The outline of the argument of the book has now been repeated several times. Hellenistic Greek mathematical practice focused on the features of the individual proof, trying to isolate it and endow it with a special aura. Thus the characteristic object of Hellenistic Greek mathematics is the particular geometrical configuration. Medieval mathematical practice focused on the features of systems of results, trying to bring them into some kind of order and completion. Thus the characteristic object of medieval mathematics is the second-order expression. In a particular geometrical configuration, the mathematician foregrounds the local, qualitative features of spatial figures. In a second-order expression, the mathematician foregrounds the global, quantitative features of mathematical relations. Thus, Hellenistic Greek mathematics – the mathematics of the aura – gave rise to the problem; medieval mathematics – the mathematics of deuteronomy – gave rise to the equation.

The comparison between the two kinds of mathematics is at its starkest when we compare Hellenistic Greek mathematics directly with advanced Arabic mathematics. This comparison is useful, then, to get a sense of the nature of the transformation. But, to look for the historical account for this transformation, we have concentrated in this book on a more subtle comparison. In this book, I have given much attention to the transitional stage of Late Antiquity, already different from Classical Hellenistic mathematics, though in ways that are less obvious. In the work of Eutocius, we saw suggestions of the direction ahead.

In this chapter I discuss the Archimedean problem in its first, “Classical” stage. In section 1.1, I show how it was first obtained by Archimedes and then, in 1.2, I offer a translation of the synthetic part of Archimedes' solution. Following that, section 1.3 makes some preliminary observations on the geometrical nature of the problem as studied by Archimedes. Sections 1.4 and 1.5 follow the parallel treatments of the same problem by two later Hellenistic mathematicians, Dionysodorus and Diocles. Putting together the various treatments, I try to offer in section 1.6 an account of the nature of Ancient geometrical problems. Why were the ancient discussions geometrical rather than algebraic – why were these problems, and not equations?

The problem obtained

In his Second Book on the Sphere and Cylinder, Archimedes offers a series of problems concerning spheres. The goal is to produce spheres, or segments of spheres, defined by given geometrical equalities or ratios. In Proposition 4 the problem is to cut a sphere so that its segments stand to each other in a given ratio. For instance, we know that to divide a sphere into two equal parts, the solution is to divide it along the center, or, in other words, at the center of the diameter. But what if want to have, say, one segment twice the other?

Does mathematics have a history? I believe it does, and in this book I offer an example. I follow a mathematical problem from its first statement, in Archimedes' Second Book on the Sphere and Cylinder, through many of the solutions that were offered to it in early Mediterranean mathematics. The route I have chosen starts with Archimedes himself and ends (largely speaking) with Omar Khayyam. I discuss the solutions offered by Hellenistic mathematicians working immediately after Archimedes, as well as the comments made by a late Ancient commentator; finally, I consider the solutions offered by Arab mathematicians prior to Khayyam and by Khayyam himself, with a brief glance forward to an Arabic response to Khayyam.

The entire route, I shall argue, constitutes history: the problem was not merely studied and re-studied, but transformed. From a geometrical problem, it became an equation.

For, in truth, not everyone agrees that mathematics has a history, while those who defend the historicity of mathematics have still to make the argument. I write the book to fill this gap: let us consider, then, the historiographical background.

My starting point is a celebrated debate in the historiography of mathematics. The following question was posed: are the historically determined features of a given piece of mathematics significant to it as mathematics? This debate was sparked by Unguru's article from 1975, “On the Need to Re-write the History of Greek Mathematics”.

In this chapter we concentrate on the fate of Archimedes' problem in one eminent work of Arabic science: Omar Khayyam's Algebra (eleventh to twelfth centuries). (This is, of course, the same Omar Khayyam famous for his Persian poetry; here we concentrate on his science.)

As we shall see below, this decision to focus on Khayyam is to a certain extent arbitrary: the problem had a significant history in the Arabic world before and after Khayyam. He does occupy a special position in the history of the problem. Our knowledge of Arabic treatments prior to him is in some cases derived from him alone (much as we know of early Greek treatments of the problem through the work of Eutocius). And while the later history of the problem adds much that is mathematically valuable, we can usefully end our survey with Khayyam. With him, as we shall see, the route from problems to equations is largely completed. It is also helpful to compare like with like: and it is therefore appropriate to have our survey – begun with the genius of Archimedes – end with the genius of Khayyam.

Our goal in this chapter, then, is to show that Khayyam's mathematics already differs essentially from Archimedes'. This should be a deep conceptual divide, along the lines suggested by Klein and Unguru. We also need to show the historical basis for this divide, in terms of changes in the practice of mathematics from the world of Archimedes to the world of Khayyam.

The texts we have read so far come not from works extant under the names of Archimedes, Dionysodorus, or Diocles. They were handed down in a single work, extant under the name of a relatively obscure scholar: Eutocius of Ascalon. In the sixth century ad, Eutocius wrote a series of mathematical commentaries, of which one, the commentary to Archimedes' Second Book on the Sphere and Cylinder, is especially rich in mathematical and historical detail. Having reached Proposition four, Eutocius noted the lacuna in Archimedes' reasoning. He has (so he tells us) uncovered Archimedes' original text, which he then incorporated into his commentary. Finally, he added into it the solutions by Dionysodorus and Diocles. This, then, is our main source for the ancient form of the problem (we also happen to have the same solution by Diocles, preserved in Arabic translation).

Was Eutocius' work a mere record of the past, or did it make some original contribution to the history of mathematics? In this chapter, I argue that, already in the work of Eutocius, we can find mathematics making the transition from problems to equations. This comes at seemingly trivial moments, of little consequence in terms of their original mathematical contribution. Eutocius, without noticing this, occasionally happens to speak of mathematical objects that are rather like our quantitative, abstract magnitudes, and not the spatial geometrical objects studied by Classical mathematicians. He stumbles across functions and equations, without ever thinking about it.

/Introduction/

Archimedes to Dositheus: greetings

Earlier you sent me a request to write the proofs of the problems, whose proposals I had myself sent to Conon; and for the most part they happen to be proved through the theorems whose proofs I had sent you earlier: <namely, through the theorem> that the surface of every sphere is four times the greatest circle of the <circles> in it, and through <the theorem> that the surface of every segment of a sphere is equal to a circle, whose radius is equal to the line drawn from the vertex of the segment to the circumference of the base, and through <the theorem> that, in every sphere, the cylinder having, <as> base, the greatest circle of the <circles> in the sphere, and a height equal to the diameter of the sphere, is both: itself, in magnitude, half as large again as the sphere; and, its surface, half as large again as the surface of the sphere, and through <the theorem> that every solid sector is equal to the cone having, <as> base, the circle equal to the surface of the segment of the sphere <contained> in the sector, and a height equal to the radius of the sphere. Now, I have sent you those theorems and problems that are proved through these theorems <above>, having proved them in this book. And as for those that are found through some other theory, <namely:> those concerning spirals, and those concerning conoids, I shall try to send quickly.

Now that the proofs of the theorems in the first book are clearly discussed by us, the next thing is the same kind of study with the theorems of the second book.

First he says in the 1st theorem:

“Let a cylinder be taken, half as large again as the given cone or Arch. 188 cylinder.” This can be done in two ways, either keeping in both the same base, or the same height. And to make what I said clearer, let a cone or a cylinder be imagined, whose base is the circle A, and its height AΓ, and let the requirement be to find a cylinder half as large again as it.

(a) Let the cylinder AΓ be laid down, (b) and let the height of the cylinder, AΓ, be produced, (c) and let ΓΔ be set out <as> half AΓ; (1) therefore ΓΔ is half as large again as AΓ. (d) So if we imagine a cylinder having, <as> base, the circle A, and, <as> height, the line AΔ, (2) it shall be half as large again as the <cylinder> set forth, AΓ; (3) for the cones and cylinders which are on the same base are to each other as the height.

TO BOOK 1

As I found that no one before us had written down a proper treatise on the books of Archimedes on Sphere and Cylinder, and seeing that this has not been overlooked because of the ease of the propositions (for they require, as you know, precise attention as well as intelligent insight), I desired, as best I could, to set out clearly those things in it which are difficult to understand; and I was more led to do this by the fact that no one had yet taken up this project, than I was deterred by the difficulty; as I was also reasoning in the Socratic manner that, with god's support, most probably we shall reach the end of my efforts. And third, I thought that, even if, through my youth, something will strike out of tune, this will be made right by your scientific comprehension of philosophy in general, and especially of mathematics; and so I dedicate it to you, Ammonius, the best of philosophers. It would be fitting that you help my effort. And if the book seems to you slight, then do not allow it to go from yourself to anyone else, but if it has not strayed completely off the mark, make your view upon it clear for, if it comes to be established by your own judgment, I shall try to explicate some other of the Archimedean treatises.