Addendum to the Life of Sir William Rowan Hamilton,
LL.D, D.C.L.
On Sir W. R. Hamilton's Irish Descent.
On the Calculus of Quaternions.
Robert Perceval Graves
Dublin: Hodges Figgis, and Co., 1891.
It has been thought desirable that the following Letters should
be printed in a form adapted to their being made a concluding
part of the Third Volume of the `Life of Sir W. R. Hamilton.'
They contain fuller proofs of the Irish descent of the great
mathematician than appeared sufficient when the narrative given
in Vol. i., pp. 3-7, was sent to the press, and may, I
think be regarded as settling the point in dispute. They were
put together in consequence of the publication of the Article on
Sir W. R. H. in that noble work the `Dictionary of National
Biography.' In reference to Sir W. R. H.'s descent that Article
advocated a story of an opposite tenour, and, besides, contained
other important errors in matters of fact. I felt it, therefore,
incumbent on me to endeavour that these errors should not go down
to posterity in a book of such high authority without an effort
at correction. I have to acknowledge the courtesy of the Editor
of the `Athenæum' in admitting to his pages the Letter which
had this object. It was replied to by Mr. R. E. Anderson, the
author of the Dictionary Article; and he has obligingly permitted
me to reprint his reply, which is required for the full
presentation of the discussion. He accorded that permission in
terms which enhanced the value of the favour, and he has allowed
me to reproduce here courteous admissions on his part which the
Editor's closure of the correspondence prevented him from
communicating to the `Athenæum.' Mr. Anderson's letter is as
follows:- `With reference to the letter you mention, pray accept
my consent, full and unqualified, to the publication, should you
think it desirable. Let me add that I am happy in this
opportunity of saying that I intended - when writing the
``Athenæum-letter'' - to acknowledge several of your
corrections, e. g. Kirkmaiden for Kirkmichael, and
other not unimportant things which ought to have been accepted on
the authority of your ``Life of Sir W. R. H.'' So much was I
conscious of this indebtedness that when the Editorial ruling
precluded any further correspondence in their columns I again and
again thought of addressing you personally to say so. In spite
of the cold judicial tone affected in a ``Dictionary-article'' I
trust that some genuine and earnest admiration of Hamilton's
genius and rare excellencies was still palpable in what I wrote.'
---------
SIR W. ROWAN HAMILTON AND THE `DICTIONARY OF NATIONAL
BIOGRAPHY.'
[Athenæum, March 21, 1891.]
1, WINTON-ROAD, DUBLIN, March, 1891.
In the volume preceding that last issued of the `Dictionary of
National Biography' is an article on Sir W. Rowan Hamilton, the
distinguished mathematician and discoverer of quaternions. As
the author of a biography of Rowan Hamilton referred to in it, I
feel I ought to place on record a correction of some errors
occurring in its opening and some subsequent paragraphs. Trusting
that you will allow me to do this in your pages, I ask you to
reprint these paragraphs, and to add the observations I am
obliged to make:-
`Hamilton, Sir William Rowan (1805-1865), mathematician, born in
Dublin at midnight, between 3 and 4 Aug., 1805, was the fourth
child of Archibald Hamilton, a solicitor there, and his wife
Sarah Hutton, a relative of Dr. Hutton the mathematician.
Archibald Hamilton was Scottish by birth, and went to Dublin when
a boy with his father, William Hamilton, who settled as an
apothecary there, and his mother, who was the daughter of the
Rev. James M`Ferrand, parish minister of Kirkmichael, Galloway.
The Rev. R. P. Graves maintains that William Rowan Hamilton was
Irish by descent, while admitting that both the paternal and
maternal grandmothers are Scottish; but the express statements of
Professor Tait and Dr. Ingleby that the paternal grandfather went
to Dublin from Scotland seem conclusive. The apothecary had also
brought a second son, James, from Scotland, who studied for the
Church, became curate of Trim, county Meath, and earned some
reputation as a linguist.'
There are in the above paragraph numerous misstatements. The
most important, to which the others are incidental, is the
assertion of the Scottish parentage of Sir W. R. Hamilton's
father. It will, therefore, be convenient to deal with this in
the first place, and to correct the others as they crop up. The
statements on this point are the following: `Archibald Hamilton
[the father of Rowan Hamilton] was Scottish by birth and went to
Dublin ... with his father, William Hamilton, who settled as
an apothecary there. ... The apothecary had also brought a
second son, James, from Scotland.' In disproof of these
statements I have to allege: (1) That in the Entrance Book of
Trinity College, Dublin, under date May 2nd, 1791, is recorded
the entrance, or matriculation, of the above James Hamilton, aged
fifteen, son of William Hamilton, apothecary, deceased, born in
the city of Dublin. It is to be remarked that the information on
which this entry was founded was furnished by James Hamilton
himself. It determines his birth to the year 1776.
If by the expression `second son', above applied to him, is meant
second in order of birth, this statement is incorrect: James was
the elder brother of Archibald. (2) That I have seen in the List
of Freemen, among the archives of the Corporation of Dublin,
under date Midsummer, 1802, the name of Archibald Hamilton, as
admitted a freeman on the qualification of birth,
i. e., as being the son of a freeman; and in the same
record under date, Midsummer, 1774, the name of his father,
William Hamilton, apothecary, as admitted on the qualification of
service: this word implies that he had served an
apprenticeship of seven years to a freeman-member of the Guild of
Apothecaries, resident in Dublin, and brings his residence in
that city back to the year 1767. In the same corporation-record,
under date 1777, is an entry of admittance as a freeman, on the
qualification of service, of Francis Hamilton, younger
brother of the above William Hamilton, who afterwards became
Alderman of Dublin, and was father of Arthur Hamilton, called in
the biography `Cousin Arthur.' In `Wilson's Dublin Directory' of
1785, in the list of merchants and traders of the city, is the
name of Roger Hamilton, apothecary, with the address, 30,
Jervis-street, which was the residence of his deceased eldest
brother, William. The family was, it thus appears, established
in Dublin. But, as confirming Irish extraction of an anterior
date, we also know that the father of William Rowan Hamilton
(concerning whom I have heard from Sir Rowan Hamilton's sister
Sydney that his Christian name was Francis) married a Miss
(believed Margaret) Blood; this lady was in her extreme old age
(she lived to be above a hundred) seen by Sir. W. R.
Hamilton's sister Eliza, as mentioned in my biography, and was of
a well-known family settled in Ireland in the reign of James I.,
members of which from the beginning of this century up to the
present time have recognized their kinship with Rowan Hamilton,
both in letters and in familiar personal intercourse. In the
cited paragraph it is stated that the Rev. R. P. Graves had
admitted `that both the paternal and maternal grandmothers are
Scottish.' This is incorrect. I prominently drew attention to
the fact that the paternal grandmother was Scottish, as perhaps
accounting for the mistake that had arisen, but I stated that the
mother of Sir W. R. Hamilton was Sarah Hutton, of a family
settled in Ireland, with which, as I learn from members now
living, no connexion whatever of relationship subsisted with
Charles Hutton, the eminent mathematician, presumably referred to
in the text of the `Dictionary.' I made no mention of, or
allusion to, her mother, the `maternal grandmother' of Sir W. R.
Hamilton. The marriage of Grace M`Ferrand, daughter of the
minister of Kirkmaiden, the Maiden Kirk of Burns (not
Kirkmichael, as in the article), took place almost certainly in
Dublin in 1774 or 1775, when she resided with Mrs. Gawen
Hamilton in Rutland-square, which is in the same parish,
St. Mary's, as Jervis-steet, the place of residence of her
husband. It was on the death of her father that she was brought
from Scotland and adopted by Mrs. Gawen Hamilton, of Killileagh
Castle, and by her taken on a continental tour before her
marriage. I have not been able to obtain a certificate of the
marriage of Grace M`Ferrand with William Hamilton; it was not
improbably celebrated with Presbyterian rites, and no records of
such marriages of the date required are now obtainable.
The facts I have certified are quite irreconcilable with the
story of the `Dictionary' article. But it may be asked how it
came to pass that Professor Tait and Dr. Ingleby made the
statement with regard to Sir W. R. Hamilton's parentage which I
have now again, and with some additional circumstances,
disproved. Let it be remembered, in the first place, that it was
impossible for either of them to possess personal knowledge of
the facts. I have been informed that Professor Tait has referred
to Mr. W. E. Hamilton, elder son of Sir William, as his
authority, and I have reason to think that Dr. Ingleby derived
his information from the same source. They would naturally
consider such an authority sufficient. And I do not suggest that
the representation of Mr. W. E. Hamilton was not given in good
faith. I believe it to have originated in impressions received
from his cousin, the Rev. James Alexander Hamilton, who had heard
his father, the Rev. James Hamilton, speak of having been in
Scotland at some early stage in his life, to which fact I have
seen one reference in the correspondence in my hands. But as
against this representation of the son I can set the superior
authority of the father. From Sir W. R. Hamilton, in his own
handwriting, still existing, are statements to the effect that
his particular branch of the Hamilton family came from Scotland
to Ireland in the reign of James I. Such a statement was
furnished by him to me in 1841 as part of the material for the
biographical sketch contributed by me to the Dublin
University Magazine, and which appeared in the January number
for 1842. It may be seen repeated by him in a letter to De
Morgan, printed in vol. iii. of the `Life,' pp. 392-3.
It was natural that Professor Tait, justly proud of the
intellectual attributes of his countrymen, should be willing to
claim for Scotland Rowan Hamilton as son of a Scottish father: I
trust he will be consoled for disappointment as to this claim by
finding that the Scottish derivation of his master in quaternions
is only put back for some generations.
It has always been a pleasure to me to give ample expression to my
sense of the warm fidelity of Professor Tait's championship of
the greatness of Hamilton's mathematical powers, and of the value
of the quaternion calculus. If I believe that at one time he
acted prematurely in announcing by advertisement his intended
`Introduction' to quaternions, without having previously gained
Hamilton's consent, I believe also that high praise is due to
him for his immediately afterwards offering to comply with
Hamilton's wishes in the matter, and for his subsequently
withholding, in a spirit of the strictest honour, the publication
of his own work till after the posthumous publication of
Hamilton's `Elements'; when, if he had construed literally a
permission given him by Hamilton, he might have anticipated that
publication (see `Life,', vol. iii. pp. 133-4, and especially
pp. 561-2). Yet estimating so highly as I do the merits of
Professor Tait in this connexion, I cannot but think that he
himself would be pained at reading some sentences in later
columns of the `Dictionary' article. I quote them with their
context:-
`In 1844, before the Royal Irish Academy, of which he was still
President, he [Sir William] formally defined the term
``quaternions,'' by which the new calculus was to be known; but
not till 1848 can the method be considered as systematically
established, when he began, in Trinity College, Dublin, the
``Lectures on Quaternions,'' which were published in 1853.
Nearly the whole of this bulky octavo, occupying 808 pages,
besides an introduction of 64 pages, can be understood only by
advanced mathematicians. But for Professor Tait of Edinburgh, who
interpreted the new science for more commonplace mathematicians,
Hamilton's merits must long have remained unrealized or
absolutely unknown. The truth is that this great book of
Hamilton's, as well as his so-called ``Elements of Quaternions,''
is frequently unpleasant in style, besides being obscure and
difficult of interpretation. ... As a whole the method is
pronounced by most mathematicians to be neither easy nor
attractive, the interpretation being hazy or metaphysical and
seldom clear and precise.'
The first of these extracts commences with a material
misstatement. It was upon the 13th of November, 1843,
that Hamilton made, at the Royal Irish Academy, the first public
communication of his discovery, fully setting forth at the time
the bases of his calculus, having previously, on the 16th of
October [see note],
the day of his discovery, made announcement of it to the Academy.
In connexion with this point I refer to the `Life,'
pp. 443-4, and, indeed, from p. 432 to p. 447; and, in
particular, to the commencement of his letter, dated October
15th, 1858, to Professor Tait, at p. 435, in which he
says, ``The quaternions started into life, or light, full
grown, on
![[Monday]](moon.gif)
the 16th of October, 1843.'' It will also be
seen by reference to the `List of Papers, Memoirs, &c.,' at the
end of vol. iii. of the `Life,' pp. 649, 650, that between 1843
and 1848, the year spoken of in the article as that of the first
systematic establishment of the method, Hamilton published
numerous papers on quaternions in the Proceedings of the
Royal Irish Academy, in the Philosophical Magazine, and in
the Cambridge and Dublin Mathematical Journal. And as for
the extraordinary statement that but for what was published on
the subject by Professor Tait, `Hamilton's merits must long
have remained unrealized or absolutedly unknown' (!), it is
enough to refer to the facts set forth abundantly in the `Life,'
that his discovery was immediately with the applause it merited
by the first mathematicians in England - Herschel, Cayley, and
others - and had become known and celebrated over Europe and in
America long before Professor Tait's first publication on
quaternions. Professor Tait's services in the field of
quaternions stood in need of no such exaggeration. With regard
to the article-writer's characterization of Hamilton's style in
his mathematical works, it is not for me to say more than that I
believe it is not justly chargeable with either obscurity or want
of precision. I believe no sentence ever proceeded from his pen
to which a second meaning could be ascribed, so habitually
careful was he to express exactly what he intended, neither more
nor less. This sometimes, doubtless, caused an inclusion of
modifications which tended to apparent overfulness, but no
mathematician would be likely to object to such accuracy.
In a subsequent paragraph of the article is the following
sentence:- `The pension of £200 which he had received
since he was knighted was afterwards continued to his widow.' If
this means that the pension was conferred in the year when he was
knighted - and, if not, why is his knighthood mentioned? - the
statement is incorrect. His knighthood was conferred in 1835,
the pension in 1843. It was continued in equal moieties to his
widow and daughter, by neither of whom was it long enjoyed.
R. P. GRAVES.
---------
THE ARTICLE `SIR WILLIAM ROWAN HAMILTON' IN THE `DICTIONARY OF NATIONAL
BIOGRAPHY.'
[Athenæum, April 4, 1891.]
March 28, 1891.
Mr. Graves letter in your last week's number seems to require
some reply, as shortly as may be.
1.
He amazes one by stating, with reference to the `maternal
grandmother of Sir W. R. Hamilton, that in his `Life' of Sir
William he had `made no mention of or allusion to her.' What are
the facts? Near the top of p. 6, vol. i., are printed the
following words:-
`The maternal grandmother of Sir W. Rowan Hamilton was of
Scottish birth. To this extent Scottish blood was in his veins.'
There are two tables of errata for vol. i. (i. 698, and
ii., p. xvi), but nowhere any hint that the phrase was not
intended to stand.
2.
The `maternal grandmother,' however, scarcely affects the main
argument. Her daughter Sarah Hutton, mother of Sir William, was
said, both by Prof. Tait and De Morgan, to be related to
Dr. Hutton the mathematician; v. loc. cit. Mr. Graves
`learns from members [of the family] now living' that there was
`no connexion whatever.' One can only reply that both Professors
had been closely trained to scrutinize and sift evidence, and
were, moreover, personally (and one rather intimately) associated
with Sir William's family.
3.
Professor Tait has repeatedly stated that the grandfather of Sir
William came from Scotland with two sons, and settled in
Dublin (N. B. Rev., September, 1866, and last ed.
`Encycl. Brit.') - a fact confirmed by Dr. Ingleby in the
Brit. Controversialist, xxii. 161. Mr. Graves, while
admitting that Sir William's elder son also held that opinion,
explains the fact by `impressions derived [by that son] from his
cousin,' the son of Rev. James Hamilton. The opinions of Sir
William's son on such a matter must surely have been largely due
to what he had learned from his own father and in his father's
family. To support the contrary theory Mr. Graves supplies two
extracts from the Dublin `List of Freemen'; and then, by
interpreting the word `birth' to mean so-and-so, and the word
`service' to mean such-and-such, he hopes to clench the argument.
One is once more reminded of the pregnant nod or shake of Lord
Burleigh's head in the play.
4.
The criticisms passed in the `Dictionary' upon Sir W. R.
Hamilton's work, and the value of his great calculus, are what
Mr. Graves chiefly founds a grievance upon. The writer said, for
example, that, but for Professor Tait, the `merits of the system
must have long remained unrealized.' The late Professor Kelland,
however, said the same thing in 1873, and no more authoritative
judge could be found - a senior wrangler, by the way, like
Professor Tait - who greatly advanced the study of quaternions by
showing their ready application to conics. Kelland's words
are:-
`The first work of Sir William Hamilton, ``Lectures on
Quaternions,'' was very dimly and imperfectly understood by me,
and I dare say by others, until Professor Tait published his
Papers on the subject. Then, and not till then, did the science
develop itself to me. Hamilton's second work is far from
elementary, whatever its title may imply. ... My knowledge of
quaternions is due exclusively to him [Professor Tait].'
Another mathematician wrote:-
`Partly on account of the metaphysical atmosphere, the method is
neither easy nor attractive.'
In 1847 Hamilton read a Paper before the British Association at
Oxford, and in a letter quoted in Mr. Graves's book we find a
short summary of some criticisms made by the mathematicians
present. Sir J. Herschel spoke of the difficulty of `mastering
the extremely abstract conception and the new algorithm which
they involve.' Airy, the Astronomer Royal, `warned all persons
if they should use the method to do so with the extremest caution
... expressions extremely difficult of geometrical
interpretation.' A third speaker dwelt particularly on the
possibility of making mistakes in using the new calculus.
Professor MacCullagh, a contemporary of Hamilton's in Dublin, was
a most able mathematician, and Mr. Graves (ii. 464) informs us
that he was not satisfied with quaternions as the foundation of
an algebra of space; adding `The same want of satisfaction with
them seems to have been felt by Professor De Morgan.'
At the close of his account of the work and brilliant talents of
Hamilton, De Morgan thus sums up judicially: `a man whose place
among mathematical discoverers we cannot venture to assign.'
The fact undoubtedly is that, as a mathematical method, the
quaternions have not made such progress as was anticipated by the
inventor and his friends. Even in Dublin University itself,
though famed as a mathematical school, there is no evidence that
Hamilton's teaching has made much mark. The Examination Papers,
1891, set during a year at Trinity College, form a volume of 587
pages, and on only one of these pages apparently does there occur
any reference to the quaternion calculus of Sir William Rowan
Hamilton.
R. E. ANDERSON.
---------
THE `DICTIONARY OF NATIONAL BIOGRAPHY' AND SIR WILLIAM ROWAN
HAMILTON.
[Athenæum, April 25, 1891.]
The word `maternal,' in the sentence quoted from my book
(vol. i., p. 6) by Mr. Anderson, was an erratum for
`paternal.' Mr. Anderson seems to have anticipated this. The
person pointed to was Grace M`Ferrand, as would be expected by a
reader of the preceding pages. I am glad to have attention drawn
to the mistake in my text, of which I was ignorant until made
aware of it by Mr. Anderson's letter. But while recognizing that
he had warrant in it for his statement, I have to add that in
point of fact the `maternal grandmother' of Sir W. R. Hamilton,
the mother of Sarah Hutton, was not of Scottish birth, but a
French lady named Guinant.
With regard to the asserted relationship of this branch of the
Hutton family to Dr. Hutton, the mathematician, and the story of
Sir W. R. Hamilton's grandfather, William Hamilton, bringing over
his two sons from Scotland, Professors Tait and De Morgan and
Dr. Ingleby were not called upon to question what was conveyed to
them on apparently sufficient authority. As to the first of
these statements, I do not blame Mr. Anderson for repeating what
I, on the authority of the family, have felt it right to correct;
but in reference to the second, I conceive that as a biographer,
meeting with contravening statements, supported by documentary
testimony accessible to all - the public records of the
Corporation of Dublin and of Dublin University, and printed
statements emanating at distant intervals from Sir William
Hamilton himself - he was bound to grapple with them in a manner
which he has shrunk from seriously attempting, either in his
`Dictionary' article or in his letter to the Athenæum.
Facts irreconcilable with the story advanced are not to be
displaced by the vague expression that opposite statements
`seem conclusive,' or by what the writer supposes to be a
witticism.
I pass now to Mr. Anderson's assertion that, but for Professor
Tait, Hamilton's `merits must long have remained unrealized.'
Mr. Anderson now drops the words which follow in his article, `or
absolutely unknown.' Professor Tait is by no one more than by
myself honoured and gratefully regarded as an expositor of
quaternions, and still more as a mathematician who has afforded
splendid examples of the successful application of the calculus
to physics. I have, therefore, read with true pleasure Professor
Kelland's generous tribute to his former pupil, extracted from
his preface to their joint publication of 1873; but this in no
way conflicts with the recorded facts, that before the appearance
in 1867 of Tait's first book upon the subject twenty-three years
had elapsed since the discovery of quaternions in 1843, and
thirteen years since the publication of Hamilton's `Lectures on
Quaternions' in 1853, preceded as it was, and followed, by his
numerous contributions to scientific periodicals; and that during
his period the discovery had been largely commented upon both in
Europe and in America. Bellavitis in Italy, in 1858, Allegret in
Paris, in 1862, had published their essays; and the impression
made by the discovery in America is sufficiently testified by the
fact that in 1864, as narrated in the `Life' (vol. iii. p. 204),
Hamilton's name was in consequence placed first on the roll of
Foreign Associates of the new National Academy of the United
States. The preceding part of Professor Kelland's preface shows
that exalted opinion which he held of the quaternion theory and
its inventor, and his conviction that the calculus should be
introduced into `an elementary course of mathematics. It belongs
to first principles, and is their crowning and completion.' Of
this Mr. Anderson gives no inkling.
In his reference to the discussion on quaternions at Oxford in
1847, he introduces Herschel as speaking of the difficulty of
`mastering the extremely abstract conceptions and the new
algorithm which they involve'; the reader would suppose that this
was a full representation of Herschel's opinion. Here is the
whole sentence (vol. ii. p. 586): he said that `his admiration of
the quaternions had increased with every resumption of his study
of them, and that although it might be difficult at first
to master the extremely abstract conceptions and the new
algorithm which they involve, yet he was well convinced it was
worth the trouble': and, again (p. 587): `It was a cornucopia of
riches, and he urged all who studied the Cartesian to study also
the quaternion theory.' Can it be said that Mr. Anderson's
fractional quotation is a fair quotation? Professor Airy's
precautionary warnings, delivered when, as he himself avowed, he
was not acquainted with the method, did not prevent him from
subsequently writing upon it (p. 682) as `the large science of
quaternions.' The `third speaker' referred to by Mr. Anderson
was Mr. Jarrett, of whom no more need be said. Again,
Mr. Anderson quotes from the `Life' (vol. ii. p. 464)
transitional opinions of MacCullagh and De Morgan as if they were
definitive. Only a few pages onward (p. 575) De Morgan
acknowledges that he is brought round to see the superiority of
quaternions over competing systems, and his conversion to the
introduction, as an inevitable necessity, of non-commutative
multiplication - the great ground of objection, but one, as
M. Laisant points out, `inherent in the nature of things'; while
as to MacCullagh, his conversion was also proved by the high
testimony he bore to the science only a few days before his death
in conversation with Dr. Stokes, as recorded in vol. ii., p. 595,
a testimony afterwards reported to myself in still stronger terms
by Dr. Stokes.
The statement in the article that `most mathematicians have
pronounced the method to be neither easy nor attractive,'
&c., has come down in the letter to the dictum of one unnamed
mathematician. Against this is to be set the testimony of Tait
that its processes are sometimes `perplexingly easy' (vol. iii,
p. 106); of Clerk Maxwell (vol. ii., p. 446), `the relations of
which [physical quantities] to each other may be expressed far
more simply by a few words of Hamilton's than by the ordinary
equations'; of M. Laisant, who speaks of its `caractère
intuitif'. The method has received the approving suffrage of
Peacock (vol. ii., p. 579 and passim), Herschel, Cayley
(who has distinguished himself by extending the science, and has
recently contributed an important chapter to the third edition of
Prof. Tait's `Elementary Treatise'), Sylvester (also an extender
of the science), W. K. Clifford, Henry Smith, Tait,
Kelland - British mathematicians of the highest grade - not to
attempt enumerating American and European writers upon the
system. How then has it happened that the method has not yet
come into more general use, when its theory is admitted to be
more direct in its action of spatial relations, more
comprehensive in its range, than the Cartesian with its
`scaffolding of non-natural co-ordinates' (Tait)? Not being a
mathematician, I have no pretension to dogmatize or prophesy, but
I may be allowed to state my belief that it is caused principally
by its necessary departure, in its adoption of non-commutative
multiplication, from processes of calculation which had become
habitual, and to point to a precedent which seems to me
instructive as a parallel case. It is about two hundred years
since the Leibnitzian differential calculus was published; it was
not till about 1815 - a hundred and fifty years afterwards - that
it came into general use in England: its introduction into
Ireland was later still. We gather from the `Encyclopædia
Britannica' that the new calculus very slowly made its way,
mainly because the English, through national prejudice, would not
accept Leibnitz's algorithm, and that of Newton was so difficult
to work that it impeded research. When Herschel, Peacock, and
others, about 1815, popularized the Leibnitzian system, the
calculus began to more more rapidly in England - but not till
then.
As a looker-on I find it hard to conceive that, when the ideas
and terms of the Quaternion theory have pervaded the whole field
of mathematics, the calculus founded on them should not also
prevail, and I attach more weight to the anticipations, in
respect of the future of quaternions, of mathematicians who, like
Tait and Laisant and Padelletti (see `Life,' vol. iii., p. 657),
have used the method for physical applications, than to the
misgivings even of accomplished mathematicians who have not done
so. I expect that it will first flourish where, as I understand
is the case in America, its rudiments, with examples, shall be
taught to the young, and where it shall be examined in
at entrance in universities.
R. P. GRAVES.
Links:
D.R. Wilkins
(dwilkins@maths.tcd.ie)
School of Mathematics
Trinity College, Dublin