# Imposition Wizard Crack 3.1.4 With License Key 2021

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### Imposition Wizard Crack 3.1.4 With License Key 2021 -

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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

39

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

or

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

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2 The Continuum in Greek-Hellenistic Antiquity

hence

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

41

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

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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

0 6

6 7

7

Fig. 2.1.14. Lunes of Hippocrates

2.1 The Greeks Shape Mathematics

43

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:

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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

45

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

'

4

6

( \ \

$ 2 (a) Definition of the quadratrix

Θ

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

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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 N N N

2

$

N N N (a) Definition conchoid

of

the

(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

47

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+

3

1

& 0

.

Fig. 2.1.18. A mechanical construction of the conchoid

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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

49

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].

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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

51

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

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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

53

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:

54

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

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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.

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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.

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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!

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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]

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$$$$$$$$ %%%%%%%% &&&&&&&& $$$$$$$$ %%%%%%%% &&&&&&&& 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.

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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.

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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

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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

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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.

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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.]:

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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

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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

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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]:

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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.)

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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

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

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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.

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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.

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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 http://www.archimedespalimpsest.org 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]:

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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!

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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

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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

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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.

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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

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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.

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