Julian Havil is the author of Gamma: Exploring Euler’s Constant, Nonplussed!: Mathematical Proof of Implausible Ideas, Impossible?: Surprising Solutions to Counterintuitive Conundrums, and John Napier: Life, Logarithms, and Legacy (all Princeton). He is a retired former master at Winchester College, England, where he taught mathematics for more than three decades.

The Irrationals

By Julian Havil

PRINCETON UNIVERSITY PRESS

Copyright © 2012 Princeton University Press
All right reserved.
ISBN: 978-0-691-14342-2

Contents

Acknowledgments………………………………………………….ixIntroduction…………………………………………………….1Chapter One Greek Beginnings……………………………………..9Chapter Two The Route to Germany………………………………….52Chapter Three Two New Irrationals…………………………………92Chapter Four Irrationals, Old and New……………………………..109Chapter Five A Very Special Irrational…………………………….137Chapter Six From the Rational to the Transcendental…………………154Chapter Seven Transcendentals…………………………………….182Chapter Eight Continued Fractions Revisited………………………..211Chapter Nine The Question and Problem of Randomness…………………225Chapter Ten One Question, Three Answers……………………………235Chapter Eleven Does Irrationality Matter?………………………….252Appendix A The Spiral of Theodorus………………………………..272Appendix B Rational Parameterizations of the Circle…………………278Appendix C Two Properties of Continued Fractions……………………281Appendix D Finding the Tomb of Roger Apry………………………….286Appendix E Equivalence Relations………………………………….289Appendix F The Mean Value Theorem…………………………………294Index…………………………………………………………..295

Chapter One

Greek Beginnings

It is terrifying to think how much research is needed to determine the truth of even the most unimportant fact. Stendhal

Sources and Apologia

The birth of irrational numbers took place in the cradle of European mathematics: the Greece of several centuries B.C.E. For this conviction and for much more we must place reliance on a few fragments of contemporary papyrus, some complete but much later manuscripts, and the scholarship of many specialists who, even between themselves, sometimes disagree in fundamental ways. Of great importance is the following passage:

Thales, who had travelled to Egypt, was the first to introduce this science [geometry] into Greece. He made many discoveries himself and taught the principles for many others to his successors, attacking some problems in a general way and others more empirically. Next after him Mamercus, brother of the poet Stesichorus, is remembered as having applied himself to the study of geometry; and Hippias of Elis records that he acquired a reputation in it. Following upon these men, Pythagoras transformed mathematical philosophy into a scheme of liberal education, surveying its principles from the highest downwards and investigating its theorems in an immaterial [abstract] and intellectual manner. He it was who discovered the doctrine of proportionals and the structure of the cosmic figures.

So begins the Eudemian Summary, which forms part of the second of two prologues to A Commentary on the First Book of Euclid’s Elements, written by Proclus (411–485 C.E.), or to give him his full accepted name, Proclus Diadochus (Proclus the Successor), for reasons we shall soon discover. Here Proclus, the last great ancient Greek philosopher, mentions Thales of Miletus (624–546 B.C.E.), who was possibly the first, as he might have been the first pure mathematician. Proclus also mentions Pythagoras of Samos (580–520 B.C.E.), who might have been the second, and with whom the story of irrational numbers, according to what evidence is available, should begin. Our reliance on Proclus is not novel and we will call on the distinguished British polymath Ivor Bulmer-Thomas to provide a proper historical perspective:

It [the Eudemian Summary] is, along with the Collection of Pappus and the commentaries of Eutocius on Archimedes, one of the three most precious sources for the early history of Greek mathematics. In his closing words Proclus expressed the hope that he would be able to go through the remaining books [of Euclid’s Elements] in the same fashion; there is no evidence that he ever did so, but as Book 1 contains definitions, postulates, and axioms underlying all the remainder we have the most important things that he would have wished to say.

We have, then, a few pages of observations written a thousand years after the events they chronicle as a principal source of reliable information about these ancient times, and these take the form of a commentary on part of another paramount source; the most influential, most studied, most copied, and most widely read mathematical work ever to be written: Euclid’s Elements. Ironically, the paucity of surviving written material is in no small part due to this iconic work, without which our knowledge of ancient Greek mathematics would be so significantly impoverished. The Greeks’ medium of record was papyrus, made from a grass-like plant originating in the Nile delta and subject to natural and often rapid decomposition, particularly in the comparatively damp Greek climate. In that climate, it was simply not a safe long-term medium of record: it rotted. Those works which were deemed worthy of the considerable expense of being preserved were copied by scribes, perhaps as faithful replicas or perhaps with changes that were thought to be appropriate; the remainder were simply left to decompose. From the same reference, again we hear from Ivor Bulmer-Thomas:

Euclid’s Elements was so immediately successful that it drove all its predecessors out of the field.

It is a simple fact that the prominence of The Elements reduced to irrelevance much that preceded it, condemning the works to obscurity and then oblivion; David Hilbert’s remark, made in the nineteenth century, really sums up the situation quite nicely:

One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.

We shall have much need of The Elements later in this chapter but the reader should be clear that we must again rely on secondary sources, since no extant version of it exists. In fact, the earliest surviving copies of the work, held at the Vatican and the Bodleian Library in Oxford, date from the ninth century C.E.; a thousand years after Euclid. That said, some much earlier fragments have been found on potsherds discovered in Egypt dating from around 225 B.C.E. and pieces of papyrus dated from 100 B.C.E.: the former containing notes on two Propositions from Book XIII, and the latter having inscribed parts of Book II.

So, with the Greek’s medium of record fatally inadequate, The Elements relegating untold numbers of earlier works to insignificance, and the effect of providence that accompanies the passing of so many years, we must accept the consequent historical difficulties: and these do not end here. Really, they have their beginning with the Greek custom, until about 450 B.C.E., of transmitting knowledge orally, continue with the fondness on the part of later commentators to exaggerate the contributions of great men and culminate with the staged destruction of the academic riches of Alexandria: the Romans (seemingly in 48 B.C.E.) razed the great Library of Alexandria with its estimated 500,000 manuscripts, the Christians (in 392 C.E.) pillaged Alexandria’s Temple of Serapis with its possible 300,000 manuscripts, and finally the Muslims burnt thousands more of its books (in about 640 C.E.). Add to these the Pythagorean custom of attributing all results to their founder, who appears never to have written anything down, and their (near) strict adherence to their canon of omerta, and we have the ingredients of the mathematical historian’s nightmare; judging veracity and objectivity of the scant available evidence is a responsibility properly undertaken only by a few specialists, upon whom our discussion must rely.

Specifically, as to our knowledge of Pythagoras, the contemporary classicist Professor Carl Huffman provides a dampening perspective:

… any chronology constructed for Pythagoras’ life is a fabric of the loosest possible weave.

He may have been a pupil of Thales and to Proclus can be added significant further material; for example, Plato (428–347 B.C.E.) mentions him as a great teacher in Book X of The Republic, which is dated somewhere around 380 B.C.E. And there are three biographies too: that of Diogenes Laertius (200–250 C.E.), who wrote as part of a ten-volume work on the lives of Greek philosophers, the volume Life of Pythagoras; the other two are those of Iamblichus of Chalcis (ca. 245–325 C.E.), On the Pythagorean Way of Life, and of his teacher, Porphyry (234–305 C.E.) with Life of Pythagoras. All were written about eight hundred years after Pythagoras’ time, but they are at least extant. The definitive modern treatise must surely be that by the German classical scholar Walter Burkert, to which we refer the interested and committed reader; we shall be content with the following thumbnail impression of Pythagoreanism, which will suit our own modest needs.

All things are number: such was the Pythagorean dictum central to their philosophy. To them, number meant the discrete positive integers, with 1 the unit by which all other numbers were measured. This meant that all pairs of numbers were each multiples of the unit; that is, all pairs of numbers were commensurable by it. Contrastingly, lengths, areas, volumes, masses, etc., were continuous quantities, the magnitudes which served the ancient Greeks in place of real numbers. Ratios of discrete were conceptually secure and those of magnitudes could be envisaged too, provided that the two values concerned were of the same type. Further, the modern statement

A:B = C:D

was meaningful, where on one side of the equality there are magnitudes of one type and on the other magnitudes of another type. This could mean lengths on one side and areas on the other, and we will see part of the utility of this a little later. Additionally, their study of musical scales revealed that philosophical coincided with musical harmony with cordant sounds found to be measured by integer ratios of lengths of strings with, for example, the octave corresponding to a ratio of length of 2 to 1 and a perfect fourth to 3 to 2, etc. This was evidence to them that the continuous could be measured by the discrete. We will allow Aristotle to summarize the situation:

Contemporaneously with these philosophers and before them, the so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also having been brought up in it they thought its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being – more than in fire and earth and water (such and such a modification of numbers being justice, another being soul and reason, another being opportunity – and similarly almost all other things being numerically expressible); since, again, they saw that the modifications and the ratios of the musical scales were expressible in numbers; since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth – the ‘counter-earth’.

With the Pythagoreans’ dogmatism, the stage was set for the crisis in Greek mathematics that was to unfold, as ‘a veritable logical scandal’, possibly the very first of the long sequence of them that continues to this day.

So, there are available sources and there are scholars who have mined them of their dependable evidence regarding these remote times. By now, however, we hope that the reader will have a proper appreciation of the intrinsic historical complications and accept that given dates are sometimes approximate and statements made only with the authority borne of a compromise of accepted wisdom.

As a final emphasis we can gain some idea of the chain connecting Thales and Pythagoras to Proclus by relying on the scholarly labour of the early twentieth-century Dutch mathematical researcher J. G. van Pesch, which includes a detailed study of the works which he deemed were accessible to and directly used by Proclus, whether or not the dependence was explicitly stated. Figure 1.1 shows the resulting timeline of individuals and consists of a mixture of familiar and not-so-familiar names, together with approximate dates. Most particularly, the name of Eudemus of Rhodes (350–290 B.C.E.) appears, an historian who is attributed with writing a long-lost history of Greek geometry covering the period prior to 335 B.C.E.; it is, in particular, this formative work that van Pesch (and others) are confident that Proclus had at his disposal and summarized: hence Eudemian Summary.

Yet, two names on van Pesch’s list are missing from the timeline, since the dates of these individuals are simply not known. The first is Carpus of Antioch, or ‘Carpus the Engineer’, to whom Proclus attributed the definition that an angle is a quantity, specifically, the distance between the containing lines or planes; he is also the person accredited by Iamblichus of Chalcis as being among the Pythagoreans who solved one of the three great problems of antiquity: that of the impossibility of squaring the circle. He appears to have lived at some time between 200 B.C.E. and 200 C.E.

The second name is that of Syrianus of Alexandria, himself a commentator on Plato and Aristotle, and through him we can learn a little about Proclus himself. The revival of Plato’s academy under the leadership of Plutarch (46–120 C.E.) brought important scholars to Athens, among whom was the philosopher Syrianus and a young man, about 20 years old and of immense promise, by the name of Proclus. Such was his promise that for a short time before his death the aged Plutarch agreed, exceptionally, to tutor Proclus. When Syrianus replaced Plutarch as leader he also replaced him as tutor to Proclus. In his turn, Proclus succeeded Syrianus as head of the academy; it was this event that brought about the addition of Diadochus (Successor) to his name. The relationship between Proclus and Syrianus was to become intellectually and emotionally immensely close, so much so that Proclus left instructions that, on his death, he be placed in the tomb already occupied by Syrianus. The tomb was located on the slopes of the Lycabettus Hill, overlooking Athens, a limestone peak of some 1000 feet and, for very good reasons, a modern tourist attraction. The level of affection is easily judged by Proclus’ decree that the following epitaph be inscribed on their joint resting place:

Proclus was I, of Lycian race, whom Syrianus Beside me here nurtured as a successor in his doctrine. This single tomb has accepted the bodies of us both; May a single place receive our two souls.

As with so much else, the tomb has long ago disappeared.

Inheritance and Legacy

With the authority of Proclus, Thales, the first of the Seven Sages of Greek tradition, brought geometry from Egypt to Greece. In particular, as the Commentary develops, Proclus attributes to him the following four geometric results:

1. A circle is bisected by any diameter.

2. The base angles of an isosceles triangle are equal.

3. The angles between two intersecting straight lines are equal.

4. Two triangles are congruent if they have two angles and one side equal.

These seem modest achievements. Yet their simplicity belies their significance, as they exhibit the germ of the deductive procedures of Greek philosophy being brought to bear on mathematical processes: the yet more ancient Egyptian and Babylonian civilizations had no thought of axiomatics, abstraction or generalization, with mathematical results having the form of mysterious individual recipes. This is not to say that some of the knowledge could not be called ‘advanced’, it is simply that the deductive method that we consider as an essential aspect of pure mathematics was entirely absent. Paradoxically, evidence abounds regarding these older civilizations; the ancient Egyptians used papyrus too, but their climate was more papyrus-friendly than Greece, and the Babylonians wrote in cuneiform on wet clay tablets, thousands of which have survived.

To gain a perspective of the magnitude of the step that had been taken by Thales we will trouble to annotate two contemporary examples, one from each of these civilizations. From the Moscow papyrus, which dates from around 1850 B.C.E., we have an Egyptian problem:

Method of calculating a [??]

If you are told [??] of 6 as height, of 4 as lower side, and of 2 as upper side.

You shall square these 4. 16 shall result.

You shall double 4. 8 shall result.

You shall square these 2. 4 shall result.

You shall add the 16 and the 8 and the 4. 28 shall result.

You shall calculate [bar.3] of 6. 2 shall result.

You shall calculate 28 times 2. 56 shall result.

Look, belonging to it is 56.

What has been found by you is correct.

Here the [bar.3] is our 1/3, which means that the calculation is 1/3 × 6 × (42+42+2²) = 56, a special case of the general formula for the volume of the frustram of a square pyramid, V = 1/3h(a²+ab+b²), with this special case shown in figure 1.2.

And from a Babylonian tablet from about 2000-1600 b.c.12:

A circle was 1 00.

I descended 2 rods.

What was the dividing line (that I reached)?

You: <> <> 2.

You will see 4.

Take away <> 4 from 20, the dividing line.

You will see 16.

Square 20, the dividing line.

You [will see] 6 40.

(Continues…)

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