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Exact Quotes (75 quotes)

[Edison] definitely ended the distinction between the theoretical man of science and the practical man of science, so that today we think of scientific discoveries in connection with their possible present or future application to the needs of man. He took the old rule-of-thumb methods out of industry and substituted exact scientific knowledge, while, on the other hand, he directed scientific research into useful channels.
In My Friend Mr. Edison (1930). Quoted in Dyson Carter, If You Want to Invent (1939), 110.
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[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity.
In Ueber das Verhältnis der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden (1896), Bd. 1, 176. Also seen translated as “In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor”, in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 20. From the original German, “Hier sehen wir die bewusste logische Thätigkeit unseres Geistes in ihrer reinsten und vollendetsten Form; wir können hier die ganze Mühe derselben kennen lernen, die grosse Vorsicht, mit der sie vorschreiten muss, die Genauigkeit, welche nöthig ist, um den Umfang der gewonnenen allgemeinen Sätze genau zu bestimmen, die Schwierigkeit, abstracte Begriffe zu bilden und zu verstehen; aber ebenso auch Vertrauen fassen lernen in die Sicherheit, Tragweite und Fruchtbarkeit solcher Gedankenarbeit.”
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[The] subjective [historical] element in geologic studies accounts for two characteristic types that can be distinguished among geologists: one considering geology as a creative art, the other regarding geology as an exact science.
In 'The Scientific Character of Geology', The Journal of Geology (Jul 1961), 69, No. 4, 453.
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Die Politik ist keine Wissenschaft, wie viele der Herren Professoren sich einbilden, sondern eine Kunst, sie ist ebensowenig.
Politics is not a science, as many of the professors imagine, but an art, it is just like that.
Original German in Otto von Bismarck, Horst Kohl, Bismarckreden: 1847-1895 (1899), 255. As quoted in translation in William Roscoe Thayer 'Cavour and Bismarck', The Atlantic (Mar 1909), 103, 343.
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~~[Attributed, authorship undocumented]~~ Mathematical demonstrations are a logic of as much or more use, than that commonly learned at schools, serving to a just formation of the mind, enlarging its capacity, and strengthening it so as to render the same capable of exact reasoning, and discerning truth from falsehood in all occurrences, even in subjects not mathematical. For which reason it is said, the Egyptians, Persians, and Lacedaemonians seldom elected any new kings, but such as had some knowledge in the mathematics, imagining those, who had not, men of imperfect judgments, and unfit to rule and govern.
From an article which appeared as 'The Usefulness of Mathematics', Pennsylvania Gazette (30 Oct 1735), No. 360. Collected, despite being without clear evidence of Franklin’s authorship, in The Works of Benjamin Franklin (1809), Vol. 4, 377. Evidence of actual authorship by Ben Franklin for the newspaper article has not been ascertained, and scholars doubt it. See Franklin documents at the website founders.archives.gov. The quote is included here to attach this caution.
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A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws.
As quoted in Philipp Frank, Modern Science and its Philosophy (1949), 15, which cites Théorie Physique; Son Objet—Son Structure (1906), 24.
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Analogy is a wonderful, useful and most important form of thinking, and biology is saturated with it. Nothing is worse than a horrible mass of undigested facts, and facts are indigestible unless there is some rhyme or reason to them. The physicist, with his facts, seeks reason; the biologist seeks something very much like rhyme, and rhyme is a kind of analogy.... This analogizing, this fine sweeping ability to see likenesses in the midst of differences is the great glory of biology, but biologists don't know it.... They have always been so fascinated and overawed by the superior prestige of exact physical science that they feel they have to imitate it.... In its central content, biology is not accurate thinking, but accurate observation and imaginative thinking, with great sweeping generalizations.
In Science is a Sacred Cow (1950), 98-100.
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Bacon himself was very ignorant of all that had been done by mathematics; and, strange to say, he especially objected to astronomy being handed over to the mathematicians. Leverrier and Adams, calculating an unknown planet into a visible existence by enormous heaps of algebra, furnish the last comment of note on this specimen of the goodness of Bacon’s view… . Mathematics was beginning to be the great instrument of exact inquiry: Bacon threw the science aside, from ignorance, just at the time when his enormous sagacity, applied to knowledge, would have made him see the part it was to play. If Newton had taken Bacon for his master, not he, but somebody else, would have been Newton.
In Budget of Paradoxes (1872), 53-54.
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Besides accustoming the student to demand, complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth. Hardly any of the non-mathematical sciences, except chemistry, has this advantage. One of the commonest modes of loose thought, and sources of error both in opinion and in practice, is to overlook the importance of quantities. Mathematicians and chemists are taught by the whole course of their studies, that the most fundamental difference of quality depends on some very slight difference in proportional quantity; and that from the qualities of the influencing elements, without careful attention to their quantities, false expectation would constantly be formed as to the very nature and essential character of the result produced.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611. [The French phrase, à peu près means “approximately”. —Webmaster]
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Descriptive geometry has two objects: the first is to establish methods to represent on drawing paper which has only two dimensions,—namely, length and width,—all solids of nature which have three dimensions,—length, width, and depth,—provided, however, that these solids are capable of rigorous definition.
The second object is to furnish means to recognize accordingly an exact description of the forms of solids and to derive thereby all truths which result from their forms and their respective positions.
From On the Purpose of Descriptive Geometry as translated by Arnold Emch in David Eugene Smith, A Source Book in Mathematics (1929), 426.
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Detection is, or ought to be, an exact science, and should be treated in the same cold unemotional manner. You have attempted to tinge it with romanticism, which produces the same effect as if you worked a love-story into the fifth proposition of Euclid.
By Sherlock Holmes to Dr. Watson, fictional characters in The Sign of Four (1890), 6.
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Doubtless the reasoning faculty, the mind, is the leading and characteristic attribute of the human race. By the exercise of this, man arrives at the properties of the natural bodies. This is science, properly and emphatically so called. It is the science of pure mathematics; and in the high branches of this science lies the truly sublime of human acquisition. If any attainment deserves that epithet, it is the knowledge, which, from the mensuration of the minutest dust of the balance, proceeds on the rising scale of material bodies, everywhere weighing, everywhere measuring, everywhere detecting and explaining the laws of force and motion, penetrating into the secret principles which hold the universe of God together, and balancing worlds against worlds, and system against system. When we seek to accompany those who pursue studies at once so high, so vast, and so exact; when we arrive at the discoveries of Newton, which pour in day on the works of God, as if a second fiat had gone forth from his own mouth; when, further, we attempt to follow those who set out where Newton paused, making his goal their starting-place, and, proceeding with demonstration upon demonstration, and discovery upon discovery, bring new worlds and new systems of worlds within the limits of the known universe, failing to learn all only because all is infinite; however we may say of man, in admiration of his physical structure, that “in form and moving he is express and admirable,” it is here, and here without irreverence, we may exclaim, “In apprehension how like a god!” The study of the pure mathematics will of course not be extensively pursued in an institution, which, like this [Boston Mechanics’ Institute], has a direct practical tendency and aim. But it is still to be remembered, that pure mathematics lie at the foundation of mechanical philosophy, and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation.
In Works (1872), Vol. 1, 180.
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Exactness cannot be established in the arguments unless it is first introduced into the definitions.
In Science et Méthode, as translated by Francis Maitland, in Science and Method (1914), 123-124.
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Examine your words well, and you will find that even when you have no motive to be false, it is a very hard thing to say the exact truth, even about your own immediate feelings—much harder than to say something fine about them which is not the exact truth.
In Adam Bede (1859, 1860), 151.
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Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.
In 'The Future of Data Analysis', Annals of Mathematical Statistics (1962), 33, No. 1, 13-14.
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Formal thought, consciously recognized as such, is the means of all exact knowledge; and a correct understanding of the main formal sciences, Logic and Mathematics, is the proper and only safe foundation for a scientific education.
In Number and its Algebra (1896), 134.
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How often might a man, after he had jumbled a set of letters in a bag, fling them out upon the ground before they would fall into an exact poem, yea, or so much as make a good discourse in prose. And may not a little book be as easily made by chance as this great volume of the world.
In The Works of the Most Reverend Dr. John Tillotson (1714), 15.
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Hurrah for positive science! long live exact demonstration!
In Leaves of Grass (1855), 28.
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I was interested in flying beginning at age 7, when a close family friend took me in his little airplane. And I remember looking at the wheel of the airplane as we rolled down the runway, because I wanted to remember the exact moment that I first went flying... the other thing growing up is that I was always interested in science.
…...
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If a project is truly innovative, you cannot possibly know its exact cost and exact schedule at the beginning. And if you do know the exact cost and the exact schedule, chances are that the technology is obsolete.
From interview with Technology Review, quoted in Douglas Martin, 'Joseph Gavin, Who Helped Put First Man on Moon, Dies at 90', New York Times (4 Nov 2010)
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If the actual order of the bases on one of the pair of chains were given, one could write down the exact order of the bases on the other one, because of the specific pairing. Thus one chain is, as it were, the complement of the other, and it is this feature which suggests how the deoxyribonucleic acid molecule might duplicate itself.
[Co-author with Francis Crick]
In 'Genetic Implications of the Structure of Deoxyribonucleic Acid', Nature (1958), 171, 965-966.
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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.
Address (28 Mar 1912), Michigan School Masters' Club, Ann Arbor, 'The Humanization of the Teaching of Mathematics. Printed in Science (26 Apr 1912). Collected in The Human Worth of Rigorous Thinking: Essays and Addresses (1916), 65-66.
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In all that has to do with the relations between man and the supernatural, we have to seek for a more than mathematical precision; this should be more exact than science.
In Gravity and Grace, (1947, 1952), 186.
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In order that the facts obtained by observation and experiment may be capable of being used in furtherance of our exact and solid knowledge, they must be apprehended and analysed according to some Conceptions which, applied for this purpose, give distinct and definite results, such as can be steadily taken hold of and reasoned from.
Philosophy of the Inductive Sciences (1840), Vol. 2, 205.
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In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. “Pauca sed matura” was Gauss’s motto.
In Presidential Address British Association for the Advancement of Science, Section A, (1890), Nature, 42, 467. [The Latin motto translates as “Few, but ripe”. —Webmaster]
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In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite.
As quoted, without citation, in Robert Jungk and James Cleugh (trans.), Brighter Than a Thousand Suns: A Personal History of the Atomic Scientists (1958), 22. The writing of poetry by Robert Oppenheimer was being remarked upon by Dirac, as he was talking with him.
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In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.
In letter to Tschirnhaus. As quoted in George F. Simmons Calculus Gems (1992), 156, citing Dirk Jan Struik, 281-282.
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In the year 1692, James Bernoulli, discussing the logarithmic spiral [or equiangular spiral, ρ = αθ] … shows that it reproduces itself in its evolute, its involute, and its caustics of both reflection and refraction, and then adds: “But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might not be inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent, simillima filia matri. Or, if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light from light, remains ὁμοούσιος with him, howsoever overshadowed. Or, if you prefer, since our spira mirabilis remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a symbol, either of fortitude and constancy in adversity, or, of the human body, which after all its changes, even after death, will be restored to its exact and perfect self, so that, indeed, if the fashion of Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, ‘Though changed, I arise again exactly the same, Eadem numero mutata resurgo.’”
In 'The Uses of Mathesis', Bibliotheca Sacra, Vol. 32, 516-516. [The Latin phrase “simillima filia matri” roughly translates as “the daughter resembles the mother”. “Spira mirabilis” is Latin for “marvellous spiral”. The Greek word (?µ???s???) translates as “consubstantial”, meaning of the same substance or essence (used especially of the three persons of the Trinity in Christian theology). —Webmaster]
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It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth.
In 'The Analyst', in The Works of George Berkeley (1898), Vol. 3, 10.
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It is hard to describe the exact route to scientific achievement, but a good scientist doesn’t get lost as he travels it.
Epigraph in Isaac Asimov’s Book of Science and Nature Quotations (1988), 290.
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It is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
…...
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Laws are important and valuable in the exact natural sciences, in the measure that those sciences are universally valid.
In Max Weber with translation by Edward Shils and Henry A. Finch (eds.), The Methodology of the Social Sciences (1949), 80.
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Life is not an exact science; it is an art.
In Samuel Butler and Henry Festing Jones (ed.), 'Reconciliation', The Note-books of Samuel Butler (1912, 1917), 351.
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Mathematicians boast of their exacting achievements, but in reality they are absorbed in mental acrobatics and contribute nothing to society.
From Complete Works on Japan’s Philosophical Thought (1956). As quoted and cited in Alan L. Mackay, A Dictionary of Scientific Quotations (1991), 185.
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Mathematics … is necessarily the foundation of exact thought as applied to natural phenomena.
In An Introduction to Mathematics (1911), 8.
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Mathematics has beauties of its own—a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty. … When this subject is properly and concretely presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant.
In The Teaching of Mathematics in the Elementary and the Secondary School (1906), 44-45.
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Mathematics is an obscure field, an abstruse science, complicated and exact; yet so many have attained perfection in it that we might conclude almost anyone who seriously applied himself would achieve a measure of success.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 82.
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Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.
(1923). Quoted, without source, in E.T. Bell, Men of Mathematics (1937), Vol. 1, li.
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Modern Science, as training the mind to an exact and impartial analysis of facts is an education specially fitted to promote sound citizenship.
From The Grammar of Science (1892), 11.
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Moreover, I am now well-informed about that part of the country to be traversed, as I can acquire the most exact information from the Nubian leaders, who are truly well-versed in regional knowledge, and thus supported in my route I could determine positions with the greatest certainty. I could not do this before, as one only learns how to interpret the statements of the Nubians by undertaking such a journey.
In August Petermann, Petermann’s Geographische Mittheilungen (1871), 16. As quoted and cited in Kathrin Fritsch, '"You Have Everything Confused And Mixed Up…!" Georg Schweinfurth, Knowledge And Cartography Of Africa In The 19th Century', History in Africa (2009), 36, 94. Fritsch comments on his changed “acknowledgement of the quality of indigenous knowledge.”
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Natural powers, principally those of steam and falling water, are subsidized and taken into human employment Spinning-machines, power-looms, and all the mechanical devices, acting, among other operatives, in the factories and work-shops, are but so many laborers. They are usually denominated labor-saving machines, but it would be more just to call them labor-doing machines. They are made to be active agents; to have motion, and to produce effect; and though without intelligence, they are guided by laws of science, which are exact and perfect, and they produce results, therefore, in general, more accurate than the human hand is capable of producing.
Speech in Senate (12 Mar 1838). In The Writings and Speeches of Daniel Webster (1903), Vol. 8, 177.
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Nature could not err, but knew exactly and infallibly. Our function was to ask her questions, to listen patiently to the answers, and to understand them correctly.
And to this end we went out into the field, time and time again. Often at brief intervals we returned to the same outcrop where our Pythia, nature, opened her mouth from time to time to utter her equivocal oracles. Time and again we studied the same stratification, or the same interpenetration of rocks, and yet each time advanced one step further, because of what we had learned on the last visit, because the previous impression had had time to settle, or because this time our eyes were a little keener and now observed what hitherto had escaped them.
In Hans Cloos, Ernst Cloos (ed.) and Curt Dietz (ed.), Conversation With the Earth (1953, 1959), 28, as translated by E.B. Garside from the original German edition, Gespräch mit der Erde (1947).
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Nature! … She is the only artist; working-up the most uniform material into utter opposites; arriving, without a trace of effort, at perfection, at the most exact precision, though always veiled under a certain softness.
As quoted by T.H. Huxley, in Norman Lockyer (ed.), 'Nature: Aphorisms by Goethe', Nature (1870), 1, 10.
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One aim of physical sciences had been to give an exact picture the material world. One achievement of physics in the twentieth century has been to prove that that aim is unattainable.
From The Ascent of Man (1973), 353.
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Only science, exact science about human nature itself, and the most sincere approach to it by the aid of the omnipotent scientific method, will deliver man from his present gloom and will purge him from his contemporary share in the sphere of interhuman relations.
In Ivan Pavlov and William Horsley Gantt (trans.), Lectures on Conditioned Reflexes (1928, 1941), Preface, 41.
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Poetry is as exact a science as geometry.
In Letters of Gustave Flaubert (1951), 81.
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Science enhances the moral value of life, because it furthers a love of truth and reverence—love of truth displaying itself in the constant endeavor to arrive at a more exact knowledge of the world of mind and matter around us, and reverence, because every advance in knowledge brings us face to face with the mystery of our own being.
In Max Planck and James Vincent Murphy (trans.), Where is Science Going?, (1932), 169.
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Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,— we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,— social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.
In Lectures on Teaching (1906), 891-92.
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That the fundamental aspects of heredity should have turned out to be so extraordinarily simple supports us in the hope that nature may, after all, be entirely approachable. Her much-advertised inscrutability has once more been found to be an illusion due to our ignorance. This is encouraging, for, if the world in which we live were as complicated as some of our friends would have us believe we might well despair that biology could ever become an exact science.
The Physical Basis of Heredity (1919), 15.
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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.
In An Examination of Sir William Hamilton’s Philosophy (1865), 531.
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The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
In 'The Mathematician', Works of the Mind (1947), 1, No. 1. Collected in James Roy Newman (ed.), The World of Mathematics (1956), Vol. 4, 2055.
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The determination of the relationship and mutual dependence of the facts in particular cases must be the first goal of the Physicist; and for this purpose he requires that an exact measurement may be taken in an equally invariable manner anywhere in the world… Also, the history of electricity yields a well-known truth—that the physicist shirking measurement only plays, different from children only in the nature of his game and the construction of his toys.
In 'Mémoire sur la mesure de force de l'électricité', Journal de Physique (1782), 21, 191. English version by Google Translate tweaked by Webmaster. From the original French, “La determination de la relation & de la dépendance mutuelle de ces données dans certains cas particuliers, doit être le premier but du Physicien; & pour cet effet, il falloit one mesure exacte qui indiquât d’une manière invariable & égale dans tous les lieux de la terre, le degré de l'électricité au moyen duquel les expéiences ont été faites… Aussi, l’histoire de l'électricité prouve une vérité suffisamment reconnue; c’est que le Physicien sans mesure ne fait que jouer, & qu’il ne diffère en cela des enfans, que par la nature de son jeu & la construction de ses jouets.”
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The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.
In J.R. Newman (ed.), 'Commentary on John Couch Adams', The World of Mathematics (1956), 820.
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The earliest of my childhood recollections is being taken by my grandfather when he set out in the first warm days of early spring with a grubbing hoe (we called it a mattock) on his shoulder to seek the plants, the barks and roots from which the spring medicine for the household was prepared. If I could but remember all that went into that mysterious decoction and the exact method of preparation, and with judicious advertisement put the product upon the market, I would shortly be possessed of wealth which might be made to serve the useful purpose of increasing the salaries of all pathologists. … But, alas! I remember only that the basic ingredients were dogwood bark and sassafras root, and to these were added q.s. bloodroot, poke and yellow dock. That the medicine benefited my grandfather I have every reason to believe, for he was a hale, strong old man, firm in body and mind until the infection came against which even spring medicine was of no avail. That the medicine did me good I well know, for I can see before me even now the green on the south hillside of the old pasture, the sunlight in the strip of wood where the dogwood grew, the bright blossoms and the delicate pale green of the leaf of the sanguinaria, and the even lighter green of the tender buds of the sassafras in the hedgerow, and it is good to have such pictures deeply engraved in the memory.
From address, 'A Medical Retrospect'. Published in Yale Medical Journal (Oct 1910), 17, No. 2, 57. [Note: q.s. in an abbreviation for quantum sufficit meaning “as much as is sufficient,” when used as a quantity specification in medicine and pharmacology. -Webmaster]
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The electrical engineer has an enormous advantage over other engineers; everything lends itself to exact calculation, and a completed machine or any of its parts may he submitted to the most searching electrical and magnetic tests, since these tests, unlike those applied by other engineers, do not destroy the body tested.
From Opening Address, Engineering Section, British Association meeting, Belfast. ,In Norman Lockyer (ed.), Nature (25 Sep 1902), 66, No. 1717, 536.
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The end of the eighteenth and the beginning of the nineteenth century were remarkable for the small amount of scientific movement going on in this country, especially in its more exact departments. ... Mathematics were at the last gasp, and Astronomy nearly so—I mean in those members of its frame which depend upon precise measurement and systematic calculation. The chilling torpor of routine had begun to spread itself over all those branches of Science which wanted the excitement of experimental research.
Quoted in Sophia Elizabeth De Morgan, Memoir of Augustus De Morgan (1882), 41
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The external world of physics has … become a world of shadows. In removing our illusions we have removed the substance, for indeed we have seen that substance is one of the greatest of our illusions. Later perhaps we may inquire whether in our zeal to cut out all that is unreal we may not have used the knife too ruthlessly. Perhaps, indeed, reality is a child which cannot survive without its nurse illusion. But if so, that is of little concern to the scientist, who has good and sufficient reasons for pursuing his investigations in the world of shadows and is content to leave to the philosopher the determination of its exact status in regard to reality.
In Introduction to The Nature of the Physical World (1928), xiv.
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THE fundamental questions in chemistry,—those questions the answers to which would convert chemistry into a branch of exact science, and enable us to predict with absolute certainty the result of every reaction—are (1) What is the nature of the forces which retain the several molecules or atoms of a compound together? and (2) How may their direction and amount be determined? We may safely say that, in the present state of the science, these questions cannot be answered; and it is extremely doubtful whether any future advances will render their solution possible.
Opening paragraph of his University of Edinburgh M.D. thesis, 'On the Theory of Chemical Combination' (1861).
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The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
In History of Mathematics (3rd Ed., 1901), 468.
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The growing complexity of civilized life demands with each age broader and more exact knowledge as to the material surroundings and greater precision in our recognition of the invisible forces or tendencies about us.
From Presidential Address (5 Dec 1896) to the Biological Society of Washington, 'The Malarial Parasite and Other Pathogenic Protozoa', Popular Science Monthly (Mar 1897), 642.
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The method of scientific investigation is nothing but the expression of the necessary mode of working of the human mind. It is simply the mode at which all phenomena are reasoned about, rendered precise and exact.
In 'Method of Discovery', On Our Knowledge of the Causes of the Phenomena of Organic Nature (1863), 56. Also in excerpt collected in in Isabel S. Gordon and Sophie Sorkin (eds.), The Armchair Science Reader (1959), 263.
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The observed phenomena of meteorology and the well-established laws of physics are the two extremes of the science of meteorology between which we trace the connection of cause and effect; in so far as we can do this successfully meteorology becomes an exact deductive science.
In 'The Meteorological Work of the U.S. Signal Service, 1870 to 1891', U.S. Department of Agriculture, Weather Bureau, Bulletin No. 11, Report of the International Meteorological Congress, Chicago, Ill., August 21-24, 1893 (1894), 242.
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The only way in which to treat the elements of an exact and rigorous science is to apply to them all the rigor and exactness possible.
Quoted in De Morgan, Trigonometry and Double Algebra (1849), Title page.
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The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language. It would defy all the power of Mercury himself to explain to a person ignorant of the science what is meant by the single phrase “functional exponent.” How much more impossible, if we may say so, would it be to explain a whole treatise like Hamilton’s Quaternions, in such a wise as to make it possible to judge of its value! But to one who has learned this language, it is the most precise and clear of all modes of expression. It discloses the thought exactly as conceived by the writer, with more or less beauty of form, but never with obscurity. It may be prolix, as it often is among French writers; may delight in mere verbal metamorphoses, as in the Cambridge University of England; or adopt the briefest and clearest forms, as under the pens of the geometers of our Cambridge; but it always reveals to us precisely the writer’s thought.
In North American Review (Jul 1857), 85, 224-225.
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The universe is what it is and can’t be changed by jiggery-pokery. It works by exact rules, like a machine. … Natural law never takes a holiday. The invariability of natural law is the cornerstone of science.
In Glory Road (1963, 1981), 115.
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The whole art of making experiments in chemistry is founded on the principle: we must always suppose an exact equality or equation between the principles of the body examined and those of the products of its analysis.
From Traité Élémentaire de Chimie (1789), 140. Translation as given in James Riddick Partington, A Short History of Chemistry (1960), 124. This is an alternate translation of part of the same passage on this page that begins “We may lay it down as an incontestible axiom…”
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There is plenty of room left for exact experiment in art, and the gate has been opened for some time. What had been accomplished in music by the end of the eighteenth century has only begun in the fine arts. Mathematics and physics have given us a clue in the form of rules to be strictly observed or departed from, as the case may be. Here salutary discipline is come to grips first of all with the function of forms, and not with form as the final result … in this way we learn how to look beyond the surface and get to the root of things.
Paul Klee
Quoted in The Bulletin of the Atomic Scientists (Feb 1959), 59, citing Bauhaus-Zeitschrijt (1928).
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We all came from the sea. And it is an interesting biological fact that all of us have, in our veins the exact same percentage of salt in our blood that exists in the ocean, and, therefore, we have salt in our blood, in our sweat, in our tears.
Speech at the America’s Cup Dinner (14 Sep 1962).
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We do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic musical.
In 'Works and Days', Society and Solitude (1883), Chap. 7, 171.
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We have very strong physical and chemical evidence for a large impact; this is the most firmly established part of the whole story. There is an unquestionable mass extinction at this time, and in the fossil groups for which we have the best record, the extinction coincides with the impact to a precision of a centimeter or better in the stratigraphic record. This exact coincidence in timing strongly argues for a causal relationship.
Referring to the theory that he, and his father (physicist Luis W. Alvarez), held that dinosaurs abruptly went extinct as a result of a 6-mile-wide asteroid or comet struck the earth. In American Geophysical Union, EOS (2 Sep 1986), as quoted and cited in John Noble Wilford, 'New Data Extend Era of Dinosaurs' New York Times (9 Nov 1986), A41.
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We love to discover in the cosmos the geometrical forms that exist in the depths of our consciousness. The exactitude of the proportions of our monuments and the precision of our machines express a fundamental character of our mind. Geometry does not exist in the earthly world. It has originated in ourselves. The methods of nature are never so precise as those of man. We do not find in the universe the clearness and accuracy of our thought. We attempt, therefore, to abstract from the complexity of phenomena some simple systems whose components bear to one another certain relations susceptible of being described mathematically.
In Man the Unknown (1935), 8.
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Were the Greeks scientists? Then so are the modern chiropractors. What they had of exact knowledge, in fact, was mainly borrowed, and most of it was spoiled in the borrowing.
From book review (of The Cambridge Ancient History, Vol. V), in the American Mercury (Oct 1927), 254-55. Collected in A Mencken Chrestomathy (1949, 1956), 215.
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What is exact about mathematics but exactness? And is not this a consequence of the inner sense of truth?
In 'Sprüche in Prosa', Natur, 6, 948.
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When we speak of the picture of nature in the exact science of our age, we do not mean a picture of nature so much as a picture of our relationships with nature. The old division of the world into objective processes in space and time and the mind in which these processes are mirrored … is no longer a suitable starting point for our understanding of modern science. Science, we find, is now focused on the network of relationships between man and nature, on the framework which makes us as living beings dependent parts of nature, and which we as human beings have simultaneously made the object of our thoughts and actions. Science no longer confronts nature as an objective observer, but sees itself as an actor in this interplay between man and nature. The scientific method of analysing, explaining and classifying has become conscious of its limitations, which arise out of the fact that by its intervention science alters and refashions the object of investigation. In other words, method and object can no longer be separated. The scientific world view has ceased to be a scientific view in the true sense of the word.
From Das Naturbild der Heutigen Physik (1955). Translated by Arnold J. Pomerans, The Physicist’s Conception of Nature (1958), 29. Compare a similar but original translation by O. T. Benfey, 'The Representation of Nature in Contemporary Physics', in Daedalus (Summer 1958), 87, No. 3, 107.
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Working is thinking, hence it is not always easy to give an exact accounting of one’s time. Usually I work about four to six hours a day. I am not a very diligent man.
In Cosmic Religion: With Other Opinions and Aphorisms (1931), 105.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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