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 civilizationHistory of Mathematics

Danielle Verburg – 0706566Tara van Zalen – 0837261

May 23, 2012

Abstract

In this paper we will discuss both the history of the Maya civilization, and the most importantaspects of their mathematics.

The most remarkable fact of their mathematics is that the Maya were one of the two civilizationsthat invented and used the concept zero. Another remarkable fact is that the Maya number systemis a base 20 system in stead of a base 10 system, as we have nowadays. We will explain how theywrote 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 preciseastronomy and calendars. Their calendar system was one of the most complex, intricate andaccurate among all the world’s ancient calendar systems.

Contents

1 History 2

2 Mathematics 32.1 The number zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The base 20 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Symbolic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Sacred numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 Further calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Head glyphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Applications of mathematics 73.1 Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Calendars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Long Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Calendar Round . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

References 10

1 History

The first signs of the Maya go back to 2600 B.C. in what was called Yucatan. The Maya civilizationhad 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 Spanishconquest, 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 whoruled all. The whole society consisted of several independent states. Those states had urban sites nearceremonial centres. Thanks to the modern techniques we know more about the Maya in this periodof time. One of the biggest city that is known by now is Tikal in the Southern Lowlands. This cityhad 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 ofonly three civilizations in the world that invented a complete writing system. They also developedastronomical and calendrical systems, to which we will pay more attention later, and writing inhieroglyphs. Also interesting to know is that the Maya built everything without metal tools. Even thepalaces, observatories and temple-pyramids. As farmers they made sure that they could trade withother peoples, for which they cleared part of the rain forest. They also made some storage for waterin the areas where water was scarce.

Around 900 A.D. the southern Maya left their cities. This was the decline of the Maya civilization. Itis still a mystery why the southern Maya left. The Maya civilization stopped around 1200 A.D. whenthe northern Maya joined the Toltec society. But as mentioned, very small groups continued to liveuntil the beginning of the sixteenth century.

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2 Mathematics

2.1 The number zero

The Maya Civilization was the first before any other culture in the world that discovered and usedthe 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 oftheir numbers were quite hard to write, and their mathematics very difficult and cumbersome. TheEuropeans eventually borrowed the number zero from the Arabs, who themselves borrowed it fromIndia.

2.2 The base 20 system

They based their system on counting all the fingers and toes in the human body. This resulted in avigesimal, 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’, sothey have zero contents.

The Maya were one of the first who used a positional system, which makes it easier to calculate andwrite 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 writetheir 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 vigesimalpositions grow up from the base. If the numbers were written horizontally, then the bars were placedvertical and the dots to the left. In addition to plain dots and bars, the ancient Maya often usedfancier 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 representsthe numbers 0 to 19. Then the first level shows how many 1s are in the number, after subtracting

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the sum of the numbers in the higher positions. The second level shows how many 20s there are in anumber. 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, 64could 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 400sthere are in the number.For higher numbers, also the fourth, fifth, sixth and all higher positions can be used. The fourthpositions 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 higherlevels written above it. All symbols in each level are multiplied by their place-value factor. The firstlevel factor is 1, the second level factor is 20, the third level factor is 400 etc. Hence the ith level haslevel factor 20i−1.

In this way really big numbers can be written using the Maya number system. In fact, there really isno 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 eventsin both past and future. Moreover, with this symbols even uneducated people could do the simplearithmetic needed for trace and commerce.

2.4 Sacred numbers

The Maya considered some numbers more sacred than others, [8]. Some of these are the followingnumbers:

• 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 toour 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 onelevel are converted into one dot on the next higher level.

An example of addition with the Maya numerals can be found in Figure 4.

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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 thelower 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 belowyou 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 theMaya 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.

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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 (oneabove 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, circulardots 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 lesssquare in shape to make them fit together more nicely. The Maya would also often decorate the barsto make them more interesting and artistic.

Besides the three common symbols and the fancier way of writing in number glyphs, the Maya alsoused head glyphs and full body glyphs for the number from 0 to 19, [1]. A few examples of the headglyphs 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 theglyphs.Also, the head glyphs were sometimes compounds, so that for example two head glyphs were mergedinto one. The head glyphs were also combined with the usual Maya number symbol.

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3 Applications of mathematics

Mathematics was a sufficiently important discipline among the Maya that it appears in Maya art suchas wall paintings. Here the mathematicians can be recognized by number scrolls which trail fromunder their arms. Interestingly, the first mathematician identified as such on a glyph was a femalefigure.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 priestslooked to the heavens for guidance. They used observatories, shadow-casting devices and observationsof 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, theMaya 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 themovements of celestial bodies. One of the most notable series of buildings is a complex formed by 4buildings which forms an astronomical observatory, see Figure 9. This complex is the first one found inthe Maya world. From this observatory the early Maya could watch the sun rise behind these buildingsand 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 springand 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, isthe Observatory at Chichen Itza in Mexico, see Figure 10. Each year, people gather there to observethe sun illuminate the stairs of a pyramid dedicated to Quetzalcoatl, the Feathered Serpent god. Atthe vernal and autumnal equinoxes, the Sun gradually illuminates the pyramid stairs and the serpenthead 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

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calendrical data into an intricate, mathematical discipline. For example they made ingenious con-structions of the Venus and eclipse tables. They also expressed pure mathematics in their calendarsby 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 ac-curate. Calculations of the congruence of the 260-day and the 365-day Maya cycles (see Sections 3.2.1and 3.2.2) is almost exactly equal to the actual solar year in the tropics, with only a 19 minute marginof error.The Maya used 24 different calendars, based on every celestial body whose movement they couldconsistently observe and record. However, for simplicity only 2 calendars were used. These calendarsare called the Long Count, which records linear time from a mythological zero point, which is approx-imately 13th of August 3114 B.C. plus or minus 2 days, onwards; and the Calendar Round, which isa 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’inand the Haab’ linear and therefore theoretically infinite and never ending. Our Gregorian calendar issimilar 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 20times 20. With this exception to the rule the Maya were approximating the closest possible number ofdays 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 withthe 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 isreckoned in days in stead of years. The Long Count has therefore advantages over our system asregards 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 thegods of the underworld will appear in December 2012. To some, that means a great Maya deity willrise up and destroy the earth.But for most scholars, however, this prediction does not signal the end of the Maya calendar, or

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the destruction of the world. It simply underscores the importance of the end of one cycle and thebeginning of a new one. Another theory is that the end of the Long Count is a miscalculation, andthe Long Count will end in 2220.

On Friday May 11, 2012 there was some news about the Maya and their ’prediction’ that the worldwould be destroyed on December 21, 2012, [7]. In 2010 scientists found some hieroglyphs of a calendarin the ruined city Xultun. After a lot of investigation they have discovered that the world will go onwith its existence. The calendar that has been found will continue for 7000 years. Even after thatthey 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 nameddays.The first day of the Tzolk’in is ”1 Imix”, the next day ”2 Ik”, the third ”3 Ak’b’al”, and so on, untilafter 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 extradays added on at the end of the year. This is almost the same as our year, with the exception thatthey didn’t make the leap year adjustments every 4 years (although they knew that the length of ayear 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 ”1Pop” reoccurs. The beginning of the month was called the ”seating” of the month, and after 19 daysPop 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 minimaltime it takes for a particular Calendar Round date to repeat is 52 years. Again, we can see the use ofmathematics arise in the calendars.The repeating cycles of creation and destruction as described in Maya mythology were a reminderof the consequences if humans neglected their obligations to the gods. Humans had an inherentresponsibility to the gods who made humanity’s continued existence possible. According to CalendarRound, each 52-year period signalled the renewed possibility of the destruction of the world. This wasseen as a frightening time when the gods and other forces of creation and chaos would do battle inthe world of mortals, determining the fate of all earthly creatures.

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References

[1] Mark Pitts (2009), The complete writing in Maya Glyphs Book 2 – Maya numbers and the Mayacalendar

[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

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“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

Yvonne van Haaren (s1186477)

Martin Swinkels

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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

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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.

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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.

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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

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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.

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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.

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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

8

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

is equal to 107x =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

9

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)

10

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.

11

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.

12

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:

13

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.

14

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

15

= 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.

16

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

values. The first real slide rule was made by William Oughtred. This

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

William Oughtred, inventor of the slide rule

17

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 10br a . The

slide rule is then used to make calculations with the ‘mantissa’ of the number (the a-part) and

the 10bmust 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.

18

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

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 theChinese already knew about it around the year 1050. At about the same time the Persians alsoalready 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 andcube roots out of numbers. In China, after the discovery of the relationship between extractingroots and the binomial coefficients of the triangle, several Chinese algebraist’s continued on thiswork to solve higher than cubic equations.

But why is it called Pascal’s triangle then? The answer to this is simple. Pascal developed manyapplications of it and he was the first one to organize all the information together in his treatise,Traite du triangle arithmetique (1653). In this he made a systematic study of the numbers in thetriangle. They have roles in mathematics as figurate numbers, combination numbers, and binomialcoefficients, and he elaborated on all these.

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

2

2 History of Blaise Pascal

Blaise Pascal (1623-1662) was born in Clermont-Ferrand (France). He had two older sisters and hismother died when he was only three years old. In 1632 Pascal’s left Clermont-Ferrand and movedto Paris. Etienne Pascal, Blaise’s father, taught Blaise at home. He decided that Blaise shouldn’tstudy mathematics before the age of 15 and let all the mathematical texts be removed from hishouse. Because of this, Blaise’s curiosity raised and he started to work on geometry himself. Whenhe was 12 years old Blaise discovered that the sum of the angles of a triangle is equal to two rightangles. 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 MarinMersenne, 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. Itcontained 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: Essayon Conic Sections. Furthermore, the reason that they moved was that Blaise’s father got a job intax computations. Blaise Pascal wanted to develop a device that would help his father in his workand 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 theoremof 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 anhonestly division of stakes in an interrupted game of change. Gombaud wanted to improve hischanges at gambling and asked Pascal the following question: two players are playing a game untilone of them wins a certain number of rounds. But then the game got interrupted before any ofthe to reaches this. How should the stakes be divided? It should be taken into account how manygames each player has won. The solution requires the combinatorial properties inherent in thenumbers in the Arithmetical Triangle, as Pascal had discovered.

From Pascal’s Traite du triangle arithmetique we will also learn about the principle of mathematicalinduction. The concept of induction already occurred in the Islamic world in the Middle Ages andin Europe in the fourteenth century, but Pascal was perhaps the first to make a explicit statementand 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 triangleto the binomial theorem, Fermat discovered a proof of the famous and important theorem on primenumbers. Because of that theorem we can now use online payment in a secure way since RSA isbased on the knowledge that Fermat’s theory gave us.

Gottfried Leibniz was one of the inventors of infinitesimal calculus, a part of mathematics thatis concerned with finding slopes of curves, areas under curves, minima and maxima and othergeometric and analytic problems. Leibniz explicitly credited Pascal’s approach on some of thesesubjects because it stimulated his own ideas on characteristic triangle of infinitesimals in hisfundamental theorem of calculus.

Pascal also contributed in the area of physics. Another ’object’ is named after Pascal, namely: thescientific unit of pressure. One pascal is equal to one newton per square meter. Moreover, Pascaldid 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 oftransmission of fluid-pressure.

Throughout his thirty-nine years, Pascal contributed to a lot of important statements as well inmathematics as in physics. The short summation above is far from complete. He was one of themost outstanding scientists of the seventeenth century. Unfortunately, due to other interests andhis 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 someof them are too remarkable not to mention, and are therefore listed in this chapter. Some ofthe properties below are also described in Pascal’s original Treatise on the Arithmetical Triangle;Traite du triangle arithmetique, 1654.

Figure 1: The original triangle in Pascal’s Treatise3(1654), where he explains his definitions parallelexponent 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’striangle to be the row containing only the number 1, then the i-th number of row j equals(

j − 1

i− 1

)=

(j − 1)!

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

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 combi-nations of a number less by unity than its parallel exponent in a number less by unity than theexponent 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 : (x + y)n =

(n

0

)xny0 +

(n

1

)xn−1y1 +

(n

2

)xn−2y2 + · · ·+

(n

n− 1

)x1yn−1 +

(n

n

)x0yn

is strongly related to Pascal’s triangle. Indeed, all binomial coefficients in the theorem are directlygiven 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 alongthe left and right edges of Pacal’s triangle contain only 1’s. Now go deeper into the triangle, tofind 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, thetetrahedral numbers, the pentatope numbers, and so on. In general, the n-th diagonal of thetriangle (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 movingone element to the right, and one element to the bottem-right. Then what you get, is a sequenceof 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 naturalnumbers, named after the Belgian mathematician Eugne Charles Catalan (1814− 1894), and usedto solve several combinatorial problems), as follows. For every odd, natural number n, the middle

element of row n, minus the element two spots to the left (or right), equals the (n+1)2 -th Catalan

number.

6. When Pascal’s triangle is embedded in a matrix {Aij : i, j ∈ N}, the number of ways to walkfrom 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 borderof the triangle, and walk along any diagonal to the bottem-right or left, for as many numbers asyou like. Then the sum of the numbers in this sequence, equals the number directly below thelowest element of the sequence, which is not in line with the sequence. This property can be provedquite 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: ”Inevery arithmetical triangle each cell is equal to the sum of all the cells of the preceding perpendicularrow 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’sdefinitions 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 isp + 1 and the base contains p + 1 cells. From proposition 2 it now follows that the k-th cell ofthe base, with k ∈ {2, 3, · · · , p}, equals the number of ways to pick parallel-exponent-minus-onethings from base-number-minus-one things, or

((p + 1)− 1

(p− k + 2)− 1

)=

(p

p− (k − 1)

)=

p!

(p− (k − 1))!(k − 1)!=

p!

(p− i)! i!= p · (p− 1)!

(p− i)! i!,

where i = k − 1 ∈ {1, 2, · · · , p− 1}.

Define n =( (p+1)−1(p−k+2)−1

)and q = (p−1)!

(p−i)! i! , then it follows that

n = p · q,

where n, p ∈ N and q ∈ Q.

Note that the right-hand side of

q =(p− 1)!/(p− i)!

i!=

(p− 1) · (p− 2) · · · (p− (i− 1))

i · (i− 1) · · · 2

has i−1 consecutive factors in both the numerator and the denominator. Since the greatest factorof the numerator, (p− 1), is greater than, or equal to, the greatest factor of the denominator, i, itfollows that

q ≥ 1.

To see that q is indeed an integer, let us assume that q ∈ Q \ Z. Now reduce

q =(p− 1) · (p− 2) · · · (p− (i− 1))

i · (i− 1) · · · 2

to lowest terms such that q = a/b, for certain (and unique) a, b ∈ N. Observe that the primefactorization of b does not contain a factor p (or a product of factors which equals p), since p isprime and i, (i− 1), · · · , 2 < p. From this, and our assumption that q ∈ Q \ Z, it follows that p · qis 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, Traite du triangle arithmetique.

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/RBPascal.html

http://www.britannica.com/EBchecked/media/57051/Pascals-hexagon-Blaise-Pascal-proved-that-for-any-hexagon-inscribed

9

Evangelista Torricelliand 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 53.1 Torricelli’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Some notes on Torricelli’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 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 parentswere Gaspare Torricelli and Caterina Angetti. The family was fairly poor. Evangelista had twoyounger brothers and no sisters. Evangelista’s parents could not provide an education for him, so theysent him to his uncle Jacopo, who was a Camaldoles monk. He made sure that Evangelista was givena 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 studiedmathematics and philosophy. After this he was in Rome for sure. He was an outstanding student andhis uncle, Jacopo, arranged for him to study with Benedetto Castelli, who was also a Camaldolesemonk. 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 privatearrangement. Torricelli was taught mathematics, mechanics, hydraulics and astronomy by Castelliand became his secretary from 1626 to 1632 as payment for the tuition. Later he took over Castelli’steaching when he was absent from Rome.While Castelli was absent from Rome, Galileo had written to him and Torricelli wrote back to explainthat Castelli was not in Rome and therefore could not answer Galileo. Ambitious as he was, he tookthe opportunity to inform Galileo about his own mathematical work and texts he had studied, includ-ing the work Dialogue Corcerning the Two Chief Systems of the World - Ptolemaic and Copernicanby Galileo. From this letter, written on September 11th 1631, it is clear that Torricelli was fascinatedby astronomy and a strong supporter of Galileo. But the trial of Galileo in 1633 scared him, so heshifted his attention onto mathematical areas. During the next nine years he was the secretary ofGiovanni 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 Operageometrica in 1644. In 1641 he asked Castelli for his opninion on De motu gravium (the second of thethree parts which basically carried on developing Galileo’s study of the parabolic motion of projec-tiles). Castelli was so impressed that he wrote to Galileo, brought him a copy of Torricelli’s manuscriptand 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 inCastelli’s place for a while and his mother died. On October 10th 1641 Torricelli arrived at Galileo’shouse 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 math-ematician to Grand Duke Ferdinando II of Tuscany. He held this post until he died of typhoid onOctober 25th 1647 in Florence.

Torricelli and Galileo

3

2 Torricelli’s Work

Another pupil of Galileo, Bonaventura Cavalieri, published Geometria indivisibilis continuorum novain 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 axisof symmetry. He also computed the area and centre of gravity of the cycloid. His most remarkableresults 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 hyper-bola between the y-axis and a fixed point on the curve round the y-axis is finite. This proof will bediscussed later. This was against the intuition of mathematicians of that time, so this result arousedgreat interest and admiration.Around 1640 Torricelli determined the isogenic centre of the triangle (the point in the triangles suchthat 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 barom-eter. In 1643 he proposed an experiment, performed by Vincenzo Viviani, that led to the developmentof 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 soundcould 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 thesquare root of the height of the liquid. This result is known as Torricelli’s theorem. Also in De motugravium 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 publishedOpera geometrica. Much is known from letters and some lectures that he gave. Hours before his deathhe 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 agreeto undertake the task, but he failed to accomplish it. Some manuscripts were lost and the remainingmaterial was published in 1919. Original material left by Torricelli got destroyed in the TorricelliMuseum in Faenza in 1944.

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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 they-axis to get a hyperbolic solid as in figure 1.

Figure 1

We call the origin point A.

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

Figure 2

Proof of lemmaFirst we construct the linesegment AC, in such a way that C is on the hyperbola and the angle betweenthe y-axis and AC and the angle between the x-axis and AC are the same (both are 45 degrees). Thisline 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 EFGH 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

2a2

Because our hyperbolic solid is symmetric around the y-axis (since we have rotated it around they-axis), it holds that

Area(Rectangle(EFGH)) = 2Area(Rectangle(AIGH))

= 21

2a2

= a2

Now we can calculate the surface area of the cylinder EFGH, as in figure 3. Note that we do notinclude the bases of the cylinder when calculating the surface area.It holds that

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

= πa2

Because we have taken EFGH to be an arbitrary cylinder, this holds for all cylinders contained inour hyperbolic solid.QED.

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

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Figure 6

We will now prove that this figure has a finit volume.First we construct the rectangle AEFG, such that AG is in the extension of the y-axis, EF is in theextension 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 extensionof 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

= π(2a

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 ourfigure.We have also just proven that for every cylinder contained in our figure, we have an unique circle thathas 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 AEFG.So the volume of our figure equals the volume of cylinder AEFG. And the volume of cylinder AEFGequals π × 2a×AE and is thus finite. So the volume of our figure is finite.QED.

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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 ofour figure is actually infinite. Maybe he thought this was obvious and did not need proving, maybesomebody else had already proven this, or maybe he proved it himself. At any rate, we could not findthis proof, and we have to ask the reader to believe us on our word that the surface area is actuallyinfinite. We will give a modern version of Torricelli’s finite volume contained by an infinite surfacearea later in this paper, and for this version we will actually proof that both the volume is finite, andthat 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 upinto 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’strumpet.

Torricelli’s Trumpet

This figure is made by taking the equation y = 1x , 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 givenby the following equation:

V = π

∫ b

a(f(x))2dx

The surface area of a solid of revolution of a function f(x) on an interval x ∈ [a, b] is given by thefollowing equation:

A = 2π

∫ b

af(x)

√1 + (f ′(x))2dx

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

V = π

∫ b

a(f(x))2dx

= π limb→∞

∫ b

1(1

x)2dx

= π limb→∞

∫ b

1

1

x2dx

= π limb→∞

[−1

x

]b1

= −π( limb→∞

1

b− 1

1)

= −π(0− 1)

= π

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

A = 2π

∫ b

af(x)

√1 + (f ′(x))2dx

= 2π limb→∞

∫ b

1

1

x

√1 + (− 1

x2)2dx

= 2π limb→∞

∫ b

1

1

x

√1 +

1

x4dx

> 2π limb→∞

∫ b

1

1

x

√1dx

= 2π limb→∞

∫ b

1

1

xdx

= 2π limb→∞

[log(x)]b1

= 2π( limb→∞

log(b)− log(1))

= 2π( limb→∞

log(b)− 0)

= 2π( limb→∞

log(b))

→ ∞

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

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4 Sources used

The image of Evangelista Torricelli on the front page is fromhttp://catalogue.museogalileo.it/gallery/EvangelistaTorricelli.htmlThe image of Torricelli with Galileo is fromhttp://www.lepla.org/en/modulus.php?name=Activities&file=m42For the biography of Torricelli we used information fromhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Torricelli.htmlFor Torricelli’s proof of a finite volume contained by an infinite surface area we used chapter 4 of thebook A Source Book in Mathematics, 1200-1800 by D.J. Struik.

12

JOHN VON NEUMANN

MAAIKE ASSENDORP & JURGEN RINKEL

Contents

1. Biograpy 11.1. Childhood and education 11.2. Life in Europe 21.3. Working in America and World War II 21.4. Later years and death 32. Work 32.1. Set Theory 32.2. Game Theory 62.3. Quantum mechanics 83. Conclusion 8References 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 halfof the twentieth century. This reputation was mainly due to the fact that he hadthe most extraordinary insights in numerous fields, in- and outside from mathemat-ics. In fact, he contributed to almost every mathematical field but topology andnumber theory, while in the meantime becoming renowned for his revolutions inphysical, economical, and even philosophical thinking. Towards the end of his lifehe would work on the first computers, thereby making an everlasting contributionto our modern society.

1. Biograpy

1.1. Childhood and education. John von Neumann was born at December28, 1903 in Budapest, Hungary (at the time part of the dual monarchy Austria-Hungary). At birth, he was called Neumann Janos1. He would change his name toJohn when he moved to the United States.2 While living in Hungary he was calledby the diminutive Jancsi, which became Johnny when he anglicized his name. VonNeumann’s family was Jewish and prosperous, which was a quite common com-bination in Budapest in the early 1900s. Max Neumann, Johnny’s father, was abanker who provided his children with a good education, hiring several teachers tolearn them their languages and achieving an enormous library. Johnny, his oldestson, was a child prodigy who was completely comfertable at the von Neumann’sintellectual environment. Contrary to other Jews in central Europe, Johnny’s lifewas assured to be richly endowed. As a child he could regularly been found at theaforementioned library, devouring many of the books and even memorizing them.In later life he became infamous for his almost photographic memory. One anecdotefrom the early 1950s illustrates this. When he was asked by a collegue how CharlesDickens’ novel A Tale of Two Cities began, von Neumann started to quote the firstdozen 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 BelaKun’s communist regime, but in 1919 that government collapsed and Miklos Hor-thy, a former admiral of the Austrian-Hungarian navy, seized power. Under hisconservative rule the so-called ’White Terror’ gripped Hungary which also victim-ized the Jews. As a result, approximately 100,000 people fled Hungary. One ofthem 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 familysuggested Johnny to study chemistry as a compromise. All agreed and Johnny en-rolled 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 col-leges, only to show up to ace his exams. He left Berlin in 1923 to continue hisstudies at the Eidgenossische Technische Hochshule in Zurich. In 1926 he recievedhis Ph.D. in Mathematics from the University of Budapest, doing this in only fiveyears. At the time he was only twenty-two years old. His dissertation was con-cerned with the axiomatization of Cantor’s set theory, of which we will come tospeak 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 dopost-doctoral work at the University of Gottingen after having earned a Rockefellergrant. Gttingen was one of the most renown universities of the time, when it cameto 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 wouldmake his contributions on the mathematical foundations of quantum mechanics. Bythe middle of 1927, his work with Hilbert was done and von Neumann decided tomove on. He went to Berlin were he was appointed as Privat docent. He reportedlywas 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 ar-ticles at the impressive rate of about a month. Some of these articles containedastonishing new ideas about the topics they concerned and that revolutionized theway mathematician looked at their subjects.In the meantime, European polics became harsh and the prospect of a new WorldWar was glooming. Several attempts to overtrow the local or national govern-ments in Germany had occured (including Hitler’s Bierkeller Putsch. Von Neumannthougth that in the future, a nationalistic Germany would wage war on Russia andthat, 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 plansfor 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 wasfalling behind on it’s European counterpart. It was Oskar Veblen who suggestedto attract John von Neumann for this purpose. Von Neumann agreed, and it wasarranged that he should come the next year, to return in the second half of 1930to give a course of lectures in Berlin.But before all this happened, he had decided to propose to Mariette Kovesi, and shehad agreed. This required that he converted to Catholicism, which von Neumanndid. Religion never had meant much to him, and this attitude would continue untilshort befor his dead. But now von Neumann would leave soon and wanted her tocome with him, so things were getting in a hurry. The original plan of marrying inJune 1930 was abandoned and they married on New Year’s Day instead. In 1935Mariette 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, leavingMarina with her mother until she became twelve, as it was felt that her educationwas better left at the hands of her father.

1.3. Working in America and World War II. Von Neumann would spend thefirst years following 1930 travelling back and forth from Europe to the States. In1933, after Hitler came to power, he gave up all his work in Germany. The sameyear the Princeton Institute of Advanced Study (IAS) was opened. This institutewas meant to house the world greatest scientists who wouldn’t have to give anycolleges and who were required to do anything but think. Von Neumann accepteda professorship there. The IAS would soon become a attractor for fleeing scientistfrom Europe. Among them were Albert Einstein and Kurt Gdel. Johnny wouldeventually end up working at the IAS until 1955 and his work is still consideredone of the greatest forthcomings of the institute (along with Gdel’s work on thecontinuum hypothesis). However, he couldn’t match his old publicity rate of nearlyone paper a month while working at the IAS, a problem that all the scientistsworking 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. Itwon’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 vonNeumann knew for sure that war was coming, so he applied for the position oflieutenant in the Reserve Corps. He passed all the exams (of course) but in the endhe was considered too old.In 1938 he travelled to European continent one last time, while Czechoslowakia wasalready under the immediate threat of Nazi Germany. He had several reasons to goback to the continent. One was that the IAS had sent him to convince Niels Bohrto come to the other side of the Atlantic, as he was considered too important toleave in the middle of a war. A second reason was that von Neumann intended tomarry 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 inDecember 1941, von Neumann took the interest of the American government forbeing a calculator on explosive weapons. In the meantime a project was startedto create the atomic bomb: the Manhattan Project. In 1943 von Neumann wasrecruited as a consultant at Los Alamos laboratory in New Mexico, where the bombwould be constructed. In 1945, von Neumann witnessed the first test and he alsoput forward several Japanes cities to bomb first, Nagasaki and Hiroshima beingamong them.A 1943 visit to England brought to von Neumann the idea of actually inventing thecomputer. He would begin with this in Philadelphia, where ENIAC, the first realelectronic computer, was created.4

1.4. Later years and death. After the War, Johnny would keep up working atthe IAS until 1955. In the meantime he had several other jobs as a consultant forthe government on different affairs. On of those affairs was nuclear detterence, inwhich he argued in favour of bombing the Soviet Union, before this country couldbe able to bomb America, attracting support from members of the government.In 1955, von Neumann became sick and was diagnosed with cancer. It is commonlythought 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 Medalof 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 settingin 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. Konig, Weyl, Brouwer(2) Zermelo, Fraenkel, Schonflies

First however, it is neccesary to know what set theory was like before the 20thcentury.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 revealedthat 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 ratio-nals and the irrationals? Georg Ferdinand Ludwig Philip Cantor sought to answerthese questions, developping what would later on be called (by Von Neumann also)’naıve set theory’.Georg Cantor began by comparing sizes of sets, using one-to-one relations betweensets. He discovered that ’infinity’ was not ’the end’, that the natural numbers (hav-ing cardinality ℵ0) were as many as the the rationals, but the real numbers werevaster (having cardinality c). He proved that while the rationals were denumerable(or equipotent with N, having cardinality ℵ0), the irrationals, were in fact, not. Inhis search for sets greater (of higher cardinality) than the set of real numbers, hefirst set out to prove that a square had more elements then an interval. However,he proved that they were equal 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 ofideas, 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 , wichtells us that there is something vaster then U . This is a paradox that followed fromCantor’s set theory, showing that it was flawed.

Now we shall return to the two groups that Von Neumann mentions. He is notthe only source that uses this division.5

(1) The first group wishes to redefine mathematics from scratch, seeing howthe definitions so far led to paradoxes. Von Neumann claims that bothWeyl 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 didnot find it neccesary to redefine it all together. In stead, they axiomatiseset theory, using the naıve version that Cantor introduced. I will focus alittle more on this group, as the article shows that Von Neumann himselfbelongs to this group.7

Interesting to note is that there are some parallels with Euclides and this lattergroup. 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? Isthere a cardinality between the natural numbers and the real numbers. He formu-lated the hypothesis that it was in fact not possible to find such an intermediateset. Though no matter how hard Cantor tried he could not find the answer8. Muchlike Euclides’ fifth postulate could not be proven.In the 19th century it was discovered by Kurt Godel that the hypothesis couldnot be disproven.9 But a little over 20 years later, Paul Cohen proved that thehypothesis could not be proven.10 These proves came later then the articles VonNeumann 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 well7Von Neumann himself also states that the article belongs to the latter category.8Dunham suggests that this has cost him a few mental break downs.91940101963

JOHN VON NEUMANN 5

2.1.2. Axiomatizing of Set Theory. The two articles that Von Neumann wrote onthe 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 ratherthen 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 elsethan) is decribed in the postulates. He says that the postulates can be formulatesthus that the set theory of Cantor followes from it, but that the paradoxes orantanomilies don’t.He mentions the fact that in stead of axiomatizing the sets, he will axiomatize thefunctions, wich are (according to him) one and the same. He means with this thata function f(x) (written as [f, x]) can be viewed as an element from a set, while aset 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 bean argument.13

After explaining how he is going to define sets or functions, Von Neumann explainsthe 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. Itdescribes which operations can be done to create the functions. He doesn’t describethe others though. I too will limit myself to these first 3 groups, due to the lengthit 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 an-other), 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 funcionshave 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 theory12German: The axiomatization of set theory13I 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 secondcoordinate respectively.)

(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 statedin 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 4thgroup) 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 Neumannwasn’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 afinite number of events (i.e. the movement of a piece at a chessboard) which lead toa result (black or white wins, or there is a draw). Every player has several optionsto play, here they are called strategies. A simple and familiar ’game’ is the divisionof 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, forboth players. These are given in Figure 1. Besides that, the results for cutter aregiven, for any combination of strategies.

The question we must ask ourselves now is: what is the optimal strategy for thecutter? Or to formulate this otherwise, what must the cutter do to maximize hisresult? As we can see, the best result would be for him to get the big piece. Thatsuggests he should cut the cake so that there are two uneven parts. But the chooserwill obviously choose the bigger piece, leaving the smaller one for the cutter. But ifthe cutter cuts the cake in two more or less even parts, the chooser will still choosethe bigger piece, but what’s left is a bigger piece for the cutter then in the firstcase. 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 hisresult is defined by that of the cutter, in the sense that the chooser gets what thecutter gets not (this is called a ’zero-sum game’) he wants to minimize the resultof 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 forplayer 1 (the ’cutter’) when player 1 chooses strategy x and player 2 chooses strat-egy y where x, y ∈ {1, 2} then the corresponding values for g can be found inFigure 2

Figure 2. A mathematical representation of the outcomes

Consequently we find:

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

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

maxx

minyg(x, y) = 0.49 = min

ymaxx

g(x, y)

8 MAAIKE ASSENDORP & JURGEN RINKEL

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

maxξ

minµh(ξ, µ) = min

µmaxξh(ξ, µ)

for every zero-sum game. An alternative formulation is that for every game thereexists an optimal strategy for both players. When both players choose this strategy,the game will eventually reach an equilibrium point. To von Neumann, this resultwas very important. Without this theorem, he said, it wouldn’t even be interestingto publish anything on game theory.In later years, von Neumann didn’t publish much about game theory, altough hecowrote the 1944 book The Theory of Games and Economic Behavior with OskarMorgenstern. This book is still seen as one of the most important textbooks ineconomics.

2.3. Quantum mechanics. We will discuss some history of Quantum Mechanicsin this last chapter. This will remain quite brief. One reason is that ’mechanics’ isscience, but not math. The other reason is that we have focussed more on abovesubjects in stead of this one.The following list is a sumarization from a chapter on quantum mechanics in thebook we used:

1900 Max Planck makes up the concept quantum, a package that transportsenergy’.

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 Gottingen, 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 Schrodinger formulates a series of wave equations for the movementsof the elektron. Heisenberg rejects these. Both models fitted the experi-ments.

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 Neumannmay not have done much in a field, he did have a great influence. By joining thetwo 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 Jaıbi

May 24, 2012

1 Introduction

“Parallel straight lines are straight lines which, being in the sameplane and being produced indefinitely in both directions, do notmeet one another in either direction.”

“That, if a straight line falling on two straight lines make theinterior angles on the same side less than two right angles, thetwo straight lines, if produced indefinitely, meet on that side onwhich are the angles less than the two right angles.”[1]

Ever since Euclid (circa 330-275 BCE) layed down planar geometry in anaxomiatic framework, there has been much controversy on the fifth postu-late1 of his Elements[1]. The 2300 year old history of the fifth postulate isone of many attempts at proving this intriguing postulate and just as manyfailures. The discussion has not been on whether the fifth postulate is trueor whether there exist geometries where it does not hold, but rather on whyEuclid formulated his statement on parallel lines as a postulate and not asa theorem, perhaps derived from a more elementary, shorter postulate.

In this paper, we will give a short overview of several attempts at provingor replacing the parallel postulate, after which we focus on the work of CarlFriedrich Gauss (1777-1855), who was the first to discover non-Euclidangeometry.

2 The Parallel Postulate

The main source of information on the earliest attempts to prove the parallelpostulate stem from the Greek philosopher Proclus (410-485)[4, p.2], whowrote a commentary on the first book of the Elements[2]. Proclus firmlystates that “this proposition should have been excluded completely fromthe postulates, for it is a theorem.”[2, p.53] He then discusses the flaws inthe 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 theMiddle Ages, Euclid’s work was studied and copied by the Arabs, who alsoinvestigated the fifth postulate, after which discussion on the parallel pos-tulate emerged in Europe again after 1550.[4, p.12] In 1763 Georg SimonKlugel wrote his dissertation on the fifth postulate, listing an overview ofall 26 historical ‘proofs’ of the parallel postulate known to him, after whichhe concludes that Euclid must have been right after in taking the postulatefor 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 theirproofs bear the same mistakes. Others replace it by other axioms, whichare neither clearer nor more secure than the Euclidian. Therefore, when allattempts are thoroughly inspected, it becomes clear that Euclid was rightin putting as axiom a proposition which cannot correctly be proven fromanother one.”[5, §I]

The approach of finding other definitions or axioms seems more interestingthan the flawed proofs given by some. Geometry professor from OxfordJohn Wallis (1616-1703) gave a lecture in 1663 in which he proposed analternative postulate which he believes to be clearer and which “does notneed a demonstration.”[7, p.674] Wallis assumes that for any given figure,there exists a similar figure of arbitrary magnitude and then shows thatEuclid’s postulate follows from this axiom.[7, pp.676-7] It was later shownby Geralomo Saccheri (1667-1733) that it is sufficient to assume that thereexists 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 postulateis equivalent to the proposition that the angles in a triangle always sum totwo right angles. Instead of introducing another axiom or a new definitionof parallel lines, Saccheri ponders on the possibility of the angle sum of atriangle not necessarily being equal to two right angles. He proofs that if inone 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 itis less, it is less in all triangles. (Prop. V-VII.) Thus three hypotheses onthe nature of geometry emerge: the right angle hypothesis, the obtuse anglehypothesis and the acute angle hypothesis. Saccheri is able to prove sometheorems on angles in a triangle and lenghts of their bases. He then dis-misses the obtuse angle hypothesis because a triangle can be constructed inwhich the sum of two angles exceeds two right angles, which contradicts I.17which was proven by Euclid without using the parallel postulate.2 (Prop.XIV.) Even though he had disproved the obtuse angle hypothesis, Sacchericontinued proving theorems which hold for all three hypotheses; however,they did not yet lead to the contradiction in the obtuse hypothesis he waslooking for. Among other theorems, Saccheri notes that in any quadrilat-eral, the angle sum will be less, equal to or more than four right anglesdepending 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 thehypothesis. (Thales’ theorem.) Then there is an intermezzo in the book,in which the contributions of Proclus, Nasir al-Din, Wallis and others tothe problem of the parallel postulate are discussed. Next, Saccheri gives

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

2

theorems on the distance between two parallel lines and finally proves thatfor the acute angle hypothesis, two straight lines run together as they nearinfinity (their perpendicular converges), from which Saccheri concludes thatthe acute angle hypothesis is “absolutely false; because [it is] repugnant tothe nature of the straight line.” (Prop. XXXIII)

Saccheri was the first to explore the possibilities of non-Euclidian geome-tries, but he did so still with the intent to prove the parallel postulate.Unfortunately, his work was forgotten until it was rediscovered by EugenioBeltrami (1835-1899). It would be almost 60 years until Gauss indepen-dently explored the possibilities of non-Euclidian geometry and concludedthat though it was very different from Euclidian geometry, it was indepen-dent and just as logically sound.

3 Gauss

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

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

3

Gauss, born on the 30th of April 1777 in Brunswick, Germany, was raised ina poor family, being the son of Gerhard (called Gebhard) Dietrich Gauss, agardener and bricklayer, and Dorothea Gauss, the daughter of a stonecutter.Despite the miserable living conditions, he exhibited such early talent thathis family and neighbors called him the “wonder child”[8]. When he was twoyears old, he gradually got his parents to tell him how to pronounce all theletters of the alphabet. Then, by sounding out combinations of letters, helearned (on his own) to read aloud.[12, §II] He also picked up the meaningsof the number symbols and learned to do arithmetical calculations. Thestory 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 sonwas following the proceedings with critical attention. Coming tothe end of his long computations, Gerhardt was startled to hearthe little boy pipe up, ’Father, the reckoning is wrong, it shouldbe ...” A check of the account showed that the figure named bythe young Gauss was correct.[10]”

At the age of seven, Gauss started elementary school, and his potential wasnoticed almost immediately. His teacher, Buttner, and his assistant, MartinBartels, were amazed when Gauss summed the integers from 1 to 100 almostinstantly. He used a very clever way to find the result. His observation wasthat: “100 + 1 = 101; 99 + 2 = 101; . . . 52 + 49 = 101; 51 + 50 = 101′′ andthus that there were ‘as many pairs’ as there were in 100. Thus he found asa 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 whowould change the world of mathematics. This did not escape from the at-tention of Buttner, who then took a special interest in the young boy: Heprovided him with additional books and organized after-school meetings toinvestigate advanced mathematical ideas with Gauss’ from then on tutorBartel [13]. This did not stay unnoticed for long and, as his name grew inthe area and he kept up his studying, the Duke of Brunswick heard abouthim. When Gauss was 15 the Duke summoned him to his castle. It wasthere that Gauss formed a lasting friendship with the Duke and received astipend that allowed him to go to college and to devote his time to study-ing. For the next 4 years Gauss spent his time learning at Caroline College.Even while studying, Gauss was formulating many of the important the-orems he would later go on to prove. He then left to enter University ofGottingen in 1795. While there he learnt and discovered a lot, but most ofhis discoveries hjad already been discovered. Discouraged with mathematicsand his lack of making any new discovery, Gauss was on his way to becom-ing a philologist. That is until he made a discovery that declared him a

4

mathematician.[9] Gauss obtained conditions for constructibility of regularpolygons and found out that the regular 17-gon was constructible by rulerand 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 math-ematics, since Euclid had shown that regular polygons, with 3, 4, 5, and 15sides and those the sides of which result from doubling the above could beconstructed geometrically with compass and ruler and no further progresshad been made since then[14]. He left the University without a degree andwent back to Brunswick where he got a degree in 1799. After the Duke ofBrunswick had agreed to continue Gauss’s stipend, he requested that Gausssubmit a doctoral dissertation to the University of Helmstedt. The subjectof his dissertation was a proof of the fundamental theorem of algebra–whichwas proven only partially before Gauss–which states that every algebraicequation with complex coefficients has complex solutions; moreover, Gaussskillfully formulated and proved this theorem without the use of complexnumbers.Between 1796−1800, Gausss mathematical thinking matured tremendously.Mathematical ideas came to him so easily and frequently that he had troublegetting 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 summerof 1801. There were seven sections, all but the last section, being devotedto number theory. In section V II, he published his discovery about theconstructablilty of the 17-gon.[14]This book is considered one of the most brilliant achievements in the his-tory of mathematics, in which he formulated systematic and widely influen-tial concepts and methods of number theory, dealing with relationships andproperties of integers which, for him, was of paramount importance in math-ematics. This included dealing with the theory of congruent numbers, law ofquadratic reciprocity for which he the first proof and algebraic numbers.In1831 (published 1832) he gave a detailed explanation of how an exact theoryof 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. Astronomywas one of the main fields that gained Gauss’ interest. Using his methodof 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 atGottingen University, a position that he held for 48 years. In this capacityhe actively pursued research in theoretical astronomy, taught courses andpublished 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 afterthe long battle during twenty centuries many mathematicians held tryingto 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] andBessel[1784-1846] on the subject.[4, §III] In 1799, he confided to WolfgangBolyai that he was doubtful of the truth of geometry conceived by Euclid.He had found too many mistakes in the calculations made by other peoplein defence of the fifth postulate that he could not believe in the Euclideansystem 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 whichwe seek, and which you assure me to doubt the truth of the ge-ometry 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 re-vealed it after the works of Lobatschewsky[1792-1856] in 1829-30 and Bolyaiin 1832 appeared. In a letter he wrote to Schumacher[May 17, 1831] hesays[4]:

In the last few weeks I have begun to put down a few of myown Meditations, which are already to some extent nearly 40years old. These I had never put in writing, so that I have beencompelled three or four times to go over the whole matter afreshin my head. Also I wished that it should not perish with me.

This marked the first written steps of Gauss’ theory of non-Euclidean ge-ometry. His notes were published posthumously in his assembled works in1870.[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 eachother, while, on the other hand, every straight line through Abetween AM and AB cuts BN, then AM is said to be parallel toBN.

Gauss supposes a straight line l passing through A. First, l lies on the lineAB and then is rotated continuously to AC on the side towards which BN isdrawn (see figure 4). In the beginning the line l cuts BN and from a singleposition on, these two do not intersect anymore. This position must be thefirst 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 linesAB and BN are supposed to be produced indefinitely in the directions ofAM and BN. This definition is different from the one Euclid gave in hisElements. If Euclid’s Postulate is rejected, there could be different linesthrough A, on the side mentioned ealier, that do not cut BN. According toEuclid, all of these would be parallel to BN. Gauss does only take the firstof these lines and defines it as being the parallel to BN.Gauss completes his definition by showing that “the parallelism of the lineAM to the line BN is independent of the points A and B, provided the sensein 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 infiniteline BN is taken (thus B’ taken on BN or produced backwards. It remainsto 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. Wedistinguish two cases: A’ being upon AM and A’ being produced backwardson 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. Drawthe line A’P through A’ between A’M and A’B in any direction. Let Q beany point on A’P. Join A and Q. Then from the definition AQ must cut BNwhich means that QP cuts BN as well. Thus AA’M is the first of the lineswhich 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 backwardsand 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 closedfigure A’ARB and thus had to cut one of its sides A’A, AR, RB or BA’. Itis obvious that this had to be the side RB. Thus for any A’P (and Q takenon A’P) between BN and AM, QA cuts BN. Therefore A’M is parallel toBN.Thus, the definition of parallels was established. Gauss also proved the reci-procity 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 changeswithout any specific reason the order of A and B and Gauss mixed the orderof the angles, that is 6 BAC = 6 CAB.The proof is stated as follow: from any point A upon (2) draw AB perpen-dicular to (1). Through A draw any line (3) between AB and (2) and letAC be a line between the same lines AB and (2) such that 6 BAC = 1

2(2, 3)(angle between the lines 3 and 2). Then there are two cases that can bedistinguished: the first case corresponds to AC cutting (1). The second onecorresponds to when AC does not cut (1).Figure 5 corresponds to the first case: Let AC cut (1) in D. Take BE = BDso that B lies between E and D. Through D, draw the line DF between DAand (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 ABEare congruent and thus AE = AD. The triangles ADG and AEH are congru-ent, 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 andsince 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 thesame argument as above it holds that 6 GAH = 6 DAE. But it also holdsthat 6 BAD < 6 BAC and thus 6 DAE < 6 (2, 3) (since 6 BAC = 1

26 (2, 3)).

Therefore 6 (2, 3) > GAH and thus (3) is a line between AH and AD and itcuts 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 sharedwith Taurinus in a letter[Nov, 1824]:

. . . the assumption that the angle sum is less than 180◦ leads toa geometry quite different from Euclid’s, logically coherent, andone that I am entirely satisfied with. It depends on a constant,which is not given a priori. The larger the constant, the closerthe geometry to Euclid’s and when the constant is infinite theyagree. The theorems are paradoxical but not self-contradictoryor illogical. [. . .] All my efforts to find a contradiction havefailed, the only thing that our understanding finds contradictoryis that, if the geometry were to be true, there would be an ab-solute (if unknown to us) measure of length [. . . ] I As a joke

9

I’ve even wished Euclidean geometry was not true, for then wewould have an absolute measure of length a priori.[16, p.63].

This is the beginning of hyperbolic geometry, something Saccheri found outbefore and rejected. This marked the beginning of the area of non-Euclideangeometry 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. Sjostedt, Le Axiome de Paralleleles de Euclides a Hilbert.Bokforlaget Natur och Kultur. Stockholm, 1968.

[3] Girolamo Saccheri, Euclides ab omni naevo vindicatus: sive ConatusGeometricus quo stabiliuntur prima ipsa universae Geometriae Prin-cipia. (Euclid Freed of Every Fleck: or A Geometric Endeavor in WhichAre Established the Foundation Principles of Universal Geometry.) Mi-lan, 1733. Original text and translation by G.H. Helsted, The OpenCourt Publishing Company, Chicago, London, 1920.

[4] Roberto Bonola, Non-Euclidian Geometry. Padua, 1906. Translationfrom the Italian by H.S. Carslaw. Sydney, 1911. Dover edition, 1955.

[5] Georg Simon Klugel, Conatuum praecipuorum theoriam parallelarumdemonstrandi recensio. Dissertation, Gotting University, 1763. Fac-simile and German translation by Martin Hellmann, Universityof Cologne, http://www.uni-koeln.de/math-nat-fak/didaktiken/

mathe/volkert/titel.

[6] C.E. Sjostedt, Le Axiome de Paralleleles de Euclides a Hilbert.Bokforlaget Natur och Kultur. Stockholm, 1968.

[7] John Wallis, De Algebra Tractatus; Historicus & Practicus. TheatroSheldoniano. 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., NewYork, 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 andFritz-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 (SociedadAdaluza de educacion matematica 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 Historyof the Mathematical Sciences, U.K http://www.springerlink.com/

content/ux2p811331w83357/

11

The seven bridges of KonigsbergAnd a brief summary of the rise of graph theory

History of mathematicsFinal paper

Authors:

H. ImhoffK. Meilgaard

Universiteit Leiden

2

Contents

1 Introduction 5

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 problem 9

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. AlthoughEuler didn’t learn any additional mathematics in school his interest grew. So he startedreading 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 obtaina general education. There he met Johann Bernoulli who discovered the great mathe-matical potentional in Euler as he gave him private tuition. In 1723 Euler finished hisstudy in philosophy and started studying theology. He didn’t like this study, but did it tosatisfy his father. Johann Bernoulli persuaded his father to give his consent to change toa mathematical study. He finished his study in 1726 and it the same year he published hisfirst paper on ”isochronus curves in a resisting medium”. And in 1726 Euler became sec-ond 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 ofphysics. 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 tothe mathematical-physical division of the Academy. Here he worked along side a lot ofcolleagues who greatly improved his work. When Bernoulli left St. Petersburg to returnto Basel in 1733 it was Euler who was appointed to the senior chair of mathematics. Notlong after that he married Katharina Gsell and had 13 children with her, from whom only5 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 mathematicalwork. 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 hisreputation was very high and he won the Grand Prize of the Paris Academy in 1738 and1740.

1.1.4 Berlin Academy

In 1744 the Berlin Academy of Sciences was finished and after two invitations from Frederikthe Great, Euler accepted and moved to Berlin to become the director of mathematicsthere. Here he worked for 25 years and wrote around 380 articles. Besides this astoundigamount of articles he also wrote entire books on variating subjects. Even while in BerlinEuler continued to receive part of his salary from Russia. For this remuneration he boughtbooks and instruments for the St Petersburg Academy, he continued to write scientificreports 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 ofGermany 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 byfire and he was able to save only himself and his mathematical manuscripts. A cataractoperation shortly after the fire, still in 1771, restored his sight for a few days but Eulerseems 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 afterhis return to St. Petersburg (when Euler was 59) he produced almost half his total worksdespite the total blindness. Euler was greatly helped by a lot of people but in particularhis son Johann Albrecht Euler who was appointed to the chair of physics at the Academyof 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: Graphtheory. 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 aboutthe notion of mathematical proof itself. The problem asks if we can use just four colors tocolor 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 Hakenpublished a computer-assisted proof. A large group of mathematicians did not acceptthis prove, because it could not be directly checked or validated by a member of themathematical community.

This discussion highlights an important historical fact about the standard of a mathe-matical 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 thedifference 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 ofKonigsberg 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 Bridgesproblem.

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 eliminatedin order to focus only on those most relevant to Eulers reformulation of the bridge crossingproblem as a purely mathematical problem.

S¯OLUTIO 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 an-other branch, previously almost unknown, which Leibniz first mentioned,calling it the geometry of position. This branch is concerned only with thedetermination of position and its properties; it does not involve measure-ments, nor calculations made with them. It has not yet been satisfactorilydetermined what kind of problems are relevant to this geometry of posi-tion, or what methods should be used in solving them. Hence, when aproblem was recently mentioned, which seemed geometrical but was soconstructed that it did not require the measurement of distances, nor didcalculation help at all, I had no doubt that it was concerned with thegeometry of position — especially as its solution involved only position,and no calculation was of any use. I have therefore decided to give herethe method which I have found for solving this kind of problem, as anexample of the geometry of position.

It is clear from this text that Euler was completely new to this type of mathematics. Andalso 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 Konigsbergin Prussia, there is an island A, called the Kneiphof ; the river whichsurrounds 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 andg. Concerning these bridges, it was asked whether anyone could arrangea route in such a way that he would cross each bridge once and onlyonce. I was told that some people asserted that this was impossible, whileothers were in doubt: but nobody would actually assert that it could bedone. From this, I have formulated the general problem: whatever be thearrangement and division of the river into branches, and however manybridges there be, can one find out whether or not it is possible to crosseach bridge exactly once?

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

3 As far as the problem of the seven bridges of Konigsberg is concerned, itcan be solved by making an exhaustive list of all possible routes, and thenfinding whether or not any route satisfies the conditions of the problem.Because of the number of possibilities, this method of solution would betoo difficult and laborious, and in other problems with more bridges itwould be impossible. Moreover, if this method is followed to its conclu-sion, many irrelevant routes will be found, which is the reason for thedifficulty of this method. Hence I rejected it, and looked for anothermethod concerned only with the problem of whether or not the specifiedroute could be found; I considered that such a method would be muchsimpler.

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

4 My whole method relies on the particularly convenient way in which thecrossing of a bridge can be represented. For this I use the capital lettersA, B, C, D, for each of the land areas separated by the river. If a travellergoes from A to B over bridge a or b, I write this as AB — where the firstletter refers to the area the traveller is leaving, and the second refers to thearea he arrives at after crossing the bridge. Thus, if the traveller leaves Band 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 lettersABD, where the middle letter B refers to both the area which is enteredin 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 reallysimilar as what Euler does here.

5 Similarly, if the traveller goes on from D to C over the bridge g, I shallrepresent these three successive crossings by the four letters ABDC, whichshould 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 separatedfrom every other by a branch of the river, the traveller must have crossedthree bridges. Similarly, the successive crossing of four bridges wouldbe represented by five letters, and in general, however many bridges thetraveller crosses, his journey is denoted by a number of letters one greaterthan the number of bridges. Thus the crossing of seven bridges requireseight letters to represent it.

Right now it seems like a very inconvenient way to denote a path. But in a minute we’llsee 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 ofletters occur the required number of times. Before I turn to the problemof finding such a sequence, it would be useful to find out whether or not itis even possible to arrange the letters in this way, for if it were possible toshow that there is no such arrangement, then any work directed towardfinding it would be wasted. I have therefore tried to find a rule which willbe useful in this case, and in others, for determining whether or not suchan arrangement can exist.

8 In order to try to find such a rule, I consider a single area A, into whichthere lead any number of bridges a, b, c, d, etc. (Fig. [1.3]). Let ustake first the single bridge a which leads into A: if a traveller crosses thisbridge, he must either have been in A before crossing, or have come intoA after crossing, so that in either case the letter A will occur once in therepresentation described above. If three bridges (a, b and c, say) lead toA, and if the traveller crosses all three, then in the representation of hisjourney the letter A will occur twice, whether he starts his journey from Aor not. Similarly, if five bridges lead to A, the representation of a journeyacross all of them would have three occurrences of the letter A. And ingeneral, if the number of bridges is any odd number, and if it is increasedby 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. Andthat it’s very similar to the arguments we would use today.

9 In the case of the Konigsberg bridges, therefore, there must be three occur-rences 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, theletter B must occur twice; similarly, D must occur twice, and C also. Soin a series of eight letters, representing the crossing of seven bridges, theletter 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 sucha journey cannot be undertaken across the seven bridges of Konigsberg.

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 crossingeach bridge once, for any arrangement of bridges, whenever the numberof bridges leading to each area is odd. For if the sum of the number oftimes 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 ofbridges, then such a journey can never be accomplished. The rule which Igave for finding the number of occurrences of the letter A from the numberof bridges leading to the area A holds equally whether all of the bridgescome from another area B, as shown in Fig. [1.3], or whether they comefrom different areas, since I was considering the area A alone, and tryingto find out how many times the letter A must occur.

In the Koningberg bridges problem, Euler only encountered odd numbered amounts ofbridges 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 describingthe journey one must consider whether or not the traveller starts hisjourney from A; for if two bridges lead to A, and the traveller starts fromA, then the letter A must occur twice, once to represent his leaving Aby one bridge, and once to represent his returning to A by the other.If, however, the traveller starts his journey from another area, then theletter A will only occur once; for this one occurrence will represent bothhis arrival in A and his departure from there, according to my method ofrepresentation.

The rule becomes a little bit more complicated because he needs to divide the evennumbered pieces of land into two different categories depending wether or not you startyour 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 occurthree 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 bridgesleading to A, then the letter A will occur four times if the journey startsfrom A, and if the traveller does not start by leaving A, then it mustoccur three times. So, in general, if the number of bridges is even, thenthe number of occurrences of A will be half of this number if the journeyis not started from A, and the number of occurrences will be one greaterthan 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 calcu-lation, 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 numberof occurrences of the letter denoting that area to be half the number ofbridges plus one, if the number of bridges is odd, and if the number ofbridges is even, to be half of it. Then, if the total of all the occurrencesis equal to the number of bridges plus one, the required journey will bepossible, and will have to start from an area with an odd number of bridgesleading to it. If, however, the total number of letters is one less than thenumber of bridges plus one, then the journey is possible starting from anarea with an even number of bridges leading to it, since the number ofletters will therefore be increased by one.

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

14 So, whatever arrangement of water and bridges is given, the followingmethod will determine whether or not it is possible to cross each of thebridges: I first denote by the letters A, B, C, etc. the various areas whichare separated from one another by the water. I then take the total numberof 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 toeach one the number of bridges leading to it. Fourthly, I indicate with anasterisk 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 oneI write half the number increased by one. Sixthly, I add together theselast numbers, and if this sum is one less than, or equal to, the numberwritten above, which is the number of bridges plus one, I conclude thatthe required journey is possible. It must be remembered that if the sumis one less than the number written above, then the journey must beginfrom one of the areas marked with an asterisk, and it must begin from anunmarked one if the sum is equal. Thus in the Konigsberg problem, I setout 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

BridgesA, 5 3B, 3 2C, 3 2D, 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 whichleads 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 requiredto determine whether one can arrange a journey which crosses each bridgeexactly once. First, therefore, I name all the areas separated by water asA, B, C, D, E, F, so that there are six of them. Next, I increase the num-ber of bridges (15) by one, and write the result (16) above the workingwhich follows.

16

A*, 8 4B*, 4 2C*, 4 2D, 3 2E, 5 3F*, 6 3

16

Thirdly, I write the letters A, B, C, etc. in a column, and write next toeach one the number of bridges which lead to the corresponding area, sothat eight bridges lead to A, four to B, and so on. Fourthly, I indicatewith 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 secondcolumn, and then I add one to the odd numbers and write down halfthe result in each case. Sixthly, I add up all the numbers in the thirdcolumn 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 itstarts 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:

E a F b B c F d A e F f C g A h C i D k A m E n A p B o E l D,

where I have written the bridges which are crossed between the corre-sponding 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 numberedamount of bridges leading to an area, are very similar to those made in present GraphTheory. 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 wererebuilt. Knigsberg, along with the rest of northern East Prussia, became part of the SovietUnion (now Russia) at the end of World War II and was renamed Kaliningrad. Now it ispossible to visit the five rebuilt bridges via an Euler path (route that begins and ends indifferent 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 followingtheorems we can use:

Theorem 2.2. A connected graph is an Eulerian graph (it contains an Eulerian cycle) ifand 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 fitthe given conditions, which it does in the case of the second, but not the first theorem. Andthus we can now draw the following conclusion: We can walk an Eulerian path accordingto 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 EulerianCycle. That would have been a great tribute to Euler. We would like to end this articlewith a few examples of rebuilding the brides that would have allowed for. It’s quite easyto construct different examples based upon Theorem 2.1 in the previous section. We justhave 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 adifferent bridge), then to B and then to D, and then back to A. We’ve now walked anEulerian cycle!

2.4 Hamilton

Another thing sprung from graph theory is the Hamilton cycle, the problem here is tocross all the vertexes once and only once. Of course we can see directly that the followingpath 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 andC 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 sowe 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. Andthat the simple mathematical puzzles can derive entirely new branches of mathematics alltogether.

Therefore it is of utter importance that we keep puzzling with the fun problems of math-ematics of our modern society, because maybe those will sprout their own branches ofmathematics 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 assignmentRoel Jongen, Matthijs Warrens

Title: Cantor and the countable versus uncountable distinction

1. Introduction

Georg Cantor was a German mathematician who greatly contributed to thedevelopment of mathematics in the second half of the 19th century, especiallyto 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 accepteda position at the University of Halle where he would spend the rest of his ca-reer. In 1872 Cantor was promoted to extraordinary professor and he movedthe focus of his work from number theory to analysis. In this period he provedthe uniqueness of the representation of a function by trigoniometric series, aproblem that had withstood attempts by various high achieving mathemati-cians. In 1873 Cantor introduced the notion of one-to-one correspondence andproved that the rational and algebraic numbers are countably infinite. A shorttime later he proved that the real numbers are uncountably infinite. These re-sults are described in his seminal paper published in 1874. Beginning in 1879,Cantor published the first of a series of articles in Mathematische Annalen onhis ideas of set theory. In these papers Cantor introduced the notions of cardi-nal and ordinal numbers, and well-ordered sets. Basically, he single-handedlycreated an extraordinary set theory in these papers. During his career Cantordid not have the support of all of his peers. Many refused to correspond orwork with Cantor because they felt that Cantor’s work contained philosophi-cal 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 whichhe introduced the distinction between countable and uncountable sets. Thepaper is organized as follows. In the next section we discuss some issues re-lated to the foundations of calculus that where raised in the 19th century. InSection 3 we discuss the notion of countably infinite and present several ex-amples. In Section 4 we discuss the notion of uncountably infinite and presentsome related results. Section 5 is used to discuss algebraic and transcenden-tal numbers. In Section 6 we discuss the importance of the countable versusuncountable 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 mathe-maticians solved problems in mathematical physics and booked various othersuccesses 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, es-pecially the ‘notion’ of infinitely small quantities. One of the key concept incalculus, the ‘limit’, was not well-defined. Since the concept of a ‘limit’ isinherently quite deep, requiring an appreciation of the real numbers that is byno means easy to come by, it took several gifted mathematicians to refine theidea 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 > 0such 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 functionat a point.

Definition 2. A function f(x) is said to be continuous in a point c iflimx→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 ofirrational numbers R\Q. The rational numbers are the fractions, the numbersthat are ratios of integers. Examples are 1

2 and 227 . The irrational numbers

are those numbers that cannot be represented by fractions. An example of anirrational number that was already known in Ancient Greece is the number√

2. As the 19th century progressed a function was found that is continuousat each irrational point in the interval [0, 1], yet discontinuous at each rationalpoint. An example of such a function is the so-called ruler function (Heuer1965, Dunham 2005, Sholapurkar 2007).

Definition 3. The ruler function f : [0, 1] → R is defined by f(

mn

):= 1

n whenmn 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 continuouselsewhere.

Proof: Let a be a rational number in [0, 1]. By the density of irrationalnumbers there is a sequence {xn}n∈N of irrational numbers that converges toa. But f(xn) = 0 for all n ∈ N, while f(a) > 0. Hence, f is discontinuous ata. Next, let b be a irrational number in [0, 1], and let ε > 0. The Archimedeanproperty of R asserts that there is an integer n such that n > 1

ε . Fix this n.Since there are only finitely many rational numbers in [0, 1] with denominatorless than n, there exist a δ > 0 such that for every rational number p

q with|b− p

q | < δ, we have q > n. This implies that for every x with |b− x| < δ, wehave f(x) < ε. Hence f is continuous at b. �

This result for the ruler function indicates that there is not a symmetry be-tween 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 andfundamental questions of the calculus rested upon profound properties of sets.The man who would show the mathematical community how to study theproperties 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 com-paring the number of elements in both of them. Cantor used the followingdefinition for infinite sets (Dunham 1990).

Definition 4. Two sets A and B are called equivalent, denoted by A ∼ B, ifthere exists a bijection between them.

Because a bijection between A and B links to every element in A one uniqueelement in B and vice versa, A and B must have the same number of elementsif 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, symmetricand transitive. Using the identity function M → M , x 7→ x, we find thatM ∼ M , that is, ∼ is reflexive. Next, suppose that M ∼ N . Then there existsa 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 thereexist bijections f : M → N and g : N → P . Now g ◦ f : M → P is a bijectionsince 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 ∼ cardinalnumbers. 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 ∼ Nsince |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 theprime numbers using the regular order <, and write pn for the n-th primenumber. 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

f : n 7→ 1 + (−1)n(2n− 1)4

3

is a bijection from N to Z. For the first small numbers of N and Z the functionf 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. Sincethe function f : n 7→ 2n is a bijection from Z to 2Z, we have 2Z ∼ Z. BecauseZ ∼ N, it follows that 2Z is countably infinite.

Example 5. Let n be a positive integer and let nZ = {0,±n,±2n, . . .} denotethe sets of multiples of n. Since the function f : m 7→ nm is a bijection fromZ 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 withnumerator −1 in the second column, all fractions with numerator 2 in thethird column and so on. We arrange the rows in such a way that all fractionsin a row match the number of the row in their denominator. In this way wefind the following array.

01 −1 2 −2 3 −3 4 . . .

12 −1

222 −2

232 −3

242 . . .

13 −1

323 −2

333 −3

343 . . .

14 −1

424 −2

434 −3

444 . . .

......

......

......

...

It is clear that this array contains each rational number at least once. Nowwe 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 . . .

l l l l l l l l l l l l l l

Q : 0 1 12 −1 2 −1

213

14 −1

3 −2 3 23 −1

415 . . .

�

Cantor denoted the cardinal number belonging to N by ℵ0. We have found inthis 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 askourselves whether it is true that any infinite set is countably infinite. Claim 4shows 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 sequencex1, x2, x3, . . ., containing all the elements of (0, 1). Now we write each xn asa sequence an1, an2, an3, . . . of the consecutive digits of its decimal expansionand 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 rowb1, 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 particularany 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 beequal 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 concludethat (0, 1) is uncountably infinite. �

Example 6. Let a, b ∈ R such that a < b. The function f : x 7→ a + (b− a)xis a bijection from (0, 1) to (a, b). Hence, (0, 1) ∼ (a, b), and it follows thatthe interval (a, b) is uncountably infinite.

Example 7. The function

f : x 7→ 2x− 1x(1− x)

is a bijection from (0, 1) to R. Hence, (0, 1) ∼ R, and it follows that R isuncountably infinite.

Claim 5. The set of irrational numbers R\Q is uncountable.

Proof: Suppose that the irrational numbers are countable. Then there existsa 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 thesequence {cn}n∈N given by

cn =

{a(n+1)/2 if n is odd,an/2 if n is even

contains any real number. Hence, we have a contradiction, and we concludethat the irrational numbers are uncountable. �

Cantor denoted the cardinal number belonging to (0, 1) by c. We have foundin 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 withinteger coefficients has a zero in the complex numbers. That is, for any poly-nomial 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. Thiswarrants 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 thatf(a) = 0. The number a is called transcendental if it is not algebraic.

The algebraic numbers appear to be quite abundant. Since a rational numberm/n with m, n ∈ Z is a zero of the linear polynomial f(x) = nx − m, wehave that all rational numbers are in fact algebraic. Other examples, arethe 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 dif-ficult to show that a given number is transcendental. The first to prove theexistence of transcendental numbers was Liouville in 1844, using continuedfractions. In 1851 Liouville presented the number

∑∞n=1 10−n! = 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 byrational numbers than can any algebraic number. In 1873 Hermite proved thatthe number e =

∑∞n=0(n!)−1, the base of the natural logarithm, is transcen-

dental. This was the first number to be proved transcendental without havingbeen specifically constructed for the purpose (Burger, Tubbs 2004, Shidlovskii1989).

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 therebysolved the ancient Greek problem concerning the quadrature of the circle. TheGreeks had sought to construct, with ruler and compass, a square with areaequal to that of a given circle. If a unit length is prescribed this amounts toconstructing 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 asmall set of transcendental numbers had been found. The algebraic numbersin contrast seemed to constitute a vast set. Then, in 1874, Cantor presentedthe 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 asp(x) = a0 + a1x + ... + anxn 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 polynomialhaving α as its root. For every N ∈ N there exist finite many polynomialshaving N as its height, so there exist finite many algebraic numbers havingN as its height. Finally, we can make a sequence of algebraic numbers bysorting them by height and by using the order of R to sort the numbers ofequal height. �

Since Example 7 showed that the real numbers R are uncountably infinite,an immediate corollary of Claim 6 is that the transcendental numbers areuncountably 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 ofmathematics. Although many notions in what we now call set theory existedbefore Cantor, Cantor’s work vastly expanded this language. Most objects thatare used in modern mathematics are defined as sets equipped with some extrastructure. Consider the following two examples from algebra and topology.

Definition 7. A group is a set G equipped with an associative map G×G → Gfor which there is an identity element, and with respect to which every elementin G has an inverse.

Definition 8. A metric space is a set S equipped with a function d : S×S → Rwhich satisfies for all x, y, z ∈ S, d(x, x) = 0 ⇔ x = 0, d(x, y) ≥ 0, and thetriangle inequality d(x, y) ≤ d(x, z) + d(y, z).

The significance of Cantor’s work lies perhaps more in the distinction betweencountable and uncountable sets. In many contexts in mathematics one mustinclude 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 alsoa complete metric space with respect to the distance function induced by theinner product. A Hilbert space is separable if and only if it admits a countableorthonormal basis.

Hence, for a separable Hilbert space there is a countable subset which getsarbitrarily close to any element of the Hilbert space.

Definition 10. A space is said to be second-countable if its topology has acountable base.

Most ‘well-behaved’ spaces in mathematics are second-countable. For exam-ple, the Euclidean space Rn 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 ifit is non-empty, closed under complementation, and closed under countableunions.

Using the notion of a σ-algebra we can define a measure of a set. In probabilitytheory the probability of an event occuring corresponds to the measure of someset 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 forall 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

µ

(⋃i∈I

Ei

)=∑i∈I

µ (Ei) ,

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 mustshow that we can cover M with a countable number of open intervals suchthat the sum of the length of the intervals is less than ε. Cover mn with theinterval (

mn −ε

2n+2,mn +

ε

2n+2

).

Thus, for element mn the length of the interval is ε/2n+1. If we sum theselengths we obtain

∞∑n=1

ε

2n+1= ε

( ∞∑n=0

(12

)n

− 1− 12

)=

ε

2< ε. �

8

References

Burger EB, Tubbs R (2004) Making Transcendence Transparent. Springer,New York.

Cantor GFLP (1874) Uber eine Eigenschaft des Inbegriffes aller reellenalgebraischen Zahlen. Journal fur 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, NewYork.

Hart KP (2011) Verzamelingenleer.

Heuer GA (1965) Functions continuous at the irrationals and discontinuousat 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 12.1 The first ’proof’ by Kempe . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Detected error: eleven years later . . . . . . . . . . . . . . . . . . . . . . . 22.3 Decisive work through the century . . . . . . . . . . . . . . . . . . . . . . 2

3 The First "Proof" 33.1 A sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 The reaction of the mathematical community . . . . . . . . . . . . . . . . . 3

4 Computer-assisted Proofs 44.1 What is a proof? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Modern role of computers in mathematics . . . . . . . . . . . . . . . . . . 4

5 Arguments for Computer-assisted Proofs 55.1 Unreliability of the Computer . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 Proof length: beyond human capacity . . . . . . . . . . . . . . . . . . . . . 55.3 Social context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Arguments against Computer assisted proofs 66.1 The Value of a Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66.2 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 astudent at University College London, where he studied Law and Mathematics with DeMorgan as his supervisor. In London, he functioned as a barrister until he moved to SouthAfrica in 1861 as a Professor of Mathematics at the South African College which laterbecame University of Cape Town. He published a few mathematical papers and becameinterested in botany. He earned a lasting name for himself with his botanical research: eventwo newly discovered types of plants were named after Guthrie.

During the time Francis Guthrie was studying law, his brother Frederick Guthrie hadbecome a student of DeMorgan. In his spare time Francis Guthrie was trying to colour themap of countries of England and noticed that four colours seemed to be sufficient to colourmost maps. He asked his brother Frederick if it was true that any map can be colouredusing four colours in such a way that regions sharing a common boundary segment receivedifferent colours. Frederick Guthrie communicated the conjecture to DeMorgan. DeMorganwasn’t able to answer this question, but communicated the problem to Hamilton, who wasn’tlikely to answer this question soon either, mainly because the problem did not grab hisattention.

De Morgan kept trying to persuade mathematicians to work on Guthrie’s problem andthus 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. Cayleyalso became acquainted with the four colour theorem and sent a paper on the colouring ofmaps to the Royal Geographical Society, which was published in 1879. This paper is thefirst printed reference of the Four Colouring conjecture, it explains where the difficulties liein attempting to prove the Conjecture.

Theorem. (Four colour Theorem): The regions of any simple planar map can be colouredwith 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 math-ematicians over a hundred years to come to a proof that still isn’t generally accepted as aproof.

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. Heannounced in Nature that he had a proof of the Four Colour Conjecture.

Kempe submitted his proof of the theorem to the American Journal of Mathematicswhere 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 wasgreatly admired for his proof. He even was elected a Fellow of the Royal Society and servedas 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 Kempewas 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 thefalse assumption that every three-connected planar graph is Hamiltonian. Neither Kempenor Tait could correct the mistake in the proof, so the Four Colour theorem became the FourColour problem for the second time in history. In this same paper Heawood did prove thatevery 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 beadapted 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 eachvertex 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 conjecturethat any planar cubic graph contains a Hamiltonian circuit.

The map colouring problem never lost Heawood’s interest. In 1898 he proved that if thenumber 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 onthe Four Colour conjecture generalizing Heawood’s work. Birkhoff introduced theories onreducibility and chromatic polynomials.

Franklin in 1922 published further examples and proved that any map with less than 25regions can be 4-coloured. Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Oreand Stemple to 39 in 1970 and Mayer to 95 in 1976.

Heesch introduced the final ideas necessary for the solution of the Four Colour conjec-ture: the method of discharging in 1969. He thought that the Four Colour conjecture couldbe 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 proofthat wasn’t found to be flawed was finally devised in 1976 by Kenneth Appel and WolfgangHaken. What was unusual about this proof, is that it made extensive use of what is calledcomputer-assistance.

The necessary groundwork for this computer-aided proof was laid by the German math-ematician Heinrich Heesch over a period of time during the 1960s and 1970s. The maintheme in this was a number of proof techniques which allowed for the reduction of theamount of different subgraphs which needed to be analysed. The main technique he devel-oped was the method of discharging, which was essential for the proof given by Appel andHaken. These techniques were devised especially for the Four Colour theorem, but he wasunable 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 exista smallest map which can only be coloured with five colours or more. They then showed thatsuch 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 containatleast one member of this set.

• A reducible configuration: A submap such that, if a map which is four colourablecontains such a submap, it can be reduced to a map which is smaller but is stillfour colourable. This means that a minimal counter-example cannot contain such asubmap, 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. Thisis were the computer played its role, and checked each of these in what took a computerover 1200 hours of time to compute. After this lengthy wait, the two mathematicians wereleft with a group of reducible configurations that were in an unavoidable set. Since everypossible map must contain at least one of these submaps, and since any map containing oneof those submaps cannot be a minimal counter-example, they could conclude that no suchthing 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 problemsand hit a nerve that is very deep engrained in most mathematicians. The proof which theyeventually published in its entirety 1989 was 741 pages long, and included 400 pages ofmicrofiche which contained the output of the computer verification. Microfiche reduces thephysical size of documents significantly, so the total length of the proof was enormous, andmuch too long to be humanly verified.

Due to the fact that this proof could not be hand checked, many mathematicians didn’tconsider it a true proof. Philosopher and mathematician Thomas Tymoczko took it even fur-ther and chose to not classify the problem as a mathematical theorem, but as an a posterioritruth 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 theevolution of the form of mathematics. Nonetheless, these mathematicians also recognizedthe 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 understandto 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, whileoften presented in natural language, it must be possible to convert it into formal predicatelogic. 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 naturallanguage. On the other hand the "formal proof", according to proof theory, is one whichtechnically could be verified by a machine as it only has to comply with the axioms of prooftheory 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 quickarithmetic speed of a computer to usually check a predefined amount of objects, and thussuch proofs are mostly proofs-by-exhaustion. The methods used by Appel and Haken weretailor-programmed for the Four Colour proof, in which they formalized the required logicalreasoning for the proof in computer language.

Figure 2: An excerpt from Metamath

Since then the use of computers in mathematical reasoning has grown and maturedmuch. 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 togetherreach 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 automatictheorem prover, which checks the basic logical steps to see if they are justified. A few ofsuch programs are Metamath, or HOL Light.

Although the field of computer-assisted proofs has grown much since 1976, and havetried to solve much of the problems people initially had with computer-assisted proofs, wewould 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. Everynew example of computer-assisted proof is very helpful to learn from. Introducing newfields in mathematics is a matter of acceptance and familiarity. Nowadays the computeris as indispensable as a pencil or paper, but this hasn’t always been the case. Especiallypeople who grew up without computers have the instinct to question the validity of theresults obtained from such elusive machines.

We need to see a wide variety of examples before we are comfortable using this tech-nique. We need to read clearly written arguments and algorithms which prove theorems viathis 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-assistedproofs. They need to be clearly formulated and carefully tested. The algorithm should bewell documented and should be able to be implemented in any computer language and anyoperating system. Multiple different implementations by different people gives confidencethat 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 andhardware. This can be resolved by providing a formal proof of correctness for the computerprogram. Such a proof was provided for the Four Colour theorem in 2005. Obtaining thesame results by using different programming languages, different compilers, and differentcomputer hardware is of course necessary.

Moreover, this complaint can be applied to humans as well. People are also likely tomake mistakes. The first so called ”proof” of the Four Colour problem by Kempe is aperfect example that humans are only human. Only after eleven years the error in the proofwas detected. Until then the proof was widely recognized and accepted.

Computers follow a pre-designed rigid program and are not distracted by moods, stressand other outside factors. So even though computer errors might be harder to detect, humansare 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 complicatedand long computations, which might be understood by humans, but by no means executablewithout the use of computers. The Four Colour theorem is one of those results. What ini-tially 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 forthose results which can be done by humans, the speed of a computer outweighs that of thehuman.

5.3. Social context

Mathematicians prove theorems in a social context. It is a socially conditioned body ofknowledge and techniques. Proofs in practice are completely different from what we wouldlike them to be in theory. It is not a formal process but a social ritual of acceptance whichallows theorems to achieve a certain amount of credibility. It all comes down to the opinionsand beliefs of those in the mathematical community: will we believe in a computer-aidedresult 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 reasonablysure that computers will be accurate, given that the computer languages, original code, andsuch are carefully checked by multiple people. You could also argue that people checkinga proof are also prone to mistakes, which we try to minimize by having large amounts ofpeople check a proof. This can be emulated in computers by having multiple algorithms indifferent languages to also minimize the chance of error.

The line between a computer generated proof which cannot be checked by hand and onethat can be checked by hand seems to be a very subjective divide. In theory a checker withenough time could sift through the pages of proof given by Appel and Haken, and actingcarefully enough they could rigorously check the validity of the proof. But in my opinionthis 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 theone hand, it is of course such that the reader will know without any doubt in his mind thatthe theorem holds. This is where the term "proof" comes from. But this is not the onlything. 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 whatcauses the proof to function, often leading to further insights or new leads towards differenttheorems. Even in somewhat non-constructivist proofs we still gain some understanding ofthe workings by seeing what causes such a contradiction.

Unfortunately a proof that can’t be read, even if we can be incredibly sure towardsits validity, will not give us unique insight into the mathematics itself. Then the role ofthe human will be to scrutinize the programs and algorithms which allow the computer tocalculate, 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 ofsuperfluous 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 "theBook", this being God’s book of all mathematical proofs in their most elegant form. Whilethere is little doubt that the current proof of the Four Colour theorem is false, it is clearly nota piece of art from the Book. If someone were to find a proof of a couple of hundred pagesthe 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 moreconcise 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 powerfulmethods of discharging. It is entirely possible that in time a humanly readable proof will befound and this will be considered the true proof of the Four Colour theorem. Such a proofwouldn’t quite be considered one from the Book, but maybe it could one from one of hisnotebooks.

7. Conclusion

In the end, mathematicians will always be searching for shorter, more elegant proofs. TheFour Colour theorem will be remembered because it actually initiated a completely newfield 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 itcarefully, in the future this technique will be accepted more and more. Because computerswill become increasingly important in the future, eventually the whole mathematic societywill accept the computer-assisted proofs.

7