View source for History of quaternions

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[[File:Inscription on Broom Bridge (Dublin) regarding the discovery of Quaternions multiplication by Sir William Rowan Hamilton.jpg|right|thumb|Quaternion plaque on [[Broom Bridge|Brougham (Broom) Bridge]], [[Dublin]], which says:<blockquote style="margin:0px;"><div>Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication<div style="white-space:nowrap;">{{math|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = <span style="font-style:italic;letter-spacing:2px;">ijk</span> = −1}}</div>& cut it on a stone of this bridge.</div></blockquote>]]

In [[mathematics]], [[quaternion]]s are a non-[[Commutativity|commutative]] number system that extends the [[complex number]]s. Quaternions and their applications to rotations were first described in print by [[Olinde Rodrigues]] in all but name in 1840,<ref>{{cite magazine | author = Simon L. Altmann | title = Hamilton, Rodrigues and the quaternion scandal |magazine=[[Mathematics Magazine]] | year = 1989 | volume = 62 | pages = 291–308 | doi = 10.2307/2689481 | jstor = 2689481 | issue = 5 }}</ref> but independently discovered by Irish mathematician Sir [[William Rowan Hamilton]] in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.

== Hamilton's discovery ==
In 1843, Hamilton knew that the [[complex number]]s could be viewed as [[point (geometry)|point]]s in a [[plane (mathematics)|plane]] and that they could be added and multiplied together using certain geometric operations.  Hamilton sought to find a way to do the same for points in [[space]]. Points in space can be represented by their coordinates, which are triples of numbers and have an obvious addition, but Hamilton had difficulty defining the appropriate multiplication.

According to a letter Hamilton wrote later to his son Archibald:
<blockquote>Every morning in the early part of October 1843, on my coming down to breakfast, your brother [[William Edwin Hamilton|William Edwin]] and yourself used to ask me: "Well, Papa, can you multiply triples?"  Whereto I was always obliged to reply, with a sad shake of the head, "No, I can only add and subtract them."</blockquote>

On October 16, 1843, Hamilton and his wife took a walk along the [[Royal Canal]] in [[Dublin]]. While they walked across Brougham Bridge (now [[Broom Bridge]]), a solution suddenly occurred to him. While he could not "multiply triples", he saw a way to do so for ''quadruples''. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:
:{{math|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = −1}}

Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted the remainder of his life to studying and teaching them. From 1844 to 1850 ''[[Philosophical Magazine]]'' communicated Hamilton's exposition of quaternions.<ref>W.R. Hamilton(1844 to 1850) [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/ On quaternions or a new system of imaginaries in algebra], [[Philosophical Magazine]], link to David R. Wilkins collection at [[Trinity College Dublin]]</ref> In 1853 he issued ''Lectures on Quaternions'', a comprehensive treatise that also described [[biquaternion]]s. The facility of the algebra in expressing geometric relationships led to broad acceptance of the method, several compositions by other authors, and stimulation of applied algebra generally. As mathematical terminology has grown since that time, and usage of some terms has changed, the traditional expressions are referred to as [[classical Hamiltonian quaternion]]s.

== Precursors ==
Hamilton's innovation consisted of expressing quaternions as an [[algebra over a field|algebra over {{math|'''R'''}}]]. The formulae for the multiplication of quaternions are implicit in the [[four squares formula]] devised by [[Leonhard Euler]] in 1748. In 1840, [[Olinde Rodrigues]] used spherical trigonometry and developed a formula closely related to quaternion multiplication in order to describe the new axis and angle of two combined rotations.<ref>{{cite arxiv | arxiv=2211.07787 | title=Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un système solide...", translation and commentary | first=Richard M. | last=Friedberg | authorlink = Richard M. Friedberg | year=2022}}</ref><ref name=CS>[[John H. Conway]] & Derek A. Smith (2003) ''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry'', [[A K Peters]], {{ISBN|1-56881-134-9}}</ref>{{rp|9}}

== Response ==
The special claims of quaternions as the algebra of [[four-dimensional space]] were challenged by [[James Cockle]] with his exhibits in 1848 and 1849 of [[tessarine]]s and [[coquaternion]]s as alternatives. Nevertheless, these new algebras from Cockle were, in fact, [[subalgebra]]s of Hamilton's [[biquaternion]]s. From Italy, in 1858 [[Giusto Bellavitis]] responded<ref>[[Giusto Bellavitis]] ( 1858) [https://archive.today/20141101193025/http://babel.hathitrust.org/cgi/pt?id=pst.000003356951;view=1up;seq=1 Calcolo dei Quaternioni di W.R. Hamilton e sua Relazione col Metodo delle Equipollenze], link from HathiTrust</ref> to connect Hamilton's vector theory with his theory of [[equipollence (geometry)|equipollence]]s of directed line segments.

[[Jules Hoüel]] led the response from France in 1874 with a textbook on the elements of quaternions. To ease the study of [[versor]]s, he introduced "biradials" to designate great circle arcs on the sphere. Then the quaternion algebra provided the foundation for [[spherical trigonometry]]  introduced in chapter 9. Hoüel replaced Hamilton's basis vectors {{math|'''i'''}}, {{math|'''j'''}}, {{math|'''k'''}} with {{math|''i''<sub>1</sub>}}, {{math|''i''<sub>2</sub>}}, and {{math|''i''<sub>3</sub>}}.

The variety of fonts available led Hoüel to another notational innovation: {{math|''A''}} designates a point, {{math|''a''}} and {{math|a}} are algebraic quantities, and in the equation for a quaternion
:<math>\mathcal{ A} = \cos \alpha + \mathbf{A} \sin \alpha ,</math>
{{math|'''A'''}} is a vector and {{math|''α''}} is an angle. This style of quaternion exposition was perpetuated by [[Charles-Ange Laisant]]<ref>[[Charles Laisant]]  (1881) [https://books.google.com/books?id=WvMGAAAAYAAJ Introduction a la Méthode des Quaternions], link from [[Google Books]]</ref> and [[Alexander Macfarlane]].<ref>A. Macfarlane (1894) ''[https://archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from [[archive.org]]</ref>

[[William K. Clifford]] expanded the types of biquaternions, and explored [[elliptic space]], a geometry in which the points can be viewed as versors. Fascination with quaternions began before the language of [[set theory]] and [[mathematical structure]]s was available. In fact, there was little [[mathematical notation]] before the [[Formulario mathematico]]. The quaternions stimulated these advances: For example, the idea of a [[vector space]] borrowed Hamilton's term but changed its meaning. Under the modern understanding, any quaternion is a vector in four-dimensional space. (Hamilton's vectors lie in the subspace with scalar part zero.)

Since quaternions demand their readers to imagine four dimensions, there is a metaphysical aspect to their invocation. Quaternions are a [[object (philosophy)|philosophical object]]. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of [[dot product]]s and [[cross product]]s in [[three-dimensional space]], for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus [[Willard Gibbs]] and [[Oliver Heaviside]] made this accommodation, for pragmatism, to avoid the distracting superstructure.<ref>[[Michael J. Crowe]] (1967) [[A History of Vector Analysis]], [[University of Notre Dame Press]]</ref>

For mathematicians the quaternion structure became familiar and lost its status as something mathematically interesting. Thus in England, when [[Arthur Buchheim]] prepared a paper on biquaternions, it was published in the [[American Journal of Mathematics]] since some novelty in the subject lingered there.  Research turned to [[hypercomplex number]]s more generally. For instance, [[Thomas Kirkman]] and [[Arthur Cayley]] considered the number of equations between basis vectors which would be necessary to determine a unique system. The wide interest that quaternions aroused around the world resulted in the [[Quaternion Society]]. In contemporary mathematics, the [[division ring]] of quaternions exemplifies an [[algebra over a field]].

== Principal publications ==
* 1853 ''Lectures on Quaternions''<ref>[http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 Lectures on Quaternions], Royal Irish Academy, weblink from [[Cornell University]] ''Historical Math Monographs''</ref>
* 1866 ''Elements of Quaternions''<ref>[https://books.google.com/books?id=fIRAAAAAIAAJ Elements of Quaternions], [[University of Dublin]] Press. Edited by [[William Edwin Hamilton]], son of the deceased author</ref>
* 1873 ''Elementary Treatise'' by [[Peter Guthrie Tait]]<ref>[https://books.google.com/books?id=TwkXAAAAYAAJ Elementary Treatise on Quaternions]</ref>
* 1874 [[Jules Hoüel]]: ''Éléments de la Théorie des Quaternions''<ref>J. Hoüel (1874) [https://books.google.com/books?id=tisLAAAAYAAJ Éléments de la Théorie des Quaternions], Gauthier-Villars publisher, link from [[Google Books]]</ref> 
* 1878 [[Abbott Lawrence Lowell]]:  Quadrics: Harvard dissertation:<ref>[[Abbott Lawrence Lowell]] (1878) [https://www.biodiversitylibrary.org/item/35671#page/248/mode/1up Surfaces of the second order, as treated by quaternions], [[Proceedings of the American Academy of Arts and Sciences]] 13:222–50, from [[Biodiversity Heritage Library]]</ref>
* 1882 Tait and [[Philip Kelland]]: ''Introduction with Examples''<ref>[https://books.google.com/books?id=xxsFAAAAMAAJ Introduction to Quaternions with Numerous Examples]</ref>
* 1885 [[Arthur Buchheim]]:  Biquaternions<ref>[https://www.jstor.org/stable/2369176  "A Memoir on biquaternions"], [[American Journal of Mathematics]] 7(4):293 to 326 from [[Jstor]] early  content</ref> 
* 1887 Valentin Balbin: (Spanish) ''Elementos de Calculo de los Cuaterniones'', Buenos Aires<ref>Gustav Plarr (1887) [http://www.nature.com/nature/journal/v37/n946/pdf/037145b0.pdf Review of Valentin Balbin's ''Elementos de Calculo de los Cuaterniones''] in [[Nature (journal)|Nature]]</ref>
* 1899 [[Charles Jasper Joly]]: ''Elements'' vol 1, vol 2 1901<ref>Hamilton (1899) ''Elements of Quaternions'' [https://archive.org/details/elementsofquater01hamiuoft volume I], (1901) [https://archive.org/details/elementsofquater02hamiuoft volume II]. Edited by [[Charles Jasper Joly]]; published by [[Longmans, Green & Co.]], now in [[Internet Archive]]</ref>
* 1901 [[Vector Analysis]] by [[Willard Gibbs]] and [[Edwin Bidwell Wilson]] (quaternion ideas without quaternions)
* 1904 [[Cargill Gilston Knott]]: third edition of Kelland and Tait's textbook<ref>[[C. G. Knott]] (editor) (1904) [https://catalog.hathitrust.org/Record/008871944 Introduction to Quaternions, 3rd edition] via [[Hathi Trust]]</ref>
* 1904 ''Bibliography'' prepared for the [[Quaternion Society]] by [[Alexander Macfarlane]]<ref>[[Alexander Macfarlane]] (1904) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03030001 Bibliography of Quaternions and Allied Systems of Mathematics], weblink from Cornell University ''Historical Math Monographs''</ref>
* 1905 C.J. Joly's ''Manual for Quaternions''<ref>[[Charles Jasper Joly]] (1905) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=00790001 ''A Manual for Quaternions''] (1905), originally published by [[Macmillan Publishers]], now from Cornell University Historical Math Monographs</ref>
* 1940 [[Julian Coolidge]] in ''A History of Geometrical Methods'', page 261, uses the coordinate-free methods of Hamilton's operators and cites A. L. Lawrence's work at Harvard. Coolidge uses these operators on [[dual quaternion]]s to describe screw displacement in [[kinematics]].

== Octonions ==
{{main|Octonion}}

[[Octonion]]s were developed independently by [[Arthur Cayley]] in 1845 <ref>Penrose 2004 pg 202</ref> and  [[John T. Graves]], a friend of Hamilton's. Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold [the three imaginary units], why should you stop there?"{{Sfn|Baez|2002|p=146}}

Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843, presenting a kind of double quaternion,<ref>See Penrose Road to Reality pg. 202 'Graves discovered that there exists a kind of double quaternion...'</ref> which he called ''octaves'', and showed that they were what we now call a [[norm (mathematics)|norm]]ed [[division algebra]].<ref>{{citation
 | last1 = Brown | first1 = Ezra
 | last2 = Rice | first2 = Adrian
 | doi = 10.1080/0025570X.2022.2125254
 | issue = 5
 | journal = Mathematics Magazine
 | mr = 4522169
 | pages = 422–436
 | title = An accessible proof of Hurwitz's sums of squares theorem
 | volume = 95
 | year = 2022}}</ref> Hamilton observed in reply that they were not [[associativity|associative]], which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it. Hamilton needed a way to distinguish between two different types of double quaternions, the associative [[biquaternion]]s and the octaves.  He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion.<ref>[https://books.google.com/books?id=TCwPAAAAIAAJ&q=octaves+hamilton+thinks  Hamilton 1853 pg 740See a hard copy of Lectures on quaternions, appendix B, half of the hyphenated word double quaternion has been cut off in the online Edition ]</ref><ref>See Hamilton's talk to the Royal Irish Academy on the subject</ref> 

Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845&nbsp;– as an appendix to a paper on a different subject. Hamilton was stung into protesting Graves' priority in discovery, if not publication; nevertheless, octonions are known by the name Cayley gave them&nbsp;– or as ''Cayley numbers''.

The major deduction from the existence of octonions was the [[Degen's eight-square identity|eight squares theorem]], which follows directly from the product rule from octonions. It had also been previously discovered as a purely algebraic identity by [[Carl Ferdinand Degen]] in 1818.{{Sfn|Baez|2002|p=146-7}} This sum-of-squares identity is characteristic of [[composition algebra]], a feature of complex numbers, quaternions, and octonions.

== Mathematical uses ==
{{main|Quaternions}}

Quaternions continued to be a well-studied ''mathematical'' structure in the twentieth century, as the third term in the [[Cayley–Dickson construction]] of [[hypercomplex number]] systems over the reals, followed by the [[octonion]]s, the [[sedenion]]s, the [[trigintaduonion]]s; they are also a useful tool in [[number theory]], particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the [[quaternion group]], is also the simplest non-commutative [[Sylow group]].

The study of [[integral quaternion]]s began with [[Rudolf Lipschitz]] in 1886, whose system was  later simplified by [[Leonard Eugene Dickson]]; but the modern system was published by [[Adolf Hurwitz]] in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions ''all four'' of whose coordinates are [[half-integers]]. Both systems are closed under subtraction and multiplication, and are therefore [[ring (mathematics)|ring]]s, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.<ref>Hardy and Wright, ''Introduction to Number Theory'', §20.6-10''n'' (pp. 315–316, 1968 ed.)</ref>

== Quaternions as rotations ==
{{main| Quaternions and spatial rotation}}

Quaternions are a concise  method of representing the [[automorphism]]s of three- and four-dimensional spaces. They have the technical advantage that [[unit quaternion]]s form the [[simply connected]] cover of the space of three-dimensional rotations.<ref name=CS/>{{rp|ch 2}}

For this reason, quaternions are used in [[computer graphics]],<ref>Ken Shoemake (1985), [https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf Animating Rotation with Quaternion Curves], ''[[Computer Graphics (publication)|Computer Graphics]]'', '''19'''(3), 245–254.  Presented at [[SIGGRAPH]] '85.</ref> [[control theory]], [[robotics]],<ref>J. M. McCarthy, 1990, [https://books.google.com/books?id=glOqQgAACAAJ&q=introduction+to+theoretical+kinematics ''Introduction to Theoretical Kinematics''], MIT Press</ref> [[signal processing]], [[Spacecraft attitude control|attitude control]], [[physics]], [[bioinformatics]], and [[orbital mechanics]]. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. ''[[Tomb Raider]]'' (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation.<ref>Nick Bobick (February 1998) "[http://www.gamasutra.com/features/19980703/quaternions_01.htm Rotating Objects Using Quaternions]", [[Game Developer (magazine)]]</ref> Quaternions have received another boost from [[number theory]] because of their relation to [[quadratic form]]s.

== Memorial ==
Since 1989, the Department of Mathematics of the [[National University of Ireland, Maynooth]] has organized a pilgrimage, where scientists (including physicists [[Murray Gell-Mann]] in 2002, [[Steven Weinberg]] in 2005, [[Frank Wilczek]] in 2007, and mathematician [[Andrew Wiles]] in 2003) take a walk from [[Dunsink Observatory]] to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.<ref>[http://www.maths.nuim.ie/links/hamilton.shtml Hamilton walk] at the [[National University of Ireland, Maynooth]].</ref>

== References ==
* {{citation|first=John C.|last= Baez|title= The Octonions|journal=Bulletin of the American Mathematical Society |series=New Series|volume= 39|issue= 2|pages= 145–205|doi=10.1090/S0273-0979-01-00934-X |year=2002|mr=1886087 |arxiv=math/0105155|s2cid= 586512}}
* [[G. H. Hardy]] and [[E. M. Wright]], ''Introduction to Number Theory''. Many editions.
* Johannes C. Familton (2015) [https://arxiv.org/abs/1504.04885 Quaternions: A History of Complex Non-commutative Rotation Groups in Theoretical Physics], Ph.D. thesis in [[Columbia University]] Department of Mathematics Education.
* {{cite journal |last=Whittaker |first=E.T. | authorlink=E. T. Whittaker |title=The sequence of ideas in the discovery of quaternions |journal=Proceedings of the Royal Irish Academy A |year=1944 |volume=50 |pages=93–98 |jstor=20520633 |url=https://www.jstor.org/stable/20520633}}

== Notes ==
{{reflist|2}}

[[Category:Historical treatment of quaternions]]