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Home > Category Index for Science Quotations > Category Index S > Category: Series

Series Quotes (26 quotes)

Derričre la série de Fourier, d'autres séries analogues sont entrées dans la domaine de l'analyse; elles y sont entrees par la męme porte; elles ont été imaginées en vue des applications.
After the Fourier series, other series have entered the domain of anylsis; they entered by the same door; they have been imagined in view of applications.
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 74.
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La théorie des séries infinies en général est justqu’à présent très mal fondée. On applique aux séries infinies toutes les opérations, come si elles aient finies; mais cela est-il bien permis? Je crois que non. Où est-il démonstré qu/on ontient la différentielle dune série infinie en prenant la différentiaella de chaque terme. Rien n’est plus facile que de donner des exemples où cela n’est pas juste.
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
Quoted in Reinhold Remmert and Robert B. Burckel, Theory of Complex Functions: Readings in Mathematics (1991), 125.
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Lyveris to-forn us
Useden to marke
For selkouthes that thei seighen,
Hir sones for to teche;
And helden it an heigh science
Hir wittes to knowe.
Ac thorugh hir science soothly
Was nevere no soule y-saved,
Ne broght by hir bokes
To blisse ne to joye;
For alle hir kynde knowynges
Come but of diverse sightes.
Patriarkes and prophetes
Repreveden hir science,
And seiden hir wordes and hir wisdomes
Nas but a folye
And to the clergie of Crist
Counted it but a trufle.

Our ancestors in olden days used to record
The strange things they saw, and teach them to their sons;
And they held it a high science, to have knowledge of such things.
But no soul was ever saved by all that science,
Nor brought by books into eternal bliss;
Their science was only a series of sundry observations.
So patriarchs and prophets disapproved of their science,
And said their so-called words of wisdom were but folly—
And compared with Christian philosophy, a contemptible thing.
In William Langland and B. Thomas Wright (ed.) The Vision and Creed of Piers Ploughman (1842), 235-236. Modern translation by Terrence Tiller in Piers Plowman (1981, 1999), 123.
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All revolutionary advances in science may consist less of sudden and dramatic revelations than a series of transformations, of which the revolutionary significance may not be seen (except afterwards, by historians) until the last great step. In many cases the full potentiality and force of a most radical step in such a sequence of transformations may not even be manifest to its author.
The Newtonian Revolution (1980), 162.
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Among the studies to which the [Rockefeller] Foundation is giving support is a series in a relatively new field, which may be called molecular biology, in which delicate modern techniques are being used to investigate ever more minute details of certain life processes.
In 'Molecular Biology', Annual Report of the Rockefeller Foundation (1938), 203-4. Reprinted in a letter to Science (6 Nov 1970), 170, 582.
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Embryonic stem cell research is at the leading edge of a series of moral hazards.
'Address to the Nation on Stem Cell Research', (9 Aug 2001) in Public Papers Of The Presidents Of The United States, George W. Bush, 2001 (2004), Book 2, 955.
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Entropy theory, on the other hand, is not concerned with the probability of succession in a series of items but with the overall distribution of kinds of items in a given arrangement.
In Entropy and Art: An Essay on Disorder and Order (1974), 19.
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Every arsenate has its corresponding phosphate, composed according to the same proportions, combined with the same amount of water of crystallization, and endowed with the same physical properties: in fact, the two series of salts differ in no respect, except that the radical of the acid in one series in phosphorus, while in the other it is arsenic.
The experimental clue he used forming his law of isomerism. Originally published in 'Om Förhĺllandet emellan chemiska sammansättningen och krystallformen hos Arseniksyrade och Phosphorsyrade Salter', (On the Relation between the Chemical Composition and Crystal Form of Salts of Arsenic and Phosphoric Acids), Kungliga Svenska vetenskapsakademiens handlingar (1821), 4. Translation as shown in Joseph William Mellor, A Comprehensive Treatise on Inorganic and Theoretical Chemistry (1922), Vol. 1, 652. A very similar translation (“the same physical properties” is replaced with “nearly equal solubilities in water and acids”) is in F. Szabadváry article on 'Eilhard Mitscherlich' in Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography (1974), Vol. 9, 424; perhaps from J.R. Partington, A History of Chemistry, Vol. 4 (1964), 210.
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I have not trodden through a conventional university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling.”
First letter to G.H. Hardy (16 Jan 1913). In Collected Papers of Srinivasa Ramanujan (1927), xxiii. Hardy notes he did “seem to remember his telling me that his friends had given him some assistance” in writing the letter because Ramanujan's “knowledge of English, at that stage of his life, could scarcely have been sufficient.”
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I'd like the [Cosmos] series to be so visually stimulating that somebody who isn't even interested in the concepts will just watch for the effects. And I'd like people who are prepared to do some thinking to be really stimulated.
Quoted by Dennis Meredith, in 'Carl Sagan's Cosmic Connection and Extraterrestrial Life-Wish', Science Digest (Jun 1979), 85, 38. Reproduced in Carl Sagan and Tom Head, Conversations With Sagan (2006), 55.
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In its earliest development knowledge is self-sown. Impressions force themselves upon men’s senses whether they will or not, and often against their will. The amount of interest in which these impressions awaken is determined by the coarser pains and pleasures which they carry in their train or by mere curiosity; and reason deals with the materials supplied to it as far as that interest carries it, and no further. Such common knowledge is rather brought than sought; and such ratiocination is little more than the working of a blind intellectual instinct. It is only when the mind passes beyond this condition that it begins to evolve science. When simple curiosity passes into the love of knowledge as such, and the gratification of the ćsthetic sense of the beauty of completeness and accuracy seems more desirable that the easy indolence of ignorance; when the finding out of the causes of things becomes a source of joy, and he is accounted happy who is successful in the search, common knowledge passes into what our forefathers called natural history, whence there is but a step to that which used to be termed natural philosophy, and now passes by the name of physical science.
In this final state of knowledge the phenomena of nature are regarded as one continuous series of causes and effects; and the ultimate object of science is to trace out that series, from the term which is nearest to us, to that which is at the farthest limit accessible to our means of investigation.
The course of nature as it is, as it has been, and as it will be, is the object of scientific inquiry; whatever lies beyond, above, or below this is outside science. But the philosopher need not despair at the limitation on his field of labor; in relation to the human mind Nature is boundless; and, though nowhere inaccessible, she is everywhere unfathomable.
The Crayfish: an Introduction to the Study of Zoölogy (1880), 2-3. Excerpted in Popular Science (Apr 1880), 16, 789-790.
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It is obvious that we know with certainty, that the Flütz [layered] and primitive mountains have been produced by a series of precipitations and depositions formed in succession; that they took place from water which covered the globe, existing always more or less generally, and containing the different substances which have been produced from them.
In New Theory of the Formation of Veins (1809), 110-1.
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Laplace would have found it child's-play to fix a ratio of progression in mathematical science between Descartes, Leibnitz, Newton and himself
The Education of Henry Adams: An Autobiography? (1918), 491.
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My experiments with single traits all lead to the same result: that from the seeds of hybrids, plants are obtained half of which in turn carry the hybrid trait (Aa), the other half, however, receive the parental traits A and a in equal amounts. Thus, on the average, among four plants two have the hybrid trait Aa, one the parental trait A, and the other the parental trait a. Therefore, 2Aa+ A +a or A + 2Aa + a is the empirical simple series for two differing traits.
Letter to Carl Nägeli, 31 Dec 1866. In Curt Stern and Eva R. Sherwood (eds.), The Origin of Genetics: A Mendel Source Book (1966), 63.
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Science has hitherto been proceeding without the guidance of any rational theory of logic, and has certainly made good progress. It is like a computer who is pursuing some method of arithmetical approximation. Even if he occasionally makes mistakes in his ciphering, yet if the process is a good one they will rectify themselves. But then he would approximate much more rapidly if he did not commit these errors; and in my opinion, the time has come when science ought to be provided with a logic. My theory satisfies me; I can see no flaw in it. According to that theory universality, necessity, exactitude, in the absolute sense of these words, are unattainable by us, and do not exist in nature. There is an ideal law to which nature approximates; but to express it would require an endless series of modifications, like the decimals expressing surd. Only when you have asked a question in so crude a shape that continuity is not involved, is a perfectly true answer attainable.
Letter to G. F. Becker, 11 June 1893. Merrill Collection, Library of Congress. Quoted in Nathan Reingold, Science in Nineteenth-Century America: A Documentary History (1966), 231-2.
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Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many.
In A Preliminary Discourse on the Study of Natural Philosophy (1830), 164-165.
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The digestive canal is in its task a complete chemical factory. The raw material passes through a long series of institutions in which it is subjected to certain mechanical and, mainly, chemical processing, and then, through innumerable side-streets, it is brought into the depot of the body. Aside from this basic series of institutions, along which the raw material moves, there is a series of lateral chemical manufactories, which prepare certain reagents for the appropriate processing of the raw material.
Speech to the Society of Russian Physicians (Dec 1874). as translated in Daniel P. Todes, Pavlov's Physiology Factory: Experiment, Interpretation, Laboratory Enterprise (2002), 155.
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The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes.
From letter (Jan 1828) to his former teacher Berndt Holmböe. In Morris Kline, Mathematics: The Loss of Certainty (1982), 170.
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The primitive history of the species is all the more fully retained in its germ-history in proportion as the series of embryonic forms traversed is longer; and it is more accurately retained the less the mode of life of the recent forms differs from that of the earlier, and the less the peculiarities of the several embryonic states must be regarded as transferred from a later to an earlier period of life, or as acquired independently. (1864)
As translated and quoted in Ernst Haeckel and E. Ray Lankester (trans.) as epigraph for Chap. 13, The History of Creation (1886), Vol. 1, 406.
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The series is divergent, therefore we may be able to do something with it.
Epigraph without citation, in Morris Kline (ed.), Mathematical Thought From Ancient to Modern Times (1990), Vol. 3, 107.
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The whole aim of practical politics is to keep the populace alarmed (and hence clamorous to be led to safety) by menacing it with an endless series of hobgoblins, all of them imaginary.
Essay, 'The Smart Set' (Dec 1921), 29. As cited in A Mencken Chrestomathy (1949, 2012), p. 29 (1949).
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There is more danger of numerical sequences continued indefinitely than of trees growing up to heaven. Each will some time reach its greatest height.
Grundgesetz der Arithmetik(1893), Vol. 2, Section 60, In P. Greach and M. Black (eds., Translations from the Philosophical Writings of Gottlob Frege (1952), 204.
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To trace the series of these revolutions, to explain their causes, and thus to connect together all the indications of change that are found in the mineral kingdom, is the proper object of a THEORY OF THE EARTH.
Illustrations of the Huttonian Theory of the Earth (1802), 2.
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Why it is that animals, instead of developing in a simple and straightforward way, undergo in the course of their growth a series of complicated changes, during which they often acquire organs which have no function, and which, after remaining visible for a short time, disappear without leaving a trace ... To the Darwinian, the explanation of such facts is obvious. The stage when the tadpole breathes by gills is a repetition of the stage when the ancestors of the frog had not advanced in the scale of development beyond a fish.
In The Works of Francis Maitland Balfour (1885), Vol. 1, 702.
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With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
From letter (Jan 1828) to his former teacher Berndt Holmböe. In Morris Kline, Mathematics: The Loss of Certainty (1982), 170.
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[It] may be laid down as a general rule that, if the result of a long series of precise observations approximates a simple relation so closely that the remaining difference is undetectable by observation and may be attributed to the errors to which they are liable, then this relation is probably that of nature.
'Mémoire sur les Inégalites Séculaires des Planètes et des Satellites' (I 785, published 1787). In Oeuvres completes de Laplace, 14 Vols. (1843-1912), Vol. 11, 57, trans. Charles Coulston Gillispie, Pierre-Simon Laplace 1749-1827: A Life in Exact Science (1997), 130.
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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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