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
Plimpton 322 (Babylonian c. 1900 BC), the Rhind Mathematical Papyrus
(Egyptian c. 2000–1800 BC) and the Moscow Mathematical Papyrus
(Egyptian c. 1890 BC). All of these texts mention the so-called
Pythagorean triples and so, by inference, the
Pythagorean theorem
Pythagorean theorem ,
which seems to be the most ancient and widespread mathematical
development after basic arithmetic and geometry.
The study of mathematics as a demonstrative discipline begins in the
6th century BC with the
Pythagoreans , who coined the term
"mathematics" from the ancient Greek μάθημα (mathema), meaning
"subject of instruction".
Greek mathematics
Greek mathematics 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
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 millennium AD in
India
India and were transmitted to the west via
Islamic mathematics
Islamic mathematics through
the work of
Muḥammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī . Islamic
mathematics , in turn, developed and expanded the mathematics known to
these civilizations. Many Greek and
Arabic
Arabic texts on mathematics were
then translated into
Latin
Latin , which led to further development of
mathematics in medieval Europe .
From ancient times through the
Middle Ages
Middle Ages , periods of mathematical
discovery were often followed by centuries of stagnation. Beginning in
Renaissance
Renaissance
Italy
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.
CONTENTS
* 1 Prehistoric mathematics
* 2
Babylonian mathematics
Babylonian mathematics
* 3
Egyptian mathematics
Egyptian mathematics
* 4
Greek mathematics
Greek mathematics
* 5
Chinese mathematics
* 6
Indian mathematics
Indian mathematics
* 7
Islamic mathematics
Islamic mathematics
* 8 Medieval European mathematics
* 9
Renaissance
Renaissance mathematics
* 10
Mathematics
Mathematics during the
Scientific Revolution
Scientific Revolution
* 10.1 17th century
* 10.2 18th century
* 11 Modern mathematics
* 11.1 19th century
* 11.2 20th century
* 11.3 21st century
* 12
Future of mathematics
* 13 See also
* 14 Notes
* 15 References
* 16 Further reading
* 16.1 General
* 16.2 Books on a specific period
* 16.3 Books on a specific topic
* 17 External links
* 17.1 Documentaries
* 17.2 Educational material
* 17.3 Bibliographies
* 17.4 Organizations
* 17.5 Journals
* 17.6 Directories
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. The idea of
the "number" concept evolving gradually over time is supported by the
existence of languages which preserve the distinction between "one",
"two", and "many", but not of numbers larger than two.
Prehistoric artifacts discovered in Africa, dated 20,000 years old or
more suggest early attempts to quantify time. The
Ishango bone ,
found near the headwaters of the
Nile
Nile river (northeastern Congo ), may
be more than 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. Peter Rudman argues that the development of the
concept of prime numbers could only have come about after the concept
of division, which he dates to after 10,000 BC, with prime numbers
probably not being understood until about 500 BC. He also writes that
"no attempt has been made to explain why a tally of something should
exhibit multiples of two, prime numbers between 10 and 20, and some
numbers that are almost multiples of 10." The Ishango bone, according
to scholar
Alexander Marshack , may have influenced the later
development of mathematics in
Egypt
Egypt as, like some entries on the
Ishango bone, Egyptian arithmetic also made use of multiplication by
2; this however, is disputed.
Predynastic
Egyptians
Egyptians of the 5th millennium BC pictorially
represented geometric designs. It has been claimed that megalithic
monuments in
England
England and
Scotland
Scotland , dating from the 3rd millennium BC,
incorporate geometric ideas such as circles , ellipses , and
Pythagorean triples in their design. All of the above are disputed
however, and the currently oldest undisputed mathematical documents
are from Babylonian and dynastic Egyptian sources.
BABYLONIAN MATHEMATICS
Main article:
Babylonian mathematics
Babylonian mathematics See also:
Plimpton 322
Plimpton 322
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Babylonian mathematics
Babylonian mathematics refers to any mathematics of the peoples of
Mesopotamia
Mesopotamia (modern
Iraq
Iraq ) from the days of the early Sumerians
through the
Hellenistic period
Hellenistic period almost to the dawn of
Christianity
Christianity .
The majority of Babylonian mathematical work comes from two widely
separated periods: The first few hundred years of the second
millennium BC (Old Babylonian period), and the last few centuries of
the first millennium BC (
Seleucid
Seleucid period). It is named Babylonian
mathematics due to the central role of
Babylon
Babylon as a place of study.
Later under the
Arab
Arab Empire , Mesopotamia, especially
Baghdad
Baghdad , once
again became an important center of study for
Islamic mathematics
Islamic mathematics .
In contrast to the sparsity of sources in
Egyptian mathematics
Egyptian mathematics , our
knowledge of
Babylonian mathematics
Babylonian mathematics is derived from more than 400 clay
tablets unearthed since the 1850s. Written in
Cuneiform script
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.
Geometry
Geometry problem on a clay tablet
belonging to a school for scribes;
Susa
Susa , first half of the 2nd
millennium BCE
Babylonian mathematics
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. It is likely the sexagesimal system was chosen
because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and
30. 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. The power
of the Babylonian notational system lay in that it could be used to
represent fractions as easily as whole numbers; thus multiplying two
numbers that contained fractions was no different than multiplying
integers, similar to our modern notation. The notational system of
the Babylonians was the best of any civilization until the Renaissance
, and its power allowed it to achieve remarkable computation accuracy
and power; for example, the Babylonian tablet YBC 7289 gives an
approximation of √2 accurate to five decimal places. The
Babylonians lacked, however, an equivalent of the decimal point, and
so the place value of a symbol often had to be inferred from the
context. By the
Seleucid
Seleucid period, the Babylonians had developed a zero
symbol as a placeholder for empty positions; however it was only used
for intermediate positions. This zero sign does not appear in
terminal positions, thus the Babylonians came close but did not
develop a true place value system.
Other topics covered by
Babylonian mathematics
Babylonian mathematics include fractions,
algebra, quadratic and cubic equations, and the calculation of regular
reciprocal pairs . The tablets also include multiplication tables and
methods for solving linear , quadratic equations and cubic equations ,
a remarkable achievement for the time. Tablets from the Old
Babylonian period also contain the earliest known statement of the
Pythagorean theorem
Pythagorean theorem . However, as with Egyptian mathematics,
Babylonian mathematics
Babylonian mathematics shows no awareness of the difference between
exact and approximate solutions, or the solvability of a problem, and
most importantly, no explicit statement of the need for proofs or
logical principles.
EGYPTIAN MATHEMATICS
Main article:
Egyptian mathematics
Egyptian mathematics Image of Problem 14 from the
Moscow Mathematical Papyrus
Moscow Mathematical Papyrus . The problem includes a diagram
indicating the dimensions of the truncated pyramid.
Egyptian mathematics
Egyptian mathematics refers to mathematics written in the Egyptian
language . From the
Hellenistic period
Hellenistic period , Greek replaced Egyptian as
the written language of Egyptian scholars. Mathematical study in Egypt
later continued under the
Arab
Arab Empire as part of
Islamic mathematics
Islamic mathematics ,
when
Arabic
Arabic became the written language of Egyptian scholars.
The most extensive Egyptian mathematical text is the Rhind papyrus
(sometimes also called the Ahmes Papyrus after its author), dated to
c. 1650 BC but likely a copy of an older document from the Middle
Kingdom of about 2000–1800 BC. It is an instruction manual for
students 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). It also shows how to solve first order linear
equations as well as arithmetic and geometric series .
Another significant Egyptian mathematical text is the Moscow papyrus
, also from the Middle Kingdom period, dated to c. 1890 BC. 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 (truncated pyramid).
Finally, the
Berlin Papyrus 6619
Berlin Papyrus 6619 (c. 1800 BC) shows that ancient
Egyptians
Egyptians could solve a second-order algebraic equation .
GREEK MATHEMATICS
Main article:
Greek mathematics
Greek mathematics The
Pythagorean theorem
Pythagorean theorem . The
Pythagoreans are generally credited with the first proof of the
theorem.
Greek mathematics
Greek mathematics refers to the mathematics written in the Greek
language from the time of
Thales of Miletus
Thales of Miletus (~600 BC) to the closure
of the Academy of
Athens
Athens in 529 AD. Greek mathematicians lived in
cities spread over the entire Eastern Mediterranean, from
Italy
Italy to
North Africa, but were united by culture and language. Greek
mathematics of the period following
Alexander the Great
Alexander the Great is sometimes
called Hellenistic mathematics.
Greek mathematics
Greek mathematics was much more sophisticated than the mathematics
that had been developed by earlier cultures. All surviving records of
pre-
Greek mathematics
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, and used
mathematical rigor to prove them.
Greek mathematics
Greek mathematics is thought to have begun with
Thales of Miletus
Thales of Miletus (c.
624–c.546 BC) and
Pythagoras of Samos
Pythagoras of Samos (c. 582–c. 507 BC). Although
the extent of the influence is disputed, they were probably inspired
by Egyptian and
Babylonian mathematics
Babylonian mathematics . According to legend,
Pythagoras traveled to
Egypt
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. He is credited
with the first use of deductive reasoning applied to geometry, by
deriving four corollaries to Thales\' Theorem . As a result, he has
been hailed as the first true mathematician and the first known
individual to whom a mathematical discovery has been attributed.
Pythagoras established the Pythagorean School , whose doctrine it was
that mathematics ruled the universe and whose motto was "All is
number". It was the
Pythagoreans who coined the term "mathematics",
and with whom the study of mathematics for its own sake begins. The
Pythagoreans are credited with the first proof of the Pythagorean
theorem , though the statement of the theorem has a long history, and
with the proof of the existence of irrational numbers . Although he
was preceded by the Babylonians and the Chinese , the Neopythagorean
mathematician
Nicomachus (60–120 AD) provided one of the earliest
Greco-Roman multiplication tables , whereas the oldest extant Greek
multiplication table is found on a wax tablet dated to the 1st century
AD (now found in the
British Museum
British Museum ). The association of the
Neopythagoreans with the Western invention of the multiplication table
is evident in its later Medieval name: the mensa Pythagorica.
One of the oldest surviving fragments of Euclid's Elements, found at
Oxyrhynchus
Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book
II, Proposition 5.
Archimedes
Archimedes used the method of exhaustion to
approximate the value of pi .
Plato
Plato (428/427 BC – 348/347 BC) is important in the history of
mathematics for inspiring and guiding others. His
Platonic Academy ,
in
Athens
Athens , became the mathematical center of the world in the 4th
century BC, and it was from this school that the leading
mathematicians of the day, such as
Eudoxus of Cnidus , came. Plato
also discussed the foundations of mathematics, clarified some of the
definitions (e.g. that of a line as "breadthless length"), and
reorganized the assumptions. The analytic method is ascribed to
Plato, while a formula for obtaining Pythagorean triples bears his
name.
Eudoxus (408–c.355 BC) developed the method of exhaustion , a
precursor of modern integration and a theory of ratios that avoided
the problem of incommensurable magnitudes . The former allowed the
calculations of areas and volumes of curvilinear figures, while the
latter enabled subsequent geometers to make significant advances in
geometry. Though he made no specific technical mathematical
discoveries,
Aristotle
Aristotle (384–c.322 BC) contributed significantly to
the development of mathematics by laying the foundations of logic .
In the 3rd century BC, the premier center of mathematical education
and research was the
Musaeum
Musaeum of
Alexandria
Alexandria . It was there that Euclid
(c. 300 BC) taught, and wrote the Elements , widely considered the
most successful and influential textbook of all time. The Elements
introduced mathematical rigor through the axiomatic method and is the
earliest example of the format still used in mathematics today, that
of definition, axiom, theorem, and proof. Although most of the
contents of the Elements were already known,
Euclid
Euclid arranged them into
a single, coherent logical framework. The Elements was known to all
educated people in the West until the middle of the 20th century and
its contents are still taught in geometry classes today. In addition
to the familiar theorems of
Euclidean geometry
Euclidean geometry , the Elements was
meant as an introductory textbook to all mathematical subjects of the
time, such as number theory , algebra and solid geometry , including
proofs that the square root of two is irrational and that there are
infinitely many prime numbers.
Euclid
Euclid also wrote extensively on other
subjects, such as conic sections , optics , spherical geometry , and
mechanics, but only half of his writings survive. Apollonius of
Perga made significant advances in the study of conic sections .
Archimedes
Archimedes (c.287–212 BC) of Syracuse , widely considered the
greatest mathematician of antiquity, used the method of exhaustion to
calculate the area under the arc of a parabola with the summation of
an infinite series , in a manner not too dissimilar from modern
calculus. He also showed one could use the method of exhaustion to
calculate the value of π with as much precision as desired, and
obtained the most accurate value of π then known, 310/71 < π <
310/70. He also studied the spiral bearing his name, obtained
formulas for the volumes of surfaces of revolution (paraboloid,
ellipsoid, hyperboloid), and an ingenious method of exponentiation
for expressing very large numbers. While he is also known for his
contributions to physics and several advanced mechanical devices,
Archimedes
Archimedes himself placed far greater value on the products of his
thought and general mathematical principles. He regarded as his
greatest achievement his finding of the surface area and volume of a
sphere, which he obtained by proving these are 2/3 the surface area
and volume of a cylinder circumscribing the sphere.
Apollonius of Perga
Apollonius of Perga (c. 262–190 BC) made significant advances to
the study of conic sections , showing that one can obtain all three
varieties of conic section by varying the angle of the plane that cuts
a double-napped cone. He also coined the terminology in use today for
conic sections, namely parabola ("place beside" or "comparison"),
"ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). His
work Conics is one of the best known and preserved mathematical works
from antiquity, and in it he derives many theorems concerning conic
sections that would prove invaluable to later mathematicians and
astronomers studying planetary motion, such as Isaac Newton. While
neither Apollonius nor any other Greek mathematicians made the leap to
coordinate geometry, Apollonius' treatment of curves is in some ways
similar to the modern treatment, and some of his work seems to
anticipate the development of analytical geometry by Descartes some
1800 years later. Title page of the 1621 edition of Diophantus'
Arithmetica, translated into
Latin
Latin by Claude Gaspard Bachet de
Méziriac .
Around the same time,
Eratosthenes of Cyrene (c. 276–194 BC)
devised the
Sieve of Eratosthenes for finding prime numbers . The 3rd
century BC is generally regarded as the "Golden Age" of Greek
mathematics, with advances in pure mathematics henceforth in relative
decline. Nevertheless, in the centuries that followed significant
advances were made in applied mathematics, most notably trigonometry ,
largely to address the needs of astronomers.
Hipparchus of Nicaea (c.
190–120 BC) is considered the founder of trigonometry for compiling
the first known trigonometric table, and to him is also due the
systematic use of the 360 degree circle. Heron of
Alexandria
Alexandria (c.
10–70 AD) is credited with Heron\'s formula for finding the area of
a scalene triangle and with being the first to recognize the
possibility of negative numbers possessing square roots. Menelaus of
Alexandria
Alexandria (c. 100 AD) pioneered spherical trigonometry through
Menelaus\' theorem . The most complete and influential trigonometric
work of antiquity is the
Almagest
Almagest of Ptolemy (c. AD 90–168), a
landmark astronomical treatise whose trigonometric tables would be
used by astronomers for the next thousand years. Ptolemy is also
credited with Ptolemy\'s theorem for deriving trigonometric
quantities, and the most accurate value of π outside of
China
China until
the medieval period, 3.1416.
Following a period of stagnation after Ptolemy, the period between
250 and 350 AD is sometimes referred to as the "Silver Age" of Greek
mathematics. During this period,
Diophantus
Diophantus made significant advances
in algebra , particularly indeterminate analysis , which is also known
as "Diophantine analysis". The study of
Diophantine equations
Diophantine equations and
Diophantine approximations is a significant area of research to this
day. His main work was the Arithmetica, a collection of 150 algebraic
problems dealing with exact solutions to determinate and indeterminate
equations . The Arithmetica had a significant influence on later
mathematicians, such as
Pierre de Fermat
Pierre de Fermat , who arrived at his famous
Last Theorem after trying to generalize a problem he had read in the
Arithmetica (that of dividing a square into two squares). Diophantus
also made significant advances in notation, the Arithmetica being the
first instance of algebraic symbolism and syncopation.
Among the last great Greek mathematicians is Pappus of Alexandria
(4th century AD). He is known for his hexagon theorem and centroid
theorem , as well as the
Pappus configuration and
Pappus graph
Pappus graph . His
Collection is a major source of knowledge on
Greek mathematics
Greek mathematics as most
of it has survived. Pappus is considered the last major innovator in
Greek mathematics, with subsequent work consisting mostly of
commentaries on earlier work. The
Haghia Sophia was designed by
the Greek mathematicians
Anthemius of Tralles and
Isidore of Miletus
Isidore of Miletus .
The first woman mathematician recorded by history was
Hypatia
Hypatia of
Alexandria
Alexandria (AD 350–415). She succeeded her father as Librarian at
the Great Library and wrote many works on applied mathematics. Because
of a political dispute, the Christian community in
Alexandria
Alexandria had her
stripped publicly and executed. Her death is sometimes taken as the
end of the era of the Alexandrian Greek mathematics, although work did
continue in
Athens
Athens for another century with figures such as
Proclus ,
Simplicius and
Eutocius . Although
Proclus and Simplicius were more
philosophers than mathematicians, their commentaries on earlier works
are valuable sources on Greek mathematics. The closure of the
neo-
Platonic Academy of
Athens
Athens by the emperor
Justinian
Justinian in 529 AD is
traditionally held as marking the end of the era of Greek mathematics,
although the Greek tradition continued unbroken in the Byzantine
empire with mathematicians such as
Anthemius of Tralles and Isidore of
Miletus , the architects of the
Haghia Sophia . Nevertheless,
Byzantine mathematics consisted mostly of commentaries, with little in
the way of innovation, and the centers of mathematical innovation were
to be found elsewhere by this time.
CHINESE MATHEMATICS
Main article:
Chinese mathematics Counting rod numerals
The
Tsinghua Bamboo Slips , containing the world's earliest decimal
multiplication table , dated 305 BC during the
Warring States
Warring States period
The Nine Chapters on the Mathematical Art , one of the earliest
surviving mathematical texts from
China
China (2nd century AD).
An analysis of early
Chinese mathematics has demonstrated its unique
development compared to other parts of the world, leading scholars to
assume an entirely independent development. The oldest extant
mathematical text from
China
China is the
Zhoubi Suanjing
Zhoubi Suanjing , variously dated
to between 1200 BC and 100 BC, though a date of about 300 BC appears
reasonable. However, the
Tsinghua Bamboo Slips , containing the
earliest known decimal multiplication table (although ancient
Babylonians had ones with a base of 60), is dated around 305 BC and is
perhaps the oldest surviving mathematical text of China.
Of particular note is the use in
Chinese mathematics of a decimal
positional notation system, the so-called "rod numerals" in which
distinct ciphers were used for numbers between 1 and 10, and
additional ciphers for powers of ten. Thus, the number 123 would be
written using the symbol for "1", followed by the symbol for "100",
then the symbol for "2" followed by the symbol for "10", followed by
the symbol for "3". This was the most advanced number system in the
world at the time, apparently in use several centuries before the
common era and well before the development of the Indian numeral
system. Rod numerals allowed the representation of numbers as large
as desired 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.
The oldest existent work on geometry in
China
China comes from the
philosophical Mohist canon c. 330 BC, compiled by the followers of
Mozi
Mozi (470–390 BC). The Mo Jing described various aspects of many
fields associated with physical science, and provided a small number
of geometrical theorems as well.
In 212 BC, the Emperor
Qin Shi Huang
Qin Shi Huang (Shi Huang-ti) commanded all
books in the Qin Empire other than officially sanctioned ones be
burned. This decree was not universally obeyed, but as a consequence
of this order little is known about ancient
Chinese mathematics before
this date. After the book burning of 212 BC, the
Han dynasty
Han dynasty (202
BC–220 AD) produced works of mathematics which presumably expanded
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 values of π . It created mathematical proof for the
Pythagorean theorem
Pythagorean theorem , and a mathematical formula for Gaussian
elimination .
Liu Hui
Liu Hui commented on the work in the 3rd century AD, and
gave a value of π accurate to 5 decimal places. Though more of a
matter of computational stamina than theoretical insight, in the 5th
century AD
Zu Chongzhi computed the value of π to seven decimal
places, which remained the most accurate value of π for almost the
next 1000 years. He also established a method which would later be
called Cavalieri\'s principle to find the volume of a sphere .
The high-water mark of
Chinese mathematics occurs in the 13th century
(latter part of the Song period ), with the development of Chinese
algebra. The most important text from that period is the Precious
Mirror of the Four Elements by Chu Shih-chieh (fl. 1280–1303),
dealing with the solution of simultaneous higher order algebraic
equations using a method similar to Horner\'s method . The Precious
Mirror also contains a diagram of Pascal\'s triangle with coefficients
of binomial expansions through the eighth power, though both appear in
Chinese works as early as 1100. 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
1238–1298).
Even after European mathematics began to flourish during the
Renaissance
Renaissance , European and
Chinese mathematics were separate
traditions, with significant Chinese mathematical output in decline
from the 13th century onwards.
Jesuit
Jesuit missionaries such as Matteo
Ricci carried mathematical ideas back and forth between the two
cultures from the 16th to 18th centuries, though at this point far
more mathematical ideas were entering
China
China than leaving.
INDIAN MATHEMATICS
Main article:
Indian mathematics
Indian mathematics See also: History of the
Hindu–Arabic numeral system
Hindu–Arabic numeral system The numerals used in the Bakhshali
manuscript , dated between the 2nd century BCE and the 2nd century CE.
Brahmi numerals (lower row) in
India
India in the 1st century CE
The earliest civilization on the Indian subcontinent is the Indus
Valley Civilization (mature phase: 2600 to 1900 BC) that flourished in
the
Indus river
Indus river basin. Their cities were laid out with geometric
regularity, but no known mathematical documents survive from this
civilization.
The oldest extant mathematical records from
India
India are the Sulba
Sutras (dated variously between the 8th century BC and the 2nd century
AD), appendices to religious texts which give simple rules for
constructing altars of various shapes, such as squares, rectangles,
parallelograms, and others. As with Egypt, the preoccupation with
temple functions points to an origin of mathematics in religious
ritual. The
Sulba Sutras
Sulba Sutras give methods for constructing a circle with
approximately the same area as a given square , which imply several
different approximations of the value of π . In addition, they
compute the square root of 2 to several decimal places, list
Pythagorean triples, and give a statement of the
Pythagorean theorem
Pythagorean theorem .
All of these results are present in Babylonian mathematics,
indicating Mesopotamian influence. It is not known to what extent the
Sulba Sutras
Sulba Sutras influenced later Indian mathematicians. As in China,
there is a lack of continuity in Indian mathematics; significant
advances are separated by long periods of inactivity.
Pāṇini
Pāṇini (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
Fibonacci numbers (called mātrāmeru).
The next significant mathematical documents from
India
India after the
Sulba Sutras
Sulba Sutras are the Siddhantas, astronomical treatises from the 4th
and 5th centuries AD (
Gupta period
Gupta period ) showing strong Hellenistic
influence. They are significant in that they contain the first
instance of trigonometric relations based on the half-chord, as is the
case in modern trigonometry, rather than the full chord, as was the
case in Ptolemaic trigonometry. Through a series of translation
errors, the words "sine" and "cosine" derive from the Sanskrit "jiya"
and "kojiya".
In the 5th century AD,
Aryabhata
Aryabhata wrote the
Aryabhatiya , a slim
volume, written in verse, intended to supplement the rules of
calculation used in astronomy and mathematical mensuration, though
with no feeling for logic or deductive methodology. Though about half
of the entries are wrong, it is in the
Aryabhatiya that the decimal
place-value system first appears. Several centuries later, the Muslim
mathematician
Abu Rayhan Biruni
Abu Rayhan Biruni described the
Aryabhatiya as a "mix of
common pebbles and costly crystals". Explanation of the sine
rule in
Yuktibhāṣā
In the 7th century,
Brahmagupta
Brahmagupta identified the
Brahmagupta
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
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. Various symbol sets are used to
represent numbers in the Hindu–
Arabic
Arabic numeral system, all of which
evolved from the Brahmi numerals . Each of the roughly dozen major
scripts of
India
India has its own numeral glyphs. In the 10th century,
Halayudha 's commentary on
Pingala 's work contains a study of the
Fibonacci sequence
Fibonacci sequence and Pascal\'s triangle , and describes the
formation of a matrix .
In the 12th century,
Bhāskara II lived in southern
India
India and wrote
extensively on all then known branches of mathematics. His work
contains mathematical objects equivalent or approximately equivalent
to infinitesimals, derivatives, the mean value theorem and the
derivative of the sine function. To what extent he anticipated the
invention of calculus is a controversial subject among historians of
mathematics.
In the 14th century,
Madhava of Sangamagrama , the founder of the
so-called Kerala School of
Mathematics
Mathematics , found the Madhava–Leibniz
series , and, using 21 terms, computed the value of π as
3.14159265359. Madhava also found the Madhava-Gregory series to
determine the arctangent, the Madhava-Newton power series to determine
sine and cosine and the Taylor approximation for sine and cosine
functions. In the 16th century,
Jyesthadeva consolidated many of the
Kerala School's developments and theorems in the Yukti-bhāṣā.
However, the Kerala School did not formulate a systematic theory of
differentiation and integration , nor is there any direct evidence of
their results being transmitted outside Kerala.
ISLAMIC MATHEMATICS
Main article:
Mathematics
Mathematics in medieval Islam See also: History of the
Hindu–Arabic numeral system
Hindu–Arabic numeral system Page from The Compendious Book on
Calculation by Completion and Balancing by Muhammad ibn Mūsā
al-Khwārizmī (c. AD 820)
The Islamic Empire established across
Persia
Persia , the
Middle East
Middle East ,
Central Asia
Central Asia ,
North Africa
North Africa , Iberia , and in parts of
India
India in the
8th century made significant contributions towards mathematics.
Although most Islamic texts on mathematics were written in
Arabic
Arabic ,
most of them were not written by Arabs , since much like the status of
Greek in the Hellenistic world,
Arabic
Arabic was used as the written
language of non-
Arab
Arab scholars throughout the Islamic world at the
time. Persians contributed to the world of
Mathematics
Mathematics alongside
Arabs.
In the 9th century, the Persian mathematician Muḥammad ibn Mūsā
al-Khwārizmī 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). 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 discussed 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-Khwārizmī originally described as al-jabr. 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."
In Egypt,
Abu Kamil extended algebra to the set of irrational numbers
, accepting square roots and fourth roots as solutions and
coefficients to quadratic equations. He also developed techniques used
to solve three non-linear simultaneous equations with three unknown
variables. One unique feature of his works was trying to find all the
possible solutions to some of his problems, including one where he
found 2676 solutions. His works formed an important foundation for
the development of algebra and influenced later mathematicians, such
as al-Karaji and Fibonacci.
Further developments in algebra were made by
Al-Karaji
Al-Karaji in his
treatise al-Fakhri, where he extends the methodology to incorporate
integer powers and integer roots of unknown quantities. Something
close to a proof by mathematical induction appears in a book written
by
Al-Karaji
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
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
Diophantus into Arabic. 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
Omar Khayyam wrote Discussions of the
Difficulties in Euclid, a book about what he perceived as flaws in
Euclid\'s Elements , especially the parallel postulate . He was also
the first to find the general geometric solution to cubic equations .
He was also very influential in calendar reform .
In the 13th century,
Nasir al-Din Tusi
Nasir al-Din Tusi (Nasireddin) made advances in
spherical trigonometry . He also wrote influential work on
Euclid
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 achievements of Muslim mathematicians during this period
include 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
Omar Khayyam and
the development of an algebraic notation by al-Qalasādī .
During the time of the
Ottoman Empire
Ottoman Empire and
Safavid Empire
Safavid Empire from the
15th century, the development of
Islamic mathematics
Islamic mathematics became stagnant.
MEDIEVAL EUROPEAN MATHEMATICS
Further information: Category:Medieval European mathematics , List of
medieval European scientists , and European science in the Middle Ages
Nicole Oresme (1323-1382), shown in this contemporary
illuminated manuscript with an armillary sphere in the foreground, was
the first to offer a mathematical proof for the divergence of the
harmonic series .
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
Plato 's Timaeus and the biblical passage (in the
Book of Wisdom
Book of Wisdom )
that God had ordered all things in measure, and number, and weight.
Boethius
Boethius provided a place for mathematics in the curriculum in the
6th century 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
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.
In the 12th century, European scholars traveled to Spain and Sicily
seeking scientific
Arabic
Arabic texts , including al-Khwārizmī 's The
Compendious Book on Calculation by Completion and Balancing ,
translated into
Latin
Latin by
Robert of Chester , and the complete text of
Euclid\'s Elements , translated in various versions by Adelard of Bath
,
Herman of Carinthia
Herman of Carinthia , and
Gerard of Cremona . These and other new
sources sparked a renewal of mathematics. See also:
Latin
Latin translations
of the 12th century
Leonardo of Pisa, now known as
Fibonacci
Fibonacci , serendipitously learned
about the Hindu–
Arabic
Arabic numerals on a trip to what is now
Béjaïa
Béjaïa ,
Algeria
Algeria with his merchant father. (Europe was still using Roman
numerals .) There, he observed a system of arithmetic (specifically
algorism ) which due to the positional notation of Hindu–Arabic
numerals was much more efficient and greatly facilitated commerce.
Leonardo wrote
Liber Abaci
Liber Abaci in 1202 (updated in 1254) introducing the
technique to Europe and beginning a long period of popularizing it.
The book also brought to Europe what is now known as the Fibonacci
sequence (known to Indian mathematicians for hundreds of years before
that) which was used as an unremarkable example within the text.
The 14th 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
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
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 will traverse in some given time a completely
equal to that which it would traverse if it were moving continuously
through the same time with the mean degree ".
Nicole Oresme at the
University of Paris
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
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.
RENAISSANCE MATHEMATICS
Portrait of Luca Pacioli
Portrait of Luca Pacioli , a painting traditionally attributed
to Jacopo de\' Barbari , 1495, (
Museo di Capodimonte
Museo di Capodimonte ). Further
information:
Mathematics
Mathematics and art
During the
Renaissance
Renaissance , the development of mathematics and of
accounting were intertwined. While there is no direct relationship
between algebra and accounting, the teaching of the subjects and the
books published often intended for the children of merchants who were
sent to reckoning schools (in
Flanders
Flanders and
Germany
Germany ) or abacus schools
(known as abbaco in Italy), where they learned the skills useful for
trade and commerce. There is probably no need for algebra in
performing bookkeeping operations, but for complex bartering
operations or the calculation of compound interest , a basic knowledge
of arithmetic was mandatory and knowledge of algebra was very useful.
Piero della Francesca
Piero della Francesca (c.1415–1492) wrote books on solid geometry
and linear perspective , including
De Prospectiva Pingendi (On
Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and
De corporibus regularibus (Regular Solids).
Luca Pacioli
Luca Pacioli 's Summa de Arithmetica, Geometria, Proportioni et
Proportionalità (Italian: "Review of
Arithmetic
Arithmetic ,
Geometry
Geometry , Ratio
and Proportion ") was first printed and published in
Venice
Venice in 1494.
It included a 27-page treatise on bookkeeping , "Particularis de
Computis et Scripturis" (Italian: "Details of Calculation and
Recording"). It was written primarily for, and sold mainly to,
merchants who used the book as a reference text, as a source of
pleasure from the mathematical puzzles it contained, and to aid the
education of their sons. In Summa Arithmetica, Pacioli introduced
symbols for plus and minus for the first time in a printed book,
symbols that became standard notation in Italian Renaissance
mathematics. Summa Arithmetica was also the first known book printed
in
Italy
Italy to contain algebra . Pacioli obtained many of his ideas from
Piero Della Francesca whom he plagiarized.
In Italy, during the first half of the 16th century, Scipione del
Ferro and
Niccolò Fontana Tartaglia discovered solutions for cubic
equations .
Gerolamo Cardano
Gerolamo Cardano published them in his 1545 book Ars Magna
, together with a solution for the quartic equations , discovered by
his student
Lodovico Ferrari . In 1572
Rafael Bombelli published his
L'
Algebra
Algebra in which he showed how to deal with the imaginary quantities
that could appear in Cardano's formula for solving cubic equations.
Simon Stevin
Simon Stevin 's book De Thiende ('the art of tenths'), first
published in Dutch in 1585, contained the first systematic treatment
of decimal notation , which influenced all later work on the real
number system .
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.
During the
Renaissance
Renaissance the desire of artists to represent the natural
world realistically, together with the rediscovered philosophy of the
Greeks, led artists to study mathematics. They were also the engineers
and architects of that time, and so had need of mathematics in any
case. The art of painting in perspective, and the developments in
geometry that involved, were studied intensely.
MATHEMATICS DURING THE SCIENTIFIC REVOLUTION
17TH CENTURY
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz .
The 17th century saw an unprecedented increase of mathematical and
scientific ideas across Europe.
Galileo
Galileo observed the moons of Jupiter
in orbit about that planet, using a telescope based on a toy imported
from Holland.
Tycho Brahe
Tycho Brahe had gathered an enormous quantity of
mathematical data describing the positions of the planets in the sky.
By his position as Brahe's assistant,
Johannes Kepler
Johannes Kepler was first
exposed to and seriously interacted with the topic of planetary
motion. Kepler's calculations were made simpler by the contemporaneous
invention of logarithms by
John Napier and
Jost Bürgi
Jost Bürgi . Kepler
succeeded in formulating mathematical laws of planetary motion. The
analytic geometry developed by
René Descartes
René Descartes (1596–1650) allowed
those orbits to be plotted on a graph, in
Cartesian coordinates
Cartesian coordinates .
Building on earlier work by many predecessors, Isaac Newton
discovered the laws of physics explaining Kepler\'s Laws , and brought
together the concepts now known as calculus . Independently, Gottfried
Wilhelm Leibniz , who is arguably one of the most important
mathematicians of the 17th century, 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
Pierre de Fermat and
Blaise Pascal
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
Leonhard Euler
Leonhard Euler by
Emanuel Handmann .
The most influential mathematician of the 18th century was arguably
Leonhard Euler
Leonhard Euler . His contributions range from founding the study of
graph theory with the
Seven Bridges of Königsberg
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 {displaystyle 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
Joseph Louis Lagrange , who did pioneering work in number theory,
algebra, differential calculus, and the calculus of variations, and
Laplace
Laplace who, in the age of
Napoleon
Napoleon , did important work on the
foundations of celestial mechanics and on statistics .
MODERN MATHEMATICS
19TH CENTURY
Carl Friedrich Gauss
Carl Friedrich Gauss .
Throughout the 19th century mathematics became increasingly abstract.
Carl Friedrich Gauss
Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did
revolutionary work on functions of complex variables , in geometry ,
and on the convergence of series , leaving aside his many
contributions to science. He also gave the first satisfactory proofs
of the fundamental theorem of algebra and of the quadratic reciprocity
law . Behavior of lines with a common perpendicular in each of
the three types of geometry
This century saw the development of the two forms of non-Euclidean
geometry , where the parallel postulate of
Euclidean geometry
Euclidean geometry no
longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky
and his rival, the Hungarian mathematician
János Bolyai
János 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
Elliptic geometry was
developed later in the 19th century by the German mathematician
Bernhard Riemann
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 generalizes
the ideas of curves and surfaces .
The 19th century saw the beginning of a great deal of abstract
algebra .
Hermann Grassmann
Hermann Grassmann in
Germany
Germany gave a first version of vector
spaces ,
William Rowan Hamilton
William Rowan Hamilton in Ireland developed noncommutative
algebra . The British mathematician
George Boole
George Boole devised an algebra
that soon evolved into what is now called
Boolean algebra
Boolean algebra , in which
the only numbers were 0 and 1.
Boolean algebra
Boolean algebra is the starting point
of mathematical logic and has important applications in computer
science .
Augustin-Louis Cauchy
Augustin-Louis Cauchy ,
Bernhard Riemann
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
Évariste Galois , a Frenchman,
proved that there is no general algebraic method for solving
polynomial equations of degree greater than four (Abel–Ruffini
theorem ). 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. On the other
hand, the limitation of three dimensions in geometry was surpassed in
the 19th century through considerations of parameter space and
hypercomplex numbers .
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
Peano ,
L. E. J. Brouwer ,
David Hilbert
David Hilbert ,
Bertrand Russell
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
London Mathematical Society in 1865, the
Société Mathématique de France
Société Mathématique de France in 1872, the Circolo Matematico di
Palermo in 1884, the
Edinburgh Mathematical Society in 1883, and the
American Mathematical Society in 1888. The first international,
special-interest society, the
Quaternion Society , was formed in 1899,
in the context of a vector controversy .
In 1897, Hensel introduced p-adic numbers .
20TH CENTURY
A map illustrating the
Four Color Theorem
Four Color Theorem
The 20th century saw mathematics become a major profession. Every
year, thousands of new Ph.D.s in mathematics were awarded, and jobs
were available in both teaching and industry. An effort to catalogue
the areas and applications of mathematics was undertaken in Klein\'s
encyclopedia .
In a 1900 speech to the
International Congress of Mathematicians
International Congress of Mathematicians ,
David Hilbert
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.
Notable historical conjectures were finally proven. In 1976, Wolfgang
Haken and
Kenneth Appel used a computer to prove the four color
theorem .
Andrew Wiles
Andrew Wiles , building on the work of others, proved
Fermat\'s Last Theorem in 1995. Paul Cohen and
Kurt Gödel
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
Kepler conjecture .
Mathematical collaborations of unprecedented size and scope took
place. An 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
Jean Dieudonné and
André Weil
André Weil , publishing under the
pseudonym "
Nicolas Bourbaki
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.
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet
orbiting a star, with relativistic precession of apsides
Differential geometry
Differential geometry came into its own when
Einstein
Einstein used it in
general relativity . Entirely 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
Grothendieck and Serre recast
algebraic geometry using sheaf theory . Large advances were made in
the qualitative study of dynamical systems that Poincaré had begun in
the 1890s.
Measure theory
Measure theory was developed in the late 19th and early
20th centuries. Applications of measures include the Lebesgue integral
,
Kolmogorov 's axiomatisation of probability theory , and ergodic
theory .
Knot theory greatly expanded.
Quantum mechanics led to the
development of functional analysis . Other new areas include Laurent
Schwartz 's distribution theory , fixed point theory , singularity
theory and
René Thom
René Thom 's catastrophe theory , model theory , and
Mandelbrot 's fractals .
Lie theory with its Lie groups and Lie
algebras became one of the major areas of study.
Non-standard analysis , introduced by
Abraham Robinson
Abraham Robinson ,
rehabillitated the infinitesimal approach to calculus, which had
fallen into disrepute in favour of the theory of limits , by extending
the field of real numbers to the Hyperreal numbers which include
infinitesimal and infinite quantities. An even larger number system,
the surreal numbers were discovered by
John Horton Conway
John Horton Conway in
connection with combinatorial games .
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
Alan Turing 's
computability theory ; complexity theory ;
Derrick Henry Lehmer 's use
of
ENIAC
ENIAC to further number theory and the
Lucas-Lehmer test ; 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 . Some of the most important methods and algorithms of the
20th century are: the simplex algorithm , the
Fast Fourier Transform ,
error-correcting codes , the
Kalman filter
Kalman filter from control theory and the
RSA algorithm of public-key cryptography .
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 some algorithm. In 1931,
Kurt Gödel
Kurt Gödel found that this was not the
case for the natural numbers plus both addition and multiplication;
this system, known as
Peano
Peano arithmetic , was in fact incompletable .
(
Peano
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
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
David Hilbert 's dream of making
all of mathematics complete and consistent needed to be reformulated.
The absolute value of the
Gamma function
Gamma function on the complex plane.
One of the more colorful figures in 20th-century mathematics was
Srinivasa Aiyangar Ramanujan
Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who
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
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.
Emmy Noether
Emmy Noether has been described by many as the most important woman
in the history of mathematics. She studied the theories of rings ,
fields , and algebras .
As in most areas of study, the explosion of knowledge in the
scientific age has led to specialization: by the end of the century
there were hundreds of specialized areas in mathematics and the
Mathematics
Mathematics Subject Classification was dozens of pages long. More and
more mathematical journals were published and, by the end of the
century, the development of the
World Wide Web
World Wide Web led to online
publishing.
21ST CENTURY
See also: List of unsolved problems in mathematics § Problems solved
since 1995
In 2000, the Clay
Mathematics
Mathematics Institute announced the seven
Millennium Prize Problems , and in 2003 the
Poincaré conjecture
Poincaré conjecture was
solved by
Grigori Perelman
Grigori Perelman (who declined to accept an award, as he was
critical of the mathematics establishment).
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 .
FUTURE OF MATHEMATICS
Main article:
Future of mathematics
There are many observable trends in mathematics, the most notable
being that the subject is growing ever larger, computers are ever more
important and powerful, the application of mathematics to
bioinformatics is rapidly expanding, the volume of data to be analyzed
being produced by science and industry, facilitated by computers, is
explosively expanding.
SEE ALSO
*
Mathematics
Mathematics portal
*
History of algebra
History of algebra
*
History of calculus
*
History of combinatorics
*
History of the function concept
*
History of geometry
History of geometry
*
History of logic
History of logic
*
History of mathematical notation
*
History of numbers
*
History of number theory
*
History of statistics
History of statistics
*
History of trigonometry
*
History of writing numbers
*
Kenneth O. May Prize
*
List of important publications in mathematics
*
Lists of mathematicians
*
List of mathematics history topics
*
Timeline of mathematics
NOTES
* ^ A B (Boyer 1991 , "
Euclid
Euclid of Alexandria" p. 119)
* ^ J. Friberg, "Methods and traditions of Babylonian mathematics.
Plimpton 322, Pythagorean triples, and the Babylonian triangle
parameter equations", Historia Mathematica, 8, 1981, pp. 277—318.
* ^ Neugebauer, Otto (1969) . The Exact Sciences in Antiquity (2
ed.).
Dover Publications . ISBN 978-0-486-22332-2 . Chap. IV
"Egyptian
Mathematics
Mathematics and Astronomy", pp. 71–96.
* ^ Heath. A Manual of Greek Mathematics. p. 5.
* ^ 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."
* ^ George Gheverghese Joseph, The Crest of the Peacock:
Non-European Roots of Mathematics, Penguin Books, London, 1991,
pp.140—148
* ^ Georges Ifrah, Universalgeschichte der Zahlen, Campus,
Frankfurt/New York, 1986, pp.428—437
* ^ 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
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
* ^ A B (Boyer 1991 , "Origins" p. 3)
* ^
Mathematics
Mathematics in (central) Africa before colonization
* ^ Williams, Scott W. (2005). "The Oldest Mathematical Object is
in Swaziland". Mathematicians of the African Diaspora. SUNY Buffalo
mathematics department. Retrieved 2006-05-06.
* ^ Marshack, Alexander (1991): The Roots of Civilization, Colonial
Hill, Mount Kisco, NY.
* ^ Rudman, Peter Strom (2007). How
Mathematics
Mathematics Happened: The First
50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4 .
* ^ Marshack, A. 1972. The Roots of Civilization: the Cognitive
Beginning of Man’s First Art, Symbol and Notation. New York:
McGraw-Hil
* ^ 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
University Press. ISBN 0-521-33381-4 .
* ^ (Boyer 1991 , "Mesopotamia" p. 24)
* ^ A B C D E F (Boyer 1991 , "Mesopotamia" p. 26)
* ^ A B C (Boyer 1991 , "Mesopotamia" p. 25)
* ^ A B (Boyer 1991 , "Mesopotamia" p. 41)
* ^ Duncan J. Melville (2003). Third Millennium Chronology, Third
Millennium Mathematics.
St. Lawrence University
St. Lawrence University .
* ^ A B (Boyer 1991 , "Mesopotamia" p. 27)
* ^ Aaboe, Asger (1998). Episodes from the Early History of
Mathematics. New York: Random House. pp. 30–31.
* ^ (Boyer 1991 , "Mesopotamia" p. 33)
* ^ (Boyer 1991 , "Mesopotamia" p. 39)
* ^ (Boyer 1991 , "Egypt" p. 11)
* ^ Egyptian Unit Fractions at MathPages
* ^ Egyptian Unit Fractions
* ^ Egyptian Papyri
* ^ Egyptian
Algebra
Algebra – Mathematicians of the African Diaspora
* ^ (Boyer 1991 , "Egypt" p. 19)
* ^ Egyptian Mathematical Papyri – Mathematicians of the African
Diaspora
* ^ Howard Eves, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 99)
* ^ 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.
* ^ (Boyer 1991 , "Ionia and the Pythagoreans" p. 43)
* ^ (Boyer 1991 , "Ionia and the Pythagoreans" p. 49)
* ^ Eves, Howard, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0 .
* ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by
Hippasus of Metapontum". The Annals of Mathematics.
* ^ James R. Choike (1980). "The Pentagram and the Discovery of an
Irrational Number". The Two-Year College
Mathematics
Mathematics Journal.
* ^ A B Jane Qiu (7 January 2014). "Ancient times table hidden in
Chinese bamboo strips". Nature . Retrieved 15 September 2014.
* ^ David E. Smith (1958), History of Mathematics,
Volume
Volume I:
General Survey of the History of Elementary Mathematics, New York:
Dover Publications (a reprint of the 1951 publication), ISBN
0-486-20429-4 , pp 58, 129.
* ^ David E. Smith (1958), History of Mathematics,
Volume
Volume I:
General Survey of the History of Elementary Mathematics, New York:
Dover Publications (a reprint of the 1951 publication), ISBN
0-486-20429-4 , p. 129.
* ^ Bill Casselman . "One of the Oldest Extant Diagrams from
Euclid". University of British Columbia. Retrieved 2008-09-26.
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 86)
* ^ A B (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 88)
* ^ Calian, George F. (2014). "One, Two, Three… A Discussion on
the Generation of Numbers" (PDF). New Europe College. Archived from
the original (PDF) on 2015-10-15.
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 87)
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 92)
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 93)
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 91)
* ^ (Boyer 1991 , "The Age of
Plato
Plato and Aristotle" p. 98)
* ^ (Boyer 1991 , "
Euclid
Euclid of Alexandria" p. 100)
* ^ A B (Boyer 1991 , "
Euclid
Euclid of Alexandria" p. 104)
* ^ Howard Eves, An Introduction to the History of Mathematics,
Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible
, has been more widely used...."
* ^ (Boyer 1991 , "
Euclid
Euclid of Alexandria" p. 102)
* ^ (Boyer 1991 , "
Archimedes
Archimedes of Syracuse" p. 120)
* ^ A B (Boyer 1991 , "
Archimedes
Archimedes of Syracuse" p. 130)
* ^ (Boyer 1991 , "
Archimedes
Archimedes of Syracuse" p. 126)
* ^ (Boyer 1991 , "
Archimedes
Archimedes of Syracuse" p. 125)
* ^ (Boyer 1991 , "
Archimedes
Archimedes of Syracuse" p. 121)
* ^ (Boyer 1991 , "
Archimedes
Archimedes of Syracuse" p. 137)
* ^ (Boyer 1991 , "Apollonius of Perga" p. 145)
* ^ (Boyer 1991 , "Apollonius of Perga" p. 146)
* ^ (Boyer 1991 , "Apollonius of Perga" p. 152)
* ^ (Boyer 1991 , "Apollonius of Perga" p. 156)
* ^ (Boyer 1991 , "Greek
Trigonometry
Trigonometry and Mensuration" p. 161)
* ^ A B (Boyer 1991 , "Greek
Trigonometry
Trigonometry and Mensuration" p. 175)
* ^ (Boyer 1991 , "Greek
Trigonometry
Trigonometry and Mensuration" p. 162)
* ^ S.C. Roy. Complex numbers: lattice simulation and zeta function
applications, p. 1 . Harwood Publishing, 2007, 131 pages. ISBN
1-904275-25-7
* ^ (Boyer 1991 , "Greek
Trigonometry
Trigonometry and Mensuration" p. 163)
* ^ (Boyer 1991 , "Greek
Trigonometry
Trigonometry and Mensuration" p. 164)
* ^ (Boyer 1991 , "Greek
Trigonometry
Trigonometry and Mensuration" p. 168)
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
178)
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
180)
* ^ A B (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
181)
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
183)
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
183-190)
* ^ Medieval Sourcebook: Socrates Scholasticus: The Murder of
Hypatia
Hypatia (late 4th Cent.) from Ecclesiastical History, Bk VI: Chap. 15
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
190-194)
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
193)
* ^ (Boyer 1991 , "Revival and Decline of Greek Mathematics" p.
194)
* ^ (Boyer 1991 , "
China
China and India" p. 201)
* ^ A B (Boyer 1991 , "
China
China and India" p. 196)
* ^ Katz 2007 , pp. 194–199
* ^ (Boyer 1991 , "
China
China and India" p. 198)
* ^ Needham, Joseph (1986). "Science and Civilisation in
China
China ".
3,
Mathematics
Mathematics and the Sciences of the Heavens and the Earth. Taipei:
Caves Books Ltd.
* ^ A B C (Boyer 1991 , "
China
China and India" p. 202)
* ^ Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009).
Calculus: Early Transcendentals (3 ed.). Jones & Bartlett Learning. p.
xxvii. ISBN 0-7637-5995-3 . Extract of page 27
* ^ A B C (Boyer 1991 , "
China
China and India" p. 205)
* ^ (Boyer 1991 , "
China
China and India" p. 206)
* ^ A B C D (Boyer 1991 , "
China
China and India" p. 207)
* ^ T. K. Puttaswamy, "The Accomplishments of Ancient Indian
Mathematicians", pp. 411–2, in Selin, Helaine ; D\'Ambrosio,
Ubiratan , eds. (2000).
Mathematics
Mathematics Across Cultures: The History of
Non-western Mathematics. Springer . ISBN 1-4020-0260-2 .
* ^ R. P. Kulkarni, "The Value of π known to Śulbasūtras
Archived 2012-02-06 at the
Wayback Machine
Wayback Machine .", Indian Journal for the
History of Science, 13 1 (1978): 32-41
* ^ J.J. Connor, E.F. Robertson. The Indian
Sulba Sutras
Sulba Sutras Univ. of
St. Andrew,
Scotland
Scotland The values for π are 4 x (13/15)2 (3.0044...),
25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), and
339/108 (3.1389).
* ^ J.J. Connor, E.F. Robertson. The Indian
Sulba Sutras
Sulba Sutras Univ. of
St. Andrew,
Scotland
Scotland
* ^ Bronkhorst, Johannes (2001). "Panini and Euclid: Reflections on
Indian Geometry". Journal of Indian Philosophy. Springer Netherlands.
29 (1–2): 43–80. doi :10.1023/A:1017506118885 .
* ^ Kadvany, John (2008-02-08). "Positional Value and Linguistic
Recursion". Journal of Indian Philosophy. 35 (5-6): 487–520. ISSN
0022-1791 . doi :10.1007/s10781-007-9025-5 .
* ^ Sanchez, Julio; Canton, Maria P. (2007). Microcontroller
programming : the microchip PIC. Boca Raton, Florida: CRC Press. p.
37. ISBN 0-8493-7189-9 .
* ^ W. S. Anglin and J. Lambek, The Heritage of Thales, Springer,
1995, ISBN 0-387-94544-X
* ^ Hall, Rachel W. (2008). "Math for poets and drummers" (PDF).
Math Horizons. 15: 10–11.
* ^ (Boyer 1991 , "
China
China and India" p. 208)
* ^ A B (Boyer 1991 , "
China
China and India" p. 209)
* ^ (Boyer 1991 , "
China
China and India" p. 210)
* ^ (Boyer 1991 , "
China
China and India" p. 211)
* ^ Boyer (1991). "The
Arabic
Arabic Hegemony". History of Mathematics. p.
226. By 766 we learn that an astronomical-mathematical work, known to
the Arabs as the Sindhind, was brought to
Baghdad
Baghdad from India. It is
generally thought that this was the Brahmasphuta Siddhanta, although
it may have been the Surya Siddhanata. A few years later, perhaps
about 775, this Siddhanata was translated into Arabic, and it was not
long afterwards (ca. 780) that Ptolemy's astrological
Tetrabiblos
Tetrabiblos was
translated into
Arabic
Arabic from the Greek.
* ^ Plofker 2009 182–207
* ^ Plofker 2009 pp 197–198; George Gheverghese Joseph, The Crest
of the Peacock: Non-European Roots of Mathematics, Penguin Books,
London, 1991 pp 298–300; Takao Hayashi, Indian Mathematics, pp
118–130 in Companion History of the History and Philosophy of the
Mathematical Sciences, ed. I. Grattan.Guinness, Johns Hopkins
University Press, Baltimore and London, 1994, p 126
* ^ Plofker 2009 pp 217–253
* ^ P. P. Divakaran, The first textbook of calculus:
Yukti-bhāṣā, Journal of Indian Philosophy 35, 2007, pp 417–433.
* ^ Pingree, David (December 1992), "Hellenophilia versus the
History of Science", Isis, 83 (4): 562,
JSTOR
JSTOR 234257 , doi
:10.1086/356288 , One example I can give you relates to the Indian
Mādhava's demonstration, in about 1400 A.D., of the infinite power
series of trigonometrical functions using geometrical and algebraic
arguments. When this was first described in English by Charles Whish,
in the 1830s, it was heralded as the Indians' discovery of the
calculus. This claim and Mādhava's achievements were ignored by
Western historians, presumably at first because they could not admit
that an Indian discovered the calculus, but later because no one read
anymore the Transactions of the Royal Asiatic Society, in which
Whish's article was published. The matter resurfaced in the 1950s, and
now we have the Sanskrit texts properly edited, and we understand the
clever way that Mādhava derived the series without the calculus; but
many historians still find it impossible to conceive of the problem
and its solution in terms of anything other than the calculus and
proclaim that the calculus is what Mādhava found. In this case the
elegance and brilliance of Mādhava's mathematics are being distorted
as they are buried under the current mathematical solution to a
problem to which he discovered an alternate and powerful solution.
* ^ Bressoud, David (2002), "Was
Calculus
Calculus Invented in India?",
College
Mathematics
Mathematics Journal, 33 (1): 2–13, doi :10.2307/1558972
* ^ Plofker, Kim (November 2001), "The 'Error' in the Indian
"Taylor Series Approximation" to the Sine", Historia Mathematica, 28
(4): 293, doi :10.1006/hmat.2001.2331 , It is not unusual to encounter
in discussions of
Indian mathematics
Indian mathematics such assertions as that 'the
concept of differentiation was understood from the time of Manjula
(... in the 10th century)' , or that 'we may consider Madhava to have
been the founder of mathematical analysis' (Joseph 1991, 293), or that
Bhaskara II may claim to be 'the precursor of Newton and Leibniz in
the discovery of the principle of the differential calculus' (Bag
1979, 294).... The points of resemblance, particularly between early
European calculus and the Keralese work on power series, have even
inspired suggestions of a possible transmission of mathematical ideas
from the Malabar coast in or after the 15th century to the Latin
scholarly world (e.g., in (Bag 1979, 285)).... It should be borne in
mind, however, that such an emphasis on the similarity of Sanskrit (or
Malayalam) and
Latin
Latin mathematics risks diminishing our ability fully
to see and comprehend the former. To speak of the Indian 'discovery of
the principle of the differential calculus' somewhat obscures the fact
that Indian techniques for expressing changes in the Sine by means of
the Cosine or vice versa, as in the examples we have seen, remained
within that specific trigonometric context. The differential
'principle' was not generalized to arbitrary functions—in fact, the
explicit notion of an arbitrary function, not to mention that of its
derivative or an algorithm for taking the derivative, is irrelevant
here
* ^ Katz, Victor J. (June 1995), "Ideas of
Calculus
Calculus in Islam and
India" (PDF),
Mathematics
Mathematics Magazine, 68 (3): 163–174,
JSTOR
JSTOR 2691411 ,
doi :10.2307/2691411
* ^ (Boyer 1991 , "The
Arabic
Arabic Hegemony" p. 230) "The six cases of
equations given above exhaust all possibilities for linear and
quadratic equations having positive root. So systematic and exhaustive
was al-Khwārizmī's exposition that his readers must have had little
difficulty in mastering the solutions."
* ^ Gandz and Saloman (1936), The sources of Khwarizmi's algebra,
Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be
called "the father of algebra" than
Diophantus
Diophantus because Khwarizmi is
the first to teach algebra in an elementary form and for its own sake,
Diophantus
Diophantus is primarily concerned with the theory of numbers".
* ^ (Boyer 1991 , "The
Arabic
Arabic Hegemony" p. 229) "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."
* ^ Rashed, R.; Armstrong, Angela (1994). The Development of Arabic
Mathematics. Springer . pp. 11–12. ISBN 0-7923-2565-6 . OCLC
29181926 .
* ^ Sesiano, Jacques (1997-07-31). "Abū Kāmil". Encyclopaedia of
the history of science, technology, and medicine in non-western
cultures. Springer. pp. 4–5.
* ^ (Katz 1998 , pp. 255–59)
* ^ F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par
Abou Bekr Mohammed Ben Alhacan Alkarkhi.
Paris
Paris .
* ^ Katz, Victor J. (1995). "Ideas of
Calculus
Calculus in Islam and India".
Mathematics
Mathematics Magazine. 68 (3): 163–74. doi :10.2307/2691411 .
* ^ Alam, S (2015). "MATHEMATICS FOR ALL AND FOREVER" (PDF). Indian
Institute of Social Reform & Research International Journal of
Research.
* ^ O\'Connor, John J. ; Robertson, Edmund F. , "Abu\'l Hasan ibn
Ali al Qalasadi", MacTutor History of
Mathematics
Mathematics archive , University
of St Andrews .
* ^ Pickover, Clifford A. (2009), The Math Book: From Pythagoras to
the 57th Dimension, 250 Milestones in the History of Mathematics,
Sterling Publishing Company, Inc., p. 104, ISBN 9781402757969 , Nicole
Oresme ... was the first to prove the divergence of the harmonic
series (c. 1350). His results were lost for several centuries, and the
result was proved again by Italian mathematician
Pietro Mengoli in
1647 and by Swiss mathematician
Johann Bernoulli
Johann Bernoulli in 1687.
* ^ 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
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
Renaissance and
Renewal in the Twelfth Century, (Cambridge: Harvard University Press,
1982).
* ^ Grant, Edward and John E. Murdoch (1987), eds.,
Mathematics
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
Geometry of Euclid" Q. 14, pp.
560–65, in Marshall Clagett, ed.,
Nicole Oresme and the Medieval
Geometry
Geometry of Qualities and Motions, (Madison: University of Wisconsin
Press, 1968).
* ^ Heeffer, Albrecht: On the curious historical coincidence of
algebra and double-entry bookkeeping, Foundations of the Formal
Sciences,
Ghent University
Ghent University , November 2009, p.7
* ^ della Francesca, Piero. De Prospectiva Pingendi, ed. G. Nicco
Fasola, 2 vols., Florence (1942).
* ^ della Francesca, Piero. Trattato d'Abaco, ed. G. Arrighi, Pisa
(1970).
* ^ della Francesca, Piero. L'opera "De corporibus regularibus" di
Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli,
ed. G. Mancini, Rome, (1916).
* ^ Alan Sangster, Greg Stoner & Patricia McCarthy: "The market for
Luca Pacioli’s Summa Arithmetica" (Accounting, Business & Financial
History Conference, Cardiff, September 2007) p. 1–2
* ^ Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A
History of the Mathematical Sciences. W.W. Norton. ISBN 0-393-32030-8
.
* ^ Kline, Morris (1953).
Mathematics
Mathematics in Western Culture. Great
Britain: Pelican. pp. 150–151.
* ^ Struik, Dirk (1987). A Concise History of
Mathematics