## On multiple algebra: an address before the Section of Mathematics and Astronomy of the American Association for the Advancement of Science at the Buffalo meeting, August, 1886.

Salem, Mass. Salem Press, 1886.

First edition, very rare offprint issue. “One of Gibbs’ most famous papers was entitled ‘On Multiple Algebra.’ It was given as an ‘Address before the Section of Mathematics and Astronomy of the American Association for the Advancement of Science, by the Vice-President’ and was published in 1886. Of all Gibbs’ writings this essay gives the best picture of his conception of the place of vector analysis within the wider fields of algebra and mathematics in general” (Crowe, p. 158). “In the year 1844 two remarkable events occurred, the publication by [William Rowan] Hamilton of his discovery of quaternions, and the publication by [Hermann Günther] Grassmann of his ‘Ausdehnungslehre.’ With the advantage of hindsight we can see that Grassmann's was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case … When Grassmann’s work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things. So it was left to the physicist Gibbs to present for the first time in his 1886 lecture the essential ideas of Grassmann and Hamilton side by side. The last words of his lecture are, ‘We begin by studying multiple algebras; we end, I think, by studying MULTIPLE ALGEBRA’” (Dyson, p. 644). In modern terms, a ‘multiple algebra’ is an ‘algebra’, i.e., a ‘vector space’ equipped with a product that is distributive with respect to addition (*a*(*b + c*) = *ab* + *ac*). The algebras considered by Grassmann and Gibbs were associative ((*ab*)*c* = *a*(*bc*)), but not necessarily commutative (*ab* = *ba*).

“Gibbs began with a brief treatment of the history of multiple algebra and considered such men as Möbius, Grassmann, Hamilton, Saint-Venant, Cauchy, Cayley, Hankel, Benjamin Peirce, and Sylvester. By this analysis Gibbs hoped among other things ‘to illustrate the fact, which I think is a general one, that the modern geometry is not only tending to results which are appropriately expressed in multiple algebra, but that it is actually striving to clothe itself in forms which are remarkably similar to the notations of multiple algebra, only less simple and general and far less amenable to analytical treatment, and therefore, that a certain logical necessity calls for throwing off the yoke under which analytical geometry has so long labored.’

The strongest praise and fullest treatment was bestowed upon Grassmann. Considering in detail a number of the products defined by Grassmann, Gibbs discussed their significance and argued that Grassmann’s system provides a rich and encompassing point of view. Gibbs concluded the paper with a discussion of applications of multiple algebra to physical science. Thus he stated: ‘First of all, geometry, and the geometrical sciences which treat of things having position in space, kinematics, mechanics, astronomy, physics, crystallography, seem to demand a method of this kind, for position in space is essentially a multiple quantity and can only be represented by simple quantities in an arbitrary and cumbersome manner. For this reason, and because our spatial intuitions are more developed than those of any other class of mathematical relations, these subjects are especially adapted to introduce the student to the methods of multiple algebra.’

“Proceeding then to specifics, Gibbs noted that through Maxwell electricity and magnetism had become associated with the methods of multiple algebra but that astronomy had so far remained aloof. Having noted that for geometrical applications multiple algebra will generally take the form of a point or a vector analysis, Gibbs stated that ‘in mechanics, kinematics, astronomy, physics, or crystallography, Grassmann’s point analysis will rarely be wanted.’ However arguments were given on behalf of the usefulness of point analysis for investigations in pure mathematics. The concluding paragraph and the concluding sentence in particular have now become classic. ‘But I do not so much desire to call your attention to the diversity of the applications of multiple algebra, as to the simplicity and unity of its principles. The student of multiple algebra suddenly finds himself freed from various restrictions to which he has been accustomed. To many, doubtless, this liberty seems like an invitation to license. Here is a boundless field in which caprice may riot. It is not strange if some look with distrust for the result of such an experiment. But the farther we advance, the more evident it becomes that this too is a realm subject to law. The more we study the subject, the more we find all that is most useful and beautiful attaching itself to a few central principles. We begin by studying *multiple algebras*; we end, I think, by studying MULTIPLE ALGEBRA.’

“Gibbs’ paper may be taken as symbolic of the ever-increasing interest expressed at that time by pure and applied mathematicians in multiple algebra. The increasing interest in vector analysis was a part, in one way the most influential part, of that trend. Gibbs was able, as some later writers were not, to view vector analysis in the broad perspective offered by multiple algebra. That Gibbs was deeply interested in multiple algebra is shown by the facts that every two or three years he gave a course in multiple algebra and that he planned to publish additional writings on multiple algebra and had actually done research in this regard” (*ibid*., pp. 158-160).

“Josiah Willard Gibbs was born in 1839: his father was at that time a professor of sacred literature at Yale University. Gibbs graduated from Yale in 1858, after he had compiled a distinguished record as a student. His training in mathematics was good, mainly because of the presence of H. A. Newton on the faculty. Immediately after graduation he enrolled for advanced work in engineering and attained in 1863 the first doctorate in engineering given in the United States. After remaining at Yale as tutor until 1866, Gibbs journeyed to Europe for three years of study divided between Paris, Berlin, and Heidelberg. Not a great deal of information is preserved concerning his areas of concentration during these years, but it is clear that his main interests were theoretical science and mathematics rather than applied science. It is known that at this time he became acquainted with Möbius’ work in geometry, but probably not with the systems of Grassmann or Hamilton. Gibbs returned to New Haven in 1869 and two years later was made professor of mathematical physics at Yale, a position he held until his death.

“His main scientific interests in his first year of teaching after his return seem to have been mechanics and optics. His interest in thermodynamics increased at this time, and his research in this area led to the publication of three papers, the last being his now classic ‘On the Equilibrium of Heterogeneous Substances,’ published in 1876 and 1878 in volume III of the *Transactions of the Connecticut Academy. *This work of over three hundred pages was of immense importance. When scientists finally realized its scope and significance, they praised it as one of the greatest contributions of the century” (ibid., p. 151).

J. Crowe, *A History of Vector Analysis*, 1967. F. Dyson, ‘Missed opportunities,’ *Bulletin of the American Mathematical Society* 78 (1972), 635-652.

Offprint from: Proceedings of the American Association for the Advancement of Science, Vol. XXXV. 8vo (246 x 155 mm), pp. 32. Original printed wrappers (a few small marginal chips), unopened.

Item #4433

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