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.'



[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] 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.




[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.




[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.



D.R. Wilkins
School of Mathematics
Trinity College, Dublin