## A History of Algebra

From the desk of Fjordman on Thu, 2009-10-22 16:47

*Arithmetica*, a landmark work in the history of algebra and number theory with the so-called Diophantine equations, is believed to have been completed around AD 250. Dirk J. Struik explains in

*A Concise History of Mathematics*, Fourth Revised Edition:

“Their skillful treatment of indeterminate equations shows that the ancient algebra of Babylon or perhaps India not only survived under the veneer of Greek civilization but also was improved by a few active men. How and when it was done is not known, just as we do not know who Diophantus was – he may have been a Hellenized Babylonian….In Diophantus we find the first systematic use of algebraic symbols. He has a special sign for the unknown, for the minus, for reciprocals. The signs are still of the nature of abbreviations rather than algebraic symbols in our sense (they form the so-called ‘syncopated’ algebra); for each power of the unknown there exists a special symbol. There is no doubt that we have here not only, as in Babylon, arithmetical questions of a definite algebraic nature, but also a well-developed algebraic notation which was greatly conducive to the solution of problems of greater complexity than were ever taken up before.”

Basic algebra was known to the ancient Egyptians, to the Babylonians in Mesopotamia in the second millennium BC, to the Chinese, the Indians and other cultures. Yet with the exception of the work Diophantus and some contributions by scholars in medieval East Asia, India and the Middle East, the history of algebra seemingly made surprisingly little progress for several thousand years until Renaissance Europe, after which modern algebra was born. The solutions to linear and quadratic equations were known to the ancient Babylonians, but the solution to the general cubic equation did not come until Renaissance Italy.

According to the book *Mathematics
Across Cultures: The History of Non-Western Mathematics*, “Throughout
these 3000 years, the Greeks, Indians, Chinese, Muslims, Hebrews and Christians
seem to have done no more than present their own versions of solutions to
linear and quadratic equations, which were well known to the Babylonians.
Diophantus, Bhaskara, Jia Xian, al-Khwarizmi, Levi ben Gerson and Leonardo of
Pisa, all provide examples of this. However, the history of algebra is subtler
than that, with small changes accumulating slowly. There were many concepts
which needed to mature before the breakthrough in algebra was ready to occur.
Independence from geometry, a comfortable symbolic notation, a library of
polynomial identities, and the tool of proof by induction, were all stepping
stones in the history of algebra.”

Bhaskara (1114-1185), also known as Bhaskara II or Bhaskaracharya, was a prominent mathematician and astronomer in medieval India. Jia Xian (ca. AD 1010-ca. 1070) in China in the mid-eleventh century invented what has become known as Pascal's triangle, which was discovered independently by Blaise Pascal in France centuries later. It was also used by other prominent Chinese mathematicians such as Zhu Shijie (ca. 1260-1320). The knowledge of Pascal's triangle is one example of how the Chinese originated, but did not follow up, inventions and discoveries that later became key elements of Western science and technology.

Levi ben Gerson, or Gersonides (1288-1344) was a Jewish rabbi, philosopher and astronomer who lived all his life in the south of France and was highly regarded in the Christian majority community. His surveying device called Jacob’s Staff, which was popular with sailors who used it for navigational purposes, was similar to a device which had been employed for several centuries in China. It is not currently known whether the idea was carried along trade routes from East Asia or whether it was an independent invention in Europe.

Arguably
the most important mathematician who ever lived in the Islamic world was
Muhammad ibn Musa al-Khwarizmi (ca. 780-850). He, or perhaps his ancestors,
came from Khwarizm, the region south of the Aral Sea now part of Uzbekistan and
Turkmenistan. He probably wasn’t an ethnic Arab and may have been born in
Central Asia, yet he spent much of his life working as a scholar in Baghdad,
synthesizing Babylonian with Greek methods. According to author David C.
Lindberg, al-Khwarizmi’s *Algebra *“contains no equations or algebraic symbols,
but only geometrical figures and Arabic prose, and it would not be recognized
as algebra by a mathematics student of the twenty-first century. Its
achievement was to deploy Euclidean geometry for the purpose of solving
problems that we would* now *state in algebraic terms (including quadratic equations).” This book
circulated in Western Europe and contributed in the long run to the development
of a true symbolic algebra there.

Next to
al-Khwarizmi, perhaps the most gifted mathematician of the medieval Middle East
was the Persian scholar and poet Omar Khayyam
(1048-1131). Ibn Warraq in the book *Leaving Islam: Apostates Speak Out* calls him the “Poet of Doubt.” Khayyam was
definitely not an orthodox Muslim and he loved wine. He compiled astronomical
tables and contributed to reform of the Persian calendar by introducing ideas
from the Hindu one. The result was superior to the Julian calendar and
comparable in accuracy with the Gregorian one. Khayyam was the first to solve
some cubic equations and the first to see the equivalence between algebra and
geometry, although further progress here did not take place in the Islamic
world.

Our numeral system dates back to India during the early post-Roman era. It came to Europe via the medieval Middle East, which is why these numbers are called “Arabic” numbers in European languages, yet even Muslims admit that they imported them from India. Labeling them “Arabic” numerals is this therefore deeply misleading. Calling them “Hindu-Arabic” number system could be accepted, but the preferred term should be “Indian numerals.”

The Maya in Mesoamerica developed a place-value number
system with the zero at least as early as Indians did in Eurasia, but this
great innovation sadly did not influence people elsewhere. According to Michael
P. Closs in *Mathematics Across Cultures*, “There
is reason to credit the Maya with the first invention of a zero symbol. It is
absent in the surviving epi-Olmec texts but is very common in the Maya
inscriptions. Zeros are found in many chronological counts in the Dresden Codex
where they occur in positional contexts just as other numerals. Most Maya
glyphs come in several variants and the same is true of the zero sign. The
zeros in the codices are identifiable as shells and are always painted red. In
most cases, the zero shells are stylized and simplified. In the inscriptions,
the most common form of the zero is shaped somewhat like a three quarter
portion of a Maltese cross.”

The Aztecs who were politically dominant in Central Mexico from the 1300s on used hand, heart and arrow symbols to represent fractional distances when calculating areas of land. Mesoamerican and especially Mayan mathematics is the one pre-Columbian scientific achievement that compares most favorably to developments in the Old World, but the mainstream development of mathematics happened in the major Eurasian civilizations and the Maya seem to have concentrated their efforts largely in the field of planetary astronomy.

The zero can be used as an empty place indicator, to show that 2106 is different from 216. The ancient Babylonians had a place-value number system with this feature, but base 60. The second use of zero is as a number itself in the form we use it now. Some historians believe that the Indian use of zero evolved from earlier innovations by Greek astronomers. Symbols for the first nine numbers of our number system have their origins in the Brahmi system of writing in India, which dates back to at least the mid-third century BC. More important than the form of the symbols is the notion of place value, and here the evidence is weaker.

The Chinese had a multiplicative system with the base 10,
probably derived from the Chinese counting
board, a checker board with rows and columns. Numbers were represented by
little rods made from bamboo or ivory. The abacus was introduced in China
around the fourteenth century. Somewhere around or before 600 AD (the place and
date remains uncertain) Indians dropped symbols for numbers higher than 9 and
began to use symbols for 1 through 9 in our familiar place-value arrangement.
The question remains why Indians dropped their own multiplicative system and
introduced the place-value system, including a symbol for zero. We currently
don’t know for sure. Historian of mathematics Victor J. Katz elaborates in his
excellent book *A History of Mathematics*,
Second Edition:

“It has been suggested, however, that the true origins of the system in India may be found in the Chinese counting board. Counting boards were portable. Certainly, Chinese traders who visited India brought them along. In fact, since southeast Asia is the border between Hindu culture and Chinese influence, it may well have been the area in which the interchange took place. Perhaps what happened was that the Indians were impressed with the idea of using only nine symbols, but they took for their symbols the ones they had already been using. They then improved the Chinese system of counting rods by using exactly the same symbols for each place value rather than alternating two types of symbols in the various places. And because they needed to be able to write numbers in some form, rather than just have them on the counting board, they were forced to use a symbol, the dot and later the circle, to represent the blank column of the counting board. If this theory is correct, it is somewhat ironic that Indian scientists then returned the favor and brought this new system back to China early in the eighth century.”

A decimal place-value system for integers definitely existed in India by the eighth century AD, possibly earlier. Although decimal fractions were used in China, in India there is no early evidence of their use. It was the Muslims who “completed the Indian written decimal place-value system by introducing these decimal fractions.”

There is evidence of the transmission of pre-Ptolemaic
Greek astronomical knowledge to India, possibly along the Roman trade routes.
The earliest known Indian work containing trigonometry dates from the fifth
century AD. The Gupta period from the fourth to seventh centuries was a golden
age for Indian civilization, with a flourishing of art and literature.
Astronomers produced a series of textbooks (*siddhanta* or “solutions”)
covering the basics of astronomy and planetary movements using Greek planetary
theory. The *Aryabhatiya* of the prominent Indian mathematical
astronomer Aryabhata
(476-550) from 499 was an important work which summarized Hindu mathematics up
to that point in time, covering arithmetic, algebra, plane trigonometry and
spherical trigonometry. Next to Aryabhata, Brahmagupta
(598-ca. 665) was the most accomplished Indian astronomer and mathematician of
this age, making advances in algorithms for square roots and the solution of
quadratic equations.

As Katz writes, “in 773 an Indian scholar visited the court
of al-Mansur in Baghdad, bringing with him a copy of an Indian astronomical
text, quite possibly Brahmagupta’s *Brahmasphutasiddhanta*. The caliph
ordered this work translated into Arabic….The earliest available arithmetic
text that deals with Hindu numbers is the *Kitab al-jam’wal tafriq bi hisab
al-Hind (Book on Addition and Subtraction after the Method of the Indians*)
by Muhammad ibn-Musa al-Khwarizmi (ca. 780-850), an early member of the House
of Wisdom. Unfortunately, there is no extant Arabic manuscript of this work,
only several different Latin versions made in Europe in the twelfth century. In
his text al-Khwarizmi introduced nine characters to designate the first nine
numbers and, as the Latin version tells us, a circle to designate zero. He
demonstrated how to write any number using these characters in our familiar
place-value notation. He then described the algorithms of addition,
subtraction, multiplication, division, halving, doubling, and determining
square roots, and gave examples of their use.”

One Latin manuscript begins with the words “Dixit
Algorismi,” or “al-Khwarizmi says.” The word “algorismi” through some
misunderstandings became a term referring to arithmetic operations and the
source of the word *algorithm*. Some Sanskrit works and terms were
introduced to Europe via Arabic translations. “Zero” derives from *sifr*,
Latinized into “zephirum.” The word *sifr* itself was an Arabic
translation of Sanskrit *sunya*, or “empty.”

Rabbi Abraham ben Meir ibn Ezra, or Abenezra (ca. 1090-1167), a Spanish-Jewish philosopher, poet and Biblical commentator, left Spain before 1140 to escape persecution of Jews and Christians by the regime of the Muslim Almohads. He wrote three treatises which helped to bring the Indian symbols to the attention of some learned people in Europe, but it took more time for Indian numerals to become fully adopted on this continent.

Leonardo of
Pisa (ca. 1170-1240), often known as Fibonacci (son of Bonaccio), was an
Italian and the first great Western mathematician after the decline of ancient
Greek science. The son of a merchant from the city of Pisa with contacts in
North Africa, Leonardo himself travelled much in the region. He is most famous
for his masterpiece the *Liber abbaci* or *Book of Calculation*.
The word *abbaci* (from* abacus*) does not refer to a computing
device but to calculation in general. The first edition appeared in 1202, and a
revised one was published in 1228. This work enjoyed a wide European readership
and contained rules for computing with the new Indian numerals. The examples
were often inspired by examples from Arabic-language treatises, but filtered
through Leonardo’s own considerable creative genius. Indian numerals faced
powerful opposition for generations but were gradually adopted during the
Renaissance period, especially by merchants. Their practical advantages
compared to the more cumbersome Roman numerals were simply too great to ignore,
although Roman numerals are still used for certain limited purposes in the West
in the twenty-first century.

Victor J. Katz sums up the state of global mathematics around the year 1300, with a special emphasis on the major Eurasian civilizations, Europe, India, China and the Middle East:

“European algebra of this time period, like its Islamic counterpart, did not consider negative numbers at all. India and China, however, were very fluent in the use of negative quantities in calculation, even if they were still hesitant about using them as answers to mathematical problems. The one mathematical subject present in Europe in this time period which was apparently not considered in the other areas was the complex of ideas surrounding motion. It was apparently only in Europe that mathematicians considered the mathematical question of the meaning of instantaneous velocity and therefore were able to develop the mean speed rule. Thus the seed was planted which ultimately grew into one branch of the subject of calculus nearly three centuries later. It appears that the level of mathematics in these four areas of the world was comparable at the turn of the fourteenth century. Although there were specific techniques available in each culture that were not available in others, there were many mathematical ideas and methods common to two or more.”

If the level of knowledge was comparable across the major regions of Eurasia by the early fourteenth century, why was modern mathematics developed in Europe? In the Islamic world, mathematical sciences and natural philosophy tended to be classified as “foreign sciences” and treated with some suspicion, not integrated into the core curriculum at places of learning. In Europe there was a growing body of universities where natural sciences were viewed more favorably and where students enjoyed much more free inquiry and legal protection. The Islamic world did not develop calculus, analytic geometry or heliocentric astronomy.

In China, the education system was a part of the imperial bureaucracy, which did not encourage studies in science or mathematics but memorization of ancient literary classics. Those who did mathematical work usually did so in isolation, independent of each other and often unknown to each other, and their work was in many cases not followed up. This does not mean that Chinese mathematicians did not make valuable contributions, but like in the Islamic world this often happened more in spite of than because of the education system.

The practical handbook *Jiuzhang
Suanshu** *(*Nine Chapters on the
Mathematical Art*) is the longest surviving Chinese
mathematical work, and prominent Chinese mathematicians, among them Liu Hui in
263 AD, published commentaries on it. Zu Chongzhi (ca. 429-500 AD) calculated π
to seven decimals, the most accurate known estimate in the world until the
Persian Jamshid al-Kashi
(ca. 1380-1429) surpassed this. The Chinese were proficient in solving many
kinds of algebraic problems. One of the most dynamic periods of mathematics in
China was the late thirteenth century, with men such as Qin Jiushao (ca.
1202-1261).

Katz states that “Chinese scholars were primarily
interested in solving problems of importance to the Chinese bureaucracy.
Although some development of better techniques evidently occurred over the
centuries, to a large extent ‘progress’ was stifled by the general Chinese
reverence for the past. Hence even incorrect methods from such works as the *Jiuzhang suanshu* were repeated through the centuries. Although the
thirteenth-century mathematicians exploited the counting board to the fullest,
its very use imposed limits. Equations remained numerical, so the Chinese were
unable to develop a theory of equations comparable to the one developed several
centuries later in the West….Finally, in the late sixteenth century, with the
arrival of the Jesuit priest Matteo Ricci (1552-1610), Western mathematics entered
China and the indigenous tradition began to disappear.”

Madhava of Sangamagramma (ca.1350-ca.1425) was an innovative mathematician and founder of the Kerala school of astronomy and mathematics in South India, which did some interesting work from the fourteenth to sixteenth centuries, but there is today little proof of transfer of knowledge to other regions. According to author John North, “Indian religious tradition was a powerful controlling force, not only of content, but also of form and of the ways of learning by rote. As a result, a typical work of eighteenth-century astronomy can be easily mistaken for one of the previous millennium. We are reminded of the situation in China, markedly different from that in the West.” There was never any strong drive in India to link astronomy with other systems of knowledge, for instance physics, as happened in Europe.

The most important Asian mathematician of the early modern
era was arguably Seki
Kowa or Seki Takakazu (ca. 1642-1708) in Japan, a rough contemporary of
Gottfried Leibniz and Isaac Newton who anticipated some European advances. Born
into a samurai family, he was a leading figure in the *wasan* (“Japanese
calculation”) movement. He was the first person to study determinants and
independently discovered Bernoulli numbers at the same time as or slightly
before the brilliant Jacob Bernoulli (1654-1705) in Switzerland. While you can
find a handful of Asian exceptions here and there, Victor J. Katz concludes
that “Nevertheless, the locus of the history of mathematics after the
fourteenth century was primarily in Europe.” If I concentrate on Europe in the
reminder of this story it is because almost all global advances in mathematics
between the fourteenth and the mid-twentieth century happened there.

By the fourteenth century a commercial revolution had begun
in Western Europe where the “new capitalists” could remain at home and hire
others to travel to various ports as their agents. This led to the creation of
international trading companies centered in major cities, and these companies
needed more sophisticated mathematics than their predecessors did because they
had to deal with letters of credit, bills of exchange, promissory notes and
interest calculations. Double-entry bookkeeping began as a way of keeping track
of these various transactions. A new class of “professional” mathematicians
grew up in response to these growing needs, the *maestri d’abbaco *or abacists. Italian abacists and merchants were instrumental in
teaching Europeans the Hindu-Arabic decimal place-value system.

Luca Pacioli
(ca. 1445-1517) was one of the last of the abacists. He was ordained a
Franciscan friar and taught mathematics at various places in Italy. He gathered
materials for some 20 years and in 1494 completed the most comprehensive
mathematics text in Europe of the time, *Summa de arithmetica, geometria,
proportioni et proportionalita*. This summary lacked
originality, but its comprehensiveness and the fact that it was printed caused
it to be widely circulated. Pacioli was well-connected. His friends included
the architect Leon Battista Alberti and above all the polymath Leonardo da
Vinci, who had a passionate interest for mathematics which was reflected in his
art and his studies of proportions and perspective.

Niccolò Tartaglia (1499-1557) was largely self-taught in mathematics and got the nickname Tartaglia (“Stammerer”) from an injury he suffered at the hands of a French soldier. He settled in Venice in 1534 as a teacher of mathematics, wrote on the application of mathematics to artillery fire and was thus a pioneer in the science of ballistics. By 1535 he had discovered a general method for solving cubic equations. Gerolamo Cardano or Cardan (1501-1576), who was trained as a physician and was a lecturer of mathematics in Milan, in 1539 contacted Tartaglia to publish his solution. Tartaglia refused at first but eventually confided in him. He extracted an oath from Cardano that he would never publish Tartaglia’s mathematical discoveries as he planned to publish them himself at a later date.

Cardano began working with the problem of cubic equations,
assisted by his gifted student Lodovico Ferrari (1522-1565). Over the next
years he worked out the solutions and their justifications to all the various
cases of the cubic. Ferrari solved the fourth degree (quartic) equation as
well. Tartaglia still hadn’t published anything and Cardano was eager that the
solutions should be made available. In 1545 he published his great work *Ars
Magna*, *Sive de Regulis
Algebraicis* (*The Great Art, or On
the Rules of Algebra*), devoted to the
solution of cubic and quartic equations. Tartaglia was furious, even though
Cardano did mention him as one of the discoverers of the method. Cardano’s
masterpiece contained much else of interest, including a solid understanding of
the use of negative numbers. Cardano’s *Ars Magna* was extremely influential in Europe and has been
dubbed the first book
of modern mathematics.

The Italian mathematician Rafael Bombelli
(1526-1572) was educated as an engineer and spent much of his adult life
working on engineering projects. He made free use of negative numbers and was
the first person to write down the rules for addition, subtraction and
multiplication of complex numbers. He had studied Cardano’s popular work.
Algebraic notation was gradually replacing the strictly verbal accounts of the
Muslims, and Bombelli contributed to this change. According to Katz, “Although
only the first three of the five parts were published in his lifetime, and
although in the questions concerning multiple roots of cubic equations Bombelli
did not achieve as much as Cardano, nevertheless Bombelli’s *Algebra* marks the high point of the Italian algebra of the
Renaissance.”

The Italian humanist Federico Commandino
(1509-1575) had studied Latin and Greek. He spent years of his life publishing
improved Latin translations with commentaries of Greek texts by Archimedes,
Ptolemy, Euclid, Aristarchus, Pappus, Apollonius and Hero of Alexandria. He
contributed greatly to the survival of these works, which were eagerly studied
by leading figures in Europe*. Some medieval
translations made before the printing press were linguistically flawed and had
not been done by skilled mathematicians, as *Commandino was.

Pappus of Alexandria (ca. 290-ca 350 AD) was the last of
the notable Greek geometers. He worked in Roman Egypt, probably as a teacher.
His compendium of
mathematics, the *Synagoge* (“Collection”), was not terribly original
but valuable to us because it preserved some Greek mathematical texts that
would otherwise have been lost. According to the *Encyclopædia
Britannica*, “Pappus’s *Synagoge* first became widely known among
European mathematicians after 1588, when a posthumous Latin translation by
Federico Commandino was printed in Italy. For more than a century afterward,
Pappus’s accounts of geometric principles and methods stimulated new
mathematical research.” His influence is conspicuous in the work of René
Descartes, Pierre de Fermat and Isaac Newton, among many others.

As these examples demonstrate, Italy during the Renaissance
period was the leading nation in Europe in science and mathematics, but the
northern peoples were making advances, too. Robert Recorde (1510-1558), a Welsh
physician and mathematician and graduate from the University of Oxford in
England, was the first author of mathematical works in Britain during the
Renaissance. His work *The Whetstone of Witte*
from 1557 introduced the equals sign =.

The Flemish mathematician and engineer Simon
Stevin (1548-1620) made substantial contributions to a change in
mathematical thinking with his well-though-out notation for decimal fractions
and his role in erasing the Aristotelian distinction between number and
magnitude. Decimals had been used by the Chinese and Muslims before, but not by
Europeans. Stevin introduced this in 1585 with *De Thiende* (*The Art of Tenths*). The Scottish scholar John Napier and others took
up his notation and developed it into that used today. Stevin also published *L'arithmétique* in French, a work containing arithmetic and algebra.

Born in the city of Bruges in Flanders, Stevin became a bookkeeper with a firm in Antwerp and traveled in Poland, Prussia, Norway and other parts of northern Europe in the 1570s. He left the southern Netherlands, then under Spanish rule, and in 1583 entered the University of Leiden. He became an advisor to Prince Maurice of Nassau (1567-1625) and was responsible for meeting the growing need of the Dutch nation for trained engineers, merchants and navigators. He wrote textbooks in Dutch rather than the traditional Latin. Inspired by Archimedes, Stevin wrote works on mechanics and arguably founded the science of hydrostatics. In 1586, before Galileo did the same, he reported that different weights fall a given distance in the same time after experiments conducted from a church tower in Delft.

Algebra in the Islamic world had been entirely rhetorical, with no symbols for the unknown. Everything was written out in words. In fifteenth century Italy some of the abacists began to use abbreviations for unknowns. Innovations spread faster after the introduction of printing.

The brilliant French algebraist François Viète (1540-1603),
Latinized as Franciscus Vieta, was a lawyer by
profession. He developed the first systematic use of algebraic symbols. In
their online biography, J. J. O'Connor and E F Robertson state
that “François Viète was a French amateur mathematician and astronomer who
introduced the first systematic algebraic notation in his book *In artem
analyticam isagoge*. He was also involved in
deciphering codes.” He demonstrated the value of symbols to represent
quantities. “While Viète had come only part way toward modern symbolism, the
crucial step of allowing letters to stand for numerical constants enabled him
to break away from the style of examples and verbal algorithms of his
predecessors. He could now treat general examples rather than specific ones and
give formulas rather than rules.” He was not the first person ever to employ
letters of the alphabet to denote numerals, but after 1590 he popularized this
concept.

Omar Khayyam had seen a relationship between geometry and
algebra, but the decisive breakthrough came with Descartes and Fermat
in France. *Just like* Viète,
Pierre de Fermat
in Toulouse was a busy lawyer.
H*is mathematical work was communicated to
friends by means of letters, often with little in the way of formal proof. He
was fluent in several languages, among them Italian, Greek and Latin, and*
made major contributions to geometric optics, modern number theory, probability
theory and analytic geometry. According to Victor J. Katz,* “Descartes, along with Thomas Harriot and Albert
Girard, reworked some of Viète’s algebraic ideas into a theory of
equations….Fermat also was responsible for the first new work in number theory
since Leonardo of Pisa, while Pascal, along with Girard Desargues, made some of
the earliest contributions to the subject of projective geometry.”*

In the margin of a Latin translation of the *Arithmetica* by Diophantus, Fermat claimed to have found a beautiful theorem
which became famous and known as Fermat’s Last Theorem. Mathematicians
struggled with it for centuries until finally in 1995 the English mathematician
Andrew Wiles (born 1953) appeared to have solved it. There are some historians
of mathematics who question
whether Fermat ever had the proof that he claimed to have.

The French philosopher and mathematician René
Descartes (1596-1650) was one of the key figures in the Scientific
Revolution. Descartes
had studied at a Jesuit college in Anjou, classical subjects, Aristotle and
mathematics as well as acting, poetry, riding and fencing. He enlisted in a
military school and in the 1620s travelled through Europe, ending up in the
Netherlands to enjoy the liberty of s society where he could be an original
thinker without fear of the Catholic Counter-Reformation. He later went for
brief visits back to France. Descartes contributed substantially to optics and
meteorology and devised a universal method of deductive reasoning based on
mathematics, his most famous statement being *Cogito ergo sum* (*I think, therefore I am*).

Viète had used vowels for the unknowns and consonants for known quantities. Descartes introduced the convention we use today of employing letters at the beginning of the alphabet (a, b, c etc.) to represent known quantities, and letters at the end of the alphabet to represent unknown ones (x, y, z). He created the Cartesian coordinate system with axes labeled x, y and z respectively, which made it possible to express positions in two or three dimensions. This was later extended to include negative numbers. Algebra could now be linked with geometry, a development which had repercussions right down to Einstein’s theory of relativity.

Some ideas of coordinate geometry had been anticipated by the Frenchman Nicole Oresme (1323-1382), but his work did not link algebra and geometry. Coordinates had been used by ancients Greeks such as Hipparchus and Ptolemy in astronomy and geography, but historians give René Descartes and Pierre de Fermat shared credit for the invention of analytic geometry. Traditionally, algebra had been treated as completely separate from geometry. This tradition began breaking down with the work of François Viète in the late sixteenth century. Both Fermat and Descartes built on Viète’s ideas. Author Marvin Jay Greenberg writes:

“Descartes was the first to publish in 1637, as an appendix (

La Géométrie, in three parts) to his very influentialDiscourse on Method, his philosophical method for finding and recognizing correct knowledge. Fermat never did publish his work; instead, he communicated his results in private letters to a few colleagues, and his work was made public only in 1679, fourteen years after he died. Curiously, although both these men were outstanding mathematicians, mathematics was not their profession. Fermat was a jurist who did mathematics as a hobby. He is best known for his work in number theory….Fermat also discovered the basic idea of the differential calculus before Newton and Leibniz. Descartes contributed to other sciences besides mathematics, but he was primarily a philosopher whose writings had a great impact on the way educated people viewed the world. Both men initially introduced their algebraic methods in order to solve problems from classical Greek geometry, recognizing that the new methods had great potential to solve other problems. Their successors over many decades realized that potential. Descartes’ stated goal was to provide general methods, using algebra, to ‘solve any problem in geometry.’”

The French natural and religious philosopher Blaise Pascal (1623-1662) was a very influential mathematician who contributed to many areas of mathematics. Descartes visited Pascal and the two argued about the existence of a vacuum, which Descartes did not believe in. Using the newly invented barometer, Pascal in 1648 observed that the pressure of the atmosphere decreases with height and deduced that a vacuum existed above the atmosphere. In mathematics he is especially famous as the co-founder of probability theory, a branch of mathematics which is of vital importance to modern physics and economics. He corresponded with Fermat in 1654 about problems related to a game of dice. Blaise Pascal soon underwent a profound religious experience and pledged his life to Christianity. He argued his case for the rational belief in God by stating that if God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.

The Dutch astronomer Christiaan Huygens
(1629-1695) learnt of the work on probability carried out by Pascal and Fermat
and in 1657 wrote a small work *De Ratiociniis in Ludo Aleae* on the calculus of probabilities, which was the first printed text
on the subject.

Abraham de
Moivre (1667-1754) was born in the Champagne region, but as a Protestant he
left Catholic France for England after the revocation of the Edict of Nantes in
1685. He became a member of the Royal Society in London and friendly with
Edmond Halley and Isaac Newton. His major work was *The Doctrine of Chances*, published in 1718 with expanded editions in 1738 and 1756. This
text was much more detailed than the work on probability by Huygens, partly
because of the rapid general advances in European mathematics since then. De
Moivre gives not only general rules but also detailed applications of these
rules.

According to author Charles Murray, “The discovery that uncertainty could be calibrated fundamentally changed human capacity to acquire and manage knowledge. In science, it led not only to the edifice of statistical analysis that is indispensable in all the hard sciences, the social sciences, engineering, and industrial processes of all sorts, but to the unraveling of mysteries that could be understood only in terms of probabilities - quantum theory is one example. In economics, the ability to analyze reality not just in terms of yes or no, but as precise numbers in between, enabled the management of risk that in turn makes possible modern economies.”