History of Mathematics Final Papers Universiteit Leiden Voorjaar 2012 Dr. John Bukowski

Contents:

1. Mathematics in the Mayan Civilization Daniëlle Verburg and Tara van Zalen

2. Logarithms – Henry Briggs, John Napier Yvonne van Haaren and Martin Swinkels

3. Pascal’s Triangle Maurits Carsouw and Tanita de Graaf

4. Evangelista Torricelli and His Proof of the Existence of a Finite Volume Contained by an Infinite Surface Area Ellen Schlebusch and Bart Verbeek

5. John von Neumann Maaike Assendorp and Jurgen Rinkel

6. Euclid’s Parallel Postulate and the Birth of Non-Euclidean Geometry Olfa Jaïbi and Jeroen van Splunder

7. The Seven Bridges of Königsberg Hent van Imhoff and Kasper Meilgaard

8. Cantor and the Countable Versus Uncountable Distinction Roel Jongen and Matthijs Warrens

9. Four Colour Problem Dorian Brown and Marieke Kortsmit

Mathematics in the Mayan civilization History of Mathematics

Dani¨elleVerburg – 0706566 Tara van Zalen – 0837261

May 23, 2012 Abstract In this paper we will discuss both the history of the Maya civilization, and the most important aspects of their mathematics. The most remarkable fact of their mathematics is that the Maya were one of the two civilizations that invented and used the concept zero. Another remarkable fact is that the Maya number system is a base 20 system in stead of a base 10 system, as we have nowadays. We will explain how they wrote down the numbers, and how to do simple arithmetic with their number system. As for the applications of their mathematical skills, the Maya are well known for their precise astronomy and calendars. Their calendar system was one of the most complex, intricate and accurate among all the world’s ancient calendar systems.

Contents

1 History 2

2 Mathematics 3 2.1 The number zero ...... 3 2.2 The base 20 system ...... 3 2.3 Symbolic representation ...... 3 2.4 Sacred numbers ...... 4 2.5 Addition and subtraction ...... 4 2.6 Further calculations ...... 5 2.7 Head glyphs ...... 6

3 Applications of mathematics 7 3.1 Astronomy ...... 7 3.2 Calendars ...... 8 3.2.1 Long Count ...... 8 3.2.2 Calendar Round ...... 9

References 10 1 History

The ﬁrst signs of the Maya go back to 2600 B.C. in what was called Yucat´an.The Maya civilization had a peak around 250 A.D in what is now Mexico, Guatamala, Belize, Honduras and El Salvador. The civilization declined around 900 A.D. but some small groups could survive until the Spanish conquest, which was in the beginning of the sixteenth century. In the classic period, 200-900 A.D. the Maya used a hierarchical system, with kings and nobles who ruled all. The whole society consisted of several independent states. Those states had urban sites near ceremonial centres. Thanks to the modern techniques we know more about the Maya in this period of time. One of the biggest city that is known by now is Tikal in the Southern Lowlands. This city had 50000 inhabitants at its peak. There has been discovered 3000 separate constructions in Tikal.

Figure 1: Overview of the land where the Maya lived.

The Maya created a civilization that was outstanding in many ways. For example they were one of only three civilizations in the world that invented a complete writing system. They also developed astronomical and calendrical systems, to which we will pay more attention later, and writing in hieroglyphs. Also interesting to know is that the Maya built everything without metal tools. Even the palaces, observatories and temple-pyramids. As farmers they made sure that they could trade with other peoples, for which they cleared part of the rain forest. They also made some storage for water in the areas where water was scarce. Around 900 A.D. the southern Maya left their cities. This was the decline of the Maya civilization. It is still a mystery why the southern Maya left. The Maya civilization stopped around 1200 A.D. when the northern Maya joined the Toltec society. But as mentioned, very small groups continued to live until the beginning of the sixteenth century.

2 2 Mathematics

2.1 The number zero

The Maya Civilization was the ﬁrst before any other culture in the world that discovered and used the concept of zero, [1]. This ”invention” of zero has only been done twice in the history of the world. The Europeans never invented the zero. The Romans, for example, never had a zero and so most of their numbers were quite hard to write, and their mathematics very diﬃcult and cumbersome. The Europeans eventually borrowed the number zero from the Arabs, who themselves borrowed it from India.

2.2 The base 20 system

They based their system on counting all the ﬁngers and toes in the human body. This resulted in a vigesimal, or base 20, system. Although they used a vigesimal system the Maya counting system required only three symbols: • a shell representing the value of 0, • a dot representing the value of 1, • and a bar representing the value of 5. The Maya used a shell for zero, since shells are often empty containers: they contain ’nothing’, so they have zero contents. The Maya were one of the ﬁrst who used a positional system, which makes it easier to calculate and write big numbers, long before any other culture. Just as when they wrote words, the Maya used a lot of variety in writing numbers. They could write their numbers both vertically as horizontally. Most of the time the numbers were written vertically. In that case the bars are placed horizontally and the dots go on top of them, so that the vigesimal positions grow up from the base. If the numbers were written horizontally, then the bars were placed vertical and the dots to the left. In addition to plain dots and bars, the ancient Maya often used fancier number glyphs. The use of the glyphs can be found in Section 2.7.

2.3 Symbolic representation

The numbers 0 to 19 were easy to write down. These numbers are just combinations of bars and dots, all on one level. The symbolic representation of the ﬁrst 21 numbers can be found in Figure 2.

Figure 2: Symbolic representation of the numbers 0 to 20.

For the numbers greater than 19, the Maya used an extra level above the ﬁrst level, which represents the numbers 0 to 19. Then the ﬁrst level shows how many 1s are in the number, after subtracting

3 the sum of the numbers in the higher positions. The second level shows how many 20s there are in a number. The two signs are separated and not placed together like the bars and dots on the first level. This is important because it has to be clear that they are in two different positions. For example, 64 could be written as 3 dots on the first level, and 3 dots on the second level, so that 64 = 3 · 1 + 3 · 20. The symbolic representation of the numbers 21 to 34 can be found in Figure 3.

Figure 3: Symbolic representation of the numbers 21 to 34.

For the numbers greater than 399, the Maya used a third position. This position shows how may 400s there are in the number. For higher numbers, also the fourth, fifth, sixth and all higher positions can be used. The fourth positions shows how many times 203 = 8000 is found in the number. Therefore, each larger Maya number is composed of sections, a lower first level with one or more higher levels written above it. All symbols in each level are multiplied by their place-value factor. The first level factor is 1, the second level factor is 20, the third level factor is 400 etc. Hence the ith level has level factor 20i−1. In this way really big numbers can be written using the Maya number system. In fact, there really is no limit to how big a number is one can write. These three symbols were used in various combinations, for example to keep track of calendar events in both past and future. Moreover, with this symbols even uneducated people could do the simple arithmetic needed for trace and commerce.

2.4 Sacred numbers

The Maya considered some numbers more sacred than others, [8]. Some of these are the following numbers: • 20: Since it represented the number of ﬁngers and toes a human being could count on, • 5: Since it represented the number of digits on a hand or foot, • 13: Since it is the number of original Maya gods, • 52: Since it is the number of years in a ”bundle”. A ”bundle” is a concept which is similar to our concept of a century, • 400: Since it is the number of Maya gods of the night.

2.5 Addition and subtraction

Addition only requires the counting of symbols, and the ability to keep symbols on their proper level, [2]. By adding the symbols of two numbers, ﬁve dots are converted into one bar, and four bars on one level are converted into one dot on the next higher level. An example of addition with the Maya numerals can be found in Figure 4.

4 Figure 4: Example of adding 4567 and 5678, which yields 10,245.

Subtraction is also not very complicated. Subtraction is just the cancelling of symbols. • If there are insuﬃcient dots in a level, then one of the bars of that level is converted to 5 dots. • If there are insuﬃcient bars, a dot from the next higher level is converted to four bars in the lower level. An example of subtraction with the Maya numerals can be found in Figure 5.

Figure 5: Example of subtracting 52,963 from 97,549, which yields 44,586.

2.6 Further calculations

Besides adding and subtracting there are of course more operations you can do. In the picture below you can see how to multiply 6 · 126. You ﬁrst have to multiply as we know it, and then add it up. Nothing really special there. For division there are examples known, but there is no evidence that the Maya really used division. So, it would be translating division from nowadays into Maya notation. That’s why we won’t pay attention to this. An example of multiplying with the Maya numerals can be found in Figure 6.

Figure 6: Example of multiplying 6 and 26, which results in 156.

5 2.7 Head glyphs

As already mentioned in Section 2.2, the Maya wrote down the number symbols also in a fancier way. An example how to write the number 6 in four diﬀerent ways can be found in Figure 7.

Figure 7: Four diﬀerent ways for writing the number 6.

At ﬁrst glance, the number glyph on the left may look like the number 8. However, the two loops (one above and one below the solid dot in the middle) do not count as dots. Similarly, the second number glyph from the left, the X does not count as dots. Only solid, circular dots count as dots; loops and X’s don’t count as such. The Maya used the loops and the X’s for artistic reasons. They made all their glyphs more or less square in shape to make them ﬁt together more nicely. The Maya would also often decorate the bars to make them more interesting and artistic. Besides the three common symbols and the fancier way of writing in number glyphs, the Maya also used head glyphs and full body glyphs for the number from 0 to 19, [1]. A few examples of the head glyphs can be found in Figure 8.

Figure 8: Examples of head glyphs as number signs.

The head glyphs were similar to other glyphs representing gods. This led to confusion in decoding the glyphs. Also, the head glyphs were sometimes compounds, so that for example two head glyphs were merged into one. The head glyphs were also combined with the usual Maya number symbol.

6 3 Applications of mathematics

Mathematics was a sufficiently important discipline among the Maya that it appears in Maya art such as wall paintings. Here the mathematicians can be recognized by number scrolls which trail from under their arms. Interestingly, the first mathematician identified as such on a glyph was a female figure. However, most of the applications of mathematics were found in the Maya astronomy and calendars.

3.1 Astronomy

The Maya are well known for their precise calendar and astronomy. The Maya astronomer priests looked to the heavens for guidance. They used observatories, shadow-casting devices and observations of the horizon to trace the complex motions of the sun, the stars and the planets, in order to observe, calculate and record this information in their chronicles: the codices. From these observations, the Maya developed calendars to keep track of celestial movements and the passage of time. The Maya built observatories at many of their cities, and aligned important structures with the movements of celestial bodies. One of the most notable series of buildings is a complex formed by 4 buildings which forms an astronomical observatory, see Figure 9. This complex is the ﬁrst one found in the Maya world. From this observatory the early Maya could watch the sun rise behind these buildings and mark the beginning of the summer and winter solstices, which are the longest and shortest days. They could also mark the vernal and autumnal equinoxes, since these days are of equal length, [9].

Figure 9: Sketch of the astronomical observatory.

In Maya cities, ceremonial buildings were precisely aligned with compass directions, [8]. At the spring and fall equinoxes, the sun might be made to cast its rays trough small openings in a Maya observatory, lighting up the observatory’s interior walls. The most famous example of an alignment that is related to the exteriors of the temples and places, is the Observatory at Chichen Itza in Mexico, see Figure 10. Each year, people gather there to observe the sun illuminate the stairs of a pyramid dedicated to Quetzalcoatl, the Feathered Serpent god. At the vernal and autumnal equinoxes, the Sun gradually illuminates the pyramid stairs and the serpent head at its base, creating the image of a snake slithering down the sacred mountain to Earth.

Figure 10: The observatory at Chichen Itza in Mexico.

The Maya were deeply concerned with astrology, but they also incorporated their astronomical and

7 calendrical data into an intricate, mathematical discipline. For example they made ingenious constructions of the Venus and eclipse tables. They also expressed pure mathematics in their calendars by determining the least common multiples of various astronomical and calendar cycles.

3.2 Calendars

Of all the world’s ancient calendar systems, the Maya systems are the most complex, intricate and accurate. Calculations of the congruence of the 260-day and the 365-day Maya cycles (see Sections 3.2.1 and 3.2.2) is almost exactly equal to the actual solar year in the tropics, with only a 19 minute margin of error. The Maya used 24 different calendars, based on every celestial body whose movement they could consistently observe and record. However, for simplicity only 2 calendars were used. These calendars are called the Long Count, which records linear time from a mythological zero point, which is approximately 13th of August 3114 B.C. plus or minus 2 days, onwards; and the Calendar Round, which is a cyclical time with two calendrical cycles, called the Tzolk’in and the Haab’. For longer periods the Maya developed the Long Count calendars, which were unlike the Tzolk’in and the Haab’ linear and therefore theoretically infinite and never ending. Our Gregorian calendar is similar in that it can be extended to refer to any date in the future or in the past. Despite the Maya mathematical system is a base 20 positional system, in Maya calendrical calculations, the Haab’ coefficient breaks the harmonic vigesimal rule being a multiple of 18 times 20 in stead of 20 times 20. With this exception to the rule the Maya were approximating the closest possible number of days to the solar years, thereby reaching a compromise of 360 days. Note that the Haab’ coefficient, which consists of 360 days, in the Long Count calendar (see Section 3.2.1) is not to be confused with the Haab’ calendar, which consists of 365 days, in the Calendar Round (see Section 3.2.2). Figure 11 represents a calendar of the Maya.

Figure 11: Calendar of the Maya.

3.2.1 Long Count

The Long Count calendar resembles our linear calendar with the exception that the Long Count is reckoned in days in stead of years. The Long Count has therefore advantages over our system as regards to precision in recording time using only one calendrical system. The calendar exists of 13 b’ak’tun, with 1 b’ak’tun equal to 144,000 days. The current 13th b’ak’tun will end on December 21, 2012. A fragment from the seventh century B.C. bears the only written reference to 2012 ever found. It contains an inscription stating that one of the gods of the underworld will appear in December 2012. To some, that means a great Maya deity will rise up and destroy the earth. But for most scholars, however, this prediction does not signal the end of the Maya calendar, or

8 the destruction of the world. It simply underscores the importance of the end of one cycle and the beginning of a new one. Another theory is that the end of the Long Count is a miscalculation, and the Long Count will end in 2220. On Friday May 11, 2012 there was some news about the Maya and their ’prediction’ that the world would be destroyed on December 21, 2012, [7]. In 2010 scientists found some hieroglyphs of a calendar in the ruined city Xult´un.After a lot of investigation they have discovered that the world will go on with its existence. The calendar that has been found will continue for 7000 years. Even after that they accept that the time will go on.

3.2.2 Calendar Round

The Tzolk’in is a cycle of 260 days, made up of the permutation of 13 numbers with twenty named days. The first day of the Tzolk’in is ”1 Imix”, the next day ”2 Ik”, the third ”3 Ak’b’al”, and so on, until after 260 different combinations ”1 Imix” occurs again. The Haab’ is a solar year of 365 days, made up of 18 named ”months” of 20 days each, with 5 extra days added on at the end of the year. This is almost the same as our year, with the exception that they didn’t make the leap year adjustments every 4 years (although they knew that the length of a year was approximately 365.25 days). The first day of the first month is ”1 Pop”, the next day ”2 Pop” and so on, until after 365 days ”1 Pop” reoccurs. The beginning of the month was called the ”seating” of the month, and after 19 days Pop is completed and the next moth (Wo) is ”seated”. The Calendar Round date records a specific date by given both its Tzolk’in and its Haab’ positions. Since the least common multiple of 260 and 365 is 18,980 days, or approximately 52 years, the minimal time it takes for a particular Calendar Round date to repeat is 52 years. Again, we can see the use of mathematics arise in the calendars. The repeating cycles of creation and destruction as described in Maya mythology were a reminder of the consequences if humans neglected their obligations to the gods. Humans had an inherent responsibility to the gods who made humanity’s continued existence possible. According to Calendar Round, each 52-year period signalled the renewed possibility of the destruction of the world. This was seen as a frightening time when the gods and other forces of creation and chaos would do battle in the world of mortals, determining the fate of all earthly creatures.

9 References

[1] Mark Pitts (2009), The complete writing in Maya Glyphs Book 2 – Maya numbers and the Maya calendar [2] W. French Anderson (1969), Arithmetic in Maya Numerals [3] Harri Kettunen & Christophe Helmke (2005), Introduction to Maya Hieroglyphs [4] http://www.digitalmeesh.com/maya/history.htm [5] http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html [6] http://www.wiskundeophdc.be/admin/upload/maja.pdf [7] http://www.grenswetenschap.nl/permalink.asp?i=8870 [8] http://www.civilization.ca/cmc/exhibitions/civil/maya/mmc01eng.shtml [9] http://www.authenticmaya.com/maya_astronomy.htm

“Napier was a great lover of astrology, but Briggs was the most satirical man against it that hath been known.”

by T Whittaker, Henry Briggs, Dictionary of National Biography Vol VI (London, 1886), 326-327

Logarithms - Henry Briggs,

John Napier

Final paper History of Mathematics, University of Leiden, May 2012

Martin Swinkels Yvonne van Haaren (s1186477)

Table of contents

1. History of Logarithms ...... 2 1.1 Introduction to Logarithm ...... 2 1.2 Early history ...... 2 1.3 The invention of modern logarithms by Napier and Briggs ...... 4 1.4 Further developments ...... 4 2. John Napier (1550-1617) ...... 6 2.1 Bibliography of Napier ...... 6 2.2 Napier’s approach to logarithm ...... 7 2.3 The impact of Napier ...... 8 3. Henry Briggs (1561-1630) ...... 10 3.1 Bibliography of Henry Briggs ...... 10 3.2 Briggs’s approach to logarithm ...... 11 3.3 The impact of Briggs ...... 11 4. Logarithm tables: Arithmetica Logarithmica (1624) ...... 12 4.1 The tables ...... 12 4.2 Example of Briggs’s difference method for log(3) ...... 14 5. Applied logarithms: the Slide Rule ...... 16 5.1 History of the Slide Rule ...... 16 5.2 How to use a slide rule ...... 16 Resources ...... 18

1 1. History of Logarithms

1.1 Introduction to Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 10000 to base 10 is 4, because 10000 is 10 to the power 4: 1000 = 104 = 10×10×10×10 . More generally, if x = by, then y is the logarithm of x to base b, and is written logb(x), so log10(10000) = 4.

Logarithms have been invented (or discovered) for the purpose of making calculations in a more efficient way, especially with large numbers and for other calculations that traditionally cost a lot of time, like taking roots. From the invention of logarithms untill the arrival of electronic calculators and computers, logarithms have saved mathematicians, technicians, astronomers and economists large amounts of time.

The basic idea behind the time saving by using logarithms is in replacement of multiplication by addition and division by subtraction, according to the well-known formulas:

Since logarithms are, nowadays, defined using the term ‘exponent’, it may be clear that it is hard to talk about this subject without mentioning the transcendental number e ( ≈ 2.71828182845…), the base of the natural logarithm.

The nature of the concept of logarithm connects it to the concept of ‘geometric series’ of number that differ by a constant factor. Example: 1, 1/3, 1/9, 1/27, 1/81, ….

1.2 Early history

The Babylonians The Babylonians sometime in 2000–1600 BC invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of squares. However it could not be used for division without an additional table of reciprocals.

Archimedes Archimedes, in the third century B.C, used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.

Assuming that the blue triangle has area 1, the total area is an infinite sum:

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives

This is a geometric series with common ratio 1/4 and the fractional part is equal to 1/3:

The sum is

A true application of ‘replacement of multiplication by addition and division by subtraction’ can be found in other writings of Archimedes. In ‘the Sand Reckoner’ he found an upper bound on the number of grains of sand required to fill a sphere large enough to contain the universe as it was known to the Greeks. Archimedes gave an estimate, which we would write as 1063 , as ten million units of the eighth order of numbers, and remarked when defining the various orders of numbers that the addition of the orders of numbers corresponded to their multiplication.

3 And although the use was implicit, we could say that Archimedes had basic understanding of logarithms and their application, although he did not give the concept a name.

Indian mathematics Around 200 AD Indian mathematicians discovered the laws of indices, which also shows basic knowledge of the concept of the connection between multiplying numbers by adding exponents.

the Jaina text named ‘Anuyoga Dwara Sutra’ states:

... the second square root multiplied by the third square root is the cube of the third square root.

The third square root was the eighth root of a number. This therefore is the formula

(√√a).(√√√a) = (√√√a)3.

Some historians studying these works believe that they see evidence for the Jaina’s having developed logarithms to base 2.

1.3 The invention of modern logarithms by Napier and Briggs

In the 16th century, in the years before the definitive invention of logarithms, a method was developed to quickly estimate the outcome of multiplication of large numbers by relating products of trigonometric functions to sums. The method was developed with contributions of different people, like Paul Wittich, Ibn Yunis and Jost Bürgi.

The first explicit development of the concept of logarithm happened in the 16th century with John Napier (1550-1617) and Henry Briggs (1561-1630). Napier invented the idea. Briggs read about it and proposed the idea of base 10 logarithms to Napier. They eventually cooperated and both contributed greatly to the development of the theory of logarithms. The details of their work are described in Chapter 2 and 3.

Jost Bürgi (1552-1632) from Switzerland, invented logarithms independently of John Napier. There is evidence that he invented them in 1588, 6 years before Napier began to work on the subject. But because Bürgi waited 20 years before publishing, Napier was generally perceived and praised as the inventor of logarithms.

1.4 Further developments

After the groundwork had been laid by Napier and Briggs, many other mathematicians worked on the subject. Because of the practical use of logarithms, it was important that extensive and accurate logarithm tables were created. In 1628 the Dutchman Adriaan Vlacq

4 published logarithm tables including the 70000 numbers for which the logarithms had not been calculated by Briggs. The calculations for these tables had been done by Ezechiel De Dekker (1603-1647).

In 1620 the English astronomer Edmund Gunter developed the first precursor of the slide rule,which was developed further by Warner in 1722. (More about slide rules in chapter 5).

The present-day notation of logarithms comes from Leonhard Euler (1707-1783), who connected them to the exponential function in the 18th century.

2. John Napier (1550-1617)

2.1 Bibliography of Napier

John Napier was born in 1550 in Merchiston Castle, near Edinburgh, Scotland. He died on April 4th, 1617 in Edinburg, Scotland.

His father, Archibald Napier, was an important man and owned several estates in Scotland. John was his eldest son.

John Napier went to St. Andrews University in 1563. He left St. Andrews University before completing a degree. There are no records of where he studied and where he acquired his knowledge of classical literature and his knowledge of higher mathematics, but most likely Napier spent some time in Italy, the Netherlands and at the University of Paris. But there is little known of the training Napier had in mathematics.

By 1571 Napier had returned to Scotland and in 1572 most of the estates of his family were handed over to him. He married at around 1573.

Besides running his estates and working on theology, study of mathematics was his hobby. In his mathematical works he writes that he often had hardly time for the calculations needed on this math work.

Napier’s work on logarithms was done while living at Gartness estate. He had conceived the general principles of logarithms in 1594 or before and he spent the next twenty year in developing their theory. In 1614 he published his description of logarithms in Latin in ‘Mirifici logarithmorum canonis descriptio’. This work was considered ‘one of the very greatest scientific discoveries that the world has seen’. Two years later, in 1616, his work was translated into English by the mathematician and cartographer Edward Wright (1561-1615) and this was published in 1616.

In the preface of the book Napier explains the idea behind the discovery, and how he hoped that his logarithm will save calculators much time and free them from the slippery errors of calculations. Other mathematicians had foreseen properties of the correspondence between an arithmetic and geometric progression, but only Napier and Jost Bürgi (1552-1632) constructed tables for the purpose of simplifying calculations. Bürgi’s work was published in incomplete form in 1620, 6 years after Descriptio.

Wright was a friend of Henry Briggs (1561-1630).This friendship may have led Briggs to visit Napier in 1615 and 1616 and further develop the decimal logarithms (see chapter 3).

Napier had problems with his health and died on the 4th of April 1617. He was buried in the old church of the parish of St Cuthbert’s, Edinburgh.

2.2 Napier’s approach to logarithm

In his interest in simplifying computations, Napier introduced a new notion of numbers, which he initially called ‘artificial numbers’. In the latter work Napier introduced the word Logarithm, from the Greek word logos (ratio) and arithmos (number). Most likely he was influenced by the work of the Greek Archimedes.

Napier’s work on Logarithms signifies ‘ratio-number’ and is based upon three ideas:

1. The idea to provide an arithmetic measure of a geometric ratio (comparing arithmetic and geometric progressions) 2. The idea to define a continuous correspondence between two progressions (to use the concept of motion) 3. The idea of doing multiplication via addition

Napier took as origin the value 10 and defined its logarithm to be 0. Any small value x was given a logarithm corresponding to the ratio between 10 and x.

We will give a short summary of Napier’s definition of a logarithm, which is very different from our now so familiar one: a. First, Napier speaks always of the logarithm of a sine, not of a number. He aimed to simplify the Trigonometric calculations and therefore it was the sine that bulked most largely. Note that at that time, the sine was a line, not a ratio, and the whole sine meant the radius of the circle whose half-chords were the sines. b. Secondly, Napier makes use of two moving points to define a logarithm:

Consider two lines AX (of unlimited length) and BY (of fixed length r). Points P and Q start to move simultaneously to the right on the line, starting at resp. A and B. Point P moves with a uniform velocity V. Point Q moves, starting with the velocity V, and moving (not uniformly) so that its velocity at any point, as D, is proportional to the distance DY from D to the end Y of the line BY. If C is the point P has reached, moving with velocity V, when Q, moving in the way described, has reached D then the number which measures AC is the logarithm of the sine (or number) which measures DY.

Napier defined AC (=y) as the logarithm of YD (=x), that is: y = Nap.log x

In Napier's terminology r, the length of BY, is the whole sine; when Q is at B, P is at A so that the logarithm of the whole sine is 0. The logarithms of numbers less than BY are positive ("abundant"); if Q were to the left of B then P would be to the left of A so that the logarithms

7 of numbers greater than the whole sine are negative ("defective"). In the Constructio r = 107 so that log 107 = 0 and log x > 0 as x < 107, log x < 0 as x > 107. 107 is based on the fact that the best tables of sines available were given to seven decimal places and he thought of the argument x as being of the form 102.sinX.

The fundamental rule is: if a : b = c : d then log a - log b = log c - log d.

In the initial definition of the use of logarithms log (1) is not zero, so the following rule goes wrong: log (ab) = log a + log b.

In the approach of Napier we first write the proportion: Say, we have ab : a = b : 1 Then log (ab) - log a = log b - log 1 or log ab = log a + log b - log 1. We use r instead of 1 ; thus let rx = ab so that x : a = b : r log x = log a + log b - log r = log a + log b. Then, when x has been found, multiply by r, which is easy if r = 107.

The definition of logarithm by Napier is not so simple in actual work as the logarithm we nowadays use. Note that a) Napier’s logarithms are not really to any base, but involve a constant 107. His definition of log (x) (y= Nap log (x) matches our 107 ln(107/x) ≈ ln (x), with ln=elog. b) In Napier’s system the sum of two logs y=y1 + y2 is not equal to the log of it’s product, but 7 is equal to 10 x =x1x2. c) Nap log 1 ≠ 0 as 107 (ln 107 – ln(x)) =0 when x=107.

Napier realized there were opportunities to improve his logarithms.

2.3 The impact of Napier

John Napier is most famous for his invention of logarithms, stated at that time ‘as one of the very greatest scientific discoveries that the world has seen’, published in

1. Mirifici Logarithmorum Canonis Descriptio (1614) The Descriptio defines a logarithm, lays down the rules for working with logs, contains a table of logs. It illustrates their use by applying them to the solution of triangles.

Other important works by Napier include:

2. Rabdologia (1617) and the Arte Logistica (1839) the ‘Napier bones’, (also called ‘Napier numbering rods’ i.e. a tool for mechanically multiplying, dividing, and taking square roots and cube roots. (the reason for publishing as said by Napier is that so many of his friends, to whom he had shown the numbering rods, were so pleased with them that they were already becoming

8 widely used, even beginning to be used in foreign countries.)

3. Mirifici Logarithmorum Canonin Constructio (1619) (this work, written several years before the Descriptio, only appeared after his death and contained notes by Briggs).

4. An inventive, technical formulae used in spherical triangles

5. The ‘Napier analogies’, i.e. two formulae used in solving spherical triangles

6. Exponential expressions for trigonometric functions

7. Introducing decimal notations for fractions

Figuur 1 Mirifici Logarithmorum Canonin Constructio (1619)

9 3. Henry Briggs (1561-1630)

3.1 Bibliography of Henry Briggs

Henry Briggs was born in 1561 in Warleywood, Yorkshire, England. He died on Jan 26th, 1630 in Oxford, England.

At grammar school, Henry Briggs became highly proficient at Greek and Latin. After completing his studies, he entered St. John’s College of Cambridge University in 1577, where he received his B.A. degree in 1581 and his M.A. degree in 1585. In 1588 he was elected as fellow of St John’s College and in 1592 he became Reader of the Physic Lecture in London founded by Dr Linacre. In this year Briggs was also appointed as an examiner and lecturer in mathematics at Cambridge. In 1596 he became the first professor of geometry at Gresham College, London. This position he hold for 23 years. Here he became close friends with James Ussher in 1609, who became archbishop of Armagh later. Besides his interest in Navigation, letters to Ussher showed that Briggs was greatly interested in astronomy, especially in studying eclipses. Both interests required heavy calculations an when he read Napier’s work Descriptio on logarithms he recognized its merits. Brigss was already involved in producing tables to aid calculation and he had published two types of tables before he read Napier’s logarithms: - A table to find the height of the pole, The magnetic declination being given (1602) - Tables for the improvement of Navigation (1610)

Briggs made a difficult journey from London to Edinburgh to see Napier in the summer of 1615 (it took at least 4 days by horse and coach in those times, nowadays it takes only 4 hours by train). Prior to his journey he had suggested to Napier in a letter that logs should be to base 10 and Briggs had begun to construct such tables.

Napier replied that he had the same idea, but he replied that he could not undertake the construction of new tables as he was ill and weak.

At their meeting Briggs and Napier discussed some simplifications to the idea and presentation of logarithms. But it was Napier who suggested to Briggs the new tables should be constructed with base 10 and he proposed a more far-reaching change than Briggs had done, namely that zero should be the log of unity, not of the whole sine: log(1)=0 and log (1010)=1010, which would result in a logarithm which is 109 times our present logarithm of base 10. (Later Briggs changed the logarithm to the one we use today). Brigss did construct such tables. He spent a month with Napier on his first visit in 1615, came a second time in 1616 and scheduled a third visit the year after but Napier died before the planned visit.

Briggs’ first work on logarithms Logarithmorum Chilias was published in London in 1617. Briggs’ master piece, Arithmetica Logarithmica, appeared in 1624. This work gave the logarithms of the natural numbers from 1 to 20,000 and 90,000 to 10,0000 computed to 14 decimal places. It also gave tables of natural sine functions to 15 decimal places and the tan and sec functions to 10 decimal places.

10 In the book Briggs suggested that the logs of the missing numbers (between 20,000 and 90,000) might be computed by a team of people and he even offered to supply specially designed paper for the purpose.

The complete tables were printed at Gouda, the Netherlands, in 1628 in an edition by Vlacq. The tables were also published in London in 1633 under the title of Trigonometria Britannica, after Briggs death.

Besides logarithms, his interest in astronomy continued and Briggs wrote on comets and other astronomical and mathematical topics. Briggs was asked to fill the chair of geometry at Oxford and worked on the ninth proposition of Euclid’s elements. Briggs resigned in 1620 from Gresham College so he could focus on his work on Euclid.

Briggs died on 26th of January 1630 in Oxford, England.

3.2 Briggs’s approach to logarithm

See for details chapter 4 of this paper.

3.3 The impact of Briggs

Briggs was the man most responsible for scientists’ acceptance of logarithms. He is of great importance in the development of mathematics, but his greatest achievement was as a contact and public relations man.

He showed a modesty from his writings and he gave full attributions to others.

Briggs gave the credit for the idea to make log(1)=0 to Napier. But Briggs deserves all the credit for his labours in the calculation of logarithms. His name should always be associated with that of Napier in any fair account of the origin of logarithms, as he was the man who made logarithms a vital tool for all scientists.

11 4. Logarithm tables: Arithmetica Logarithmica (1624)

4.1 The tables

Pictures of the book ‘ArithmeticaLogarithmica’ at the University Library of Leiden

Briggs presented in Arithmetica logarithmica (1624) the different methods he used to compute decimal logarithms and build his tables, each method improving computing efficiency and reducing calculation time :

CH 6 'Continued means' (repeated square roots) method CH 8 Difference method CH 11 Proportional parts (interpolation) in the linear region CH 12 Sub tabulation (1st method) using second differences, CH 13 Sub tabulation (2nd method - finite difference method) CH 14 Radix method

We will outline the most important aspects of his methods used to calculate his logarithms.

Besides a table consisting of the logarithms from 1 to 20,000 and from 90,000 to 100,000, accurate to 14 decimal places, the book also contained explanations and examples on how the table had been computed – but without proofs for the validity of his methods. Briggs methods for computing logarithms are fundamentally different from those of Napier. Briggs used a whole range of different tricks.

In his strategy, Briggs used the following discoveries:

12 a) The logarithm of the square root of a number t is equal to 1/2 times the logarithm of that number. b) For x a small number log(1 + x) ~ k x , where k is a constant. He calculated the following value for k: 0,43429448190325180...

Notice that in modern terminology b) can be expressed in the following way: The 1 degree polynomial approximating to the logarithmic function in the point xo = 1 is a good approximation to the logarithm in the neighbourhood of that point.

Briggs’ idea was to bring that number t, to which he wanted to calculate the logarithm, down to a number close to 1 by taking repeated square roots. The logarithm of the resulting number he could then find by applying b) above. To get back to log(t) he needed to multiply by 2n, where n is the number of times he extracted square roots. He used this method to calculate log(2).

For log (2) he applied the square root 54 times! This required an enormous amount of calculations. Fortunately, Briggs had invented a special difference method to reduce the amount of calculations needed by reducing the number of direct square roots. His ingenious “difference method” is presented in the chapter 8.

Using his difference method and other ingenious tricks he was able to calculate the logarithm of a number of primes. Knowing the logarithms of these primes he could rather easily calculate the logarithms to a large number of composite numbers, using the product rule for logarithms.

In chapter 12 and 13 Briggs deals with interpolation methods (i.e. methods to calculate function values of x-values, which are distributed evenly between values with known function values). What he does in chapter 12 actually corresponds to the method now called Newton- forward. However his real great contribution is the unusual method described in chapter 13, which show his excellent understanding of Numerical Analysis.

Without actually applying it, Briggs describes how his difference method can be used to calculate the rest of his table of logarithms between 20,000 and 90,000: - Notice that 20,000 = 5 . 4,000 ; 20,005 = 5 . 4,001 ; 20,010 = 5 . 4,002 ; ... ; 90,000 = 5 . 18,000. Because he had already calculated the logarithms of the first 20,000 natural numbers, the logarithms of 20000, 20005, 20010, 20015, 20020,.. , 90000 could easily be calculated by using the product rule. - Finally he explained how the logarithms of the remaining intermediate natural numbers could be calculated using his special difference method.

Briggs never finished his table. The Dutchman Adrain Vlacq (1600–1666 (maybe 1667)) finished the table and it was published in 1628. The accuracy in Vlacq’s table was however “only” 10 decimals, against Briggs’ 14 decimals.

13 4.2 Example of Briggs’s difference method for log(3)

Log(3)≈0.4771212547 on our calculator.

We will explain Briggs’s difference method and use the example of log(3).

Take number 3. Take of this number 7 repeated square roots up to the result 1.008619847 (see Column A in the table on the left). We aim to expand down the table for a better accuracy.

Here are the details of the difference method:

Briggs noticed -in the repeated square root (col A) - that the fractional parts of the numbers are roughly halved as he scrolled down the column. He had the genial idea of expanding the differences to the right, by subtracting the fraction parts, e. g. line 4:

forming the column B (see below.)

Briggs remarked that the numbers of this column B are decreasing roughly by a factor of a quarter and he continued to the right and discovered –for the other columns - that the decreasing factors were:

Briggs continued his table until the differences in the last column diminished to zero.

Then he computed the missing numbers from right to left and filled the rest of the table, using the same procedure (in red).

, up to the first entry in the first table

= in column A.

This procedure can be repeated one line at a time reducing drastically the number of calculations.

One line further we get (10th square root) = 1.0021480308 in column A , and x = 0.0021480308.

This gives the final result: 10log 3 ≈ 29·kx, with k= 0.43429448190325180..., which is 10log 3 ≈ 0.47763549

Compared to our log(3) ≈ 0.4771212547 we have an accuracy of 3 decimals.

Briggs carried his calculation to 30 decimal places, a precision that requires 50 repeated square roots.

15 5. Applied logarithms: the Slide Rule

5.1 History of the Slide Rule

In 1620 the English astronomer Edmund Gunter made the first mechanical device for doing multiplication and division without using tables, based on the work of Napier and Briggs, creating a 2 foot long stick with numbers spaced at intervals proportionate to their log William Oughtred, values. The first real slide rule was made by William Oughtred. This inventor of the slide rule device consisted of separate ‘sliding’ parts with logarithmic scales on them.

From its invention until the advent of electronic calculators in the 1960’s, the slide rule was improved upon by many different people. In 1722 Warner introduced the two- and three- decade scales. In 1755 an inverted scale was introduced by Everard. In 1815, Roget created the ‘log log’ slide rule with a scale that displayed the logarithm of the logarithm, enabling the user to do calculations with roots and exponents, especially useful for working with fractional powers. Later even more advanced slide rules were made, including extra scales for trigonometric functions.

In 1859 a French artillery lieutenant, Amedee Mannheim, created the ‘modern’ form of the slide rule. Because of the industrial revolution, slide rules became very popular in Europe for use in engineering. In 1881 the slide rule became popular in the United States after Thacher introduced a cylindrical rule. Astronomical work also required fine computations, and in the 19th century in Germany a steel slide rule about 2 meters long was used at an observatory, with a microscope attached, giving it accuracy to six decimal places.

5.2 How to use a slide rule

Until the invention of the electronic calculator, many different types and forms of slide rules have been constructed for many different types of calculation. The basic use of the slide rule however, is multiplication and division. In the picture below, one can see how the numbers 2 and 3 are multiplied by adding the logs. When we multiply 2 by 3, the upper sliding part is put with the starting position (left value 1) on the ‘2’ position of the lower part. Then we look along the upper part at the ‘3’ position and see which value on the lower part is in that

16 position. We can see that this is the number 6, so 2 x 3 = 6. In the exact same position we can for example divide 6 by 3 and get 2.

A slide rule is not as simple to use as a modern calculator. One must for example first take out all powers of 10 from the numbers that are being multiplied or divided. In fact the scientific number notation must be used, where a numbers r is written as ra10b . The slide rule is then used to make calculations with the ‘mantissa’ of the number (the a-part) and the 10b must be added and corrected afterwards

There are many tricks for circumventing the limits of a slide rule. For example multiplying 2 by 7 seems impossible in the picture above, because 7 on the upper part is off the scale of the lower part in our picture. In such cases, the user can slide the upper part to the left until its right index aligns with the 2, effectively dividing by 10. See picture below.

17 Resources

1. http://www-history.mcs.st-and/ac/uk/biogIndex.html 2. Work by Ian Bruce: napier: http://www.17centurymaths.com/contents/napier/ Translation of Arithmetica Logarithmica: http://www-history.mcs.st-andrews.ac.uk/Miscellaneous/Briggs/Chapters/Ch4.pdf 3. LOCOMAT project: The Loria collection of mathematical tables http://locomat.loria.fr/locomat/reconstructed.html 4. Work by Dennis Roegel (2010): Napier’s ideal construction of the logarithms http://hal.inria.fr/docs/00/54/39/34/PDF/napier1619construction.pdf 5. Work by Jacques Laporte: http://www.jacques-laporte.org/The%20method%20of%20Henry%20briggs.htm 6. ‘Revisiting the History of Logarithms’. In Frank Swetz et al, Learn from the Masters! MAA, 1995 7. “A Manual of the Slide Rule Its History, Principle and Operation – D. van Nostrand, 1930”, available through: http://books.google.nl/books?id=6scOAAAAQAAJ&pg=PA6&dq=slide+rule&hl=nl&sa =X&ei=N7WeT86nM8eVOpzRmPsB&redir_esc=y#v=onepage&q=slide%20rule&f=fal se

18 Pascal’s triangle by Maurits Carsouw and Tanita de Graaf

May 23, 2012 1 History of Pascal’s triangle

Long before Blaise Pascal was born ”Pascal’s triangle” was already known. We know that the Chinese already knew about it around the year 1050. At about the same time the Persians also already discovered it. Here you see the Chinese representation of Pascal’s triangle1.

In both cases the mathematicians used this triangle for the same purposes: extracting square and cube roots out of numbers. In China, after the discovery of the relationship between extracting roots and the binomial coefficients of the triangle, several Chinese algebraist’s continued on this work to solve higher than cubic equations. But why is it called Pascal’s triangle then? The answer to this is simple. Pascal developed many applications of it and he was the first one to organize all the information together in his treatise, Traitédu triangle arithmétique (1653). In this he made a systematic study of the numbers in the triangle. They have roles in mathematics as figurate numbers, combination numbers, and binomial coefficients, and he elaborated on all these.

1Illustration from Georges Ifrah, The Universal History of Numbers from Prehistory to the Invention of the Computer.

2 2 History of Blaise Pascal

Blaise Pascal (1623-1662) was born in Clermont-Ferrand (France). He had two older sisters and his mother died when he was only three years old. In 1632 Pascal’s left Clermont-Ferrand and moved to Paris. Etienne´ Pascal, Blaise’s father, taught Blaise at home. He decided that Blaise shouldn’t study mathematics before the age of 15 and let all the mathematical texts be removed from his house. Because of this, Blaise’s curiosity raised and he started to work on geometry himself. When he was 12 years old Blaise discovered that the sum of the angles of a triangle is equal to two right angles. And when his father found out about this, he gave his son a copy of Euclid’s Elements. During his life in France, his father took him to meetings for mathematical discussion run by Marin Mersenne, who was an important link for transmitting mathematical ideas widely at that time. At the age of sixteen, Pascal presented a paper to one of Mersenne’s meetings in June 1639. It contained a number of geometry theorems, including Pascal’s mystic hexagon.

In 1639 the family moved from Paris to Rouen. In Rouen published Blaise his first work: Essay on Conic Sections. Furthermore, the reason that they moved was that Blaise’s father got a job in tax computations. Blaise Pascal wanted to develop a device that would help his father in his work and therefore he was one of the first to invent the calculation machine2. Pascal played an important role on many other subjects as well, such as on probability theory, mathematical induction, an important theorem on prime numbers, on the fundamental theorem of calculus and on the examination of vacuum. Pascal was the first to connect binomial coefficients with combinatorial coefficients in probability. Pascal became interested in this subject since Antoine Gombaud asked him a question about an honestly division of stakes in an interrupted game of change. Gombaud wanted to improve his changes at gambling and asked Pascal the following question: two players are playing a game until one of them wins a certain number of rounds. But then the game got interrupted before any of the to reaches this. How should the stakes be divided? It should be taken into account how many games each player has won. The solution requires the combinatorial properties inherent in the numbers in the Arithmetical Triangle, as Pascal had discovered. From Pascal’s Traitédu triangle arithmétiquewe will also learn about the principle of mathematical induction. The concept of induction already occurred in the Islamic world in the Middle Ages and in Europe in the fourteenth century, but Pascal was perhaps the first to make a explicit statement and justification for this method of proving theorems.

2Hamrick, Kathy B. (1996-10). ”The History of the Hand-Held Electronic Calculator”.

3 Pierre de Fermat was a correspondent of Pascal, and because of the connections of Pascal’s triangle to the binomial theorem, Fermat discovered a proof of the famous and important theorem on prime numbers. Because of that theorem we can now use online payment in a secure way since RSA is based on the knowledge that Fermat’s theory gave us. Gottfried Leibniz was one of the inventors of infinitesimal calculus, a part of mathematics that is concerned with finding slopes of curves, areas under curves, minima and maxima and other geometric and analytic problems. Leibniz explicitly credited Pascal’s approach on some of these subjects because it stimulated his own ideas on characteristic triangle of infinitesimals in his fundamental theorem of calculus. Pascal also contributed in the area of physics. Another ’object’ is named after Pascal, namely: the scientific unit of pressure. One pascal is equal to one newton per square meter. Moreover, Pascal did some experiments on atmospheric pressure and in 1647 he had proved that a vacuum existed. And Pascal has a physical law named after him: Pascal’s law. Also known as the principle of transmission of fluid-pressure. Throughout his thirty-nine years, Pascal contributed to a lot of important statements as well in mathematics as in physics. The short summation above is far from complete. He was one of the most outstanding scientists of the seventeenth century. Unfortunately, due to other interests and his short live, we will never know how much more he could have accomplished.

4 3 Applications and properties of Pascal’s triangle

Pascal’s triangle has a lot of interesting properties. It is impossible to name all of them, but some of them are too remarkable not to mention, and are therefore listed in this chapter. Some of the properties below are also described in Pascal’s original Treatise on the Arithmetical Triangle; Trait´edu triangle arithm´etique,1654.

Figure 1: The original triangle in Pascal’s Treatise3(1654), where he explains his deﬁnitions parallel exponent and perpendicular exponent. Every diagonal is called a base.

1. We can identify the numbers of Pascal’s triangle as follows. Deﬁne the ﬁrst row of Pascal’s triangle to be the row containing only the number 1, then the i-th number of row j equals

j − 1 (j − 1)! = , j ≥ 2, i ∈ {1, ··· , j}. i − 1 (j − i − 1)!i!

This property is written in Pascal’s original Treatise as the second proposition.

Figure 2: Proposition 2, translation: ”The number of any cell is equal to the number of combinations of a number less by unity than its parallel exponent in a number less by unity than the exponent of its base”.

3Included pictures from the original treatise are from website http://www.lib.cam.ac.uk/deptserv/rarebooks/PascalTraite/.

5 2. From the ﬁrst property it follows that the binomial theorem

n n n n n ∀n ∈ :(x + y)n = xny0 + xn−1y1 + xn−2y2 + ··· + x1yn−1 + x0yn N 0 1 2 n − 1 n is strongly related to Pascal’s triangle. Indeed, all binomial coefficients in the theorem are directly given by the (n + 1)-th row of Pascal’s triangle. 3. Let us take a look at certain diagonals of the triangle. First observe that the two diagonals along the left and right edges of Pacal’s triangle contain only 1’s. Now go deeper into the triangle, to find that the the two diagonals next to the edge diagonals contain the natural numbers (in order). Continuing like this, each time observing two new diagonals, gives us the triangular numbers, the tetrahedral numbers, the pentatope numbers, and so on. In general, the n-th diagonal of the triangle (counting from the edge diagonal inwards) contains the (n − 1)-simplex numbers. 4. This property links Pascal’s triangle to the Fibonacci numbers. Start with the first (left) number 1 of an arbitrary row of Pascal’s triangle, and construct the following numbers by moving one element to the right, and one element to the bottem-right. Then what you get, is a sequence of finitely many elements, who’s sum is a Fibonacci number.

Figure 3: The sum 1 + 3 + 6 + 4 + 5 + 1 + 1 equals the eighth Fibonacci number 21.

5. Pascal’s triangle also corresponds to the Catalan numbers (deﬁned by a sequence of natural numbers, named after the Belgian mathematician Eugne Charles Catalan (1814 − 1894), and used to solve several combinatorial problems), as follows. For every odd, natural number n, the middle (n+1) element of row n, minus the element two spots to the left (or right), equals the 2 -th Catalan number.

6. When Pascal’s triangle is embedded in a matrix {Aij : i, j ∈ N}, the number of ways to walk from the upper left entry A11 to a square Akk, moving only to the right and downwards with k ∈ N, equals Akk. This is illustrated in the following ﬁgure.

Figure 4: One of the twenty paths from A11 to A44.

6 7. Create a sequence of numbers of the triangle as follows. Start with any number 1 in the border of the triangle, and walk along any diagonal to the bottem-right or left, for as many numbers as you like. Then the sum of the numbers in this sequence, equals the number directly below the lowest element of the sequence, which is not in line with the sequence. This property can be proved quite easily using property 1. In fact, this property is the third consequence (see the picture below) in Pascal’s original treatise.

Figure 5: On the bottom of the right page, Pascal states his third consequence. Translation: ”In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding perpendicular row from its own parallel row to the ﬁrst, inclusive.”.

7 4 Theorem

In this chapter we use proposition 2 from Pascal’s Treatise on the Arithmetical Triangle (P.T.A.T.), to prove one of the triangle’s most interesting properties, in a modern way. Hereby we use Pascal’s deﬁnitions of a base and a parallel exponent (see ﬁgure 1, chapter 3). Theorem If the second cell of a base (the direct neighbor of ’1’) is prime, then all the cells of that base (except for the two 1s) are multiples of that prime. Proof Assume that the second cell of an arbitrary base is a prime number p, then the base number is p + 1 and the base contains p + 1 cells. From proposition 2 it now follows that the k-th cell of the base, with k ∈ {2, 3, ··· , p}, equals the number of ways to pick parallel-exponent-minus-one things from base-number-minus-one things, or

(p + 1) − 1 p p! p! (p − 1)! = = = = p · , (p − k + 2) − 1 p − (k − 1) (p − (k − 1))!(k − 1)! (p − i)! i! (p − i)! i! where i = k − 1 ∈ {1, 2, ··· , p − 1}.

(p+1)−1 (p−1)! Deﬁne n = (p−k+2)−1 and q = (p−i)! i! , then it follows that n = p · q, where n, p ∈ N and q ∈ Q. Note that the right-hand side of (p − 1)!/(p − i)! (p − 1) · (p − 2) ··· (p − (i − 1)) q = = i! i · (i − 1) ··· 2 has i − 1 consecutive factors in both the numerator and the denominator. Since the greatest factor of the numerator, (p − 1), is greater than, or equal to, the greatest factor of the denominator, i, it follows that q ≥ 1.

To see that q is indeed an integer, let us assume that q ∈ Q \ Z. Now reduce (p − 1) · (p − 2) ··· (p − (i − 1)) q = i · (i − 1) ··· 2 to lowest terms such that q = a/b, for certain (and unique) a, b ∈ N. Observe that the prime factorization of b does not contain a factor p (or a product of factors which equals p), since p is prime and i, (i − 1), ··· , 2 < p. From this, and our assumption that q ∈ Q \ Z, it follows that p · q is not an integer, but this contradicts p · q = n ∈ Z.

In conclusion, we have that the k-th cell of base p + 1, with k ∈ {2, 3, ··· , p}, equals n(k) = n with

n = p · q, which is indeed a multiple of p.

8 5 References

Marvin R. O’Connell, Blaise Pascal, Reasons of the Heart. Blaise Pascal, Trait´edu triangle arithm´etique. David Pengelley, Pascal’s Treatise on the Arithmetical Triangle: Mathematical Induction, Combi- nations, the Binomial Theorem and Fermat’s Theorem. http://www-history.mcs.st-andrews.ac.uk/Biographies/Pascal.html http://www.britannica.com/EBchecked/topic/445406/Blaise-Pascal http://pages.csam.montclair.edu/ kazimir/history.html http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RBP ascal.html http://www.britannica.com/EBchecked/media/57051/Pascals-hexagon-Blaise-Pascal-proved-that- for-any-hexagon-inscribed

9 Evangelista Torricelli and his prove of the existence of a ﬁnite volume contained by an inﬁnite surface area

Ellen Schlebusch (0932701) & Bart Verbeek (0947687)

May 24, 2012

1 Contents

1 Torricelli’s Life 3

2 Torricelli’s Work 4

3 A Finite Volume Contained by an Inﬁnite Surface Area 5 3.1 Torricelli’s Proof ...... 5 3.2 Some notes on Torricelli’s Proof ...... 10 3.3 A Modern Proof ...... 10

4 Sources used 12

2 1 Torricelli’s Life

Evangelista Torricelli was born on October 15th 1608 in Romagna. Evangelista Torricelli’s parents were Gaspare Torricelli and Caterina Angetti. The family was fairly poor. Evangelista had two younger brothers and no sisters. Evangelista’s parents could not provide an education for him, so they sent him to his uncle Jacopo, who was a Camaldoles monk. He made sure that Evangelista was given a sound education until he entered a Jesuit school. From 1624 to 1626 Torricelli studied at the Jesuit College, in either Faenza or Rome, where he studied mathematics and philosophy. After this he was in Rome for sure. He was an outstanding student and his uncle, Jacopo, arranged for him to study with Benedetto Castelli, who was also a Camaldolese monk. Castelli taught at the University of Sapienza in Rome. There is no evididence of an enroll- ment of Torricelli at the university and it is pretty sure that he was taught by Castelli as a private arrangement. Torricelli was taught mathematics, mechanics, hydraulics and astronomy by Castelli and became his secretary from 1626 to 1632 as payment for the tuition. Later he took over Castelli’s teaching when he was absent from Rome. While Castelli was absent from Rome, Galileo had written to him and Torricelli wrote back to explain that Castelli was not in Rome and therefore could not answer Galileo. Ambitious as he was, he took the opportunity to inform Galileo about his own mathematical work and texts he had studied, including the work Dialogue Corcerning the Two Chief Systems of the World - Ptolemaic and Copernican by Galileo. From this letter, written on September 11th 1631, it is clear that Torricelli was fascinated by astronomy and a strong supporter of Galileo. But the trial of Galileo in 1633 scared him, so he shifted his attention onto mathematical areas. During the next nine years he was the secretary of Giovanni Ciampoli and possibly a number of other professors. By 1641 Torricelli had completed much of the work which he was to publish in three parts as Opera geometrica in 1644. In 1641 he asked Castelli for his opninion on De motu gravium (the second of the three parts which basically carried on developing Galileo’s study of the parabolic motion of projectiles). Castelli was so impressed that he wrote to Galileo, brought him a copy of Torricelli’s manuscript and suggested that he should take Evangelista as an assistant. Galileo was keen to have Torricelli as an assistant, but there was some delay. Torricelli gave lectures in Castelli’s place for a while and his mother died. On October 10th 1641 Torricelli arrived at Galileo’s house in Arcetri. He lived there with Galileo and his other assistant Viviani for only a few months, since Galileo died in January 1642. Torricelli was appointed to succeed Galileo as the court mathematician to Grand Duke Ferdinando II of Tuscany. He held this post until he died of typhoid on October 25th 1647 in Florence.

Torricelli and Galileo

3 2 Torricelli’s Work

Another pupil of Galileo, Bonaventura Cavalieri, published Geometria indivisibilis continuorum nova in 1635. Torricelli used a combination of this work and old methods to discover his results. Torricelli examined the three dimensional figures obtained by rotating a regular polygon about an axis of symmetry. He also computed the area and centre of gravity of the cycloid. His most remarkable results resulted from his extension of Cavalieri’s method of indivisibles to cover curved indivisibles. With this he showed that the volume of the rotation of the unlimited area of a rectangular hyperbola between the y-axis and a fixed point on the curve round the y-axis is finite. This proof will be discussed later. This was against the intuition of mathematicians of that time, so this result aroused great interest and admiration. Around 1640 Torricelli determined the isogenic centre of the triangle (the point in the triangles such that the sum of the distances from this point to the vertices is minimal). Torricelli was the first person to create a sustained vacuum and to discover the principle of the barometer. In 1643 he proposed an experiment, performed by Vincenzo Viviani, that led to the development of the barometer. Before he created a vacuum, the existence of a vacuum was a centuries old question. He attempted to examine the vacuum which he was able to create. He wanted to find out if sound could travel in a vacuum and if insects were able to live in it. He was not able to find this out. In De motu gravium Torricelli proved that the flow of liquid through an opening is proportional to the square root of the height of the liquid. This result is known as Torricelli’s theorem. Also in De motu gravium he developed Galileo’s work of the parabolic trajectory of horizontally launched projectiles. He gave a similar theory for projectiles launched at any angle. There is not much left of Torricelli’s mathematical and scientific work, because he only published Opera geometrica. Much is known from letters and some lectures that he gave. Hours before his death he tried to ensure that someone would prepare unpublished manuscripts and letters for publication. His friend, Ludovico Serenai, was trusted with this material. After some rejections Viviani did agree to undertake the task, but he failed to accomplish it. Some manuscripts were lost and the remaining material was published in 1919. Original material left by Torricelli got destroyed in the Torricelli Museum in Faenza in 1944.

4 3 A Finite Volume Contained by an Inﬁnite Surface Area

3.1 Torricelli’s Proof

Consider a hyperbola with the x-axis and y-axis as asymptotes. We rotate this ﬁgure around the y-axis to get a hyperbolic solid as in ﬁgure 1.

Figure 1

We call the origin point A.

Lemma Any cylinder DEFG contained in our ﬁgure as in ﬁgure 2, has surface area πa2, with a a constant.

Figure 2

Proof of lemma First we construct the linesegment AC, in such a way that C is on the hyperbola and the angle between the y-axis and AC and the angle between the x-axis and AC are the same (both are 45 degrees). This line segment is called the semi-axis. We deﬁne the length of AC to be equal to a. See also ﬁgure 3.

5 Figure 3

Let us now consider the square ABCD as in ﬁgure 4.

Figure 4

This is a square, because the diagonal AC bisects the angle BAD. So we have AD = DC. From the Pythagorean theorem we know AC2 = AB2 + DC2, so we have:

AC2 = AB2 + DC2 a2 = AB2 + DC2 = 2AB2 = 2Area(Square(ABCD))

Now let us consider an arbitrary rectangle EF GH as in ﬁgure 5.

6 Figure 5

Because one of the properties is that multiplying the x-value with the y-value gives a constant, we have

Area(Rectangle(AIGH)) = AH × AI = AB × AD = Area(Square(ABCD)) 1 = a2 2 Because our hyperbolic solid is symmetric around the y-axis (since we have rotated it around the y-axis), it holds that

Area(Rectangle(EF GH)) = 2Area(Rectangle(AIGH)) 1 = 2 a2 2 = a2

Now we can calculate the surface area of the cylinder EF GH, as in ﬁgure 3. Note that we do not include the bases of the cylinder when calculating the surface area. It holds that

Area(Cylinder(EF GH)) = π × EF × GH = πArea(Rectangle(EF GH)) = πa2

Because we have taken EF GH to be an arbitrary cylinder, this holds for all cylinders contained in our hyperbolic solid. QED.

Now consider our hyperbolic solid with a cylinder BCDE contained in it. We consider the solid ﬁgure consisting of this cylinder BCDE and the part of the hyperbolic solid that lies above it. See also ﬁgure 6.

7 Figure 6

We will now prove that this figure has a finit volume. First we construct the rectangle AEF G, such that AG is in the extension of the y-axis, EF is in the extension of DE and AG = EF = 2a. See also figure 7.

Figure 7

Now we consider an arbitrary rectangle HIJK in our ﬁgure. We also draw the line KL in the extension of JK, such that KL = 2a. See also ﬁgure 8.

8 Figure 8

Now we consider the circle with diameter KL, that is perpendicular to the x-axis. See also ﬁgure 9.

Figure 9

For this circle the following holds:

Area(Circle(KL)) = π(radius)2 2a = π( )2 2 = πa2

Recall that the surface area of the cylinder HIJK equals πa2 as well. Notice that the area’s of all cylinders contained in our figure together make up the volume of our figure. We have also just proven that for every cylinder contained in our figure, we have an unique circle that has the same area as the surface area of the cylinder. The the area’s of all these circles together make up the volume of the cylinder AEF G. So the volume of our figure equals the volume of cylinder AEF G. And the volume of cylinder AEF G equals π × 2a × AE and is thus finite. So the volume of our figure is finite. QED.

9 3.2 Some notes on Torricelli’s Proof

First of all we would like to note that we could not find a proof by Torricelli that the surface area of our figure is actually infinite. Maybe he thought this was obvious and did not need proving, maybe somebody else had already proven this, or maybe he proved it himself. At any rate, we could not find this proof, and we have to ask the reader to believe us on our word that the surface area is actually infinite. We will give a modern version of Torricelli’s finite volume contained by an infinite surface area later in this paper, and for this version we will actually proof that both the volume is finite, and that the surface area is infinite. Next we would like to point out the similarities between Torricelli’s proof and modern integrals. Although integrals did not exist during Torricelli’s time, he does us the idea of splitting something up into infinitely small pieces (recall for example the Riemann sum, which uses the same principle).

3.3 A Modern Proof

For a modern proof, using calculus, we will look at the threedimensional ﬁgure called Torricelli’s trumpet.

Torricelli’s Trumpet

1 This ﬁgure is made by taking the equation y = x , and then taking this equation for x ≥ 1, rotated around the x-axis. We can now use easy calculus to calculate the surface area and the volume of this ﬁgure. First, recall that the volume of a solid of revolution of a function f(x) on an interval x ∈ [a, b] is given by the following equation: Z b V = π (f(x))2dx a The surface area of a solid of revolution of a function f(x) on an interval x ∈ [a, b] is given by the following equation: Z b p A = 2π f(x) 1 + (f 0(x))2dx a

10 1 For Torricelli’s trumpet, we have f(x) = x , a = 1 and b → ∞. For the volume, this results in:

Z b V = π (f(x))2dx a Z b 1 = π lim ( )2dx b→∞ 1 x Z b 1 = π lim 2 dx b→∞ 1 x 1 b = π lim − b→∞ x 1 1 1 = −π( lim − ) b→∞ b 1 = −π(0 − 1) = π

The surface area of Torricelli’s trumpet is a little bit harder to calculate, but we can estimate it:

Z b p A = 2π f(x) 1 + (f 0(x))2dx a Z b r 1 1 2 = 2π lim 1 + (− 2 ) dx b→∞ 1 x x Z b 1 r 1 = 2π lim 1 + 4 dx b→∞ 1 x x Z b 1 √ > 2π lim 1dx b→∞ 1 x Z b 1 = 2π lim dx b→∞ 1 x b = 2π lim [log(x)]1 b→∞ = 2π( lim log(b) − log(1)) b→∞ = 2π( lim log(b) − 0) b→∞ = 2π( lim log(b)) b→∞ → ∞

So the surface area is larger than something which approaches infinity, so the surface area approaches infinity. So Torricelli’s trumpet has an infinite surface area, but a finite volume (namely π). QED.

11 4 Sources used

The image of Evangelista Torricelli on the front page is from http://catalogue.museogalileo.it/gallery/EvangelistaTorricelli.html The image of Torricelli with Galileo is from http://www.lepla.org/en/modulus.php?name=Activities&file=m42 For the biography of Torricelli we used information from http://www-history.mcs.st-andrews.ac.uk/Biographies/Torricelli.html For Torricelli’s proof of a finite volume contained by an infinite surface area we used chapter 4 of the book A Source Book in Mathematics, 1200-1800 by D.J. Struik.

12 JOHN VON NEUMANN

MAAIKE ASSENDORP & JURGEN RINKEL

Contents 1. Biograpy 1 1.1. Childhood and education 1 1.2. Life in Europe 2 1.3. Working in America and World War II 2 1.4. Later years and death 3 2. Work 3 2.1. Set Theory 3 2.2. Game Theory 6 2.3. Quantum mechanics 8 3. Conclusion 8 References 8

0 JOHN VON NEUMANN 1

This is an article about the Hungarian-American mathematician John von Neu- mann has the reputation of being one of the leading mathematicians in the first half of the twentieth century. This reputation was mainly due to the fact that he had the most extraordinary insights in numerous fields, in- and outside from mathematics. In fact, he contributed to almost every mathematical field but topology and number theory, while in the meantime becoming renowned for his revolutions in physical, economical, and even philosophical thinking. Towards the end of his life he would work on the first computers, thereby making an everlasting contribution to our modern society.

1. Biograpy 1.1. Childhood and education. John von Neumann was born at December 28, 1903 in Budapest, Hungary (at the time part of the dual monarchy Austria- Hungary). At birth, he was called Neumann János1. He would change his name to John when he moved to the United States.2 While living in Hungary he was called by the diminutive Jancsi, which became Johnny when he anglicized his name. Von Neumann’s family was Jewish and prosperous, which was a quite common combination in Budapest in the early 1900s. Max Neumann, Johnny’s father, was a banker who provided his children with a good education, hiring several teachers to learn them their languages and achieving an enormous library. Johnny, his oldest son, was a child prodigy who was completely comfertable at the von Neumann’s intellectual environment. Contrary to other Jews in central Europe, Johnny’s life was assured to be richly endowed. As a child he could regularly been found at the aforementioned library, devouring many of the books and even memorizing them. In later life he became infamous for his almost photographic memory. One anecdote from the early 1950s illustrates this. When he was asked by a collegue how Charles Dickens’ novel A Tale of Two Cities began, von Neumann started to quote the first dozen pages. After the First World War, Austria-Hungary was split in several nations and Hun- gary became a independent state. For a short time it was under control of Bela Kun’s communist regime, but in 1919 that government collapsed and MiklósHor- thy, a former admiral of the Austrian-Hungarian navy, seized power. Under his conservative rule the so-called ’White Terror’ gripped Hungary which also victim- ized the Jews. As a result, approximately 100,000 people fled Hungary. One of them would be John von Neumann. When he approached college age, he wished to study mathematics. His father dis- approved, saying there was no money in that profession. A friend of the family suggested Johnny to study chemistry as a compromise. All agreed and Johnny enrolled at the University of Budapest in 1921. His college career was very complex, because it spanned three countries. He also enrolled at the University of Berlin, while simultaneously staying in Budapest, where he never actually attended colleges, only to show up to ace his exams. He left Berlin in 1923 to continue his studies at the Eidgenossische Technische Hochshule in Zurich. In 1926 he recieved his Ph.D. in Mathematics from the University of Budapest, doing this in only five years. At the time he was only twenty-two years old. His dissertation was concerned with the axiomatization of Cantor’s set theory, of which we will come to speak later on.

1In Magyar (’Hungarian’) the family name always comes ﬁrst. 2In this article we will consequently use the English version of his name and for every individual the common order of ﬁrst name - family name, whether Hungarian or not. 2 MAAIKE ASSENDORP & JURGEN RINKEL

1.2. Life in Europe. After receiving his Ph.D., von Neumann went on to do post-doctoral work at the University of G¨ottingen after having earned a Rockefeller grant. Gttingen was one of the most renown universities of the time, when it came to mathematics. It was the home base of David Hilbert, one of the greatest mathe- maticans at the turn of the twentieth century. It was also here that Johnny would make his contributions on the mathematical foundations of quantum mechanics. By the middle of 1927, his work with Hilbert was done and von Neumann decided to move on. He went to Berlin were he was appointed as Privat docent. He reportedly was the youngest man ever to hold that position. In 1929 he moved to Hamburg, to occupy the same position. During these years, von Neumann published his articles at the impressive rate of about a month. Some of these articles contained astonishing new ideas about the topics they concerned and that revolutionized the way mathematician looked at their subjects. In the meantime, European polics became harsh and the prospect of a new World War was glooming. Several attempts to overtrow the local or national govern- ments in Germany had occured (including Hitler’s Bierkeller Putsch. Von Neumann thougth that in the future, a nationalistic Germany would wage war on Russia and that, would that indeed be the case, his fatherland, Hungary, would side with them3. This prospect didn’t look all too good to von Neumann, so he started to make plans for leaving the continent. Luckily, in 1929 he was oﬀered a lectureship at Princeton University in New Jersey. The institute desperately wanted to modernize America’s mathematics, for it was falling behind on it’s European counterpart. It was Oskar Veblen who suggested to attract John von Neumann for this purpose. Von Neumann agreed, and it was arranged that he should come the next year, to return in the second half of 1930 to give a course of lectures in Berlin. But before all this happened, he had decided to propose to Mariette Kovesi, and she had agreed. This required that he converted to Catholicism, which von Neumann did. Religion never had meant much to him, and this attitude would continue until short befor his dead. But now von Neumann would leave soon and wanted her to come with him, so things were getting in a hurry. The original plan of marrying in June 1930 was abandoned and they married on New Year’s Day instead. In 1935 Mariette gave birth to their only child, a daughter, who was named Marina. Un- fortunately, the marriage wouldn’t last long. In 1937, the couple divorced, leaving Marina with her mother until she became twelve, as it was felt that her education was better left at the hands of her father.

1.3. Working in America and World War II. Von Neumann would spend the ﬁrst years following 1930 travelling back and forth from Europe to the States. In 1933, after Hitler came to power, he gave up all his work in Germany. The same year the Princeton Institute of Advanced Study (IAS) was opened. This institute was meant to house the world greatest scientists who wouldn’t have to give any colleges and who were required to do anything but think. Von Neumann accepted a professorship there. The IAS would soon become a attractor for ﬂeeing scientist from Europe. Among them were Albert Einstein and Kurt Gdel. Johnny would eventually end up working at the IAS until 1955 and his work is still considered one of the greatest forthcomings of the institute (along with Gdel’s work on the continuum hypothesis). However, he couldn’t match his old publicity rate of nearly one paper a month while working at the IAS, a problem that all the scientists working there encountered. During von Neumann’s time at Princeton, he met the British mathematician Alan

3Which would eventually prove to be the case JOHN VON NEUMANN 3

Turing, with whom he co-operated on the foundations of computer programming. It won’t be an exaggeration to say that this might be his most influential co-operation. But there was more to come when the Second World War approached. In 1937 von Neumann knew for sure that war was coming, so he applied for the position of lieutenant in the Reserve Corps. He passed all the exams (of course) but in the end he was considered too old. In 1938 he travelled to European continent one last time, while Czechoslowakia was already under the immediate threat of Nazi Germany. He had several reasons to go back to the continent. One was that the IAS had sent him to convince Niels Bohr to come to the other side of the Atlantic, as he was considered too important to leave in the middle of a war. A second reason was that von Neumann intended to marry again, this time with Klari Dan, who was by then in the middle of a divorce. She approved, and the couple would stay together until von Neumann’s death. In September 1939 the war finally broke out and with the United States joining in December 1941, von Neumann took the interest of the American government for being a calculator on explosive weapons. In the meantime a project was started to create the atomic bomb: the Manhattan Project. In 1943 von Neumann was recruited as a consultant at Los Alamos laboratory in New Mexico, where the bomb would be constructed. In 1945, von Neumann witnessed the first test and he also put forward several Japanes cities to bomb first, Nagasaki and Hiroshima being among them. A 1943 visit to England brought to von Neumann the idea of actually inventing the computer. He would begin with this in Philadelphia, where ENIAC, the first real electronic computer, was created.4 1.4. Later years and death. After the War, Johnny would keep up working at the IAS until 1955. In the meantime he had several other jobs as a consultant for the government on different affairs. On of those affairs was nuclear detterence, in which he argued in favour of bombing the Soviet Union, before this country could be able to bomb America, attracting support from members of the government. In 1955, von Neumann became sick and was diagnosed with cancer. It is commonly thought that this is a result of his involvement with the tests for the atomic bomb, as many other participants would end up dying from cancer. In 1956, he was con- fined to a wheel chair. His last public appearance was when he recieved the Medal of Freedom from the hands of president Eisenhower. Near the end of his life, von Neumann seriously converted to Catholicism. On Feb- ruary 8, 1957, John von Neumann died at the Walter Reed Hospital at Washington, D.C.

2. Work 2.1. Set Theory. 2.1.1. Historical setting. John von Neumann himself describes the historical setting in which he wrote his articles. He does so in ’Eine Axiomatisierung der Mengen- lehre’, however, he obviously didn’t write is as history, but as present. He descibes two groups of people and their contributions and views on the subject. (1) Russell, J. König,Weyl, Brouwer (2) Zermelo, Fraenkel, Schönflies First however, it is neccesary to know what set theory was like before the 20th century. Set theory was developped due to certain questions that arose from calculus. How

4Actually, this was only thought to be the first. Years later the British government revealed that they had had a computer some time earlier. 4 MAAIKE ASSENDORP & JURGEN RINKEL functions behave with certain sets? And, What is the difference between the rationals and the irrationals? Georg Ferdinand Ludwig Philip Cantor sought to answer these questions, developping what would later on be called (by Von Neumann also) ’na¨ıve set theory’. Georg Cantor began by comparing sizes of sets, using one-to-one relations between sets. He discovered that ’infinity’ was not ’the end’, that the natural numbers (having cardinality ℵ0) were as many as the the rationals, but the real numbers were vaster (having cardinality c). He proved that while the rationals were denumerable (or equipotent with N, having cardinality ℵ0), the irrationals, were in fact, not. In his search for sets greater (of higher cardinality) than the set of real numbers, he first set out to prove that a square had more elements then an interval. However, he proved that they wéreequal in stead. Finally he discovered that for every set A, the power set had higher cardinality, or: ∀A : R(A) > A As beautiful as these results sounded, they led to a paradox: Define a set U that contains everything. ’Everything’ being, every set, every set of ideas, of numers, of subsets, of... etc. This universal set U is defined to be greater, vaster then all else - as it contains everything. But we also have R(U) > U, wich tells us that there is something vaster then U. This is a paradox that followed from Cantor’s set theory, showing that it was flawed.

Now we shall return to the two groups that Von Neumann mentions. He is not the only source that uses this division.5 (1) The first group wishes to redefine mathematics from scratch, seeing how the definitions so far led to paradoxes. Von Neumann claims that both Weyl and Brouwer a greak part of mathematics and set theory discard as ’useless’. Von Neumann6 does not agree with their vision. (2) The second group realised that the old definitions led to paradoxes, but did not find it neccesary to redefine it all together. In stead, they axiomatise set theory, using the na¨ıve version that Cantor introduced. I will focus a little more on this group, as the article shows that Von Neumann himself belongs to this group.7 Interesting to note is that there are some parallels with Euclides and this latter group. Both axiomatizes something that was allready known, for the sake of con- sistency. But there was one more similarity. Cantor posed the following question: Is there a group A, such that ℵ0 < A < c? Is there a cardinality between the natural numbers and the real numbers. He formulated the hypothesis that it was in fact not possible to find such an intermediate set. Though no matter how hard Cantor tried he could not find the answer8. Much like Euclides’ fifth postulate could not be proven. In the 19th century it was discovered by Kurt Gödelthat the hypothesis could not be disproven.9 But a little over 20 years later, Paul Cohen proved that the hypothesis could not be proven.10 These proves came later then the articles Von Neumann wrote about it though.

5Although I will use what he wrote, I will not fully depend on it. The tone Von Neumann uses in his article makes it clear that he has an opinion about this. 6If my knowlegde of the German language severs me well 7Von Neumann himself also states that the article belongs to the latter category. 8Dunham suggests that this has cost him a few mental break downs. 91940 101963 JOHN VON NEUMANN 5

2.1.2. Axiomatizing of Set Theory. The two articles that Von Neumann wrote on the subject are: • Eine Axiomatisierung der Mengenlehre11, 1925 • Die Axiomatisierung der Mengenlehre12, 1928 I will focus mostly on the first of the two. The first I noticed was that Von Neumann defines and axiomatizes funcions rather then sets. He begins with saying that in the article a ’set’ is merely a ’thing’ (or ’Ding’) of which we know nothing more (and don’t want to know anything else than) is decribed in the postulates. He says that the postulates can be formulates thus that the set theory of Cantor followes from it, but that the paradoxes or antanomilies don’t. He mentions the fact that in stead of axiomatizing the sets, he will axiomatize the functions, wich are (according to him) one and the same. He means with this that a function f(x) (written as [f, x]) can be viewed as an element from a set, while a set can be viewed as function with two values. He makes a distinction between two sections. • Arguments (Argumente, I-Dinge) • Functions (Functionen, II-Dinge) • Both (I,II-Dinge) [x, y] would be the function x for the argument y.[x, y] itself would once again be an argument.13 After explaining how he is going to define sets or functions, Von Neumann explains the way the axioms are grouped. The first axiom group is for the initial definitions. The second and third axiom group is about how sets or functions react together. It describes which operations can be done to create the functions. He doesn’t describe the others though. I too will limit myself to these first 3 groups, due to the length it would take to discuss them all in detail.

I will now give an overview of the postulates and what they mean. ”Will equip ourselves with I-Dinge, II-Dinge, A,B (Dinge, not equal to one another), de operations [x, y] and (x, y). We have:14 I. (1) A, B are I-Dinge (2) [x, y] makes sense if and only if x is a II-Ding, and y is an I-Ding. It itself is an I-Ding. (3) (x, y) makes sense if and only if x, y are both I-Dinge. It itself is again an I-Ding. (4) Let a, b be II-Dinge. Then if for all I-Dinge x we have [a, x] = [b, x], then a = b. The ﬁrst three are deﬁnitions of the operators, the last says that if two funcions have the same value for all arguments, they must be the same.

II. (1) There exists a II-Ding a, such that always [a, x] = x. (This postulate gives the existence of the identity function.

11German: An axiomatization of set theory 12German: The axiomatization of set theory 13I will further on put down the formal postulates. 14These postulates may be phrased weirdly from time to time. I did my best to translate them from German as good as I could. 6 MAAIKE ASSENDORP & JURGEN RINKEL

(2) Let u be an I-Ding. Then there exists a II-Ding a such that allways [a, x] = u. (a is a constant function.) (3) There exists a II-Ding a such that always [a, (x, y)] = x. (4) There exists a II-Ding a such that always [a, (x, y)] = y. (This tells us that one could find funtions that give the first or second coördinaterespectively.) (5) There exists a II-ding a such that always (When x is a I,II-Ding) [a, (x, y)] = [x, y]. (x must be a I,II-Ding, otherwise the expression would make sense, as stated in I.2 and I.3.) (6) a, b are II-Dinge. There exists a II-Ding c such that [c, x] = ([a, x], [b, x]). (Writing in a more familiar notation we would get c(x) = (a(x), b(x).) (7) a, b are II-Dinge. There exists a II-ding c such that [c, x] = [a, [b, x]]. (We would say c = a ◦ b.) The above axioms are formulated such that the functions become a group. III. (1) There exists a II-Ding a such that x = y with [a, (x, y)] 6= A is the same. (2) Let a be a II-Ding. There exists a II-Ding b such that then and only then [b, x] 6= A when for all y [a, (x, y)] = A. (3) Let a be a II-Ding. There exists a II-Ding b such that always, if for a single y [a, (x, y)] 6= A,[b, x] = y. These are rules as to how one could construct functions.

After this third group, the article continues with describing I,II-Dinge (the 4th group) and axioms for infinity (5th group). 2.2. Game Theory. 2.2.1. Two player games. A key article in the field of game theory was von Neu- mann’s 1928 paper Zur Theorie der Gesellschaftsspiele15. Although von Neumann wasn’t the founder of game theory, his contributions to it are, as in other field, farreaching. In this article we will stick to two-player games, even though von Neu- mann’s article considers games of more players. Von Neumann examines games in their widest sent, that is when it can be seen as a finite number of events (i.e. the movement of a piece at a chessboard) which lead to a result (black or white wins, or there is a draw). Every player has several options to play, here they are called strategies. A simple and familiar ’game’ is the division of a piece of cake by two people. To do this fair, one ’player’ cuts the cake in two, while the other chooses a half. Now, there are several options, called strategies, for both players. These are given in Figure 1. Besides that, the results for cutter are given, for any combination of strategies.

The question we must ask ourselves now is: what is the optimal strategy for the cutter? Or to formulate this otherwise, what must the cutter do to maximize his result? As we can see, the best result would be for him to get the big piece. That suggests he should cut the cake so that there are two uneven parts. But the chooser will obviously choose the bigger piece, leaving the smaller one for the cutter. But if the cutter cuts the cake in two more or less even parts, the chooser will still choose the bigger piece, but what’s left is a bigger piece for the cutter then in the ﬁrst case. So the optimal strategy for the cutter is to cut the cake as evenly as possible.

15German: On the Theory of Games JOHN VON NEUMANN 7

Figure 1. The strategies for each player when cutting the cake, and their outcome.

Similarly, we may ask what the optimal strategy for the chooser could be. As his result is deﬁned by that of the cutter, in the sense that the chooser gets what the cutter gets not (this is called a ’zero-sum game’) he wants to minimize the result of the cutter. As we have seen, this can be done by choosing the bigger piece.

Let’s make this more abstract. We deﬁne a function g(x, y) to be the result for player 1 (the ’cutter’) when player 1 chooses strategy x and player 2 chooses strategy y where x, y ∈ {1, 2} then the corresponding values for g can be found in Figure 2

Figure 2. A mathematical representation of the outcomes

Consequently we ﬁnd:

g(1, 1) = 0.49 g(1, 2) = 0.51 g(2, 1) = 0.1 g(2, 2) = 0.9

As we said, player 1 wants to maximize the outcome of and player 2 wants to minimize. Now we ask, what are maxx miny g(x, y) and miny maxx g(x, y)? We have already seen that player 1 and 2 have to choose x = 1, y = 1 to accomplish this.

max min g(x, y) = 0.49 = min max g(x, y) x y y x 8 MAAIKE ASSENDORP & JURGEN RINKEL

2.2.2. The MinMax-theorem. In the foregoing example we saw that MaxMin and MinMax are equal. The question von Neumann now raises is whether this will be the case for more games. In his article, von Neumann now goes on to consider games of two players where player 1 choose strategy ξ and player 2 chooses strategy µ. These strategies are comprise several events, because the game will have several turns. Now we take h(ξ, µ) to be the result for player 2. Now the MinMax-theorem says that

max min h(ξ, µ) = min max h(ξ, µ) ξ µ µ ξ for every zero-sum game. An alternative formulation is that for every game there exists an optimal strategy for both players. When both players choose this strategy, the game will eventually reach an equilibrium point. To von Neumann, this result was very important. Without this theorem, he said, it wouldn’t even be interesting to publish anything on game theory. In later years, von Neumann didn’t publish much about game theory, altough he cowrote the 1944 book The Theory of Games and Economic Behavior with Oskar Morgenstern. This book is still seen as one of the most important textbooks in economics.

2.3. Quantum mechanics. We will discuss some history of Quantum Mechanics in this last chapter. This will remain quite brief. One reason is that ’mechanics’ is science, but not math. The other reason is that we have focussed more on above subjects in stead of this one. The following list is a sumarization from a chapter on quantum mechanics in the book we used: 1900 Max Planck makes up the concept quantum, a package that transports energy’. 1909 Ernest Rutherford is the first one to split an atom. 1913 Niels Bohr discovers rings of electrons and connect it to the quanta of Planck. 1925 Werner von Heisenberg (in Göttingen,where Von Neumann would work a few years later) formulates equations that describe the structure of atoms. Together with Max Born and Pascual Joardan he develops this further. Not all scientist were as enthousiastic. 1926 Erwin Schrödingerformulates a series of wave equations for the movements of the elektron. Heisenberg rejects these. Both models fitted the experiments. 1927 Heisenberg formulates his uncertainty principle. 1928 Von Neumann joins the two sides using axiomas for the quantummechanics. He made use of so called infinite Hilbert spaces. We included the part on quantum mechanics to illustrate how John von Neumann may not have done much in a field, he did have a great influence. By joining the two sides, quantum mechanics leaped forward.

3. Conclusion References [1] John von Neumann, Eine Axiomatisierung der Mengenlehre. Dissertation, unpublished. 1925 [2] John von Neumann, Zur Theorie der Gesellschaftsspiele. In Mathematische Annalen. Berlin: Springer Verlag, 1928 JOHN VON NEUMANN 9

[3] Norman Macrae, John von Neumann: The Scientiﬁc Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. United States of America: American Mathematical Society, 1999. Orignally published: New York: Pantheon Books, 1992 [4] William Poundstone, Prisoner’s Dilemma. New York: Doubleday, 1992 Euclid’s Parallel Postulate and the Birth of Non-Euclidean Geometry

Jeroen van Splunder & Olfa Ja¨ıbi

May 24, 2012 1 Introduction

“Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”[1] Ever since Euclid (circa 330-275 BCE) layed down planar geometry in an axomiatic framework, there has been much controversy on the fifth postu- late1 of his Elements[1]. The 2300 year old history of the fifth postulate is one of many attempts at proving this intriguing postulate and just as many failures. The discussion has not been on whether the fifth postulate is true or whether there exist geometries where it does not hold, but rather on why Euclid formulated his statement on parallel lines as a postulate and not as a theorem, perhaps derived from a more elementary, shorter postulate. In this paper, we will give a short overview of several attempts at proving or replacing the parallel postulate, after which we focus on the work of Carl Friedrich Gauss (1777-1855), who was the first to discover non-Euclidan geometry.

2 The Parallel Postulate

The main source of information on the earliest attempts to prove the parallel postulate stem from the Greek philosopher Proclus (410-485)[4, p.2], who wrote a commentary on the first book of the Elements[2]. Proclus firmly states that “this proposition should have been excluded completely from the postulates, for it is a theorem.”[2, p.53] He then discusses the flaws in the alternative definition of parallel lines by Posidonius (1st century BCE) and the ‘proof’ of the fifth postulate by the famous astronomer Ptolemy, after which Proclus gives his own flawed ‘proof’.[4, p.6][2, pp.57-67] In the Middle Ages, Euclid’s work was studied and copied by the Arabs, who also investigated the fifth postulate, after which discussion on the parallel postulate emerged in Europe again after 1550.[4, p.12] In 1763 Georg Simon Klügelwrote his dissertation on the fifth postulate, listing an overview of all 26 historical ‘proofs’ of the parallel postulate known to him, after which he concludes that Euclid must have been right after in taking the postulate for granted.[5] He writes in the introduction to his dissertation, that “most

1In some editions, the parallel postulate is grouped under the axioms as axiom 11.

1 proponents of rigorous proof throw it out of the list of axioms, but their proofs bear the same mistakes. Others replace it by other axioms, which are neither clearer nor more secure than the Euclidian. Therefore, when all attempts are thoroughly inspected, it becomes clear that Euclid was right in putting as axiom a proposition which cannot correctly be proven from another one.”[5, §I] The approach of finding other definitions or axioms seems more interesting than the flawed proofs given by some. Geometry professor from Oxford John Wallis (1616-1703) gave a lecture in 1663 in which he proposed an alternative postulate which he believes to be clearer and which “does not need a demonstration.”[7, p.674] Wallis assumes that for any given figure, there exists a similar figure of arbitrary magnitude and then shows that Euclid’s postulate follows from this axiom.[7, pp.676-7] It was later shown by Geralomo Saccheri (1667-1733) that it is sufficient to assume that there exists at least one equi-angular pair of non-congruent triangles.[6, p.85] In 1773, Saccheri published a work in which he looked at the Parallel Postu- late in a novel way.[3] Saccheri is aware of the fact that the Parallel postulate is equivalent to the proposition that the angles in a triangle always sum to two right angles. Instead of introducing another axiom or a new definition of parallel lines, Saccheri ponders on the possibility of the angle sum of a triangle not necessarily being equal to two right angles. He proofs that if in one triangle the angle sum is two right angles, then it so in every triangle. Likewise, if the sum is more in one triangle, it is in all triangles and if it is less, it is less in all triangles. (Prop. V-VII.) Thus three hypotheses on the nature of geometry emerge: the right angle hypothesis, the obtuse angle hypothesis and the acute angle hypothesis. Saccheri is able to prove some theorems on angles in a triangle and lenghts of their bases. He then dis- misses the obtuse angle hypothesis because a triangle can be constructed in which the sum of two angles exceeds two right angles, which contradicts I.17 which was proven by Euclid without using the parallel postulate.2 (Prop. XIV.) Even though he had disproved the obtuse angle hypothesis, Saccheri continued proving theorems which hold for all three hypotheses; however, they did not yet lead to the contradiction in the obtuse hypothesis he was looking for. Among other theorems, Saccheri notes that in any quadrilat- eral, the angle sum will be less, equal to or more than four right angles depending on the hypothesis used, and that the angle inscribed in a semi- circle is less than, equal to or more than a right angle depending on the hypothesis. (Thales’ theorem.) Then there is an intermezzo in the book, in which the contributions of Proclus, Nasir al-Din, Wallis and others to the problem of the parallel postulate are discussed. Next, Saccheri gives

2By adapting the Euclidian notion of straight lines of inﬁnite length, elliptic geometry can be constructed, in which the angle sum can exceeds two right angles.

2 theorems on the distance between two parallel lines and finally proves that for the acute angle hypothesis, two straight lines run together as they near infinity (their perpendicular converges), from which Saccheri concludes that the acute angle hypothesis is “absolutely false; because [it is] repugnant to the nature of the straight line.” (Prop. XXXIII) Saccheri was the first to explore the possibilities of non-Euclidian geometries, but he did so still with the intent to prove the parallel postulate. Unfortunately, his work was forgotten until it was rediscovered by Eugenio Beltrami (1835-1899). It would be almost 60 years until Gauss independently explored the possibilities of non-Euclidian geometry and concluded that though it was very different from Euclidian geometry, it was independent and just as logically sound.

3 Gauss

Carl Friedrich Gauss was, almost surely, the ﬁrst mathematician to have a clear view of a geometry independent of the Fifth Postulate, among other inﬂuent discoveries he made during his life.

Figure 1: Portrait of Carl Friedrich Gauss in 1840 painted by Christian Albrecht Jensen

3 Gauss, born on the 30th of April 1777 in Brunswick, Germany, was raised in a poor family, being the son of Gerhard (called Gebhard) Dietrich Gauss, a gardener and bricklayer, and Dorothea Gauss, the daughter of a stonecutter. Despite the miserable living conditions, he exhibited such early talent that his family and neighbors called him the “wonder child”[8]. When he was two years old, he gradually got his parents to tell him how to pronounce all the letters of the alphabet. Then, by sounding out combinations of letters, he learned (on his own) to read aloud.[12, §II] He also picked up the meanings of the number symbols and learned to do arithmetical calculations. The story as told by Eric T. Bell: One Saturday Gebhardt Gauss was making out the weekly pay- roll for the laborers under his charge, unaware that his young son was following the proceedings with critical attention. Coming to the end of his long computations, Gerhardt was startled to hear the little boy pipe up, ’Father, the reckoning is wrong, it should be ...” A check of the account showed that the figure named by the young Gauss was correct.[10]” At the age of seven, Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner,and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 almost instantly. He used a very clever way to find the result. His observation was that: “100 + 1 = 101; 99 + 2 = 101; ... 52 + 49 = 101; 51 + 50 = 10100 and thus that there were ‘as many pairs’ as there were in 100. Thus he found as a result: 50 × 101 = 5050 which was correct and astonished his teachers[12, §II]. These two events, which, for Gauss himself were special[12], made it, among other persons around him, clear: A genius was born, someone who would change the world of mathematics. This did not escape from the attention of Büttner,who then took a special interest in the young boy: He provided him with additional books and organized after-school meetings to investigate advanced mathematical ideas with Gauss’ from then on tutor Bartel [13]. This did not stay unnoticed for long and, as his name grew in the area and he kept up his studying, the Duke of Brunswick heard about him. When Gauss was 15 the Duke summoned him to his castle. It was there that Gauss formed a lasting friendship with the Duke and received a stipend that allowed him to go to college and to devote his time to studying. For the next 4 years Gauss spent his time learning at Caroline College. Even while studying, Gauss was formulating many of the important theorems he would later go on to prove. He then left to enter University of Göttingenin 1795. While there he learnt and discovered a lot, but most of his discoveries hjad already been discovered. Discouraged with mathematics and his lack of making any new discovery, Gauss was on his way to becoming a philologist. That is until he made a discovery that declared him a

4 mathematician.[9] Gauss obtained conditions for constructibility of regular polygons and found out that the regular 17-gon was constructible by ruler and compass. This convinced him that he was meant to be a mathematician. This was the most major advance in this field since the time of Greek mathematics, since Euclid had shown that regular polygons, with 3, 4, 5, and 15 sides and those the sides of which result from doubling the above could be constructed geometrically with compass and ruler and no further progress had been made since then[14]. He left the University without a degree and went back to Brunswick where he got a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss’s stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. The subject of his dissertation was a proof of the fundamental theorem of algebra–which was proven only partially before Gauss–which states that every algebraic equation with complex coefficients has complex solutions; moreover, Gauss skillfully formulated and proved this theorem without the use of complex numbers. Between 1796−1800, Gausss mathematical thinking matured tremendously. Mathematical ideas came to him so easily and frequently that he had trouble getting them all down on paper.[9] He contributed to different fields. One of his greatest contributions is cer- tainly the book Disquisitiones Arithmeticae that he published in the summer of 1801. There were seven sections, all but the last section, being devoted to number theory. In section VII, he published his discovery about the constructablilty of the 17-gon.[14] This book is considered one of the most brilliant achievements in the history of mathematics, in which he formulated systematic and widely influential concepts and methods of number theory, dealing with relationships and properties of integers which, for him, was of paramount importance in mathematics. This included dealing with the theory of congruent numbers, law of quadratic reciprocity for which he the first proof and algebraic numbers.In 1831 (published 1832) he gave a detailed explanation of how an exact theory of complex numbers can be developed with the aid of representation in the x, y plane.[14] He also contributed to many other fields of science. Astronomy was one of the main fields that gained Gauss’ interest. Using his method of least squares he could determine the orbit of several newly discovered as- teroids. This lead him to get the position of Director of the Observatory at GöttingenUniversity, a position that he held for 48 years. In this capacity he actively pursued research in theoretical astronomy, taught courses and published a lot of astronomical papers.[12, §II] Beside those two main fields, Gauss played an important role in physics, working on topics like electroc- ity and magnetism, optics, (the flow of) fluids, in geodesy and in geometry, differential as well as non-Euclidean. This field of mathematics raised after the long battle during twenty centuries many mathematicians held trying to prove the Fifth Postulate. This intrigued Gauss who attempted to solve

5 the problem as well. Gauss corresponded a lot with W.Bolyai[1775-1856], Olbers[1758-1840], Schumacher[1780-1850], Gerling[1788-1864], Taurinus[1794-1874] and Bessel[1784-1846] on the subject.[4, §III] In 1799, he confided to Wolfgang Bolyai that he was doubtful of the truth of geometry conceived by Euclid. He had found too many mistakes in the calculations made by other people in defence of the fifth postulate that he could not believe in the Euclidean system anymore. In a letter to W.Bolyai[Dec 17, 1799] he says: As for me, I have already made some progress in my work. How- ever the path I have chosen does not lead at all to the goal which we seek, and which you assure me to doubt the truth of the geometry itself.[4] He thus developed a view of a geometry independent of the Fifth Postulate, but this remained for quite fifty years concealed in his mind. He only revealed it after the works of Lobatschewsky[1792-1856] in 1829-30 and Bolyai in 1832 appeared. In a letter he wrote to Schumacher[May 17, 1831] he says[4]: In the last few weeks I have begun to put down a few of my own Meditations, which are already to some extent nearly 40 years old. These I had never put in writing, so that I have been compelled three or four times to go over the whole matter afresh in my head. Also I wished that it should not perish with me. This marked the first written steps of Gauss’ theory of non-Euclidean geometry. His notes were published posthumously in his assembled works in 1870.[15]

4 Gauss’s Contribution to Non-Euclidean Geom- etry

Gauss starts by giving a definition of parallelism[4, §34]: If the coplanar straight lines AM, BN, do not intersect each other, while, on the other hand, every straight line through A between AM and AB cuts BN, then AM is said to be parallel to BN. Gauss supposes a straight line l passing through A. First, l lies on the line AB and then is rotated continuously to AC on the side towards which BN is drawn (see figure 4). In the beginning the line l cuts BN and from a single position on, these two do not intersect anymore. This position must be the first line of the lines that do not cut BN. This follows from the definition of

6 Figure 2: The definition of parallels as given by Gauss in his book the parallel AM. In his definition, Gauss assumes starting points A and B, though the lines AB and BN are supposed to be produced indefinitely in the directions of AM and BN. This definition is different from the one Euclid gave in his Elements. If Euclid’s Postulate is rejected, there could be different lines through A, on the side mentioned ealier, that do not cut BN. According to Euclid, all of these would be parallel to BN. Gauss does only take the first of these lines and defines it as being the parallel to BN. Gauss completes his definition by showing that “the parallelism of the line AM to the line BN is independent of the points A and B, provided the sense in which the lines are to be produced infinitely remain the same.” It is clear that the same parallel is obtained if another point B’ on the infinite line BN is taken (thus B’ taken on BN or produced backwards. It remains to be shown that it is independent of A, that is, if AM is parallel to BN, then for any point A’ on the infinite line AM, A’M is parrallel to BN. We distinguish two cases: A’ being upon AM and A’ being produced backwards on the ‘extension’ of AM. The following figure corresponds to the first case:

Figure 3: Case of A’ being upon AM

7 Instead of A, let A’ be the starting point upon AM as in the figure. Draw the line A’P through A’ between A’M and A’B in any direction. Let Q be any point on A’P. Join A and Q. Then from the definition AQ must cut BN which means that QP cuts BN as well. Thus AA’M is the first of the lines which do not cut BN and A’M is parallel to BN. The second case is described in figure 4: Take the point A’ upon AM produced backwards. Draw the line A’P

Figure 4: Case of A’ being produced on AM backwards through A’ between A’B and A’M in any direction. Extend A’P backwards and take any point Q on it. By definition, QA must cut BN. Take the in- tersection to be the point R. Then it follows that A’P lies within the closed figure A’ARB and thus had to cut one of its sides A’A, AR, RB or BA’. It is obvious that this had to be the side RB. Thus for any A’P (and Q taken on A’P) between BN and AM, QA cuts BN. Therefore A’M is parallel to BN. Thus, the definition of parallels was established. Gauss also proved the reciprocity of the Parallelism: “if a line (1) is parallel to a line (2) , then line (2) is parallel to line (1). The following proof is from his notes Werke. In his work[15], Gauss changes without any specific reason the order of A and B and Gauss mixed the order of the angles, that is 6 BAC = 6 CAB. The proof is stated as follow: from any point A upon (2) draw AB perpendicular to (1). Through A draw any line (3) between AB and (2) and let 6 1 AC be a line between the same lines AB and (2) such that BAC = 2 (2, 3) (angle between the lines 3 and 2). Then there are two cases that can be distinguished: the first case corresponds to AC cutting (1). The second one corresponds to when AC does not cut (1). Figure 5 corresponds to the first case: Let AC cut (1) in D. Take BE = BD so that B lies between E and D. Through D, draw the line DF between DA and (1) such that 6 ADF = 6 AED. The line DF cuts (2) in G. Take H on (1) such that EH = DG and draw the line AH. The triangles ABD and ABE are congruent and thus AE = AD. The triangles ADG and AEH are congruent, thus 6 EAH = 6 DAG. Then it holds that 6 EAH = 6 DAE = 6 (2, 3) This means that AH is indentical to the line (3) which cuts (1) in H and since 3 is any line between (2) and AB, then it holds that (2) is parallel to (1).

8 Figure 5: Case 1: When AC cuts (1)

The second case is drawn in ﬁgure 6. Its proof goes as follows: sup-

Figure 6: Case 1: When AC does not cut (1) pose AC does not cut (1) and let D be any point on (1). Then with the same argument as above it holds that 6 GAH = 6 DAE. But it also holds 6 6 6 6 6 1 6 that BAD < BAC and thus DAE < (2, 3) (since BAC = 2 (2, 3)). Therefore 6 (2, 3) > GAH and thus (3) is a line between AH and AD and it cuts DH. Thus (2) is parallel to (1).

Gauss defined parallelism in a new way, leaving the Fifth Postulate aside. This openend his mind to a whole new way of thinking, something he shared with Taurinus in a letter[Nov, 1824]: . . . the assumption that the angle sum is less than 180◦ leads to a geometry quite different from Euclid’s, logically coherent, and one that I am entirely satisfied with. It depends on a constant, which is not given a priori. The larger the constant, the closer the geometry to Euclid’s and when the constant is infinite they agree. The theorems are paradoxical but not self-contradictory or illogical. [. . .] All my efforts to find a contradiction have failed, the only thing that our understanding finds contradictory is that, if the geometry were to be true, there would be an absolute (if unknown to us) measure of length [. . . ] I As a joke

9 I’ve even wished Euclidean geometry was not true, for then we would have an absolute measure of length a priori.[16, p.63]. This is the beginning of hyperbolic geometry, something Saccheri found out before and rejected. This marked the beginning of the area of non-Euclidean geometry supported by the works of Bolyai and Lobachevsky.

References

[1] Euclid, Elements. Sir Thomas Little Heath. New York, Dover, 1956 [2] Proclos, Commentary on the First Book of the Elements by Euclid. Facsimile of Grynaeus’ edition (1533) and a translation in Interlingua. In: C.E. Sjöstedt, Le Axiome de Paralleleles de Euclides a Hilbert. BokförlagetNatur och Kultur. Stockholm, 1968. [3] Girolamo Saccheri, Euclides ab omni naevo vindicatus: sive Conatus Geometricus quo stabiliuntur prima ipsa universae Geometriae Prin- cipia. (Euclid Freed of Every Fleck: or A Geometric Endeavor in Which Are Established the Foundation Principles of Universal Geometry.) Mi- lan, 1733. Original text and translation by G.H. Helsted, The Open Court Publishing Company, Chicago, London, 1920. [4] Roberto Bonola, Non-Euclidian Geometry. Padua, 1906. Translation from the Italian by H.S. Carslaw. Sydney, 1911. Dover edition, 1955. [5] Georg Simon Klügel, Conatuum praecipuorum theoriam parallelarum demonstrandi recensio. Dissertation, GöttingUniversity, 1763. Fac- simile and German translation by Martin Hellmann, University of Cologne, http://www.uni-koeln.de/math-nat-fak/didaktiken/ mathe/volkert/titel. [6] C.E. Sjöstedt, Le Axiome de Paralleleles de Euclides a Hilbert. BokförlagetNatur och Kultur. Stockholm, 1968. [7] John Wallis, De Algebra Tractatus; Historicus & Practicus. Theatro Sheldoniano. Oxford, 1693. [8] Eves, Howard W. 1969. In Mathematical Circles: A Selection of Math- ematical Stories and Anecdotes. Vol. 2, Quadrants III and IV. Boston: Prindle, Weber & Schmidt. (p. 112, item 319) [9] Holistic Numerical Methods, Transforming Numerical Methods Ed- ucation for the STEM Undergraduate, University of Florida, USA. http://numericalmethods.eng.usf.edu/anecdotes/gauss.html

10 [10] Eric Temple Bell, Men Of Mathematics , Simon Schuster, Inc., New York, 1937 [11] The MacTutor History of Mathematics archive, School of Mathe- matics and Statistics, University of St Andrews Scotland, http:// www-history.mcs.st-and.ac.uk/Biographies/Gauss.html [12] G. Waldo Dunnington with additional material from Jeremy Gray and Fritz-Egbert Dohse, Carl Friedrich Gauss: Titan of Science, New York, 1995 [13] Michael John Bradley, The Foundations of Mathematics: 1800 to 1900 (Pioneers in Mathematics), New York, 2006 [14] Andaluzian Mathematical Society Of Education Thales (Sociedad Adaluza de educaci´onmatem´atica Thales http://thales.cica.es/ rd/Recursos/rd99/ed99-0289-02/biografias/cfgauss.html [15] Gauss, Werke, Bd. VII. http://www.wilbourhall.org/pdfs/Carl_ Friedrich_Gauss_Werke___8.pdf [16] Jeremy Gray, emphGauss and Non-Euclidean Geometry, Mathemat- ics and Its Applications, 2006, Volume 581, Centre for the History of the Mathematical Sciences, U.K http://www.springerlink.com/ content/ux2p811331w83357/

11 The seven bridges of K¨onigsberg And a brief summary of the rise of graph theory

History of mathematics Final paper

Authors: H. Imhoﬀ K. Meilgaard

Universiteit Leiden 2 Contents

1 Introduction5 1.1 The life of Euler...... 5 1.1.1 Early Life...... 5 1.1.2 University of Basel...... 5 1.1.3 St. Petersburg Academy of Sciences...... 5 1.1.4 Berlin Academy...... 6 1.1.5 Back to St. Petersburg...... 6 1.2 Konigsberg...... 6 1.2.1 The ﬁrst problem of graph theory...... 6

2 The Konigsberg Bridges problem9 2.1 Euler’s proof of the Konigsberger Bridges problem...... 9 2.2 The new bridges problem...... 15 2.3 How they should have rebuilt the bridges...... 16 2.4 Hamilton...... 16

3 Conclusion 17

4 References 19

3 4 Chapter 1

Introduction

1.1 The life of Euler

1.1.1 Early Life

Leonhard Euler grew up in Riehen near Basel, where he was born on April 15th 1707. His father, Paul Euler, a Protestant minister taught him his ﬁrst mathematics. Although Euler didn’t learn any additional mathematics in school his interest grew. So he started reading mathematical text on his own and took private lessons.

1.1.2 University of Basel

In 1720 when he was just 14 years old Euler was sent to the University of Basel to obtain a general education. There he met Johann Bernoulli who discovered the great mathematical potentional in Euler as he gave him private tuition. In 1723 Euler finished his study in philosophy and started studying theology. He didn’t like this study, but did it to satisfy his father. Johann Bernoulli persuaded his father to give his consent to change to a mathematical study. He finished his study in 1726 and it the same year he published his first paper on ”isochronus curves in a resisting medium”. And in 1726 Euler became second for the Grand Prize of the Paric Academy on the best arrangement of masts on a ship.

1.1.3 St. Petersburg Academy of Sciences

On April 5th Euler left Basel, because he knew he wasn’t get appointed to the chair of physics. So he moved to St. Petersberg and joined the St. Petersburg Academy of Sciences. Through the request of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical division of the Academy. Here he worked along side a lot of colleagues who greatly improved his work. When Bernoulli left St. Petersburg to return to Basel in 1733 it was Euler who was appointed to the senior chair of mathematics. Not long after that he married Katharina Gsell and had 13 children with her, from whom only 5 survied their infancy. Euler wrote a lot of papers during his time in St. Petersburg, and ﬁnished his book ”Mechanica” wich started Euler on the way to mayor mathematical work. In 1735 Euler started having health problems. He almost died of a fever and had

5 CHAPTER 1. INTRODUCTION

eye problems due to extensive cartographic work. Despite his health issues by 1740 his reputation was very high and he won the Grand Prize of the Paris Academy in 1738 and 1740.

1.1.4 Berlin Academy

In 1744 the Berlin Academy of Sciences was ﬁnished and after two invitations from Frederik the Great, Euler accepted and moved to Berlin to become the director of mathematics there. Here he worked for 25 years and wrote around 380 articles. Besides this astoundig amount of articles he also wrote entire books on variating subjects. Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration he bought books and instruments for the St Petersburg Academy, he continued to write scientiﬁc reports for them, and he educated young Russians.

1.1.5 Back to St. Petersburg

In 1766 Euler returned to St Petersburg after some problems he had with the king of Germany considering his presidency of Berlin Academy. Soon after his return to Russia, Euler became almost entirely blind after an illness. In 1771 his home was destroyed by ﬁre and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the ﬁre, still in 1771, restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind. Because of his remarkable memory he was able to continue with his work. Amazingly after his return to St. Petersburg (when Euler was 59) he produced almost half his total works despite the total blindness. Euler was greatly helped by a lot of people but in particular his son Johann Albrecht Euler who was appointed to the chair of physics at the Academy of St. Petersburg in 1766.

1.2 Konigsberg

1.2.1 The ﬁrst problem of graph theory

The Konigsberg Bridge Problem doesn’t look more important than an interesting puzzle. However this puzzle was the ﬁrst problem in an entirely new area of mathematics: Graph theory. Graph theory has since then grown to have applications in all kinds of sciences, like physics, biology and social sciences.

Another problem in graph theory: the Four Color Problem even raised questions about the notion of mathematical proof itself. The problem asks if we can use just four colors to color every planar map, such that two connecting regions do not have the same color. The problem was ﬁrst formulated by Augustus De Morgan in 1852 in a letter to Hamilton. A problem rose when in 1976 two mathematicians Kenneth Appel, and Wolfgang Haken published a computer-assisted proof. A large group of mathematicians did not accept this prove, because it could not be directly checked or validated by a member of the mathematical community. This discussion highlights an important historical fact about the standard of a mathematical proof. The standards are dependant on time and culture. In this paper we will

6 The Konigsberger Bridges problem. . . 1.2. KONIGSBERG

show an old and a modern version of the Konigsberg Bridge Problem to emphasize the diﬀerence in these proofs.

H. van Imhoﬀ, K. Meilgaard, Leiden University 7 8 Chapter 2

The Konigsberg Bridges problem

Informally we can sate the problem as: ”Is it possible to plan a stroll through the town of Konigsberg which crosses each of the towns seven bridges exactly once?”.

In 1736 an article appeared in the ”Commentarii Academiae Scientiarum Imperialis Petropoli- tanae”. In this article Euler makes a mathematical formulation of the Konigsberger Bridges problem.

2.1 Euler’s proof of the Konigsberger Bridges problem

In what follows, we take our translation from [1, pp. 3 - 8], with some portions eliminated in order to focus only on those most relevant to Eulers reformulation of the bridge crossing problem as a purely mathematical problem.

SOLUTIO PROBLEMATIS AD GEOMETRIAM SITUS PERTINENTIS ¯ 1 In addition to that branch of geometry which is concerned with magni- tudes, and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz ﬁrst mentioned, calling it the geometry of position. This branch is concerned only with the determination of position and its properties; it does not involve measure- ments, nor calculations made with them. It has not yet been satisfactorily determined what kind of problems are relevant to this geometry of position, or what methods should be used in solving them. Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did not require the measurement of distances, nor did calculation help at all, I had no doubt that it was concerned with the geometry of position — especially as its solution involved only position, and no calculation was of any use. I have therefore decided to give here the method which I have found for solving this kind of problem, as an example of the geometry of position.

It is clear from this text that Euler was completely new to this type of mathematics. And also that he doesn’t really see the point of it yet.

9 CHAPTER 2. THE KONIGSBERG BRIDGES PROBLEM

2 The problem, which I am told is widely known, is as follows: in K¨onigsberg in Prussia, there is an island A, called the Kneiphof ; the river which surrounds it is divided into two branches, as can be seen in Fig. [1.2], and these branches are crossed by seven bridges, a, b , c , d , e , f and g. Concerning these bridges, it was asked whether anyone could arrange a route in such a way that he would cross each bridge once and only once. I was told that some people asserted that this was impossible, while others were in doubt: but nobody would actually assert that it could be done. From this, I have formulated the general problem: whatever be the arrangement and division of the river into branches, and however many bridges there be, can one ﬁnd out whether or not it is possible to cross each bridge exactly once?

He immediately generalizes the problem like any great mathimatician would do for the following reason:

3 As far as the problem of the seven bridges of Königsberg is concerned, it can be solved by making an exhaustive list of all possible routes, and then finding whether or not any route satisfies the conditions of the problem. Because of the number of possibilities, this method of solution would be too difficult and laborious, and in other problems with more bridges it would be impossible. Moreover, if this method is followed to its conclusion, many irrelevant routes will be found, which is the reason for the difficulty of this method. Hence I rejected it, and looked for another method concerned only with the problem of whether or not the specified route could be found; I considered that such a method would be much simpler.

He is only concerned in the question IF there exist a path, not in ﬁnding exactly which path that is.

4 My whole method relies on the particularly convenient way in which the crossing of a bridge can be represented. For this I use the capital letters A, B, C, D, for each of the land areas separated by the river. If a traveller goes from A to B over bridge a or b, I write this as AB — where the ﬁrst letter refers to the area the traveller is leaving, and the second refers to the area he arrives at after crossing the bridge. Thus, if the traveller leaves B and crosses into D over bridge f, this crossing is represented by BD, and

10 The Konigsberger Bridges problem. . . 2.1. EULER’S PROOF OF THE KONIGSBERGER BRIDGES PROBLEM

the two crossing AB and BD combined I shall denote by the three letters ABD, where the middle letter B refers to both the area which is entered in the ﬁrst crossing and to the one which is left in the second crossing.

In present Graph Theory we use ordered pairs to represent the paths which is really similar as what Euler does here.

5 Similarly, if the traveller goes on from D to C over the bridge g, I shall represent these three successive crossings by the four letters ABDC, which should be taken to mean that the traveller, starting in A, crosses to B, goes on to D, and ﬁnally arrives in C. Since each land area is separated from every other by a branch of the river, the traveller must have crossed three bridges. Similarly, the successive crossing of four bridges would be represented by ﬁve letters, and in general, however many bridges the traveller crosses, his journey is denoted by a number of letters one greater than the number of bridges. Thus the crossing of seven bridges requires eight letters to represent it.

Right now it seems like a very inconvenient way to denote a path. But in a minute we’ll see how he applies it.

7 The problem is therefore reduced to finding a sequence of eight letters, formed from the four letters A, B, C and D, in which the various pairs of letters occur the required number of times. Before I turn to the problem of finding such a sequence, it would be useful to find out whether or not it is even possible to arrange the letters in this way, for if it were possible to show that there is no such arrangement, then any work directed toward finding it would be wasted. I have therefore tried to find a rule which will be useful in this case, and in others, for determining whether or not such an arrangement can exist.

8 In order to try to find such a rule, I consider a single area A, into which there lead any number of bridges a, b, c, d, etc. (Fig. [1.3]). Let us take first the single bridge a which leads into A: if a traveller crosses this bridge, he must either have been in A before crossing, or have come into A after crossing, so that in either case the letter A will occur once in the representation described above. If three bridges (a, b and c, say) lead to A, and if the traveller crosses all three, then in the representation of his journey the letter A will occur twice, whether he starts his journey from A or not. Similarly, if five bridges lead to A, the representation of a journey across all of them would have three occurrences of the letter A. And in general, if the number of bridges is any odd number, and if it is increased by one, then the number of occurrences of A is half of the result.

H. van Imhoﬀ, K. Meilgaard, Leiden University 11 CHAPTER 2. THE KONIGSBERG BRIDGES PROBLEM

Here you start to see what Euler’s methodology is towards solving this problem. And that it’s very similar to the arguments we would use today.

9 In the case of the Königsberg bridges, therefore, there must be three occurrences of the letter A in the representation of the route, since five bridges (a, b, c, d, e) lead to the area A. Next, since three bridges lead to B, the letter B must occur twice; similarly, D must occur twice, and C also. So in a series of eight letters, representing the crossing of seven bridges, the letter A must occur three times, and the letters B, C and D twice each - but this cannot happen in a sequence of eight letters. It follows that such a journey cannot be undertaken across the seven bridges of Königsberg.

With this simple argument Euler has already solved the Koningsberg bridges problem. But obviously he wants to generalize this strategy.

10 It is similarly possible to tell whether a journey can be made crossing each bridge once, for any arrangement of bridges, whenever the number of bridges leading to each area is odd. For if the sum of the number of times each letter must occur is one more than the number of bridges, then the journey can be made; if, however, as happened in our example, the number of occurrences is greater than one more than the number of bridges, then such a journey can never be accomplished. The rule which I gave for finding the number of occurrences of the letter A from the number of bridges leading to the area A holds equally whether all of the bridges come from another area B, as shown in Fig. [1.3], or whether they come from different areas, since I was considering the area A alone, and trying to find out how many times the letter A must occur.

In the Koningberg bridges problem, Euler only encountered odd numbered amounts of bridges leading to each area. So he needs to ﬁnd a rule for the even numbered ones aswell.

11 If, however, the number of bridges leading to A is even, then in describing the journey one must consider whether or not the traveller starts his journey from A; for if two bridges lead to A, and the traveller starts from A, then the letter A must occur twice, once to represent his leaving A by one bridge, and once to represent his returning to A by the other. If, however, the traveller starts his journey from another area, then the letter A will only occur once; for this one occurrence will represent both his arrival in A and his departure from there, according to my method of representation.

The rule becomes a little bit more complicated because he needs to divide the even numbered pieces of land into two diﬀerent categories depending wether or not you start your journey there.

12 If there are four bridges leading to A, and if the traveller starts from A, then in the representation of the whole journey, the letter A must occur three times if he is to cross each bridge once; if he begins his walk in

12 The Konigsberger Bridges problem. . . 2.1. EULER’S PROOF OF THE KONIGSBERGER BRIDGES PROBLEM

another area, then the letter A will occur twice. If there are six bridges leading to A, then the letter A will occur four times if the journey starts from A, and if the traveller does not start by leaving A, then it must occur three times. So, in general, if the number of bridges is even, then the number of occurrences of A will be half of this number if the journey is not started from A, and the number of occurrences will be one greater than half the number of bridges if the journey does start at A.

Combined with the following observation it is possible to determine, with an easy calculation, wether a journey exists or not.

13 Since one can start from only one area in any journey, I shall deﬁne, corresponding to the number of bridges leading to each area, the number of occurrences of the letter denoting that area to be half the number of bridges plus one, if the number of bridges is odd, and if the number of bridges is even, to be half of it. Then, if the total of all the occurrences is equal to the number of bridges plus one, the required journey will be possible, and will have to start from an area with an odd number of bridges leading to it. If, however, the total number of letters is one less than the number of bridges plus one, then the journey is possible starting from an area with an even number of bridges leading to it, since the number of letters will therefore be increased by one.

With the result it is suﬃcient to count for each area how many bridges lead to it and how many occurances it make in the representation of a path. And then to add all of these up and compare them to the total amount of bridges plus one. We will see how this is done exactly in the next part.

14 So, whatever arrangement of water and bridges is given, the following method will determine whether or not it is possible to cross each of the bridges: I ﬁrst denote by the letters A, B, C, etc. the various areas which are separated from one another by the water. I then take the total number of bridges, add one, and write the result above the working which follows. Thirdly, I write the letters A, B, C, etc. in a column, and write next to each one the number of bridges leading to it. Fourthly, I indicate with an asterisk those letters which have an even number next to them. Fifthly, next to each even one I write half the number, and next to each odd one I write half the number increased by one. Sixthly, I add together these last numbers, and if this sum is one less than, or equal to, the number written above, which is the number of bridges plus one, I conclude that the required journey is possible. It must be remembered that if the sum is one less than the number written above, then the journey must begin from one of the areas marked with an asterisk, and it must begin from an unmarked one if the sum is equal. Thus in the K¨onigsberg problem, I set out the working as follows:

H. van Imhoﬀ, K. Meilgaard, Leiden University 13 CHAPTER 2. THE KONIGSBERG BRIDGES PROBLEM

Number of bridges 7, which gives 8

Bridges A, 5 3 B, 3 2 C, 3 2 D, 3 2

Since this gives more than 8, such a journey can never be made.

15 Suppose that there are two islands A and B surrounded by water which leads to four rivers as shown in Fig. [1.4]. Fifteen bridges (a, b, c, d, etc.) cross the rivers and the water surrounding the islands, and it is required to determine whether one can arrange a journey which crosses each bridge exactly once. First, therefore, I name all the areas separated by water as A, B, C, D, E, F, so that there are six of them. Next, I increase the number of bridges (15) by one, and write the result (16) above the working which follows.

16 A*, 8 4 B*, 4 2 C*, 4 2 D, 3 2 E, 5 3 F*, 6 3 16 Thirdly, I write the letters A, B, C, etc. in a column, and write next to each one the number of bridges which lead to the corresponding area, so that eight bridges lead to A, four to B, and so on. Fourthly, I indicate with an asterisk those letters which have an even number next to them. Fifthly, I write in the third column half the even numbers in the second column, and then I add one to the odd numbers and write down half the result in each case. Sixthly, I add up all the numbers in the third column in turn, and I get the sum 16; since this is equal to the number (16) written above, it follows that the required journey can be made if it starts from area D or E, since these are not marked with an asterisk. The

14 The Konigsberger Bridges problem. . . 2.2. THE NEW BRIDGES PROBLEM

journey can be done as follows:

EaFbBcFdAeFfCgAhCiDkAmEnApBoElD,

where I have written the bridges which are crossed between the corresponding capital letters.

This concludes Euler’s quest to ﬁnd a general method to solve bridge-type problems. You can clearly see that the types of reasonings Euler does, with odd and even numbered amount of bridges leading to an area, are very similar to those made in present Graph Theory. Now we use the deﬁnition of the order of a node to describe the same thing.

2.2 The new bridges problem

All seven bridges were destroyed by an Allied bombing raid in 1944 and only five were rebuilt. Knigsberg, along with the rest of northern East Prussia, became part of the Soviet Union (now Russia) at the end of World War II and was renamed Kaliningrad. Now it is possible to visit the five rebuilt bridges via an Euler path (route that begins and ends in different places), but there is still no Euler tour (begin and end at the same place).

We’ll now state the following theorem about this graph: Theorem 2.1. The new graph contains an Eulerian cycle but not an Eulerian trail.

Thanks to modern mathematics and Euler’s original article we now have the following theorems we can use: Theorem 2.2. A connected graph is an Eulerian graph (it contains an Eulerian cycle) if and only if the degree of each vertex is even.

As a second we have: Theorem 2.3. A connected graph is a semi-Eulerian graph (it contains an Eulerian path) if and only if the there are at max two vertices with uneven degree.

Now with these two theorems we can easily check if all the vertices of the new problems ﬁt the given conditions, which it does in the case of the second, but not the ﬁrst theorem. And thus we can now draw the following conclusion: We can walk an Eulerian path according to Theorem 2.2, but we can’t walk an Eulerian cycle according to Theorem 2.1. QED

H. van Imhoﬀ, K. Meilgaard, Leiden University 15 CHAPTER 2. THE KONIGSBERG BRIDGES PROBLEM

2.3 How they should have rebuilt the bridges

It is deﬁnitely a pity that they haven’t rebuilt the bridges so that you can walk an Eulerian Cycle. That would have been a great tribute to Euler. We would like to end this article with a few examples of rebuilding the brides that would have allowed for. It’s quite easy to construct diﬀerent examples based upon Theorem 2.1 in the previous section. We just have to make sure all that the degree of each vertex is even:

In this example for instance, we can start at A, then go to C and back to A (crossing a diﬀerent bridge), then to B and then to D, and then back to A. We’ve now walked an Eulerian cycle!

2.4 Hamilton

Another thing sprung from graph theory is the Hamilton cycle, the problem here is to cross all the vertexes once and only once. Of course we can see directly that the following path would deliver us a Hamilton cyle in both examples: Recall ﬁg 1.2:

We can see that if we start at A, we can go to B, and from B to D and from D to C and C to A, and thus we have a Hamilton path. Exactly the same path can still be made if bridges b and d wouldn’t be there! And so we are done.

16 The Konigsberger Bridges problem. . . Chapter 3

Conclusion

We’ve seen that neither the seven, nor the ﬁve bridges problem could be solved. And that the simple mathematical puzzles can derive entirely new branches of mathematics all together.

Therefore it is of utter importance that we keep puzzling with the fun problems of mathematics of our modern society, because maybe those will sprout their own branches of mathematics one day.

17 18 Chapter 4

References

[1]: Graph Theory: 1736 to 1936, Biggs, N., Lloyd, E., Wilson, R., Clarendon Press, Oxford, 1976. [2]:Early Writings on Graph Theory: Euler Circuits and The Konigsberg Bridge Problem, by Janet Heine Barnett

19 History of Mathematics - Paper assignment Roel Jongen, Matthijs Warrens Title: Cantor and the countable versus uncountable distinction

1. Introduction Georg Cantor was a German mathematician who greatly contributed to the development of mathematics in the second half of the 19th century, especially to the field of set theory. Cantor received his doctorate in 1867 from the Uni- versity of Berlin for a thesis on number theory. A short time later he accepted a position at the University of Halle where he would spend the rest of his career. In 1872 Cantor was promoted to extraordinary professor and he moved the focus of his work from number theory to analysis. In this period he proved the uniqueness of the representation of a function by trigoniometric series, a problem that had withstood attempts by various high achieving mathematicians. In 1873 Cantor introduced the notion of one-to-one correspondence and proved that the rational and algebraic numbers are countably infinite. A short time later he proved that the real numbers are uncountably infinite. These results are described in his seminal paper published in 1874. Beginning in 1879, Cantor published the first of a series of articles in Mathematische Annalen on his ideas of set theory. In these papers Cantor introduced the notions of cardinal and ordinal numbers, and well-ordered sets. Basically, he single-handedly created an extraordinary set theory in these papers. During his career Cantor did not have the support of all of his peers. Many refused to correspond or work with Cantor because they felt that Cantor’s work contained philosophical errors. Nowadays Cantor’s work is thought to be brilliant (Dunham 1990, Gillispie 1970). In this paper we consider the importance of Cantor’s 1874 paper, in which he introduced the distinction between countable and uncountable sets. The paper is organized as follows. In the next section we discuss some issues related to the foundations of calculus that where raised in the 19th century. In Section 3 we discuss the notion of countably infinite and present several examples. In Section 4 we discuss the notion of uncountably infinite and present some related results. Section 5 is used to discuss algebraic and transcendental numbers. In Section 6 we discuss the importance of the countable versus uncountable distinction for modern mathematics.

2. The notions of limit and continuity The invention of calculus in the second half of the 17th century can be at- tributed to both Leibniz and Newton. Throughout the 18th century mathematicians solved problems in mathematical physics and booked various other successes with this new calculus. But at the end of the 18th century more and

1 more mathematicians became uneasy over the foundations of the calculus, especially the ‘notion’ of infinitely small quantities. One of the key concept in calculus, the ‘limit’, was not well-defined. Since the concept of a ‘limit’ is inherently quite deep, requiring an appreciation of the real numbers that is by no means easy to come by, it took several gifted mathematicians to refine the idea of ‘limit’ in the 19th century. Due to the work of Cauchy and Weierstrass, a means was finally found to avoid the ‘notion’ of infinitely small quantities (Dunham 1990).

Definition 1. We have limx→a f(x) = L if for any ε > 0, there exists a δ > 0 such that, if 0 < |x − a| < δ, then |f(x) − L| < ε. Using the notion of a limit we can define the notion of continuity of a function at a point. Definition 2. A function f(x) is said to be continuous in a point c if limx→c f(x) = f(c). As mathematicians examined the calculus using these rigorous definitions, they made several unsettling discoveries concerning the set of real numbers, especially two subsets of the reals, the sets of rational numbers Q, and the set of irrational numbers R\Q. The rational numbers are the fractions, the numbers 1 22 that are ratios of integers. Examples are 2 and 7 . The irrational numbers are those numbers that cannot be represented by fractions. An example of an √irrational number that was already known in Ancient Greece is the number 2. As the 19th century progressed a function was found that is continuous at each irrational point in the interval [0, 1], yet discontinuous at each rational point. An example of such a function is the so-called ruler function (Heuer 1965, Dunham 2005, Sholapurkar 2007). m 1 Definition 3. The ruler function f : [0, 1] → R is defined by f n := n when m n is a fraction in lowest terms, f(0) := 1, and f(x) := 0 when x is irrational. Claim 1. The ruler function is discontinuous at each a ∈ Q and is continuous elsewhere. Proof: Let a be a rational number in [0, 1]. By the density of irrational numbers there is a sequence {xn} of irrational numbers that converges to n∈N a. But f(xn) = 0 for all n ∈ N, while f(a) > 0. Hence, f is discontinuous at a. Next, let b be a irrational number in [0, 1], and let ε > 0. The Archimedean 1 property of R asserts that there is an integer n such that n > ε . Fix this n. Since there are only finitely many rational numbers in [0, 1] with denominator p less than n, there exist a δ > 0 such that for every rational number q with p |b − q | < δ, we have q > n. This implies that for every x with |b − x| < δ, we have f(x) < ε. Hence f is continuous at b. This result for the ruler function indicates that there is not a symmetry between the rational and irrational numbers. The two sets are not interchange- able, but to the mathematicians in the first half of the 19th century it was

2 not clear what was going on. It appeared that some of the important and fundamental questions of the calculus rested upon profound properties of sets. The man who would show the mathematical community how to study the properties of sets was Georg Cantor (Dunham 1990).

3. Countable sets If we compare two finite sets, we can tell which of the two is larger by comparing the number of elements in both of them. Cantor used the following definition for infinite sets (Dunham 1990). Definition 4. Two sets A and B are called equivalent, denoted by A ∼ B, if there exists a bijection between them. Because a bijection between A and B links to every element in A one unique element in B and vice versa, A and B must have the same number of elements if they are equivalent. Claim 2. The relation ∼ is an equivalence relation Proof: Let M,N,P be sets. We have to prove that ∼ is reflexive, symmetric and transitive. Using the identity function M → M, x 7→ x, we find that M ∼ M, that is, ∼ is reflexive. Next, suppose that M ∼ N. Then there exists a bijection f : M → N. By definition f −1 is a bijection N → M, so N ∼ M. Thus, ∼ is symmetric. Finally, suppose that M ∼ N and N ∼ P . Then there exist bijections f : M → N and g : N → P . Now g ◦ f : M → P is a bijection since it is a composition of two bijections. Hence, M ∼ P , and it follows that ∼ is transitive. Cantor called the equivalence classes of the equivalence relation ∼ cardinal numbers. For a set M we will write |M| to denote its cardinal number. Example 1. Let M = {0, 1}, N = {±1} and P = {1, 2, 3}. We have M ∼ N since |M| = |N| = 2. P is not equivalent to either M or N because |P | = 3. For studying infinite sets Cantor proposed the following definition.

Definition 5. A set is called countably infinite if it is equivalent to N = {1, 2,...}. A set is called countable if it is either finite or countably infinite.

Example 2. Let P denote the set of prime numbers. Suppose we order the prime numbers using the regular order <, and write pn for the n-th prime number. The function f : n 7→ pn is a bijection from N to P. Hence, N ∼ P, and it follows that P is countably inﬁnite. Example 3. Let Z = {0, ±1, ±2,...} denote the set of integers. The function 1 + (−1)n(2n − 1) f : n 7→ 4

3 is a bijection from N to Z. For the ﬁrst small numbers of N and Z the function f does the following.

N : 1 2 3 4 5 6 7 8 9 ··· l l l l l l l l l

Z : 0 1 −1 2 −2 3 −3 4 −4 ···

Hence, N ∼ Z, and it follows that Z is countably infinite. Example 4. Let 2Z = {0, ±2, ±4,...} denote the set of even integers. Since the function f : n 7→ 2n is a bijection from Z to 2Z, we have 2Z ∼ Z. Because Z ∼ N, it follows that 2Z is countably infinite. Example 5. Let n be a positive integer and let nZ = {0, ±n, ±2n, . . .} denote the sets of multiples of n. Since the function f : m 7→ nm is a bijection from Z to nZ, we have nZ ∼ Z, and it follows that nZ is countably infinite. Claim 3. The set of rational numbers Q is countably infinite. Proof: For this proof we arrange the rational numbers in an array. To do this, we place all fractions with numerator 1 in the first column, all fractions with numerator −1 in the second column, all fractions with numerator 2 in the third column and so on. We arrange the rows in such a way that all fractions in a row match the number of the row in their denominator. In this way we find the following array.

0 1 −1 2 −2 3 −3 4 ... 1 1 2 2 3 3 4 2 − 2 2 − 2 2 − 2 2 ... 1 1 2 2 3 3 4 3 − 3 3 − 3 3 − 3 3 ... 1 1 2 2 3 3 4 4 − 4 4 − 4 4 − 4 4 ......

It is clear that this array contains each rational number at least once. Now we can construct a bijection between N and Q by weaving through the array, omitting the fractions which we have passed already in another representation. This gives the following bijective map.

N : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... lllllllll lllll 1 1 1 1 1 2 1 1 Q : 0 1 2 −1 2 − 2 3 4 − 3 −2 3 3 − 4 5 ...

Cantor denoted the cardinal number belonging to N by ℵ0. We have found in this section that |N| = |P| = |Z| = |2Z| = |nZ| = |Q| = ℵ0.

4 4. Uncountable sets After the surprising result that so many sets are countable, we could ask ourselves whether it is true that any inﬁnite set is countably inﬁnite. Claim 4 shows that the answer to this question is negative. Claim 4. The unit interval (0, 1) is uncountable. Proof: Suppose that (0, 1) is countable. Then there exists a sequence x1, x2, x3,..., containing all the elements of (0, 1). Now we write each xn as a sequence an1, an2, an3,... of the consecutive digits of its decimal expansion and arrange it in an array as follows.

x1 = a11 a12 a13 a14 ··· x2 = a21 a22 a23 a24 ··· x3 = a31 a32 a33 a34 ··· x4 = a41 a42 a43 a44 ··· ...... xn = an1 an2 an3 an4 ··· ......

Now consider the real number b whose decimal expansion is given by the row b1, b2, b3,.... Here b1 is chosen in such a way that it is not equal to 0, 9 or a11, b2 is chosen in such a way that it is unequal to 0, 9 or a22 and in particular any bn is chosen in such a way that it is unequal to 0, 9 or ann. It is clear that b is a real number, and since we prohibit its decimals to be equal to 0 or 9, it can’t be equal to 0.0000... = 0 or 0.99999... = 1. So b ∈ (0, 1) holds. On the other hand it is evident that b is unequal to any xn, since its n- th decimal place is different. Hence, we have a contradiction, and we conclude that (0, 1) is uncountably infinite. Example 6. Let a, b ∈ R such that a < b. The function f : x 7→ a + (b − a)x is a bijection from (0, 1) to (a, b). Hence, (0, 1) ∼ (a, b), and it follows that the interval (a, b) is uncountably infinite. Example 7. The function 2x − 1 f : x 7→ x(1 − x) is a bijection from (0, 1) to R. Hence, (0, 1) ∼ R, and it follows that R is uncountably infinite.

Claim 5. The set of irrational numbers R\Q is uncountable. Proof: Suppose that the irrational numbers are countable. Then there exists a sequence {an}n∈N, containing each irrational number exactly once. Let {bn}n∈N be a sequence which contains each rational number exactly once. Because R is the disjoint union of the rational and the irrational numbers,

5 each real number is either a unique element of {an}n∈N or of {bn}n∈N. So the sequence {cn}n∈N given by ( a(n+1)/2 if n is odd, cn = an/2 if n is even contains any real number. Hence, we have a contradiction, and we conclude that the irrational numbers are uncountable. Cantor denoted the cardinal number belonging to (0, 1) by c. We have found in this section that |(0, 1)| = |(a, b)| = |R\Q| = |R| = c.

5. Algebraic and transcendental numbers The fundamental theorem of algebra tells us that a non-zero polynomial with integer coefficients has a zero in the complex numbers. That is, for any polynomial with integer coefficients there is a a ∈ C such that f(a) = 0. Transcen- dence theory is concerned with the converse question, given a number a ∈ C, is there a polynomial f with integer coefficients such that f(a) = 0. This warrants the following definition, which is limited to real numbers.

Definition 6. a ∈ R is called algebraic if there is a f ∈ Z[x]\{0} such that f(a) = 0. The number a is called transcendental if it is not algebraic. The algebraic numbers appear to be quite abundant. Since a rational number m/n with m, n ∈ Z is a zero of the linear polynomial f(x) = nx − m, we have that all rational numbers√ √ are in fact algebraic. Other examples, are the irrational numbers 2 and 3 5, since they are zeros of the polynomials f(x) = x2 − 2 and g(x) = x3 − 5 respectively. Since transcendental numbers are defined by what they are not, it can be difficult to show that a given number is transcendental. The first to prove the existence of transcendental numbers was Liouville in 1844, using continued P∞ −n! fractions. In 1851 Liouville presented the number n=1 10 = 0.1100010..., the first decimal example of a transcendental number. It is a so-called Liou- ville number, a class of numbers that can be more closely approximated by rational numbers than can any algebraic number. In 1873 Hermite proved that P∞ −1 the number e = n=0(n!) , the base of the natural logarithm, is transcendental. This was the first number to be proved transcendental without having been specifically constructed for the purpose (Burger, Tubbs 2004, Shidlovskii 1989). Building on Hermite’s result, Lindemann showed that π, the ratio of the cir- cumference to the diameter of a circle, is transcendental in 1882. He thereby solved the ancient Greek problem concerning the quadrature of the circle. The Greeks had sought to construct, with ruler and compass, a square with area equal to that of a given circle. If a unit length is prescribed this amounts to √ constructing two points in the plane at a distance π apart. In 1837 Wantzel

6 showed that the constructible numbers are a subset of the algebraic numbers. √ Lindemann showed that π is however transcendental (Burger, Tubbs 2004, Shidlovskii 1989). In 1873, when Hermite proved that the number e is transcendental, only a small set of transcendental numbers had been found. The algebraic numbers in contrast seemed to constitute a vast set. Then, in 1874, Cantor presented the following astounding result (Hart 2011). Claim 6. The set of algebraic numbers is countably inﬁnite. Proof: Let α be an algebraic number. Then there exists a polynomial p ∈ Z[x]\{0} such that p(α) = 0. Write this polynomial as n p(x) = a0 + a1x + ... + anx and call the natural number

N = n + |a0| + |a1| + ... + |an| the height of p. Define the height of α as the smallest height of a polynomial having α as its root. For every N ∈ N there exist finite many polynomials having N as its height, so there exist finite many algebraic numbers having N as its height. Finally, we can make a sequence of algebraic numbers by sorting them by height and by using the order of R to sort the numbers of equal height. Since Example 7 showed that the real numbers R are uncountably infinite, an immediate corollary of Claim 6 is that the transcendental numbers are uncountably infinite, that is, almost all real numbers are in fact transcendental.

6. The countable versus uncountable distinction At some levels the language of set theory provides the vocabulary for all of mathematics. Although many notions in what we now call set theory existed before Cantor, Cantor’s work vastly expanded this language. Most objects that are used in modern mathematics are deﬁned as sets equipped with some extra structure. Consider the following two examples from algebra and topology. Deﬁnition 7. A group is a set G equipped with an associative map G×G → G for which there is an identity element, and with respect to which every element in G has an inverse.

Definition 8. A metric space is a set S equipped with a function d : S×S → R which satisfies for all x, y, z ∈ S, d(x, x) = 0 ⇔ x = 0, d(x, y) ≥ 0, and the triangle inequality d(x, y) ≤ d(x, z) + d(y, z). The significance of Cantor’s work lies perhaps more in the distinction between countable and uncountable sets. In many contexts in mathematics one must include some sort of countability assumption to exclude pathological cases. Consider the following examples from functional analysis and topology.

7 Definition 9. A Hilbert space is a complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. A Hilbert space is separable if and only if it admits a countable orthonormal basis. Hence, for a separable Hilbert space there is a countable subset which gets arbitrarily close to any element of the Hilbert space. Definition 10. A space is said to be second-countable if its topology has a countable base. Most ‘well-behaved’ spaces in mathematics are second-countable. For exam- n ple, the Euclidean space R with its usual topology is second-countable. Another place where the notion of countability shows up is measure theory, which lies at the foundations of probability theory and modern calculus (Dun- ham 2005). Definition 11. Let S be a set. A subset Σ ⊂ P(S) is called a σ-algebra if it is non-empty, closed under complementation, and closed under countable unions. Using the notion of a σ-algebra we can define a measure of a set. In probability theory the probability of an event occuring corresponds to the measure of some set corresponding to that event. Definition 12. Let Σ be a σ-algebra over a set S. A function µ : S → R ∪ {±∞} is called a measure if it is non-negative, that is, µ(E) ≥ 0 for all E ∈ Σ, if it is countable additive, that is, if for all countable collections

{Ei}i∈I of pairwise disjoint sets in Σ we have ! [ X µ Ei = µ (Ei) , i∈I i∈I and if µ(∅) = 0. We end this paper by showing that the measure of a countable set is zero. Claim 7. A countable set has measure zero.

Proof: Let ε > 0 and let M = {m1, m2, m3,...} be a countable set. We must show that we can cover M with a countable number of open intervals such that the sum of the length of the intervals is less than ε. Cover mn with the interval ε ε m − , m + . n 2n+2 n 2n+2 n+1 Thus, for element mn the length of the interval is ε/2 . If we sum these lengths we obtain ∞ ∞ n ! X ε X 1 1 ε = ε − 1 − = < ε. 2n+1 2 2 2 n=1 n=0

8 References

Burger EB, Tubbs R (2004) Making Transcendence Transparent. Springer, New York. Cantor GFLP (1874) Uber¨ eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angwandte Mathematik, 77, 258-262. Dunham W (1990) Journey through Genius: The Great Theorems of Math- ematics. John Wiley & Sons. Dunham W (2005) Touring the Calculus Gallery. The American Mathe- matical Monthly, 112, 1-19. Gillispie CC (ed) (1970) Dictionary of Scientific Biography. Scribner, New York. Hart KP (2011) Verzamelingenleer. Heuer GA (1965) Functions continuous at the irrationals and discontinuous at the rationals. The American Mathematical Monthly, 72, 370-373. Shidlovskii AB (1989) Transcendental numbers. De Gruyter, Berlin. Sholapurkar VM (2007) On a theorem of Vito Volterra. Resonance, 12, 76-79.

9 D.S. Brown (0737666), M. Kortsmit (0740675)

Four colour Problem

History of Mathematics: Final Paper

May 24, 2012

Mathematisch Instituut, Universiteit Leiden CONTENTS CONTENTS

Contents

1 Introduction 1

2 Proof attempts 1 2.1 The ﬁrst ’proof’ by Kempe ...... 1 2.2 Detected error: eleven years later ...... 2 2.3 Decisive work through the century ...... 2

3 The First "Proof" 3 3.1 A sketch of the proof ...... 3 3.2 The reaction of the mathematical community ...... 3

4 Computer-assisted Proofs 4 4.1 What is a proof? ...... 4 4.2 Modern role of computers in mathematics ...... 4

5 Arguments for Computer-assisted Proofs 5 5.1 Unreliability of the Computer ...... 5 5.2 Proof length: beyond human capacity ...... 5 5.3 Social context ...... 6

6 Arguments against Computer assisted proofs 6 6.1 The Value of a Proof ...... 6 6.2 In Search of Elegance ...... 6

7 Conclusion 7

i 2 PROOF ATTEMPTS

1. Introduction

The Four Colour problem was first noticed by Francis Guthrie in 1852. Guthrie was a student at University College London, where he studied Law and Mathematics with De Morgan as his supervisor. In London, he functioned as a barrister until he moved to South Africa in 1861 as a Professor of Mathematics at the South African College which later became University of Cape Town. He published a few mathematical papers and became interested in botany. He earned a lasting name for himself with his botanical research: even two newly discovered types of plants were named after Guthrie. During the time Francis Guthrie was studying law, his brother Frederick Guthrie had become a student of DeMorgan. In his spare time Francis Guthrie was trying to colour the map of countries of England and noticed that four colours seemed to be sufficient to colour most maps. He asked his brother Frederick if it was true that any map can be coloured using four colours in such a way that regions sharing a common boundary segment receive different colours. Frederick Guthrie communicated the conjecture to DeMorgan. DeMorgan wasn’t able to answer this question, but communicated the problem to Hamilton, who wasn’t likely to answer this question soon either, mainly because the problem did not grab his attention. De Morgan kept trying to persuade mathematicians to work on Guthrie’s problem and thus several mathematicians did. Charles Peirce in the USA attempted to prove the Con- jecture in the 1860’s but that only resulted in his lifelong interest in the problem. Cayley also became acquainted with the four colour theorem and sent a paper on the colouring of maps to the Royal Geographical Society, which was published in 1879. This paper is the first printed reference of the Four Colouring conjecture, it explains where the difficulties lie in attempting to prove the Conjecture.

Theorem. (Four colour Theorem): The regions of any simple planar map can be coloured with only four colours, in such a way that any two adjacent regions have diﬀerent colours.

The Four Colour theorem may seem simple, but the proof deﬁnitely isn’t. It took mathematicians over a hundred years to come to a proof that still isn’t generally accepted as a proof. This paper will give a short glance at the history of the proof of the Four Colour theorem, and discusses the philosophical aspects of this proof.

2. Proof attempts

2.1. The ﬁrst ’proof’ by Kempe

In 1879 the ﬁrst attempt to prove the theorem was done by Kempe. Kempe was a Lon- don barrister as well, and he had studied mathematics under Cayley at Cambridge. He announced in Nature that he had a proof of the Four Colour Conjecture. Kempe submitted his proof of the theorem to the American Journal of Mathematics where it was published in 1879. His method involves creating ”chains of two colours” within a graph, in order to predict possible combinations of such colourings. Kempe was greatly admired for his proof. He even was elected a Fellow of the Royal Society and served as its treasurer for many years. He was knighted in 1912.

1 2.2 Detected error: eleven years later 2 PROOF ATTEMPTS

He published improved versions of his proof twice. The second had the interest of Tait, Professor of Natural Philosophy at Edinburgh. Tait showed in 1879 that the proof by Kempe was incorrect, and he came up with his own proof.

2.2. Detected error: eleven years later In 1890, 11 years later, Percy John Heawood, a lecturer at Durham, published a paper called "Map colouring theorem" containing a defect in the proof of Tait which was based on the false assumption that every three-connected planar graph is Hamiltonian. Neither Kempe nor Tait could correct the mistake in the proof, so the Four Colour theorem became the Four Colour problem for the second time in history. In this same paper Heawood did prove that every map can be 5-coloured.

2.3. Decisive work through the century In 1891, Petersen realised that Tait’s methods show that the Four Colour problem could be adapted to a conjecture on colouring the edges of a graph.

Figure 1: Map to graph

Since the first textbook on graph theory was published in 1969, Peterson was very pro- gressive with his idea about graphs. In his work, he used a definition by Hamilton: Definition. (Hamiltonion circuit): a Hamiltionian circuit is a closed walk containing each vertex exactly once) or by a collection of mutually disjoint subcircuits of even length. Peterson went on to show that the Four Colour problem is equivalent to the conjecture that any planar cubic graph contains a Hamiltonian circuit. The map colouring problem never lost Heawood’s interest. In 1898 he proved that if the number of edges around each region is divisible by 3 then the regions are 4-colourable. Renewed interest in the USA was due to Veblen who published a paper in 1912 on the Four Colour conjecture generalizing Heawood’s work. Birkhoff introduced theories on reducibility and chromatic polynomials. Franklin in 1922 published further examples and proved that any map with less than 25 regions can be 4-coloured. Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Ore and Stemple to 39 in 1970 and Mayer to 95 in 1976. Heesch introduced the final ideas necessary for the solution of the Four Colour conjecture: the method of discharging in 1969. He thought that the Four Colour conjecture could be solved by considering a set of around 8900 configurations.

2 3 THE FIRST "PROOF"

3. The First "Proof"

After years of failed attempts by some of the greatest mathematicians for their time, a proof that wasn’t found to be flawed was finally devised in 1976 by Kenneth Appel and Wolfgang Haken. What was unusual about this proof, is that it made extensive use of what is called computer-assistance. The necessary groundwork for this computer-aided proof was laid by the German mathematician Heinrich Heesch over a period of time during the 1960s and 1970s. The main theme in this was a number of proof techniques which allowed for the reduction of the amount of different subgraphs which needed to be analysed. The main technique he developed was the method of discharging, which was essential for the proof given by Appel and Haken. These techniques were devised especially for the Four Colour theorem, but he was unable to obtain the necessary supercomputer time to put his work to use.

3.1. A sketch of the proof The proof starts by assuming the theorem is false, and use this to deduce a counter-example. In their case this meant assuming that not all maps are four colourable, and there must exist a smallest map which can only be coloured with five colours or more. They then showed that such a smallest counter-example couldn’t exist. This was done by utilizing two concepts: • An unavoidable set: A set of submaps such that every possible map must contain atleast one member of this set. • A reducible configuration: A submap such that, if a map which is four colourable contains such a submap, it can be reduced to a map which is smaller but is still four colourable. This means that a minimal counter-example cannot contain such a submap, as this would contradict its minimality. In their proof they then showed that there are essentially 1,936 reducible configurations, and each of these configurations were checked to see if any of these were unavoidable. This is were the computer played its role, and checked each of these in what took a computer over 1200 hours of time to compute. After this lengthy wait, the two mathematicians were left with a group of reducible configurations that were in an unavoidable set. Since every possible map must contain at least one of these submaps, and since any map containing one of those submaps cannot be a minimal counter-example, they could conclude that no such thing could exist, thereby proving the Four Colour theorem.

3.2. The reaction of the mathematical community Although this was clearly a significant result, it was one which caused a number of problems and hit a nerve that is very deep engrained in most mathematicians. The proof which they eventually published in its entirety 1989 was 741 pages long, and included 400 pages of microfiche which contained the output of the computer verification. Microfiche reduces the physical size of documents significantly, so the total length of the proof was enormous, and much too long to be humanly verified. Due to the fact that this proof could not be hand checked, many mathematicians didn’t consider it a true proof. Philosopher and mathematician Thomas Tymoczko took it even further and chose to not classify the problem as a mathematical theorem, but as an a posteriori truth and putting it next to the realms of the empirical sciences.

3 4 COMPUTER-ASSISTED PROOFS

Others recieved the proof in a more welcoming fashion, and considered it part of the evolution of the form of mathematics. Nonetheless, these mathematicians also recognized the limitations of such a non-veriﬁable proof.

4. Computer-assisted Proofs

4.1. What is a proof? In order to meaningfully enter into this discussion, we must first define what we understand to be a proof. In general a mathematical proof is a logical argument which is used to justify, rigorously, a mathematical statement called a theorem. For this logical argument, while often presented in natural language, it must be possible to convert it into formal predicate logic. A natural language proof, which is often called an "informal proof" in proof theory, must be verified by readers and can contain some ambiguity which is inherent to natural language. On the other hand the "formal proof", according to proof theory, is one which technically could be verified by a machine as it only has to comply with the axioms of proof theory and the rules of logic.

4.2. Modern role of computers in mathematics The idea of computer-assisted proofs was one that naturally evolved together with the in- creasing computation strength of computers. These are proofs which make use of the quick arithmetic speed of a computer to usually check a predeﬁned amount of objects, and thus such proofs are mostly proofs-by-exhaustion. The methods used by Appel and Haken were tailor-programmed for the Four Colour proof, in which they formalized the required logical reasoning for the proof in computer language.

Figure 2: An excerpt from Metamath

Since then the use of computers in mathematical reasoning has grown and matured much. The ﬁeld can be approximately split into three groups of programs. We have interac- tive theorem provers, such as COQ, which try to work together with human input to together reach proofs. On the other hand we have automatic theorem provers, like SPASS, which try

4 5 ARGUMENTS FOR COMPUTER-ASSISTED PROOFS to prove theorems by assuming the opposite of a theorem and exhaustively combine ex- isting theorems till it finds a contradiction. The last kind is what is called an automatic theorem prover, which checks the basic logical steps to see if they are justified. A few of such programs are Metamath, or HOL Light. Although the field of computer-assisted proofs has grown much since 1976, and have tried to solve much of the problems people initially had with computer-assisted proofs, we would like to bring up some of the discussion caused by their original paper in 1976.

5. Arguments for Computer-assisted Proofs

Computer-assisted proofs are revolutionary mathematics: a brand new technique. Every new example of computer-assisted proof is very helpful to learn from. Introducing new fields in mathematics is a matter of acceptance and familiarity. Nowadays the computer is as indispensable as a pencil or paper, but this hasn’t always been the case. Especially people who grew up without computers have the instinct to question the validity of the results obtained from such elusive machines. We need to see a wide variety of examples before we are comfortable using this technique. We need to read clearly written arguments and algorithms which prove theorems via this new method in order to appreciate their elegance and set standards for publication. To gain acceptance, we need to see a wide variety of examples of computer-assisted proofs. They need to be clearly formulated and carefully tested. The algorithm should be well documented and should be able to be implemented in any computer language and any operating system. Multiple different implementations by different people gives confidence that the proof is correct

5.1. Unreliability of the Computer Computers may seem like a black box with bugs and errors in their programs, compilers and hardware. This can be resolved by providing a formal proof of correctness for the computer program. Such a proof was provided for the Four Colour theorem in 2005. Obtaining the same results by using different programming languages, different compilers, and different computer hardware is of course necessary. Moreover, this complaint can be applied to humans as well. People are also likely to make mistakes. The first so called ”proof” of the Four Colour problem by Kempe is a perfect example that humans are only human. Only after eleven years the error in the proof was detected. Until then the proof was widely recognized and accepted. Computers follow a pre-designed rigid program and are not distracted by moods, stress and other outside factors. So even though computer errors might be harder to detect, humans are more likely to make mistakes in their proofs.

5.2. Proof length: beyond human capacity The length of some proofs is beyond the scope of human computation, but perfectly accept- able by machine standards. The kind of results we are interested in involve complicated and long computations, which might be understood by humans, but by no means executable without the use of computers. The Four Colour theorem is one of those results. What initially took 1200 hours of computer work, would be practically impossible for one single

5 5.3 Social context 6 ARGUMENTS AGAINST COMPUTER ASSISTED PROOFS mathematician to do; which explains why the proof is not doable manually. And even for those results which can be done by humans, the speed of a computer outweighs that of the human.

5.3. Social context

Mathematicians prove theorems in a social context. It is a socially conditioned body of knowledge and techniques. Proofs in practice are completely diﬀerent from what we would like them to be in theory. It is not a formal process but a social ritual of acceptance which allows theorems to achieve a certain amount of credibility. It all comes down to the opinions and beliefs of those in the mathematical community: will we believe in a computer-aided result or not?

6. Arguments against Computer assisted proofs

6.1. The Value of a Proof

As is argued above, for the checking or generation of repetitive proofs we can be reasonably sure that computers will be accurate, given that the computer languages, original code, and such are carefully checked by multiple people. You could also argue that people checking a proof are also prone to mistakes, which we try to minimize by having large amounts of people check a proof. This can be emulated in computers by having multiple algorithms in diﬀerent languages to also minimize the chance of error. The line between a computer generated proof which cannot be checked by hand and one that can be checked by hand seems to be a very subjective divide. In theory a checker with enough time could sift through the pages of proof given by Appel and Haken, and acting carefully enough they could rigorously check the validity of the proof. But in my opinion this theoretical approach is of no use, since in reality no mathematician will take the time, or has the sanity to undertake such a thing. From here we must ask ourself why it is we want a proof for such a statement? On the one hand, it is of course such that the reader will know without any doubt in his mind that the theorem holds. This is where the term "proof" comes from. But this is not the only thing. From a proof we also want to learn what the machinery behind such a theorem is, and what causes such a beast to function. In a constructivist proof we often see exactly what causes the proof to function, often leading to further insights or new leads towards diﬀerent theorems. Even in somewhat non-constructivist proofs we still gain some understanding of the workings by seeing what causes such a contradiction. Unfortunately a proof that can’t be read, even if we can be incredibly sure towards its validity, will not give us unique insight into the mathematics itself. Then the role of the human will be to scrutinize the programs and algorithms which allow the computer to calculate, but this gives us little greater understanding.

6.2. In Search of Elegance

As a general rule in mathematics, from two proofs of diﬀerent lengths of the same theorem, the shorter one is often taken. This kind of Occam’s razor is that we want to get rid of superﬂuous information to leave just the essence of the proof for the viewer.

6 7 CONCLUSION

Paul Erdös famously joked that his most favourite and elegant proofs came from "the Book", this being God’s book of all mathematical proofs in their most elegant form. While there is little doubt that the current proof of the Four Colour theorem is false, it is clearly not a piece of art from the Book. If someone were to ﬁnd a proof of a couple of hundred pages the computer-assisted proof would quickly be forgotten in favor of a more elegant approach. This leads to the point that such proof, while having their uses, is less desirable than a more concise and transparent proof. And although the theorem was technically proven in 1976, since that time people have continued to try and shorten the proof by using more powerful methods of discharging. It is entirely possible that in time a humanly readable proof will be found and this will be considered the true proof of the Four Colour theorem. Such a proof wouldn’t quite be considered one from the Book, but maybe it could one from one of his notebooks.

7. Conclusion

In the end, mathematicians will always be searching for shorter, more elegant proofs. The Four Colour theorem will be remembered because it actually initiated a completely new ﬁeld in mathematics. It forced researchers to look back and question the notion of proof. Although it is good to question the reliability of computer-assisted proofs and think about it carefully, in the future this technique will be accepted more and more. Because computers will become increasingly important in the future, eventually the whole mathematic society will accept the computer-assisted proofs.