Fellows Of The Royal Society

"I am not surprised that men are not thankful to me; but I wonder that they are not grateful to God for the good which he has made me the instrument of conveying to my fellow-creatures."

"A sincere acquiescence in the dispensations of Providence will check discomposure of mind beyond any thing. It will produce a calm in the midst of a storm."

"The highest powers in our nature are our sense of moral excellence, the principle of reason and reflection, benevolence to our creatures and our love of the Divine Being."

"It should not forgotten, however, that it was his "Observations on the natural history of the Cuckoo" (1788) that had won him membership in the Royal Society 10 years before. His work on bird migration, although done at about the same time, was not published until after his death in 1823. Both of these papers were landmarks in ornithological history, for no one prior to Jenner had approached these problems, of brood parasitism and migration, in anywhere near so comprehensive a fashion or with such searching questions."

"Like the ski resort full of girls hunting for husbands and husbands hunting for girls the situation is not as symmetrical as it might seem."

"Behold a universe so immense that I am lost in it. I no longer know where I am. I am just nothing at all. Our world is terrifying in its insignificance."

"The geometrical spirit is not so tied to geometry that it cannot be detached from it and transported to other branches of knowledge. A work of morals or politics or criticism, perhaps even of eloquence, would be better (other things being equal) if it were done in the style of a geometer. The order, clarity, precision and exactitude which have been apparent in good books for some time might well have their source in this geometric spirit. ...Sometimes one great man gives the tone to a whole century; Descartes], to whom one might legitimately be accorded the glory of having established a new art of reasoning, was an excellent geometer."

"At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée."

"Men are not willing to suffer the decision of things to be too easie, and therefore they mingle their own prejudices with truths, and so create greater perplexities than are Naturally found therein; and those scruples, which our selves frame, give us the most pain to untangle."

"It is more reasonable to remove error from truth, than to venerate error because it is mix'd with truth."

"We can never add more truth to what is true already, nor make that true which is false."

"But why then did the Ancient Priestesses always answer in Verse? ...To this Plutarch replies... That even the Ancient Priestesses did now and then speak in Prose. And besides this, in Old times all People were born Poets. ...[T]hey had no sooner drank a little freely, but they made Verses; they had no sooner cast their eyes on a Handsom Woman, but they were all Poesy, and their very common discourse fell naturally into Feet and Rhime: So that their Feasts and their Courtships were the most delectable things in the World. But now this Poetick Genius has deserted Mankind: and tho' our passions be as ardent... yet Love at present creeps in humble prose. ...Plutarch gives us another reason ...that the Ancients wrote always in Verse, whether they treated of Religion, Morality, Natural Philosophy or Astrology. Orpheus and Hesiod, whom every body acknowledges for Poets, were Philosophers also: and Parmenides, Xenophanes, Empedocles, Eudoxus, and Thales... [the] Philosophers, were Poets too. It is very strange indeed that Poetry should be elder Brother to Prose... but it is very probable... precepts... were shap'd into measured lines, that they might be the more easily remembred: and therefore all their Laws and their rules of Morality were in Verse. By this we may see that Poetry had a much more serious beginning than is usually imagin'd, and that the Muses have of late days mightily deviated from their original Gravity."

"Now, the Priests who belonged to the Temples, scorn'd to use the same Customs in common with these Gypsies; for they thought themselves to be a nobler and graver sort of Fortune-tellers; which makes a mighty difference, I'll assure you, in this great affair."

"[A]bout the time of Alexander the Great, a little before Pyrrhus's days, there appear'd in Greece certain great Sects of Philosophers, such as the Peripateticks and Epicureans, who made a mock of Oracles. The Epicureans especially made sport with the paltry Poetry that came from Delphos. For the Priests hammered out their Verses as well as they could, and they often times committed faults against the common Rules of Prosodia. Now those Fleering Philosophers were mightily concerned that Apollo, the very God of Poetry, should come so far behind Homer, who was but a meer mortal, and was beholding to the same Apollo for his inspirations."

"It was to little purpose to excuse the matter, by saying, that the badness of the Verses was a kind of Testimony that they were made by a God, who nobly scorn'd to be tyed up to rules and to be confined to the Beauty of a Style. For this made no impression upon the Philosophers; who, to turn this answer into ridicule, compared it to the Story of a Painter, who being hired to draw the Picture of a Horse tumbling on his Back upon the ground, drew one running full speed: and when he was told, that this was not such a Picture as was bespoke, he turned it upside down, and then ask'd if the Horse did not tumble upon his back now. Thus these Philosophers jeered such Persons, who by a way of arguing that would serve both ways, could equally prove that the Verses were made by a God, whether they were good or bad."

"So that at length the Priests of Delphos being quite baffled with the railleries of those learned Wits, renounced all Verses, at least as to the speaking them from the Tripos; for there were still some Poets maintain'd in the Temple, who at leisure turned into Verse, what the Divine fury had inspired the Pythian Priestess withal in Prose. It was very pretty, that Men could not be contented to take the Oracle just as it came piping hot from the Mouth of their God. But perhaps, when they had come a great way for it, they thought it would look silly to carry home an Oracle in Prose."

"[T]he intellectual changes of Louis XIV's reign touch the history of science—especially as they represent the extension of the scientific method into other realms of thought. ...we meet the beginnings of the criticism of the French monarchy... acute criticism from... the French intelligentsia who could claim to understand the... state better than the king himself. ...The funeral orations of Fontenelle call attention to an aspect of this movement... [i.e.,] the initial effect of the new scientific movement on political thought. ...The first result ...as Fontenelle makes clear, was the insistence that politics requires the inductive method, the collection of information, the accumulation of concrete data and statistics. ...He describes ...how Vauban... travelled over France, accumulating data, seeing the condition[s]... for himself, studying commerce and the possibilities of commerce... gaining a knowledge of local conditions. Vauban, says Fontenelle, did more than anybody else to call mathematics out of the skies... [he] put statistics to the service of modern political economy and first applied the rational and experimental method in matters of finance. ...Fontenelle tells us that ...Sir , the author of Political Arithmetic, showed how much of the knowledge requisite for government reduces itself to mathematical calculation."

"Fontenelle provides an example of the transfer of the scientific spirit, and the application of methodical doubt, in... the History of Oracles. In a sense he is one of the predursors of the comparative method in the history of religion—the collection of myths of all lands to throw light on the development of human reason. ...he recommends the study of primitive tribes in our own day ...He treats myths as... a natural product, subject to scientific analysis—not the fruits of conscious imposture but the characteristic of a certain stage in human development. The human mind he regards as... the same in all times and ages, but subject to local influences... Here is a self-conscious attempt to show how the scientific method could receive extended application and could be transferred from the examination of purely material phenomena even into... human studies."

"His wit, his Learning, his Knowledge of Mankind, his exquisite Taste in all that is Polite, the Fire of his Imagination, the uncommon Felicity of his Eloquence, and the ready Turn of his Expression, are Reasons which the Publick will think very natural to direct me in this Address to Your Lordship."

"A mere nothing, a tiny fibre, something that could never be found by the most delicate anatomy, would have made of Erasmus and Fontenelle two idiots, and Fontenelle himself speaks of this very fact in one of his best dialogues."

"Although I am not aware of having omitted any thing that is requisite to the full explanation of the subject, yet I cannot flatter myself that it will be thoroughly understood from this Work alone. For, in general it may be laid down as true, that no doctrine, of novelty and intricacy, can be completely taught by a single Treatise. It seems to be indispensably necessary for the student, that the subject should be put under several points of view: that if not apprehended under one, it may be under another."

"There is another point... and that is the method of demonstration by geometrical figures. In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise [footnote: Methodus inveniendi, &c.], he introduces geometrical figures, but almost entirely, for the purpose of illustration: and finally, in the tenth volume of the Novi Comm. Petrop. as Lagrange had done in the Miscellanea Taurinensea, he expounds the calculus, in its most refined state, entirely without the aid of diagrams and their properties. A similar history will belong to every other method of calculation, that has been advanced to any degree of perfection."

"The Authors who write near the beginnings of science, are, in general the most instructive: they take the reader more along with them, shew him the real difficulties, and, which is a main point, teach him the subject, the way by which they themselves learned it."

"On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"In 1810 a work was published in Cambridge under the following title—A Treatise on Isoperimetrical Problems and the Calculus of Variations. By Robert Woodhouse... This work details the history of the Calculus of Variations from its origin until the close of the eighteenth century, and has obtained a high reputation for accuracy and clearness. During the present century some of the most eminent mathematicians have endeavored to enlarge the boundaries of the subject, and it seemed probable that a survey of what had been accomplished would not be destitute of interest and value. Accordingly the present work has been undertaken... As the early history of the Calculus of Variations had been already so ably written, it was unnecessary to go over it again; but it seemed convenient to commence with a short account of the two works of Lagrange and a work of Lacroix..."

"Taylor's method... has no recommendation from its neatness and perspicuity, but is justly censured by John Bernoulli for its obscure conciseness."

"The methods of the Bernoullis and of Taylor, were held, at the time of their invention, to be most complete and exact. Several imperfections, however, belong to them. They do not apply to problems involving three or more properties; nor do they extend to cases involving differentials of a higher order than the first: for instance, they will not solve the problem, in which a curve is required, that with its radius of curvature and evolute shall contain the least area. Secondly, they do not extend to cases, in which the analytical expression contains, besides x, y, and their differentials, integral expressions; for instance, they will not solve the second case proposed in James Bernoulli's Programma if the Isoperimetrical condition be excluded; for then the arc s, an integral, since it =\int \!dx \sqrt(1+\frac{dy^2}{dx^2}), is not given. Thirdly, they do not extend to cases, in which the differential function, expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever."

"To history we shall adhere no farther, than is sufficient to preserve an unbroken series of methods gradually becoming more exact and extensive; the series beginning with the first rude, though perfectly just, method of James Bernoulli, and ending with Lagrange's exquisite and refined Calculus of Variations."

"The 'language theory' is inadequate as a description of the nature of mathematics."

"Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations."

"Logical analysis is indispensable for an examination of the strength of a mathematical structure, but it is useless for its conception and design. The great advances in mathematics have not been made by logic but by creative imagination."

"The function of logic in mathematics is critical rather than constructive."

"There is the definition [of mathematics], boldly proposed by Pierce that 'Mathematics is the science which draws necessary conclusions', and more explicitly formulated by Russel that 'Pure Mathematics is the class of all propositions of the form "p implies q"... it was... the purpose of Russell's treatise to provide a complete, exact and convincing justification of this definition... instead, he and Whitehead collaborated to give a magisterial account of the Principia Mathematica."

"At each stage of in the advance of mathematical thought the outstanding characteristics are novelty and originality. That is why mathematics is such a delight to study, such a challenge to practise and such a puzzle to define."

"The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction."

"The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life. We turn, then, from the attempt to characterize the material object of mathematics to an attempt to determine its formal object, i.e., its methodology."

"From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics."

"This book... is primarily and essentially an account of the discovery or invention of mathematical concepts, and the historical material is... divided and classified, neither chronologically nor biographically, but philosophically."

"The brilliant summaries by Bourbaki (1969) and Weyl (1951)... set a high standard of exposition, but... there is room for a history of mathematical ideas which will demand less mathematical expertise and offer a more detailed account of the motivation of research."

"The professional mathematician can scarcely avoid specialization and needs to transcend his private interests and take a wide synoptic view of the whole landscape of contemporary mathematics. His scientific colleagues are continually seeking enlightenment on the relevance of mathematical abstractions. The undergraduate needs a guidebook to the topography of the immense and expanding world of mathematics. There seems to be only one way to satisfy these varied interests... a concise historical account of the main currents... Only by a study of the development of mathematics can its contemporary significance be understood."

"Ninety per cent of all the mathematics we know has been discovered (or invented), in the last hundred years... the advances made in each of some dozen directions are converging into one single discipline uniting algebra, topology and analysis."

"During this century mathematics has been transformed..."

"The history of mathematics throws little light on the psychology of mathematical invention."

"What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy?"

"Most mathematicians are by nature Platonists who cheerfully, unreflectingly and habitually employ such loaded phrases as 'We assume there exists...' or 'Therefore there exists...' an entity with such and such characteristics. Challenged by the realist they would probably reply that since the truths of mathematics are absolute, universal and eternal it is hard indeed to deny them an existence independent of human intelligence."

"For the great majority of mathematicians, mathematics is... a whole world of invention and discovery—an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor."

"Mathematics has also been developed as a philosophy, in the sense in which this term is defined by A.N. Whitehead as 'the endeavor to frame a coherent, logical and necessary system of general ideas in terms of which every element of our experience can be interpreted'. Substitute 'mathematics' for 'experience' and we have an admirable description of its speculative and philosophic development. ...Philosophy of mathematics... has its paradoxes and antimonies, and also diverse schools of thought..."

"As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field... have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems... still remain as challenges."

"Mathematical activity has taken the forms of a science, a philosophy and an art."

"An English Court cannot judge by the light of nature."