/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.

GOAL OF THE TRANSLATION

The extraordinary influence of Archimedes over the scientific revolution was due in the main to Latin and Greek–Latin versions handwritten and then printed from the thirteenth to the seventeenth centuries. Translations into modern European languages came later, some languages served better than others. There are, for instance, three useful French translations of the works of Archimedes, of which the most recent, by C. Mugler – based on the best text known to the twentieth century – is still easily available. A strange turn of events prevented the English language from possessing until now any full-blown translation of Archimedes. As explained by T. L. Heath in his important book, The Works of Archimedes, he had set out there to make Archimedes accessible to contemporary mathematicians to whom – so he had thought – the mathematical contents of Archimedes' works might still be of practical (rather than historical) interest. He therefore produced a paraphrase of the Archimedean text, using modern symbolism, introducing consistency where the original is full of tensions, amplifying where the text is brief, abbreviating where it is verbose, clarifying where it is ambiguous: almost as if he was preparing an undergraduate textbook of “Archimedean Mathematics.” All this was done in good faith, with Heath signalling his practices very clearly, so that the book is still greatly useful as a mathematical gloss to Archimedes. (For such a mathematical gloss, however, the best work is likely to remain Dijksterhuis' masterpiece from 1938 (1987), Archimedes.)

/Introduction: general/

Archimedes to Dositheus: greetings.

Earlier, I have sent you some of what we had already investigated then, writing it with a proof: that every segment contained by a straight line and by a section of the right-angled cone is a third again as much as a triangle having the same base as the segment and an equal height. Later, theorems worthy of mention suggested themselves to us, and we took the trouble of preparing their proofs. They are these: first, that the surface of every sphere is four times the greatest circle of the <circles> in it. Further, 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 circle which is the base of the segment. Next to these, that, in every sphere, the cylinder having a base equal to the greatest circle of the <circles> in the sphere, and a height equal to the diameter of the sphere, is, itself, half as large again as the sphere; and its surface is <half as large again> as the surface of the sphere.

In nature, these properties always held for the figures mentioned above. But these <properties> were unknown to those who have engaged in geometry before us – none of them realizing that there is a common measure to those figures.

TRADITION AND INNOVATION

We now move from Part I to Part II of the dialogue. At 390d11–e5, Socrates and Hermogenes agree on the following proposition:

Hermogenes is not resistant, but needs this newly emerging principle of the correctness of names to be explained and illustrated to him. Socrates, reasserting his own ignorance of the topic, refuses the requested explanation, but offers in its place a joint investigation of the matter. This disavowal of knowledge is, not untypically of Plato's Socrates, partly ironic. In the event he will not just pour forth a veritable flood of illustrations, but will make it clear, by rejecting etymologies he disagrees with, that he is not entirely ignorant of the practice of etymology. Indeed, he will pretend to have been inspired by Euthyphro whom he says he heard – presumably etymologising – earlier this very morning, even though none of the actual etymologies he will propose or consider is attributed to Euthyphro. In fact the question of his familiarity or unfamiliarity with existing etymological practice is kept teasingly unsettled throughout.

INTRODUCTION

This chapter's title has a double reference. First, after completing his main etymological survey, Socrates sets out to halt a regress, which threatens to derive each name from component names, and those from sub-components, and so on ad infinitum. He does so by sketching a theory of primary, non-derivative sounds from which compound names are built up. The limit which he here sets is an entirely beneficent one, designed to save the etymological method from incoherence. Second, in Part III Socrates, in discussion now with Cratylus, scrutinises the method in order to see whether it can yield the truths which Cratylus believes can be extracted from words. This time the limits are negative: names can never be relied on as totally accurate depictions. But in no way does this latter limitation threaten to undermine etymology as an exegetical device, only as a philosophical one. So, at any rate, I shall argue, in this chapter and the final one.

HALTING THE REGRESS

At 421c, Socrates has at last completed his survey of the Greek philosophical vocabulary, emphasising above all the theme of fluidity. Hermogenes now quizzes Socrates about the small words which repeatedly recur in these etymologies, choosing, as suggested by the flux theme, the components ion, ‘going’, rheon, ‘flowing’, and, on the opposite side, doun, ‘binding’. These are not all monosyllables, and nothing excludes the possibility that they might be further dissected into their components.

THE PRINCIPLES OF ETYMOLOGY

We have seen Socrates nearing his final verdict on the nature–convention debate. His demonstration that names must rely on some degree of convention in order to succeed in signifying things was not, it turned out, any kind of vindication of Hermogenes' common-sense conventionalist thesis. It was simply one element in his refutation of Cratylus' thesis that names map onto reality with a perfect precision which makes their study the ideal guide to truth. It is that refutation of Cratylus, and the matching advocacy of a different route to truth, that will occupy us as we reach the dialogue's climax. But, on the way to that finale, one of my tasks will be to gather together the series of other Platonic lessons that the dialogue has brought to light.

First let me try to characterise Socrates' final position on the correctness of names. When Socrates argued that convention must play a part, he included the following words: ‘I myself too like the idea that so far as possible names do resemble their objects’ (435c2–3). The expressions ‘I … like the idea’ (literally ‘it pleases (areskei) me that …’) and ‘so far as possible’ have sometimes led to the impression that Socrates is either presenting little more than a pipedream (wouldn't it be nice if names really were vocal portraits?), or else more formally setting out the norms for an ideal language. There is actually no good reason to think so.

PLATO AND THE DIALOGUE

Why did Plato write dialogues? His motive for favouring this format has sometimes been construed as a kind of radical self-distancing: as the mere dramatist of the conversations rather than a participant in them, Plato enables himself to suppress his own authorial voice, avoiding any degree of commitment that might obviate further thought by himself or the reader. I am reluctant to go all the way with this. Plato is an overwhelming presence in his dialogues. Most of his readers over two and half millennia have found it hard not to speak of, think of, and criticise the ideas and arguments defended in the dialogues as Plato's own, and we too should feel no embarrassment about talking that way.

Plato's real reason for persisting with the dialogue form is, I think, a very different one, his growing belief – more than once made explicit in his later work – that conversation, in the form of question and answer, is the structure of thought itself. When we think, what we are doing is precisely to ask and answer questions internally, and our judgements are the outcome of that same process. Hence it seems that what Plato dramatises as external conversations can be internalised by us, the readers, as setting the model for our own processes of philosophical reasoning. More important still is the converse, that these same question-and-answer sequences can legitimately be read by us as Plato thinking aloud.

FLUX AND BEING

In his theological and cosmological etymologies, Socrates' very first choice among divine names is Hestia (401c1–d7).

Hestia's theological primacy is evident in her being the first deity you sacrifice to.

ETYMOLOGY IN THE PHILEBUS

I want to start this chapter from somewhere unexpected – not Plato's Cratylus, but his late dialogue the Philebus. In the opening part of this dialogue, Socrates recommends a method which he calls ‘a gift to mankind from the gods’, maybe transmitted, along with fire, ‘through some Prometheus’, to our forebears, who were themselves superior to us and lived closer to the gods (16c5–8). The Prometheus in question has long been suspected of being Pythagoras, and at all events the method, as sketched by Socrates, is likely to be of Pythagorean inspiration. But the allusion to the mythical figure Prometheus, especially as portrayed by Aeschylus in Prometheus Bound, remains direct and significant. For Aeschylus' and Plato's Prometheus have it in common that they both passed to mankind, along with the gift of fire, all of the arts, prominently including the understanding of number.

The gift transmitted by Plato's Prometheus is based on number in the following way: between the single genus from which a scientific investigation might start, and its infinite range of individual members, the true scientist will be concerned above all with systematic enumeration of the intervening kinds or species. Asked for an explanation, Socrates illustrates the method with three examples, all concerned with the classification of sounds. The first, literacy, need not detain us now, but the second, musical expertise, deserves close attention (17c11–e6).