In order to realise the magnitude of the revolution which Hamilton has wrought in the application of symbols to mathematical investigation, it is necessary to think of what Hamilton did beside the mighty advance made by Descartes.  To describe the character of the quaternion calculus would be unsuited to the pages of this work, but we may quote an interesting letter, written by Hamilton from his deathbed, twenty-two years later, to his son Archibald, in which he has recorded the circumstances of the discovery:—­

“Indeed, I happen to be able to put the finger of memory upon the year and month—­October, 1843—­when having recently returned from visits to Cork and Parsonstown, connected with a meeting of the British Association, the desire to discover the laws of multiplication referred to, regained with me a certain strength and earnestness which had for years been dormant, but was then on the point of being gratified, and was occasionally talked of with you.  Every morning in the early part of the above-cited month, 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 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 undercurrent 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 which contains the solution of the problem, but, of course, the inscription has long since mouldered away.  A more durable notice remains, however, on the Council Books of the Academy for that day (October 16, 1843), which records the fact that I then asked for and obtained leave to read a Paper on ‘Quaternions,’ at the First General Meeting of the Session; which reading took place accordingly, on Monday, the 13th of November following.”

Writing to Professor Tait, Hamilton gives further particulars of the same event.  And again in a letter to the Rev. J. W. Stubbs:—­