Everything about History Of Mathematics totally explained
The area of study known as the
history of mathematics is primarily an investigation into the origin of new discoveries in
mathematics, to a lesser extent an investigation into the standard 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 ca. 1900 BC), the
Moscow Mathematical Papyrus (
Egyptian mathematics ca. 1850 BC), the
Rhind Mathematical Papyrus (Egyptian mathematics
ca. 1650 BC), and the
Shulba Sutras (
Indian mathematics ca. 800 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.
Egyptian and Babylonian mathematics were then further developed in
Greek and Hellenistic mathematics, which is generally considered to be one of the most important for greatly expanding both the method and the subject matter of mathematics. The mathematics developed in these ancient civilizations were then further developed and greatly expanded in
Islamic mathematics. Many Greek and Arabic texts on mathematics were then
translated into Latin in medieval Europe and further developed there.
One striking feature of the history of ancient and medieval mathematics is that bursts of mathematical development 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 ever increasing pace, and this continues to the present day.
Early mathematics
Long before the earliest written records, there are drawings that do indicate a knowledge of mathematics and of measurement of time based on the stars. For example,
paleontologists have discovered
ochre rocks in a cave in
South Africa adorned with scratched
geometric patterns dating back to c. 70,000 BC. Also
prehistoric artifacts discovered in Africa and
France, dated between
35,000 BC and
20,000 BC, indicate early attempts to
quantify time.
Evidence exists that early counting involved women who kept records of their monthly biological cycles; twenty-eight, twenty-nine, or thirty scratches on bone or stone, followed by a distinctive scratching on the bone or stone, for example. Moreover, hunters had the concepts of
one,
two, and
many, as well as the idea of
none or
zero, when considering herds of animals.
The
Ishango Bone, found in the area of the headwaters of the
Nile River (northeastern
Congo), dates as early as
20,000 BC. One common interpretation is that the bone is the earliest known demonstration of
sequences of
prime numbers and
Ancient Egyptian multiplication.
Predynastic Egyptians of the 5th millennium BC pictorially represented
geometric spatial designs. It has been claimed that
Megalithic monuments in
England and
Scotland from the 3rd millennium BC, incorporate geometric ideas such as
circles,
ellipses, and
Pythagorean triples in their design.
The earliest known mathematics in
ancient India dates back to circa 3000-2600 BC in the
Indus Valley Civilization (
Harappan civilization) of
North India and
Pakistan, which developed a system of
uniform weights and measures that used the
decimal system, a surprisingly advanced
brick technology which utilised
ratios, streets laid out in perfect
right angles, and a number of geometrical shapes and designs, including
cuboids,
barrels,
cones,
cylinders, and drawings of concentric and intersecting
circles and
triangles. Mathematical instruments discovered include an accurate decimal ruler with small and precise subdivisions, a shell instrument that served as a
compass to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees, a shell instrument used to measure 8–12 whole sections of the horizon and sky, and an instrument for measuring the positions of stars for navigational purposes. The
Indus script hasn't yet been deciphered; hence very little is known about the written forms of
Harappan mathematics. Archeological evidence has led some historians to believe that this civilization used a
base 8 numeral system and possessed knowledge of the ratio of the length of the
circumference of the circle to its
diameter, thus a value of
π.
Dating from the
Shang Dynasty (1600—1046 BC), the earliest extant Chinese mathematics consists of numbers scratched on tortoise shell
(External Link
) (External Link). These numbers use a decimal system, so that the number 123 is written (from top to bottom) as the symbol for 1 followed by the symbol for a hundred, then the symbol for 2 followed by the symbol for ten, 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 isn't certain, but the earliest written reference was in AD 190 in the
Supplementary Notes on the Art of Figures written by Xu Yue.
Ancient Near East (c. 1800-500 BC)
Mesopotamia
Babylonian mathematics refers to any mathematics of the peoples 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, Iraq/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
Pythagorean triples (see
Plimpton 322). The tablets also include multiplication tables,
trigonometry 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 was written using a
sexagesimal (base-60)
numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. 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're 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's 56. You will find it right."
The
Rhind papyrus (c. 1650 BC
(External Link)) 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 (see
(External Link)), 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)
(External Link). It also shows how to solve first order
linear equations
(External Link) as well as
arithmetic and
geometric series (External Link).
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
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
(External Link) (External Link)) shows that ancient Egyptians could solve a second-order
algebraic equation (External Link).
Ancient Indian mathematics (c. 900 BC — AD 200)
Vedic mathematics begins in the early Iron Age, with the
Shatapatha Brahmana (c. 9th century BC), which approximates the value of
π to 2 decimal places.
(External Link), 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 five decimal places; gave the method for
squaring the circle; solved
linear equations and
quadratic equations; developed
Pythagorean triples algebraically and gave a statement and numerical
proof of the
Pythagorean theorem.
Pāṇini (c. 5th century BC) formulated the
grammar rules for
Sanskrit. His notation was similar to modern mathematical notation, and used metarules,
transformations, and
recursions with such sophistication that his grammar had the
computing power equivalent to a
Turing machine.
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 the
binomial theorem. Pingala's work also contains the basic ideas of
Fibonacci numbers (called
mātrāmeru). The
Brāhmī script was developed at least from the
Maurya dynasty in the 4th century BC, with recent archeological evidence appearing to push back that date to around 600 BC. The
Brahmi numerals date to the 3rd century BC.
Between 400 BC and AD 200,
Jaina mathematicians began studying mathematics for the sole purpose of mathematics. They were the first to develop
transfinite numbers,
set theory,
logarithms, fundamental laws of
indices,
cubic equations,
quartic equations,
sequences and progressions,
permutations and combinations, squaring and extracting
square roots, and finite and
infinite powers. The
Bakhshali Manuscript written between 200 BC and AD 200 included solutions of linear equations with up to five unknowns, the solution of the quadratic equation, arithmetic and geometric progressions, compound series, quadratic indeterminate equations,
simultaneous equations, and the use of
zero and
negative numbers. Accurate computations for irrational numbers could be found, which includes computing square roots of numbers as large as a million to at least 11 decimal places.
Greek and Hellenistic mathematics (c. 550 BC—AD 300)
Greek mathematics refers to mathematics written in
Greek between about 600 BCE and 450 CE. 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 is sometimes called Hellenistic mathematics.
Greek mathematics was much 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 ideas of
Egypt,
Mesopotamia and perhaps
India. According to legend, Pythagoras travelled 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 discovered 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 (ca. 230 BC) was used to discover prime numbers.
Some say the greatest of Greek mathematicians, if not of all time, was
Archimedes (287—212 BC) of
Syracuse. He used the
method of exhaustion to calculate the
area under the arc of a
parabola with the
summation of an infinite series, and gave a remarkably accurate approximation of
Pi. He also defined the
spiral bearing his name, formulas for the
volumes of
surfaces of revolution and an ingenious system for expressing very large numbers.
Classical Chinese mathematics (c. 500 BC—AD 1300)
China, in 212 BC, the Emperor
Qin Shi Huang (Shi Huang-ti) commanded that all books outside of Qin state to be burned. While this order wasn't universally obeyed, as a consequence little is known with certainty 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/or 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 of c. 330 BC, compiled by the followers of
Mozi (470 BC-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 179 AD, 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 Pythagoras'
Pythagorean theorem, and mathematical formula for
Gaussian elimination. The work was commented on by
Liu Hui in the 3rd century AD.
In addition, the mathematical works of the Han astronomer and inventor
Zhang Heng (78-139 AD) 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 to 31
octaves. This would later lead to the discovery of
53 equal temperament, and wasn't 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 which was described in ancient times and perfected by
Yang Hui (1238–1398 AD).
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.
In the thousand years following the Han dynasty, starting in the
Tang dynasty and ending in the
Song dynasty, Chinese mathematics thrived at a time when European mathematics didn't exist. Developments first made in China, and only much later known in the
West, include
negative numbers, the
binomial theorem,
matrix methods for solving systems of
linear equations and the
Chinese remainder theorem. The Chinese also developed
Pascal's triangle and the
rule of three long before it was known in Europe. Besides Zu Chongzhi, some of the most important figures of Chinese mathematics during this period include
Yi Xing,
Shen Kuo,
Qin Jiushao,
Zhu Shijie, and others. The scientist Shen Kuo used problems involving
calculus,
trigonometry,
metrology,
permutations, and once computed the possible amount of terrain space that could be used with specific battle formations, as well as the longest possible military campaign given the amount of food carriers could bring for themselves and soldiers.
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.
Classical Indian mathematics (c. 400—1600)
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, corresponds 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 to Arabic and
Latin during the
Middle Ages.
Aryabhata in 499 introduced the
versine function, produced the first
trigonometric tables of sine, developed techniques and
algorithms of
algebra,
infinitesimals,
differential equations, and obtained whole number solutions to linear equations by a method equivalent to the modern method, 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 computed the value of
π to the fourth decimal place as 3.1416.
Madhava later in the 14th century computed the value of π to the eleventh decimal place 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 (around 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,
tests of convergence,
iterative methods for solutions of
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 became stagnant from the late 16th century onwards due to subsequent political turmoil.
Islamic mathematics (c. 800—1500)
The
Islamic
Arab Empire established across 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, they were not all 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. Some of the most important Islamic mathematicians were
Persian.
Muḥammad ibn Mūsā al-Ḵwārizmī, a 9th century Persian mathematician and astronomer to the
Caliph of Baghdad, 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 the Arab mathematician 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 preservation of ancient algebraic methods and for his original contributions to the field. Further developments in
algebra were made by
Abu Bakr al-Karaji (953—1029) in his treatise
al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. In the
10th century,
Abul Wafa translated the works of
Diophantus into Arabic and developed the
tangent function.
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."
Ibn al-Haytham was the first mathematician to derive the formula for the sum of the
fourth powers, and using the method of induction, he developed a method for determining the general formula for the sum of any integral
powers, which was fundamental to the development of
integral calculus.
Omar Khayyam, the 12th century
poet, was also a mathematician, and wrote
Discussions of the Difficulties in Euclid, a book about flaws in
Euclid's Elements, especially the
parallel postulate, and thus he 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. The Persian mathematician
Nasir al-Din Tusi (Nasireddin) in the 13th century 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 sine,
al-Kindi's introduction of
cryptanalysis and
frequency analysis,
al-Karaji's introduction of algebraic
calculus and
proof by
mathematical induction, the development of
analytic geometry and the earliest general formula for
infinitesimal and
integral calculus 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, and numerous 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. This parallels the stagnation of mathematics when the Romans conquered the Hellenistic world.
John J. O'Connor and Edmund F. Robertson wrote in the
MacTutor History of Mathematics archive:
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 biblical passage that God had "ordered all things in measure, and number, and weight" (
Wisdom 11:21).
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
Geometry. 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 travelled to Spain and Sicily
seeking scientific Arabic texts, including
al-Khwarizmi's
al-Jabr wa-al-Muqabilah, 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 area that contributed to the development of mathematics concerned the analysis 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 hadn't 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 [abody]
if ... it were moved uniformly at the same degree of speed with which it's moved in that given instant".
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (which we'd solve by a simple
integration), stating that "a moving body uniformly acquiring or losing that increment [ofspeed] 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 [ofspeed]".
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
Geometry, 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 circa 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 1478
Treviso Arithmetic and then the first real mathematics book
Euclid's Elements printed and published by
Ratdolt 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' table of sines and cosines was published in 1533.
By century's end, thanks to
Regiomontanus (1436—1476) and
François Vieta (1540—1603), 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 mathematicians,
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
As we've seen, knowledge of the natural numbers, 1, 2, 3,..., as preserved in monolithic structures, is older than any surviving written text. The earliest civilizations -- in Mesopotamia, Egypt, India and China -- knew arithmetic.
One way to view the development of the various number systems of modern mathematics is to see new numbers studied and investigated to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1? In India and China, and much later in Germany, negative numbers were developed to answer the question: what do you get when you subtract a larger number from a smaller?
Another natural question is: what kind of a number is the square root of two? The Greeks knew that it wasn't a fraction, and this question may have played a role in the development of
continued fractions. But a better answer came with the invention of decimals, developed by
John Napier (1550 - 1617) and perfected later by
Simon Stevin. Using decimals, and an idea that anticipated the concept of the
limit, Napier also studied a new constant, which
Leonhard Euler (1707 - 1783) named
e.
Euler was very influential in the standardization of other mathematical terms and notations. He named the square root of minus 1 with the symbol
i. He also popularized the use of the Greek letter
to stand for the ratio of a circle's circumference to its diameter. He then derived one of the most remarkable identities in all of mathematics:
»
(see
Euler's Identity.)
19th century
Throughout the 19th century mathematics became increasingly abstract. In this century lived
Carl Friedrich Gauss (1777 - 1855). Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on
functions 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 discovered
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.
Also in the nineteenth century
William Rowan Hamilton developed
noncommutative algebra.
In addition to new directions in mathematics, older mathematics were given a stronger logical foundation, especially in the case of
calculus, in work by
Augustin-Louis Cauchy and
Karl Weierstrass.
A new form of algebra was developed in the nineteenth century called
Boolean algebra, named after the British mathematician
George Boole. It was a system in which the only numbers were 0 and 1, a system which today has important applications in
computer science.
Also, for the first time, the limits of mathematics were explored.
Niels Henrik Abel, a Norwegian, and
Évariste Galois, a Frenchman, proved that there's 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.
Towards the end of the 19th century,
Georg Cantor invented the
set theory, which has become the common language of different mathematical branches. The introduction of
infinite set set off a debate on
foundations of mathematics.
The 19th century also saw the founding of the first 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.
Before the 20th century, there were very 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. There were a few who found meager livelihoods teaching at a university, like Fourier.
Niels Henrik Abel, unable to obtain a position, died in poverty of malnutrition and tuberculosis at the age of twenty-six.
20th century
The profession of mathematician became much more important in the 20th century. Every year, hundreds of new Ph.D.s in mathematics are awarded, and jobs are available both in teaching and industry. Mathematical development has grown at an exponential rate, with too many new developments for a survey to even touch on any but a few of the most profound.
In 1900,
David Hilbert presented a list of
23 unsolved problems in mathematics at the
International Congress of Mathematicians. These problems spanned many areas of mathematics and have formed a central focus for much of 20th century mathematics. Today ten have been resolved, seven are partially resolved and two problems are still open. The remaining four are too loose to be stated as resolved or not.
In the 1910s,
Srinivasa Aiyangar Ramanujan (1887-1920) developed over 3000 theorems, including properties of
highly composite numbers, the
partition function and its
asymptotics, and
mock theta functions. He also made major breakthroughs and discoveries in the areas of
gamma functions,
modular forms,
divergent series,
hypergeometric series and
prime number theory.
In 1931,
Kurt Gödel published his two
incompleteness theorems which state the limit of mathematical logic. It put an end to David Hilbert's dream of a complete and consistent mathematical system.
Famous conjectures of the past yielded to new and more powerful techniques.
Wolfgang Haken and
Kenneth Appel used a computer to prove the
four color theorem in 1976.
Andrew Wiles, working alone in his office for years, proved
Fermat's last theorem in 1995. Mathematical collaborations of unprecedented size and scope took place. The
classification of finite simple groups (also called the "enormous theorem") spanned tens of thousands of pages in 500-odd journal articles written by about 100 authors, published mostly between 1955 and 1983.
Entire new areas of mathematics such as
mathematical logic,
topology,
complexity theory, and
game theory changed the kinds of questions that could be answered by mathematical methods.
The French
Bourbaki Group attempted to bring all of mathematics into a coherent rigorous whole, publishing under the
pseudonym Nicolas Bourbaki. Their extensive work had a controversial influence on mathematical education.
There were also new investigations of limitations to mathematics.
Kurt Gödel proved that in any mathematical system that includes the integers, there are true statements that
cannot be proved.
Paul Cohen proved the
independence of the
continuum hypothesis from the
standard axioms of set theory.
Further Information
Get more info on 'History Of Mathematics'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://history_of_mathematics.totallyexplained.com">History of mathematics Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
