The area of study known as the
history of
mathematics is primarily an investigation into the origin
of discoveries in
mathematics and, to a
lesser extent, an investigation into the mathematical methods and
notation of the past.
Before the modern age and the worldwide spread of knowledge,
written examples of new mathematical developments have come to
light only in a few locales. The most ancient mathematical texts
available are
Plimpton 322
(
Babylonian mathematics c.
1900 BC), the
Moscow
Mathematical Papyrus (
Egyptian mathematics c. 1850 BC), and
the
Rhind Mathematical
Papyrus (Egyptian mathematics c. 1650 BC). All of these
texts concern the so-called
Pythagorean theorem, which seems to be
the most ancient and widespread mathematical development after
basic arithmetic and geometry.
The
Greek and Hellenistic
contribution greatly refined the methods (especially through the
introduction of deductive reasoning and
mathematical rigor in
proofs) and expanded the subject matter
of mathematics.
Chinese
mathematics made early contributions, including a
place value system. The
Hindu-Arabic numeral system and
the rules for the use of its operations, in use throughout the
world today, likely evolved over the course of the first millenium
AD in
India and was transmitted
to the west via Islamic mathematics.
Islamic mathematics, in turn, developed
and expanded the mathematics known to these civilizations. Many
Greek and Arabic texts on mathematics were then
translated into
Latin, which led to further development of mathematics in
medieval Europe.
From ancient times through the
Middle
Ages, bursts of mathematical creativity were often followed by
centuries of stagnation.
Beginning in Renaissance Italy in the 16th
century, new mathematical developments, interacting with new
scientific discoveries, were made at an increasing pace that continues through
the present day.
Prehistoric mathematics
The origins of mathematical thought lie in the concepts of number,
magnitude, and form. Modern studies of animal cognition have shown
that these concepts are not unique to humans. Such concepts would
have been part of everyday life in hunter-gatherer societies. That
the concept of number evolved gradually over time is evident in
that some languages today preserve the distinction between "one",
"two", and "many", but not of numbers larger than two.
The oldest
known mathematical object is the Lebombo
bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC. It
consists of 29 distinct notches deliberately cut into a baboon's
fibula. There is evidence that women used counting to keep track of
their
menstrual cycles; 28 to 30
scratches on bone or stone, followed by a distinctive marker.
Also
prehistoric artifact discovered in Africa and
France, dated between 35,000 and
20,000 years old, suggest early
attempts to quantify
time.
The
Ishango bone, found near the headwaters
of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series
of tally marks carved in three columns running the length of the
bone. Common interpretations are that the Ishango bone shows
either the earliest known demonstration of
sequences of
prime
numbers or a six month lunar calendar.
Predynastic Egyptians of the 5th
millennium BC pictorially represented
geometric designs.
It has been claimed that megalithic monuments in England and Scotland, dating from
the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and
Pythagorean triples in their
design.
Ancient Near East (3rd millenium BC–500 BC)
Mesopotamia
Babylonian mathematics refers to any mathematics
of the people of Mesopotamia (modern
Iraq) from the days of the early Sumerians until the beginning of the Hellenistic period. It is named Babylonian
mathematics due to the central role of Babylon as a place
of study, which ceased to exist during the Hellenistic
period. From this point, Babylonian mathematics merged with
Greek and Egyptian mathematics to give rise to
Hellenistic mathematics.
Later under the
Arab Empire, Mesopotamia, especially
Baghdad, once again became an important center of study for
Islamic
mathematics.
In contrast to the sparsity of sources in
Egyptian mathematics, our knowledge of
Babylonian mathematics is derived from more than 400 clay tablets
unearthed since the 1850s. Written in
Cuneiform script, tablets were inscribed
whilst the clay was moist, and baked hard in an oven or by the heat
of the sun. Some of these appear to be graded homework.
The earliest evidence of written mathematics dates back to the
ancient
Sumerians, who built the earliest
civilization in Mesopotamia. They developed a complex system of
metrology from 3000 BC. From around 2500
BC onwards, the Sumerians wrote
multiplication tables on clay tablets
and dealt with
geometrical exercises and
division problems. The
earliest traces of the Babylonian numerals also date back to this
period.
The majority of recovered clay tablets date from 1800 to 1600 BC,
and cover topics which include fractions, algebra, quadratic and
cubic equations, and the calculation of
regular reciprocal pairs (see
Plimpton
322). The tablets also include multiplication tables and
methods for solving
linear and
quadratic equations. The
Babylonian tablet YBC 7289 gives an approximation to √2 accurate to
five decimal places.
Babylonian mathematics were written using a
sexagesimal (base-60)
numeral system. From this derives the modern
day usage of 60 seconds in a minute, 60 minutes in an hour, and 360
(60 x 6) degrees in a circle, as well as the use of seconds and
minutes of arc to denote fractions of a degree. Babylonian advances
in mathematics were facilitated by the fact that 60 has many
divisors. Also, unlike the Egyptians, Greeks, and Romans, the
Babylonians had a true place-value system, where digits written in
the left column represented larger values, much as in the
decimal system. They lacked, however, an equivalent
of the decimal point, and so the place value of a symbol often had
to be inferred from the context.
Egypt
Egyptian mathematics refers to mathematics written in the
Egyptian language. From the
Hellenistic period,
Greek replaced Egyptian as the written
language of
Egyptian scholars, and from
this point Egyptian mathematics merged with Greek and Babylonian
mathematics to give rise to
Hellenistic mathematics.
Mathematical study in
Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian
scholars.
The oldest mathematical text discovered so far is the
Moscow papyrus, which is an
Egyptian Middle Kingdom papyrus dated c.
2000–1800 BC. Like many ancient mathematical texts, it consists of
what are today called
word problems or
story
problems, which were apparently intended as entertainment. One
problem is considered to be of particular importance because it
gives a method for finding the volume of a
frustum: "If you are told: A truncated pyramid of 6
for the vertical height by 4 on the base by 2 on the top. You are
to square this 4, result 16. You are to double 4, result 8. You are
to square 2, result 4. You are to add the 16, the 8, and the 4,
result 28. You are to take one third of 6, result 2. You are to
take 28 twice, result 56. See, it is 56. You will find it
right."
The
Rhind papyrus (c. 1650 BC
[1829]) is another major Egyptian mathematical text,
an instruction manual in arithmetic and geometry. In addition to
giving area formulas and methods for multiplication, division and
working with unit fractions, it also contains evidence of other
mathematical knowledge, including
composite and
prime
numbers;
arithmetic,
geometric and
harmonic means; and simplistic understandings
of both the
Sieve of
Eratosthenes and
perfect number
theory (namely, that of the number 6)
[1830].
It also shows how to solve first order
linear equations [1831] as well as
arithmetic and
geometric series [1832].
Also, three geometric elements contained in the Rhind papyrus
suggest the simplest of underpinnings to
analytical geometry: (1) first and
foremost, how to obtain an approximation of \pi accurate to within
less than one percent; (2) second, an ancient attempt at
squaring the circle; and (3) third, the
earliest known use of a kind of
cotangent.
Finally, the
Berlin papyrus (c. 1300
BC
[1833] [1834]) shows that ancient Egyptians could
solve a second-order
algebraic
equation [1835].
Greek and Hellenistic mathematics (c. 600 BC–300 AD)
Pythagoras of Samos
Greek mathematics refers to mathematics written in the
Greek language between about 600 BC and AD
300. Greek mathematicians lived in cities spread over the entire
Eastern Mediterranean, from Italy to North Africa, but were united
by culture and language. Greek mathematics of the period following
Alexander the Great is sometimes
called Hellenistic mathematics.
Thales of Miletus
Greek mathematics was more sophisticated than the mathematics that
had been developed by earlier cultures. All surviving records of
pre-Greek mathematics show the use of inductive reasoning, that is,
repeated observations used to establish rules of thumb. Greek
mathematicians, by contrast, used deductive reasoning. The Greeks
used logic to derive conclusions from definitions and axioms.
Greek mathematics is thought to have begun with
Thales (c. 624–c.546 BC) and
Pythagoras (c. 582–c. 507 BC). Although the
extent of the influence is disputed, they were probably inspired by
the mathematics of
Egypt,
Mesopotamia and
India. According to legend, Pythagoras
traveled to Egypt to learn mathematics, geometry, and astronomy
from Egyptian priests.
Thales used
geometry to solve problems such
as calculating the height of pyramids and the distance of ships
from the shore. Pythagoras is credited with the first proof of the
Pythagorean theorem, though the
statement of the theorem has a long history.In his commentary on
Euclid,
Proclus states
that Pythagoras expressed the theorem that bears his name and
constructed
Pythagorean triples
algebraically rather than geometrically.
The Academy of
Plato had the motto, "Let none unversed in geometry enter
here".
The
Pythagoreans proved the existence
of irrational numbers.
Eudoxus (408–c.355
BC) developed the
method of
exhaustion, a precursor of modern
integration.
Aristotle
(384—c.322 BC) first wrote down the laws of
logic.
Euclid (c. 300 BC) is the
earliest example of the format still used in mathematics today,
definition, axiom, theorem, proof. He also studied
conics. His book,
Elements, was known to all
educated people in the West until the middle of the 20th century.
In addition to the familiar theorems of geometry, such as the
Pythagorean theorem,
Elements includes a proof that the square root of two is
irrational and that there are infinitely many prime numbers. The
Sieve of Eratosthenes (c. 230
BC) was used to discover prime numbers.
Archimedes (c.287–212 BC) of Syracuse used the method of
exhaustion to calculate the area under the
arc of a parabola with the summation of an infinite series, and
gave remarkably accurate approximations of Pi. He also studied the
spiral bearing his name, formulas for the
volumes of
surfaces of revolution, and an
ingenious system for expressing very large numbers.
Chinese mathematics (c. 2nd millenium BC–1300 AD)
The Nine Chapters on the
Mathematical Art.
The earliest extant Chinese mathematics dates from the
Shang Dynasty (1600–1046 BC), and consists of
numbers scratched on a tortoise shell
[1836] [1837]. These numbers were represented by means
of a decimal notation. For example, the number 123 is written (from
top to bottom) as the symbol for 1 followed by the symbol for 100,
then the symbol for 2 followed by the symbol for 10, then the
symbol for 3. This was the most advanced number system in the world
at the time, and allowed calculations to be carried out on the
suan pan or (Chinese abacus). The
date of the invention of the
suan pan is not certain, but
the earliest written mention dates from AD 190, in Xu Yue's
Supplementary Notes on the Art of Figures.
In
China, the Emperor Qin Shi
Huang (Shi Huang-ti) commanded in 212 BC that all books in Qin
Empire other than officially sanctioned ones should be
burned. This decree was not universally obeyed, but as a
consequence of this order little is known about ancient Chinese
mathematics.
From the
Western Zhou Dynasty
(from 1046 BC), the oldest mathematical work to survive the
book burning is the
I Ching, which uses the 8 binary 3-
tuples (trigrams) and 64 binary 6-
tuples (hexagrams) for philosophical, mathematical,
and mystical purposes. The binary tuples are composed of broken and
solid lines, called yin (female) and yang (male), respectively (see
King Wen sequence).
The oldest existent work on
geometry in
China comes from the philosophical
Mohist
canon c. 330 BC, compiled by the followers of
Mozi (470–390 BC). The
Mo Jing described
various aspects of many fields associated with physical science,
and provided a small wealth of information on mathematics as
well.
After the book burning, the
Han dynasty
(202 BC–220 AD) produced works of mathematics which presumably
expand on works that are now lost. The most important of these is
The Nine
Chapters on the Mathematical Art, the full title of which
appeared by AD 179, but existed in part under other titles
beforehand. It consists of 246 word problems involving agriculture,
business, employment of geometry to figure height spans and
dimension ratios for
Chinese pagoda
towers, engineering,
surveying, and
includes material on
right triangles
and
π. It also made use of
Cavalieri's principle on volume more
than a thousand years before Cavalieri would propose it in the
West. It created mathematical proof for the
Pythagorean theorem, and a mathematical
formula for
Gaussian
elimination.
Liu Hui commented on the
work by the 3rd century AD.
In addition, the mathematical works of the Han astronomer and
inventor
Zhang Heng (AD 78–139) had a
formulation for
pi as well, which differed from
Liu Hui's calculation. Zhang Heng used his formula of pi to find
spherical volume. There was also the written work of the
mathematician and
music theorist
Jing Fang (78–37 BC); by using the
Pythagorean comma, Jing observed
that 53
just fifths approximates 31
octaves. This would later lead to the
discovery of
53 equal
temperament, and was not calculated precisely elsewhere until
the German
Nicholas Mercator did
so in the 17th century.
The Chinese also made use of the complex combinatorial diagram
known as the
magic square and
magic circles, described in
ancient times and perfected by
Yang Hui (AD
1238–1398).
Zu Chongzhi (5th century) of the
Southern and Northern
Dynasties computed the value of π to seven decimal places,
which remained the most accurate value of π for almost 1000
years.
Even after European mathematics began to flourish during the
Renaissance, European and Chinese
mathematics were separate traditions, with significant Chinese
mathematical output in decline, until the
Jesuit missionaries such as
Matteo Ricci carried mathematical ideas back
and forth between the two cultures from the 16th to 18th
centuries.
Indian mathematics (c. 800 BC–1600 AD)
The
earliest civilization on the Indian subcontinent is the Indus Valley Civilization that
flourished between 2600 and 1900 BC in the Indus river basin. Their cities were laid out with
geometric regularity, but no known mathematical documents survive
from this civilization.
Vedic mathematics began in India in the early Iron Age. The
Shatapatha Brahmana (c.
9th century BC), which approximates the value of
π, and the
Sulba Sutras
(c. 800–500 BC) were
geometry texts that
used
irrational numbers,
prime numbers, the
rule of three and
cube roots; computed the
square root of 2 to one part in one hundred
thousand; gave the method for constructing a
circle with approximately the same area as a
given square, solved
linear and
quadratic equations; developed
Pythagorean triples
algebraically, and gave a statement and numerical
proof of the
Pythagorean theorem.
(c. 5th century BC) formulated the rules for Sanskrit grammar. His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion. Pingala (roughly 3rd-1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system. His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru).
The
Surya Siddhanta (c.
400) introduced the
trigonometric functions of
sine,
cosine, and inverse sine,
and laid down rules to determine the true motions of the
luminaries, which conforms to their actual positions in the sky.
The cosmological time cycles explained in the text, which was
copied from an earlier work, correspond to an average
sidereal year of 365.2563627 days, which is
only 1.4 seconds longer than the modern value of 365.25636305 days.
This work was translated into to Arabic and Latin during the Middle
Ages.
Aryabhata, in 499, introduced the
versine function, produced the first Indian
trigonometric tables of sine, developed
techniques and
algorithms of
algebra,
infinitesimals, and
differential equations, and obtained
whole number solutions to linear equations by a method equivalent
to modern methods, along with accurate
astronomical calculations based on a
heliocentric system of
gravitation. An
Arabic
translation of his
Aryabhatiya was available from the 8th
century, followed by a Latin translation in the 13th century. He
also gave a value of π corresponding to 62832/20000 = 3.1416. In
the 14th century,
Madhava of
Sangamagrama found the
Madhava–Leibniz series, and, using 21
terms, computed the value of π as 3.14159265359.
In the 7th century,
Brahmagupta
identified the
Brahmagupta
theorem,
Brahmagupta's
identity and
Brahmagupta's
formula, and for the first time, in
Brahma-sphuta-siddhanta, he
lucidly explained the use of
zero as both
a
placeholder and
decimal digit, and explained the
Hindu-Arabic numeral system. It
was from a translation of this Indian text on mathematics (c. 770)
that Islamic mathematicians were introduced to this numeral system,
which they adapted as
Arabic
numerals. Islamic scholars carried knowledge of this number
system to Europe by the 12th century, and it has now displaced all
older number systems throughout the world. In the 10th century,
Halayudha's commentary on
Pingala's work contains a study of the
Fibonacci sequence and
Pascal's triangle, and describes the
formation of a
matrix.
In the 12th century,
Bhaskara first
conceived
differential
calculus, along with the concepts of the
derivative,
differential coefficient, and differentiation.
He also stated
Rolle's theorem (a
special case of the
mean value
theorem), studied
Pell's
equation, and investigated the derivative of the sine function.
From the 14th century, Madhava and other
Kerala School mathematicians further developed
his ideas. They developed the concepts of
mathematical analysis and
floating point numbers, and concepts
fundamental to the overall development of
calculus, including the mean value theorem, term by
term
integration, the relationship of an
area under a curve and its antiderivative or integral, the
integral test for convergence,
iterative methods for solutions to
non-linear equations, and a number of
infinite series,
power series,
Taylor
series, and trigonometric series. In the 16th century,
Jyeshtadeva consolidated many of the Kerala
School's developments and theorems in the
Yuktibhasa, the
world's first differential calculus text, which also introduced
concepts of
integral
calculus.
Mathematical progress in India stagnated from the late 16th century
to the 20th century, due to political turmoil.
Islamic mathematics (c. 800–1500)
The
Islamic Empire established across
Persia, the
Middle East, Central Asia, North
Africa, Iberia, and in parts of India in the 8th century made significant
contributions towards mathematics. Although most Islamic
texts on mathematics were written in
Arabic, most of them were not written by
Arabs, since much like the status of Greek in
the Hellenistic world, Arabic was used as the written language of
non-Arab scholars throughout the Islamic world at the time.
Persians contributed to the world of
Mathematics alongside Arabs.
In the 9th century, wrote several important books on the
Hindu-Arabic numerals and on methods for solving equations. His
book
On the Calculation with Hindu Numerals, written about
825, along with the work of
Al-Kindi, were
instrumental in spreading
Indian
mathematics and
Indian
numerals to the West. The word
algorithm is derived from the Latinization of
his name, Algoritmi, and the word
algebra from the title of one of his works,
Al-Kitāb
al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (
The
Compendious Book on Calculation by Completion and Balancing).
Al-Khwarizmi is often called the "father of algebra", for his
fundamental contributions to the field. He gave an exhaustive
explanation for the algebraic solution of quadratic equations with
positive roots, and he was the first to teach algebra in an
elementary form and for its own
sake. He also introduced the fundamental method of "
reduction" and "balancing",
referring to the transposition of subtracted terms to the other
side of an equation, that is, the cancellation of like terms on
opposite sides of the equation. This is the operation which
Al-Khwarizmi originally described as
al-jabr. "It is not
certain just what the terms
al-jabr and
muqabalah
mean, but the usual interpretation is similar to that implied in
the translation above. The word
al-jabr presumably meant
something like "restoration" or "completion" and seems to refer to
the transposition of subtracted terms to the other side of an
equation; the word
muqabalah is said to refer to
"reduction" or "balancing" - that is, the cancellation of like
terms on opposite sides of the equation." His algebra was also no
longer concerned "with a series of
problems
to be resolved, but an
exposition
which starts with primitive terms in which the combinations must
give all possible prototypes for equations, which henceforward
explicitly constitute the true object of study." He also studied an
equation for its own sake and "in a generic manner, insofar as it
does not simply emerge in the course of solving a problem, but is
specifically called on to define an infinite class of
problems."
Further developments in algebra were made by
Al-Karaji in his treatise
al-Fakhri,
where he extends the methodology to incorporate integer powers and
integer roots of unknown quantities. The first known
proof by
mathematical induction appears in a
book written by Al-Karaji around 1000 AD, who used it to prove the
binomial theorem,
Pascal's triangle, and the sum of
integral cubes. The
historian of mathematics, F. Woepcke,
praised Al-Karaji for being "the first who introduced the
theory of
algebraic calculus." Also in the 10th century,
Abul Wafa translated the works of
Diophantus into Arabic and developed the
tangent function.
Ibn al-Haytham was the first mathematician to
derive the formula for the sum of the
fourth
powers, using a method that is readily generalizable for
determining the general formula for the sum of any integral powers.
He performed an integration in order to find the volume of a
paraboloid, and was able to generalize
his result for the integrals of
polynomials up to the
fourth degree. He thus came close to
finding a general formula for the
integrals
of polynomials, but he was not concerned with any polynomials
higher than the fourth degree.
In the late 11th century,
Omar Khayyam
wrote
Discussions of the Difficulties in Euclid, a book
about flaws in
Euclid's
Elements, especially the
parallel postulate, and laid the
foundations for
analytic geometry
and
non-Euclidean geometry.
He was also the first to find the general geometric solution to
cubic equations. He was also very
influential in
calendar
reform.
In the late 12th century,
Sharaf al-Dīn al-Tūsī
introduced the concept of a
function, and he was the first to
discover the
derivative of
cubic polynomials. His
Treatise on
Equations developed concepts related to differential calculus,
such as the derivative function and the
maxima and minima of curves, in order to
solve cubic equations which may not have positive solutions.
In the 13th century,
Nasir al-Din
Tusi (Nasireddin) made advances in
spherical trigonometry. He also wrote
influential work on
Euclid's
parallel postulate. In the 15th century,
Ghiyath al-Kashi computed the value
of
π to the 16th decimal place. Kashi also
had an algorithm for calculating
nth roots, which was a
special case of the methods given many centuries later by
Ruffini and
Horner.
Other notable Muslim mathematicians included
al-Samawal,
Abu'l-Hasan al-Uqlidisi,
Jamshid al-Kashi,
Thabit ibn Qurra,
Abu
Kamil and
Abu Sahl
al-Kuhi.
Other achievements of Muslim mathematicians during this period
include the development of
algebra and
algorithms (see
Muhammad ibn
Mūsā al-Khwārizmī), the development of
spherical trigonometry, the addition
of the
decimal point notation to the
Arabic numerals, the discovery of
all the modern
trigonometric
functions besides the sine,
al-Kindi's
introduction of
cryptanalysis and
frequency analysis, the
development of
analytic geometry
by
Ibn al-Haytham, the beginning of
algebraic geometry by
Omar Khayyam, the first refutations of
Euclidean geometry and the
parallel postulate by
Nasīr al-Dīn
al-Tūsī, the first attempt at a
non-Euclidean geometry by Sadr
al-Din, the development of an
algebraic notation by
al-Qalasādī,
and many other advances in algebra,
arithmetic, calculus,
cryptography,
geometry,
number theory and
trigonometry.
During the time of the
Ottoman Empire
from the 15th century, the development of Islamic mathematics
became stagnant.
Medieval European mathematics (c. 500–1400)
Medieval European interest in mathematics was driven by concerns
quite different from those of modern mathematicians. One driving
element was the belief that mathematics provided the key to
understanding the created order of nature, frequently justified by
Plato's
Timaeus and the
apocryphal biblical passage (in the
Book of Wisdom) that God had
ordered all things in measure, and number, and
weight.
Early Middle Ages (c. 500–1100)
Boethius provided a place for mathematics
in the curriculum when he coined the term
quadrivium to describe the study of
arithmetic, geometry, astronomy, and music. He wrote
De
institutione arithmetica, a free translation from the Greek of
Nicomachus's
Introduction to
Arithmetic;
De institutione musica, also derived from
Greek sources; and a series of excerpts from
Euclid's
Elements. His works were
theoretical, rather than practical, and were the basis of
mathematical study until the recovery of Greek and Arabic
mathematical works.
Rebirth of mathematics in Europe (1100–1400)
In the 12th century, European scholars traveled to Spain and Sicily
seeking
scientific Arabic texts, including
al-Khwarizmi's
The
Compendious Book on Calculation by Completion and
Balancing, translated into Latin by
Robert of Chester, and the complete text
of
Euclid's Elements,
translated in various versions by
Adelard of Bath,
Herman of Carinthia, and
Gerard of Cremona.
These new sources sparked a renewal of mathematics.
Fibonacci, writing in the
Liber Abaci, in 1202 and updated in 1254,
produced the first significant mathematics in Europe since the time
of
Eratosthenes, a gap of more than a
thousand years. The work introduced
Hindu-Arabic numerals to Europe, and
discussed many other mathematical problems.
The fourteenth century saw the development of new mathematical
concepts to investigate a wide range of problems. One important
contribution was development of mathematics of local motion.
Thomas Bradwardine proposed that
speed (V) increases in arithmetic proportion as the ratio of force
(F) to resistance (R) increases in geometric proportion.
Bradwardine expressed this by a series of specific examples, but
although the logarithm had not yet been conceived, we can express
his conclusion anachronistically by writing:V = log (F/R).
Bradwardine's analysis is an example of transferring a mathematical
technique used by
al-Kindi and
Arnald of Villanova to quantify the
nature of compound medicines to a different physical problem.
One of the 14th-century
Oxford
Calculators,
William
Heytesbury, lacking
differential calculus and the concept
of
limits, proposed to measure
instantaneous speed "by the path that
would be
described by [a body]
if... it were moved
uniformly at the same degree of speed with which it is moved in
that given instant".
Heytesbury and others mathematically determined the distance
covered by a body undergoing uniformly accelerated motion (today
solved by
integration), stating that "a
moving body uniformly acquiring or losing that increment [of speed]
will traverse in some given time a [distance] completely equal to
that which it would traverse if it were moving continuously through
the same time with the mean degree [of speed]".
Nicole Oresme at the University
of Paris and the Italian Giovanni di Casali independently provided
graphical demonstrations of this relationship, asserting that the
area under the line depicting the constant acceleration,
represented the total distance traveled. In a later
mathematical commentary on Euclid's
Elements, Oresme made
a more detailed general analysis in which he demonstrated that a
body will acquire in each successive increment of time an increment
of any quality that increases as the odd numbers. Since Euclid had
demonstrated the sum of the odd numbers are the square numbers, the
total quality acquired by the body increases as the square of the
time.
Early modern European mathematics (c. 1400–1600)
In Europe at the dawn of the
Renaissance, mathematics was still limited by
the cumbersome notation using
Roman
numerals and expressing relationships using words, rather than
symbols: there was no plus sign, no equal sign, and no use of
x as an unknown.
In 16th century European mathematicians began to make advances
without precedent anywhere in the world, so far as is known today.
The first of these was the general solution of
cubic equations, generally credited to
Scipione del Ferro c.
1510, but
first published by Johannes
Petreius in Nuremberg in Gerolamo
Cardano's Ars magna, which also included the solution
of the general quartic equation
from Cardano's student Lodovico
Ferrari.
From this point on, mathematical developments came swiftly,
contributing to and benefiting from contemporary advances in the
physical sciences. This progress
was greatly aided by advances in
printing.
The earliest
mathematical
books printed were
Peurbach's
Theoricae nova
planetarum (1472), followed by a book on commercial
arithmetic, the
Treviso
Arithmetic (1478), and then the first extant book on
mathematics, Euclid's
Elements, printed and published by
Ratdolt in 1482.
Driven by the demands of navigation and the growing need for
accurate maps of large areas,
trigonometry grew to be a major branch of
mathematics.
Bartholomaeus
Pitiscus was the first to use the word, publishing his
Trigonometria in 1595. Regiomontanus's table of sines and
cosines was published in 1533.
By century's end, thanks to
Regiomontanus (1436–76) and
Simon Stevin (1548–1620), among others,
mathematics was written using Hindu-Arabic numerals and in a form
not too different from the notation used today.
17th century
The 17th century saw an unprecedented explosion of mathematical and
scientific ideas across Europe.
Galileo, an
Italian, observed the moons of Jupiter in orbit about that planet,
using a telescope based on a toy imported from Holland.
Tycho Brahe, a Dane, had gathered an enormous
quantity of mathematical data describing the positions of the
planets in the sky. His student,
Johannes Kepler, a German, began to work
with this data. In part because he wanted to help Kepler in his
calculations,
John Napier, in Scotland,
was the first to investigate
natural
logarithms. Kepler succeeded in formulating mathematical laws
of planetary motion. The
analytic
geometry developed by
René
Descartes (1596–1650), a French mathematician and philosopher,
allowed those orbits to be plotted on a graph, in
Cartesian coordinates.
Building on earlier work by many predecessors,
Isaac Newton, an Englishman, discovered the
laws of physics explaining
Kepler's
Laws, and brought together the concepts now known as
calculus. Independently,
Gottfried Wilhelm Leibniz, in
Germany, developed calculus and much of the calculus notation still
in use today. Science and mathematics had become an international
endeavor, which would soon spread over the entire world.
In addition to the application of mathematics to the studies of the
heavens, applied mathematics began to expand into new areas, with
the correspondence of
Pierre de
Fermat and
Blaise Pascal. Pascal
and Fermat set the groundwork for the investigations of
probability theory and the corresponding
rules of
combinatorics in their
discussions over a game of
gambling.
Pascal, with his
wager, attempted to
use the newly developing probability theory to argue for a life
devoted to religion, on the grounds that even if the probability of
success was small, the rewards were infinite. In some sense, this
foreshadowed the development of
utility
theory in the 18th–19th century.
18th century
The most influential mathematician of the 1700s was arguably
Leonhard Euler.
His contributions
range from founding the study of graph
theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical
terms and notations. For example, he named the square root
of minus 1 with the symbol
i, and he popularized the use
of the Greek letter \pi to stand for the ratio of a circle's
circumference to its diameter. He made numerous contributions to
the study of topology, graph theory, calculus, combinatorics, and
complex analysis, as evidenced by the multitude of theorems and
notations named for him.
Other important European mathematicians of the 18th century
included
Joseph Louis
Lagrange, who did pioneering work in number theory, algebra,
differential calculus, and the calculus of variations, and
Laplace who, in the age of
Napoleon did important work on the foundations of
celestial mechanics and on
statistics.
19th century
Behavior of lines with a common perpendicular in each of the three
types of geometry
Throughout the 19th century mathematics became increasingly
abstract. In the 19th century lived
Carl Friedrich Gauss (1777–1855).
Leaving aside his many contributions to science, in pure
mathematics he did revolutionary work on
function of
complex variables, in
geometry, and on the convergence of
series. He gave the first satisfactory
proofs of the
fundamental
theorem of algebra and of the
quadratic reciprocity law.
This century saw the development of the two forms of
non-Euclidean geometry, where the
parallel postulate of
Euclidean geometry no longer holds.The
Russian mathematician
Nikolai Ivanovich Lobachevsky
and his rival, the Hungarian mathematician
Janos Bolyai, independently defined and studied
hyperbolic geometry, where
uniqueness of parallels no longer holds. In this geometry the sum
of angles in a triangle add up to less than 180°.
Elliptic geometry was developed later in
the 19th century by the German mathematician
Bernhard Riemann; here no parallel can be
found and the angles in a triangle add up to more than 180°.
Riemann also developed
Riemannian
geometry, which unifies and vastly generalizes the three types
of geometry, and he defined the concept of a
manifold, which generalize the ideas of
curves and
surfaces.
The 19th century saw the beginning of a great deal of
abstract algebra.
Hermann Grassmann in Germany gave a first
version of
vector spaces,
William Rowan Hamilton in Ireland
developed
noncommutative
algebra. The British mathematician
George Boole devised an algebra that soon
evolved into what is now called
Boolean
algebra, in which the only numbers were 0 and 1 and in which,
famously, 1 + 1 = 1. Boolean algebra is the
starting point of
mathematical
logic and has important applications in
computer science.
Augustin-Louis Cauchy,
Bernhard Riemann, and
Karl Weierstrass reformulated the calculus
in a more rigorous fashion.
Also, for the first time, the limits of mathematics were explored.
Niels Henrik Abel, a Norwegian,
and
Évariste Galois, a
Frenchman, proved that there is no general algebraic method for
solving polynomial equations of degree greater than four. Other
19th century mathematicians utilized this in their proofs that
straightedge and compass alone are not sufficient to
trisect an arbitrary angle, to
construct the side of a cube twice the volume of a given cube, nor
to construct a square equal in area to a given circle.
Mathematicians had vainly attempted to solve all of these problems
since the time of the ancient Greeks.
Abel and Galois's investigations into the solutions of various
polynomial equations laid the groundwork for further developments
of
group theory, and the associated
fields of
abstract algebra. In the
20th century physicists and other scientists have seen group theory
as the ideal way to study
symmetry.
In the later 19th century,
Georg Cantor
established the first foundations of
set
theory, which enabled the rigorous treatment of the notion of
infinity and has become the common language of nearly all
mathematics. Cantor's set theory, and the rise of
mathematical logic in the hands of
Peano,
L.
E. J. Brouwer,
David Hilbert,
Bertrand Russell, and
A.N. Whitehead,
initiated a long running debate on the
foundations of mathematics.
The 19th century saw the founding of a number of national
mathematical societies: the
London Mathematical Society in
1865, the
Société
Mathématique de France in 1872, the
Circolo Mathematico di
Palermo in 1884, the
Edinburgh Mathematical
Society in 1883, and the
American Mathematical Society
in 1888.
20th century
The 20th century saw mathematics become a major profession. Every
year, thousands of new Ph.D.s in mathematics are awarded, and jobs
are available in both teaching and industry. In earlier centuries,
there were few creative mathematicians in the world at any one
time. For the most part, mathematicians were either born to wealth,
like
Napier, or supported by wealthy
patrons, like
Gauss. A few, like
Fourier, derived meager livelihoods from
teaching in universities.
Niels Henrik
Abel, unable to obtain a position, died in poverty of
malnutrition and tuberculosis at the age of twenty-six.
As in most areas of study, the explosion of knowledge in the
scientific age has led to specialization: there are now hundreds of
specialized areas in mathematics and the latest
Mathematics Subject
Classification runs to 46 pages. More and more
mathematical journals were published
and, by the end of the century, the development of the
world wide web led to online
publishing.
In a 1900 speech to the
International Congress
of Mathematicians,
David Hilbert
set out a list of
23 unsolved
problems in mathematics. These problems, spanning many areas of
mathematics, formed a central focus for much of 20th century
mathematics. Today, 10 have been solved, 7 are partially solved,
and 2 are still open. The remaining 4 are too loosely formulated to
be stated as solved or not.
Famous historical conjectures were finally proved. In 1976,
Wolfgang Haken and
Kenneth Appel used a computer to prove the
four color theorem.
Andrew Wiles, building on the work of others,
proved
Fermat's Last Theorem
in 1995.
Paul Cohen and
Kurt Gödel proved that the
continuum hypothesis is
independent of (could neither be proved
nor disproved from) the
standard axioms of set
theory. In 1998
Thomas
Callister Hales proved the
Kepler
conjecture.
Mathematical collaborations of unprecedented size and scope took
place. A famous example is the
classification of finite
simple groups (also called the "enormous theorem"), whose proof
between 1955 and 1983 required 500-odd journal articles by about
100 authors, and filling tens of thousands of pages. A group of
French mathematicians, including
Jean Dieudonné and
André Weil, publishing under the
pseudonym "
Nicolas
Bourbaki," attempted to exposit all of known mathematics as a
coherent rigorous whole. The resulting several dozen volumes has
had a controversial influence on mathematical education.
Differential geometry came
into its own when
Einstein used it in
general relativity. Entire new
areas of mathematics such as
mathematical logic,
topology, and
John von
Neumann's
game theory changed the
kinds of questions that could be answered by mathematical methods.
All kinds of
structures were
abstracted using axioms and given names like
metric spaces,
topological spaces etc. As mathematicians
do, the concept of an abstract structure was itself abstracted and
led to
category theory.
Grothendieck and
Serre recast
algebraic geometry using
sheaf theory. Large advances were made
in the qualitative study of
dynamical systems that
Poincaré had began in the 1890s.
Measure theory was developed in the late 19th
and early 20th century. Applications of measures include the
Lebesgue integral,
Kolmogorov's axiomatisation of
probability theory, and
ergodic theory.
Knot
theory greatly expanded. Other new areas include
functional analysis,
Laurent Schwarz's
distribution theory,
fixed point theory,
singularity theory and
René Thom's
catastrophe theory,
model theory, and
Mandelbrot's
fractals.
The development and continual improvement of
computers, at first mechanical analog machines and
then digital electronic machines, allowed
industry to deal with larger and larger amounts of
data to facilitate mass production and distribution and
communication, and new areas of mathematics were developed to deal
with this:
Alan Turing's
computability theory;
complexity theory;
Claude Shannon's
information theory;
signal processing;
data analysis;
optimization and other areas of
operations research. In the
preceding centuries much mathematical focus was on
calculus and continuous functions, but the rise of
computing and communication networks led to an increasing
importance of
discrete concepts
and the expansion of
combinatorics
including
graph theory. The speed and
data processing abilities of computers also enabled the handling of
mathematical problems that were too time-consuming to deal with by
pencil and paper calculations, leading to areas such as
numerical analysis and
symbolic computation.
At the same time, deep insights were made about the limitations to
mathematics. In 1929 and 1930, it was proved the truth or falsity
of all statements formulated about the
natural numbers plus one of addition and
multiplication, was
decidable, i.e. could
be determined by algorithm. In 1931,
Kurt Gödel found that this was not the case
for the natural numbers plus both addition and multiplication; this
system, known as
Peano arithmetic,
was in fact
incompletable.
(Peano arithmetic is adequate for a good deal of
number theory, including the notion of
prime number.) A consequence of Gödel's
two
incompleteness theorems
is that in any mathematical system that includes Peano arithmetic
(including all of
analysis and
geometry), truth necessarily outruns proof,
i.e. there are true statements that
cannot be proved within the system.
Hence mathematics cannot be reduced to mathematical logic, and
David Hilbert's dream of making all of
mathematics complete and consistent died.
One of the more colorful figures in 20th century mathematics was
Srinivasa Aiyangar
Ramanujan (1887–1920) who, despite being largely self-educated,
conjectured or proved over 3000 theorems, including properties of
highly composite numbers,
the
partition
function and its
asymptotics, and
mock theta functions. He
also made major investigations in the areas of
gamma functions,
modular forms,
divergent series,
hypergeometric series and
prime number theory.
Paul Erdős published more papers
than any other mathematician in history, working with hundreds of
collaborators. Mathematicians have a game equivalent to the
Kevin Bacon Game, which leads to
the
Erdős number of a
mathematician. This describes the "collaborative distance" between
a person and Paul Erdős, as measured by joint authorship of
mathematical papers.
21st century
In 2000,
the Clay
Mathematics Institute announced the Millennium Prize Problems, and in
2003 the Poincaré
conjecture was solved by Grigori
Perelman.
Most mathematical journals now have online versions as well as
print versions, and many online-only journals are launched. There
is an increasing drive towards
open access publishing, first
popularized by the
arXiv.
See also
References
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Plimpton 322, Pythagorean triples, and the Babylonian triangle
parameter equations", Historia Mathematica, 8, 1981, pp.
277—318.
- O. Neugebauer, "The Exact Sciences in Antiquity", Chap. IV
"Egyptian Mathematics and Astronomy", 2nd ed., Dover, New York,
1969, pp. 71—96.
- Sir Thomas L. Heath, A Manual of Greek Mathematics,
Dover, 1963, p. 1: "In the case of mathematics, it is the Greek
contribution which it is most essential to know, for it was the
Greeks who first made mathematics a science."
- Robert Kaplan, "The Nothing That Is: A Natural History of
Zero", Allen Lane/The Penguin Press, London, 1999
- "The ingenious method of expressing every possible number using
a set of ten symbols (each symbol having a place value and an
absolute value) emerged in India. The idea seems so simple nowadays
that its significance and profound importance is no longer
appreciated. Its simplicity lies in the way it facilitated
calculation and placed arithmetic foremost amongst useful
inventions. the importance of this invention is more readily
appreciated when one considers that it was beyond the two greatest
men of Antiquity, Archimedes and Apollonius." - Pierre Simon
Laplace
http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html
- A. P. Juschkewitsch, "Geschichte der Mathematik im
Mittelalter", Teubner, Leipzig, 1964
- http://mathworld.wolfram.com/LebomboBone.html
- An old mathematical object
- Mathematics in (central) Africa before
colonization
- Marshack, Alexander (1991): The Roots of Civilization,
Colonial Hill, Mount Kisco, NY.
- Thom, Alexander, and Archie Thom, 1988, "The metrology and
geometry of Megalithic Man", pp 132-151 in C.L.N. Ruggles, ed.,
Records in Stone: Papers in memory of Alexander Thom.
Cambridge Univ. Press. ISBN 0-521-33381-4.
- Duncan J. Melville (2003). Third Millennium Chronology, Third
Millennium Mathematics. St. Lawrence University.
- Egyptian Unit Fractions at MathPages
- Howard Eves, An Introduction to the History of
Mathematics, Saunders, 1990, ISBN 0030295580
- Martin Bernal, "Animadversions on the Origins of Western
Science", pp. 72–83 in Michael H. Shank, ed., The Scientific
Enterprise in Antiquity and the Middle Ages, (Chicago:
University of Chicago Press) 2000, p. 75.
- Eves, Howard, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0.
- Howard Eves, An Introduction to the History of
Mathematics, Saunders, 1990, ISBN 0030295580 p. 141: "No work,
except The Bible,
has been more widely used...."
- [1]. The values given are 25/8 (3.125), 900/289
(3.11418685...), 1156/361 (3.202216...), and 339/108 (3.1389), the
last of which is correct (when rounded) to two decimal places
- The Indian Sulbasutras. The method constructs a
square of side 13/15 times the diameter of the given circle
(corresponds to taking π=3.00444), so it is not a very good
approximation.
- Rachel W. Hall. Math
for poets and drummers. Math Horizons
15 (2008) 10-11.
- The History of Algebra. Louisiana State University.
- "The six cases of equations given above exhaust all
possibilities for linear and quadratic equations having positive
root. So systematic and exhaustive was al-Khwarizmi's exposition
that his readers must have had little difficulty in mastering the
solutions."
- Gandz and Saloman (1936), The sources of al-Khwarizmi's
algebra, Osiris i, pp. 263–77: "In a sense, Khwarizmi is more
entitled to be called "the father of algebra" than Diophantus
because Khwarizmi is the first to teach algebra in an elementary
form and for its own sake, Diophantus is primarily concerned with
the theory of numbers".
- Victor J. Katz (1998). History of Mathematics: An
Introduction, pp. 255–59. Addison-Wesley. ISBN 0321016181.
- F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par
Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
- Victor J. Katz (1995), "Ideas of Calculus in Islam and India",
Mathematics Magazine 68 (3): 163–74.
- J. L. Berggren (1990). "Innovation and Tradition in Sharaf
al-Din al-Tusi's Muadalat", Journal of the American Oriental
Society 110 (2), pp. 304–09.
- Wisdom, 11:21
- Caldwell, John (1981) "The De Institutione Arithmetica
and the De Institutione Musica", pp. 135–54 in Margaret
Gibson, ed., Boethius: His Life, Thought, and Influence,
(Oxford: Basil Blackwell).
- Folkerts, Menso, "Boethius" Geometrie II, (Wiesbaden:
Franz Steiner Verlag, 1970).
- Marie-Thérèse d'Alverny, "Translations and Translators", pp.
421–62 in Robert L. Benson and Giles Constable, Renaissance and
Renewal in the Twelfth Century, (Cambridge: Harvard University
Press, 1982).
- Guy Beaujouan, "The Transformation of the Quadrivium", pp.
463–87 in Robert L. Benson and Giles Constable, Renaissance and
Renewal in the Twelfth Century, (Cambridge: Harvard University
Press, 1982).
- Grant, Edward and John E. Murdoch (1987), eds., Mathematics
and Its Applications to Science and Natural Philosophy in the
Middle Ages, (Cambridge: Cambridge University Press) ISBN
0-521-32260-X.
- Clagett, Marshall (1961) The Science of Mechanics in the
Middle Ages, (Madison: University of Wisconsin Press), pp.
421–40.
- Murdoch, John E. (1969) "Mathesis in Philosophiam
Scholasticam Introducta: The Rise and Development of the
Application of Mathematics in Fourteenth Century Philosophy and
Theology", in Arts libéraux et philosophie au Moyen Âge
(Montréal: Institut d'Études Médiévales), at pp. 224–27.
- Clagett, Marshall (1961) The Science of Mechanics in the
Middle Ages, (Madison: University of Wisconsin Press), pp.
210, 214–15, 236.
- Clagett, Marshall (1961) The Science of Mechanics in the
Middle Ages, (Madison: University of Wisconsin Press), p.
284.
- Clagett, Marshall (1961) The Science of Mechanics in the
Middle Ages, (Madison: University of Wisconsin Press), pp.
332–45, 382–91.
- Nicole Oresme, "Questions on the Geometry of Euclid"
Q. 14, pp. 560–65, in Marshall Clagett, ed., Nicole Oresme and
the Medieval Geometry of Qualities and Motions, (Madison:
University of Wisconsin Press, 1968).
- Eves, Howard, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0, p. 379, "...the concepts of
calculus...(are) so far reaching and have exercised such an impact
on the modern world that it is perhaps correct to say that without
some knowledge of them a person today can scarcely claim to be well
educated."
- Mathematics Subject Classification 2010
- Maurice Mashaal, 2006. Bourbaki: A Secret Society of
Mathematicians. American Mathematical
Society. ISBN 0821839675, ISBN 978-0821839676.
Further reading
- Boyer, C. B., A History of Mathematics, 2nd ed. rev.
by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991
pbk ed. ISBN 0-471-54397-7).
- Eves, Howard, An Introduction to the History of
Mathematics, Saunders, 1990, ISBN 0-03-029558-0,
- Hoffman, Paul,
The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical
Truth. New York: Hyperion, 1998 ISBN 0-7868-6362-5.
- van der Waerden, B. L., Geometry and Algebra in Ancient
Civilizations, Springer, 1983, ISBN 0387121595.
- O'Connor, John J. and Robertson, Edmund F. The MacTutor History of Mathematics Archive.
(See also
MacTutor History
of Mathematics archive.) This website contains biographies,
timelines and historical articles about mathematical concepts; at
the School of Mathematics and Statistics, University
of St. Andrews, Scotland. (Or see the alphabetical list of history topics.)
- Burton, David M. The History of Mathematics: An
Introduction. McGraw Hill: 1997.
- Katz, Victor J. A History of Mathematics: An
Introduction, 2nd Edition. Addison-Wesley: 1998.
- Kline, Morris. Mathematical Thought from Ancient to Modern
Times.
- .
- .
External links
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