A Brief History of Mathematics
I.
Ancient Period
Prehistoric people probably first counted with their fingers. They also had various methods for
recording such quantities as the number of animals in a herd or the days since the full moon. To represent
such amounts, they used a corresponding number of pebbles, knots in a cord, or marks on wood, bone, or
stone. They also learned to use regular shapes when they molded pottery or carved arrowheads. As
civilizations developed, the need for numeration systems, measurement techniques and arithmetic
procedures also arose.
By about 3000 B.C., mathematicians of ancient Egypt used an additive base ten system that was
without place values. The Egyptians developed geometric formulas for finding the area and volume of
simple figures. Their mathematics had many practical applications, ranging from surveying fields after
the annual floods to making the intricate calculations necessary to build the pyramids.
By 2100 B.C., the people of ancient Babylonia had developed a sexagesimal numeration system;
a system based on groups of sixty. The system had important uses in astronomy and also in commerce,
because sixty can be divided easily and works well with a calendar. It was also notable for the use of
place value to represent numbers of any size. The system survives today in the way we measure time and
angles. The Babylonians also went beyond the Egyptians in algebra and geometry. They found solutions
to quadratic equations and developed techniques for calculating square roots.
Chinese mathematics originally developed to aid record keeping, land surveying, and building.
By the 100's B.C., the Chinese had devised a decimal system of numbers that included fractions, zero, and
negative numbers. They solved arithmetic problems with the aid of special sticks called counting rods.
The Chinese also used these devices to solve equations—even groups of simultaneous equations in
several unknowns.
Perhaps the best-known early Chinese mathematical work is the Chiu Chang Suan Shu (Nine
Chapters on the Mathematical Art, a handbook of practical problems that was compiled in the first two
centuries B.C. In 263 A.D., the Chinese mathematician Liu Hui wrote a commentary on the book. Among
Liu Hui's greatest achievements was his analysis of a mathematical statement called the Gou-Gu theorem.
The theorem, known as the Pythagorean theorem in the West, describes a special relationship that exists
between the sides of a right triangle. Liu Hui also calculated the value of pi more accurately than ever
before. He did so by using a figure with 3,072 equal sides to approximate a circle.
II.
Greek Period (600 B.C. to 400 A.D.)
Ancient Greek scholars introduced the concepts of logical deduction and proof to create a
systematic theory of mathematics. According to tradition, one of the first to provide mathematical proofs
based on deduction was the philosopher Thales, who worked in geometry about 600 B.C. He was a
Greek merchant whose travels brought him in contact with the mathematics of Babylonia and Egypt.
Until this time, geometry had consisted strictly of measuring techniques (in fact the word geometry means
“earth measurement”). However, Thales made abstract, general statements, such as “when two lines
intersect, they create pairs of equal angles,” and attempted to justify those statements logically.
The Greek philosopher Pythagoras, who lived about 550 B.C., explored the nature of numbers,
believing that everything could be understood in terms of whole numbers or their ratios. His followers
explored number patterns and discovered irrational numbers. They had a significant influence on Greek
philosophy and the promotion of mathematics for its own sake.
Around 300 B.C, Euclid, one of the foremost Greek mathematicians, organized the geometrical ideas of
the previous three hundred years into a systematic, logical structure in his work The Elements of
Geometry. In this book an entire system of geometry is constructed by means of abstract definitions,
accepted facts (postulates) and logical deductions. It had an enormous impact on mathematical thought
and became the model for the development of a mathematical system.
During the 200's B.C., the Greek mathematician and physicist Archimedes used the “method of
exhaustion” to find many formulas for the volume and surface areas of solids and to calculate a highly
accurate value for pi (the ratio of a circle's circumference to its diameter). He was also famous for
creating many engineering devices, such as “Archimedes’ screw” and discovering some of the
fundamental laws of physics.
Working at about the same time, the Greek mathematician Appolonius of
Perga, known as the “Great Geometer” wrote an eight volume work in which he
investigated the curves obtained by taking cross sections of a double cone. These
curves, the circle, ellipse, parabola and hyperbola, are called conic sections.
One of the last great Greek scholars, Ptolemy, applied geometry and
trigonometry to astronomy about A.D. 150. In a book, known as the Almagest, he
presented a scheme for the motions of the heavenly bodies. He claimed that the earth
was stationary and that it was in the center of a larger sphere around which the sun,
stars, moon and planets revolved at uniform rates of speed. This model became the
accepted theory of the solar system throughout the middle ages, both in the European and the Islamic
worlds.
Pythagoras of Crotona
Thales of Miletus
Apollonius of Perga
Archimedes of Syracuse
Euclid and Ptolemy
of Alexandria
Eratosthenes of Cyrene
The World of Greek Mathematics
III. Hindu-Arabian Period (200 B.C. to 1200 A.D. )
Mathematics and the sciences entered a long period of stagnation with the decline of the Greek
and Roman civilizations. This inactivity was uninterrupted until after the Islamic religion and the
resulting Islamic culture were founded by the prophet Muhammad in A.D. 622. Within a century, the
Islamic empire stretched from Spain, Sicily, and Northern Africa to India.
Islamic culture encouraged the development of the sciences as well as the arts. Arab scholars
translated many Greek and Hindu works in mathematics and the sciences, including Apollonius's work on
conic sections. It is likely that much of the Greeks' work in science and mathematics would have been lost
if not for these Arab scholars.
The Arab mathematician Mohammed ibn Musa al-Khowarizmi wrote two important books
around A.D. 830, each of which was translated into Latin in the twelfth century. Much of the
mathematical knowledge of medieval Europe was derived from the Latin translations of al-Khowarizmi's
two works. Al-Khowarizmi's first book, on arithmetic, was titled Algorithmi de numero Indorum (or alKhowarizmi on Indian Numbers). The Latin translation of this book introduced to Europe the Hindu
number system and the simpler calculation techniques (such as the procedures for multiplication and long
division) that system allows. This system is now called the Hindu-Arabic number system. The book's title
is the origin of the word algorithm, which means a procedure for solving a certain type of problem, such
as the procedure for long division.
Al-Khowarizmi's second book, Al-Jabr w'al Muqabatah, discussed linear and
quadratic equations. In fact, the word algebra comes from the title of this second book. This title, which
translates literally as Restoration and Opposition, refers to the solving of an equation by adding the same
thing to each side of the equation (which "restores the balance" of the equation) and simplifying the result
by canceling opposite terms (which is the title's "opposition"). For example (using modern symbolic
algebra):
6x = 5x+ 11
6x + -5x
= 5x + 11 + -5x
"al-jabr" or restoration of balance
x = 11
al-muqabalah" or opposition
The quote below, from a translation of Al-Jabr w'al Muqabalah, demonstrates several important
features of al-Khowarizmi 's algebra. First, it is entirely verbal, as was the algebra of Apollonius—there is
no symbolic algebra at all. Second, this algebra differs from that of Apollonius in that it is not based on
proportions. Third, the terminology betrays the algebra's connections with geometry. When alKhowarizmi refers to "a square," he is actually referring to the area of a square; when he refers to "a
root," he is actually referring to the length of one side of the square (hence the modern phrase "square
root"). Modern symbolic algebra uses the notations x2 and x in place of "a square" and "a root."
The quote from Al-Jabr w'al Muqabalah is on the left; a modern version of
the same instructions is on the right. You might recognize this modern version from intermediate algebra,
where it is called "completing the square."
The following is an example of squares and
roots equal to numbers: a square and 10 roots
are equal to 39 units.
The question therefore in this type of equation
is about as follows: what is the square which
combined with ten of its roots will give a sum
total of 39? The manner of solving this type of
equation is to take one half of the roots just
mentioned. Now the roots in the problem
before us are 10. Therefore take 5, which
multiplied by itself gives 25, an amount which
you add to 39, giving 64. Having taken then
the square root of this which is 8, subtract from
it the half of the roots, 5, leaving 3. The
number three therefore represents one root of
this square, which itself, of course, is 9. Nine
therefore gives that square.
x2 + 10x = 39
Solve for x2
½ ·10 = 5
52 = 25
x2 + 10x + 25 = 39 + 25
x2 + 10x + 25 = 64
(x + 5)2 = 64
x+5=8
x+5-3=8-3
x=3
x2 = 9
Al-Khowarizmi, like Apollonius, understood numbers to be lengths of line segments, areas, and
volumes. He did not recognize negative numbers, because neither a line nor an area nor a volume can be
represented by a negative number.
Arab astronomers of the 900's made major contributions to trigonometry. During the 1000s, an
Arab physicist known as Alhazen applied geometry to optics. The Persian poet and astronomer Omar
Khayyam wrote an important book on algebra about 1100. In the 1200s, Nasiral-Din al-Tusi, a Persian
mathematician, created ingenious mathematical models for use in astronomy.
IV.
Period of Transmission (1000 AD – 1500 AD)
Contest between abacus
and newer methods
The Moors (Moslems from North Africa) entered Spain in A.D.
711 and built universities in Toledo, Cordoba, and Seville. This culture
was the only major exception to the mathematical and scientific
stagnation in Europe that started with the end of the Roman Empire and
continued through the Middle Ages. By the twelfth century, many Arabic
mathematical and scientific works (including al-Khowarizmi's two
books), as well as Greek and Hindu works, were translated into Latin,
often by Jewish scholars in Spain. Greek works of literature and
philosophy were also translated.
In 1202, Leonardo of Pisa (Fibonacci), an Italian mathematician,
published a book on algebra, the Arab number system, and pen and paper
arithmetic techniques that helped promote this system. Hindu-Arabic
numerals gradually replaced Roman numerals and the use of an abacus in
Europe.
During the 1400s and 1500's, European explorers sought new overseas trade routes, stimulating
the application of mathematics to navigation and commerce. As trade expanded, Arab and Greek
knowledge was transmitted throughout Europe. In 1453 the Turks conquered Constantinople, the last
remaining center of Greek culture. Many Eastern scholars moved from Constantinople to Europe,
bringing Greek knowledge and manuscripts with them. Around the same time, Gutenberg invented the
movable-type printing press, which greatly increased the availability of scientific information in the form
of both new works and translations of ancient works. Translations of Euclid's work, Ptolemy’s Almagest
and some of Apollonius's work on geometry were printed, as was the Franciscan monk Luca Pacioli's
Summa de arithmetica, geometrica, proportioni et proportionalita, which was a summary of the
arithmetic, geometry, algebra, and double-entry bookkeeping known at that time.
V. Early Modern Period (1500 AD – 1800 AD)
Development of Algebra and Analytic Geometry
For the Arab mathematicians, algebra was a set of specific techniques that could be used to solve
specific equations. There was little generalization, and there was no way to write an equation to represent
an entire class of equations, as we would now write x2 + bx + c = 0 to represent all quadratic equations.
There were only ways to write specific equations such as 3x2 + 5x +7 = 0. Thus, it was impossible to
write a formula like the quadratic formula [if ax2 + bx + c = 0, then x = (-b ±
)/2a]. It was only
possible to give an example, such as al-Khowarizmi's example of completing the square.
In the late sixteenth century, algebra matured into a much more powerful tool. It became more
symbolic. Exponents were introduced; what had been written as "cubus," "A cubus" or "AAA" could now
be written as "A3." The symbols + . - , and = were also introduced.
Francois Viete, a French lawyer who studied mathematics as a hobby, began using vowels to
represent variables and consonants to represent constants. This allowed mathematicians to represent the
entire class of quadratic equations by writing "A2 + BA = C" (where the vowel A is the variable and the
consonants B and C are the constants) and made it possible to discuss general techniques that could be
used to solve classes of equations. All these notational changes were slow to gain acceptance. No one
mathematician adopted all the new notations. Viete's algebra was quite verbal. He did not even adopt the
symbol + until late in his life.
In 1637, the famous French philosopher and mathematician Rene Descartes published La
Geometrie , a work that explored the relationship between algebra and geometry in a way unforeseen by
Apollonius and al-Khowarizmi. Descartes showed how to interpret algebraic operations and solve
quadratic equations geometrically. He also showed that algebra could be applied to geometric problems.
This approach is now called “analytic geometry”. To the readers of Descartes it was an amazing method
that combined algebra and geometry in new and unique ways. However, it did not especially resemble our
modern analytic geometry, which consists of ordered pairs, x and y axes, and a correspondence between
algebraic equations and their graphs. Descartes used an x axis, but he did not have a y axis. Although he
knew that an equation in two unknowns determines a curve, he had very little interest in sketching curves:
he never plotted a new curve directly from its equation.
In 1629, eight years before Descartes's La Geometrie, the French lawyer and amateur
mathematician Pierre de Fermat attempted to recreate one of the lost works of Apollonius on conic
sections using references to that work made by other Greek mathematicians. Fermat applied Viete's
algebra to Apollonius's work and created an analytic geometry much more similar to the modern one than
was Descartes's. Fermat emphasized the sketching of graphs of equations. He showed a parallelism
between certain types of equations and certain types of graphs. For example, he showed that the graph of
"d planum p. a planum aequetur b in e” (d2 + a2 = be) is always a parabola.
Modern analytic geometry is thus considered to be an invention of both Descartes and Fermat.
Descartes's algebra was more modern and sophisticated than Fermat’s or any of his contemporaries.
Fermat, on the other hand, developed the important relation between geometric shapes and a coordinate
graph. Together they are credited with developing analytic geometry to the point where calculus could be
invented.
Problems:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
What type of mathematics are the Greeks known for?
What are the conic sections?
Which Greek mathematician is known for his work on conic sections?
What influential book did Ptolemy write? Why is it important?
How did Thales influence Greek mathematics?
The mathematics and science of the Greeks could well have been lost if it were not for a
certain culture. Which culture saved this Greek knowledge, expanded it, and reintroduced
it to Europe?
What were the subjects of al-Khowarizmi's two books?
Why is our modern number system called the Hindu-Arabic number system? What is
important about this system?
Al-Khowarizmi described his method of solving quadratic equations with an example; he
did not generalize his method into a formula. What characteristic of the mathematics of
his time limited him to this form of a description? What change in mathematics lifted this
limitation? To whom is that change due? Approximately how many years after alKhowarizmi did this change occur?
What is analytic geometry?
What did Descartes contribute to analytic geometry?
What did Fermat contribute to analytic geometry?
What did Viete contribute to algebra?