Berlin is the capital and largest city of Germany by both area and population. Its 3,748,148 inhabitants make it the second most populous city proper of the European Union after London; the city is one of Germany's 16 federal states. It is surrounded by the state of Brandenburg, contiguous with its capital, Potsdam; the two cities are at the center of the Berlin-Brandenburg capital region, which is, with about six million inhabitants and an area of more than 30,000 km², Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions. Berlin straddles the banks of the River Spree, which flows into the River Havel in the western borough of Spandau. Among the city's main topographical features are the many lakes in the western and southeastern boroughs formed by the Spree and Dahme rivers. Due to its location in the European Plain, Berlin is influenced by a temperate seasonal climate. About one-third of the city's area is composed of forests, gardens, rivers and lakes; the city lies in the Central German dialect area, the Berlin dialect being a variant of the Lusatian-New Marchian dialects.
First documented in the 13th century and situated at the crossing of two important historic trade routes, Berlin became the capital of the Margraviate of Brandenburg, the Kingdom of Prussia, the German Empire, the Weimar Republic, the Third Reich. Berlin in the 1920s was the third largest municipality in the world. After World War II and its subsequent occupation by the victorious countries, the city was divided. East Berlin was declared capital of East Germany. Following German reunification in 1990, Berlin once again became the capital of all of Germany. Berlin is a world city of culture, politics and science, its economy is based on high-tech firms and the service sector, encompassing a diverse range of creative industries, research facilities, media corporations and convention venues. Berlin serves as a continental hub for air and rail traffic and has a complex public transportation network; the metropolis is a popular tourist destination. Significant industries include IT, biomedical engineering, clean tech, biotechnology and electronics.
Berlin is home to world-renowned universities, orchestras and entertainment venues, is host to many sporting events. Its Zoological Garden is one of the most popular worldwide. With the world's oldest large-scale movie studio complex, Berlin is an popular location for international film productions; the city is well known for its festivals, diverse architecture, contemporary arts and a high quality of living. Since the 2000s Berlin has seen the emergence of a cosmopolitan entrepreneurial scene. Berlin lies in northeastern Germany, east of the River Saale, that once constituted, together with the River Elbe, the eastern border of the Frankish Realm. While the Frankish Realm was inhabited by Germanic tribes like the Franks and the Saxons, the regions east of the border rivers were inhabited by Slavic tribes; this is why most of the villages in northeastern Germany bear Slavic-derived names. Typical Germanised place name suffixes of Slavic origin are -ow, -itz, -vitz, -witz, -itzsch and -in, prefixes are Windisch and Wendisch.
The name Berlin has its roots in the language of West Slavic inhabitants of the area of today's Berlin, may be related to the Old Polabian stem berl-/birl-. Since the Ber- at the beginning sounds like the German word Bär, a bear appears in the coat of arms of the city, it is therefore a canting arm. Of Berlin's twelve boroughs, five bear a Slavic-derived name: Pankow, Steglitz-Zehlendorf, Marzahn-Hellersdorf, Treptow-Köpenick and Spandau. Of its ninety-six neighborhoods, twenty-two bear a Slavic-derived name: Altglienicke, Alt-Treptow, Buch, Gatow, Kladow, Köpenick, Lankwitz, Lübars, Marzahn, Prenzlauer Berg, Schmöckwitz, Stadtrandsiedlung Malchow, Steglitz and Zehlendorf; the neighborhood of Moabit bears a French-derived name, Französisch Buchholz is named after the Huguenots. The earliest evidence of settlements in the area of today's Berlin are a wooden beam dated from 1192, remnants of a house foundation dated to 1174, found in excavations in Berlin Mitte; the first written records of towns in the area of present-day Berlin date from the late 12th century.
Spandau is first mentioned in 1197 and Köpenick in 1209, although these areas did not join Berlin until 1920. The central part of Berlin can be traced back to two towns. Cölln on the Fischerinsel is first mentioned in a 1237 document, Berlin, across the Spree in what is now called the Nikolaiviertel, is referenced in a document from 1244. 1237 is considered the founding date of the city. The two towns over time formed close economic and social ties, profited from the staple right on the two important trade routes Via Imperii and from Bruges to Novgorod. In 1307, they formed an alliance with a common external policy, their internal administrations still being separated. In 1415, Frederick I became the elector of the Margraviate of Brandenburg, which he ruled until 1440. During the 15th century, his successors established Berlin-Cölln as capital of the margraviate, subsequent members of the Hohenzol
Amstelveen is a municipality in the province of North Holland, Netherlands with a population of 89,918. It is a suburban part of the metropolitan area of Amsterdam; the municipality of Amstelveen consists of the historical villages of Bovenkerk and Nes aan de Amstel. The name Amstelveen comes from the Amstel, a local river, veen, meaning fen, peat, or moor. KLM has its headquarters in Amstelveen. Nowadays Amstelveen is best known for its Museum of Modern Art. During the French occupation between 1810 and 1814, it was the capital of a canton in the French department Zuyderzée, until 1964 the municipality of Amstelveen was called Nieuwer-Amstel, it is technically a large dorp. The symbol adopted for Amstelveen was not based on the Amsterdam symbol of three crosses, with one additional cross for distinction, but rather Amsterdam has one cross less to indicate Amstelveen's and Ouder-Amstel's seniority; the Thijssepark, was the first heempark in the Netherlands and is one of sixteen heemparks or heemgroen in Amstelveen.
Designed by landscape architect C. P. Broerse. P. Thijsse, it was developed between 1940 and 1972 and covers an area of 5 hectares, is situated just south of the Amsterdamse Bos. Amstelveen was chosen as an unlikely host of a match in the 1999 Cricket World Cup, for which the Netherlands had not qualified. South Africa played Kenya in the match. Former Dutch prime minister Jan Peter Balkenende started his political career as member of the council for Amstelveen; as a result of the vicinity of Schiphol, its links to Amsterdam, Amstelveen has grown and become a cosmopolitan mix of many cultures living in Amstelveen. In the early 20th century Amstelveen was a small rural village; the turf industry had collapsed, so the revenues of it were gone. The village was somewhat isolated, because waterway were connected; the main source of income was livestock farming with some arable, but the horticulture and floriculture were emerging. In 1852 the Haarlemmermeer polder was reclaimed and the "Fort at the Schiphol" was created as a defense for the capital Amsterdam.
Forts were in those days more named after rivers. "Fort at the Schiphol" was a separating ditch between Aalsmeer and Amstelveen, named after a piece of land from Amstelveen. Fort Schiphol, became a military airport in 1916. Four years Schiphol became a civilian airport. Schiphol Fort was demolished in 1934; the demolition was necessary for the construction of the provincial road from Amstelveen to Schiphol, with a swing bridge over the circular canal of the Haarlemmermeer. The development of Schiphol attracted many people; the headquarters of KLM was established there. Amstelveen once was the fastest growing city in the Netherlands and has now grown to 81,003 inhabitants. After World War II Amstelveen caught a portion of Amsterdam's housing shortage, was a member of the municipality of Schiphol. Amsterdams Planning was to introduce Amstelveen as a metropolitan area, with its urban and green areas. Amstelveen remained an independent and self-conscious municipality and adopted a policy that reflected many attractive new residential areas.
Amstelveens landscaping and added art attracted much international attention. Amstelveen in 2003 was voted the most attractive city of the Netherlands to live in. Amstelveen is in the top three on the national list of best cities to live in. With Amstelveen city center receiving the number one award for best shopping center Netherlands in 2013, 2014 and 2015 KLM Netherlands has its head office in Amstelveen. Air France-KLM is represented by the KLM head office. In addition Amstelveen has the international headquarters of KPMG auditing firm. Annual Revenue -$1B - $2.5B Amstelveen has a metro and tram connection with Amsterdam. The metro is a larger version of the tram, it has a point to point bus connection to other villages and Amsterdam Airport Schiphol as well as a local network. The metro line passing through Amstelveen now runs to Amsterdam Centraal station; the extension and renovation of the line includes its conversion into a tram line. Shopping Amstelveen city centre. Which has received the number one award for best shopping center Netherlands in 2013, 2014 and 2015 Aan de Poel.
A fine-dining restaurant, awarded one Michelin star for the period 2009–2012. Since 2013 it carries two Michelin stars. Aan de Poel is located on the shore of lake De Poel. Cobra Museum is located in the centre of Amstelveen. Museum Jan van der Togt, is centrehousing a glass art collection Birthplace of the artist Jan Cornelis Hofman. Birthplace of the actress Famke Janssen. Birthplace of the DJ and artist Martin Garrix A statue of Rembrandt overlooks the river Amstel, south east from the Amstelpark, where a windmill open to visitors is situated An Electric Museum-tram line still connects Amstelveen to
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis, he was the founder of the mathematical philosophy of intuitionism. Early in his career, Brouwer proved a number of theorems in the emerging field of topology; the most important were his fixed point theorem, the topological invariance of degree, the topological invariance of dimension. Among mathematicians the best known is the first one referred to now as the Brouwer Fixed Point Theorem, it is a simple corollary to the second, concerning the topological invariance of degree, the best known among algebraic topologists. The third theorem is the hardest. Brouwer proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.
In 1912, at age 31, he was elected a member of the Royal Netherlands Academy of Sciences. He was an Invited Speaker of the ICM in 1908 at Rome and in 1912 at Cambridge, UK. Brouwer founded intuitionism, a philosophy of mathematics that challenged the then-prevailing formalism of David Hilbert and his collaborators, who included Paul Bernays, Wilhelm Ackermann, John von Neumann. A variety of constructive mathematics, intuitionism is a philosophy of the foundations of mathematics, it is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning. Brouwer was a member of the Significs Group, it formed part of the early history of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism can not be disentangled from the intellectual milieu of that group. In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life and Mysticism, described by the mathematician Martin Davis as "drenched in romantic pessimism".
Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Brouwer "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions". In 1908: "... Brouwer, in a paper entitled'The untrustworthiness of the principles of logic', challenged the belief that the rules of the classical logic, which have come down to us from Aristotle have an absolute validity, independent of the subject matter to which they are applied"."After completing his dissertation, Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess". Hilbert—the formalist with whom the intuitionist Brouwer would spend years in conflict—admired the young man and helped him receive a regular academic appointment at the University of Amsterdam, it was that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism ".
He was combative as a young man. He was involved in a public and demeaning controversy in the 1920s with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal, he became isolated. Dutch mathematician and historian of mathematics, Bartel Leendert van der Waerden attended lectures given by Brouwer in years, commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on—and only on—the foundations of his intuitionism, it seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, he judged everything he had done before, his greatest output, false according to his philosophy."About his last years, Davis remarks: "...he felt more and more isolated, spent his last years under the spell of'totally unfounded financial worries and a paranoid fear of bankruptcy and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house."
Jean van Heijenoort, 1967 3rd printing 1976 with corrections, A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 pbk; the original papers are prefaced with valuable commentary. 1923. L. E. J. Brouwer: "On the significance of the principle of excluded middle in mathematics in function theory." With two Addenda and corrigenda, 334-45. Brouwer gives brief synopsis of his belief that the law of excluded middle cannot be "applied without reservation in the mathematics of infinite systems" and gives two examples of failures to illustrate his assertion. 1925. A. N. Kolmogorov: "On the principle of excluded middle", pp. 414–437. Kolmogorov disputes a few.
Oliver Heaviside FRS was an English self-taught electrical engineer and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications and science for years to come. Heaviside was born in London, at 55 Kings Street, he was a short and red-headed child, suffered from scarlet fever when young, which left him with a hearing impairment. A small legacy enabled the family to move to a better part of Camden when he was thirteen and he was sent to Camden House Grammar School, he was a good student, placed fifth out of five hundred students in 1865, but his parents could not keep him at school after he was 16, so he continued studying for a year by himself and had no further formal education.
Heaviside's uncle by marriage was Sir Charles Wheatstone, an internationally celebrated expert in telegraphy and electromagnetism, the original co-inventor of the first commercially successful telegraph in the mid-1830s. Wheatstone took a strong interest in his nephew's education and in 1867 sent him north to work with his own, older brother Arthur, managing one of Wheatstone's telegraph companies in Newcastle-upon-Tyne. Two years he took a job as a telegraph operator with the Danish Great Northern Telegraph Company laying a cable from Newcastle to Denmark using British contractors, he soon became an electrician. Heaviside continued to study while working, by the age of 22 he published an article in the prestigious Philosophical Magazine on'The Best Arrangement of Wheatstone's Bridge for measuring a Given Resistance with a Given Galvanometer and Battery' which received positive comments from physicists who had unsuccessfully tried to solve this algebraic problem, including Sir William Thomson, to whom he gave a copy of the paper, James Clerk Maxwell.
When he published an article on the duplex method of using a telegraph cable, he poked fun at R. S. Culley, the engineer in chief of the Post Office telegraph system, dismissing duplex as impractical. In 1873 his application to join the Society of Telegraph Engineers was turned down with the comment that "they didn't want telegraph clerks"; this riled Heaviside, who asked Thomson to sponsor him, along with support of the society's president he was admitted "despite the P. O. snobs". In 1873 Heaviside had encountered Maxwell's newly published, famous, two-volume Treatise on Electricity and Magnetism. In his old age Heaviside recalled: I remember my first look at the great treatise of Maxwell's when I was a young man… I saw that it was great and greatest, with prodigious possibilities in its power… I was determined to master the book and set to work. I was ignorant. I had no knowledge of mathematical analysis and thus my work was laid out for me, it took me several years before I could understand as much as I could.
I set Maxwell aside and followed my own course. And I progressed much more quickly… It will be understood that I preach the gospel according to my interpretation of Maxwell. Undertaking research from home, he helped develop transmission line theory. Heaviside showed mathematically that uniformly distributed inductance in a telegraph line would diminish both attenuation and distortion, that, if the inductance were great enough and the insulation resistance not too high, the circuit would be distortionless while currents of all frequencies would have equal speeds of propagation. Heaviside's equations helped further the implementation of the telegraph. From 1882 to 1902, except for three years, he contributed regular articles to the trade paper The Electrician, which wished to improve its standing, for which he was paid £40 per year; this was hardly enough to live on, but his demands were small and he was doing what he most wanted to. Between 1883 and 1887 these averaged 2–3 articles per month and these articles formed the bulk of his Electromagnetic Theory and Electrical Papers.
In 1880, Heaviside researched the skin effect in telegraph transmission lines. That same year he patented, in the coaxial cable. In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations; the four re-formulated Maxwell's equations describe the nature of electric charges, magnetic fields, the relationship between the two, namely electromagnetic fields. Between 1880 and 1887, Heaviside developed the operational calculus using p for the differential operator, giving a method of solving differential equations by direct solution as algebraic equations; this caused a great deal of controversy, owing to its lack of rigour. He famously said, "Mathematics is an experimental science, definitions do not come first, but on." On another occasion he asked somewhat more defensively, "Shall I refuse my dinner because I do not understand the process of digestion?"In 1887, Heaviside worked with his brother Arthur on a paper entitled "The Bridge System of Telephony".
However the paper was blocked by Arthur's
Leiden University, founded in the city of Leiden, is the oldest university in the Netherlands. The university was founded in 1575 by William, Prince of Orange, leader of the Dutch Revolt in the Eighty Years' War; the Dutch Royal Family and Leiden University have a close association: Queen Juliana, Queen Beatrix and King Willem-Alexander are former students. The university came into particular prominence during the Dutch Golden Age, when scholars from around Europe were attracted to the Dutch Republic due to its climate of intellectual tolerance and Leiden's international reputation. During this time Leiden was home to such figures as René Descartes, Christiaan Huygens, Hugo Grotius, Baruch Spinoza and Baron d'Holbach. Leiden University has over 50 departments; the university is a member of the Coimbra Group, the Europaeum and the League of European Research Universities. Leiden University houses international research institutes; the University is associated with ten leaders and Prime Ministers of the Netherlands including the current Prime Minister Mark Rutte, nine foreign leaders, among them the 6th President of the United States John Quincy Adams, a Secretary General of NATO, a President of the International Court of Justice, a Prime Minister of the United Kingdom and sixteen recipients of the Nobel Prize.
In 1575, the emerging Dutch Republic did not have any universities in its northern heartland. The only other university in the Habsburg Netherlands was the University of Leuven in southern Leuven under Spanish control; the scientific renaissance had begun to highlight the importance of academic study, so Prince William founded the first Dutch university in Leiden, to give the Northern Netherlands an institution that could educate its citizens for religious purposes, but to give the country and its government educated men in other fields. It is said the choice fell on Leiden as a reward for the heroic defence of Leiden against Spanish attacks in the previous year; the name of Philip II of Spain, William's adversary, appears on the official foundation certificate, as he was still the de jure count of Holland. Philip II replied by forbidding any subject to study in Leiden. Located in the convent of St Barbara, the university moved to the Faliede Bagijn Church in 1577 and in 1581 to the convent of the White Nuns, a site which it still occupies, though the original building was destroyed by fire in 1616.
The presence within half a century of the date of its foundation of such scholars as Justus Lipsius, Joseph Scaliger, Franciscus Gomarus, Hugo Grotius, Jacobus Arminius, Daniel Heinsius and Gerhard Johann Vossius made Leiden university into a regarded institution that attracted students from across Europe in the 17th century. Renowned philosopher Baruch Spinoza was based close to Leiden during this period and interacted with numerous scholars at the university; the learning and reputation of Jacobus Gronovius, Herman Boerhaave, Tiberius Hemsterhuis and David Ruhnken, among others, enabled Leiden to maintain its reputation for excellence down to the end of the 18th century. At the end of the nineteenth century, Leiden University again became one of Europe's leading universities. At the world’s first university low-temperature laboratory, professor Heike Kamerlingh Onnes achieved temperatures of only one degree above absolute zero of −273 degrees Celsius. In 1908 he was the first to succeed in liquifying helium and can be credited with the discovery of the superconductivity in metals.
The University Library, which has more than 5.2 million books and fifty thousand journals has a number of internationally renowned special collections of western and oriental manuscripts, printed books, prints, photographs and atlases. It houses the largest collections worldwide on the Caribbean; the research activities of the Scaliger Institute focus on these special collections and concentrate on the various aspects of the transmission of knowledge and ideas through texts and images from antiquity to the present day. In 2005 the manuscript of Einstein on the quantum theory of the monatomic ideal gas was discovered in one of Leiden's libraries; the portraits of many famous professors since the earliest days hang in the university aula, one of the most memorable places, as Niebuhr called it, in the history of science. In 2012 Leiden entered into a strategic alliance with Delft University of Technology and Erasmus University Rotterdam in order for the universities to increase the quality of their research and teaching.
The university is the unofficial home of the Bilderberg Group, a meeting of high-level political and economic figures from North America and Europe. The university has no central campus; some buildings, like the Gravensteen, are old, while buildings like Lipsius and Gorlaeus are much more modern. Among the institutions affiliated with the university are The KITLV or Royal Netherlands Institute of Southeast Asian and Caribbean Studies, the observatory 1633; the anatomical and pathological laboratories of the university are modern, the museums of geology and mineralogy have been restored. T
Josiah Willard Gibbs
Josiah Willard Gibbs was an American scientist who made important theoretical contributions to physics and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous inductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics, explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs worked on the application of Maxwell's equations to problems in physical optics; as a mathematician, he invented modern vector calculus. In 1863, Yale awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics from 1871 until his death. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history".
In 1901, Gibbs received what was considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics". Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century New England and the great international impact of his ideas. Though his work was entirely theoretical, the practical value of Gibbs's contributions became evident with the development of industrial chemistry during the first half of the 20th century. According to Robert A. Millikan, in pure science, Gibbs "did for statistical mechanics and for thermodynamics what Laplace did for celestial mechanics and Maxwell did for electrodynamics, made his field a well-nigh finished theoretical structure". Gibbs was born in Connecticut, he belonged to an old Yankee family that had produced distinguished American clergymen and academics since the 17th century. He was the fourth of five children and the only son of Josiah Willard Gibbs Sr. and his wife Mary Anna, née Van Cleve.
On his father's side, he was descended from Samuel Willard, who served as acting President of Harvard College from 1701 to 1707. On his mother's side, one of his ancestors was the Rev. Jonathan Dickinson, the first president of the College of New Jersey. Gibbs's given name, which he shared with his father and several other members of his extended family, derived from his ancestor Josiah Willard, Secretary of the Province of Massachusetts Bay in the 18th century; the elder Gibbs was known to his family and colleagues as "Josiah", while the son was called "Willard". Josiah Gibbs was a linguist and theologian who served as professor of sacred literature at Yale Divinity School from 1824 until his death in 1861, he is chiefly remembered today as the abolitionist who found an interpreter for the African passengers of the ship Amistad, allowing them to testify during the trial that followed their rebellion against being sold as slaves. Willard Gibbs was educated at the Hopkins School and entered Yale College in 1854 at the age of 15.
At Yale, Gibbs received prizes for excellence in mathematics and Latin, he graduated in 1858, near the top of his class. He remained at Yale as a graduate student at the Sheffield Scientific School. At age 19, soon after his graduation from college, Gibbs was inducted into the Connecticut Academy of Arts and Sciences, a scholarly institution composed of members of the Yale faculty. Few documents from the period survive and it is difficult to reconstruct the details of Gibbs's early career with precision. In the opinion of biographers, Gibbs's principal mentor and champion, both at Yale and in the Connecticut Academy, was the astronomer and mathematician Hubert Anson Newton, a leading authority on meteors, who remained Gibbs's lifelong friend and confidant. After the death of his father in 1861, Gibbs inherited enough money to make him financially independent. Recurrent pulmonary trouble ailed the young Gibbs and his physicians were concerned that he might be susceptible to tuberculosis, which had killed his mother.
He suffered from astigmatism, whose treatment was still unfamiliar to oculists, so that Gibbs had to diagnose himself and grind his own lenses. Though in years he used glasses only for reading or other close work, Gibbs's delicate health and imperfect eyesight explain why he did not volunteer to fight in the Civil War of 1861–65, he was not conscripted and he remained at Yale for the duration of the war. In 1863, Gibbs received the first Doctorate of Philosophy in engineering granted in the US, for a thesis entitled "On the Form of the Teeth of Wheels in Spur Gearing", in which he used geometrical techniques to investigate the optimum design for gears. In 1861, Yale had become the first US university to offer a Ph. D. degree and Gibbs's was only the fifth Ph. D. granted in the US in any subject. After graduation, Gibbs was appointed as tutor at the College for a term of three years. During the first two years, he taught Latin and during the third year, he taught "natural philosophy". In 1866, he patented a design for a railway brake and read a paper before the Connecticut Academy, entitled "The Proper Magnitude of the Units of Length", in which he proposed a scheme for rationalizing the system of units of measurement used in mechanics.
After his term as tutor ended, Gibbs traveled to Europe with his sisters. They spent the winter of 1866–67 i