Mathematicians From England

"Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful."

"A mechanistic model has the following advantages: 1. It contributes to our scientific understanding of the phenomenon under study. 2. It usually provides a better basis for extrapolation (at least to conditions worthy of further experimental investigation if not through the entire range of all input variables). 3. It tends to be parsimonious (i.e, frugal) in the use of parameters and to provide better estimates of the response"

"An innovative discussion of building empirical models and the fitting of surfaces to data. Introduces the general philosophy of response surface methodology, and details least squares for response surface work, factorial designs at two levels, fitting second-order models, adequacy of estimation and the use of transformation, occurrence and elucidation of ridge systems, and more. Some results are presented for the first time. Includes real-life exercises, nearly all with solutions."

"The penalty for scientific irrelevance is, of course, that the statistician's work is ignored by the scientific community."

"A man in daily muddy contact with field experiments could not be expected to have much faith in any direct assumption of independently distributed normal errors."

"Of the many references to Newton in 18th-century electrical writings only a small number were to the Principia, the greater part by far were to the Opticks. This was true not alone of the electrical writings but also in other fields of experimental enquiry. ...[The Opticks] would allow the reader to roam, with great Newton as his guide, through the major unresolved problems of science and even the relation of the whole world of nature to Him who had created it. ...in the Opticks Newton did not adopt the motto... —Hypotheses non fingo; I frame no hypotheses—but, so to speak, let himself go, allowing his imagination full reign and by far exceeding the bounds of experimental evidence."

"Opticks was out of harmony with the ideas of 19th-century physics. ...an exposition of the "wrong" (i.e., corpuscular) theory of light,—even though it also contained many of the basic principles of the "correct" (i.e., wave) theory. Not only had Newton erred in his choice... but also he apparently had found no insuperable difficulty in simultaneously embracing features of two opposing theories. ...by adopting a combination of the two theories at once, he had violated one of the major canons of 19th-century physics... Today our point of view is influenced by the theory of photons and matter waves, or the... complementarity of Neils Bohr; and we may read with a new interest Newtons ideas on the interaction of light and matter or his explanation of the corpuscular and undulatory aspects of light."

"My quotations from Newton suggest the motive which induced him to take a stand against the use of hypotheses, namely, the danger of becoming involved in disagreeable controversies. ...Newton could no more dispense with hypotheses in his own cogitations than an eagle can dispense with flight. Nor did Newton succeed in avoiding controversy."

"When Newton saw an apple fall, he found In that slight startle from his contemplation ... A mode of proving that the earth turn'd round In a most natural whirl, called 'gravitation'."

"The history of mathematics and mechanics for a hundred years subsequent to Newton appears primarily as a period devoted to the assimilation of his work and the application of his laws to more varied types of phenomena. So far as objects were masses, moving in space and time under the impress of forces as he had defined them, their behaviour was now, as a result of his labours, fully explicable in terms of exact mathematics."

"A student of the history of physical science will assign to Newton a further importance which the average man can hardly appreciate. ...the separation ...of positive scientific inquiries from questions of ultimate causation."

"There is a basic incompatibility between any organization and freedom of thought. Suppose Newton had founded a Church of Newtonian Physics and refused to show his formula to anyone who doubted the tenets of Newtonian Physics?"

"Newton is known for humbly declaring that he had achieved his great breakthroughs by 'standing on the shoulders of giants.' Though this may be true in part, it is largely humbug. Newton was hardly humble, and it would be just as true to say that he achieved greatness by stamping on the shoulders of giants. When others, such as Robert Hooke and Gottfried Leibniz, made breakthroughs in fields he was also researching, Newton fought ferociously to deny them credit for their work."

"If Sir Isaac Newton had not been distinguished as a mathematician and a natural philosopher, he would have enjoyed a high reputation as a theologian."

"[T]he life and writings of Sir Isaac Newton abound with the richest counsel. Here the philosopher will learn the art by which alone he can acquire an immortal name. The moralist will trace the lineaments of a character adjusted to all the symmetry of which our imperfect nature is susceptible; and the Christian will contemplate with delight the high-priest of science quitting the study of the material universe,—the scene of his intellectual triumphs,—to investigate with humility and patience the mysteries of his faith."

"The landscape has been so totally changed, the ways of thinking have been so deeply affected, that it is very hard to get hold of what it was like before... It is very hard to realize how total a change in outlook he has produced."

"How does the world recognizes England, the United Kingdom, as the country that gave birth to the modern age? It was not Newton but Galilei who opened the Moderna age."

"Here lies Isaac Newton, Knight, Who, by a Vigour of Mind almost supernatural, First demonstrated The Motions and Figures of the Planets, The Paths of the Comets, and the Tides of the Ocean. He diligently investigated The different Refrangibilities of the Rays of Light, And the Properties of the Colours to which they give rise. An assiduous, sagacious, and faithful Interpreter Of Nature, Antiquity, and the Holy Scriptures, He asserted his Philosophy of the Majesty of God, And exhibited in his conduct the Simplicity of the Gospel. Let mortals rejoice That there has existed such and so great An Ornament of Human Nature."

"And from my pillow, looking forth by light Of moon or favouring stars, I could behold The antechapel where the statue stood Of Newton, with his prism and silent face, The marble index of a mind for ever Voyaging through strange seas of Thought, alone."

"He was unhappy with the relativity of motion, even though it is a consequence of his equations, and to escape it he postulated the existence of "absolute" space, with respect to which true rest and motion are defined."

"It is one of the most intriguing facts in the history of science that the two most influential theories concerning the stars—Newton's theory of gravitation and Eddington's theory of stellar construction—were each developed so successfully although Newton was ignorant of the origin of gravitation and Eddington of the origin of stellar energy."

"During the Middle Ages the universe was regarded as finite, with the earth at its centre. The idea was abandoned during the Scientific Renaissance, and the universe came to be pictured as an indefinitely large number of stars scattered throughout infinite Euclidean space. This conception appeared to be a necessary consequence of the theory of gravitation; for, as Newton pointed out, a finite material universe in infinite space would tend to concentrate in one massive lump."

"Due to the genius and labours of Newton almost all the problems presented by the motions of the planets had been mastered. Newton had shown for all time that these motions could be completely accounted for if it were assumed that the same laws of nature, and in particular gravity, operated in the celestial realm as well as in the terrestrial. Although the old Aristotelian distinction between the corrupt earth and the incorruptible heavens was thus finally abandoned, the stellar realm still lay beyond the range of scientific investigation. The natural step, taken by Digges and Bruno, of likening the stars to the sun and scattering them throughout space was still only a step of the imagination."

"Such, then, is the great Newtonian induction of universal gravitation, and such its history. It is indisputably and incomparably the greatest scientific discovery ever made, whether we look at the advance which it involved, the extent of the truth disclosed, or the fundamental and satisfactory nature of this truth."

"The reader will recollect that we are here speaking of the Principia as a mechanical treatise only... As a work on dynamics, its merit is, that it contains a wonderful store of refined and beautiful mathematical artifices, applied to solve all the most general problems which the subject offered. It can hardly be said to contain any new inductive discovery respecting the principles of mechanics; for though Newton's "Axioms or Laws of Motion," which stand at the beginning of the book, are a much clearer and more general statement of the grounds of mechanics than had yet appeared, it can hardly be said that they contain any doctrines which had not been previously stated or taken for granted by other mathematicians."

"When he [Newton] uttered his Hypotheses non fingo he was saying in a very abbreviated, and hence cryptic way: In induction, I do not invent hypotheses, and in deduction I do not demonstrate from them. More fully, he meant that the inductive side of scientific method has a beginning, a middle, and an end and all must be complete before any deductive system is set up. The beginning consists in '"hinting several things" or making "conjectures" about the causes of phenomena...because they are "plausible consequences" drawn from the facts...they are not derived, like Descartes' conclusions, merely by the Light of Reason or intuition. Although hypothetical in character, Newton did not call them "hypotheses". The middle consists of examining these "hints" and improving them by observations and the tests of experiment. The end is defined by his remark: "and if no exception occur from phenomena, the conclusion may be pronounced generally" and considered "proved" as a "general law of nature". "Afterwards,", the deduction proceeds by assuming the conclusions established as principles, and from them demonstrating the phenomena...The peculiar character of this method, the stress upon experience and the rejection of hypotheses of the Cartesian kind, may be briefly described in Berkeley's words: "It is one thing to arrive at general laws of nature from a contemplation of of the phenomena, and another to frame an hypothesis, and from thence deduce the phenomena (S, 229)."

"The mechanical philosophy is a case of being victimized by metaphor. I choose Descartes and Newton as excellent examples of metaphysicians of mechanism malgré eux, that is to say, as unconscious victims of the metaphor of the great machine. Together they have founded a church, more powerful than that founded by Peter and Paul, whose dogmas are now so entrenched that anyone who tries to reallocate the facts is guilty of more than heresy."

"Newton did not show the cause of the apple falling, but he shewed a similitude between the apple and the stars. By doing so he turned old facts into new knowledge; and was well content if he could bring diverse phenomenon under "two or three Principles of Motion" even "though the Causes of these Principles were not yet discovered.""

"By analyzing the measurements of , Johannes Kepler established that planetary motions weren't circles but ellipses... Through his telescopes, Galileo saw that the Sun had its perfection tarnished by ugly black spots. And the Moon wasn't a perfect sphere but looked like a place, complete with mountains and giant craters. So why didn't it fall down? Isaac Newton finally answered... by exploring... [a radical] idea... that heavenly objects obey the same laws as objects here on Earth. ...Newton ...realized that ...the fate of a horizontally fired cannon ball depends on its speed: it crashes to the ground only if its speed is below some magic value. ...[W]ith ever higher speeds, they'll travel farther ...before landing ...until ...they keep their height over the ground ...constant and never land, merely orbiting ...just like the Moon! Since he knew the strength of gravity near the Earth's surface... he was able to calculate the magic speed... 7.9 kilometers per second. Assuming the Moon... was obeying the same laws... he could similarly predict what speed it needed... Moreover, since the Moon took one month to travel around a circle whose circumference Aristarchos had figured out, Newton already knew its speed... Now he made a remarkable discovery: if he assumed that the force of gravity weakened like the inverse square... then this magical speed that would give the Moon a circular orbit exactly matched its measured speed! He had discovered the law of gravity... applying not merely here on Earth, but in the heavens as well. ...People boldly extrapolated not only to the macrocosmos... but also to the microcosmos, finding that many properties... could be explained by applying to... atoms... The scientific revolution had begun."

"After dinner, the weather being warm, we went into the garden, & drank tea under the shade of some apple trees; only he & myself. Amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. "Why should that apple always descend perpendicularly to the ground," thought he to him self; occasion'd by the fall of an apple, as he sat in a contemplative mood. "Why should it not go sideways, or upwards? but constantly to the earths centre? Assuredly, the reason is, that the earth draws it. There must be a drawing power in matter. The sum of the drawing power in the matter must be in the earth's center, not in any side of the earth. Therefore does this apple fall perpendicularly or toward the center. If matter thus draws matter; it must be in proportion to its quantity. Therefore the apple draws the earth, as well as the earth draws the apple.""

"Despite Newton's belated appreciation of Euclid's geometry, he set it aside as an undergraduate and immediately turned to Descartes' Geometrie, a much more difficult text. Newton read a few pages... and immediately got stuck. ...The second time through, he progressed a page or two further before running into more difficulties. Again, he read it from the beginning, this time getting further still. He continued this process until he mastered Descartes' text. Had Newton mastered Euclid first, Descartes' analytic geometry would have been much easier to understand. Newton later advised others not to make the same mistake. But Descartes had ignited Newton's interest in mathematics, an interest that bordered on obsession."

"The view of space that exists independent of any relationship is called the absolute view. It was Newton's view, but it has been definitely repudiated by the experiments that have verified Einstein's theory of general relativity. ...There are unfortunately not a few good professional physicists who still think about the world as if space and time had an absolute meaning."

"The weight of a smallish apple is, pleasingly, about 1 newton, or 1 N. ...Newton probably weighed about 700 newtons."

"Newton proposed that the particles of the air (we would call them molecules), were motionless in space and were held apart by repulsive forces between them... He assumed that the repulsive force was inversely proportional to the distance between the particles...He showed that, on the basis of this assumption, a collection of static particles in a box would behave exactly as Boyle had found. His model led directly to Boyle's law. Probably the greatest scientist ever, Newton managed to get the right answer from a model that was wrong in every possible way."

"Un genio es alguien que descubre que la piedra que cae y la luna que no cae representan un solo y mismo fenómeno."

"Dr. Pemberton tells us a that the first thoughts, which gave rise to Newton's Principia, occurred to him when he had retired from Cambridge into Lincolnshire, in 1666, on account of the plague. Voltaire had his information from Mrs. Catharine Barton, Newton's favourite niece, who married Conduitt, a member of the Royal Society, and one of his intimate friends: from having spent a great portion of her life in his society, she was good authority for such an anecdote, and she related that some fruit, falling from a tree, was the accidental cause of this direction to Newton's speculations."

"Were it possible to trace the succession of ideas in the mind of Sir Isaac Newton, during the time that he made his greatest discoveries, I make no doubt but our amazement at the extent of his genius would a little subside. But if, when a man publishes discoveries, he, either through design, or through habit, omit the intermediate steps by which he himself arrived at them; it is no wonder that his speculations confound others... [W]here we see him most in the character of an experimental philosopher, as in his optical inquiries... we may easily conceive that many persons, of equal patience and industry... might have done what he did. And were it possible to see in what manner he was first led to those speculations, the very steps by which he pursued them, the time that he spent in making experiments, and all the unsuccessful and insignificant ones that he made in the course of them; as our pleasure of one kind would be increased, our admiration would probably decrease. Indeed he himself used candidly to acknowledge, that if he had done more than other men, it was owing rather to a habit of patient thinking, than to any thing else. ...[T]he interests of science have suffered by the excessive admiration and wonder, with which several first rate philosophers are considered; and... an opinion of the greater equality of mankind, in point of genius, and powers of understanding, would be of real service in the present age."

"Sir Isaac Newton, having perhaps the greatest scientific mind of all time, accepted the books of Book of Daniel and Revelation as revelations from God, being very detailed and accurate representations of the history of the world's dominating kingdoms, and prophesying both the first and second coming of Christ. He understood that the scriptures taught that the true Church of Jesus Christ had been lost, and he awaited three separate future events: 1) the restoration of the gospel by an angel, 2) the re-establishment of the true church, and 3) the rise of a new world kingdom led by the Savior himself, which will crush the kingdoms of the world as the stone pulverized the statue to powder. He saw the whole purpose of these revelations is not to satisfy man's curiosity about the future, but to be a testimony of the foreknowledge of God after they are all fulfilled in the last days. He proposed that the revelations can be understood by discovering rules governing their consistent imagery, but only after they have been fulfilled, unless an interpretation is given with the revelation. Truly Newton's genius was remarkable, and we could learn much from his insights and systematic methods."

"Nature and Nature's laws lay hid in night: God said, Let Newton be! — and all was light."

"There is a traditional story about Newton: as a young student, he began the study of geometry, as was usual in his time, with the reading of the Elements of Euclid. He read the theorems, saw that they were true, and omitted the proofs. He wondered why anybody should take pains to prove things so evident. Many years later, however, he changed his opinion and praised Euclid. The story may be authentic or not ..."

"The first thoughts, which gave rise to his Principia, he had, when he retired from Cambridge in 1666 on account of the plague. As he sat alone in a garden, he fell into a speculation on the power of gravity; that as this power is not found sensibly diminished at the remotest distance from the centre of the earth to which we can rise, neither at the tops of the loftiest buildings, nor even on the summits of the highest mountains, it appeared to him reasonable to conclude that this power must extend much further than was usually thought: why not as high as the moon? said he to himself."

"When I had the honour of his conversation, I endeavoured to learn his thoughts upon mathematical subjects, and something historical concerning his inventions, that I had not been before acquainted with. I found, he had read fewer of the modern mathematicians, than one could have expected; but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook. I have often heard him censure the handling geometrical subjects by algebraic calculations; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Des Cartes had given to the treatise, wherein he shews, how the geometer may assist his invention by such kind of computations. He frequently praised , Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius's book De sectione rationis for giving us a clearer notion of that analysis than we had before."

"Newton's exegesis merged with a prophetic tradition that helped create during the nineteenth and twentieth centuries the religious and political climates that paved the way for the resettlement of Jews in Palestine – the longed-for vision of the Restoration. Newton would have approved."

"Newton had a profound interest in things Jewish. ...Newton owned five of the works of Maimonides... He also possessed Christian Knorr von Rosenroth’s Kabbala denudata (1677)... along with an edition of the first century Jewish philosopher Philo. His writings reveal that he used the Talmud, the learning of which he accessed through Maimonides and other sources in his library."

"At the end of the [19th] century no extension or analogue of the Newtonian gravitation formula has been generally accepted, and it still stands there as almost the only firmly established mathematical relation, expressive of a property of all matter, to which the progress of more than two centuries has added nothing, from which it has taken nothing away."

"We cannot... regard Newton's statement as an appeal to experience and observation, but rather as a deduction of the third law of motion from the first."

"The fact that a magnet draws iron towards it was noticed by the ancients, but no attention was paid to the force with which the iron attracts the magnet. Newton, however, by placing the magnet in one vessel and the iron in another, and floating both vessels in water so as to touch each other, showed experimentally that as neither vessel was able to propel the other along with itself through the water, the attraction of the iron on the magnet must be equal and opposite to that of the magnet on the iron, both being equal to the pressure between the two vessels."

"It is an observed fact that bodies of equal mass, placed in the same position relative to the earth, are attracted equally towards the earth whatever they are made of; but this is not a doctrine of abstract dynamics founded on axiomatic principles, but a fact discovered by observation, and verified by the careful experiments of Newton on the times of oscillation of hollow wooden balls suspended by strings of the same length, and containing gold, silver, lead, glass, sand, common salt, wood, water, and wheat. ...measuring the length of a pendulum which swings seconds."

"We shall find it more conducive to scientific progress to recognise, with Newton, the ideas of time and space as distinct, at least in thought, from that of the material system whose relations these ideas serve to co-ordinate."

"Newton has... acted contrary to his expressed intention only to investigate actual facts. No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience. All our principles of mechanics are... experimental knowledge concerning the relative positions and motions of bodies. ...No one is warranted in extending these principles beyond the boundaries of experience. In fact, such an extension is meaningless, as no one possesses the requisite knowledge to make use of it."