Quotes about case
page 17

Ulysses S. Grant photo

“My lord, I have heard that your father was a military man. Was that the case?”

Ulysses S. Grant (1822–1885) 18th President of the United States

To Arthur Richard Wellesley, 2nd Duke of Wellington, son of Field Marshall Arthur Wellesley, 1st Duke of Wellington, as quoted in Handy-book of Literary Curiosities (1892) by William Shepard Walsh, p. 511.

Marc Randazza photo
Robert Maynard Hutchins photo
Thom Yorke photo

“I am fused, just in case I blow out. I am glued, just in case I crack out.”

Thom Yorke (1968) English musician, philanthropist and singer-songwriter

"Blow Out"
Lyrics, Pablo Honey (1993)

Alphonse Karr photo

“If one wants to abolish the death penalty in this case, Messrs. murderers should take the first step: they do not kill, we will not kill them.”

Alphonse Karr (1808–1890) French critic, journalist, and novelist

Si l'on veut abolir la peine de mort en ce cas, que MM. les assassins commencent: qu'ils ne tuent pas, on ne les tuera pas.
http://books.google.com/books?id=5RAoAQAAIAAJ&q=%22Si+l'on+veut+abolir+la+peine+de+mort+en+ce+cas+que%22+%22MM+les+assassins+commencent+qu'ils+ne+tuent+pas+on+ne+les+tuera+pas%22&pg=PA304#v=onepage
Les Guêpes, January 1849, vi.

Thomas Little Heath photo
Ai Weiwei photo
Valentino Braitenberg photo
Duncan Gregory photo

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation.
p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52
Examples of the processes of the differential and integral calculus, (1841)

Siddharth Katragadda photo

“A little arrogance in the hands of the capable is well earned - and in most case, deserved.”

Siddharth Katragadda (1972) Indian writer

page 79
Dark Rooms (2002)

Christopher Hitchens photo
Max Born photo

“The continuity of our science has not been affected by all these turbulent happenings, as the older theories have always been included as limiting cases in the new ones.”

Max Born (1882–1970) physicist

As quoted in Beyond Positivism and Relativism : Theory, Method, and Evidence (1996) by Larry Laudan, p. 259

Koenraad Elst photo
Hugh Macmillan, Baron Macmillan photo
Werner Heisenberg photo

“Modern positivism…expresses criticism against the naïve use of certain terms… by the general postulate that the question whether a given sentence has any meaning… should always be thoroughly and critically examined. This… is derived from mathematical logic. The procedure of natural science is pictured as an attachment of symbols to the phenomena. The symbols can, as in mathematics, be combined according to certain rules… However, a combination of symbols that does not comply with the rules is not wrong but conveys no meaning.
The obvious difficulty in this argument is the lack of any general criterion as to when a sentence should be considered meaningless. A definite decision is possible only when the sentence belongs to a closed system of concepts and axioms, which in the development of natural science will be rather the exception than the rule. In some case the conjecture that a certain sentence is meaningless has historically led to important progress… new connections which would have been impossible if the sentence had a meaning. An example… sentence: "In which orbit does the electron move around the nucleus?"”

Werner Heisenberg (1901–1976) German theoretical physicist

But generally the positivistic scheme taken from mathematical logic is too narrow in a description of nature which necessarily uses words and concepts that are only vaguely defined.
Physics and Philosophy (1958)

Augustus De Morgan photo
Rousas John Rushdoony photo
Kurt Schwitters photo
Frederick Winslow Taylor photo
Richard Nixon photo

“I recognize that this additional material I am now furnishing may further damage my case.”

Richard Nixon (1913–1994) 37th President of the United States of America

After the court-ordered release of the White House tapes (5 August 1974)
1970s

Lionel Tertis photo
J. B. S. Haldane photo

“The time has gone by when a Huxley could believe that while science might indeed remould traditional mythology, traditional morals were impregnable and sacrosanct to it. We must learn not to take traditional morals too seriously. And it is just because even the least dogmatic of religions tends to associate itself with some kind of unalterable moral tradition, that there can be no truce between science and religion.
There does not seem to be any particular reason why a religion should not arise with an ethic as fluid as Hindu mythology, but it has not yet arisen. Christianity has probably the most flexible morals of any religion, because Jesus left no code of law behind him like Moses or Muhammad, and his moral precepts are so different from those of ordinary life that no society has ever made any serious attempt to carry them out, such as was possible in the case of Israel and Islam. But every Christian church has tried to impose a code of morals of some kind for which it has claimed divine sanction. As these codes have always been opposed to those of the gospels a loophole has been left for moral progress such as hardly exists in other religions. This is no doubt an argument for Christianity as against other religions, but not as against none at all, or as against a religion which will frankly admit that its mythology and morals are provisional. That is the only sort of religion that would satisfy the scientific mind, and it is very doubtful whether it could properly be called a religion at all.”

J. B. S. Haldane (1892–1964) Geneticist and evolutionary biologist

Daedalus or Science and the Future (1923)

Clive Staples Lewis photo
Glenn Beck photo

“And it was from America. Progressive movement in America. Eugenics. In case you don't know what Eugenics led us to: the Final Solution. A master race! A perfect person. …. The stuff that we are facing is absolutely frightening. So I guess I have to put my name on yes, I hope Barack Obama fails. But I just want his policies to fail; I want America to wake up.”

Glenn Beck (1964) U.S. talk radio and television host

Powers
Ryan
Beck: Stem-cell research will lead directly to the search for a new ‘master race.’
2009-03-09
ThinkProgress
http://thinkprogress.org/2009/03/09/beck-eugenics/
The Glenn Beck Program
Premiere Radio Networks
2009-03-09
on President Obama overturning the ban on federally funded stem cell research
2000s, 2009

“A general proof is difficult because of the lack of general criteria for the existence of steady states, but it can be given for special cases.”

Ludwig von Bertalanffy (1901–1972) austrian biologist and philosopher

Source: General System Theory (1968), 5. The Organism Considered as Physical System, p. 132

Marie-Louise von Franz photo
Doris Lessing photo

“Think wrongly, if you please, but in all cases think for yourself.”

Doris Lessing (1919–2013) British novelist, poet, playwright, librettist, biographer and short story writer

Interview with Amanda Craig, "Grand dame of letters who's not going quietly," The Times, London (23 November 2003)
Variant: Think wrongly, if you please, but in all cases think for yourself.

“[Neoinstitutional Economics…] theory has made an indispensable contribution in recent times to advances of understanding in this area. But it seems to me that in the economics of institutions theory is now outstripping empirical research to an excessive extent. No doubt the same could be said of other fields in economics, but there is a particular point about this one. Theoretical modelling may or may not be more difficult in this field than in others, but empirical work is confronted by a special difficulty. Because economic institutions are complex, they do not lend themselves easily to quantitative measurement. Even in the respects in which they do, the data very often are not routinely collected by national statistical offices. As a result, the statistical approach which has become the bread and butter of applied economics is not straightforwardly applicable. Examples of it do exist, the literature on the economics of slavery being perhaps the most fully developed - not surprisingly because slavery is an institution that is sharply defined. But to a large extent the empirical literature has consisted of case-studies which are interesting but not necessarily representative, together with a certain amount on legal court cases, which are almost certainly not representative. Is this the best we can do? There is a challenge here on the empirical side to economists to see what is the best way forward.”

R. C. O. Matthews (1927–2010) British economist
Dexter S. Kimball photo
Hermann Hesse photo
Alvin Plantinga photo

“Well, I don't think there are any methodological conflicts either. As for those social conflicts, those aren't conflicts—in my opinion—between science and religion. They're conflicts between Christians and atheists or Christians and secularists: Christians want to do things one way, secularists want to do things another way. But that's not a science/religion conflict at all. You might as well say it's a science/secularism conflict. In each case, each group wants to do science and then use it in a certain way.”

Alvin Plantinga (1932) American Christian philosopher

[2011-12-13, Interview with Alvin Plantinga on Where the Conflict Really Lies, Paul, Pardi, Philosophy News, http://www.philosophynews.com/post/2011/12/13/Interview-with-Alvin-Plantinga-on-Where-the-Conflict-Really-Lies.aspx]
Posed question: Are you mainly trying to show that there's no logical conflict even though there might be a methodological conflict?

Olaudah Equiano photo

“Soon after this the blacks who brought me on board went off, and left me abandoned to despair. I now saw myself deprived of all chance of returning to my native country, or even the least glimpse of hope of gaining the shore, which I now considered as friendly; and I even wished for my former slavery in preference to my present situation, which was filled with horrors of every kind, still heightened by my ignorance of what I was to undergo. I was not long suffered to indulge my grief; I was soon put down under the decks, and there I received such a salutation in my nostrils as I had never experienced in my life: so that, with the loathsomeness of the stench, and crying together, I became so sick and low that I was not able to eat, nor had I the least desire to taste any thing. I now wished for the last friend, death, to relieve me; but soon, to my grief, two of the white men offered me eatables; and, on my refusing to eat, one of them held me fast by the hands, and laid me across I think the windlass, and tied my feet, while the other flogged me severely. I had never experienced any thing of this kind before; and although, not being used to the water, I naturally feared that element the first time I saw it, yet nevertheless, could I have got over the nettings, I would have jumped over the side, but I could not; and, besides, the crew used to watch us very closely who were not chained down to the decks, lest we should leap into the water: and I have seen some of these poor African prisoners most severely cut for attempting to do so, and hourly whipped for not eating. This indeed was often the case with myself.”

Olaudah Equiano (1745–1797) African abolitionist

Chap. II
The Interesting Narrative of the Life of Olaudah Equiano, or Gustavus Vassa, the African (1789)

Vivek Wadhwa photo

“By picking this particular fight [with the FBI], Apple is doing the technology industry a big disservice. … Apple will very likely lose this case in the courts and suffer a public relations disaster.”

Vivek Wadhwa American academic

Apple picked the wrong battle in privacy war http://dallasnews.com/opinion/commentary/2016/02/18/vivek-wadhwa-apple-picked-the-wrong-battle-in-privacy-war in The Dallas Morning News (18 February 2016)

Maneka Gandhi photo

“In all cases that have been investigated, the only time a state government gives permission for nilgai and wild boar shooting is when it is requested by vips, hotel and tourism people or friends of politicians.”

Maneka Gandhi (1956) Indian politician and activist

Criticising Punjab state for giving hunting licences to VIPs, as quoted in "VIP Hunters Get Licence To Kill In Punjab" http://archive.tehelka.com/story_main17.asp?filename=ts042206VIP_hunters.asp, Tehelka (22 April 2006)
2001-2010

François Fénelon photo

“Peace treaties signed by the vanquished are not freely signed. Men sign with a knife at their throat, they sign in spite of themselves, in order to avoid still greater losses; they sign as men surrender their purse when it is a case of your money or your life.”

François Fénelon (1651–1715) Catholic bishop

Les traités de paix ne couvrent rien, lorsque vous êtes le plus fort, & que vous réduisez vos voisins à signer le traité pour éviter de plus grands maux: alors il signe comme un particulier donne sa bourse à un voleur qui lui tient le pistolet sur la gorge.
Directions pour la conscience d'un roi (Paris: Estienne, 1775) p. 60; translation by A. Lentin, cited from Margaret Lucille Kekewich (ed.) Princes and Peoples: France and the British Isles, 1620-1714 (Manchester: Manchester University Press, 1994) p. 226. (c. 1694).

Harold Wilson photo
Otto Neurath photo
Bill Downs photo

“Though Mitra’s case was different because it was a heart attack, I shudder to think what would have happened if an accident occurs in one of these studios. With no professional medical practitioner in attendance, things became difficult.”

Arin Paul (1980) Indian film director

Interview in Indian Express on Studios of Calcutta http://www.expressindia.com/latest-news/actor-kunal-dies-of-heart-attack-while-on-shoot/413868/(2009)

Jayant Narlikar photo
Koenraad Elst photo
Thomas Warton photo
Bram van Velde photo
Elon Musk photo
Saddam Hussein photo
Ralph Ellison photo

“…there must be possible a fiction which, leaving sociology and case histories to the scientist, can arrive at the truth about the human condition, here and now, with all the bright magic of the fairy tale.”

Ralph Ellison (1914–1994) American novelist, literary critic, scholar and writer

"Brave Words for a Startling Occasion" (1953), in The Collected Essays, ed. John F. Callahan (New York: Modern Library, 1995), p. 153.

Daniel Dennett photo

“Surely just about everybody has faced a moral dilemma and secretly wished, "If only somebody — somebody I trusted — could just tell me what to do!" Wouldn't this be morally inauthentic? Aren't we responsible for making our own moral decisions? Yes, but the virtues of "do it yourself" moral reasoning have their limits, and if you decide, after conscientious consideration, that your moral decision is to delegate further moral decisions in your life to a trusted expert, then you have made your own moral decision. You have decided to take advantage of the division of labor that civilization makes possible and get the help of expert specialists.We applaud the wisdom of this course in all other important areas of decision-making (don't try to be your own doctor, the lawyer who represents himself has a fool for a client, and so forth). Even in the case of political decisions, like which way to vote, the policy of delegation can be defended. … Is the a dereliction of [one's] dut[y] as a citizen? I don't think so, but it does depend on my having good grounds for trusting [the delegate's] judgment. … That why those who have an unquestioning faith in the correctness of the moral teachings of their religion are a problem: if they themselves haven't conscientiously considered, on their own, whether their pastors or priests or rabbis or imams are worthy of this delegated authority over their own lives, then they are in fact taking a personally immoral stand.This is perhaps the most shocking implication of my inquiry, and I do not shrink from it, even though it may offend many who think of themselves as deeply moral. It is commonly supposed that it is entirely exemplary to adopt the moral teachings of one's own religion without question, because -- to put it simply — it is the word of God (as interpreted, always, by the specialists to whom one has delegated authority). I am urging, on the contrary, that anybody who professes that a particular point of moral conviction is not discussable, not debatable, not negotiable, simply because it is the word of God, or because the Bible says so, or because "that is what all Muslims [Hindus, Sikhs… ] [sic] believe, and I am a Muslim [Hindu, Sikh… ]" [sic], should be seen to be making it impossible for the rest of us to take their views seriously, excusing themselves from the moral conversation, inadvertently acknowledging that their own views are not conscientiously maintained and deserve no further hearing.”

Breaking the Spell (2006)

“The coordination of information technology management presents a challenge to firms with dispersed IT practices. Decentralization may bring flexibility and fast response to changing business needs, as well as other benefits, but decentralization also makes systems integration difficult, presents a barrier to standardization, and acts as a disincentive toward achieving economies of scale. As a result, there is a need to balance the decentralization of IT management to business units with some centralized planning for technology, data, and human resources.
Here we explore three major mechanisms for facilitating inter-unit coordination of IT management: structural design approaches, functional coordination modes, and computer-based communication systems. We define these various mechanisms and their interrelationships, and we discuss the relative costs and benefits associated with alternative coordination approaches.
To illustrate the cost-benefit tradeoffs of coordination approaches, we present a case study in which computer-based communication systems were used to support team-based coordination of IT management across dispersed business units. Our analysis reveals possibilities for future approaches to IT coordination in large, dispersed organizations.”

Gerardine DeSanctis (1954–2005) American organizational theorist

Gerardine DeSanctis and Brad M. Jackson (1994) "Coordination of information technology management: Team-based structures and computer-based communication systems." Journal of Management Information Systems Vol 10 (4). p. 85-110. Abstract

Francis George photo
Sir Frederick Pollock, 1st Baronet photo

“You need not cite cases that are familiar.”

Sir Frederick Pollock, 1st Baronet (1783–1870) British lawyer and Tory politician

Reg. v. Baldry (1852), 5 Cox, C. C. 525.

Brian Clevinger photo

“The pattern of sex differences found in our species mirrors that found in most mammals and in many other animals. As such, considerations of parsimony suggest that the best explanation for the human differences will invoke evolutionary forces common to many species, rather than social forces unique to our own. When we find the standard pattern of differences in other, less culture-bound creatures, we inevitably explain this in evolutionary terms. It seems highly dubious, when we find exactly the same pattern in human beings, to say that, in the case of this one primate species, we must explain it in terms of an entirely different set of causes — learning or cumulative culture — which coincidentally replicates the pattern found throughout the rest of the animal kingdom. Anyone who wishes to adopt this position has a formidable task in front of them. They must explain why, in the hominin lineage uniquely, the standard evolved psychological differences suddenly became maladaptive, and thus why natural selection “wiped the slate clean” of any biological contribution to these differences. They must explain why natural selection eliminated the psychological differences but left the correlated physical differences intact. And they must explain why natural selection would eliminate the psychological differences and leave it all to learning, when learning simply replicated the same sex differences anyway. How could natural selection favor extreme flexibility with respect to sex differences if that flexibility was never exercised and was therefore invisible to selection?”

Source: The Ape that Thought It Was a Peacock: Does Evolutionary Psychology Exaggerate Human Sex Differences? (2013), pp. 142-143

Ulysses S. Grant photo
Jimmy Fallon photo

“Actually, it's tough, because he's not really screwing up. He seems to be doing a good job, but we're there just in case — the first time he does anything.”

Jimmy Fallon (1974) American TV Personality

In answer the question "Is it easy to write jokes [on Saturday Night Live] about President Bush?", posed in the summer of 2001
Smith, Kerry L. (2001-07-05), "Jimmy Fallon". Rolling Stone. (872):153

“I often pray, though I’m not really sure Anyone’s listening; and I phrase it carefully, just in case He’s literary.”

Mignon McLaughlin (1913–1983) American journalist

The Complete Neurotic's Notebook (1981), Unclassified

Christopher Hitchens photo

“Taking the points in order, it's fairly easy to demonstrate that Saddam Hussein is a bad guy's bad guy. He's not just bad in himself but the cause of badness in others. While he survives not only are the Iraqi and Kurdish peoples compelled to live in misery and fear (the sheerly moral case for regime-change is unimpeachable on its own), but their neighbors are compelled to live in fear as well.

However—and here is the clinching and obvious point—Saddam Hussein is not going to survive. His regime is on the verge of implosion. It has long passed the point of diminishing returns. Like the Ceausescu edifice in Romania, it is a pyramid balanced on its apex (its powerbase a minority of the Sunni minority), and when it falls, all the consequences of a post-Saddam Iraq will be with us anyway. To suggest that these consequences—Sunni-Shi'a rivalry, conflict over the boundaries of Kurdistan, possible meddling from Turkey or Iran, vertiginous fluctuations in oil prices and production, social chaos—are attributable only to intervention is to be completely blind to the impending reality. The choices are two and only two—to experience these consequences with an American or international presence or to watch them unfold as if they were none of our business.”

Christopher Hitchens (1949–2011) British American author and journalist

2002-11-07
Machiavelli in Mesopotamia
Slate
1091-2339
http://www.slate.com/articles/news_and_politics/fighting_words/2002/11/machiavelli_in_mesopotamia.html: On the 2003 invasion of Iraq
2000s, 2002

Alain-Fournier photo
Thomas Jefferson photo
William Westmoreland photo
Lysander Spooner photo
Adam Smith photo
Joseph Alois Schumpeter photo
William Ewart Gladstone photo
Karl Kraus photo
Leo Igwe photo
Benjamin Peirce photo
Charles Dickens photo
Aron Ra photo
Michelangelo Antonioni photo

“I’m sorry, counselor, I’d rather blow the goddamn case.”

While on trial and his lawyer asked him to stop wearing $350 suits, which were hurting his case with the jury, and to switch to clothes from the plain pipe rack.
as quoted in The Mafia Encyclopedia, written by Carl Sifakis, p. 125 http://books.google.com/books?id=jgCpxTpPCPcC&pg=PA125&lpg=PA125&dq=frank+costello+I%27d+rather+blow+the+case&source=bl&ots=RDPUk332jm&sig=nFLz8Y5VFBvsBva3YH2pO_8cyTw&hl=en&sa=X&ei=xsupUqXfLsjgyQGH14DYAQ&ved=0CDcQ6AEwAw#v=onepage&q=frank%20costello%20I%27d%20rather%20blow%20the%20case&f=false

Jane Roberts photo
Daniel Lyons photo

“We had hoped a tablet from Apple would do something new, something we've never seen before. That's not the case. Jobs and his team kept using words like "breakthrough" and "magical," but the iPad is neither.”

Daniel Lyons (1960) American writer

Why the iPad is a Letdown http://mag.newsweek.com/2010/01/27/in-ipad-we-trust.html in Newsweek (27 January 2010)

Buckminster Fuller photo

“The eternal is omniembracing and permeative; and the temporal is linear. This opens up a very high order of generalizations of generalizations. The truth could not be more omni-important, although it is often manifestly operative only as a linear identification of a special-case experience on a specialized subject.”

Buckminster Fuller (1895–1983) American architect, systems theorist, author, designer, inventor and futurist

1005.52 http://www.rwgrayprojects.com/synergetics/s10/p0520.html#1005.50
1970s, Synergetics: Explorations in the Geometry of Thinking (1975), "Synergy" onwards

Amir Taheri photo

“After weeks of dancing around the issue, the Obama administration has expressed concern about “heightened military activity” by Russia in Syria. But what if we are facing something more than “heightened military activity?” What if Moscow is preparing to give Syria the full Putin treatment? For years, Russia has been helping Syrian despot Bashar al-Assad cling to a diminishing power structure in a shrinking territorial base without trying to impose an overall strategy. Now, however, there are signs that Russia isn’t content to just support Assad. It wants to control Syria. The Putin treatment is reserved for countries in Russia’s “near neighborhood” that try to break out of Moscow’s orbit and deprive it of strategic assets held for decades. In such cases, unable to restore its past position, Russia tries to create a new situation in which it keeps a sword dangling above the head of the recalcitrant nation. Russia’s military intervenes directly and indirectly, always with help from a segment of the local population concerned. Russia starts by casting itself as protector of an ethnic, linguistic or religious minority that demands its military intervention against a central power vilified with labels such as “fascist” and “terrorist.””

Amir Taheri (1942) Iranian journalist

Putin is turning the Syrian coast into another Crimea http://nypost.com/2015/09/19/putin-is-turning-the-syrian-coast-into-another-crimea/, New York Post (September 19, 2015).
New York Post

Greg Egan photo
A.A. Milne photo

“Piglet took Pooh's arm, in case Pooh was frightened.”

Source: The House at Pooh Corner (1928), Chapter Four.

Daniel Handler photo
Rob Enderle photo
William A. Dembski photo
Richard Dawkins photo
Neil Armstrong photo

“Space has not changed but technology has, in many cases, improved dramatically. A good example is digital technology where today's cell phones are far more powerful than the computers on the Apollo Command Module and Lunar Module that we used to navigate to the moon and operate all the spacecraft control systems.”

Neil Armstrong (1930–2012) American astronaut; first person to walk on the moon

On the differences between the present and the time of the space race which existed during the Cold War years, in an interview at The New Space Race (August 2007)

Moshe Chaim Luzzatto photo
Lysander Spooner photo
Oswald Mosley photo
David Graeber photo

“Honor is the same as credit; it's one's ability to keep ones promises, but also, in the case of a wrong, to "get even."”

David Graeber (1961) American anthropologist and anarchist

Source: Debt: The First 5,000 Years (2011), Chapter Seven, "Honor and Degradation", p. 193

Jacob Leupold photo
Stephen Harper photo
Kurt Lewin photo
Bernhard Riemann photo
Bernhard Riemann photo

“Let us imagine that from any given point the system of shortest lines going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. …the square of the line-element is \sum (dx)^2 for infinitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order… an infinitesimal, therefore, of the fourth order; so that we obtain a finite quantity on dividing this by the square of the infinitesimal triangle, whose vertices are (0,0,0,…), (x1, x2, x3,…), (dx1, dx2, dx3,…). This quantity retains the same value so long as… the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is flat, i. e., when the squared line-element is reducible to \sum (dx)^2, and may therefore be regarded as the measure of the deviation of the manifoldness from flatness at the given point in the given surface-direction. Multiplied by -¾ it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. …The measure-relations of a manifoldness in which the line-element is the square root of a quadric differential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e. g., the fourth root of a quartic differential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from flatness in the squared line-element is an infinitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called flatness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them…”

Bernhard Riemann (1826–1866) German mathematician

On the Hypotheses which lie at the Bases of Geometry (1873)