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The Canon: Diophantus I

Some fraction of these articles I’ll dedicate to exploring a part of the mathematical canon: who the author was, what they were on about, and why their work is worth reading not just for technical specialists, but for everyone. Finally, I’ll conclude with some suggestions about the kairos for the author: when they can be incorporated into a plan of reading or a curriculum.

Rather than start with Euclid, perhaps the best known of the classical mathematicians, I’m going to begin with Diophantus’s Arithmetic, Book I.

Diophantus in Context

What do we know about Diophantus the person? Essentially nothing. Various authorities place him either during the reign of Antoninus Pius (2nd century) or Julian (4th century). We can narrow down his work by looking at whom he quotes and who does not quote him, which places him in the middle of the third century, towards the twilight of classical mathematics   [Heath, 2].

What biography we have comes in the form of an epigrammatic riddle:

His boyhood lasted 1/6 of his life; his beard grew after 1/12 more; after 1/7 more he married, and his son was born five years later; the son lived to half the father’s age, and the father died four years after his son [Heath, 3].

Much of his work is lost. What remains is perplexing.

The Role of the Particular in Diophantus

To divide a given number into two having a given difference. Given number 100, given difference 40 [Heath, 131].

Have a go at it. What two numbers add up to 100, and when you compare them, differ by 40? So begins the first book of Diophantus. Categorically similar puzzles round out the first book, increasing in complexity and difficulty.

The conventional wisdom in modern mathematics textbooks states that students learn by proceeding from the general concept to the particular application. Diophantus upends that convention. He proceeds from the particular to, well… where he proceeds to is a matter of scholarly debate. Many historians of mathematics relegate Diophantus to the silver age of classical mathematics: witty and playful, but with no general method to instruct us. Indeed, comparing Diophantus to Euclid reveals that Diophantus makes peculiarly particular arguments in a way that Euclid does not. If Euclid established that deductive proof was the currency of mathematics, Diophantus’s cheque bounces.

I want to suggest, though, that the particularity of Diophantus offers us something, and that he does offer a path towards general truths that merits his inclusion in the canon. Let’s take a look at how Diophantus explains that first puzzle:

Given number 100, given difference 40. Lesser number required x.

Therefore  2x + 40 = 100,

x = 30

The required numbers are 70, 30.

Terse. Laconic, even. An Algebra I teacher might not give him full credit, as he hasn’t “shown his work”. It might be more clear to say “(x) + (x+40) = 100” to show the relationship of the two numbers. Diophantus leaves out a few steps in reducing his equation. But the core of it presages the development of algebra several hundred years later. And the terseness offers opportunity for active and engaged reading. Could his method work for different numbers? What if the given number was 120 and the given difference 30?

In fact, Diophantus explicitly encourages that play with other particulars. By the fifth puzzle, Diophantus begins to add “necessary conditions” to his enunciation of the puzzle: “The latter given number must be such that it lies between the numbers arising when the given fractions respectively are taken of the first given number” [Heath 132]. The addition of that necessary condition makes no sense if we stick to the particular numbers Diophantus offers us: he has already chosen numbers that meet the condition. So he must intend for us to choose other numbers and play with his method. Even though Diophantus does not offer us a general method or solution the way Euclid does, he points us towards generality, and offers us the tools to seek it on our own.

The General in Diophantus

One way that I’ve used Diophantus with students is to gradually move from the particular to the general. We might take that first puzzle and try it with a few combinations of numbers: Given number 54, given difference 7, given number 12, difference 3, and so on. Eventually we might think to ourselves, what if we know the given number, but don’t know the given difference? What could we say then?

Well, we could look at where the given difference appears in Diophantus’s method. Maybe we can call the given difference d, and we’d have (x) + (x + d) = 100 or 2x + d = 100 . And maybe we can call the given number n, which would give us (x) + (x + d) = n or 2x + d = n. Employing the algebraic tools that Diophantus uses, we could rewrite that  x = n/2 – d/2 and we’ve got a general solution that reveals something quite interesting: if we start in the middle and subtract half the difference we get the lesser number, and if we do the same but add half the difference we get the greater number. That quite abstract idea emerges naturally from playing with the particular that Diophantus offers.

Diophantus in the Good Life

What are we to make of Diophantus, then? First, he offers us a series of puzzles to play with and methods to resolve them. He encourages us to start with particularity and seek out generality in an organic way. That vision, starkly different from modern textbooks, offers a playful door into more advanced mathematics. The first book, in particular, has a very low floor: anyone who has learned to add, subtract, multiply, and divide in elementary school can engage with it and enjoy his puzzles.

Diophantus offers no practical use for his puzzles: they are not “word problems”. He won’t help us build a house or manage our finances. But he will offer us a window into a world of imagination and play. He reminds me most of the poetae novi, Catullus, Horace, and their kin. Those “new poets” emphasized wit and playfulness, tackling subjects more earthy and trivial than the epic poets. Homer and Virgil invite you to ponder the ruin of cities and nature of the gods. Catullus invites you to play with a napkin thief. Diophantus, I think, writes in the spirit of the latter. His goal is playful cleverness, and he delivers on it admirably.

What if I want to find three numbers such that the sum of any chosen pair exceeds the left-out third by a given number? Diophantus has the answer.


[Thomas L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra]

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