## A History of Mathematical Astronomy - Part 4

From the desk of Fjordman on Sat, 2010-01-16 23:04

The Swiss
mathematician, clockmaker and astronomer Joost Bürgi (1552-1632) independently
invented a system of logarithms which he published in 1620, but Napier has the
priority due to his publication in 1614 of his *Mirifici Logarithmorum
Canonis Descriptio*. Napier’s system was extended and improved by his admirer Henry
Briggs (1561-1630), an English professor of geometry at Gresham College in
London, who visited Napier at Edinburgh in 1615. Briggs is
especially known for his publication of tables of logarithms to the base 10,
first one in 1617 and later the *Arithmetica Logarithmica* in 1624.

Adriaan Vlacq (1600-1667), a book publisher born in Gouda in the Netherlands, extended the earlier work of Briggs and in 1628 published the first full table of logs from 1 to 100,000, calculated to ten places. The Slovenian mathematician Jurij Vega (1754-1802) attended school in Ljubljana and became professor of mathematics at the Austrian Imperial Artillery School in Vienna. As an artillery officer he fought against the Turks near Belgrade in 1788, but he is best remembered for his accurate tables of logarithms based on those of Vlacq.

Basel in Switzerland had been a free city and a center of learning for centuries. The sciences flourished there as it did in the Dutch and Flemish cities. The Bernoulli merchant family were originally religious refugees of the Protestant faith fleeing from persecution by the Spanish rulers of the southern Netherlands who had come to Basel from Antwerp via Amsterdam

The Bernoullis were to become the world’s most successful
mathematical family. The founding member and arguably also the most gifted
representative of this famous scientific family was the brilliant mathematician
Jacob
Bernoulli (1654-1705), whose father was an important citizen of Basel.
Jacob Bernoulli was one of the early students of probability theory, on which
subject he wrote the *Ars Conjectandi* (*The
Art of Conjecturing*), published
posthumously by his nephew in 1713. This work contains the theorem of Bernoulli
and the Bernoulli
numbers, which are among the most interesting number sequences in
mathematics, particularly important in number theory. As author Dirk J. Struik
states:

“Indeed it is difficult to find in the whole history of science a family with a more distinguished record. This record begins with two mathematicians, Jakob (James, Jacques) and Johann (John, Jean) Bernoulli. Jakob studied theology, Johann studied medicine; but when Leibniz’s papers in the

Acta eruditorumappeared both men decided to become mathematicians. They became the first important pupils of Leibniz. In 1687 Jakob accepted the chair of mathematics at Basel University, where he taught until his death in 1705. In 1695 Johann became professor at Groningen; upon his older brother’s death he succeeded him in the chair at Basel, where he remained for forty-three more years. Jakob began his correspondence with Leibniz in 1687. Then, in a constant exchange of ideas with Leibniz and with each other – often with bitter rivalry between them – the two brothers began to discover the treasures contained in Leibniz’s pioneering venture. The list of their results is long and includes not only much of the material now contained in our elementary texts on differential and integral calculus, but also material on the integration of many ordinary differential equations.”

Jacob’s younger brother Johann Bernoulli (1667-1748) was born and died in Basel, but spent some time teaching at the University of Groningen in the Netherlands. Johann was tutored in mathematics by his brother and developed a mastery of the new Leibnizian calculus. Johann’s life was always full of controversy. Both brothers are considered the inventors of the calculus of variations, but their relationship changed from one of collaborators to one of often bitter rivals. Johann Bernoulli became the mentor of his even more brilliant pupil, Leonhard Euler.

The Bernoulli family produced good mathematicians for
generations. Johann’s son Daniel
Bernoulli (1700-1782) was born in Groningen but spent decades teaching as a professor at the University
of Basel in Switzerland. He
stayed with Euler in St. Petersburg, and his *Hydrodynamica* from 1736 contains Bernoulli’s law
on hydraulic pressure and a kinetic theory of gases. This work inspired Euler
in his studies of the dynamics of fluids.

The Swiss scholar Leonhard Euler (1707-1783) is widely recognized as one of the greatest mathematicians of all time. He was certainly the most productive measured in published pages, although the eccentric and prolific Hungarian Jewish mathematician Paul Erdős (1913-1996) beat him in the number of published papers. Erdős, too, was interested in combinatorics, graph theory and number theory and the “contributions which Erdös made to mathematics were numerous and broad. However, basically Erdös was a solver of problems, not a builder of theories.” The mathematical community was obviously much smaller in the eighteenth century than it is today; it would be difficult for even the most gifted scholar to have such a wide-ranging influence and dominate the entire, now very diverse field of mathematics as Euler did in his time, but his achievements are nevertheless impressive.

Euler graduated with honors from the University of Basel and got a post at the newly formed St. Petersburg Academy of Sciences, created by Tsar Peter the Great (1672-1725) as part of his modernization efforts of the Russian state. The forceful and ruthless Peter traveled unofficially with a group of 250 Russian officials on a tour of Western European capitals and was particularly impressed with the power and dynamism of the Dutch and the English. Sweden was at this time a regional power in the Baltic Sea which had to be defeated. The Swedes fought well during the Great Northern War (1700-21), but Peter the Great implemented sweeping military reforms and eventually gained the upper hand thanks to his country’s much larger population base. At the Battle of Poltava in the Ukraine in 1709, Tsar Peter won a decisive victory over the forces of King Charles XII (1682-1718) of Sweden.

In 1703, Peter the Great decided to build a new and magnificent capital city and founded Saint Petersburg next to the Baltic Sea, which remained the capital of the Russian Empire until Soviet times, after the Russian Revolution in 1917. Russia had now definitely become a European Great Power to be reckoned with and moved a little closer to the European mainstream with the influx of Western Enlightenment ideas, although these never fully penetrated Russian society. However, serfdom “became more oppressive” under his rule, and modernization efforts were undertaken with forced labor and achieved at great human cost.

Euler left St. Petersburg in 1741 to take up a post which he had been offered by Frederick the Great (1712-1786) of Prussia at the Berlin Academy of Sciences, founded on the advice of Leibniz. He returned to Russia in 1766 at the invitation of Empress Catherine the Great (1729-1796), whose succession to the throne marked a return to the Westernizing policies.

Leonhard Euler was extremely prolific and made contributions
to almost every field of mathematics that existed in his day, from hydraulics
to artillery, but a large part of his activity was devoted to astronomy. The
tremendous prestige of his textbooks settled forever many moot questions of
notation on calculus and algebra. Lagrange, Laplace and Gauss followed Euler in
their works. Euler was able to dictate articles and letters to his sons until
the day of his death in 1783. As *The Oxford Guide to the History of Physics
and Astronomy* states:

“Yet productivity was perhaps the least important of Euler’s claims to mathematical distinction. One of his great contributions was his clarity….He contributed to every branch of mathematics of his day except probability. He achieved much in the realm of number theory. He arguably founded graph theory and combinatorics when he solved the

KönigsbergBridge problem in 1736….in addition Euler contributed to ordinary and partial differential equations, the calculus of variations, and differential geometry….Euler made major contributions to every branch of mechanics. The motion of mass points, celestial mechanics, the mechanics of continuous media (mechanics of solids and nonviscous fluids, theories of materials, hydrodynamics, hydraulics, elasticity theory, the motion of a vibrating string, and rigid-body kinematics and dynamics), ballistics, acoustics, vibration theory, optics, and ship theory all received something important from him. If Beethoven did not need to hear to compose music, Euler did not need to see to create mathematics. He began to go blind in one eye in 1738 and became totally blind thirty years later. This only increased his productivity, since total blindness relieved him of academic chores like proofreading and eliminated unwanted visual distractions. Euler did not miss eyes for another reason; he had a prodigious memory.”

An important insight in the history of science was formulated by the French mathematician, biologist and astronomer Pierre-Louis de Maupertuis (1698-1759). Maupertuis had studied in Paris and in Basel with Johann Bernoulli. He became a leading member of the Berlin Academy of Sciences in 1741 and in 1744 enunciated the principle of least action. He hoped that the principle might unify the laws of the universe and prove the existence of God.

Basically,
this concept says that nature is lazy, for instance that light always
travels in straight lines. It turned out to be hugely important in quantum
mechanics. According to scholar Alan Gabbey this principle “enjoyed an improved
mathematical treatment by William Rowan Hamilton (1834, 1835), whose
transformation of *Lagrange’s equations was
modified and generalized by *Carl Gustav Jacobi in the form now
known as the Hamilton-Jacobi Equation (1837). In turn, the Hamilton-Jacobi
Equation found fruitful application in the establishment of the quantum
mechanics of Louis de Broglie (1923) and Erwin Schrödinger (1926).”

Euler further developed the principle of least action and “pointed the way for the work of Joseph Lagrange (1736-1813), which in turn provided the basis for a mathematical description of the quantum world in the twentieth century.” Lagrange was born Giuseppe Lodovico Lagrangia in Turin, Italy, where he lived during the early years of his life. He replaced Euler when the latter left Berlin for Saint Petersburg in 1766 and spent twenty productive years in that city. By 1786 he moved to France where he eventually became known as Joseph-Louis Lagrange. Lagrange was a much better mathematician than de Maupertuis and provided the concept of least action with a thorough mathematical foundation. He also worked with analysis and analytical and celestial mechanics. He is particularly famous for defining the Lagrange points, which are of great practical significance in space exploration today.

Throughout the eighteenth century, Newtonian theory advanced. After having mastered the main orbits of the planets around the Sun, European scholars began to focus on the smaller, but not negligible, effects called “perturbations” caused by the gravity of other bodies. For example, the planets Jupiter and Saturn modify the motions of each other about the Sun. Euler helped to develop the mathematical techniques needed to compute perturbation effects.

The French
mathematical astronomer Pierre-Simon Laplace (1749-1827) extended the
work of his predecessors in his multi-volume *Mécanique Céleste* (*Celestial Mechanics*) (1799-1825). He was born in the Calvados area in Normandy next to the English
Channel, a region famous for its cider, an alcoholic beverage made from
fermented apple juice. His father was in the cider business and fairly
well-off. Laplace was preoccupied with issues related to probability throughout
his life and used his considerable mathematical skills to analyze the orbits of
the planets and moons in our Solar System and their influence on each other.

In his 1796
book *Exposition du Système du Monde*, Laplace summarized for lay people the general
knowledge about astronomy in his day and advanced his version of the “nebular
hypothesis,” the idea that the Solar System formed from a nebula or cloud of
gas and dust. All the planets move around the Sun in the same direction and
plane, which is a powerful indication that they were formed at roughly the same
time through the same physical processes. Comets had very different orbits,
which indicated that their origin was different.

Laplace was a survivor who managed to keep his life through the Reign of Terror under Maximilien Robespierre (1758-1794) during the French Revolution, unlike his chemist friend Antoine Lavoisier who was beheaded. Newton himself had suggested that divine intervention might be necessary after some centuries to keep the system stable. Pierre-Simon Laplace showed that the Solar System is reasonably stable, although some of his conclusions were modified in the late twentieth century. In the words of science historian John North:

“Leibniz had taunted Samuel Clarke with the imperfection of a Newtonian universe, which he said would need winding up by God from time to time, so implying that God was an inferior artisan. The stability question was a true test of mathematical prowess. Laplace made much use of Lagrange’s method of introducing variations into the six elements of a planet’s orbit – the eccentricity, direction of aphelion, and other parameters that define it – and in 1773 he was able to prove that, even if one planet’s elements are perturbed by another planet, its mean distance from the Sun will not change appreciably, even over millennia. Over the next few years Laplace followed this with more complex theorems relating the distances, eccentricities, and angles of the orbital planes, and again these seemed to point in the same direction: the solar system is highly stable….In more recent studies, the frictional effects of the tides have been introduced into the account, and again it has been found necessary to qualify Laplace’s claims, but the skeleton of his analysis remains, a remarkable testimony to the achievements of Newton’s great successors in the century following his death.”

Johann Friedrich Carl Gauss (1777-1855), one of the greatest mathematicians of all time, was born in 1777 to a poor family in Brunswick, Germany. He was a child prodigy and his skills eventually gained him the patronage of the Duke of Brunswick, which granted him the opportunity to study at the University of Göttingen. In 1800 Ceres, the first known asteroid, was discovered by the Italian astronomer Giuseppe Piazzi (1746-1826) but was soon lost from sight. Reflecting his interest in astronomy, Gauss developed a new method for calculating orbits which succeeded in tracking it. This feat earned him a rising Europe-wide fame.

In 1807 he accepted a post at the observatory in Göttingen. He began a thorough geodesic survey of the city of Hanover in 1818 which was not completed until 1832. At Göttingen he became acquainted with physics professor Wilhelm Weber. The two shared an interest in electricity and magnetism which they explored together. Gauss’ laws describing magnetic and electric fluxes served as part of the foundation on which the Scottish mathematical physicist James Clerk Maxwell (1831-1879) developed his electromagnetic theory. Gauss outlived two wives and two of his six children and suffered from depression at times. Victor J. Katz writes:

“The patronage of the duke lasted until he was killed in battle against France in 1806 and the duchy was occupied by the French army. Fortunately for science, the French general had been given explicit orders to look out for Gauss’s welfare. Thus Gauss was able to stay in Brunswick until he accepted a position at

Göttingenin the following year as Professor of Astronomy and Director of the Observatory. Gauss remained atGöttingenfor the remainder of his life, doing research in pure and applied mathematics as well as astronomy and geodesy. Gauss was never particularly happy with teaching classes, because most of the students were uninterested in, and ill-prepared for, mathematics, but he was willing to work privately with any actively interested student who approached him. Compared to his predecessor Euler and his French contemporary Cauchy, Gauss ultimately published little, his collected works occupying only (!) 12 volumes. Nevertheless, his mathematical papers in various fields are of such profundity that they have influenced the progress of the subject to the present day.”

Augustin-Louis Cauchy (1789-1857) brought a new insistence on precision and rigor to mathematics. His life began in Paris a month after the French Revolution started with the Storming of the Bastille prison on 14 July 1789. The learned Cauchy family once had Laplace and the chemist Claude Louis Berthollet as neighbors. A staunch Roman Catholic, he made an appeal to the pope on behalf of the stricken people during the famine in Ireland in 1846. He was a prolific writer in mathematical physics and astronomy and established the calculus on the basis of the limit concept familiar today. Cauchy wrote 789 papers, a quantity exceeded only by Euler and Arthur Cayley. Many mathematical theorems have been named after him.

Cauchy’s notion of convergence had been developed in essence before by the Bohemian mathematician and theologian Bernard Bolzano (1781(1781-10-05)-1848), associated with the University of Prague, and by the Portuguese mathematician José Anastácio da Cunha (1744-1787). However, they worked in secondary languages, not in the mathematical centers of France and Germany, and it was out of Cauchy’s work that today’s notions developed. Cunha worked with algebraic analysis and differential calculus and witnessed how the Great Lisbon Earthquake in 1755 reduced the city to rubble before it was rebuilt in a beautiful way. In 1778 he was sentenced by the Inquisition to three years in prison for heretical views and for being a follower of Voltaire. Prison ruined his health and probably contributed to his premature death.

The great astronomer William Herschel (1738-1822) was born in Hanover, Germany, where his father was an oboist and brought up his sons to be musicians. William eventually ended up as an organist in England, and his sister Caroline soon joined him there. His interest in music led him to mathematics and from there on to astronomy. While this was an unusual career path it was not a unique one; there has been a perceived connection between music, mathematics and astronomy in Europe at least since the Pythagoreans in ancient Greece.

William Herschel is credited with the discovery of Uranus in 1781, the first planet to be identified with the telescope, and naturally became famous after this. He discovered it with a fine telescope of his own making. He was a keen observer of nebulae and believed that the Milky Way, which he described in greater detail than anybody had done before him, was composed of millions of individual stars. Through his unprecedented observational work in the late eighteenth century he “reached farther into space than anyone had before and began to outline the structure of our galaxy, but his speculative cosmology did not attract disciples.”

The German astronomer Johann Elert Bode (1747-1826), director of the Berlin Observatory, determined the new planet’s orbit and gave it the name Uranus. Bode collected most known observations of it and found that it had been observed before William Herschel, among others by the great English astronomer John Flamsteed, yet nobody had realized that it was a planet.

Caroline Herschel (1750-1848) became William’s valued assistant in England and the first notable woman astronomer. She was granted a salary from the king, just like her brother, and could with some justification be viewed as a professional astronomer. She personally discovered eight comets and together with the Scottish science writer Mary Somerville (1780-1872) became the first honorary woman member of the Royal Society in London in 1835.

Uranus can be spotted by a person with good eyesight under ideal conditions, yet no ancient culture had identified it as a planet as far as we know. Neptune, on the other hand, is so distant and faint that it cannot be seen through naked-eye observations. Galileo had apparently noticed it through his telescope as early as 1612 but had mistaken it for a star. Neptune was recorded several more times without being recognized as a planet, among others by Jérôme Lalande (1732-1807), a French astronomer who made accurate tables of planetary positions and taught men such as astronomer Jean Baptiste Joseph Delambre (1749-1822). The same goes for the English scholar John Herschel (1792-1871), son of Wilhelm Herschel, in 1830.

Neptune was eventually discovered following detailed mathematical calculations in the 1840s undertaken to explain certain anomalies that had been detected in the orbit of Uranus. In 1845 the mathematical physicist François Arago (1786-1853), director of the Paris Observatory, persuaded the French mathematical astronomer Urbain Le Verrier (1811-1877) to start work on the problem. He was appointed a teacher of astronomy at the École Polytechnique in Paris in 1837. Le Verrier had carefully studied the orbit of the innermost planet Mercury, which exhibits anomalies that could not be explained before the general theory of relativity (spacetime curves around massive objects such as the Sun) and were thought to be caused by a hypothetical planet between Mercury and the Sun called Vulcan. Le Verrier explained the irregular orbit of Uranus by assuming the presence of another, even more distant planet.

On Sept. 23, 1846, after only an hour of searching, the German astronomer Johann Galle (1812-1910) at the Berlin Observatory found the new planet very close to where it had been predicted to exist by Le Verrier. Unknown to Le Verrier, similar calculations based on studies of the orbit of Uranus were made at the same time by the English mathematician and astronomer John Couch Adams (1819-1892). Adams was initially self-taught in mathematics but later gained admission to the University of Cambridge. He had sent some of his calculations to the Cambridge Observatory, but these had not been properly followed up.

This situation triggered a long and fruitless debate over who should receive credit for the discovery of Neptune. Many astronomers credit both Urbain Le Verrier and John Couch Adams as co-discoverers because they had worked simultaneously and independently on the same problem and Adams had in fact begun his work first. Others credit Le Verrier alone as the actual discovery by Galle was made from his calculations, not those done by Adams.

The discovery of Neptune was rightly hailed as a great triumph for science. Two centuries after Newton, European mathematicians could calculate orbits with such precision that they could scientifically predict the existence of a planet. However, while the predictions of Le Verrier and Adams were reasonably correct in 1846 they would have been significantly less so a few years later since they didn’t get everything right regarding Neptune’s obit. As with many scientific discoveries, this one was based on good scholarly work and a little bit of luck.

Mathematics has historically been overwhelmingly created by men, often by very young men. The French mathematician Évariste Galois (1811-1832) died from wounds suffered in a duel before his twenty-first birthday, but had nevertheless contributed notable works whose full importance was grasped years after his premature death. As one online biography states, “Galois’ complete works fill only 60 pages, but he will be remembered.” In 1830 he developed group theory, which was to prove of great importance for mathematical physics, especially in the development of quantum mechanics during the twentieth century.

The
Norwegian mathematician Niels Henrik Abel (1802-1829) died in poverty while still only
in his twenties, yet he, too, had already left his mark on mathematical
history. He suffered from tuberculosis, but continued to develop
groundbreaking mathematics as his health deteriorated. Dirk J. Struik from *A Concise History of
Mathematics*, Fourth
Revised Edition:

“Abel proved the impossibility of solving the general quintic equation by means of radicals – a problem which had puzzled mathematicians from the time of Bombelli and Viète (a proof of 1799 by the Italian Paolo Ruffini was considered by Poisson and other mathematicians as too vague). Abel now obtained a stipend which enabled him to travel to Berlin, Italy, and France. But, tortured by poverty most of his life and unable to get a position worthy of his talents, Abel established few personal mathematical contacts and died (1829) soon after his return to his native land….Abel’s investigations on elliptic functions were conducted in a short but exciting competition with Jacobi. Gauss in his private notes had already found that the inversion of elliptic integrals leads to single-valued, doubly periodic functions, but he never published his ideas. Legendre, who had spent so much effort on elliptic integrals, had missed this point entirely and was deeply impressed when, as an old man, he read Abel’s discoveries.”

The Frenchman Adrien-Marie Legendre (1752-1833) was one of the leading mathematicians in Europe at the turn of the nineteenth century. While he made many valuable personal contributions, some of his work was later perfected by others, among them Abel, Jacobi and Galois. Carl Gustav Jacobi (1804-1851) was a German (Prussian) mathematician, born of Jewish parents, who studied in Potsdam and at the University of Berlin to be able to teach mathematics, Greek and Latin. In 1829 Jacobi met Legendre and other French mathematicians such as Fourier when he made a visit to Paris, and he visited Gauss in Göttingen. He became an influential and inspiring teacher and made contributions to the theory of elliptic functions.

William Rowan Hamilton (1805-1865) lived his whole life in Dublin,
Ireland. He could read Hebrew, Latin and Greek at the age of five. By 1822 his
mathematical abilities had advanced to such an extent that he discovered a
significant error in Laplace’s treatise *Celestial Mechanics*.* *He formed
a lasting friendship with the English Romantic poet William Wordsworth
(1770-1850). Hamilton made major contributions to optics, and Hamiltonian
mechanics became even more appreciated as the discipline of quantum mechanics
began to take shape in the twentieth century. In 1843 he introduced
quaternions, algebra with hyper-complex numbers. His many years of work on a
theory of quaternions “would greatly impact the development of the modern
system of vector analysis and is sometimes utilized today for computer
graphics, attitude control systems, and the control theory used in engineering.”

Richard Dedekind (1831-1916) is especially remembered for his redefinition of irrational numbers in terms of Dedekind cuts. Dedekind was, like Gauss before him, born in Brunswick (Braunschweig) in north-central Germany. He became the last pupil of Carl Friedrich Gauss in 1852 before the latter retired from teaching. He thereafter studied at the University of Berlin where he was a contemporary of Bernhard Riemann (1826-1866). In 1858 he began teaching at the Polytechnic in Zürich, Switzerland. He was a friend of the mathematician Lejeune Dirichlet (1805-1859) and was fond of holidays in Switzerland, the Austrian Tyrol or the Black Forest region in southern Germany. He met Georg Cantor (1845-1918), the creator of set theory, in 1874 while staying in the scenic town of Interlaken in the Swiss Alps.

J. J. O’Connor and E. F. Robertson have given a brief summary of mathematical history:

“Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th century. Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann. Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass….Lie’s work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory. The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm’s work led to Hilbert and the development of functional analysis.”

Hermann Grassmann (1809-1877) was a German polymath educated in theology, languages, philosophy and literature at the University of Berlin. He was more famous in his day as a linguist and scholar of Sanskrit and Indo-European languages than as a mathematician, somewhat to his personal disappointment. His significant mathematical contributions to the development of a general calculus of vectors were only fully appreciated after his death.

The Norwegian mathematician Marius Sophus Lie (1842-1899), who studied at the University of Christiania (Oslo) but later spent much of his time in Germany where he collaborated with Felix Klein, is renowned for Lie algebra and Lie groups, which today play a role in quantum physics and quantum mechanics. Erik Ivar Fredholm (1866-1927) from Sweden did work on spectral theory and was professor of theoretical physics at the University of Stockholm. He founded modern integral equation theory, and his mathematical efforts inspired David Hilbert.

The leading English mathematician Arthur Cayley
(1821-1895) while
training to be a lawyer went to Dublin to hear William Rowan Hamilton lecture
on quaternions. He practiced law in London until 1863, but eventually focused
on his mathematical talents. In 1863 he was appointed professor of mathematics
at Cambridge. He published over 900 papers covering nearly every aspect of
modern mathematics, including non-Euclidean and n-dimensional geometry.
According to the *Encyclopædia Britannica*
online, “Cayley
made important contributions to the algebraic theory of curves and surfaces,
group theory, linear algebra, graph theory, combinatorics, and elliptic
functions. He formalized the theory of matrices.”

The German mathematician Karl Weierstrass
(1815-1897) was a pioneer of modern analysis and theory of functions and added
rigor to many problems in mathematics. His father wanted him to study law to
secure a position in the Prussian civil service, but Weierstrass spent his time
at the University of Bonn on fencing and beer-drinking. He then studied
mathematics at the University of Münster and began teaching at small schools.
In 1853 he wrote up an original result and sent it to a well-read professional
journal, *Crelle’s Journal*. His fame
spread quickly after that. When the Russian-born Sonya Kovalevskaya
(1850-1891), one of the earliest woman mathematicians of some note, came to
Germany from Russia to study mathematics, Weierstrass privately tutored her.
Many students benefited from his teaching.