16 Oct

The discovery of quaternions

If I may be allowed to speak of myself in connexion with the subject, I might do so in a way which would bring you in, by referring to an ante-quaternionic time, when you were a mere child […] Every morning in the early part of the above-cited month [ October, 1843], on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, “Well, Papa, can you multiply triplets”? Whereto I was always obliged to reply, with a sad shake of the head: “No, I can only add and subtract them.”

But on the 16th day of the same month – which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that 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, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse – unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, ijk; namely,
i2 = j2 = k2 = ijk = -1
which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.

Letter dated August 5, 1865 from Sir W. R. Hamilton to Rev. Archibald H. Hamilton.

The classic account of the discover of quaternions. Hamilton also notes in this letter that the Council Books of the Academy record that he had obtained leave to read a paper on quaternions, which reading took place on the 13th November 1843.

In a 1858 letter to Prof Tait, Hamilton claimed he entered the equation in a notebook, in possible embarrassment at his “mathematical vandalism” (see Letters describing the Discover of Quaternions on the TCD website). This was the culmination of a project that Hamilton had started with his “Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time”, published in 1837 (pdf from TCD).

Those of a philosophical bent may hear echoes of Kant in that long title. Kant argued that space and time were “pure intuitions”: not concepts in the mind, but structuring our experience. Mathematics was the expression of these pure intuitions, with geometry relating to space and algebra to time. This is underscored by Hamilton’s assertion in the first section of the work that:
“The argument for the conclusion that the notion of time may be unfolded into an independent Pure Science, or that a Science of Pure Time is possible, rests chiefly on the existence of certain a priori intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on the actual deduction of such a Science from those principles, which the author conceives that he has begun. “

The obvious explanation is that Hamilton was inspired by Kant in taking this view. However, as Michael J. Crowe points out1 Hamilton does not mention Kant in the Essay. In addition, in 1827 in Account of a Theory of Systems of Rays (pdf, TCD) he wrote that “The sciences of Space and Time (to adopt here a view of Algebra which I have elsewhere ventured to propose) became intimately intertwined and indissolubly connected with each other.” This expresses a similar view regarding the roots of algebra and was, per Crowe, four years before Hamilton started reading Kant.

It seems Hamilton came to the same conclusion as Kant separately. In 1834 in a letter to Lord Adare, Hamilton states that he enjoys reading Kant but “a large part of my pleasure consists in recognising through Kant’s works, opinions, or rather views, which have been long familiar to myself, although far more clearly and systematically expressed”. Hamilton also suggests a debt to George Berkeley, with whose work Hamilton was well acquainted: “Kant is, I think, much more indebted than he owns, or, perhaps knows, to Berkeley, whom he calls by a sneer, ‘GUTEM Berkeley’. . . as it were, ‘good soul, well meaning man'”2.  


  1. Michael J. Crowe (1994) A History of Vector Analysis: The Evolution of the Idea of a Vectorial System Courier Corporation, pp. 24-5
  2. Letter quoted in R. S. Ball (1895) The Great Astronomers available online at Project Gutenberg

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