History of Mathematics - Chapter Summaries

History of MathematicsChapter Summaries

Part One Ancient MathematicsChapter 1: Egypt and MesopotamiaChapter 2: The Beginnings of Mathematics in GreeceChapter 3: EuclidChapter 4: Archimedes and ApolloniusChapter 5: Mathematical Methods in Hellenistic TimesChapter 6: The Final Chapters of Greek MathematicsPart Two Medieval MathematicsChapter 7: Ancient and Medieval ChinaChapter 8: Ancient and Medieval IndiaChapter 9: The Mathematics of IslamChapter 10: Mathematics of Medieval EuropeChapter 11: Mathematics Around the WorldPart Three Early Modern MathematicsChapter 12: Algebra in the RenaissanceChapter 13: Mathematical Methods in the RenaissanceChapter 14: Algebra, Geometry, and Probability in the Seventeenth CenturyChapter 15: The Beginnings of CalculusChapter 16: Newton and LeibnizPart Four Modern MathematicsChapter 17: Analysis in the Eighteenth CenturyChapter 18: Probability and Statistics in the Eighteenth CenturyChapter 19: Algebra and Number Theory in the Eighteenth CenturyChapter 20: Geometry in the Eighteenth CenturyChapter 21: Algebra and Number Theory in the Nineteenth CentruyChapter 22: Analysis in the Nineteenth CenturyChapter 23: Probability and Statistics in the Nineteenth CenturyChapter 24: Geometry in the Nineteenth CenturyChapter 25: Aspects of the Twentieth Century and Beyond

Part One Ancient MathematicsChapter 1: Egypt and Mesopotamia

Mesopotamia Slightly older civilization than the Egyptian. Had their beginning around 4500 BCE. Consisted of city-states. Needed numerate scribes. Writing started around 3000 BCE out of the need to manage labor and trade. Largest unit was 60 of the smallest unit. We found a lot of clay tablets, teaching us about Mesopotamian mathematics.

Egypt First dynasty dates from about 3100 BCE. Writing started around the same time the pyramids were built. There was a limited number of signs. The same symbol could mean various things. Due to Greek influence, Egyptian writing forms began to disappear. Around 1800, Jean Champollion managed to translate it with the help of the Rosetta stone. Not many papyri with Egyptian mathematics still exist. Egyptian mathematics was practical in nature, dealing with accounting and architecture.

General Notes Much of the mathematical texts we found of these civilizations relate to teaching. Many of the examples in such texts, those dealing with quadratic equations, for example, had no applied context, but were used to train the minds of future leaders.

Chapter 2: The Beginnings of Mathematics in Greece

A new attitude towards mathematics appeared in Greece before the fourth century BCE. Starting here, mathematical proofs became important. This change, beginning around 600 BCE, was related to the differences between the Greeks and Egyptians/Babylonians. Each Greek city-state was ruled by law and therefore encouraged its citizens to argue and debate. Perhaps out of this, the necessity for mathematical proofs emerged.

2.1 The Earliest Greek Mathematics There are virtually no texts of Greek mathematics from the first millennium BCE. The information dates from much later, hence do not always agree with each other.2.1.1. Greek Numbers The Greek numbers were written using their alphabet In the tunnel on the island of Samos numbers are inscribed on the walls. The numbers were likely written to keep tabs on the distances dug. In Greece mathematics was the province of a leisured class.

2.1.2 Thales The earliest Greek mathematician mentioned in book I of Euclids Elements is Thales of Miletus. Stories of Thales include the prediction of a solar eclipse, measuring the distance to a ship at sea and determining the height of a pyramid using the length of its shadow.2.1.3 Pythagoras and His School Stories about Pythagoras of Samos state that he spent time in Egypt and Babylonia. He settled in Crotona, Italy, where he gathered a group called the Pythagoreans and formed a school. One important mathematical doctrine was that Number is the substance of all things. In other words, that numbers (positive integers) form the basic organizing principle of the universe. Problems of interest included the construction of Pythagorean triples (three numbers a, b and c satisfying a+b=c). In Greek mathematics, a number was a multitude composed of units. Therefore, 1 was not seen as a number. It turned out that numbers were not the substance of all things, as the side and diagonal of a square are incommensurable (have no common measure). This was discovered around 430 BCE.2.1.4 Squaring the Circle and Doubling the Cube These two geometric problem exemplified the idea of proof. However, both problems are impossible. Hippocrates of Chios (not the famous physician) was among the first to attack these problems.

2.2. The Time of Plato Plato (429-347 BCE) founded an academy in Athens around 385 BCE. Scholars conducted seminars in mathematics and philosophy and conducted research. A story states that over the entrance Let no one ignorant of geometry enter here was inscribed. The mathematical syllabus for students is described in Platos most famous work, The Republic. The mathematical part of this education consisted of arithmetic, plane geometry, solid geometry, astronomy and harmonics.

2.3 AristotleAristotle (384-322 BCE) studied at Platos academy from the time he was 18 until Platos death. Afterwards he was invited to the court of Philip II of Macedon to undertake the education of his son Alexander. Later he returned to Athens to found his own school, the Lyceum. Aristotles strongest influence was in the area of logic.2.3.1 Logic Aristotle believed that logical arguments should be built out of syllogisms. A syllogism consists of certain statements that are taken as true and certain other statements that are then necessarily true. Syllogisms allowed for old knowledge to imply new knowledge. However, one has to start with truths that are accepted without argument. Postulates: Basic truths that are peculiar to each particular science Axioms: Basic truths that are peculiar to all sciences. Aristotle only admitted for the most basic ideas a postulate as definition. Outside of Aristotle, Greek mathematicians never used syllogisms. The basic form of arguments used in mathematical proof was based on propositions. A proposition is a statement that can be either true or false.2.3.2 Number versus Magnitude Aristotle introduced the distinction between numbers and magnitude. Aristotle rejected the Pythagorean doctrine that all was number. The primary distinction between numbers and magnitudes is that a magnitude cannot be composed of indivisible elements. Aristotle clarified this idea by the following definitions: Things are in succession if there is nothing of their own kind intermediate between them. Things are continuous when they touch and when the touching limits of each become the same.2.3.3. Zenos Paradoxes One of the reasons Aristotle had such an extended discussion of the notions of infinity, indivisibles, continuity and discreteness was the he wanted to refute the paradoxes of Zeno. The paradoxes of Zeno were an attempt to show that the then current notions of motion were not clear. The four paradoxes exhaust the four possibilities of divisibility/indivisibility of space and time.

Chapter 3: Euclid

Euclid Generally assumed to have flourished around 300 BCE Beside this date there is nothing known about the author of the Elements Historians believe he was active at the Museum and Library at Alexandria The Elements Most important mathematical text of Greek times and probably all times Extremely dull:o no examples, no motivation, no remarks, no calculationso only definitions, axioms, theorems and proofs Excited and motivated many famous mathematicianso provided them with a model of how pure mathematics should be written Euclids is the only version of Elements to survive, since it was complete (it contained Aristotles work) and well-organized Copies were made regularly from the time of Euclid; various editors added comments or put in new lemmas. Theon of Alexandria was responsible for one important new edition, most of the extant manuscripts of Euclids Elements are copies of this edition. Work of 13 books First six books: treatment of two-dimensional geometric magnitudes (Book V: treatment of proportion theory for magnitudes)o Books VII-IX: theory of numbers (keeping with Aristotles instructions to separate the study of magnitude and number) (Book VII: treatment of proportion theory for numbers)o Book X: link between concepts of magnitudes and numberso Book XI: three-dimensional geometric objectso Book XII: method of exhaustiono Book XIII: constructed the five regular polyhedra Much of the ancient mathematics is included, but the methodology is entirely different Aristotle suggested a scientific work needs to begin with definitions and axioms, so many books started with that Proved one result after the other, based on the previous results and/or axioms Proofs were written out in natural language Euclid assumed that if he proved a result for a particular conguration representing the hypotheses of the theorem and illustrated in a diagram, he had proved the result generally Euclid never discussed his philosophy of proof, he just proved things

The Babylonians and Euclid Debate over whether Euclids geometric algebra was a transformation of the Babylonian quasi-algebraic results into formal geometry. Many solutions of Euclid mirror the Babylonian solutions We dont know whether Babylonian algebra was transmitted in some form to Greece by the fourth century BCE. Number Theory Euclid did not consider numbers as types of magnitudes. Included Theaetetus definitions and proofs of number theory of Pythagoreans. A number is a multitude of units! For Euclid, as for the Pythagoreans earlier, 1 is not a number.Irrational magnitudes Book X (longest and best organized) is considered the most important of the Elements.o Much of this book is attributed to Theaetetus What did the Greeks learn from the Egyptians? No documentation of transmission from Egypt to Greece before the 3rd century BCE. The Greeks in general stated that they had learned from Egypto Many Greek documents say that geometry was rst invented by the Egyptians and then passed on to the Greekso The Egyptians invented the results, not the method of proof The evidence of direct Egyptian influence on Greek mathematics is relatively strong.

Chapter 4: Archimedes and Apollonius

Archimedes of Syracuse (c.287-212 BCE): Was born in Syracuse (Italy). Son of an astronomer Phidias.Studied in Alexandria, Egypt. More biographical information of him survived than any other Greek mathematician, found in Plutarchs biography. Probably spent his youth in Alexandria, he is credited with invention of the Archimedean screw: a machine for raising water used for irrigation. Spent most of his life in Syracuse. He was repeatedly called to solve practical problems for Hiero and successor. Devoted to mathematics: made him forget his food and neglect his person. As military engineer he kept the Roman army under Marcellus at bay for months. He was killed by an enraged soldier whom commands Archimedes refused to follow until he had worked out some problem with a diagram. He was so busy with his investigation, he didnt notice the incursion of the Romans nor the capture of the city.

Archimedess work: Used the limit methods of Eudoxus and succeeded in applying them to determine areas and volumes of new figures. Treatises (originally letters sent to people he knew) who presented mathematical models of certain aspects and applied his physical principles to the invention of various mechanical devices. First mathematician to derive quantitative results from the creation of mathematical models of physical problems on earth. First to prove the law of lever and its application to finding centers of gravity.He never gave a definition of the term center of gravity even though he used it often. He developed the mathematical principles of the lever stating seven postulates he would assume. The first postulate is an example of the Principle of Insufficient Reason. First to prove the basic principle of hydrostatics and some of its applications. There are no surviving manuscripts of his work dating from anywhere near the time of composition. There was an editon of some of his works prepared by Eutocius in the sixth century somewhere near Byzantium. There is a Latin translation by Moerbeke of the second oldest extant Archimedes manuscript..

Story told by Vitruvius about Archimedes: Hiero, the King of Syracuse contracted a contractor to make a crown of gold as dedication to the immortal Gods. The contractor replaced certain amount of gold for an equal amount of silver. Archimedes, in charge for finding a way to expose the theft, took a bath. He noticed that the however much he immersed his body in the tub, that much water spilled over the sides. He rushed home naked shouting: Eureka! ( I found it).(To find the crowns volume he had to immerse the crown in a vessel full of water and measure the spillage. Because if the gold was substituted for the same amount of silver, the crown would occupy a larger space, since he knew that gold is more dense than silver.)

Apollonius (250-175 BCE): Was born in Perga (Asia Minor). He was instrumental in extending the domain of analysis to new and more difficult geometric construction problems. Few details are known about his life, most of the reliable information comes from the prefaces to his magnum opus, the Conics: eight books developing the properties of conics, central in developing new solutions to problems as the duplication of the cube and the trisection of the angle. Seven of the eight books of the Conics survive and represent in some sense the culmination of Greek mathematics. Went to Alexandria as a youth to study with successors of Euclid and probably remained there for most of his life, studying, teaching and writing. Became famous for his work on astronomy, mathematical work, most is known today only by titles and summaries in works of later authors. Discovered and proved hundreds of theorems without modern algebraic symbolism. There are no surviving manuscripts of his work dating from anywhere near the time of composition.

Preface to Book 1 of Apolloniuss Conics: The third book contains many theorems for the construction of solid loci and determination of limits, most of them new. Euclid had not worked out the construction of the three-line and four-line locus, but a part of it, since the construction couldnt have been completed without Apollonius additional discoveries.

Chapter 5: Mathematical Methods in Hellenistic Times

Claudius Ptolemy wrote a major work answering Platos challenge Extensively criticized, yet never replaced for 1400 years Used earlier ideas from plane and spherical geometry and devised new ways perform the calculations necessary to make his book useful Heavily used in astrology Primary reason of astronomy was the solving of problems with the calender Greeks created a mathematical model of the universe Major contributors to the development of mathematical astronomy:o Eudoxus in 4th century BCEo Apollonius late in the 3rd century BCEo Hipparchus in the 2nd century BCEo Menelaus around 100 CEo Ptolemy Astronomy before Ptolemy Most important heavenly bodies were the sun and the moono Obvious that both rose in the east and set in the west, actual movements were more subtle Observed everywhere that the sun cycle repeated itself at intervals Stonehenge was built to observe the suns position The motions of the moon determined the monthso The Egyptians and Babylonians both used the phases of the moon to establish the months of their years The ability to predict Eclipses was an important function of the priestly classes The calendrical situation in Metopotamia was differento Priests wanted to accomodate the calender to the sun and the moono Their months alternated between 29 and 30 dayso They added an extra month every several yearso Babylonians were able to make relatively accurate predictions of the recurrence of various celestial problems (sunrise/sunset and lunar eclipses) Basic model contained two spheres: sphere of the earth and the sphere of the starso Greeks were convinced of the earth sphericity by for example the shadow of the earth on the moon during a lunar eclipse Greeks were convinced that that the earth was stationary in the middle of the celestial sphereo The earth was considered immovable, so the celestrial sphere must have been moving with the fixed-on stars attached to ito The wanderers sun, moon, Mercury, Venus, Mars, Jupiter and Saturn were loosely attached to the sphere, they also had their own motionso Greeks tried to make sense of this, but were limited in their solutions by a philosophical consideration Eudoxes Famous for his work on ratios and the method of exhaustion Largely responsible for turning astronomy into a mathematical science Probably the inventor of the two-sphere model and the modifications necessary to account for the motions of the sun, moon and planets Claudius Ptolemy (c. 100-178 CE) Nothing known about his personal life Made numerous observations of the heavens from locations near Alexandria Wrote several important books Most famous for a work in 13 books that contained a complete mathematical description of the Greek model of the universe with parameters for the various motions of the sun, moon, and planetso Replaced all earlier works on this subjecto Most inuential astronomical work from the time it was written until the 16th centuryo All subsequent astronomical works were based on Ptolemys masterpieceo Became known as megisti syntaxis, Islamics called it al-magisti, since then it has been known as Almagest

Chapter 6: The Final Chapters of Greek Mathematics

Rule of Rome (beginning 31 BCE) Alexandria remained an important Greek mathematical centre in this chapter are 4 mathematiciens discussed that worked onpure mathematical instead of the applied mathematics in chapter 5Nicomachus Very little is known about his life In his work he uses a lot pythagorean ideas It is likely that he studied in Alexandria (centre of nea-Pythagorean) He wrote the book: Introduction to Arithmetic (late 1ste century) This book explains pythagorean number philosophie He wrote 2 other books that didnt survive He used no proofs only examples

Diophantus He lived in Alexandria He wrote Arithmetica this was divided in 13 books (mid 3th century) He introduced symbolics and dealt with powers higher than the tirth

Pappus He live in Alexandria in the early 4th century. He was one of the last in the Greek tradition. He is best known for his collection, 8 works on various topics of him combined.

Part Two Medieval MathematicsChapter 7: Ancient and Medieval China

Oracle bones were curious pieces of bones inscribed with very ancient writing. These bones are the source of our knowledge of early Chinese number systems. In 1984, a tomb of an official was opened and among the books was discovered a mathematics text. This work, called the Suan Shu Shu (Book of Numbers and Computation), is the earliest extant text of Chinese mathematics. China was ruled by different dynasties. Despite numerous wars and dynastic conflicts, a true Chinese culture was developing throughout most of east Asia, with a common language and common values. The Chinese government encouraged the study of mathematics, but hardly no new methods were introduced. There was no particular incentive for mathematical creativity. Although there were some creative mathematicians. Qin Jiushao (1202-1261) wrote his Mathematical Treatise in Nine Sections. Qin lived a fascinating life with corruption and love affairs. Li Ye (1192-1279) wrote Sea Mirror of Circle Measurements. Li Ye dealt with the properties of circles inscribed in right triangles, but he was chiefly concerned with the setting up and solution of algebraic equations for dealing with these properties. Yang Hui wrote A Detailed Analysis of the Arithmetical Rules in the Nine Sections. In contrast to Lis work, Yang Hui gave detailed account of his methods. Zhu Shijie wrote Precious Mirror of the Four Elements, where he was able to work with up to four unknowns. There is not much known about mathematical transmission to and from China before the sixteenth century. In all cases of similarities, there are sufficient differences in detail to rule out direct copying from one civilization to the other Chapter 8: Ancient and Medieval India

The first evidence of mathematics in India is from a civilization formed along the Ganges River y Aryan tribes late in the second millennium BCE. From about the eighth century there were monarchical states with a highly stratified social system headed by the king and the priests (brahims). The literature of the brahims was oral and expressed in lengthy verses called Vedas. The Sulbasutras (mathematical works) are the sources for our knowledge of ancient Indian mathematics. In 327 BCE Alexander the Great started conquering small north eastern Indian kingdoms and Greek influence began to spread into India. Alexander came not just as a conqueror interested in plunder but also on a mission to civilize the East. After Alexanders death his Indian provinces were reconquered by Chandragutpa. After his death Ashoka succeeded the throne. He left records of his reign in edicts carved on pillars throughout his kingdom, containing some of the earliest written evidence of Indian numerals. Under the rule of the Guptas in the fourth century northern India reached a high point of culture with the flowering of art and medicine and the opening of universities. The earliest identifiable Indian Mathematician was Aryabhata. His chief work was the Aryabhatiya, although concentrating mainly on astronomy, it also contained a wide range of mathematical topics. The ninth-century mathematician Mahavira composed the earliest Sanskrit textbook entirely to mathematics, rather than having mathematic as an adjunct to astronomy. The most influential mathematics texts were two works by Bhaskara II from the twelfth century, the Lilavati and the Bijaganita, on arithmetic and algebra, respectively. In the Vijayanagara empire in southern India the mathematics school of Madhava became established. From the fourteenth to the sixteenth centuries there was a sequence of transmissions from teacher to pupil in this region, which resulted in the writing of proofs of many results. Overall it seems that whoever ruled the country needed astronomers to help with calendrical questions and to give astrological advice. Thus, much of Indian mathematics is recorded in astronomical works, but there are mathematicians who went beyond the strict requirements of practical problem solving to develop new areas of mathematics that they found of interest. The Decimal Place Value SystemOur modern decimal place value system is usually referred to as the Hindu-Arabic system because of its supposed origins in India and its transmission to the West via the Arabs. Symbols for the first nine numbers of our system have their origins in the Brahmi system of writing in India (mid-third century BCE). Probably in the eight century these digits were picked up by the Moslems. A century later they appeared in Spain and still later in Italy and the rest of Europe.Notion of place value: In India, although there were number symbols for the first nine digits, there were also symbols to represent 10 through 90. Larger numbers were represented by combining the symbol for 100 or 1000 with a symbol for one of the first 9 numbers. Around the year 600, the Indians vidently dropped the symbols for numbers higher than 9 and began to use their symbols for 1 through 9 in our familiar place value arrangement. The earliest (662) reference to this use comes from Severus Sebokht, a Syrian priest, with the remark that the Hindus have a valuable Method of calculation done by means of nine signs.A symbol for zero: Severus did not mention a sign for zero. However, in the Bakhshali manuscript the numbers are written using the place value system and with a dot to represent zero. In other Indian works from the same period, numbers were written as words (moon for 1, eye for 2 etc.) to accommodate the poetic nature of the documents. The question why the Indians at the beginning of the seventh century dropped their earlier system and introduced the place value system including a symbol for zero remains. It has been suggested that the true origins of the system in India come from the Chinese counting board.

Chapter 9: The Mathematics of Islam

The Islam The civilization originated out of Arabia, and had their beginning in the first half of the seventh century. In less than a century after Muhammads capture of Mecca in 630, the Islamic armies propagated their religion in a large part of Africa, Asia, and parts of Spain, but got halted their by Charles Martel. In 766 the caliph al-Manr founded his new capital of Baghdad, a city that soon became a flourishing commercial and intellectual center The caliph Hrn al-Rashd, who ruled from 786 to 809, established a library in Baghdad. Many greek texts on math and science were translated there to Arabic and the successor of Hrn, the caliph al-Mamn, established a research institute, the Bayt al-Hikma (House of Wisdom), which was to last over 200 years. the Islamic scholars there absorbed the mathematical knowledge translated to Arabic from the Greek, the Hindus, and the Babylonian scribes. Islamic culture in general regarded secular knowledge not as in conflict with holy knowledge, but as a way to it and thus encouraged learning. In the eleventh century the status of mathematical thought in Islam was beginning to change, to many Islamic religious leaders, the foreign sciences were potentially subversive to the faith and certainly superfluous to the needs of life, either here or hereafter. Thus, although there were significant mathematical achievements in Islam through the fifteenth century, gradually science became less important. A complete history of mathematics of medieval Islam cannot yet be written, since so many of these Arabic manuscripts lie unstudied and even unread in libraries throughout the world.

Decimal Arithmetic The decimal place value system had spread from India at least as far as Syria by the mid seventh century. There were two systems in use. The merchants in the marketplace generally used a form of finger reckoning. In this system, calculations were generally carried out mentally. Numbers were expressed in words, and fractions were expressed in the Babylonian scale of sixty. When numbers had to be written, a ciphered system was used in which the letters of the Arabic alphabet denoted numbers.

Algebra They took the material already developed by the Babylonians, combined it with the classical Greek heritage of geometry, and produced a new algebra. One of the earliest Islamic algebra texts, written about 825 by al-Khwrizm, was entitled Alkitb al-mutaar f isb al-jabr wa-l-muqbala (The Condensed Book on the Calculation of al-Jabr and al-Muqabala) with al-Jabr meaning restoring and al-muqabala meaning comparing. al-Khwrizm was interested in writing a practical manual, not a theoretical one.

Transmission of Islamic Mathematics By the fifteenth century, Islamic scientific civilization was in a state of decline.

Chapter 10: Mathematics of Medieval Europe

5th 10th centuryThe general level of culture in Europe was very low.The practical need of mathematics was little (because the feudal estates were relatively self-sufficient and trade was almost non-existent). But the study of the so-called quadrivium (arithmetic, geometry, music and astronomy) was required for an educated man (even in the Roman Catholic culture).St Augustine (354-430) considered this study important, because he believed that in this way we can get to know God.The availability of the texts for the study of these subjects was limited. Virtually the only schools in existence were connected with the monasteries.Monks copied Greek and Latin manuscripts and thus preserved much ancient learning.New intellectual developments sprung forth in the 6th to the 8th centuries due to missionaries. Charlemagne, a future Holy Roman Emperor, wanted mathematics to be part of the curriculum in Church schools, mainly because the problem of the calendar.Alcuin of York (735-804) helped Charlemagne to establish more schools.

10th centuryIn the 10th century, a revival of interest in mathematics began with the work of Gerbert d Aurillac (945-1003), who became Pope in 999.Gerbert reorganized the cathedral school and successfully reintroduced the study of mathematics. He dealt with basic arithmetic, geometry, mensuration and astronomy. 12th centuryThe mathematical heritage was only brought into Europa through the work of translators.Europian scholars (most of them in Toledo) began the translate the (mostly) Arabic translations into Latin.Because of the flourishing Jewish community, more and more translations went as follows: first by Spanish Jew from Arabic into Spanish, and then by Christian scholar from Spanish into Latin. Euclids Elements was translated into Latin early in the twelfth century. Before then, Abraham bar Hiyya (d. 1136) wrote a treatise on Mensuration and Calculation, with a summary of some important definitions, axioms and theorems from Euclid. An overview of translations can be found in the book on page 327, sidebar 10.1 Leonardo (also known as Fibonacci) of Pisa (c. 1170-1240)In his early life he spent much time on Arabic and mathematics under Moslem teachers. Later he travelled and absorbed the mathematical knowledge of the Islamic world. When he returned to Pisa he wrote his knowledge down and books which are preserved include: Liber Abbaci (1202, 1228), the Practica geometriae (1220), and the Liber quadratorum (1225). Levi ben Gerson (1288-1344)He was a mathematician, astronomer, philosopher, and biblical commentator. His best-known contribution to astronomy is his invention of the Jacob Staff, which was used for centuries to measure the angular separation between heavenly bodies.The works of Levi ben Gerson were read, although there are no references of his work. Marin Mersenne wrote about combinatorics, a subject which Levi ben Gerson had written about. UniversitiesDuring the late twelfth century Europe saw the beginning of the universities. There is not an exact date to assign the beginning of these institutions because they were formed as societies, or guilds.The earliest of these institutions were in Paris, Oxford and Bologna. In Paris, the university grew out of the cathedral school of Notre Dame.

--- More on universities can be found in the lecture notes --- In the medieval period much of the till then available works were not studied and their new ideas had to be rediscovered centuries later.The Hundred Years War caused a marked decline in learning in France and England. Only a few new ideas were therefore in Italy and Germany generated in the Renaissance.

Chapter 11: Mathematics Around the World

Mathematics around the world in medieval timesMost of the other civilizations were nonliterate.The Mayans, however, did have written language.Spaniards arrived in the early 16th century but they did not succeed in destroying the Mayan culture completely.

Part Three Early Modern MathematicsChapter 12: Algebra in the Renaissance

The Renaissance- 14th to 17th century. Bridge between the Middle Ages and Modern history.

Importation in Italy in the beginning of the Renaissance increased the need for mathematics.

"Professional methematicians, the maestri d' abbaco (abbacists) appeared in the early 14th centruy, didn't preform the mathematics of the quadrivium, just mathetmatics needed for calculation and problem solving. Wrote texts from which they taught the necessary mathematics tothe skns of the merchants in new schools created for this purpose.

Algebraists in the Renaissance based their work on Islamic algebras first translated into Latin in the twelfth century.

By the middle of the sixteenth century, virtually all of the survivig works of Greek mathematics, newly translated into Latin from theGreek manuscripts that had been stored in Constantinople, were available to European mathematicians.

The Italian abacists of the 14th century were instrumental in teaching the merchants the "new" Hindu-Arabic decimal place value system and the algorithms for using it. There was great resistance to this change. For many years, account books were still kept in Roman numerals. It was believed that the Hindu-Arabic numerals could be altered too easily, and thus was too risky to depend of for large transactions.

Early in the fifteenth century abacists begun to substitute abbreviations for unknowns- cosa-thing x- censo-square x^2- cubo-cube x^3- radice-root x^1/2

Some authers used the abbreviations c, ce, cu and R. Combinations of the abbreviations were used for higher powers.- ce ce - x^4- ce cu - x^5Etc

Near the end of the fifteenth centruy, Luca Pacioli introduced the abbreviations p (with bar over it) and m(with bar over it)to represent plus and meno.

Modern algebraic symbolism was not fully formed until the mid-seventeenth century.

Michael Stifel (1487-1567): made a wrong prediction of the end of the world and was discharged from his parish and for a time placed under house arrest. Was given another parish in 1535 and devoted himself to the study of mathematics. Wrote "Deutsche Arithmetica" (1545) and "Arithmetica Integra" (1546)

Robert Recorde (1510-1558): studied in Oxford. Wrote textbooks in the form of a dialogue between master and pupil.

Pedro Nunes (1502-1578): studied in Salamanca. Was of Jewish origin but not persecuted in the Inquisition. Translated his "Libro de Algebra" into Spanish and had it printed in the Netherlands in 1567. Also a poet. Was not persecuted because (probably) one of his students became the Inquisitor General.

Gerolamo Cardano (1501-1576): lectured in mathematics in Milan and wrote a textbook on arithmetic. was finally admitted into Milan's College of Physicians (wasn't before due to his illegitemate birth). Helped the archbishop by discovering his allergy to feathers. Predicted a long life for Edward VI but was wrong. Wrote an autobiography "De Propria Vida"

Rafael Bombelli (1526-1572): educated as an engineer. Largest project was reclaimong arable land from marshes, wrote an algebraic treatise (a written work dealing formally and systematically with a subject)

Sixteenth century in Italy: revival of Greek mathematics. The translators were not expert mathematicians so some of their translations were unintelligeble (translated a few centuries earlier).

Federigo Commandino (1509-1574): single handedly prepared Latin translations of birtually all of the know works of Archimedes, Apollonius etc.

European mathematicians began to serach for the "methods of analysis" used by the ancient Greeks.

Franois Vite (1540-1603): received a law degree from the University of Poitiers. Acted as a Cryptanalyst of intercepted messages between King Henri III's enemies. Was denounced by some who thought the decipherment could only have been made by sorcery. Wrote "The Analytic Art" replaced the search for solutions (of algebraic equations) with a detailed study of the structure of these equations.

Simon Stevin (1548-1620): major mathematical contribution was the creations of a notation for decimal fractions. Also played a fundamental role in erasing the Aristotelian distinction between number andmagnitude. Wrote "the Art of Tenths" and "l'Arithmtique" (a work containing arithmetic and algebra)

Decimal fractions: not used in Europe in the late Middle Ages or in the Renaissance. Steving was probably influenced by Islamic development when he made his notation for decimal fractions. Stevin in his book promosed to show that all operations using his new system could be preformed exactly as if one were using whole numbers.

Notation of decimal fractions: e.g. 8(0)9(1)3(2)7(3)=8.937 (1): prime (2): second (3):third (0): commencment.

Stevin begun "l'Arithmetique" with two definitions:1: Arithmetic is the science of numbers2. Number is that which explains the quantith of each thingStevin made the point that number represents quantity, any type of quantity at all. Numbet is no longer to be only a collections of units, as defined by Euclid. Unity is a number. The Greeks had rejected this notion. To them unity was only a generator of a number. He did distinguish between numbers that are commensurable and incommensurable but all these quantities are numbers to him.

Chapter 13: Mathematical Methods in the Renaissance

Geometry was a central aspect of mathematics in the Renaissance.

Vernacular versions of the "Elements" began appearing in the 16th century.

John Dee (1527-1608) wrote the "Mathematical Preface" to Billengsley's English translations of "Elements". Studied at Cambridge University. Was a mystic. He studied and wrote about how various symbols could be combined in certain figures, the proper understanding of which would enable the reader to understand the hidden secrets of the phsycial world. Was accused of practising black magicwhich caused him to lose his royal patronage (he was a court astrologer to Queen Elizabeth). He died in poverty.

Painters in the fifteenth century: begun attempting to derive a mathematical basis for displaying 3D objects on a 2D surface. Answers came from geometry (of perspective). Durer taught perspective.

In the 15th and 156th century, Europeans were exploring the rest of the world so methods of navigation were of central importance.

Johannes Muller (1436-1476) a.k.a. Regiomontanus wrote the first puretrigonometry text in Europe n Triangles of Every Kind. Preface to learning astronomy. Made a new translations of Ptolemys Algamest directly from Greek.

Many faults were found in Ptolemys work: predictions of lunar eclipses were greatly in error, Ptolemys geographywas in error, etc. the way was prepared for believeing that the fundamentals of his astronomy could be wrong.

Greek philosophers had proposed a sun-centered (heliocentric) system in which the Earth moves.

Nicolaus Copernicus (1473-1543): wrote a treatise called On the Revolutions of the Heavenly Spheres. This book sets forth the first mathematical description of the motions of the heavens based on the assumption that the earth moves. Conceived of the system of the universe as a series of nested spheres containing the planets.

Copernicus ideas received strong oppositions from the Protestants.

Tycho Brahe (1546-1601): was the first astronomer to realize the necessity for making continuous observations of the various planets. Devised a model of the universe intermediate between that of Ptolemy and Copernicus in which all of the planets except the earth traveled around the sun while the whole system revolved around the central inmovable earth.

Johannes Kepler (1571-1630) German: used Brahes observations to construct a new heliocentric theory that could accurately predict heavenly events without the elaborate machinery of epicycles. Believed Copernicus theory in essence represented the correct system of the world.

Keplers first law of planetary motion: a planet travels in an ellipse around the sun with the sun at one focus.

Napier is primarily responsible for the introduction of our modern notation for decimal fractions.

Galileo Galilei (1564-1642): considered to be the founder of modern physics. Was responsible in large measure for reformulating the laws of motion considered first by the Greeks and later by certain medieval scholars. His most important work dealing with the naturalaccelerated motion of freefall and the violentmotion of a projectile were published in 1638 in his Discourses and Mathematical Demonstrations Concerning Two New Sciencesf. Was brought before the Inquisition in Rome for believing the earth moves. He was forced to confess his error. He was then sentenced to house imprisonment and forbidden to publish any more books. He did, however , manage to publish Discourse...Two New Sciences. Presented a postulate to the effect that the velocity acquired by an object sliding down an inclined plane (without friction) depends only on the height of the plane and not the angle of inclination. Concluded that the path of swiftest descent is a circular arc. This was erroneous. It was in fact a cycloid. Mathematical modelling was Galileos most fundamental contribution to the mutual development of mathematics and physics.

Chapter 14: Algebra, Geometry, and Probability in the Seventeenth Century

Mathematics in the 17th centuryPrinting well-established and good communication let to acceleration of development of math.

Algebra Vites ideas on algebra let the basis for algebra in the 17th century Notation of math in symbols was developed by a.o. William Oughtred, Thomas Harriot and Albert Girard William Oughtred (1575-1660): Cleric, lived in England, most important work: Clavis mathematicae (Key of Mathematics) Thomas Harriot (1560-1621):Lived in England, most important work: Treatise on equations, transformed Vites algebra in modern form, did not publish it himself

Analytic geometry Started in 1637 by both Ren Descartes and Pierre de Fermat Fermats Introduction to Plane and Solid Loci and Descartes The Geometry relate algebra and geometry They made use of the Greek classics, in particular Domain of Analysis of Pappus An important step was using coordinates to study the relationship between geometry and algebra Fermat and Descartes developed different approaches to the subject, because of their different view on mathematics One of those differences was that Descartes started with a curve and derived its algebraic equation and Fermat started with an algebraic equation and derived its curve Pierre de Fermat (16011665):Born in wealthy family in France, studied at University of Toulouse (undergraduate education) and at Orlans (civil law). Was jurist for many years, mathematics was his hobby. He never published his ideas so its often unclear what proofs he used. Ren Descartes (15961650):Born in France in a noble family. Meditated a lot and doubted a lot of things he learned. He traveled a lot in his youth to get life experience. His goal in life was to create a philosophy to discover truth about the world. He wrote a major treatise on physics: Discourse on Method. His mathematical works were hard to read, were written in French instead of Latin and had gaps in the arguments. After Van Schooten translated Descartes work and wrote commentaries on it, it was fully understood. Jan de Witt (16231672):Born in Holland, was appointed to grand pensionary of Holland (prime minister)Studied with Van Schooten the works of Descartes and Fermat. Wrote elementacurvarum linearum (Elements of Curves) which was about conic sections. Blaise Pascal (16231662):Born in France, soon introduced to Mersenne, who was the head of a group prominent mathematicians. He invented a calculating machine and investigated the action of fluids under the pressure of air. He was much interested in religious matters.

Chapter 15: The Beginnings of Calculus

Before the invention of the calculus, a lot of problems relating to maxima and minima, areas, tangents and volumes where already tacklet. However, they were all special cases and required ingenious constructions. The advent of analytic geometry played a role in the invention of calculus: because of it, mathematics had a new way of describing curves and solids: though equations (not yet through functions!).

Several people made important stepts towards the discovery of calculus: De Roberval (1602-1675) could determine tangents through geometrical constructions, but he had no algebraic algorithm. Fermat as well as Descartes had, but they were by no means simple. Fermat came close to inventing calculus, but he did not understand the inverse relationship between the intergral and the derivative. In fact, he was only interested in finding the tangent line, not its slope (the derivative). Hudde (1628-1704) & De Sluse (1622-1685) discovered simpler algorithms in the 1650s. The idea to break a region up into very small pieces, whose individual area or volume is known, came from the Greek. Kepler used this procedure to discover the laws of planetary motion. Bonaventura Cavalieri (1598-1647) was the first to develop a full theory of indivisibles. Isaac Barrow (see below) and James Gregory were among the mathematicians who could relate tangents to areas. However, they used a geometrical style in their books and talks (which they were taught in university). Their approaches did not result in computational methods.

Isaac Barrow (1630-1677) Got kicked out of university for his royalist sympathies. Toured Europe afterwards, studying mathematics. He probably died due to an overdose of drugs.

Evangelista Torricelli (1608-1647) Studied at Galileo's house. Continued Galileo's work on motion and grinding lenses for more powerful telescopes. Discovered the principle of the barometer.

Descartes wrote that the human mind could not determine the lengths of curved lines. The bitch was proved wrong by several mathematicians (Neile, Wren, Huygens), most notably by Hendrick van Hauraet, who discovered a general procedure. He was born in 1634. He studied mathematics in Leiden. The death of his father (who had been a cloth-merchant) made him rich, which allowed him to travel and study without worry, but he died young (before his 30th birthday).

Chapter 16: Newton and Leibniz

I recommend to read the biographies of Newton and Leibniz (the blue boxes, pages 545 and 566), because its really hard to determine whats important, so it is hard to summarize but it doesnt take much time to just read it.Isaac Newton and Gottfried Leibniz were contemporaries in the last half of the 17th century.Newton developed the concept of fluxion and fluent, Leibniz of differential and integral. Both were related to the two basic problems of calculus: extrema and area. Newton (1643-1727) succeeded, over the course of a brief few years in the 1660s, in consolidating and generalizing all the material on tangents and areas developed by his 17th-century predecessors into the magnicent problem solving tool exhibited in the thousand-page calculus textbooks of our own day. Newton wrote papers on calculus, but never published them (they were published years later). Newton took breaks from mathematics to work on astronomy, physics etc. He believed that power series were central to expanding the field of analysis. Newton developed the calculus well before the physics, so using it for physics was not his sole purpose. In his physical arguments based on geometry, he usually followed 3 steps: 1. establish a result for finite regions, 2. assume the result will remain true in infinitesimal regions of the same type, 3. use the infinitesimal result to conclude something about the original figure. Occasionally in the Principia and elsewhere, Newton showed that he could translate his geometry into analysis. Although Newton did not invent the calculus to do celestial mechanics, he did use the ideas and results of his theory of uxions for his most important physical work. The ideas he generated in his rooms at Cambridge and at home in Woolsthorpe in the mid-1660s proved critical when he began to work out his system of the world in the mid-1680s (and wrote it down in the Principia). Contrary to what Newton wants to make us believe, his ideas of the Principia were not developed in the 1660s. He started thinking about the problem of gravity then, but was only able to put it together in the 1680s, publishing the Principia in 1687. His peculiar gift was the power of holding in his mind a purely mental problem until he had seen straight through it (He could concentrate really well for a really long time). Newtons influence:The Principia, arguably the most important text of the Scientic Revolution, was the work that dened the study of physics for the next 200 years. But Newtons calculus had relatively little inuence because only parts appeared in print many years after they were written. In fact, it was Gottfried Wilhelm Leibniz work (8 to 10 years after Newton) that constituted the basis of the rst publication of the ideas of the calculus. Leibniz (1646-1716) Was brought to the frontiers of mathematical research by Christiaan Huygens during his stay in Paris from 1672 until 1676. After he read material such as van Schootens edition of Descartes Geometry and the works of Pascal, he was able to begin the investigations that led to his own invention of the differential and integral calculus (around 1676). Around 1686 he began to publish his results in short notes in a German scientific journal that he helped to found. Like Newton in the 1670s, Leibniz wanted to justify his work by appealing to Greek standards. Leibniz technique of manipulating with infinitesimal differentials became a very useful one, especially for his immediate followers, Johann and Jakob Bernoulli. Newton and Leibniz (plagiarism): They discovered essentially the same rules and procedures that we call calculus today, but their approaches were entirely different: Newton used the ideas of velocity and distance, while Leibniz used those of differences and sums. Leibniz was accused of plagiarism by English mathematics for the following reasons:o he had read some of Newtons material during his stays in Londono he had received two letters from Newton where he discussed some results Newton was accused of plagiarism by Johann and Jakob Bernoulli because:o his work was not published until the 18th century Leibniz was found guilty by a commission of the Royal Society, where Newton was president, which caused the communication between England and the Continent to cease. Leibniz method and notation were easier, so progress in analysis was faster on the Continent. England stayed behind throughout the whole 18th century. Differences between English and Continental approaches were clear in first calculus textso Marquis de lHospital in Franceo Charles Hayes and Humphry Ditton in EnglandlHospital (1661-1704): Served in his youth as an army officer. In about 1690, he became interested in the new calculus that was beginning to appear in journal articles of Leibniz and the Bernoulli brothers. lHospital asked Johann Bernoulli to lecture him on the subjects. Bernoulli sent him material (and new discoveries) and promised not to let anyone else see them. lHospital published a text with mostly Johanns discoveries in it in 1699 He used the same rules and proves as Leibniz and the Bernoullis, but he dealt with algebraic curves instead of transcendental curves.

Part Four Modern MathematicsChapter 17: Analysis in the Eighteenth Century

Analysis in the 18th century

The major figure in the development of analysis in the eighteenth century was the most prolific mathematician in history, Leonhard Euler.

Jacob Bernoulli (1654-1705) Swiss: taught himself mathematics. Travelled to France, the Netherlands and England. Professor of physics then of mathematics at the University of Basel.

Johann Bernoulli (1667-1748) Swiss: unsuccessful businessman, then studied medicine. Studied work of Leibniz with his brother, mastered it and made their own contributions. Professor at RUG until his brother's death, the professor at the University of Basel.

Leonhard Euler (1707-1783) Swiss: graduated from the University of Basel at 15. Convinced Johann Bernoulli to tutor him privately. On his urging, Peter the Great of Russia created the St. Petersburg Academy of Sciences. Had 13 children. Became almost blind in 1771. Performed calculations in his head and dictated his articles to his children. Died suddenly when playing with one of his grandchildren.

Alexis Clairaut (1713-1765): genius. Mastered L'Hopital's Analyse by 10. Researched curves by 13 and elected to the Paris Academy of Sciences at age 18. Studied celestial mechanics and pedagogy.

Jean d'Alembert (1717-1783): abandoned as an infant on the steps of a Perisian church. Was a lawyer. Studied math on his own. Published several papers in the area of differential equations. Admitted to the Paris Academy in 1741. Wrote treatises on dynamics and fluid mechanics.

Continent: calculus of differentials

Britain: calculus of fluxions

In the middle third of the 18th century many calculus texts were written including texts for the layman and Latin texts for universities.

Thomas Simpson (1710-1771) English: had a rift with his father because he wanted him to be a weaver but he wanted a better education. Taught himself mathematics. Professor of Mathematics at the Royal Military Academy at Woolwich.

Colin Maclaurin (1698-1746) Scottish: went to the University of Glasgow at 11 years. At 19 years he was appointed to a chair of mathematics at the University of Arberdeen. Took a three year tour of Europe as tutor to the son of a wealthy lord. Taught after at Glasgow by Newton's recommendation. Helped fortify Edinburgh in 1745 against the forces of Bonnie Prince Charlie but city fell. Died at 48.

Euler's "Integral Calculus": pure analysis, does not deal with application to geometry of physics. No mention of vibrating string problem which led Euler to "invent" the trigonometric functions in the 1730s.

Eulers "Differential Calculus": no tangent lines or normal lines, not tangent planes, no study of curvature.

After the French Revolution (1789-1799) there was a great need for educating a new class of students who were entering the sciences. This need inspired the writing of many new texts in the vernacular.

Joseph-Louis Lagrange (1736-1813) French: Born in Turin, Italy. At age of 19 became a professor of mathematics at the Royal Artillery School in Turin. Read Euler's book on the calculus of variations. Wrote to Euler with an explanation of a better method he had found for deriving the central equation of the subject.

Chapter 18: Probability and Statistics in the Eighteenth Century

Mathematics and Statistics in the Eighteenth CenturyEuler developed the mathematics of lotteries. He made, for example, a detailed study of lotteries which he presented at the Berlin Academy of Sciences.

Theoretical Probability Probability in its beginnings was closely related to the notion of an aleatory contract: a contract providing for the exchange of a present certain value for a future uncertain one. Included policies in which a certain sum of money was paid now in exchange for an unknown sum to be returned at a later date under certain conditions. The risk involved had to be quantified in order for the contract to be fair. The early practitioners (of certain types of games) were able to work out efficient ways of counting successes and failures and thus to determine the expectation or probability a priori. It was, in most realistic situations, much more difficult to quantify risk, that is, to determine the degree of belief that a reasonable man would have. Jakob Bernoulli wanted to be able to quantify risk in situations where it was impossible to enumerate all possibilities. He used some statistics to do this.

Jakob Bernoulli and the Ars ConjectandiJakob Bernoulli gave a scientific proof of the principle that the more observations one made of a given situation, the better one would be able to predict future occurrences. This proof was presented in Bernoullis Law of Large Numbers. This proof was placed in Bernoullis important text on probability, the Ars Conjectandi (Arts of Conjecturing), published 8 years after his death in 1705.

Abraham De Moivre (1667-1754) Born into a Protestant family in France. Studied physics as well as the the standard mathematics curriculum beginning with Euclid. Moved to England in 1688, after being in prison for more than two years. There, he mastered Newtons theory of fluxions and began his own original work. Was elected to the Royal Society in 1697, but never achieved a university position. Made his living by tutoring and by solving problems arising from games of chance and annuities for gamblers and speculators.

Pierre-Simon de Laplace (1749-1827) Born in Normandy. Entered the university of Caen in 1766 to begin preparation for a career in the Church. Went to Paris in 1768 to continue his studies. Got a position in mathematics at the cole Militaire, where he taught elementary mathematics to aspiring cadets. Legend has it that he examined, and passed, Napoleon there in 1785. Won election to the Academy of Sciences in 1773. His most important accomplishments were in the field of celestial mechanics. During the period from 1799 to 1825, he produced his five-volume Trait de mcanique cleste (Treatise on Celestial Mechanics) about Newtons law of gravitation among other things. He produced his Thorie analytique des probabilits (Analytic Theory of Probability) in 1812. Was rewarded with the title of marquis. At his death he was eulogized as the Newton of France.

Chapter 19: Algebra and Number Theory in the Eighteenth Century

Algebra and Number Theory in the Eighteenth Century That Euler in 1742 believed the truth of the fundamental theorem of algebra was confirmed by two letters he wrote to Nicolaus Bernoulli Christian Goldbach Algebra meant the solving of equations Few major new developments in algebra took place Especially systematization of earlier work has been done: more general procedures sought by mathematicians such as Newton MacLaurin EulerNewton Decided in 1683 to comply with the rules of the Lucasian professorship and wrote up the lectures A successor of Newton (Willian Whiston) prepared the lectures for publication (Arthmetica Universalis) in 1707 Arthemtica Universalis became very popular despite of the fact that Newton made no attempt to justify any of his statements about the arithmetic algorithms (for example the multiplication rule) his listeners probably just only needed techniques for manipulation Versions of the word problems still appear in algebra texts today Example: If a scribe can copy out 15 sheets in 8 days, how many scribes of the same output are needed to copy 405 sheets in 9 days? Newton solved also much more difficult problems, but have never put it into a truly polished form. Because he was no longer heavily involved in math in 1707.Maclaurin For Maclaurin algebra is not abstract, but simply generalized arithmetic In his Treatise of Algebra: algorithms for calculation attempts to explain the reasoning behind the algorithms demonstration how to calculate with positive and negative quantities a negative quantity is no less real than a positive one

Chapter 20: Geometry in the Eighteenth Century

Gaspard Monge (1746-1818) Founded the field of descriptive geometry while working on military fortifications.

The French Revelution & Mathematics Education In the 18th century, mathematicans were mainly associated with academies. These academies were founded by monarchs (for prestige and so they could get advice in military matters).

The revolutionary government of France introduced the decimal system for lengths (the metric system), areas (100 grads = 360 degrees) and the calendar (a month consisted of three ten day periods). Most military schools and universities were closed during the revolution, as they were centers of Royalist suport. The Parisian cole Polytechnique was founded in 1794 as a substitute, for the advancement of science. At first, the school failed, because the students were not prepared for Monge's difficult curriculum, because he fell ill and because of food shortages and the severe winter. Improvements were made, prospective students were tested before they were admitted and France's best mathematicians came to teach (e.g. Lagrange & Laplace). The setup of the school became the standard for other colleges. Napoleon took control of France in 1799. He restored the Gregorian calendar in 1806.

Chapter 21: Algebra and Number Theory in the Nineteenth Centruy

Algebra in 1800 meant the solving of equations. By 1900 this changed to the study of elements with well-defined operations which satisfy specified axioms. 1801: Prince of mathematics Carl Friedrich Gauss wrote the book Disquisitiones Arthemticea which is about the basics of number theory. Studying further lead to Gaussian integers (complex numbers of form a+bi).

1827: Influenced by Gauss study of solutions of cyclotomic equations and study of permutations by Augustin-Louis Cauchy. Niels Hendrik found the solutions of general equations with and order higher than degree 5.

1843: William Rowan Hamilton discovered the quaternions partly in attempt to determine a physically meaningful algebra in 3D space. Quaternions were 4D however so only part of the quaternions could be used.

1846: Ernst Kummer tried to generalize the properties of the Gaussian integers and came to the realizations that some of the most important properties fail to hold. So Kummer created ideal complex numbers by 1846. (this lead to Richard Dedekind in 1870s to define ideals which have to property of unique factorization into primes)

1850: James Sylvester introduced the term matrix. Arthur Carley developed the algebra of matrices Cauchy began with his study of eigenvalues.

Carl Friedrich Gauss (1777-1855): Gauss lived and went to school in Brunswick. (were a teacher gave the task to 100 pupils to sum the first 100 integers, Gauss immediately wrote 5050). Apart from his book in number theory at the same time he developed a new method for calculating orbits which lead to several asteroids to be discovered. In 1806 Brunswick was occupied by the French. The French general had been given orders to look out for his welfare. (Sophie Germain insured this). Sophie Germain (1776-1831): She mastered mathematics through calculus on her own and studied lecture notes (She was not allowed to attend) from several math classes. Had a corresponded with Gauss under pseudonym M. le Blanc and suggested to the French general which occupied Brunswick to insure Gauss safety. Ernst Kummer (1810-1893): Born in Sorau (the Germany now Poland) he first studied theology but soon switched to mathematics. Along with Karl Weirerstrass he established Germanys first ongoing seminar in pure mathematics. Which attracted a lot of attention and helped to make Berlin one of the most important world centres of mathematics in late 19th and early 20th century. Niels Hendrik Abel (1802-1829): Born near Stavanger in Norway. He became interested in fifth-degree equations and believed that he was able to solve it using radicals. When he was asked to provide numerical examples however he realized that his method was incorrect. He continued on the solvability question until he managed to prove its impossibility.

William Rowan Hamilton (1805-1865): Born in Dublin but educated by his uncle in the town of Trim. By the time he was ten he spoke fluently in: Latin, Greek, modern European languages, Hebrew, Persian, Arabic and Samskrit. His first original work was in optics. In fact today he is more famous for his work in dynamics than mathematics.

Chapter 22: Analysis in the Nineteenth Century

Toward the end of the 18th century (the French revolutions restricted the mathematics) there was an increasing necessity for mathematics to teach rather than research.Cauchy developed calculus on the basis of limits. Although it was discovered earlier by Newton he was the first to translate it in Arithmetic terms). He used limits to define continuity and convergence of sequences. (published in 1821)

Cauchy notion of convergence was also in essence developed by both Bernhard Bolzano and Jose Anastacio however both were not appreciated or read in France and Germany because works of these latter two appeared in the far corners of Europe.

One of Cauchys important results, that the sum of an infinite series of continuous functionsis continuous, assuming this sum exists, turned out be false. Although counterexamples were discovered as early as 1826 they were first in detail by Joseph Fourier in his work onheat conduction. (these series are now known as the Fourier series).Fouriers works stimulated Peter Lejeune-Dirichlet to study in more detail the notion of a function and Bernhard Riemann to develop the concept today known as the Riemann integral. Some unresolved questions in the work of Cauchy and Bolzano led several mathematiciansin the second half of the century to consider the structure of the real number system Richard Dedekind and Georg Cantor began the detailed study of infinite sets

Augustin-Louis Cauchy (1789-1857): He was born in the capital in the year that the French Revolution began. He received an excellent classical education and studied engineering. He worked as an engineer until he showed a strong interested in pure mathematics and was encouraged by Laplace and Lagrange to leave engineering. He became one of the most respected members of the French mathematical community. And he wrote so many papers that the journal of the Paris Academy was forced to limit the contributions of any one person. Cauchy got around that by making his own journal. When the last Bourbon king was overthrown in 1830 he refused to take the oath of allegiance to the new king and went into self-imposed exile in Italy and Prague. He only returned when taking to oath was no longer necessary. Jean Baptiste Joseph Fourier (17681830): Orphaned at the age of nine he was placed in the local military school where he showed talent in mathematics. He was arrested for defending victims of the Terror in 1794. And after his release he was appointed as an assistant to Lagrange and Monge.

Karl Weierstrass (1815-1897): Born in Westphalia, Germany. He briefly studied public finance and administration however he left without getting a degree because of his interest in mathematics and going to taverns. To earn a living he got a teaching certificate and began to teach mathematics, physics, German botany, geography, history, gymnastics and calligraphy at various gymnasia for 14 years. He was awarded a doctorate after writing a series of brilliant papers in Crelles journal. Because of his declining health he taught while seated with an advanced student writing on the blackboard. His clear lectures won him a European-wide reputations Sofia Kovalevskaya (18501891): Sofias room was papered with lecture notes of calculus. She grew to like mathematics but could not pursue her studies in Russia since women were not yet allowed to attend University. Because her family would not allow her to study on her own at a European university she solved the problem with a marriage of convenience with Vladimir Kovalevsky, a publisher of scientific and political works. Sofia studied privately with Weierstrass and, after writing several publishable mathematics papers, the most significant being on the theory of partial differential equations, received her doctorate in 1874 from the University of Gottingen, a university that was willing to grant doctorates in absentia. When her husband died in 1833, Sofia secured a position as a professor at the University of Stockholm, a first for a woman. She had it difficult as a single mother, as she wrote in a letter Were I a man, Id choose myself a beautiful little housewife who would free me from all this.

Chapter 23: Probability and Statistics in the Nineteenth Century

The beginning of the application of statistical methods in various field: agriculture and social sciences. Which lead to the development of standard statistical techniques in the 19th and 20th century. De Moivre developed the normal curve y= Ae^(-kx^2) for the errors of measurements.

Adolphe Quetelet (1796-1874): a Belgian mathematician, astronomer, meteorologist, sociologist and statistician. He used to normal curve to develop the Average man. He did this by compiling vast numbers of statistics covering not only physical characteristics but also moral characteristics. He noticed that many characteristics could be plotted in terms of a normal curve.

Francis Galton (1822-1911) used the idea of normal distribution in biology and tried to mathematize Charles Darwin theory of evolution. He was curious why the same normal curve persisted generation after generation. In 1875 he conducted an experiment of a type of sweet pea. He studied the offspring of pea seed with seven sizes. And it turned out that the sizes in each set were normally distributed.

Chapter 24: Geometry in the Nineteenth Century

In the nineteenth century, various new kinds of geometries appeared.

The importance of pure geometry was lessened by the growing importance of analysis. However, applications of analysis led to new geometrical ideas.

In 1827, Gauss (1777-1855) wrote a paper, General Investigations of Curved Surfaces, in which he carried forward the work of Euler on surfaces. He applied many basic notions of surface theory, like the notion of curvature. He also came up with ideas on new (non-Euclidean) geometries.

It took nearly 40 years for ideas of non-Euclidean geometry to make an impression. It was with the work of Riemann and Hermann von Helmholtz that the meaning of these ideas took hold.

In 1854, Riemann held a lecture on geometry. It contained few mathematical details He stated that the general geometrical notion was a manifold. The usual space of Euclidean geometry is a special case of a three-dimensional manifold, with the metric ds2 = dx2+dy2+dz2 Agreeing with Gauss, he stated that the precise nature of physical space could only be determined by experience.His lecture was divided into three partsPart 1Dealt with n-dimensional manifolds, constructed inductively, beginning with one dimension.Part 2Dealt with the idea of a metric relation on manifolds: a way of determining lenghts. Transforming from a given metric to another is in general not possible. Riemann called the class of manifolds with the Euclidean metric "flat".Part 3He gave a description of how his ideas related to our usual concept of three-dimensional Euclidean space.

Gauss was quite impressed with Riemann's work, but it was only published in 1868, after Riemann's death. It was met with widespread acclaim. William Clifford and Hermann von Helmholtz extended Riemann's work, which helped bringing it to the attention of the wider community.

Hermann Grassmann (1809-1877) most of his life in Poland. At the University of Berlin he mostly studied philology and theology, and after leaving the university he returned to Poland to pursue work in mathematics and physics

In 1868, Helmholtz came up with a set of hypotheses that provided a basis for a study of geometry. - n-dimensional space is a manifold - Rigid bodies exist - Rigid bodies can move freely Taking n=3 in these hypotheses leads to his concept of physical space. He had some ideas regarding curvature that gives three possibilities for physical space. Measure of curvature positive --> spherical space Measure of curvature negative --> pseudospherical space Measure of curvature zero --> Euclidean spaceIn the early 1870s, William Clifford also attempted to determine the postulates of physical space. His speculations on this subject made Riemann's ideas on the theory of manifolds applicable in physics. There were also advances in projective geometry, by Jean-Victor Poncelet, Michel Chasles and Julius Plcker.Graph theory received new emphasis with the posing of the four-color problem.Geometry now also dealt with dimension greater than three. A work by Hermann Grassmann in 1844 on this was not appreciated until the end of the century.Mathematicians felt that it was time to redo the foundations of geometry. David Hilbert (1862-1943) came up with a new set of axioms for Euclidean space that helped developing the new geometries. David Hilbert (18621943) was one of the last of the universal mathematicians, who contributed greatly to many areas of mathematics. He attended the university in Knigsberg, Russia. After he was called by Felix Klein to Gottingen, he soon became one of the major reasons for that universitys surpassing Berlin as the preeminent university for mathematics in Germany, and probably the world.

Chapter 25: Aspects of the Twentieth Century and Beyond

Mathematical output of the 20th century far exceeds that of all previous centuries put together Most of the mathematics taught to undergraduate students dates from 19th century or earlier 20th century Problems in the foundations of mathematics at the beginning of the 20th century:o Cantors work on infinite sets continued to cause problems early in the 20th century. The key to solving these problems was the axiom of choice (used implicitly for many years until it was explicitly stated in 1904)o In 1931 Kurt Gdel established his Incompleteness Theorems Growth of (point-set and combinatorial) topologyo Point-set: roots in Cantors work on the theory of sets of real numberso Combinatorial: roots in Riemanns attempts to integrate complex functions in regions with holes Subsequent algebraization of topologyo Growth of algebraic techniques in all areas of mathematicso Continued in the theory of categories and functors Statistics exploded in importanceo Development of techniques for designing experiments and testing hypotheseso Development of electronic computers in the second half of the 20th century Hilberts 1900 Address to the International Congress of Mathematicians In 1900, Hilbert was asked to talk at the International Congress of Mathematics, he talked about 23 problems that encompassed virtually all branches of mathematicso His problems were proved to be central in the 20th century, many have been solved Grace Chisholm Young (1868-1944) Educated at home and then entered Girton College, Cambridge (the first institution where women could receive a university education) Attained a superior score at Cambridge Tripos Exam in 1892 Earned her PhD in 1895, being the first woman to receive a German doctorate in mathematics through regular procedure Married William Young in 1896, a mathematician who had been her tutor Had a major role in the production of 200 books in William Youngs name. Leonhard Eugene Dickson (1874-1954) First recipient of doctorate in mathematics at the University of Chicago Wrote hundreds of articles and some 18 bookso Most important book: three-volume History of the Theory of Numbers Emmy Noether (1882-1935) Middle-class German-Jewish girl In 1900, after studying French and English, she was qualified to teach at schools Her interest shifted from languages to mathematicso She taught mathematics at University of Erlangen Was asked to come to Gttingen by David Hilberto She taught courses under his name since she officially wasnt allowed to Couldnt continue her work after 1933 since she was Jewish. Women in Mathematics Up until recently, very few women have participated in the discipline of mathematics Without a supportive background, women could not enter the field Even the women that managed to achieve a reasonable knowledge of mathematics were often not able to participate in the mathematical community Over the last several decades, it became possible for women who want to be mathematicians to achieve that aim, even without a family member as a role model Women are gradually entering positions of inuence in the mathematical community The Prehistory of Computers Some Islamic scientists in the middle ages used certain instruments to help in their own calculations, particularly in astronomy Wilhelm Schikard (1592-1635), around 1623, designed and built a machine that performed addition and subtraction automatically. He built one for Kepler, but it was destroyed in a fire before it could be used. Around 1643, Pascal constructed an adding and substracting machine. In 1671, Leibniz constructed a machine that also did multiplication and division. Babbages difference engine and analytical engine Leibnizs machine nor the improved models were used that much, since mathematical practitioners continued to do calculations by hand since the machine wasnt faster. For complicated computations, tables were used, particularly of logarithms and trigonometric functions. Around 1821, Charles Babbage had the idea to use the steam engine to drive the machine, so the speed and the accuracy of the computation would be increased. Babbages aim was to attach the machine to a device for printing plates, but he never succeeded. Babbage developed another machine, a general-purpose calculating machine, his Analytical Engineo Contained many of the features of todays computerso He never had the financial resources to actually construct it Ada Byron King Lovelace (1815-1852) Raised by her mother, who was a mathematics student herself Tutored by well-known mathematicians Became interested in Charles Babbages Difference Engine Major mathematical work is a heavily annotated translation of a paper. Alan Turing (1912-1954) was interested in determining what a computation is and whether a given computation can in fact be carried out To answer this, he developed the (universal) Turing machine, which can calculate any number or function that can be calculated by any special machine, provided that it is given the appropriate instructions. He led the succesfull effort to crack the German enigma code in Buckinghamshire during World War II. After the war, he continued his interest in automatic computing machines and worked on a computer. In 1952, he was arrested for gross indecency since he was a homosexual. Due to the penalty for his crime, he committed suicide in 1954 with a poisoned apple.