classes ::: Science,
children :::
branches ::: Euclid
see also :::

Instances - Definitions - Quotes - Chapters - Wordnet - Webgen


object:Euclid
subject class:Science


questions, comments, suggestions/feedback, take-down requests, contribute, etc
contact me @ integralyogin@gmail.com or
join the integral discord server (chatrooms)
if the page you visited was empty, it may be noted and I will try to fill it out. cheers


OBJECT INSTANCES [0] - TOPICS - AUTHORS - BOOKS - CHAPTERS - CLASSES - SEE ALSO - SIMILAR TITLES

TOPICS
SEE ALSO


AUTH

BOOKS
Evolution_II
Plotinus_-_Complete_Works_Vol_01
Process_and_Reality
The_Yoga_Sutras
Toward_the_Future

IN CHAPTERS TITLE

IN CHAPTERS CLASSNAME

IN CHAPTERS TEXT
01.07_-_Blaise_Pascal_(1623-1662)
02.03_-_National_and_International
05.07_-_The_Observer_and_the_Observed
1.01_-_What_is_Magick?
1.02_-_The_Pit
1.02_-_The_Three_European_Worlds
1.04_-_THE_APPEARANCE_OF_ANOMALY_-_CHALLENGE_TO_THE_SHARED_MAP
1.04_-_The_First_Circle,_Limbo__Virtuous_Pagans_and_the_Unbaptized._The_Four_Poets,_Homer,_Horace,_Ovid,_and_Lucan._The_Noble_Castle_of_Philosophy.
1.14_-_The_Limits_of_Philosophical_Knowledge
1.50_-_A.C._and_the_Masters;_Why_they_Chose_him,_etc.
1f.lovecraft_-_At_the_Mountains_of_Madness
1f.lovecraft_-_The_Call_of_Cthulhu
1f.lovecraft_-_The_Dreams_in_the_Witch_House
1f.lovecraft_-_The_Trap
1.poe_-_Eureka_-_A_Prose_Poem
1.ww_-_Book_Fifth-Books
2.01_-_On_Books
Blazing_P3_-_Explore_the_Stages_of_Postconventional_Consciousness
BOOK_II._--_PART_II._THE_ARCHAIC_SYMBOLISM_OF_THE_WORLD-RELIGIONS
Phaedo
The_Act_of_Creation_text
Theaetetus

PRIMARY CLASS

SIMILAR TITLES
Euclid

DEFINITIONS

Euclid
(Named after the Greek geometer, fl ca 300 BC.) A
{Pascal} descendant for development of verifiable system
software. No {goto}, no {side effects}, no global
assignments, no functional arguments, no nested procedures, no
floats, no {enumeration types}. Pointers are treated as
indices of special arrays called collections. To prevent
{aliasing}, Euclid forbids any overlap in the list of actual
parameters of a procedure. Each procedure gives an imports
list, and the compiler determines the identifiers that are
implicitly imported. Iterators.
Ottawa Euclid is a variant.
["Report on the Programming Language Euclid", B.W. Lampson et
al, SIGPLAN Notices 12(2):1-79, Feb 1977].
(1998-11-23)


Euclid: (c. 400 B.C.) Of Megara, founder of the Megarian School. He was chiefly interested in the theory of refutation. See Megarians.

Euclidean algorithm: An algorithm for finding the HCF of two positive integers.

Euclidean Algorithm
{Euclid's Algorithm}


Euclidean construction: Geometric construction using only straight edges and compasses. Also known as geometric construction or simply a straightedge and compass construction..

Euclidean geometry: The geometry system defined by the following axioms and "notions".

Euclidean norm
The most common {norm}, calculated by summing
the squares of all coordinates and taking the square root.
This is the essence of {Pythagoras's theorem}. In the
infinite-dimensional case, the sum is infinite or is replaced
with an integral when the number of dimensions is
{uncountable}.
(2004-02-15)


Euclidean norm ::: (mathematics) The most common norm, calculated by summing the squares of all coordinates and taking the square root. This is the essence of Pythagoras's theorem. In the infinite-dimensional case, the sum is infinite or is replaced with an integral when the number of dimensions is uncountable.(2004-02-15)

Euclidean norm: The usual norm for finding the magnitude of a vector in Rn - the square root of the sum of the orthogonal components of a vector.

Euclid ::: (language) (Named after the Greek geometer, fl ca 300 BC.) A Pascal descendant for development of verifiable system software. No goto, no side list, and the compiler determines the identifiers that are implicitly imported. Iterators.Ottawa Euclid is a variant.[Report on the Programming Language Euclid, B.W. Lampson et al, SIGPLAN Notices 12(2):1-79, Feb 1977]. (1998-11-23)

Euclid of Megara identified the good and the One. The many are unreal. Not to be confused with the great geometer who lived at Alexandria (c. 300 B.C.), author of the Elements in 13 books. -- M.F.

Euclid's Algorithm ::: (algorithm) (Or Euclidean Algorithm) An algorithm for finding the greatest common divisor (GCD) of two numbers. It relies on the identity gcd(a, b) = gcd(a-b, b) 96, 36 -> 60, 36 -> 24, 36 -> 24, 12 -> 12, 12 so the GCD of 132 and 168 is 12.This algorithm requires only subtraction and comparison operations but can take a number of steps proportional to the difference between the initial numbers (e.g. gcd(1, 1001) will take 1000 steps). (1997-06-30)

Euclid's Algorithm
(Or "Euclidean Algorithm") An {algorithm} for
finding the {greatest common divisor} (GCD) of two numbers.
It relies on the identity
gcd(a, b) = gcd(a-b, b)
To find the GCD of two numbers by this algorithm, repeatedly
replace the larger by subtracting the smaller from it until
the two numbers are equal. E.g. 132, 168 -> 132, 36 -> 96, 36
-> 60, 36 -> 24, 36 -> 24, 12 -> 12, 12 so the GCD of 132 and
168 is 12.
This algorithm requires only subtraction and comparison
operations but can take a number of steps proportional to the
difference between the initial numbers (e.g. gcd(1, 1001) will
take 1000 steps).
(1997-06-30)


Euclid's postulates were:

Euclid's Theorem: A theorem which states that there are infinitely many primes.

euclidian ::: n. --> Related to Euclid, or to the geometry of Euclid.

euclid ::: n. --> A Greek geometer of the 3d century b. c.; also, his treatise on geometry, and hence, the principles of geometry, in general.



QUOTES [2 / 2 - 222 / 222]


KEYS (10k)

   1 Mortimer J Adler
   1 Alfred Korzybski

NEW FULL DB (2.4M)

   15 Euclid
   14 Fyodor Dostoyevsky
   9 Mario Livio
   7 Anonymous
   6 Roger Penrose
   4 Walter Isaacson
   4 Neal Stephenson
   4 Albert Einstein
   3 James Gleick
   3 Benoit Mandelbrot
   3 Augustus De Morgan
   3 Arthur Conan Doyle
   2 Simon Singh
   2 Samuel Johnson
   2 Oliver Heaviside
   2 Mortimer J Adler
   2 Michelle McNamara
   2 Liu Cixin
   2 Jon Krakauer
   2 John D Barrow

1:To The Works Of:
   Aristotle, Cassius J. Keyser, Eric T. Bell, G. W. Leibnitz, Eugen Bleuler, J. Locke, Niels Bohr, Jacques Loeb, George Boole, H. A. Lorentz, Max Born, Ernst Mach, Louis De Brogue, J. C. Maxwell, Georg Cantor, Adolf Meyer, Ernst Cassirer, Hermann Minkowsja, Charles M. Child, Isaac Newton, C. Darwin, Ivan Pavlov, Rene Descartes, Giuseppe Peano, P. A. M. Dirac, Max Planck, A. S. Eddington, Plato, Albert Einstein, H. Poincare, Euclid, M. Faraday, Sigmund Freud, Josiah Royce, Karl F. Gauss, G. Y. Rainich, G. B. Riemann, Bertrand Russell, Thomas Graham, Ernest Rutherford, Arthur Haas, E. Schrodinger, Wm. R. Hamilton, C. S. Sherrington, Henry Head, Socrates, Werner Heisenberg, Arnold Sommerfeld, C. Judson Herrick, Oswald Veblen, E. V. Huntington, Wm. Alanson White, Smith Ely Jeluffe, Alfred N. Whitehead, Ludwig Wittgenstein
   Which Have Creatly Influenced My Enquiry
   This System Is Dedicated ~ Alfred Korzybski, Science and Sanity,
2:Reading list (1972 edition)[edit]
1. Homer - Iliad, Odyssey
2. The Old Testament
3. Aeschylus - Tragedies
4. Sophocles - Tragedies
5. Herodotus - Histories
6. Euripides - Tragedies
7. Thucydides - History of the Peloponnesian War
8. Hippocrates - Medical Writings
9. Aristophanes - Comedies
10. Plato - Dialogues
11. Aristotle - Works
12. Epicurus - Letter to Herodotus; Letter to Menoecus
13. Euclid - Elements
14.Archimedes - Works
15. Apollonius of Perga - Conic Sections
16. Cicero - Works
17. Lucretius - On the Nature of Things
18. Virgil - Works
19. Horace - Works
20. Livy - History of Rome
21. Ovid - Works
22. Plutarch - Parallel Lives; Moralia
23. Tacitus - Histories; Annals; Agricola Germania
24. Nicomachus of Gerasa - Introduction to Arithmetic
25. Epictetus - Discourses; Encheiridion
26. Ptolemy - Almagest
27. Lucian - Works
28. Marcus Aurelius - Meditations
29. Galen - On the Natural Faculties
30. The New Testament
31. Plotinus - The Enneads
32. St. Augustine - On the Teacher; Confessions; City of God; On Christian Doctrine
33. The Song of Roland
34. The Nibelungenlied
35. The Saga of Burnt Njal
36. St. Thomas Aquinas - Summa Theologica
37. Dante Alighieri - The Divine Comedy;The New Life; On Monarchy
38. Geoffrey Chaucer - Troilus and Criseyde; The Canterbury Tales
39. Leonardo da Vinci - Notebooks
40. Niccolò Machiavelli - The Prince; Discourses on the First Ten Books of Livy
41. Desiderius Erasmus - The Praise of Folly
42. Nicolaus Copernicus - On the Revolutions of the Heavenly Spheres
43. Thomas More - Utopia
44. Martin Luther - Table Talk; Three Treatises
45. François Rabelais - Gargantua and Pantagruel
46. John Calvin - Institutes of the Christian Religion
47. Michel de Montaigne - Essays
48. William Gilbert - On the Loadstone and Magnetic Bodies
49. Miguel de Cervantes - Don Quixote
50. Edmund Spenser - Prothalamion; The Faerie Queene
51. Francis Bacon - Essays; Advancement of Learning; Novum Organum, New Atlantis
52. William Shakespeare - Poetry and Plays
53. Galileo Galilei - Starry Messenger; Dialogues Concerning Two New Sciences
54. Johannes Kepler - Epitome of Copernican Astronomy; Concerning the Harmonies of the World
55. William Harvey - On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals
56. Thomas Hobbes - Leviathan
57. René Descartes - Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy
58. John Milton - Works
59. Molière - Comedies
60. Blaise Pascal - The Provincial Letters; Pensees; Scientific Treatises
61. Christiaan Huygens - Treatise on Light
62. Benedict de Spinoza - Ethics
63. John Locke - Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education
64. Jean Baptiste Racine - Tragedies
65. Isaac Newton - Mathematical Principles of Natural Philosophy; Optics
66. Gottfried Wilhelm Leibniz - Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology
67.Daniel Defoe - Robinson Crusoe
68. Jonathan Swift - A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal
69. William Congreve - The Way of the World
70. George Berkeley - Principles of Human Knowledge
71. Alexander Pope - Essay on Criticism; Rape of the Lock; Essay on Man
72. Charles de Secondat, baron de Montesquieu - Persian Letters; Spirit of Laws
73. Voltaire - Letters on the English; Candide; Philosophical Dictionary
74. Henry Fielding - Joseph Andrews; Tom Jones
75. Samuel Johnson - The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets
   ~ Mortimer J Adler,

*** NEWFULLDB 2.4M ***

1:Non-euclidian sofa ~ Charles Stross
2:A line is length without breadth. ~ Euclid
3:There is no Royal Road to Geometry. ~ Euclid
4:And the whole is greater than the part. ~ Euclid
5:Sire, there is no royal road to geometry. ~ Euclid
6:Euclid for children is barbarous. ~ Oliver Heaviside
7:Better balance, less pain and less restless leg syndrome. ~ Euclid
8:A prime number is one (which is) measured by a unit alone. ~ Euclid
9:The laws of nature are but the mathematical thoughts of God. ~ Euclid
10:A triangle with four points is what Euclid rides into hell. ~ Steve Martin
11:Give him threepence, since he must make a gain out of what he learns. ~ Euclid
12:The early study of Euclid made me a hater of geometry. ~ James Joseph Sylvester
13:What has been affirmed without proof can also be denied without proof. ~ Euclid
14:Things which are equal to the same thing are also equal to one another. ~ Euclid
15:From Euclid to Newton there were straight lines. The modern age analyzes the wavers. ~ Saul Bellow
16:Handwriting is a spiritual designing, even though it appears by means of a material instrument. ~ Euclid
17:I have in later years taken to Euclid, Whitehead, Bertrand Russell, in an elemental way. ~ Carl Sandburg
18:Todo porque se le mezclan las realidades y los recuerdos de una manera sumamente no-euclidiana. ~ Julio Cort zar
19:The sacred writings excepted, no Greek has been so much read and so variously translated as Euclid. ~ Augustus De Morgan
20:She wanted to run her hands over him as he whispered the impassioned corollaries of non-Euclidean geometry. ~ Sherry Thomas
21:Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. ~ Eric Temple Bell
22:1. An 'unit' is that by virtue of which each of the things that exist is called one.
2. A 'number' is a multiple composed of units. ~ Euclid
23:A machine is as distinctively and brilliantly and expressively human as a violin sonata or a theorem in Euclid. —GREGORY VLASTOS ~ Ray Kurzweil
24:Euclid alone has looked on Beauty bare. ~ Edna St. Vincent Millay, "Euclid alone has looked on Beauty bare", published in American Poetry 1922.
25:I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier. ~ Thomas Jefferson
26:There is no other Royal path which leads to geometry. ~ Euclid to Ptolemy I. See Proclus' Commentaries on Euclid's Elements, Book II, Chapter IV.
27:The once-surprising existence of non-Euclidean models of Euclid's first four axioms can be seen as a sort of mathematical joke. ~ John Allen Paulos
28:En general es el método analítico el que yo propongo para la exposición de la matemática, en lugar del sintético utilizado por Euclides. ~ Anonymous
29:In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. ~ Euclid
30:While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics. ~ Mario Livio
31:The victory over Euclidean space was not achieved by isolated individuals, but by a field of young rebels opposed to all absolutes. ~ Marshall McLuhan
32:Que lo que Euclides demostró es así hemos de admitirlo forzados por el principio de contradicción: pero de por qué es así, no nos enteramos. ~ Anonymous
33:In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points. ~ Madeleine L Engle
34:They even dare to dream that two parallel lines, which according to Euclid cannot meet on Earth, may perhaps meet somewhere in infinity. ~ Fyodor Dostoyevsky
35:As to writing another book on geometry [to replace Euclid] the middle ages would have as soon thought of composing another New Testament. ~ Augustus De Morgan
36:Euclid avoids it [the treatment of the infinite]; in modern mathematics it is systematically introduced, for only then is generality obtained. ~ Arthur Cayley
37:A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. ~ Euclid
38:If Euclid's point, though incapable of being drawn by any human agency, has an imperishable value, my picture has its own for mankind to live. ~ Mahatma Gandhi
39:THE APOSTOLIC CAFÉ had expanded its offerings to include Euclid’s Grill—casual buffet by day, fine dining by night, served in the former nave. ~ Neal Stephenson
40:[The Euclidean algorithm is] the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. ~ Donald Knuth
41:The pits and tangles are more than blemishes distorting the classic shapes of Euclidian geometry. They are often the keys to the essence of a thing ~ James Gleick
42:The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task. ~ G H Hardy
43:By the end of the nineteenth century, the best that was achieved was a proof that Euclidean geometry was consistent as long as arithmetic was consistent. ~ Ted Chiang
44:I see now the virtue in madness, for this country knows no law nor any boundary. I pity the poor shades confined to the Euclidean prison that is sanity ~ Grant Morrison
45:Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1. ~ Simon Singh
46:The question: 'is the Euclidean geometry true?' has no significance for Poincaré, for these is no such thing as one geometry being more true than another. ~ Fulton J Sheen
47:His hair, from much running of fingers
through it, radiates in all directions and surrounds his head
like a halo of glory, or like the second Corollary of Euclid
I. 32. ~ Lewis Carroll
48:When a king asked Euclid, the mathematician, whether he could not explain his art to him in a more compendious manner? he was answered, that there was no royal way to geometry. ~ Samuel Johnson
49:He prescribed Euclidean geometry, followed by a dose of trigonometry and algebra. That should cure anyone, they both thought, from having too many artistic or romantic passions. ~ Walter Isaacson
50:Euclid, who was still, when I was young, the sole acknowledged text-book of geometry for boys, lived in Alexandria, about 300 B.C., a few years after the death of Alexander and Aristotle. ~ Anonymous
51:One afternoon late in October of the year 1697, Euclide Auclair, the philosopher apothecary of Quebec, stood on the top of Cap Diamant gazing down the broad, empty river far beneath him. ~ Willa Cather
52:The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry. ~ Eric Bell
53:The study of Euclid put him into a compassionate and tranquil frame of mind, and illuminated, among other things, that his thinking and feeling had recently been crippled by confusion and despair. ~ John Cheever
54:Mother Nature did not attend high school geometry courses or read the books of Euclid of Alexandria. Her geometry is jagged, but with a logic of its own and one that is easy to understand. ~ Nassim Nicholas Taleb
55:I would say, if you like, that the party is like an out-moded mathematics...that is to say, the mathematics of Euclid. We need to invent a non-Euclidian mathematics with respect to political discipline. ~ Alain Badiou
56:every possible Euclidean and non-Euclidean permutation of lights; ten thousand lights, ten million lights, the ten billion lights of Great Yu, each one a voice calling, “I is what I am! Notice me! Notice me! ~ Ian McDonald
57:The Ottoman Turks were about to capture Constantinople, unleashing on Italy a migration of fleeing scholars with bundles of manuscripts containing the ancient wisdom of Euclid, Ptolemy, Plato, and Aristotle. ~ Walter Isaacson
58:It is shocking that young people should be addling their brains over mere logical subtleties in Euclid's Elements, trying to understand the proof of one obvious fact in terms of something equally .. obvious. ~ Oliver Heaviside
59:In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it came to us from Euclid. ~ Nikolai Lobachevsky
60:…if geometry were as much opposed to our passions and present interests as is ethics, we should contest it and violate I but little less, notwithstanding all the demonstrations of Euclid and Archimedes… ~ Gottfried Wilhelm Leibniz
61:Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience. ~ Arthur Cayley
62:they head down the alley to the White Hen, a small convenience store on Euclid, about a block and a half away. They went to the White Hen all the time, sometimes three or four times a day, for a Kit Kat or a Coke. ~ Michelle McNamara
63:You have attempted to tinge detection with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid." - Holmes to Watson, The Sign of Four ~ Arthur Conan Doyle
64:WALKED THE SAME HALF MILE TO ST. EDMUND’S EVERY DAY, A LEFT on Randolph, a right on Euclid, a left on Pleasant. The girls wore gray plaid jumpers and white shirts; the boys, a mustard-colored collared shirt and slacks. ~ Michelle McNamara
65:Euclid manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does. ~ Lucio Russo
66:At that stage of my youth, death remained as abstract a concept as non-Euclidean geometry or marriage. I didn't yet appreciate its terrible finality or the havoc it could wreak on those who'd entrusted the deceased with their hearts. ~ Jon Krakauer
67:I was interviewed on the Israeli radio for five minutes and I said that more than 2000 years ago, Euclid proved that there are infinitely many primes. Immediately the host interrupted me and asked, 'Are there still infinitely many primes?' ~ Noga Alon
68:Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. ~ G H Hardy
69:That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. ~ Euclid
70:Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. ~ Simon Singh
71:His conclusions were as infallible as so many propositions of Euclid. So startling would his results appear to the uninitiated that until they learned the processes by which he had arrived at them they might well consider him as a necromancer. ~ Arthur Conan Doyle
72:There is no royal road to geometry. (μή εἶναι βασιλικήν ἀτραπόν ἐπί γεωμετρίαν, Non est regia [inquit Euclides] ad Geometriam via) ~ Euclid, alleged reply when Ptolemy I Soter asked him if there was a shorter road to learning geometry than through Euclid's Elements.
73:The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. ..."Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean geometry. ~ Hans Reichenbach
74:Entretanto se impuso el racionalismo nacido en oposición al empirismo, y conforme a él elaboró Euclides la matemática, es decir, basando solamente los axiomas, y por necesidad, en la evidencia intuitiva (φαινομενον), y todo lo demás en razonamientos (νοουμενον). ~ Anonymous
75:Euclid 's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument. ~ George Polya
76:The Greek period inspired the greatest flowering of knowledge in human history, producing the forefathers of the entire Western intellectual tradition, including Socrates, Plato, Aristotle, Pythagoras and Euclid. It changed the world in ways both subtle and profound. ~ Matthew Syed
77:He thought that when he had healed sufficiently, and withdrawn from the capital, he might write the magus a letter and open a correspondence on Euclid, or Thales, or the new idea from the north, that the sun and not the Earth might be the centre of the universe. ~ Megan Whalen Turner
78:...the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors. ~ Hans Reichenbach
79:We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later. ~ Arthur Koestler
80:pretender inferir exclusivamente mediante una deducción lógica a partir del principio de contradicción la necesidad de las relaciones espaciales cognoscible en la intuición es como si al señor inmediato de un país otro se lo quisiera dar en feudo. Mas eso es lo que Euclides ha hecho. ~ Anonymous
81:Detection is, or ought to be, an exact science, and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid. ~ Arthur Conan Doyle
82:We think of Euclid as of fine ice; we admire Newton as we admire the peak of Teneriffe. Even the intensest labors, the most remote triumphs of the abstract intellect, seem to carry us into a region different from our own-to be in a terra incognita of pure reasoning, to cast a chill on human glory. ~ Walter Bagehot
83:APODICTICAL  (APODI'CTICAL)   adj.[from    evident truth; demonstration.]Demonstrative; evident beyond contradiction. Holding an apodictical knowledge, and an assured knowledge of it; verily, to persuade their apprehensions otherwise, were to make Euclid believe, that there were more than one centre in ~ Samuel Johnson
84:I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry. ~ Henri Poincare
85:Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently a mere trifle remains. But a careful analysis shows that in this seeming trifle lies the crux of the matter; usually it contains either the proposition that is being proved or a postulate equivalent to it. ~ Johann Heinrich Lambert
86:At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness. ~ Bertrand Russell
87:Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. ~ George Polya
88:But I still read Shaw on a regular basis. What I love is the nakedness of the polemic and the irresistible good humour. For me, 'Major Barbara' is the greatest of all the plays in that it starts from the rational and proceeds to the ecstatic in a spectacular way, and leaves you very confused if you cling to Euclidean logic. ~ Tony Kushner
89:Yet there have been and still are mathematicians and philosophers who doubt whether the whole universe, or to speak more widely, the whole of being, was only created in Euclid's geometry. They even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. ~ Fyodor Dostoyevsky
90:result: every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for ‘distances’, the ‘distance’ being represented physically by means of the convention of two marks on a rigid body. ~ Albert Einstein
91:It was in Alexandria that the circumference of the earth was first measured, the sun fixed at the center of the solar system, the workings of the brain and the pulse illuminated, the foundations of anatomy and physiology established, the definitive editions of Homer produced. It was in Alexandria that Euclid had codified geometry. ~ Stacy Schiff
92:Regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities, but some shapes in nature are so complicated that they are equally complicated at the big scale and come closer and closer and they don't become any less complicated. ~ Benoit Mandelbrot
93:The text also just grows increasingly garbled. For instance, here it says that our new subway system will streamline the rush-hour commute, but about halfway down, it's a series of nearly indecipherable glyphs our experts insist hint at "non-Euclidian emotions" and "appeasement" (though we think this may be a euphemism for "fares"). ~ Joseph Fink
94:Nadie se ha despedido llorando de la geometría de Euclides o del sistema periódico de los átomos. Nadie se echa a llorar porque va a ser desconectado de internet o de la tabla de multiplicar. Es del mundo de lo que uno se despide, de la vida, del cuento. Y al mismo tiempo, uno se despide de una pequeña selección de seres queridos. ~ Jostein Gaarder
95:The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel. ~ Benoit Mandelbrot
96:There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. ~ Nikolai Ivanovich Lobachevsky, as quoted in George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer (1998 [1975]), p. 225; also in Stanley Gudder, A Mathematical Journey, McGraw-Hill (1976), p. 36.
97:I told myself, "Lincoln, you can never make a lawyer if you do not understand what demonstrate means." So I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what "demonstrate" means, and went back to my law studies. ~ Abraham Lincoln
98:When the nineteenth-century Hungarian mathematician János Bolyai invented non-Euclidean geometry, his father urged him to publish his findings immediately, before someone else landed on the same idea, saying, “When the time is ripe for certain things, they appear at different places, in the manner of violets coming to light in early spring. ~ Elizabeth Gilbert
99:It would be foolish to give credit to Euclid for pangeometrical conceptions; the idea of geometry deifferent from the common-sense one never occurred to his mind. Yet, when he stated the fifth postulate, he stood at the parting of the ways. His subconscious prescience is astounding. There is nothing comperable to it in the whole history of science. ~ George Sarton
100:Another example of a class of well-defined mathematical problems that have no algorithmic solution is the tiling problem. This is formulated as follows: given a set of polygonal shapes, decide whether those shapes will tile the plane; that is, is it possible to cover the entire Euclidean plane using only these particular shapes, without gaps or overlaps? ~ Roger Penrose
101:Non- Euclidean calculus and quantum physics are enough to stretch any brain; and when one mixes them with folklore, and tries to trace a strange background of multi-dimensional reality behind the ghoulish hints of Gothic tales and the wild whispers of the chimney-corner, one can hardly expect to be wholly free from mental tension. (Dreams In The Witch-House) ~ H P Lovecraft
102:Like a young heir, come a little prematurely to a large inheritance, we shall wanton and run riot until we have brought our reputation to the brink of ruin, and then, like him, shall have to labor with the current of opinion, when COMPELLED perhaps, to do what prudence and common policy pointed out, as plain as any problem in Euclid, in the first instance. ~ George Washington
103:Non- Euclidean calculus and quantum physics are enough to stretch any brain; and when one mixes them with folklore, and tries to trace a strange background of multi-dimensional reality behind the ghoulish hints of Gothic tales and the wild whispers of the chimney-corner, one can hardly expect to be wholly free from mental tension.
(Dreams In The Witch-House) ~ H P Lovecraft
104:a kind of splendid confusion; it is something both shining and shapeless, at once a blaze and a blur. But the circle of the moon is as clear and unmistakable, as recurrent and inevitable, as the circle of Euclid on a blackboard. For the moon is utterly reasonable; and the moon is the mother of lunatics and has given to them all her name. CHAPTER III.—The Suicide ~ G K Chesterton
105:Every night as I gazed up at the window I said softly to myself the word paralysis. It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism. But now it sounded to me like the name of some maleficent and sinful being. It filled me with fear, and yet I longed to be nearer to it and to look upon its deadly work. ~ James Joyce
106:At the age of eleven, I began Euclid, with my brother as my tutor. ... I had not imagined that there was anything so delicious in the world. After I had learned the fifth proposition, my brother told me that it was generally considered difficult, but I had found no difficulty whatsoever. This was the first time it had dawned on me that I might have some intelligence. ~ Bertrand Russell
107:A marveilous newtrality have these things mathematicall and also a strange participation between things supernaturall, imortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, compounded and divisible. ~ John Dee, The mathematicall praeface to the Elements of geometrie of Euclid of Megara (1570) as editor of Euclid's Elements, translated by Henry Billingsley.
108:The Greek excellence in mathematics was largely a direct consequence of their passion for knowledge for its own sake, rather than merely for practical purposes. A story has it that when a student who learned one geometrical proposition with Euclid asked, "But what do I gain from this?" Euclid told his slave to give the boy a coin, so that the student would see an actual profit. ~ Mario Livio
109:No very good sense can be given to the idea that the elements of Euclidean geometry may be found in nature because either everything is found in nature or nothing is. Euclidean geometry is a theory, and the elements of a theory may be interpreted only in terms demanded by the theory itself. Euclid’s axioms are satisfied in the Euclidean plane. Nature has nothing to do with it. ~ David Berlinski
110:It is curious to observe the triumph of slight incidents over the mind; and what incredible weight they have in forming and governing our opinions, both of men and things, that trifles light as air shall waft a belief into the soul, and plant it so immovable within it, that Euclid's demonstrations, could they be brought to batter it in breach, should not all have power to overthrow it! ~ Laurence Sterne
111:Who amongst them realizes that between the Differential Calculus and the dynastic principle of politics in the age of Louis XIV, between the Classical city-state and the Euclidean geometry, between the space perspective of Western oil painting and the conquest of space by railroad, telephone and long range weapon, between contrapuntal music and credit economics, there are deep uniformities? ~ Oswald Spengler
112:The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. ~ John Maynard Keynes
113:I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid - a term used in this work to denote all of standard geometry - Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous." ~ Benoit Mandelbrot
114:[it was not a] circle—just a concrete platform with a pay phone and a sign that read EUCLID CIRCLE. I thought Euclid would have been mad.
“That’s so typical of your attitude,” Svetlana said. “You always think everyone is angry. Try to have some perspective. It’s over two thousand years after his death, he’s in Boston for the first time, they’ve named something after him—why should his first reaction be to get pissed off? ~ Elif Batuman
115:The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around. ~ Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly 87 (2), February 1980, pp. 81-90.
116:Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance). In my opinion, one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all. ~ Albert Einstein
117:True, both his eyes, in themselves, must simultaneously act; but is his brain so much more comprehensive, combining, and subtle than man's, that he can at the same moment of time attentively examine two distinct prospects, one on one side of him, and the other in an exactly opposite direction? If he can, then is it as marvellous a thing in him, as if a man were able simultaneously to go through the demonstrations of two distinct problems in Euclid. ~ Herman Melville
118:Women don’t use knives,’ Griffoni answered, reciting it as though she were Euclid listing another axiom. Although he agreed with her, Brunetti was curious about the basis for her belief. ‘You offering proof of that?’ ‘Kitchens,’ she said laconically. ‘Kitchens?’ ‘The knives are kept in the kitchen, and their husbands pass through there every day, countless times, yet very few of them get stabbed. That’s because women don’t use knives, and they don’t stab people. ~ Donna Leon
119:The science of the church is neglected for the study of geometry, and they lose sight of Heaven while they are employed in measuring the earth. Euclid is perpetually in their hands. Aristotle and Theophrastus are the objects of their admiration; and they express an uncommon reverence for the works of Galen. Their errors are derived from the abuse of the arts and sciences of the infidels, and they corrupt the simplicity of the gospel by the refinements of human reason. ~ Edward Gibbon
120:At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, “out of a curiosity to see what there was in it.” He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus. ~ Carl Sagan
121:In gravitational fields there are no such thing as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. ...
For this reason non-rigid reference-bodies are used, which are, as a whole, not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion...
This non-rigid reference-body, ... might appropriately be termed a "reference mollusc,"... ~ Albert Einstein
122:Did chemistry theorems exist? No: therefore you had to go further, not be satisfied with the quia, go back to the origins, to mathematics and physics. The origins of chemistry were ignoble, or at least equivocal: the dens of the alchemists, their abominable hodgepodge of ideas and language, their confessed interest in gold, their Levantine swindles typical of charlatans and magicians; instead, at the origin of physics lay the strenuous clarity of the West-Archimedes and Euclid. ~ Primo Levi
123:The anceints devoted a lifetime to the study of arithmetic; it required days to extract a square root or to multiply two numbers together. Is there any harm in skipping all that, in letting the school boy learn multiplication sums, and in starting his more abstract reasoning at a more advanced point. Where would be the harm in letting the boy assume the truth of many propositions of the first four books of Euclid, letting him assume their truth partly by faith, partly by trial? ~ John Perry
124:It was possible, I knew, to live on two planes at once - to have one's feet planted in reality but pointed in the direction of progress. It was what I had done as a kid on Euclid Avenue, what my family - and marginalized people more generally - had always done. You get somewhere by building that better reality, if at first only in your own mind. Or as Barack had put it that night, you may live in the world as it is, but you can still work to create the world as it should be. ~ Michelle Obama
125:After introducing Sharko and executing his regulation about-face, the second lieutenant left the cop alone with the colonel, who was busy signing papers. The policeman estimated that the commanding officer must have been about his age and build, minus the pudginess and taller by an inch or two. His faultless gray crew cut further amplified the Euclidean geometry of his face. On his dark uniform, a small badge read COLONEL CHASTEL in red letters. “I’ll ask you to wait a few more seconds. ~ Franck Thilliez
126:This picture of matter curving space and curvaceous space dictating how matter and light will move has several striking features. It brings the non-Euclidean geometries that we talked about in the last chapter out from the library of pure mathematics into the arena of science. The vast collection of geometries describing spaces that are not simply the flat space of Euclid are the ones that Einstein used to capture the possible structures of space distorted by the presence of mass and energy. ~ John D Barrow
127:Faith in God, then, is not at all the same as the kind of logical certainty that we attain in Euclidean geometry. God is not the conclusion to a process of reasoning, the solution to a mathematical problem. To believe in God is not to accept the possibility of his existence because it has been “proved” to us by some theoretical argument, but it is to put our trust in One whom we know and love. Faith is not the supposition that something might be true, but the assurance that someone is there. ~ Kallistos Ware
128:Supporters of the "modified Platonic view" of mathematics like to point out that, over the centuries, mathematicians have produced (or "discovered") numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unexpectedly feeding into physics, but there are many more. ~ Mario Livio
129:The new painters do not propose, any more than did their predecessors, to be geometers. But it may be said that geometry is to the plastic arts what grammar is to the art of the writer. Today, scholars no longer limit themselves to the three dimensions of Euclid. The painters have been lead quite naturally, one might say by intuition, to preoccupy themselves with the new possibilities of spatial measurement which, in the language of the modern studios, are designated by the term fourth dimension. ~ Guillaume Apollinaire
130:There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid. ~ Augustus De Morgan, "Short Supplementary Remarks on the First Six Books of Euclid's Elements" (Oct, 1848) Companion to the Almanac for 1849 as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements Vol.1, Introduction and Books I, II. Preface, p. v.
131:The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system. ~ David Berlinski
132:With the neatness of a Euclidean demonstration, the energy of the sun was now united with the smaller concentrations of energy at man's command: thus the Sun God had in effect undergone a human incarnation, and his priests at last commanded a commensurate authority. Theirs is a Calvinist theology, only slightly revised, in which the mass of men are predestined to awful damnation, and only the elect-that is, the technocratic elite-will be saved. In short, the eschatology of Jehovah's Witnesses, brought up to date. ~ Lewis Mumford
133:But then, Cap'n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken treasure related shapes that the cereal aestheticians were probably clamoring for, they came up with this hard-to-pin-down striated pillow formation. ~ Neal Stephenson
134:But then, Cap'n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken treasure related shapes that the cereal aestheticians were probably clamoring for, they came up with this hard -to-pin-down striated pillow formation. ~ Neal Stephenson
135:I walked downhill to the rental place, my backpack ten pounds heavier than it was this morning because of three huge textbooks: one on government from world history class; one from English class called Catastrophes of New England: 1650 to 1875; and a much-used book from my last class of the day, Non-Euclidean Geometry. The class was taught by Mr. Gint, a pale, balding man who barely looked at us. The entire class period he sat at his desk with a protractor and pencil, drawing pictures and muttering to himself. ~ Daryl Gregory
136:... in fact any good mind properly taught can think like Euclid and like Walt Whitman. The Renaissance, as we saw, was full of such minds, equally competent as poet and as engineers. The modern notion of "the two cultures," incompatible under one skull, comes solely from the proliferation of specialties in science; but these also divide scientists into groups that do not understand one another, the cause being the sheer mass of detail and the diverse terminologies. In essence the human mind remains one, not 2 or 60 different organs. ~ Jacques Barzun
137:Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions. The surface of the earth is finite in extent but it doesn’t have a boundary or edge: if you sail off into the sunset, you don’t fall off the edge or run into a singularity. (I know, because I have been round the world!) ~ Stephen Hawking
138:Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance). In my opinion, one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all. ~ Albert Einstein
139:About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. ~ John Aubrey
140:I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry. ~ Henri Poincare
141:Einstein créa donc sa théorie de la relativité restreinte en 1905 et, celle-ci n'étant pas en accord avec les lois de la gravitation de Newton, il poussa ses recherches plus loin pour formuler, en 1915, sa théorie de la relativité générale qui allait jusqu'à mettre en doute les axiomes d'Euclide sur l'espace. Chez Einstein, le plus remarquable ne résidait pas dans telle ou telle expérience, seulement, comme Alan sut le comprendre, dans sa capacité à douter, à prendre ses propres idées au sérieux et à les mener à leur conclusion logique, aussi troublante qu'elle puisse paraître. ~ Andrew Hodges
142:Let me tell you how at one time the famous mathematician Euclid became a physician. It was during a vacation, which I spent in Prague as I most always did, when I was attacked by an illness never before experienced, which manifested itself in chilliness and painful weariness of the whole body. In order to ease my condition I took up Euclid's Elements and read for the first time his doctrine of ratio, which I found treated there in a manner entirely new to me. The ingenuity displayed in Euclid's presentation filled me with such vivid pleasure, that forthwith I felt as well as ever. ~ Bernard Bolzano
143:If the ancients had been able to see it as I see it now, Mr. Palomar thinks, they would have thought they had projected their gaze into the heaven of Plato's ideas, or in the immaterial space of the postulates of Euclid; but instead, thanks to some misdirection or other, this sight has been granted to me, who fear it is too beautiful to be true, too gratifying to my imaginary universe to belong to the real world. But perhaps it is this same distrust of our senses that prevents us from feeling comfortable in the universe. Perhaps the first rule I must impose on myself is this: stick to what I see. ~ Italo Calvino
144:Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction; but made one help out the other as he best could. ~ Augustus De Morgan, The Differential and Integral Calculus (1836)
145:Spengler's book is rich in these "morphological relationships" between dissimilar activities that prove the coherent spirit of each culture and epoch. So there was a common spirit int eh ancient Greek polis and in Euclidean geometry, as there was also between the differential calculus and the state of Louis XIV. Chronological "contemporaneity" was misleading. It should be replaced by an understanding of how different events play similar roles in expressing the culture spirit. Thus he sees his own kind of "contemporaneity" in the Trojan War and the Crusades, in Homer and the songs of the Nibelungs. ~ Daniel J Boorstin
146:I believe like a child that suffering will be healed and made up for, that all the humiliating absurdity of human contradictions will vanish like a pitiful mirage, like the despicable fabrication of the impotent and infinitely small Euclidean mind of man, that in the world's finale, at the moment of eternal harmony, something so precious will come to pass that it will suffice for all hearts, for the comforting of all resentments, for the atonement of all the crimes of humanity, for all the blood that they've shed; that it will make it not only possible to forgive but to justify all that has happened. ~ Fyodor Dostoyevsky
147:I believe like a child that suffering will be healed and made up for, that all the humiliating absurdity of human contradictions will vanish like a pitiful mirage, like the despicable fabrication of the impotent and infinitely small Euclidian mind of man, that in the world's finale, at the moment of eternal harmony, something so precious will come to pass that it will suffice for all hearts, for the comforting of all resentments, for the atonement of all the crimes of humanity, of all the blood they've shed; that it will make it not only possible to forgive but to justify all that has happened with men- ~ Fyodor Dostoyevsky
148:we must not forget that the restful experience of enjoyable beauty is not limited to the contemplation of sensible objects. We can experience it as well in the contemplation of purely intelligible objects—the contemplation of truths we understand. “Mathematics,” wrote Bertrand Russell, “rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere … without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music …” Or, as the poet Edna St. Vincent Millay wrote in the opening line of her sonnet on Euclid, “Euclid alone has looked on beauty bare. ~ Mortimer J Adler
149:Even today, I am in total awe of the following wondrous chain of ideas and interconnections. Guided throughout by principles of symmetry, Einstein first showed that acceleration and gravity are really two sides of the same coin. He then expanded the concept to demonstrate that gravity merely reflects the geometry of spacetime. The instruments he used to develop the theory were Riemann's non-Euclidean geometries-precisely the same geometries used by Felix Klein to show that geometry is in fact a manifestation of group theory (because every geometry is defined by its symmetries-the objects it leaves unchanged). Isn't this amazing? ~ Mario Livio
150:Euclid Alone
Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
~ Edna St. Vincent Millay
151:I have a childlike conviction that the sufferings will be healed and smoothed over, that the whole offensive comedy of human contradiction will disappear like a pitiful mirage, a vile concoction of man's Euclidean mind, feeble and puny as an atom, and that ultimately, at the world's finale, in that moment of eternal harmony, there will occur and be revealed something so precious that it will suffice for all hearts, to allay all indignation, to redeem all human villainy, all bloodshed; it will suffice not only to make forgiveness possible, but also to justify everything that has happened with men -- let this, let off of this, come true and be revealed. ~ Fyodor Dostoyevsky
152:The Romans were too practical-minded to appreciate Euclid; the first of them to mention him is Cicero, in whose time there was probably no Latin translation; indeed there is no record of any Latin translation before Boethius (ca. A.D. 480). The Arabs were more appreciative: a copy was given to the caliph by the Byzantine emperor about A.D. 760, and a translation into Arabic was made under Harun al Rashid, about A.D. 800. The first still extant Latin translation was made from the Arabic by Adelard of Bath in A.D. 1120. From that time on, the study of geometry gradually revived in the West; but it was not until the late Renaissance that important advances were made. ~ Anonymous
153:As a monarch who should care more for the outlying colonies he knows on the map or through the report of his vicegerents, than for the trunk of his empire under his eyes at home, are we not more concerned about the shadowy life that we have in the hearts of others, and that portion in their thoughts and fancies which, in a certain far-away sense, belongs to us, than about the real knot of our identity - that central metropolis of self, of which alone we are immediately aware - or the diligent service of arteries and veins and infinitesimal activity of ganglia, which we know (as we know a proposition in Euclid) to be the source and substance of the whole? ~ Robert Louis Stevenson
154:All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors, and learned in what had been before them. There is not one exception. I do not say that every man has made direct acquantance with the whole of his mental ancestry... But... it is remarkable how many of the greatest names in all departments of knowledge have been real antiquaries in their several subjects. I may cite among those... in science, Aristotle, Plato, Ptolemy, Euclid, Archimedes, Roger Bacon, Copernicus, Francis Bacon, Ramus, Tycho Brahe, Galileo, Napier, Descartes, Leibnitz, Newton, Locke. ~ Augustus De Morgan, A Budget of Paradoxes (1872)
155:There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply. ~ Carl Sagan
156:This disdain for science and scholarship baffled Muslim commentators, who had great respect for Ptolemy and Euclid, for Homer and Aristotle. Some had little doubt what was to blame. Once, wrote the historian al-Mas  ūdī, the ancient Greeks and the Romans had allowed the sciences to flourish; then they adopted Christianity. When they did so, they ‘effaced the signs of [learning], eliminated its traces and destroyed its paths’. 92 Science was defeated by faith. It is almost the precise opposite of the world as we see it today: the fundamentalists were not the Muslims, but the Christians; those whose minds were open, curious and generous were based in the east –and certainly not in Europe. ~ Peter Frankopan
157:At that stage of my youth, death remained as abstract a concept as non-Euclidean geometry or marriage. I didn't yet appreciate its terrible finality or the havoc it could wreak on those who'd entrusted the deceased with their hearts. I was stirred by the dark mystery of mortality. I couldn't resist stealing up to the edge of doom and peering over the brink. The hint of what was concealed in those shadows terrified me, but I caught sight of something in the glimpse, some forbidden and elemental riddle that was no less compelling than the sweet, hidden petals of a woman's sex.
In my case - and, I believe, in the case of Chris McCandless - that was a very different thing from wanting to die. ~ Jon Krakauer
158:En vista de que ni siquiera esto soy capaz de comprender, he decidido no intentar comprender a Dios. Confieso humildemente mi incapacidad para resolver estas cuestiones. En esencia, mi mentalidad es la de Euclides: una mentalidad terrestre. ¿Para qué intentar resolver cosas que no son de este mundo? Te aconsejo que no te tortures el cerebro tratando de resolver estas cuestiones, y menos aún el problema de la existencia de Dios. ¿Existe o no existe? Estos puntos están fuera del alcance de la inteligencia humana, que sólo tiene la noción de las tres dimensiones. Por eso yo admito sin razonar no sólo la existencia de Dios, sino también su sabiduría y su finalidad para nosotros incomprensible. ~ Fyodor Dostoyevsky
159:My earliest memories are of CP4 — that's a Kähler manifold that looks locally like a vector space with four complex directions, though the global topology's quite different. But I didn't really grow up there; I was moved around a lot when I was young, to keep my perceptions flexible. I only used to spend time in anything remotely like this" — he motioned at the surrounding more-or-less-Euclidean space — for certain special kinds of physics problems. And even most Newtonian mechanics is easier to grasp in a symplectic manifold; having a separate visible coordinate for the position and momentum of every degree of freedom makes things much clearer than when you cram everything together in a single three-dimensional space. ~ Greg Egan
160:My earliest memories are of CP4 – that’s a Kähler manifold that looks locally like a vector space with four complex dimensions, though the global topology’s quite different. But I didn’t really grow up there; I was moved around a lot when I was young, to keep my perceptions flexible. I only used to spend time in anything remotely like this” – he motioned at the surrounding, more-or-less-Euclidean space – “for certain special kinds of physics problems. And even most Newtonian mechanics is easier to grasp in a symplectic manifold; having a separate, visible coordinate for the position and momentum of every degree of freedom makes things much clearer than when you cram everything together in a single, three-dimensional space. ~ Greg Egan
161:Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. ~ Mario Livio
162:We’ve got nothing to do with right-wing, left-wing or any other half-assed political category. If you work within the system, you come to one of the either/or choices that were implicit in the system from the beginning. You’re talking like a medieval serf, asking the first agnostic whether he worships God or the Devil. We’re outside the system’s categories. You’ll never get the hang of our game if you keep thinking in flat-earth imagery of right and left, good and evil, up and down. If you need a group label for us, we’re political non-Euclideans. But even that’s not true. Sink me, nobody of this tub agrees with anybody else about anything, except maybe what the fellow with the horns told the old man in the clouds: Non serviam. ~ Robert Shea
163:In Steven Spielberg’s film Lincoln, the screenwriter Tony Kushner has the great emancipator explain Euclid’s axiom in the context of a discussion on the equality of the races: “Euclid’s first common notion is this: Things which are equal to the same thing are equal to each other. That’s a rule of mathematical reasoning. It’s true because it works. Has done and always will do. In his book Euclid says this is self-evident. You see, there it is, even in that 2,000-year-old book of mechanical law it is a self-evident truth.” Although Lincoln never actually uttered those words, there is every reason to think that he would have made just such an argument because it’s precisely what is implied in his 1854 argument that A is interchangeable with B. ~ Michael Shermer
164:Certainly she can't and won't measure what is measureless, what neither terminates nor repeats, what is beyond even the transcendental of π - though HE doesn't think so - what is beyond polynomials and quadratic formulas, beyond the rational and irrational, the humanist and the logical, beyond the minds of the Cantors and the Dedekinds, the Renaissance philosophers and the Indian Tantrists, what falls instead into the realm of gods and kinds, of myth, of dawn of man, of the mystery of mankind - that there is a space inside her designed solely for him and despite clear Euclidian impossibilities not only does everything, in plenary excess, cleave like it's meant to, but it makes her feel what math cannot explain, what science cannot explain. What nothing can explain. ~ Paullina Simons
165:These estimates may well be enhanced by one from F. Klein (1849-1925), the leading German mathematician of the last quarter of the nineteenth century. 'Mathematics in general is fundamentally the science of self-evident things.' ... If mathematics is indeed the science of self-evident things, mathematicians are a phenomenally stupid lot to waste the tons of good paper they do in proving the fact. Mathematics is abstract and it is hard, and any assertion that it is simple is true only in a severely technical sense—that of the modern postulational method which, as a matter of fact, was exploited by Euclid. The assumptions from which mathematics starts are simple; the rest is not. ~ Eric Temple Bell
166:Captain West advanced to meet me, and before our outstretched hands touched, before his face broke from repose to greeting and the lips moved to speech, I got the first astonishing impact of his personality. Long, lean, in his face a touch of race I as yet could only sense, he was as cool as the day was cold, as poised as a king or emperor, as remote as the farthest fixed star, as neutral as a proposition of Euclid. And then, just ere our hands met, a twinkle of--oh--such distant and controlled geniality quickened the many tiny wrinkles in the corner of the eyes; the clear blue of the eyes was suffused by an almost colourful warmth; the face, too, seemed similarly to suffuse; the thin lips, harsh-set the instant before, were as gracious as Bernhardt's when she moulds sound into speech. ~ Jack London
167:And so will I here state just plainly and briefly that I accept God. But I must point out one thing: if God does exist and really created the world, as we well know, he created it according to the principles of Euclidean geometry and made the human brain capable of grasping only three dimensions of space. Yet there have been and still are mathematicians and philosophers-among them some of the most outstanding-who doubt that the whole universe or, to put it more generally, all existence was created to fit Euclidean geometry; they even dare to conceive that two parallel lines that, according to Euclid, never do meet on earth do, in fact, meet somewhere in infinity. And so my dear boy, I’ve decided that I am incapable of understanding of even that much, I cannot possibly understand about God. ~ Fyodor Dostoyevsky
168:The nuggets themselves are pillow-shaped and vaguely striated to echo piratical treasure chests. Now, with a flake-type of cereal, Randy’s strategy would never work. But then, Cap’n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken-treasure-related shapes that the cereal-aestheticians were probably clamoring for, they came up with this hard-to-pin-down striated pillow formation. The important thing, for Randy’s purposes, is that the individual pieces of Cap’n Crunch are, to a very rough approximation, shaped ~ Neal Stephenson
169:There have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely the whole of existence, was only created in Euclid's geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can't understand even that, I can't expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidean earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with an idea of only three dimensions. ~ Fyodor Dostoyevsky
170:This is the remarkable paradox of mathematics," observed commentator John Tierney. "No matter how determinedly its practitioners ignore the world, they consistently produce the best tools for understanding it. The Greeks
decide to study, for no good reason, a curve called an ellipse, and 2,000 years later astronomers discover that it describesthe way the planets move around the sun. Again, for no good reason, in 1 854 a German mathematician, Bernhard Riemann, wonders what would happen if he discards one of the hallowed postulates of Euclid's plane geometry. He builds a seemingly ridiculous assumption that it's not possible to draw two lines parallel to each other. His non-Euclidean geometry replaces Euclid's plane with a bizarre abstraction called curved space, and then, 60 years later, Einstein announces that this is the shape of the universe. ~ Paul Hoffman
171:In Fyodor Dostoyevsky’s great novel The Brothers Karamazov, there is a scene in which two people are talking about suffering. Ivan Karamazov is talking about there being any possibility that we can make sense of suffering, and here’s what he says: “I believe like a child that suffering will be healed and made up for, that all the humiliating absurdity of human contradictions will vanish like a pitiful mirage, like the despicable fabrication of the impotent and infinitely small Euclidean mind of man, that in the world’s finale, at the moment of eternal harmony, something so precious will come to pass that it will suffice for all hearts, for the comforting of all resentments, for the atonement of all the crimes of humanity, of all the blood that they’ve shed; that it will make it not only possible to forgive but to justify all that has happened.”11 ~ Timothy J Keller
172:Este indubitabil că, atunci când e vorba de matematică, știință, filozofie, artă și literatură, oricine a crescut într-o civilizație occindentală sau a Orientului Mijlociu este un elev al vechilor greci. Fraza lui Goethe – “dintre toate poapoarele, grecii au visat cel mai bine visul vieții” – nu este decât un mic omagiu în fața eforturilor de pionierat pe care grecii le-au depus în toate ramurile cunoașterii și ale științei inventate și botezate de ei. […] Măiestria grecilor în matematică a fost în mare măsură urmarea directă a pasiunii lor pentru cunoaștere ca scop în sine, mai degrabă decât pentru scopuri practice. O istorioară pretinde că, atunci când un elev căruia Euclid îi explica o propoziție geometrică a întrebat “Dar ce câștig eu de aici?”, Euclid i-ar fi poruncit sclavului să-i dea băiatului un bănuț, așa încât să poată vedea și el un profit efectiv. ~ Mario Livio
173:It’s not God that I do not accept, you understand, it is this world of God’s, created by God, that I do not accept and cannot agree to accept. With one reservation: I have a childlike conviction that the sufferings will be healed and smoothed over, that the whole offensive comedy of human contradictions will disappear like a pitiful mirage, a vile concoction of man’s Euclidean mind, feeble and puny as an atom, and that ultimately, at the world’s finale, in the moment of eternal harmony, there will occur and be revealed something so precious that it will suffice for all hearts, to allay all indignation, to redeem all human villainy, all bloodshed; it will suffice not only to make forgiveness possible, but also to justify everything that has happened with men—let this, let all of this come true and be revealed, but I do not accept it and do not want to accept ~ Fyodor Dostoyevsky
174:There was no hope for him this time: it was the third stroke. Night after night I had passed the house (it was vacation time) and studied the lighted square of window: and night after night I had found it lighted in the same way, faintly and evenly. If he was dead, I thought, I would see the reflection of candles on the darkened blind, for I knew that two candles must be set at the head of a corpse. He had often said to me: I am not long for this world and I had thought his words idle. Now I knew they were true. Every night as I gazed up at the window I said softly to myself the word paralysis. It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism. But now it sounded to me like the name of some maleficent and sinful being. It filled me with fear, and yet I longed to be nearer to it and to look upon its deadly work. ~ James Joyce
175:I wonder if you've ever considered how strange it is that the educational and character-shaping structures of our culture expose us but a single time in our lives to the ideas of Socrates, Plato, Euclid, Aristotle, Herodotus, Augustine, Machiavelli, Shakespeare, Descartes, Rousseau, Newton, Racine, Darwin, Kant, Kierkegaard, Tolstoy, Schopenhauer, Goethe, Freud, Marx, Einstein, and dozens of others of the same rank, but expose us annually, monthly, weekly, and even daily to the ideas of persons like Jesus, Moses, Muhammad, and Buddha. Why is it, do you think, that we need quarterly lectures on charity, while a single lecture on the laws of thermodynamics is presumed to last us a lifetime? Why is the meaning of Christmas judged to be so difficult of comprehension that we must hear a dozen explications of it, not once in a lifetime, but every single year, year after year after year? ~ Daniel Quinn
176:Modern mathematics contains much more than that, of course. It includes set theory, for example, created by Georg Cantor in 1874, and “foundations,” which another George, the Englishman George Boole, split off from classical logic in 1854, and in which the logical underpinnings of all mathematical ideas are studied. The traditional categories have also been enlarged to include big new topics—geometry to include topology, algebra to take in game theory, and so on. Even before the early nineteenth century there was considerable seepage from one area into another. Trigonometry, for example, (the word was first used in 1595) contains elements of both geometry and algebra. Descartes had in fact arithmetized and algebraized a large part of geometry in the seventeenth century, though pure-geometric demonstrations in the style of Euclid were still popular—and still are— for their clarity, elegance, and ingenuity. ~ Anonymous
177:On what may be the last page he wrote in his notebooks, Leonardo drew four right triangles with bases of differing lengths (fig. 143). Inside of each he fit a rectangle, and then he shaded the remaining areas of the triangle. In the center of the page he made a chart with boxes labeled with the letter of each rectangle, and below it he described what he was trying to accomplish. As he had done obsessively over the years, he was using the visualization of geometry to help him understand the transformation of shapes. Specifically, he was trying to understand the formula for keeping the area of a right triangle the same while varying the lengths of its two legs. He had fussed with this problem, explored by Euclid, repeatedly over the years. It was a puzzle that, by this point in his life, as he turned sixty-seven and his health faded, might seem unnecessary to solve. To anyone other than Leonardo, it may have been. ~ Walter Isaacson
178:By all accounts mathematics is mankind’s most successful intellectual undertaking. Every problem of mathematics gets solved, sooner or later. Once solved, a mathematical problem is forever finished: no later event will disprove a correct solution. As mathematics progresses, problems that were difficult become easy and can be assigned to schoolchildren.Thus Euclidean geometry is taught in the second year of high school. Similarly, the mathematics learned by my generation in graduate school is now taught at the undergraduate level, and perhaps in the not too distant future, in the high schools. Not only is every mathematical problem solved, but eventually every mathematical problem is proved trivial. The quest for ultimate triviality is characteristic of the mathematical enterprise. ~ Gian-Carlo Rota, "The Pernicious Influence of Mathematics upon Philosophy" 18 Unconventional Essays on the Nature of Mathematics (2006) p. 223, ed. Reuben Hersh.
179:To The Works Of:
   Aristotle, Cassius J. Keyser, Eric T. Bell, G. W. Leibnitz, Eugen Bleuler, J. Locke, Niels Bohr, Jacques Loeb, George Boole, H. A. Lorentz, Max Born, Ernst Mach, Louis De Brogue, J. C. Maxwell, Georg Cantor, Adolf Meyer, Ernst Cassirer, Hermann Minkowsja, Charles M. Child, Isaac Newton, C. Darwin, Ivan Pavlov, Rene Descartes, Giuseppe Peano, P. A. M. Dirac, Max Planck, A. S. Eddington, Plato, Albert Einstein, H. Poincare, Euclid, M. Faraday, Sigmund Freud, Josiah Royce, Karl F. Gauss, G. Y. Rainich, G. B. Riemann, Bertrand Russell, Thomas Graham, Ernest Rutherford, Arthur Haas, E. Schrodinger, Wm. R. Hamilton, C. S. Sherrington, Henry Head, Socrates, Werner Heisenberg, Arnold Sommerfeld, C. Judson Herrick, Oswald Veblen, E. V. Huntington, Wm. Alanson White, Smith Ely Jeluffe, Alfred N. Whitehead, Ludwig Wittgenstein
   Which Have Creatly Influenced My Enquiry
   This System Is Dedicated ~ Alfred Korzybski, Science and Sanity,
180:However, the answer can be inferred from the WMAP data, by measuring the sizes of the temperature fluctuations—the hot and cold (light and dark) splotches in Figure 3 ([>]). Before WMAP was launched, theorists had already worked out how big the physical sizes of the strongest fluctuations should be. Converting that into apparent angular size in the sky depends on the geometry of space: if the universe is positively curved, it would make the angles appear larger, while negative curvature would make them smaller. If the universe is geometrically flat (that is, has Euclidean geometry), the angular size of the strongest hot and cold fluctuations should be about 1° across. The results that flowed back from the satellite were definitive.14 The fluctuations were very close to 1° in size, a result confirmed by ground-based and balloon-based experiments. Cosmologists then declared that to within observational accuracy of about 2 percent, space is flat.15 ~ Paul Davies
181:Some persons fancy that bias and counter-bias are favorable to the extraction of truth–that hot and partisan debate is the way to investigate. This is the theory of our atrocious legal procedure. But Logic puts its heel upon this suggestion. It irrefragably demonstrates that knowledge can only be furthered by the real desire for it, and that the methods of obstinacy, of authority and every mode of trying to reach a foregone conclusion, are absolutely of no value. These things are proved. The reader is at liberty to think so or not as long as the proof is not set forth, or as long as he refrains from examining it. Just so, he can preserve, if he likes, his freedom of opinion in regard to the propositions of geometry; only, in that case, if he takes a fancy to read Euclid, he will do well to skip whatever he finds with A, B, C, etc., for, if he reads attentively that disagreeable matter, the freedom of his opinion about geometry may unhappily be lost forever. ~ Charles Sanders Peirce
182:Baccalaureate
A year or two, and grey Euripides,
And Horace and a Lydia or so,
And Euclid and the brush of Angelo,
Darwin on man, Vergilius on bees,
The nose and Dialogues of Socrates,
Don Quixote, Hudibras and Trinculo,
How worlds are spawned and where the dead gods go,-All shall be shard of broken memories.
And there shall linger other, magic things,-The fog that creeps in wanly from the sea,
The rotton harbor smell, the mystery
Of moonlit elms, the flash of pigeon wings,
The sunny Green, the old-world peace that clings
About the college yard, where endlessly
The dead go up and down. These things shall be
Enchantment of our heart's rememberings.
And these are more than memories of youth
Which earth's four winds of pain shall blow away;
These are earth's symbols of eternal truth,
Symbols of dream and imagery and flame,
Symbols of those same verities that play
Bright through the crumbling gold of a great name.
~ Archibald MacLeish
183:Pe măsură ce urcăm scara cerurilor, priveliştile sînt mai neaşteptate. Printre constelaţii şiroiuri de galaxii, stele novae, pitice şi albe, uitînd de predici mînioase, volume de teologie şi argumente apologetice, depăşind izvoare veşnice de hidrogen – reglate de spiritul profesorului Hoyle -, lăsînd în urmă judecători, constructori, socotitori, profeţi, gravi filosofi şi geometri neeuclidieni. Sufletul urcă mereu mai sus, curăţîndu-se, pînă la staţia terminus: locul de lumină şi verdeţă, pajiştea înflorată, mişunînd de căţei mici şi dolofani şi de pisicuţe albe cu fundă, acolo unde răsună acordurile divertsmentelor lui Mozart şi se ostenesc îngerii cu aripi ai lui Liliom să ofere necontenit dulceaţuri şi şerbet, acolo unde se află Dumnezeu cel adevărat, al pruncilor lăsaţi – în sfîrşit – să vină, oricît de bătrîni de ani sau de împovăraţi de grele amintiri, să vadă: pe Tatăl cu barba albă, la mijloc, pe Hristos purtător de stigmate şi cruce în dreapta, pe Duhul curăţitor şi alinător în stînga. ~ Nicolae Steinhardt
184:Oh, with my pathetic, earthly, Euclidean mind, I know only that there is suffering, that none are to blame, that all things follow simply and directly from one another, that everything flows and finds its level - but that is all just Euclidean gibberish, of course I know that, and of course I cannot consent to live by it! What do I care that none are to blame and that I know it - I need retribution, otherwise I will destroy myself. And retribution not somewhere and sometime in infinity, but here and now, on earth, so that I see it myself. I have believed, and I want to see for myself, and if I am dead by that time, let them resurrect me, because it will be too unfair if it all takes place without me. Is it possible that I've suffered so that I, together with my evil deeds and sufferings, should be manure for someone's future harmony? I want to see with my own eyes the hind lie down with the lion, and the murdered man rise up and embrace his murderer. I want to be there when everyone suddenly finds out what it was all for. ~ Fyodor Dostoyevsky
185:Sullivan filled the moody silence by announcing that my digression on numbers was somewhat less than Euclidean in its sweep and purity; that one of my main faults was a tendency to get blinded by the neon of an idea, never reaching truly inside it; that to follow a number to infinity was not necessarily to arrive at God. With her fork she bisected a crisp slice of bacon, a piece so brittle the fork barely had to touch it; she then halved the two fragments, then the smaller four, then the resulting eight, and so on, working with the quietly fanatical precision of all those people whose job it is to divide small things into smaller things, who live on the rim of insanity; finally there was nothing left of the slice but a hundred decimal points. Did the bacon represent the insignificance of numbers; the futile quest for infinity; the indivisible nature of God as opposed to the fractional promiscuity of numbers? Was it all a lesson in prime matter and substantial form: Were the bits of bacon supposed to be numbers and the fried egg God? ~ Don DeLillo
186:This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn't just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, once could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that both do not hold, and obtain an entirely different theory. Similarly, assuming the validity of one of the two axioms and the negation of the other would have led to yet two other set theories.

This was the non-Euclidean crisis revisited, only worse. The fundamental role of set theory as the potential basis for the whole of mathematics made the problem for the Platonists much more acute. If indeed one could formulate many set theories simply by choosing a different collection of axioms, didn't this argue for mathematics being nothing but a human invention? The formalists' victory looked virtually assured. ~ Mario Livio
187:Non-Euclidean' became a byword for non-absolute knowledge. It also served to illustrate most vividly the gap between mathematics and the natural world. Mathematics was much bigger than physical reality. There were mathematical systems that described aspects of Nature, but there were others that did not. Later, mathematicians would use these discoveries about geometry to discover that there were other logics as well. Aristotle's system was, like Euclid's, just one of many possibilities. Even the concept of truth was not absolute. What is false in one logical system can be true in another. In Euclid's geometry of flat surfaces, parallel lines never meet, but on curved surfaces they can. These discoveries revealed the difference between mathematics and science. Mathematics was something bigger than science, requiring only self-consistency to be valid. It contained all possible patterns of logic. Some of those patterns were followed by parts of Nature; others were not. Mathematics was open-ended, uncompleteable, infinite; the physical universe was smaller. ~ John D Barrow
188:The one created thing which we cannot look at is the one thing in the light of which we look at everything. Like the sun at noonday, mysticism explains everything else by the blaze of its own victorious invisibility. Detached intellectualism is (in the exact sense of a popular phrase) all moonshine; for it is light without heat, and it is secondary light, reflected from a dead world. But the Greeks were right when they made Apollo the god both of imagination and of sanity; for he was both the patron of poetry and the patron of healing. Of necessary dogmas and a special creed I shall speak later. But that transcendentalism by which all men live has primarily much the position of the sun in the sky. We are conscious of it as of a kind of splendid confusion; it is something both shining and shapeless, at once a blaze and a blur. But the circle of the moon is as clear and unmistakable, as recurrent and inevitable, as the circle of Euclid on a blackboard. For the moon is utterly reasonable; and the moon is the mother of lunatics and has given to them all her name. ~ G K Chesterton
189:The cosmic sculptor had felt compelled to dot pupils onto the universe, yet had a tremendous terror of granting it sight. This balance of fear and desire resulted in the tininess of the stars against the hugeness of space, a declaration of caution above all. “See how the stars are points? The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically.” “But there’s nothing concrete to study in your cosmic sociology, Dr. Ye. Surveys and experiments aren’t really possible.” “That means your ultimate result will be purely theoretical. Like Euclidean geometry, you’ll set up a few simple axioms at first, then derive an overall theoretic system using those axioms as a foundation.” “It’s all fascinating, but what would the axioms of cosmic sociology be?” “First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant.” The ~ Liu Cixin
190:Those two axioms are solid enough from a sociological perspective … but you rattled them off so quickly, like you’d already worked them out,” Luo Ji said, a little surprised. “I’ve been thinking about this for most of my life, but I’ve never spoken about it with anyone before. I don’t know why, really.… One more thing: To derive a basic picture of cosmic sociology from these two axioms, you need two other important concepts: chains of suspicion, and the technological explosion.” “Interesting terms. Can you explain them?” Ye Wenjie glanced at her watch. “There’s no time. But you’re clever enough to figure them out. Use those two axioms as a starting point for your discipline, and you might end up becoming the Euclid of cosmic sociology.” “I’m no Euclid. But I’ll remember what you said and give it a whirl. I might come to you for guidance, though.” “I’m afraid there won’t be that opportunity.… In that case, you might as well just forget I said anything. Either way, I’ve fulfilled my duty. Well, Xiao Luo, I’ve got to go.” “Take care, Professor.” Ye Wenjie went off through the twilight to her final meet-up. The ~ Liu Cixin
191:Now we can see what makes mathematics unique. Only in mathematics is there no significant correction-only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day.

Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever.

Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like A History of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago.

Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak. ~ Isaac Asimov
192:It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other. ~ Roger Penrose
193:Mathematics
I've really done enough of sums,
I've done so very many,
That now instead of doing sum
I'd rather not do any.
I've toiled until my fingers are
With writing out of joint;
And even now of Decimals
I cannot see the point.
Subtraction to my weary mind
Brings nothing but distraction,
And vulgar and improper I
Consider every fraction.
"Practice makes perfect," so they say.
It may be true. The fact is
That I unhappily am not
Yet perfect in my Practice.
Discount is counted troublesome
By my unlearned pate;
For cubic root I entertain
A strongly rooted hate.
The heathen worship stocks and stones;
My pious soul it shocks
To be instructed thus to take
An Interest in Stocks.
Of Algebra I fear I have
A very vague impression;
I study hard, but fail to make
Harmonical Progression.
In Euclid too I always climb
The Asses' Bridge with pain;
A superficies to me
Is anything but plane.
"Apply yourself," my master said,
When I my woes confided,
"And, when you multiply, bestow
Attention undivided."
Oh, if one master tries so hard
Tyrannical to be,
How out of all Proportion I
Should find a Rule of Three.
~ Arthur Clement Hilton
194:I am a bug, and I recognise in all humility that I cannot understand why the world is arranged as it is. Men are themselves to blame, I suppose; they were given paradise, they wanted freedom, and stole fire from heaven, though they knew they would become unhappy, so there is no need to pity them. With my pitiful, earthly, Euclidian understanding, all I know is that there is suffering and that there are none guilty; that cause follows effect, simply and directly; that everything flows and finds its level—but that's only Euclidian nonsense, I know that, and I can't consent to live by it! What comfort is it to me that there are none guilty and that cause follows effect simply and directly, and that I know it?—I must have justice, or I will destroy myself. And not justice in some remote infinite time and space, but here on earth, and that I could see myself. I have believed in it. I want to see it, and if I am dead by then, let me rise again, for if it all happens without me, it will be too unfair. Surely I haven't suffered simply that I, my crimes and my sufferings, may manure the soil of the future harmony for somebody else. I want to see with my own eyes the hind lie down with the lion and the victim rise up and embrace his murderer. I want to be there when everyone suddenly understands what it has all been for. ~ Fyodor Dostoyevsky
195:How did nature manage to evolve such complicated architecture? Mandelbrot's point is that the complications exist only in the context of traditional Euclidean geometry. As fractals, branching structures can be described with transparent simplicity, with just a few bits of information. Perhaps the simplest transformations that gave rise to the shapes devised by Koch, Peano, and Sierpinski have their analogue in the coded instructions of an organism's genes. DNA surely cannot specify the vast number of bronchi, bronchioles, and alveoli or the particular spatial structure of the resulting tree, but it can specify a repeating process of bifurcation and development. Such processes suit nature's purposes. When E.I. Dupont de Nemours & Company and the United States Army finally began to produce a synthetic match for goose down, it was by finally realizing that the phenomenal air-trapping ability of the natural product came from the fractal nodes and branches of down's key protein, keratin. Mandelbrot glided matter-of-factly from pulmonary and vascular trees to real botanical trees, trees that need to capture sun and resist wind, with fractal branches and fractal leaves. And theoretical biologists began to speculate that fractal scaling was not just common but universal in morphogenesis. They argued that understanding how such patterns were encoded and processed had become a major challenge to biology. ~ James Gleick
196:I do not think there is a demonstrative proof (like Euclid) of Christianity, nor of the existence of matter, nor of the good will and honesty of my best and oldest friends. I think all three are (except perhaps the second) far more probable than the alternatives. The case for Christianity in general is well given by Chesterton…As to why God doesn't make it demonstratively clear; are we sure that He is even interested in the kind of Theism which would be a compelled logical assent to a conclusive argument? Are we interested in it in personal matters? I demand from my friend trust in my good faith which is certain without demonstrative proof. It wouldn't be confidence at all if he waited for rigorous proof. Hang it all, the very fairy-tales embody the truth. Othello believed in Desdemona's innocence when it was proved: but that was too late. Lear believed in Cordelia's love when it was proved: but that was too late. 'His praise is lost who stays till all commend.' The magnanimity, the generosity which will trust on a reasonable probability, is required of us. But supposing one believed and was wrong after all? Why, then you would have paid the universe a compliment it doesn't deserve. Your error would even so be more interesting and important than the reality. And yet how could that be? How could an idiotic universe have produced creatures whose mere dreams are so much stronger, better, subtler than itself? ~ C S Lewis
197:Spațiile lui Riemann sunt lipsite de orice fel de omogenitate. Fiecare dintre ele se caracterizează prin forma expresiei care definește pătratul distanței dintre două puncte infinit învecinate. ... Rezultă de aici că doi observatori aflați învecinați pot să repereze într-un spațiu riemannian punctele care se află în imediata lor vecinătate, dar nu pot, fără stabilirea unei noi convenții, să se repereze unul față de celălalt. Fiecare vecinătate este deci ca o mică bucată de spațiu euclidian, dar racordarea dintre o vecinătate și următoarea nu e definită și poate fi făcută într-o infinitate de moduri. Spațiul riemannian cel mai general se prezintă, astfel, ca o colecție amorfă de bucăți juxtapuse fără a fi legate’ (Albert Lautmann, Les schèmas de structure, Hermann, 1938, pp.23, 34-35); această mulțime poate fi definită independent de orice referire la o metrică, prin condiții de frecvență sau, mai curând, de acumulare valabile pentru un ansamblu de vecinătăți, condiții total diferite de cele care determină spațiile metrice și tăieturile lor (chiar dacă un raport între cele două feluri de spațiu trebuie să decurgă de aici). Pe scurt, dacă urmăm frumoasa descriere a lui Lautmann, spațiul riemannian este un pur patchwork. Are conexiuni sau raporturi tactile. Are valori ritmice care nu se regăsesc în altă parte, chiar dacă pot fi traduse într-un spațiu metric. Eterogen, în variație continuă, este un spațiu neted, în măsura în care este amorf, nu omogen. (Gilles Deleuze et Félix Guattari) ~ Gilles Deleuze
198:Although the collection of known mathematical structures is large and exotic, and even more remain to be discovered, every single mathematical structure can be analyzed to determine its symmetry properties, and many have interesting symmetry. Intriguingly, one of the most important discoveries in physics has been that our physical reality also has symmetries built into it: for example, the laws of physics have rotational symmetry, which means that there's not special direction in our Universe that you can call "up." They also appear to have translation (sideways shifting) symmetry, meaning that there's no special place that we can call the center of space. Many of these spaces just mentioned have beautiful symmetries, some of which match the observed symmetries of our physical world. For example, Euclidean space has both rotational symmetry (meaning that you can't tell the difference if the space gets rotated) and translational symmetry (meaning that you can't tell the difference if the space gets shifted sideways). The four-dimensional Minkowski space has even more symmetry: you can't even tell the difference if you do a type of generalized rotation between the space and time dimensions-and Einstein showed that this explains why time appears to slow down if you travel near the speed of light, as mentioned in the last chapter. Many more subtle symmetries of nature have been discovered in the last century, and these symmetries form the foundations of Einstein's relativity theories, quantum mechanics, and the standard model of particle physics. ~ Max Tegmark
199:Ibn Sina was born in a tiny settlement called Afshanah, outside the village of Kharmaythan, and soon after his birth his family moved to the nearby city of Bukhara. While he was still a small boy his father, a tax collector, arranged for him to study with a teacher of Qu’ran and a teacher of literature, and by the time he was ten he had memorized the entire Qu’ran and absorbed much of Muslim culture. His father met a learned vegetable peddler named Mahmud the Mathematician, who taught the child Indian calculation and algebra. Before the gifted youth grew his first facial hairs he had qualified in law and delved into Euclid and geometry, and his teachers begged his father to allow him to devote his life to scholarship. He began the study of medicine at eleven and by the time he was sixteen he was lecturing to older physicians and spending much of his time in the practice of law. All his life he would be both jurist and philosopher, but he noted that although these learned pursuits were given deference and respect by the Persian world in which he lived, nothing mattered more to an individual than his well-being and whether he would live or die. At an early age, fate made Ibn Sina the servant of a series of rulers who used his genius to guard their health, and though he wrote dozens of volumes on law and philosophy—enough to win him the affectionate sobriquet of Second Teacher (First Teacher being Mohammed)—it was as the Prince of Physicians that he gained the fame and adulation that followed him wherever he traveled. In Ispahan, where he had gone at ~ Noah Gordon
200:For what is it you and I are trying to do now? What I'm trying to do is to attempt to explain to you as quickly as possible the most important thing about me, that is to say, what sort of man I am, what I believe in what I hope for - that's it isn't it? And that's why I declare that I accept God plainly and simply. But there's this that has to be said: if God really exists and if he really has created the world, then, as we all know, he created it in accordance with the Euclidean geometry, and he created the human mind with the conception of only the three dimensions of space. And yet there have been and there still are mathematicians and philosophers, some of them indeed men of extraordinary genius, who doubt whether the whole universe, or, to put it more wildly, all existence was created only according to Euclidean geometry and they even dare to dream that two parallel lines which, according to Euclid can never meet on earth, may meet somewhere in infinity. I, my dear chap, have come to the conclusion that if I can't understand even that, then how can I be expected to understand about God? I humbly admit that I have no abilities for settling such questions. And I advise you too, Aloysha, my friend, never to think about it, and least of all about whether there is a God or not. All these problems which are entirely unsuitable to a mind created with the idea of only three dimensions. And so I accept God, and I accept him not only without reluctance, but what's more, I accept his divine wisdom and his purpose- which are completely beyond our comprehension. ~ Fyodor Dostoyevsky
201:A computational procedure is said to have a top-down organization if it has been constructed according to some well-defined and clearly understood fixed computational procedure (which may include some preassigned store of knowledge), where this procedure specifically provides a clear-cut solution to some problem at hand. (Euclid's algorithm for finding the highest common factor of two natural numbers, as described in ENM, p. 31, is a simple example of a top-down algorithm.) This is to be contrasted with a bottom-up organization, where such clearly defined rules of operation and knowledge store are not specified in advance, but instead there is a procedure laid down for the way that the system is to 'learn' and to improve its performance according to its 'experience'. Thus, with a bottom-up system, these rules of operation are subject to continual modification. One must allow that the system is to be run many times, performing its actions upon a continuing input of data. On each run, an assessment is made-perhaps by the system itself-and it modifies its operations, in the lifht of this assessment, with a view to improving this quality of output. For example, the input data for the system might be a number of photographs of human faces, appropriately digitized, and the system's task is to decide which photographs represent the same individuals and which do not. After each run, the system's performance is compared with the correct answers. Its rules of operation are then modified in such a way as to lead to a probable improvement in its performance on the next run. ~ Roger Penrose
202:I am trying to explain as quickly as possible my essential nature, that is, what manner of man I am, what I believe in, and for what I hope, that's it, isn't it? And therefore I tell you that I accept God honestly and simply. But you must note this: If God exists and if He really did create the world, then, as we all know, He created it according to the geometry of only three dimensions in space. Yet there have been some very distinguished ones, who doubt whether the whole universe, or to speak more generally the whole of being, was only created in Euclid's geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can't understand even that, I can't expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidian earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with a conception of only three dimensions. And so I accept God and am glad to, and what's more I accept His wisdom, His purpose - which are utterly beyond our ken; I believe in the underlying order and the meaning of life; I believe in the eternal harmony in which they say we shall one day be blended. I believe in the Word to Which the universe is striving, and Which Itself was "with God", and Which Itself is God and so on, and so on, to infinity. ~ Fyodor Dostoyevsky
203:Q5. Have not I merely shown that it is possible to outdo just a particular algorithmic procedure, A, by defeating it with the computation Cq(n)? Why does this show that I can do better than any A whatsoever?

The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will know of Euclid's argument that there is no largest prime number. This, also, is a reductio ad absurdum. Euclid's argument is as follows. Suppose, on the contrary, that there is a largest prime; call it p. Now consider the product N of all the primes up to p and add 1:

N=2*3*5*...*p+1.

N is certainly larger than p, but it cannot be divisible by any of the prime numbers 2,3,5...,p (since it leaves the remainder 1 on division); so either N is the required prime itself or it is composite-in which case it is divisible by a prime larger than p. Either way, there would have to be a prime larger than p, which contradicts the initial assumption that p is the largest prime. Hence there is no largest prime. The argument, being a reductio ad absurdum, does not merely show that a particular prime p can be defeated by finding a larger one; it shows that there cannot be any largest prime at all. Likewise, the Godel-Turing argument above does not merely show that a particular algorithm A can be defeated, it shows that there cannot be any (knowably sound) algorithm at all that is equivalent to the insights that we use to ascertain that certain computations do not stop. ~ Roger Penrose
204:To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us-humans had not choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town! ~ Mario Livio
205:Joining the world of shapes to the world of numbers in this way represented a break with the past. New geometries always begin when someone changes a fundamental rule. Suppose space can be curved instead of flat, a geometer says, and the result is a weird curved parody of Euclid that provides precisely the right framework for the general theory of relativity. Suppose space can have four dimensions, or five, or six. Suppose the number expressing dimension can be a fraction. Suppose shapes can be twisted, stretched, knotted. Or, now, suppose shapes are defined, not by solving an equation once, but by iterating it in a feedback loop.

Julia, Fatou, Hubbard, Barnsley, Mandelbrot-these mathematicians changed the rules about how to make geometrical shapes. The Euclidean and Cartesian methods of turning equations into curves are familiar to anyone who has studied high school geometry or found a point on a map using two coordinates. Standard geometry takes an equation and asks for the set of numbers that satisfy it. The solutions to an equation like x^2 + y^2 = 1, then, form a shape, in this case a circle. Other simple equations produce other pictures, the ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number goes in, and so on, points hopping from place to place. A point is plotted not when it satisfies the equation but when it produces a certain kind of behavior. One behavior might be a steady state. Another might be a convergence to a periodic repetition of states. Another might be an out-of-control race to infinity. ~ James Gleick
206:Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof.

Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did!

Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence. ~ Mario Livio
207:Most white Americans believe elections should be a choice of policies rather than expressions of racial identity. If Americans vote for a candidate because of his racial agenda, representative government is crippled. Democratic systems operate well only when politicians recognize that even if their opponents’ approaches may be different, all parties are trying to work for the good of the country as a whole. When politics fracture along racial lines, it becomes easy to assume that elected officials work for narrow, ethnic interests, and political contests become very bitter.
The ultimate logic of politics in a racially fractured electorate is a system of quotas in which seats in elective bodies are set aside in proportion to the racial composition of the population. This is the formula hopelessly divided countries such as Lebanon and immediate post-white-rule Zimbabwe and South Africa hit upon. It could be the solution for other divided countries such as Iraq, Sudan, Fiji, Malaysia, or Sri Lanka, where politics is a perpetual squabble over ethnic interests.
There is already implied support for proportional racial representation in the federal approach to voter districts. The US Department of Justice has long required that congressional districts be gerrymandered to create black and Hispanic majorities that are expected to vote along racial lines and send one of their own to Congress. The department also routinely sues cities that choose their governing bodies in at-large elections. If, for example, a city is 30 percent black but has no blacks on the city council because all candidates must appeal to the entire city, voting must be switched to a ward system, with wards drawn so that blacks—by voting for people like themselves have approximately 30 percent of the council seats. In 2006, the Justice Department used precisely this argument to threaten Euclid, Ohio, with litigation if it did not replace its at-large elections with a system of eight separate wards.
In 2010, Hispanics made the same argument when they sued the city of Compton: They claimed that an at-large voting system shut them out and kept the city council all black. ~ Jared Taylor
208:There is one are of work that should be mentioned here, referred to as 'automatic theorem proving'. One set of procedures that would come under this heading consists of fixing some formal system H, and trying to derive theorems within this system. We recall, from 2.9, that it would be an entirely computational matter to provide proofs of all the theorems of H one after the other. This kind of thing can be automated, but if done without further thought or insight, such an operation would be likely to be immensely inefficient. However, with the employment of such insight in the setting up of the computational procedures, some quite impressive results have been obtained. In one of these schemes (Chou 1988), the rules of Euclidean geometry have been translated into a very effective system for proving (and sometimes discovering) geometrical theorems. As an example of one of these, a geometrical proposition known as V. Thebault's conjecture, which had been proposed in 1938 (and only rather recently proved, by K.B. Taylor in 1983), was presented to the system and solved in 44 hours' computing time.

More closely analogous to the procedures discussed in the previous sections are attempts by various people over the past 10 years or so to provide 'artificial intelligence' procedures for mathematical 'understanding'. I hope it is clear from the arguments that I have given, that whatever these systems do achieve, what they do not do is obtain any actual mathematical understanding! Somewhat related to this are attempts to find automatic theorem-generating systems, where the system is set up to find theorems that are regarded as 'interesting'-according to certain criteria that the computational system is provided with. I do think that it would be generally accepted that nothing of very great actual mathematical interest has yet come out of these attempts. Of course, it would be argued that these are early days yet, and perhaps one may expect something much more exciting to come out of them in the future. However, it should be clear to anyone who has read this far, that I myself regard the entire enterprise as unlikely to lead to much that is genuinely positive, except to emphasize what such systems do not achieve. ~ Roger Penrose
209:One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a two-dimensional universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has no meaning to them. Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them) gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our planet heading straight off over the seas would eventually return from the other horizon. The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it.10 By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well. In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,” the physicist Max Born has declared.11 Yes, ~ Walter Isaacson
210:We feel that our actions are voluntary when they follow a decision, and involuntary when they happen without decision. But if decision itself were voluntary, every decision would have to be preceded by a decision to decide–an infinite regression which fortunately does not occur. Oddly enough, if we had to decide to decide, we would not be free to decide. We are free to decide because decision “happens.” We just decide without having the faintest understanding of how we do it. In fact, it is neither voluntary nor involuntary. To “get the feel” of this relativity is to find another extraordinary transformation of our experience as a whole, which may be described in either of two ways. I feel that I am deciding everything that happens, or, I feel that everything, including my decisions, is just happening spontaneously. For a decision–the freest of my actions-just happens like hiccups inside me or like a bird singing outside me. Such a way of seeing things is vividly described by a modern Zen master, the late Sokei-an Sasaki: One day I wiped out all the notions from my mind. I gave up all desire. I discarded all the words with which I thought and stayed in quietude. I felt a little queer–as if I were being carried into something, or as if I were touching some power unknown to me … and Ztt! I entered. I lost the boundary of my physical body. I had my skin, of course, but I felt I was standing in the center of the cosmos. I spoke, but my words had lost their meaning. I saw people coming towards me, but all were the same man. All were myself! I had never known this world. I had believed that I was created, but now I must change my opinion: I was never created; I was the cosmos; no individual Mr. Sasaki existed.7 It would seem, then, that to get rid of the subjective distinction between “me” and “my experience”–through seeing that my idea of myself is not myself–is to discover the actual relationship between myself and the “outside” world. The individual, on the one hand, and the world, on the other, are simply the abstract limits or terms of a concrete reality which is “between” them, as the concrete coin is “between” the abstract, Euclidean surfaces of its two sides. Similarly, the reality of all “inseparable opposites”–life and death, good and evil, pleasure and pain, gain and loss–is that “between” for which we have no words. ~ Alan W Watts
211:Hymne
Salut, divin triomphe! entre dans nos murailles;
Rends-nous ces guerriers illustrés
Par le sang de Désille et par les funérailles
De tant de Français massacrés.
Jamais rien de si grand n'embellit ton entrée;
Ni quand l'ombre de Mirabeau
S'achemina jadis vers la voûte sacrée
Où la gloire donne un tombeau;
Ni quand Voltaire mort et sa cendre bannie
Rentrèrent aux murs de Paris,
Vainqueurs du fanatisme et de la calomnie
Prosternés devant ses écrits.
Un seul jour peut atteindre à tant de renommée,
Et ce beau jour luira bientôt:
C'est quand tu conduiras Jourdan à notre armée,
Et Lafayette à l'échafaud.
Quelle rage à Coblentz! quel deuil pour tous ces princes,
Qui, partout diffamant nos lois,
Excitent contre nous et contre nos provinces
Et les esclaves et les rois!
Ils voulaient nous voir tous à la folie en proie.
Que leur front doit être abattu!
Tandis que parmi nous quel orgueil, quelle joie
Pour les amis de la vertu,
Pour vous tous, ô mortels, qui rougissez encore
Et qui savez baisser les yeux,
De voir des échevins que la Râpée honore
Asseoir sur un char radieux
Ces héros que jadis sur les bancs des galères
Assit un arrêt outrageant,
Et qui n'out égorgé que très peu de nos frères
Et volé que très peu d'argent!
Eh bien, que tardez-vous, harmonieux Orphées?
Si sur la tombe des Persans
Jadis Pindare, Eschyle, ont dressé des trophées,
Il faut de plus nobles accents.
Quarante meurtriers, chéris de Robespierre,
Vont s'élever sur nos autels.
Beaux-arts, qui faites vivre et la toile et la pierre,
58
Hâtez-vous, rendez immortels
Le grand Collot d'Herbois, ses clients helvétiques,
Ce front que donne à des héros
La vertu, la taverne et le secours des piques.
Peuplez le ciel d'astres nouveaux,
O vous, enfants d'Eudoxe et d'Hipparque et d'Euclide,
C'est par vous que les blonds cheveux
Qui tombèrent du front d'une reine timide
Sont tressés en célestes feux;
Par vous l'heureux vaisseau des premiers Argonautes
Flotte encor dans l'azur des airs.
Faites gémir Atlas sous de plus nobles hôtes,
Comme eux dominateurs des mers.
Que la nuit de leurs noms embellisse ses voiles,
Et que le nocher aux abois
Invoque en leur galère, ornement des étoiles,
Les Suisses de Collot d'Herbois.
~ Andre Marie de Chenier
212:Vale la pena soffermarci su quest’incubo [della fine della letteratura e delle arti], per come Borges ce lo racconta in una sua conversazione sui sogni e gli incubi.
Il terribile sogno è del poeta inglese William Wordsworth e si trova nel secondo [rectius: quinto] libro del poema The Prelude — un poema autobiografico, come dice il sottotitolo. Fu pubblicato nel 1850, l’anno stesso della morte del poeta. Allora non si pensava, come invece oggi, a un possibile cataclisma cosmico che annientasse ogni grande opera umana, se non l’umanità interamente.
Ma Wordsworth ne ebbe la preoccupazione e, in sogno, la visione.
Ed ecco come Borges l’assume e riassume nel suo discorso: “Nel sogno la sabbia lo circonda, un Sahara di sabbia nera. Non c’è acqua, non c’è mare. Sta al centro del deserto — nel deserto si sta sempre al centro — ed è ossessionato dal pensiero di come fare per sfuggire al deserto, quando vede qualcuno vicino a lui. Stranamente, è un arabo della tribù dei beduini, che cavalca un cammello e ha nella mano destra una lancia.
Sotto il braccio sinistro ha una pietra; nella mano una conchiglia. L’arabo gli dice che ha la missione di salvare le arti e le scienze e gli avvicina la conchiglia all’orecchio; la conchiglia è di straordinaria bellezza. Wordsworth ci dice che ascoltò la profezia (‘in una lingua che non conoscevo ma che capii’): una specie di ode appassionata, che profetizzava che la Terra era sul punto di essere distrutta dal diluvio che l’ira di Dio mandava. L’arabo gli dice che è vero, che il diluvio si avvicina, ma che egli ha una missione: salvare l’arte e le scienze. Gli mostra la pietra. La pietra, stranamente, è la Geometria di Euclide pur rimanendo una pietra. Poi gli avvicina la conchiglia, che è anche un libro: è quello che gli ha detto quelle cose terribili. La conchiglia è, anche, tutta la poesia del mondo, compreso, perche' no?, il poema di Wordsworth.
Il beduino gli dice: ‘Devo salvare queste due cose, la pietra e la conchiglia, entrambi libri’. Volge il viso all’indietro, e vi è un momento in cui Wordsworth vede che il volto del beduino cambia, si riempie di orrore. Anche lui si volge e vede una gran luce, una luce che ha inondato metà del deserto. Questa luce è quella dell’acqua del diluvio che sta per sommergere la Terra. Il beduino si allontana e Wordsworth vede che è anche don Chisciotte, che il cammello è anche Ronzinante e che allo stesso modo che la pietra è il libro e la conchiglia il libro, il beduino è don Chisciotte e nessuna delle due cose ed entrambe nello stesso tempo”...
l’immagine di don Chisciotte che si allontana invincibilmente richiama quella dipinta da Daumier, forse contemporaneamente. E ci è lecito, in aura borgesiana, chiederci se il poeta e il pittore non abbiano fatto lo stesso sogno. ~ Leonardo Sciascia
213:My task is to explain to you as quickly as possible my essence, that is, what sort of man I am, what I believe in, and what I hope for, is that right? And therefore I declare that I accept God pure and simple. But this, however, needs to be noted: if God exists and if he indeed created the earth, then, as we know perfectly well, he created it in accordance with Euclidean geometry, and he created human reason with a conception of only three dimensions of space. At the same time there were and are even now geometers and philosophers, even some of the most outstanding among them, who doubt that the whole universe, or, even more broadly, the whole of being, was created purely in accordance with Euclidean geometry; they even dare to dream that two parallel lines, which according to Euclid cannot possibly meet on earth, may perhaps meet somewhere in infinity. I, my dear, have come to the conclusion that if I cannot understand even that, then it is not for me to understand about God. I humbly confess that I do not have any ability to resolve such questions, I have a Euclidean mind, an earthly mind, and therefore it is not for us to resolve things that are not of this world. And I advise you never to think about it, Alyosha my friend, and most especially about whether God exists or not. All such questions are completely unsuitable to a mind created with a concept of only three dimensions. And so, I accept God, not only willingly, but moreover I also accept his wisdom and his purpose, which are completely unknown to us; I believe in order, in the meaning of life, I believe in eternal harmony, in which we are all supposed to merge, I believe in the Word for whom the universe is yearning, and who himself was 'with God,' who himself is God, and so on and so forth, to infinity. Many words have been invented on the subject. It seems I'm already on a good path, eh? And now imagine that in the final outcome I do not accept this world of God's, created by God, that I do not accept and cannot agree to accept. With one reservation: I have a childlike conviction that the sufferings will be healed and smoothed over, that the whole offensive comedy of human contradictions will disappear like a pitiful mirage, a vile concoction of man's Euclidean mind, feeble and puny as an atom, and that ultimately, at the world's finale, in the moment of eternal harmony, there will occur and be revealed something so precious that it will suffice for all hearts, to allay all indignation, to redeem all human villainy, all bloodshed; it will suffice not only to make forgiveness possible, but also to justify everything that has happened with men--let this, let all of this come true and be revealed, but I do not accept it and do not want to accept it! Let the parallel lines even meet before my own eyes: I shall look and say, yes, they meet, and still I will not accept it. ~ Fyodor Dostoyevsky
214:The Heathen Pass-Ee
Which I wish to remark,
And my language is plain,
That for plots that are dark
And not always in vain,
The heathen Pass-ee is peculiar,
And the same I would rise to explain.
I would also premise
That the term of Pass-ee
Most fitly applies,
As you probably see,
To one whose vocation is passing
The ‘ordinary B.A. degree’.
Tom crib was his name,
And I shall not deny
In regard to the same
What that name might imply,
But his face it was trustful and childlike,
And he had the most innocent eye.
Upon April the First,
The Little-Go fell,
And that was the worst
Of the gentleman’s sell,
For he fooled the examining Body
In a way I’m reluctant to tell.
The candidate came
And Tom Crib soon appeared;
It was Euclid,, The same
Was ‘the subject he feared’,
But he smiled as he sat by the table
With a smile that was wary and weird.
Yet he did what he could,
And the papers he showed
Were remarkably good,
And his countenance glowed
10
With pride when I met him soon after
As he walked down the Trumpington Road.
We did not find him out,
Which I bitterly grieve,
For I’ve not the least doubt
That he’d placed up his sleeve
Mr. Toodhunter’s excellent Euclid,
The same with intent to deceive
But I shall not forget
How the next day at two
As stiff paper was sett
By Examiner U……..
On Euripides’ tragedy, Bacchae.
A subject Tom ‘partially knew’.
But the knowledge displayed
By that heathen Pass-ee.
And the answers he made
Were quite frightful to see,
For he rapidly floored the whole paper
By about twenty minutes to three.
Then I looked up at U…..
And he gazed upon me.
I oberserved ‘This won’t do.’
He replies, ‘Goodness me!
We are fooled by this artful young person’,
And he sent for that heathen Pass-ee.
The scene that ensued
Was disgraceful to view,
For the floor it was strewed
With a tolerable few
Of the ‘tips’ that Tom Crib had been hiding
For the ‘subject he partially knew’
On the cuff of his shirt
He had managed to get
What we hoped had been dirt,
11
But which proved, I regret,
To be notes on the rise of the Drama,
A question invariably set.
In his various coats
We proceeded to seek,
Where we found sundry notes
And-with sorrow I speak—
One of Bohn’s publications, so useful
To the student of Latin or Greek.
In the crown of his cap
Were the Furies and Fates,
And a delicate map
Of the Dorian States
And we found in his palms which were hollow,
What are frequent in palms,-that is dates.
Which is why I remark,
And my language is plain,
That for plots that are dark
And not always in vain,
The heathen Pass-ee is peculiar,
Which the same I am free to maintain.
~ Arthur Clement Hilton
215:A More Ancient Mariner
The swarthy bee is a buccaneer,
A burly velveted rover,
Who loves the booming wind in his ear
As he sails the seas of clover.
A waif of the goblin pirate crew,
With not a soul to deplore him,
He steers for the open verge of blue
With the filmy world before him.
His flimsy sails abroad on the wind
Are shivered with fairy thunder;
On a line that sings to the light of his wings
He makes for the lands of wonder.
He harries the ports of Hollyhocks,
And levies on poor Sweetbriar;
He drinks the whitest wine of Phlox,
And the Rose is his desire.
He hangs in the Willows a night and a day;
He rifles the Buckwheat patches;
Then battens his store of pelf galore
Under the taughtest hatches.
He woos the Poppy and weds the Peach,
Inveigles Daffodilly,
And then like a tramp abandons each
For the gorgeous Canada Lily.
There's not a soul in the garden world
But wishes the day were shorter,
When Mariner B. puts out to sea
With the wind in the proper quarter.
Or, so they say! But I have my doubts;
For the flowers are only human,
And the valor and gold of a vagrant bold
Were always dear to woman.
11
He dares to boast, along the coast,
The beauty of Highland Heather,How he and she, with night on the sea,
Lay out on the hills together.
He pilfers every port of the wind,
From April to golden autumn;
But the theiving ways of his mortal days
Are those his mother taught him.
His morals are mixed, but his will is fixed;
He prospers after his kind,
And follows an instinct compass-sure,
The philosophers call blind.
And that is why, when he comes to die,
He'll have an earlier sentence
Than someone I know who thinks just so,
And then leaves room for repentance.
He never could box the compass round;
He doesn't know port from starboard;
But he knows the gates of the Sundown Straits,
Where the choicest goods are harbored.
He never could see the Rule of Three,
But he knows the rule of thumb
Better than Euclid's, better than yours,
Or the teachers' yet to come.
He knows the smell of the hydromel
As if two and two were five;
And hides it away for a year and a day
In his own hexagonal hive.
Out in the day, hap-hazard, alone,
Booms the old vagrant hummer,
With only his whim to pilot him
Throught the splendid vast of summer.
He steers and steers on the slant of the gale,
12
Like the fiend or Vanderdecken;
And there's never an unknown course to sail
But his crazy log can reckon.
He drones along with his rough sea-song
And the throat of a salty tar,
This devil-may-care, till he makes his lair
By the light of a yellow star.
He looks like a gentleman, lives like a lord,
And makes like a Trojan hero;
Then loafs all winter upon his hoard,
With the mercury at zero.
~ Bliss William Carman
216:Retroduction To American History
Cats walk the floor at midnight; that enemy of fog,
The moon, wraps the bedpost in receding stillness; sleep
Collects all weary nothings and lugs away the towers,
The pinnacles of dust that feed the subway.
What stiff unhappy silence waits on sleep
Struts like an officer; tongues next-door bewitch
Themselves with divination; I like a melancholy oaf
Beg the nightly pillow with impossible loves.
And abnegation folds hands, crossed like the knees
Of the complacent tailor, stitches cloaks of mercy
To the backs of obsessions.
Winter like spring no less
Tolerates the air; the wild pheasant meets innocently
The gun; night flouts illumination with meagre impudence.
In such serenity of equal fates, why has Narcissus
Urged the brook with questions? Merged with the element
Speculation suffuses the meadow with drops to tickle
The cow's gullet; grasshoppers drink the rain.
Antiquity breached mortality with myths.
Narcissus is vocabulary. Hermes decorates
A cornice on the Third National Bank. Vocabulary
Becomes confusion, decoration a blight; the Parthenon
In ..Tennessee stucco, art for the sake of death. Now
(The bedpost receding in stillness) you brush your teeth
'Hitting on all thirty-two;' scholarship pares
The nails of Catullus, sniffs his sheets, restores
His 'passionate underwear;' morality disciplines the other
Person; every son-of-a-bitch is Christ, at least Rousseau;
Prospero serves humanity in steam-heated universities, three
Thousand dollars a year. Simplicity, Flamineo, is obscene;
Sunlight topples indignant from the hill.
In every railroad station everywhere every lover
Waits for his train. He cannot hear. The smoke
Thickens. Ticket in hand, he pumps his body
Toward lower six, for one more terse ineffable trip,
His very eyeballs fixed in disarticulation. The berth
Is clean; no elephants, vultures, mice or spiders
59
Distract him from nonentity: his metaphors are dead.
More sanitation is enough, enough remains: dreams
Do not end lucidities beyond the stint of thought.
For intellect is a mansion where waste is without drain;
A corpse is your bedfellow, your great-grandfather dines
With you this evening on a cavalry horse. Intellect
Connives with heredity, creates fate as Euclid geometry
By definition:
The sunlit bones in your house
Are immortal in the titmouse,
They trip the feet of grandma
Like an afterthought each day.
These unseen sunlit bones,
They may be in the cat
That startles them in grandma
But look at this or that
They meet you every way.
For Pelops' and Tantalus' successions were at once simpler,
If perplexed, and less subtle than you think. Heredity
Proposes love, love exacts language, and we lack
Language. When shall we speak again? When shall
The sparrow dusting the gutter sing? When shall
This drift with silence meet the sun? When shall I wake?
~ Allen Tate
217:It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other.

This, however, is a completely misleading interpretation of what Godel has demonstrated for us. He has taught us that the very notion of a formal axiomatic system is inadequate for capturing even the most basic of mathematical concepts. When we use the term 'arithmetic' without further qualification, we indeed mean the ordinary arithmetic which operates with the ordinary natural numbers 0,1,2,3,4,...(and perhaps their negatives) and not with some kind of 'supernatural' numbers. We may choose, if we wish, to explore the properties of formal systems, and this is certainly a valuable part of mathematical endeavour. But it is something different from exploring the ordinary properties of the ordinary natural numbers. The situation is, in some ways, perhaps not so very unlike that which occurs with geometry. The study of non-Euclidean geometries is something mathematically interesting, with important applications (such as in physics, see ENM Chapter 5 especially Figs 5.1 and 5.2, and also 4.4), but when the term 'geometry' is used in ordinary language (as distinct from when a mathematician or theoretical physicist might use that term), we do indeed mean the ordinary geometry of Euclid. There is a difference, however, in that what a logician might refer to as 'Euclidean geometry' can indeed be specified (with some reservations) in terms of a particular formal system, whereas, as Godel has shown, ordinary 'arithmetic' cannot be so specified.

Rather than showing that mathematics (most particularly arithmetic) is an arbitrary pursuit, whose direction is governed by the whim of Man, Godel demonstrated that it is something absolute, there to be discovered rather than invented (cf. 1.17). We discover for ourselves what the natural numbers are, and we do not have trouble in distinguishing them from any sort of supernatural numbers. Godel showed that no system of 'man-made' rules can, by themselves, achieve this for us. Such a Platonic viewpoint was important to Godel, and it will be important also for us in the later considerations of this book (8.7). ~ Roger Penrose
218:Reading list (1972 edition)[edit]
1. Homer – Iliad, Odyssey
2. The Old Testament
3. Aeschylus – Tragedies
4. Sophocles – Tragedies
5. Herodotus – Histories
6. Euripides – Tragedies
7. Thucydides – History of the Peloponnesian War
8. Hippocrates – Medical Writings
9. Aristophanes – Comedies
10. Plato – Dialogues
11. Aristotle – Works
12. Epicurus – Letter to Herodotus; Letter to Menoecus
13. Euclid – Elements
14. Archimedes – Works
15. Apollonius of Perga – Conic Sections
16. Cicero – Works
17. Lucretius – On the Nature of Things
18. Virgil – Works
19. Horace – Works
20. Livy – History of Rome
21. Ovid – Works
22. Plutarch – Parallel Lives; Moralia
23. Tacitus – Histories; Annals; Agricola Germania
24. Nicomachus of Gerasa – Introduction to Arithmetic
25. Epictetus – Discourses; Encheiridion
26. Ptolemy – Almagest
27. Lucian – Works
28. Marcus Aurelius – Meditations
29. Galen – On the Natural Faculties
30. The New Testament
31. Plotinus – The Enneads
32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine
33. The Song of Roland
34. The Nibelungenlied
35. The Saga of Burnt Njál
36. St. Thomas Aquinas – Summa Theologica
37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy
38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales
39. Leonardo da Vinci – Notebooks
40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy
41. Desiderius Erasmus – The Praise of Folly
42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres
43. Thomas More – Utopia
44. Martin Luther – Table Talk; Three Treatises
45. François Rabelais – Gargantua and Pantagruel
46. John Calvin – Institutes of the Christian Religion
47. Michel de Montaigne – Essays
48. William Gilbert – On the Loadstone and Magnetic Bodies
49. Miguel de Cervantes – Don Quixote
50. Edmund Spenser – Prothalamion; The Faerie Queene
51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis
52. William Shakespeare – Poetry and Plays
53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences
54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World
55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals
56. Thomas Hobbes – Leviathan
57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy
58. John Milton – Works
59. Molière – Comedies
60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises
61. Christiaan Huygens – Treatise on Light
62. Benedict de Spinoza – Ethics
63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education
64. Jean Baptiste Racine – Tragedies
65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics
66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology
67. Daniel Defoe – Robinson Crusoe
68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal
69. William Congreve – The Way of the World
70. George Berkeley – Principles of Human Knowledge
71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man
72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws
73. Voltaire – Letters on the English; Candide; Philosophical Dictionary
74. Henry Fielding – Joseph Andrews; Tom Jones
75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets ~ Mortimer J Adler
219:Reading list (1972 edition)[edit]
1. Homer - Iliad, Odyssey
2. The Old Testament
3. Aeschylus - Tragedies
4. Sophocles - Tragedies
5. Herodotus - Histories
6. Euripides - Tragedies
7. Thucydides - History of the Peloponnesian War
8. Hippocrates - Medical Writings
9. Aristophanes - Comedies
10. Plato - Dialogues
11. Aristotle - Works
12. Epicurus - Letter to Herodotus; Letter to Menoecus
13. Euclid - Elements
14.Archimedes - Works
15. Apollonius of Perga - Conic Sections
16. Cicero - Works
17. Lucretius - On the Nature of Things
18. Virgil - Works
19. Horace - Works
20. Livy - History of Rome
21. Ovid - Works
22. Plutarch - Parallel Lives; Moralia
23. Tacitus - Histories; Annals; Agricola Germania
24. Nicomachus of Gerasa - Introduction to Arithmetic
25. Epictetus - Discourses; Encheiridion
26. Ptolemy - Almagest
27. Lucian - Works
28. Marcus Aurelius - Meditations
29. Galen - On the Natural Faculties
30. The New Testament
31. Plotinus - The Enneads
32. St. Augustine - On the Teacher; Confessions; City of God; On Christian Doctrine
33. The Song of Roland
34. The Nibelungenlied
35. The Saga of Burnt Njal
36. St. Thomas Aquinas - Summa Theologica
37. Dante Alighieri - The Divine Comedy;The New Life; On Monarchy
38. Geoffrey Chaucer - Troilus and Criseyde; The Canterbury Tales
39. Leonardo da Vinci - Notebooks
40. Niccolò Machiavelli - The Prince; Discourses on the First Ten Books of Livy
41. Desiderius Erasmus - The Praise of Folly
42. Nicolaus Copernicus - On the Revolutions of the Heavenly Spheres
43. Thomas More - Utopia
44. Martin Luther - Table Talk; Three Treatises
45. François Rabelais - Gargantua and Pantagruel
46. John Calvin - Institutes of the Christian Religion
47. Michel de Montaigne - Essays
48. William Gilbert - On the Loadstone and Magnetic Bodies
49. Miguel de Cervantes - Don Quixote
50. Edmund Spenser - Prothalamion; The Faerie Queene
51. Francis Bacon - Essays; Advancement of Learning; Novum Organum, New Atlantis
52. William Shakespeare - Poetry and Plays
53. Galileo Galilei - Starry Messenger; Dialogues Concerning Two New Sciences
54. Johannes Kepler - Epitome of Copernican Astronomy; Concerning the Harmonies of the World
55. William Harvey - On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals
56. Thomas Hobbes - Leviathan
57. René Descartes - Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy
58. John Milton - Works
59. Molière - Comedies
60. Blaise Pascal - The Provincial Letters; Pensees; Scientific Treatises
61. Christiaan Huygens - Treatise on Light
62. Benedict de Spinoza - Ethics
63. John Locke - Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education
64. Jean Baptiste Racine - Tragedies
65. Isaac Newton - Mathematical Principles of Natural Philosophy; Optics
66. Gottfried Wilhelm Leibniz - Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology
67.Daniel Defoe - Robinson Crusoe
68. Jonathan Swift - A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal
69. William Congreve - The Way of the World
70. George Berkeley - Principles of Human Knowledge
71. Alexander Pope - Essay on Criticism; Rape of the Lock; Essay on Man
72. Charles de Secondat, baron de Montesquieu - Persian Letters; Spirit of Laws
73. Voltaire - Letters on the English; Candide; Philosophical Dictionary
74. Henry Fielding - Joseph Andrews; Tom Jones
75. Samuel Johnson - The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets
   ~ Mortimer J Adler,
220:Is it possible that the Pentateuch could not have been written by uninspired men? that the assistance of God was necessary to produce these books? Is it possible that Galilei ascertained the mechanical principles of 'Virtual Velocity,' the laws of falling bodies and of all motion; that Copernicus ascertained the true position of the earth and accounted for all celestial phenomena; that Kepler discovered his three laws—discoveries of such importance that the 8th of May, 1618, may be called the birth-day of modern science; that Newton gave to the world the Method of Fluxions, the Theory of Universal Gravitation, and the Decomposition of Light; that Euclid, Cavalieri, Descartes, and Leibniz, almost completed the science of mathematics; that all the discoveries in optics, hydrostatics, pneumatics and chemistry, the experiments, discoveries, and inventions of Galvani, Volta, Franklin and Morse, of Trevithick, Watt and Fulton and of all the pioneers of progress—that all this was accomplished by uninspired men, while the writer of the Pentateuch was directed and inspired by an infinite God? Is it possible that the codes of China, India, Egypt, Greece and Rome were made by man, and that the laws recorded in the Pentateuch were alone given by God? Is it possible that Æschylus and Shakespeare, Burns, and Beranger, Goethe and Schiller, and all the poets of the world, and all their wondrous tragedies and songs are but the work of men, while no intelligence except the infinite God could be the author of the Pentateuch? Is it possible that of all the books that crowd the libraries of the world, the books of science, fiction, history and song, that all save only one, have been produced by man? Is it possible that of all these, the bible only is the work of God? ~ Robert G Ingersoll
221:The Cock And The Bull
You see this pebble-stone? It’s a thing I bought
Of a bit of a chit of a boy i’ the mid o’ the day —
I like to dock the smaller parts-o’-speech,
As we curtail the already cur-tail’d cur
(You catch the paronomasia, play ’po’ words?),
Did, rather, i’ the pre-Landseerian days.
Well, to my muttons. I purchased the concern,
And clapt it i’ my poke, having given for same
By way o’ chop, swop, barter or exchange —
‘Chop’ was my snickering dandiprat’s own term —
One shilling and fourpence, current coin o’ the realm.
O-n-e one and f-o-u-r four
Pence, one and fourpence — you are with me, sir? —
What hour it skills not: ten or eleven o’ the clock,
One day (and what a roaring day it was
Go shop or sight-see — bar a spit o’ rain!)
In February, eighteen sixty nine,
Alexandrina Victoria, Fidei
Hm — hm — how runs the jargon? being on throne.
Such, sir, are all the facts, succinctly put,
The basis or substratum — what you will —
Of the impending eighty thousand lines.
‘Not much in ’em either,’ quoth perhaps simple Hodge.
But there’s a superstructure. Wait a bit.
Mark first the rationale of the thing:
Hear logic rivel and levigate the deed.
That shilling — and for matter o’ that, the pence —
I had o’ course upo’ me — wi’ me say —
(Mecum’s the Latin, make a note o’ that)
When I popp’d pen i’ stand, scratch’d ear, wip’d snout,
(Let everybody wipe his own himself)
Sniff’d — tch! — at snuffbox; tumbled up, he-heed,
Haw-haw’d (not hee-haw’d, that’s another guess thing
Then fumbled at, and stumbled out of, door,
I shoved the timber ope wi’ my omoplat;
And in vestibulo, i’ the lobby to-wit,
(Iacobi Facciolati’s rendering, sir,)
Donn’d galligaskins, antigropeloes,
57
And so forth; and, complete with hat and gloves,
One on and one a-dangle i’ my hand,
And ombrifuge (Lord love you!), case o’ rain,
I flopp’d forth, ’sbuddikins! on my own ten toes,
(I do assure you there be ten of them,)
And went clump-clumping up hill and down dale
To find myself o’ the sudden i’ front o’ the boy.
Put case I hadn’t ’em on me, could I ha’ bought
This sort-o’-kind-o’-what-you-might-call toy,
This pebble-thing, o’ the boy-thing? Q.E.D.
That’s proven without aid from mumping Pope,
Sleek porporate or bloated Cardinal.
(Isn’t it, old Fatchaps? You’re in Euclid now.)
So, having the shilling — having i’ fact a lot —
And pence and halfpence, ever so many o’ them,
I purchased, as I think I said before,
The pebble (lapis, lapidis, -di, -dem, -de —
What nouns ’crease short i’ the genitive, Fatchaps, eh?)
O’ the boy, a bare-legg’d beggarly son of a gun,
For one-and-fourpence. Here we are again.
Now Law steps in, bigwigg’d, voluminous-jaw’d;
Investigates and re-investigates.
Was the transaction illegal? Law shakes head.
Perpend, sir, all the bearings of the case.
At first the coin was mine, the chattel his.
But now (by virtue of the said exchange
And barter) vice versa all the coin,
Per juris operationem, vests
I’ the boy and his assigns till ding o’ doom;
(In sæcula sæculo-o-o-orum;
I think I hear the Abate mouth out that.)
To have and hold the same to him and them… .
Confer some idiot on Conveyancing.
Whereas the pebble and every part thereof,
And all that appertaineth thereunto,
Quodcunque pertinet ad eam rem,
(I fancy, sir, my Latin’s rather pat)
Or shall, will, may, might, can, could, would or should,
(Subaudi cætera — clap we to the close —
For what’s the good of law in a case o’ the kind)
58
Is mine to all intents and purposes.
This settled, I resume the thread o’ the tale.
Now for a touch o’ the vendor’s quality.
He says a gen’lman bought a pebble of him,
(This pebble i’ sooth, sir, which I hold i’ my hand) —
And paid for ’t, like a gen’lman, on the nail.
‘Did I o’ercharge him a ha’penny? Devil a bit.
Fiddlepin’s end! Get out, you blazing ass!
Gabble o’ the goose. Don’t bugaboo-baby me!
Go double or quits? Yah! tittup! what’s the odds?’
— There’s the transaction view’d i’ the vendor’s light.
Next ask that dumpled hag, stood snuffling by,
With her three frowsy blowsy brats o’ babes,
The scum o’ the kennel, cream o’ the filth-heap — Faugh!
Aie, aie, aie, aie! ?t?t?t?t?t??,
(’Stead which we blurt out Hoighty toighty now) —
And the baker and candlestickmaker, and Jack and Gill,
Blear’d Goody this and queasy Gaffer that.
Ask the schoolmaster. Take schoolmaster first.
He saw a gentleman purchase of a lad
A stone, and pay for it rite, on the square,
And carry it off per saltum, jauntily,
Propria quæ maribus, gentleman’s property now
(Agreeably to the law explain’d above),
In proprium usum, for his private ends.
The boy he chuck’d a brown i’ the air, and bit
I’ the face the shilling: heaved a thumping stone
At a lean hen that ran cluck clucking by,
(And hit her, dead as nail i’ post o’ door,)
Then abiit — what’s the Ciceronian phrase? —
Excessit, evasit, erupit — off slogs boy;
Off like bird, avi similis — (you observed
The dative? Pretty i’ the Mantuan!) — Anglice,
Off in three flea skips. Hactenus, so far,
So good, tam bene. Bene, satis, male — ,
Where was I with my trope ’bout one in a quag?
I did once hitch the syntax into verse:
Verbum personale, a verb personal,
Concordat — ay, ‘agrees,’ old Fatchaps — cum
59
Nominativo, with its nominative,
Genere, i’ point o’ gender, numero,
O’ number, et persona, and person. Ut,
Instance: Sol ruit, down flops sun, et and,
Montes umbrantur, out flounce mountains. Pah!
Excuse me, sir, I think I’m going mad.
You see the trick on ’t though, and can yourself
Continue the discourse ad libitum.
It takes up about eighty thousand lines,
A thing imagination boggles at;
And might, odds-bobs, sir! in judicious hands,
Extend from here to Mesopotamy.
~ Charles Stuart Calverley
222:The Dunciad: Book Iv
Yet, yet a moment, one dim ray of light
Indulge, dread Chaos, and eternal Night!
Of darkness visible so much be lent,
As half to show, half veil, the deep intent.
Ye pow'rs! whose mysteries restor'd I sing,
To whom time bears me on his rapid wing,
Suspend a while your force inertly strong,
Then take at once the poet and the song.
Now flam'd the Dog Star's unpropitious ray,
Smote ev'ry brain, and wither'd every bay;
Sick was the sun, the owl forsook his bow'r.
The moon-struck prophet felt the madding hour:
Then rose the seed of Chaos, and of Night,
To blot out order, and extinguish light,
Of dull and venal a new world to mould,
And bring Saturnian days of lead and gold.
She mounts the throne: her head a cloud conceal'd,
In broad effulgence all below reveal'd;
('Tis thus aspiring Dulness ever shines)
Soft on her lap her laureate son reclines.
Beneath her footstool, Science groans in chains,
And Wit dreads exile, penalties, and pains.
There foam'd rebellious Logic , gagg'd and bound,
There, stripp'd, fair Rhet'ric languish'd on the ground;
His blunted arms by Sophistry are borne,
And shameless Billingsgate her robes adorn.
Morality , by her false guardians drawn,
Chicane in furs, and Casuistry in lawn,
Gasps, as they straighten at each end the cord,
And dies, when Dulness gives her page the word.
Mad Mathesis alone was unconfin'd,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now running round the circle finds it square.
But held in tenfold bonds the Muses lie,
Watch'd both by Envy's and by Flatt'ry's eye:
191
There to her heart sad Tragedy addres'd
The dagger wont to pierce the tyrant's breast;
But sober History restrain'd her rage,
And promised vengeance on a barb'rous age.
There sunk Thalia, nerveless, cold, and dead,
Had not her sister Satire held her head:
Nor couldst thou, Chesterfield! a tear refuse,
Thou weptst, and with thee wept each gentle Muse.
When lo! a harlot form soft sliding by,
With mincing step, small voice, and languid eye;
Foreign her air, her robe's discordant pride
In patchwork flutt'ring, and her head aside:
By singing peers upheld on either hand,
She tripp'd and laugh'd, too pretty much to stand;
Cast on the prostrate Nine a scornful look,
Then thus in quaint recitativo spoke.
'O
Cara! Cara!
silence all that train:
Joy to great Chaos! let Division reign:
Chromatic tortures soon shall drive them hence,
Break all their nerves, and fritter all their sense:
One trill shall harmonize joy, grief, and rage,
Wake the dull Church, and lull the ranting Stage;
To the same notes thy sons shall hum, or snore,
And all thy yawning daughters cry,
encore
Another Phoebus, thy own Phoebus, reigns,
Joys in my jigs, and dances in my chains.
But soon, ah soon, Rebellion will commence,
If Music meanly borrows aid from Sense.
Strong in new arms, lo! Giant Handel stands,
Like bold Briarerus, with a hundred hands;
To stir, to rouse, to shake the soul he comes,
And Jove's own thunders follow Mars's drums.
Arrest him, Empress, or you sleep no more-'
She heard, and drove him to th' Hibernian shore.
And now had Fame's posterior trumpet blown,
192
And all the nations summoned to the throne.
The young, the old, who feel her inward sway,
One instinct seizes, and transports away.
None need a guide, by sure attraction led,
And strong impulsive gravity of head:
None want a place, for all their centre found
Hung to the Goddess, and coher'd around.
Not closer, orb in orb, conglob'd are seen
The buzzing bees about their dusky Queen.
The gath'ring number, as it moves along,
Involves a vast involuntary throng,
Who gently drawn, and struggling less and less,
Roll in her Vortex, and her pow'r confess.
Not those alone who passive own her laws,
But who, weak rebels, more advance her cause.
Whate'er of dunce in college or in town
Sneers at another, in toupee or gown;
Whate'er of mongrel no one class admits,
A wit with dunces, and a dunce with wits.
Nor absent they, no members of her state,
Who pay her homage in her sons, the Great;
Who false to Phoebus bow the knee to Baal;
Or, impious, preach his Word without a call.
Patrons, who sneak from living worth to dead,
Withhold the pension, and set up the head;
Or vest dull Flattery in the sacred gown;
Or give from fool to fool the laurel crown.
And (last and worst) with all the cant of wit,
Without the soul, the Muse's hypocrite.
There march'd the bard and blockhead, side by side,
Who rhym'd for hire, and patroniz'd for pride.
Narcissus, prais'd with all a Parson's pow'r,
Look'd a white lily sunk beneath a show'r.
There mov'd Montalto with superior air;
His stretch'd-out arm display'd a volume fair;
Courtiers and Patriots in two ranks divide,
Through both he pass'd, and bow'd from side to side:
But as in graceful act, with awful eye
Compos'd he stood, bold Benson thrust him by:
193
On two unequal crutches propp'd he came,
Milton's on this, on that one Johnston's name.
The decent knight retir'd with sober rage,
Withdrew his hand, and closed the pompous page.
But (happy for him as the times went then)
Appear'd Apollo's mayor and aldermen,
On whom three hundred gold-capp'd youths await,
To lug the pond'rous volume off in state.
When Dulness, smiling-'Thus revive the Wits!
But murder first, and mince them all to bits;
As erst Medea (cruel, so to save!)
A new edition of old Aeson gave;
Let standard authors, thus, like trophies born,
Appear more glorious as more hack'd and torn,
And you, my Critics! in the chequer'd shade,
Admire new light through holes yourselves have made.
Leave not a foot of verse, a foot of stone,
A page, a grave, that they can call their own;
But spread, my sons, your glory thin or thick,
On passive paper, or on solid brick.
So by each bard an Alderman shall sit,
A heavy lord shall hang at ev'ry wit,
And while on Fame's triumphal Car they ride,
Some Slave of mine be pinion'd to their side.'
Now crowds on crowds around the Goddess press,
Each eager to present their first address.
Dunce scorning dunce beholds the next advance,
But fop shows fop superior complaisance,
When lo! a spector rose, whose index hand
Held forth the virtue of the dreadful wand;
His beaver'd brow a birchen garland wears,
Dropping with infant's blood, and mother's tears.
O'er every vein a shud'ring horror runs;
Eton and Winton shake through all their sons.
All flesh is humbl'd, Westminster's bold race
Shrink, and confess the Genius of the place:
The pale boy senator yet tingling stands,
And holds his breeches close with both his hands.
194
Then thus. 'Since man from beast by words is known,
Words are man's province, words we teach alone.
When reason doubtful, like the Samian letter,
Points him two ways, the narrower is the better.
Plac'd at the door of learning, youth to guide,
We never suffer it to stand too wide.
To ask, to guess, to know, as they commence,
As fancy opens the quick springs of sense,
We ply the memory, we load the brain,
Bind rebel Wit, and double chain on chain,
Confine the thought, to exercise the breath;
And keep them in the pale of words till death.
Whate'er the talents, or howe'er design'd,
We hang one jingling padlock on the mind:
A Poet the first day, he dips his quill;
And what the last? A very Poet still.
Pity! the charm works only in our wall,
Lost, lost too soon in yonder house or hall.
There truant Wyndham every Muse gave o'er,
There Talbot sunk, and was a wit no more!
How sweet an Ovid, Murray was our boast!
How many Martials were in Pult'ney lost!
Else sure some bard, to our eternal praise,
In twice ten thousand rhyming nights and days,
Had reach'd the work, and All that mortal can;
And South beheld that Masterpiece of Man.'
'Oh' (cried the Goddess) 'for some pedant Reign!
Some gentle James, to bless the land again;
To stick the Doctor's chair into the throne,
Give law to words, or war with words alone,
Senates and courts with Greek and Latin rule,
And turn the council to a grammar school!
For sure, if Dulness sees a grateful day,
'Tis in the shade of arbitrary sway.
O! if my sons may learn one earthly thing,
Teach but that one, sufficient for a king;
That which my priests, and mine alone, maintain,
Which as it dies, or lives, we fall, or reign:
May you, may Cam and Isis, preach it long!
'The Right Divine of Kings to govern wrong'.'
195
Prompt at the call, around the Goddess roll
Broad hats, and hoods, and caps, a sable shoal:
Thick and more thick the black blockade extends,
A hundred head of Aristotle's friends.
Nor wert thou, Isis! wanting to the day,
Though Christ Church long kept prudishly away.
Each staunch polemic, stubborn as a rock,
Each fierce logician, still expelling Locke,
Came whip and spur, and dash'd through thin and thick
On German Crousaz, and Dutch Burgersdyck.
As many quit the streams that murm'ring fall
To lull the sons of Marg'ret and Clare Hall,
Where Bentley late tempestuous wont to sport
In troubled waters, but now sleeps in Port.
Before them march'd that awful Aristarch;
Plow'd was his front with many a deep remark:
His hat, which never vail'd to human pride,
Walker with rev'rence took, and laid aside.
Low bowed the rest: He, kingly, did but nod;
So upright Quakers please both man and God.
'Mistress! dismiss that rabble from your throne:
Avaunt-is Aristarchus yet unknown?
Thy mighty scholiast, whose unwearied pains
Made Horace dull, and humbl'd Milton's strains.
Turn what they will to verse, their toil is vain,
Critics like me shall make it prose again.
Roman and Greek grammarians! know your better:
Author of something yet more great than letter;
While tow'ring o'er your alphabet, like Saul,
Stands our Digamma, and o'ertops them all.
'Tis true, on words is still our whole debate,
Disputes of
Me
or
Te
, of
aut
or
at
To sound or sink in
196
cano
, O or A,
Or give up Cicero to C or K.
Let Freind affect to speak as Terence spoke,
And Alsop never but like Horace joke:
For me, what Virgil, Pliny may deny,
Manilius or Solinus shall supply:
For Attic Phrase in Plato let them seek,
I poach in Suidas for unlicens'd Greek.
In ancient sense if any needs will deal,
Be sure I give them fragments, not a meal;
What Gellius or Stobaeus hash'd before,
Or chew'd by blind old Scholiasts o'er and o'er.
The critic eye, that microscope of wit,
Sees hairs and pores, examines bit by bit:
How parts relate to parts, or they to whole,
The body's harmony, the beaming soul,
Are things which Kuster, Burman, Wasse shall see,
When man's whole frame is obvious to a
Flea
'Ah, think not, Mistress! more true dulness lies
In Folly's cap, than Wisdom's grave disguise.
Like buoys, that never sink into the flood,
On learning's surface we but lie and nod.
Thine is the genuine head of many a house,
And much Divinity without a Nous.
Nor could a Barrow work on every block,
Nor has one Atterbury spoil'd the flock.
See! still thy own, the heavy canon roll,
And metaphysic smokes involve the pole.
For thee we dim the eyes, and stuff the head
With all such reading as was never read:
For thee explain a thing till all men doubt it,
And write about it, Goddess, and about it:
So spins the silkworm small its slender store,
And labours till it clouds itself all o'er.
'What tho' we let some better sort of fool
Thrid ev'ry science, run through ev'ry school?
Never by tumbler through the hoops was shown
197
Such skill in passing all, and touching none.
He may indeed (if sober all this time)
Plague with dispute, or persecute with rhyme.
We only furnish what he cannot use,
Or wed to what he must divorce, a Muse:
Full in the midst of Euclid dip at once,
And petrify a Genius to a Dunce:
Or set on metaphysic ground to prance,
Show all his paces, not a step advance.
With the same cement ever sure to bind,
We bring to one dead level ev'ry mind.
Then take him to develop, if you can,
And hew the block off, and get out the man.
But wherefore waste I words? I see advance
Whore, pupil, and lac'd governor from France.
Walker! our hat' -nor more he deign'd to say,
But, stern as Ajax' spectre, strode away.
In flow'd at once a gay embroider'd race,
And titt'ring push'd the Pedants off the place;
Some would have spoken, but the voice was drown'd
By the French horn, or by the op'ning hound.
The first came forwards, with as easy mien,
As if he saw St. James's and the Queen.
When thus th' attendant Orator begun,
Receive, great Empress! thy accomplish'd Son:
Thine from the birth, and sacred from the rod,
A dauntless infant! never scar'd with God.
The Sire saw, one by one, his Virtues wake:
The Mother begg'd the blessing of a Rake.
Thou gav'st that Ripeness, which so soon began,
And ceas'd so soon, he ne'er was Boy, nor Man,
Thro' School and College, thy kind cloud o'ercast,
Safe and unseen the young AEneas past:
Thence bursting glorious, all at once let down,
Stunn'd with his giddy Larum half the town.
Intrepid then, o'er seas and lands he flew:
Europe he saw, and Europe saw him too.
There all thy gifts and graces we display,
Thou, only thou, directing all our way!
To where the Seine, obsequious as she runs,
Pours at great Bourbon's feet her silken sons;
198
Or Tyber, now no longer Roman, rolls,
Vain of Italian Arts, Italian Souls:
To happy Convents, bosom'd deep in vines,
Where slumber Abbots, purple as their wines:
To Isles of fragrance, lilly-silver'd vales,
Diffusing languor in the panting gales:
To lands of singing, or of dancing slaves,
Love-whisp'ring woods, and lute-resounding waves.
But chief her shrine where naked Venus keeps,
And Cupids ride the Lyon of the Deeps;
Where, eas'd of Fleets, the Adriatic main
Wafts the smooth Eunuch and enamour'd swain.
Led by my hand, he saunter'd Europe round,
And gather'd ev'ry Vice on Christian ground;
Saw ev'ry Court, hear'd ev'ry King declare
His royal Sense, of Op'ra's or the Fair;
The Stews and Palace equally explor'd,
Intrigu'd with glory, and with spirit whor'd;
Try'd all hors-d' uvres, all Liqueurs defin'd,
Judicious drank, and greatly-daring din'd;
Dropt the dull lumber of the Latin store,
Spoil'd his own Language, and acquir'd no more;
All Classic learning lost on Classic ground;
And last turn'd Air, the Eccho of a Sound!
See now, half-cur'd, and perfectly well-bred,
With nothing but a Solo in his head;
As much Estate, and Principle, and Wit,
As Jansen, Fleetwood, Cibber shall think fit;
Stol'n from a Duel, follow'd by a Nun,
And, if a Borough chuse him, not undone;
See, to my country happy I restore
This glorious Youth, and add one Venus more.
Her too receive (for her my soul adores)
So may the sons of sons of sons of whores,
Prop thine, O Empress! like each neighbour Throne,
And make a long Posterity thy own.
Pleas'd, she accepts the Hero, and the Dame,
Wraps in her Veil, and frees from sense of Shame.
Then look'd, and saw a lazy, lolling sort,
Unseen at Church, at Senate, or at Court,
Of ever-listless Loit'rers, that attend
No Cause, no Trust, no Duty, and no Friend.
199
Thee too, my Paridel! she mark'd thee there,
Stretch'd on the rack of a too easy chair,
And heard thy everlasting yawn confess
The Pains and Penalties of Idleness.
She pity'd! but her Pity only shed
Benigner influence on thy nodding head.
But Annius, crafty Seer, with ebon wand,
And well-dissembl'd Em'rald on his hand,
False as his Gems and canker'd as his Coins,
Came, cramm'd with Capon, from where Pollio dines.
Soft, as the wily Fox is seen to creep,
Where bask on sunny banks the simple sheep,
Walk round and round, now prying here, now there;
So he; but pious, whisper'd first his pray'r.
Grant, gracious Goddess! grant me still to cheat,
O may thy cloud still cover the deceit!
Thy choicer mists on this assembly shed,
But pour them thickest on the noble head.
So shall each youth, assisted by our eyes,
See other C‘sars, other Homers rise;
Thro' twilight ages hunt th'Athenian fowl,
Which Chalcis Gods, and mortals call an Owl,
Now see an Attys, now a Cecrops clear,
Nay, Mahomet! the Pigeon at thine ear;
Be rich in ancient brass, tho' not in gold,
And keep his Lares, tho' his house be sold;
To headless Ph be his fair bride postpone,
Honour a Syrian Prince above his own;
Lord of an Otho, if I vouch it true;
Blest in one Niger, till he knows of two.
Mummius o'erheard him; Mummius, Fool-renown'd,
Who like his Cheops stinks above the ground,
Fierce as a startled Adder, swell'd, and said,
Rattling an ancient Sistrum at his head.
Speak'st thou of Syrian Princes? Traitor base!
Mine, Goddess! mine is all the horned race.
True, he had wit, to make their value rise;
From foolish Greeks to steal them, was as wise;
More glorious yet, from barb'rous hands to keep,
When Sallee Rovers chac'd him on the deep.
Then taught by Hermes, and divinely bold,
Down his own throat he risqu'd the Grecian gold;
200
Receiv'd each Demi-God, with pious care,
Deep in his Entrails — I rever'd them there,
I bought them, shrouded in that living shrine,
And, at their second birth, they issue mine.
Witness great Ammon! by whose horns I swore,
(Reply'd soft Annius) this our paunch before
Still bears them, faithful; and that thus I eat,
Is to refund the Medals with the meat.
To prove me, Goddess! clear of all design,
Bid me with Pollio sup, as well as dine:
There all the Learn'd shall at the labour stand,
And Douglas lend his soft, obstetric hand.
The Goddess smiling seem'd to give consent;
So back to Pollio, hand in hand, they went.
Then thick as Locusts black'ning all the ground,
A tribe, with weeds and shells fantastic crown'd,
Each with some wond'rous gift approach'd the Pow'r,
A Nest, a Toad, a Fungus, or a Flow'r.
But far the foremost, two, with earnest zeal,
And aspect ardent to the Throne appeal.
The first thus open'd: Hear thy suppliant's call,
Great Queen, and common Mother of us all!
Fair from its humble bed I rear'd this Flow'r,
Suckled, and chear'd, with air, and sun, and show'r,
Soft on the paper ruff its leaves I spread,
Bright with the gilded button tipt its head,
Then thron'd in glass, and nam'd it Caroline:
Each Maid cry'd, charming! and each Youth, divine!
Did Nature's pencil ever blend such rays,
Such vary'd light in one promiscuous blaze?
Now prostrate! dead! behold that Caroline:
No Maid cries, charming! and no Youth, divine!
And lo the wretch! whose vile, whose insect lust
Lay'd this gay daughter of the Spring in dust.
Oh punish him, or to th' Elysian shades
Dismiss my soul, where no Carnation fades.
He ceas'd, and wept. With innocence of mien,
Th'Accus'd stood forth, and thus address'd the Queen.
Of all th'enamel'd race, whose silv'ry wing
Waves to the tepid Zephyrs of the spring,
Or swims along the fluid atmosphere,
Once brightest shin'd this child of Heat and Air.
201
I saw, and started from its vernal bow'r
The rising game, and chac'd from flow'r to flow'r.
It fled, I follow'd; now in hope, now pain;
It stopt, I stopt; it mov'd, I mov'd again.
At last it fix'd, 'twas on what plant it pleas'd,
And where it fix'd, the beauteous bird I seiz'd:
Rose or Carnation was below my care;
I meddle, Goddess! only in my sphere.
I tell the naked fact without disguise,
And, to excuse it, need but shew the prize;
Whose spoils this paper offers to your eye,
Fair ev'n in death! this peerless Butterfly.
My sons! (she answer'd) both have done your parts:
Live happy both, and long promote our arts.
But hear a Mother, when she recommends
To your fraternal care, our sleeping friends.
The common Soul, of Heav'n's more frugal make,
Serves but to keep fools pert, and knaves awake:
A drowzy Watchman, that just gives a knock,
And breaks our rest, to tell us what's a clock.
Yet by some object ev'ry brain is stirr'd;
The dull may waken to a Humming-bird;
The most recluse, discreetly open'd, find
Congenial matter in the Cockle-kind;
The mind, in Metaphysics at a loss,
May wander in a wilderness of Moss;
The head that turns at super-lunar things,
Poiz'd with a tail, may steer on Wilkins' wings.
'O! would the sons of men once think their eyes
And reason given them but to study flies !
See Nature in some partial narrow shape,
And let the Author of the Whole escape:
Learn but to trifle; or, who most observe,
To wonder at their Maker, not to serve.'
'Be that my task' (replies a gloomy clerk,
Sworn foe to Myst'ry, yet divinely dark;
Whose pious hope aspires to see the day
When Moral Evidence shall quite decay,
And damns implicit faith, and holy lies,
Prompt to impose, and fond to dogmatize):
'Let others creep by timid steps, and slow,
On plain experience lay foundations low,
202
By common sense to common knowledge bred,
And last, to Nature's Cause through Nature led.
All-seeing in thy mists, we want no guide,
Mother of Arrogance, and Source of Pride!
We nobly take the high Priori Road,
And reason downward, till we doubt of God:
Make Nature still encroach upon his plan;
And shove him off as far as e'er we can:
Thrust some Mechanic Cause into his place;
Or bind in matter, or diffuse in space.
Or, at one bound o'erleaping all his laws,
Make God man's image, man the final Cause,
Find virtue local, all relation scorn
See all in self , and but for self be born:
Of naught so certain as our reason still,
Of naught so doubtful as of soul and will .
Oh hide the God still more! and make us see
Such as Lucretius drew, a god like thee:
Wrapp'd up in self, a god without a thought,
Regardless of our merit or default.
Or that bright image to our fancy draw,
Which Theocles in raptur'd vision saw,
While through poetic scenes the Genius roves,
Or wanders wild in academic groves;
That Nature our society adores,
Where Tindal dictates, and Silenus snores.'
Rous'd at his name up rose the bousy Sire,
And shook from out his pipe the seeds of fire;
Then snapp'd his box, and strok'd his belly down:
Rosy and rev'rend, though without a gown.
Bland and familiar to the throne he came,
Led up the youth, and call'd the Goddess Dame .
Then thus, 'From priestcraft happily set free,
Lo! ev'ry finished Son returns to thee:
First slave to words, then vassal to a name,
Then dupe to party; child and man the same;
Bounded by Nature, narrow'd still by art,
A trifling head, and a contracted heart.
Thus bred, thus taught, how many have I seen,
Smiling on all, and smil'd on by a queen.
Marked out for honours, honour'd for their birth,
203
To thee the most rebellious things on earth:
Now to thy gentle shadow all are shrunk,
All melted down, in pension, or in punk!
So K-- so B-- sneak'd into the grave,
A monarch's half, and half a harlot's slave.
Poor W-- nipp'd in Folly's broadest bloom,
Who praises now? his chaplain on his tomb.
Then take them all, oh take them to thy breast!
Thy Magus , Goddess! shall perform the rest.'
With that, a Wizard old his Cup extends;
Which whoso tastes, forgets his former friends,
Sire, ancestors, himself. One casts his eyes
Up to a Star , and like Endymion dies:
A Feather , shooting from another's head,
Extracts his brain, and principle is fled,
Lost is his God, his country, ev'rything;
And nothing left but homage to a king!
The vulgar herd turn off to roll with hogs,
To run with horses, or to hunt with dogs;
But, sad example! never to escape
Their infamy, still keep the human shape.
But she, good Goddess, sent to ev'ry child
Firm impudence, or stupefaction mild;
And straight succeeded, leaving shame no room,
Cibberian forehead, or Cimmerian gloom.
Kind self-conceit to somewhere glass applies,
Which no one looks in with another's eyes:
But as the flatt'rer or dependant paint,
Beholds himself a patriot, chief, or saint.
On others Int'rest her gay liv'ry flings,
Int'rest that waves on party-colour'd wings:
Turn'd to the sun, she casts a thousand dyes,
And, as she turns, the colours fall or rise.
Others the siren sisters warble round,
And empty heads console with empty sound.
No more, Alas! the voice of Fame they hear,
The balm of Dulness trickling in their ear.
Great C--, H--, P--, R--, K--,
Why all your toils? your Sons have learn'd to sing.
How quick ambition hastes to ridicule!
The sire is made a peer, the son a fool.
204
On some, a Priest succinct in amice white
Attends; all flesh is nothing in his sight!
Beeves, at his touch, at once to jelly turn,
And the huge boar is shrunk into an urn:
The board with specious miracles he loads,
Turns hares to larks, and pigeons into toads.
Another (for in all what one can shine?)
Explains the
Seve
and
Verdeur
of the vine.
What cannot copious sacrifice atone?
Thy truffles, Perigord! thy hams, Bayonne!
With French libation, and Italian strain,
Wash Bladen white, and expiate Hays's stain.
Knight lifts the head, for what are crowds undone.
To three essential partridges in one?
Gone ev'ry blush, and silent all reproach,
Contending princes mount them in their coach.
Next, bidding all draw near on bended knees,
The Queen confers her Titles and Degrees .
Her children first of more distinguish'd sort,
Who study Shakespeare at the Inns of Court,
Impale a glowworm, or vertú profess,
Shine in the dignity of F.R.S.
Some, deep Freemasons, join the silent race
Worthy to fill Pythagoras's place:
Some botanists, or florists at the least,
Or issue members of an annual feast.
Nor pass'd the meanest unregarded, one
Rose a Gregorian, one a Gormogon.
The last, not least in honour or applause,
Isis and Cam made Doctors of her Laws.
Then, blessing all, 'Go, Children of my care!
To practice now from theory repair.
All my commands are easy, short, and full:
My sons! be proud, be selfish, and be dull.
Guard my prerogative, assert my throne:
This nod confirms each privilege your own.
The cap and switch be sacred to his Grace;
205
With staff and pumps the Marquis lead the race;
From stage to stage the licens'd Earl may run,
Pair'd with his fellow charioteer the sun;
The learned Baron butterflies design,
Or draw to silk Arachne's subtle line;
The Judge to dance his brother Sergeant call;
The Senator at cricket urge the ball;
The Bishop stow (pontific luxury!)
An hundred souls of turkeys in a pie;
The sturdy Squire to Gallic masters stoop,
And drown his lands and manors in a soupe .
Others import yet nobler arts from France,
Teach kings to fiddle, and make senates dance.
Perhaps more high some daring son may soar,
Proud to my list to add one monarch more;
And nobly conscious, princes are but things
Born for first ministers, as slaves for kings,
Tyrant supreme! shall three Estates command,
And make one mighty Dunciad of the Land!
More she had spoke, but yawn'd-All Nature nods:
What mortal can resist the yawn of gods?
Churches and Chapels instantly it reach'd;
(St. James's first, for leaden Gilbert preach'd)
Then catch'd the schools; the Hall scarce kept awake;
The Convocation gap'd, but could not speak:
Lost was the nation's sense, nor could be found,
While the long solemn unison went round:
Wide, and more wide, it spread o'er all the realm;
Even Palinurus nodded at the helm:
The vapour mild o'er each committee crept;
Unfinish'd treaties in each office slept;
And chiefless armies doz'd out the campaign;
And navies yawn'd for orders on the main.
O Muse! relate (for you can tell alone,
Wits have short memories, and Dunces none),
Relate, who first, who last resign'd to rest;
Whose heads she partly, whose completely blest;
What charms could faction, what ambition lull,
The venal quiet, and entrance the dull;
Till drown'd was sense, and shame, and right, and wrongO sing, and hush the nations with thy song!
206
In vain, in vain-the all-composing hour
Resistless falls: The Muse obeys the Pow'r.
She comes! she comes! the sable throne behold
Of Night primeval, and of Chaos old!
Before her, Fancy's gilded clouds decay,
And all its varying rainbows die away.
Wit shoots in vain its momentary fires,
The meteor drops, and in a flash expires.
As one by one, at dread Medea's strain,
The sick'ning stars fade off th' ethereal plain;
As Argus' eyes by Hermes' wand oppress'd,
Clos'd one by one to everlasting rest;
Thus at her felt approach, and secret might,
Art after Art goes out, and all is Night.
See skulking Truth to her old cavern fled,
Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before,
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence,
And Metaphysic calls for aid on Sense !
See Mystery to Mathematics fly!
In vain! they gaze, turn giddy, rave, and die.
Religion blushing veils her sacred fires,
And unawares Morality expires.
Nor public Flame, nor private , dares to shine;
Nor human Spark is left, nor Glimpse divine !
Lo! thy dread Empire, Chaos! is restor'd;
Light dies before thy uncreating word:
Thy hand, great Anarch! lets the curtain fall;
And universal Darkness buries All.
~ Alexander Pope

IN CHAPTERS



   4 Integral Yoga
   3 Philosophy
   2 Occultism
   1 Psychology
   1 Poetry
   1 Fiction


   3 Nolini Kanta Gupta
   2 Plato
   2 Aleister Crowley


   2 Magick Without Tears
   2 Collected Works of Nolini Kanta Gupta - Vol 01


01.07 - Blaise Pascal (1623-1662), #Collected Works of Nolini Kanta Gupta - Vol 02, #Nolini Kanta Gupta, #Integral Yoga
  
   "The zeal for the Lord hath eaten me up." Such has indeed been the case with Pascal, almost literally. The fire that burned in him was too ardent and vehement for the vehicle, the material instrument, which was very soon used up and reduced to ashes. At twenty-four he was already a broken man, being struck with paralysis and neuras thenia; he died at the comparatively early age of 39, emulating, as it were, the life career of his Lord the Christ who died at 33. The Fire martyrised the body, but kindled and brought forth experiences and realisations that save and truths that abide. It was the Divine Fire whose vision and experience he had on the famous night of 23 November 1654 which brought about his final and definitive conversion. It was the same fire that had blazed up in his brain, while yet a boy, and made him a precocious genius, a marvel of intellectual power in the exact sciences. At 12 this prodigy discovered by himself the 32nd proposition of Euclid, Book I. At sixteen he wrote a treatise on conic sections. At nineteen he invented a calculating machine which, without the help of any mathematical rule or process, gave absolutely accurate results. At twenty-three he published his experiments with vacuum. At twenty-five he conducted the well-known experiment from the tower of St. Jacques, proving the existence of atmospheric pressure. His studies in infinitesimal calculus were remarkably creative and original. And it might be said he was a pioneer in quite a new branch of mathematics, viz., the mathematical theory of probability. We shall see presently how his preoccupation with the mathematics of chance and probability coloured and reinforced his metaphysics and theology.
  

02.03 - National and International, #Collected Works of Nolini Kanta Gupta - Vol 01, #Nolini Kanta Gupta, #Integral Yoga
  
   We have just passed through another, a far greater, a catastrophic Kurukshetra, the last Act (Shanti Parvam) of which we are negotiating at the present moment. The significance of this cataclysm is clear and evident if we only allow ourselves to be led by the facts and not try to squeeze the facts into the groove of our past prejudices and set notions. All the difficulties that are being encountered on the way to peace and reconstruction arise mainly out of the failure to grasp what Nature has forced upon us. It is as simple as the first axiom of Euclid: Humanity is one and all nations are free and yet interdependent members of that one and single organism. No nation can hope henceforth to stand in its isolated grandeurnot even America or Russia. Subject or dependent nations too who are struggling to be free will be allowed to work out their freedom and independence, on condition that the same is worked out in furtherance and in collaboration with the ideal of human unity. That ideal has become dynamic and insistent the more man refuses to accept it, the more he will make confusion worse confounded.
   ***

05.07 - The Observer and the Observed, #Collected Works of Nolini Kanta Gupta - Vol 01, #Nolini Kanta Gupta, #Integral Yoga
  
   Now we come to the sanctum, the Shekinah, of the problem. For there is a still deeper mystery. And pre-eminently it is an Einsteinian discovery. It is not merely the measuring ray of light, not merely the beam in the eye of the observer that is the cause of interference: the very mind behind the eye is involved in a strange manner. The mind is not a tabula rasa, it comes into the field with certain presuppositionsaxioms and postulates, as it calls themdue to its angle of vision and perhaps to the influence upon it of immediate sense perception. It takes for granted, for example, that light travels in a straight line, that parallels do not meet, indeed all the theorems and deductions of Euclidean geometry. There is a strong inclination in the mind to view things as arranged according to that pattern. Einstein has suggested that the spherical scheme can serve as well or even better our observations. Riemann's non- Euclidean geometry has assumed momentous importance in contemporary scientific enquiry. It is through that scheme that Einstein proposes to find the equation that will subsume the largest number of actual and possible or potential facts and bring about the reconciliation of such irreconcilables as wave and particle, gravitation and electricity.
  

1.01 - What is Magick?, #Magick Without Tears, #Aleister Crowley, #Occultism
  
  First let me go all Euclidean, and rub your nose in the Definition, Postulate and Theorems given in my comprehensive (but, alas! too advanced and too technical) Treatise on the subject.[1] Here we are!
  

1.02 - The Pit, #A Garden of Pomegranates - An Outline of the Qabalah, #Israel Regardie, #Occultism
  Victorians so simple, objective, and intelligible-such as matter, energy, space, etc.-have completely failed to resist analysis. A few modern thinkers, seeing clearly the absolute debacle in which the old positivist science was bound to lead them, the breaking up of this icy expanse of frozen thought, determined at all costs to find a modus vivendi for
  Athena. This necessity was emphasized in the most surprising way by the result of the Michelson-Morley experiments, when Physics itself calmly and frankly offered a contradiction in terms. It was not the metaphysicians this time who were picking holes in a vacuum. It was the mathematicians and the physicists who found the ground completely cut away from under their feet. It was not enough to replace the geometry of Euclid by those of Riemann and Lobatchevsky and the mechanics of Newton by those of Einstein, so long as any of the axioms of the old thought and the definitions of its terms survived. They deliberately abandoned positivism and materialism for an indeterminate mysticism, creating a new mathematical philosophy and a new logic, wherein infinite-or rather transfinite-ideas might be made commensurable with those of ordinary thought in the forlorn hope that all might live happily ever after. In short, to use a Qabalistic nomenclature, they found it incumbent upon themselves to adopt for inclusion of terms of Ruach (intellect) concepts which are proper only to Neschamah (the organ and faculty of direct spiritual apperception and intuition). This same process took place in Philosophy years earlier. Had the dialectic of Hegel been only. half understood, the major portion of philosophical speculation from the Schoolmen to
  Kant's perception of the Antinomies of Reason would have been thrown overboard.

1.02 - The Three European Worlds, #The Ever-Present Origin, #unset, #Zen
  
  Despite, or indeed because of, Euclidean geometry, there is no evidence of an awareness of qualitative and objectified space in early antiquity or in the epoch preceding the Renaissance.
  
  This has been indirectly confirmed by von Kaschnitz-Weinberg, who has documented two opposing yet complementary structural elements of ancient art as it emerged from the Megalithic (stone) age. The first, Dolmen architecture, entered the Mediterranean region primarily from Northern and Western Europe and was especially influential on Greek architecture. It is phallic in nature and survives in the column architecture in Greece, as in the Par thenon. Space is visible here simply as diastyle or the intercolumnar space, whose structure is determined by the vertical posts and the horizontal lintels and corresponds to Euclidean cubic space.
  
  --
  
  This overwhelming new discovery and encounter, this elemental irruption of the third dimension and transformation of Euclidean plane surfaces, is so disorienting that it at first brought about an inflation and inundation by space. This is clearly evident in the numerous experimental representations of perspective. We will have occasion to note a parallel confusion and disorder in the painting of the period alter1800when we consider the new dimension of emergent consciousness in our own day. But whereas the preoccupation of the Early Renaissance was with the concretion of space, our epoch is concerned with the concretion of time. And our fundamental point of departure, the attempt to concretize time and thus realize and become conscious of the fourth dimension, furnishes a means whereby we may gain an all-encompassing perception and knowledge of our epoch.
  
  --
  
  Before returning to Leonardo, we must mention two facts which demonstrate better than any description the extent of fascination with the problem of perspective during the later Part of the fifteenth century when perspective becomes virtually normative (as in Ghiberti's modification of Vitruvius). In his DivinaProporzione, Luca Pacioli - the learned mathematician, translator of Euclid, co-worker with Pierodella Francesca, and friend of Leonardo - celebrated perspective as the eighth art; and when Antonio del Pollaiuolo built a memorial to perspective on one of his papal tombs in St. Peters some ten years later (in the 1490s), he boldly added perspective as the eighth free art to the other seven.
  

1.04 - THE APPEARANCE OF ANOMALY - CHALLENGE TO THE SHARED MAP, #Maps of Meaning, #Jordan Peterson, #Psychology
  A paradigmatic structure provides for determinate organization of (unlimited) information, according to
  limited principles. The system of Euclidean geometry provides a classic example. The individual who
  wishes to generate a desired outcome of behavior, as a consequence of the application of Euclidean
  principles, is bound by necessity to accept certain axioms on faith. These axioms follow:
  --
  remembered, or understood, for geometry to prove useful that gives rise to the internally consistent and
  logical Euclidean structure we are all familiar with. What constitutes truth, from within the perspective of
  this structure, can be established by reference to these initial postulates. However, the postulates themselves
  --
  a given structure appears necessarily predicated on unconscious presuppositions the presupposition that
  space has three dimensions, in the case of Euclidean geometry (a presupposition which is clearly
  questionable).
  It appears, in many cases, that the assumptions of explicit semantic statements take episodic or imagistic
  form. The Euclidean postulates, for example, appear based upon observable facts (images of the world
  of experience as interpreted). Euclid grounded his explicit abstract (semantic) system in observable
  absolutes. It can be concretely demonstrated, for example, that any two points drawn in the sand can be
  --
  geometry itself, but they appear true, and will be accepted as such, as a consequence of practical example.
  What this means is that belief in Euclidean presumptions is dependent upon acceptance of practical
  experience as sufficient certainty. The Euclidean draws a line in the sand, so to speak, and says the
  questions stop here.
  --
  cannot be articulated explicitly. 379
  The Euclidean draws a line connecting two points in the sand, and accepts on faith the sufficiency of that
  behavioral demonstration and the evident certainty of its outcome (in part, because no alternative
  conceptualization can presently be imagined). Euclidean geometry worked and was considered complete
   for centuries, because it allowed for the prediction and control of all those experienceable phenomena that
  --
  years ago, we did not know how to act concretely, or think abstractly, in a manner that would produce some
  situation whose nature could not be described by Euclid. That is no longer the case. Many alternative, and
  more inclusive geometries have been generated during the course of the last century. These new systems

1.14 - The Limits of Philosophical Knowledge, #The Problems of Philosophy, #Bertr and Russell, #Philosophy
  
  The mathematicians, however, have not been content with showing that space as it is commonly supposed to be is possible; they have shown also that many other forms of space are equally possible, so far as logic can show. Some of Euclid's axioms, which appear to common sense to be necessary, and were formerly supposed to be necessary by philosophers, are now known to derive their appearance of necessity from our mere familiarity with actual space, and not from any _a priori_ logical foundation. By imagining worlds in which these axioms are false, the mathematicians have used logic to loosen the prejudices of common sense, and to show the possibility of spaces differing--some more, some less--from that in which we live. And some of these spaces differ so little from Euclidean space, where distances such as we can measure are concerned, that it is impossible to discover by observation whether our actual space is strictly Euclidean or of one of these other kinds.
  

1.50 - A.C. and the Masters; Why they Chose him, etc., #Magick Without Tears, #Aleister Crowley, #Occultism
  
  I know this sounds mad; but it's true. Well, then, I set myself to repair the omission with Part III; this should be a really complete treatise on the Art and Science of magick, and it should be worked out from the beginning, a logical sequence like Euclid. Hence Axiom, Postulate and Theorems. I supposed even then that I could cover the field with another volume comparable in size with the former two.
  

1f.lovecraft - At the Mountains of Madness, #Lovecraft - Poems, #unset, #Zen
   bizarrerie, endless variety, preternatural massiveness, and utterly
   alien exoticism. There were geometrical forms for which an Euclid could
   scarcely find a namecones of all degrees of irregularity and

1.poe - Eureka - A Prose Poem, #Poe - Poems, #unset, #Zen
  
  "Do you know, my dear friend," says the writer, addressing, no doubt, a contemporary -"Do you know that it is scarcely more than eight or nine hundred years ago since the metaphysicians first consented to relieve the people of the singular fancy that there exist but two practicable roads to Truth? Believe it if you can! It appears, however, that long, long ago, in the night of Time, there lived a Turkish philosopher called Aries and surnamed Tottle." [Here, possibly, the letter-writer means Aristotle; the best names are wretchedly corrupted in two or three thousand years.] "The fame of this great man depended mainly upon his demonstration that sneezing is a natural provision, by means of which over-profound thinkers are enabled to expel superfluous ideas through the nose; but he obtained a scarcely less valuable celebrity as the founder, or at all events as the principal propagator, of what was termed the de ductive or a priori philosophy. He started with what he maintained to be axioms, or self-evident truths: -and the now well-understood fact that no truths are self -evident, really does not make in the slightest degree against his speculations: -it was sufficient for his purpose that the truths in question were evident at all. From axioms he proceeded, logically, to results. His most illustrious disciples were one Tuclid, a geometrician," [meaning Euclid] "and one Kant, a Dutchman, the originator of that species of Transcendentalism which, with the change merely of a C for a K, now bears his peculiar name.
  
  --
  
  "And now, by the logic of their own propounder, let us proceed to test any one of the axioms propounded. Let us give Mr. Mill the fairest of play. We will bring the point to no ordinary issue. We will select for investigation no common-place axiom -no axiom of what, not the less preposterously because only impliedly, he terms his secondary class -as if a positive truth by definition could be either more or less positively a truth: -we will select, I say, no axiom of an unquestionability so questionable as is to be found in Euclid. We will not talk, for example, about such propositions as that two straight lines cannot enclose a space, or that the whole is greater than any one of its parts. We will afford the logician every advantage. We will come at once to a proposition which he regards as the acme of the unquestionable -as the quintessence of axiomatic undeniability. Here it is: -'Contradictions cannot both be true that is, cannot coexist in nature.' Here Mr. Mill means, for instance, -and I give the most forcible instance conceivable -that a tree must be either a tree or not a tree -that it cannot be at the same time a tree and not a tree: -all which is quite reasonable of itself and will answer remarkably well as an axiom, until we bring it into collation with an axiom insisted upon a few pages before -in other words -words which I have previously employed -until we test it by the logic of its own propounder. 'A tree,' Mr. Mill asserts, 'must be either a tree or not a tree.' Very well: -and now let me ask him, why. To this little query there is but one response: -I defy any man living to invent a second. The sole answer is this: 'Because we find it impossible to conceive that a tree can be anything else than a tree or not a tree.' This, I repeat, is Mr. Mill's sole answer: -he will not pretend to suggest another: -and yet, by his own showing, his answer is clearly no answer at all; for has he not already required us to admit, as an axiom, that ability or inability to conceive is in no case to be taken as a criterion of axiomatic truth? Thus all -absolutely his argumentation is at sea without a rudder. Let it not be urged that an exception from the general rule is to be made, in cases where the 'impossibility to conceive' is so peculiarly great as when we are called upon to conceive a tree both a tree and not a tree. Let no attempt, I say, be made at urging this sotticism; for, in the first place, there are no degrees of 'impossibility,' and thus no one impossible conception can be more peculiarly impossible than another impossible conception: -in the second place, Mr. Mill himself, no doubt after thorough deliberation, has most distinctly, and most rationally, excluded all opportunity for exception, by the emphasis of his proposition, that, in no case, is ability or inability to conceive, to be taken as a criterion of axiomatic truth: -in the third place, even were exceptions admissible at all, it remains to be shown how any exception is admissible here. That a tree can be both a tree and not a tree, is an idea which the angels, or the devils, may entertain, and which no doubt many an earthly Bedlamite, or Transcendentalist, does.
  
  --
  
  I maintain, first, that only in the mode described is it conceivable that Matter could have been diffused so as to fulfil at once the conditions of irradiation and of generally equable distribution. I maintain, secondly, that these conditions themselves have been imposed upon me, as necessities, in a train of ratiocination as rigorously logical as that which establishes any demonstration in Euclid; and I maintain, thirdly, that even if the charge of "hypothesis" were as fully sustained as it is, in fact, unsustained and untenable, still the validity and indisputability of my result would not, even in the slightest particular, be disturbed.
  

2.01 - On Books, #Evening Talks With Sri Aurobindo, #unset, #Zen
  
   Disciple: He says that Euclidian geometry is not applicable to the material world. That is to say, space is not flat, a three-dimensional analogue of a two-dimensional flat surface. Euclidian figures like the square and solid and straight line are abstract, not real or actual. He also says that material space is "boundless but not infinite".
  

Blazing P3 - Explore the Stages of Postconventional Consciousness, #unset, #Sri Aurobindo, #Integral Yoga
  under paper that holds iron filings. This paradigm made it possible for Einstein to use
  notions of curved space to describe spacetime to replace Euclidean geometry. The waves were
  bent by the mass of objects so that the rings no longer fit in a flat plane. From there modern

BOOK II. -- PART II. THE ARCHAIC SYMBOLISM OF THE WORLD-RELIGIONS, #The Secret Doctrine, #H P Blavatsky, #Theosophy
  knowledge increases daily in mankind, "but that intellectual capacity does not increase with it," is
  shown when the intellect, if not the physical knowledge, of the Euclids, Pythagorases, Paninis,
  Kapilas, Platos, and Socrates, is compared with that of the Newtons, Kants, and the modern Huxleys
  --
  1.)
  "Before the mathematical numbers," says Proclus (in Quinto Libro, Euclid), "there are the Selfmoving numbers; before the figures apparent -- the vital figures, and before producing the material
  worlds which move in a Circle, the Creative Power produced the invisible Circles."

Phaedo, #unset, #Sri Aurobindo, #Integral Yoga
  
  During the voyage of the sacred ship to and from Delos, which has occupied thirty days, the execution of Socrates has been deferred. (Compare Xen. Mem.) The time has been passed by him in conversation with a select company of disciples. But now the holy season is over, and the disciples meet earlier than usual in order that they may converse with Socrates for the last time. Those who were present, and those who might have been expected to be present, are mentioned by name. There are Simmias and Cebes (Crito), two disciples of Philolaus whom Socrates 'by his enchantments has attracted from Thebes' (Mem.), Crito the aged friend, the attendant of the prison, who is as good as a friendthese take part in the conversation. There are present also, Hermogenes, from whom Xenophon derived his information about the trial of Socrates (Mem.), the 'madman' Apollodorus (Symp.), Euclid and Terpsion from Megara (compare Theaet.), Ctesippus, Antis thenes, Menexenus, and some other less-known members of the Socratic circle, all of whom are silent auditors. Aristippus, Cleombrotus, and Plato are noted as absent. Almost as soon as the friends of Socrates enter the prison Xanthippe and her children are sent home in the care of one of Crito's servants. Socrates himself has just been released from chains, and is led by this circumstance to make the natural remark that 'pleasure follows pain.' (Observe that Plato is preparing the way for his doctrine of the alternation of opposites.) 'Aesop would have represented them in a fable as a two-headed creature of the gods.' The mention of Aesop reminds Cebes of a question which had been asked by Evenus the poet (compare Apol.): 'Why Socrates, who was not a poet, while in prison had been putting Aesop into verse?''Because several times in his life he had been warned in dreams that he should practise music; and as he was about to die and was not certain of what was meant, he wished to fulfil the admonition in the letter as well as in the spirit, by writing verses as well as by cultivating philosophy. Tell this to Evenus; and say that I would have him follow me in death.' 'He is not at all the sort of man to comply with your request, Socrates.' 'Why, is he not a philosopher?' 'Yes.' 'Then he will be willing to die, although he will not take his own life, for that is held to be unlawful.'
  
  --
  
  Other persons, Menexenus, Ctesippus, Lysis, are old friends; Evenus has been already satirized in the Apology; Aeschines and Epigenes were present at the trial; Euclid and Terpsion will reappear in the Introduction to the Theaetetus, Hermogenes has already appeared in the Cratylus. No inference can fairly be drawn from the absence of Aristippus, nor from the omission of Xenophon, who at the time of Socrates' death was in Asia. The mention of Plato's own absence seems like an expression of sorrow, and may, perhaps, be an indication that the report of the conversation is not to be taken literally.
  
  --
  
  PHAEDO: Yes, there were; Simmias the Theban, and Cebes, and Phaedondes; Euclid and Terpison, who came from Megara.
  

The Act of Creation text, #The Act of Creation, #unset, #Zen
  numbers, or analytical geometry, would have been discovered within
  a few generations after Euclid.
  
  --
  
  To p. 240), It took two thousand yean until Archimedes and Euclid were
  rediscovered. It took four hundred years until the Occamites' work on impetus
  --
  Leonardo and Durer, saw beauty wherever mortal flesh testified to
  the immortal axioms of Euclidean geometry.
  
  However, the ideal to which the bloated Venus of Willendorf testi-
  fies with her pendulous breasts and enormous hips, is not Euclid, but
  the goddess of Fertility. Our whole manner of perceiving the human
  --
  the spirituality in the transfigured face of the Byzantine madonna; the
  harmonious resolution of the body into Euclidean forms, or a patch-
  work of coloured blobs. Whichever aspect is dominant, its matrix
  --
  
  That vocabulary and its Euclidean grammar of proportion
  
  --
  versal laws of form and colour harmony, the norms of beauty hidden
  in the mysteries of the golden section or anchored in Euclid's anxioms.
  Everything has two aspects,' wrote Chirico, 'the current aspect,

Theaetetus, #unset, #Sri Aurobindo, #Integral Yoga
  
  The Theaetetus is one of the narrated dialogues of Plato, and is the only one which is supposed to have been written down. In a short introductory scene, Euclides and Terpsion are described as meeting before the door of Euclides' house in Megara. This may have been a spot familiar to Plato (for Megara was within a walk of Athens), but no importance can be attached to the accidental introduction of the founder of the Megarian philosophy. The real intention of the preface is to create an interest about the person of Theaetetus, who has just been carried up from the army at Corinth in a dying state. The expectation of his death recalls the promise of his youth, and especially the famous conversation which Socrates had with him when he was quite young, a few days before his own trial and death, as we are once more reminded at the end of the dialogue. Yet we may observe that Plato has himself forgotten this, when he represents Euclides as from time to time coming to Athens and correcting the copy from Socrates' own mouth. The narrative, having introduced Theaetetus, and having guaranteed the au thenticity of the dialogue (compare Symposium, Phaedo, Parmenides), is then dropped. No further use is made of the device. As Plato himself remarks, who in this as in some other minute points is imitated by Cicero (De Amicitia), the interlocutory words are omitted.
  
  --
  
  Terpsion, who has come to Megara from the country, is described as having looked in vain for Euclides in the Agora; the latter explains that he has been down to the harbour, and on his way thither had met Theaetetus, who was being carried up from the army to Athens. He was scarcely alive, for he had been badly wounded at the battle of Corinth, and had taken the dysentery which prevailed in the camp. The mention of his condition suggests the reflection, 'What a loss he will be!' 'Yes, indeed,' replies Euclid; 'only just now I was hearing of his noble conduct in the battle.' 'That I should expect; but why did he not remain at Megara?' 'I wanted him to remain, but he would not; so I went with him as far as Erineum; and as I parted from him, I remembered that Socrates had seen him when he was a youth, and had a remarkable conversation with him, not long before his own death; and he then prophesied of him that he would be a great man if he lived.' 'How true that has been; how like all that Socrates said! And could you repeat the conversation?' 'Not from memory; but I took notes when I returned home, which I afterwards filled up at leisure, and got Socrates to correct them from time to time, when I came to Athens'...Terpsion had long intended to ask for a sight of this writing, of which he had already heard. They are both tired, and agree to rest and have the conversation read to them by a servant...'Here is the roll, Terpsion; I need only observe that I have omitted, for the sake of convenience, the interlocutory words, "said I," "said he"; and that Theaetetus, and Theodorus, the geometrician of Cyrene, are the persons with whom Socrates is conversing.'
  
  --
  
   Euclid and Terpsion meet in front of Euclid's house in Megara; they enter the house, and the dialogue is read to them by a servant.
  
  --
  
  TERPSION: Quite right, Euclid.
  

WORDNET



--- Overview of noun euclid

The noun euclid has 1 sense (no senses from tagged texts)
                    
1. Euclid ::: (Greek geometer (3rd century BC))




--- Synonyms/Hypernyms (Ordered by Estimated Frequency) of noun euclid

1 sense of euclid                          

Sense 1
Euclid
   INSTANCE OF=> geometer, geometrician
     => mathematician
       => scientist
         => person, individual, someone, somebody, mortal, soul
           => organism, being
             => living thing, animate thing
               => whole, unit
                 => object, physical object
                   => physical entity
                     => entity
           => causal agent, cause, causal agency
             => physical entity
               => entity




--- Hyponyms of noun euclid
                                    




--- Synonyms/Hypernyms (Ordered by Estimated Frequency) of noun euclid

1 sense of euclid                          

Sense 1
Euclid
   INSTANCE OF=> geometer, geometrician










--- Coordinate Terms (sisters) of noun euclid

1 sense of euclid                          

Sense 1
Euclid
  -> geometer, geometrician
   HAS INSTANCE=> Euclid










--- Grep of noun euclid
euclid
euclid's axiom
euclid's fifth axiom
euclid's first axiom
euclid's fourth axiom
euclid's postulate
euclid's second axiom
euclid's third axiom
euclidean axiom
euclidean geometry
euclidean space





IN WEBGEN [10000/105]

https://military.wikia.org/wiki/Theodore_Roosevelt#Emergence_as_a_national_figure
https://religion.wikia.org/wiki/Creation_myth#Emergence
https://religion.wikia.org/wiki/Five_Wisdoms#Emergence_of_Pa.C3.B1ca-j.C3.B1.C4.81na
https://religion.wikia.org/wiki/History_of_Buddhism#Emergence_of_the_Vajrayana_.285th_century.29
https://religion.wikia.org/wiki/History_of_Christian_theology#Emergence_of_Christian_theology
https://religion.wikia.org/wiki/Jewish_denominations#Emergence_of_trans-_and_post-denominational_Judaism
https://religion.wikia.org/wiki/List_of_creation_myths#Emergence
https://religion.wikia.org/wiki/Mami_Wata#Reemergence_in_contemporary_times
https://religion.wikia.org/wiki/Quimbanda#The_emergence_of_Quimbanda
https://religion.wikia.org/wiki/Talk:Buddhism/Revised#Emergence_of_the_Vajray.C4.81na
https://religion.wikia.org/wiki/Vestments_controversy#The_emergence_of_separatism_and_Presbyterianism
Integral World - The Self-Creating Universe, The Emergence of a New Worldview, John Clarke
Integral World - Ken Wilber and "Moronic" Evolution, The Religion of Tomorrow and the misunderstanding of Emergence, David Lane
Integral World - Buddha Nature, Inner Illumination and Evolutionary Emergence, Barclay Powers
Integral World - Ken Wilber's Evolutionary View Gets a Trim With Ockham's Razor, Part III: The Emergence of Consciousness Over the Course of Evolution, Don Salmon
Integral World - Ken Wilber's Evolutionary View Gets a Trim With Ockham's Razor, Part IV: The emergence of consciousness in birds, mammals and primates, Don Salmon
Integral World - Over the Rainbow: Civilizations, Spiral Dynamics, and the Emergence of a Global Holon
Integral World - An absolutely obvious look at how evolution actually operates, Wilber and Whitehead on the emergence of novelty, Frank Visser
Integral World - Demystifying Emergence, A Comparison of Harold Morowitz and Ken Wilber, Frank Visser
Integral World - "Entire populations simply show up", Ken Wilber on Emergence and Speciation, Frank Visser
Integral World - "It is Part of this Miracle of Emergence", Ken Wilber Argues for his Mysterianism, Frank Visser
Integral World - The Emergence of Integral Consciousness in Europe, Dennis Wittrock
The Emergence of a New Activism
Themes of Emergence: Awareness, Responsibility, and Mutuality
selforum - emergence of integral yoga
selforum - emergence of matter life thought
selforum - emergence is central to hayekian
wiki.auroville - Stewardship_for_New_Emergence
Psychology Wiki - Emergence
Stanford Encyclopedia of Philosophy - logic-firstorder-emergence
https://tvtropes.org/pmwiki/pmwiki.php/Fanfic/EmergenceRWBY
https://tvtropes.org/pmwiki/pmwiki.php/Main/MermaidArcEmergence
https://en.wikiquote.org/wiki/Emergence
Wikipedia - Alliance of Movements for the Emergence of Niger -- Political party in Niger
Wikipedia - Alternative Forces for Renewal and Emergence -- Political party in Mali
Wikipedia - Anthropic principle -- Philosophical premise that all scientific observations presuppose a universe compatible with the emergence of sentient organisms that make those observations
Wikipedia - Category:Emergence
Wikipedia - Charlot Jeudy -- Haitian activist involved in the emergence of LGBT rights Haiti
Wikipedia - Cradle of civilization -- Locations identified as the sites of the emergence of civilization
Wikipedia - Emergence from Chaos
Wikipedia - Emergence International -- LGBT support organization
Wikipedia - Emergence: The Connected Lives of Ants, Brains, Cities, and Software -- Book by Steven Berlin Johnson
Wikipedia - Emergence (TV series) -- 2019 American mystery/thriller drama television series
Wikipedia - Emergence -- Phenomenon in complex systems where interactions produce effects not directly predictable from the subsystems
Wikipedia - Evolutionary origin of religions -- Emergence of religious behavior discussed in terms of natural evolution
Wikipedia - Hot spring -- Spring produced by the emergence of geothermally heated groundwater
Wikipedia - Impact Wrestling Emergence -- 2020 Impact Wrestling special
Wikipedia - Neocatastrophism -- Hypothesis that life-exterminating events such as gamma-ray bursts have acted as a galactic regulation mechanism in the Milky Way upon the emergence of complex life
Wikipedia - Noogenesis -- Emergence and evolution of intelligence
Wikipedia - Rinse FM -- London-based radio station, critical in the emergence of dubstep and grime
Wikipedia - Scientific Revolution -- Events that marked the emergence of modern science in the early modern period
Wikipedia - Season of the Emergence
Wikipedia - Spontaneous order -- Spontaneous emergence of order out of seeming chaos
World Ninja War Jiraiya (1988 - 1989) - The only non-armored hero of this era of shows, this program features an actual ninja master from a historically known ninja clan preparing his son, daughter, and youngest child, along with a family relative and a police officer with ninja roots to combat the re-emergence of a centuries-old demon sa...
Adventure, Fantasy | TV Series ::: Connections -- 9 episodes -- Epic drama set thousands of years before the events of J.R.R. Tolkien's 'The Hobbit' and 'The Lord of the Rings' follows an ensemble cast of characters, both familiar and new, as they confront the long-feared re-emergence of evil to Middle-earth. Creators: Patrick McKay, John D. Payne Stars: Cynthia Addai-Robinson, Maxim Baldry, Ian Blackburn
Emergence ::: TV-14 | 43min | Drama, Mystery, Sci-Fi | TV Series (20192020) -- A police chief investigates a case involving a young child found near the site of a mysterious accident. Creators: Tara Butters, Michele Fazekas
Hellboy Animated: Blood and Iron (2007) ::: 6.7/10 -- Unrated | 1h 15min | Animation, Action, Adventure | TV Movie 10 March -- Hellboy Animated: Blood and Iron Poster Hellboy (Ron Perlman) and his team face off against a new supernatural threat, while Professor Bruttenholm (Sir John Hurt) must investigate the possible reemergence of a vampire he had slain decades prior. Directors: Victor Cook, Tad Stones Writers: Mike Mignola (based on the Dark Horse comic book "Hellboy" created by), Mike Mignola (story by) | 2 more credits
Longtime Companion (1989) ::: 7.6/10 -- R | 1h 36min | Drama, Romance | 1 May 1990 (Portugal) -- The emergence and devastation of the AIDS epidemic is chronicled in the lives of several gay men living during the 1980s. Director: Norman Ren Writer: Craig Lucas
Next ::: TV-14 | Crime, Drama, Sci-Fi | TV Series (2020) -- A former tech CEO joins an FBI Cybersecurity Agent to stop the emergence of a rogue Artificial Intelligence. Creator: Manny Coto
Split (2016) ::: 7.3/10 -- PG-13 | 1h 57min | Horror, Thriller | 20 January 2017 (USA) -- Three girls are kidnapped by a man with a diagnosed 23 distinct personalities. They must try to escape before the apparent emergence of a frightful new 24th. Director: M. Night Shyamalan Writer:
Surface ::: TV-14 | 1h | Adventure, Drama, Mystery | TV Series (20052006) -- A marine biologist, an insurance salesman and a teen-aged boy find their lives fundamentally changed by the emergence of a new, and often dangerous, species of sea life, while government agents work to keep the affair under wraps. Creators:
Temple Grandin (2010) ::: 8.3/10 -- TV-PG | 1h 47min | Biography, Drama | TV Movie 6 February 2010 -- A biopic of Temple Grandin, an autistic woman who has become one of the top scientists in the humane livestock handling industry. Director: Mick Jackson Writers: Temple Grandin (based on the book: "Emergence"), Margaret Scariano (based on the book: "Emergence") | 3 more credits Stars:
https://eq2.fandom.com/wiki/The_Footsteps_of_Dartain:_Emergence
https://evolutionaryspirituality.fandom.com/wiki/Emergence
https://ffxiclopedia.fandom.com/wiki/Periapt_of_Emergence
https://future.fandom.com/wiki/Scenario:_Emergence_of_India
https://grey-goo.fandom.com/wiki/Emergence
https://humanscience.fandom.com/wiki/Emergence_of_the_Conscious_Individual
https://memory-alpha.fandom.com/wiki/Emergence_(episode)
https://memory-alpha.fandom.com/wiki/Rite_of_Emergence
https://memory-beta.fandom.com/wiki/Emergence_(episode)
https://sintek.fandom.com/wiki/SiN_Episodes:_Emergence
https://starwars.fandom.com/wiki/Eighteenth_Emergence
https://starwars.fandom.com/wiki/Emergences
https://starwars.fandom.com/wiki/Eriadu_system_Emergence
https://starwars.fandom.com/wiki/First_Emergence
https://starwars.fandom.com/wiki/Fortieth_Emergence
https://starwars.fandom.com/wiki/Lost_Tribe_of_Sith_emergence
https://starwars.fandom.com/wiki/Ringlite_system_Emergence
https://starwars.fandom.com/wiki/Thirty-ninth_Emergence
https://starwars.fandom.com/wiki/Trymant_system_Emergence
https://thegifted.fandom.com/wiki/Emergence
https://thenhl.fandom.com/wiki/Emergence_of_the_NHL
Date A Live Movie: Mayuri Judgment -- -- Production IMS -- 1 ep -- Light novel -- Harem Romance Sci-Fi -- Date A Live Movie: Mayuri Judgment Date A Live Movie: Mayuri Judgment -- War(Date), resumes. -- The peaceful days of Itsuka Shidou returned without having to save the Spirits from the relentless battle days. But, the peaceful days were interrupted by the emergence of a mysterious sphere emitting spiritual wave――as well as a mysterious girl who showed herself at the same time. What is her purpose of keeping an eye on Shidou……? The mission that had been entrusted to Shidou under the tense situation was, -- -- “Date all of the Spirits, and make them fall in love?!” -- -- ――With the fate of humanity at stake, the war(date) resumes! -- -- (Source: Date A Live Wiki) -- Movie - Aug 22, 2015 -- 148,796 7.35
Gakusen Toshi Asterisk -- -- A-1 Pictures -- 12 eps -- Light novel -- Action Comedy Ecchi Fantasy Harem Romance School Sci-Fi Supernatural -- Gakusen Toshi Asterisk Gakusen Toshi Asterisk -- In the previous century, an unprecedented disaster known as the Invertia drastically reformed the world. The powers of existing nations declined significantly, paving the way for a conglomerate called the Integrated Empire Foundation to assume control. But more importantly, the Invertia led to the emergence of a new species of humans who are born with phenomenal physical capabilities—the Genestella. Its elite are hand-picked across the globe to attend the top six schools, and they duel amongst themselves in entertainment battles called Festas. -- -- Ayato Amagiri is a scholarship transfer student at the prestigious Seidoukan Academy, which has recently been suffering from declining performances. Through a series of events, he accidentally sees the popular Witch of Resplendent Flames, Julis-Alexia von Riessfeld, half-dressed! Enraged, Julis challenges him to a duel for intruding on her privacy. After said duel is voided by the student council president, Ayato reveals that he has no interest in Festas. Instead, he has enrolled in the academy to investigate the whereabouts of his missing elder sister. But when a more devious plot unravels, Ayato sets out to achieve victory, while being surrounded by some of the most talented Genestella on the planet. -- -- -- Licensor: -- Aniplex of America -- 507,111 6.88
God Eater -- -- ufotable -- 13 eps -- Game -- Action Fantasy Military Sci-Fi -- God Eater God Eater -- The year is 2071. Humanity has been pushed to the brink of extinction following the emergence of man-eating monsters called "Aragami" that boast an immunity to conventional weaponry. They ravaged the land, consuming almost everything in their path and leaving nothing in their wake. To combat them, an organization named Fenrir was formed as a last-ditch effort to save humanity through the use of "God Eaters"—special humans infused with Oracle cells, allowing them to wield the God Arc, the only known weapon capable of killing an Aragami. One such God Eater is Lenka Utsugi, a New-Type whose God Arc takes the form of both blade and gun. -- -- Now, as one of Fenrir's greatest weapons, Lenka must master his God Arc if he is to fulfill his desire of wiping out the Aragami once and for all. The monsters continue to be born en masse while the remnants of humanity struggle to survive the night. Only God Eaters stand between the Aragami and complete and total annihilation of the human race. -- -- -- Licensor: -- Aniplex of America -- 443,037 7.27
Koukaku Kidoutai: SAC_2045 -- -- Production I.G, Sola Digital Arts -- 12 eps -- Manga -- Action Military Sci-Fi Police Mecha Seinen -- Koukaku Kidoutai: SAC_2045 Koukaku Kidoutai: SAC_2045 -- The year is 2045, and artificial intelligence has undergone tremendous developments. Governments use them as weapons, engaging in never-ending warfare to sustain their economies. The mercenary group GHOST, headed by Major Motoko Kusanagi, is no stranger to this landscape. However, the rapid innovation of this technology causes a new threat to loom over the horizon. -- -- Having left Public Security Section 9, Kusanagi and her group are involved in many operations worldwide related to these proxy wars. But a seemingly simple job of locating an arms dealer drags GHOST into a hidden conflict against cybernetically enhanced individuals, who have inexplicably gained extreme intelligence and physical abilities. Dubbed "post-humans," their emergence sets off a chain of events leading to the reunion of Section 9. Armed with a new mission, it is up to Kusanagi and her reestablished team to prevent global chaos at the hands of these post-humans. -- -- ONA - Apr 23, 2020 -- 26,133 6.67
Mobile Suit Gundam Wing: Endless Waltz -- -- Sunrise -- 3 eps -- Original -- Action Drama Mecha Military Sci-Fi Space -- Mobile Suit Gundam Wing: Endless Waltz Mobile Suit Gundam Wing: Endless Waltz -- In the year After Colony 196, one year after the conclusion of the intergalactic civil war, a state of stasis prevails over the Earth and its colonies. Seeing no further use for their Gundam mobile suits, war heroes Duo Maxwell, Heero Yuy, Trowa Barton, and Quatre Raberba Winner decide to destroy these weapons by launching them into the sun's surface. -- -- Before the Gundam reach their destination, the universal peace is shattered by the emergence of Mariemaia Khushrenada—the only child of the former tyrannical aristocrat Treize. Mariemaia abducts diplomat Relena Peacecraft and announces plans to launch "Operation Meteor," with the intention of posthumously fulfilling Treize's world domination plot. -- -- With the help of former enemy Zechs Marquise and his mobile suit Tallgeese, the heroic pilots must reacquire their mobile suits to wage one final battle against the Khushrenada dynasty, including fighting against their former ally Wufei Chang, now aligning himself with Mariemaia's ambitions. -- -- -- Licensor: -- Bandai Entertainment -- OVA - Jan 25, 1997 -- 41,721 7.76
Sky Girls -- -- J.C.Staff -- 26 eps -- Original -- Mecha Military Sci-Fi -- Sky Girls Sky Girls -- In the near future, Earth is at war with aliens called Worms. The war leaves the population of Earth decimated, with young men the main casualties. The Worms were beaten back but their re-emergence heralds the creation of a mecha unit called the Sonic Divers, piloted by 3 young girls to counter this threat. -- TV - Jul 6, 2007 -- 21,726 7.02
Sky Girls OVA -- -- J.C.Staff -- 1 ep -- - -- Mecha Military Sci-Fi -- Sky Girls OVA Sky Girls OVA -- In the near future, Earth is at war with aliens called Worms. The war leaves the population of Earth decimated, with young men the main casualties. The Worms were beaten back but their re-emergence heralds the creation of a mecha unit called the Sonic Divers, piloted by 3 young girls to counter this threat. -- -- (Source: ANN) -- OVA - Aug 25, 2006 -- 6,733 6.63
SSSS.Gridman -- -- Trigger -- 12 eps -- Original -- Action Sci-Fi Mecha -- SSSS.Gridman SSSS.Gridman -- Yuuta Hibiki wakes up in the room of Rikka Takarada and notices two things: he has no memories, and he can hear a mysterious voice calling his name from a nearby room. On further inspection, he finds a robot—which introduces itself as Hyper Agent Gridman—behind the screen of an old computer. Much to Yuuta's surprise, Rikka cannot hear Gridman, nor can she see the ominous monsters looming over a thick fog as it envelopes the town outside. -- -- Another giant monster materializes in the city and proceeds to wreak havoc. Amidst the confusion, Yuuta is once again drawn to the old computer and merges with Gridman. Suddenly, he appears in the middle of the battle and is forced to fight the monster. Together with Rikka and fellow classmate Shou Utsumi, Yuuta forms the "Gridman Alliance" to defeat the monsters plaguing the city and find whoever is responsible for their emergence. -- -- -- Licensor: -- Funimation -- 213,180 7.17
https://commons.wikimedia.org/wiki/File:1_Emergence_-_Michael_David_Crawford_-_Geometric_Visions_1994_-_Ogg_Frog_q6.ogg
Alliance of Movements for the Emergence of Niger
Emergence
Emergence delirium
Emergence (disambiguation)
Emergence (Neil Sedaka album)
Emergence of agriculture in the Philippines
Emergence (Star Trek: The Next Generation)
Emergence (Symphony Number One album)
Emergence: The Connected Lives of Ants, Brains, Cities, and Software
Impact Wrestling Emergence
KwaZulu-Natal Elimination and Prevention of Re-emergence of Slums Act, 2007
The Emergence of Modern Turkey
The Emergence of the American University


change font "color":
change "background-color":
change "font-family":
change "padding":
change "table font size":
last updated: 2021-08-18 18:34:44
354486 site hits