Mathematicians From England

"From the 3rd century onwards, orthodox Christianity, based on a Hebrew story and worshipping the Jew Jesus, also led many campaigns of anti-Semitism."

"Non-Newtonian calculi... have considerable potential as alternative approaches to traditional problems."

"Mathematics occupies a peculiar position in cultural life today. 'Everybody knows' that it is one of the most basic, and also ancient, types of knowledge; yet it is not part of normal cultural discourse, and few people know much about its historical development, or even that it has a history."

"Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics."

"To know him was to know one of the historic figures of all time, one of the immortals; and when he was really moved to speak, his eloquence equalled his genius."

"As a public teacher of mere striplings, I am often amazed by the facility and absence of resistance with which the principles of the infinitesimal calculus are accepted and assimilated by the present race of learners. When I was young, a boy of sixteen or seventeen who knew his infinitesimal calculus would have been almost pointed at in the streets as a prodigy, like Dante, as a man who had seen hell. Now-a-days, our Woolwich cadets at the same age, talk with glee of tangents and asymptotes and points of contrary flexure and discuss questions of double maxima and minima, or ballistic pendulums, or motion in a resisting medium, under the familiar and ignoble name of sums."

"It has been said that to appreciate what virtue and morals mean, men must live virtuous and moral lives. It is equally true, that a knowledge of the objects of science is not to be attained by any scheme of definitions, however carefully contrived. He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer."

"Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence."

"It always seems to me absurd to speak of a complete proof of a theorem being rigorously demonstrated. An incomplete proof is no proof and a mathematical truth not rigorously demonstrated is not demonstrated at all. I do not mean to deny that there are mathematical truths, morally certain, which defy and will probably to the end of time to defy proof, as, e.g. that every indecomposable integer polynomial function must represent an infinitude of primes. I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitation of our faculties in regard to time, which like space may be in its essence poly-dimensional, and that this and such sort of truths would become self-evident to a being whose mode of perception is according to superficially as distinguished from our own limitation to linearly extended time."

"As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention."

"Number, place, and combination . . . the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred."

"The object of pure Physic[s] is the unfolding of the laws of the intelligible world; the object of pure Mathematic[s] that of unfolding the laws of human intelligence."

"It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock."

"Grattan-Guiness's uniformly interesting and valuable account of the interwoven development of logic and related fields of mathematics . . . between 1870 and 1940 presents a significantly revised analysis of the history of the period. . . . [His] book is important because it supplies what has been lacking: a full account of the period from a primary mathematical perspective."

"Grattan-Guinness has achieved a synthesis here of remarkable historical and mathematical scope and sensitivity. His book is a 'must read' for historians of science and mathematics, as well as mathematicians."

"Mathematics is one of the most basic -- and most ancient -- types of knowledge. Yet the details of its historical development remain obscure to all but a few specialists."

"The great historian of logic and mathematics Ivor Grattan-Guinness passed away about a month ago, aged 73. ... Ivor's work and scholarship spans over an impressive range of topics and areas, and is bound to continue to influence many generations of scholars to come. It is a great loss."

"Opinions derived from long experience are exceedingly valuable, and outweigh all others, while they are consistent with facts and with each other; but they are worse than useless when they lead, as in this instance, to directly opposite opinions."

"No longer was light analogous to the discharge of a blunderbuss, but rather to the pulsating flight of birds."

"Every new discovery in science brings with it a host of new problems, just as the invention of the automobile brought with it gas stations, roads, garages, mechanics, and a thousand other subsidiary details."

"Professor Airy, in his thirty-fourth year, became . Thirty-eight years have since elapsed. Under his directions... the organization of the establishment at Greenwich has been completely transformed. ...He has contrived to establish newer and sounder methods of calculation and publication. He has introduced, constructed, mounted, and employed, a series of novel instruments for the advancement of astronomic research. Perhaps the finest transit-circle at present anywhere to be found is the one he there constructed in 1860, the circles being no less than six feet in diameter, and the telescope affixed between the two graduated disks being twelve feet long, and having an object-glass of as many as eight inches in aperture. Through this splendid apparatus the altitude of the stars, as well as the time of meridian passage, is now unerringly marked at the great national observatory. But the greatest of all the instruments established by him at Greenwich is a large, first-class ..."

"Already... he began those remarkable improvements in the method of calculating and publishing the observations which eventually became the law at Greenwich and at all the other great observatories. ...at Cambridge he superintended the construction and mounting... of a series of renowned astronomical instruments. In that observatory, he brought into use a noble specimen of the equatorial, being that peculiar description of telescope which has its fixed axis so directed to the pole of the heavens that the tube may be made to follow any star by a single motion. There, moreover, he brought into effective employment a mural circle of admirable construction, bearing a telescope which revolves in the plane of the meridian, the whole being rigidly bound into some immovable structure of ponderous masonry."

"Prof. Airy, once elevated to that position... he for nearly ten years—namely from 1827 to 1836—delivered with admirable effect, a series of public lectures on experimental philosophy, by which his scientific reputation was considerably advanced. ...it was one of the earliest means of effectively illustrating the marvelous phenomena constituting the now almost universally adopted undulatory theory of light. Two years after Prof. Airy's induction... the estimation in which he was held at the university was still further signalized by his election to the Plumian Professorship. ...he at once obtained, by right of his position, the supreme command of the ."

"From the Colchester Grammar-School, when eighteen years of age, he went, in 1819, to Trinity College, Cambridge. Three years afterward he was elected to a scholarship. In 1823, on his graduating B. A., young Airy came out as Senior Wrangler. In 1824 he obtained his fellowship at Trinity. His degree of M. A. was taken in 1826, and he was simultaneously elected, though only then in his twenty-fifth year, as Lucasian Professor at Cambridge. Illustrious philosophers like Barrow and Newton had preceeded him... Latterly, however, the office had become, in a great measure, purely honorary, and might also be said to have degenerated into a sinecure."

"Sir George Biddell Airy, English Astronomer Royal from 1836 till 1881, died on January 2d, after a few months' illness, in the ninety-first year of his age. A sketch of his life and works up to that time, with a portrait, were given in The Popular Science Monthly for May, 1873. He after that made the preparations for the equipment of the British expedition for the observation of the of 1874, a subject on which he had been engaged since 1836. He retired from his office in the Greenwich Observatory in 1881, after forty-five years of service."

"Newton pointed out and assigned generally, not only the nature and the magnitude of the periodical forces which are concerned in producing the tides, but likewise indicated their true character as undulations, in one very remarkable proposition, as well as in a special explanation of... the tides of the Port of Batsha. The equilibrium theory of Daniel Bernoulli adopted the first part of Newton's views but altogether neglected the second. ...Professor Airy ...has pronounced the theory proposed by La Place in the Mécanique Céleste,—if viewed with reference to the boldness and comprehensive character of its design rather than to the success of its execution—"as one of the most splendid works of the greatest mathematician of the past age." The problem, however, was not considered by [La Place] in the most general form which it is capable of receiving. He assumed the earth to be entirely covered by water, and its depth to be uniform, at least throughout the same parallel of latitude, and he neglected the resistance both of the particles of the fluid amongst each other, and of that which arises from the irregular surfaces in the channels over which the tide is transmitted. He was consequently obliged to omit the consideration of the tides in canals, rivers, and narrow seas, which constitute some of the most interesting, and by no means the most unmanageable, of the problems which later, and even in some respects more simple, investigations of the oscillations of the sea have brought within the control of analysis. Imperfect, however, as the results of this theory were as it came from the hand of its author, their importance cannot easily be estimated too highly. Dr. Young adopted the general principles which they involved, though he has subjected them to a totally different treatment; and Professor Airy, who has materially simplified the investigations which it contains, by rejecting some conditions which they included, such as the density of the sea, by which they were made needlessly difficult and complicated, has not only verified the more remarkable of the conclusions at which La Place arrived, but has also made important use of his methods in his own theory of waves and tides, which is by far the most complete and comprehensive that has ever yet appeared."

"[A]s in the times of Flamsteed and Halley, the earnest zeal of men of science occasionally led to much controversy and bitterness... Airy was by no means exempt... He was a man of keen sensitiveness, though it was combined with great steadiness of temper, and he never hesitated to attack theories and methods that he considered to be scientifically wrong. This led to differences with Ivory, Challis, South, Cayley, , and others; but however much he might differ from them he was always personally courteous, and the disputes generally went no farther than as regarded the special matter in question. Almost all these controversial discussions were carried on openly, and were published in the Athenaeum, the , or elsewhere; for he printed nearly everything that he wrote, and was very careful in the selection of the most suitable channels for publication. He regarded it as a duty to popularize as much as possible the work done at the Observatory, and to take the public into his confidence. And this he effected by articles communicated to newspapers, lectures, numerous Papers written for scientific societies, reports, debates, and critiques."

"His nature was essentially cheerful, and literature of a witty and humourous character had a great charm for him. He was very fond of music and knew a great number of songs; and he was well acquainted with the theory of music: but he was no performer. He did not sketch freehand but made excellent drawings with his ."

"He was extremely well versed in mechanics, and in the principles and theory of construction, and took the greatest interest in large engineering works. This led to much communication with Stephenson, Brunel, and other Engineers, who consulted him freely on the... great works on which they were engaged: in particular he rendered much assistance in connection with the construction of the over the Menai Straits."

"Antiquities and Architecture were very favourite subjects with him. He had visited most of the camps and castles in the United Kingdom and was never tired of tracing their connection with ancient military events: and he wrote several Papers on this subject, especially those relating to the Roman Invasions of Britain. Ecclesiastical Architecture he was very fond of: he had visited nearly all the cathedrals and principal churches in England, and many on the Continent..."

"He eagerly... mastered the Physical Astronomy in the most thorough manner, as was evidenced by his Papers collected in his "Mathematical Tracts," his investigation of the Long Inequality of the Earth and Venus, and many other works. As Plumian Professor he had charge of the small Observatory at Cambridge, where he did a great deal of the observing and reduction work himself, and became thoroughly versed in the practical working of an Observatory. The result of this was immediately seen in the improved methods which he introduced at Greenwich, and which were speedily imitated at other Observatories. Optics and the Undulatory Theory of Light had been very favourite subjects with him, and he had written and lectured frequently upon them. In the construction of the new and powerful telescopes and other optical instruments required from time to time this knowledge was very essential, for in its instrumental equipment the Greenwich Observatory was entirely remodelled during his tenure of office. And in many of the matters referred to him, as for instance that of the Lighthouses, a thorough knowledge of Optics was most valuable. He had made a great study of the theory and construction of clocks, and this knowledge was invaluable to him at Greenwich in the establishment of new and more accurate astronomical clocks, and especially in the improvement of chronometers. He had carefully studied the theory of pendulums, and had learned how to use them in his experiments in the Cornish mines. This knowledge he afterwards utilized very effectively at the Harton Pit in comparing the density of the Earth's crust with its mean density; and it was very useful to him in connection with geodetic surveys and experiments on which he was consulted. And his mechanical knowledge was useful in almost everything."

"[H]is custom was to work in his official room from 9 to about 2.30... He then took a brisk walk and dined at about 3.30. ...He... had tea, and from about 7 to 10 he worked in the same room with his family. He would never retire to a private room, and regarded the society of his family as highly beneficial in "taking the edge off his work." His powers of abstraction were remarkable: nothing seemed to disturb him; neither music, singing, nor miscellaneous conversation. He would then play a game or two at cards, read a few pages of a classical or historical book, and retire at 11."

"His courtesy was unfailing: no amount of trouble could shake it. Whether it was the Secretary of the Admiralty, or a servant girl wanting her fortune told: whether a begging-letter for money, or miscellaneous invitations: all had their answer in the most clear and courteous language. But he would not grant personal interviews when he could avoid it: they took up too much of his time."

"He was made for work and could not long be happy without it. Whatever subject he was engaged upon, he kept his object clearly in view, and made straight for it, aiming far more at clearness and directness than at elegance... or symmetry of arrangement."

"He never shirked arithmetical work: the longest and most laborious reductions had no terrors for him, and he was remarkably skilful with the various mathematical expedients for shortening and facilitating arithmetical work of a complex character. This power of handling arithmetic was of great value to him in the Observatory... He regarded it as a duty to finish off his work, whatever it was, and the writer well remembers his comment on the mathematics of one of his old friends, to the effect that "he was too fond of leaving a result in the form of three complex equations with three unknown quantities.""

"[A] very important feature of his [mathematical] investigations was the thoroughness of them. He was never satisfied with leaving a result as a barren mathematical expression. He would reduce it, if possible, to a practical and numerical form, at any cost of labour: and would use any approximations which would conduce to this result, rather than leave the result in an unfruitful condition."

"His nature was eminently practical, and any subject which had a distinctly practical object, and could be advanced by mathematical investigation, possessed interest for him. And his dislike of mere theoretical problems and investigations was proportionately great. He was continually at war with some of the resident Cambridge mathematicians on this subject. ...and conducted an interesting and acrimonious private correspondence with Professor Cayley on the same ..."

"The ruling feature of his character was undoubtedly Order. ...He seems not to have destroyed a document of any kind whatever: counterfoils of old cheque-books, notes for tradesmen, circulars, bills, and correspondence of all sorts were carefully preserved in the most complete order... To a high appreciation of order he attributed in a great degree his command of mathematics, and sometimes spoke of mathematics as nothing more than a system of order carried to a considerable extent. In everything he was methodical and orderly, and he had the greatest dread of disorder creeping into the routine work of the Observatory, even in the smallest matters."

"His eye-sight was peculiar, and required correction by spectacles the lenses of which were ground to peculiar curves according to formulae which he himself investigated: with these spectacles he saw extremely well, and he commonly carried three pairs, adapted to different distances: he took great interest in the changes that took place in his eye-sight and wrote several Papers on the subject."

"The history of the early part of his life was written in great detail and contained a large quantity of family matter which was evidently not intended for publication. This part of the Autobiography has been compressed. The history of the latter part of his life was not written by himself at all, and has been compiled from his Journal and other sources. In both these cases, and occasionally in short paragraphs throughout the narrative, it has been found convenient to write the history in the third person."

"[E]very subject of a distinctly practical nature, which could be advanced by mathematical knowledge, had an interest for him... Amongst such subjects were Tides and Tidal Observations, Clockwork, and the Strains in Beams and Bridges. A certain portion of his time was also given to Lectures, generally on current astronomical questions, for he held it as his duty to popularize the science as far as lay in his power. And he... took a very active part in the discussions and business of the [Royal Astronomical] Society. He also did much work for the Royal Society and... for the British Association."

"There was... much work on important subjects more or less connected with his official duties—such as geodetical survey work, the establishment of time-balls at different places, longitude determinations, observation of eclipses, and the determination of the density of the Earth. Lastly, there was a great deal of time and work given to questions... on which the Government asked his assistance in the capacity of general scientific adviser: such were the Correction of the Compass in iron ships, the Railway Gauge Commission, the Commission for the Restoration of the Standards of Length and Weight, the Maine Boundary Lighthouses, the Westminster Clock, the London University, and many other questions."

"His real business life commenced after he became Astronomer Royal, and from that time forward, during the 46 years that he remained in office, he was so entirely wrapped up in the duties of his post that the history of the Observatory is the history of his life."

"The Illustrated Review... to which we are indebted for the preceding statements, remarks that, since the death of Sir John Herschel... Sir George Airy, the Astronomer Royal, is the admitted master of the sublime science. There are other eminent English astronomers—as John Hinde, the discoverer of many asteroids, and John Adams, also a Cambridge Senior Wrangler and the rival of Urban Leverrier, who groped his way by mathematical calculation to the discovery of the hitherto unknown planet Neptune. If incidents as brilliant and remarkable as these are wanting in the history of Sir George Airy, his claims to respect are equally valuable, solid, and enduring."

"For his successful optical theories he has... the Copley Gold Medal of the . The of the same society has been given to him in recompense for his tidal investigations. Twice the Gold Medal of the has been his—first, for his discovery of an inequality of long period in the movements of Venus and the earth; secondly, in return for his reduction of the planetary observations. He has been enrolled among the most honored members of the Royal Astronomical Society, of the , and of the Institute of Civil Engineers. For many years he has been among the foreign correspondents to the Institute of France, as well as of several other scientific academies on the Continent. On May 17, 1872, Sir George was gazetted a Knight of the Bath."

"The writings of the Astronomer Royal are numerous. He has contributed largely to the Cambridge Transactions and the Philosophical Transactions. His pen has notably illustrated the memoirs of the Astronomical Society. He has written abundantly for the ', and still more abundantly, under his reversed initials, A. B. G., in the columns of the Athenœum. His principle works, however, are...: "Gravitation," published in 1837, was written originally for the "Penny Cyclopædia." "Mathematical Tracts" have reached a fourth edition, as have also his "Ipswich Lectures on Astronomy." In 1861 appeared his treatise on "Errors of Observation;" in 1869 his treatise on "Sound," and in 1870 his treatise on "Magnetism." Sir George Airy's well-known work on "Trigonometry" was published in 1855. Another work of his, entitled "Figure of the Earth," has yet to be named, as well as the luminous paper on "Tides and Waves," contributed by him, first of all, to the "." Even while simply Professor of Astronomy at Cambridge his "Astronomical Observations," issuing... between 1829 and 1838, extended in nine quarto volumes, and were adopted at once as models for that class of publication."

"Sir George Airy has been repeatedly called into council on matters of grave difficulty by the government. He was chairman of the royal commission empowered to supervise... contriving new standards of length and of weight... He was consulted... in respect to the bewildering disturbance of the magnetic compass in iron-built ships of war. Thereupon he contrived an ingenious system of mechanical construction, through a combination of magnets and iron. ...and the system was generally adopted. He conducted the astronomical observations necessary to the drawing of the boundary-line now traceable on the map of the New World between the Canadas and the United States. During the battle of the gauges in the railway world Sir George Airy strenuously advocated the narrow gauge..."

"In 1854 he approximated more nearly than any previous investigation... the weight of the earth, through a series of experiments on the relative vibration of a pendulum at the top and bottom of Harton Coal-pit."

"During Sir George Airy's rule at the observatory he has... thrown considerable light on ancient chronology by his ingenious calculation of some of the most renowned of historical eclipses."

"I had long wished for some opportunity of endeavouring to explain... the principles on which the instruments of an Observatory are constructed, (omitting all details, so far as they are merely subsidiary,) and the principles on which the observations made with these instruments are treated... Such an opportunity appeared to present itself in the course of Lectures which I engaged to give to the Members of the and their friends."