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{{Short description|none}}
], ], which says: <br><small>Here as he walked by<br> on the 16th of October 1843<br> Sir William Rowan Hamilton<br> in a flash of genius discovered<br> the fundamental formula for quaternion multiplication<br> i² = j² = k² = i j k = &minus;1<br> & cut it on a stone of this bridge.</small>]] ], ], 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 ], ]s are a non-] number system that extends the ]s. Quaternions and their applications to rotations were first described in print by ] in all but name in 1840,<ref>{{cite magazine | author = Simon L. Altmann | title = Hamilton, Rodrigues and the quaternion scandal |magazine=] | year = 1989 | volume = 62 | pages = 291–308 | doi = 10.2307/2689481 | jstor = 2689481 | issue = 5 }}</ref> but independently discovered by Irish mathematician Sir ] 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.
⚫ ]s were introduced by Irish mathematician Sir ] in 1843. Hamilton knew that the ]s could be viewed as ]s in a ], and he was looking for a way to do the same for points in ]. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and multiply triples of numbers. But he had been stuck on the problem of division: He did not know how to take the quotient of two points in space.


== Hamilton's discovery ==
⚫ On October 16, 1843, Hamilton and his wife took a walk along the ] in ]. While they walked across Brougham Bridge (now ]), a solution suddenly occurred to him. He could not divide triples, but he could divide ''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:
⚫ In 1843, Hamilton knew that the ]s could be viewed as ]s in a ] 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 ]. 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:
:<math>i^2 = j^2 = k^2 = ijk = -1.</math>
<blockquote>Every morning in the early part of October 1843, on my coming down to breakfast, your brother ] 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 ] in ]. While they walked across Brougham Bridge (now ]), 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:
Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted the remainder of his life to studying and teaching them. He founded a school of "quaternionists" and popularized them in several books. The last and longest, ''Elements of Quaternions'', had 800 pages and was published shortly after his death.
:{{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 '']'' communicated Hamilton's exposition of quaternions.<ref>W.R. Hamilton(1844 to 1850) , ], link to David R. Wilkins collection at ]</ref> In 1853 he issued ''Lectures on Quaternions'', a comprehensive treatise that also described ]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 ]s.
After Hamilton's death, his pupil ] continued promoting quaternions. At this time, quaternions were a mandatory examination topic in Dublin, and in some American universities they were the only advanced mathematics topic taught. Topics in physics and geometry that would now be described using vectors, such as ] in space and ], were described entirely in terms of quaternions. There was even a professional research association, the ], exclusively devoted to the study of quaternions.


== Precursors ==
From the mid 1880s, quaternions began to be displaced by ], which had been developed by ] and ]. Vector analysis described the same phenomena as quaternions, so it borrowed ideas and terms liberally from the classical quaternion literature. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side effect of this transition is that works on ] are difficult to comprehend for many modern readers because they use familiar terms from vector analysis in unfamiliar and fundamentally different ways.
Hamilton's innovation consisted of expressing quaternions as an ]. The formulae for the multiplication of quaternions are implicit in the ] devised by ] in 1748. In 1840, ] 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>] & Derek A. Smith (2003) ''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry'', ], {{ISBN|1-56881-134-9}}</ref>{{rp|9}}


== Response ==
However, quaternions have had a revival in the late 20th century, primarily due to their utility in describing spatial rotations. Representations of rotations by quaternions are more compact and faster to compute than representations by matrices. For this reason, quaternions are used in ],<ref>] (1985), , '']'', '''19'''(3), 245-254. Presented at ] '85. <br />'']'' (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth 3D rotation. See eg Nick Bobick, "", ''Game Developer'' magazine, February 1998</ref> ], ], ], ], ], and ]. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. Quaternions have received another boost from ] because of their relation to ].
The special claims of quaternions as the algebra of ] were challenged by ] with his exhibits in 1848 and 1849 of ]s and ]s as alternatives. Nevertheless, these new algebras from Cockle were, in fact, to be found inside Hamilton's ]s. From Italy, in 1858 ] responded<ref>] ( 1858) , link from HathiTrust</ref> to connect Hamilton's vector theory with his theory of ]s of directed line segments.


] led the response from France in 1874 with a textbook on the elements of quaternions. To ease the study of ]s, he introduced "biradials" to designate great circle arcs on the sphere. Then the quaternion algebra provided the foundation for ] 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>}}.
⚫ Since 1989, the Department of Mathematics of the ] has organized a pilgrimage, where scientists (including physicists ] in 2002, ] in 2005, and mathematician ] in 2003) take a walk from ] to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.


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
⚫ ==References==
:<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 ]<ref>] (1881) , link from ]</ref> and ].<ref>A. Macfarlane (1894) '''', B. Westerman, New York, weblink from ]</ref>

] expanded the types of biquaternions, and explored ], a geometry in which the points can be viewed as versors. Fascination with quaternions began before the language of ] and ]s was available. In fact, there was little ] before the ]. The quaternions stimulated these advances: For example, the idea of a ] 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 ]. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of ]s and ]s in ], for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus ] and ] made this accommodation, for pragmatism, to avoid the distracting superstructure.<ref>] (1967) ], ]</ref>

For mathematicians the quaternion structure became familiar and lost its status as something mathematically interesting. Thus in England, when ] prepared a paper on biquaternions, it was published in the ] since some novelty in the subject lingered there. Research turned to ]s more generally. For instance, ] and ] 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 ]. In contemporary mathematics, the ] of quaternions exemplifies an ].

== Principal publications ==
* 1853 ''Lectures on Quaternions''<ref>, Royal Irish Academy, weblink from ] ''Historical Math Monographs''</ref>
* 1866 ''Elements of Quaternions''<ref>, ] Press. Edited by ], son of the deceased author</ref>
* 1873 ''Elementary Treatise'' by ]<ref></ref>
* 1874 ]: ''Éléments de la Théorie des Quaternions''<ref>J. Hoüel (1874) , Gauthier-Villars publisher, link from ]</ref>
* 1878 ]: Quadrics: Harvard dissertation:<ref>] (1878) , ] 13:222–50, from ]</ref>
* 1882 Tait and ]: ''Introduction with Examples''<ref></ref>
* 1885 ]: Biquaternions<ref>, ] 7(4):293 to 326 from ] early content</ref>
* 1887 Valentin Balbin: (Spanish) ''Elementos de Calculo de los Cuaterniones'', Buenos Aires<ref>Gustav Plarr (1887) in ]</ref>
* 1899 ]: ''Elements'' vol 1, vol 2 1901<ref>Hamilton (1899) ''Elements of Quaternions'' , (1901) . Edited by ]; published by ], now in ]</ref>
* 1901 ] by ] and ] (quaternion ideas without quaternions)
* 1904 ]: third edition of Kelland and Tait's textbook<ref>] (editor) (1904) via ]</ref>
* 1904 ''Bibliography'' prepared for the ] by ]<ref>] (1904) , weblink from Cornell University ''Historical Math Monographs''</ref>
* 1905 C.J. Joly's ''Manual for Quaternions''<ref>] (1905) (1905), originally published by ], now from Cornell University Historical Math Monographs</ref>
* 1940 ] 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 ]s to describe screw displacement in ].

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

]s were developed independently by ] in 1845 <ref>Penrose 2004 pg 202</ref> and ], 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 , 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 ]ed ].<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 ], 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 ]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></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 ], which follows directly from the product rule from octonions. It had also been previously discovered as a purely algebraic identity by ] in 1818.{{Sfn|Baez|2002|p=146-7}} This sum-of-squares identity is characteristic of ], 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 ] of ] systems over the reals, followed by the ]s, the ]s, the ]s; they are also a useful tool in ], particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the ], is also the simplest non-commutative ].

The study of ]s began with ] in 1886, whose system was later simplified by ]; but the modern system was published by ] 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 ]. Both systems are closed under subtraction and multiplication, and are therefore ]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 ]s of three- and four-dimensional spaces. They have the technical advantage that ]s form the ] cover of the space of three-dimensional rotations.<ref name=CS/>{{rp|ch 2}}

For this reason, quaternions are used in ],<ref>Ken Shoemake (1985), , '']'', '''19'''(3), 245–254. Presented at ] '85.</ref> ], ],<ref>J. M. McCarthy, 1990, , MIT Press</ref> ], ], ], ], and ]. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions. '']'' (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) "", ]</ref> Quaternions have received another boost from ] because of their relation to ]s.

== Memorial ==
⚫ Since 1989, the Department of Mathematics of the ] has organized a pilgrimage, where scientists (including physicists ] in 2002, ] in 2005, ] in 2007, and mathematician ] in 2003) take a walk from ] to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.<ref> at the ].</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}}
* ] and ], ''Introduction to Number Theory''. Many editions.
* Johannes C. Familton (2015) , Ph.D. thesis in ] Department of Mathematics Education.

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

]

Latest revision as of 00:09, 28 December 2024

Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says:
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 multiplicationi = j = k = ijk = −1& cut it on a stone of this bridge.

In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, 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 numbers could be viewed as points in a 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:

Every morning in the early part of October 1843, on my coming down to breakfast, your brother 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."

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:

i = j = k = 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. In 1853 he issued Lectures on Quaternions, a comprehensive treatise that also described biquaternions. 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 quaternions.

Precursors

Hamilton's innovation consisted of expressing quaternions as an algebra over 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.

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 tessarines and coquaternions as alternatives. Nevertheless, these new algebras from Cockle were, in fact, to be found inside Hamilton's biquaternions. From Italy, in 1858 Giusto Bellavitis responded to connect Hamilton's vector theory with his theory of equipollences 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 versors, 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 i, j, k with i1, i2, and i3.

The variety of fonts available led Hoüel to another notational innovation: A designates a point, a and a are algebraic quantities, and in the equation for a quaternion

A = cos ⁡ α + A sin ⁡ α , {\displaystyle {\mathcal {A}}=\cos \alpha +\mathbf {A} \sin \alpha ,}

A is a vector and α is an angle. This style of quaternion exposition was perpetuated by Charles-Ange Laisant and Alexander Macfarlane.

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 structures 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 philosophical object. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of dot products and cross products 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.

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 numbers 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

Octonions

Main article: Octonion

Octonions were developed independently by Arthur Cayley in 1845 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 , why should you stop there?"

Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843, presenting a kind of double quaternion, which he called octaves, and showed that they were what we now call a normed division algebra. Hamilton observed in reply that they were not 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 biquaternions 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.

Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845 – 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 – or as Cayley numbers.

The major deduction from the existence of octonions was the 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. This sum-of-squares identity is characteristic of composition algebra, a feature of complex numbers, quaternions, and octonions.

Mathematical uses

Main article: 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 octonions, the sedenions, the trigintaduonions; 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 quaternions 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 rings, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.

Quaternions as rotations

Main article: Quaternions and spatial rotation

Quaternions are a concise method of representing the automorphisms of three- and four-dimensional spaces. They have the technical advantage that unit quaternions form the simply connected cover of the space of three-dimensional rotations.

For this reason, quaternions are used in computer graphics, control theory, robotics, signal processing, 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. Quaternions have received another boost from number theory because of their relation to quadratic forms.

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.

References

Notes

  1. Simon L. Altmann (1989). "Hamilton, Rodrigues and the quaternion scandal". Mathematics Magazine. Vol. 62, no. 5. pp. 291–308. doi:10.2307/2689481. JSTOR 2689481.
  2. W.R. Hamilton(1844 to 1850) On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, link to David R. Wilkins collection at Trinity College Dublin
  3. Friedberg, Richard M. (2022). "Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un système solide...", translation and commentary". arXiv:2211.07787.
  4. ^ John H. Conway & Derek A. Smith (2003) On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, ISBN 1-56881-134-9
  5. Giusto Bellavitis ( 1858) Calcolo dei Quaternioni di W.R. Hamilton e sua Relazione col Metodo delle Equipollenze, link from HathiTrust
  6. Charles Laisant (1881) Introduction a la Méthode des Quaternions, link from Google Books
  7. A. Macfarlane (1894) Papers on Space Analysis, B. Westerman, New York, weblink from archive.org
  8. Michael J. Crowe (1967) A History of Vector Analysis, University of Notre Dame Press
  9. Lectures on Quaternions, Royal Irish Academy, weblink from Cornell University Historical Math Monographs
  10. Elements of Quaternions, University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author
  11. Elementary Treatise on Quaternions
  12. J. Hoüel (1874) Éléments de la Théorie des Quaternions, Gauthier-Villars publisher, link from Google Books
  13. Abbott Lawrence Lowell (1878) 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
  14. Introduction to Quaternions with Numerous Examples
  15. "A Memoir on biquaternions", American Journal of Mathematics 7(4):293 to 326 from Jstor early content
  16. Gustav Plarr (1887) Review of Valentin Balbin's Elementos de Calculo de los Cuaterniones in Nature
  17. Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co., now in Internet Archive
  18. C. G. Knott (editor) (1904) Introduction to Quaternions, 3rd edition via Hathi Trust
  19. Alexander Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, weblink from Cornell University Historical Math Monographs
  20. Charles Jasper Joly (1905) A Manual for Quaternions (1905), originally published by Macmillan Publishers, now from Cornell University Historical Math Monographs
  21. Penrose 2004 pg 202
  22. Baez 2002, p. 146.
  23. See Penrose Road to Reality pg. 202 'Graves discovered that there exists a kind of double quaternion...'
  24. Brown, Ezra; Rice, Adrian (2022), "An accessible proof of Hurwitz's sums of squares theorem", Mathematics Magazine, 95 (5): 422–436, doi:10.1080/0025570X.2022.2125254, MR 4522169
  25. 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
  26. See Hamilton's talk to the Royal Irish Academy on the subject
  27. Baez 2002, p. 146-7.
  28. Hardy and Wright, Introduction to Number Theory, §20.6-10n (pp. 315–316, 1968 ed.)
  29. Ken Shoemake (1985), Animating Rotation with Quaternion Curves, Computer Graphics, 19(3), 245–254. Presented at SIGGRAPH '85.
  30. J. M. McCarthy, 1990, Introduction to Theoretical Kinematics, MIT Press
  31. Nick Bobick (February 1998) "Rotating Objects Using Quaternions", Game Developer (magazine)
  32. Hamilton walk at the National University of Ireland, Maynooth.
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