Oliver Thomas looks at what ancient Greek Mathematics looks like in a modern setting.

Which ideas from ancient Greece does everyone in the UK study in school? The strongest claim, to my mind, belongs to the theorems that have found a place in the compulsory maths curriculum, such as the basics of prime factorisation, ratios and geometry from Euclid’s Elements, or Archimedes’ formulas for the volume and surface area of the sphere. Here one may be strongly tempted to let the strangeness of the ancient world fade from view because of the dazzle of these discoveries more durable than marble. But the superficial appearance of similarity can be misleading, as with any other texts and ideas from antiquity.

A good example is the mathematics of Diophantus, who worked in Egypt under the Roman empire (perhaps 2nd or early 3rd century CE). His main work is the Arithmetika, the Greek root of our word ‘arithmetic’, which originally meant ‘Things related to finding values’. The work was designed to help students solve problems for which we would use algebra today. And as a result Diophantus was studied by many of the great mathematicians of the medieval Arab world and early modern Europe. Fermat’s Last Theorem started life when Fermat was scribbling in the margins of his copy of the Arithmetika, around 1640. Even today, ‘Diophantine equations’ are named after him. But his way of writing out mathematics, and even his whole way of thinking about solutions, were very different from current ideas.

In algebra, we learn how to use any letter as a symbol for an unknown quantity, and using subscripts like x1, x2, y1, y2, … we can have as many of these symbols as we could need. Diophantus, by contrast, operated with just a single named unknown: in manuscripts the symbol most often looks like a final sigma (ς) or a tall y, and was a shorthand for ‘(the) number’, which is often written out in full too. This image has an empty alt attribute; its file name is picture1.jpg

­Fig. 1: Arithmetika 2.9 in the oldest manuscript (Madrid, Biblioteca Nacional de Espana 4678), with lots of annotation by medieval readers. Notice how different this looks on the page from a modern mathematical proof, which is mostly symbols. © BNE.

Moreover, mathematicians today might use these symbols to stand for any positive or negative number, or for stranger things like ‘complex numbers’ whose properties they have had to define specially. By contrast, Diophantus gave primacy just to positive whole numbers (now called ‘natural’ numbers, because nature presents us with groups of 1, 2, 3 etc.), and the ratios between them – only these count as ‘numbers’ in his solutions. Collectively, they are known as positive rational numbers.

As a result of having only one unknown symbol, and a more limited idea of ‘numbers’, Diophantus’ conception of what a problem looks like was necessarily different too. Algebra teaches us how to deal systematically with equations that have any number of unknown symbols in them, with the aim of finding as many solutions as possible. But Diophantus almost always set problems where the challenge is just to find one solution, always involving positive rational numbers.

The key tool he has for achieving this is the cunning substitution. Diophantus looks for special cases, where all sorts of extra relationships between the unknown quantities are assumed. These assumptions wouldn’t get you any marks in an algebra exam today, but they enabled Diophantus to translate each problem entirely into the terms of his one named unknown, and therefore to find a special solution.

Let’s see how this works with the following example (Arithmetika 2.9): ‘Redivide a given number, which is composed of two squares, into two other squares.’ After this general form, Diophantus specifies the given number to be 13 = 2x2 + 3x3. We are asked to find a pair of squares of positive rational numbers other than (4, 9) whose sum is 13.

The basic approach is to fill in each gap in ‘13 = ___ squared + ___ squared’ with an expression in terms of the named unknown, which we’ll write as ς, in such a way that we can be confident of reaching a solution. How do we do this?

If we replace the gaps for example with ς and ς+1, we will end up with 13 = ς2 + (ς2+2ς+1), a quadratic equation in modern terms. Diophantus did know how to solve these by doing what we now call ‘completing the square’. But at this fairly early stage in his work he is teaching us hacks to make the solution as easy as possible. And one of his key hacks is that one should try to make everything cancel except for the multiples of two neighbouring powers of ς. In our case, that means designing substitutions so that 13 appears to the right of the equals sign.

We use the fact that 13 = 22 + 32 to our advantage. Replace the gaps instead with ς+2 and ς-3, where ς is non-zero. Mathematically, there is no reason to think that the numbers have to be 5 apart, and if our task were to find lots of solutions, it would be silly to make such an assumption. But remember, we only need to find one solution. These substitutions give us 13 = (ς2+4ς+4) + (ς2-6ς+9) = 2ς2-2ς+13, so we cancel the 13 and find 2ς2 = 2ς, from which we can infer that ς = 1.

This is progress, but we actually hit a stumbling-block. When we plug this value back in, we just return to the solution 13 = 9+4, and we were challenged to find a different pair of squares. Luckily, we can avoid this while still ensuring that 13 gets cancelled, simply by tweaking our attempted substitution ς-3 to 2ς-3. If you rerun the working, you should get 13 = 5ς2-8ς+13, so ς = 8/5, and 13 = (18/5)2 + (1/5)2.

Diophantus’ technique of substitution is somewhat hit-and-miss. But it is also remarkably powerful. For one thing, there is nothing special about the number 13 here. If we had to split say 61 = 5x5 + 6x6 into two different squares, the substitutions ς+5 and 2ς-6 would work. And Diophantus also knew that his method was flexible: 2ς-6 could be replaced by 3ς-6 to generate a different solution.

If we shift back into modern notation, as a tool for evaluating Diophantus’ method, he proposed that we solve a2+b2 (where a and b are given) = x2+y2 (where x and y are unknown positive rationals) by substituting x = ς+a, y = cς-b, where c is a rational constant that we can choose with some restrictions. In our main example, we had a=2, b=3; c=1 did not work, but c=2 did. If we multiply out we get a2+b2 = (1+c2)ς2 + 2(a-bc)ς + a2+b2, leading to ς = 2(bc-a)/(c2+1), i.e. a2+b2 = [(2bc-a+ac2)/(c2+1)]2 + [(bc2-2ac-b)/(c2+1)]2. Unless (2bc-a+ac2)/(c2+1) = ±a or ±b, we’ll get a viable solution.

And in fact Diophantus’ method produces every possible solution to his problem given the right choice of c. This can be seen using another modern perspective. x2+y2 = 13 can be represented as a graph, namely the circle centred on the origin that passes through the point (2, 3). We are looking for points on the upper right quadrant of this circle (where x and y are positive) with rational coordinates. The substitutions x = ς+2, y = cς-3, for various values of c, can also be represented on the graph: since ς = x-2, we have y = cx – (2c+3), which is a straight line of gradient c; and when ς=0 these lines all pass through (2, -3).

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Fig. 2: The graph x2+y2 = 13 (green), with point A = (2, -3) and the disallowed solutions B1 = (2,3) and B2 = (3,2); the blue line corresponds to c=3, i.e. the substitutions ς+2, 3ς-3, meeting the circle at C at (17/5, 6/5), showing that (17/5)2 + (6/5)2 = 13. Graph created using www.geogebra.org.

So we have a family of substitutions, each of which, when mapped on a graph, involves setting out from the point (2, -3) on our circle, and proceeding at a rational gradient (c) until we meet the circle again. We have seen that that point will have rational coordinates – already a surprising fact. And conversely, every acceptable point (x, y) on the circle will be hit by setting out from (2, -3) on one of those rational gradients, namely (y+3)/(x-2).

Diophantus’ method has enough built-in flexibility to provide every solution to x2+y2 = 13 where x and y are positive rationals, but it takes the language of later algebra to describe this set of solutions in general terms. The readiness with which Diophantus prompts this sort of updating has secured his legacy, and many people read Diophantus’ Arithmetika today because they want to see the aspects of his work that are mathematically ‘meaty’ by current standards. That has led to a tradition of translating Diophantus, and other ancient mathematicians, into modern notation and terminology. Of course, I’ve done that in parts of this article, while also drawing attention to how this alters Diophantus’ own conceptions. (But I couldn’t include every aspect: for example how Diophantus thinks and writes about square numbers is its own issue.)

By contrast, many historians of mathematics insist that any translation into modern notation is bound to deform what was distinctive about the original thinking. Some refuse even to use symbols like = or +. Other translators adopt various middle courses, such as deciding that + corresponds well to the Greek word that means ‘together with’, but that one shouldn’t introduce extra symbols for constants or variables. Factor in debates about whether or not an ancient concept really corresponds to a modern one, and you can imagine the variety of possible approaches.

Mathematical texts, then, certainly pose the age-old translation problems: to modernise or not, and if so, how? As usual, there isn’t a single right answer, other than meeting the needs of a real target audience. But with maths those potential audience needs are very varied, to the point where extremely literal translations are a respectable practice. By contrast, classicists translating literary, historical or even philosophical texts take it for granted that some degree of compromise is necessary between faithfulness to literal meaning and faithfulness to how the original audience perceived the text’s style, which didn’t involve being constantly bombarded by awkwardness.

A third important aspect of ‘faithfulness’ to the original is capturing its practical effect on the audience. What I mean might be clearer if we consider for a moment the choices faced by a translator of ancient comedy. One pull is to capture the meaning of the original. Another is to capture the tone of each line (chatty, coarse, mock-serious… whatever it might be). A third is to create a translation that will cause a modern audience to laugh – and that might well mean introducing modern (anachronistic) jokes.

Returning to Diophantus, it is curious how his original target audience – school-level students, who know some of the basic operations but aren’t experts yet – is poorly served by most translations. The best edition of the Arithmetika comes with a double translation, one as literal as possible, and the other into the notation and concepts of university-level algebraic geometry. Neither approach is a serious attempt to capture Diophantus’ pedagogical aims and efficacy.

In my view, this is a meaningful thing to attempt with Diophantus’ work, and that of many other Greek mathematicians. Their work can still be useful, but not without being connected to the modern curriculum. However, that doesn’t mean neglecting their historical strangeness completely. In Diophantus’ case, treating his work by the standards of modern algebra produces rather repetitive problems, with partial solutions whose failings any teacher would have to waste class-time explaining. However, focusing on his artistry with the simplifying substitution may be more promising. The way Diophantus thought laterally about what he could achieve with his toolkit, especially by finding clever ways to simplify a problem, is still a useful mathematical skill, occasionally in algebra (especially inequalities), but particularly in calculus and combinatorics. And with the aims of Classics for All in mind, I think this is the most promising approach for attracting a wide range of school students to the intriguing and impressive achievements of Classical science, and for justifying some expansion of where the ancient world features in the curriculum.

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Oliver Thomas is Assistant Professor of Classics at the University of Nottingham. His edition of the Homeric Hymn to Hermes was published last year by Cambridge University Press. He is currently working on a book about Diophantus of the kind outlined above, and would love to hear from anyone interested in incorporating ancient maths in UK schools. Contact him at: Oliver.Thomas@nottingham.ac.uk