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

the beginning of modern vector analysis. 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 electr.…