see also:

• a brief summary of mathematical principles and tools
• History of Western Philosophy

• counting numbers
  • used to solve x - 5 = 0
  • c3000BC, ancient Sumerians used an astronomical sexagesimal system which used base of 60:
    • allowed counting to 60 with 2 hands by using thumb of one hand to count each of the 3 finger bones on each finger of that hand giving total of 12 and then using the 5 digits of the other hand to record multiples of these to get to 60
    • Babylonians however counted in 6 units of 10 to get to 60
    • also allowed for easier fractions as 60 has twelve factors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers.
    • Hellenistic Greeks limited their use of sexagesimal numbers to the fractional part of a number
    • the early shekel was one-sixtieth of a mana though the Greeks later moved this relationship into the more base-10 compatible ratio of a shekel being one-fiftieth of a mina
  • 1550BC Egyptian fraction expansions of different forms for prime and composite numbers
  • Bernoulli numbers (a sequence of rational numbers which occur frequently in number theory) were known from antiquity from the sums of integer powers but there were no real formulae documented
  • Pythagorus (c532BC) founded a school of mathematics - all things are numbers
  • 300BC Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime.
  • in the 18th C, Euler:
    • led to the development of the prime number theorem
    • proved that the sum of the reciprocals of the primes diverges
    • discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function
• fractions (quotients)
  • used to solve x * 5 = 3
  • ie how do you share something amongst x number of people
  • c2000BC ancient Egyptians used least common multiples with unit fractions
  • c1000BC ancient Egyptians developed Egyptian fractions which use a finite sum of distinct unit fractions to represent a fraction
  • common or vulgar fractions a/b where a,b are real numbers and b cannot be zero;
  • astronomical sexagesimal fraction system eg. 1⁄32 = 0;1,52,30
    • still used for measuring angles, geographic coordinates, electronic navigation, and time
  • 150BC Jain mathematicians in India wrote the Sthananga Sutra, which contains work on the theory of numbers, arithmetical operations, and operations with fractions.
• irrational numbers (surds, square roots, etc)
  • those that cannot be expressed as the ratio of two integers
  • their decimal expansion does not terminate, nor end with a repeating sequence
  • all square roots of natural numbers, other than of perfect squares, are irrational
  • some mathematical constants are irrational
  • square roots are needed to solve binomial equations such as Pythagoras Theorem
  • square roots are also needed to solve many other equations such as:
    • Fibonacci sequence
  • c1900-1650BC Babylonians approximated the square root of 2, the length of the diagonal of a unit square
  • π
    • Archimedes' constant (the ratio of a circle's circumference to its diameter)
    • Ptolemy approximated the value of π (half the circumference of a unit radius circle) as 3;8,30 in sexagesimal fraction notation
    • in the 18thC, Euler popularized the Greek letter π (lowercase pi) to denote this constant
    • Tau has been defined as 2π
  • the golden ratio phi φ
    • an irrational number that is a solution to the quadratic equation x² − x − 1 = 0 with a value of (1+sqrt(5))/2 = 1.618033..
    • 5thC BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it was a fequent number found in geometry)
  • constant e, the base of the natural logarithm
    • in the 18thC, Euler, gave the current definition of the , still known as Euler's number
• negative counting numbers took a long time to be accepted in Western cultures
  • used to solve x + 5 = 0
  • ie. do you owe something to someone or do they owe you?
  • negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (Jiu zhang suan-shu), which dates from the period of the Han Dynasty (202 BC – AD 220) and the negative numbers were counted with black rods while the positives were counted with red rods (opposite to later Western bookkeeping)
  • Western civilisations regarded negative numbers as absurd
  • in the 3rdC AD, the mathematician Liu Hui established rules for the addition and subtraction of negative numbers and perhaps the Chinese philosophy of duality made this concept more acceptable than to Western civilisations
  • during the 7thC AD, negative numbers were used in India to represent debts.
  • in the 9thC AD, the Islamic mathematician, Al-Khwarizmi in his Al-jabr wa'l-muqabala (from which we get the word “algebra”) did not use negative numbers or negative coefficients but within the next century they were used.
  • by the 12thC, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions.
  • In general, until middle of the 19thC, western civilisations refused to accept negative numbers as solutions and ignored negative roots to quadratic equations, however, in 13thC, Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits or losses, and in 1545, the Italian, Gerolamo Cardano, in his Ars Magna, provided the first satisfactory treatment of negative numbers in Europe, but did not allow negative numbers in his consideration of cubic equations (although he was to invent complex numbers to solve them)
  • in Western book-keeping they were marked in red;
• complex numbers (z = x + yi where z = complex, x and y are real and i = sqrt(-1);)
  • 16th century Italian mathematicians to solve cubic equations
  • in the 18thC, Euler, used i to denote the imaginary component

• these have many purposes:
  • as a notation to represent related data or vectors
  • solving simultaneous equations via linear equations
  • manipulating 2D and 3D space via linear transformations
• 10th-2ndC BC: Chinese text The Nine Chapters on the Mathematical Art is the first example of the use of array methods to solve simultaneous equations including the concept of determinants.
• in 1545 Italian mathematician Gerolamo Cardano brought the method to Europe when he published Ars Magna
• 1683AD, Japanese mathematician Seki Kōwa developed idea of determinants.
• 18thC: Gabriel Cramer did some work on matrices and determinants
• in 19thC, Cauchy was the first to prove general statements about determinants and in 1829, that the eigenvalues of symmetric matrices are real.
• 1858: Cayley–Hamilton theorem proposed and demonstrated
• 1906: Andrey Markov developed and publishes stochastic matrix and his Markov chain / Markovian process
• 1913: English mathematician Cullis was the first to use modern bracket notation for matrices and demonstrated the first significant use of the notation A = [ai,j] to represent a matrix where ai,j refers to the ith row and the jth column

• 1900-1600BC Babylonians used Pythagorean triples to accurately map land allocations
  • Pythagorean triples are three whole numbers in which the sum of the squares of the first two equals the square of the third eg. 3,4 and 5; 8,15 and 17; 5,12 and 13;
• Thales's theorem
  • if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle.
  • this had been known by Indian and Babylonian mathematicians for special cases
  • it is said that Thales proved this using his own results that the base angles of an isosceles triangle are equal (Pons Asinorum), and that the sum of angles in a triangle is equal to 180° (Triangle angle sum)
• Pythagoras's theorem
  • Pythagorus was born c570BC
  • his theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
• Euclidean geometry
  • a mathematical system attributed to the 3rdC BC Alexandrian Greek mathematician Euclid (born c.325BC, died c270BC), which he described in his textbook on geometry: the Elements which collated prior knowledge into one book and his mathematical works probably coincided with the establishment of the Library of Alexandria which was probably built during the reign of Ptolemy II Philadelphus 285-246BC
  • Euclidean geometry has two fundamental types of measurements: angle and distance
  • see https://en.wikipedia.org/wiki/Euclidean_geometry
  • Congruence of triangles
  • Euclid proved the area of a plane figure is proportional to the square of any of its linear dimensions, and the volume of a solid to the cube in various special cases such as the area of a circle and the volume of a parallelepipedal solid
  • Archimedes' constant / pi (the ratio of a circle's circumference to its diameter or pi as it is now known)
  • Archimedes proved that a sphere has 2/3 the volume of the circumscribing cylinder
  • Ptolemy's theorem if a quadrilateral is inscribable in a circle then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.
  • circumference of the earth estimated very accurately to be 40,000km by Eratosthenes in 254BC who realised that at summer solstice a vertical pole in the town Syene (modern day Aswan) cast no shadow so he set a similar pole up in Alexandria and this did cast a shadow of 7.2deg (and hence the angular distance between the two cities), he then arranged for “bematists”, professional surveyors, to walk from Alexandria to Syene and measure that distance which came to 5000 stadia (800km). Circumference = 800km x 360deg / 7.2deg = 40,000km ¹⁾
• Cartesian coordinates
  • in the 17th century, René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra and becoming the foundation of analytic geometry and linear algebra, by inventing his Cartesian coordinates (x,y coordinates for a 2D plane and x,y,z for a 3D plane). Use of matrices as a linear transformation of the geometric plane and the vectors within that plane whilst keeping the origin of the vector the same. The determinant of the matrix transformation equates to the scaling factor of the area (if a 3D vector, then the volume) of the vector. A determinant of zero means area of a 2D vector is zero as it is now a 1D line in other words it reduces the number of dimensions by 1. A negative determinant has the effect of flipping or inverting the 2D plane. The rank value of the transformation equates to how many dimensions there are in the output. The inverse matrix of the transformation reverses the transformation. If one exists, the axis of rotation (ie. vectors which do not come off their span) is called the eigenvector and the scaling factor of stretch, the eigenvalue.

• stonehenge and similar to give solstice information and the solar year
• lunisolar calendars - attempts to match lunar cycles into the solar year
• sun dials to give time of day
• Babylonians:
  • sidereal month - time it takes the Moon to pass twice a “fixed” star
  • synodic month - time interval between two consecutive occurrences of a particular phase
  • anomalistic month - average time the Moon takes to go from perigee to perigee—the point in the Moon's orbit when it is closest to Earth
  • draconic month - the period in which the Moon returns to the same node of its orbit;
  • Saros cycle of eclipses - 223 synodic (lunar) months or approx. approximately 6585.3211 days (18 years and 10-12 days depending on leap years)
    • Three periodicities related to lunar orbit, the synodic month, the draconic month, and the anomalistic month coincide almost perfectly each saros cycle.
• 6-7thC BC: Roman calendar which had intercalarus years with a short Feb of 23-4 days (28 days in common years which were 355 days) and an extra intercalarus month of 27-28 days in intercalarus years (377-378 days)
• 6thC BC: incorrect geocentric model of the universe proposed by Anaximander and main concept of everything rotating around earth was perpetuated, and standardised by Claudius Ptolemaeus in 2ndC AD as the Ptolemaic model until Copernican revolution changed the concept to a heliocentric model
• 5thC BC: Metonic cycle of the moon phases of approx. 19 years (6940 days) by which time the moon phases occur at same time of year named after Meton of Athens
• 5thC BC: ancient Greek philosopher Parmenides
  • devised a mathematical model of how the moon and planets moved and this model was presumably used to create what appears to be the first mechanical computer - the intricate geared Antikythera Mechanism found on a Roman ship ²⁾
• 3rdC BC: heliocentric concept proposed by Aristarchus of Samos but seemingly lost and not accepted again until 16-17thC AD
• 45BC: Julian calendar introduced as a reform of the Roman calendar
  • a 3 normal years of 365 days and then one leap year of 366 days
  • removed the extra intercalarus month
  • Jan, Aug and Dec 31 days not 29, Feb 28 and 29 in leap year, Apr, Jun, Sep, Nov 30 days not 29 days
  • still used in parts of the Eastern Orthodox Church and in parts of Oriental Orthodoxy as well as by the Berbers
• 2ndC AD: Claudius Ptolemaeus standardised the geocentric model which became the Ptolemaic model
• 5thC AD: Indian mathematician Aryabhata, correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view, that the sky rotated. This was not accepted by Western world until 16thC with Copernicus
• 1252: Alfonsine tables of ephemerides - new Spanish tables based on earlier astronomical works and observations by Islamic astronomers and contemporary Spanish and Jewish astronomers
• 1543: Nicolaus Copernicus published De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres) triggered the Copernican Revolution of the heliocentric concept of the solar system
• 1551:Erasmus Reinhold published his Prutenic Tables astronomical ephemeris
• 1601: Johannes Kepler - suggested elliptical orbits and gravitation model of orbits and lunar link with tides which had been noted by sailors; inverse-square law governing the intensity of light;
• 1627: Johannes Kepler publishes his Rudolphine Tables of ephemirides
• 1582: Gregorian calendar introduced by Pope Gregory XIII to replace the Julian calendar to reduce the average length of the year from 365.25 days to 365.2425 days and thus corrected the Julian calendar's drift against the solar year which meant that the Julian calendar gained a day every 128 years.
• 18-19thc AD: realisation that the sun is not the centre of the universe

• Sumerian astronomers studied angle measure, using a division of circles into 360 degrees
• 1900-160BC Babylonians had trigonometric tables
• trigonometry principles emerged in Hellenistic word in 3rdC BC:
  • mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically.
  • a chord of a circle is a line segment whose endpoints are on the circle.
• 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.
• in 2ndC AD, astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords using a Ptolemy used a circle whose diameter is 120 parts) in Book 1, chapter 11 of his Almagest. This remained in use for the next 1200yrs although has a slight difference to modern sine usage which was first documented in perhaps 5thC AD in the Surya Siddhanta
• 1533: Gemma Frisius described for the first time the method of triangulation still used today in surveying
• 1595, Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria
• in the 18thC, Leonhard Euler fully incorporated complex numbers into trigonometry (e^ix = cosx +isinx) and created graph theory
• 17-18thC: development of trigonometric series based on works of James Gregory and Colin Maclaurin

• the use of symbols in place of unknown numbers or variables
• ancient Babylonians developed formulae to calculate solutions for problems
• Hellenistic Greek geometric algebra:
  • work on geometry such as by Euclid, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations
  • 3rdC AD, Diophantus, author of a series of books called Arithmetica dealt with solving algebraic equations, and have led, in number theory, to the modern notion of Diophantine equation which is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones (an integer solution is such that all the unknowns take integer values), and was one of the first mathematicians to introduce symbolism into algebra and established the theory of equations.
• algebra independent of geometry and arithmetic:
  • 780-850AD, Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic and established rules for manipulating and solving equations.
  • 1070AD, Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebra, laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe.
  • Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations and developed the concept of a function.
  • a number of Middle Eastern and Eastern mathematicians solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods
• 13thC: Fibonacci's solution of a cubic equation represents the beginning of a revival in European algebra.
• 15thC: Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī introduced algebraic symbolism and computed sum of n squared and sum of n cubed, and used the method of successive approximation to determine square roots
• 17thC: René Descartes invented analytic geometry and introduced modern algebraic notation
• 1683AD: Japanese mathematician Seki Kōwa developed the idea of a determinant (Gottfried Leibniz was to independently develop this 10 yrs later)
• 1770: Joseph-Louis Lagrange studied permutations and introduced Lagrange resolvents.
• late 18thC: Paolo Ruffini was the first person to develop the theory of permutation groups in the context of solving algebraic equations.
• in the 18thC, Leonhard Euler:
  • introduces f(x) notation for functions
  • elaborated the theory of higher transcendental functions by introducing the gamma function
  • introduced a new method for solving quartic equations
  • created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions.
• 19thC:
  • abstract (“modern”) algebra was developed - general theories and formalisation of algebraic structures and use of groups and set theory
  • George Peacock was the founder of axiomatic thinking in arithmetic and algebra
  • Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic
  • Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space
  • Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).

• the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
• formally developed in the 17th C AD
• ancient analysis
  • 433BC: Ācārya Bhadrabāhu uses the sum of a sum of a geometric series in his Kalpasūtra
  • Zeno's paradox of the dichotomy implicitly uses an infinite geometric sum
  • Archimedes' explicit use of infinitesimals in The Method of Mechanical Theorems
  • 3rdC AD: Chinese mathematician Liu Hui used the method of exhaustion to find the area of a circle
• medieval
  • 14thC AD, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent.
• 17thC AD
  • Johannes Kepler early 1600s:
    • contributed to the development of infinitesimal methods and numerical analysis, including iterative approximations, infinitesimals, and the early use of logarithms and transcendental equations
  • Descartes (1637 Cartesian system) and Fermat (method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves) independently developed analytic geometry
  • Newton and Leibniz independently developed infinitesimal calculus

• 1614, John Napier, publicly propounded the method of logarithms in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms)
• 1617, Henry Briggs compiles the 1st log table (using 10 as the base)
• 1647, Grégoire de Saint-Vincent published his results on solving the quadrature for the hyperbola leading to the invention of the natural logarithm
• 1675, Leibniz adopts the notation Log y
• 1714, Roger Cotes introduces complex numbers into logarithms with log(cosb + isinb) = ib
• in the 18thC, Leonhard Euler gave the current definition of the constant e, the base of the natural logarithm, still known as Euler's number and introduced the use of the exponential function and logarithms in analytic proofs

• development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
• ancient world had various concepts of calculus
  • ancient Egyptian methods to calculate volume and area
  • c408-355BC: Eudoxus used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes
  • c287–212 BC: Archimedes developed this idea further, inventing heuristics which resemble the methods of integral calculus.
  • 3rdC AD: Liu Hui in China independently discovered the method of exhaustion in order to find the area of a circle
  • 5thC AD: Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri's principle to find the volume of a sphere
  • c965-1040AD: Hasan Ibn al-Haytham, Latinized as Alhazen, derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.
• 17thC AD:
  • Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus
  • Bonaventura Cavalieri, essentially re-created Archimedes' The Method which had been lost and unknown to Cavalieri
  • calculus of finite differences developed
  • 1670: Isaac Barrow, and James Gregory prove the second fundamental theorem of calculus
• 17thC: modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other)
  • 1687: Newton was the first to apply calculus to general physics in his Principia Mathematica
  • Leibniz developed much of the notation used in calculus today and discovered the product rule and chain rule
• 1715: Brook Taylor's Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point.
• in the 18thC, Leonhard Euler and his family friends, the Bernoullis develop:
  • infinitesimal calculus
  • power series
  • invented the calculus of variations
  • found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis
  • formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations.
• 1748: Maria Gaetana Agnesi publishes one of the first and most complete works on both infinitesimal and integral calculus
• 1785: following on from Euler's work, Pierre-Simon Laplace evolves the integral form of his Laplace Transform and in 1809, applied it to extended solutions of Fourier's series method to solve the heat diffusion equation which had been developed in 1807.

• 17thC: Blaise Pascal(1623-1662) considered the case where probability of success = 1/2 as a fore-runner to the binomial distribution. Jacob Bernoulli (1654-1705) derived the binomial distribution when considered the case where p = r/(r + s) where p is the probability of success and r and s are positive integers.
• 1738: de Moivre publishes The Doctrine of Chances the study of the coefficients in the binomial expansion of (a + b)n and is accredited with the discovery of the normal distribution although at this point lacked the concept of the probability density function
• mid 18thC: Thomas Bayes (1702–1761) proved a special case of what is now called Bayes' theorem
• late 18thC: Pierre-Simon Laplace developed the Bayesian interpretation of probability and also developed his Laplacian distribution and in 1782, was the first to calculate the value of the integral ∫ e−t2 dt = √π, providing the normalization constant for the normal distribution, and in 1810 presented to the Academy the fundamental central limit theorem
• 1823: Gauss published his monograph Theoria combinationis observationum erroribus minimis obnoxiae where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution.
• 1875–6: Friedrich Robert Helmert describes the chi distribution
• 1900: English mathematician Karl Pearson independently re-discovers the chi distribution in the context of goodness of fit, for which he developed his Pearson's chi-squared test
• c1935: Fisher's Exact Test, Fisher's notion of a null hypothesis, F-distribution and hypergeometric distribution
• late 20thC: Debabrata Basu's likelihood principle.