Hamilton's quaternions
A number system with one real and three imaginary units. Conceived as an algebra that would handle three-dimensional space the way complex numbers handle the plane.
It was found by the Irishman William Rowan Hamilton on 16 October 1843 during a walk. A British school formed around quaternions: Maxwell, too, used quaternions in his book of 1873. But by 1910 physics had moved to the vector analysis of Gibbs and Heaviside, obtained from quaternions by discarding their central feature: the union of scalar and vector parts into a single product.
Crucially, quaternions were never refuted; they were displaced by a more convenient notation, while the algebra itself stayed valid. In the twentieth century they returned as a working tool for describing rotations: in computer animation after Shoemake's 1985 paper, then in robotics and in spacecraft attitude control, where they suffer no gimbal lock.
Why this date
The dating follows Hamilton's letter to John T. Graves of 17 October 1843, which describes the discovery as having happened the day before, that is 16 October 1843; the letter was printed in Philosophical Magazine XXV (1844), 489-95, where the paper 'On quaternions; or on a new system of imaginaries in algebra' also appeared (1844), 10-13. Crowe dates the discovery to the same day. The year is counted from the discovery and first announcement, not from the later books.
What grew out of it
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Vector analysis and the vector form of the field equations
improved · Confidence: high
The vector analysis of Gibbs and Heaviside was obtained by directly simplifying the quaternion calculus: both came to it through quaternion texts and both discarded the union of scalar and vector parts into a single product, after which quaternions were displaced from physics without ever being refuted.
Dispute: Quaternions were never shown to be false: they were displaced by a more convenient notation, while the algebra itself stayed valid. Historians differ on whether vector analysis was a direct descendant of quaternions. Crowe documents the actual route: both Gibbs and Heaviside came to vectors through the quaternion texts of Tait and Maxwell. Heaviside himself, by contrast, claimed that vector analysis could have been derived by any mathematician directly from the mechanics of Cartesian mathematics, without quaternions at all, and on that reckoning the dependence would be indirect only.
Link sources
- Oliver Heaviside, quoted in Bruce J. Hunt, Oliver Heaviside: A First-Rate Oddity, Physics Today 65 (11), 48-54 (2012) Verified 20 Sep 2026
I dropped out the quaternion altogether, and kept to pure scalars and vectors, using a very simple vectorial algebra in my papers from 1883 onward.
- J. Willard Gibbs, quoted in Michael J. Crowe, A History of Vector Analysis (lecture text), p. 12 (2002) Verified 20 Sep 2026
where Quaternion notations are considerably used, I became convinced that to master those subjects, it was necessary for me to commence by mastering those methods.
- Michael J. Crowe, A History of Vector Analysis (lecture text), p. 16 (2002) Verified 20 Sep 2026
by 1910, the Gibbs-Heaviside, i.e., the modern system had won out in the struggle for existence
- Simon L. Altmann, Hamilton, Rodrigues, and the Quaternion Scandal, Mathematics Magazine 62 (5), 291-308, p. 306 (1989)
nothing that they did, however, succeeded in preventing the rise of vectors and the consequent decline of quaternions
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Oliver Heaviside, quoted in Bruce J. Hunt, Oliver Heaviside: A First-Rate Oddity, Physics Today 65 (11), 48-54 (2012) Verified 20 Sep 2026
Node sources
- W. R. Hamilton, Letter to John T. Graves on Quaternions (17 October 1843), School of Mathematics, Trinity College Dublin (1843) Verified 20 Sep 2026
the notion that we must admit, in some sense, a fourth dimension of space
- Michael J. Crowe, A History of Vector Analysis (lecture text) (2002) Verified 20 Sep 2026
On 16 October 1843, his search ends with his discovery of mathematical entities he calls 'quaternions.'
- Michael J. Crowe, A History of Vector Analysis (lecture text), p. 16 (2002) Verified 20 Sep 2026
by 1910, the Gibbs-Heaviside, i.e., the modern system had won out in the struggle for existence
- Simon L. Altmann, Hamilton, Rodrigues, and the Quaternion Scandal, Mathematics Magazine 62 (5), 291-308, p. 306 (1989)
nothing that they did, however, succeeded in preventing the rise of vectors and the consequent decline of quaternions
- Quaternion, Encyclopaedia Britannica (2024) Verified 24 Sep 2026
have been mostly replaced in practice by operations with matrices and vectors
- Ken Shoemake, Animating rotation with quaternion curves, SIGGRAPH '85, Computer Graphics 19 (3), 245-254 (1985)
The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper.
- Danail Brezov, The Tragic Downfall and Peculiar Revival of Quaternions, Mathematics 13 (4), 637 (2025)
Quaternions were to be torn out of physics and nearly forgotten for about a century
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.