# Josiah Willard Gibbs

J. Willard Gibbsw
Josiah Willard Gibbs
Born February 11, 1839
New Haven, Connecticut
Died April 28, 1903 (aged 64)
New Haven, Connecticut
Residence United States
Nationality United States
Alma mater Yale University
Known for Statistical mechanics
Statistical ensemble
Phase space
Gibbs entropy
Ergodic hypothesis
Enthalpy
Gibbs free energy
Gibbs' phase rule
Vector calculus
Cross product
Gibbs phenomenon
Gibbs-Helmholtz equation
Gibbs-Duhem equation
Gibbs algorithm
Gibbs distribution
Gibbs state
Gibbs-Thomson effect
Gibbs isotherm
Gibbs-Donnan effect
Gibbs lemma
Awards Rumford Prize (1880), Copley Medal (1901)
Scientific career
Fields Physics, chemistry, mathematics
Institutions Yale University
Doctoral advisor Hubert Anson Newton
Doctoral students Edwin Bidwell Wilson, Irving Fisher, Henry Andrews Bumstead, Lynde Wheeler
Influences Rudolf Clausius, Gustav Kirchhoff, Hermann von Helmholtz, James Clerk Maxwell, Ludwig Boltzmann
Signature
150px

Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term which he coined), explaining the laws of thermodynamics in terms of the statistical properties of large ensembles of particles. As a mathematician, he invented modern vector calculus (independently of Oliver Heaviside).

In 1863, Yale University awarded Gibbs the first American doctorate in engineering.[1] After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics. 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."[2] In 1901 Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London,[2] "for his contributions to mathematical physics."[3]

## Biography

### Family background

Gibbs was the fourth of the five children, and the only son, of Josiah Willard Gibbs, a linguist and theologian who was professor of sacred literature at Yale Divinity School from 1824 to 1861, and his wife Mary Anna, née Van Cleve. The father, an active abolitionist, is now remembered chiefly for finding an interpreter for the African passengers of the ship Amistad, which allowed them to testify during the trial that followed their rebellion against being sold as slaves.[4]

Gibbs belonged to a long line of American academics and clergymen that stretched back to the 17th century. On his father's side, he descended from Samuel Willard, who served as acting President of Harvard University from 1701 to 1707. On his mother's side, one of his ancestors was the Rev. Jonathan Dickinson, who was the first president of the College of New Jersey (later Princeton University). His 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 in the 18th century.[5]

### Early years

Gibbs in his youth.

Gibbs was educated at the Hopkins School and entered Yale College in 1854, at the age of 15. He graduated in 1858 near the top of his class, and was awarded prizes for excellence in mathematics and Latin. He remained at Yale as a graduate student at the Sheffield Scientific School. After the death of his father in 1861, Gibbs inherited enough money to make him financially independent.

Gibbs suffered from recurrent pulmonary trouble as a young man and his doctors were concerned that he might be susceptible to tuberculosis, which had killed his mother. This, as well as a defect in his eyesight, probably explain why he did not volunteer to fight in the Civil War of 1861-65.[6] His name was never reached by the Connecticut draft and he remained at Yale for the duration of the war.

In 1863 he received the first Ph.D. degree in engineering 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.[1][7] 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 Natural Philosophy (i.e., physics).[5] In 1866 he patented a design for a railway brake.[8]

After his term as tutor ended, Gibbs travelled to Europe with his sisters, spending the winter of 1866-67 in Paris, where he attended lectures at the Sorbonne and the Collège de France. From there he went to Berlin, where he attended the lectures of Magnus, and to Heidelberg, where he was exposed to the scientific work of Kirchhoff and Helmholtz. At the time, German academics were the leading authorities in chemistry, thermodynamics, and natural science in general. Gibbs returned to Yale in June of 1869. Except for his customary summer vacations in the Adirondacks or other nearby mountains, those three years were almost the only time that Gibbs ever spent outside New Haven.[5]

File:Thermodynamicist Willard Gibbs.jpg
Gibbs during his time as a tutor at Yale[9]

Upon his return from Europe, Gibbs briefly taught French to Yale engineering students.[10] It was probably also around this time that he worked on a new design for a steam-engine governor, which seems to have been his last significant investigation in mechanical engineering.[11] In 1871 he was appointed Professor of Mathematical Physics at Yale, the first such professorship in the United States. His position was unpaid, a situation common in Germany and otherwise not unusual at the time, as Gibbs, who had independent means, had yet to publish anything.

### Middle years

Maxwell's sketch of the lines of constant temperature and pressure, made in preparation for his construction of a solid model based on Gibbs's thermodynamic surface for water

Gibbs's first published work, which appeared in 1873 when he was already 34 years old, was on the geometric representation of thermodynamic quantities. That work appeared in the Transactions of the Connecticut Academy, which had few readers capable of understanding Gibbs's work, but he shared reprints with his correspondents in Europe and received a particularly favorable response from Maxwell, who made three plaster casts illustrating Gibbs's construct with his own hands and mailed one to Gibbs. That model is still on display at the Yale physics department.[12]

Between 1875 and 1878 Gibbs wrote a series of papers applying his graphical techniques of thermodynamic analysis to multi-phase chemical systems. These were collected as a long monograph titled "On the Equilibrium of Heterogeneous Substances," which is now deemed to be one of the greatest scientific achievements of the 19th century and one of the foundations of both physical chemistry and statistical mechanics. In that work Gibbs rigorously and ingeniously applied the concepts of thermodynamics to the interpretation of physico-chemical phenomena, successfully explaining and interrelating what had previously been a mass of isolated facts.

It is universally recognised that its publication was an event of the first importance in the history of chemistry... Nevertheless it was a number of years before its value was generally known, this delay was due largely to the fact that its mathematical form and rigorous deductive processes make it difficult reading for anyone, and especially so for students of experimental chemistry whom it most concerns...
— J J O'Connor and E F Robertson[13]

Gibbs continued to work without pay until 1880, when the new Johns Hopkins University in Baltimore, Maryland offered him a position paying $3,000 per year. In response, Yale gave him a salary of$2,000, which he was content to accept.[14]

### Later years

From 1880 to 1884, Gibbs worked on developing the exterior algebra of Hermann Grassmann into a vector calculus well-suited to the needs of physicists. This work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside. In the course of this effort, Gibbs introduced the concept of dyadics and he sought to convince other physicists of the convenience of his approach over the quaternionic calculus of William Rowan Hamilton, which was then in widespread use by British scientists. This led, in the early 1890s, to a controversy with Peter Guthrie Tait and others in the pages of Nature.[5]

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 into a textbook, Vector Analysis, published in 1901.[5] That book helped to popularize the notation that is widely used today in electrodynamics and fluid mechanics (see del operator). In other mathematical work, he re-discovered the Gibbs phenomenon in the theory of Fourier series (which, unbeknownst to him and to later scholars, had been found three decades before by an obscure English mathematician, Henry Wilbraham).[15]

From 1882 to 1889, Gibbs wrote five papers on physical optics, in which he investigated birefringence and other optical phenomena and defended Maxwell's electromagnetic theory of light against the mechanical theories of Kelvin and others.[5] In his work on optics just as much as in his work on thermodynamics, Gibbs deliberately avoided speculating about the microscopic structure of matter, which proved a wise course in view of the revolutionary developments in quantum mechanics that began around the time of his death.[16]

Gibbs created both the term "statistical mechanics" and many of the key concepts of the corresponding mathematical description of physical systems, including the notions of chemical potential (1876), statistical ensemble (1878), and phase space (1901). His derivation of the phenomenological laws of thermodynamics from the statistical properties of systems with many particles was presented in his highly-influential textbook Elementary Principles in Statistical Mechanics, published in 1902, a year before his death.

Gibbs had few students and his retiring personality and intense focus on his scientific work were such that he was generally unavailable personally.[13] He did supervise the doctoral thesis on mathematical economics written by Irving Fisher in 1891 (Fisher later financed the publication of Gibbs's Collected Works). Gibbs's 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."[17]

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. Gibbs died in New Haven, aged 64, the victim of an acute intestinal obstruction.[18] He is buried in Grove Street Cemetery.

In an obituary published in the American Journal of Science, Gibbs's former student Henry A. Bumstead referred to Gibbs's personal character:

Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realising the ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life.
— H. A. Bumstead, 1903[5]

## Major scientific contributions

### Chemical thermodynamics

Gibbs's graph of the thermodynamic free energy, showing a plane perpendicular to the axis of v (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface of dissipated energy. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy) respectively. AD and AE are respectively the energy and entropy of the body in its initial state; AB and AC its available energy (Helmholtz free energy) and its capacity for entropy (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume). The figure appears in a paper published in 1873.

Gibbs's papers from the 1870s introduced the idea of expressing the internal energy U of a system in terms of the entropy S, in addition to the usual state variables of V (volume), p (pressure), and T (temperature).[19] He also introduced the concept of the chemical potential $\displaystyle \mu$ , defined by the change in energy associated with the variation in the number N of molecules of a given chemical species. Thus, it was Gibbs who first combined the first and second laws of thermodynamics by expressing the infinitesimal change in the energy a system in the form:

$\displaystyle \mathrm{d}U = T\mathrm{d}S - p \,\mathrm{d}V + \sum_i \mu_i \,\mathrm{d} N_i\,$

where the sum in the last term is over the different chemical species. By Legendre-transforming this expression, he defined the concepts of enthalpy and "free energy" (now universally known as the "Gibbs free energy"), a thermodynamic potential which is especially useful to chemists since it determines whether a reaction will proceed spontaneously at a fixed temperature and pressure. In the same way he also obtained what is now known as the "Gibbs–Duhem equation."[19][20]

The publication of the paper "On the Equilibrium of Heterogeneous Substances" (1875-8) is now regarded as a landmark in the development of physical chemistry. That paper formulated the phase rule for the number of variables that can be controlled in a heterogeneous mixture in equilibrium (see also phase diagram). It also developed a rigorous mathematical theory for various transport phenomena, including electrochemical processes and the Marangoni effect in fluid mixtures.

### Vector analysis

Diagram showing the magnitude and direction of the cross product of two vectors, in the notation introduced by Gibbs

Gibbs introduced the now common notation for the dot product and the cross product of two vectors, and he was largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics. As Gibbs had advocated in the 1880s and 1890s, Hamilton's quaternions were eventually all but abandoned by physicists in favor of the approach developed by him and, independently, by Oliver Heaviside. Gibbs also applied his vector methods to the determination of planetary and comet orbits, and he developed the concept of mutually reciprocal triads of vectors which later proved to be of importance in crystallography.[21]

### Statistical mechanics

Together with James Clerk Maxwell and Ludwig Boltzmann, Gibbs is considered one of the founders of statistical mechanics. It was Gibbs who coined the term "statistical mechanics" to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of large ensembles of particles. He introduced the concept of phase space and used it to define the microcanonical, canonical, and grand canonical ensembles, thus obtaining a more general formulation of the statistical properties of many-particle systems than what had been achieved previously by Maxwell and Boltzmann.[22] Gibbs's theoretical framework was so carefully constructed that it could be carried over almost intact after the discovery that the microscopic laws of nature obey the rules of quantum mechanics, rather than the classical mechanics known to Gibbs and his contemporaries.[13] 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 mechanics.[23]

### 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 by applying Maxwell's equations to the the theory of optical processes such as birefringence, dispersion, and optical activity.[5][16] 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 aether). Gibbs also stressed that the absence of a longitudinal electromagnetic wave, which is needed to account for the observed properties of light, 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.[16]

## 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.[13] His seminal work on thermodynamics and statistical mechanics 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 USA and even less so in Europe. In fact, when Gibbs submitted his long paper on the equilibrium of heterogeneous substances to the Academy, both Elias Loomis and Hubert Anson 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 equations and mathematical symbols in the paper. Funds for the purpose were contributed both by members of the university and by local business and professional men in New Haven.[24]

Mathematician Gian-Carlo Rota, while casually browsing the mathematical stacks of Sterling Library, stumbled on a handwritten mailing list attached to some of Gibbs's course notes. It listed over two hundred notable scientists of his day, including Poincaré, Hilbert, Boltzmann, and Mach. One may conclude that Gibbs's work was better known among the scientific elite of his day than the published material suggests.[25] Gibbs did succeed in interesting his European correspondents in that work, which 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. His phase rule was experimentally validated by the works of Dutch chemist H. W. Bakhuis Roozeboom, who showed how to apply it in a variety of situations, thereby assuring it of widespread use.

Gibbs, though never well known outside his field, did receive the major honors then possible for an academic scientist in the US: he was elected to the National Academy of Sciences in 1879 and was awarded the 1880 Rumford Prize from the American Academy of Arts and Sciences for his work on chemical thermodynamics.[26] Gibbs also received honorary doctorates from Princeton and from Williams College in the US, and from the universities of Erlangen and Christiania (now Oslo) in Europe.[5]

In his later years he was a tall, dignified gentleman, with a healthy stride and ruddy complexion, performing his share of household chores, approachable and kind (if unintelligible) to students. Gibbs was highly esteemed by his friends, but American science was too preoccupied with practical questions to make much use of his profound theoretical work during his lifetime. He lived out his quiet life at Yale, deeply admired by a few able students but making no immediate impress on American science commensurate with his genius.
— J. G. Crowther, 1937

Gibbs was inducted as a foreign member of the Royal Society of London in 1897 and received the Society's Copley Medal in 1901.[13] At the time, that was considered the highest international honor in the natural sciences.[2] Gibbs was also a corresponding member of the Prussian and French Academies of Science.[5]

## Influence

Gibbs's impact on modern science is obvious in physical chemistry and statistical mechanics, both of which he greatly helped to found. When Dutch physicist J. D. van der Waals received the 1910 Nobel Prize "for his work on the equation of state for gases and liquids" he acknowledged the great influence of Gibbs's work on that subject.[27] Max Planck received the 1918 Nobel Prize for his work on quantum mechanics, particularly his 1900 paper on the quantization of black-body radiation (see Planck's law). That work was based largely on the thermodynamics of Kirchhoff, Boltzmann, and Gibbs. Planck said 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."[28]

At the beginning of the 20th century, Gilbert N. Lewis and Merle Randall used and extended Gibbs's work on chemical thermodynamics and presented their results in the 1923 textbook Thermodynamics and the Free Energy of Chemical Substances, one of the two founding books in chemical thermodynamics. William Giauque received a bachelor of science degree in chemistry from Berkeley in 1920. At first he wanted to become a chemical engineer, but soon developed an interest in fundamental chemical research under Lewis's influence. In 1934, Giauque became a full professor of chemistry at Berkeley. In 1949, he won the Nobel Prize in Chemistry for his studies in the properties of matter at temperatures close to absolute zero, studies guided by the third law of thermodynamics.

Gibbs's work on ensembles and on the ergodic hypothesis, as presented in his 1902 textbook on statistical mechanics, has had a considerable impact in both theoretical physics and in pure mathematics. According to mathematical physicist Arthur Wightman

It is one of the striking features of the work of Gibbs, noticed by every student of thermodynamics and statistical mechanics, that his formulations of physical concepts were so felicitiously chosen that they have survived 100 years of turbulent development in theoretical physics and mathematics.
— A. S. Wightman, 1990[29]

Gibbs also had an indirect influence on mathematical economics and on general equilibrium theory. He supervised the thesis of Irving Fisher, who received the first Ph.D. in economics from Yale in 1891.[30] One of Gibbs's protegés was Edwin Bidwell Wilson, who in turn was a mentor to leading American economist and Nobel Laureate Paul Samuelson.[31] In 1947, Samuelson published Foundations of Economic Analysis, based on his Harvard University doctoral dissertation. Samuelson explicitly acknowledged the influence of the classical thermodynamic methods of Gibbs.[31] In 2003, Samuelson described Gibbs as "Yale's great physicist."[32]

Mathematician Norbert Wiener cited Gibbs as a major influence on his conception of cybernetics and explained in 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."[33]

## Commemoration

File:Gibbs-US-stamp.jpg
US postage stamp honoring Willard Gibbs, issued in 2005

In 1910, the American Chemical Society posthumously established the Willard Gibbs Medal, through the initiative of William A. Converse, a former chairman and secretary of the Chicago Section.[34] The American Mathematical Society established the Josiah Willard Gibbs Lectureship in 1923 to increase public awareness of mathematics and its applications.[35]

In 1945, Yale University created the J. Willard Gibbs Professorship in Theoretical Chemistry, held until 1973 by Lars Onsager, who won the 1968 Nobel Prize in chemistry. This appointment was a very fitting one, as Onsager, like Gibbs, was involved primarily in the application of new mathematical ideas to problems in physical chemistry, especially statistical mechanics. Yale's J. W. Gibbs Laboratory and J. Willard Gibbs Assistant Professorship in Mathematics are also named in his honor. On February 28, 2003, Yale held a symposium on the centennial of his death.[36] Rutgers University has a J. Willard Gibbs Professorship of Thermomechanics presently held by Bernard D. Coleman.[37]

In 1950, Gibbs was elected to the Hall of Fame for Great Americans. The United States Navy oceanographic research ship USNS Josiah Willard Gibbs (T-AGOR-1), in service from 1958 to 1971, was named for Gibbs. On May 4, 2005, theUnited States Postal Serviceissued the American Scientists commemorative postage stamp series designed by artist Victor Stabin, depicting Gibbs, John von Neumann, Barbara McClintock, and Richard Feynman.[38]

### In literature

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

Willard Gibbs is the type of the imagination at work in the world. His story is that of an opening up which has had its effect on our lives and our thinking; and, it seems to me, it is the emblem of the naked imagination —which is called abstract and impractical, but whose discoveries can be used by anyone who is interested, in whatever "field"— an imagination which for me, more than that of any other figure in American thought, any poet, or political, or religious figure, stands for imagination at its essential points.
— Muriel Rukeyser, 1949[39]

Both Gibbs and Rukeyser's biography of him figure prominently in the poetry collection True North (1997) by Stephanie Strickland. An official and more technically-oriented biography of Gibbs was published by his former student Lynde Wheeler in 1951.[40]

In fiction, Gibbs appears as the mentor to character Kit Traverse in Thomas Pynchon's novel Against the Day (2006). That novel also prominently discusses the birefringence of Iceland spar, an optical phenomenon investigated by Gibbs.

## References

1. Wheeler, Lynde Phelps (1998 [1951]). Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. p. 32. ISBN 1-881987-11-6. Check date values in: |date= (help)
2. Biography at the American Physical Society
3. Copley Medal, Royal Society
4. Douglas Linder, Josiah W. Gibbs, in Biographies of Players in the Amistad Affair
5. H. A. Bumstead, "Josiah Willard Gibbs," in The Collected Works of J. Willard Gibbs (1928)
6. Wheeler, Lynde Phelps (1998 [1951]). Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. p. 30. ISBN 1-881987-11-6. Check date values in: |date= (help)
7. The Early Work of Willard Gibbs in Applied Mechanics, (New York: Henry Schuman, 1947).
8. US Patent No. 53,971, "Car Brake," Apr. 17, 1866. See The Early Work of Willard Gibbs in Applied Mechanics, (New York: Henry Schuman, 1947), pp. 51-62.
9. Wheeler, Lynde Phelps (1998 [1951]). Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. p. 44. ISBN 1-881987-11-6. Check date values in: |date= (help)
10. M. J. Klein, "The Physics of J. Willard Gibbs in His Time," Proceedings of the Gibbs Symposium (1990), pp. 1-22.
11. O. Mayr, "Victorian Physicists and Speed Regulation: An Encounter Between Science and Technology," Notes and Records of the Royal Society of London, 26, 205-228 (1971).
12. "Thermodynamic Case Study: Gibbs' Thermodynamic Graphical Method"
13. J J O'Connor and E F Robertson, "Josiah Willard Gibbs", The MacTutor History of Mathematics archive
14. Wheeler, Lynde Phelps (1998 [1951]). Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. p. 91. ISBN 1-881987-11-6. Check date values in: |date= (help)
15. Hewitt, E.; Hewitt, R. E. (1979). "The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis". Archive for History of Exact Sciences. 21 (2): 129–160. doi:10.1007/BF00330404.
16. Wheeler, Lynde Phelps (1998 [1951]). "VIII. Professor of Mathematical Physics: Optics". Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. ISBN 1-881987-11-6. Check date values in: |date= (help)
17. Wilson (1931), p. 405
18. Wilson (1931), pp. 411-2
19. M. J. Klein, "Gibbs, Josiah Willard," in Complete Dictionary of Scientific Biography, vol. 5, (Detroit: Charles Scriber's Sons, 2008), pp. 386-393
20. Wheeler, Lynde Phelps (1998 [1951]). "V. Professor of Mathematical Physics: Thermodynamics". Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. ISBN 1-881987-11-6. Check date values in: |date= (help)
21. U.Shmueli, "Reciprocal Space in Crystallography," International Tables for Crystallography, vol. B, ch. 1.1, pp. 2-9 (2006)
22. Wheeler, Lynde Phelps (1998 [1951]). "X. Professor of Mathematical Physics: Statistical Mechanics". Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. ISBN 1-881987-11-6. Check date values in: |date= (help)
23. See, e.g., Kerson Huang, Statistical Mechanics, 2nd ed., (New York: John Wiley & Sons, 1987), sec. 6.6.
24. M. Rukeyser, Willard Gibbs, pp. 225-6.
25. G. C. Rota, Indiscrete Thoughts, (Boston: Birkhäuser, 1996), p. 25.
26. Müller, Ingo (2007). A History of Thermodynamics - the Doctrine of Energy and Entropy. Springer. ISBN 978-3-540-46226-2.
27. J. D. van der Waals, "The Equation of State for Gases and Liquids," Nobel Lectures, 12 Dec. 1910
28. Capri, Anton (2007). Quips, Quotes, and Quanta: An Anecdotal History of Physics. World Scientific Publishing Company. ISBN 978-9812709202.
29. A. S. Wightman, "On the Prescience of J. Willard Gibbs," in Proceedings of the Gibbs Symposium: Yale University, May 15-17, 1989, (American Mathematical Society and American Institute of Physics, 1990), pp. 23-38.
30. Shiller, Robert (2011). "The Yale Tradition in Macroeconomics," (pg. 31). Economic Alumni Conference.
31. Paul Samuelson, "Maximum Principles in Analytical Economics", Nobel Prize Lecture, 1970
32. Paul Samuelson, "How I Became an Economist", Nobel Prize website, 5 September 2003
33. N. Wiener, The Human Use of Human Beings, (Boston: Houghton Mifflin, 1950), p. 11.
34. Willard Gibbs Medal, American Chemical Society
35. Josiah Willard Gibbs Lectures, American Mathematical Society
36. J. Willard Gibbs and his Legacy: A Double Centennial - Yale University (2003).
37. J. Willard Gibbs Professor of Thermomechanics - Rutgers University.
38. "Yale scientist featured in new stamp series", Yale Bulletin & Calendar, May 20, 2005
39. Rukeyser, M. (1949). "Josiah Willard Gibbs". Physics Today. 2 (2): 6–13, 27. doi:10.1063/1.3066422.
40. Wheeler, Lynde Phelps (1998 [1951]). Josiah Willard Gibbs: The History of a Great Mind. Ox Bow Press. ISBN 1-881987-11-6. Check date values in: |date= (help)

## Bibliography

### Secondary

• Bibliography on J. W. Gibbs, The MacTutor History of Mathematics archive
• H. A. Bumstead, "Josiah Willard Gibbs," American Journal of Science, 16, 187-202 (1903), reprinted with some additions in both The Collected Papers (1906) and The Collected Works of J. Willard Gibbs (1928).
• 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).
• J. G. Crowther, Famous American Men of Science, (Freeport, NY: Books for Libraries Press, 1969 [1937]). ISBN 0836900405
• F. G. Donnan, A. E. Hass and P. M. M. Duhem, A Commentary on the Scientific Writings of J. Willard Gibbs, (New York: Arno, 1980 [1936]). ISBN 0405125445
• 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, 372-393 (1909).
• M. J. Klein, "Gibbs, Josiah Willard," in Complete Dictionary of Scientific Biography, vol. 5, (Detroit: Charles Scriber's Sons, 2008), pp. 386-393.
• K. Meinke and J. V. Tucker, "Universal Algebra," in Handbook of Logic in Computer Science, vol. I, eds. S. Abramsky, D. Gabbay and T. S. E. Maibaum, (Oxford: Oxford University Press), pp. 189-411. ISBN 0198537611
• M. Rukeyser, Willard Gibbs: American Genius, (Woodbridge, CT: Ox Bow Press, 1988 [1942]). ISBN 0918024579
• R. J. Seeger, J. Willard Gibbs, American mathematical physicist par excellence, (Oxford and New York: Pergamon Press, 1974). ISBN 0080180132
• L. P. Wheeler, Josiah Willard Gibbs, The History of a Great Mind, (Woodbridge, CT: Ox Bow Press, 1998 [1951]). ISBN 1881987116
• E. B. Wilson, "Reminiscences of Gibbs by a student and colleague", Bulletin of the American Mathematical Society, 37, 401-416 (1931).
• Dictionary of American Naval Fighting Ships: San Carlos