"Philosophia Perennis--the phrase was coined by Leibniz; but the thing--
the metaphysic that recognizes a divine Reality substantial to the world of
things and lives and minds; the psychology that finds in the soul something
similar to, or even identical with, divine Reality; the ethic that places man's
final end in the knowledge of the immanent and transcendent Ground of all
being--the thing is immemorial and universal."
Aldous Huxley. (1944). The Perennial Philosophy
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Ten cycles of the moon the Roman year comprised:
This number then was held in high esteem,
Because, perhaps, on fingers we are wont to count,
Or that a woman in twice five months brings forth,
Or else that numbers wax till ten they reach
And then from one begin their rhythm anew.
—Ovid, Fasti, III.
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"The main duty of the historian of mathematics,
as well as his fondest privilege, is
to explain the humanity of mathematics, to
illustrate its greatness, beauty and dignity,
and to describe how the incessant efforts
and accumulated genius of many generations
have built up that magnificent monument,
the object of our most legitimate
pride as men, and of our wonder, humility
and thankfulness, as individuals. The study
of the history of mathematics will not
make better mathematicians but gentler
ones, it will enrich their minds, mellow
their hearts, and bring out their finer
qualities."
-G. Sarton
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"The major pitfalls in investment are buried deep in the emotional recesses of the individual," he replied. "In other words, the enemy is not the fluctuating circumstances in the world; it is the individual himself." As a Chinese and a philosopher, he was aware of Taoist and Zen cultural characteristics, such as "tranquillity as the foundation of wisdom" (Platform Sutra), "the illusiveness of all phenomenal forms (manifestations)" (Diamond Sutra), "reversal as the movement of the Tao" (Tao Te Ching), and "fasting of the mind" (Chuang Tzu).
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Alexander, Jane. Chairman, National Endowment for the Arts (1993-1997)
"When we teach a child to sing or play the flute, we teach her how to listen. When we teach her to draw, we teach her to see. When we teach a child to dance, we teach him about his body and about space, and when he acts on a stage, he learns about character and motivation. When we teach a child design, we reveal the geometry of the world. When we teach children about the folk and traditional arts and the great masterpieces of the world, we teach them to celebrate their roots and find their own place in history."
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Archimedes. 287-212 BC Greek mathematician, engineer, and physicist.
Soldier, stand away from my diagram.
Supposedly spoken by Archimedes to the Roman soldier who killed him
Perhaps the best indication of what Archimedes truly loved most is his request that his tombstone include a cylinder circumscribing a sphere, accompanied by the inscription of his amazing theorem that the sphere is exactly two-thirds of the circumscribing cylinder in both surface area and volume!"
Laubenbacher and Pengelley, p. 95
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Aristotle. 384-22 BC. Greek philosopher.
There are some who, because the point is the limit and extreme of the line, the line of the plane, and the plane of the solid, think there must be real things of this sort.
We do not know a truth without knowing its cause.
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Banach Stefan. 1892-1945. Polish mathematician who founded modern functional analysis.
A mathematician is a person who can find analogies between theorems, a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.
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Cartier-Bresson, Henri. b1908. French photographer, painter and draughtsman
For me photography is to place one's head, heart and eye along the same line of sight. It is a way of life. This attitude requires concentration, sensitivity, a discipline of mind and a sense of geometry.
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Cocteau, Jean. 1891?-1963. French modernist author.
The composer opens the cage door for arithmetic,
the draftsman gives geometry its freedom.
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H.S.M. Coxeter. 1907-2003. The twentieth century's preeminent classical geometer and mathematician of polyhedra
In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles and spheres are seen to behave in strange but precisely determined ways.
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Crelle, August. 1780-1856 German civil engineer and mathematician.
It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?"
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Descartes, René. 1596-1650. French mathematician and philosopher.
Cogito, ergo sum. (I think, therefore I am.)
Thus what I thought I had seen with my eyes, I actually grasped solely with the faculty of judgement, which is in my mind.
These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered.
Discours de la Méthode. 1637.
A inspiração, por sua vez, vinha da geometria, na qual partia-se de conceitos simples para descrever progressivamente entidades mais complexas:
“Ces longues chaînes de raisons, toutes simples et faciles, dont les géomètres ont coutume de se servir pour parvenir à leurs plus difficiles démonstrations, m'avoient donné occasion de m'imaginer que toutes les choses qui peuvent tomber sous la connoissance des hommes s'entresuivent en même façon[...]” (Discurso, parte 4)
(“Estas longas cadeias de razão, todas simples e fáceis, sobre as quais os geômetras costumam se servir para chegar às mais difíceis demonstrações, me levaram a imaginar que todas as coisas que podem penetrar na consciência dos homens são ligadas da mesma maneira [...]”)
"As longas cadeias de raciocínios tão simples e fáceis, de que os geômetras costumam servir-se para chegar às suas mais difíceis demonstrações proporcionaram-me o desejo de imaginar que todas as coisas, a respeito das quais o homem pode ter conhecimento, se seguem do mesmo modo, desde que ele se abstenha de aceitar por verdadeira uma coisa que não o seja e que respeite sempre a ordem necessária para deduzir uma coisa da outra, nada haverá tão distante que não se chegue a alcançar por fim, nem tão oculto que não se possa descobrir".
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Dürer, Albrecht. 1471-1528. German artist.
And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art.
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Course in the Art of Measurement
M.C. Escher. 1898-1972. Artist, and leading exponent of the art of tessellation.
The geometry of space translates to a reoccurring theme in my creations: the tessellation. A tessellation is an arrangement of closed shapes that completely covers the plane without overlapping and without leaving gaps. The regular division of the plane had been considered solely in theory prior to me, some say. I diverged from traditional approaches, and chose instead to find solutions visually. Where other mathematicians used notebooks, I preferred to use a canvas.
To gain access to a greater number of designs, I used transformational geometry techniques including reflections, glide reflections, translations, and rotations. The result is a ´mathematical tessellation of artistic proportions´
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Euclid. About 325 BC-265 BC.
Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by study of the Elements, whereupon Euclid answered that there was no royal road to geometry.
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Commentary on Euclid's Elements I. Proctus Diadochus. AD 410-485.
According to Stobaeus, “some one who had begun to read geometry with Euclid, when he had learnt the first theorem, asked Euclid, ‘But what shall I get by-learning these things?’ Euclid called his slave and said ‘Give him threepence, since he must make gain out of what he learns.’”
Euclid, Elements (ed. Thomas L. Heath)
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Euripides. 480?-406 BC Greek dramatist
Mighty is geometry; joined with art, resistless.
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Fontenelle, Bernard de. 1657 - 1757. French mathematician and philosopher.
The geometrical method is not so rigidly confined to geometry itself that it cannot be applied to other branches of knowledge as well. A work of morality, politics, criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry.
Preface sur l'Utilité des Mathématiques et de la Physique, 1729.
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Freud Sigmund. 1856-1939. Austrian physician and pioneer psychoanalyst.
I have an infamously low capability for visualizing spatial relationships which made the study of geometry and all subjects derived from it impossible to me.
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Galileo Galilei, 1564 - 1642. Italian astronomer, mathematician, and physicist.
The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
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Opere Il Saggiatore
Aubrey, John. 1626-1697. English antiquarian.
[About Thomas Hobbes (1588-1679. English philosopher):]
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", said 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.
In O. L. D*** (ed.) Brief Lives, Oxford: Oxford University Press, 1960.
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Ibn Khaldun, 1332-1406. Arab historian
Geometry enlightens the intellect and sets one's mind right. All its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence. It has been assumed that the following statement was written upon Plato's door: "No one who is not a geometrician may enter our house."
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Kepler Johannes. 1571-1630. German astronomer and mathematician.
Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God.
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Conversation with the Sidereal Messenger (an open letter to Galileo Galilei)
Where there is matter, there is geometry.
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Lafayette Marquis de. 1757-1834. French military, political, and revolutionary leader
How have I loved liberty? With the enthusiasm of religion, with the rapture of love, with the conviction of geometry. That is how I have always loved liberty.
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Lagrange, Joseph Louis. 1736-1813. French mathematician.
As long as algebra and geometry have been separated, their progress have been slow and their uses limited, but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.
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Leonardo da Vinci. 1452-1519. Florentine artist, engineer, musician, and scientist.
Nessuna humana investigazione si pio dimandara vera scienzia s'essa non passa per le matematiche dimonstrazione.
No human investigation can be called real science if it cannot be demonstrated mathematically.
Treatise on Painting.
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While I thought I was learning how to live, I have been learning how to die.
Quoted in Des MacHale, Wisdom (London, 2002).
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Lincoln, Abraham, 1809-65. 16th U.S. President
"He studied and nearly mastered the Six-books of Euclid (geometry) since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring."
Abraham Lincoln from Short Autobiography of 1860.
If you have ever studied geometry, you remember that by a course of reasoning, Euclid proves that all the angles in a triangle are equal to two right angles. Euclid has shown you how to work it out. Now, if you undertake to disprove that proposition, and to show that it is erroneous, would you prove it to be false by calling Euclid a liar?
Political Debates Between Lincoln and Judge Douglas. Fourth Joint Debate at Charleston, 1858
There are two ways of establishing a proposition. One is by trying to demonstrate it upon reason, and the other is, to show that great men in former times have thought so and so, and thus to pass it by the weight of pure authority. Now, if Judge Douglas will demonstrate somehow that this is popular sovereignty,—the right of one man to make a slave of another, without any right in that other, or anyone else to object,—demonstrate it as Euclid demonstrated propositions,—there is no objection. But when he comes forward, seeking to carry a principle by bringing it to the authority of men who themselves utterly repudiate that principle, I ask that he shall not be permitted to do it.
Speech of Hon. Abraham Lincoln. At Columbus, Ohio, September, 1859
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Mandelbrot, Benoit. 1924-. Mathematician born in Warsaw. Fractal geometer.
It's ironic that fractals, many of which were invented as examples of pathological behavior, turn out to be pathological at all. In fact they are the rule in the universe. Shapes, which are not fractal, are the exception. I love Euclidean geometry, but it is quite clear that it does not give a reasonable presentation of the world. Mountains are not cones, clouds are not spheres, trees are not cylinders, neither does lightning travel in a straight line. Almost everything around us is non-Euclidean.
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Napoleon Bonaparte. 1769-1821. French Emperor
There are axioms in probity, in honesty, in justice, just as much as there are axioms in geometry; and the truths of morality are no more at the mercy of a vote than are the truths of algebra.
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Isaac Newton, 1642–1727, English mathematician and natural philosopher
It is the glory of geometry that from so few principles, fetched from without, it is able to accomplish so much.
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Pappus of Alexandria. ca 290-350. Greek geometer
Bees. . . by virtue of a certain geometrical forethought . . . know that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material
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Pedersen, Jean. Professor of Mathematics at Santa Clara University
Geometry is a skill of the eyes and the hands as well as of the mind.
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Plato. ca 429-347 BC. Greek philosopher.
"Let no man ignorant of geometry enter here." Inscribed above the door Plato's Academy in Athens.
Geometry will draw the soul toward truth and create the spirit of philosophy
The knowledge of which geometry aims is the knowledge of the eternal.
Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is not unhappily allowed to fall down.
The Republic, VII, 52.
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Plutarch. ca 46-127. Greek essayist and biographer.
[about Archimedes:]
... being perpetually charmed by his familiar siren, that is, by his geometry, he neglected to eat and drink and took no care of his person; that he was often carried by force to the baths, and when there he would trace geometrical figures in the ashes of the fire, and with his finger draws lines upon his body when it was anointed with oil, being in a state of great ecstasy and divinely possessed by his science.
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In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Poincaré, Jules Henri. 1854-1912. French mathematician and physicist.
...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
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Science and Method.
Polya George. 1887-1985.
The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.
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Mathematical discovery (New York, 1981)
If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem, it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that theorem is true, you can start proving it.
How to Solve It (Princeton, 1945)
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Pushkin, Aleksander Sergeevich. 1799-1837. Russian author.
Inspiration is needed in geometry, just as much as in poetry.
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Likhtenshtein
Pythagoras. ca.560-ca.480 BC. Greek philosopher and mathematician
There is geometry in the humming of the strings. There is music in the spacings of the spheres.
Geometry is knowledge of the eternally existent.
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Regiomontanus, Johann. 1436-1476.
You, who wish to study great and wonderful things, who wonder about the movement of the stars, must read these theorems about triangles. Knowing these ideas will open the door to all of astronomy and to certain geometric problems.
De triangulis omnimodis
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Riemann Bernhard. 1826-1866. German mathematician and educator.
If only I had the theorems! Then I should find the proofs easily enough.
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Russell Bertrand. 1872-1970. English pacifist, mathematician, philosopher, and author (Nobel, 1950).
At the age of eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love.
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Valéry, Paul. 1871-1945. French poet and critic.
In the physical world, one cannot increase the size or quantity of anything without changing its quality. Similar figures exist only in pure geometry.
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Voltaire. François Marie Arouet. 1694-1778. French philosopher and author.
There are no sects in geometry.
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Whitehead, Alfred North. 1861-1947, British mathematician, logician and philosopher
I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry. I do not apologize, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are.
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Wigner, Eugene Paul. 1902-1995. Hungarian-born Amer. physicist (Nobel, 1963)
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, 1960
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Wittgenstein, Ludwig. 1889-1951. Austrian philosopher.
We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.
Tractatus Logico Philosophicus, New York, 1922.
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Insights on the Shortest Distance
TSDB2P = The Shortest Distance Between 2 Points
1. In Geometry TSDB2P is a straight line.
2. In marriage TSDB2P is love.
3. In mountain climbing TSDB2P is in the heart.
4. In sickness TSDB2P is relief.
5. In deep poverty TSDB2P is realizing you have plenty to give.
6. In a career TSDB2P is integrity.
7. In parenting TSDB2P allowing them to grow from their own mistakes.
8. In a friendship TSDB2P is trust.
9. In learning TSDB2P is a mind awaiting discovery.
10. In personal growth TSDB2P is learning your lesson the first time.
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Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible of proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
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