# The start of something: though most Americans have likely not heard of Josiah Willard Gibbs, he was an American physicist upon whose works much of science and math revolve.

"I must study politics and war," John Adams once wrote, "that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain."Many of America's founding generation, such as Adams, Jefferson, Madison, and many others, were men and women of refinement and culture, who understood that the blessings of liberty would allow America to grow into an independent civilization, complete with all the trappings of wealth and progress. At the time of America's founding, and for several decades thereafter. Western Europe was the center of scientific and technological progress, but by the mid-19th century, that had begun to change. America's pre-eminent universities were beginning to come into their own, while more and more prominent American inventors were driving the progress of Western civilization. By the beginning of the 20th century, the world's foremost inventor, Thomas Edison, was American, and other prodigies, such as Nikola Tesla, were crossing the Atlantic to take advantage of the opportunities afforded by American liberties. After World War I, many of Europe's best scientific minds, such as Albert Einstein, also immigrated to America, and by the mid-20th century, America was indisputably the world leader in science, engineering, and invention. All of this has come about, as Adams long ago implied, as the effects of liberty have propagated through society, allowing unprecedented economic growth and corresponding greater and greater opportunities for men and women of ideas to drive scientific progress.

Early American Prodigy

One of the very first of America's scientific prodigies came from obscure beginnings in 19th-century Connecticut. Although his name is not as familiar as Einstein's, his contributions to theoretical physics, chemistry, and mathematics were very nearly as significant; yet he lived nearly his entire life in New Haven, far from the great European centers of learning, with no tools besides his formidable powers of reason and no colleagues other than occasional letters from admirers on the other side of the Atlantic.

New Haven, Connecticut, enjoyed one of the greatest concentrations of talent in the entire Western world in the 19th century. In a time of unprecedented progress in science, engineering, and manufacturing, New Haven, despite its relatively small size, produced, in a very brief span of time, some of the most notable innovators of the age. One New Haven inventor, Eli Whitney, invented the cotton gin, while another, Ithiel Town, developed new methods of bridge construction, which facilitated America's westward expansion. Samuel Morse, a graduate of Yale University and an accomplished painter, helped to develop the telegraph and invented Morse code, while Charles Goodyear, yet another New Haven inventor, revolutionized the rubber industry through his discovery of the process of vulcanization. A Yale professor, Benjamin Silliman, Jr., laid the theoretical foundation for the modern petroleum industry with his study of Pennsylvania oil and various methods for refining it. But perhaps the greatest of all of New Haven's 19th-century savants was Josiah Willard Gibbs, one of America's first scientists of international rank and among the most brilliant theoretical physicists of all time.

Born into a long line of university-educated men and women (unusual in the early 19th century, especially in America) in 1839, Gibbs had a Yale-educated philologist and theologian for a father and a dedicated amateur ornithologist and botanist for a mother, who instilled in her children a knowledge of local birdlife, among other things.

Gibbs (known familiarly as Willard, presumably to distinguish him from his father) grew up in the heady company of many Yale families who--along with their counterparts in Harvard--constituted the bulk of New England's intellectual elite in early America. He was by all accounts a shy and thoughtful youth, little interested in the ordinary social pursuits of his peers, and only ever manifesting interest in one young lady, during his last year in grammar school, who evidently spurned his affections.

In 1854, Gibbs enrolled in Yale after passing the oral entrance exams typical of university admission at that time. From the outset, he was an exceptional student, receiving prestigious awards for excellence in Latin and mathematics while still in his freshman year. In his sophomore year, he was again recognized for his mathematical aptitude, while in his junior year, he was again rewarded for his talent in Latin. In his senior year, he received two scholarships, received a first prize in mathematics, and delivered the Latin oration at commencement. As with grammar school, Gibbs the college student apparently had little inclination for social life, preferring the rarefied company of books and college professors.

American science and mathematics in the mid-19th century were very much in their infancy, with but few American academics competent to teach the cutting-edge theory effusing from their counterparts on the other side of the Atlantic. Many topics now familiar to undergraduate math students were still unknown in the United States (and in some cases, such as the "method of Frobenius" familiar to all modern students of ordinary differential equations, had yet to be discovered). On the other hand, much of differential and integral calculus and the classical physics of Newton, Lagrange, and Hamilton were well established. The twin 20thcentury revolutions in physics--quantum mechanics and relativity--were yet decades in the future, and Gibbs was not to live long enough to witness their arrival.

Harvard University had one mathematician and theoretical physicist of international rank, Benjamin Peirce (whose brilliant son Charles was later to become a correspondent of Gibbs), but the professors at Yale, including competent though now-forgotten names such as Chester S. Lyman in engineering and Hubert A. Newton in mathematics, were hardly of the caliber of a Faraday, Maxwell, or Boltzmann. This was in part because America in the 19th century was (and to a large extent still is) a land of inventors and engineers rather than theorists. Gibbs, very much in the spirit of his contemporaries, opted to pursue graduate studies at Yale in engineering rather than in physics or mathematics.

In 1863, Gibbs was awarded one of three doctors of philosophy at Yale. This was only the third year that the new degree was conferred, with Yale the first American university to offer a Ph.D. In 1861,one doctorate in astronomy had been awarded; all other Ph.D.s up to and including 1863 were in classics, except for Gibbs'; his was thus the second Ph.D. in science and the first in engineering to be awarded in the history of the United States.

European Stint

Following graduation, Gibbs was appointed as a tutor at Yale for a three-year term, during which he mostly taught Latin. Despite his obvious aptitude for mathematics, he opted to teach natural science as his other subject. His earliest research papers, produced during this period, show a dogged determination to contribute to practical inventions, and concerned such subjects as railway car brakes and hydraulic turbines. Gibbs the theoretical physicist had not yet come into view, although in his early writings can already be discerned characteristics of thought that were to become his hallmarks: a minutely reasoned, economically worded style of reasoning, an insistence on precise definitions, and a penchant for geometrical approaches to the problems he considered.

By the time Gibbs' three-year stint as a Yale tutor drew to a close, he had exhausted the intellectual potential of the materials available to him at the time on American soil. He longed to study in Europe, where the great advances in mathematics and physics were taking place. His father had passed away in 1861 and left Gibbs and his two older sisters, Anna and Julia, with a modest inheritance, including a sizeable house on New Haven's High Street, which Gibbs' father had built for his family in 1846, and which Willard called home for the rest of his life.

In August 1866, Gibbs and his two sisters, having found renters for the family home, set off for what would become an extended stay in Western Europe. They sojourned first in Paris, where Gibbs was anxious to make the acquaintance of the illustrious mathematicians and physicists at the Sorbonne and the College de France. Gibbs spent two semesters in Paris working at a feverish pace (he carried 16 hours of lecture appointments per week--what we would now call credits--in his first semester, resulting in severe exhaustion by semester's end that required a brief escape to the Riviera). But there can be no doubt that, in that comparatively brief period, Gibbs gained a degree of sophistication in higher mathematics and physics that would have been impossible back in America. With his prodigious powers of synthesis and memory, Gibbs imbibed the doctrines of the great French luminaries in mathematics and science, including Laplace, Lagrange, Fourier, Fresnel, Poisson, Ampere, and Legendre--men whose lives had mostly coincided with the terrible upheavals of the French Revolution, the Napoleonic Era, and the Bourbon Restoration, and yet who had somehow managed to accomplish some of the greatest advances in the history of science.

In the summer of 1867, Gibbs and his two sisters made their way to Berlin. Here they met up with a former Yale classmate of Gibbs, Addison Van Name (who was Yale University's librarian until 1905 and also an accomplished linguist responsible for the first studies of "creole" languages such as Haitian French). Van Name had been engaged to Gibbs' sister Julia for many years, and the two finally wed in Berlin in August 1867, enjoyed a short honeymoon, and then returned to New Haven, leaving Gibbs to complete his European studies in the company of his other sister, Anna.

After a year of studies in Berlin, where Gibbs continued to study mathematics and all branches of theoretical physics, the Gibbs siblings moved on to the great university town of Heidelberg, where Willard studied for a third and final year. Few details of his months in Heidelberg have survived, but it seems likely that he studied with the likes of world-renowned physicists Gustav Kirchhoff, Robert Bunsen, and Hermann von Helmholtz and mathematician Georg Cantor, all of whom were at Heidelberg at the time.

Return Home

In June of 1869, Willard and Anna returned to New Haven, where Gibbs was to remain the rest of his life. And a modest and unassuming life it was, in keeping with Gibbs' character. According to those who knew him, he possessed none of the eccentricities or flamboyant cravings for attention typical of so many exceptionally gifted people. Gibbs was kindly and unassuming, and for many years after he embarked on a series of theoretical breakthroughs that earned him renown among his colleagues at home and abroad, many of his friends and acquaintances had no idea that the dapper, bearded, church-going son of a Yale philologist was in fact one of the most brilliant scientists of the century. Recalled one of Gibbs' students:

To me he always appeared ... perfectly friendly and approachable, ready to talk on any subject, and always equable, he exhibited a flattering welcome to every friend.... He laughed readily and possessed a lively sense of humor.

Gibbs never married but was fond of children, often taking groups of young people hiking, camping, and berry picking in the wilder parts of New England and the Adirondacks. One man privileged to spend time with Gibbs as a boy recalled:

Gibbs was one of those that made the summer of 1873 an astonishing delight. I did not know, then, that he was an epoch-making philosopher. But all of us younger people had sense enough to see that his mind traveled on serene heights beyond our reach. We climbed, with him, the mild peaks that surrounded us; we rode, with him, on rough farm wagons, to picnic ... and ... we went with him, to church on Sunday. Our church was a birch-grove; and he and I shared, for the singing of hymns, one hymn book.... We, younger people, took pleasure in being with him; but we had no suspicion at that time, that he was one of the greatest philosophers; that he was in the class of Newton and Darwin, and greater than Plato.

Of his religious or political convictions Gibbs said little, but it is plain that he had them, being an active churchgoer his entire life and a strong supporter of Grover Cleveland, arguably the last U.S. president to respect constitutional limits on presidential power. But Gibbs' consuming interest was science, and to this he devoted the remainder of his life, with an intensity that few have ever been able to match, following his return from Europe.

Yale University lost little time appointing Gibbs professor of mathematical physics after his return to New Haven, although the position was unpaid for many years. Yet it was in his new station as a bona fide professor, teaching the most advanced topics in math and science to Yale's brightest students, where Gibbs' genius began to manifest itself. It should be noted that the totality of Gibb's scholarly oeuvre, in contrast to that of most university professors in today's "publish or perish" academic climate, was comparatively brief, consisting of a few papers and a single book (but one of his "papers" was hundreds of pages long, a far cry from the brief and often trivial jottings served up by many modern peer-reviewed academic journals).

Gibbs first studied the fledgling science of thermodynamics. Pioneered by the great French scientist and industrialist Sadi Carnot at the beginning of the 19th century, thermodynamics was concerned with the properties of heat and its transfer. Carnot, who was primarily interested in engines that could exploit thermal energy from internal combustion to perform external work, laid the foundation for the eventual development of the gasoline engine and refrigeration. By Gibbs' time, the notion of heat energy was explicable in terms of pressure, temperature, and entropy (roughly, the amount of "disorder" in a system). Both the first and second laws of thermodynamics, as we now call them, were understood; that is, that energy may change form, but is always conserved (the first law) and that the total amount of entropy in a closed system never decreases (the second law; in other words, all processes in the natural world tend toward disorder, the maximum value of which is defined as "thermodynamic equilibrium"). According to the science of thermodynamics at the time Gibbs began his investigations, the change in intrinsic energy in a system could be defined either in terms of the change in heat energy minus the change in mechanical energy, or as the product of the pressure and the change in volume subtracted from the product of the temperature and the change in entropy. These were neat formulations, but they had one glaring defect: They were only valid for bodies of uniform composition. As such, they did not apply, for example, to two chemicals being mixed together, where uniformity will only be achieved once thermodynamic equilibrium is reached.

A Man of Talents

It was Gibbs' first great achievement to generalize the equation defining the total energy of a thermodynamic system to account for non-uniform (heterogeneous) bodies and substances. He published three great papers on thermodynamics, culminating in the magisterial (if difficult, even for specialists) On the Equilibrium of Heterogeneous Substances, a closely reasoned, 400-page tour de force published by the Connecticut Academy of Arts and Sciences in two parts, in 1876 and 1878. In effect, Gibbs' solution was fairly straightforward. The original prime equation of thermodynamics for homogeneous bodies said in effect that a change in energy equals temperature times change in entropy minus pressure times change in volume. For heterogeneous substances, Gibbs introduced the concept of the "chemical potential" for a given component of a substance. The new generalized equation equated the change in total energy to the sum of the terms of the prime equation and the products of the chemical potentials and changes in mass of every component measured. Of course, a major difficulty then became how to find the chemical potential of a given substance, a topic that occupied a large part of Gibbs' paper and for which he developed an entire intricate methodology that revolutionized thermodynamics and gave birth to the theory of chemical equilibrium.

Gibbs' path-breaking work on thermodynamics has found application in fields as diverse as metallurgy, petrology, chemistry, and volcanology. It is central to the entire field of electrochemistry. It is used routinely in manufacturing, such as in the creation of new alloys and of harder cement and concrete, and in the synthesis of ammonia. It is no exaggeration to say that On the Equilibrium of Heterogeneous Substances is one of the most significant scientific treatises of all time--nearly on a par with Newton's Principia Mathematica--and that all of its potential ramifications for modern civilization (particularly for chemical engineering) are still being worked out.

It was Heterogeneous Substances that first brought Gibbs' genius to the attention of the rest of the scientific world. The monograph caught the attention of no less a figure than James Clerk Maxwell, the father of classical electromagnetic theory. Though Maxwell and Gibbs apparently never corresponded and did not know one another, the former immediately recognized the genius of the latter and disseminated On the Equilibrium of Homogeneous Substances among all his colleagues in England and on the European continent.

Even as professional accolades for his accomplishments poured in from abroad, Gibbs' fertile mind was already considering an entirely new set of problems. Fundamental to classical physics are the notions of the trajectory and the field, both of which require mathematical descriptors that can specify direction as well as magnitude. Such entities are called vectors (in contrast to scalars, which are simple numbers or magnitudes with no direction). Vector analysis is a standard component of undergraduate linear algebra and calculus (not to mention physics). But most modem students are blissfully unaware that, in its infancy, the science of vectors and vector spaces was notoriously difficult. The problem lay not so much with inherent difficulties as with the choice of notation. In Gibbs' day, what we now call vectors were represented by "quaternions," a system devised by British physicist William Rowan Hamilton. It was Gibbs who first devised notation for the "dot product" (or scalar product) of two vectors, conventionally defined as the product of the magnitudes of two vectors and the cosine of the angle between them, and the cross product (or vector product) as the product of the magnitudes of two vectors and the sine of the angle between them; the vector resulting from a cross product will be perpendicular ("normal") to the plane defined by the two original vectors (the well-known physical property of torque, for example, is expressed as a cross product). Gibbs also reformulated other key definitions of vector calculus, such as the divergence (div) and curl. Curiously, Gibbs was not particularly interested in publishing these contributions in and of themselves, instead incorporating them into his other work. Long after Gibbs' death, however, E. B. Wilson published in 1929 a textbook on vector analysis "founded upon the lectures of J. Willard Gibbs" that cemented Gibbs' legacy in this area. As a result, his methods and notation gradually came to be preferred by other physicists and mathematicians and are universal today.

Although he made contributions to other areas of physics, such as optics, Gibbs' other monumental contribution to science was his invention of statistical mechanics. In the late 19th century, the precise nature of heat and entropy continued to defy mathematical formulation. Despite the heroic advances in thermodynamics not only by Gibbs, but also by the likes of Boltzmann, Clapeyron, Maxwell, and others, the theory of heat transfer and entropy could not be accommodated by the deterministic, time-reversible framework of classical mechanics. In other words, the phenomenon of heat transfer appeared not to be congruent with the trajectory-based accounts of other types of physical behavior. Moreover, the transfer of heat in nature always seemed to run in only one direction; time, in the thermodynamic universe, appeared not to be reversible (in accordance with the second law of thermodynamics which stipulates that entropy always increases), whereas in the physics of Newton, Lagrange, and Hamilton, all equations were time-symmetric, i.e., they worked equally well no matter which direction the arrow of time pointed. For example, if we make a video of a ball rising and falling along some trajectory, we can run the video in either direction without being able to discern an arrow of time. But any process involving entropy is always directed', a video of two chemicals unmixing themselves or of a broken glass spontaneously jumping back onto a table and reassembling itself are immediately recognizable as time in reverse. In real life, such things simply do not take place (for reasons which are still not entirely clear).

While it was understood that changes in heat represented changes in the energy of all the particles (molecules or atoms) making up a system--the faster the particles move, the more heat the system possesses--there was, before Gibbs, no way to mathematically formulate this state of affairs. Whereas the motion of a projectile through space or the action of a spring on a weight, for example, could be precisely formulated thanks to the mathematics of Newton, Lagrange, and Hamilton, there existed no corresponding mathematics to describe what was believed to be the impossibly complex mechanics of a vast array of particles, each with its own energy and momentum.

Gibbs' second magnum opus, Elementary Principles in Statistical Mechanics, Developed With Especial Reference to the Rational Foundation of Thermodynamics, which appeared in 1902, changed all that. Using statistical methods of his own devising, Gibbs invented an entirely new field of mathematical physics, statistical mechanics, which allowed, for the first time, a precise formulation of the mechanics of a grouping of particles of any size (an "ensemble," in Gibbs' terminology). In other words, Gibbs' statistical mechanics permitted him to derive the first and second laws of thermodynamics from the laws of classical mechanics, if a statistical approach was adopted. Although it came on the eve of the double revolution in physical theory, relativity, and quantum mechanics, Gibbs' statistical mechanics has borne well the tests of time, and remains a standard component of physical science. Moreover, many of the statistical methods that he pioneered in Elementary Principles in Statistical Mechanics were used to great effect in the development of quantum mechanics. Einstein himself, unaware of Gibbs' work, published three papers on statistical mechanics before reading a German translation of Elementary Principles', he subsequently declared that Gibbs' treatment of the subject was far superior to his own, and that, had he known of it, he would not have published anything on the subject.

Aside from his own discoveries, Gibbs nurtured several leading figures in America's next generation of scientists, including Lee De Forest and Lynde Wheeler, both of whom contributed to the development of the radio.

Josiah Willard Gibbs died the year after publication of Elementary Principles in Statistical Mechanics, in April 1903. As befitted a man of so modest a temperament, he was buried two days later following a simple ceremony, his headstone recording only that he had been "professor of mathematical physics in Yale University."

Gibbs was among the last of the great classical physicists. Three years before his death, German physicist Max Planck had introduced the idea of the quantum, an elementary unit of which all amounts of energy were held to be multiples. Planck's breakthrough led to the formulation of quantum mechanics over the next three decades. Two years after Gibbs' death, Einstein exploited the power of a mathematical operation called the "Lorentz transformation" to produce an entirely new type of physical theory, special relativity, which explained the observed invariance of the speed of light. It does not appear that Gibbs was aware of the work of Planck or Lorentz, or if he was, that he appreciated its significance.

But his brilliance as a scientist is uncontested. More remarkable was his nurturing on American, not European, soil, in an era where the United States still had scant resources for a would-be theoretical scientist. But Gibbs was one of the first to embody John Adams' vision of an America where theoretical science as well as practical labor could flourish as a consequence of the liberty her citizens enjoyed. Nowadays, of course, American physicists and chemists are awarded the lion's share of Nobel Prizes, while mathematicians at American institutions have been recipients of every round of Fields Medal awards since 1970. Yet if it is true that every great thinker has, with Newton, been obliged to stand on the shoulders of giants, then all of these must be reckoned as standing on the shoulders of Josiah Willard Gibbs.

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Title Annotation: | HISTORY--PAST AND PERSPECTIVE |
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Author: | Scaliger, Charles |

Publication: | The New American |

Geographic Code: | 1USA |

Date: | Nov 17, 2014 |

Words: | 4123 |

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