Henry John Stephen Smith: Form

“If we except the great name of Newton (and the exception is one that the great Gauss himself would have been delighted to make) it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute vigorousness in demonstration, which the ancient Greeks themselves might have envied. It may be admitted, without any disparagement to the eminence of such great mathematicians as Euler and Cauchy that they were so overwhelmed with the exuberant wealth of their own creations, and so fascinated by the interest attaching to the results at which they arrived, that they did not greatly care to expend their time in arranging their ideas in a strictly logical order, or even in establishing by irrefragable proof propositions which they instinctively felt, and could almost see to be true. With Gauss the case was otherwise. It may seem paradoxical, but it is probably nevertheless true that it is precisely the effort after a logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity and unnecessary difficulty. The fact is that there is neither obscurity nor difficulty in his writings, as long as we read them in the submissive spirit in which an intelligent schoolboy is made to read his Euclid. Every assertion that is made is fully proved, and the assertions succeed one another in a perfectly just analogical order… But when we have finished the perusal, we soon begin to feel that our work is but begun, that we are still standing on the threshold of the temple, and that there is a secret which lies behind the veil and is as yet concealed from us. No vestige appears of the process by which the result itself was obtained, perhaps not even a trace of the considerations which suggested the successive steps of the demonstration. Gauss says more than once that for brevity, he gives only the synthesis, and suppresses the analysis of his propositions. Pauca sed matura—few but well matured… If, on the other hand, we turn to a memoir of Euler's, there is a sort of free and luxuriant gracefulness about the whole performance, which tells of the quiet pleasure which Euler must have taken in each step of his work; but we are conscious nevertheless that we are at an immense distance from the severe grandeur of design which is characteristic of all Gauss's greater efforts.”

Henry John Stephen Smith

As quoted by Alexander Macfarlane, Lectures on Ten British Physicists of the Nineteenth Century (1916) p. 95, https://books.google.com/books?id=43SBAAAAIAAJ&pg=PA95 "Henry John Stephen Smith (1826-1883) A Lecture delivered March 15, 1902"

Pleasure , Secret , Age , Feeling

“The first demonstration (Disq. Arith., Arts. 125-145) which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal… and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished.”

Henry John Stephen Smith

Report on the Theory of Numbers (1859) Part I, p. 59.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

Art , Mathematics

“Legendre's Law of Quadratic Reciprocity' … is …the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. …[W]e find in the 'Opuscula Analytica' of Euler… a memoir… which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler… expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined… that he had obtained a satisfactory demonstration.”

Henry John Stephen Smith

Report on the Theory of Numbers (1859) Part I, pp. 56-57.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

Truth , Law , Imagination , Science