Jean-Robert Argand was a highly influential mathematician. A self-taught mathematician born in Geneva, Switzerland, Argand and his family moved to Paris in 1806, where he privately published a landmark essay on the representation of imaginary quantities. It described a method of graphing complex numbers via analytical geometry, which became called the Argand diagram, and was the first essay to propose the idea of modulus to indicate the magnitude of vectors and complex numbers, and the notation for vectors.

In 1814 Argand published Réflexions sur la nouvelle théorie d'analyse (Reflections on the new theory of analysis), which proved the fundamental theorem of algebra. This was the first complete proof of the theorem. He died on 13 August 1822 in Paris.

Friedrich Wilhelm Bessel was a German mathematician, astronomer and the systematizer of the Bessel functions. Born in Minden-Ravensberg, Bessel produced a refinement on the orbital calculations for Halley’s Comet, later becoming an assistant at Lilienthal Observatory, where he worked on James Bradley’s stellar observations to produce precise positions for 3,222 stars.

In January 1810, Bessel was appointed director of the Königsberg Observatory by King Frederick William III of Prussia, where he was able to pin down the position of over 50,000 stars. His work at the Königsberg Observatory won him the Lalande Prize from the French Academy of Sciences in 1811.

Bessel was the first person to use parallax in calculating the distance to a star in 1838. His announcement of Sirius’s ‘dark companion’ in 1844 was the first correct claim of a previously unobserved star by positional measurement, and eventually led to the discovery of Sirius B. While studying the dynamics of ‘many-body’ gravitational systems, he developed Bessel functions. Critical for the solution of certain differential equations; these functions are widely used in both classical and quantum physics to this day. Bessel is also responsible for the correction to the formula for the sample variance estimator named in his honour. He died from cancer in 1846 in Königsberg.

George Boole (2 November 1815–8 December 1864) was an English mathematician, philosopher and logician that worked in the fields of differential equations and algebraic logic. Best known as the author of The Laws of Thought, Boole is also the inventor of the prototype of what is now called Boolean logic, which became the basis of the modern digital computer.

At 16, Boole took up a junior teaching position in Doncaster. By the time he was 19, Boole had established his own school at Lincoln, before going on to run a boarding school. In 1849 he was appointed as the first professor of mathematics at Queen’s College, Cork, in Ireland. Boole later went on to be elected a Fellow of the Royal Society in 1857, and received honorary degrees of LLD from the University of Dublin and Oxford University. Boole died of an attack of fever on 8 December 1864.

Robert Boyle, FRS was a natural philosopher, chemist, physicist and inventor. Regarded today as the first modern chemist, he is best known for Boyle’s law, which describes the inversely proportional relationship between the absolute pressure and volume of a gas, providing the temperature is kept constant within a closed system. The Sceptical Chymist is seen as a cornerstone book in the field of chemistry.

Boyle was born in Lismore Castle, in County Waterford, Ireland, the son of an Earl, and received private tutoring in Latin, Greek and French before being sent to Eton College in England. He later travelled, then went on to devote his life to scientific research, taking a prominent place in the group known as the ‘Invisible College’, who devoted themselves to the growth of the ‘new philosophy’.

In 1691 Robert Boyle died from paralysis.

Baron Augustin-Louis Cauchy was a French mathematician who became an early pioneer of analysis.

More concepts and theorems have been named after Cauchy than any other mathematician (in elasticity alone there are 16 concepts and theorems named for Cauchy). A prolific writer, he wrote approximately 800 research articles. Cauchy covered notable subjects, including the theory of series, in which he developed the notion of convergence and discovered many of the basic formulas for q-series; he developed the theory of numbers and complex quantities, and the theory of groups and substitutions, the theory of functions, differential equations and determinants. Cauchy was the first to define complex numbers as pairs of real numbers.

Cauchy also contributed significant research in mechanics, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He introduced a 3 × 3 symmetric matrix of numbers that is now known as the Cauchy stress tensor. In elasticity, Cauchy originated the theory of stress, and his results are nearly as valuable as those of Poisson.

Anders Celsius was a Swedish astronomer that went on to become professor of astronomy at Uppsala University, later going on to found the Uppsala Astronomical Observatory in 1741. In 1742 he proposed the Celsius temperature scale which takes his name.

AThe son of an astronomy professor, Celsius was a talented mathematician from an early age, and studied at Uppsala University, where his father was a teacher. In 1730 he became a professor of astronomy there.

ACelsius was the first to suggest a connection between the aurora borealis and changes in the magnetic field of the Earth. At Nuremberg in 1733 he published a collection of 316 observations of the aurora borealis made by himself and others. In astronomy, Celsius began the first attempt to measure the magnitude of starlight with a tool other than the human eye.

AHe proposed the Celsius temperature scale in a paper to the Royal Society of Sciences in Uppsala, the oldest Swedish scientific society, founded in 1710. His thermometer was calibrated with a value of 100°C for the freezing point of water and 0°C for the boiling point. In 1744 he died from tuberculosis.

Jacques Alexandre César Charles was a French inventor, scientist, mathematician and balloonist.

Jacques Charles and the Robert brothers launched the world’s first hydrogen filled balloon on 27 August 1783, from the Champ de Mars (now the site of the Eiffel Tower). It was filled with hydrogen that had been made by pouring a quarter of a tonne of sulphuric acid onto half a tonne of scrap iron. This gas was fed into the balloon via lead pipes.

On 1 December 1783, Charles undertook a solo balloon flight. This time it ascended rapidly to an altitude of about 3,000 metres, where he saw the sun again. He began suffering from aching pain in his ears so he ‘valved’ to release gas, and descended to land gently about 3 km away at Tour du Lay. Charles never flew again, but a hydrogen balloon came to be called a Charlière in his honour. Charles developed several useful inventions, including a valve to let hydrogen out of balloons and other devices, such as the hydrometer and reflecting goniometer, and improved the Gravesand heliostat and Fahrenheit's aerometer. In addition he confirmed Benjamin Franklin's electrical experiments.

Charles’ law (also known as the law of volumes) describes how gases expand when heated. It was first published by natural philosopher Joseph Louis Lussac in 1802, but he credited it to unpublished work by Jacques Charles, and named the law in his honour. Charles’ Law states that under constant pressure, the volume of an ideal gas is proportional to its absolute temperature. The volume of a gas at constant pressure increases linearly with the absolute temperature of the gas.

The formula he created was V₁/T₁ = V₂/T₂.

Gabriel Cramer was a Swiss mathematician born in Geneva. At 18 he received his doctorate and at 20 he was co-chair of mathematics. His articles cover a wide range of subjects, including the study of geometric problems, the history of mathematics, philosophy, and the date of Easter. He published an article on the aurora borealis in the Philosophical Transactions of the Royal Society of London and he also wrote an article on law where he applied probability to demonstrate the significance of having independent testimony from two or three witnesses rather than from a single witness.

Cramer’s most famous book, Introduction à l'analyse des lignes courbes algébraique,is a work which Cramer modelled on Newton’s memoir on cubic curves, and he highly praises a commentary on Newton’s memoir written by Stirling.

Abraham de Moivre was a French mathematician famous for de Moivre’s formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers at the time. He also first discovered Binet’s formula, the closed-form expression for Fibonacci numbers linking the nth power of φ to the nth Fibonacci number.

Throughout his life de Moivre remained poor. He continued studying the fields of probability and mathematics until his death in 1754, and several additional papers were published after his death. He pioneered the development of analytic geometry and the theory of probability by expanding upon the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family.

De Moivre also published an article called ‘Annuities upon Lives’, in which he revealed the normal distribution of the mortality rate over a person’s age. From this he produced a simple formula for approximating the revenue produced by annual payments based on a person’s age. This is similar to the types of formulas used by insurance companies today.

Augustus De Morgan was a British mathematician and logician. He formulated De Morgan’s laws and introduced the term ‘mathematical induction’.

Augustus De Morgan was born in 1806, in India, but the family moved to England when he was seven months old. In 1823, at the age of 16, he entered Trinity College, Cambridge, later going on to become Professor of Mathematics at the newly established London University (now University College London).

De Morgan published a number of works, including The Differential and Integral Calculus, and made them cheaply and easily available to anyone. He was the first president of the London Mathematical Society. Five years after his resignation from University College De Morgan died of nervous prostration on 18 March 1871.

René Descartes was a French philosopher, mathematician and writer. He has been dubbed the ‘Father of Modern Philosophy’, and much subsequent Western philosophy is a result of his writings, which are studied closely to this day, in particular, his Meditations on First Philosophy. Descartes’ influence in mathematics is also apparent; the Cartesian coordinate system – allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system – was named after him.

Descartes is perhaps best known for the philosophical statement ‘Cogito ergo sum’ (I think, therefore I am), found in part IV of Discourse on the Method.

One of Descartes’ most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c. He also pioneered the standard notation that uses superscripts to show the powers or exponents, such as the 4 used in x4 to indicate squaring of squaring. He was also the first person to assign a place for algebra in our system of knowledge, and believed that algebra was a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. René Descartes died on 11 February 1650 in Stockholm, Sweden, of suspected pneumonia.

Leonhard Euler was a pioneering Swiss mathematician and physicist who made important discoveries in infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation.

Euler worked in almost all areas of mathematics, as well as continuum physics, lunar theory and other areas of physics. He is the only mathematician to have two numbers named after him – the immensely important Euler’s number in calculus, and the Euler–Mascheroni Constant γ (gamma), sometimes referred to as ‘Euler's constant’.

Euler proved Newton's identities, Fermat’s little theorem, Fermat’s theorem on sums of two squares, and he made distinct contributions to Lagrange’s four-square theorem. He also invented the totient function φ(n), which is the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat’s little theorem to what is now known as Euler’s theorem.

Euler also made contributions in optics. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christian Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light. Euler suffered a brain haemorrhage on 18 September 1783 and died a few hours later.

Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. Fourier is also credited with the discovery of the greenhouse effect.

Fourier made important contributions to dimensional analysis. The other physical contribution was Fourier’s proposal of his partial differential equation for conductive diffusion of heat. This equation is now taught to every student of mathematical physics. In the 1820s Fourier calculated that an object the size of the Earth, and at its distance from the sun, should be considerably colder than the planet actually is if warmed only by the effects of incoming solar radiation. His consideration of the possibility that the Earth's atmosphere might act as an insulator of some kind is widely recognized as the first proposal of what is now known as the greenhouse effect. He died in his bed on 16 May 1830.

Ferdinand Georg Frobenius was a German mathematician best known for his contributions to the theory of elliptic functions, differential equations and to group theory. He is known for determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions and for developing the theory of biquadratic forms. He also lent his name to Frobenius manifolds – differential-geometric objects in modern mathematical physics.

Group theory was one of Frobenius’ principal interests in the second half of his career. His proof of the first Sylow theorem is frequently used today. Frobenius created the theory of group characters and group representations, which are fundamental tools for studying the structure of groups. This work led to the notion of Frobenius reciprocity and the definition of what are now called Frobenius groups.

Johann Carl Friedrich Gauss was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, statistics, electrostatics, astronomy and optics.

JIn 1796 Gauss discovered a construction of the heptadecagon and simplified manipulations in number theory. He also became the first to prove the quadratic reciprocity law. This law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.

JGauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae, which introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

JThe discovery of Ceres led Gauss to publish his theory of the motion of planetoids disturbed by large planets. This introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used to this day in order to minimize the impact of measurement error. Gauss proved the method under the assumption of normally distributed errors. In 1818 Gauss carried out a geodesic survey of the Kingdom of Hanover, linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that can be used to measure positions by using a mirror to reflect sunlight over great distances.

JIn 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoff’s circuit laws in electricity. They constructed the first electromechanical telegraph in 1833, which connected the observatory.

JHe died in Göttingen, Germany in 1855.

Joseph Henry was an American scientist who discovered the electromagnetic phenomenon of self-inductance. He also discovered mutual inductance independently of Michael Faraday, although Faraday was the first to publish his results. The SI unit of inductance, the henry, is named in his honour.

Heinrich Rudolf Hertz was the first person to conclusively prove the existence of electromagnetic waves. The scientific unit of frequency was named the hertz in his honour.

In some of his more advanced experiments, Hertz measured the velocity of electromagnetic radiation and found it to be the same as the light’s velocity. He also established beyond any doubt that light is a form of electromagnetic radiation.

His experiments expanded the field of electromagnetic transmission, and he also found that radio waves could be transmitted through different types of materials but were reflected by others, leading in the distant future to radar.

His discoveries would later be more fully understood by others and be part of the new ‘wireless age’.

Robert Hooke FRS was an English natural philosopher, architect and polymath.

Robert Hooke was born in 1635 in Freshwater on the Isle of Wight. On his father’s death in 1648, Robert was left £40, a sum that enabled him to buy an apprenticeship. Hooke studied at Wadham College during the Protectorate, where he became one of a tightly knit group of ardent Royalists centred around John Wilkins. Here he was employed as an assistant to Thomas Willis and to Robert Boyle, for whom he built the vacuum pumps used in Boyle’s gas law experiments. He built some of the earliest Gregorian telescopes, observed the rotations of Mars and Jupiter and, based on his observations of fossils, was an early advocate of biological evolution. He investigated the phenomenon of refraction, deducing the wave theory of light, and was the first to suggest that matter expands when heated and that air is made of small particles separated by relatively large distances. He performed pioneering work in the field of surveying and map-making and was involved in the work that led to the first modern plan-form map.

In 1660, Hooke discovered the law of elasticity which bears his name and which describes the linear variation of tension with extension in an elastic spring.

In 1665 Hooke published Micrographia, a book describing microscopic and telescopic observations, and some original work in biology. During this period Hooke coined the term cell for describing biological organisms.

Micrographia also contains Hooke’s ideas on combustion. His experiments led him to conclude that combustion involves a substance that is mixed with air, a statement with which modern scientists would agree, but that was not widely understood at the time. Hooke went on to conclude that respiration also involves a specific component of the air.

Maurice Karnaugh is an American physicist, famous for the Karnaugh map used in Boolean algebra.

He studied mathematics and physics at City College of New York (1944–1948) and transferred to Yale University to complete his BSc (1949), MSc (1950) and PhD.

Karnaugh worked at Bell Labs for much of his career, developing the Karnaugh map (1954) as well as patents for PCM encoding and magnetic logic circuits and coding. He later worked at IBM’s Federal Systems Division in Gaithersburg (1966–1970) and at the IBM Thomas J. Watson Research Center (1970–1989), studying multistage interconnection networks.

Gustav Robert Kirchhoff was a German physicist who contributed to the understanding of spectroscopy, electrical circuits and the emission of black-body radiation by heated objects.

Two sets of independent concepts in both circuit theory and thermal emission are named ‘Kirchhoff’s laws’ after him, as well as a law of thermochemistry.

Kirchhoff formulated his circuit laws, which are still used today, while still a student. In 1857 he calculated that an electric signal in a resistanceless wire travels at the speed of light. He proposed his law of thermal radiation in 1859, then gave a proof in 1861. Together with Robert Bunsen, Kirchhoff discovered caesium and rubidium in 1861.

He also contributed greatly to the field of spectroscopy by formalizing three laws that describe the spectral composition of light emitted by incandescent objects. He also contributed to optics, carefully solving Maxwell’s equations to provide a solid foundation for Huygen’s principle (and correcting it in the process). Kirchhoff died in 1887.

Martin Wilhelm Kutta studied at the University of Breslau from 1885 to 1890, in which time he wrote a thesis that contains the now-famous Runge–Kutta method for solving ordinary differential equations.

He is best known for the Runge–Kutta method for solving ordinary differential equations and the Zhukovsky–Kutta (Joukowski–Kutta) theorem giving the lift on an aerofoil. Kutta also went on to make further important contributions to aerodynamics.

Two further topics which Kutta worked on were research on glaciers and the history of mathematics. Kutta died on Christmas day in 1944.

Pierre-Simon, marquis de Laplace was a French mathematician and astronomer who formulated Laplace’s equation, and pioneered the Laplace transform which appears in many branches of mathematical physics. The Laplacian differential operator is also named after him. He was also one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.

While he conducted much research in physics, another major theme of his life’s endeavours was probability theory. Laplace set out a mathematical system of inductive reasoning based on probability. He died in Paris in 1827.

Adrien-Marie Legendre was a French mathematician. The moon crater Legendre is named after him.

Legendre developed the least squares method, which is used in linear regression, signal processing, statistics, and curve fitting; this was published in 1806 as an appendix to his book on the paths of comets. In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss, and as a result of this the Legendre symbol is named after him. He also did pioneering work on the distribution of primes, and on the application of analysis to number theory.

Gottfried Wilhelm Leibniz (sometimes von Leibniz) was a German mathematician and philosopher.

Leibniz developed infinitesimal calculus independently of Isaac Newton, and his Law of Continuity and Transcendental Law of Homogeneity only found mathematical use in the twentieth century. In 1685 he was the first to describe a pinwheel calculator, and also invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers.

Leibniz was the first to see that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system.

Guillaume François Antoine, Marquis de l'Hôpital was a French mathematician. His name is firmly associated with l’Hôpital’s rule for calculating limits involving indeterminate forms 0/0 and ∞/∞.

In 1693, l’Hôpital was elected to the French academy of sciences and went on to serve twice as its vice-president. Among his accomplishments were the determination of the arc length of the logarithmic graph, one of the solutions to the brachistochrone problem and the discovery of a turning point singularity on the involute of a plane curve near an inflection point.

Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series are named after him.

Maclaurin used Taylor series to characterize maxima, minima and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. The Taylor series expanded around 0 is sometimes known as the Maclaurin series.

Independently from Euler and using the same methods, Maclaurin discovered the Euler–Maclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirling’s formula and to derive the Newton–Cotes numerical integration formulas which includes Simpson’s rule as a special case.

John Napier of Merchiston is best known as the discoverer of logarithms. The inventor of the so-called ‘Napier’s bones’, Napier also made common the use of the decimal point in arithmetic and mathematics.

The computational advance available via logarithms made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics and physics, and also in astrology. Napier also improved Simon Stevin’s decimal notation; Arab lattice multiplication, used by Fibonacci, was made more convenient by the introduction of Napier’s bones, a multiplication tool he invented using a set of numbered rods.

Sir Isaac Newton PRS MP was an English polymath. Philosophiæ Naturalis Principia Mathematica, published in 1687, lays the foundations for much of classical mechanics used today. Newton showed that the motions of objects are governed by the same set of natural laws, by demonstrating the consistency between Kepler’s laws of planetary motion and his theory of gravitation.

Newton developed a theory of colour based on the observation that a prism decomposes white light into the many colours that form the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound. In mathematics, Newton shares the credit with Leibniz for the development of differential and integral calculus.

Newton was appointed Lucasian Professor of Mathematics in 1669. From 1670 to 1672, Newton lectured on optics. During this period he investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multicoloured spectrum into white light. He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected, scattered or transmitted, it stayed the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton’s theory of colour.

From this work, he concluded that the lens of any refracting telescope would suffer from the dispersion of light into colours (chromatic aberration). As a proof of the concept, he constructed the first known functional reflecting telescope, today known as a Newtonian telescope, which involved solving the problem of a suitable mirror material and shaping technique.

The Principia was published on 5 July 1687. In this work, Newton stated the three universal laws of motion that enabled many of the advances of the Industrial Revolution.

Newton’s three laws of motion (stated in modernized form): Newton’s First Law (also known as the Law of Inertia) states that an object at rest tends to stay at rest and that an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force. Newton’s Second Law states that an applied force on an object equals the rate of change of its momentum with time. The SI unit of force is the newton, named in Newton’s honour. Newton’s Third Law states that for every action there is an equal and opposite reaction. This means that any force exerted onto an object has a counterpart force that is exerted in the opposite direction back onto the first object. He died in his sleep in London on 20 March 1727 and was buried in Westminster Abbey.

Georg Simon Ohm was a Bavarian physicist and mathematician. Ohm found that there is a direct proportionality between the potential difference (voltage) applied across a conductor and the resultant electric current. This relationship is known as Ohm’s law. It first appeared alongside his complete theory of electricity, in which he stated his law for electromotive force. Ohm died in Munich in 1854, and is buried in the Alter Südfriedhof.

Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity.

Collection, his best-known work, is a compendium of mathematics in eight volumes. It covers a wide range of topics, including geometry, recreational mathematics, doubling the cube, polygons and polyhedra.

The characteristics of Pappus’s Collection are that it contains an account, systematically arranged, of the most important results obtained by his predecessors, and, second, notes about and extending previous discoveries.

Blaise Pascal was a French polymath. A child prodigy educated by his father, Pascal’s earliest work was in the natural and applied sciences, where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum.

Pascal went on to become an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of 16, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

Siméon Denis Poisson, was a French mathematician, geometer and physicist. His work on the theory of electricity and magnetism virtually created a new branch of mathematical physics, and his study of celestial mechanics discussed the stability of the planetary orbits.

In pure mathematics, his most important works were his series of memoirs on definite integrals and his discussion of Fourier series. The Poisson distribution in probability theory is named after him.

Pythagoras of Samos was an Ionian Greek philosopher and mathematician. Pythagoras made influential contributions to philosophy in the late sixth century BC. He is best known for the Pythagorean theorem, which states that in a right-angled triangle the area of the square of the hypotenuse (the side opposite the right-angle) is equal to the sum of the areas of the squares of the other two sides – that is, a² + b² = c².

Benjamin Olinde Rodrigues, more commonly known as Olinde Rodrigues, was a French banker, mathematician and social reformer.

In 1840 he published a result on transformation groups, a discovery of the quaternions, three years prior to William Rowan Hamilton’s. In his own time, his work on mathematics was largely ignored, and he has only been rediscovered late in the twentieth century.

Rodrigues is remembered for three results: Rodrigues’ rotation formula for vectors; the Rodrigues formula about series of orthogonal polynomials; and the Euler–Rodrigues parameters.

He died in Paris in 1851.

Carl David Tolmé Runge was a German mathematician, physicist and spectroscopist. He was co-developer of the Runge–Kutta method in the field of numerical analysis. The Runge crater on the Moon is named after him.

Thomas Simpson FRS was the British mathematician who invented Simpson’s rule to approximate definite integrals.

Simpson, born in Market Bosworth, Leicestershire, taught himself mathematics, then turned to astrology after seeing a solar eclipse. He also dabbled in divination and caused fits in a girl after ‘raising a devil’ from her. After this incident he and his wife had to flee to Derby. They later moved to London, where he taught mathematics at the Royal Military Academy, Woolwich.

Brook Taylor became highly involved with mathematics at St John’s College Cambridge. He graduated with an LLB in 1709 but by this time he had already solved the problem of the centre of oscillation of a body. In 1712 Taylor was elected to the Royal Society.

Two of Taylor’s books that appeared in 1715, Methodus incrementorum directa et inversa and Linear Perspective are extremely important in the history of mathematics. Second editions appeared in 1717 and 1719, respectively.

Taylor added a new branch to mathematics now called the ‘calculus of finite differences’, invented integration by parts and discovered the celebrated series known as Taylor’s expansion.

John Wallis was an English mathematician partially credited for the development of infinitesimal calculus, and is also credited with introducing the symbol ∞ for infinity. He made significant contributions to trigonometry, calculus, geometry and the analysis of infinite series. In his Opera Mathematica I (1695) Wallis introduced the term ‘continued fraction’.

Generally credited as the originator of the idea of the number line where numbers are represented geometrically in a line with the positive numbers increasing to the right and negative numbers to the left, in 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. Algebra is noteworthy as containing the first systematic use of formulae.

Thomas Young was an English polymath. He is famous for having partly deciphered Egyptian hieroglyphics (specifically the Rosetta Stone). Young made notable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony and Egyptology.

At the age of 14 Young had learned Greek and Latin and was acquainted with French, Italian, Hebrew, German, Chaldean, Syriac, Samaritan, Arabic, Persian, Turkish and Amharic.

Young’s modulus relates the stress (pressure) in a body to its associated strain (change in length as a ratio of the original length). For the first time it allowed prediction of the strain in a component subject to a known stress (and vice versa). Young’s modulus depends only on the material, not its geometry, thus allowing a revolution in engineering strategies.

Young has also been called the founder of physiological optics. In 1793 he explained the mode in which the eye accommodates itself to vision at different distances as depending on changes of the curvature of the crystalline lens, being the first to describe astigmatism and hypothesized that colour perception depends on the presence in the retina of three kinds of nerve fibres.