| The Abstractor | Elementary Notions | A Fourth Dimension | A Division Algebra | Closure | An Example of Hamilton's Work | Bibliography | Back to the front page A Fourth Dimension
Joseph Johnson
After realizing that using And here dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating triplets [letter to friend (via van der Waerden 10)]... But on the 16th day of the month (October 1843) - which happened to be a Monday and a Council day of the Royal Irish Academy - I was walking in to attend and preside...along the Royal Canal...and then, yet an undercurrent of thought was going on in my mind, which gave at last a result, whereof...I felt at once the importance. An electric circuit seemed to close, and a spark flashed forth, the herald (as I foresaw immediately) of many long years to come of definitely directed thought and work...I pulled out on the spot a pocket-book, which still exists, and made the entry there and then. Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on a stone of Brougham Bridge...the fundamental formula with the symbolsHamilton invented a new symbol, k, which was symmetric to i and j. The somewhat complicated definition of
multiplication is as follows:
(w+xi+yj+zk)(w'+x'i+y'j+z'k)=(ww'-xx'-yy'-zz')+(wx'+xw'+yz'-zy')i+(wy'-xz'+yw'+zx')j+(wz'+xy'-yx'+zw')k.
Note that the following rules make the definition infinitely more simple:
Of course, Hamilton redefined the definitions of equality and addition as well.
Hamilton may have been influenced by a provocative paper written by Arthur Cayley published in that same year. In the
Cambridge Mathematical Journel, Cayley expounded his ``Analytic Geometry of
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