by DK
See also: Euclid’s Elements • Trigonometry • Cubic equations
ARYABHATA
476–550 CE
A Hindu mathematician and astronomer, Aryabhata worked in Kusumapara, an Indian center of learning. His verse treatise Aryabhatiya contains sections on algebra and trigonometry, including an approximation for pi (π) of 3.1416, accurate to four decimal places. Aryabhata also correctly believed pi to be irrational. He calculated Earth’s circumference as a distance close to the current accepted figure. He also defined some trigonometric functions, produced complete and accurate sine and cosine tables, and calculated solutions to simultaneous quadratic equations.
See also: Quadratic equations • Calculating pi • Trigonometry • Algebra
BHASKARA I
C. 600–C. 680
Little is known about Bhaskara I, although he may have been born in the Saurastra region on India’s west coast. He became one of the most important scholars of the astronomy school founded by Aryabhata, and wrote a commentary, Aryabhatiyabhasya, on Aryabhata’s earlier Aryabhatiya treatise. Bhaskara I was the first person to write numbers in the Hindu-Arabic decimal system with a circle for zero. In 629, he also found a remarkably accurate approximation of the sine function.
See also: Trigonometry • Zero
IBN AL-HAYTHAM
C. 965–C. 1040
Also known as Alhazen, Ibn al-Haytham was an Arab mathematician and astronomer, born in Basra, now in Iraq, who worked at the court of the Fatimid Caliphate in Cairo. He was a pioneer of the scientific method that maintained that hypotheses should be tested by experiment and not just assumed to be true. Among his achievements, he established the beginnings of a link between algebra and geometry, building on the work of Euclid and trying to complete the lost eighth volume of Apollonius of Perga’s Conics.
See also: Euclid’s Elements • Conic sections
BHASKARA II
1114–85
One the greatest of the medieval Indian mathematicians, Bhaskara II was born in Vijayapura, Karnataka, and is believed to have become the head of the astronomical observatory at Ujjain in Madhya Pradesh. He introduced some preliminary concepts of calculus; established that dividing by zero yields infinity; found solutions to quadratic, cubic, and quartic equations (including negative and irrational solutions); and suggested ways to unlock Diophantine equations of the second order (to the power of two), which would not be solved in Europe until the 1700s.
See also: Quadratic equations • Diophantine equations • Cubic equations
NASIR AL-DIN AL-TUSI
1201–74
Born in Tus, the Persian mathematician al-Tusi devoted his life to study after he lost his father at a young age. He became one of the great scholars of his day, making important discoveries in math and astronomy. He established trigonometry as a discipline, and in his Commentary on the Almagest—an introduction to trigonometry—described methods for calculating sine tables. Although taken prisoner by invading Mongols in 1255, al-Tusi was appointed a scientific advisor by his captors and later established an astronomical observatory in the Mongol capital Maragheh, now in Iran.
See also: Trigonometry
KAMAL AL-DIN AL-FARISI
C. 1260–C. 1320
Al-Farisi was born in Tabriz, Persia (now Iran). He was a student of polymath Qutb al-Din al-Shirazi, himself a pupil of Nasir al-Din al-Tusi (see above), and, like them, was a member of the Maragheh school of mathematician–astronomers. His explorations of number theory included amicable numbers and factorization. He also applied the theory of conic sections (circles, ellipses, parabolas, and hyperbolas) to solve optical problems, and explained that the different colors of a rainbow were produced by the refraction of light.
See also: Conic sections • The binomial theorem
NICOLE ORESME
C. 1320–82
Born in Normandy, France, probably to a peasant family, Oresme studied at the College of Navarre, where pupils from poor backgrounds were subsidized by the royal estate. He later became dean of Rouen Cathedral. Oresme devised a coordinate system with two axes to represent the change of one quality with respect to another—for example, how temperature changes with distance. He worked on fractional exponents and infinite series and was the first to prove the divergence of harmonic series, but his proof was lost and the theory was not proven again until the 1600s. He also argued that Earth could be rotating in space, rather than the Church-approved view that the celestial bodies circled around Earth.
See also: Algebra • Coordinates • Calculus
NICCOLÒ FONTANA TARTAGLIA
1499–1557
As a child, Tartaglia was attacked by French soldiers invading Venice. He survived, but with serious facial injuries and a speech impediment, which earned him the nickname “Tartaglia,” or stutterer. Essentially self-taught, he became a civil engineer, designing fortifications. Tartaglia realized that an understanding of the trajectory of cannonballs was critical for his designs, which led him to pioneer the study of ballistics. His published mathematical works included a formula for solving cubic equations, an encyclopedic math treatment—Treatise on Numbers and Measures—and translations of Euclid and Archimedes.
See also: The Platonic solids • Trigonometry • Cubic equations • The complex plane
GEROLAMO CARDANO
1501–76
A contemporary of Niccolò Tartaglia, Cardano was born in Lombardy and became an outstanding physician, astronomer, and biologist, as well as a renowned mathematician. He studied at the universities of Pavia and Padua in what is now Italy, was awarded a doctorate in medicine, and worked as a physician before becoming a teacher of mathematics. Cardano published a solution to cubic and quartic equations, acknowledged the existence of imaginary numbers (based on the square root of –1), and is alleged to have forecast the exact date of his own death.
See also: Algebra • Cubic equations • Imaginary and complex numbers
JOHN WALLIS
1616–1703
Although Wallis studied medicine at Cambridge University and was later ordained a priest, he retained the interest in arithmetic he first developed as a schoolboy in Kent, England. A supporter of the Parliamentarian cause, Wallis deciphered Royalist dispatches during the English Civil War. In 1644, he was appointed professor of geometry at the University of Oxford and became a champion of arithmetic algebra. His contributions toward the development of calculus include originating the idea of the number line, introducing the symbol for infinity, and developing standard notation for powers. He was one of the small group of scholars whose meetings led to the establishment of the Royal Society of London in 1662.
See also: Conic sections • Algebra • The binomial theorem • Calculus
GUILLAUME DE L’HÔPITAL
1661–1704
Born in Paris, l’Hôpital was interested in math from a young age and was elected to the French Academy of Sciences in 1693. Three years later, he published the first textbook on infinitesimal calculus: Analyse des infiniment petits pour l’intelligence des lignes courbes (Analysis of the Infinitesimally Small for the Understanding of Curved Lines). Although l’Hôpital was an accomplished mathematician, many of his ideas were not original. In 1694, he had offered the Swiss mathematician Johann Bernoulli 300 livres a year for information on his latest discoveries and an agreement that he would not share them with other mathematicians. Many of these ideas were published by l’Hôpital in Infinitesimal Calculus.
See also: Calculus
JEAN LE ROND D’ALEMBERT
1717–83
The illegitimate son of a celebrated Paris hostess, d’Alembert was brought up by a glazier’s wife. Funded by his estranged father, he studied law and medicine, then turned to mathematics. In 1743, he stated that Newton’s third law of motion is as true for freely moving bodies as it is for fixed bodies (d’Alembert’s principle). He also developed partial differential equations, explained the variations in the orbits of Earth and other planets, and researc
hed integral calculus. Like other French philosophes, such as Voltaire and Jean-Jacques Rousseau, d’Alembert believed in the supremacy of human reason over religion.
See also: Calculus • Newton’s laws of motion • The algebraic resolution of equations
MARIA GAETANA AGNESI
1718–99
Born in Milan, then under Austrian Hapsburg rule, Agnesi was a child prodigy who, as a teenager, lectured friends of her father on a wide range of scientific subjects. In 1748, Agnesi became the first woman to write a math textbook, the two-volume Instituzioni analitiche (Analytical Institutions), which covered arithmetic, algebra, trigonometry, and calculus. Two years later, recognizing her achievement, Pope Benedict XIV awarded her the chair of mathematics and natural philosophy at the University of Bologna, making her the first woman professor of math at any university. The equation describing a particular bell-shaped curve called the “witch of Agnesi” is named in her honor, although “witch” was a mistranslation from the Italian word for “curve.”
See also: Trigonometry • Algebra • Calculus
JOHANN LAMBERT
1728–77
Lambert was a Swiss-German polymath, born in Mulhouse (now in France), who taught himself math, philosophy, and Asian languages. He worked as a tutor before becoming a member of the Munich Academy in 1759 and the Berlin Academy five years later. Among his mathematical achievements, he provided rigorous proof that pi is an irrational number, and introduced hyperbolic functions into trigonometry. He produced theorems on conic sections, simplified the calculation of the orbits of comets, and created several new map projections. Lambert also invented the first practical hygrometer, used to measure the humidity of air.
See also: Calculating pi • Conic sections • Trigonometry
GASPARD MONGE
1746–1818
The son of a merchant, by the age of 17, Monge was teaching physics in Lyon, France. He later worked as a draftsman at the École Royale, Mézières, and in 1780 became a member of the Academy of Sciences. Monge was active in public life, embracing the ideals of the French Revolution. He was appointed Minister of the Marine in 1792, and also worked to reform France’s education system, helping to found the École Polytechnique in Paris in 1794 and contributing to the founding of the metric system of measurement in 1795. Described as the “father of engineering drawing,” Monge invented descriptive geometry, the mathematical basis of technical drawing, and orthographic projection.
See also: Decimals • Projective geometry • Pascal’s triangle
ADRIEN-MARIE LEGENDRE
1752–1833
Legendre taught physics and math at the École Militaire in Paris from 1775 to 1780. During this period, he also worked on the Anglo-French Survey, using trigonometry to calculate the distance between the Paris Observatory and London’s Royal Greenwich Observatory. During the French Revolution, he lost his private fortune, but in 1794 he published Eléments de géométrie (Elements of Geometry), which remained a key geometry textbook for the next century, and he was then appointed a math examiner at the École Polytechnique. In number theory, he conjectured the quadratic reciprocity law and the prime number theorem. He also produced the least-squares method for estimating a quantity based on consideration of measurement errors, and gave his name to three forms of elliptic integrals—the Legendre transform, transformation, and polynomials.
See also: Calculus • The fundamental theorem of algebra • Elliptic functions
SOPHIE GERMAIN
1776–1831
During the chaos of the French Revolution, 13-year-old Sophie Germain was confined to her wealthy father’s house in Paris and began to study the mathematics books in his library. As a woman she was ineligible to study at the École Polytechnique, but she obtained lecture notes and corresponded with the mathematician Joseph-Louis Lagrange. In her work on number theory, Germain also corresponded with Adrien-Marie Legendre (see above) and Carl Gauss, and her ideas on Fermat’s last theorem helped Legendre to prove the theorem where n = 2. In 1816, she was the first woman to win a prize from the Academy of Sciences in Paris, for a paper on the elasticity of metal plates.
See also: The fundamental theorem of algebra • Proving Fermat’s last theorem
NIELS ABEL
1802–29
Abel was a Norwegian mathematician who died tragically young. After graduating from the University of Christiana (now Oslo) in 1822, he traveled widely in Europe, visiting leading mathematicians. He returned to Norway in 1828, but died from tuberculosis the following year at the age of 26, days before a letter arrived offering him a prestigious math professorship at the University of Berlin. Abel’s most important mathematics contribution was to prove that there is no general algebraic formula for solving all quintic (fifth-degree) equations. To make his proof, he invented a type of group theory where the order of the elements within a group is immaterial. This is now known as an abelian group. The annual Abel Prize for mathematics is awarded in his honor.
See also: The fundamental theorem of algebra • Elliptic functions • Group theory
JOSEPH LIOUVILLE
1809–82
Born in northern France, Liouville graduated from the École Polytechnique, Paris, in 1827 and took up a teaching post there in 1838. His academic work spanned number theory, differential geometry, mathematical physics, and astronomy, and in 1844 he was the first to prove the existence of transcendental numbers. Liouville wrote more than 400 papers and in 1836 founded the Journal de Mathématiques Pures et Appliquées (Journal of Pure and Applied Mathematics), the world’s second-oldest mathematical journal, which is still published monthly.
See also: Calculus • The fundamental theorem of algebra • Non-Euclidean geometries
KARL WEIERSTRASS
1815–97
Born in Westphalia, Germany, Weierstrass developed an interest in mathematics at an early age. His parents wanted their son to have a career in administration, so he was sent to study law and economics at his university, but left without gaining a degree. He then trained as a teacher, ultimately becoming a professor of mathematics at the Humboldt University of Berlin. Weierstrass was a pioneer in the development of mathematical analysis and in the modern theory of functions, and rigorously reformulated calculus. An influential teacher, he included among his pupils the young Russian émigré and pioneering mathematician Sofya Kovalevskaya.
See also: Calculus • The fundamental theorem of algebra
FLORENCE NIGHTINGALE
1820–1910
Named after her Italian birthplace, Florence Nightingale was a British social reformer and pioneer of modern nursing, who based much of her work on the use of statistics. In 1854, after the outbreak of the Crimean War, Nightingale went to work among wounded soldiers at The Barrack Hospital in Scutari, Turkey. There, she campaigned tirelessly for better hygiene, earning the nickname “The lady with the lamp.” Back in Britain, Nightingale became an innovator in the use of graphs to display statistical data. She developed the Coxcomb chart, a variation on the pie chart, using circle segments of different sizes to display variations in data, such as the causes of mortality among soldiers. Her actions helped to establish a Royal Commission on health in the army in 1856. In 1907, she was the first woman to receive the Order of Merit, Britain’s highest civilian honor.
See also: The birth of modern statistics
ARTHUR CAYLEY
1821–95
Born in Richmond, Surrey, Cayley was probably the leading British pure mathematician of the 1800s. Graduating from Trinity College, Cambridge, he embarked on a career as a conveyancing lawyer. In 1860, however, he gave up his lucrative law practice to take up a pure math professorship at Cambridge, on a far more modest salary. Cayley was a pioneer of group theory and matrix algebra, devised the theories of higher singularities and invariants, worked in higher-dimensional geometry, and extended the quaternions of William Hamilton to create octonions.
See also: Non-Euclidean geometries • Group theory • Quate
rnions • Matrices
RICHARD DEDEKIND
1831–1916
Dedekind was one of Carl Gauss’s students at the University of Göttingen, Germany. After graduating, he worked as an unsalaried lecturer before teaching at the Zurich Polytechnic, Switzerland. Returning to Germany, in 1862 he started work at the Technical High School in Braunschweig, where he remained for the rest of his working life. He proposed the Dedekind cut, now a standard definition of real numbers, and defined concepts of set theory, such as similar sets and infinite sets.
See also: The fundamental theorem of algebra • Group theory • Boolean algebra
MARY EVEREST BOOLE
1832–1916
Mary Everest’s love of math began young when she studied the books in the study of her clergyman father, whose friends included polymath Charles Babbage, the inventor of the Difference Engine. At 18, Mary met renowned mathematician George Boole (who, like her, was self-taught) in Ireland. They married five years later, but George died soon after the birth of their fifth child. In 1864, with five daughters to raise and no financial support, Mary returned to London, where she worked as a librarian at Queen’s College, a girls’ school, and later gained a reputation as an eminent children’s teacher. She also wrote books that made math more accessible to young students, including Philosophy and Fun of Algebra (1909).