A classical Christian education movement is underway in America, which warms my heart. But many of the schools involved, while teaching ancient languages and great works of literature and history, teach mathematics using contemporary textbooks, typically something like Saxon or Singapore Math1. I’d like to propose that they use Euclid’s Elements instead. Here’s why.
Firstly, it’s the obvious choice. Euclid was the undisputed classic of beginner’s mathematical education more or less from when it was written until modern times2. In the course of modernity it began to be displaced by alternative textbooks, but as late as 1879 Charles Dodgson (Lewis Carroll of Alice in Wonderland fame, in his real life guise as an Oxford mathematics lecturer) wrote a fictional polemic pitting the ghost of Euclid against ‘his modern rivals’, with Euclid winning out. The real turning point, as with so many other things, came only after World War II, symbolized by the slogan “Down with Euclid! Death to triangles!” coined by a member of the influential modernist and hyper-formalist mathematical group Bourbaki. In short, if you read Homer, Herodotus, Sophocles, and Plato, you should be reading Euclid too.
A brief summary for those who haven’t read it. The Elements is about 500 pages long and divided into 13 books. They cover, in order, basic plane geometry (books 1-6), number theory (7-10)3, and solid geometry (11-13). These are, as a famous mathematician recently said, the core of mathematics, despite all the advances made in the millennia since4. Along the way the work proves many of the most famous theorems of elementary mathematics: Pythagoras’ theorem; that any angle in a semicircle is right; the existence of infinitely many prime numbers and the fact that all whole numbers decompose uniquely into them; that there are ‘irrational numbers’5; and the volume of the cone. He also shows how to construct, with only a straight-edge and compasses, the ‘golden ratio’, most of the regular polygons that can be constructed, and finally, to cap the book off, all five regular polyhedra:
However, Euclid’s style is totally different from contemporary school mathematics textbooks, which at this point have almost given up on presenting a logical, connected exposition of their subject matter; instead they seem to be a mishmash of unmotivated rules, contrived ‘real world’ examples, boring and repetitive exercises, and logically unfounded generalizations (don’t ask your algebra textbook why manipulations that apply to rational numbers will also apply to irrational ones!). The Elements, instead, is an attempt at a consistent, rigorous set of proofs of mathematical theorems and problems starting only with basic, obvious assumptions like “the whole is greater than the part” and “all right angles are equal”.6
Moreover, it’s pretty clear the abandonment of Euclid everywhere has led to the present dire state of mathematics education, where young people not only don’t know very much mathematics, but worse, don’t like it. Contrast this with the following story from the biography of Thomas Hobbes:
He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library, Euclid’s Elements lay open, and ‘twas the 47th proposition of book I7. He read the proposition. ‘By God’, said he, ‘this is impossible!’ So he read the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. And so on, that at last he was demonstratively convinced of that truth. This made him in love with geometry.
This has also been the experience of many others down the ages8.
In terms of broader culture, the influence of Euclid has been immense. The history of mathematics could be written as a series of steps taken from Euclid onwards: presupposed by the great ancient mathematicians, Euclidean methods then underwrote the development of calculus in the early modern period; the great mathematician Gauss extended Euclid’s theory of polygonal constructions as the culmination of his magnum opus9, while he and others also showed that (contrary to thousands of years of speculation) Euclid had been correct to assume that a certain postulate related to parallel lines was necessary, and that if this postulate was denied new ‘non-Euclidean geometries’ could be found; and when in the last two centuries mathematicians attempted to give new foundations to mathematics, with Dedekind’s theory of real numbers and Hilbert’s axiomatization of geometry, it was again Euclid that inspired them. Moreover, works of philosophy like Plato’s and Aristotle’s are peppered with references to mathematics taken from or similar to Euclid’s, and Euclid of course occupies a prominent position in Raphael’s famous panorama of Greek philosophy.
That sounds great, you’re thinking, but don’t students need to know modern math? Of course, there’s no reason Euclid can’t be followed by the standard contemporary courses on algebra, calculus, and so on. But one should be careful in assuming that modern mathematics is unproblematically superior or even conceptually equal to ancient mathematics, or that the differences between them don’t have philosophical or even theological ramifications. Consider this quotation from another famous mathematician:
Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”10
The great semi-Christian thinker Simone Weil put it philosophically: ‘The very institution of algebra corresponds to a fundamental error concerning the human mind’: ‘Modern algebra: substitution of the sign for the signified’— ‘The method is in the signs, not in the mind.’ Thus, “God is always a geometer”11, not an algebraist. So it isn’t at all clear that modern mathematics is value-neutral, or superior to ancient mathematics!
But haven’t errors been discovered in Euclid’s proofs? Many have claimed to spot errors in Euclid’s theorems, reasoning, or definitions over the ages. Some of these are real, though minor and easily correctable, mistakes; some, perhaps, are not mistakes but simply depend on unarticulated philosophical assumptions. But contemporary mathematics, based on theories devised only in the last two centuries, is not so secure itself. Take this quote from the mathematician mentioned at the beginning:
Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn’t make them right. Set theorists construct many alternate and mutually contradictory “mathematical universes” such that if one is consistent, the others are too. This leaves very little confidence that one or the other is the right choice or the natural choice. Gödel’s incompleteness theorem implies that there can be no formal system that is consistent, yet powerful enough to serve as a basis for all of the mathematics that we do.
Modern mathematics has devised an ever-expanding universe of numbers—negative, imaginary, (ab)surd, irrational—whose very names testify to their paradoxical ontological status and the resistance traditionally-oriented mathematicians displayed towards them (usually contested by pointing to their pragmatic usefulness in calculation). These were capped off by the so-called “real” numbers, which are currently defined as “infinite sets of infinite sequences of infinite equivalence classes of sets of empty sets”. Contemporary mathematics has even gone so far as to define the ‘basic’, whole numbers — 1, 2, 3, 4, 5… — as “nested sets of nothing”, a kind of mathematical nihilism. The reader may judge for themselves if these types of numbers are truly more ‘real’ than Euclid’s magnitudes and multitudes. But even if all these innovations are legitimate, Euclid approaches mathematics with a directness and intuitive immediacy that is ideal for beginners.
Sounds great! So how then can I teach him? The whole book is easy to procure, and students can demonstrate propositions to each other at a blackboard or whiteboard as is done successfully at various Great Books colleges12. Homeschoolers or teachers who just want to dip into the Elements can also get just the first book, about a hundred pages, to try out the most famous part.13 Using a physical straight-edge and compass to try out the constructions, and a model kit to build the regular solids, is a pleasant ancillary activity. And if your students happen to study Greek, they can even try translating short passages for themselves14. The riches of studying Euclid are immense: a beautiful core of mathematical knowledge that students can share with the greatest minds of Western history. I hope you enjoy it.
Some however, like the Chesterton Academy schools, already include some Euclid in their curriculum!
Of course, during the middle ages he was studied only by a few Arab and then Latin intellectuals—but those who did learn mathematics, learnt it through Euclid.
In fact book 10 is not really about number theory but rather an elaborate treatment of ‘irrational’ magnitudes; I’ve included it in this category for simplicity, and because the theory of commensurable and incommensurable magnitudes given at the beginning of the book is treated in parallel to the theory of relatively prime and composite numbers given in the three previous books.
From his article On proof and progress in mathematics:
Mathematicians generally feel that they know what mathematics is, but find it difficult to give a good direct definition. It is interesting to try. For me, “the theory of formal patterns” has come the closest, but to discuss this would be a whole essay in itself.
Could the difficulty in giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality? Along these lines we might say that mathematics is the smallest subject satisfying the following:
Mathematics includes the natural numbers and plane and solid geometry.
Mathematics is that which mathematicians study.
Mathematicians are those humans who advance human understanding of
mathematics.
What Euclid more precisely calls incommensurable magnitudes.
In fact, this rigorous ideal is really only fully attempted in the first book, after which the level of rigor drops somewhat, but it is the ideal of the opening that has been most influential.
That is, Pythagoras’ theorem:
Another example, from the autobiography of Bertrand Russell: At the age of eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.
The Disquisitiones Arithmeticae; he was so proud of this result he even asked for the 17-sided polygon he constructed to be engraved on his tombstone.
Or this slightly more abstract formulation of Hermann Weyl’s: In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.
A saying attributed by Plutarch to Plato, and quoted by Weil.
One might want to skip some propositions in the long book 10, and consider substituting the translation ‘unity’ for ‘unit’ in the number theory books: the latter tends to give the impression of an arbitrary length repeatedly concatenated together, whereas the former emphasizes the quality of “oneness” that Euclid means to highlight. Also, in general, keep in mind that Euclid means something different by ‘equal’ than contemporary mathematicians do.
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As in so many other domains, ancient Greek mathematical terminology is wonderfully transparent and direct; for example, the Greek words for “acute” and “obtuse” angles literally mean “sharp” and “blunt”, and the word “parallel” literally means “next to each other” (παράλληλος: παρά “next to” + ἀλλήλους “each other”—itself made up of ἄλλος “other” + ἄλλος “other”).
Physicist as I am, I am tickled by the fact that Einstein himself credited Euclid (which he read on his own as a teenager) for sort of introducing him to the power of logical reasoning. From just a few starting premises we can derive all of that?
(That said, it has not been my personal experience, anyway, that modern geometry textbooks are as bad as you say. There are good ones out there that build a logical progression for the students, even if they are not Euclid.)