### 1. Brief *Vita*

The Irish mathematician, physicist and astronomer William Rowan Hamilton, not to be confused with the Scottish philosopher Sir William Hamilton (1788–1856), was born in Dublin on August 3 or 4, 1805. He died after a relatively quiet life on September 2, 1865 in Dunsink, now belonging to the Greater Dublin Area.

His father was Archibald Rowan Hamilton, a Dublin solicitor. Both Hamilton and his father took one of their Christian names from the most important of his father’s clients, the famous Irish patriot Archibald Hamilton Rowan. As a child William was widely recognized as a prodigy. His knowledge of languages is legendary. He is said to have spoken Hebrew, Latin and Greek at the age of five, at the age of ten a couple of more languages including Persian, Syriac, and Sanskrit (Hankins 1980, 13). During his school days his interest moved to mathematics and science. Already at the age of seventeen he reflected on the relation between the laws of arithmetic and their applications. He stated that all the branches of arithmetic were applied in a much more extensive manner than was contemplated by the inventors of these number systems: “By the introduction of negative and fractional quantities, operations that diminish are included under Addition and Multiplication, and others that increase under Subtraction and Division” (Graves 1882-1891, I, 102).

Hamilton entered Trinity College, Dublin, in 1823. There he was highly honored, being twice awarded an *optime*, an extremely rare judgment of excellence, the first in classics, the second in mathematics. In his second college year he submitted a paper “On Caustics” to the Royal Irish Academy dealing with a special kind of light reflection. The paper was regarded as too abstract and general to be published. A second paper, “Theory of Systems of Rays,” presented to the Academy on April 23, 1827, was immediately recommended for publication. It provided an application of algebra to optics. It was considered “the single most important paper that the Royal Irish Academy had received since its foundation or very likely ever since” (O’Donnell 1983, 60), although its author was only an undergraduate.

Hamilton was still approaching his B.A. exams when he was appointed Astronomer Royal of Ireland and Director of Dunsink Observatory in 1827. This position was connected with a professorship of astronomy at Trinity College, Dublin. He was active in the British Association for the Advancement of Science, being responsible for bringing the annual meeting of the association to Dublin in 1835. On this occasion he was knighted by the Lord Lieutenant (*cf*. Morrell and Thackray 1982, 147), the first Irish scientist to receive this honor. Two years later, in 1837, he became President of the Royal Irish Academy (until 1845), the countries’ most prestigious scientific position. In 1863 he was elected a Foreign Associate of the newly founded National Academy of Science of the United States.

Already his first publications show his interest for applications of algebra which finally led him to new algebras going beyond the traditional view on algebra as dealing with magnitudes in general. His first paper “On Caustics” (1824) sketches an application of algebra to the properties of light, developed in the “Theory of Systems of Rays” (1824, published 1828) and elaborated in three “Supplements” (1830–1833). He introduced a “characteristic function” intended to provide “The most complete and simple definition that could be given of the application of analysis to optics.” The characteristic function contains “the whole of mathematical optics” (Hamilton 1931–2000, I, 17, 168). His approach had the advantage of being independent from the particle-wave dualism of light. This theory enabled him to predict conical refraction in bi-refringent crystals (1832), experimentally verified by Humphrey Lloyd the same year.

His further work in theoretical physics led him to apply his idea of a characteristic function to problems in mechanics. The most important paper in this field was his “On a General Methods in Dynamics” (1834). There he introduced a “Principle Function” and the “Principle of Stationary Action” (“Hamilton Principle”) as new fundamental laws. This extremal principle, in connection with Hermann von Helmholtz’ Principle of Conversation of Energy, was to become an extremely fruitful tool for all of theoretical physics. With its help it was possible to solve all problems of movement of conservative mechanical systems with two partial differential equations. Hamilton’s ideas were taken up and extended by Carl Gustav Jacob Jacobi which led to the Hamilton-Jacobi Equation now indispensable in quantum mechanics and, in the judgment of Erwin Schrödinger (1945, 82), a cornerstone in modern physics.

Among Hamilton’s further innovations were the invention of the hodograph (1846), a plot of the velocity of a particle as a function of time, used for describing planetary orbits. Velocity is represented by a vector from the origin to a point on the hodograph. In 1856 Hamilton created the Icosian Calculus (“Hamilton Game”) dealing with the problem of finding a Hamiltonian circuit along the edges of a dodecahedron, i.e. a path such that every vertex is visited once, no edge is visited twice, and the ending point is the same as the starting point.

Hamilton himself, however, regarded the research on new algebras as his most important innovation in mathematics.

### 2. Towards New Algebras

Hamilton’s contributions to algebra have to be seen in the context of a long discussion on the nature of number and operations on them which had taken place in Great Britain since the seventeenth century (*cf*. Pycior 1976). The powerful extensions of the number system contradicted the view of algebra as the science of number in general, numbers interpreted as standing for magnitudes. Negative, imaginary, and complex numbers were regarded as being constructed by impossible operations with numbers, they were therefore regarded as “impossible numbers.”

Hamilton’s approach was to give up the concept of magnitude and to found his theory on the assumption that algebra was the science of pure time. His first major work on algebra was entitled “Theory of Conjugate Functions or Algebraic Couples, with a Preliminary and Elementary Essay on Algebra as Science of Pure Time” (1837). This paper was divided into three parts. The general introductory remarks were written last. The second section with the essay “On Algebra as the Science of Pure Time” from 1835, the “Theory of Conjugate Functions or Algebraic Couples” formed the third section, composed mostly in 1833. In this last section he developed complex numbers as is still done today, as ordered pairs of real numbers. But he also attempted to establish the metaphysical basis underlying algebra. Hamilton himself reported (1931–2000, III, 117n)that it was Kant’s *Critique of Pure Reason* (1787) that stimulated him to write this work. He had already considered his conception before he read Kant, but he later recognized the closeness of his ideas to those of the German philosopher. His definition of the relations of algebra as successive states of some changing thing or thought, and numbers as names or nouns of algebra which help to remember those successive states and to distinguish them from one another (Notebook entry “Metaphysical Remarks on Algebra,” 1827; *cf*. Hankins 1980, 258-59), had its analogue in Kant’s claim in the *Prolegomena *(1783, §10) that arithmetic creates its concepts by successively adding units in time. In his foundational work he was also inspired by Martin Ohm’s *Versuch eines vollkommen consequenten Systems der Mathematik* (1822). The combinatorial analyst Ohm had aimed at a Euclidean-style foundation of all of mathematics based on a distinction between number (or unnamed number) and quantity (or named number) coming close in his theory to the later British symbolical algebra.

Hamilton’s original goal was to investigate the algebra of triplets, i.e., three dimensional complex numbers, a plan that suggests itself as the next following step after having formulated the algebra of couples. It promised to provide a natural mathematics of the three-dimensional space. For years Hamilton struggled with this theory, in particular with the problem of multiplying triplets (proved to be impossible by Ferdinand Georg Frobenius thirteen years after Hamilton’s death).

The solution was to give up triplets and use quadruplets instead—a solution Hamilton found unexpectedly. On October 16, 1843, Hamilton and his wife were walking into Dublin along the Royal Canal to attend a meeting of the Royal Irish Academy. Suddenly he felt, in a flash of inspiration, a primitive solution of his problem. He himself reported, that when passing Brougham Bridge he could not resist cutting with a knife the fundamental formula into a stone of the bridge. He essentially discovered that he could use an obvious system of triplets which had to be extended to a quaternion of four numbers. The basic principle was the “multiplication assumption,” *i*^{2} = *j*^{2} = *k*^{2} = *ijk* = –1. It forced him to give up the law of commutativity because the following laws hold: *ij* = *–ji* = *k*, *jk* = –*kj* = *i*, *ki* = –*ik* = *j*. But exactly this insight—that he could sacrifice commutativity and still gain a meaningful and consistent algebra—was the epoch-making step forward. The quaternions were the first well-known consistent and significant number system which did not obey the laws of ordinary arithmetic. Like the discovery of non-Euclidean geometry before, it “broke bonds set by centuries of mathematical thought” (Crowe 1967, 31).

Hamilton wrote a great number of books and papers developing his new theory over the next ten years. In 1848 he gave a series of lectures on quaternions at Trinity College which developed into his comprehensive book *Lectures on Quaternions* (1853). His attempts to write a brief treatise failed. It grew into his *Elements of Quaternions*, even bigger than the *Lectures*, and unfinished at the time of his death in 1865. The first edition appeared in 1866. It consisted of three books, the first related to the conception of a vector considered as a directed line in a space of three dimensions, the second introduces a first conception of a quaternion considered as the quotient of two vectors, the third book deals with products and powers of vectors considered as a second principle form of the conception of quaternions in geometry.

Quaternions never gained the importance Hamilton had expected. They opened, however, the way for new fruitful directions in mathematical research, e.g., the study of hypercomplex number systems. Moreover, the geometrical properties led to modern vector analysis, and to the more general concept of a linear vector space.

Although Hamilton pushed the development of a structural view on mathematics, he never adopted this view. His algebras were provided with a metaphysical foundation. Having read George Berkeley, Immanuel Kant and Ruggiero Giuseppe Boscovich he tended towards an idealistic interpretation of the universe, rejecting a purely mechanical way of understanding. Samuel Taylor Coleridge became influential for his thought in respect to the idea of trinity or triadic order of the categories of all possible knowledge.

These philosophical influences found their expression in his work on the foundation of algebra. His idealistic approach was opposed to the formal logical approach of the Cambridge Analytical Society around George Peacock (249), although both approaches led into the same direction. The view on algebra changed from algebra as extension of arithmetic, as the science of magnitudes or quantities in general, to algebra as science of operations not necessarily compatible with operations with magnitudes. It offered a more radical solution to the problem of impossible numbers which induced a broad search for general, later new algebras.

The most important movement originated in Cambridge, in the school around George Peacock who codified his symbolical algebra in his *Treatise on Algebra* (1830) and further propagated it in his famous report for the British Association for the Advancement of Science (Peacock 1834, especially 198-207). The starting point of symbolical algebra is the “Principle of the Permanence of Equivalent Forms” which states “Whatever form is algebraically equivalent to another when expressed in general symbols, must continue to be equivalent, whatever those symbols denote” (1834, 198) or, in its converse form:

Whatever equivalent form is discoverable in arithmetical algebra considered as the science of suggestion, when the symbols are general in their form, though specific in their value, will continue to be an equivalent form when the symbols are general in their nature as well as in their form (184, 199).

This principle led to a sharp distinction between arithmetical and symbolical algebra. It was now possible to assign “impossible quantities” their undisputed place in symbolical algebra. Nevertheless, this formal approach adheres to the paradigm of numbers as names of quantities. The formal systems produced were not so general that non-commutative structures were embraced as well. A step in this direction was done with the calculus of operations formulated by Duncan Farquharson Gregory and developed by George Boole. Generalization was reached by applying the calculus to other objects than numbers (understood in the traditional sense), here operations with signs in general (*cf*. Gregory 1840).

A further step was done by George Boole with his algebraization of logic (Boole 1854) which led to an algebraic structure not compatible with the algebra of real numbers, but only with the algebra of 0 and 1, due to the validity of “Boole’s Law,” the Law of Duality *aa* = *a*. A similar approach could be found in German theories of formal structures which proceeded by generalization of the notion of number. As mentioned above, Martin Ohm distinguished between number and quantity. Ernst Schröder’s early “absolute algebra” kept the notion of number largely open. “Number” refers to all objects which are able to constitute a manifold such as “proper names, concepts, judgements, algorithms [sets of derived formulas], numbers [of arithmetic], symbols for magnitudes and operations, points and systems of points, or some geometrical objects, quantities of substances, etc.” (Schröder 1874, 3). Schröder’s work was deeply influenced by Hermann Günther Grassmann’s *Ausdehnungslehre*. This calculus of extension was preceded by a “general theory of forms” dealing with those truths which can be related to all branches of mathematics in the same way, presupposing only the general concepts of equality and difference, combination and separation (*cf*. Grassmann 1844, 1). In these later attempts Peacock’s Principle of the Permanence of Equivalent Forms was given up. So more general algebras became possible, i.e. structures going beyond an algebra of magnitudes.

In effect, these approaches led to results in the spirit of Hamilton’s efforts, although Hamilton differed in his attempt to base his algebras on metaphysical foundations. His primitive intuition was time, not number. Therefore Hamilton was able to create different kinds of numbers, and different kinds of algebraic operations. It opened, thus, much more possibilities than Peacock’s Principle of Permanence of Equivalent Forms.

### 3. Hamilton and Whitehead

Whitehead never became tired of stressing the importance of Hamilton for his first book, *A Treatise on Universal Algebra* (1898). He conceded that the ideas were largely founded on the two versions of Hermann Grassmann’s *Ausdehnungslehre* (1844, 1862), but he stressed that William Rowan Hamilton’s *Quaternions* (1853) and a preliminary paper of 1844 and Boole’s *Symbolic Logic* of 1859 were almost equally influential on his thoughts. The last two references are unclear. Maybe he refers to Hamilton’s long series of papers “On Quaternions” of which three parts were published in 1844 (Hamilton 1844a), or a paper entitled “On a New Species of Imaginary Quantities Connected with the Theory of Quaternions” (1844b). The hint on Boole obviously refers to *An Investigation of the Laws of Thoughts* (Boole 1854). Whitehead states that his subsequent work on mathematical logic came from these sources (*ESP* 10). He regarded Hamilton as one of the founding fathers of modern mathematics and mathematical logic. In a short note on *Process and Reality* he stated:

The modern phases of mathematics or mathematical logic are not modern at all, but arise out of a great past: Grassmann, Sir William Hamilton [i.e., William Rowan Hamilton,] […] Boole, De Morgan, and to go back to the origin of all such efforts, the great Leibniz (

ESP119).

It was the tradition of formal, symbolic, absolute, or general algebra, in which Whitehead positioned his *Universal Algebra*, intended as a “thorough investigation of the various systems of Symbolic Reasoning allied to ordinary Algebra” of which Hamilton’s quaternions, Grassmann’s calculus of extension, and Boole’s Symbolic Logic are the main examples (*UA* v). These algebras were intended to exhibit “both as systems of symbolism, and also as engines for the investigation of the possibilities of thought and reasoning connected with the abstract general idea of space” (*UA* v). A detailed study of quaternions and their comparison with matrices and the general theory of linear algebra was announced for the second volume of the *Universal Algebra*, which never appeared. Like the British symbolical algebraicists, Whitehead did not follow Hamilton’s metaphysical considerations. For him universal algebra “has the same claim to be a serious subject of mathematical study as any other branch of mathematics.” The significance of mathematics in its widest signification is the development of all types of formal, necessary and deductive reasoning—“formal” signifying that the meaning of a proposition does not form part of the investigation (*UA* vi). Although treated formally, ordinary algebra “has relation to almost every event, phenomenal or intellectual” which can occur (*UA* viii). This gives us the significance of ordinary algebra and thus motivates us to investigate new algebras provided they represent in their interpretation “interesting generalizations of important systems of ideas” (*UA* viii). Whitehead characterized these new algebras as being “not essentially concerned with number and quantity; and this bold extension beyond the traditional domain of pure quantity forms their peculiar interest (*UA* viii). At the end of his preface Whitehead again credits Hamilton.

Although the presented volume did not deal with quaternions, “Hamilton must be regarded as the founder of the science of Universal Algebra. He and [Augustus] De Morgan […] are the first to express quite clearly the general possibilities of algebraic symbolism” (*UA* x).

### Works Cited and Further Readings

#### Writings by W.R. Hamilton

1837. “Theory of Conjugate Functions or Algebraic Couples, with a Preliminary Essay on as the Science of Pure Time,” Irish Academy Transactions, XVII, 293-422.

1844a. “On Quaternions,” *Philosophical Magazine* 15, 10-13, 241-246, 489-495.

1844b. “On a New Species of Imaginary Quantities Connected with the Theory of Quaternions,” *Irish Academy Proceedings* 2, 424-34.

1853. *Lectures on Quaternions *(Dublin, Hodges and Smith).

1866.* Elements of Quaternions*, edited by William Edwin Hamilton (London, Longmans, Green). 2nd ed. 1899, 1901.

1931–2000. *The Mathematical Papers of William Rowan Hamilton*. 4 Vols., edited by Heini Halberstam, *et* *al*. (Cambridge, Cambridge University Press).

#### Papers of W.R. Hamilton

Most of Hamilton’s legacy of letters and notebooks are lodged at the archives of Trinity College, Dublin.

#### Further Readings

Anderson, Robert Edward [R.E.A.]. 1921-22 (1890). “Hamilton, Sir William Rowan,” *The Dictionary of National Biography*. Reprint Vol. VIII (Oxford, Oxford University Press), 1119-22.

Boole, George. 1954. *An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities* (London, Walton & Maberly).

Crowe, Michael J. 1967. *A History of Vector Analysis. The Evolution of the Idea of a Vectorial System* (Notre Dame, Notre Dame University Press). Reprint 1994, New York, Dover Publications.

Grassmann, Hermann Günther. 1844. *Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert* (Leipzig, Otto Wigand).

_____. 1862. *Die Ausdehnungslehre. Vollständig und in strenger Form* (Berlin, Th. Chr. Fr. Enslin).

Graves, Robert Perceval. 1882–1992. *Life of Sir William Rowan Hamilton*, 3 Vols. (Dublin, Hodges, Figgis & Co.).

Gregory, Duncan Farquharson. 1840. “On the Real Nature of Symbolical Algebra,” *Transactions of the Royal Society of Edinburgh* 14, 208-216.

Hankins, Thomas L. 1972. “Hamilton, William Rowan,” in *Dictionary of Scientific Biography* Vol. VI, edited by Charles Coulston Gillispie (New York, Scribner’s Sons), 85-93.

_____. 1980. *Sir William Rowan Hamilton* (Baltimore, Johns Hopkins University Press).

Kant, Immanuel. 1783. *Prolegomena zu einer jeden künftigen Metaphysik die als Wissenschaft wird auftreten können* (Riga, Hartknoch).

_____. 1787. *Kritik der reinen Vernunft*, 2nd Ed. (Riga, Hartknoch).

Morrell, Jack and Arnold Thackray. 1982. *Gentlemen of Science. Early Years of the British Association for the Advancement of Science *(Oxford, Oxford University Press).

O’Donnell, Seán. 1983. *William Rowan Hamilton. Portrait of a Prodigy* (Dublin, Boole Press).

Ohm, Martin. 1822. *Versuch eines vollkommen consequenten Systems der Mathematik*, 2 Vols. (Berlin, Reimer).

Peacock, George. 1830. *A Treatise on Algebra* (Cambridge, J. & J.J. Deighton and London, G.F. & J. Rivington).

_____. 1834. “Report on the Recent Progress and Present State of Certain Branches of Analysis,” *Report of the Third Meeting of the British Association for the Advancement of Science held at Cambridge in 1833* (London, John Murray), 185-352.

Peckhaus, Volker. 1997. *Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert* (Berlin, Akademie Verlag), Chapters 5 and 6.

Pycior, Helena M. 1976. *The Role of Sir William Rowan Hamilton in the Development of British Modern Algebra* (Ph.D. thesis, Cornell University).

Schröder, Ernst. 1874. *Über die formalen Elemente der absoluten Algebra*. Supplement to the Calender of the Pro- and Realgymnasium in Baden-Baden for 1873/74 (Stuttgart, Schweizerbart’sche Buchdruckerei).

Schrödinger, Erwin. 1945. “The Hamilton Postage Stamp: An Announcement by the Irish Minister of Posts and Telegraphs,” in *A Collection of Papers in Memory of Sir William Rowan Hamilton*, edited by David Eugene Smith (New York, Scripta Mathematica).

### Author Information

Volker Peckhaus

Institut für Humanwissenschaften: Philosophie

Universität Paderborn, D-33098 Paderborn

www.uni-paderborn.de

volker.peckhaus@upb.de

### How to Cite this Article

Peckhaus, Volker, “William Rowan Hamilton (1805–1865)”, last modified 2008, *The Whitehead Encyclopedia*, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/bios/historico-speculative-context/william-rowan-hamilton/>.