not take more space to explain just what happened in these last crucial days of Jesus after the crucifixion.

That Luke does not take the space to provide more detail may be an indication of the disciples own state of mind at the time. That is, Luke presents, in a quite literary fashion, the puzzlement of the events from the point of view of the disciples. None of them had expected the resurrection and they struggled to understand just what was happening. Their confusion is evident from the very beginning. When the women find the stone rolled away and the tomb empty, “they were perplexed about this.” (Luke 24:4) When the women tell the disciples and others about this, they disbelieved the women. There is no blind credulity here that is often attributed to religious believers. “These words seemed to them an idle tale, and they did not believe them.”(Luke 24:11) So convinced were they that the women’s story was an “idle tale” that only Peter went to investigate. He finds the tomb empty as the women had described and went away “marveling” because he still did not understand what was going on. Much like Joseph seeking to leave Mary when he learned she was pregnant, the disciples displayed the kind of common sense and healthy skepticism that critics of religious believers tout as incompatible with religious belief. Yet here it is present in the most devout of Jesus’ followers.

Even when Jesus appears directly to the disciples, they still do not understand. In fact, they think he is a spirit, a vision from beyond the grave. It is this encounter, along with Jesus breaking of the bread with the two he met on the road, that reveals the deep mystery of the Incarnation. For just what kind of being are they encountering? He appears and disappears at a blink, yet he has a physical body and eats and drinks. He is their beloved Lord, back from the grave and standing before them in the flesh. “Why are you troubled, and why do doubts arise in your hearts? See my hands and my feet, that it is I myself. Touch me, and see. For spirit does not have flesh and bones as you see that I have.” (Luke 24:38-39) At this, Luke says, they “disbelieved for joy.” (Luke 24:40) It is too much for the disciples to take, it is too good to be true, and so they doubt. Again, this is a reasonable, human response to the situation. The joy is beyond their comprehension, it simply cannot be. And so Jesus takes the time to eat with them, showing them that he is not some disembodied spirit, and to explain to them all that had happened as “he opened their minds to understand the Scriptures.” (Luke 24:45) On their (and our) own, we are not able to understand. It takes the work of God to overcome our limitations and doubts.

The notion of Jesus’ resurrection and resurrected body is as stupefying to us as it was to the disciples. With the hope and joy in the final resurrection we are given a glimpse here of eternity, but the nature of this resurrection and resurrected body are still a mystery. Just what sort of body did Jesus have, and just what sort of body will we receive? Paul explores these notions at great length in I Corinthians 15. Resurrection means something more than mere resuscitation, a notion that was not foreign to Jewish beliefs. With resurrection we are given a new body, one at the same time continuous with the present body, and yet transformed and recreated. Heaven is not a place of disembodied life, but a life fully embodied and transformed. Such passages as these ought to go far in dispelling the notions of Gnosticism. To be human is to be embodied.

These are no “idle tales,”—they represent the full hope and joy of the Christian. The joy that is beyond both our deserving and our comprehension. Luke shows us the commonality between us and the disciples, and the reasonableness which ought to typify us. The Christian is not called to blind credulity, nor to pessimistic skepticism. For the former who lead them astray, the later would keep them from the truth. And the truth is in the promise of the resurrection, the promise of the fulfillment of human nature. There is no escapist philosophy here, but a philosophy of joy and hope grounded in truth.

Summer Reading List

This has been an incredibly busy semester for me. In addition to my full time teaching load of Philosophy, Theology, Logic, & Rhetoric, I took a course on Mathematical & Scientific Reasoning (affectionately called the “Numbers Class”), which stretched me as much as any class I’ve ever taken. I also had the amazing opportunity to play Prospero in Topeka Civic Theatre & Academy’s production of The Tempest. Topeka has one of the strongest theatre communities I’ve ever seen and I am blessed beyond words to be able to be a part of these productions (back in 2012 I played Claudius in the production of Hamlet). With all this going on I wasn’t able to keep my blog updated, I was only able to write one essay that I’m working on adapting for a blog post here. It was on the value of studying Euclid.

In the meantime, I’ve figured out my summer reading list, which should give me ample opportunities for reflection and writing over the summer.

First up, I’m taking a course on Christian Humanism, which includes the following books:

  • J.I. Packer & Thomas Howard, Christianity: The True Humanism
  • R. William Franklin, et al., The Case for Christian Humanism
  • Joseph Shaw, et al., Readings in Christian Humanism
  • Craig W. Kallendorf, Humanist Educational Treatises
  • Jens Zimmermann, Incarnational Humanism
  • Walker Percy, Lost in the Cosmos: The Last Self-Help Book

Secondly, I doing an independent tutorial on the question “What is Human Nature,” which includes the following selections:

  • Aristotle, Nicomachean Ethics, Books I, VI, & X
  • Athanasius, On the Incarnation
  • Augustine, On the Trinity, Books IX-XIV
  • Thomas Aquinas, Summa Theologica, Prima pars, Questions 75-89
  • John Calvin, Institutes of the Christian Religion, Book II, Chapters 1-4
  • Reinhold Niebuhr, The Nature and Destiny of Man
  • René Girard, I See Satan Fall Like Lightning

In between these readings, or more likely when I’ve finished these around mid-July, I’ll relax with some fiction:

  • Ray Bradbury, Dandelion Wine (the first book of the summer—a new tradition)
  • Cormac McCarthy, The Road
  • Neil Gaiman, American Gods
  • Wendell Berry, Jayber Crow
  • Annie Dillard, Pilgrim at Tinker Creek
  • Dostoevsky, The Double

For now, I’m just celebrating the end of the school year. Papers & exams are graded, report cards are updated, and now to Commencement Ceremonies to say goodbye to the Seniors.