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Impearls: Archimedes and the Infinite

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Earthdate 2002-12-06

Archimedes and the Infinite

To paraphrase Yoda in George Lucas's Star Wars:  “Always in motion is the past.”

It might be thought that our knowledge of the ancient world (at least the better known periods and aspects of it) would have mostly shaken into a settled shape by now.  It's true that much information about the past is well and reliably known; in vast areas of concern, however (intellectual history, as an example), there remain crucial gaps, some of which are only now being partially filled in, with details sometimes importantly different from what had been presumed to be there before.

Mathematician and philosopher of science Jacob Bronowski put it like this: 1

It is absurd to ask why the future should turn out to chime with our knowledge of the past.  This puts the question upside down and makes nonsense of it.  What we have learnt from the past is knowledge only because the future proves it to be true.

This principle is exemplified in the appreciation of one of antiquity's most brilliant minds, Archimedes (one of a handful through history whose achievements may be said to lie on a par with those of modern giants Newton and Einstein).  The story of the recovery of Archimedes' great work Method of Mechanical Theorems in 1906 after more than 2,000 years is remarkable enough, but the tale is not yet ended!  Now after nearly another 100 years further progress has recently been made, and the results are illuminating.
 

Life and work

Born circa 290-280 B.C. in the Greek city of Syracuse in Sicily, dying in the Roman sack of that city in 212-211 B.C. during the Second Punic War, Archimedes is most famous today as the great ancient Greek inventor and mathematician.  In antiquity, Archimedes was also well known as a superb astronomer.

In a recent piece, “Proof, Amazement, and the Unexpected” in the journal Science, Stanford professor Reviel Netz characterizes Archimedes' mathematical contributions. 2

The bath anecdote does not give us the true measure of the man.  In On Floating Bodies, Archimedes made the following, astonishingly subtle deduction:  In a stable body of liquid, each column of equal volume must have equal weight; otherwise, liquid would flow from the heavier to the lighter.  The same must hold true even if some solid body is immersed in such a column of liquid.  In other words, if we have a column of liquid with a solid body immersed in it, the aggregate weight of the liquid and the body must be equal to that of a column of liquid of the same total volume.  It follows that the immersed body must lose weight: it must lose a weight equal to the weight of the volume of water it has displaced.  (This is why we feel lighter in the bath.)  This fundamental theorem was proved by Archimedes, with perfect rigor, in On Floating Bodies, Proposition 7.  Now that's something to cry “eureka” about.

Austere and technical as they are, Archimedes' treatises are just as striking as the anecdotes about him.  In the treatises three motives run together: proof, amazement, and the juxtaposition of the unexpected.  Proof and amazement are related, because Archimedes amazes us by proving that something very surprising is in fact true.  Amazement and the juxtaposition of the unexpected are related, because the amazing result is usually seen in the equality or equivalence of two seemingly separate domains.

Archimedes very rarely makes arguments that merely appear intuitive — and, crucially, when he does, he says so explicitly.  He sets out as postulates some very subtle assumptions.  For instance, in the introduction to the First Book on Sphere and Cylinder, Archimedes asserts that if two lines are concave to the same direction, and one encloses the other, the enclosing line is greater than the enclosed and so, for instance, the line is the shortest distance between two points.  He took enormous care to distinguish what can be proved from what cannot.  By turning seemingly obvious observations into explicit postulates, Archimedes was able to set out truly incontrovertible proofs.

Books such as Euclid's Elements have come down to us by way of numerous Greek and Arabic manuscripts, but, as Carl Boyer points out in his excellent History of Mathematics, the connecting link to Archimedes' works is thin.  Boyer writes: 3

Almost all copies are from a single Greek original which was in existence in the early sixteenth century and itself was copied from an original of about the ninth or tenth century. … There have been times when few or even none of Archimedes' works were known.  In the days of Eutocius, a first-rate scholar and skillful commentator of the sixth century, only three of the many Archimedean works were generally known….
 

Misconceptions of his work — and Archimedes' Method

The dearth in availability of Archimedes' works over much of the past two millennia has, as one might expect, led to errors in the appreciation of the body of his work.  Boyer writes:

His other treatises are gems of logical precision, with little hint of the preliminary analysis that may have led to the definitive formulations.  So thoroughly without motivation did his proofs appear to some writers of the seventeenth century that they suspected Archimedes of having concealed his method of approach in order that his work might be admired the more.

How unwarranted such an ungenerous estimate of the great Syracusan was became clear in 1906 with the discovery of the manuscript containing The Method.  Here Archimedes had published, for all the world to read, a description of the preliminary “mechanical” investigations that had led to many of his chief mathematical discoveries.  He thought that his “method” in these cases lacked rigor, since it assumed an area, for example, to be a sum of line segments.

The Method, as we have it, contains most of the text of some fifteen propositions sent in the form of a letter of Eratosthenes, mathematician and librarian at the university of Alexandria.  The author opened by saying that it is easier to supply a proof of a theorem if we first have some knowledge of what is involved; as an example he cites the proofs of Eudoxus of the cone and pyramid, which had been facilitated by the preliminary assertions, without proof, made by Democritus.  Then, Archimedes announced that he himself had a “mechanical” approach that paved the way for some of his proofs.  The very first theorem that he discovered by this approach was the one on the area of a parabolic segment; in Proposition 1 of The Method the author describes how he arrived at this theorem by balancing lines as one balances weights in mechanics.

Gerald Toomer of Brown University assesses Archimedes' Method, as it was known during the twentieth century. 4

Method Concerning Mechanical Theorems describes the process of discovery in mathematics.  It is the sole surviving work from antiquity and one of the few from any period that deals with this topic.  In it Archimedes recounts how he used a “mechanical” method to arrive at some of his key discoveries, including the area of a parabolic segment and the surface area and volume of a sphere.  The technique consists of dividing each of two figures, one bounded by straight lines and the other by a curve, into an infinite but equal number of infinitesimally thin strips, then “weighing” each corresponding pair of these strips against each other on a notional balance, and summing them to find the ratio of the two whole figures.  Archimedes emphasizes that, though useful as a heuristic method, this procedure does not constitute a rigorous proof.
 

The Method: discovery and recovery, lost and found again

The story of the recovery of Archimedes' Method of Mechanical Theorems is a terrific example, in my view, of how our comprehension of the past can even at this late date be dramatically changed by new discoveries, which may hinge on the merest chance.  Quoting Boyer again:

The work containing such marvelous results of more than 2000 years ago was recovered almost by accident in 1906.  The indefatigable Danish scholar J. L. Heiberg had read that at Constantinople there was a palimpsest of mathematical content.  (A palimpsest is a parchment the original writing on which as been only imperfectly washed off and replaced with a new and different text.)  Close inspection showed him that the original manuscript had contained something by Archimedes, and through photographs he was able to read most of the Archimedean text.  The manuscript consisted of 185 leaves, mostly of parchment but a few of paper, with the Archimedean text copied in a tenth-century hand.  An attempt — fortunately, none too successful — had been made to expunge this text in order to use the parchment for a Euchologion (a collection of prayers and liturgies used in the Eastern Orthodox Church) written in about the thirteenth century.  The mathematical text contained On the Sphere and Cylinder, most of the work On Spirals, part of the Measurement of a Circle and of On the Equilibrium of Planes, and On Floating Bodies, all of which have been preserved in other manuscripts; most important of all, the palimpsest gives us the only surviving copy of The Method.

In a sense, the palimpsest is symbolic of the contribution of the Medieval Age.  Intense preoccupation with religious concerns very nearly wiped out one of the most important works of the greatest mathematician of antiquity; yet in the end it was medieval scholarship that inadvertently preserved this, and much besides, which might otherwise have been lost.

No sooner had Archimedes' Method been almost miraculously recovered in 1906, but it was lost again (or stolen), disappearing for most of the rest of the century.  Fortunately for Archimedean scholarship of the twentieth century, the palimpsest had been photographed before being lost, allowing Heiberg to perform his remarkable feat. 

In 1998 (!) the nearly priceless document reappeared at a New York auction house.  Netz chronicles the manuscript's recent history:

Archimedes' treatise on the Method of Mechanical Theorems, which itself tends to turn up in unexpected places, was his most remarkable work.  It was lost until the great philologist Heiberg discovered it in a palimpsest (a scraped and overwritten parchment) in Istanbul in 1906.  Heiberg had discovered a 10th-century copy of the treatise, which had been used as the fabric for a 13th-century prayer-book.  Heiberg was able to read much, but not all of the faint traces.  Shortly after this astonishing discovery, the manuscript was lost or stolen, but in 1998 it resurfaced at a Christie's auction sale at New York.  It sold for two million dollars.  The anonymous owner generously supports the conservation and imaging now taking place at the Walters Art Museum, Baltimore. 5
 

Archimedes and the Infinite

Netz proceeds to the heart of the matter: the remarkable results recently obtaining from the re-recovery of the original manuscript in 1998.

As we should expect of Archimedes, the results of our recent research on the palimpsest are indeed unexpected.

Since 1906, it has been known that in the Method of Mechanical Theorems, Archimedes combined concepts of straight, curved, physical, and geometrical.  Above all, anticipating the calculus, he combined finite and infinite….

So much we have known for a century.  In a visit to Baltimore in 2001, Ken Saito from Osaka Prefecture University and I examined a hitherto unread piece of the Method of Mechanical Theorems.  We could hardly believe our eyes:  It turned out that Archimedes was looking for rigorous ways of establishing the calculus.

Modern scholarship always assumed that mathematics has undergone a fundamental conceptual shift during the Scientific Revolution in the 16th century.  It has always been thought that modern mathematicians were the first to be able to handle infinitely large sets, and that this was something the Greek mathematicians never attempted to do.  But in the palimpsest we found Archimedes doing just that.  He compared two infinitely large sets and stated that they have an equal number of members.  No other extant source for Greek mathematics has that.
 

References

1 J. Bronowski, The Common Sense of Science, Harvard University Press, Cambridge, Massachusetts, 1963; pp. 117-118.

2 Reviel Netz (Assistant Professor of Classics, Stanford University), “Proof, Amazement, and the Unexpected” (link requires subscription or pay per view), Science (1 Nov 2002), Vol. 298, No. 5595, pp. 967-968.

3 Carl B. Boyer (Professor of Mathematics, Brooklyn College), A History of Mathematics, Second Edition, revised by Uta C. Merzbach, John Wiley & Sons, Inc., New York, 1991, ISBN 0-471-09763-2 or 0-471-54397-7 (pbk); pp. 136-137, 139.

4 Gerald J. Toomer (Professor of the History of Mathematics, Brown University, Providence, Rhode Island), “Archimedes,” Encyclopædia Britannica, CD 1997, Encyclopaedia Britannica, Inc.

5 See “Eureka! Archimedes Palimpsest at the Walters Art Gallery.”



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