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Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
was instrumental in transforming physical chemistry into a rigorous inductive science. Together with
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
and
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
, he created
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
(a term that he coined), explaining the
laws of thermodynamics The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...
as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
to problems in physical optics. As a mathematician, he invented modern
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
(independently of the British scientist Oliver Heaviside, who carried out similar work during the same period). In 1863, Yale awarded Gibbs the first American
doctorate A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' ...
in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
from 1871 until his death in 1903. 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 Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
as "the greatest mind in American history." In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the
Copley Medal The Copley Medal is an award given by the Royal Society, for "outstanding achievements in research in any branch of science". It alternates between the physical sciences or mathematics and the biological sciences. Given every year, the medal is t ...
of the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, ...
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 New England is a region comprising six states in the Northeastern United States: Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont. It is bordered by the state of New York to the west and by the Canadian provinces ...
and the great international impact of his ideas. Though his work was almost 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 thermodynamics what
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure."

Biography

Family background

Gibbs was born in New Haven, Connecticut. He belonged to an old
Yankee The term ''Yankee'' and its contracted form ''Yank'' have several interrelated meanings, all referring to people from the United States. Its various senses depend on the context, and may refer to New Englanders, residents of the Northern United St ...
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 (later
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
). Gibbs's given name, which he shared with his father and several other members of his extended family, derived from his ancestor Josiah Willard, who had been Secretary of the
Province of Massachusetts Bay The Province of Massachusetts Bay was a colony in British America which became one of the thirteen original states of the United States. It was chartered on October 7, 1691, by William III and Mary II, the joint monarchs of the kingdoms of E ...
in the 18th century.Bumstead 1928 His paternal grandmother, Mercy (Prescott) Gibbs, was the sister of Rebecca Minot Prescott Sherman, the wife of American founding father Roger Sherman; and he was the second cousin of Roger Sherman Baldwin, see the Amistad case below. The elder Gibbs was generally 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 Abolitionism, or the abolitionist movement, is the movement to end slavery. In Western Europe and the Americas, abolitionism was a historic movement that sought to end the Atlantic slave trade and liberate the enslaved people. The Britis ...
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.

Education

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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...
, and 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 primarily of members of the Yale faculty.Rukeyser 1988, p. 104 Relatively few documents from the period survive and it is difficult to reconstruct the details of Gibbs's early career with precision.Wheeler 1998, pp. 23–24 In the opinion of biographers, Gibbs's principal mentor and champion, both at Yale and in the Connecticut Academy, was probably 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.Rukeyser 1998, pp. 120, 142 Recurrent
pulmonary The lungs are the primary organs of the respiratory system in humans and most other animals, including some snails and a small number of fish. In mammals and most other vertebrates, two lungs are located near the backbone on either side ...
trouble ailed the young Gibbs and his physicians were concerned that he might be susceptible to
tuberculosis Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, i ...
, which had killed his mother. He also suffered from astigmatism, whose treatment was then still largely unfamiliar to oculists, so that Gibbs had to diagnose himself and grind his own lenses.Wheeler 1998, pp. 29–31Rukeyser 1988, p. 143 Though in later years he used
glasses Glasses, also known as eyeglasses or spectacles, are vision eyewear, with lenses (clear or tinted) mounted in a frame that holds them in front of a person's eyes, typically utilizing a bridge over the nose and hinged arms (known as temples ...
only for reading or other close work, Gibbs's delicate health and imperfect eyesight probably explain why he did not volunteer to fight in the
Civil War A civil war or intrastate war is a war between organized groups within the same state (or country). The aim of one side may be to take control of the country or a region, to achieve independence for a region, or to change government poli ...
of 1861–65.Wheeler 1998, p. 30 He was not conscripted and he remained at Yale for the duration of the war.Rukeyser 1998, p. 134 In 1863, Gibbs received the first
Doctorate of Philosophy A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
(PhD) 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
gear A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic p ...
s.Wheeler 1998, p. 32 In 1861, Yale had become the first US university to offer a PhD degree and Gibbs's was only the fifth PhD granted in the US in any subject.

Career, 1863–73

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" (i.e., physics). In 1866, he patented a design for a
railway brake A railway brake is a type of brake used on the cars of railway trains to enable deceleration, control acceleration (downhill) or to keep them immobile when parked. While the basic principle is similar to that on road vehicle usage, operational ...
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.Wheeler 1998, appendix II After his term as tutor ended, Gibbs traveled to Europe with his sisters. They spent the winter of 1866–67 in Paris, where Gibbs attended lectures at the Sorbonne and the Collège de France, given by such distinguished mathematical scientists as
Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
and
Michel Chasles Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician. Biography He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth C ...
. Having undertaken a punishing regimen of study, Gibbs caught a serious cold and a doctor, fearing tuberculosis, advised him to rest on the Riviera, where he and his sisters spent several months and where he made a full recovery. Moving to
Berlin Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
, Gibbs attended the lectures taught by mathematicians
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
and
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
, as well as by chemist Heinrich Gustav Magnus. In August 1867, Gibbs's sister Julia was married in Berlin to Addison Van Name, who had been Gibbs's classmate at Yale. The newly married couple returned to New Haven, leaving Gibbs and his sister Anna in Germany. In
Heidelberg Heidelberg (; Palatine German: ') is a city in the German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914, of which roughly a quarter consisted of student ...
, Gibbs was exposed to the work of physicists
Gustav Kirchhoff Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He ...
and
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...
, and chemist
Robert Bunsen Robert Wilhelm Eberhard Bunsen (; 30 March 1811 – 16 August 1899) was a German chemist. He investigated emission spectra of heated elements, and discovered caesium (in 1860) and rubidium (in 1861) with the physicist Gustav Kirchhoff. The ...
. At the time, German academics were the leading authorities in the natural sciences, especially chemistry and
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
. Gibbs returned to Yale in June 1869 and briefly taught French to engineering students. It was probably also around this time that he worked on a new design for a steam-engine
governor A governor is an administrative leader and head of a polity or political region, ranking under the head of state and in some cases, such as governors-general, as the head of state's official representative. Depending on the type of political ...
, his last significant investigation in mechanical engineering. In 1871, he was appointed Professor of Mathematical Physics at Yale, the first such professorship in the United States. Gibbs, who had independent means and had yet to publish anything, was assigned to teach graduate students exclusively and was hired without salary.Rukeyser 1988, pp. 181–182

Career, 1873–80

Gibbs published his first work in 1873. His papers on the geometric representation of thermodynamic quantities appeared in the ''Transactions of the Connecticut Academy''. These papers introduced the use of different type phase diagrams, which were his favorite aids to the imagination process when doing research, rather than the mechanical models, such as the ones that Maxwell used in constructing his electromagnetic theory, which might not completely represent their corresponding phenomena. Although the journal had few readers capable of understanding Gibbs's work, he shared reprints with correspondents in Europe and received an enthusiastic response from
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
at
Cambridge Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...
. Maxwell even made, with his own hands, a clay model illustrating Gibbs's construct. He then produced two plaster casts of his model and mailed one to Gibbs. That cast is on display at the Yale physics department. Maxwell included a chapter on Gibbs's work in the next edition of his ''Theory of Heat'', published in 1875. He explained the usefulness of Gibbs's graphical methods in a lecture to the
Chemical Society The Chemical Society was a scientific society formed in 1841 (then named the Chemical Society of London) by 77 scientists as a result of increased interest in scientific matters. Chemist Robert Warington was the driving force behind its creation. ...
of London and even referred to it in the article on "Diagrams" that he wrote for the ''
Encyclopædia Britannica The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various tim ...
''. Prospects of collaboration between him and Gibbs were cut short by Maxwell's early death in 1879, aged 48. The joke later circulated in New Haven that "only one man lived who could understand Gibbs's papers. That was Maxwell, and now he is dead." Gibbs then extended his thermodynamic analysis to multi-phase chemical systems (i.e., to systems composed of more than one form of matter) and considered a variety of concrete applications. He described that research in a monograph titled " On the Equilibrium of Heterogeneous Substances", published by the Connecticut Academy in two parts that appeared respectively in 1875 and 1878. That work, which covers about three hundred pages and contains exactly seven hundred numbered mathematical equations,Cropper 2001, p. 109 begins with a quotation from Rudolf Clausius that expresses what would later be called the first and second
laws of thermodynamics The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...
: "The
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of he ...
of the world is constant. The
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
of the world tends towards a maximum." Gibbs's monograph rigorously and ingeniously applied his thermodynamic techniques to the interpretation of physico-chemical phenomena, explaining and relating what had previously been a mass of isolated facts and observations.Wheeler 1998, ch. V The work has been described as "the '' Principia'' of thermodynamics" and as a work of "practically unlimited scope". It solidly laid the foundation for physical Chemistry. Wilhelm Ostwald, who translated Gibbs's monograph into German, referred to Gibbs as the "founder of chemical energetics". According to modern commentators, Gibbs continued to work without pay until 1880, when the new
Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consiste ...
in
Baltimore, Maryland Baltimore ( , locally: or ) is the most populous city in the U.S. state of Maryland, fourth most populous city in the Mid-Atlantic, and the 30th most populous city in the United States with a population of 585,708 in 2020. Baltimore was ...
offered him a position paying $3,000 per year. In response, Yale offered him an annual salary of$2,000, which he was content to accept.

Career, 1880–1903

From 1880 to 1884, Gibbs worked on developing the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
into a
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
well-suited to the needs of physicists. With this object in mind, Gibbs distinguished between the dot and
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
s of two vectors and introduced the concept of dyadics. Similar work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside. Gibbs sought to convince other physicists of the convenience of the vectorial approach over the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
ic calculus of
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ...
, which was then widely used by British scientists. This led him, in the early 1890s, to a controversy with Peter Guthrie Tait and others in the pages of ''
Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
''. Gibbs's lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by
Edwin Bidwell Wilson Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...
into a textbook, ''
Vector Analysis Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
'', published in 1901. That book helped to popularize the "
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denot ...
" notation that is widely used today in electrodynamics and
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
. In other mathematical work, he re-discovered the "
Gibbs phenomenon In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuo ...
" in the theory of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ' ...
(which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician,
Henry Wilbraham Henry Wilbraham (25 July 1825 – 13 February 1883) was an English mathematician. He is known for discovering and explaining the Gibbs phenomenon nearly fifty years before J. Willard Gibbs did. Gibbs and Maxime Bôcher, as well as nearly every ...
). From 1882 to 1889, Gibbs wrote five papers on physical optics, in which he investigated
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefri ...
and other optical phenomena and defended Maxwell's electromagnetic theory of light against the mechanical theories of
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did import ...
and others. In his work on optics, just as much as in his work on thermodynamics, Gibbs deliberately avoided speculating about the microscopic structure of matter and purposefully confined his research problems to those that can be solved from broad general principles and experimentally confirmed facts. The methods that he used were highly original and the obtained results showed decisively the correctness of Maxwell's electromagnetic theory. Gibbs coined the term ''statistical mechanics'' and introduced key concepts in the corresponding mathematical description of physical systems, including the notions of
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
(1876), and statistical ensemble (1902). Gibbs's derivation of the laws of thermodynamics from the statistical properties of systems consisting of many particles was presented in his highly influential textbook ''Elementary Principles in Statistical Mechanics'', published in 1902, a year before his death. Gibbs's retiring personality and intense focus on his work limited his accessibility to students. His principal protégé was Edwin Bidwell Wilson, who nonetheless explained that "except in the classroom I saw very little of Gibbs. He had a way, toward the end of the afternoon, of taking a stroll about the streets between his study in the old Sloane Laboratory and his home—a little exercise between work and dinner—and one might occasionally come across him at that time."Wilson 1931 Gibbs did supervise the doctoral thesis on mathematical economics written by
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt d ...
in 1891. After Gibbs's death, Fisher financed the publication of his ''Collected Works''. Another distinguished student was
Lee De Forest Lee de Forest (August 26, 1873 – June 30, 1961) was an American inventor and a fundamentally important early pioneer in electronics. He invented the first electronic device for controlling current flow; the three-element " Audion" triode ...
, later a pioneer of radio technology. Gibbs died in New Haven on April 28, 1903, at the age of 64, the victim of an acute intestinal obstruction. A funeral was conducted two days later at his home on 121 High Street, and his body was buried in the nearby Grove Street Cemetery. In May, Yale organized a memorial meeting at the Sloane Laboratory. The eminent British physicist J. J. Thomson was in attendance and delivered a brief address.

Personal life and character

Gibbs never married, living all his life in his childhood home with his sister Julia and her husband Addison Van Name, who was the Yale librarian. Except for his customary summer vacations in the
Adirondacks The Adirondack Mountains (; a-də-RÄN-dak) form a massif in northeastern New York with boundaries that correspond roughly to those of Adirondack Park. They cover about 5,000 square miles (13,000 km2). The mountains form a roughly circula ...
(at Keene Valley, New York) and later at the White Mountains (in Intervale, New Hampshire),Seeger 1974, pp. 15–16 his sojourn in Europe in 1866–69 was almost the only time that Gibbs spent outside New Haven. He joined Yale's College Church (a
Congregational church Congregational churches (also Congregationalist churches or Congregationalism) are Protestant churches in the Calvinist tradition practising congregationalist church governance, in which each congregation independently and autonomously runs i ...
) at the end of his freshman year and remained a regular attendant for the rest of his life.Wheeler, 1998, p. 16 Gibbs generally voted for the Republican candidate in presidential elections but, like other " Mugwumps", his concern over the growing corruption associated with machine politics led him to support
Grover Cleveland Stephen Grover Cleveland (March 18, 1837June 24, 1908) was an American lawyer and politician who served as the 22nd and 24th president of the United States from 1885 to 1889 and from 1893 to 1897. Cleveland is the only president in American ...
, a conservative Democrat, in the election of 1884. Little else is known of his religious or political views, which he mostly kept to himself. Gibbs did not produce a substantial personal correspondence and many of his letters were later lost or destroyed. Beyond the technical writings concerning his research, he published only two other pieces: a brief obituary for Rudolf Clausius, one of the founders of the mathematical theory of thermodynamics, and a longer biographical memoir of his mentor at Yale, H. A. Newton. In Edward Bidwell Wilson's view, According to Lynde Wheeler, who had been Gibbs's student at Yale, in his later years Gibbs He was a careful investor and financial manager, and at his death in 1903 his estate was valued at $100,000 (roughly$ today). For many years, he served as trustee, secretary, and treasurer of his alma mater, the Hopkins School.Wheeler, 1998, p. 144 US President Chester A. Arthur appointed him as one of the commissioners to the National Conference of Electricians, which convened in
Philadelphia Philadelphia, often called Philly, is the largest city in the Commonwealth of Pennsylvania, the sixth-largest city in the U.S., the second-largest city in both the Northeast megalopolis and Mid-Atlantic regions after New York City. Sinc ...
in September 1884, and Gibbs presided over one of its sessions. A keen and skilled horseman, Gibbs was seen habitually in New Haven driving his sister's
carriage A carriage is a private four-wheeled vehicle for people and is most commonly horse-drawn. Second-hand private carriages were common public transport, the equivalent of modern cars used as taxis. Carriage suspensions are by leather strapping ...
. In an obituary published in the '' American Journal of Science'', Gibbs's former student Henry A. Bumstead referred to Gibbs's personal character:

Major scientific contributions

Chemical and electrochemical thermodynamics

Gibbs's papers from the 1870s introduced the idea of expressing the internal energy ''U'' of a system in terms of the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
''S'', in addition to the usual state-variables of volume ''V'', pressure ''p'', and temperature ''T''. He also introduced the concept of the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
$\mu$ of a given chemical species, defined to be the rate of the increase in ''U'' associated with the increase in the number ''N'' of molecules of that species (at constant entropy and volume). Thus, it was Gibbs who first combined the first and second
laws of thermodynamics The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...
by expressing the infinitesimal change in the internal energy, d''U'', of a
closed system A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed. In ...
in the form: :$\mathrmU = T\mathrmS - p \,\mathrmV + \sum_i \mu_i \,\mathrm N_i\,$ where ''T'' is the absolute temperature, ''p'' is the pressure, d''S'' is an infinitesimal change in entropy and d''V'' is an infinitesimal change of volume. The last term is the sum, over all the chemical species in a chemical reaction, of the chemical potential, ''μ''i, of the ''i''th species, multiplied by the infinitesimal change in the number of moles, d''N''i of that species. By taking the Legendre transform of this expression, he defined the concepts of
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
, ''H'' and
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and ...
, ''G''. :$G_ = H - TS$ This compares to the expression for
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ene ...
, ''A''. :$A_ = U-TS\,$ When the Gibbs free energy for a chemical reaction is negative the reaction will proceed spontaneously. When a chemical system is at equilibrium, the change in Gibbs free energy is zero. An
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
is simply related to the free energy change when the reactants are in their standard states. :$\Delta G^\ominus=-RT \ln K^\ominus$
Chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
is usually defined as partial molar Gibbs free energy. :$\mu_i=\left\left(\frac\right\right)_$ Gibbs also obtained what later came to be known as the " Gibbs–Duhem equation". In an electrochemical reaction characterized by an
electromotive force In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical '' tran ...
ℰ and an amount of transferred charge Q, Gibbs's starting equation becomes :$\mathrmU = T\mathrmS - p \,\mathrmV + \mathcal\mathrmQ$. The publication of the paper " On the Equilibrium of Heterogeneous Substances" (1874–78) is now regarded as a landmark in the development of
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
. In it, Gibbs developed a rigorous mathematical theory for various transport phenomena, including
adsorption Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which ...
,
electrochemistry Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outc ...
, and the Marangoni effect in fluid mixtures. He also formulated the phase rule :$F\;=\;C\;-\;P\;+\;2$ for the number ''F'' of variables that may be independently controlled in an equilibrium mixture of ''C'' components existing in ''P'' phases. Phase rule is very useful in diverse areas, such as metallurgy, mineralogy, and petrology. It can also be applied to various research problems in physical chemistry.

Statistical mechanics

Together with James Clerk Maxwell and
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
, Gibbs founded "statistical mechanics", a term that he coined to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of ensembles of all possible physical states of a system composed of many particles. He introduced the concept of " phase of a mechanical system". He used the concept to define the microcanonical, canonical, and
grand canonical ensemble In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibriu ...
s; all related to the Gibbs measure, thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him. Gibbs generalized Boltzmann's statistical interpretation of
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
$S$ by defining the entropy of an arbitrary ensemble as : $S = -k_\text\,\sum_i p_i \ln \,p_i$, where $k_\text$ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas con ...
, while the sum is over all possible microstates $i$, with $p_i$ the corresponding probability of the microstate (see Gibbs entropy formula). This same formula would later play a central role in
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts Insti ...
's
information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 194 ...
and is therefore often seen as the basis of the modern information-theoretical interpretation of thermodynamics. According to Henri Poincaré, writing in 1904, even though Maxwell and Boltzmann had previously explained the
irreversibility In science, a process that is not reversible is called irreversible. This concept arises frequently in thermodynamics. All complex natural processes are irreversible, although a phase transition at the coexistence temperature (e.g. melting of ic ...
of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his ''Elementary Principles of Statistical Mechanics''." Gibbs's analysis of irreversibility, and his formulation of Boltzmann's
H-theorem In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules. L. Boltzmann,Weitere Studien über das W ...
and of the
ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., t ...
, were major influences on the mathematical physics of the 20th century. Gibbs was well aware that the application of the equipartition theorem to large systems of classical particles failed to explain the measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter". Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable microstates, could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries. His resolution of the so-called " Gibbs paradox", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics.

Vector analysis

British scientists, including Maxwell, had relied on Hamilton's
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s in order to express the dynamics of physical quantities, like the electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Following W. K. Clifford in his '' Elements of Dynamic'' (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathemat ...
, so that the use of quaternions involved mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. In his Yale classroom notes he defined distinct dot and cross products for pairs of vectors and introduced the now common notation for them. Through the 1901 textbook ''
Vector Analysis Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
'' prepared by E. B. Wilson from Gibbs notes, he was largely responsible for the development of the
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
techniques still used today in electrodynamics and fluid mechanics. While he was working on vector analysis in the late 1870s, Gibbs discovered that his approach was similar to the one that Grassmann had taken in his "multiple algebra".Letter by Gibbs to Victor Schlegel, quoted in Wheeler 1998, pp. 107–109 Gibbs then sought to publicize Grassmann's work, stressing that it was both more general and historically prior to Hamilton's quaternionic algebra. To establish priority of Grassmann's ideas, Gibbs convinced Grassmann's heirs to seek the publication in Germany of the essay "Theorie der Ebbe und Flut" on
tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ...
s that Grassmann had submitted in 1840 to the faculty at the
University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiativ ...
, in which he had first introduced the notion of what would later be called a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
( linear space).Wheeler 1998, pp. 113–116 As Gibbs had advocated in the 1880s and 1890s, quaternions were eventually all but abandoned by physicists in favor of the vectorial approach developed by him and, independently, by Oliver Heaviside. Gibbs applied his vector methods to the determination of planetary and comet
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such ...
s. He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The w ...
.

Physical optics

Though Gibbs's research on physical optics is less well known today than his other work, it made a significant contribution to classical
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
by applying
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
to the theory of optical processes such as
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefri ...
, dispersion, and
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circul ...
.Wheeler 1998, ch. VIII In that work, Gibbs showed that those processes could be accounted for by Maxwell's equations without any special assumptions about the microscopic structure of matter or about the nature of the medium in which electromagnetic waves were supposed to propagate (the so-called luminiferous ether). Gibbs also stressed that the absence of a longitudinal electromagnetic wave, which is needed to account for the observed properties of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
, is automatically guaranteed by Maxwell's equations (by virtue of what is now called their " gauge invariance"), whereas in mechanical theories of light, such as Lord Kelvin's, it must be imposed as an ''ad hoc'' condition on the properties of the aether. In his last paper on physical optics, Gibbs concluded that Shortly afterwards, the electromagnetic nature of light was demonstrated by the experiments of
Heinrich Hertz Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The unit ...
in Germany.

Scientific recognition

Gibbs worked at a time when there was little tradition of rigorous theoretical science in the United States. His research was not easily understandable to his students or his colleagues, and he made no effort to popularize his ideas or to simplify their exposition to make them more accessible. His seminal work on thermodynamics was published mostly in the ''Transactions of the Connecticut Academy'', a journal edited by his librarian brother-in-law, which was little read in the US and even less so in Europe. When Gibbs submitted his long paper on the equilibrium of heterogeneous substances to the academy, both
Elias Loomis Elias Loomis (August 7, 1811 – August 15, 1889) was an American mathematician. He served as a professor of mathematics and natural philosophy at Western Reserve College (now Case Western Reserve University), the University of the City of New Y ...
and H. A. Newton protested that they did not understand Gibbs's work at all, but they helped to raise the money needed to pay for the typesetting of the many mathematical symbols in the paper. Several Yale faculty members, as well as business and professional men in New Haven, contributed funds for that purpose.Rukeyser 1998, pp. 225–226 Even though it had been immediately embraced by Maxwell, Gibbs's graphical formulation of the laws of thermodynamics only came into widespread use in the mid 20th century, with the work of László Tisza and Herbert Callen. According to James Gerald Crowther, On the other hand, Gibbs did receive the major honors then possible for an academic scientist in the US. He was elected to the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
in 1879 and received the 1880
Rumford Prize Founded in 1796, the Rumford Prize, awarded by the American Academy of Arts and Sciences, is one of the oldest scientific prizes in the United States. The prize recognizes contributions by scientists to the fields of heat and light. These terms ...
from the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, ...
for his work on chemical thermodynamics. He was also awarded honorary doctorates by Princeton University and
Williams College Williams College is a private liberal arts college in Williamstown, Massachusetts. It was established as a men's college in 1793 with funds from the estate of Ephraim Williams, a colonist from the Province of Massachusetts Bay who was kille ...
. In Europe, Gibbs was inducted as honorary member of the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...
in 1892 and elected Foreign Member of the Royal Society in 1897. He was elected as corresponding member of the Prussian and French Academies of Science and received honorary doctorates from the universities of
Dublin Dublin (; , or ) is the capital and largest city of Ireland. On a bay at the mouth of the River Liffey, it is in the province of Leinster, bordered on the south by the Dublin Mountains, a part of the Wicklow Mountains range. At the 201 ...
,
Erlangen Erlangen (; East Franconian: ''Erlang'', Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative district Erlangen), and with 116,062 inha ...
, and Christiania (now Oslo). The Royal Society further honored Gibbs in 1901 with the
Copley Medal The Copley Medal is an award given by the Royal Society, for "outstanding achievements in research in any branch of science". It alternates between the physical sciences or mathematics and the biological sciences. Given every year, the medal is t ...
, then regarded as the highest international award in the natural sciences, noting that he had been "the first to apply the second law of thermodynamics to the exhaustive discussion of the relation between chemical, electrical and thermal energy and capacity for external work." Gibbs, who remained in New Haven, was represented at the award ceremony by Commander Richardson Clover, the US naval attaché in London. In his autobiography, mathematician
Gian-Carlo Rota Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...
tells of casually browsing the mathematical stacks of Sterling Library and stumbling on a handwritten mailing list, attached to some of Gibbs's course notes, which listed over two hundred notable scientists of his day, including Poincaré, Boltzmann,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, and Ernst Mach. From this, Rota concluded that Gibbs's work was better known among the scientific elite of his day than the published material suggests. Lynde Wheeler reproduces that mailing list in an appendix to his biography of Gibbs.Wheeler 1998, appendix IV That Gibbs succeeded in interesting his European correspondents in his work is demonstrated by the fact that his monograph "On the Equilibrium of Heterogeneous Substances" was translated into German (then the leading language for chemistry) by Wilhelm Ostwald in 1892 and into French by Henri Louis Le Châtelier in 1899.

Influence

Gibbs's most immediate and obvious influence was on physical chemistry and statistical mechanics, two disciplines which he greatly helped to found. During Gibbs's lifetime, his phase rule was experimentally validated by Dutch chemist H. W. Bakhuis Roozeboom, who showed how to apply it in a variety of situations, thereby assuring it of widespread use. In industrial chemistry, Gibbs's thermodynamics found many applications during the early 20th century, from electrochemistry to the development of the
Haber process The Haber process, also called the Haber–Bosch process, is an artificial nitrogen fixation process and is the main industrial procedure for the production of ammonia today. It is named after its inventors, the German chemists Fritz Haber and ...
for the synthesis of
ammonia Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogen ...
. When Dutch physicist J. D. van der Waals received the 1910
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
"for his work on the equation of state for gases and liquids" he acknowledged the great influence of Gibbs's work on that subject.
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretica ...
received the 1918 Nobel Prize for his work on quantum mechanics, particularly his 1900 paper on
Planck's law In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. A ...
for quantized
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous sp ...
. That work was based largely on the thermodynamics of Kirchhoff, Boltzmann, and Gibbs. Planck declared that Gibbs's name "not only in America but in the whole world will ever be reckoned among the most renowned theoretical physicists of all times." The first half of the 20th century saw the publication of two influential textbooks that soon came to be regarded as founding documents of
chemical thermodynamics Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. Chemical thermodynamics involves not only laboratory measurem ...
, both of which used and extended Gibbs's work in that field: these were ''Thermodynamics and the Free Energy of Chemical Processes'' (1923), by Gilbert N. Lewis and Merle Randall, and ''Modern Thermodynamics by the Methods of Willard Gibbs'' (1933), by Edward A. Guggenheim. Gibbs's work on statistical ensembles, as presented in his 1902 textbook, has had a great impact in both theoretical physics and in pure mathematics. According to mathematical physicist Arthur Wightman, Initially unaware of Gibbs's contributions in that field,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
wrote three papers on statistical mechanics, published between 1902 and 1904. After reading Gibbs's textbook (which was translated into German by Ernst Zermelo in 1905), Einstein declared that Gibbs's treatment was superior to his own and explained that he would not have written those papers if he had known Gibbs's work. Gibbs's early papers on the use of graphical methods in thermodynamics reflect a powerfully original understanding of what mathematicians would later call " convex analysis", including ideas that, according to Barry Simon, "lay dormant for about seventy-five years". Important mathematical concepts based on Gibbs's work on thermodynamics and statistical mechanics include the Gibbs lemma in
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, the Gibbs inequality in
information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 194 ...
, as well as Gibbs sampling in
computational statistics Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computi ...
. The development of vector calculus was Gibbs's other great contribution to mathematics. The publication in 1901 of E. B. Wilson's textbook ''
Vector Analysis Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
'', based on Gibbs's lectures at Yale, did much to propagate the use of vectorial methods and notation in both mathematics and theoretical physics, definitively displacing the quaternions that had until then been dominant in the scientific literature. At Yale, Gibbs was also mentor to Lee De Forest, who went on to invent the
triode A triode is an electronic amplifying vacuum tube (or ''valve'' in British English) consisting of three electrodes inside an evacuated glass envelope: a heated filament or cathode, a grid, and a plate (anode). Developed from Lee De Forest's ...
amplifier and has been called the "father of radio".Seeger 1974, p. 18 De Forest credited Gibbs's influence for the realization "that the leaders in electrical development would be those who pursued the higher theory of waves and oscillations and the transmission by these means of intelligence and power." Another student of Gibbs who played a significant role in the development of radio technology was Lynde Wheeler. Gibbs also had an indirect influence on mathematical economics. He supervised the thesis of
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt d ...
, who received the first PhD in economics from Yale in 1891. In that work, published in 1892 as ''Mathematical Investigations in the Theory of Value and Prices'', Fisher drew a direct analogy between Gibbsian equilibrium in physical and chemical systems, and the
general equilibrium In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
of markets, and he used Gibbs's vectorial notation. Gibbs's protégé Edwin Bidwell Wilson became, in turn, a mentor to leading American economist and Nobel Laureate
Paul Samuelson Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he " ...
. In 1947, Samuelson published ''
Foundations of Economic Analysis ''Foundations of Economic Analysis'' is a book by Paul A. Samuelson published in 1947 (Enlarged ed., 1983) by Harvard University Press. It is based on Samuelson's 1941 doctoral dissertation at Harvard University. The book sought to demonstrate a ...
'', based on his doctoral dissertation, in which he used as epigraph a remark attributed to Gibbs: "Mathematics is a language." Samuelson later explained that in his understanding of prices his "debts were not primarily to Pareto or Slutsky, but to the great thermodynamicist, Willard Gibbs of Yale." Mathematician
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher ...
cited Gibbs's use of probability in the formulation of statistical mechanics as "the first great revolution of twentieth century physics" and as a major influence on his conception of
cybernetics Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson ma ...
. Wiener explained in the preface to his book '' The Human Use of Human Beings'' that it was "devoted to the impact of the Gibbsian point of view on modern life, both through the substantive changes it has made to working science, and through the changes it has made indirectly in our attitude to life in general."

Commemoration

When the German physical chemist
Walther Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the ...
visited Yale in 1906 to give the Silliman lecture, he was surprised to find no tangible memorial for Gibbs. Nernst donated his \$500 lecture fee to the university to help pay for a suitable monument. This was finally unveiled in 1912, in the form of a bronze bas-relief by sculptor Lee Lawrie, installed in the Sloane Physics Laboratory.Seeger 1974, p. 21 In 1910, the
American Chemical Society The American Chemical Society (ACS) is a scientific society based in the United States that supports scientific inquiry in the field of chemistry. Founded in 1876 at New York University, the ACS currently has more than 155,000 members at all ...
established the
Willard Gibbs Award The Willard Gibbs Award, presented by thChicago Sectionof the American Chemical Society, was established in 1910 by William A. Converse (1862–1940), a former Chairman and Secretary of the Chicago Section of the society and named for Professor ...
for eminent work in pure or applied chemistry. In 1923, the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
endowed the
Josiah Willard Gibbs Lectureship The Josiah Willard Gibbs Lectureship (also called the Gibbs Lecture) of the American Mathematical Society is an annually awarded mathematical prize, named in honor of Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, ...
, "to show the public some idea of the aspects of mathematics and its applications". In 1945, Yale University created the J. Willard Gibbs Professorship in Theoretical Chemistry, held until 1973 by
Lars Onsager Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian-born American physical chemist and theoretical physicist. He held the Gibbs Professorship of Theoretical Chemistry at Yale University. He was awarded the Nobel Prize in ...
. Onsager, who much like Gibbs, focused on applying new mathematical ideas to problems in physical chemistry, won the 1968 Nobel Prize in chemistry. In addition to establishing the Josiah Willard Gibbs Laboratories and the J. Willard Gibbs Assistant Professorship in Mathematics, Yale has also hosted two symposia dedicated to Gibbs's life and work, one in 1989 and another on the centenary of his death, in 2003.
Rutgers University Rutgers University (; RU), officially Rutgers, The State University of New Jersey, is a public land-grant research university consisting of four campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's College, and was ...
endowed a J. Willard Gibbs Professorship of Thermomechanics, held as of 2014 by Bernard Coleman. Gibbs was elected in 1950 to the Hall of Fame for Great Americans. The oceanographic research ship USNS ''Josiah Willard Gibbs'' (T-AGOR-1) was in service with the
United States Navy The United States Navy (USN) is the maritime service branch of the United States Armed Forces and one of the eight uniformed services of the United States. It is the largest and most powerful navy in the world, with the estimated tonnag ...
from 1958 to 1971. Gibbs crater, near the eastern limb of the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width ...
, was named in the scientist's honor in 1964. Edward Guggenheim introduced the symbol ''G'' for the Gibbs free energy in 1933, and this was used also by Dirk ter Haar in 1966.Seeger 1974, p. 96 This notation is now universal and is recommended by the
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
. In 1960, William Giauque and others suggested the name "gibbs" (abbreviated gbs.) for the unit of entropy
calorie The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of o ...
per
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
, but this usage did not become common, and the corresponding SI unit
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
per kelvin carries no special name. In 1954, a year before his death, Albert Einstein was asked by an interviewer who were the greatest thinkers that he had known. Einstein replied: " Lorentz", adding "I never met Willard Gibbs; perhaps, had I done so, I might have placed him beside Lorentz." Author
Bill Bryson William McGuire Bryson (; born 8 December 1951) is an American–British journalist and author. Bryson has written a number of nonfiction books on topics including travel, the English language, and science. Born in the United States, he has b ...
in his bestselling
popular science ''Popular Science'' (also known as ''PopSci'') is an American digital magazine carrying popular science content, which refers to articles for the general reader on science and technology subjects. ''Popular Science'' has won over 58 awards, incl ...
book '' A Short History of Nearly Everything'' ranks Gibbs as "perhaps the most brilliant person that most people have never heard of". In 1958, USS ''San Carlos'' was renamed USNS ''Josiah Willard Gibbs'' and re-designated as an oceanographic research ship.

In literature

In 1909, the American historian and novelist Henry Adams finished an essay entitled "The Rule of Phase Applied to History", in which he sought to apply Gibbs's phase rule and other thermodynamic concepts to a general theory of human history.
William James William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States. James is considered to be a leading thinker of the la ...
, Henry Bumstead, and others criticized both Adams's tenuous grasp of the scientific concepts that he invoked, as well as the arbitrariness of his application of those concepts as metaphors for the evolution of human thought and society. The essay remained unpublished until it appeared posthumously in 1919, in ''The Degradation of the Democratic Dogma'', edited by Henry Adams's younger brother Brooks. In the 1930s, feminist poet Muriel Rukeyser became fascinated by Willard Gibbs and wrote a long poem about his life and work ("Gibbs", included in the collection ''A Turning Wind'', published in 1939), as well as a book-length biography (''Willard Gibbs'', 1942). According to Rukeyser: In 1946, '' Fortune'' magazine illustrated a cover story on "Fundamental Science" with a representation of the thermodynamic surface that Maxwell had built based on Gibbs's proposal. Rukeyser called this surface a "statue of water"Rukeyser 1988, p. 203 and the magazine saw in it "the abstract creation of a great American scientist that lends itself to the symbolism of contemporary art forms." The artwork by Arthur Lidov also included Gibbs's mathematical expression of the phase rule for heterogeneous mixtures, as well as a
radar Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles ...
screen, an
oscilloscope An oscilloscope (informally a scope) is a type of electronic test instrument that graphically displays varying electrical voltages as a two-dimensional plot of one or more signals as a function of time. The main purposes are to display repetiti ...
waveform, Newton's apple, and a small rendition of a three-dimensional phase diagram. Gibbs's nephew, Ralph Gibbs Van Name, a professor of physical chemistry at Yale, was unhappy with Rukeyser's biography, in part because of her lack of scientific training. Van Name had withheld the family papers from her and, after her book was published in 1942 to positive literary but mixed scientific reviews, he tried to encourage Gibbs's former students to produce a more technically oriented biography. Rukeyser's approach to Gibbs was also sharply criticized by Gibbs's former student and protégé Edwin Wilson. With Van Name's and Wilson's encouragement, physicist Lynde Wheeler published a new biography of Gibbs in 1951.Wheeler 1998, pp. ix–xiii Both Gibbs and Rukeyser's biography of him figure prominently in the poetry collection ''True North'' (1997) by Stephanie Strickland. In fiction, Gibbs appears as the mentor to character Kit Traverse in
Thomas Pynchon Thomas Ruggles Pynchon Jr. ( , ; born May 8, 1937) is an American novelist noted for his dense and complex novels. His fiction and non-fiction writings encompass a vast array of subject matter, genres and themes, including history, music, sci ...
's novel '' Against the Day'' (2006). That novel also prominently discusses the birefringence of Iceland spar, an optical phenomenon that Gibbs investigated.

Gibbs stamp (2005)

In 2005, the
United States Postal Service The United States Postal Service (USPS), also known as the Post Office, U.S. Mail, or Postal Service, is an independent agency of the executive branch of the United States federal government responsible for providing postal service in the ...
issued the ''American Scientists'' commemorative
postage stamp A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail), who then affix the stamp to the f ...
series designed by artist Victor Stabin, depicting Gibbs,
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cov ...
,
Barbara McClintock Barbara McClintock (June 16, 1902 – September 2, 1992) was an American scientist and cytogeneticist who was awarded the 1983 Nobel Prize in Physiology or Medicine. McClintock received her PhD in botany from Cornell University in 1927. Ther ...
, and
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the supe ...
. The first day of issue ceremony for the series was held on May 4 at Yale University's Luce Hall and was attended by John Marburger, scientific advisor to the president of the United States, Rick Levin, president of Yale, and family members of the scientists honored, including physician John W. Gibbs, a distant cousin of Willard Gibbs. Kenneth R. Jolls, a professor of chemical engineering at
Iowa State University Iowa State University of Science and Technology (Iowa State University, Iowa State, or ISU) is a public land-grant research university in Ames, Iowa. Founded in 1858 as the Iowa Agricultural College and Model Farm, Iowa State became one of th ...
and an expert on graphical methods in thermodynamics, consulted on the design of the stamp honoring Gibbs. The stamp identifies Gibbs as a "thermodynamicist" and features a diagram from the 4th edition of Maxwell's ''Theory of Heat'', published in 1875, which illustrates Gibbs's thermodynamic surface for water. Microprinting on the collar of Gibbs's portrait depicts his original mathematical equation for the change in the energy of a substance in terms of its entropy and the other state variables.

Outline of principal work

*
Physical chemistry Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical ...
: free energy,
phase diagram A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseou ...
, phase rule, transport phenomena *
Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
: statistical ensemble,
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
,
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
, Gibbs entropy, Gibbs paradox *
Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
: ''
Vector Analysis Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
'', convex analysis,
Gibbs phenomenon In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuo ...
*
Electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
:
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
,
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefri ...

* Concentration of measure in physics * Thermodynamics of crystal growth *
Governor (device) A governor, or speed limiter or controller, is a device used to measure and regulate the speed of a machine, such as an engine. A classic example is the centrifugal governor, also known as the Watt or fly-ball governor on a reciprocating stea ...
* List of notable textbooks in statistical mechanics *
List of theoretical physicists The following is a partial list of notable theoretical physicists. Arranged by century of birth, then century of death, then year of birth, then year of death, then alphabetically by surname. For explanation of symbols, see Notes at end of this ar ...
* List of things named after Josiah W. Gibbs * Timeline of United States discoveries * Timeline of thermodynamics

References

Primary

* L. P. Wheeler, E. O. Waters and S. W. Dudley (eds.),''The Early Work of Willard Gibbs in Applied Mechanics'', (New York: Henry Schuman, 1947). . This contains previously unpublished work by Gibbs, from the period between 1863 and 1871. * J. W. Gibbs, " On the Equilibrium of Heterogeneous Substances", ''Transactions of the Connecticut Academy of Arts and Sciences'', 3, 108–248, 343–524, (1874–1878). Reproduced in both ''The Scientific Papers'' (1906)
pp. 55–353
and ''The Collected Works of J. Willard Gibbs'' (1928), pp. 55–353. * E. B. Wilson, '' Vector Analysis, a text-book for the use of students of Mathematics and Physics, founded upon the Lectures of J. Willard Gibbs'', (New Haven: Yale University Press, 1929 901. * J. W. Gibbs, '' Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics'', (New York: Dover Publications, 1960 902. Gibbs's other papers are included in both: * ''The Scientific Papers of J. Willard Gibbs,'' in two volumes, eds. H. A. Bumstead and R. G. Van Name, (Woodbridge, CT: Ox Bow Press, 1993 906. . For scans of the 1906 printing, se
vol. I
an
vol. II
* '' The Collected Works of J. Willard Gibbs'', in two volumes, eds. W. R. Longley and R. G. Van Name, (New Haven: Yale University Press, 1957 928. For scans of the 1928 printing, se
vol. I
an
vol. II

Secondary

* . Reprinted with some additions in both ''The Scientific Papers'', vol. I
pp. xiii–xxviiii
(1906) and ''The Collected Works of J. Willard Gibbs'', vol. I, pp. xiii–xxviiii (1928). Also available her

* D. G. Caldi and G. D. Mostow (eds.), ''Proceedings of the Gibbs Symposium, Yale University, May 15–17, 1989'', (American Mathematical Society and American Institute of Physics, 1990). * W. H. Cropper, "The Greatest Simplicity: Willard Gibbs", in ''Great Physicists'', (Oxford: Oxford University Press, 2001), pp. 106–123. * M. J. Crowe, ''A History of Vector Analysis: The Evolution of the Idea of a Vectorial System'', (New York: Dover, 1994 967. * J. G. Crowther, ''Famous American Men of Science'', (Freeport, NY: Books for Libraries Press, 1969 937. * F. G. Donnan and A. E. Hass (eds.), ''A Commentary on the Scientific Writings of J. Willard Gibbs'', in two volumes, (New York: Arno, 1980 936. . Onl
vol I.
is currently available online. * P. Duhem
''Josiah-Willard Gibbs à propos de la publication de ses Mémoires scientifiques''
(Paris: A. Herman, 1908). * C. S. Hastings
"Josiah Willard Gibbs"
''Biographical Memoirs of the National Academy of Sciences'', 6, 373–393 (1909). * M. J. Klein, "Gibbs, Josiah Willard", in ''Complete Dictionary of Scientific Biography'', vol. 5, (Detroit: Charles Scribner's Sons, 2008), pp. 386–393. * M. Rukeyser, ''Willard Gibbs: American Genius'', (Woodbridge, CT: Ox Bow Press, 1988 942. * R. J. Seeger, ''J. Willard Gibbs, American mathematical physicist'' par excellence, (Oxford and New York: Pergamon Press, 1974). * L. P. Wheeler, ''Josiah Willard Gibbs, The History of a Great Mind'', (Woodbridge, CT: Ox Bow Press, 1998 951. * A. S. Wightman, "Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics". Published as an introduction to R. B. Israel, ''Convexity in the Theory of Lattice Gases'', (Princeton, NJ: Princeton University Press, 1979), pp. ix–lxxxv. * E. B. Wilson,
"Reminiscences of Gibbs by a student and colleague"
''Bulletin of the American Mathematical Society'', 37, 401–416 (1931).