The mathematicians

Augustin Louis Cauchy

Cauchy pioneered the study of analysis and the theory of permutation groups. He also researched in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics.

Cauchy stated as a military engineer and in 1810 went to Cherbourg to work on Napoleon's English invasion fleet. In 1813 he returned to Paris and, after persuasion from Lagrange and Laplace, devoted himself to mathematics.

He held various posts in Paris at Faculté des Sciences, the Collège de France and École Polytechnique. In 1816 he won the Grand Prix of the French Academy of Science.

He pioneered the study of analysis and the theory of substitution groups (now called permutation groups). Cauchy proved in 1811 that the angles of a convex polyhedron are determined by its faces. In 1814 he published the memoir on definite integrals that became the basis of the theory of complex functions.

His other contributions include researches in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics.

Numerous terms in mathematics bear his name:- the Cauchy integral theorem, in the theory of complex functions; the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations; the Cauchy-Riemann equations and Cauchy sequences.

Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series and he also gave a rigorous definition of an integral. His text "Cours d'analyse" in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. The 4-volume text "Exercises d'analyse et de physique mathematique" published between 1840 and 1847 proved extremely important.

He produced 789 mathematics papers but was disliked by most of his colleagues. He displayed self-righteous obstinacy and an aggressive religious bigotry. An ardent royalist he spent some time in Italy after refusing to take an oath of allegiance. He left Paris after the revolution of 1830 and after a short time in Switzerland he accepted an offer from the King of Piedmont of a chair in Turin where he taught from 1832. In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his son.

Cauchy returned to Paris in 1838 and regained his position at the Academy but not his teaching position because he refused to take an oath of allegiance. When Louis Philippe was overthrown in 1848 Cauchy regained his chair at the Sorbonne. He held this post until his death.
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Jean Baptiste Joseph Fourier

Fourier studied the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions.

Fourier trained for the priesthood but did not take his vows. Instead took up mathematics studying (1794) and later teaching mathematics at the new Eacute;cole Normale.

In 1798 he joined Napoleon's army in its invasion of Egypt as scientific advisor. He helped establish educational facilities in Egypt and carried out archaeological explorations. He returned to France in 1801 and was appointed prefect of the department of Isere by Napoleon.

He published "Theacuteorie analytique de la chaleur" in 1822 devoted to the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions. In this he introduced the representation of a function as a series of sines or cosines now known as Fourier series.

Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.

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Carl Friedrich Gauss

Gauss's dissertation gave the first proof of the fundamental theorem of algebra. At the age of 24 he published his theory of numbers, one of the most brilliant achievements in the history of mathematics.

A child prodigy, Gauss taught himself to read and to count by the age of three. Recognising Gauss's talent, the Duke of Brunswick in 1792 provided him with money to allow him to pursue his education. He attended Caroline College from 1792 to 1795 and at this time Gauss formulated the least-squares method and a conjecture on the distribution of primes. This conjecture was proved by Jacques Hadamard in 1896.

In 1795 Gauss went to Göttingen where he discovered the fundamental theorem of quadratic residues.

Gauss developed the concept of complex numbers and in 1799 the University of Helmstedt granted Gauss a Ph.D. for a dissertation that gave the first proof of the fundamental theorem of algebra. In his dissertation Gauss severly criticized Legendre, Laplace and other major mathematicians of the day for their lack of rigour.

At the age of 24 he published "Disquisitiones arithmeticae", his theory of numbers, one of the most brilliant achievements in the history of mathematics. The construction of regular polyhedra occur in this work as do integer congruences and the law of quadratic reciprocity.

He also calculated orbits for the minor planets Ceres and Pallas. The asteroid Ceres had been briefly observed in January 1801 but had then, after it had been tracked for 41 days, was lost in the brightness of the Sun. Gauss computed the orbit using his least squares method and correctly predicted where and when Ceres would reappear. After this he accepted a position as astronomer at the Göttingen Observatory.

In 1820 Gauss invented the heliotrope, an instrument with a movable mirror which reflected the Sun's rays. It is used in geodesy. During the late 1820s, in collaboration with the physicist Wilhelm Weber who he met while the guest of Alexander von Humboldt in Berlin, Gauss explored many areas of physics doing basic research in electricity and magnetism, mechanics, acoustics, and optics. In 1833 he constructed the first telegraph.

When in his 80th year a fellow mathematician met him and described him as follows:

... a venerable, fine old fellow, with a contented manly expression. There is an extraordinary aspect of power about him and his every word. He is about 80 years of age, but not a trace of superannuation is seen about him.

Gauss made a careful study of foreign papers in the reading room at Göttingen and in particular made a systematic study of the financial news. This stood him in very good stead since he was able to gain a considerable personal fortune through his dealings on the stock exchange. He died a very rich man.
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David Hilbert

Hilbert received his Ph.D. from the University of Königsberg and was a member of staff there from 1886 to 1895 In 1895 he was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his life.

Hilbert's first work was on invariant theory, in 1888 he proved his famous Basis Theorem. First he gave an existence proof but, after Cayley, Gordan, Lindemann and others were baffled, in 1892 Hilbert produced a constructive proof which satisfied everyone.

In 1893 while still at Königsberg he began a work "Zahlbericht" on algebraic number theory. The "Zahlbericht" (1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas. The ideas of the present day subject of 'Class field theory' are all contained in this work.

Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance.

He published "Grundlagen der Geometrie" in 1899 putting geometry on a formal axiomatic setting. His famous 23 Paris problems challenged (and still today challenges) mathematicians to solve fundamental questions.

In 1915 Hilbert discovered the correct field equation for general relativity before Einstein but never claimed priority.

In 1934 and 1939 two volumes of "Grundlagen der Mathematik" were published which were intended to lead to a 'proof theory' a direct check for the consistency of mathematics. Göde's paper of 1931 showed that this aim is impossible.

Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.

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George Gabriel Stokes

Stokes established the science of hydrodynamics with his law of viscosity (1851), describing the velocity of a small sphere through a viscous fluid.

Stokes published papers on the motion of incompressible fluids in 1842-43 and on the friction of fluids in motion and the equilibrium and motion of elastic solids in 1845.

In 1849 Stokes was appointed Lucasian Professor of Mathematics at Cambridge. In 1851 Stokes was elected to the Royal Society and was secretary of the Society from 1854 to 1884 when he was elected president.

He investigated the wave theory of light, named and explained the phenomenon of fluorescence in 1852, and in 1854 theorised an explanation of the Fraunhofer lines in the solar spectrum. He suggested these were caused by atoms in the outer layers of the Sun absorbing certain wavelengths. However when Kirchhoff later published this explanation Stokes disclaimed any prior discovery.

Stokes developed mathematical techniques for application to physical problems, founded the science of geodesy, and greatly advanced the study of mathematical physics in England. His mathematical and physical papers were published in 5 volumes, the first 3 of which Stokes edited himself in 1880, 1883 and 1891. The last 2 were edited by Sir Joseph Larmor in 1887 and 1891.

The Scientists

Albert Einstein

March 14, 1879 - April 18, 1955
Physicist and Mathematician
Nobel Laureate for Physics 1921

"There are only two ways to live your life.
One is as though nothing is a miracle.
The other is as if everything is."
- Albert Einstein -

Albert Einstein was a German-born theoretical physicist who is widely considered one of the greatest physicists of all time.

While best known for the theory of relativity (and specifically mass-energy equivalence, E=mc2), he was awarded the 1921 Nobel Prize in Physics for his 1905 (Annus Mirabilis) explanation of the photoelectric effect and "for his services to Theoretical Physics". In popular culture, the name "Einstein" has become synonymous with great intelligence and genius. Einstein was named Time magazine's "Man of the Century."

He was known for many scientific investigations, among which were: his special theory of relativity which stemmed from an attempt to reconcile the laws of mechanics with the laws of the electromagnetic field, his general theory of relativity which extended the principle of relativity to include gravitation, relativistic cosmology, capillary action, critical opalescence, classical problems of statistical mechanics and problems in which they were merged with quantum theory, leading to an explanation of the Brownian movement of molecules; atomic transition probabilities, the probabilistic interpretation of quantum theory, the quantum theory of a monatomic gas, the thermal properties of light with a low radiation density which laid the foundation of the photon theory of light, the theory of radiation, including stimulated emission; the construction of a unified field theory, and the geometrization of physics.

Einstein was born on March 14, 1879, to a Jewish family, in Ulm, Württemberg, Germany. His father was Hermann Einstein, a salesman who later ran an electrochemical works, and his mother was Pauline née Koch. They were married in Stuttgart-Bad Cannstatt.

At his birth, Albert's mother was reputedly frightened that her infant's head was so large and oddly shaped. Though the size of his head appeared to be less remarkable as he grew older, it's evident from photographs of Einstein that his head was disproportionately large for his body throughout his life, a trait regarded as "benign macrocephaly" in large-headed individuals with no related disease or cognitive deficits. His parents also worried about his intellectual development as a child due to his initial language delay and his lack of fluency until the age of nine, though he was one of the top students in his elementary school.

In 1880, shortly after Einstein's birth the family moved to Munich, where his father and his uncle founded a company manufacturing electrical equipment (Elektrotechnische Fabrik J. Einstein & Cie). This company provided the first lighting for the Oktoberfest as well as some cabling in the suburb of Schwabing.

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Niels Bohr

Niels Henrik David Bohr was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in Copenhagen. He was also part of the team of physicists working on the Manhattan Project. Bohr married Margrethe Nørlund in 1912, and one of their sons, Aage Niels Bohr, grew up to be an important physicist who, like his father, received the Nobel prize, in 1975. Bohr has been described as one of the most influential physicists of the 20th century.

Bohr was born in Copenhagen, Denmark in 1885. His father, Christian Bohr, a devout Lutheran, was professor of physiology at the University of Copenhagen (it is his name which is given to the Bohr shift or Bohr effect), while his mother, Ellen Adler Bohr, came from a wealthy Jewish family prominent in Danish banking and parliamentary circles. His brother was Harald Bohr, a mathematician and Olympic footballer who played on the Danish national team. Niels Bohr was a passionate footballer as well, and the two brothers played a number of matches for the Copenhagen-based Akademisk Boldklub.

In 1903 Bohr enrolled as an undergraduate at Copenhagen University, initially studying philosophy and mathematics. In 1905, prompted by a gold medal competition sponsored by the Danish Academy of Sciences and Letters, he conducted a series of experiments to examine the properties of surface tension, using his father's laboratory in the university, familiar to him from assisting there since childhood. His essay won the prize, and it was this success that decided Bohr to abandon philosophy and adopt physics.

As a student under Christian Christiansen he received his doctorate in 1911. As a post-doctoral student, Bohr first conducted experiments under J. J. Thomson at Trinity College, Cambridge. He then went on to study under Ernest Rutherford at the University of Manchester in England. On the basis of Rutherford's theories, Bohr published his model of atomic structure in 1913, introducing the theory of electrons traveling in orbits around the atom's nucleus, the chemical properties of the element being largely determined by the number of electrons in the outer orbits. Bohr also introduced the idea that an electron could drop from a higher-energy orbit to a lower one, emitting a photon (light quantum) of discrete energy. This became a basis for quantum theory.

Niels Bohr and his wife Margrethe Nørlund had six children. Two died young, and most of the others went on to lead successful lives. One, Aage Niels Bohr, also became a very successful physicist; like his father, he won a Nobel Prize in 1975.

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Marie Curie

Marie Curie was a physicist and chemist of Polish upbringing and, subsequently, French citizenship. She was a pioneer in the field of radioactivity, the first person honored with two Nobel Prizes, and the first female professor at the University of Paris.

Her achievements include the creation of a theory of radioactivity (a term coined by her), techniques for isolating radioactive isotopes, and the discovery of two new elements, polonium and radium. It was also under her personal direction that the world's first studies were conducted into the treatment of neoplasms ("cancers"), using radioactive isotopes.

While an actively loyal French citizen, she never lost her sense of Polish identity. She named the first new chemical element that she discovered (1898) "polonium" for her native country, and in 1932 she founded a Radium Institute in her home town Warsaw, headed by her physician-sister Bronislawa.

In 1896 Henri Becquerel discovered that uranium salts emitted rays that resembled X-rays in their penetrating power. He demonstrated that this radiation, unlike phosphorescence, did not depend on an external source of energy but seemed to arise spontaneously from uranium itself. Becquerel had in fact discovered radioactivity. Marie decided to look into uranium rays as a possible field of research for a thesis. She used a clever technique to investigate samples.

Fifteen years earlier, her husband and his brother had invented the electrometer, a device for measuring extremely low electrical currents. Using the Curie electrometer, she discovered that uranium rays caused the air around a sample to conduct electricity. Her first result, using this technique, was the finding that the activity of the uranium compounds depended only on the amount of uranium present. She had shown that the radiation was not the outcome of some interaction between molecules but must come from the atom itself. In scientific terms, this was the most important single piece of work that she carried out.

Marie's systematic studies had included two uranium minerals, pitchblende and torbernite. Her electrometer showed that pitchblende was four times as active as uranium itself, and chalcolite twice as active. She concluded that, if her earlier results relating the amount of uranium to its activity were correct, then these two minerals must contain small amounts of some other substance far more active than uranium itself.

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Max Planck

Max Karl Ernst Ludwig Planck was born on 23 April in Kiel Germany. He was the sixth child in a family devoted to the church and state. His father was a prominent jurist and professor of law at the University of Kiel. At the age of 9, his father received a post at the University of Munich, and Planck attended the Maximilian Gymnasium. While there Planck succeeded very well in all subjects and he gained an interest in physics and mathematics finally graduating at the age of 17.

He found it difficult to make a decision on what career he was going to aim for, finally settling for physics rather than music or classical philogy, since he believed he had his greatest originality within that physics. He was an excellent pianist and found great pleasure in playing, having the gift of absolute pitch. Another passion of his was hiking, mountain climbing and taking long walks as regularly as possible.

In 1874 Planck entered the University of Munich, and was unimpressed with his physics professor there, Professor Phillip von Jolly. He did, however, find intellectual stimulation through self study. Planck was deeply impressed by the law of conservation of energy and he became convinced that the second law of termodynamics was an absolute law of nature.

Planck based his doctoral dissertation on the second law of thermodynamics and in July 1879, at the young age of 21 he received his doctoral degree. Following this he completed his qualifying dissertation at Munich and he employed as a lecturer (Privatdozent). With the help of his father in 1885 he took up the position of associate professor at the University of Kiel becoming a full professor in 1892 at the University of Berlin when Kirchoff died. Planck lectured on all branches of theoretical physics and had nine doctoral students study under him.

Planck was intrigued by the law discovered by his colleague Wilhelm Wien in 1896. He made several attempts at deriving this law, starting from the second law of thermodynamics as a base Experimental evidence was coming to light which showed that Wiens law broke down completely at low frequencies but was perfectly viable at high frequencies.

Plank guessed that, since the entropy of radiation depended mathematically on it's energy in the high frequency range due to Wiens law, and that because he knew what the dependance was in the low frequency region, he should somehow combine these two properties in some simple manner resulting in a formula relating frequency, to the energy of radiation. The formula was hailed a great success, but Planck noted that it was just a formula; a lucky guess which still had to be derived from first principles inorder to give it a proper scientific standing.

In 1900, at the age of 42, Planck achieved this, but in the process he had to abandon one of his greatest beliefs - that the second law of thermodynamics was an absolute law of nature. He was forced to accept Ludwig Boltzmann's statistical explanation for the second law. Planck also had to assume that the black body oscillators could only absorb and emit energy in discrete amounts of energy - packets of energy which he called quanta. Only by carrying out a statistical analysis of these quanta of energy could Planck derive his formula.

Each quanta contained an energy directly proportional to a constant, h, multiplied by the frequency of oscillation of the particular blackbody oscillator associated with that quanta. Using his formula he calculated a value for Boltzmanns constant, Avogadros numer, the charge of the electron as well as the constant h. As time passed others came to realise that because of the finite, non-zero value of h, the world at atomic dimensions could not be explained with classical mechanics. The quantum age had truly begun!

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Wilhelm Conrad Rontgen

Rontgen, Wilhelm Conrad - born March 27, 1845, Lennep, Prussia [now Remscheid, Ger.] d. Feb. 10, 1923, Munich - German physicist who was a recipient of the first Nobel Prize for Physics, in 1901, for his discovery of X rays, which heralded the age of modern physics and revolutionized diagnostic medicine.

Rontgen studied at the Polytechnic in Zürich and then was professor of physics at the universities of Strasbourg (1876-79), Giessen (1879-88), Wurzburg (1888-1900), and Munich (1900-20). His research also included work on elasticity, capillary action of fluids, specific heats of gases, conduction of heat in crystals, absorption of heat by gases, and piezoelectricity.

In 1895, while experimenting with electric current flow in a partially evacuated glass tube (cathode-ray tube), Röntgen observed that a nearby piece of barium platinocyanide gave off light when the tube was in operation.

He theorized that when the cathode rays (electrons) struck the glass wall of the tube, some unknown radiation was formed that traveled across the room, struck the chemical, and caused the fluorescence.

Further investigation revealed that paper, wood, and aluminum, among other materials, are transparent to this new form of radiation. He found that it affected photographic plates, and, since it did not noticeably exhibit any properties of light, such as reflection or refraction, he mistakenly thought the rays were unrelated to light.

In view of its uncertain nature, he called the phenomenon X-radiation, though it also became known as Rontgen radiation.

He took the first X-ray photographs, of the interiors of metal objects and of the bones in his wife's hand.