## Elements of Vector Analysis. [Offered with:] Autograph letter from Gibbs to John Monroe Van Vleck.

New Haven: Tuttle, Morehouse & Taylor, 1881.

First edition, first issue (see below), of this extremely rare pamphlet which “marks the beginning of modern vector analysis” (Crowe, p. 150). “Nearly all branches of classical physics and many areas of modern physics are now presented in the language of vectors, and the benefits derived thereby are many. Vector analysis has likewise proved a valuable aid for many problems in engineering, astronomy and geometry” (*ibid*. p. v). The genesis of the present work was described in Gibbs’s own words in an 1888 letter to Victor Schlegel: “My first acquaintance with quaternions was in reading Maxwell’s E & M [i.e. *Treatise on Electricity and Magnetism*, 1873] where quaternion notations are considerably used. I became convinced that to master those subjects, it was necessary for me to commence by mastering those methods. At the same time I saw, that although the methods were called quaternionic the idea of the quaternion was quite foreign to the subject. In regard to the product of vectors, I saw that there were two important functions (or products) called the vector part & the scalar part of the product, but that the union of the two to form what was called the (whole) product did not advance the theory as an instrument of geom[etric] investigation. Again with respect to the operator as applied to a vector I saw that the vector part & the scalar part of the result represented important operations, but their union (generally to be separated afterwards) did not seem a valuable idea … I therefore began to work out *ab initio*, the algebra of the two kinds of multiplication, the three differential operations applied to a scalar, & the two operations to a vector … This I ultimately printed but never published, although I distributed a good many copies among such persons as I though might possibly take an interest in it” (*ibid*. pp. 152-3). This is the first issue; a second issue, with two additional chapters, was published in 1884. Both issues of Gibbs’s pamphlet are extremely rare in commerce. We have been unable to locate any copy of this first issue in auction records, and only two of the second issue: the Horblit copy (Christie’s, 16 February 1994) and the Richard Green copy (Christie’s, 17 June 2008).

“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” (Dyson). “During the 1880’s Gibbs seems to have concentrated on optics and particularly on Maxwell’s electromagnetic theory of light … Gibbs’s reading Maxwell’s *Treatise on Electricity and Magnetism* led him to a study of quaternions, since Maxwell had used the quaternion notation to a limited extent in that work. Gibbs decided, however, that quaternions did not really provide the mathematical language appropriate for theoretical physics, and he worked out a simpler and more straightforward vector analysis” (DSB). From Schlegel’s letter, we learn that “Gibbs commenced his search for a vector analysis ‘with some knowledge of Hamilton’s methods’ and ended up with methods that were ‘nearly those of Hamilton’ … Gibbs also stated that he was not ‘conscious that Grassmann exerted any particular influence on my V-A.’ This is to be expected since Gibbs had begun searching for a new vector system ‘long before my acquaintance with Grassmann.’ When (1877 or later) Gibbs finally began to read Grassmann, he found a kindred spirit. Although Gibbs admitted he had never been able to read through either of Grassmann’s books, he did recognize Grassmann’s priority and warmly praised his ideas on numerous occasions” (Crowe, pp. 153-4).

“In 1879 Gibbs gave a course in vector analysis with applications to electricity and magnetism, and in 1881 he arranged for the private printing of the first half of his *Elements of Vector Analysis*; the second half appeared in 1884. In an effort to make his system known, Gibbs sent out copies of this work to more than 130 scientists and mathematicians. Many of the leading scientists of the day received copies, for example, Michelson, Newcomb, J. J. Thomson, Rayleigh, FitzGerald, Stokes, Kelvin, Cayley, Tait, Sylvester, G. H. Darwin, Heaviside, Helmholtz, Clausius, Kirchhoff, Lorentz, Weber, Felix Klein, and Schlegel. Though the work was not given the advertisement that a regular publication would have had, such a selective distribution must have aided in making it known.

“Some idea of the form of Gibbs’ *Elements of Vector Analysis* may be obtained from Gibbs' introductory paragraph: ‘The fundamental principles of the following analysis are such as are familiar under a slightly different form to students of quaternions. The manner in which the subject is developed is somewhat different from that followed in treatises on quaternions, since the object of the writer does not require any use of the conception of the quaternion, being simply to give a suitable notation for those relations between vectors, or between vectors and scalars, which seem most important, and which lend themselves most readily to analytical transformations, and to explain some of these transformations. As a precedent for such a departure from quaternionic usage, Clifford's *Kinematic *may be cited. In this connection, the name of Grassmann may also be mentioned, to whose system the following method attaches itself in some respects more closely than to that of Hamilton.’

“Although Gibbs mentioned only Clifford and Grassmann in the introductory paragraph, the previously cited letter makes it clear that his chief debt was not to either Clifford or Grassmann but to the quaternionists. In the discussion of Gibbs’ book this point will be illustrated; specifically it will be suggested that Gibbs was strongly influenced by the content and form of presentation found in the second edition of Tait’s *Treatise on Quaternions.*

“Chapter I, ‘Concerning the Algebra of Vectors,’ began with such definitions as ‘vector’, ‘scalar,’ and ‘vector analysis’. In much of the symbolism introduced, Gibbs followed the quaternion traditions: for example, Gibbs represented vectors by Greek eletters, their components by means of *i*, *j*, and *k* … In dealing with vector products Gibbs introduced the ‘direct product,’ written α.β, and the ‘skew product,’ written α x β. These are the now-current scalar (dot) and vector (cross) products … Chapter I concluded with a treatment of methods for solving vectorial equations, and chapter II was entitled ‘Concerning the Differential and Integral Calculus of Vectors.’ Herein Gibbs introduced the operator , proved the related transformation theorems, and gave an extended treatment of the mathematics of potential theory”. The part of Gibbs' booklet printed in 1881 terminated near the end of chapter II; the remainder was printed in 1884.

The present first issue terminated at the end of Chapter II. It was issued without a separate title page. In 1884 a second issue appeared, with a variant title “Elements of vector analysis, arranged for the use of students in physics”. To the second issue was added a further 47 pages, comprising Chapters III and IV, which dealt with linear vector functions, or “dyads” in Gibbs’s terminology. (See the Harvard library catalogue for comments on the differences between the two issues.) Gibbs's version of the vector analysis was not formally published until 1901, when one of his students, Edwin B. Wilson, prepared a textbook based on Gibbs's lectures.

“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).

Dibner 117; DSB V: 391; Crowe, *A History of Vector Analysis*, 1967.

8vo (245 x153 mm), pp 36. Stitched as issued (with original stitching), upper inner corner of final leaf torn away (no text loss), and tear to second final leaf (again not effecting the text).

Item #4853

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Price:
$19,500.00
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