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hence we have shown G0 − G1 > 12 G0 and this is nothing but G1

1 1 (A(C) − A(Pn )) = Gn , 2 2

2.1 The Greeks Shape Mathematics


where A(C) − A(Pn ) is the sum of the areas of the 2n+1 segments of the circle which are cut off by the edges of Pn . As a final example of the method of exhaustion let us consider the following theorem which can be found in Euclid’s Elements as Proposition 2 in Book XII [Euclid 1956, Vol. 3, p. 371]: Theorem: If C1 and C2 are circles with radii r1 and r2 , respectively, then it holds A(C1 ) r2 = 12 . (2.1) A(C2 ) r2 (‘Circles are to one another as the squares on the diameters.’) The proof of this theorem is accomplished via the method of double reductio ad absurdum. There can only be three possibilities since either A(C1 ) r2 = 12 , A(C2 ) r2 We state assumption 1:

or A(C1 ) A(C2 )

A(C1 ) r2 < 12 , A(C2 ) r2


A(C1 ) r2 > 12 . A(C2 ) r2

hence we assume

A(C1 )r22 =: S. r12

Then the number ε := A(C2 ) − S would be positive, i.e. ε > 0. Following the theorem on the exhaustion of the area of a circle above there exists a polygon P inscribed in circle C2 so that A(C2 ) − A(P ) < ε = A(C2 ) − S holds, i.e. A(P ) > S. We now inscribe a polygon P corresponding to P into C1 . We now have (cp. fig. 2.1.12) A(Q) r2 = 12 , A(P ) r2

Fig. 2.1.12. Regular polygons in circles


2 The Continuum in Greek-Hellenistic Antiquity


A(Q) r2 A(C1 ) A(C1 ) = 12 = A(C )r2 = . 1 2 A(P ) r2 S 2 r1

But from this it follows S A(C1 ) = > 1, A(P ) A(Q) hence S > A(P ), and this contradicts our assumption A(P ) > S. Thus assumption 1 must be wrong and this we have proven with the method of reductio ad absurdum. r2

1) 1 Now we state assumption 2: A(C A(C2 ) > r22 and show that a contradiction follows as above. We have now completed the double reductio ad absurdum and only r12 1) the case A(C A(C2 ) = r 2 remains as correct possibility. 2

2.1.5 The Problem of Horn Angles The mathematics of the Greeks was ruled by Archimedean number systems since the days of Eudoxus, i.e. number systems in which the Archimedean axiom holds: To any two positive quantities x < y a natural number n can always be found so that n · x > y holds. But already Eudoxus knew that also other number systems – so-called non-Archimedean number systems – were conceivable [Becker 1998, p. 104]. Such a system of quantities which was already known to the Greeks were cornicular angles or horn angles [Thiele 2003, p. 1f.]. These are angles between two circles touching each other or between a circle and its tangent as shown in figure 2.1.13. In Book III of Euclid’s Elements we find Proposition 16 [Euclid 1956, Vol. II, p. 37] on cornicular angles: ‘The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle.’ Between the circumference and the tangent simply no further straight line can be drawn which remains outside the circle. Cornicular angles comprise a nonArchimedean system of numbers since in the case of any two cornicular angles the Archimedean axiom does obviously not hold. If we define cornicular angles as angles between the tangents then every cornicular angle is simply zero and the Archimedean number system is again restored. As Körle writes [Körle 2009, p. 29]:

2.1 The Greeks Shape Mathematics


α α Fig. 2.1.13. Cornicular or horn angle

‘For us there simply are no cornicular angles. Their problem is of a psychologigal nature. One did not know how to defend oneself against the idea that something had to fill this gap. They could not be argued away but became invalid with the notion of the limit. By any stretch of imagination concerning the interpretation there remains only the quantity zero for cornicular angles. The controversey concerning cornicular angles remained for a long time and even Leibniz was concerned with them.’ (Für uns gibt es schlichtweg keine Kontingenzwinkel. Ihr Problem ist psychologischer Natur. Man wusste sich nicht gegen die Vorstellung zu wehren, irgendwas müsse jene Öffnung doch ausfüllen. Wegdiskutieren ließen sie sich nicht, gegenstandslos wurden sie mit dem Begriff des Grenzwerts. Bei bestem Willen zur Interpretation bliebe den Kontingenzwinkeln nur die Größe Null. Die Kontroverse um sie hielt lange an, noch Leibniz beschäftigte sich mit ihnen.) We shall see later that analysis is also possible in non-Archimedean number systems. In such systems different cornicular angles can be of different sizes!

2.1.6 The Three Classical Problems of Antiquity We have already reported on the quadrature of the circle and its authorship by the imprisoned Anaxagoras. We have to mention two other problems which, together with the quadrature of the circle, have played an important role in the history of analysis. All three problems became know as the ‘classical problems of mathematics’; compare [Alten et al. 2005] and [Scriba/Schreiber 2000]. The Trisection of the Angle: Given an angle α. Construct an exact trisection of this angle by means of straightedge and compass. Heath [Heath 1981, p. 235] has suspected that this problem originated at a time when one was able to construct the pentagon by means of straightedge


2 The Continuum in Greek-Hellenistic Antiquity

and compass and wanted to construct further polygons. The construction of a regular polygon of 10 edges in fact requires the trisection of an angle. As the quadrature of the circle is an unsolvable problem the trisection of the angle also is unsolvable as modern algebra has revealed. Generation of mathematicians have tried to solve the problem nevertheless. The Doubling of the Cube: Given a cube with volume V . Construct a cube with double volume by means of straightedge and compass. Legend has it that the Deloians, inhabitants of the Cycladic island of Delos in the Aegean Sea, were haunted by the plague. The oracle was asked for advice and suggested the doubling of the cube-shaped altar in the temple of the Deloians. As this was unsuccessful the Deloians asked the great philosopher Plato who answered that the god did not actually require a new altar but that he had posed this problem to put the Deloians to shame because they were not interested in mathematics at all and despised geometry, cp. [Heath 1981, p. 245f.]. Concerning the Quadrature of the Circle The attempt to square the circle seems to have been a temptation for the Greek mathematicians. An approach not based on the method of exhaustion was developed by Hippocrates of Chios (middle or second half of the 5th c BC) which has had an impact even on the maths books used in schools in recent times. The method relies on the ‘lunes of Hippocrates’. Hippocrates thereby followed the less ambitious task of computing the area of surfaces which are bounded by parts of circles. If an isosceles triangle is drawn inside a half circle as shown in figure 2.1.14 this triangle is right-angled after the theorem of Thales. Drawing two further half circles about the legs of the triangle then circular shaped areas M appear which remind on lunes. With the notations in figure 2.1.14 we introduce the areas


0 6

6 7


Fig. 2.1.14. Lunes of Hippocrates

2.1 The Greeks Shape Mathematics


C1 := M + S C2 := 2 · (S + T ). Now we know from the theorem on page 37 that ratios of areas of circles are like the ratios of the square of the radii. This, of course, holds true also in the case of the areas of half circles. If the radius of the large half circle is r then√it follows from Pythagoras’ theorem that the radii of the small circles are 22 r. The squares of these radii are r2 and 12 r2 , respectively. Hence for the areas of the half circles C1 und C2 it follows: C1 1 = . C2 2 Inserting the definition of C2 it follows 2(T + S) = 2C1 , hence T + S = C1 . By definition we have C1 = M + S, i.e. T + S = C1 = M + S and it is shown that M = T holds true. The area of one of the lunes hence is exactly the area of the triangle T . There is no doubt that Hippocrates’ results encouraged not only himself but also others to go on and finally tackle the quadrature of the circle. More complicated lunes can thus be found in the repertoire of the method of lunes, cp. [Baron 1987, p. 32f.], [Scriba/Schreiber 2000, p. 47f.]. Heath [Heath 1981, Vol.1, p. 225f.] cites some ancient authors reporting on some of the mathematical developments which arose from the many futile attempts to solve the three great problems by means of a construction with straightedge and compass. Iamblichus wrote concerning the quadrature of the circle: ‘Archimedes effected it by means of the spiral-shaped curve, Nicomedes by means of the curve known by the special name quadratrix [...], Apollonius by means of a certain curve which he himself calls “sister of the cochloid” but which is the same as Nicomedes’s curve, and finally Carpus by means of a certain curve which he simply calls (the curve arising) “from a double motion”.’ Pappus of Alexandria (about 290–about 350) is cited with: ‘for the squaring of the circle Dinostratus, Nicomedes and certain other and later geometers used a certain curve which took its name from its property; for those geometers called it quadratrix.’ Proclus (412–485) wrote concerning the trisection of the angle:


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.1.15. Sphinx and pillar of Pompeius in Alexandria; the town with the largest library of antiquity and many scholars. Pappus of Alexandria was one of them [Photo: H.-W. Alten]

‘Nicomedes trisected any rectilineal angle by means of the conchoidal curves, the construction, order and properties of which he handed down, being himself the discoverer of their peculiar character. Others have done the same thing by means of the quadratrices of Hippias and Nicomedes ... Others again, starting from the spirals of Archimedes, divided any given rectilineal angle in any given ratio.’ Proclus then goes on and mentions explicitly the mathematicians who explained the properties of different curves: ‘thus Apollonius shows in the case of each of the conic curves what is its property, and similarly Nicomedes with the conchoids, Hippias with the quadrices, and Perseus with the spiric curves.’ The three great problems are unsovable in their original formulation, i.e. with a construction by means of straightedge and compass alone. However, they can be solved with the help of the curves mentioned above. Therefore it is worth looking at at least two of those curves in the context of the trisection of the angle.

2.1 The Greeks Shape Mathematics


Concerning the Trisection of the Angle Hippias of Elis (5th c BC) was born in Western Greece. He is attributed with the discovery of the quadratrix which is defined pointwise by means of a mechanical model as shown in figure 2.1.16(a). %

& 5




( \ \

$ 2 (a) Definition of the quadratrix


7 Θ/3

2 $ (b) Trisection of an angle by means of the quadratrix

Fig. 2.1.16. The quadratrix – an auxiliary curve to trisect an angle

Let a square OACB with edge length 1 and an inscribed quarter circle BRA be given. We imagine the edge BC moving with constant speed down to OA. Simultaneously the edge OB rotates with constant speed about the point O in radial direction also towards OA. Both edges are assumed to reach OA at exactly the same time. At any time in between BC has reached DE and OB is at the position OR. The point of intersection Q is the defined to be a point of the quadratrix. Now follow the movement of the point R. Its coordinates are described by x = cos Θ y = sin Θ, where the angle Θ changes from Θ = π/2 (=90◦ ) to Θ = 0 as shown in figure 2.1.16(b). Edge BC moves with constant speed in y-direction. Let us denote this speed by vy and connect the movement of BC with the angle Θ by y = vy · Θ. If Θ = 0 then also y = 0. If Θ = π/2 then y = 1. These conditions lead to the equation π y = 1 = vy · , 2


2 The Continuum in Greek-Hellenistic Antiquity

hence the speed of BC has to be vy = 2/π. Hence the relation between angle and y-coordinate is πy Θ= . 2 But y sin Θ = = tan Θ, x cos Θ so that tan(πy/2) = y/x follows, hence x = y · cot

πy . 2

This then is the equation of the quadratrix which, of course, was unknown to Hippias. Since the cotangent appears in the equation it is a transcendent function. The trisection can then be accomplished as follows. To an angle Θ there corresponds a certain value of y, cp. figure 2.1.16(b), and since Θ and y are connected via Θ = πy/2 we only need to intersect a horizontal line at height y/3 with the quadratrix (the point of intersection is T ) in order to trisect the angle Θ. The use of the quadratrix to square a circle is much more involved, cp. [Heath 1981, Vol.1, p. 227f.]. The second curve mentioned in the citations above is the conchoid of Nicomedes (about 280–about 210 BC), also called ‘shell curve’ because its outer branches resemble the shape of conch shells. It seems a number of different curves were known as cocleoids and the conchoid is just one of them. D




N N N (a) Definition conchoid



(b) Trisection of an angle by means of a conchoid

Fig. 2.1.17. The conchoid – a further auxiliary curve to trisect the angle

2.1 The Greeks Shape Mathematics


It is an algebraic curve and can be constructed mechanically. Choose two positive numbers a and k. In a Cartesian coordinate system draw a vertical line at distance a to the origin. Then a point of the conchoid is defined as follows. Draw a line from O to the vertikal line and extend it by a line segment of length k. At the end of this line segment lies a point of the conchoid, cp. figure 2.1.17(a). One can describe the conchoid of Nicomedes either in Cartesian form y2 =

x2 (k + a − x)(k − a + x) (x − a)2

or in polar form

a , cos Θ where r denotes the length of the line segment from O to a point of the conchoid lying under the angle Θ measured counterclockwise from the x-axis. The trisection of an arbitrary angle α can now be accomplished as follows. As shown in figure 2.1.17(b) one leg of the angle is put on the horizontal axis and the vertical axis is shifted so that the second leg from O to the point of intersection with the vertical line has length k/2. A parallel line to the horizontal axis through this point of intersection B results in the intersection point T on the conchoid. Connecting T with the origin results in the trisection of the angle α. In case of the conchoid there is also a mechanical construction shown in figure 2.1.18. A pointer with tip point P slides in a groove N of a horizontal rail with a pin C. Perpendicular to this rail and firmly attached to it is a lug with a fixed pin K to support the groove of the pointer. Moving the pointer its tip point will describe a conchoid. As further methods Archimedes and later Pappus of Alexandria described two insertion methods, known as ‘neusis’, which were popular in ancient Greek geometry. Details can be found in [Scriba/Schreiber 2000, p. 45f.]. r=k+



& 0


Fig. 2.1.18. A mechanical construction of the conchoid


2 The Continuum in Greek-Hellenistic Antiquity

Concerning the Doubling of the Cube Let a cube with edge length a be given so that its volume will be V = a3 . If a new cube with double the volume has to be constructed than the new edge length x has to satisfy x3 = 2 · a3 , or √ 3 x = 2 · a. Modern algebra as developed only in the 19th century tells us that this number √ x can not be constructed by means of straightedge and compass since 3 2 is not a constructable number. The honour having given the first rigorous proof of the unsolveability of the problems of doubling the cube and trisecting the angle is due to the French mathematician Pierre Laurent Wantzel (1814–1848) who published his proofs in 1837 in Liouvilles’s ‘Journal de Mathématiques Pures et Appliquées’ [Cajori 1918]. We do not want to describe the many ingenious attempts of the Greeks to tackle this problem but Hippocrates of Chios earned the honour of immortality due to a groundbreaking discovery. He succeeded to reduce the problem of doubling the cube to a problem of the determination of two mean proportionals. A cube with edge length 2a obviously does not satisfy our conditions since it has twice the edge length of the cube we started with but its volume is 8·a3 . Nevertheless the doubling of the edge length must have got stuck somehow in Hippocrates’s mind. He looked for two numbers in between the lengths a and 2a which are called two mean proportionals. Here x is a mean proportional of two numbers a and b if a : x = x : b. √ Solving this proportionalty for x results in x2 = a · b or x = a · b, hence the mean proportional is nothing but the geometric mean of a and 2a defined by a : x = x : 2a, √

hence x = 2 · a. There is √ no way to arrive at a solution of our problem of doubling the cube, i.e. x = 3 2 · a, by means of just one mean proportional. It is highly likely that this perception was already of pythagorean origin since Plato writes in his dialogue Timaios [Plato 1929, p. 59, 32A-B]: ‘But it is not possible that two things alone should be conjoined without a third; for their must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of nay three numbers, cubic or square, is such that as the first term is to it, so is it to the last term, – and again, conversely, as the last term is to the middle, so

2.1 The Greeks Shape Mathematics


is the middle to the first, – then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. Now if the body of the All had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms; but now it is otherwise: for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two.’ Hippocrates was also concerned with a solid, namely a cube. One mean proportional hence did not suffice. Therefore he sought two mean proportionals x and y of a and 2a so that a : x = x : y = y : 2a holds. From these proportionalities three equations follow, namely x2 = a · y,

y 2 = 2a · x,

x · y = 2a2 .

Solving the last equation for y and inserting the result into the first equation yields the equation of the doubling of the cube, √ 3 x3 = 2 · a3 ⇒ x = 2 · a. It is obvious that Hippocrates did not succeed coming closer to the doubling of the cube. But the insight that the problem of doubling the cube is completely equivalent to finding two mean proportionals of two line segments can only be named a strike of genius! Of course further Greek mathematicians tried to solve the problem of doubling the cube and achieved impressive advances in the development of their mathematical methods. We have to name Diocles (about 240–about 180 BC) and the cissoid (ivy curve) named after him, with which two mean proportionals can be constructed geometrically. In this context we also have to mention Archytas of Tarentum (428–347 BC) who presented a remarkable three-dimensional construction to determine the edge length x of the sought new cube. He succeeded in intersecting no less than three bodies of revolution the unique point of intersection being the sought x. In the treatment of the problem of doubling the cube by means of Hippocrates’s two mean proportionals Menaechmus (380–320 BC) discovered the conic sections. However, conic sections got their name and were analysed only later by Apollonius. Details concerning the constructions mentioned above can be found in [Alten et al. 2005], [Scriba/Schreiber 2000] und in [Heath 1981, Vol. I].


2 The Continuum in Greek-Hellenistic Antiquity

Remarks The insight and creativity of the Greek mathematicians is impressive even from a modern point of view. Although they were able to solve the problems of squaring the circle, doubling the cube and trisecting the angle by means of certain curves like the conchoid, the conic sections and the quadratrix with arbitrary order of accuracy the actual task – exact construction with straightedge and compass alone – proved to be not solvable at all. However, even today, after the development of analysis, algebra and number theory, there are some amateur mathematicians who believe that they have succeeded in solving at least one of the three great problems of antiquity or in proving the rationality of π. Splendid examples can be found in Underwood Dudley’s book [Dudley 1987]. Every attempt to put a stop to the game of these pseudomathematicians is unfortunately doomed to failure; they simply either do not understand the problem definition, or necessary notions and mathematical knowledge are missing. Often they invent approximate methods leading to astonishingly good results in a finite number of steps but refuse to acknowledge that the exact solution would only result after an infinity of steps (and therefore can not be the solution the Greeks sought for).

2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles The discovery of incommensurable quantities, i.e. the existence of irrational numbers, may have unsettled the Greek mathematicians and may even have been responsible for the Greeks withdrawing to geometry. The irrational was at the same time the unspeakable, incomprehensible, nonpictorial [Lasswitz 1984, Vol. 1, p. 175]. However, a philosophical quarrel concerning items of being (existence) has shattered analysis almost more and this shock can be felt even today. We do not want to dive too deeply into philosophical problems but refer the reader to the literature, e.g. [von Fritz 1971]. However, a few words may certainly be in order for the sake of a better understanding of the mathematical and historical background.

2.2.1 The Eleatics Due to warlike struggles with the Persians at the Ionian coast took some Greeks to the South-Italian west coast in the year 545 BC. There they founded the settlement Elea which today is Velia. In Elea there evolved a community of philosophers called the Eleatics. The poet and natural philosopher Xenophanes (about 570–about 475 BC) is seen as its founding

2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles


Fig. 2.2.1. Parmenides; Zeno of Elea [Photo: Sailko]

father. One of the truelly great Eleatics was Parmenides (about 540/535– about 483/475 BC) who introduced a new thinking into Greek philosophy. While philosophers before Parmenides were keen on understanding the world Parmenides now introduced the claim of absolute certainty of non-empirical theories into philosophical thinking whereby these theories can not serve directly to describe the world [Parmenides 2016, p. 4ff.]. The ‘existing’ (‘being’) became a central point of Parmenides’ philosophy and the being (or the logos, the one, or god [De Crescenzo 1990, Vol. 1, p. 112]) is something unique, a whole, and an immovable. There is no void and no ‘becoming’; ‘nonbeing’ is inconceivable. Since the being is immobile Parmenides obviously doubted the possibility of movement at all – we only see apparent movement of human beings whereas the actual being is static – and this is splendidly acknowledged by his most famous pupil, Zeno of Elea (about 490–about 430 BC). Plato in his dialogue Parmenides [Plato 1939, p. 205] reports in 128d that Zeno wanted to come to his teacher’s defence against the accusation of absurd consequences if movement would be rejected. But what is behind all that mathematically?

2.2.2 Atomism and the Theory of the Continuum Almost all we know about Zeno has come down to us in the writings of the great philosopher Aristotle who shaped the thinking in the Western world for many centuries. Of all philosophers from Thales to some who lived in Socrates’s days no written lore is extant. These philosophers are called the Pre-Socratics. The only material we have has come to us in form of fragments [Early Greek Philosophy 2016] which were only written down by philosophers


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.2.2. Democritus of Abdera; Detail of a banknote (100 Greek drachma 1967)

of later generations. In Aristotle’s Physics the great philosopher dedicated a whole book – Book VI – to the problem of the continuum [Aristotle 1995, p. 390–407]. It is there where Zeno gets a chance to speak. Two schools of thought concerning the passage of time and the structure of space were popular with the Greek thinkers: atomism and the theory of the continuum, respectively. The philosophers Leucippus (5th c BC) and his pupil Democritus (about 460–about 370 BC) are regarded as having invented atomism. Following their thoughts everything consists of infinitely small quantities, the ‘atoms’ (atomon = indivisible). We must not, however, confuse our modern understanding of atoms with what Democritus understood when he talked about atoms. Democritus’s ‘atoms’ in their primal meaning were probably still further divisible but concerning our discussion of their mathematical implications we should imagine an atom as being a point lying in a straight line. This point is an atom and following Democritus the whole straight line is made up of infinitely many points. Aristotle and many others overwhelmingly rejected this theory of atoms that in part can be ascribed to Zeno as we shall see. Following the theory of the continuum a straight line is a ‘continuum’ which is arbitrarily divisible. Even if a continuum is divided arbitrarily often there always remains a continuum which is still further divisible. Never will the process of division results in a point, however! A point can therefore not be an element of a straight line! In Book V of his Physics Aristotle introduces the notions of ‘together’, ‘apart’, ‘contact’, ‘succession’, continuity’, and others [Aristotle 1995, p. 383] ‘Let us now proceed to say what it is to be together and apart, in contact, between, in succession, contiguous, and continuous, and to show in what circumstances each of these terms is naturally applicable.

2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles


Things are said to be together in place when they are in one primary place and to be apart when they are in different places. Things are said to be in contact when their extremities are together.’ And he goes on [Aristotle 1995, p. 383]: ‘A thing is in succession when it is after the beginning in position or in form or in some other respect in which it is definitely so regarded, and when further there is nothing of the same kind as itself between it and that to which it is in succession, e.g. a line or lines if it is a line, a unit or units if it is a unit, a house if it is a house (there is nothing to prevent something of a different kind being between). [...] A thing that is in succession and touches is contiguous. The continuous is a subdivision of the contiguous: things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are two.’ This definition is used in Book VI to give a mortal blow to the idea of atomism [Aristotle 1995, p. 390f.]: ‘Now if the terms ‘continuous’, ‘in contact’, and ‘in succession’ are understood as defined above – things being continuous if their extremities are one, in contact if their extremities are together, and in succession if there is nothing of their own kind intermediate between them – nothing that is continuous can be composed of indivisibles: e.g. a line cannot be composed of points, the line being continuous and the point indivisible.’ At the beginning of his Elements Euclid defines [Euclid 1956, Vol. I, p. 153]: 1. A point is that which has no part. 2. A line is breadthless length. Thereby he cleverly avoided any subtle discussions. Using ‘breadthless length’ as a paraphrase for a line simply excludes any form of critique concerning atoms or the continuum. But if a point has no part, so Aristotle asked, how then can a line be built from points? In which sense should two points on a line then be adjacent? Questions like these have fascinated thinkers up to our present days. We remind our readers of the mathematician Hermann Weyl (1885–1955) wrote on the continuum already in 1917 [Weyl 1917] (English translation see [Weyl 1994]) and discussed philosophical problems of mathematics still in 1946 when he was in old age. In [Weyl 2009, p. 41] he looks at the continuum from a modern point of view and builds a bridge to modern analysis when he writes:


2 The Continuum in Greek-Hellenistic Antiquity ‘The individual natural numbers form the subject of number theory, the possible sets (or the infinite sequences) of natural numbers are the subject of the theory of the continuum.’

He also cites Anaxagoras to characterise the nature of the continuum [Weyl 2009, p. 41]: ‘Among the small there is no smallest, but always something smaller. For what is cannot cease to be no matter how far it is being subdivided.’ This citation refers to the so-called Fragment 3 of Anaxagoras [Schofield 1980, p. 80]: ‘For of the small there is no least but always a lesser (for what is cannot not be)’ and is seen in connection with Zeno’s paradoxes which we shall discuss in the following.

2.2.3 Indivisibles and Infinitesimals Among other reasons atomists and the supporters of the continuum collided was because infinity was concerned, cp. [Heuser 2008, p. 59ff.]. Democritus and the atomists stated nothing less than the existence of an actual infinity since a line (or even a line segment only) consists of an actual infinity of points, hence atoms. Aristotle and many others already in the life time of Zeno rejected the existence of an actual infinity and postulated the ‘potential infinity’ so that the process of division can always be continued. This dispute, how ‘ancient’ it may seem to us, has not ceased even today! It was only Georg Cantor (1845–1918) who introduced the actual infinity rigorously into mathematics by the launch of set theory. In Cantor’s mathematics a line (e.g. the real number line) is indeed called a ‘continuum’, but Cantor’s continuum is defined by means of single points! Aristotle as well as Democritus would shudder! Only in the 1960s the idea of the continuum aroused again like phoenix from the ashes with the invention of nonstandard analysis. We shall report on this development at the end of this book. Concerning Democritus it is said [Edwards 1979] that he found the volume formula 1 V = A·h 3 for the cone and the tetrahedron, where A denotes the base area and h the height of these solids. However, this was proven only by Eudoxus. Democritus imagined that solids consisted of infinitely many slices of zero thickness. Today we call these slices ‘indivisibles’. It is to be noted, however, that

2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles


Democritus certainly did not have an idea concerning a theory of indivisibles [Heath 1981, Vol. I, p. 181]. A point, a line, and a surface are indivisibles in one-, two-, and three-dimensional space, respectively, since one of their dimensions is nil. On the other end of the spectrum and quite contrary to the ideas of the atomists the supporters of the continuum believed that solids consisted of continua – hence solids again – which were themselves arbitrarily divisible again. Those slices of finite thickness are today called ‘infinitesimals’. The proof of Democritus’s volume formula by Eudoxus in Euclid’s Elements XII.5 relies on a subdivision of the tetrahedron but we have every reason to believe that Democritus arrived at his result when he imagined a pyramid being built up from infinitely many indivisibles (plane cuts parallel to the base area as in figure 2.2.3(a)). Baertel van der Waerden (1903–1996) in his book Science Awakening [van der Waerden 1971, p. 138] cites Plutarch (compare also [Heath 1981, Vol. I, p. 179f.]) who attributed the following argument to Democritus: ‘If a cone is cut by surfaces parallel to the base, then how are the sections, equal or unequal? If they were unequal [(and, we might add mentally, if the slices are considered as cylinders)]3 , then the cone would have the shape of a staircase; but if they were equal, then all sections will be equal, and the cone will look like a cylinder, made up of equal circles; but this is entirely nonsensical.’ Hence Democritus seemed to have very obscure ideas of a solid being an accumulation of two-dimensional cuts as Eberhard Knobloch has pointed out in [Knobloch 2000]. Knobloch even called it a ‘pseudo-problem’ [Knobloch 2000, p. 86]. The intersection of a plane with the cone results in two cuts A and B; one belonging to the lower frustum of the cone and the other belonging

(a) Tetraeder of indivisibles

(b) Tetraeder of infinitesimals

Fig. 2.2.3. Indivisible and infinitesimal 3

Remark by van der Waerden.


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.2.4. Sunset above the tetrahedron in Bottrop. It was designed by the architect Wolfgang Christ (Bauhaus University Weimar) and errected in 1995 with a viewing platform [Photo: H. Wesemüller-Kock]

to the upper cone. These two cuts may be identical, A = B, without implying the identity of all cuts. Only a further slice at another height resulting in two cuts C and D and the use of the law of transitivity would lead from A = B and C = D to A = C and would show that the cone actually is a cylinder. Democritus’s arguments are therefore of a non-rigorous, physical type, while the arguments of Archimedes would be of a mathematical rigorous type. However it is only a small step now to assume that Democritus already had the principle of Cavalieri (Bonaventura Cavalieri (1598–1647)) at his disposal: Principle of Cavalieri: Suppose two solids are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross sections of equal area, then the two regions have equal volumes. It is evident that Democritus could also have seen that a prism with triangular base area A can be dissected into three tetrahedra of equal size and that, following the principle of Cavalieri, the volumes of the three pyramids would exactly match the volume of the prism. The transfer of this argument to the cone would also have been fairly easy for an atomist like Democritus [Heath 1981, Vol. I, p. 180]; he certainly would have argued that the cone could be constructed from the tetrahedron by infinite addition of lateral surfaces.

2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles


2.2.4 The Paradoxes of Zeno What is the role played by Zeno? Remember that he wanted to defend his teacher Parmenides and that he wanted to show that all movement is nothing but an illusion. Aristotle reports four paradoxes of Zeno which we will now start to discuss. The most widely known paradox is that of Achilles and the tortoise but the dichotomy, the flying arrow, and the stadium have become immortal through the reports of Aristotle. Achilles and the tortoise: The fast runner Achilles is asked to compete against a tortoise. Since Achilles is much faster than the tortoise the latter gets a considerable lead. Now Zeno states: Achilles will never catch up with the tortoise! He reasons as follows: when Achilles will be at the starting point of the tortoise the latter will be a short distance in front of him. When Achilles reaches this point the tortoise is again a (very) short distance in front of him, and so on. To illustrate the argument let us assume that the tortoise gets a lead of 10m and that Achilles is 10 times faster than the tortoise. Achilles needs only 1 second to run a distance of 10m. If the competition has started Achilles will be in the starting position of the tortoise after 1 second. During this second the tortoise is 1m ahead of Achilles. Now Achilles has to run 1m to arrive at the position of the tortoise (he needs only 1/10 seconds for this), but then the latter is still 10cm in front of him. When Achilles has covered the 10cm the tortoise will still be 1cm ahead of him, and so on ad infinitum. The dichotomy: Zeno states: one can not move from a point A to a point B, A 6= B. For to get from A to B one has to cover half the distance first. To cover half of the distance one has to cover a quarter of the distance; to cover a quarter one has to cover one eighth of the distance, and so on. Hence an infinity of distances has to be covered to come from A to B and this is not possible in finite time. Therefore movement is impossible. The flying arrow: Following Zeno an arrow shot from a bow does not fly. Suppose the arrow is flying and freeze time at a certain point during the flight. At this point of time the tip of the arrow is at a fixed point of space and its velocity is zero (because at this point of time the arrow is fixed). Since this point of time can be chosen anywhere during the flight of the arrow the arrow has everywhere nil velocity. Hence the arrow does not fly at all. The stadium: Imagine being an observer of an ancient chariot race between two chariots manned by 8 persons each, cp. figure 2.2.6. One of the chariots is manned by persons B, the other with persons C, while further 8 persons A are watching from a fixed stand. The B-chariot moves to the right while the C-chariot moves to the left with the same velocity. When the B-chariot has travelled one A-position then B and C-chariot have travelled two positions!


2 The Continuum in Greek-Hellenistic Antiquity

Following Zeno this means, however, that half of the elapsed time is equal to the elapsed time and this contradiction again implies that movement is not possible at all. Now one can immediately argue against the paradox of the stadium that Zeno obviously was not aware of the notion of relative velocity which would resolve the paradox. The paradox of Achilles and the tortoise is also not a real problem for modern mathematicians since the whole distance travelled (in meters) by Achilles is k ∞  X 1 10 + 1 + 0.1 + 0.01 + . . . = 10 + 10 k=0

and this infinite series is a convergent geometric series achieving the value of 10/9. The distance 10 + 10/9 = 11.11111 . . . is exactly the distance at which Achilles would overtake the tortoise. But this is not the point here! Firstly we are using here knowledge of the 19th century, and secondly such arguments fail to explain the actual problem. Let us concentrate on Achilles and the tortoise for the sake of illustration: the actual question raised by Zeno is the question concerning the structure of space (in this case the structure of the racecourse) and time. If we take the real numbers as a basis the paradox can be resolved, but who tells us that the real space and the real time can in fact be modelled by real numbers? In an essay of the year 1992 the late Jochen Höppner [Höppner 1992, p. 59–69] has examined this

Fig. 2.2.5. Stadium in Delphi. The stadium is also an ancient measure of length of 600 feet. Depending on the local measure of a foot this is between 165 and 195m (Olympia 192.28m, Delphi 177.35m) [Photo: J. Mars]

2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles


$$$$$$$$ %%%%%%%% &&&&&&&& $$$$$$$$ %%%%%%%% &&&&&&&& Fig. 2.2.6. Concerning the paradox of the stadium

subtle point in detail and has taken other structures as the basis of Achilles’s racecourse than the real numbers. Besides an imagined Minkowski space, a probabilistic racecourse, and a representation with obstacles Höppner also examines the Cantor set as a racecourse. The Cantor set is the set of those numbers between 0 and 1 which have representations in the ternary numeral system (i.e. only with digits 0, 1, 2) where no digit 1 appears. This set is often called ‘Cantor dust’. It can be constructed by recursively removing the middle thirds starting with [0, 1]. The Cantor set has length zero but still contains denumerable points, hence it is a set ‘as large as’ the starting set [0, 1]! On this set neither Achilles nor the tortoise can move at all – as Höppner writes: ‘they sink irrecoverably in the Cantor dust’ (... versinken sie unrettbar im Cantor-Staub). If we now look at the paradoxes of Zeno from the point of view of the difference between atomism and the theory of the continuum then we recognise two groups of paradox: Achilles and the tortoise and the dichotomy are aimed against the assumption of a continuum and show that serious problems occur if a continuum is assumed (i.e. arbitrary divisibility). The flying arrow and the stadion are aimed against the assumption of atomism and show that this assumption also leads to severe problems. Imagining time being build up from atoms, hence as a collection of points of time, then, as Zeno says, we can look at the arrow in one of these points of time and we find it standing still. This observation does not seem to be in accordance with the movement of the arrow! Imagining on the other hand the racecourse in the paradox of Achilles and the tortoise being a continuum the runner would have to pass an infinity of parts of that continuum getting progressively smaller. This, says Zeno, can not be done in finite time.


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.2.7. Marble bust of Aristotle (National Museum Rome). Roman copy after the Greek bronze original by Lysippos (330 BC). The alabaster mantle was added in modern times [Photo: Jastrow 2006]

Translating this into the language of analysis we stand here at the cradle of two different views having an effect even today. The great Leibniz (1646–1716) will turn out to be a mathematician of the infinitesimal and he therefore is not troubled by computing with infinitely small quantities. Great Isaac Newton (1643–1727, 1642–1726 old style) became inclined to atomism in his thoughts on physics and even thought about light in an atomistic way. How did the fundamental ideas of the continuum and of atomism find its way into the Western cultural hemisphere? We owe this to Aristotle, to his translators, and to the overwhelming interest of medieval Christian scholastic philosophers in Aristotle. We will have to report on that later. Zeno and his paradoxes are discussed controversially even now. The great English mathematician and philosopher Bertrand Russell (1872–1970) got so excited about the ideas of Zeno that he saw Zeno as a precursor of the mathematics of the 19th century, in particular of Karl Weierstraß [Russell 1903, p. 346ff.]. This certainly drives things too far. At the other end of the spectrum stands Baertel van der Waerden who wanted to marginalise the role of Zeno. He argued in [van der Waerden 1940, p. 141ff.] that atomism developed only after Zeno and as a counter-reaction against the Eleatics. Furthermore he claimed that the Pythagoreans never showed a verifiable interest in infinitesimal methods. This opinion also takes it too far in the other direction.

2.3 Archimedes


It is certain that the question concerning the structure of a straight line lies at the root of analysis and that all researchers like Newton and Leibniz and their successors were influenced by it. However, the Aristotelian continuum cannot be reconciled with our recent set theoretic opinion of the real line being a ‘continuum’. We also have to remark that the continuum in Aristotle’s writings is always closely linked to the problem of movement, cp. [Wieland 1965]. It is this thinking about the nature of movement which will bring the question of the continuum into Christian scholastics.

2.3 Archimedes As splendid as the mathematics of the ancient Greeks may seem; the star outshining everything else – a universal genius – was Archimedes (about 287– 212 BC). His domain was the town of Syracuse on Sicily which then belonged to the Greek realm. Probably Archimedes was even born in Syracuse.

2.3.1 Life, Death, and Anecdotes We know disconcertingly little about the life of this genius. However, some anecdotes have come to us where we have to be very cautious concerning their truthfulness. When he discovered the law of the lever he is said to have stated: ‘Give me a place to stand on, and I will move the Earth’ (quoted by Pappus of Alexandria). Even more famous is the story concerning the crown of King Hiero II (about 306–215 BC). Archimedes was situated at the court of this king and probably even his relative. Hiero II is said to have ordered a second crown as an exact copy of the original one and, although both crowns were of the same weight, was wondering whether the goldsmith had betrayed him concerning the mass of gold in the copy. Archimedes was assigned to examine the case. In order to allow for relaxed thinking he went to a bathhouse and laid down in the warm water. In the tub all of a sudden the idea of the ‘Archimedean principle’ is said to have come to him: every body displaces exactly as much water as it has volume. He immediately jumped out of the bath and ran home naked, crying ‘Eureka! Eureka!’ (‘I have found [it]!’). The Roman architect and engineer Marcus Vitruvius Pollio (Vitruvius) (1st c BC) describes this event in [Vitruvius 1914, Book IX, p. 253f.]: 9. In the case of Archimedes, although he made many wonderful discoveries of diverse kinds, yet of them all, the following, which I shall relate, seems to have been the result of a boundless ingenuity. Hiero, after gaining the royal power in Syracuse, resolved, as a


2 The Continuum in Greek-Hellenistic Antiquity consequence of his successful exploits, to place in a certain temple a golden crown which he had vowed to the immortal gods. He contracted for its making at a fixed price, and weighed out a precise amount of gold to the contractor. At the appointed time the latter delivered to the king’s satisfaction an exquisitely finished piece of handiwork, and it appeared that in weight the crown corresponded precisely to what the gold had weighed. 10. But afterwards a charge was made that gold had been abstracted and an equivalent weight of silver had been added in the manufacture of the crown. Hiero, thinking it an outrage that he had been tricked, and yet not knowing how to detect the theft, requested Archimedes to consider the matter. The latter, while the case was still on his mind, happened to go to the bath, and on getting into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, without a moment’s delay, and transported with joy, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, `Ενρηκα, ενρηκα΄. 11. Taking this as the beginning of his discovery, it is said that he made two masses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim, and dropped the mass of silver into it. As much water ran out as was equal in bulk to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water. 12. After this experiment, he likewise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight. Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over for the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.

And also further Archimedean inventions are concerned with water. Even today the Archimedean screw shown in the left part of figure 2.3.2 is used to

2.3 Archimedes


Fig. 2.3.1. Archimedes [Oil painting by Domenico Fetti, 1620] (Gemäldegalerie Alter Meister, Staatliche Kunstsammlungen Dresden)

pump water from a lower reservoire to a higher level, for example on rice fields in Asia. One can study these screws even on modern playgrounds where they appear in form of tubes bent into screwshaped form as shown in the right part of figure 2.3.2.


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.3.2. Archimedian screw [Chambers’s Encyclopedia Vol. I. Philadelphia: J. B. Lippincott & Co. 1871, S. 374]

The most famous of all Archimedean anecdotes is woven around his death. Our knowledge comes firstly from Plutarch (about 45–about 125) and secondly from Titus Livius (Livy) (about 59 BC–about AD 17 ). Plutarch in [Plutarch 2004, p. 437–523] describes the life of the Roman consul and general Marcus Claudius Marcellus, called Marcellus (about 268–208 BC) who besieged Syracuse with his troops from land and sea in 214 BC. The siege happened during the Second Punic War (218–201 BC) and was directed agains the Carthaginians. Although King Hiero II was a supporter of the Romans until his death his successor turned to the side of the Carthaginians what made Syracuse a target for Roman attacks. Plutarch reports about an artillery on eight galleys bound together with which Marcellus attacked. But [Plutarch 2004, p. 471]: ‘... all this proved to be of no account in the eyes of Archimedes and in comparison with the engines of Archimedes. To these he had by no means devoted himself as work worthy of his serious effort, but most of them were mere accessories of a geometry practised for amusement, since in the bygone days Hiero the king had eagerly desired and at last persuaded him to turn this art somewhat from abstract notions to material things, and by applying his philosophy somehow to the needs which make themselves felt, to render it more evident to the common mind.’ Marcellus and his troops now had to experience the war machines of Archimedes at first hand and also here the genius of Archimedes shone. Besides the law of the lever Archimedes employed pulleys and therewith built machines never seen before. These machines made it difficult for the Romans to conquer Syracuse. A pulley allowed a claw tied to a long rope to be brought under the bow of a ship; the ship was lifted up and then dropped back so that it broke. Archimedes is said to even have constructed parabolic mirrors with which Roman ships could be set to fire. Livy reports [Livy 1940, Book XXIV, p. 283ff.]:

2.3 Archimedes


Fig. 2.3.3. Archimedes’s contribution to the defence of Syracuse, collage of modern depictions (among others of the Renaissance). The lack of authentic drawings caused artists to produce fantasy images


2 The Continuum in Greek-Hellenistic Antiquity ‘And an undertaking begun with so vigorous an assault would have met with success if one man had not been at Syracuse at that time. It was Archimedes, an unrivalled observer of the heavens and the stars, more remarkable, however, as inventor and contriver of artillery and engines of war, by which the least pains he frustrated whatever the enemy undertook with vast efforts. The walls, carried along uneven hills, mainly high positions and difficult to approach, but some of them low and accessible from level ground, were equipped by him with every kind of artillery, as seemed suited to each place. The wall of Achradina, which, as has been said already, is washed by the sea, was attacked by Marcellus with sixty five-bankers. From most of the ships archers and slingers, also light-armed troops, whose weapon is difficult for the inexpert to return, allowed hardly anyone to stand on the wall without being wounded; and these men kept their ships at a distance from the wall, since range is needed for missile weapons. Other fivebankers, paired together, with the inner oars removed, so that side was brought close to side, were propelled by the outer banks of oars like a single ship, and carried towers of several stories and in addition engines for battering walls. To meet this naval equipment Archimedes disposed artillery of different sizes on the walls. Against ships at a distance he kept discharging stones of great weight; nearer vessels he would attack with lighter and all the more numerous missile weapons. Finally, that his own men might discharge their bolts at the enemy without exposures to wounds, he opened the wall from bottom to top with numerous loopholes about a cubit wide, and through these some, without being seen, shot at the enemy with arrows, others from small scorpions. As for the ships which came closer, in order to be inside the range of his artillery, against these an iron grapnel, fastened to a stout chain, would be thrown on to the bow by means of a swing-beam projecting over the wall. When this sprung backward to the ground owing to the shifting of a heavy leaden weight, it would set the ship on its stern, bow in the air. Then, suddenly released, it would dash the ship, falling, as it were, from the wall, into the sea, to the great alarm of the sailors, and with the result that, even if she fell upright, she would take considerable water. Thus the assault from the sea was baffled, and all hope shifted to a plan to attack from the land with all their forces. But that side also had been provided with the same complete equipment of artillery, at the expense and the pains of Hiero during many years, by the unrivalled art of Archimedes.’

Finally Syracuse fell after a siege of two years. Now Marcellus wanted to talk to Archimedes whom he had learned to admire. The soldier ordered to fetch Archimedes found him absorbed in thoughts over a drawing. Archimedes refused to go with the soldier until he had finished a certain proof and was thereupon slayed with the sword by the angry soldier. But Plutarch even

2.3 Archimedes


Fig. 2.3.4. Copperplate on the title page of the Latin edition of the Thesaurus opticus by Alhazen (Ibn Al-Haytham). Archimedes sets fire to Roman ships by mean of parabolic mirrors

gave two further versions of this murder. In the second version a soldier is said to have killed him immediately while in the third version Archimedes wanted to follow the soldier to Marcellus but wanted to take along with him some of his mechanical models to present them to Marcellus. The soldier panicked because he never saw such models before and assumed that they were weapons which Archimedes could use against him; therefore he killed Archimedes. When Marcellus heard about the death of Archimedes he was grief-stricken and turned his back to the murderer. Livy writes [Livy 1940, Book XXV, p. 461]:


2 The Continuum in Greek-Hellenistic Antiquity ‘While many shameful examples of anger and many of greed were being given, the tradition is that Archimedes, in all the uproar which the alarm of a captured city could produce in the midst of plundering soldiers dashing about, was intent upon the figures which he had traced in the dust and was slain by a soldier, not knowing who he was; that Marcellus was grieved at this, and his burial duly provided for; and that his name and memory were an honour and a protection to his relatives, search even being made for them.’

Many artists took up the death of Archimedes as a motif. Figure 2.3.5 shows a mosaic from the Städtische Galerie Liebieghaus in Frankfurt am Main. It was long assumed that it dates back to antiquity but today experts assume it being either a forgery or a copy from the 18th century. It belongs to the treasure trove of anecdotes that the last sentence spoken by Archimedes before the sword of the soldier pierced him was ‘Noli turbare circulos meos’ (Don’t disturb my circles). However, neither Plutarch nor Livy noted such a sentence. Only Valerius Maximus, a Latin writer of the first century AD, let Archimedes say: ‘Noli obsecro istum disturbar’ (Please do not disturb this), cp. [Stein 1999, p. 3]. In the 12th century this turns into ‘Lad, stay away from my drawing’ (Bursche, bleib’ von meiner Zeichnung weg). Hence we have to dismiss this sentence to the realm of fantasy.

Fig. 2.3.5. Death of Archimedes (Mosaic Städtische Galerie Frankfurt a. M.)

2.3 Archimedes


Without exaggeration it can be said that Archimedes certainly was the greatest engineer and physicist of antiquity; but he is a giant when it comes to mathematics and analysis in particular. However, mankind had pure luck that writings of Archimedes have come upon us at all!

2.3.2 The Fate of Archimedes’s Writings In an unprecedented bloodshed in April 1204 the city of Constantinople fell and went down. Christian crusaders who actually wanted to ‘release’ Jerusalem misappropriated the most radiant European town; they defiled the Hagia Sophia, plundered, pillaged, raped, and – they destroyed and displaced books which had been collected in Constantinople for centuries. Among them there were also three books by Archimedes; the so-called codices A, B, and C. Codices A and B found their way to Sicily but after the battle of Benevento in 1266 they were sold to the pope [Dijksterhuis 1987, p. 37]. Codex B was mentioned for the last time in 1311. After that codex A disappeared; in 1491 it was in the possession of the Italian humanist Giorgo Valla. After his death is was bought by the Prince of Capri, Alberto Pio, then went into possession of his nephew Cardinal Rodolfo Pio in 1550 who died in 1564. After that year the trace of codex A vanished. The Renaissance masters draw their knowledge of the Archimedean writings from codices A and B. Only codex C remained lost. Already with the writings contained in codices A and B Archimedes could be identified as a great mathematician and physicist, but it is codex C that catapulted Archimedes into the heaven of immortals and gave him a place of honour at the side of Newton and Leibniz. The history of codex C is a crime story – no, a thriller – which Arthur Conan Doyle could not have thought up better. The story is described in [Netz/Noel 2007] and we want to follow it in their main features. In the summer holiday of the year 1906 the Danish philologist Johan Ludvig Heiberg (1854–1928) travelled to Constantinople to examine a strange manuscript in the Metochion (ecclesiastical embassy church). Before that he got the information about a palimpsest from a catalogue of 1899 which immediately enthralled him. Palimpsests are parchments – tanned goatskin – which are reused after they had been already written on. Since parchment was an expensive raw material it takes no wonder that authors and writers fell back to already inscribed older parchments. They scraped the old inscription, cut the parchment into a new format, and inscribed it again with their writings. The author of the aforementioned catalogue, a certain Papadopoulos-Kerameus, did not enjoy a permanent position but was paid according to the number of pages he wrote for the catalogue. Therefore he delivered quite extensive descriptions. He not only described the new text on the palimpsest but also the imperfectly scraped parts of the original parchment which he could still read. Philologist


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.3.6. Manuscript from the Archimedes palimpsest [Auction catalogue of Christie’s, New York 1998]

2.3 Archimedes


Heiberg immediately recognised that the original text on the parchment could only have been a writing of Archimedes. Attempts to get the palimpsest to Copenhagen by the help of diplomatic channels failed so that the scholar had to undertake the task of travelling with it himself. In Constantinople Heiberg could meet his greatest hopes: he had located the lost codex C! The New York Times headlined on the 16th July 1907: ‘Big Literary Find in Constantinople – Savant Discovers Books by Archimedes, Copied about 900 A. D.’. The whole newspaper article is reproduced in [Stein 1999, p. 28]. Already from older translations of writings of Archimedes his genius could be seen but how he came up with his mathematical theorems he left in the dark. The palimpsest now contained a letter by Archimedes to his friend Eratosthenes of Cyrene. This letter became falsely4 known as the The Method of Archimedes Treating of Mechanical Theorems in which the master explained how he derived his theorems – by means of an ingenious method of indivisibles which we will have to discuss in some detail. Heiberg deciphered the palimpsest as good as it was possible by naked eye and magnifying glass. He published a translation of the Method in a scientific journal and compiled a completely new edition of the works of Archimedes between 1910 and 1915 – based on the codices A and B which are extinct today and codex C which he just had found again. This new edition by Heiberg became the basis of the English translation by Sir Thomas Heath (1861–1940) [Heath 2002] which made the works of Archimedes internationally known. The palimpsest contained seven more or less complete works: 1. On the equilibrium of planes or the centres of gravity of planes, 2. On floating bodies, 3. The method, 4. On spirals, 5. On the sphere and cylinder, 6. Measurement of a circle, 7. Stomachion (A fragment on a tangram-like game). Three further books have been preserved from other sources, transcriptions, and extracts of codices A and B, respectively: 1. Quadrature of the parabola, 2. The sand-reckoner, 3. On conoids and speroids. 4

As Eberhard Knobloch, himself being a renowned philologist, told me there is no Greek word in the title meaning ‘method’ but ‘access’ instead, [Knobloch 2010]. However, it is too late to change the title – the work is known as The method all over the world.


2 The Continuum in Greek-Hellenistic Antiquity

Fig. 2.3.7. Eratosthenes of Cyrene

Except of the Stomachion which is of no interest to us these works comprise all of the works of Archimedes concerning mathematics and physics we know today. They can be found in [Heath 2002]. The history of codex C is not finished here, however. In 1938 the manuscripts and books of the Metochion were removed to Athens under the eyes of the Turks; however, codex C was not among the lot. Research described in [Netz/Noel 2007] revealed that the palimpsest found its way into a private collection of a French collector. After his death in 1956 his daughter who inherited the palimpset became interested in it in the 1960s. Around 1970 this daughter seemed to have realised the importance of the manuscript in her possession because she was looking at some of the pages being cleared from fungal infestation. She unsuccessfully tried to sell the palimpset for quite some time until it appeared at an auction at Christie’s in New York in the year 1998. The estimated price was given as 800 000 US Dollars. The Greek ministry of education and cultural affairs was one of the bidders but there was also a middleman of an unknown bidder. Eventually the Greek ministry had to drop out and the palimpset was sold for the unbelievable sum of 2 200 000 US Dollars to the great unknown.

2.3 Archimedes


However, the palimpset had suffered a lot since the days of Heiberg. There was fungal infestations all over and damages from humidity so serious that even the parts which could be read by Heiberg by the naked eye were devastatingly damaged. The unknown buyer remains unknown even to this day (Bill Gates has credibly declared that it is not him), but fortunately he offered the palimpset for use in science. It is now on loan in the Walters Art Museum in Baltimore where it is conserved and examined by means of the latest methods in image processing. I can only strongly recommend the web page which was built accompanying the works on the palimpset. The discovery of the palimpset and its first publication by Heiberg as well as the recent results of the research group in Baltimore concerning the resurfaced palimpset have clearly shown Archimedes’s important role in the history of analysis. Now is the time for us to present some of his works. 2.3.3 The Method: Access with Regard to Mechanical Theorems Following Heiberg The Method of Archimedes was called The Method of Archimedes Treating of Mechanical Problems but we have already remarked that the word ‘method’ does not actually appear in the title. Instead, one should rename this work of Archimedes The Access instead of calling it The Method. Hence the correct title would be [Knobloch 2010] Access with regard to mechanical theorems (Zugang hinsichtlich der mechanischen Sätze). The Access in the edition of Heath begins with the words [Heath 2002, p. 12, Appendix after p. 326]: ‘Archimedes to Eratosthenes greeting. I sent you on a former occasion some of the theorems discovered by me, merely writing out the enunciations and inviting you to discover the proofs, which at the moment I did not give.’ Then he starts off telling Eratosthenes that he is going to deliver the proofs in the following. At the beginning he repeats some theorems on the centre of gravity for which he had given proofs already in his work On the equilibrium of planes or the centres of gravity of planes [Heath 2002, p. 189ff.]. In this work particularly we find the law of the lever which is derived in a purely axiomatic manner by Archimedes. The whole theory of the lever rests on only three axioms [Heath 2002, p. 189]:


2 The Continuum in Greek-Hellenistic Antiquity 1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance. 2. If, when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made. 3. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken.

Eventually Archimedes proves the law of the lever in Proposition 6 and 7 [Heath 2002, p. 192]. In our words: A large weight G being a distance D away from the pivot of the lever is in equilibrium with a smaller weight g at distance d from the pivot, if (2.2)

D:d=g:G holds.

With the help of this mechanical method Archimedes now goes on and weighs indivisibles!

* J


$ '

% G

Fig. 2.3.8. The law of the lever

Weighing the Area Under a Parabola To illustrate Archimedes’s ‘Access’ we study his computation of the area of a parabolic segment as shown in figure 2.3.9(a). Archimedes even considered a parabolic segment lying arbitrarily in the plane but the simpler case suffices us. In the parabolic segment we place a triangle ABC where BD is the axis of symmetry. Drawing the tangent to the parabola at point C and erecting the perpendicular in A we denote the point of intersection of tangent and perpendicular

2.3 Archimedes

75 6










1 %


2 ' $ & (a) Auxiliary construction

; $ ' (b) The process of weighing


Fig. 2.3.9. Weighing a parabolic segment

by Z. Extending the segment BC gives the point K while the extension of DB results in point E as shown in figure 2.3.9(a). Archimedes proved and employed the following facts we will take for granted: 1. K lies exactly in the middle of AZ. 2. B lies exactly in the middle of DE. 3. The area of the triangle AKC is just half of the area of triangle AZC. 4. B lies exactly in the middle of KC. 5. The area of the triangle ABC is just half of the area of triangle AKC. From this fact it follows that the area of the triangle ACZ is exactly four times as large as the area of triangle ABC. Now Archimedes used a particular property of the parabola. If we draw a line parallel to to BD, say M X in figure 2.3.9(b), then MX AC = (2.3) OX AX always holds. We do not prove this property but refer the reader to the explanations in [Stein 1999]. We now extend the segment CK to the point T which is defined by the fact that the segments KT and CK are of equal length. This is our lever or


2 The Continuum in Greek-Hellenistic Antiquity

balance beam and the point K is the pivot point. Then we shift the segment OX into the point T so that SH runs through T . Note that SH and OX share the same length. We have thus shifted the indivisible OX which is a part of the parabolic segment to the other side of the lever. Now it follows from the intercept theorem AC KC = AX KN and together with (2.3) this becomes MX KC = . OX KN Since T K = KC we also have MX TK = . OX KN But HS is nothing more than the segment OX shifted to T , hence it also has to hold MX TK = . (2.4) SH KN And this is the actual outrageous! Archimedes treats the line segment M X and SH as weights which are fixed at distances T K and KN , respectively, from the pivot. Their respective ‘weights’ are proportional to their lengths. Equation (2.4) obviously is nothing else than the law of the lever! The two segments M X and SH are obviously in balance on our lever. Since we have not stated any particular condition concerning the point X this balance holds for all segments M X and SH = OX, regardless of where the point X is chosen between A and C. Since SH = OX is an indivisible of the parabola and M X an indivisible of the triangle ACZ the area of the parabola has to be in balance with the area of the triangle if they are located at their respective distances from the pivot point. If we imagine the whole triangle shifted, the centre of gravity of the parabolic segment is located in the point T . But where is the centre of gravity of the triangle ACZ? it is located on the median KC in a distance of two thirds of K and Archimedes knew that, of course. But this means that the lever arm of the triangle is only one third as long as the lever arm T K of the parabola. Since triangle and parabola are in balance the area of the triangle has to be thrice as large as the area of the parabolic segment since the triangle ‘weighs’ thrice as much. Hence Archimedes arrived at the result: The area of the parabolic segment is one third of the area of the triangle ACZ. It is now not difficult to see that the triangle ABC inscribed in the parabola is only one quarter as large as the triangle ACZ. This gives: The area of the parabolic segment is four thirds the area of the triangle ABC.

2.3 Archimedes


The Volume of a Paraboloid of Rotation The method of ‘weighing’ indivisibles naturally works in the case of solids. We consider the simple parabola y = x2 on a segment [−a, a] of the x-axis and ask for the volume which arise when this parabola rotates around the y-axis. This solid obviously is a paraboloid of rotation of height a. As can be seen in figure 2.3.10 we place our paraboloid on the right side of a lever with pivot point A. The segments AH and AD are assumed to be of equal length. We imagine the paraboloid being enclosed in a circular cylinder with volume Vol(cylinder) = base area × height. The height of the cylinder is AD, its base area π · BD2 , and its centre of gravity is point K located exactly in the middle of AD. Our y-axis now points to the right since we have rotated the paraboloid by 90◦ . The segments from A up to the paraboloid are hence our y-values. Therefore AD BD2 = , 2 OS AS since AD is just the y-value of the parabola if the x-value is BD = M S. Then also M S2 AD = OS 2 AS has to hold and hence AS · M S 2 = AD · OS 2 .

Fig. 2.3.10. The paraboloid in the cylinder on the lever


2 The Continuum in Greek-Hellenistic Antiquity

Since AD = AH due to our premises we can also write AS · M S 2 = AH · OS 2 . But we are working with solids and not with areas and our indivisibles are not lines but discs. Hence actually we have   AS · π · M S 2 = AH · π · OS 2 , i.e. the cross sections of the cylinder at S balance the cross sections of the paraboloid at H. If we assume (like Archimedes did) that a solid consists of indivisibles then we arrive at the equation AH · Vol(paraboloid) = AK · Vol(cylinder). Now it is AK = 12 AD and AH = AD so that AD · Vol(paraboloid) =

1 AD · Vol(cylinder) 2

follows and hence: The volume of the paraboloid of revolution is exactly half of the volume of the including cylinder.

2.3.4 The Quadrature of the Parabola by means of Exhaustion We may justifiably assume that Archimedes did not accept the method of weighing indivisibles himself, cp. [Cuomo 2001]. The method simply was too outrageous. Therefore only classical proofs based on the double reductio ad absurdum are contained in the works of Archimedes. In his work Quadrature of the parabola he gave one further proof of the area of a parabolic segment which is fundamentally different from the one he gave in the Method (Access). He employed a proof by exhaustion in that he filled the parabolic segment with triangles. The first triangle is constructed as shown in figure 2.3.11. The parabolic segment is bounded by the chord AC. Let B be the point on the parabola at which the slope of the tangent equals the slope of the chord. Then ABC comprises the first triangle of the exhaustion. We construct further triangles following the same pattern. In the next step the triangle BCP appears together with its ‘sister triangle’ over the segment AB. The point P is defined to be the point on the parabola at which the slope of the tangent equals the slope of the segment BC. Joining the point B with the midpoint D of the segment AC and drawing a parallel line to BD through P defines the points M and Y . A parallel line to AC through P defines the point N as shown in the right part of figure 2.3.11.

2.3 Archimedes






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Episode 310- Chris White: Electric Jesus

May 28 2021 66 mins  

Our guest for episode #310 is filmmaker Chris White about his new film, Electric Jesus. CHRIS WHITE has written and directed three micro-budget features: showbiz comedy CINEMA PURGATORIO (2014, co-writer, director, actor), and broken family dramas GET BETTER (2012, co-writer, co-director, actor) and TAKEN IN (2011, writer, director). He co-wrote the screenplay for SIX LA LOVE STORIES (2016), and has written and directed for the multi-award-winning, web series phenomenon, Star Trek Continues. White has write-directed many acclaimed short films-collecting his most recent for a 5-film, "southern gothic comedy" anthology called UNBECOMING (2016, writer, director). About his fantastic new movie, Electric Jesus, Chris writes: "I was raised by devout Southern Baptist parents and fully immersed in (and committed to) Evangelical Christian youth culture—which included Sunday School, Bible studies, summer camps, retreats, choir tours, ski trips—all of it set to an ‘80s Christian rock soundtrack. This immersive religious culture is difficult to explain to many of my friends today—but it’s even more difficult to explain why I loved it. The fact that something so alien to most of the world is so vivid in my memory...and kind of embarrassing to talk about now... It makes me feel odd. Just listen to a Christian hair metal anthem of the era—let’s say Stryper’s “To Hell With the Devil”—and you’ll start to understand. Honestly, I don’t know whether to laugh or cry when I revisit that time in my mind, but either way, there’s no looking away. Turns out, my Christian friends and I were a lot more like all of you than I’d thought. Who can’t relate to being young and wistful, devoted to a big unifying idea…and in love? Who doesn’t remember the moment or the moments when you saw your youth, your naiveté...your hope slip away? We’ve all been young. We’ve all had plans and dreams and loves that didn’t work out the way we’d hoped. And from time to time, we think about it. We remember. ELECTRIC JESUS was born out of years of looking back, reconstructing, re-discovering moments and memories I’d long since left behind that suddenly fascinated me. I missed being a Christian youth group kid. I missed the certainty, the comfort…I missed Jesus. But then, as I wrote and eventually as we shot the film, a bigger revelation came to me. I’d been operating under the illusion that my churchy teen years were all about me—that I’d been the sole protagonist in an origin story about me…that coming of age was something that happened to me, while everyone else was just kinda along for the co-stars. So that’s what ELECTRIC JESUS came to be about to me: believing in something so much it all gets too big to fail, all the while completely missing the existentially huge story that’s happening right under your nose. And only being able to realize that several decades later when it’s too late to do it over."


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3000 Years of Analysis: Mathematics in History and Culture 3030582213, 9783030582210

Table of contents :
About the Author
Preface of the Author
Preface of the Editors
Advice to the reader
1 Prologue: 3000 Years of Analysis
1.1 What is ‘Analysis’?
1.2 Precursors of ˇ
1.3 The of the Bible
1.4 Volume of a Frustum of a Pyramid
1.5 Babylonian Approximation of 2
2 The Continuum in Greek-Hellenistic Antiquity
2.1 The Greeks Shape Mathematics
2.1.1 The Very Beginning: Thales of Miletus and his Pupils
2.1.2 The Pythagoreans
2.1.3 The Proportion Theory of Eudoxus in Euclid’s Elements
2.1.4 The Method of Exhaustion – Integration in the Greek Fashion
2.1.5 The Problem of Horn Angles
2.1.6 The Three Classical Problems of Antiquity
Concerning the Quadrature of the Circle
Concerning the Trisection of the Angle
Concerning the Doubling of the Cube
2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles
2.2.1 The Eleatics
2.2.2 Atomism and the Theory of the Continuum
2.2.3 Indivisibles and Infinitesimals
2.2.4 The Paradoxes of Zeno
2.3 Archimedes
2.3.1 Life, Death, and Anecdotes
2.3.2 The Fate of Archimedes’s Writings
2.3.3 The Method: Access with Regard to Mechanical Theorems
Weighing the Area Under a Parabola
The Volume of a Paraboloid of Rotation
2.3.4 The Quadrature of the Parabola by means of Exhaustion
2.3.5 On Spirals
2.3.6 Archimedes traps
2.4 The Contributions of the Romans
Approaches to Analysis in the Greek Antiquity
3 How Knowledge Migrates – From Orient to Occident
3.1 The Decline of Mathematics and the Rescue by the Arabs
3.2 The Contributions of the Arabs Concerning Analysis
3.2.1 Avicenna (Ibn S¯in¯a): Polymath in the Orient
3.2.2 Alhazen (Ibn al-Haytham): Physicist and Mathematician
3.2.3 Averroes (Ibn Rushd): Islamic Aristotelian
Contributions of Islamic Scholars to Analysis
4 Continuum and Atomism in Scholasticism
4.1 The Restart in Europe
4.2 The Great Time of the Translators
4.3 The Continuum in Scholasticism
4.3.1 Robert Grosseteste
4.3.2 Roger Bacon
4.3.3 Albertus Magnus
4.3.4 Thomas Bradwardine
Life in the 14th Century: The Black Death
Concerning Infinity
Bradwardine’s Continuum
Latitudes of Form: The Merton Rule as First Law of Motion
4.3.5 Nicole Oresme
Summation of Infinite Serie
Latitudes of Form and the Merton Rule
The Doctrine of Proportions
4.4 Scholastic Dissenters
4.5 Nicholas of Cusa
4.5.1 The Mathematical Works
Contributions to Analyis in the European Middle Ages
5 Indivisibles and Infinitesimals in the Renaissance
5.1 Renaissance: Rebirth of Antiquity
5.2 The Calculators of Barycentres
5.3 Johannes Kepler
5.3.1 New Stereometry of Wine Barrels
5.4 Galileo Galilei
5.4.1 Galileo’s Treatment of the Infinite
Aristotle’s Wheel
Galilei and Indivisibles
The Cardinality of the Square Numbers
5.5 Cavalieri, Guldin, Torricelli, and the High Art of Indivisibles
5.5.1 Cavalieri’s Method of Indivisibles
5.5.2 The Criticism of Guldin
5.5.3 The Criticism of Galilei
5.5.4 Torricelli’s Apparent Paradox
5.5.5 De Saint-Vincent and the Area under the Hyperbola
The Geometric Series of Saint-Vincent
Horn Angles at Saint-Vincent
The Area Under the Hyperbola Following Saint-Vincent
Analysis and Astronomy during the Renaissance
6 At the Turn from the 16th to the 17th Century
6.1 Analysis in France before Leibniz
6.1.1 France at the turn of the 16th to the 17th Century
6.1.2 René Descartes
The Circle Method of Descartes
6.1.3 Pierre de Fermat
The Quadrature of Higher Parabolas
Fermat’s Method of Pseudo-Equality
6.1.4 Blaise Pascal
The Integration of xp
The Characteristic Triangle
Further Works Concerning Analysis
6.1.5 Gilles Personne de Roberval
The Area Under the Cycloid
The Quadrature of xp
6.2 Analysis Prior to Leibniz in the Netherlands
6.2.1 Frans van Schooten
6.2.2 René François Walther de Sluse
6.2.3 Johannes van Waveren Hudde
6.2.4 Christiaan Huygens
6.3 Analysis Before Newton in England
6.3.1 The Discovery of Logarithms
6.3.2 England at the Turn from the 16th to the 17th Century
6.3.3 John Napier and His Logarithms
The Construction of Napier’s Logarithms
Napier’s Kinematic Model
The Early Meaning of Napier’s Logarithms
6.3.4 Henry Briggs and His Logarithms
The Construction Idea of Briggsian Logarithms
The Successive Extraction of Roots
Was Briggs’ Difference Calculus Stolen From Bürgi?
The Early Invention of the Binomial Theorem
6.3.5 England in the 17th Century
6.3.6 John Wallis and the Arithmetic of the Infinite
Wallis and the Establishing of the Royal Society
Wallis’ Mathematics at Oxford
6.3.7 Isaac Barrow and the Love of Geometry
Barrows Mathematics
6.3.8 The Discovery of the Series of the Logarithms by Nicholas Mercator
6.3.9 The First Rectifications: Harriot and Neile
Thomas Harriot
William Neile
6.3.10 James Gregory
6.4 Analysis in India
Development of Analysis in the 16th/17th Century
7 Newton and Leibniz – Giants and Opponents
7.1 Isaac Newton
7.1.1 Childhood and Youth
7.1.2 Student in Cambridge
7.1.3 The Lucasian Professor
7.1.4 Alchemy, Religion, and the Great Crisis
7.1.5 Newton as President of the Royal Society
7.1.6 The Binomial Theorem
7.1.7 The Calculus of Fluxions
7.1.8 The Fundamental Theorem
7.1.9 Chain Rule and Substitutions
7.1.10 Computation with Series
7.1.11 Integration by Substitution
7.1.12 Newtons Last Works Concerning Analysis
7.1.13 Newton and Differential Equations
7.2 Gottfried Wilhelm Leibniz
7.2.1 Childhood, Youth, and Studies
7.2.2 Leibniz in the Service of the Elector of Mainz
7.2.3 Leibniz in Hanover
7.2.4 The Priority Dispute
7.2.5 First Achievements with Difference Sequences
7.2.6 Leibniz’s Notation
7.2.7 The Characteristic Triangle
7.2.8 The Infinitely Small Quantities
7.2.9 The Transmutation Theorem
7.2.10 The Principle of Continuity
7.2.11 Differential Equations with Leibniz
7.3 First Critical Voice: George Berkeley
Development of the Infinitesimal Calculus and the Priority Dispute
8 Absolutism, Enlightenment, Departure to New Shores
8.1 Historical Introduction
8.2 Jacob and John Bernoulli
8.2.1 The Calculus of Variations
8.3 Leonhard Euler
8.3.1 Euler’s Notion of Function
8.3.2 The Infinitely Small in Euler’s View
8.3.3 The Trigonometric Functions
8.4 Brook Taylor
8.4.1 The Taylor Series
8.4.2 Remarks Concerning the Calculus of Differences
8.5 Colin Maclaurin
8.6 The Beginnings of the Algebraic Interpretation
8.6.1 Lagrange’s Algebraic Analysis
8.7 Fourier Series and Multidimensional Analysis
8.7.1 Jean Baptiste Joseph Fourier
8.7.2 Early Discussions of the Wave Equation
8.7.3 Partial Differential Equations and Multidimensional Analysis
8.7.4 A Preview: The Importance of Fourier Series for Analysis
Mathematicians and their Works Concerning the Analysis of the 18th Century
9 On the Way to Conceptual Rigour in the 19th Century
From the French Revolution to the German Empire
Science and Engineering in the Industrial Revolution
9.1 From the Congress of Vienna to the German Empire
9.2 Lines of Developments of Analysis in the 19th Century
9.3 Bernhard Bolzano and the Pradoxes of the Infinite
9.3.1 Bolzano’s Contributions to Analysis
9.4 The Arithmetisation of Analysis: Cauchy
9.4.1 Limit and Continuity
9.4.2 The Convergence of Sequences and Series
9.4.3 Derivative and Integral
9.5 The Development of the Notion of Integral
9.6 The Final Arithmetisation of Analysis: Weierstraß
9.6.1 The Real Numbers
9.6.2 Continuity, Differentiability, and Convergence
9.6.3 Uniformity
9.7 Richard Dedekind and his Companions
9.7.1 The Dedekind Cuts
Substantial Results in Analysis 1800-1872
10 At the Turn to the 20th Century: Set Theory and the Search for the True Continuum
General History 1871 to 1945
Technology and Natural sciences between 1871 and 1945
10.1 From the Establishment of the German Empire to the Global Catastrophes
10.2 Saint George Hunts Down the Dragon: Cantor and Set Theory
10.2.1 Cantor’s Construction of the Real Numbers
10.2.2 Cantor and Dedekind
10.2.3 The Transfinite Numbers
10.2.4 The Reception of Set Theory
10.2.5 Cantor and the Infinitely Small
10.3 Searching for the True Continuum: Paul Du Bois-Reymond
10.4 Searching for the True Continuum: The Intuitionists
10.5 Vector Analysis
10.6 Differential Geometry
10.7 Ordinary Differential Equantions
10.8 Partial Differential Equations
10.9 Analysis Becomes Even More Powerful: Functional Analysis
10.9.1 Basic Notions of Functional Analysis
10.9.2 A Historical Outline of Functional Analysis
Development of Analysis in the 19th and 20th Century
11 Coming to full circle: Infinitesimals in Nonstandard Analysis
General History From the End of WW II to Today
Developments in Natural Sciences and Technology
11.1 From the Cold War up to today
11.1.1 Computer and Sputnik Shock
11.1.2 The Cold War and its End
11.1.3 Bologna Reform, Crises, Terrorism
11.2 The Rebirth of the Infinitely Small Numbers
11.2.1 Mathematics of Infinitesimals in the ‘Black Book’
11.2.2 The Nonstandard Analysis of Laugwitz and Schmieden
11.3 Robinson and the Nonstandard Analysis
11.4 Nonstandard Analysis by Axiomatisation: The Nelson Approach
11.5 Nonstandard Analysis and Smooth Worlds
Development of Nonstandard Analysis
12 Analysis at Every Turn
List of Figures
Index of persons
Subject index

Citation preview

Thomas Sonar

3000 Years of Analysis Mathematics in History and Culture

Thomas Sonar

3000 Years of Analysis Mathematics in History and Culture

Thomas Sonar Institut für Analysis Computational Mathematics Technische Universität Braunschweig Braunschweig, Germany

Translated by Thomas Sonar Braunschweig, Germany

Morton Patricia Oxford, UK

Keith William Morton Oxford, UK

Editor: Project Group “History of Mathematics” of Hildesheim University, Hildesheim, Germany H.W. Alten (deceased), K.-J. Förster, K.-H. Schlote, H. Wesemüller-Kock

Original published by Springer-Verlag GmbH Deutschland 2016, 3000 Jahre Analysis ISBN 978-3-030-58221-0 ISBN 978-3-030-58223-4 (eBook) © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: © Helmut Schwigon This book is published under the imprint Birkhäuser, by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to Eberhard Knobloch in gratefulness.

‘I conceive that these things, king Gelon, will appear incredible to the great majority of people who have not studied mathematics ...’ Archimedes [Archimedes 2002, p. 360]

About the Author

Thomas Sonar was born 1958 in Sehnde next to Hannover. After studying Mechanical Engineering at the University of Applied Sciences (‘Fachhochschule’) Hannover he became a laboratory engineer in the Laboratory for Control Theory of the same University for a short time, and founded an engineering office. He then studied mathematics at the University of Hannover (now Leibniz University), after which he worked from 1987 until 1989 at the German Aerospace Establishment DLR (then DFVLR) in Brunswick for the orbital glider project HERMES as a scientific assistant. Next he went to the University of Stuttgart to work as a PhD student under Prof. Dr. Wolfgang Wendland while spending some time studying under Prof. Keith William Morton, at the Oxford Computing Laboratory. His PhD thesis was defended in 1991 and Thomas Sonar went to Göttingen to work as a mathematician (‘Hausmathematiker’) at the Institute for Theoretical Fluid Mechanics of the DLR; there he developed and coded the first version of the TAU-code for the numerical computation of compressible fluid fields, which is now widely used. In 1995 the postdoctoral lecture qualification for mathematics was obtained from the TU (then TH) Darmstadt on the basis of a habilitation treatise. From 1996 until 1999 Thomas Sonar was full professor of Applied Mathematics at the University of Hamburg and is professor for Technical and Industrial Mathematics at the Technical University of Brunswick since 1999 where he is currently the head of a work group on partial differential equations. In 2003 he declined an offer of a professorship at the Technical University of Kaiserslautern connected with a leading position in the Fraunhofer Institute for Industrial Mathematics ITWM. In the same year Sonar founded the centre of continuing education for mathematics teachers (‘Mathelok’) at the TU Brunswick which stays active with regular events for pupils also. VII


About the Author

Early in his career Thomas Sonar developed an interest in the history of mathematics, publishing in particular on the history of navigation and of logarithms in early modern England, and conducted the widely noticed exhibitions in the ‘Gauss year’ 2005 and in the ‘Euler year’ 2007 in Brunswick. Further publications concern Euler’s analysis, his mechanics and fluid mechanics, the history of mathematical tables, William Gilbert’s magnetic theory, the history of ballistics, the mathematician Richard Dedekind, and the death of Gottfried Wilhelm Leibniz. In 2001 Sonar published a book on Henry Briggs’ early mathematical works after intense research in Merton College, Oxford. In 2011 his book 3000 Jahre Analysis (3000 years of analysis) was published in this series and in December 2014 he edited the correspondence of Richard Dedekind and Heinrich Weber. Altogether Thomas Sonar has published approximately 150 articles and 15 books – partly together with colleagues. He has established a regular lecture on the history of mathematics at the TU Brunswick and has for many years held a lectureship on this topic at the University of Hamburg. Many of his publications also concern the presentation of mathematics and the history of mathematics to a wider public and the improvement of the teaching of mathematics at secondary schools. Thomas Sonar is a member of the Gesellschaft für Bildung und Wissen e.V. (Society for Education and Knowledge), the Braunschweiger Wissenschaftliche Gesellschaft (Brunswick Scientific Society), a corresponding member of the Academy of Sciences in Hamburg, and an honorary member of the Mathematische Gesellschaft in Hamburg (Mathematical Society in Hamburg).

Preface of the Author As an author I was very glad that the German edition of this book which was first published in 2011 was very well received. Indeed, a second edition with corrections and additions was published in 2016. After the English translation The History of the Priority Dispute between Newton and Leibniz of my book Die Geschichte des Prioritätsstreits zwischen Leibniz und Newton was published by Birkhauser in 2018 and my revered ‘language editor’ Pat Morton did not want to retire but was keen to start on another translation I began translating the second edition of 3000 Jahre Analysis. Here it is. The German edition of this book has some precursors in a series of books on the history of mathematics: ‘6000 Jahre Mathematik’, ‘5000 Jahre Geometrie’, ‘4000 Jahre Algebra’, of which only the volume on geometry has been translated into English up to now. It seemed logical to add a volume on the history of analysis to this series and thereby making the history of analysis available to interested non-specialists and a broader audience. The current volume stands out in the series for the following reason. All books in the series were designed to present scientifically reliable facts in a readable form to convey the delight of mathematics and its historical development. But while a cultural history of mathematics can be presented without much mathematical details, while geometry can be described in the history of its constructions in beautiful drawings, and while the history of algebra, at least until the 19th century, can be developed from quite elementary mathematical reflections, this concept naturally has to fail in the case of analysis. In essence analysis is the science of the infinite; namely the infinitely large as well as the infinitely small. Its roots lie already in the fragments of the Pre-Socratic philosophers and their considerations of the ‘continuum’, as well as in the burning question of whether space and time are made ‘continuously’ or made of ‘atoms’. Thin threads of the roots of analysis reach even back to the realms of the Pharaohs and the Babylonians from which the Greek received some of their knowledge. But not later than with Archimedes (about 287–212 BC) analysis reached a maturity which asks for the active involvement of my readership. Not by any stretch of imagination can one grasp the meaning of the Archimedean analysis withouth studying some examples thoroughly and to comprehend the mathematics behind them with pencil and paper. Although after Archimedes this knowledge was buried in the dark again it came back to life at latest with the Renaissance where analysis progressed in giant steps; and again this science calls for the attention of the reader! To put it somehow poetically: Analysis turns out to be a demanding beloved and one has to succumb to her in order to gain some understanding. But have no fear! My remarks are not meant to discourage you; on the contrary: they are meant to increase the excitement concerning the contents of this book. You are required to think from time to time, but then deep and satisfying insights into one of the most important disciplines of mathematics IX


Preface of the Author

wait as a reward. Without analysis the Technical Revolution and the developments of our highly engineered world relating thereto would have been unthinkable. There are several books on the history of analysis on the market and the reader deserves a few remarks concerning the position of this book in relation to others. I do not claim to publish the latest and hitherto unknown research results. However, the present book differs significantly from others. First of all historical developments in the settings are given much attention as is usual for books in our German book series. Furthermore I have put weight on the Pre-Socratic philosophers and the Christian middle ages in which the discussion of the nature of the continuum had been decisive. Finally the common clamp encompassing all areas described in this book is the infinite. This clamp allows me not to surrender to the unbelievable breadth of developments in the 20th century; functional analysis, measure theory, theory of integration, and so on, but rather to end in the nonstandard analysis in which we again find infinitely small and large quantities and in which the continuum of the Pre-Socratic philosophers is honoured again. In this sense we come to full circle which connects Zeno of Elea (about 490–about 430 BC), Thomas Bradwardine (about 1290–1349), Isaac Newton (1643– 1727), Gottfried Wilhelm Leibniz (1646–1716), Leonhard Euler (1707–1783), Karl Weierstraß(1815–1897), Augustin Louis Cauchy (1789–1857), and finally Abraham Robinson (1918–1974) and Detlef Laugwitz (1932–2000). It is also due to this encompassing clamp that I have included the development of set theory in the history of analysis which is unusual. In the light of the history of the handling of infinity set theory certainly belongs here. This book has been made possible by the project group ‘History of Mathematics’ of the University of Hildesheim, Germany, which I want to thank with gratitude. In particular I have to thank the late Heinz-Wilhelm Alten, my friend Klaus-Jürgen Förster, and Karl-Heinz Schlote for their confidence in me. Heiko Wesemüller-Kock has taken care of the design of this book in his usual, professional manner. One can only sense the enormity of his task if one has drawn pictures and sketches, modified or corrected existing diagrams, and designed hundreds of legends of pictures by oneself. The results of his extensive work can be seen in this book and in all other books in our series. Without the publisher, who encouraged this translation, the book would not have come to life. I have to thank Mrs. Sarah Annette Goob and Mrs. Sabrina Hoecklin of Birkhauser Publishers in particular. However, I am not a trained historian. My continuing and long-standing interest in history has helped a lot, of course, but reliable books like the ‘Der große Ploetz’ [Ploetz 2008] or the wonderful little volumes of Reclam Publishers starting their titles with ‘Kleine Geschichte ...’ or ‘Kleine ... Geschichte’ [Maurer 2002], [Altgeld 2001], [Dirlmeier et al. 2007], [Haupt et al. 2008] were indispensable. In case of doubts however, only an informed historian is of real help and I am very lucky that my friend and colleague

Preface of the Author


Gerd Biegel of the ‘Institut für Braunschweigische Regionalgeschichte’ was at my side although permantly suffering from an overload of work. While we smoked many a cigar and drank innumerable cups of espressos he provided insights into many historical contexts. Although in the meantime he finished his studies and is currently working on his PhD thesis my LATEX-wizard Jakob Schönborn, who already cared for the second German edition of this book, the first German edition of Die Geschichte des Prioritätsstreits zwischen Leibniz und Newton, and its English version The History of the Priority Dispute between Newton and Leibniz stood at my side to also supervise the LATEXnical side of this book. I can only thank him wholeheartedly for his commitment to this book project! I am particularly grateful to Prof. Dr. Eberhard Knobloch, not only for his precious time he sacrified while proofreading the German edition but also for numerous constructive criticism and hints concerning correct translations from ancient Greek and Latin. Since he is a true role model not only for me but for a whole generation of scientists this book is dedicated to him. I am most grateful to the wonderful Patricia (Pat) Morton who offered again to turn my ‘Germanic English’ into her lovely Oxford English. Without her encouragement to continue our work on book projects I would have dared to even start working on this book. A book like this costs time; much time! My decision to write this book therefore had serious consequences in particular for my wife Anke. I had to spend a lot of time in my study and in libraries while life went on without me. A lot of money was spend to buy new and second-hand books to enrich my private library on the history of analysis. All this as well as the now meter deep piles of books and manuscripts in our living room, on couches and chairs and on the floor my beloved wife Anke has put up with and she has reacted with humour and only with a few biting remarks. After the two volumes ‘6000 Jahre Mathematik’ by Hans Wußing had been published and were presented to the public during a small ceremony at the town hall of Hildesheim, my wife got to the heart of it: ‘My husband has a mistress who is 6000 years old, and he loves her dearly!’ For this and since she bears with me and my old mistress I thank her with all my heart. Thomas Sonar

Preface of the Editors With great pride we can present here the English translation of the second edition of the German book ‘3000 Jahre Analysis’ for which our author deserves gratitude and appreciation. Concerning the contents of this books it has to be noted that it is not just a translation of the German book. Again some sections have been reworked and some new material has been carefully added. We have to thank again Heiko Wesemüller-Kock and Anne Gottwald for their ernormous work concerning the pictures and the gathering of publication rights at different license suppliers. Their work ensures that the illustrations appearing in this book share the same high quality as in all other books in our series. We are also greatful to Birkhauser Publishers and in particular to our partner Sarah Annette Goob who supported us actively. My gratitude also extends to further persons which Thomas Sonar has already mentioned in his preface. After ‘5000 Years of Geometry’ and ‘The History of the Priority Dispute between Newton and Leibniz’ the ‘3000 Years of Analysis’ is the third book of our German book series appearing in the English language. We wish this book to also become a real success enjoying a wide distribution. May this book find many readers and may it convey an impression of the beauty and meaning of mathematics in our culture. It may perhaps even arouse their interest in mathematics.

Hildesheim, July 2020, for the editors Karl-Heinz Schlote

Klaus-Jürgen Förster

Project group ’History of Mathematics’ at the University of Hildesheim


Advice to the reader Parentheses contain additional insertions, biographical details, or references to figures. Squared brackets contain •

omissions and insertions in quotations

references to the literature within the text

references to sources in legends of figures

In the figure legends squared brackets mark the author/creator of the particular work. Further specifications appear in common paranthesis. Figures are numbered following chapters and sections, e.g. Fig. 10.1.4 means the fourth figure in section 10.1 of chapter 10. The original titles of books and journals appear in italic type, likewise quotations. Further reading or explanations of only shortly described circumstances are marked by references like ‘(cp. more detailed in. . . )’. Literally or textually quoted literature as well as further reading can be found in the bibliography.


Contents 1

Prologue: 3000 Years of Analysis . . . . . . . . . . . . . . . . . . . . . . . . .


1.1 What is ‘Analysis’ ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


1.2 Precursors of π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


1.3 The π of the Bible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


1.4 Volume of a Frustum of a Pyramid . . . . . . . . . . . . . . . . . . . . . . . 10 √ 1.5 Babylonian Approximation of 2 . . . . . . . . . . . . . . . . . . . . . . . . . 14 2

The Continuum in Greek-Hellenistic Antiquity . . . . . . . . . . . 17 2.1 The Greeks Shape Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.1 The Very Beginning: Thales of Miletus and his Pupils . 21 2.1.2 The Pythagoreans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 The Proportion Theory of Eudoxus in Euclid’s Elements 30 2.1.4 The Method of Exhaustion – Integration in the Greek Fashion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.5 The Problem of Horn Angles . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.6 The Three Classical Problems of Antiquity . . . . . . . . . . 41 2.2 Continuum versus Atoms – Infinitesimals versus Indivisibles . 50 2.2.1 The Eleatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.2 Atomism and the Theory of the Continuum . . . . . . . . . . 51 2.2.3 Indivisibles and Infinitesimals . . . . . . . . . . . . . . . . . . . . . . 54 2.2.4 The Paradoxes of Zeno . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.3 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.1 Life, Death, and Anecdotes . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.2 The Fate of Archimedes’s Writings . . . . . . . . . . . . . . . . . 69 2.3.3 The Method: Access with Regard to Mechanical Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3.4 The Quadrature of the Parabola by means of Exhaustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.5 On Spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.3.6 Archimedes traps π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.4 The Contributions of the Romans . . . . . . . . . . . . . . . . . . . . . . . . 88



Contents How Knowledge Migrates – From Orient to Occident . . . . . 91 3.1 The Decline of Mathematics and the Rescue by the Arabs . . . 92 3.2 The Contributions of the Arabs Concerning Analysis . . . . . . . . 98 3.2.1 Avicenna (Ibn S¯ın¯ a): Polymath in the Orient . . . . . . . . . 98 3.2.2 Alhazen (Ibn al-Haytham): Physicist and Mathematician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.2.3 Averroes (Ibn Rushd): Islamic Aristotelian . . . . . . . . . . . 105


Continuum and Atomism in Scholasticism . . . . . . . . . . . . . . . . 109 4.1 The Restart in Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 The Great Time of the Translators . . . . . . . . . . . . . . . . . . . . . . . 120 4.3 The Continuum in Scholasticism . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.3.1 Robert Grosseteste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.3.2 Roger Bacon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.3.3 Albertus Magnus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.3.4 Thomas Bradwardine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.3.5 Nicole Oresme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Scholastic Dissenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.5 Nicholas of Cusa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.5.1 The Mathematical Works . . . . . . . . . . . . . . . . . . . . . . . . . . 152


Indivisibles and Infinitesimals in the Renaissance . . . . . . . . . 155 5.1 Renaissance: Rebirth of Antiquity . . . . . . . . . . . . . . . . . . . . . . . . 157 5.2 The Calculators of Barycentres . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.3 Johannes Kepler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.3.1 New Stereometry of Wine Barrels . . . . . . . . . . . . . . . . . . 191 5.4 Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.4.1 Galileo’s Treatment of the Infinite . . . . . . . . . . . . . . . . . . 203 5.5 Cavalieri, Guldin, Torricelli, and the High Art of Indivisibles . 208 5.5.1 Cavalieri’s Method of Indivisibles . . . . . . . . . . . . . . . . . . . 212 5.5.2 The Criticism of Guldin . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.5.3 The Criticism of Galilei . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5.5.4 Torricelli’s Apparent Paradox . . . . . . . . . . . . . . . . . . . . . . 222 5.5.5 De Saint-Vincent and the Area under the Hyperbola . . 224

Contents 6


At the Turn from the 16th to the 17th Century . . . . . . . . . . 233 6.1 Analysis in France before Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.1.1 France at the turn of the 16th to the 17th Century . . . 235 6.1.2 René Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.1.3 Pierre de Fermat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.1.4 Blaise Pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.1.5 Gilles Personne de Roberval . . . . . . . . . . . . . . . . . . . . . . . 270 6.2 Analysis Prior to Leibniz in the Netherlands . . . . . . . . . . . . . . . 275 6.2.1 Frans van Schooten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 6.2.2 René François Walther de Sluse . . . . . . . . . . . . . . . . . . . . 277 6.2.3 Johannes van Waveren Hudde . . . . . . . . . . . . . . . . . . . . . . 279 6.2.4 Christiaan Huygens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 6.3 Analysis Before Newton in England . . . . . . . . . . . . . . . . . . . . . . . 284 6.3.1 The Discovery of Logarithms . . . . . . . . . . . . . . . . . . . . . . 284 6.3.2 England at the Turn from the 16th to the 17th Century285 6.3.3 John Napier and His Logarithms . . . . . . . . . . . . . . . . . . . 288 6.3.4 Henry Briggs and His Logarithms . . . . . . . . . . . . . . . . . . 295 6.3.5 England in the 17th Century . . . . . . . . . . . . . . . . . . . . . . . 307 6.3.6 John Wallis and the Arithmetic of the Infinite . . . . . . . 310 6.3.7 Isaac Barrow and the Love of Geometry . . . . . . . . . . . . . 318 6.3.8 The Discovery of the Series of the Logarithms by Nicholas Mercator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 6.3.9 The First Rectifications: Harriot and Neile . . . . . . . . . . . 330 6.3.10 James Gregory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 6.4 Analysis in India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339


Newton and Leibniz – Giants and Opponents . . . . . . . . . . . . . 345 7.1 Isaac Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.1.1 Childhood and Youth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.1.2 Student in Cambridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 7.1.3 The Lucasian Professor . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.1.4 Alchemy, Religion, and the Great Crisis . . . . . . . . . . . . . 362 7.1.5 Newton as President of the Royal Society . . . . . . . . . . . . 366 7.1.6 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 7.1.7 The Calculus of Fluxions . . . . . . . . . . . . . . . . . . . . . . . . . . 370


Contents 7.1.8 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . 373 7.1.9 Chain Rule and Substitutions . . . . . . . . . . . . . . . . . . . . . . 375 7.1.10 Computation with Series . . . . . . . . . . . . . . . . . . . . . . . . . . 375 7.1.11 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . 377 7.1.12 Newtons Last Works Concerning Analysis . . . . . . . . . . . 378 7.1.13 Newton and Differential Equations . . . . . . . . . . . . . . . . . 379

7.2 Gottfried Wilhelm Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 7.2.1 Childhood, Youth, and Studies . . . . . . . . . . . . . . . . . . . . . 380 7.2.2 Leibniz in the Service of the Elector of Mainz . . . . . . . . 383 7.2.3 Leibniz in Hanover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 7.2.4 The Priority Dispute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 7.2.5 First Achievements with Difference Sequences . . . . . . . . 398 7.2.6 Leibniz’s Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 7.2.7 The Characteristic Triangle . . . . . . . . . . . . . . . . . . . . . . . . 404 7.2.8 The Infinitely Small Quantities . . . . . . . . . . . . . . . . . . . . . 406 7.2.9 The Transmutation Theorem . . . . . . . . . . . . . . . . . . . . . . 412 7.2.10 The Principle of Continuity . . . . . . . . . . . . . . . . . . . . . . . . 416 7.2.11 Differential Equations with Leibniz . . . . . . . . . . . . . . . . . 418 7.3 First Critical Voice: George Berkeley . . . . . . . . . . . . . . . . . . . . . . 419 8

Absolutism, Enlightenment, Departure to New Shores . . . . 423 8.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 8.2 Jacob and John Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 8.2.1 The Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . 438 8.3 Leonhard Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 8.3.1 Euler’s Notion of Function . . . . . . . . . . . . . . . . . . . . . . . . . 454 8.3.2 The Infinitely Small in Euler’s View . . . . . . . . . . . . . . . . 457 8.3.3 The Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 460 8.4 Brook Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 8.4.1 The Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 8.4.2 Remarks Concerning the Calculus of Differences . . . . . . 465 8.5 Colin Maclaurin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 8.6 The Beginnings of the Algebraic Interpretation . . . . . . . . . . . . . 466 8.6.1 Lagrange’s Algebraic Analysis . . . . . . . . . . . . . . . . . . . . . . 467 8.7 Fourier Series and Multidimensional Analysis . . . . . . . . . . . . . . 469


XIX 8.7.1 Jean Baptiste Joseph Fourier . . . . . . . . . . . . . . . . . . . . . . 469 8.7.2 Early Discussions of the Wave Equation . . . . . . . . . . . . . 472 8.7.3 Partial Differential Equations and Multidimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 8.7.4 A Preview: The Importance of Fourier Series for Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473


On the Way to Conceptual Rigour in the 19th Century . . . 479 9.1 From the Congress of Vienna to the German Empire . . . . . . . . 483 9.2 Lines of Developments of Analysis in the 19th Century . . . . . . 490 9.3 Bernhard Bolzano and the Pradoxes of the Infinite . . . . . . . . . . 491 9.3.1 Bolzano’s Contributions to Analysis . . . . . . . . . . . . . . . . 493 9.4 The Arithmetisation of Analysis: Cauchy . . . . . . . . . . . . . . . . . . 497 9.4.1 Limit and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 9.4.2 The Convergence of Sequences and Series . . . . . . . . . . . . 503 9.4.3 Derivative and Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 9.5 The Development of the Notion of Integral . . . . . . . . . . . . . . . . 507 9.6 The Final Arithmetisation of Analysis: Weierstraß . . . . . . . . . . 515 9.6.1 The Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 9.6.2 Continuity, Differentiability, and Convergence . . . . . . . . 518 9.6.3 Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 9.7 Richard Dedekind and his Companions . . . . . . . . . . . . . . . . . . . . 523 9.7.1 The Dedekind Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

10 At the Turn to the 20th Century: Set Theory and the Search for the True Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 10.1 From the Establishment of the German Empire to the Global Catastrophes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 10.2 Saint George Hunts Down the Dragon: Cantor and Set Theory545 10.2.1 Cantor’s Construction of the Real Numbers . . . . . . . . . . 555 10.2.2 Cantor and Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 10.2.3 The Transfinite Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 565 10.2.4 The Reception of Set Theory . . . . . . . . . . . . . . . . . . . . . . 569 10.2.5 Cantor and the Infinitely Small . . . . . . . . . . . . . . . . . . . . 570 10.3 Searching for the True Continuum: Paul Du Bois-Reymond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 10.4 Searching for the True Continuum: The Intuitionists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573


Contents 10.5 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 10.6 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 10.7 Ordinary Differential Equantions . . . . . . . . . . . . . . . . . . . . . . . . . 584 10.8 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 10.9 Analysis Becomes Even More Powerful: Functional Analysis . 588 10.9.1 Basic Notions of Functional Analysis . . . . . . . . . . . . . . . . 589 10.9.2 A Historical Outline of Functional Analysis . . . . . . . . . . 592

11 Coming to full circle: Infinitesimals in Nonstandard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 11.1 From the Cold War up to today . . . . . . . . . . . . . . . . . . . . . . . . . . 605 11.1.1 Computer and Sputnik Shock . . . . . . . . . . . . . . . . . . . . . . 607 11.1.2 The Cold War and its End . . . . . . . . . . . . . . . . . . . . . . . . 609 11.1.3 Bologna Reform, Crises, Terrorism . . . . . . . . . . . . . . . . . . 611 11.2 The Rebirth of the Infinitely Small Numbers . . . . . . . . . . . . . . . 612 11.2.1 Mathematics of Infinitesimals in the ‘Black Book’ . . . . 613 11.2.2 The Nonstandard Analysis of Laugwitz and Schmieden 616 11.3 Robinson and the Nonstandard Analysis . . . . . . . . . . . . . . . . . . . 619 11.4 Nonstandard Analysis by Axiomatisation: The Nelson Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 11.5 Nonstandard Analysis and Smooth Worlds . . . . . . . . . . . . . . . . . 621 12 Analysis at Every Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Index of persons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

1 Prologue: 3000 Years of Analysis

© Springer Nature Switzerland AG 2021 T. Sonar, 3000 Years of Analysis,



1 Prologue: 3000 Years of Analysis

since 3000 BC

Nomads from the north immigrate to southern Mesopotamia. Sumerian city states emerge and the cuneiform writing on clay tablets. The realms at the Nile unite. Emergence of hieroglyphs about 2707–2170 Old Kingdom in Egypt. Emergence of pyramids; the step pyramid at Saqqara, the bent pyramid at Dahshur, the great pyramids of Khufu, Chefren and Mykerinos 2170–2020 First interim period in Egypt about 2235–2094 Realm of Akkad in Mesopotamia founded by Sargon of Akkad about 2137–1781 Middle Kingdom in Egypt. Mathematical papyri 1850 Presumable time of origin of the Moscow Papyrus 1793–1550 Second interim period in Egypt 1650 Ahmes writes the Rhind Mathematical Papyrus 2000–1595 Ancient Babylonic period in Mesopotamia. Emergence of the first legislative texts of mankind under King Hammurabi (about 1700) 1675 In Mesopotamia a clay tablet is inscribed with the length of the diagonal in a square about 1550–1070 New Kingdom in Egypt. Temple of Hatshepsut and royal tombs in Thebes. Temple of Amun in Karnak. Sun worship of Akhenaten in Amarna 1279–1213 Ramsses II, Temple of Abu Simbel 1070–525 Third interim period and late period in Egypt about 1700–609 Assyrian realm. Mathematical cuneiform texts; zikkurates about 750–620 Neo-Assyrian realm, the first great empire in the history of the world; residences in Nimrud and Nineveh 625–539 Neo-Babylonian realm, heyday of astrology and astronomy 539 Cyrus the Great conquers Babylon 525 Persians conquer Egypt 332 Alexander the Great conquers Egypt Remark: There are differing chronologies in the literature

1.1 What is ‘Analysis’ ?


Fig. 1.0.2. Egypt and Mesopotamia in the pre-Christian era

1.1 What is ‘Analysis’ ? Three thousand years of analysis? Did analysis not emerge in the 17th century by Newton and Leibniz? To answer this question satisfactorily we should look at a definition of ‘analysis’ first. On the internet the following definition1 can be found: ‘Mathematical analysis formally developed in the 17th century ...’ There you go! According to this definition analysis would be approximately 400 years old, but beware: The definition goes on: 1 analysis


1 Prologue: 3000 Years of Analysis ‘... but many of its ideas can be traced back to earlier mathematicians.’

But how far do we have to ‘trace back’ ? Good old reliable Encyclopaedia Britannica defines ‘Analysis (mathematics)’ as ‘a branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration. Since the discovery of the differential and integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz at the end of the 17th century, analysis has grown into an enormous and central field of mathematical research, with applications throughout the sciences and in areas such as finance, economics, and sociology.’ I do not have any problems whatsoever to follow this definition! Analysis is concerned with the mathematics of continuous changes from which problems of tangents, quadrature problems (i.e. the computation of areas below crooked curves), and eventually the actual differential and integral calculus of Newton and Leibniz developed. In a narrower sense analysis is but the mathematical branch of infinite processes and of ‘infinitely small quantities’ and this sense should be the ribbon accompanying us on our journey through history as a kind of Ariadne’s thread. However, this is not possible consistently. The notion of ‘function’ is certainly central to analysis but for a start has nothing to do with infinitely small quantities. Nevertheless a discussion of the concept of functions certainly belongs to the history of analysis. √ How come the 3000 years? Well, special numbers like π or 2 play a certain role and such numbers (or the approximations thereof) can in fact be found in ancient Egypt and in the cultural region of Mesopotamia.

1.2 Precursors of π Already in the famous Papyrus Rhind2 an approximate computation of the area of a circle can be found. Papyrus Rhind was written by a scribe named Ahmes about the year 1650 BC who wrote that he only copied mathematical problems which were at least 200 years older. In Problem 48 of his papyrus Ahmes depicted a circle which is inscribed in a square. We can infer from the calculations following that the square of edge length of 9 units results in an area of 81 square units, and that the circle with diameter 9 units has an area of 64 square units. In Problem 50 a precise instruction to compute a circle area can be found [Gericke 2003, p. 55]: 2

Named after the Scotsman Alexander Henry Rhind who bought the papyrus in 1858 in Luxor.

1.2 Precursors of π


Fig. 1.2.1. The start of the Papyrus Rhind. The Papyrus is 5,5 m long and has a height of 32 cm. It contains problems concerning mathematical themes which nowadays would be called algebra, fractional arithmetic, geometry and trigonometry. It is itself a copy of an original from the 12th dynasty (19th century BC). Scribe Ahmes copied this original about 1650 BC in hieratic writing. (Department of Ancient Egypt and Sudan, British Museum EA 10057, London [Photo: Paul James Cowie])

‘Example of the computation of a circular field of (diameter) 9. What is the amount of its area? Take 1/9 away from it (the diameter). The remainder is 8. Multiply 8 by 8. It becomes 64.’ (Beispiel der Berechnung eines runden Feldes vom (Durchmesser) 9. Was ist der Betrag seiner Fläche? Nimm 1/9 von ihm (dem Durchmesser) weg. Der Rest ist 8. Multipliziere 8 mal 8. Es wird 64.) This calculation rule allows us to conclude that the Egyptians used πEgypt /4 = (8/9)2 as the value for π/4. Since they did neither know the nature nor the role of π we may ask ourselves how this value was achieved. One possibility would be the use of a grid. Circumscribe a square with edge length d around a circle with diameter d units and divide the square into 9 evenly spaced subsquares as shown in figure 1.2.2 (left). The area of the square would grossly overestimate the area of the circle, hence we divide the four subsquares in the corners of the square into two triangles each and count only one each as contributing to the area as in figure 1.2.2 (right). Therefore 5 subsquares and 4 triangles remain and the area of the circle is approximated by  2  2 d 1 d 7 Acircle ≈ 5 · +4· = d2 . 3 2 3 9


1 Prologue: 3000 Years of Analysis

G Fig. 1.2.2. Approximation of the area of a circle from the outside

However, Ahmes gives the approximation Acircle

64 2 ≈ d = 81

8 d 9

2 .

2 He apparently enlarged the (correct) approximation 79 d2 = 63 81 d by an area 1 2 of 81 d to finally arrive at square numbers in numerator and denominator! But did he? Somewhat frustrated Otto Neugebauer (1899 – 1990) commented [Neugebauer 1969a, p. 124]:

‘And it is not understandable how one comes from this term [ 79 d2 for the area of the circle] to the Egyptian formula. Without new sources it therefore makes little sense to express presumptions concerning this formula since the obvious way obviously does not lead directly to the desired result’ (Und es ist nicht einzusehen, wie man von diesem Ausdruck zu der ägyptischen Formel hinüberkommen kann. Ohne neues Textmaterial hat es also wenig Sinn, über die Entstehungsgeschichte dieser Formel Vermutungen zu äußern, da der naheliegende Weg offenbar nicht direkt zum Ziel führt.) Since the true area of a circle is given by Acircle = πr2 = (π/4)d2 the ancient Egyptians worked with the approximate value πEgypt = 3.16049 which is by no means a bad approximation! At least the relative error is only πEgypt − π ≈ 0.00601643, π hence about 0.6%!

1.2 Precursors of π


In the TV production ‘The Story of Maths’ [Du Sautoy 2008] which is well worth watching, mathematician Marcus du Sautoy (b. 1965) gave another explanation of how the Egyptians might have come up with their formula for the area of a circle. Following his explanation the approximation πEgypt /4 = (8/9)2 stems from an ancient Egyptian board game in which spheres filling hemispherical depressions in a wooden board have to be moved around. Using these spheres a circle can be formed having a diameter corresponding to 9 spheres. Redistributing the spheres so that they form a square then this square happens to have an edge length of 8. If du Sautoy’s interpretation is correct then we have here an early attempt to ‘square the circle’. This problem, also called ‘quadrature of the circle’, will occupy us later on.

Fig. 1.2.3. Queen Neferarti (19th dynasty, wife of Ramesses II) playing the game Senet. The rules of the game Senet could be roughly reconstructed. The rules of other games like ‘Hounds and Jackals’ or the Snake Game are mostly unknown (Wall painting in the burial chamber of Nefertari, West Thebes)


1 Prologue: 3000 Years of Analysis

1.3 The π of the Bible The ancient Egyptian value for π was already much more accurate than the ‘biblical’ value. In the first Book of Kings, Chapter 7:23 we read ‘And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and its height was five cubits: and a line of thirty cubits did compass it round about.’ And in the Second Book Chronicles, Chapter 4:2 we find ‘Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.’ Hence the form of the sea is indeed a circle with diameter d = 10 cubits and circumference of U = 30 cubits. Since the relation between circumference and diameter of every circle is U = πd we arrive at πBible =

U 30 = = 3. d 10

This was the value which was also used by the Babylonians and Edwards (b. 1937) in [Edwards 1979, p. 4, Ex.5] gave an attempt to explain it which I find appealing. Instead of approximating the area of a circle in the Egyptian manner one could have come up with the idea of not only circumscribing a square to the circle but also to inscribe another square as in figure 1.3.1. Then the area of the circle should be approximated by the arithmetic mean of the areas of the squares. The area of the circumscribed square apparently is A1 := d2 = 4r2 . According to Pythagoras’ theorem which was well √ known in Mesopotamia it follows that the edge length of the inner square is r2 + r2 = √ 2r, hence the area of the inscribed square turns out to be A2 := 2r2 . Since the area of the circumscribed square overestimates the area of the circle while the area of the inscribed square underestimates it one can hope that the arithmetic mean might yield a useful approximation to the area of the circle: A1 + A2 = 3r2 . ACircle ≈ 2 And in fact here the biblical value of π appears! But that seems not to be the end of the story as far as the Babylonians are concerned. According to Beckmann (1924 – 1993) [Beckmann 1971, p. 21 f.] and Neugebauer [Neugebauer 1969b, p. 46 f.] clay tablets were excavated in 1936 some 200 miles east of Babylon at Susa including computations concerning some geometrical figures. One of the tablets was concerned with a regular hexagon inscribed in a circle and stated that the ratio of the perimeter of the hexagon to the circumference of the circumscribed circle would be

1.3 The π of the Bible


Fig. 1.3.1. Approximating the area of the circle from within

57 36 + 2. 60 60 The Babylonians knew that the perimeter of the hexagon is 6r if the radius is denoted by r, see figure 1.3.2. The ratio sought therefore is 6r/C if C denotes the circumference of the circle. Since π = C/2r we conclude that

57 36 3 = + π 60 602

and hence π = 31/8 = 3.125. This shows that also the Babylonians knew better approximations to π than just 3.

C r r r

Fig. 1.3.2. Approximating the area of the circle by means of a regular hexagon


1 Prologue: 3000 Years of Analysis

1.4 Volume of a Frustum of a Pyramid In the so called Moscow Mathematical Papyrus located at the Pushkin Museum in Moscow one finds Problem 14 which almost points to one of the basic tasks of analysis. In this problem the volume of a frustum of a pyramid is computed.

Fig. 1.4.1. Computation of the volume of a frustum of a pyramid (Moscow Mathematical Papyrus) in hieratic writing and in hieroglyphs

For the master builders of the pyramids this calculation must have been of particular importance since the pyramids were built in layers. A pyramid is therefore nothing more than the sum of frusta of pyramids with a pyramidal part on top. We do not want to speculate here how the Egyptians arrived at their (correct) formula of the volume of a frustum of a pyramid but refer our readers to the corresponding sections in [Gillings 1982] (see also [Scriba/Schreiber 2000, p. 14 ff.], [Wußing 2008, p. 99 f.]). Although Problem 14 is only concerned with the computation using concrete numbers the Egyptians must have been aware of the correct formula

Fig. 1.4.2. A symmetric and a right-angled pyramid with identical base areas and identical heights share the same volume

1.4 Volume of a Frustum of a Pyramid


Fig. 1.4.3. Decomposition of a cube into six symmetric pyramids of half the height with the tip points in the centre (left), and in three right-angled pyramids (right)

V =

 h 2 a + ab + b2 3


for the volume where a and b are the edge lengths of the two deck areas and h denotes the height of the frustum. Neugebauer [Neugebauer 1969a, p. 126] calls this a ‘gem’ (Glanzstück) of Egyptian mathematics. Pointing towards mathematical analysis is the method with which the volume of a pyramid was probably actually computed (in [Gillings 1982] yet another method can be found). For this purpose the Egyptian scribes considered a pyramid in which the top point is located exactly above one of the edge points. Three of those right-angled (or ‘oblique’) pyramids with identical height and base edge together form a cube with identical height and base edge; i.e. the volume of each of the pyramids is a third of the volume of the cube. One can alternatively build a cube from 6 congruent symmetric pyramids with half the height of their base edges. Place one of these pyramids top point to top point above another and fill the free space with the remaining four pyramids as

Fig. 1.4.4. The calculation of a frustum of a pyramid can graphically be understood by division into its geometric basic forms: 1 cuboid in the middle, 4 prisms at the sides, and 4 right-angled pyramids at the edges; in case of the right-angled pyramid the same cuboid but only 2 prisms of twice the volume at the sides and 1 rightangled pyramid of fourfold volume at the edge, so that both frusta have the identical volume of V = h3 (a2 + ab + b2 )


1 Prologue: 3000 Years of Analysis

Fig. 1.4.5. Step Pyramid of Pharao Djoser in Saqqara (about 2600 BC) [Photo: H.-W. Alten]

shown in figure 1.4.3 (left). Imagining now the right-angled pyramid in figure 1.4.2 cut into very many thin slices parallel to the base area and shifting these slices then a symmetric pyramid of the same volume results where its top point is now above the centre of the square base area. The same is valid in case of the frusta of pyramids in figure 1.4.4, of course. Indeed pyramids have emerged in ancient Egypt from many layers. Already the Mastabas of the kings of the first two dynasties (about 3000 – 2700 BC) show these layers. King Djoser, second king of the 3rd dynasty, ordered his original three-stage Mastaba to be increased by three further stages where each of the stages consists of many thin layers of stone cuboids. Hence emerged the famous Step Pyramid of Djoser about 2680 BC in Saqqara. Under the rule of King Sneferu of the 4th dynasty the transition from layers of frusta towards the abstract geometrical form of the pyramid took place. That seems to have been a kind of a great gamble since in the first phase of the building up to a height of approximately 49 meters the construction by means of inwardly inclined layers proved unstable. This was the result of a too steep slope angle of approximately 58 degrees as well as the inclination of the layers. In the second phase the base area was enlarged, the slope angle was decreased to 54 degrees, but the techniques of the inwardly inclined layers was kept. As this also turned out to lead to instabilities the slope angle was further decreased to 43 degrees in a third phase and horizontal layers were put on the present frustum of a pyramid. Hence emerged the Bent Pyramid of Sneferu about

1.4 Volume of a Frustum of a Pyramid


Fig. 1.4.6. The Bent Pyramid of Pharao Sneferu at Dahshur [Photo: H. Wesemüller-Kock]

2615 BC – prototype of the Great Pyramids of Khufu, Khafra, and Menkaura, as well as the other pyramids of the Old Kingdom, all with horizontal layer build-up. Let us again cite Neugebauer [Neugebauer 1969a, p. 126]: ‘What is surprising with this formula [(1.1)] is twofold: for one thing it is its symmetrical form, on the other hand its mathematical correctness which in particular in the case of this formula, if it should be derived correctly, necessarily requires infinitesimal considerations, i.e. leaves the framework of elementary geometry behind.’ (Was an dieser Formel überrascht, ist vor allem zweierlei: einerseits die symmetrische Gestalt, andererseits die mathematische Korrektheit, die ja gerade bei dieser Formel, falls sie auch korrekt abgeleitet werden sollte, bekanntlich mit Notwendigkeit Infinitesimalbetrachtungen verlangt, d.h. über den Rahmen der Elementargeometrie hinausführt.) If the idea of slicing a spatial figure into (principally infinitely many) horizontal layers really was in the heads of the ancient Egyptian scribes then they have anticipated an idea central to analysis, namely the so-called ‘principle of Cavalieri’. Millennia after the building of the Egyptian pyramids Bonaventura Cavalieri (1598–1647) discovered the principle of computing


1 Prologue: 3000 Years of Analysis

Fig. 1.4.7. Layer structure of the pyramid of Khufu (Giza, Cairo) [Photo: H.-W. Alten]

areas and volumes by ‘summing’ so-called ‘indivisibles’. Later this technique was replaced by the notion of definite integrals where (infinitely many) infinitesimal layers are ‘summed’.

1.5 Babylonian Approximation of


In the Yale Babylonian Collection a Babylonian cuneiform clay tablet can be found under the archive number YBC 7289 on which the length of the diagonal of a unit square was approximately computed as an example for using Pythagoras’ theorem, cp. [Alten et al. 2005, p. 41, Abb. 1.3.9], see figure 1.5.1. This cuneiform tablet stems from about 1675 BC. Transcribing the cuneiform text as in figure 1.5.1 results in the numbers a := 30 b := 1, 24, 51, 10 c := 42, 25, 35, written in the sexagesimal system, i.e. a number system with base 60. In this system it is not easy to find the right position of each figure since the Babylonians wrote either 1, 2, 3 ≡ 1 · 602 + 2 · 601 + 3 · 600 = 3723 or 1, 2, 3 ≡ 1 · 601 + 2 · 600 + 3 · 60−1 = 62.0166 . . . depending on the context. Cuneiform experts nowadays follow a suggestion of Otto Neugebauer and write 1, 2, 3

1.5 Babylonian Approximation of



in the first case and 1, 2; 3 in the second place; the ‘;’ marking clearly the position of the ‘sexagesimal point’ and the ‘,’ separating different figures. The separation marks are necessary since there are 59 figures in a sexagesimal system and 587 may mean 5, 8, 7 or 58, 7. If one is used to compute with sexagesimal numbers it is apparent that with the numbers a, b, c above c = a·b is valid. Interpreting the numbers as being the edge length a of the square 2 and √ the length of the diagonal c Pythagoras’ theorem tells us c√ = 2a2 , hence c = 2a. The number b thus should be an approximation of 2 and indeed we compute 1; 24, 51, 10 ≡ 1 · 600 + 24 · 60−1 + 51 · 60−2 + 10 · 60−3 = 1.414213 so that the square (1; 24, 51, 10)2 = 1; 59, 59, 59, 38, 1, 40 is fairly close to 2, cp. [Aaboe 1998]. Apparently the Babylonians knew very well that the length √ of the diagonal of a square√ is the 2-fold of the edge length and √ they had superb approximations of 2 at hand. Irrational numbers like 2 or π were very important in the development of analysis and it is just this class of

√ Fig. 1.5.1. Concerning the computation of 2: a) Cuneiform table YBC 7289 of the Yale Babylonian Collection, b) Reproduction of the text YBC 7289 after Resnikoff, c) The text in Indian-Arabic numerals in the sexagesimal system [Photo: William A. Casselman]


1 Prologue: 3000 Years of Analysis

numbers making an analysis in real numbers possible at all. We have no hint of a deeper understanding of irrational numbers in the Mesopotamian culture area and this must not surprise us. A true understanding of the structure of the real number system will come not before the end of the 19th century. Have we justified the ‘3000 years’ of the title? Khufu’s pyramid was built about 2600 BC. If we concede that the ancient Egyptians already had analytical methods for the computation of volumes of pyramids at their disposal then analysis would be 5000 years old. If we accept that the actual beginning of analysis (in our modern understanding) can be found – as so many other things – in ancient Greece, then the discovery of irrational numbers by Hippasus of Metapontum about the year 500 BC surely is important for the development of analysis. In that case analysis would be 2500 years old. The true answer to the question of the age of analysis is: we simply don’t know! This is the reason why we have chosen 3000 years as a kind of compromise.

2 The Continuum in Greek-Hellenistic Antiquity

© Springer Nature Switzerland AG 2021 T. Sonar, 3000 Years of Analysis,


18 3500 BC

2 The Continuum in Greek-Hellenistic Antiquity

First traces of minoic settlements on Crete. Large influence on the Aegean and on south-western areas of Asia Minor 2nd c. BC Indo-German tribes of the Acheans and Ionians immigrate to the southern Balkan peninsula and merge with the proto-Greek tribe of the Thracians. The Mycenaean culture develops under Minoian influence about 1630 Volcanic eruption of Santorini. Beginning of the destruction of the Minoian culture from 1200 The so-called Sea Peoples devestate the Mediterranean region. The Minoian culture disappears about 1000 Dorian invasion. The tribe of the Dorians gains predomonance in the Peleponnes. Further merging of all Greek tribes. Foundation of cities like Miletus or Ephesos 1200–750 ‘Dark age’ between the end of the Mycanaean culture and the beginning of the archaic time 750–500 Archaic time. Colonisation of the Meditteranian region. Foundation of the Greek Poleis (city-states) about 550 Sparta founds the Peloponnesian League about 500–494 The Ionian Revolt leads to conflicts with the Persian Empire under Darius I 497 The Greek defeat the Persian army in the Battle of Plataea 490 Greek victory at Marathon. Athen massively rearmaments 480 Battle of Thermopylae against the Persians under Darius’ son Xerxes I Sept. 480 Decisive Battle of Salamis. The Persian fleet is whitewashed 478/477 Athen founds the Delian League. Development of the Attic democracy on the basis of the reforms of Solon and Cleisthenes 431–404 Peloponnesian War Between the Delian League and Sparta ends with the voctory of the Spartans 395–387 In the Corinthian War Sparta has to stand up against an alliance of the city-states of Athen, Thebes, Corinth, and Argos 371 Eventually the Spartans succumb to Thebes in the Battle of Leuctra. Thebes becomes the Greek centre of power for a short while about 382–336 Philip II of Macedon. Macedonia achieve predominance in Greece 356–323 Alexander the Great, son of Philip. Victory over the Persian armies, advance to India. Beginning of the Hellenistic age 218–201 2nd Punic War against Carthage 200–197 2nd Macedonian War ends with the defeat of the Macedonians 168 Battle of Pydna. Macedonia is finally beaten by the Romans 146 Greek is completely incorporated into the Roman Empire 30 Ptolemaic Egypt, the last hellenistic enclave, is annexed by Rome

2 The Continuum in Greek-Hellenistic Antiquity



2 The Continuum in Greek-Hellenistic Antiquity

2.1 The Greeks Shape Mathematics Already when Egypt and Mesopotamia were blossoming a further advanced civilisation developed on the island Crete. This civilisation founded the Minoan culture. At the beginning of the 2nd millennium BC Indo-German tribes had immigrated to the southern Balkan peninsula. Under the influence of the Minoan culture the culture of the Mycenaeans developed between about 1600 and 1000 BC. After the so-called Sea Peoples devastated large parts of the Mediterranean region about 1200 BC the Dorian Invasion set in about 1000 BC. The tribe of the Dorians left their native region in northwest Greece (Dalmatia) and spread over the Peloponnese. Further Greek tribes merged and eventually formed a conglomerate of peoples which we today call ‘the Greek’. A ‘dark age’ about 750–500 BC in which the whole Mediterranian region was colonised by Greek peoples was followed by the foundation of the Peloponnesian League founded by Sparta in about 550 BC. This initiated the way into the ‘classic Greek period’ which transformed into the Hellenistic age only with Alexander the Great.

Fig. 2.1.1. Throne Hall in the Palace of Knossos on Crete [Photo: H.-W. Alten]

2.1 The Greeks Shape Mathematics


2.1.1 The Very Beginning: Thales of Miletus and his Pupils At the time of Thales, Miletus in Asia Minor was one of the large Ionian trading towns but already the year of birth of Thales is uncertain. Following Diogenes Laertius who wrote his Lives of Eminent Philosophers [Diogenes Laertius 1931] probably as late as the first half of the third century AD Thales lived between 640 and 562 BC; Gericke [Gericke 2003] mentions 624–548/545 BC as biographical data. Meanwhile the dates of his birth about the year 624 BC and his death about 546 BC seem to be accepted. It is certain that Thales travelled Egypt and that he brought along Egyptian mathematical knowledge to Asia Minor, hence to the cultural area of the Greeks. We do not have any written documents by Thales but his abilities were praised early on so that he was called the first of the seven wise men of antiquity. If we believe in tradition then Thales must have been a Jack of all trades. It is said that he predicted a solar eclipse in the year 585 BC and thereby terminated a long-standing war between the Medes and the Lydians. Schramm has clarified that no paramount knowledge of astronomy was necessary to predict regular eclipses of the moon [Schramm 1994, p. 572f.] since time grids could be found already in Babylonian cuneiform tables. However, this is not true in case of eclipses of the sun and we have to banish this story to the realm of myths. We follow Neugebauer [Neugebauer 1969b, p. 142]: ‘There exists no cycle for solar eclipses visible at a given place [...] No Babylonian theory for predicting a solar eclipse existed at 600 B.C. [...]’,


2.1.2. Thales of Miletus and Detail from the Gate of Miletus (Vorderasiatisches Museum, SMB [Photo: H.-W. Alten])


2 The Continuum in Greek-Hellenistic Antiquity

and even sharper [Neugebauer 1975, p. 604]: ‘Hence there is no justification for considering the story of the “Thales eclipse” as a piece of evidence for Babylonian influence on earliest Greek astronomy. All available sources point to no such contacts until three centuries later.’ Very recently Otta Wenskus (b. 1955) has taken up the case of the ‘Thales eclipse’ again in [Wenskus 2016] and showed clearly that we are concerned with a tale and not with a true story. It is said that Thales was also working as an engineer and that he had a reputation for proving theorems. The latter statement is somehow explosive! The mathematics of the Egyptians and the Mesopotamians was characterised by a strict culture of solving mathematical problems, i.e. pupils learned to calculate via concrete examples. We do not have any evidence that general mathematical theorems were stated (let alone proved) in the archaic cultures. Only concrete problems were solved and even the Pythagorean theorem stating that in any right-angled triangle the square of the hypotenuse c equals the sum of the squares of the two legs a and b was not known in the form of the formula c2 = a2 + b2 but only in form of concrete (pythagorean) number triples like 52 = 42 + 32 . That the ancient Greek felt the necessity of a derivation by means of deduction can simply not be overestimated. Thales is therefore seen by Aristotle (384–322 BC) as the inventor of the natural philosophy way of reasoning [Mansfeld 1999]. This means in particular that Thales demythologised the phenomena of nature and opened them to rational explanations. Thales marks the beginning of the triumphal way of mathematics as a deductive science and with Thales mathematics became an important part of Greek philosophy. Even the word ‘mathematics’ is of Greek origin and means ‘that what belongs to learning’. Hence mathematics was seen as a fundamental part of classical education.

Fig. 2.1.3. Anaxagoras and Anaximenes on coins

2.1 The Greeks Shape Mathematics


The most famous pupils of Thales were Anaximander (b. 611 BC) and Anaximenes (b. 570 BC). Both expanded the natural philosophy of their master but both do surely not play any role in the prehistory of analysis. Only Anaxagoras (500–428 BC) being a pupil of Anaximenes contributed to the history of analysis and that he did so fiercely that his work occupied mathematicians until well into the 19th century! Anaxagoras stemmed from Klazomenai (Klazomenae), an Ionian town which lay about 40 km west of the modern Turkish city of İzmir. With his cosmological theories he rubbed his contemporaries up the wrong way. He stated that the sun was a blazing hot fiery mass of iron which led to him being prosecuted for godlessness and he was sent to jail. Diogenes Laertius [Diogenes Laertius 1972, p. 143] gives us two different versions concerning the outcome of the trial. Since the great statesman Pericles (c. 495 – 429 BC) was a pupil of Anaxagoras he could spare him from the worst so that Anaxagoras got away with a fine and was banished. In the other version Anaxagoras was additionally accused of treason and sentenced to death. You may choose the version which appeals most to you. As things might have been Anaxagoras seemed to have suffered from boredom in prison and hence came up with the following task: The quadrature of the circle: Given a circle with radius r. Construct a coextensive square. The word ‘construct’ contains some dynamite here since shortly after Anaxagoras it went without saying that ‘construct’ would mean a construction with compass and straightedge alone. The straightedge must not show any marks and has to be thought of as an idealised abstract instrument. The compass must not show any reference scale either and is an abstract tool, too. In addition, it is thought of as a collapsible compass; every time you move it away from the paper it will collapse so that it may not serve as a divider. Since a collapsible compass is cumbersome to use (think of transferring a given length to another part of your drawing paper) Euclidhad already in the 3rd century BC in his famous Elements proved a ‘circle equivalence theorem’ as Proposition 2 in Book I [Euclid 1956, Vol. I, p. 244] which eventually allows the use of an ordinary compass not collapsing every time it is lifted from the paper. The quadrature of the circle has occupied generations of mathematicians and the problem vaporised only in the 19th century when Ferdinand Lindemann (1852 – 1939) proved that the number π is a transcendental irrational number, i.e. is not the root of an equation of type an xn + an−1 xn−1 + . . . + a1 x + a0 = 0 with rational coefficients a0 , a1 , . . . , an . This finally proved that the quadrature of the circle was simply impossible. We shall come back to Anaxagoras since he played a further important role in the history of analysis, but let us mention two remarkable things here:


2 The Continuum in Greek-Hellenistic Antiquity

(1) The problem of the quadrature of the circle finally turns out to be a problem concerning properties of irrational numbers; (2) The problem of the existence of irrational numbers seems to have played an important role in the mathematics of the Greek. It is this point we want to illuminate now.

2.1.2 The Pythagoreans In his overview of the history of Greek philosophy Luciano de Crescenzo (b. 1928) called Pythagoras (about 570 – about 496 BC) simply a ‘superstar’ [De Crescenzo 1990, Band 1, p. 61 ff.]. Pythagoras was born about 570 BC on the Ionian island of Samos lying close to the coastal town of Miletus. Following Diogenes Laertius [Diogenes Laertius 1931, S. 323] Pythagoras travelled Egypt and Mesopotamia where he learned mathematics; probably also the famous theorem which now bears his name, the Pythagorean theorem. Iamblichus (c. 250–c. 325) in his Life of Pythagoras [Guthrie 1987, p. 57 ff.] wrote that Pythagoras was kidnapped by soldiers of the Persian King Cambyses (probably the older one with this name) to Babylon. There he is said to have finished his studies of arithmetic, music, and ‘all other sciences’ [Guthrie 1987, p. 61]. When Pythagoras returned to Samos he found the island under the control of the tyrant Polycrates. Although it is said that he acted as teacher of Polycrates’ son he eventually went into exile to the city of Kroton (today Crotone in Calabria) in Southern Italy. There he is said to have established a true aristocracy (=rule of the best) and to have been legislating. Besides these political activities he founded a school in Kroton of which he was the headmaster. However, many thought his ideas inflammatory and hence he had to leave Kroton. He found refuge about 510 BC in Metapontum (now Metaponto) where he died about 496 BC. The Dorian temple of Hera at Metapontum was viewed later by the Romans as ‘school of Pythagoras’. This ‘school’ is frequently called a sect or secret society since strange rules had to be obeyed: -

abstain from beanes,


never break the bread,


eat not the heart,


do not primp by torch-light,


stir up the bed as soon as you are risen; do not leave in it any print of the body,

and many more [Guthrie 1987, p. 159 ff.]. The group must have been very effective: for a long time Pythagoreans were active in leading political positions in large regions of the Greek sphere of influence. We are more concerned with the mathematics of this secret society which was passed from

2.1 The Greeks Shape Mathematics


Fig. 2.1.4. Pythagoras of Samos, medieval wooden figure in the choir stalls of the Ulm Minster [Photo: H. Wesemüller-Kock]

the master down to his pupils. There were at least two kinds of Pythagoreans as we know from Iamblichus: the mathematikoi and the akousmatikoi. The former were those who actively were concerned with mathemata – learning topics – and the latter were the ones who only listened to what they heard (akousmata). Today we would say that the former were mathematicians while the latter were admirer of mathematics. The Pythagorean motto was: ‘All is number!’ We can be sure that ‘number’ meant the natural numbers, N := {1, 2, 3, . . .}. To appreciate the enthusiasm for natural numbers one has to know that Pythagoras left a mathematical theory of harmony. This theory of harmony probably arose from Pythagoras playing on the monochord which, according to Diogenes Laertius, he also invented. One string was stretched over a scale with twelve equally spaced parts. The choice of this division seems smart since 12 is divisible by 2, 3, 4, and 6 and therefore tones of strongly shortened strings could be compared with each other [van der Waerden 1979, p. 370]. The number 12, its half, two thirds and three quarters of it, 12, 9, 8, 6, played a central role in the thinking of the Pythagoreans. They form the proportion


2 The Continuum in Greek-Hellenistic Antiquity 12 : 9 = 8 : 6;

the number 9 being the arithmetic mean of 12 and 6 (9 = (12 + 6)/2), and 8 is the harmonic mean of 12 and 6 (9 = 2/(1/12 + 1/6)). Of course such relations were what the numerologicaly inclined Pythagoreans had waited for! If two tones appearing at the same time sounded pleasantly the Pythagoreans called them symphon. The octave, the pure fifth and the fourth are such tone differences which are symphon. Now we can assign the relation 2:1 to the octave, 3:2 to the fifth, and 4:3 to the fourth. Starting from these basic ratios the Pythagoreans calculated numerous further ones, cp. [van der Waerden 1979, p. 367 f.]. That a tone can be ‘measured’ in comparison to another one in form of a ratio was also found by the Pythagoreans in the case of lengths, areas, and volumes. Two quantities a and b were called commensurable (jointly measurable) if they both are multiples of a third number c, i.e. Two quantities a and b are commensurable :⇔ if there exists a quantity c and two numbers m and n so that a = mc and b = nc holds. This may be expressed differently, namely as a proportion: Two quantities a and b are commensurable :⇔ if a : b = m : n holds for two natural numbers m and n. We have to note that in the eyes of the Greek, comparable quantities always had to be of the same kind, i.e. true to scale or dimension. It was not allowed to compare an area with a length or a volume with an area, but only lengths with lengths, et cetera. The Pythagoreans had arrived at positive rational numbers via the proportions, i.e. in modern words they arrived at the set Q+ of the positive fractions1 . Other numbers did not exist for them – they were not allowed to exist: the natural numbers (‘all is number’) and the proportions comprised the very nature of reality. They were the epitome of being. If ‘all’ is number then everything is commensurable, hence measurable on the same scale. Nothing in their world was incommensurable! However, this fundamental conviction of the Pythagoreans was shattered by one of their own members! The Pythagorean Hippasus of Metapontum, dated by van der Waerden to have lived between 520 and 480 BC2 [van der Waerden 1979, p. 74], is said to have shown the existence of incommensurable quantities and thereby he triggered the first fundamental crisis in the history of mathematics! Very often one reads that his existence proof was carried out at the secret symbol of the 1


However, there is no evidence that the ancient Greek knew how to compute with fractions; they knew only proportions! K. v. Fritz dates his lifetime around 450 BC [von Fritz 1971]

2.1 The Greeks Shape Mathematics




(a) The pentagram inscribed in a pentagon

(b) Concerning infinite reciprocal subtraction at the pentagram

Fig. 2.1.5. Symbol of the Pythagoreans: The pentagram

Pythagoreans: the pentagram. This would have raised Hippasus’ sacrilege to the level of monstrosity. Actually, the edge of a pentagram a and the edge b of the circumscribed pentagon in figure 2.1.5(b) are incommensurable, i.e. for all natural numbers m and n we have a:b= 6 m : n; hence there is no common measure c so that a = mc and b = nc holds simultaneously. To show the incommensurability of a and b Hippasus probably used the ingenious method of reciprocal subtraction which was known long before his times [Scriba/Schreiber 2000, p. 36 ff.]. The method of reciprocal subtraction is nothing else than the well known Euclidean algorithm in geometric outfit. Let us describe the method in form of an algorithm in pseudocode for simplicity: •

As long as a 6= b do: -

if a > b set a := a − b;


otherwise set b := b − a;

print a.

The algorithm terminates after finitely many steps if and only if a and b are commensurable. If a and b are incommensurable the algorithm does not terminate! In the first step of the infinite reciprocal subtraction we subtract b from a. Noting that the edge of the pentagon and the edge of the pentagram are parallel as depicted in figure 2.1.6(a) the remaining part of a will be a1 . This remaining part is now subtracted from b and we get b1 as shown in figure 2.1.6(b). By a parallel shift into the interior of the pentagram we now see that a1 is just the edge length of a smaller pentagram while b1 is the edge length of the associated circumscribed pentagon. Hence already in the second step of the infinite reciprocal subtraction we arrive at a situation identical to








2 The Continuum in Greek-Hellenistic Antiquity





(a) First step of the reciprocal subtraction


(b) Preparations for the second step

Fig. 2.1.6. Infinite reciprocal subtraction to prove incommensurability of edge b and diagonal a of a pentagon

the point of departure! The infinite reciprocal subtraction can therefore not terminate in a finite number of steps and the incommesurability of a and b is proven. Whether Hippasus in fact chose the sacred symbol of the Pythagoreans or if he did not prove anything √ at all we do not know. At least the diagonal in the unit square has length 2 after Pythagoras’ theorem and also in this simpler case the method of infinite reciprocal subtraction would √ have been the method of choice to prove the incommensurability of 1 and 2. It belongs to the folklore of the history of mathematics that the discovery of incommensurable quantities should have shattered the Pythagoreans to the bones. Iamblichus tells us [Guthrie 1987, p. 116]: ‘It is accordingly reported that he who first divulged the theory of commensurable and incommensurable quantities to those unworthy to receive it, was by the Pythagoreans so hated that they not only expelled him from their common association, and from living with him, but also for him constructed a [symbolic] tomb, as for one who had migrated from the human into another life. It is also reported that the Divine Power was so indignant with him who divulged the method of inscribing in a sphere the dodecahedron, one of the so-called solid figures, the composition of the icostagonus. But according to others this is what happened to him who revealed the doctrine of irrational and incommensurable quantities.’

2.1 The Greeks Shape Mathematics


A dodecahedron is a solid built from 12 pentagons and here is where the link to Hippasus comes in if we read Iamblichus [Guthrie 1987, p. 79]: ‘As to Hippasus, however, they acknowledge that he was one of the Pythagoreans, but that he met the doom of the impious in the sea in consequence of having divulged and explained the method of forming a sphere from twelve pentagons; [...] ’ And here is the full folklore: Hippasus of Metapontum discovers the existence of incommensurable numbers. This is a sacrilege so hideous that his own Pythagorean friends devise a devious assassination in that they sink the ship carrying Hippasus so that Hippasus drowned. We have good reasons to mistrust this folklore tradition. Leonid Zhmud argues in [Zhmud 1997, p. 173] that the notion arrhetos used by Plato (428/427 – 348/347 BC) in his dialog Hippias major for irrational numbers indeed translates as ‘secret’ or ‘unspeakable enigma’, but that it just means not more, than ‘not expressible in numbers’. Zhmud explains the reports on Hippasus of Metapontum as coming from translating errors of earlier commentators. Substantiated is only a quarrel between Hippasus and the Pythagoreans which was politically motivated. It is also possible that the discovery of irrational numbers constituted no violation of unwritten Pythagorean laws at all! Another modern critic was David Fowler (1937 – 2004) [Fowler 1999, p. 289 ff.] who presented a painstaking research on the history and analysis of the folklore of the foundational crisis. Although Fowler’s book [Fowler 1999] can also not give definitive answers it offers some interesting and thought-provoking ideas not being in the focus of historians of mathematics before. Where did the folklore of the fundamental crisis due to the discovery of irrational numbers come from? At the height of the really shattering crisis concerning the foundations of mathematics at the beginning of the 20th century mathematician Helmut Hasse (1898 – 1979) and logician and philosopher Heinrich Scholz (1884 – 1956) wrote a paper on the foundational crisis of Greek mathematics [Hasse/Scholz 1928]. It is this paper which most likely triggered the story of the mathematical crisis of the Greeks which is still alive today. Be that as it may the discovery of the incommensurability of certain lengths seems to have brought the Pythagorean program of arithmetising geometry to a halt. It may be that this halt may also have been responsible for the good shape of geometry in Greek mathematics while Arithmetic and Algebra were only treated shabbily. At least the great philosopher Plato declared in his Theaitetos 147d–148a and his Laws 819d–822d that he felt it a shame for the Greek not to have taken up the problem of incommensurable quantities, see [Popper 2006, p. 329].


2 The Continuum in Greek-Hellenistic Antiquity

2.1.3 The Proportion Theory of Eudoxus in Euclid’s Elements The Elements of Euclid [Euclid 1956] comprise a superb summary of Greek mathematics. This famous work divided into 13 books is attributed to Euclid of Alexandria (about 300 BC) although we barely know anything about his existence. Book X contains an axiomatic treatment of commensurable and incommensurable quantities and explains the method of (infinite) reciprocal subtraction. At the end of Book X we find the proof that the diagonal in the unit square and its edge are incommensurable. This proof conducted on a square is the original one if we follow Kurt von Fritz [von Fritz 1971, p. 562]. Even more interesting is Book V, however, containing the proportion theory of Eudoxus of Cnidus. Eudoxus is thought to have been the most ingenious mathematician in Plato’s academy to which he held close contact. At the foundation of modern analysis lies what we now call the Archimedean axiom: Axiom: To every ever so small positive real number ε there exists a natural number n so that: 1 0< 0 were this smallest number a natural number n could be found such that 1/n would fit in between 0 and δ. Hence 1/n would be even smaller than δ. This axiom is the genuine Greek answer to the question of the infinite small: There are no infinitely small numbers! Hence all problems would have been solved but of course the idea of the infinitely small was much too interesting to leave it alone. We can restate the Archimedean axiom equivalently in the following way: To any two quantities y > x > 0 there exists a natural number N so that N · x > y holds. It is this form in which the Archimedean axiom has found its way into Euclid’s Elements. The axiom ought to have been called Eudoxus’ axiom. At the beginning of Book V we find Definition 4 [Euclid 1956, Vol. II, p. 114]: ‘4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.’ And it goes on: ‘5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.’ Today we would formulate this as follows: Definition 5 Eudoxus/Euclid: We define a : b = c : d if and only if for all natural numbers m, n it holds: If n · a > m · b



If n · a = m · b



If n · a < m · b


n · c < m · d.

Now it looks as if Eudoxus had done nothing but to blur the seemingly clear definition of proportions or ratios and had left an incomprehensible definition. Nothing is further from the truth! This Definition is a breakthrough of a special kind. What happens in case of two incommensurable quantities a and b? Then Eudoxus’ definition divides the rational numbers m/n into two disjoint set; namely a set U for which the first possibility in Definition 5 above holds. Members of U are those m/n for which n · a > m · b,


m a < n b


2 The Continuum in Greek-Hellenistic Antiquity

holds. Since a and b are assumed to be incommensurable the middle possibility in Definition 5 above is out of the question, but not the last possibility. Here now is a set O defined containing all those m/n for which n · a < m · b,


m a > n b

holds. Hence we have established the dissection Q=U ∪O of the rational numbers into a lower set U and an upper set O, both sets being disjoint: U ∩ O = ∅. Each number in U is smaller than any number in O. At the ‘interface’ between U and O a new number may be defined which obviously has to be an irrational one. We had to wait well until the second half of the 19th century before Eudoxus’ theory of proportions could be utilised for the construction of the real numbers. This fundamental step was finally carried out by the mathematician Richard Dedekind (1831–1916) from Brunswick, Germany. The foundation of his ‘Dedekind cuts’ is the disjoint dissection of the rational numbers by means of Eudoxus’ definition of proportionals as described above! We shall come back to Dedekind and his cuts in section 9.7.1. Another advance in the definition of proportionality is that now quantities of ‘different kind’ can be compared. Now a and b can be lengths and c and d volumes; their rations are now comparable. But for which tasks has Eudoxus used the Archimedean/Eudoxus’ axiom? This becomes clear also in Book V of Euclid’s Elements when a:c=b:c



should be proven (Proposition 9, [Euclid 1956, Vol. II, p. 153 f.]). We follow [Edwards 1979, p. 14]: Assume a > b. Following the Archimedean axiom there exists a natural number N so that N · (a − b) > c holds. Furthermore there exists a smallest natural number M with the property M · c > N · b, but it also holds N · b ≥ (M − 1) · c since M is the smallest natural number satisfying M c > N b. Adding the inequalities N (a − b) > c and N b ≥ (M − 1)c yields N a > M c, but N b < M c according to our assumption! This contradicts the definition of proportionality. Hence our assumption a > b must have been wrong.

2.1 The Greeks Shape Mathematics


Fig. 2.1.10. Euclid (Statue in the Oxford University Museum of Natural History) [Photo: Thomas Sonar]


2 The Continuum in Greek-Hellenistic Antiquity

This kind of proof is called reductio ad absurdum since an assumption leads to a contradiction. Now only two possibilities are left: either it is a < b or a = b. Following the principle of tertium non datur (principle of the excluded third) we only need to falsify a < b which works analogously to the proof above. All in all the proof is of the kind of a double reductio ad absurdum. We shall come back to this technique shortly.

2.1.4 The Method of Exhaustion – Integration in the Greek Fashion With the Archimedean axiom Eudoxus also brought a method for the computation of areas to life: the method of exhaustion. The name method of exhaustion was invented by Grégoire de Saint-Vincent (cp. section 7.2), who coined it in 1647 [Jahnke 2003a, p. 18]. The area of a curvilinearly bounded figure is filled by polygons and a sequence of such polygons is studied which exhaust the figure better and better. The foundation of this method can be found in Book X of Euclid’s Elements in Proposition 1 [Euclid 1956, Vol. III, p. 14]: ‘Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually. there will be left some magnitude which will be less than the lesser magnitude set out.’ Reading carefully we see that this is an obvious reformulation of the Archimedean axiom. In modern terms we start with two positive quantities G0 and ε and then calculate intermediate quantities G1
G1 follows and we have accomplished the first step. In the second step we start with nε > G1 . Since (n − 1)ε ≥ ε holds it follows nε ≥ 2ε, or nε ε≤ . 2 Therefore we get nε (n − 1)ε > G1 − ε ≥ G1 − . 2 Division by 2 and combining terms yields 3 ε G1 (n − 1)ε + ≥ , 4 4 2 and since (n − 1)ε ≥ ε it follows (n − 1)ε ≥

1 G1 > G 2 , 2

so that the second step is also accomplished. Proceeding in this way eventually leads to ε > Gn in step n. We can now apply this version of the Archimedean axiom to the computation of areas by means of exhaustion. Theorem (Application of the method of exhaustion to the calculation of the area of a circle): Given a circle C with area A(C) and a number ε > 0. Then there exists an inscribed regular polygonP with the property A(C) − A(P ) < ε.

In other words, increasing the number of edges of the polygons inscribed in the circle leads eventually to a polygon the area of which differs from the area of the circle not more than a given arbitrary small positive ε. The method of exhaustion now allows proving the theorem without falling back on a rigorous limit definition. To this end we start with an inscribed square P0 = EF GH as in figure 2.1.11(a). To reference the Archimedean axiom we write G0 := A(C) − A(P0 )


2 The Continuum in Greek-Hellenistic Antiquity


( &



(¶ .




)¶ )





(a) Initial polygon P0 in the circle C

(b) Second step: P1

Fig. 2.1.11. The method of exhaustion exemplified at a circle

for the difference of areas between circle C and square P0 . Doubling the number of edges results in an octagone P1 as the next regular polygon. Further doubling the number of edges leads to a sequence P0 , P1 , P2 , . . . , Pn , . . . where the regular polygon Pn has exactly 2n+2 edges. If we now could show that for Gn := A(C) − A(Pn ) the inequality

1 Gn 2 is valid, then it would follow from the above version of the Archimedean axiom that Gn < ε holds for sufficiently large n. Gn+1
2 · A(segment EKF ) 1 1 = · 4 · A(segment EKF ) = (A(C) − A(P0 )); {z } 2 2 Imposition Wizard Crack 3.1.4 With License Key 2021

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