Oct. 29, 1885 | 
not too much to say that I felt a7/ ovce 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 z, 7, 4 ;—namely, 
2? = 7? = k = 77k = — 1, which contains the solution of 
the fZroblem, but of course, as an inscription, has long 
since mouldered away. A more durable notice remains, on 
the council books of the Academy of that day—October 
16th, 1843—which records the fact, that I then asked for 
and obtained leave to read a paper on gwaternions at the 
first general meeting of the session, which reading took 
place accordingly, on Monday, November 13.” 
Among the most distinguished disciples of Hamilton is 
Prof. Tait, though eve ie has admitted that he has not 
read the whole of Hamilton’s “tremendous volumes” 
(lives there indeed the man who has?). Another account 
of the discovery is found in a letter to Prof. Tait on 
October 15, 1858 (p. 435) :— : 
“To-morrow will be the fifteenth birthday of the 
quaternions. They started into life full-grown on the 16th 
of October, 1843, as I was walking with Lady Hamilton to 
Dublin, and came up to Brougham Bridge—which my 
boys have since called Quaternion Bridge. I pulled out 
a pocket-book, which still exists, and made an entry, on 
which at the very moment | felt that it might be worth 
my while to expend the labour of at least ten or fifteen 
years to come. But then it is fair to say that this was 
because I felt a Zroblem to have been at that moment 
solved, an intellectual want relieved which had haunted 
me for at least fifteen years before.” 
The unmathematical reader may naturally ask the 
nature of this notable discovery which Hamilton made at 
“ Quaternion ” Bridge. 
It would seem that at this moment he solved the long- 
studied problem of the multiplication of directed straight 
lines, or vectors as he called them. Let @ denote a 
straight line of determined length and direction. Let 6 
denote another straight line at right angles to a, and 
radiating from the same origin; then the product aé 
denotes a third straight line from the same origin perpen- 
dicular to the plane of a and 4; the ‘product da, however, 
denotes the perpendicular line on the other side of the 
plane, so that da = — ad. This formula is eminently 
characteristic of the method, showing as it does that 
vector multiplication is non-commutative. It is, however, 
remarkable that the associative principle obtains in 
quaternions no less than in ordinary algebra; thus if 
a, 6,c be three vectors, or more, generally quaternions, 
then aX c=axX bc. This theorem, though “xe in 
quaternions, is still so far from being obvious that it 
implies the truth of an elaborate geometrical theorem. 
If we could single out one point of special significance 
in the invention of quaternions it would be found in the 
dual interpretation of the symbol of a vector. Thus if 
the letter 7 denotes a vector or directed straight line of 
unit length, then the same symbol may also mean an 
operation of rotation through a right angle around the 
vector as an axis. In the formule of quaternions the 
symbols denoting vectors can be interpreted in this dual 
manner. A quaternion may be regarded as the operating 
factor which applied to one vector transforms it into 
NATORE 
621 
another. This operation requires two quantities to 
specify the plane of the vectors—one to specify the angle 
between them and one the ratio of their lengths in all 
four quantities are required, whence the name quaternion. 
An interesting letter (p. 536) to the Rev. John W. 
Stubbs, Fellow of Trinity College, dated October 19, 
1846, gives a sketch of the points which Hamilton thought 
specially novel in his theory :— 
“ But did the thought of establishing such a system, in 
which geometrically opposite factors—namely, two lines 
(or areas) which are opposite IN SPACE give ALWAYS a 
positive product—ever come into anybody’s head, till I 
was led to it in October, 1843, by trying to extend my old 
theory of algebraic couples, and of algebra as the science 
of pure time? As to my regarding geometrical addition 
of lines as equivalent fo composition of motions (and as 
performed by the same rules), that is indeed esseztial in 
my theory, but zof feculiar to it; on the contrary I am 
only one of many who have been led to this view of 
addition.” 
A few years later Hamilton commenced the delivery of 
lectures on quaternions in Trinity College. His own 
words are (p. 605) :— 
“Tt was on Wednesday, June 21, 1848, that I delivered 
my first lecture on quaternions to a very respectable 
audience, among the persons composing which were the 
Rev. George Salmon, Fellow of Trinity College, Dublin, 
and author of a lately-published treatise on Algebraic 
Geometry, and Arthur Cayley, Fellow of Trinity College, 
Cambridge, who first, except myself, has publicly used 
the quaternions.” 
These lectures, rewritten and greatly expanded, formed 
his first and classical volume—* Lectures on Quaternions.” 
(Dublin, 1853.) 
The publication of this work drew from Hamilton’s 
many scientific friends cordial letters of congratulation. 
His old and intimate friend, Sir John Herschel, thus 
writes on July 21, 1853 (p. 681) :— 
“ Now most heartily let me congratulate you on getting 
out your book—on having found utterance ore rotundo 
for all that labouring and seething mass of thought which 
has been from time to time sending out sparkles, and 
gleams, and smokes, and shaking the soil about you—but 
now breaks into a good honest eruption with a lava 
stream and a shower of fertilising ashes. I don’t mean to 
say that there is not a good deal of cloud (albeit full of 
electric fire)—the good old ‘stupendo e orgoglioso pino’ 
of the fiery outbreak surrounding the bright jet, the true 
product—but the cloud clears as the wind drifts and 
leaves the hill conspicuous. 
“ Metaphor and simile apart, there is work for a twelve- 
month to any man to read such a book, and for half a 
lifetime to digest it, and I am quite glad to see it brought 
to a conclusion.” 
The intercourse, both social and scientific, between 
Hamilton and Sir John Herschel gives many interesting 
pages to this volume. Thus, for instance, we find (p. 492) 
an account of a meeting between these philosophers at 
the house of their common friend, Dr. Peacock, the Dean 
of Ely. On Sunday they attended service in the Cathedral 
in company with Prof. James D. Forbes, and Hamilton 
recorded the incident in a sonnet which he recited to his 
friends. The next morning he received an acknowledg- 
ment in kind from Herschel. We quote here the two 
poems: that of Hamilton (p. 493) bears the title “In 
Ely Cathedral ” :— 
