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Frobenius's theorem states that the only finite-dimensional, associative division algebras over $\mathbb R$ are: $\mathbb R, \mathbb C, \mathbb H$ (where the last of these are the quaternions). So one might be led to think that:

  • Hamilton discovered the notion of a finite-dimensional unital algebra over $\mathbb R$. These things are sometimes called hypercomplex number systems.

  • Hamilton discovered that some of these algebras are associative, and some are not.

  • Hamilton discovered that some algebras are division algebras, while most are not.

Putting these together, Hamilton might have set about to find a larger finite-dimensional, associative, division algebra than $\mathbb C$, and eventually arrived at $\mathbb H$. This would be quite impressive as I think that the above three notions did not get studied until after Hamilton presented the quaternions as the first non-trivial hypercomplex number system.

You see, some explanation is badly needed for why Hamilton dismissed straightforward generalisations of $\mathbb C$ like $\mathbb C \oplus \mathbb R$ in his project to generalise $\mathbb C$, and instead settled on a non-commutative 4-dimensional algebra whose defining relation is the arguably cryptic $i^2 = j^2 = k^2 = ijk = -1$. What criteria was he looking to satisfy?

PS: A bonus question would be to explain how Gauss independently discovered the quaternions. Should this be asked as a separate question?

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    $\begingroup$ I don’t think it is correct to say that Hamilton discovered that there are both associative and non-associative algebras. It is true that Hamilton introduced the term associative in an 1844 update to his 1843 paper on quaternions. This update was coincidental with Hamilton’s colleague Graves’ discovery of octonions - a non-associative algebra, unlike Hamilton's quaternions which are associative. This suggests that Graves may have been the first to discover that there were both associative and non-associative algebras. $\endgroup$
    – nwr
    Nov 3, 2021 at 23:01
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    $\begingroup$ Here is a reference for Hamilton's introduction of the term associative. $\endgroup$
    – nwr
    Nov 3, 2021 at 23:03
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    $\begingroup$ None of the above. Hamilton was not thinking in terms of these latter day abstractions. Some were, at best, an afterthought, and the rest did not move him at all. He was looking for a way to "multiply triples", and it hit him how to multiply quadruples instead when he was walking across the Brougham Bridge in October 1843. In excitement he carved the formulas into the stone with his knife, reportedly, and there is a memorial plaque there to mark the occasion. $\endgroup$
    – Conifold
    Nov 4, 2021 at 1:02
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    $\begingroup$ The question is misguided to begin with, Hamilton was interested in geometric interpretation, not abstract algebra. Thinking up refined options made of concepts developed after the fact and then looking for explanations why discoverers "chose" one rather than another is one of the reasons we have in circulation so many concocted myths. History does not unfold according to "rational reconstructions" that people find "natural" two centuries later, it is a messy affair, and it does not offer neat explanations for posterity's benefit. $\endgroup$
    – Conifold
    Nov 4, 2021 at 4:11
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    $\begingroup$ If you are interested in how Hamilton himself described his thought process in notebooks, letters and lectures see van der Waerden, Hamilton's Discovery of Quaternions. He wanted to multiply triplets term by term and have the length of the product vector equal the product of the lengths. In trying to understand historical developments it generally helps to forget about modern concepts, context and what they make "most natural", and look at what was current at the time. The past linked things differently than a backprojection from the present. $\endgroup$
    – Conifold
    Nov 4, 2021 at 21:46

4 Answers 4

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When I have time later I might return to this and try to provide additional exposition directly addressing your question, maybe quote something relevant from Crowe's A History of Vector Analysis (a book I've been reading the past 2 months; I'm now a little more than half-way through it). However, for now I thought it would be useful to provide some older books (and a couple of papers by Hamilton) to look at. As for the books, besides glancing through their table of contents, reading their prefaces, etc., I also recommend reading reviews of these books, which are easy to find by googling the author's last name and the book's title in google-books (date-restrict from the publication date to about 5 years later). I’ve included links to the book reviews I know about.

Every link below is freely available on the internet, at least where I am. Probably all of the books are also freely available online at the Internet Archive (easiest way: google the book's title, maybe author's last name if the title is fairly generic, along with the "word" archive.org), but probably not all of the reviews can be found elsewhere online (freely or otherwise). If you really want to dive into the older literature on quaternions, you’ll want to make use of Macfarlane’s Bibliography of Quaternions and Allied Systems of Mathematics (1904).

1843 On a new species of imaginary quantities connected with a theory of quaternions by Hamilton

1844 Additional researches in the theory of quaternions by Hamilton

1853 Lectures on Quaternions by Hamilton

review in The North American Review (a review known as one of the most over-the-top-with-praise review ever of a math book)

1866 Elements of Quaternions by Hamilton

review in Philosophical Magazine

1867 An Elementary Treatise on Quaternions by Tait

1873 An Elementary Treatise on Quaternions by Tait (2nd edition)

review in Philosophical Magazine

1876 Introduction to Quaternions by Kelland/Tait

review in Nature

1881 Elements of Quaternions by Hardy

review in The Ohio Educational Monthly; review in Wisconsin Journal of Education; review in Science; review in The Westminster Review; review in Popular Science Monthly

1882 Introduction to Quaternions by Kelland/Tait (2nd edition)

1890 An Elementary Treatise on Quaternions by Tait (3rd edition)

1894 The Outlines of Quaternions by Hime

review in Minutes of Proceedings of the Royal Artillery Institution; review in Educational Times; review in Mathematical Gazette; review in The Academy (begins middle of middle column); review in Philosophical Magazine; review in Nature; review in American Mathematical Monthly; review in Science (begins middle of right column)

1896 A Primer of Quaternions by Hathaway

review in The Literary Era; review in Physical Review; review in Science (top-most paragraph on left column of p. 700); review in Electrical Engineering

1899 Elements of Quaternions by Hamilton (2nd edition, Volume I, edited by Joly)

1901 Elements of Quaternions by Hamilton (2nd edition, Volume II, edited by Joly)

review in Nature (Vol. I); review in Philosophical Magazine (Vol. I & II); review in Science (Vol. I & II); review in Nature (Vol. II)

1904 Introduction to Quaternions by Kelland/Tait (3rd edition; prepared by Knott)

review in Nature; review in The Cambridge Review; review in Mathematical Gazette

1904 Bibliography of Quaternions and Allied Systems of Mathematics by Macfarlane

review in Nature; review in Philosophical Magazine; review in Bulletin of the American Mathematical Society

1905 A Manual of Quaternions by Joly

review in Mathematical Gazette; review in Bulletin of the American Mathematical Society; review in Nature; review in The School World; review in Technics

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You are right when you "think that the above three notions did not get studied until after Hamilton presented the quaternions".

Hamilton wanted to generalize complex numbers to have a tool to represent rotations of 3-space, similar to representation of rotations of the plane by complex numbers of absolute value 1. Since rotations in 3 space do not commute, it was clear that commutativity of multiplication had to be rejected. He found first that there is no reasonable generalization in dimension 3, and then found a proper generalization in dimension 4.

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  • $\begingroup$ Unlikely. The problem is that a quaternion is not a rotation of 3D space. The reason being that that $q$ and $-q$ act as the same rotation on 3D space. $\endgroup$
    – wlad
    Nov 3, 2021 at 22:47
  • $\begingroup$ Sorry, but this disagrees with the paper I found on the topic and my understanding of the relationship between the algebra $\mathbb H$ and the Lie group $SO(3)$. Hence my downvote. $\endgroup$
    – wlad
    Nov 3, 2021 at 22:52
  • $\begingroup$ What you say is correct, but nevertheless quaternions are useful in representing rotations, and I am sure this is what Hamilton had in mind. $\endgroup$ Nov 3, 2021 at 22:53
  • $\begingroup$ The criterion $|wz| = |w| |z|$ has some connection to rotations, I suppose. The connection being that if $q$ has unit length, and $r$ is a n-dimensional vector represented as an element of our hypothetical normed unital algebra, then $|qr| = |r|$; so by the polarisation identity $q$ must be an orthogonal transformation of n-dimensional space, and by further consideration, moreso a rotation. $\endgroup$
    – wlad
    Nov 3, 2021 at 22:59
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    $\begingroup$ I would like to rescind my downvote. I may have misread this answer. $\endgroup$
    – wlad
    Nov 3, 2021 at 23:05
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In the paper Hamilton's Discovery of Quaternions by B. L. VAN DER WAERDEN, the author lists the criteria Hamilton was looking to satisfy from his algebra:

  • They must define a finite-dimensional unital algebra over $\mathbb R$. This led him to adopt the notation $a + bi + cj$ for the elements of such an algebra.
  • Assume a hypothetical 3-dimensional number system where the numbers are of the form $z = a + bi + cj$. Hamilton defined $|z|$ to be $\sqrt{a^2 + b^2 + c^2}$ by generalising the same operation over the complex numbers. He wanted this operation to satisfy the identity $|wz| = |w| |z|$.

The combination of these criteria define precisely the normed unital algebras. By Hurwitz's theorem, the only algebras satisfying these criteria are $\mathbb R, \mathbb C, \mathbb H, \mathbb O$, where the last symbol denotes the octonions.

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    $\begingroup$ Reading over van der Waerden's paper, this answer is dubious. He actually lists the criteria as follows: "He required, first, that it be possible to multiply out term by term; and second, that the length of the product of the vectors be equal to the product of the lengths." Putting it in modern terms is silly when the whole purpose is to get historical insight into his process. If you want to understand Hamilton's reasoning, you have to think like Hamilton. If you want your own derivation, there's no need to consider Hamilton at all. $\endgroup$ Jan 27 at 15:01
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In B. L. van der Waerden's article (linked in your own answer), Hamilton's Discovery of Quaternions, the history of Hamilton and quaternions is laid out clearly, based on Hamilton's publications and correspondence. In brief (although that article is already brief), first note that Hamilton "knew and used the geometric representation of complex numbers," but in publications "he emphasized the definition of complex numbers as the couple $(a,b)$ which followed definite rules for addition and multiplication." He posed the problem:

"To find how number-triplets $(a,b,c)$ are to be multiplied in analogy to couples $(a,b)$."

Van der Waerden continues:

In analogy to the complex numbers $(a + ib)$ Hamilton wrote his triplets as $(a + bi + cj)$. He represented his unit vectors $1$, $i$, $j$ as mutually perpendicular "directed segments" of unit length in space. Later Hamilton himself used the word vector, which I also shall use in the following. Hamilton then sought to represent products such as $(a + bi + cj)(x + yi + zj)$ again as vectors in the same space. He required, first, that it be possible to multiply out term by term; and second, that the length of the product of the vectors be equal to the product of the lengths. This latter rule was called the "law of the moduli" by Hamilton.

(emphasis mine)

Which answers the question. The rest of the article explains how Hamilton determined the necessity for four dimensions instead of three based on these criteria, and how he arrived at the relations for quaternions.

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