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William Rown Hamilton, Lectures on quaternions (1853)
CHAPTER 35
WILLIAM ROWAN HAMILTON, LECTURES ON QUATERNIONS (1853) Albert C. Lewis
This, the first book devoted to quaternions, appeared ten years after their discovery by Hamilton. Later, many of his most useful concepts were separated from their quaternion context and were reformulated as a part of vector analysis. The key work in this transformation was E.B. Wilson’s Vector analysis (1901). First publication. Lectures on quaternions containing a systematic statement of a new mathematical method of which the principles were communicated in 1843 to the Royal Irish Academy: and which has since formed the subject of successive courses of lectures delivered in 1848 and subsequent years, in the halls of Trinity College, Dublin: with numerous illustrative diagrams, and with some geometrical and physical applications, Dublin: Hodges and Smith; London: Whittaker and Co.; Cambridge: Macmillan and Co., 1853. lxxii + 736 pages. Reprint. Cornell Library Digital Collections (http://historical.library.cornell.edu). [Preface only in Mathematical papers, vol. 3.] Related articles: Grassmann (§32), Heaviside (§49). 1 FROM PRODIGY TO SAGE Sir William Rowan Hamilton (1805–1865), born and raised in Ireland, was one of the most brilliant students to have passed through Trinity College, Dublin. He mastered many languages, ancient and modern. He appears to have been largely self-taught in mathematics though guided by a tutor. At Trinity from 1823 to 1827 he was exposed to the newest mathematics that emanated mainly from France, especially by P.S. Laplace, J.L. Lagrange, S.D. Poisson, and S.F. Lacroix. He had barely completed his intended program of studies when he was offered the prestigious appointment of Astronomer Royal of Ireland. Though he was not inclined to the practical aspects required for the job, he gained the aid of his four Landmark Writings in Western Mathematics, 1640–1940 I. Grattan-Guinness (Editor) © 2005 Elsevier B.V. All rights reserved. 460
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sisters who lived with him at the Dunsink Observatory and who managed the household and performed many if not most of the astronomical duties; and the post provided more time for his mathematical researches than the conventional teaching position for which he originally aimed. Hamilton’s initial fame beyond Dublin came from work on systems of light rays done already in 1824. The experimental verification of his prediction of the conical refraction of light in 1833 brought him the highest honors, including a knighthood in 1835. Another enduring work from this early period of his life was what came to be called the ‘Hamiltonian function’, which has proven of fundamental importance in physics. It was also during this period that he began the pursuit of the elusive triple number system that led to his discovery of quaternions in 1843 and to the book featured here. Though quaternions, with their promise of so many fruitful applications, was to take much of his attention in later life, he devoted himself to many other matters. From 1837 to 1846 Hamilton was president of the Royal Irish Academy. Far from being an exclusively honorary post, this entailed an intimate involvement in its administration and in the development of wide-ranging projects relating to Irish history and culture. After he resigned this post he was widely praised for his achievements in the Academy, not least for his diplomatic skill at resolving disputes among members. He was spurred by the discovery of the planet Neptune in 1846 to study perturbation theory. One result was his invention of the hodograph, an elegant geometrical representation of planetary paths which was taken up with some interest by William Thomson (Lord Kelvin) among others. Hamilton later learned that A.F. Möbius had discovered the notion earlier, as he acknowledged in the Lectures (p. 614). In 1856 Hamilton became interested in an entirely different subject, which he termed ‘the Icosian Calculus’. This was an algebra capable of describing the paths connecting the vertices of a dodecahedron. From this came the general idea of determining what have come to be known as ‘Hamilton circuits’. Further details can be found in the principal secondary sources on his life [Graves, 1882–1891] and [Hankins, 1980]. 2 THE ORIGIN OF QUATERNIONS During the 1830s Hamilton maintained an interest in a problem that a number of mathematicians regarded as one of the most important unsolved issues of the time: How can the system of number pairs, represented by complex numbers, be extended to triples of numbers in such a way as to preserve the same operational properties? For example, a complex number z = a + bi can be represented in the Euclidean plane by the directed line segment√ from the origin to the point with real number coordinates (a, b). On multiplying z by i (= −1 ), the result would be the segment from the origin to (−b, a) which can be regarded as the result of rotating the original segment 90◦ counterclockwise about the origin. The problem could thus be put in a geometrical and somewhat more general fashion: How can this mathematical operation, represented by rotation about a point in the plane, be extended to rotation about a line in three dimensions? Expressed this way the answer turns out to be that four numbers, not three, are required. In his early work Hamilton assumed, not unnaturally, that the task was one of finding the appropriate system of triples of numbers, and it was only after many unsuccessful ef-
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forts that the possibility of a quadruple of numbers presented itself. In one of the famous moments in the history of mathematics, the idea came to him—as he carefully recorded immediately afterwards—while on Brougham Bridge in Dublin, on his way to attend a Council meeting of the Academy on 16 October 1843. His construction consisted of adjoining three entities, i, j, k, to the real numbers. These can be multiplied according to the rules which he jotted down at the time in a notebook: i 2 = j 2 = k 2 = −1,
ij = k,
j k = i,
ki = j
and ij k = −1.
(1)
The hypercomplex number can be formed as a + bi + cj + dk, where a, b, c, and d are real numbers. Though this new number, Hamilton’s quaternion, suggests that it might best be suited for a four-dimensional spatial representation—and indeed in the early 20th century there were attempts to make use of it in relativity theory—Hamilton himself exploited it for a wide range of three-dimensional applications, his original motivation. Before taking up Hamilton’s development of quaternions in the Lectures proper, mention should be made of his path of discovery which he described at length and in several places, including in the Preface of the Lectures. In his view the discovery is intimately tied to his notion of algebra as the science of pure time. As he describes it in the Preface, he tended to approach the whole subject less in a ‘symbolical’ fashion than in a ‘scientific’ fashion. Influenced by ‘the Kantian parallelism between the intuitions of Time and Space’, and by geometry as the science of space, he felt that viewing algebra as the science of pure time had a high suggestive value that could easily lead to a purely symbolical calculus if and when one chose to follow that symbolical route. Hamilton’s detailed documentation of his creative path has been the basis of several historical analyses, some of which attempt to use it to draw lessons about the nature of mathematical discovery in general [Hankins, 1980, ch. 6; Pickering, 1995, ch. 4]. There are a number of predecessors for Hamilton’s work whom he acknowledges in the Preface. The most significant one for later developments is ‘the very original and remarkable work’ of H.G. Grassmann whose Ausdehnungslehre or calculus of extension of 1844 (§32) he read just as the Lectures was being completed. Hamilton noted that, though Grassmann had a non-commutative multiplication of directed lines, he was not in possession of quaternions since he admitted to not succeeding in extending the complex numbers to three dimensions or in building a theory of angles in space (Preface, p. 62). It seems that Hamilton had quaternions predominately on his mind as he read Grassmann and overlooked the fact that, as later readers recognized, Grassmann’s response to these issues was considerably more general than his own. 3 THE LECTURES After the inspiration of October 1843 Hamilton published a number of very substantial papers over the next ten years describing the new entities, including two series of papers, one in eighteen installments in the Philosophical magazine and another, left incomplete after ten installments, in the Cambridge and Dublin mathematical journal. (These papers are reprinted in the edition [Hamilton, Papers].) In 1848 he conducted a series of lectures at Trinity College, Dublin, and these formed the basis for his Lectures volume of 1853. While
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this was a most productive time for Hamilton with respect to developing and propagating the quaternions, he was also undergoing some traumatic personal experiences: his favorite sister, Eliza, died in 1851, and he had been following closely the health and well being of his greatest love in life, Catherine Disney Barlow, who attempted suicide in 1848. One of his biographers, Thomas Hankins, paints a picture of someone who was clearly deeply affected by these and other personal setbacks but who could appear totally unfazed by them when it came to carrying on with his work. The contents of Hamilton’s Lectures are summarised in Table 1. His path from consideration of progression in time, through working with number triplets, to quaternions, is given with substantial technical detail in the Preface. In fact, the Lectures themselves have been, as he puts it, ‘drawn up in a more popular style than this Preface’ and were intended, at least initially, to be fairly faithful to what was actually presented by Hamilton ‘in successive years, in the Halls of this University’. However, as the table of contents reveals, the lengths of the ‘Lectures’ increased steadily, culminating in Lecture VII, which is over 300 pages long. As he admitted, substantially more ‘calculation’ was added than would actually have been presented in the lecture hall. Nevertheless, he maintains that ‘something of Table 1. Contents by Lectures of Hamilton’s book. Parentheses around the Preface’s page numbers distinguish them from the main body. Square brackets indicate an unnumbered page. Lecture Page Sect. Preface, pp. ([1])–(64) Contents, pp. [ix]–lxxi Lec. I 1 i Lec. II 33 vi
Art.
Lec. III Lec. IV
74 130
xi xxvi
79 121
Lec. V
186
xxxvi
175
Lec. VI
241
xlvi
251
Lec. VII
381
lxi
394
App. A
701
App. B
717
App. C
731
Errata, unnumbered leaf
1 37
Contents (sample topics) Time, number triplets, quaternions. [No pages correspond to numbers i to viii.] Addition and subtraction of lines and points. Multiplication and division in geometry; squares and products of i, j, k. The quaternion; tensor and versor. √ Powers and roots of quaternions; −1 as a partially indeterminate symbol. Multiplication of three lines in space; value of ij k and kj i. General associative property of multiplication; spherical representations. Addition and subtraction; distributive principle of multiplication. [End cxvii, 689.] Gauche (i.e. non-planar) polygons inscribed in second-order surfaces. [Paper published in 1850.] Gauche polygons inscribed in second-order surfaces. [Paper published in 1849.] A ‘rapid outline of the quaternion analysis’. [End 736.] Thirty-seven errata, most quite minor.
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the style of actual lecturing has been here and there retained’ throughout (Preface, p. 63). Apparently to help the reader, if not the author, to gain control over the growing mass of material, Section numbers came into play at a later stage, ‘too late to be incorporated into the text’ as Hamilton writes (p. lxxi). Thus the 117 Sections, which act as an intermediate level of division between the 7 Lectures and 689 articles, occur only in the Table. There is no index, but the copious table of contents serves as an analytic guide. This glimpse into what was evidently a struggle to keep the book under control illustrates a general trait: as biographer Hankins put it, ‘Hamilton could not keep his published works on quaternions within reasonable bounds’ [Hankins, 1980, 365]. Hamilton’s previous course of lectures was an introduction to astronomy and his first lecture in this new series—evidently the only one in the Lectures to adhere at all closely to what he might have actually said in the lecture hall—makes the transition by drawing upon relative positions of planets to introduce the notion of a difference of points as an ordinal expression of relative position. ‘And because, according to the foregoing illustrations, this sign or mark (Minus) directs us to DRAW, or to conceive as drawn, a straight line connecting the two points, which are proposed to be compared as to their relative positions, it might, perhaps, on this account be called the SIGN OF TRACTION’ (p. 10). This sentence succinctly exhibits something of Hamilton’s style. Also, it shows his care in giving new concepts correspondingly new names even at the risk of overloading the reader. In this case, in the interest of reducing the number of new terms, ‘subtraction’ is soon used instead. In the next dozen pages, however, more terms come into play—almost at the rate of one per page—such as ‘vection’, ‘revection’, ‘provection’, and ‘transvection’ to describe various possible motions of a point along a line. Besides the subtraction of two points, the other key notion of the first Lecture is the geometrical meaning of sum of a line and a point. If B − A is conceived as the line from A to B, then the sum (B − A) + A results in the point B. The second Lecture concerns a general division and multiplication that are the analytic and synthetic cardinal operations corresponding to the ordinal operations of subtraction and addition introduced in the previous Lecture. The term ‘cardinal’ comes from the analogy that, given an expression such as β = n + n (Greek letters will represent directed line segments), we can ordinarily regard the quotient β ÷ n as the cardinal number 2. The defining expressions are: β ÷ α = q and q × α = β. This last equation shows how the quotient q can be regarded as an operator that produces one directed line segment from another. If q is a ‘tensor’, or signless number, then it affects only the length of α. If q is a sign (+ or −) it changes the direction of α. If it is a real number then q may have the effect of changing both the direction and length. If it is a ‘vector-unit’ (or ‘quadrantal versor’), i, j, k, then the effect is to turn α right-handedly through 90 degrees in a plane perpendicular to the vector-unit. Hamilton points out that a multiplication of a vector-unit, say i, by itself results in a rotation of 180◦ , i.e. the same as multiplying by −1 (reversing its direction) or i 2 = −1. Continuing with further examples in Lecture III, Hamilton broadens the conception of multiplication to include any two vectors (directed line segments) and also introduces exponentiation of vectors. He shows that these operations, as well as the quotient, q, of two vectors described above, can be characterized by four numbers, namely the tensor (a pure number or scalar, written as Tq) and three directions. This entitles the result to be called a
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quaternion. Most of the previous examples can thus be recognized at this stage as special cases of quaternions. Lectures IV and V begin to reveal the nature of the quaternion itself and further properties of quaternion multiplication. Demonstrations of the non-commutative and associative properties of multiplication are given for more general cases, though not yet in the most general case. (A demonstration of its distributivity over addition is also promised, but this is delayed until Lecture VII as Hamilton feels the need to explore multiplication more before taking up addition.) His arguments throughout are geometrical. He makes use of ‘arcual constructions’ in which ‘representative arcs’ and ‘representative angles’ on the sphere are ‘intimately connected’ with versors though ‘distinct from them’. This approach may have been deemed particularly appropriate since these Lectures followed on ones devoted to astronomy where spherical geometry figured prominently. Thus the product of two versors is represented ‘by the external vertical angle of a spherical triangle, whose base angles, taken in a determined order, represent those two versors themselves’ (p. 385). In Lecture VI, for example, he describes the ‘symbol of operation’ q( )q −1 ‘in which q may be said to be the operating quaternion, as denoting the operation of causing the arc which represents the operand quaternion, and whose symbol is supposed to be inserted within the parentheses, to move along the DOUBLED ARC of the operator, without any change of either length or inclination (like the equator on the ecliptic in precession)’ (p. xxviii). Lecture VI also contains the general proof of associativity of multiplication. Finally Lecture VII introduces addition of the various entities thus far introduced. For example, the addition of a scalar and a vector is shown to be a quaternion. Hamilton first justifies this for the case of a unit scalar and a unit vector by considering 1 + k. If each term is multiplied on the right by i the results are i and ki = j . Thus 1 + k = (i + ki) ÷ i = (i + j ) ÷ i,
(2)
and this last expression has a meaning that has been already established, namely the quotient of two vectors which has been shown to be a quaternion (pp. 387–388). Conversely, it is shown that a quaternion, q, is decomposable into a scalar and a vector. The operations of taking the scalar and vector are written as Sq and Vq respectively. The vector of the product of two vectors is shown to have length equal to the area of the parallelogram formed by the two vectors and a direction perpendicular to the plane of the parallelogram (the modern cross product). The product changes sign if the factors are interchanged (pp. 416–417). It is only in art. 450 that the ‘quadronomial form’ is formally introduced whereby a quaternion can be expressed in general as a sum of four terms, q = w + ix + jy + kz, where w, x, y, and z are numbers. If a second quaternion is written as q = w + ix + jy + kz then their sum or difference is formed by the following: q ± q = (w ± w ) + i(x ± x ) + j (y ± y ) + k(z ± z ).
(3)
The two quaternions are equal if and only if the system of four equations holds: w = w , x = x , y = y , and z = z . In spite of this introduction of an algebraic approach, the presentation remains geometrically oriented. Many illustrations of the use of quaternions to represent geometric figures and their intersections, in particular the conics and their surfaces
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of revolution such as the ellipsoid, are given. Relations to trigonometry and goniometry (functions of angles) are developed. In the middle of Lecture VII is a brief account of the form in which Hamilton originally discovered and expressed the quaternions (arts. 530–536). The ‘quadronomial’ form was central in that early stage and the geometrical connections developed gradually from it. He cites here the contribution made by his friend J.T. Graves in propagating some of these early results. Functions of quaternions are also introduced in Lecture VII, including exponential and logarithmic. Hamilton describes what is now referred to as the nabla or del operator, which he had introduced in 1846, and the Laplace operator: 2 d d d d2 d2 d +j + k , and ∇ 2 = − + + . (4) ∇ =i dx dy dz dx 2 dy 2 dz2 √ Hamilton constructs what he called ‘biquaternions’, entities of the form q + −1q √ where q and q are ‘real quaternions’ and the −1 is ‘the old and ordinary imaginary of algebra’ (p. 638). Two non-zero biquaternions may have a product of zero. (The term ‘biquaternion’ was to be used later by W.K. Clifford in a different sense.) The Lecture includes discussion of connections of quaternions with coordinates, determinants, trigonometry, series, linear and quadratic equations, differentials, integration, and continued fractions. Additional examples are given of quaternion representations of the differential geometry of curves and surfaces in three-dimensions. 4
RECEPTION AND SUBSEQUENT DEVELOPMENT
The Lectures probably did not sell many copies, but at least Hamilton had his printing costs largely covered by a grant of £300 from Trinity College. In itself the work probably cannot be regarded as a significant influence. Many, if not most, of the topics covered in the Lectures were previously published by Hamilton in journal articles. Though the Lectures did go beyond these publications, the fact that the subject matter was not regarded as new may help to explain why no special note was taken of it in the literature when it first appeared. Hamilton realized that the Lectures were, in spite of his original intentions, not suitable as an introduction for the beginner and that a new plan was called for. We know that one of England’s most renowned scientists of the time, John Herschel, in spite of repeated efforts to make his way through it, only managed the first three Lectures [Hankins, 1980, 359–360]. Hamilton thus started on the Elements of quaternions, which grew as he worked on it from a small manual to a tome of over 800 pages when it finally came to print after his death, thanks to his son William Edwin Hamilton. A second edition, with notes and appendices by his colleague C.J. Joly, appeared in two volumes in 1899 and 1901. The Lectures was also supplanted by P.G. Tait’s Elementary treatise on quaternions in 1867. Tait (1831–1901) was educated in mathematics at Cambridge University and, though his main interest was in physics, became Hamilton’s closest follower and advocate. His treatise appeared in two further editions and was translated into French and German. An even more elementary Introduction to quaternions appeared in 1873 as a joint work with P. Kelland and went through several editions. In 1905 Joly felt there was a need for A manual of
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quaternions wherein he reluctantly ‘abandoned Hamilton’s methods of establishing the laws of quaternions’ while diplomatically recognizing that Hamilton’s Lectures ‘have a charm all their own’ [Joly, 1905, v]. The Lectures thus stand less as an influence than as an historically important record of Hamilton’s intentions at a key stage of development of his quaternions. 5 QUATERNIONS VERSUS VECTORS: J.W. GIBBS AND E.B. WILSON A major impetus to the propagation of quaternions came from James Clerk Maxwell’s use of them in his Treatise on electricity and magnetism (1873) (§44). It was from reading Maxwell that the British scientist Oliver Heaviside (1850–1925) and the U.S. mathematical physicist Josiah Willard Gibbs (1839–1903) of Yale University came independently to critically study quaternions and to develop an alternate system, vector analysis. Gibbs was also influenced by Grassmann’s calculus of extension, first published in 1844 [Gibbs, 1891]. His lithographed pamphlet Elements of vector analysis (1881–1884) was privately printed but received rather wide circulation, even abroad in Europe. Heaviside’s work, initially published in the journal Electrician in 1882 and 1883, was less well known (compare §49), and Gibbs became the main target of the quaternion supporters. Their theme was set by Tait who, in 1890 in the Preface to the third edition of his Elementary treatise, stated that ‘Gibbs must be ranked as one of the retarders of Quaternion progress, in virtue of his pamphlet on Vector analysis; a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann’. The controversy between the vector and quaternion camps is unusual in the history of mathematics in its intensity and international scope, comparable to the dispute between the followers of Isaac Newton and G.W. Leibniz over the origins and best form of the calculus. In addition to many publications from both sides, quaternionists founded an International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics, which published bulletins between 1900 and 1913 [Crowe, 1967]. It should be noted that the dispute was not over the crediting of discoveries; Gibbs and other vector adherents claimed only to have a better way of achieving the same useful applications. In particular they noted that the functional usefulness of many of Hamilton’s operators, such as the scalar and vector operators, S and V, could be obtained more easily without introducing the quaternion. The modest size (83 pages) and compact style of writing in Gibbs’s work stand in contrast to Hamilton’s overwhelming prolixity. Furthermore, Gibbs never took the time to develop his pamphlet into a textbook in spite of the increasing popularity of his system. Instead this task fell to a former student at Yale, Edwin Bidwell Wilson (1879–1964). Wilson had studied quaternions as an undergraduate at Harvard University under J.M. Peirce. In building upon what were in effect Gibbs’s lecture notes, Wilson also drew upon other works, including Heaviside’s, to produce a book of 436 pages that set the pattern, with respect to notation and use, for virtually all subsequent works in vector analysis. His Vector analysis appeared, with a preface by Gibbs, in 1901 and was soon followed by several further printings. Initially published by Scribner’s in New York, after Yale University Press was founded in 1908 it produced the second edition in 1909, incorporating corrections, and
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several subsequent printings up to the Dover reprint of 1960. In spite of this substantial enlargement over Gibbs’s booklet, Wilson’s work was still designed as a textbook and not a treatise on the state of the subject. Each of its seven chapters included exercises. A comparison with the content of Hamilton’s Lectures can only be made indirectly since Wilson’s principal source, Gibbs, appears to have been informed about quaternions mainly through Tait’s Treatise [Crowe, 1967, 155–158]. Wilson opens with definitions of vector and scalar quantities and the basic operations between them, paying careful attention to naming and symbolizing conventions. He uses, for example, bold letters (or ‘Clarendon type’ as he terms it) for a vector and ordinary type for the same letter for its scalar magnitude. (Heaviside had employed this practice and name earlier: see §49.2) Three mutually perpendicular unit vectors, i, j, k and vectors as linear combinations of these are introduced. Thus a connection to the Cartesian rectangular coordinate system is immediately established. In Chapter II the direct and skew products of vectors appear, written A · B and A × B for vectors A and B, which have since taken on the names of dot and cross product respectively. Their correspondence to Hamilton’s scalar and vector components of the product of two quaternions a and b, Sab and Vab, would have been obvious to a reader versed in quaternions. Chapters III and IV deal with the differential and integral calculus of vectors, and define the notions of derivative, divergence, curl, and scalar and vector potentials. Hamilton is credited with the introduction of the ∇ symbol for derivative—one of the few passages where Hamilton’s work is explicitly mentioned. Linear functions of vectors are the subject of Chapter V. A key concept is the ‘dyad’ defined as a juxtaposition of two vectors, as in ab. Taking the dot product on the right with a vector r produces another vector r = ab · r that is, in this example, the product of a vector a and a scalar. The dyad plays a key role in the remaining two chapters which concern applications in mathematical physics and geometry, the main motivation for the subject as far as Gibbs and Wilson were concerned. There is a Section on the propagation of light in crystals that may have helped make a link to the ongoing discussions resulting from the Michelson–Morley experiments in the United States on the nature of the aether. Rotations and strains are represented by dyadic expressions (i.e. linear combinations of dyads). In particular, a dyadic reducible to the form i i + j j + k k , where each of the triples i , j , k and i, j, k are right-handed rectangular systems of unit vectors, represents a rotation and is called a ‘versor’. Here and elsewhere Hamilton’s terminology is echoed. One Section of the last chapter is devoted to the representation of quadric surfaces by means of dyadics. Another Section, on curvature of surfaces, is exceptional in that it makes more use of pure vectors than of dyadics. Wilson included all the topics covered by Gibbs except applications to crystallography and the theory of orbits, topics to which Gibbs devoted much attention. Nevertheless, as he describes in his reminiscences of Gibbs [Wilson, 1931], they had virtually no interaction regarding the preparation of the book. Of the four reviews of Wilson’s work cited in [Crowe, 1967, 229], only one was unfavorable, claiming that the work should have been quaternionic. That reviewer asserted that the dyad’s strong operational resemblance to the quaternion undermined any claim that the ‘new’ methods were really new. Furthermore, the dyad lacked the ‘geometric significance’ of the quaternion [Knott, 1902]. Most readers appear to have understood that indeed very little of this was essentially new. However, dyads were soon encompassed in the theory of
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matrices while the role of quaternions in mathematics evolved into something somewhat less grand than Hamilton envisioned. BIBLIOGRAPHY Crowe, M.J. 1967. A history of vector analysis: The evolution of the idea of a vectorial system, Notre Dame: University of Notre Dame Press. [Rev. repr. New York: Dover, 1985.] Gibbs, J.W. Papers. The scientific papers of J. Willard Gibbs, 2 vols., London: Longmans, Green. [Repr. New York: Dover, 1961.] Gibbs, J.W. 1881–1884. Elements of vector analysis arranged for the use of students in physics, New Haven, Connecticut: privately printed by Tuttle, Morehouse & Taylor. [Repr. in [Gibbs, Papers], vol. 2, 17–90.] Gibbs, J.W. 1891. ‘Quaternions and the “Ausdehnungslehre” ’, Nature, 44, 79–82. [Repr. in [Gibbs, Papers], vol. 2, 161–168.] Graves, R.P. 1882–1891. Life of Sir William Rowan Hamilton, including selections from his poems, correspondence, and miscellaneous writings, 3 vols., Dublin: Hodges, Figgis; London: Longmans, Green. Hamilton, W.R. Papers. The mathematical papers of Sir William Rowan Hamilton, 4 vols. (ed. various), Cambridge: Cambridge University Press, 1931–2000. Hamilton, W.R. 1866. Elements of quaternions (ed. W.E. Hamilton), London: Longmans, Green. [2nd ed. (ed. C.J. Joly), 2 vols., 1899–1901; repr. New York: Chelsea, 1969.] Hankins, T.L. 1980. Sir William Rowan Hamilton, Baltimore and London: The Johns Hopkins University Press. Joly, C.J. 1905. A manual of quaternions, London and New York: Macmillan. Knott, C.G. 1902. Review of [Wilson, 1901], Philosophical magazine, (6) 4, 614–622. Pickering, A. 1995. The mangle of practice: time, agency, and science, Chicago: University of Chicago Press. Tait, P.G. 1867. Elementary treatise on quaternions, Oxford: Clarendon Press. Wilson, E.B. 1901. Vector analysis; a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs, New York: Scribner’s; London: Edward Arnold. [Repr. New York: Dover, 1960.] Wilson, E.B. 1931. ‘Reminiscences of Gibbs by a student and colleague’, The scientific monthly, 32, 210–227.
WILLIAM ROWAN HAMILTON, LECTURES ON QUATERNIONS (1853) Albert C. Lewis
This, the first book devoted to quaternions, appeared ten years after their discovery by Hamilton. Later, many of his most useful concepts were separated from their quaternion context and were reformulated as a part of vector analysis. The key work in this transformation was E.B. Wilson’s Vector analysis (1901). First publication. Lectures on quaternions containing a systematic statement of a new mathematical method of which the principles were communicated in 1843 to the Royal Irish Academy: and which has since formed the subject of successive courses of lectures delivered in 1848 and subsequent years, in the halls of Trinity College, Dublin: with numerous illustrative diagrams, and with some geometrical and physical applications, Dublin: Hodges and Smith; London: Whittaker and Co.; Cambridge: Macmillan and Co., 1853. lxxii + 736 pages. Reprint. Cornell Library Digital Collections (http://historical.library.cornell.edu). [Preface only in Mathematical papers, vol. 3.] Related articles: Grassmann (§32), Heaviside (§49). 1 FROM PRODIGY TO SAGE Sir William Rowan Hamilton (1805–1865), born and raised in Ireland, was one of the most brilliant students to have passed through Trinity College, Dublin. He mastered many languages, ancient and modern. He appears to have been largely self-taught in mathematics though guided by a tutor. At Trinity from 1823 to 1827 he was exposed to the newest mathematics that emanated mainly from France, especially by P.S. Laplace, J.L. Lagrange, S.D. Poisson, and S.F. Lacroix. He had barely completed his intended program of studies when he was offered the prestigious appointment of Astronomer Royal of Ireland. Though he was not inclined to the practical aspects required for the job, he gained the aid of his four Landmark Writings in Western Mathematics, 1640–1940 I. Grattan-Guinness (Editor) © 2005 Elsevier B.V. All rights reserved. 460
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sisters who lived with him at the Dunsink Observatory and who managed the household and performed many if not most of the astronomical duties; and the post provided more time for his mathematical researches than the conventional teaching position for which he originally aimed. Hamilton’s initial fame beyond Dublin came from work on systems of light rays done already in 1824. The experimental verification of his prediction of the conical refraction of light in 1833 brought him the highest honors, including a knighthood in 1835. Another enduring work from this early period of his life was what came to be called the ‘Hamiltonian function’, which has proven of fundamental importance in physics. It was also during this period that he began the pursuit of the elusive triple number system that led to his discovery of quaternions in 1843 and to the book featured here. Though quaternions, with their promise of so many fruitful applications, was to take much of his attention in later life, he devoted himself to many other matters. From 1837 to 1846 Hamilton was president of the Royal Irish Academy. Far from being an exclusively honorary post, this entailed an intimate involvement in its administration and in the development of wide-ranging projects relating to Irish history and culture. After he resigned this post he was widely praised for his achievements in the Academy, not least for his diplomatic skill at resolving disputes among members. He was spurred by the discovery of the planet Neptune in 1846 to study perturbation theory. One result was his invention of the hodograph, an elegant geometrical representation of planetary paths which was taken up with some interest by William Thomson (Lord Kelvin) among others. Hamilton later learned that A.F. Möbius had discovered the notion earlier, as he acknowledged in the Lectures (p. 614). In 1856 Hamilton became interested in an entirely different subject, which he termed ‘the Icosian Calculus’. This was an algebra capable of describing the paths connecting the vertices of a dodecahedron. From this came the general idea of determining what have come to be known as ‘Hamilton circuits’. Further details can be found in the principal secondary sources on his life [Graves, 1882–1891] and [Hankins, 1980]. 2 THE ORIGIN OF QUATERNIONS During the 1830s Hamilton maintained an interest in a problem that a number of mathematicians regarded as one of the most important unsolved issues of the time: How can the system of number pairs, represented by complex numbers, be extended to triples of numbers in such a way as to preserve the same operational properties? For example, a complex number z = a + bi can be represented in the Euclidean plane by the directed line segment√ from the origin to the point with real number coordinates (a, b). On multiplying z by i (= −1 ), the result would be the segment from the origin to (−b, a) which can be regarded as the result of rotating the original segment 90◦ counterclockwise about the origin. The problem could thus be put in a geometrical and somewhat more general fashion: How can this mathematical operation, represented by rotation about a point in the plane, be extended to rotation about a line in three dimensions? Expressed this way the answer turns out to be that four numbers, not three, are required. In his early work Hamilton assumed, not unnaturally, that the task was one of finding the appropriate system of triples of numbers, and it was only after many unsuccessful ef-
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forts that the possibility of a quadruple of numbers presented itself. In one of the famous moments in the history of mathematics, the idea came to him—as he carefully recorded immediately afterwards—while on Brougham Bridge in Dublin, on his way to attend a Council meeting of the Academy on 16 October 1843. His construction consisted of adjoining three entities, i, j, k, to the real numbers. These can be multiplied according to the rules which he jotted down at the time in a notebook: i 2 = j 2 = k 2 = −1,
ij = k,
j k = i,
ki = j
and ij k = −1.
(1)
The hypercomplex number can be formed as a + bi + cj + dk, where a, b, c, and d are real numbers. Though this new number, Hamilton’s quaternion, suggests that it might best be suited for a four-dimensional spatial representation—and indeed in the early 20th century there were attempts to make use of it in relativity theory—Hamilton himself exploited it for a wide range of three-dimensional applications, his original motivation. Before taking up Hamilton’s development of quaternions in the Lectures proper, mention should be made of his path of discovery which he described at length and in several places, including in the Preface of the Lectures. In his view the discovery is intimately tied to his notion of algebra as the science of pure time. As he describes it in the Preface, he tended to approach the whole subject less in a ‘symbolical’ fashion than in a ‘scientific’ fashion. Influenced by ‘the Kantian parallelism between the intuitions of Time and Space’, and by geometry as the science of space, he felt that viewing algebra as the science of pure time had a high suggestive value that could easily lead to a purely symbolical calculus if and when one chose to follow that symbolical route. Hamilton’s detailed documentation of his creative path has been the basis of several historical analyses, some of which attempt to use it to draw lessons about the nature of mathematical discovery in general [Hankins, 1980, ch. 6; Pickering, 1995, ch. 4]. There are a number of predecessors for Hamilton’s work whom he acknowledges in the Preface. The most significant one for later developments is ‘the very original and remarkable work’ of H.G. Grassmann whose Ausdehnungslehre or calculus of extension of 1844 (§32) he read just as the Lectures was being completed. Hamilton noted that, though Grassmann had a non-commutative multiplication of directed lines, he was not in possession of quaternions since he admitted to not succeeding in extending the complex numbers to three dimensions or in building a theory of angles in space (Preface, p. 62). It seems that Hamilton had quaternions predominately on his mind as he read Grassmann and overlooked the fact that, as later readers recognized, Grassmann’s response to these issues was considerably more general than his own. 3 THE LECTURES After the inspiration of October 1843 Hamilton published a number of very substantial papers over the next ten years describing the new entities, including two series of papers, one in eighteen installments in the Philosophical magazine and another, left incomplete after ten installments, in the Cambridge and Dublin mathematical journal. (These papers are reprinted in the edition [Hamilton, Papers].) In 1848 he conducted a series of lectures at Trinity College, Dublin, and these formed the basis for his Lectures volume of 1853. While
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this was a most productive time for Hamilton with respect to developing and propagating the quaternions, he was also undergoing some traumatic personal experiences: his favorite sister, Eliza, died in 1851, and he had been following closely the health and well being of his greatest love in life, Catherine Disney Barlow, who attempted suicide in 1848. One of his biographers, Thomas Hankins, paints a picture of someone who was clearly deeply affected by these and other personal setbacks but who could appear totally unfazed by them when it came to carrying on with his work. The contents of Hamilton’s Lectures are summarised in Table 1. His path from consideration of progression in time, through working with number triplets, to quaternions, is given with substantial technical detail in the Preface. In fact, the Lectures themselves have been, as he puts it, ‘drawn up in a more popular style than this Preface’ and were intended, at least initially, to be fairly faithful to what was actually presented by Hamilton ‘in successive years, in the Halls of this University’. However, as the table of contents reveals, the lengths of the ‘Lectures’ increased steadily, culminating in Lecture VII, which is over 300 pages long. As he admitted, substantially more ‘calculation’ was added than would actually have been presented in the lecture hall. Nevertheless, he maintains that ‘something of Table 1. Contents by Lectures of Hamilton’s book. Parentheses around the Preface’s page numbers distinguish them from the main body. Square brackets indicate an unnumbered page. Lecture Page Sect. Preface, pp. ([1])–(64) Contents, pp. [ix]–lxxi Lec. I 1 i Lec. II 33 vi
Art.
Lec. III Lec. IV
74 130
xi xxvi
79 121
Lec. V
186
xxxvi
175
Lec. VI
241
xlvi
251
Lec. VII
381
lxi
394
App. A
701
App. B
717
App. C
731
Errata, unnumbered leaf
1 37
Contents (sample topics) Time, number triplets, quaternions. [No pages correspond to numbers i to viii.] Addition and subtraction of lines and points. Multiplication and division in geometry; squares and products of i, j, k. The quaternion; tensor and versor. √ Powers and roots of quaternions; −1 as a partially indeterminate symbol. Multiplication of three lines in space; value of ij k and kj i. General associative property of multiplication; spherical representations. Addition and subtraction; distributive principle of multiplication. [End cxvii, 689.] Gauche (i.e. non-planar) polygons inscribed in second-order surfaces. [Paper published in 1850.] Gauche polygons inscribed in second-order surfaces. [Paper published in 1849.] A ‘rapid outline of the quaternion analysis’. [End 736.] Thirty-seven errata, most quite minor.
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the style of actual lecturing has been here and there retained’ throughout (Preface, p. 63). Apparently to help the reader, if not the author, to gain control over the growing mass of material, Section numbers came into play at a later stage, ‘too late to be incorporated into the text’ as Hamilton writes (p. lxxi). Thus the 117 Sections, which act as an intermediate level of division between the 7 Lectures and 689 articles, occur only in the Table. There is no index, but the copious table of contents serves as an analytic guide. This glimpse into what was evidently a struggle to keep the book under control illustrates a general trait: as biographer Hankins put it, ‘Hamilton could not keep his published works on quaternions within reasonable bounds’ [Hankins, 1980, 365]. Hamilton’s previous course of lectures was an introduction to astronomy and his first lecture in this new series—evidently the only one in the Lectures to adhere at all closely to what he might have actually said in the lecture hall—makes the transition by drawing upon relative positions of planets to introduce the notion of a difference of points as an ordinal expression of relative position. ‘And because, according to the foregoing illustrations, this sign or mark (Minus) directs us to DRAW, or to conceive as drawn, a straight line connecting the two points, which are proposed to be compared as to their relative positions, it might, perhaps, on this account be called the SIGN OF TRACTION’ (p. 10). This sentence succinctly exhibits something of Hamilton’s style. Also, it shows his care in giving new concepts correspondingly new names even at the risk of overloading the reader. In this case, in the interest of reducing the number of new terms, ‘subtraction’ is soon used instead. In the next dozen pages, however, more terms come into play—almost at the rate of one per page—such as ‘vection’, ‘revection’, ‘provection’, and ‘transvection’ to describe various possible motions of a point along a line. Besides the subtraction of two points, the other key notion of the first Lecture is the geometrical meaning of sum of a line and a point. If B − A is conceived as the line from A to B, then the sum (B − A) + A results in the point B. The second Lecture concerns a general division and multiplication that are the analytic and synthetic cardinal operations corresponding to the ordinal operations of subtraction and addition introduced in the previous Lecture. The term ‘cardinal’ comes from the analogy that, given an expression such as β = n + n (Greek letters will represent directed line segments), we can ordinarily regard the quotient β ÷ n as the cardinal number 2. The defining expressions are: β ÷ α = q and q × α = β. This last equation shows how the quotient q can be regarded as an operator that produces one directed line segment from another. If q is a ‘tensor’, or signless number, then it affects only the length of α. If q is a sign (+ or −) it changes the direction of α. If it is a real number then q may have the effect of changing both the direction and length. If it is a ‘vector-unit’ (or ‘quadrantal versor’), i, j, k, then the effect is to turn α right-handedly through 90 degrees in a plane perpendicular to the vector-unit. Hamilton points out that a multiplication of a vector-unit, say i, by itself results in a rotation of 180◦ , i.e. the same as multiplying by −1 (reversing its direction) or i 2 = −1. Continuing with further examples in Lecture III, Hamilton broadens the conception of multiplication to include any two vectors (directed line segments) and also introduces exponentiation of vectors. He shows that these operations, as well as the quotient, q, of two vectors described above, can be characterized by four numbers, namely the tensor (a pure number or scalar, written as Tq) and three directions. This entitles the result to be called a
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quaternion. Most of the previous examples can thus be recognized at this stage as special cases of quaternions. Lectures IV and V begin to reveal the nature of the quaternion itself and further properties of quaternion multiplication. Demonstrations of the non-commutative and associative properties of multiplication are given for more general cases, though not yet in the most general case. (A demonstration of its distributivity over addition is also promised, but this is delayed until Lecture VII as Hamilton feels the need to explore multiplication more before taking up addition.) His arguments throughout are geometrical. He makes use of ‘arcual constructions’ in which ‘representative arcs’ and ‘representative angles’ on the sphere are ‘intimately connected’ with versors though ‘distinct from them’. This approach may have been deemed particularly appropriate since these Lectures followed on ones devoted to astronomy where spherical geometry figured prominently. Thus the product of two versors is represented ‘by the external vertical angle of a spherical triangle, whose base angles, taken in a determined order, represent those two versors themselves’ (p. 385). In Lecture VI, for example, he describes the ‘symbol of operation’ q( )q −1 ‘in which q may be said to be the operating quaternion, as denoting the operation of causing the arc which represents the operand quaternion, and whose symbol is supposed to be inserted within the parentheses, to move along the DOUBLED ARC of the operator, without any change of either length or inclination (like the equator on the ecliptic in precession)’ (p. xxviii). Lecture VI also contains the general proof of associativity of multiplication. Finally Lecture VII introduces addition of the various entities thus far introduced. For example, the addition of a scalar and a vector is shown to be a quaternion. Hamilton first justifies this for the case of a unit scalar and a unit vector by considering 1 + k. If each term is multiplied on the right by i the results are i and ki = j . Thus 1 + k = (i + ki) ÷ i = (i + j ) ÷ i,
(2)
and this last expression has a meaning that has been already established, namely the quotient of two vectors which has been shown to be a quaternion (pp. 387–388). Conversely, it is shown that a quaternion, q, is decomposable into a scalar and a vector. The operations of taking the scalar and vector are written as Sq and Vq respectively. The vector of the product of two vectors is shown to have length equal to the area of the parallelogram formed by the two vectors and a direction perpendicular to the plane of the parallelogram (the modern cross product). The product changes sign if the factors are interchanged (pp. 416–417). It is only in art. 450 that the ‘quadronomial form’ is formally introduced whereby a quaternion can be expressed in general as a sum of four terms, q = w + ix + jy + kz, where w, x, y, and z are numbers. If a second quaternion is written as q = w + ix + jy + kz then their sum or difference is formed by the following: q ± q = (w ± w ) + i(x ± x ) + j (y ± y ) + k(z ± z ).
(3)
The two quaternions are equal if and only if the system of four equations holds: w = w , x = x , y = y , and z = z . In spite of this introduction of an algebraic approach, the presentation remains geometrically oriented. Many illustrations of the use of quaternions to represent geometric figures and their intersections, in particular the conics and their surfaces
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of revolution such as the ellipsoid, are given. Relations to trigonometry and goniometry (functions of angles) are developed. In the middle of Lecture VII is a brief account of the form in which Hamilton originally discovered and expressed the quaternions (arts. 530–536). The ‘quadronomial’ form was central in that early stage and the geometrical connections developed gradually from it. He cites here the contribution made by his friend J.T. Graves in propagating some of these early results. Functions of quaternions are also introduced in Lecture VII, including exponential and logarithmic. Hamilton describes what is now referred to as the nabla or del operator, which he had introduced in 1846, and the Laplace operator: 2 d d d d2 d2 d +j + k , and ∇ 2 = − + + . (4) ∇ =i dx dy dz dx 2 dy 2 dz2 √ Hamilton constructs what he called ‘biquaternions’, entities of the form q + −1q √ where q and q are ‘real quaternions’ and the −1 is ‘the old and ordinary imaginary of algebra’ (p. 638). Two non-zero biquaternions may have a product of zero. (The term ‘biquaternion’ was to be used later by W.K. Clifford in a different sense.) The Lecture includes discussion of connections of quaternions with coordinates, determinants, trigonometry, series, linear and quadratic equations, differentials, integration, and continued fractions. Additional examples are given of quaternion representations of the differential geometry of curves and surfaces in three-dimensions. 4
RECEPTION AND SUBSEQUENT DEVELOPMENT
The Lectures probably did not sell many copies, but at least Hamilton had his printing costs largely covered by a grant of £300 from Trinity College. In itself the work probably cannot be regarded as a significant influence. Many, if not most, of the topics covered in the Lectures were previously published by Hamilton in journal articles. Though the Lectures did go beyond these publications, the fact that the subject matter was not regarded as new may help to explain why no special note was taken of it in the literature when it first appeared. Hamilton realized that the Lectures were, in spite of his original intentions, not suitable as an introduction for the beginner and that a new plan was called for. We know that one of England’s most renowned scientists of the time, John Herschel, in spite of repeated efforts to make his way through it, only managed the first three Lectures [Hankins, 1980, 359–360]. Hamilton thus started on the Elements of quaternions, which grew as he worked on it from a small manual to a tome of over 800 pages when it finally came to print after his death, thanks to his son William Edwin Hamilton. A second edition, with notes and appendices by his colleague C.J. Joly, appeared in two volumes in 1899 and 1901. The Lectures was also supplanted by P.G. Tait’s Elementary treatise on quaternions in 1867. Tait (1831–1901) was educated in mathematics at Cambridge University and, though his main interest was in physics, became Hamilton’s closest follower and advocate. His treatise appeared in two further editions and was translated into French and German. An even more elementary Introduction to quaternions appeared in 1873 as a joint work with P. Kelland and went through several editions. In 1905 Joly felt there was a need for A manual of
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quaternions wherein he reluctantly ‘abandoned Hamilton’s methods of establishing the laws of quaternions’ while diplomatically recognizing that Hamilton’s Lectures ‘have a charm all their own’ [Joly, 1905, v]. The Lectures thus stand less as an influence than as an historically important record of Hamilton’s intentions at a key stage of development of his quaternions. 5 QUATERNIONS VERSUS VECTORS: J.W. GIBBS AND E.B. WILSON A major impetus to the propagation of quaternions came from James Clerk Maxwell’s use of them in his Treatise on electricity and magnetism (1873) (§44). It was from reading Maxwell that the British scientist Oliver Heaviside (1850–1925) and the U.S. mathematical physicist Josiah Willard Gibbs (1839–1903) of Yale University came independently to critically study quaternions and to develop an alternate system, vector analysis. Gibbs was also influenced by Grassmann’s calculus of extension, first published in 1844 [Gibbs, 1891]. His lithographed pamphlet Elements of vector analysis (1881–1884) was privately printed but received rather wide circulation, even abroad in Europe. Heaviside’s work, initially published in the journal Electrician in 1882 and 1883, was less well known (compare §49), and Gibbs became the main target of the quaternion supporters. Their theme was set by Tait who, in 1890 in the Preface to the third edition of his Elementary treatise, stated that ‘Gibbs must be ranked as one of the retarders of Quaternion progress, in virtue of his pamphlet on Vector analysis; a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann’. The controversy between the vector and quaternion camps is unusual in the history of mathematics in its intensity and international scope, comparable to the dispute between the followers of Isaac Newton and G.W. Leibniz over the origins and best form of the calculus. In addition to many publications from both sides, quaternionists founded an International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics, which published bulletins between 1900 and 1913 [Crowe, 1967]. It should be noted that the dispute was not over the crediting of discoveries; Gibbs and other vector adherents claimed only to have a better way of achieving the same useful applications. In particular they noted that the functional usefulness of many of Hamilton’s operators, such as the scalar and vector operators, S and V, could be obtained more easily without introducing the quaternion. The modest size (83 pages) and compact style of writing in Gibbs’s work stand in contrast to Hamilton’s overwhelming prolixity. Furthermore, Gibbs never took the time to develop his pamphlet into a textbook in spite of the increasing popularity of his system. Instead this task fell to a former student at Yale, Edwin Bidwell Wilson (1879–1964). Wilson had studied quaternions as an undergraduate at Harvard University under J.M. Peirce. In building upon what were in effect Gibbs’s lecture notes, Wilson also drew upon other works, including Heaviside’s, to produce a book of 436 pages that set the pattern, with respect to notation and use, for virtually all subsequent works in vector analysis. His Vector analysis appeared, with a preface by Gibbs, in 1901 and was soon followed by several further printings. Initially published by Scribner’s in New York, after Yale University Press was founded in 1908 it produced the second edition in 1909, incorporating corrections, and
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several subsequent printings up to the Dover reprint of 1960. In spite of this substantial enlargement over Gibbs’s booklet, Wilson’s work was still designed as a textbook and not a treatise on the state of the subject. Each of its seven chapters included exercises. A comparison with the content of Hamilton’s Lectures can only be made indirectly since Wilson’s principal source, Gibbs, appears to have been informed about quaternions mainly through Tait’s Treatise [Crowe, 1967, 155–158]. Wilson opens with definitions of vector and scalar quantities and the basic operations between them, paying careful attention to naming and symbolizing conventions. He uses, for example, bold letters (or ‘Clarendon type’ as he terms it) for a vector and ordinary type for the same letter for its scalar magnitude. (Heaviside had employed this practice and name earlier: see §49.2) Three mutually perpendicular unit vectors, i, j, k and vectors as linear combinations of these are introduced. Thus a connection to the Cartesian rectangular coordinate system is immediately established. In Chapter II the direct and skew products of vectors appear, written A · B and A × B for vectors A and B, which have since taken on the names of dot and cross product respectively. Their correspondence to Hamilton’s scalar and vector components of the product of two quaternions a and b, Sab and Vab, would have been obvious to a reader versed in quaternions. Chapters III and IV deal with the differential and integral calculus of vectors, and define the notions of derivative, divergence, curl, and scalar and vector potentials. Hamilton is credited with the introduction of the ∇ symbol for derivative—one of the few passages where Hamilton’s work is explicitly mentioned. Linear functions of vectors are the subject of Chapter V. A key concept is the ‘dyad’ defined as a juxtaposition of two vectors, as in ab. Taking the dot product on the right with a vector r produces another vector r = ab · r that is, in this example, the product of a vector a and a scalar. The dyad plays a key role in the remaining two chapters which concern applications in mathematical physics and geometry, the main motivation for the subject as far as Gibbs and Wilson were concerned. There is a Section on the propagation of light in crystals that may have helped make a link to the ongoing discussions resulting from the Michelson–Morley experiments in the United States on the nature of the aether. Rotations and strains are represented by dyadic expressions (i.e. linear combinations of dyads). In particular, a dyadic reducible to the form i i + j j + k k , where each of the triples i , j , k and i, j, k are right-handed rectangular systems of unit vectors, represents a rotation and is called a ‘versor’. Here and elsewhere Hamilton’s terminology is echoed. One Section of the last chapter is devoted to the representation of quadric surfaces by means of dyadics. Another Section, on curvature of surfaces, is exceptional in that it makes more use of pure vectors than of dyadics. Wilson included all the topics covered by Gibbs except applications to crystallography and the theory of orbits, topics to which Gibbs devoted much attention. Nevertheless, as he describes in his reminiscences of Gibbs [Wilson, 1931], they had virtually no interaction regarding the preparation of the book. Of the four reviews of Wilson’s work cited in [Crowe, 1967, 229], only one was unfavorable, claiming that the work should have been quaternionic. That reviewer asserted that the dyad’s strong operational resemblance to the quaternion undermined any claim that the ‘new’ methods were really new. Furthermore, the dyad lacked the ‘geometric significance’ of the quaternion [Knott, 1902]. Most readers appear to have understood that indeed very little of this was essentially new. However, dyads were soon encompassed in the theory of
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matrices while the role of quaternions in mathematics evolved into something somewhat less grand than Hamilton envisioned. BIBLIOGRAPHY Crowe, M.J. 1967. A history of vector analysis: The evolution of the idea of a vectorial system, Notre Dame: University of Notre Dame Press. [Rev. repr. New York: Dover, 1985.] Gibbs, J.W. Papers. The scientific papers of J. Willard Gibbs, 2 vols., London: Longmans, Green. [Repr. New York: Dover, 1961.] Gibbs, J.W. 1881–1884. Elements of vector analysis arranged for the use of students in physics, New Haven, Connecticut: privately printed by Tuttle, Morehouse & Taylor. [Repr. in [Gibbs, Papers], vol. 2, 17–90.] Gibbs, J.W. 1891. ‘Quaternions and the “Ausdehnungslehre” ’, Nature, 44, 79–82. [Repr. in [Gibbs, Papers], vol. 2, 161–168.] Graves, R.P. 1882–1891. Life of Sir William Rowan Hamilton, including selections from his poems, correspondence, and miscellaneous writings, 3 vols., Dublin: Hodges, Figgis; London: Longmans, Green. Hamilton, W.R. Papers. The mathematical papers of Sir William Rowan Hamilton, 4 vols. (ed. various), Cambridge: Cambridge University Press, 1931–2000. Hamilton, W.R. 1866. Elements of quaternions (ed. W.E. Hamilton), London: Longmans, Green. [2nd ed. (ed. C.J. Joly), 2 vols., 1899–1901; repr. New York: Chelsea, 1969.] Hankins, T.L. 1980. Sir William Rowan Hamilton, Baltimore and London: The Johns Hopkins University Press. Joly, C.J. 1905. A manual of quaternions, London and New York: Macmillan. Knott, C.G. 1902. Review of [Wilson, 1901], Philosophical magazine, (6) 4, 614–622. Pickering, A. 1995. The mangle of practice: time, agency, and science, Chicago: University of Chicago Press. Tait, P.G. 1867. Elementary treatise on quaternions, Oxford: Clarendon Press. Wilson, E.B. 1901. Vector analysis; a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs, New York: Scribner’s; London: Edward Arnold. [Repr. New York: Dover, 1960.] Wilson, E.B. 1931. ‘Reminiscences of Gibbs by a student and colleague’, The scientific monthly, 32, 210–227.