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What is the origin of this notation? Who coined them and for what purpose?

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I'm retagging your question, since it is really about history (and not about algebraic geometry) – Yemon Choi Oct 24 '11 at 4:16
Just a guess: Brougham Bridge, Dublin, Oct. 16 1843, W.R. Hamilton, quaternions. – Theo Buehler Oct 24 '11 at 4:43
While this is a mildly interesting historical question, I do not think MO should become a repository for asking 'where does this common notation come from?' style questions. Searching on google for "history of mathematical notation" gives a number of interesting pages. – David Roberts Oct 24 '11 at 6:41
...but admittedly not the answer to this question. – David Roberts Oct 24 '11 at 6:42

Maybe it originates from Hamilton's quaternions $\mathbb{H}$, which has a basis $1,i,j,k$ as a real vector space, and the multiplications there, namely, $i\cdot j=k, j\cdot k=i, k\cdot i=j$ correspond exactly to the wedge product in $\mathbb{R}^3$. So $\mathbb{R}^3$ can be viewed as the imanginary part of $\mathbb{H}$.

Anyway, this is just my understanding or my guess.

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For further discussion of this perspective, see… – Dan Kneezel Oct 24 '11 at 5:05
Hamilton's original paper on quaternions starts thus: "Let an expression of the form Q = w + ix + jy + kz be called a quaternion,..." So I think that your guess might be right. – José Figueroa-O'Farrill Oct 24 '11 at 10:59
One could then speculate that there were the complex numbers with $i$ an imaginary number, and when Hamilton needed three more of them, he just used the next letters in the alphabet. – Julien Puydt Oct 24 '11 at 11:13
Apparently, it was E. B. Wilson's book, Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics Founded upon the Lectures of J. Willard Gibbs (1901), that propagated Hamilton's convention. See . – Joseph O'Rourke Oct 24 '11 at 11:31
Interestingly the Nabla-operator was also first used in Quaternionic Analysis, and even the modern Symbol originates with W.(?) Tait who after Sir Hamiltons death was wildely considered to be the leader of the "quaternionic school"- look if your local library has his "elements of the quaternions" (or something like that so?! from 1880- quite fascinating to browse through today... Back then it was wildely consideed to be superior to vector-calculus in some circles now a lot of people don't even know what quaternions are... – sisn Oct 25 '11 at 0:01

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