In the year
AD 312, on the eve of a battle against would-be rivals for the Roman Imperial
throne, Constantine
had a dream that instructed him to place the *chi*-*rho*, the
Christian symbol formed by superimposing the first two letters of the Greek
name *Christos*, on the shields of his soldiers. When he won
the battle and became Emperor, he issued an edict of tolerance for Christian
believers. Later, on his deathbed, Constantine himself became a Christian,
placing it in a position of prominence in the Empire from which it would
influence the history of the Western world to this day.

In 324, Constantine
moved the seat of the Empire to the Greek town of Byzantium
in the east of the empire, renaming it Constantinople
after himself. His was one of the last strong governments of the
Roman Empire. The tenuous union of the eastern and western halves
of the empire during the fourth century continued to fray, so that by the
year 400 it
had split in two for good. The Goths entered Rome
in 476, bringing down the Western Empire. This marks the start
of the Middle Ages, when Greek culture was effectively cut off from the
West. Tribal governments held sway, giving way to feudal society
and the slow development over centuries of what would eventually become
the familiar nation-states of Europe.

Meanwhile in
the East, the Byzantine Empire preserved Greek culture. Alexandria
was still home to the great Museum of antiquity; at the turn of the fifth
century, this was where Hypatia,
the noted female philosopher, had written commentaries on Archimedes' *Measurement
of the Circle*, Apollonius'
*Conics*, and Ptolemy's *Almagest*.
But the Byzantine Empire was also a Christian empire, and the pagan Hypatia
was to meet an untimely death in 415 at the hands of a Christian mob, who
thought her a witch for her scholarship in philosophy, mathematics and
astronomy. At about this time, the Library at Alexandria was burned
and many (though not all) texts were lost. The great mathematical
tradition of the Greeks had come to an end.

While mathematical
inquiry languished in the European Middle Ages, it flourished in other
parts of the world. In China,
we find texts in which various types of problems in surveying and astronomy
are solved that required the development of geometric and arithmetical
methods. In India, trigonometry was developed in order to deal with
astronomical calcuations, the same motivation that drove Ptolemy to build
his table of chords. The tradition in India, however, was to tabulate
half-chords of angles; it is from this that we inherited the basic idea
of the sine. Also in India, by around the year 600, we have evidence
that a decimal place-value numeration system was in use. This numeration
scheme was eventually transmitted westward into Europe by Islamic scholars.

Muhammad
the Prophet (570 - 632) founded Islam in the Arabian peninsula; by
661, the armies of the Muslim caliphs had already spread
to Persia in the east and Egypt in the west, and would soon overrun all
of North Africa and Spain. Constantinople was taken and lost more
than once over the next century, and Muslim armies were finally held back
from further European conquests by their loss at the Battle of Tours to
Frankish forces under Charles Martel in 732. In the early 800's the
caliph al-Ma'mun founded the *Bayt al-Hikma* (House
of Wisdom), an institute of higher learning and scholarship, in Baghdad,
where Arabic translations of Greek and Indian works in natural philosophy,
mathematics and astronomy were made. Here and elsewhere throughout
the Muslim empire, the mathematics of the ancients was studied and improved,
and the Western world is indebted to these Arabic scholars for being largely
responsible for the later transmission of this body of knowledge into Europe.

Muhammad
ibn-Musa al-Khwarizmi (780? - 850?) was one of the earliest scholars
at the *Bayt al-Hikma*; his most famous work was entitled *Al-kitab
al-muhtasar fi hisab al-jabr w'al-muqabala* (*The Condensed Book of
Calclation by Restoration and Comparison*). In it he describes
rules for solving problems involving an unknown quantity, and it represents
the first true work of **algebra** ever written. In fact, Latin
scholars who learned of this work centuries later identified the methods
found in this book by the transliterated words in the title: *algebra*
and
*almucabala*; eventually only the first of these terms was retained.
(Other scholars used the term *algorismus*, from the Latinized form
of the author's name. Today, the word "algorithm" is used to describe
any well-defined procedure for calculation.)

Europe began
to rouse itself from its cultural slumber by the beginning of the second
millenium. It was at this time that the first universities were established
(in Bologna in 1088, Paris in 1150, Oxford in 1167) in the Scholastic tradition.
At these schools, students learned the curriculum of the seven
liberal arts: the Greek **quadrivium** (four-fold way) of Plato's
Academy, which consisted of geometry, arithmetic, music and astronomy;
and the Roman **trivium** (three-fold way), which included the more
practical disciplines of grammar, rhetoric and logic. The schools
gave degrees in theology and philosophy, in canon or Roman law, and in
medicine. While the universities modeled themselves on the monastic
schools, they took students from amongst the families of the aristocrats
and the burgeoning merchant class. (After all, these were the only
ones that could provide tuition-paying students!)

One member of
this merchant class was Leonardo
of Pisa, whose father made a fortune in the shipping trades between
ports throughout the Mediterranean basin. Leonardo (most commonly
known today by the nickname Fibonacci) is recognized as an important mathematician
of the late Middle Ages, and he profited from having learned his mathematics
from Arab scholars. Upon returning to Pisa, he then wrote the *Liber
Abbaci*(*The Book of Calculation*), a work in Latin that introduced
these ideas to students in Europe, as well as a book titled
*Practica
Geometriae* (*The Practice of Geometry*) that relates some Euclidean
geometry, some Arabic algebra, and a little trigonometry (including a brief
table of chords), and another titled *Liber Quadratorum* (*The Book
of Squares*), in which he solves some problems like "find a square number
from which, when five is added or subtracted, always arises a square number".
Fibonacci was a man whose work provided a mathematical link between two
cultures, the Muslim East and the Christian West, as well as an indirect
link between European scholarship and the heritage of the Ancient Greeks
that had been lost since the fall of Rome.

Before the
differential and integral calculus could be formalized, mathematicians
developed a symbolic language in which to express it, the language of symbolic
algebra. It is important to note that before the fourtenth century
this language did not exist. The Islamic algebraists had begun to
formulate algebraic rules for solving problems involving unknowns, but
these rules were expressed entirely in rhetorical form. Even early
European algebraists followed this practice. For instance, Jordanus
de Nemore (1225 - 1260), a contemporary of Fibonacci's who taught at
Paris in the thirteenth century, wrote an early work in algebra called
*De
numeris datis* (*On Given Numbers*). In it he poses and solves
a simple problem:

This language is easily translated into modern symbolism: if 10 is divided into the partsIf a number is divided into two parts whose difference is given, then each of the parts is determined.Namely, the lesser part and the difference make the greater. Thus the lesser part with itself and the difference make the whole. Subtract therefore the difference from the whole and there will remain double the lesser given number. When divided [by two], the lesser part will be determined; and therefore also the greater part. For example, let 10 be divided in two parts of which the difference is 2. When this is subtracted from 10 there remains 8, whose half is 4, which is thus the lesser part. The other is 6.

In France, Nicholas Chuquet (1445 - 1488) exemplifies the movement to the inclusion of symbols in algebra. In his

In Germany, Christoff
Rudolff (1499 - 1555) wrote a book in 1525 called simply *Coss*.
The Italian algebraists had identified the unknown value in a problem as
*cosa*
(the thing), and Rudolff adopted the equivalent German word
*coss*.
He employs a symbolic coding of the unknown quantity and its powers, but
uses different symbols for each power.

In England,
Robert
Recorde (1510 - 1558) published *The Whetstone of Witte* (1557),
in which the equal
sign appears for the first time: "*I will sette as I doe often in
worke use, a paire of paralleles, or Gemow lines of one lengthe, thus ====,
bicause noe 2 thynges can be moare equalle.*"

In Italy, the
most striking developments were not notational but mathematical: Girolamo
Cardano (1501 - 1576), a Milanese physician, published a very influential
algebra book in 1545, *Ars Magna, sive de regulis algebraicis* (*The
Great Art, or on the rules of algebra*). In this book appears
for the first time a general procedure for solving cubic and quartic equations;
the solutions to linear and quadratic equations had been known since early
antiquity, and partial solutions to certain kinds of cubic equations had
been solved by Islamic mathematicians, but Cardano managed to study the
cubic equation in great generality and extend these results to the case
of the quartic. The story of how he obtained these procedures is
quite fascinating, but takes us far afield from our discussion here.
(An account of this story can be found here.)

Finally, François
Viète (1540 - 1603), a lawyer and member of the court of Kings
Henry III and IV of France, and who later achieved fame as a cryptanalyst
for the crown in its political stuggles against Philip II of Spain, composed
a number of treatises which were collected in a work titled *The Analytic
Art*. Viète mastered most of the algebraic techniques of
his predecessors, but to a degree not seen before, he lays out a recognizably
modern language for operation with these techniques. He adopted the
convention of representing quantities with letters of the alphabet, which
he called **symbolic logistic**, using vowels for the unknowns and consonants
for the given quantities of the problem. He used the German symbols
+ and -,
but continued to use words to represent many operations: *in* for
multiplication, *quad* for the square, *cub* for the cube, etc.
(The translation of his work which follows has replaced most of this with
more modern notation.) He talks about the process of translating
a geometric problem into algebraic notation (**zetetics**), manipulating
the symbols according to algebraic rules (**poristics**), and obtaining
a solution by these means (**exegetics**); these terms have not been
retained. By relying heavily on this symbolic notation, Viète
was able to apply his algebraic explorations to a much wider class of problems
simply by setting the variables equal to new values. This marked
a new level of mathematical analysis which would bear fruit far beyond
the geometric problem solving in which he was interested. Viète
sees great promise in these approaches to analysis: he writes that "the
analytic art...appropriates to itself by right the proud problem of problems,
which is THERE IS NO PROBLEM THAT CANNOT BE SOLVED."

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last modified 10/16/02

Copyright (c) 2000. Daniel E. Otero