Mathematicians From England

"And now to explain colours. I suppose that as bodies excite sounds of various tones and consequently vibrations, in the air of various bignesses, so when rays of light by impinging on the stiff refracting superficies excite vibrations in the ether, these rays excite vibrations of various bignesses... therefore, the ends of the capillamenta of the optic nerve which front or face the retina being such refracting superficies, when the rays impinge on them they must there excite these vibrations, which vibrations (like those of sound in a trumpet) will run along the pores or crystalline pith of the capillamenta through the optic nerves into the sensorium (which light itself cannot do), and there, I suppose, affect the sense with various colours, according to their bigness and mixture—the biggest with the strongest colours, reds and yellows; the least with the weakest, blues and violets; middle with green; and a confusion of all with white, much after the manner, that in the sense of hearing, nature makes use of aereal vibrations of several bignesses to generate sounds of divers tones; for the analogy of nature is to be observed."

"And so, supposing that light impinging on a refracting or reflecting ethereal superficies puts it into a vibrating motion, that physical superficies being by the perpetual applause of rays always kept in a vibrating motion, and the ether therein continually expanded and compressed by turns, if a ray of light impinge on it when it is much compressed, I suppose it is then too dense and stiff to let the ray through, and so reflects it; but the rays that impinge on it at other times, when it is either expanded by the interval between two vibrations or not too much compressed and condensed, go through and are refracted."

"Were I to assume an hypothesis, it should be this, if propounded more generally, so as not to assume what light is further than that it is something or other capable of exciting vibrations of the ether. First, it is to be assumed that there is an ethereal medium, much of the same constitution as air, but far rarer, subtiller, and more strongly elastic. ...In the second place, it is to be supposed that the ether is a vibrating medium, like air, only the vibrations much more swift and minute; those of air made by a man's ordinary voice succeeding at more than half a foot or a foot distance, but those of ether at a less distance than the hundredth-thousandth part of an inch. And as in air the vibrations are some larger than others, but yet all equally swift... so I suppose the ethereal vibrations differ in bigness but not in swiftness. ...In the fourth place, therefore, I suppose that light is neither ether nor its vibrating motion, but something of a different kind propagated from lucid bodies. They that will may suppose it an aggregate of various peripatetic qualities. Others may suppose it multitudes of unimaginable small and swift corpuscles of various sizes springing from shining bodies at great distances one after the other, but yet without any sensible interval of time. ...To avoid dispute and make this hypothesis general, let every man here take his fancy; only whatever light be, I would suppose it consists of successive rays differing from one another in contingent circumstances, as bigness, force, or vigour, like as the sands on the shore... and, further, I would suppose it diverse from the vibrations of the ether. ...Fifthly, it is to be supposed that light and ether mutually act upon one another. ...æthereal vibrations are therefore the best means by which such a subtile agent as light can shake the gross particles of solid bodies to heat them."

"I have that honour for him as to believe that he wrote good sense; and therefore take that sense to be his which is the best."

"Who is a liar, saith John, but he that denyeth that Jesus is the Christ? He is Antichrist that denyeth the Father & the Son. And we are authorized also to call him God: for the name of God is in him. Exod. 23.21. And we must believe also that by his incarnation of the Virgin he came in the flesh not in appearance only but really & truly , being in all things made like unto his brethren (Heb. 2 17) for which reason he is called also the son of man."

"Whence are you certain that ye Ancient of Days is Christ? Does Christ anywhere sit upon ye Throne?"

"God created everything by number, weight and measure."

"Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things."

"Oh, Diamond! Diamond! thou little knowest what mischief thou hast done!"

"We account the Scriptures of God to be the most sublime philosophy. I find more sure remarks of authenticity in the Bible than in any profane history whatever."

"In default of any other proof, the thumb would convince me of the existence of a God."

"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

"I have studied these things — you have not."

"In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since. What Mr Hugens has published since about centrifugal forces I suppose he had before me. At length in the winter between the years 1676 and 1677 I found the Proposition that by a centrifugal force reciprocally as the square of the distance a Planet must revolve in an Ellipsis about the center of the force placed in the lower umbilicus of the Ellipsis and with a radius drawn to that center describe areas proportional to the times. And in the winter between the years 1683 and 1684 this Proposition with the Demonstration was entered in the Register book of the R. Society. And this is the first instance upon record of any Proposition in the higher Geometry found out by the method in dispute. In the year 1689 Mr Leibnitz, endeavouring to rival me, published a Demonstration of the same Proposition upon another supposition, but his Demonstration proved erroneous for want of skill in the method."

"I keep the subject constantly before me, and wait 'till the first dawnings open slowly, by little and little, into a full and clear light."

"It is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact, as it must be, if gravitation in the sense of Epicurus, be essential and inherent in it. And this is one reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left open to the consideration of my readers."

"When I wrote my treatise about our System, I had an eye upon such principles as might work with considering men for the belief of a Deity and nothing can rejoice me more than to find it useful for that purpose. But if I have done the public any service this way, 'tis due to nothing but industry and a patient thought."

"It seems to me, that if the matter of our sun and planets and all the matter of the universe, were evenly scattered throughout all the heavens, and every particle had an innate gravity towards all the rest, and the whole of space throughout which this matter was scattered was but finite, the matter on [toward] the outside of this space would, by its gravity, tend towards all the matter on the inside, and, by consequence, fall down into the middle of the whole space, and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one another throughout all that infinite space."

"1. Fidelity & Allegiance sworn to the King is only such a fidelity and obedience as is due to him by the law of the land; for were that faith and allegiance more than what the law requires, we would swear ourselves slaves, and the King absolute; whereas, by the law, we are free men, notwithstanding those Oaths. 2. When, therefore, the obligation by the law to fidelity and allegiance ceases, that by the Oath also ceases..."

"Bullialdus wrote that all force respecting the Sun as its center & depending on matter must be reciprocally in a duplicate ratio of the distance from the center."

"I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis, and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy."

"If I have seen further it is by standing on ye sholders of Giants."

"The best and safest method of philosophizing seems to be, first to enquire diligently into the properties of things, and to establish these properties by experiment, and then to proceed more slowly to hypothesis for the explanation of them. For hypotheses should be employed only in explaining the properties of things, but not assumed in determining them, unless so far as they may furnish experiments."

"Amicus Plato — amicus Aristoteles — magis amica veritas"

"No monument should stand over [my] grave, only an apple-tree, in memory of the three apples; the two of Eve and Paris, which made hell out of earth, and that of Newton, which elevated the earth again into the circle of heavenly bodies."

"Now I a fourfold vision see, And a fourfold vision is given to me ; 'Tis fourfold in my supreme delight, And threefold in soft Beulah's night, And twofold Always. May God us keep From Single vision, & Newton's sleep !"

"Kepler succeeded in showing that the planets move along elliptic paths and that the sun lies at a focus of each of these s... Each planet moves so that a straight line drawn to connect it with the sun sweeps out equal areas in equal times. ...The discoveries ...enabled Newton to formulate the laws of mechanics in general and those of gravitation in particular. ...He was able to develop Kepler's laws into a comprehensive physical theory only because he managed first to create the necessary mathematical tools... differential and integral calculus, the basic mathematical techniques for dealing with variable quantities, such as the movement of bodies in the course of time. ...[H]e succeeded in drawing from Kepler's empirical laws the principles of motion that applied [to] every instant of time and thus shaped planetary motion into complete orbits."

"Newton's own motto, "hypotheses non fingo" was, in a sense, disregarded by Newton himself: he rejected hypotheses only where they violated his own "regula philosophandi", that is to say, his principle of their strict parsimony. In terms of present-day methodology, we reject hypotheses as scientifically meaningless if they are incapable even of indirect test; and we reject them as superfluous or as implausible if they are too complex and artificial to conform with well established canons of inductive probability. But freedom of scientific theorizing must be preserved wherever the conditions of meaningfulness and of economy appear to be satisfied."

"The greatest scientist who ever lived was Isaac Newton...[about Principia Mathematica] By all odds it's the greatest scientific book ever written or ever will be written, I think."

"According to Sir Isaac Newton's Calculations, the last Comet that made its Appearance in 1680, imbib'd so much Heat by its Approaches to the Sun, that it would have been two thousand times hotter than red hot Iron, had it been a Globe of that Metal; and that supposing it as big as the Earth, and at the same Distance from the Sun, it would be fifty thousand Years in cooling, before it recovered its natural Temper. In the like manner, if an Englishman considers the great Ferment into which our Political World is thrown at present, and how intensely it is heated in all its Parts, he cannot suppose that it will cool again in less than three hundred Years. In such a Tract of Time it is possible that the Heats of the present Age may be extinguished, and our several Classes of great Men represented under their proper Characters. Some eminent Historian may then probably arise that will not write recentibus odiis (as Tacitus expresses it) with the Passions and Prejudices of a contemporary Author, but make an impartial Distribution of Fame among the Great Men of the present Age."

"Newton and Locke are examples of the deep sagacity which may be acquired by long habits of thinking and study."

"Les hommes construisent trop de murs et pas assez de ponts."

"Tact is the knack of making a point without making an enemy."

"Atheism is so senseless. When I look at the solar system, I see the earth at the right distance from the sun to receive the proper amounts of heat and light. This did not happen by chance."

"The foundation of all religion is the belief in a God, and that He exists in certain relation with His creatures. Such belief necessarily leads to the consciousness of some obligation towards the Deity ; and this consciousness suggests the duty of worship ; and in the selection of the form of this worship originates the various creeds which distinguish and distract mankind. There is a sort of geography of religion ; and I regret to think that the majority of mankind take their creed from the clime in which they happen to be born ; and that many, and not an inconsiderable portion of mankind, suffer the sacred torch to burn out altogether, in their contact with the world, and then vainly imagine that they can recover the sacred fire by striking a park out of dogmatic theology."

"Miracles may be, for anything we know to the contrary, phenomena of a higher order of God's laws, superior to, and, under certain conditions, controlling the inferior order known to us as the ordinary laws of nature."

"There is nothing in the nature of a miracle that should render it incredible: its credibility depends upon the nature of the evidence by which it is supported. An event of extreme probability will not necessarily command our belief unless upon a sufficiency of proof; and so an event which we may regard as highly improbable may command our belief if it is sustained by sufficient evidence. So that the credibility or incredibility of an event does not rest upon the nature of the event itself, but depends upon the nature and sufficiency of the proof which sustains it."

"Mill, in speaking of Hume's celebrated principle, "that nothing is credible which is contradictory to experience, or at variance with the laws of nature," calls it a very plain and harmless proposition, being, in effect, nothing more than that whatever is contradictory to a complete induction is incredible. Admit the existence of a Deity, and the possibility of a miracle is the natural consequence. No doubt our examination of the evidence which sustains an unusual phenomenon should be most carefully conducted; but we must not measure the credibility or incredibility of an event by the narrow sphere of our own experience, nor forget that there is a Divine energy which overrides what we familiarly call the laws of nature."

"We must be careful to discriminate between our own incapacity to test truth and the necessary improbability of an event. It is plain that from our ignorance of the remote spheres of God's action we cannot judge of His works removed from our experience; but a fact is not necessarily doubtful because it cannot be reached by our ordinary senses. To recapitulate, we may lay down the following propositions: 1. That there is no real physical distinction between miracles and any other operations of the Divine energy : that we regard them differently is because we are familiar with one order of events and not the other. 2. There is nothing incredible in a miracle, and the credibility of a miraculous event is to be measured only by the evidence which sustains it. And although the extraordinary character of a phenomenon may render the event itself improbable, it does not, therefore, necessarily render it either incredible or untrue."

"Propose to any Englishman any principle, or any instrument, however admirable, and you will observe that the whole effort of the English mind is directed to find a difficulty, a defect, or an impossibility in it. If you speak to him of a machine for peeling a potato, he will pronounce it impossible; if you peel a potato with it before his eyes, he will declare it useless because it will not slice a pineapple. Impart the same principle or show the same machine to an American or to one of our Colonists and you will observe that the whole effort of his mind is to find some new application of the principle, some new use for the instrument."

"Errors using inadequate data are much less than those using no data at all."

"The Economy of Manufactures established Babbage's position as a political economist and its influence is well attested, particularly on John Stuart Mill and Karl Marx. Babbage's pioneering discussion of the effect of technical development on the size of industrial organizations was followed by Mill and the prediction of the continuing increase in the size of factories, often cited as one of Marx's successful economic predictions, in fact derives from Babbage's analysis... Babbage wrote with many talents: a natural philosopher and mechanical engineer, his knowledge of factory and workshop practice was encyclopaedic; he was well-versed in relevant business practice; and he was without rival as a mathematician among contemporary British political economists."

"Babbage was one of the founders of the Cambridge Analytical Society whose purpose he stated was to advocate "the principles of pure d-ism as opposed to the dof-age of the university."

"We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves."

"If we define a miracle as an effect of which the cause is unknown to us, then we make our ignorance the source of miracles! and the universe itself would be a standing miracle. A miracle might be perhaps defined more exactly as an effect which is not the consequence or effect of any known laws of nature."

"Mr. Herschel ... brought with him the calculations of the computers, and we commenced the tedious process of verification. After a time many discrepancies occurred, and at one point these discordances were so numerous that I exclaimed, "I wish to God these calculations had been executed by steam," to which Herschel replied, "It is quite possible.""

"If this were true, the population of the world would be at a stand-still. In truth, the rate of birth is slightly in excess of death. I would suggest that the next edition of your poem should read: “Every moment dies a man, every moment 1 1/16 is born.” Strictly speaking, the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 will be sufficiently accurate for poetry."

"It is therefore not unreasonable to suppose that some portion of the neglect of science in England, may be attributed to the system of education we pursue. A young man passes from our public schools to the universities, ignorant of almost every branch of useful knowledge; and at these latter establishments … classical and mathematical pursuits are nearly the sole objects proposed to the student's ambition."

"If we look at the fact, we shall find that the great inventions of the age are not, with us at least, always produced in universities. The doctrines of "definite proportions," and of the "chemical agency of electricity,"—principles of a high order, which have immortalized the names of their discoverers,—were not produced by the meditations of the cloister: nor is it in the least a reproach to those valuable institutions to mention truths like these. Fortunate circumstances must concur, even to the greatest, to render them eminently successful. It is not permitted to all to be born, like Archimedes, when a science was to be created; nor, like Newton, to find the system of the world "without form and void;" and, by disclosing gravitation, to shed throughout that system the same irresistible radiance as that with which the Almighty Creator had illumined its material substance. It can happen to but few philosophers, and but at distant intervals, to snatch a science, like Dalton, from the chaos of indefinite combination, and binding it in the chains of number, to exalt it to rank amongst the exact. Triumphs like these are necessarily "few and far between;", nor can it be expected that that portion of encouragement, which a country may think fir to bestow on science, should be adapted to meet such instances. Too extraordinary to be frequent, they must be left, if they are to be encouraged at all, to some direct interference of the governemeɳt. The dangers to be apprehended from such a specific interference, would arise from one, or several of the following circumstance:—That class of society, from whom the government is selected, might not possess sufficient knowledge either to judge themselves, or know upon whose judgment to rely. Or the number of persons devoting themselves to science, might not be sufficiently large to have due weight in the expression of public opinion. Or, supposing this class to be large, it might not enjoy, in the estimation of the world, a sufficiently high character for independence. Should these causes concur in any country, it might become highly injurious to commit the encouragement of science to any department of the government. This reasoning does not appear to have escaped the penetration of those who advised the abolition of the late Board of Longitude. The question whether it is good policy in the government of a country to encourage science, is one of which those who cultivate it are not perhaps the most unbiased judges. In England, those who have hitherto pursued science, have in general no very reasonable grounds of complaint; they knew, or should have known, that there was no demand for it, that it led to little honour, and to less profit. That blame has been attributed to the government for not fostering the science of the country is certain; and, as far as regards past administrations, is, to a great extent, just; with respect to the present ministers, whose strength essentially depends on public opinion, it is not necessary that they should precede, and they cannot remain long insensible to any expression of the general feeling. But supposing science were thought of some importance by any administration, it would be difficult in the present state of things to do much in its favour; because, on the one hand, the higher classes in general have not a profound knowledge of science, and, on the other, those persons whom they have usually consulted, seem not to have given such advice as to deserve the confidence of government. It seems to be forgotten, that the money allotted by government to purposes of science ought to be expended with the same regard to prudence and economy as in the disposal of money in the affairs of private life."

"The object of the present volume is to point out the effects and the advantages which arise from the use of tools and machines ;—to endeavour to classify their modes of action ;—and to trace both the causes and the consequences of applying machinery to supersede the skill and power of the human arm."