Hamilton's report on Jerrard's Mathematical Researches
Sir William Rowan Hamilton presented a report to the Sixth Meeting of the British Association for the Advancement of Science, held at Bristol in August 1836, entitled Inquiry into the Validity of a Method recently proposed by George B Jerrard, Esq. for Transforming and Resolving Equations of Elevated Degrees. We give below the minute of the British Association for the Advancement of Science concerning this report which appears in the Proceedings of the British Association:
Jerrard's Mathematical Researches.
Prof Sir William Hamilton read his report on Mr George B Jerrard's mathematical researches, connected with the general solution of algebraic equations. He wished, in the first place, to inform the Section, that no part of the grant of £80 had been expended, which the Association had so liberally placed at his disposal for the purpose of procuring the assistance of persons competent to verify, by numerical computations, the method of Mr Jerrard. The reason that he had not deemed it necessary to resort to this expense was, that he had, at a very early period after the meeting of the British Association in Dublin, satisfied his own mind that the method of Mr Jerrard entirely failed in accomplishing the solution of equations of the fifth and sixth degree; and he trusted that he should be able to lay before the Section, with as much clearness as the abstruse nature of the subject would admit of, the principal steps of a demonstration, which, to the mind of the learned Professor himself, at least, carried a complete conviction, that the method of Mr Jerrard was not applicable until the equation, as a minor limit, had reached the seventh degree.
In order that he might carry the Section fully along with him, Professor Hamilton stated, that it would be necessary to give again a rather detailed account of the peculiarities of the very ingenious notation, devised by Mr Jerrard, for denoting certain algebraic processes, resorted to in the application of his method. The Professor then proceeded to detail to the Section the several steps of Mr Jerrard's method, clearly marking the steps previously known to analysts, and such as Mr Jerrard had the merit of originating. The principal peculiarity of formulae seemed to be, that in an equation, transferred in a particular manner for the purpose of eliminating the coefficients of the original equation, the coefficients were so ingeniously obtained as to be entirely independent of the degree of the original equation, and therefore to be of a similar form in all possible equations, the solutions of which were sought. As soon as he had prepared these formulae, the Professor proceeded to demonstrate to the Section, that from the very nature of their connection with the original equation, they must fail in giving its solution, where it only rose to the fourth dimension, because he showed that this would involve the solution of no equation of the sixth degree, as a preliminary step. Equations, however, of this degree had been long solved, and it was only, therefore, in connection with the generality of Mr Jerrard's method, that its failure, as regarded them, was of any consequence.
He then proceeded to give a similar demonstration of its failure, as regarded equations of the fifth and of the sixth degree; and during his discussion of this step of his demonstration he took occasion to show that Mr Jerrard's method had succeeded in reducing equations of the fifth degree to tables of double entry - a discovery, upon the value of which he enlarged considerably, and highly eulogized and complimented the author; insomuch, that he stated that if the method had accomplished nothing but this alone, Mr Jerrard would have received the congratulations of the scientific world. He then proceeded to show, that unless the index of the equation reached as a minor limit, the number seven at least, a certain intermediate equation, concerned in the elimination, would be met with, along with a multiple of it, which, therefore, would not give a number of distinct results sufficient to complete eliminations; but, beyond that degree, he stated that he had satisfied himself that Mr Jerrard's method would afford solutions of equations, which, even if they should, from their complexity, or other causes, be useless to the practical or merely arithmetical algebraist, yet to those engaged in prosecuting inquiries involving purely symbolic algebra, he felt confident they would afford facilities and general methods of investigation, hitherto almost unlooked for and unexpected.
Mr Babbage complimented Sir W Hamilton upon the very lucid exposition which he had given of a subject which he characterized as bordering upon the very extremest limits of human knowledge, and congratulated Mr Jerrard upon the success with which he had contrived so effectually to distinguish between the symbols of operation and those of quantity, in expressing the results of elimination. Engaged, as it was well known he was, in a branch of practical numerical science, he could not suffer himself to be supposed to look with indifference upon a discovery which, if it should even fail in affording any practically important assistance to his particular branch, must yet he admitted to afford the strongest promise of advantage to the more purely abstract branch of algebraic investigation. Professor Peacock observed, that during the progress of the discussion of this question he had not failed to remark the many advantages which must result to algebra from Mr Jerrard's method, from the collateral improvements to which the prosecution of his principal object had led, partly in suggesting new, and hitherto unexplored, methods of elimination, and partly by leading to a notation, which so clearly distinguished between the marks of quantity and the observations and changes which were to be resorted to in reference to them; but as to the result itself, he need characterise it no higher, when he added, that it was an advance in the science, which it did not appear that the celebrated Lagrange had ever contemplated, and which was not approached by the result of Tschirnhaus.