| 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 ij=-ji followed the ``Rule of the Modulus," and noting that ij could not be a real number, Hamilton had ``one of those flashes of understanding that occasionally occur after long deliberation on a problem [Hankins 91]." All at once, the appropriate definition for multiplication came to him. Hamilton: 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 symbols i,j,k [letter to son (via van der Waerden 12)]. Hamilton 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 n Dimensions [Fraleigh 236]." In any case, the multiplication achieved what Hamilton sought to immediately accomplish. He had invented a multi-dimensional number system that was both consistent and elegant. | The Abstractor | Elementary Notions | A Fourth Dimension | A Division Algebra | Closure | An Example of Hamilton's Work | Bibliography | Back to the front page