Where were the Stoics in the late Middle Ages? The short answer is: everywhere and nowhere.

Stoicism is not a sport for gentlemen; it requires far too much rigorous intellectual work. Most of Western history consists of gentlemen's centuries. But there were the couple of centuries, the fourth and the third b.c., in which the ancient philosophical schools were created, and there were the three centuries from a.d. 1100 to 1400, when medieval scholasticism flourished – centuries that produced a considerable number of tough men ready to chew their way through all the tedious logical stuff that disgusts a gentleman and to make all the nice distinctions that a gentleman can never understand but only ridicule, distinctions necessary to work out a coherent, and perhaps even consistent, picture of the structure of the world. In this respect, a good scholastic and a good Stoic are kindred spirits. As an attitude to doing philosophy, Stoicism is everywhere in the late Middle Ages.

Also, bits and snippets of Stoic doctrine were available in a large number of ancient writings and could inspire or be integrated in scholastic theory. Further, some works with a high concentration of Stoicism were widely known, notably Cicero's Paradoxes of the Stoics, De Officiis, and Seneca's Letters to Lucilius. Finally, Saint Paul was a crypto-Stoic ethicist, and so were several of the church fathers.

Descartes wrote little in the way of moral philosophy, but he regarded the topic as of the utmost importance. As he describes it in the preface to the French edition of his Principles of Philosophy, the uppermost branch of the tree of philosophy is occupied by la morale, “the highest and most perfect moral system, which presupposes a complete knowledge of the other sciences and is the ultimate level of wisdom” (AT IXB 14/CSM I 186). This moral system would be more than just the final element of Descartes' philosophy; it also defines the end, or telos, of his ordered reconstruction of knowledge. In an important sense, the prior, theoretical parts of his philosophy are established for the sake of the practical benefits that follow from them. To those skeptical about what philosophy has to offer, Descartes confidently replies that the difference between his principles and those of other philosophers, “as well as the long chain of truths that can be deduced from them, will finally make them realize how important it is to continue in the search for these truths, and to what a high level of wisdom, and to what perfection and felicity of life, these truths can bring us” (AT IXB 20/CSM I 190).

Descartes is usually seen as the quintessential modern philosopher, yet in his broader conception of philosophy's goal we find repeated a central ancient theme: philosophical knowledge is valued not only for its own sake, but as the basis of the best sort of human life – one in which we realize the greatest perfection and happiness.

A successful rehabilitation of Stoic ethics will have to defeat the idea that there is something deeply wrong, and perhaps even psychologically impossible, about the kind of emotional life that Stoics recommend. The image of the austere, dispassionate, detached, tranquil, and virtually affectless sage – an image destined to be self-refuting – has become a staple of anti-Stoic philosophy, literature, and popular culture. It has been constructed from incautious use of the ancient texts and is remarkably resistant to correction. Reminders that the ancient Stoics insisted that there are good emotions are typically brushed aside by asserting that the ancient catalog of such emotions is peculiar; that the emotions in even that peculiar catalog are not accorded much significance by Stoics; and that the ruthless emotional therapy practiced by Epictetus is a reliable guide to the sort of emotional life Stoics want all of us to cultivate – namely, a life of desiccated affect and discardable attachments.

Both Stoics and anti-Stoics alike have developed an unwholesome fascination with a picture of the Stoic sage drawn for extreme circumstances. We persist, in high art and low journalism, in telling and retelling stories of good people who resolutely endure horrors – injustice, torture, disease, disability, and suffering. Those of us who are attracted to Stoicism often find such stories inspiring, and even anti-Stoics give them grudging admiration. But our fascination with them can be seriously misleading.

THE STATESMEN'S BIBLE

A child born this year in the United States has a life expectancy of 76.4 years. A child born in Sierra Leone can expect to live 34.7 years. Most adults in the United States and Europe are literate, although illiteracy remains a disturbing problem, correlated with poverty. Some developing countries attain nearly our overall rate of literacy: Sri Lanka, for example, has 90 percent adult literacy, the Philippines 94.6 percent, Jordan 86.6 percent. In many nations, however, a person's chance of learning to read (and, hence, to qualify for most well-paying jobs) is far lower. In India, only 37 percent of women and 65 percent of men are literate, in Bangladesh 26 percent of women and 49 percent of men, in Niger 6 percent of women and 20 percent of men. Clean water, health services, sanitation, maternal health and safety, adequate nutrition – all these basic human goods are distributed very unevenly around the world. The accident of being born in one country rather than another pervasively shapes the life chances of every child who is born. Being female, being lower-class, living in a rural area, and membership in an ethnic or racial or religious minority also affect life chances within every nation. But, on the whole, differences of wealth and opportunity among nations eclipse these differences.

The Stoic Discourses of Epictetus are conspicuously marked throughout by the figure of Socrates. No other philosopher, not even Zeno or Diogenes, is named nearly so frequently. Epictetus views Socrates as the single figure who best authorizes and exemplifies everything he is trying to give his students in terms of philosophical methodology, self-examination, and a life model for them to imitate. This strikingly explicit coincidence between Epictetus' objectives and Socrates makes the Stoicism of the Discourses particularly distinctive.

In order to take the measure of this point, we need to start from the role of Socrates in the preceding Stoic tradition. The earliest Stoic philosophers had drawn so heavily on Plato's and, to a lesser extent, Xenophon's Socrates that members of the school were happy to be called Socratics. The details cover numerous Stoic doctrines in ethics, moral psychology, and theology, including the priority of the soul's good over everything else, the unity of the virtues, the identity of virtue with knowledge, and divine providence.

In 1949 Max Pohlenz, the doyen of early twentieth-century German scholarship on Stoicism, published an article, “Paulus und die Stoa,” in which he discussed the first few chapters of the apostle's letter to the Romans and the Christian historian Luke's account of Paul's speech on the Areopagus in chapter 17 of the Acts of the Apostles. Pohlenz was asking about the Stoic credentials of various ideas in the two texts. He concluded that in Paul there was nothing that went directly back to Stoicism. Instead, any Stoic-sounding ideas had come to Paul through Jewish traditions that would rather reflect some form of middle Platonism. In Luke, by contrast, there is a direct reminiscence of Posidonius.

In 1989 Abraham J. Malherbe, Buckingham Professor of the New Testament at Yale Divinity School, published a book, Paul and the Popular Philosophers. He argued that in a number of individual passages in the letters, Paul was interacting directly with specific motifs derived from the moral exhortation of philosophers like the Cynics, Stoics, and Epicureans. Paul need not have read, for instance, Chrysippus. But he had an easy familiarity with the moral discourse of the “popular philosophers” of his own time, as exemplified to us by his near contemporaries Seneca, Dio Chrysostom, and Epictetus.

In Paul and the Stoics (2000), I argued that Paul is relying on central ideas in Stoicism even when he states the core of his own theological thought. This development should be of some interest to students of Stoicism.

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.