Application Quotes (164 quotes)

*Ath.*There still remain three studies suitable for freemen. Calculation in arithmetic is one of them; the measurement of length, surface, and depth is the second; and the third has to do with the revolutions of the stars in reference to one another … there is in them something that is necessary and cannot be set aside, … if I am not mistaken, [something of] divine necessity; for as to the human necessities of which men often speak when they talk in this manner, nothing can be more ridiculous than such an application of the words.

*Cle.*And what necessities of knowledge are there, Stranger, which are divine and not human?

*Ath.*I conceive them to be those of which he who has no use nor any knowledge at all cannot be a god, or demi-god, or hero to mankind, or able to take any serious thought or charge of them.

— Plato

*Dilbert:*It took weeks but I’ve calculated a new theory about the origin of the universe. According to my calculations it didn’t start with a “Big Bang” at all—it was more of “Phhbwt” sound. You may be wondering about the practical applications of the “Little Phhbwt” theory.

*Dogbert:*I was wondering when you’ll go away.

*Il n’existe pas de sciences appliquées, mais seulement des applications de la science.*

There are no such things as applied sciences, only applications of science.

*[About Pierre de Fermat]*It cannot be denied that he has had many exceptional ideas, and that he is a highly intelligent man. For my part, however, I have always been taught to take a broad overview of things, in order to be able to deduce from them general rules, which might be applicable elsewhere.

*[When questioned on his longevity]*First of all, I selected my ancestors very wisely. ... They were long-lived, healthy people. Then, as a chemist, I know how to eat, how to exercise, keep my blood circulating. ... I don't worry. I don't get angry at people. I don't worry about things I can't help. I do what I can to make the world a better place to live, but I don't complain if things aren't right. As a scientist I take the world as I find it.

*[About celebrating his 77th birthday by swimming a half mile in 22 minutes]*I used swim fins and webbed gloves because a man of intelligence should apply his power efficiently, not just churn the water.

A discovery must be, by definition, at variance with existing knowledge. During my lifetime, I made two. Both were rejected offhand by the popes of the field. Had I predicted these discoveries in my applications, and had those authorities been my judges, it is evident what their decisions would have been.

A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.

A troubling question for those of us committed to the widest application of intelligence in the study and solution of the problems of men is whether a general understanding of the social sciences will be possible much longer. Many significant areas of these disciplines have already been removed by the advances of the past two decades beyond the reach of anyone who does not know mathematics; and the man of letters is increasingly finding, to his dismay, that the study of mankind proper is passing from his hands to those of technicians and specialists. The aesthetic effect is admittedly bad: we have given up the belletristic “essay on man” for the barbarisms of a technical vocabulary, or at best the forbidding elegance of mathematical syntax.

About 6 or 8 years ago My Ingenious friend Mr John Robinson having [contrived] conceived that a fire engine might be made without a Lever—by Inverting the Cylinder & placing it above the mouth of the pit proposed to me to make a model of it which was set about by having never Compleated & I [being] having at that time Ignorant little knoledge of the machine however I always thought the Machine Might be applied to [more] other as valuable purposes [than] as drawing Water.

Abstruse mathematical researches … are … often abused for having no obvious physical application. The fact is that the most useful parts of science have been investigated for the sake of truth, and not for their usefulness. A new branch of mathematics, which has sprung up in the last twenty years, was denounced by the Astronomer Royal before the University of Cambridge as doomed to be forgotten, on account of its uselessness. Now it turns out that the reason why we cannot go further in our investigations of molecular action is that we do not know enough of this branch of mathematics.

Algebra applies to the clouds.

Almost all the greatest discoveries in astronomy have resulted from what we have elsewhere termed Residual Phenomena, of a qualitative or numerical kind, of such portions of the numerical or quantitative results of observation as remain outstanding and unaccounted for, after subducting and allowing for all that would result from the strict application of known principles.

Although with the majority of those who study and practice in these capacities [engineers, builders, surveyors, geographers, navigators, hydrographers, astronomers], secondhand acquirements, trite formulas, and appropriate tables are sufficient for ordinary purposes, yet these trite formulas and familiar rules were originally or gradually deduced from the profound investigations of the most gifted minds, from the dawn of science to the present day. … The further developments of the science, with its possible applications to larger purposes of human utility and grander theoretical generalizations, is an achievement reserved for a few of the choicest spirits, touched from time to time by Heaven to these highest issues. The intellectual world is filled with latent and undiscovered truth as the material world is filled with latent electricity.

An inventor is an opportunist, one who takes occasion by the hand; who, having seen where some want exists, successfully applies the right means to attain the desired end. The means may be largely, or even wholly, something already known, or there may be a certain originality or discovery in the means employed. But in every case the inventor uses the work of others. If I may use a metaphor, I should liken him to the man who essays the conquest of some virgin alp. At the outset he uses the beaten track, and, as he progresses in the ascent, he uses the steps made by those who have preceded him, whenever they lead in the right direction; and it is only after the last footprints have died out that he takes ice-axe in hand and cuts the remaining steps, few or many, that lift him to the crowning height which is his goal.

An inventor is one who can see the applicability of means to supply demand five years before it is obvious to those skilled in the art.

Archimedes was not free from the prevailing notion that geometry was degraded by being employed to produce anything useful. It was with difficulty that he was induced to stoop from speculation to practice. He was half ashamed of those inventions which were the wonder of hostile nations, and always spoke of them slightingly as mere amusements, as trifles in which a mathematician might be suffered to relax his mind after intense application to the higher parts of his science.

As a rule, software systems do not work well until they have been used, and have failed repeatedly, in real applications.

As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.

As science is more and more subject to grave misuse as well as to use for human benefit it has also become the scientist's responsibility to become aware of the social relations and applications of his subject, and to exert his influence in such a direction as will result in the best applications of the findings in his own and related fields. Thus he must help in educating the public, in the broad sense, and this means first educating himself, not only in science but in regard to the great issues confronting mankind today.

Astronomy was not studied by Kepler, Galileo, or Newton for the practical applications which might result from it, but to enlarge the bounds of knowledge, to furnish new objects of thought and contemplation in regard to the universe of which we form a part; yet how remarkable the influence which this science, apparently so far removed from the sphere of our material interests, has exerted on the destinies of the world!

But, contrary to the lady’s prejudices about the engineering profession, the fact is that quite some time ago the tables were turned between theory and applications in the physical sciences. Since World War II the discoveries that have changed the world are not made so much in lofty halls of theoretical physics as in the less-noticed labs of engineering and experimental physics. The roles of pure and applied science have been reversed; they are no longer what they were in the golden age of physics, in the age of Einstein, Schrödinger, Fermi and Dirac.

By an application of the theory of relativity to the taste of readers, today in Germany I am called a German man of science, and in England I am represented as a Swiss Jew. If I come to be regarded as a

*bête noire*the descriptions will be reversed, and I shall become a Swiss Jew for the Germans and a German man of science for the English!
By no process of sound reasoning can a conclusion drawn from limited data have more than a limited application.

Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”

Developing countries can leapfrog several stages in the development process through the application of bio-technology in agriculture.

During the school period the student has been mentally bending over his desk; at the University he should stand up and look around. For this reason it is fatal if the first year at the University be frittered away in going over the old work in the old spirit. At school the boy painfully rises from the particular towards glimpses at general ideas; at the University he should start from general ideas and study their applications to concrete cases.

Electricity is but yet a new agent for the arts and manufactures, and, doubtless, generations unborn will regard with interest this century, in which it has been first applied to the wants of mankind.

Engineering is the application of scientific and mathematical principles to practical ends such as the design, manufacture, and operation of efficient and economical structures, machines, processes, and systems.

Engineering is the art of directing the great sources of power in nature for the use and the convenience of people. In its modern form engineering involves people, money, materials, machines, and energy. It is differentiated from science because it is primarily concerned with how to direct to useful and economical ends the natural phenomena which scientists discover and formulate into acceptable theories. Engineering therefore requires above all the creative imagination to innovate useful applications of natural phenomena. It seeks newer, cheaper, better means of using natural sources of energy and materials.

Engineering is the conscious application of science to the problems of economic production.

Engineering is the practice of safe and economic application of the scientific laws governing the forces and materials of nature by means of organization, design and construction, for the general benefit of mankind.

Engineering is the professional and systematic application of science to the efficient utilization of natural resources to produce wealth.

Engineers apply the theories and principles of science and mathematics to research and develop economical solutions to practical technical problems. Their work is the link between scientific discoveries and commercial applications. Engineers design products, the machinery to build those products, the factories in which those products are made, and the systems that ensure the quality of the product and efficiency of the workforce and manufacturing process. They design, plan, and supervise the construction of buildings, highways, and transit systems. They develop and implement improved ways to extract, process, and use raw materials, such as petroleum and natural gas. They develop new materials that both improve the performance of products, and make implementing advances in technology possible. They harness the power of the sun, the earth, atoms, and electricity for use in supplying the Nation’s power needs, and create millions of products using power. Their knowledge is applied to improving many things, including the
quality of health care, the safety of food products, and the efficient operation of financial systems.

Every well established truth is an addition to the sum of human power, and though it may not find an immediate application to the economy of every day life, we may safely commit it to the stream of time, in the confident anticipation that the world will not fail to realize its beneficial results.

First, as concerns the

Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that

Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.

*success*of teaching mathematics. No instruction in the high schools is as difficult as that of mathematics, since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions. The interest of young people is won much more easily, if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually. For this reason it is psychologically quite correct to follow this course.Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that

*correct thinking based on true premises secures mastery over the outer world*. To accomplish this the outer world must receive its share of attention from the very beginning.Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.

For just as musical instruments are brought to perfection of clearness in the sound of their strings by means of bronze plates or horn sounding boards, so the ancients devised methods of increasing the power of the voice in theaters through the application of the science of harmony.

FORTRAN —’the infantile disorder’—, by now nearly 20 years old, is hopelessly inadequate for whatever computer application you have in mind today: it is now too clumsy, too risky, and too expensive to use. PL/I —’the fatal disease’— belongs more to the problem set than to the solution set. It is practically impossible to teach good programming to students that have had a prior exposure to BASIC: as potential programmers they are mentally mutilated beyond hope of regeneration. The use of COBOL cripples the mind; its teaching should, therefore, be regarded as a criminal offence. APL is a mistake, carried through to perfection. It is the language of the future for the programming techniques of the past: it creates a new generation of coding bums.

FORTRAN, ‘the infantile disorder’, by now nearly 20 years old, is hopelessly inadequate for whatever computer application you have in mind today: it is now too clumsy, too risky, and too expensive to use.

From that night on, the electron—up to that time largely the plaything of the scientist—had clearly entered the field as a potent agent in the supplying of man's commercial and industrial needs… The electronic amplifier tube now underlies the whole art of communications, and this in turn is at least in part what has made possible its application to a dozen other arts. It was a great day for both science and industry when they became wedded through the development of the electronic amplifier tube.

Good design is not an applied veneer.

He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws.

I am among the most durable and passionate participants in the scientific exploration of the solar system, and I am a long-time advocate of the application of space technology to civil and military purposes of direct benefit to life on Earth and to our national security.

I am of the decided opinion, that mathematical instruction must have for its first aim a deep penetration and complete command of abstract mathematical theory together with a clear insight into the structure of the system, and doubt not that the instruction which accomplishes this is valuable and interesting even if it neglects practical applications. If the instruction sharpens the understanding, if it arouses the scientific interest, whether mathematical or philosophical, if finally it calls into life an esthetic feeling for the beauty of a scientific edifice, the instruction will take on an ethical value as well, provided that with the interest it awakens also the impulse toward scientific activity. I contend, therefore, that even without reference to its applications mathematics in the high schools has a value equal to that of the other subjects of instruction.

I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events. That investigation deserves the attention of mathematicians because of the analysis required. It is primarily there that the approximation of formulas that are functions of large numbers has its most important applications. The investigation will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of the errors still to be apprehended. Lastly, the investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends. It is on the regularity of the main outcomes of events taken in large numbers that various institutions depend, such as annuities, tontines, and insurance policies. Questions about those subjects, as well as about inoculation with vaccine and decisions of electoral assemblies, present no further difficulty in the light of my theory. I limit myself here to resolving the most general of them, but the importance of these concerns in civil life, the moral considerations that complicate them, and the voluminous data that they presuppose require a separate work.

I do not intend to go deeply into the question how far mathematical studies, as the representatives of conscious logical reasoning, should take a more important place in school education. But it is, in reality, one of the questions of the day. In proportion as the range of science extends, its system and organization must be improved, and it must inevitably come about that individual students will find themselves compelled to go through a stricter course of training than grammar is in a position to supply. What strikes me in my own experience with students who pass from our classical schools to scientific and medical studies, is first, a certain laxity in the application of strictly universal laws. The grammatical rules, in which they have been exercised, are for the most part followed by long lists of exceptions; accordingly they are not in the habit of relying implicitly on the certainty of a legitimate deduction from a strictly universal law. Secondly, I find them for the most part too much inclined to trust to authority, even in cases where they might form an independent judgment. In fact, in philological studies, inasmuch as it is seldom possible to take in the whole of the premises at a glance, and inasmuch as the decision of disputed questions often depends on an aesthetic feeling for beauty of expression, or for the genius of the language, attainable only by long training, it must often happen that the student is referred to authorities even by the best teachers. Both faults are traceable to certain indolence and vagueness of thought, the sad effects of which are not confined to subsequent scientific studies. But certainly the best remedy for both is to be found in mathematics, where there is absolute certainty in the reasoning, and no authority is recognized but that of one’s own intelligence.

I have no patience with the doctrine of “pure science,”—that science is science only as it is uncontaminated by application in the arts of life: and I have no patience with the spirit that considers a piece of work to be legitimate only as it has direct bearing on the arts and affairs of men. We must discover all things that are discoverable and make a record of it: the application will take care of itself.

I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary—being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived. And these are the causes that operate in perfect harmony. Each new scientific conception gives occasion to new applications of deductive reasoning; but those applications may be only possible through the methods and the processes which belong to an earlier stage.

I suspect that the changes that have taken place during the last century in the average man's fundamental beliefs, in his philosophy, in his concept of religion. in his whole world outlook, are greater than the changes that occurred during the preceding four thousand years all put together. ... because of science and its applications to human life, for these have bloomed in my time as no one in history had had ever dreamed could be possible.

I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.

If science could get rid of consciousness, it would have disposed of the only stumbling block to its universal application.

If you ask ... the man in the street ... the human significance of mathematics, the answer of the world will be, that mathematics has given mankind a metrical and computatory art essential to the effective conduct of daily life, that mathematics admits of countless applications in engineering and the natural sciences, and finally that mathematics is a most excellent instrumentality for giving mental discipline... [A mathematician will add] that mathematics is the exact science, the science of exact thought or of rigorous thinking.

In every section of the entire area where the word science may properly be applied, the limiting factor is a human one. We shall have rapid or slow advance in this direction or in that depending on the number of really first-class men who are engaged in the work in question. ... So in the last analysis, the future of science in this country will be determined by our basic educational policy.

In geometric and physical applications, it always turns out that a quantity is characterized not only by its tensor order, but also by symmetry.

In science men have discovered an activity of the very highest value in which they are no longer, as in art, dependent for progress upon the appearance of continually greater genius, for in science the successors stand upon the shoulders of their predecessors; where one man of supreme genius has invented a method, a thousand lesser men can apply it. … In art nothing worth doing can be done without genius; in science even a very moderate capacity can contribute to a supreme achievement.

In the 1940s when I did my natural sciences degree in zoology it was very much laboratory-based. … I was not keen on the idea of spending the rest of my life in the lab. I also don’t think I would have been particularly good at it. I don't think I have as analytical a mind or the degree of application that one would need to become a first-rate research scientist.

In the application of inductive logic to a given knowledge situation, the total evidence available must be used as a basis for determining the degree of confirmation.

In the twenties the late Dr. Glenn Frank, an eminent social scientist, developed a new statement of the scientific code, which has been referred to as the “Five Fingers of the Scientific Method.” It may be outlined as follows:

*find*the facts;*filter*the facts;*focus*the facts;*face*the facts;*follow*the facts. The facts or truths are found by experimentation; the motivation is material. The facts are filtered by research into the literature; the motivation is material. The facts are focused by the publication of results; again the motivation is material. Thus the first three-fifths of the scientific method have a material motivation. It is about time scientists acknowledge that there is more to the scientific convention than the material aspect. Returning to the fourth and fifth fingers of Dr. Frank's conception of the scientific method, the facts should be faced by the proper interpretation of them for society. In other words, a scientist must assume social responsibility for his discoveries, which means that he must have a moral motivation. Finally, in the fifth definition of the scientific method, the facts are to be followed by their proper application to everyday life in society, which means moral motivation through responsibility to society.
It has been recognized that hydrogen bonds restrain protein molecules to their native configurations, and I believe that as the methods of structural chemistry are further applied to physiological problems it will be found that the significance of the hydrogen bond for physiology is greater than that of any other single structural feature.

It is a fraud of the Christian system to call the sciences

*human invention*; it is only the application of them that is human.
It is no paradox to say that in our most theoretical moods we may be nearest to our most practical applications.

It is not of the essence of mathematics to be conversant with the ideas of number and quantity. Whether as a general habit of mind it would be desirable to apply symbolic processes to moral argument, is another question.

It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty—balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on

*a priori*grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic*Applications de l’analyse à la géométrie*; Lazare Carnot, author of the celebrated works,*Géométrie de position*, and*Réflections sur la Métaphysique du Calcul infinitesimal*; Fourier, immortal creator of the*Théorie analytique de la chaleur*; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service.
It is the constant attempt in this country [Canada] to make fundamental science responsive to the marketplace. Because technology needs science, it is tempting to require that scientific projects be justified in terms of the worth of the technology they can be expected to generate. The effect of applying this criterion is, however, to restrict science to developed fields where the links to technology are most evident. By continually looking for a short-term payoff we disqualify the sort of science that … attempts to answer fundamental questions, and, having answered them, suggests fundamentally new approaches in the realm of applications.

It is the technologist who is transforming at least the outward trappings of modern civilization and no hard and fast line can or should be drawn between those who apply science, and in the process make discoveries, and those who pursue what is sometimes called basic science.

It is very desirable to have a word to express the Availability for work of the heat in a given magazine; a term for that possession, the waste of which is called Dissipation. Unfortunately the excellent word Entropy, which Clausius has introduced in this connexion, is applied by him to the negative of the idea we most naturally wish to express. It would only confuse the student if we were to endeavour to invent another term for our purpose. But the necessity for some such term will be obvious from the beautiful examples which follow. And we take the liberty of using the term Entropy in this altered sense ... The entropy of the universe tends continually to zero.

It is very remarkable that while the words

*Eternal, Eternity, Forever*, are constantly in our mouths, and applied without hesitation, we yet experience considerable difficulty in contemplating any definite term which bears a very large proportion to the brief cycles of our petty chronicles. There are many minds that would not for an instant doubt the God of Nature to have*existed from all Eternity*, and would yet reject as preposterous the idea of going back a million of years in the History of*His Works*. Yet what is a million, or a million million, of solar revolutions to an Eternity?
It needs scarcely be pointed out that in placing Mathematics at the head of Positive Philosophy, we are only extending the application of the principle which has governed our whole Classification. We are simply carrying back our principle to its first manifestation. Geometrical and Mechanical phenomena are the most general, the most simple, the most abstract of all,— the most irreducible to others, the most independent of them; serving, in fact, as a basis to all others. It follows that the study of them is an indispensable preliminary to that of all others. Therefore must Mathematics hold the first place in the hierarchy of the sciences, and be the point of departure of all Education whether general or special.

Later scientific theories are better than earlier ones for solving puzzles in the often quite different environments to which they are applied. That is not a relativist's position, and it displays the sense in which I am a convinced believer in scientific progress.

Magic is a faculty of wonderful virtue, full of most high mysteries, containing the most profound contemplation of most secret things, together with the nature, power, quality, substance and virtues thereof, as also the knowledge of whole Nature, and it doth instruct us concerning the differing and agreement of things amongst themselves, whence it produceth its wonderful effects, by uniting the virtues of things through the application of them one to the other.

Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it.

Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skilful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures.

Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect; mathematics furnishes the instrument, the tool of thought, which we wield in this realm.

MATHEMATICS … the general term for the various applications of mathematical thought, the traditional field of which is number and quantity. It has been usual to define mathematics as “the science of discrete and continuous magnitude.”

More than ever before in the history of science and invention, it is safe now to say what is possible and what is impossible. No one would claim for a moment that during the next five hundred years the accumulated stock of knowledge of geography will increase as it has during the last five hundred In the same way it may safely be affirmed that in electricity the past hundred years is not likely to be duplicated in the next, at least as to great, original, and far-reaching discoveries, or novel and almost revolutionary applications.

Mr. [Granville T.] Woods says that he has been frequently refused work because of the previous condition of his race, but he has had great determination and will and never despaired because of disappointments. He always carried his point by persistent efforts. He says the day is past when colored boys will be refused work only because of race prejudice. There are other causes. First, the boy has not the nerve to apply for work after being refused at two or three places. Second, the boy should have some knowledge of mechanics. The latter could be gained at technical schools, which should be founded for the purpose. And these schools must sooner or later be established, and thereby, we should be enabled to put into the hands of our boys and girls the actual means of livelihood.

My original decision to devote myself to science was a direct result of the discovery which has never ceased to fill me with enthusiasm since my early youth—the comprehension of the far from obvious fact that the laws of human reasoning coincide with the laws governing the sequences of the impressions we receive from the world about us; that, therefore, pure reasoning can enable man to gain an insight into the mechanism of the latter. In this connection, it is of paramount importance that the outside world is something independent from man, something absolute, and the quest for the laws which apply to this absolute appeared to me as the most sublime scientific pursuit in life.

My picture of the world is drawn in perspective and not like a model to scale. The foreground is occupied by human beings and the stars are all as small as three-penny bits. I don't really believe in astronomy, except as a complicated description of part of the course of human and possibly animal sensation. I apply my perspective not merely to space but also to time. In time the world will cool and everything will die; but that is a long time off still and its present value at compound discount is almost nothing.

Nearly all the great inventions which distinguish the present century are the results, immediately or remotely, of the application of scientific principles to practical purposes, and in most cases these applications have been suggested by the student of nature, whose primary object was the discovery of abstract truth.

Our civilization is an engineering civilization, and the prosperous life of the large population, which our earth now supports has become possible only by the work of the engineer. Engineering, however, is the application of science to the service of man, and so to-day science is the foundation, not only of our prosperity, but of our very existence, and thus necessarily has become the dominant power in our human society.

Our delight in any particular study, art, or science rises and improves in proportion to the application which we bestow upon it. Thus, what was at first an exercise becomes at length an entertainment.

Our intelligence has brought us far, but it has also brought us to the brink of total destruction. It cuts both ways, its application sometimes terrifies us, but it also reveals a humbling, sobering perspective of our cosmological home.

Our time is distinguished by wonderful achievements in the fields of scientific understanding and the technical application of those insights. Who would not be cheered by this? But let us not forget that human knowledge and skills alone cannot lead humanity to a happy and dignified life. Humanity has every reason to place the proclaimers of high moral standards and values above the discoverers of objective truth. What humanity owes to personalities like Buddha, Moses, and Jesus ranks for me higher than all the achievements of the inquiring constructive mind.

Physical chemistry is all very well, but it does not apply to organic substances.

Physical science enjoys the distinction of being the most fundamental of the experimental sciences, and its laws are obeyed universally, so far as is known, not merely by inanimate things, but also by living organisms, in their minutest parts, as single individuals, and also as whole communities. It results from this that, however complicated a series of phenomena may be and however many other sciences may enter into its complete presentation, the purely physical aspect, or the application of the known laws of matter and energy, can always be legitimately separated from the other aspects.

Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.

Probably among all the pursuits of the University, mathematics pre-eminently demand self-denial, patience, and perseverance from youth, precisely at that period when they have liberty to act for themselves, and when on account of obvious temptations, habits of restraint and application are peculiarly valuable.

Proposals for forming a Public Institution for diffusing the knowledge of Mechanical Inventions, and for teaching, by Philosophical Lectures and Experiments, the application of Science to the common purposes of life.

Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. … Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics.

Pure mathematics … reveals itself as nothing but symbolic or formal logic. It is concerned with implications, not applications. On the other hand, natural science, which is empirical and ultimately dependent upon observation and experiment, and therefore incapable of absolute exactness, cannot become strictly mathematical. The certainty of geometry is thus merely the certainty with which conclusions follow from non-contradictory premises. As to whether these conclusions are true of the material world or not, pure mathematics is indifferent.

Science affects the average man and woman in two ways already. He or she benefits by its application driving a motor-car or omnibus instead of a horse-drawn vehicle, being treated for disease by a doctor or surgeon rather than a witch, and being killed with an automatic pistol or shell in place of a dagger or a battle-axe.

Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.

Science is not a system of certain, or -established, statements; nor is it a system which steadily advances towards a state of finality... And our guesses are guided by the unscientific, the metaphysical (though biologically explicable) faith in laws, in regularities which we can uncover—discover. Like Bacon, we might describe our own contemporary science—'the method of reasoning which men now ordinarily apply to nature'—as consisting of 'anticipations, rash and premature' and as 'prejudices'.

Science only offers three kinds of interest: 1. Technical applications. 2. A game of chess. 3. A road to God. (Attractions are added to the game of chess in the shape of competitions, prizes, and medals.

Scientific subjects do not progress necessarily on the lines of direct usefulness. Very many applications of the theories of pure mathematics have come many years, sometimes centuries, after the actual discoveries themselves. The weapons were at hand, but the men were not able to use them.

Technology means the systematic application of scientific or other organized knowledge to practical tasks.

The air of caricature never fails to show itself in the products of reason applied relentlessly and without correction. The observation of clinical facts would seem to be a pursuit of the physician as harmless as it is indispensable. [But] it seemed irresistibly rational to certain minds that diseases should be as fully classifiable as are beetles and butterflies. This doctrine … bore perhaps its richest fruit in the hands of Boissier de Sauvauges. In his

*Nosologia Methodica*published in 1768 … this Linnaeus of the bedside grouped diseases into ten classes, 295 genera, and 2400 species.
The anxious precision of modern mathematics is necessary for accuracy, … it is necessary for research. It makes for clearness of thought and for fertility in trying new combinations of ideas. When the initial statements are vague and slipshod, at every subsequent stage of thought, common sense has to step in to limit applications and to explain meanings. Now in creative thought common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones, in other words it can only act by suppressing originality.

The application of algebra to geometry…, far more than any of his metaphysical speculations, immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.

The calculus is the greatest aid we have to the appreciation of physical truth in the broadest sense of the word.

The chemists work with inaccurate and poor measuring services, but they employ very good materials. The physicists, on the other hand, use excellent methods and accurate instruments, but they apply these to very inferior materials. The physical chemists combine both these characteristics in that they apply imprecise methods to impure materials.

The deflation of some of our more common conceits is one of the practical applications of astronomy.

The discovery of the conic sections, attributed to Plato, first threw open the higher species of form to the contemplation of geometers. But for this discovery, which was probably regarded in Plato’s tune and long after him, as the unprofitable amusement of a speculative brain, the whole course of practical philosophy of the present day, of the science of astronomy, of the theory of projectiles, of the art of navigation, might have run in a different channel; and the greatest discovery that has ever been made in the history of the world, the law of universal gravitation, with its innumerable direct and indirect consequences and applications to every department of human research and industry, might never to this hour have been elicited.

The establishment of the periodic law may truly be said to mark a line in chemical science, and we anticipate that its application and and extension will be fraught With the most important consequences. It reminds us how important above all things is the correct determination of the fundamental constants of our science—the atomic weights of the elements, about which in many cases great uncertainty prevails; it is much to be desired that this may not long remain the case. It also affords the strongest encouragement to the chemist to persevere in the search for new elements.

The first acquaintance which most people have with mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical proposition which everyone will have heard of. … The first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body.

The follow-on space shuttle program has fallen far short of the Apollo program in its appeal to human aspirations. The launching of the Hubble Space Telescope and the subsequent repair and servicing missions by skilled crews are highlights of the shuttle’s service to science. … Otherwise, the shuttle’s contribution to science has been modest, and its contribution to utilitarian applications of space technology has been insignificant.

The genius of Leonardo da Vinci imagined a flying machine, but it took the methodical application of science by those two American bicycle mechanics to create it.

The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.

The history of psychiatry to the present day is replete with examples of loose thinking and a failure to apply even the simplest rules of logic. “A Court of Statistical Appeal” has now been equated with scientific method.

— Myre Sim

The hybridoma technology was a by-product of basic research. Its success in practical applications is to a large extent the result of unexpected and unpredictable properties of the method. It thus represents another clear-cut example of the enormous practical impact of an investment in research which might not have been considered commercially worthwhile, or of immediate medical relevance. It resulted from esoteric speculations, for curiosity’s sake, only motivated by a desire to understand nature.

The incessant call in this country for practical results and the confounding of mechanical inventions with scientific discoveries has a very prejudicial influence on science. … A single scientific principle may include a thousand applications and is therefore though if not of immediate use of vastly more importance even in a practical view.

The instinct to command others, in its primitive essence, is a carnivorous, altogether bestial and savage instinct. Under the influence of the mental development of man, it takes on a somewhat more ideal form and becomes somewhat ennobled, presenting itself as the instrument of reason and the devoted servant of that abstraction, or political fiction, which is called the public good. But in its essence it remains just as baneful, and it becomes even more so when, with the application of science, it extends its scope and intensifies the power of its action. If there is a devil in history, it is this power principle.

The life and soul of science is its practical application, and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in the earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind.

The life work of the engineer consists in the systematic application of natural forces and the systematic development of natural resources in the service of man.

The man imbued with the proper spirit of science does not seek for immediate pecuniary reward from the practical applications of his discoveries, but derives sufficient gratification from his pursuit and the consciousness of enlarging the bounds of human contemplation, and the magnitude of human power, and leaves to others to gather the golden fruit he may strew along his pathway.

The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831-32. Faraday was asked: “What is the use of this discovery?” He answered: “What is the use of a child—it grows to be a man.” Faraday’s child has grown to be a man and is now the basis of all the modern applications of electricity.

The origin of a science is usually to be sought for not in any systematic treatise, but in the investigation and solution of some particular problem. This is especially the case in the ordinary history of the great improvements in any department of mathematical science. Some problem, mathematical or physical, is proposed, which is found to be insoluble by known methods. This condition of insolubility may arise from one of two causes: Either there exists no machinery powerful enough to effect the required reduction, or the workmen are not sufficiently expert to employ their tools in the performance of an entirely new piece of work. The problem proposed is, however, finally solved, and in its solution some new principle, or new application of old principles, is necessarily introduced. If a principle is brought to light it is soon found that in its application it is not necessarily limited to the particular question which occasioned its discovery, and it is then stated in an abstract form and applied to problems of gradually increasing generality.

Other principles, similar in their nature, are added, and the original principle itself receives such modifications and extensions as are from time to time deemed necessary. The same is true of new applications of old principles; the application is first thought to be merely confined to a particular problem, but it is soon recognized that this problem is but one, and generally a very simple one, out of a large class, to which the same process of investigation and solution are applicable. The result in both of these cases is the same. A time comes when these several problems, solutions, and principles are grouped together and found to produce an entirely new and consistent method; a nomenclature and uniform system of notation is adopted, and the principles of the new method become entitled to rank as a distinct science.

Other principles, similar in their nature, are added, and the original principle itself receives such modifications and extensions as are from time to time deemed necessary. The same is true of new applications of old principles; the application is first thought to be merely confined to a particular problem, but it is soon recognized that this problem is but one, and generally a very simple one, out of a large class, to which the same process of investigation and solution are applicable. The result in both of these cases is the same. A time comes when these several problems, solutions, and principles are grouped together and found to produce an entirely new and consistent method; a nomenclature and uniform system of notation is adopted, and the principles of the new method become entitled to rank as a distinct science.

The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. … Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.

The pre-Darwinian age had come to be regarded as a Dark Age in which men still believed that the book of Genesis was a standard scientific treatise, and that the only additions to it were Galileo's demonstration of Leonardo da Vinci’s simple remark that the earth is a moon of the sun, Newton’s theory of gravitation, Sir Humphry Davy's invention of the safety-lamp, the discovery of electricity, the application of steam to industrial purposes, and the penny post.

The process of discovery is very simple. An unwearied and systematic application of known laws to nature, causes the unknown to reveal themselves. Almost any

*mode*of observation will be successful at last, for what is most wanted is method.
The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.

The results of mathematics are seldom directly applied; it is the definitions that are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equations, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply mathematics, you have to live the life of differential equations. When you live this life, you can then go back to molecular biology with a new set of eyes that will see things you could not otherwise see.

The science of

*calculation*… becomes continually more necessary at each step of our progress, and … must ultimately govern the whole of the applications of science to the arts of life.
The significant thing about the Darbys and coke-iron is not that the first Abraham Darby “invented” a new process but that five generations of the Darby connection were able to perfect it and develop most of its applications.

The steam-engine in its manifold applications, the crime-decreasing gas-lamp, the lightning conductor, the electric telegraph, the law of storms and rules for the mariner's guidance in them, the power of rendering surgical operations painless, the measures for preserving public health, and for preventing or mitigating epidemics,—such are among the more important practical results of pure scientific research, with which mankind have been blessed and States enriched.

The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics;

*that*is*mixed*mathematics.
The study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.

The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes

“In order to solve this differential equation you look at it till a solution occurs to you.”

“This principle is so perfectly general that no particular application of it is possible.”

“Geometry is the science of correct reasoning on incorrect figures.”

“My method to overcome a difficulty is to go round it.”

“What is the difference between method and device? A method is a device which you used twice.”

*a*, he says*b*, he means*c*, but it should be*d*. Some of his sayings are handed down from generation to generation:“In order to solve this differential equation you look at it till a solution occurs to you.”

“This principle is so perfectly general that no particular application of it is possible.”

“Geometry is the science of correct reasoning on incorrect figures.”

“My method to overcome a difficulty is to go round it.”

“What is the difference between method and device? A method is a device which you used twice.”

The transition from a paradigm in crisis to a new one from which a new tradition of normal science can emerge is far from a cumulative process, one achieved by an articulation or extension of the old paradigm. Rather it is a reconstruction of the field from new fundamentals, a reconstruction that changes some of the field's most elementary theoretical generalizations as well as many of its paradigm methods and applications. During the transition period there will be a large but never complete overlap between the problems that can be solved by the old and by the new paradigm. But there will also be a decisive difference in the modes of solution. When the transition is complete, the profession will have changed its view of the field, its methods, and its goals.

The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines but of its methods. Mathematics will ever remain the past perfect type of the deductive method in general; and the applications of mathematics to the simpler branches of physics furnish the only school in which philosophers can effectually learn the most difficult and important of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex. These grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education, and regarding with Plato, one who is … as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy

The whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects; and it is only so far, therefore, as we can discover and register resemblances that we can turn our observations to account.

The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.

There are still psychologists who, in a basic misunderstanding, think that gestalt theory tends to underestimate the role of past experience. Gestalt theory tries to differentiate between and-summative aggregates, on the one hand, and gestalten, structures, on the other, both in sub-wholes and in the total field, and to develop appropriate scientific tools for investigating the latter. It opposes the dogmatic application to all cases of what is adequate only for piecemeal aggregates. The question is whether an approach in piecemeal terms, through blind connections, is or is not adequate to interpret actual thought processes and the role of the past experience as well. Past experience has to be considered thoroughly, but it is ambiguous in itself; so long as it is taken in piecemeal, blind terms it is not the magic key to solve all problems.

There cannot be a greater mistake than that of looking superciliously upon practical applications of science. The life and soul of science is its practical application .

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made. the application of the result to some fact of nature is absolutely certain.

There is no such thing as a special category of science called applied science; there is science and its applications, which are related to one another as the fruit is related to the tree that has borne it.

There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.

There is not wholly unexpected surprise, but surprise nevertheless, that mathematics has direct application to the physical world about us.

There is nothing opposed in Biometry and Mendelism. Your husband [W.F.R. Weldon] and I worked that out at Peppards [on the Chilterns] and you will see it referred in the

*Biometrika*memoir. The Mendelian formula leads up to the “ancestral law.” What we fought against was the slovenliness in applying Mendel's categories and asserting that such formulae apply in cases when they did not.
Those who consider James Watt only as a great practical mechanic form a very erroneous idea of his character: he was equally distinguished as a natural philosopher and a chemist, and his inventions demonstrate his profound knowledge of those sciences, and that peculiar characteristic of genius, the union of them for practical application.

Though, probably, no competent geologist would contend that the European classification of strata is applicable to all other parts of the globe, yet most, if not all geologists, write as though it were so.

To ask what qualities distinguish good from routine scientific research is to address a question that should be of central concern to every scientist. We can make the question more tractable by rephrasing it, “What attributes are shared by the scientific works which have contributed importantly to our understanding of the physical world—in this case the world of living things?” Two of the most widely accepted characteristics of good scientific work are generality of application and originality of conception. . These qualities are easy to point out in the works of others and, of course extremely difficult to achieve in one’s own research. At first hearing novelty and generality appear to be mutually exclusive, but they really are not. They just have different frames of reference. Novelty has a human frame of reference; generality has a biological frame of reference. Consider, for example, Darwinian Natural Selection. It offers a mechanism so widely applicable as to be almost coexistent with reproduction, so universal as to be almost axiomatic, and so innovative that it shook, and continues to shake, man’s perception of causality.

To come very near to a true theory, and to grasp its precise application, are two different things, as the history of science teaches us. Everything of importance has been said before by someone who did not discover it.

To emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations, ... let me take one more instance, by choosing a subject in which the purely mathematical interest is deemed supreme, the theory of functions of a complex variable. That at least is a theory in pure mathematics, initiated in that region, and developed in that region; it is built up in scores of papers, and its plan certainly has not been, and is not now, dominated or guided by considerations of applicability to natural phenomena. Yet what has turned out to be its relation to practical issues? The investigations of Lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation; which is merely the general geometrical method of regarding functional relations in that theory. Again, the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, made in the last twenty years, can all be, and now are all, I believe, deduced from similar considerations by interpreting functional relations between complex variables. In the dynamics of a rotating heavy body, the only substantial extension of our knowledge since the time of Lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion. Further, under the title of conjugate functions, the theory has been applied to various questions in electrostatics, particularly in connection with condensers and electrometers. And, lastly, in the domain of physical astronomy, some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas, the principles, the methods, and the results of the theory of functions. … the refined and extremely difficult work of Poincare and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects, developments which were made without a thought of such applications.

To him who devotes his life to science, nothing can give more happiness than increasing the number of discoveries, but his cup of joy is full when the results of his studies immediately find practical applications.

To some men knowledge of the universe has been an end possessing in itself a value that is absolute: to others it has seemed a means of useful applications.

To the manufacturer, chemistry has lately become fruitful of instruction and assistance. In the arts of brewing, tanning, dying, and bleaching, its doctrines are important guides. In making soap, glass, pottery, and all metallic wares, its principles are daily applied, and are capable of a still more useful application, as they become better understood.

Two extreme views have always been held as to the use of mathematics. To some, mathematics is only measuring and calculating instruments, and their interest ceases as soon as discussions arise which cannot benefit those who use the instruments for the purposes of application in mechanics, astronomy, physics, statistics, and other sciences. At the other extreme we have those who are animated exclusively by the love of pure science. To them pure mathematics, with the theory of numbers at the head, is the only real and genuine science, and the applications have only an interest in so far as they contain or suggest problems in pure mathematics.

Of the two greatest mathematicians of modern tunes, Newton and Gauss, the former can be considered as a representative of the first, the latter of the second class; neither of them was exclusively so, and Newton’s inventions in the science of pure mathematics were probably equal to Gauss’s work in applied mathematics. Newton’s reluctance to publish the method of fluxions invented and used by him may perhaps be attributed to the fact that he was not satisfied with the logical foundations of the Calculus; and Gauss is known to have abandoned his electro-dynamic speculations, as he could not find a satisfying physical basis. …

Newton’s greatest work, the

The country of Newton is still pre-eminent for its culture of mathematical physics, that of Gauss for the most abstract work in mathematics.

Of the two greatest mathematicians of modern tunes, Newton and Gauss, the former can be considered as a representative of the first, the latter of the second class; neither of them was exclusively so, and Newton’s inventions in the science of pure mathematics were probably equal to Gauss’s work in applied mathematics. Newton’s reluctance to publish the method of fluxions invented and used by him may perhaps be attributed to the fact that he was not satisfied with the logical foundations of the Calculus; and Gauss is known to have abandoned his electro-dynamic speculations, as he could not find a satisfying physical basis. …

Newton’s greatest work, the

*Principia*, laid the foundation of mathematical physics; Gauss’s greatest work, the*Disquisitiones Arithmeticae*, that of higher arithmetic as distinguished from algebra. Both works, written in the synthetic style of the ancients, are difficult, if not deterrent, in their form, neither of them leading the reader by easy steps to the results. It took twenty or more years before either of these works received due recognition; neither found favour at once before that great tribunal of mathematical thought, the Paris Academy of Sciences. …The country of Newton is still pre-eminent for its culture of mathematical physics, that of Gauss for the most abstract work in mathematics.

We are, of course, extremely concerned that if this did, in fact, happen, that there is going to be a tremendous public outcry, and we will be concerned with what the Congress does, … Obviously, we are concerned about there being a backlash against the medical applications of this technology, which have, of course, the potential to cure millions of patients.

We do not learn by inference and deduction, and the application of mathematics to philosophy, but by direct intercourse and sympathy.

What is a good definition? For the philosopher or the scientist, it is a definition which applies to all the objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.

What is important is the gradual development of a theory, based on a careful analysis of the ... facts. ... Its first applications are necessarily to elementary problems where the result has never been in doubt and no theory is actually required. At this early stage the application serves to corroborate the theory. The next stage develops when the theory is applied to somewhat more complicated situations in which it may already lead to a certain extent beyond the obvious and familiar. Here theory and application corroborate each other mutually. Beyond lies the field of real success: genuine prediction by theory. It is well known that all mathematized sciences have gone through these successive stages of evolution.

When you are identifying science of the motion of water, remember to include under each subject its application and use, so that the science will be useful.

Whether we like it or not, the ultimate goal of every science is to become trivial, to become a well-controlled apparatus for the solution of schoolbook exercises or for practical application in the construction of engines.

While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice.

While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established, and that further advances are to be sought chiefly in the rigorous applications of these principles to all the phenomena which come under our notice. It is here that the science of measurement shows its importance—where the quantitative results are more to be desired than qualitative work. An eminent physicist has remarked that the future truths of Physical Science are to be looked for in the sixth place of decimals.

Will fluorine ever have practical applications?

It is very difficult to answer this question. I may, however, say in all sincerity that I gave this subject little thought when I undertook my researches, and I believe that all the chemists whose attempts preceded mine gave it no more consideration.

A scientific research is a search after truth, and it is only after discovery that the question of applicability can be usefully considered.

It is very difficult to answer this question. I may, however, say in all sincerity that I gave this subject little thought when I undertook my researches, and I believe that all the chemists whose attempts preceded mine gave it no more consideration.

A scientific research is a search after truth, and it is only after discovery that the question of applicability can be usefully considered.

Without theory, practice is but routine born of habit. Theory alone can bring forth and develop the spirit of invention. ... [Do not] share the opinion of those narrow minds who disdain everything in science which has not an immediate application. ... A theoretical discovery has but the merit of its existence: it awakens hope, and that is all. But let it be cultivated, let it grow, and you will see what it will become.

Would you have a man reason well, you must use him to it betimes; exercise his mind in observing the connection between ideas, and following them in train. Nothing does this better than mathematics, which therefore, I think should be taught to all who have the time and opportunity, not so much to make them mathematicians, as to make them reasonable creatures; for though we all call ourselves so, because we are born to it if we please, yet we may truly say that nature gives us but the seeds of it, and we are carried no farther than industry and application have carried us.

[For the] increase of knowledge and … the useful application of the knowledge gained, … there never is a sudden beginning; even the cloud change which portends the thunderstorm begins slowly.

[My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who] devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis.

[The surplus of basic knowledge of the atomic nucleus was] largely used up [during the war with the atomic bomb as the dividend.] We must, without further delay restore this surplus in preparation for the important peacetime job for the nucleus - power production. ... Many of the proposed applications of atomic power - even for interplanetary rockets - seem to be within the realm of possibility provided the economic factor is ruled out completely, and the doubtful physical and chemical factors are weighted heavily on the optimistic side. ... The development of economic atomic power is not a simple extrapolation of knowledge gained during the bomb work. It is a new and difficult project to reach a satisfactory answer. Needless to say, it is vital that the atomic policy legislation now being considered by the congress recognizes the essential nature of this peacetime job, and that it not only permits but encourages the cooperative research-engineering effort of industrial, government and university laboratories for the task. ... We must learn how to generate the still higher energy particles of the cosmic rays - up to 1,000,000,000 volts, for they will unlock new domains in the nucleus.

[Young] was afterwards accustomed to say, that at no period of his life was he particularly fond of repeating experiments, or even of very frequently attempting to originate new ones; considering that, however necessary to the advancement of science, they demanded a great sacrifice of time, and that when the fact was once established, that time was better employed in considering the purposes to which it might be applied, or the principles which it might tend to elucidate.

… on these expanded membranes [butterfly wings] Nature writes, as on a tablet, the story of the modifications of species, so truly do all changes of the organisation register themselves thereon. Moreover, the same colour-patterns of the wings generally show, with great regularity, the degrees of blood-relationship of the species. As the laws of nature must be the same for all beings, the conclusions furnished by this group of insects must be applicable to the whole world.