J Willard Gibbs' father, also called Josiah Willard Gibbs, was professor of sacred literature at Yale University. In fact the Gibbs family originated in Warwickshire, England and moved from there to Boston in 1658. However Gibbs is said to have taken after his mother in physical appearance.
Gibbs was educated at the local Hopkins Grammar School where he was described as friendly but withdrawn. His total commitment to academic work together with rather delicate health meant that he was little involved with the social life of the school. In 1854 he entered Yale College where he won prizes for excellence in Latin and Mathematics.
Remaining at Yale, Gibbs began to undertake research in engineering, writing a thesis in which he used geometrical methods to study the design of gears. When he was awarded a doctorate from Yale in 1863 it was the first doctorate of engineering to be conferred in the United States. After this he served as a tutor at Yale for three years, teaching Latin for the first two years and then Natural Philosophy in the third year. He was not short of money however since his father had died in 1861 and, since his mother had also died, Gibbs and his two sisters inherited a fair amount of money.
From 1866 to 1869 Gibbs studied in Europe. He went with his sisters and spent the winter of 1866-67 in Paris, followed by a year in Berlin and, finally spending 1868-69 in Heidelberg. In Heidelberg he was influenced by Kirchhoff and Helmholtz.
Gibbs returned to Yale in June 1869 and, two years later in 1871, he was appointed professor of mathematical physics at Yale. However, as Crowther points out in :-
He returned more a European than an American scientist in spirit-one of the reasons why general recognition in his native country came so slowly.
Rather surprisingly his appointment to the chair at Yale came before he had published any work. Perhaps it is also surprising that Gibbs did not publish his first work until 1873 when he was 34 years old. Few scientists who produce such innovative work as Gibbs did are 34 years of age before producing signs of their genius.
Gibbs' important 1873 papers were Graphical Methods in the Thermodynamics of Fluids and A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces. In 1876 Gibbs published the first part of the work for which he is most famous On the Equilibrium of Heterogeneous Substances, publishing the second part of this work in 1878.
The first of these papers describes diagrams in thermodynamics. Bumstead (see  or ) writes:-
Of the new diagrams which he first described in this paper, the simplest, in some respects, is that in which entropy and temperature are taken as coordinates... the work or heat of any cycle is proportional to its area in any part of the diagram ... it has found most important applications in the study of the steam engine.
The second paper extended the diagrams into three dimensions and this work impressed Maxwell so much that he constructed a three dimensional model of Gibbs's thermodynamic surface and, shortly before his death, sent the model to Gibbs.
However the third paper is the most remarkable. Bumstead in  or  writes:-
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 any one, and especially so for students of experimental chemistry whom it most concerns...
Gibbs' work on vector analysis was also of major importance in pure mathematics. He first produced printed notes for the use of his own students in 1881 and 1884 and it was not until 1901 that a properly published version appeared prepared for publication by one of his students. Using ideas of Grassmann, Gibbs produced a system much more easily applied to physics than that of Hamilton.
He applied his vector methods to give a method of finding the orbit of a comet from three observations. The method was applied to find the orbit of Swift's comet of 1880 and involved less computation than Gauss's method.
A series of five papers by Gibbs on the electromagnetic theory of light were published between 1882 and 1889. His work on statistical mechanics was also important, providing a mathematical framework for quantum theory and for Maxwell's theories. In fact his last publication was Elementary Principles in Statistical Mechanics and this work is a beautiful account putting the foundations of statistical mechanics on a firm foundation.
Except for his early years and the three years in Europe, Gibbs spent his whole life living in the same house which his father had built only a short distance from the school Gibbs had attended, the College at which he had studied and the University where he worked the whole of his life. Crowther, in , sums up his life as follows:-
[Gibbs] remained a bachelor, living in his surviving sister's household. 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 U.S. 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 U.S. science commensurate with his genius.
Bumstead describes Gibbs' personal character in the following glowing terms:-
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.
The American Mathematical Society named a lecture series in honour of Gibbs. An annual lecture has been given by a distinguished mathematician most years since 1923. You can see a list of the lecturers.
Article by: J J O'Connor and E F Robertson