Mathematicians From Ireland

"The evidence is closing in more and more rigorously that the medium which transmits electrical and radiant effects must either completely accompany matter in bulk in its movements or else be entirely independent of such movements."

"The researches by which Sir Joseph Larmor will chiefly be remembered belong to the decade 1892–1901, which is now recognized as a transition period in physics. ... Before the end of the decade X-rays, electrons and radio-activity had again set experimental physics in feverish progress, to be followed later by revolutionary changes in the foundations of physical theory. But at the time when Larmor started on his main work there was little to inspire new ideas. ... Classical physics was indeed near the end of its tether. Of those who yet contrived to make substantial progress at this difficult stage—who brought classical physics finally to the point where new methods became inevitable—two names stand out prominently, Lorentz and Larmor. Their work had much in common, so that it is sometimes difficult to assess their contributions separately. Larmor’s reputation has perhaps been overshadowed by that of Lorentz. But on any estimate, Larmor’s achievements rank high; and his place in science is secure as one who re-kindled the dying embers of the old physics to prepare the advent of the new."

"A very large number of optical phenomena have been examined by various experimenters with a view to detecting an influence on them of the Earth's velocity of translation. The only such influence that has been announced is that found by Fizeau on the displacement of the plane of polarization of light, produced by transmission through a pile of glass plates: according to Fizeau's own view the experiment was uncertain owing to the numerous disturbing causes that had to be guarded against; and this doubt as to the feasibility of the observations has been fully shared by Maxwell and most authorities who have considered the matter."

"From remote ages the great question with which, since Newton's time, we have been familiar under the somewhat misleading antithesis of contact versus distance, has engaged speculation,—how is that portions of matter can interact on each other which seem to have no means of connexion between them. Can a body act where it is not?"

"The physical properties of fluid media, as regards change of state, and as regards capillary phenomena, have been closely illustrated in theory by consideration of a model medium, subject to internal expansive pressure, of kinetic or other origin, which is counteracted by the contractive effect of mutual attraction between its molecules—the latter force extending through much the greater, though for ordinary purposes still insensible, range. For liquids, the difference between these two much larger quantities, of different types, constitutes the transmitted hydrostatic pressure."

"The direct knowledge of matter that mankind can acquire is a knowledge of the average behaviour and relations of the crowd of molecules."

"The delusion of invincibility can never grow up in the mind of anyone except one who has never met a strong antagonist."

"In early times of Christianity, even those who used animal food themselves came to think of the vegetarian as one who lived a higher life, and approached more nearly to Christian perfection."

"Is it not reasonable to think that by far the greater part is solid and dark, and that this immense globe is encompassed with a thin covering of that resplendent substance from which the sun would seem to derive the whole of his vivifying heat and energy?"

"Comets encountering these precincts must be perplexed to decide between the two potentates claiming their allegiance, and perhaps on occasions pay their court to each in turns, throwing out tails, as they do so, in all sorts of anomalous and contradictory directions."

"Thus, not merely what it can do, but the rate at which it can do it, has to be considered in estimating the value of photography as an ally in astronomy."

"The planets revolved in circles because it was in their nature to do so, just as laudanum sends to sleep because it possesses a virtus dormitiva."

"He could see through the vanity and folly of a friend, and yet retain a never changing affection for him. Of his own life, he seldom or never spoke; he was not an egotist, and his own sayings or doings did not seem to interest him afterwards."

"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."

"We must confine ourselves to what we may term the great highways of the science; and... we must wholly pass by many outlying researches of great interest and importance, as we propose rather to exhibit in a clear light the most fundamental and indispensable theories, than to embarrass the treatment of a subject, already sufficiently complex, with a multitude of details, which, however important in themselves, are not essential to the comprehension of the whole."

"The problem of the direct determination of the primitive roots of a prime number is one of the 'cruces' of the Theory of Numbers. Euler, who first observed the peculiarity of these numbers, has yet left us no rigorous proof of their existence; though assuming their existence, he succeeded in accurately determining their number. The defect in his demonstration was first supplied by Gauss, who has also proposed an indirect method for finding a primitive root."

"'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."

"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."

"I do not know what Henry Smith may be at the subjects of which he professes to know something; but I never go to him about a matter of scholarship, in a line where he professes to know nothing, without learning more from him than I can get from any one else."

"Though not a poet or creative genius, he was... possessed of greater natural abilities than any one else whom I have known at Oxford. He had the clearest and most lucid mind, and a natural experience of the world and of human character hardly ever to be found in one so young. He took up all subjects at the right end; he knew whereabouts the truth lay even when he was imperfectly acquainted with the facts. And he was the most amiable and good-natured of young men. I might apply to him the words in which Plato describes the youthful Athenian Mathematician, Theaetetus, where he says: "In all my acquaintance, which is very large, I never knew any one who was his equal in natural gifts. He had a quickness of apprehension which was almost unrivalled, and he was exceedingly gentle. There was a union of qualities in him which I have never seen in any other, and should scarcely have thought possible, for quick wits have generally quick tempers... but he moved surely and smoothly and successfully in the paths of knowledge and enquiry. He flowed on silently like a river of oil. At his age it was wonderful. He was also surprisingly liberal about money, though his fortune was only moderate. (Theætetus 144)."

"In those days he was almost equally a lover of Classics and Mathematics. ...Even in the last years of his life he was in the habit of taking with him Greek books to read during the Vacation."

"His mathematical speculations could have been shared by a very few, not more than two or three, of his contemporaries at Oxford. Yet he did not withdraw himself from business or society. He was not the silent philosopher who is lost in reverie, or who, while acknowledged to be a mathematical genius, is pointed at by mankind as a poor and eccentric mortal. He was a thorough man of the world and greatly liked by everybody."

"He was very desirous to promote the interests of Natural Science in Oxford, and was in favour of some measure which would have made the knowledge of a portion of some one of the Natural Sciences the condition of obtaining a degree. The teachers of these sciences had long been fighting a battle against the older traditions of the University; they had now become the study of a few, but he clearly saw that they could never truly flourish until an interest in them was more generally diffused... But he was also the best friend that the older studies... for he could speak with authority, and he was firmly convinced that in education Science should not supersede Literature. He deplored equally the want of literary culture which he observed in many scientific men, and the gross ignorance of the most general facts of Science which prevails in the world at large, especially at English Universities and Public Schools. In a similar spirit he was anxious to encourage at Oxford the study of Medicine and also of Engineering, thinking that they would supply a missing link between the Physical Sciences and the older studies of the University."

"He was indulgent to the failings of young men, and felt a humane pity for persons who had lost their character. He was one of whom it might be said that 'he would have stood by a friend, not only in adversity, but in disgrace.'"

"[H]e had a difficulty in deciding between classics and mathematics, and there is a story to the effect that he finally solved the difficulty by tossing up a penny. He certainly used the expression: but the reasons which determined his choice in favor of mathematics were first, his weak sight, which made thinking preferable to reading, and secondly the opportunity..."

"Whewell, the master of Trinity College Cambridge, wrote The Plurality of Worlds... Whewell pointed out what he called law of waste traceable in the Divine economy; and his argument was that the other planets were waste effects, the Earth the only oasis in the desert of our system, the only world inhabited by intelligent beings; Sir David Brewster... wrote a fiery answer entitled "More worlds than one, the creed of the philosopher and the hope of the Christian." In 1855 Smith wrote an essay on this subject... in which the fallibility both of men of science and of theologians was impartially exposed. It was his first and only effort at popular writing."

"Though addicted to the theory of numbers, he was not in any sense a recluse; on the contrary he entered with zest into every form of social enjoyment in Oxford... He had the rare power of utilizing stray hours of leisure, and it was in such odd times that he accomplished most of his scientific work. After attending a picnic in the afternoon, he could mount to those serene heights in the theory of numbers... Then he could of a sudden come down from these heights to attend a dinner, and could conduct himself there, not as a mathematical genius lost in reverie and pointed out as a poor and eccentric mortal, but on the contrary as a thorough man of the world greatly liked by everybody."

"In 1858 he was selected by [the British Association] to prepare a report upon the Theory of Numbers. It was prepared in five parts, extending over the years 1859-1865. It is neither a history nor a treatise, but something intermediate. The author analyzes with remarkable clearness and order the works of mathematicians for the preceding century upon the theory of congruences, and upon that of binary quadratic forms. He returns to the original sources, indicates the principle and sketches the course of the demonstrations, and states the result, often adding something of his own. The work has been pronounced to be the most complete and elegant monument ever erected to the theory of numbers, and the model of what a scientific report ought to be."

"So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma."

"It is very difficult for us, placed as we have been from earliest childhood in a condition of training, to say what would have been our feelings had such training never taken place."

"The properties of bodies were investigated by several distinguished French mathematicians on the hypothesis that they are systems of molecules in equilibrium. The somewhat unsatisfactory nature of the results... produced... a reaction in favour of the opposite method of treating bodies as if they were... continuous. This method, in the hands of Green, Stokes, and others, has led to results the value of which does not at all depend on what theory we adopt as to the ultimate constitution of bodies."

"It may not sound very consistent with any such professed humility on my part, if I say to you that, after having served for the Quaternions during fourteen years, and having (as America seems to think) won my Rachel—to be my own by an intellectual marriage—I now wish to wind up several scientific projects, from which those quaternions had for a long time diverted me; and feel as if I were entering, or had already entered, on a new harvest of labour and reputation. As to Fame, if it have not been won or earned already, it is not likely that any future exertion will make it mine. But as to the Labour; that is a thing within everybody's power to judge of, even for himself. I have very long admired Ptolemy's description of his great astronomical Master, Hipparchus... "a labour-loving and truth-loving man."—Be such my epitaph!"

"The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention... And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning... It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional: extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might (no doubt) be represented by points upon a line, yet I thought that their simple successiveness was better conceived by comparing them with moments of time, divested, however, of all reference to cause and effect; so that the "time" here considered might be said to be abstract, ideal, or pure, like that "space" which is the object of geometry. In this manner I was led, many years ago, to regard Algebra as the Science of Pure Time: and an Essay, containing my views respecting it as such, was published in 1835. ...[I]f the letters A and B were employed as dates, to denote any two moments of time, which might or might not be distinct, the case of the coincidence or identity of these two moments, or of equivalence of these two dates, was denoted by the equation,B = Awhich symbolic assertion was thus interpreted as not involving any original reference to quantity, nor as expressing the result of any comparison between two durations as measured. It corresponded to the conception of simultaneity or synchronism; or, in simpler words, it represented the thought of the present in time. Of all possible answers to the general question, "When," the simplest is the answer, "Now:" and it was the attitude of mind, assumed in the making of this answer, which (in the system here described) might be said to be originally symbolized by the equation above written."

"Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be."

"To admire is, to me, questionless, the highest pleasure of life."

"Hamilton was apt to work by fits and starts. He has been known several times to work fourteen hours in one day, standing nearly all the while; but there were intervals of comparative inaction... Sometimes a letter was written and copied which was not sent for months, and then only the first sheet, with promise of the rest. It has even happened that the letter was knowingly never forwarded at all, and that when, long after, he found reason to wish to send it, he could not find it and sent the copy instead."

"Hamilton was not only an Irishman, but Irish: and this with curious oppositions of character. He was a non-combatant: there was too much kindness in his disposition to allow any fight to show itself. Impulsive and enthusiastic, with strong opinions and new views, he was never engaged in a scientific controversy... William Rowan Hamilton's preservative was his dread of wounding the feelings of others. In his youth, "Defender of the Absent" was his nickname. ...He had a morbid fear of being a plagiarist; and the letters which he wrote to those who had treated like subjects with himself sometimes contained curious and far-fetched misgivings about his own priority. But, with all this, there was a touch of the national temperament in him... an Irishman who never gets into a row may give quick but quiet symptoms of opposition of opinion, and of what, were it more than a rudiment, would be called pugnacity."

"Hamilton was a man who combined different talents to an extent which is often attributed, by exaggeration, to the possessor of one powerful faculty: but in his case there is abundant evidence. He was scholar, poet, metaphysician, mathematician, and natural philosopher. Highly imaginative and fluent of tongue, he was an orator in all that he knew; even in mathematics, to the details of which he could give almost a rhetorical cast in a letter. In metaphysics he was very well read, and could talk in a way which suggested a comparison to Southey, and a difference. Hamilton one day preached to Southey on this subject until the latter remarked, as they passed a ploughman, "If you had been Coleridge, you would have talked to that ploughman just as you have been talking to me.""

"He used to carry on, long trains of algebraic and arithmetical calculations in his mind, during which he was unconscious of the earthly necessity of eating; we used to bring in a 'snack' and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards."

"The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived... A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations."

"To the scientists of 1850, Hamilton's principle was the realization of a dream. ...from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. ...they made striking progress ...But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics."

"It still remained to be seen whether the laws of motion, as dependent on moving forces, could also be consistently transferred to spherical or pseudospherical space. This investigation has been carried out by Professor Lipschitz of Bonn. It is found that the comprehensive expression for all the laws of dynamics, , may be directly transferred to spaces of which the measure of curvature is other than zero. Accordingly, in this respect also, the disparate systems of geometry lead to no contradiction."

"… an undercurrent of thought was going on in my mind which gave at last a result, whereof it is not too much to say that I felt at once 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 which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away."

"The Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contemplation; according as ease of operation, or symmetry of expression, or clearness of thought, (the agere, the fari, or the sapere,) is eminently prized and sought for."