Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, physical properties of minerals and mineralized artifacts. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization. Early writing on mineralogy on gemstones, comes from ancient Babylonia, the ancient Greco-Roman world and medieval China, Sanskrit texts from ancient India and the ancient Islamic World. Books on the subject included the Naturalis Historia of Pliny the Elder, which not only described many different minerals but explained many of their properties, Kitab al Jawahir by Persian scientist Al-Biruni; the German Renaissance specialist Georgius Agricola wrote works such as De re metallica and De Natura Fossilium which began the scientific approach to the subject. Systematic scientific studies of minerals and rocks developed in post-Renaissance Europe; the modern study of mineralogy was founded on the principles of crystallography and to the microscopic study of rock sections with the invention of the microscope in the 17th century.
Nicholas Steno first observed the law of constancy of interfacial angles in quartz crystals in 1669. This was generalized and established experimentally by Jean-Baptiste L. Romé de l'Islee in 1783. René Just Haüy, the "father of modern crystallography", showed that crystals are periodic and established that the orientations of crystal faces can be expressed in terms of rational numbers, as encoded in the Miller indices. In 1814, Jöns Jacob Berzelius introduced a classification of minerals based on their chemistry rather than their crystal structure. William Nicol developed the Nicol prism, which polarizes light, in 1827–1828 while studying fossilized wood. James D. Dana published his first edition of A System of Mineralogy in 1837, in a edition introduced a chemical classification, still the standard. X-ray diffraction was demonstrated by Max von Laue in 1912, developed into a tool for analyzing the crystal structure of minerals by the father/son team of William Henry Bragg and William Lawrence Bragg.
More driven by advances in experimental technique and available computational power, the latter of which has enabled accurate atomic-scale simulations of the behaviour of crystals, the science has branched out to consider more general problems in the fields of inorganic chemistry and solid-state physics. It, retains a focus on the crystal structures encountered in rock-forming minerals. In particular, the field has made great advances in the understanding of the relationship between the atomic-scale structure of minerals and their function. To this end, in their focus on the connection between atomic-scale phenomena and macroscopic properties, the mineral sciences display more of an overlap with materials science than any other discipline. An initial step in identifying a mineral is to examine its physical properties, many of which can be measured on a hand sample; these can be classified into density. Hardness is determined by comparison with other minerals. In the Mohs scale, a standard set of minerals are numbered in order of increasing hardness from 1 to 10.
A harder mineral will scratch a softer, so an unknown mineral can be placed in this scale by which minerals it scratches and which scratch it. A few minerals such as calcite and kyanite have a hardness that depends on direction. Hardness can be measured on an absolute scale using a sclerometer. Tenacity refers to the way a mineral behaves when it is broken, bent or torn. A mineral can be brittle, sectile, flexible or elastic. An important influence on tenacity is the type of chemical bond. Of the other measures of mechanical cohesion, cleavage is the tendency to break along certain crystallographic planes, it is described by the orientation of the plane in crystallographic nomenclature. Parting is the tendency to break along planes of weakness due to twinning or exsolution. Where these two kinds of break do not occur, fracture is a less orderly form that may be conchoidal, splintery, hackly, or uneven. If the mineral is well crystallized, it will have a distinctive crystal habit that reflects the crystal structure or internal arrangement of atoms.
It is affected by crystal defects and twinning. Many crystals are polymorphic, having more than
Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, both classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics, written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences, he remained in France until the end of his life. He was involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, became Senator in 1799.
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations, he proved. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series, he studied the three-body problem for the Earth and Moon and the movement of Jupiter's satellites, in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, presented the so-called mechanical "principles" as simple results of the variational calculus.
Born as Giuseppe Lodovico Lagrangia, Lagrange was of French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, married an Italian, his mother was from the countryside of Turin. He was raised as a Roman Catholic, his father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, Lagrange seems to have accepted this willingly, he studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident.
Alone and unaided he threw himself into mathematical studies. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange proved to be a problematic professor with his oblivious teaching style, abstract reasoning, impatience with artillery and fortification-engineering applications. In this Academy one of his students was François Daviet de Foncenex. Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions.
Lagrange wrote several letters to Leonhard Euler between 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and simplifying Euler's earlier analysis. Lagrange applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was impressed with Lagrange's results, it has been stated that "with characteristic courtesy he withheld a paper he had written, which covered some of the same ground, in order that the young Italian might have time to complete his work, claim the undisputed invention of the new calculus". Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. In 1758, with the aid of his pupils, Lagrange established a society, subsequently incorporated as the Turin Aca
Kingdom of Prussia
The Kingdom of Prussia was a German kingdom that constituted the state of Prussia between 1701 and 1918. It was the driving force behind the unification of Germany in 1871 and was the leading state of the German Empire until its dissolution in 1918. Although it took its name from the region called Prussia, it was based in the Margraviate of Brandenburg, where its capital was Berlin; the kings of Prussia were from the House of Hohenzollern. Prussia was a great power from the time it became a kingdom, through its predecessor, Brandenburg-Prussia, which became a military power under Frederick William, known as "The Great Elector". Prussia continued its rise to power under the guidance of Frederick II, more known as Frederick the Great, the third son of Frederick William I. Frederick the Great was instrumental in starting the Seven Years' War, holding his own against Austria, Russia and Sweden and establishing Prussia's role in the German states, as well as establishing the country as a European great power.
After the might of Prussia was revealed it was considered as a major power among the German states. Throughout the next hundred years Prussia went on to win many battles, many wars; because of its power, Prussia continuously tried to unify all the German states under its rule, although whether Austria would be included in such a unified German domain was an ongoing question. After the Napoleonic Wars led to the creation of the German Confederation, the issue of more unifying the many German states caused revolution throughout the German states, with each wanting their own constitution. Attempts at creation of a federation remained unsuccessful and the German Confederation collapsed in 1866 when war ensued between its two most powerful member states and Austria; the North German Confederation, which lasted from 1867 to 1871, created a closer union between the Prussian-aligned states while Austria and most of Southern Germany remained independent. The North German Confederation was seen as more of an alliance of military strength in the aftermath of the Austro-Prussian War but many of its laws were used in the German Empire.
The German Empire lasted from 1871 to 1918 with the successful unification of all the German states under Prussian hegemony, this was due to the defeat of Napoleon III in the Franco-Prussian War of 1870–71. The war united all the German states against a common enemy, with the victory came an overwhelming wave of nationalism which changed the opinions of some of those, against unification. In 1871, Germany unified into a single country, minus Austria and Switzerland, with Prussia the dominant power. Prussia is considered the legal predecessor of the unified German Reich and as such a direct ancestor of today's Federal Republic of Germany; the formal abolition of Prussia, carried out on 25 February 1947 by the fiat of the Allied Control Council referred to an alleged tradition of the kingdom as a bearer of militarism and reaction, made way for the current setup of the German states. However, the Free State of Prussia, which followed the abolition of the Kingdom of Prussia in the aftermath of World War I, was a major democratic force in Weimar Germany until the nationalist coup of 1932 known as the Preußenschlag.
The Kingdom left a significant cultural legacy, today notably promoted by the Prussian Cultural Heritage Foundation, which has become one of the largest cultural organisations in the world. In 1415 a Hohenzollern Burgrave came from the south to the March of Brandenburg and took control of the area as elector. In 1417 the Hohenzollern was made an elector of the Holy Roman Empire. After the Polish wars, the newly established Baltic towns of the German states, including Prussia, suffered many economic setbacks. Many of the Prussian towns could not afford to attend political meetings outside of Prussia; the towns were poverty stricken, with the largest town, having to borrow money from elsewhere to pay for trade. Poverty in these towns was caused by Prussia's neighbours, who had established and developed such a monopoly on trading that these new towns could not compete; these issues led to feuds, trade competition and invasions. However, the fall of these towns gave rise to the nobility, separated the east and the west, allowed the urban middle class of Brandenburg to prosper.
It was clear in 1440 how different Brandenburg was from the other German territories, as it faced two dangers that the other German territories did not, partition from within and the threat of invasion by its neighbours. It prevented partition by enacting the Dispositio Achillea, which instilled the principle of primogeniture to both the Brandenburg and Franconian territories; the second issue was resolved through expansion. Brandenburg was surrounded on every side by neighbours whose boundaries were political. Any neighbour could consume Brandenburg at any moment; the only way to defend herself was to absorb her neighbours. Through negotiations and marriages Brandenburg but expanded her borders, absorbing neighbours and eliminating the threat of attack; the Hohenzollerns were made rulers of the Margraviate of Brandenburg in 1518. In 1529 the Hohenzollerns secured the reversion of the Duchy of Pomerania after a series of conflicts, acquired its eastern part following the Peace of Westphalia. In 1618 the Hohenzollerns inherited the Duchy of Prussia, since 1511 ruled by Hohenzollern Albrecht of Brandenburg Prussia, who in 1525 converted the Teutonic Order ruled state to a Protestant Duchy by accepting fiefdom of the crown of Poland.
It was ruled in a personal union with Brandenburg
Unification of Germany
The unification of Germany into a politically and administratively integrated nation state occurred on 18 January 1871, in the Hall of Mirrors at the Palace of Versailles in France. Princes of the German states, excluding Austria, gathered there to proclaim William I of Prussia as German Emperor after the French capitulation in the Franco-Prussian War. Unofficially, the de facto transition of most of the German-speaking populations into a federated organization of states had been developing for some time through alliances formal and informal between princely rulers, but in fits and starts; the self-interests of the various parties hampered the process over nearly a century of autocratic experimentation, beginning in the era of the Napoleonic Wars, which prompted the dissolution of the Holy Roman Empire in 1806, the subsequent rise of German nationalism. Unification exposed tensions due to religious, linguistic and cultural differences among the inhabitants of the new nation, suggesting that 1871 only represented one moment in a continuum of the larger unification processes.
The Holy Roman Emperor had been called "Emperor of all the Germanies". In the empire, higher nobility were referred to as "Princes of Germany" or "Princes of the Germanies"—for the lands once called East Francia had been organized and governed as pocket kingdoms since before the rise of Charlemagne. In the mountainous terrain of much of the territory, isolated peoples developed cultural, educational and religious differences over such a lengthy time period. By the nineteenth century and communications improvements brought these regions closer together; the Holy Roman Empire, which had included more than 500 independent states, was dissolved when Emperor Francis II abdicated during the War of the Third Coalition. Despite the legal and political disruption associated with the end of the Empire, the people of the German-speaking areas of the old Empire had a common linguistic and legal tradition further enhanced by their shared experience in the French Revolutionary Wars and Napoleonic Wars. European liberalism offered an intellectual basis for unification by challenging dynastic and absolutist models of social and political organization.
Economically, the creation of the Prussian Zollverein in 1818, its subsequent expansion to include other states of the German Confederation, reduced competition between and within states. Emerging modes of transportation facilitated business and recreational travel, leading to contact and sometimes conflict among German speakers from throughout Central Europe; the model of diplomatic spheres of influence resulting from the Congress of Vienna in 1814–15 after the Napoleonic Wars endorsed Austrian dominance in Central Europe. The negotiators at Vienna took no account of Prussia's growing strength within and among the German states and so failed to foresee that Prussia would rise to challenge Austria for leadership of the German peoples; this German dualism presented two solutions to the problem of unification: Kleindeutsche Lösung, the small Germany solution, or Großdeutsche Lösung, the greater Germany solution. Historians debate whether Otto von Bismarck—Minister President of Prussia—had a master plan to expand the North German Confederation of 1866 to include the remaining independent German states into a single entity or to expand the power of the Kingdom of Prussia.
They conclude that factors in addition to the strength of Bismarck's Realpolitik led a collection of early modern polities to reorganize political, economic and diplomatic relationships in the 19th century. Reaction to Danish and French nationalism provided foci for expressions of German unity. Military successes—especially those of Prussia—in three regional wars generated enthusiasm and pride that politicians could harness to promote unification; this experience echoed the memory of mutual accomplishment in the Napoleonic Wars in the War of Liberation of 1813–14. By establishing a Germany without Austria, the political and administrative unification in 1871 at least temporarily solved the problem of dualism. 1797: The French First Republic annexed the Left Bank of the Rhine as a result of the War of the First Coalition. 1802: Previous annexations by France confirmed following its victory in the War of the Second Coalition. 1804: Francis I of Austria declared the new Austrian Empire as a reaction to Napoleon Bonaparte's proclamation of the First French Empire in 1804.
1806: As a result of the War of the Third Coalition, Napoleon I annexed some territories East of the Rhine, replaced the Holy Roman Empire by the Confederation of the Rhine as a French client-state. 1807: Prussia lost one half of its territory following the War of the Fourth Coalition. 1815: After the defeat of Napoleon, the Congress of Vienna reinstated the Germanic states into the German Confederation under the leadership of the Austrian Empire. 1819: The Carlsbad Decrees suppressed any form of pan-Germanic activities to avoid the creation of a'German state'. 1834: The Prussian-led custom union evolved into the Zollverein that included all Confederation states except the Austrian Empire. 1848: Revolts across the German Confederation, such as in Berlin and Frankfurt, forced King Frederick William IV of Prussia to grant a constitution to the Confederation. In the meantime, the Frankfurt Parliament was set up in 1848 and attempted to pro
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. The word "crystallography" derives from the Greek words crystallon "cold drop, frozen drop", with its meaning extending to all solids with some degree of transparency, graphein "to write". In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography. X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on physical measurements of their geometry; this involved measuring the angles of crystal faces relative to each other and to theoretical reference axes, establishing the symmetry of the crystal in question. This physical measurement is carried out using a goniometer; the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net.
The pole to each face is plotted on the net. Each point is labelled with its Miller index; the final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the diffraction patterns of a sample targeted by a beam of some type. X-rays are most used; this is facilitated by the wave properties of the particles. Crystallographers explicitly state the type of beam used, as in the terms X-ray crystallography, neutron diffraction and electron diffraction; these three types of radiation interact with the specimen in different ways. X-rays interact with the spatial distribution of electrons in the sample. Electrons are charged particles and therefore interact with the total charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the strong nuclear forces, but in addition, the magnetic moment of neutrons is non-zero, they are therefore scattered by magnetic fields. When neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels.
However, the material can sometimes be treated to substitute deuterium for hydrogen. Because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of a small object is made using a lens to focus the beam, similar to a lens in a microscope. However, the wavelength of visible light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves. Therefore, obtaining information about the spatial arrangement of atoms requires the use of radiation with shorter wavelengths, such as X-ray or neutron beams. Employing shorter wavelengths implied abandoning microscopy and true imaging, because there exists no material from which a lens capable of focusing this type of radiation can be created. Scientists have had some success focusing X-rays with microscopic Fresnel zone plates made from gold, by critical-angle reflection inside long tapered capillaries. Diffracted X-ray or neutron beams cannot be focused to produce images, so the sample structure must be reconstructed from the diffraction pattern.
Sharp features in the diffraction pattern arise from periodic, repeating structure in the sample, which are very strong due to coherent reflection of many photons from many spaced instances of similar structure, while non-periodic components of the structure result in diffuse diffraction features - areas with a higher density and repetition of atom order tend to reflect more light toward one point in space when compared to those areas with fewer atoms and less repetition. Because of their ordered and repetitive structure, crystals give diffraction patterns of sharp Bragg reflection spots, are ideal for analyzing the structure of solids. Coordinates in square brackets such as denote a direction vector. Coordinates in angle brackets or chevrons such as <100> denote a family of directions which are related by symmetry operations. In the cubic crystal system for example, <100> would mean, or the negative of any of those directions. Miller indices in parentheses such as denote a plane of the crystal structure, regular repetitions of that plane with a particular spacing.
In the cubic system, the normal to the plane is the direction, but in lower-symmetry cases, the normal to is not parallel to. Indices in curly brackets or braces such as denote a family of planes and their normals which are equivalent in cubic materials due to symmetry operations, much the way angle brackets denote a family of directions. In non-cubic materials, <hkl> is not perpendicular to. Some materials that have been analyzed crystallographically, such as proteins, do not occur as crystals; such molecules are placed in solution and allowed to crystallize through vapor diffusion. A drop of solution containing the molecule and precipitants is sealed in a container with a reservoir containing a hygroscopic solution. Water in the drop diffuses to the reservoir increasing the concentration and allowing a crystal to form. If the concentration were to rise more the molecule would precipitate out of solution, resulting in disorderly granules rather than an orderly and hence usable crystal. Once a crystal is obtained, data can be collected using a beam of radiation.
Although many universities that engage in crystallographic research have their own X-ray producing equipment, synchrotrons are used as X-ray sources, bec
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics. Kummer was born in Brandenburg, he was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay, published a year later. Kummer was married in 1840 to Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of Felix Mendelssohn and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet, his second wife, Bertha Cauer, was a maternal cousin of Ottilie. Overall, he had 13 children, his daughter Marie married the mathematician Hermann Schwarz. Kummer retired from teaching and from mathematics in 1890 and died three years in Berlin. Kummer made several contributions to mathematics in different areas; the Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group. Kummer proved Fermat's last theorem for a considerable class of prime exponents.
His methods were closer to p-adic ones than to ideal theory as understood though the term'ideal' was invented by Kummer. He studied what were called Kummer extensions of fields: that is, extensions generated by adjoining an nth root to a field containing a primitive nth root of unity; this is a significant extension of the theory of quadratic extensions, the genus theory of quadratic forms. As such, it is still foundational for class field theory. Kummer, Ernst Eduard, André, ed. Collected papers. Volume 1: Contributions to Number Theory, New York: Springer-Verlag, ISBN 978-0-387-06835-0, MR 0465760 Kummer, Ernst Eduard, André, ed. Collected papers. Volume II: Function theory and miscellaneous, New York: Springer-Verlag, ISBN 978-3-540-06836-5, MR 0465761 Kummer configuration Kummer's congruence Kummer series Kummer theory Kummer's theorem, on prime-power divisors of binomial coefficients Kummer's function Kummer ring Kummer sum Kummer variety Eric Temple Bell, Men of Mathematics and Schuster, New York: 1986.
R. W. H. T. Hudson, Kummer's Quartic Surface, rept. 1990. "Ernst Kummer," in Dictionary of Scientific Biography, ed. C. Gillispie, NY: Scribners 1970–90. O'Connor, John J.. "Ernst Kummer", MacTutor History of Mathematics archive, University of St Andrews. Works by or about Ernst Kummer at Internet Archive Biography of Ernst Kummer Ernst Kummer at the Mathematics Genealogy Project