Geometry Quotes

Math Education

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Abel, Niels Henrik. 1802-1829. Norwegian mathematician

When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils."

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."

Alger, William R. 1823-1905. U.S. minister, writer.

The line of life is a ragged diagonal between duty and desire.

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

The works of Archimedes are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader. A History of Greek Mathematics. 1921. Heath, Sir Thomas L. Heath.

Aristophanes. ca. 446 BC-385 BC. Greek comic poet.

The geometer Meton: "With the straight ruler I set to work to inscribe a square within this circle; in its centre will be the market-place, into which all the straight streets will lead, converging to this centre like a star, which, although only orbicular, sends forth its rays in a straight line from all sides."

THE BIRDS by Aristophanes, Part 15.

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.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.

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.

Bell, Eric Temple. 1883-1960. Scottish-American mathematician and professor at Caltech.

"With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics." Coxeter and Greitzer 1967, Geometry Revisited, p. 1.

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. The Search For Truth.

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. In H. Eves Return to Mathematical Circles., Boston: Prindle, Weber and Schmidt, 1988.

Bennis, Warren G. b.1925. American writer, educator.

Leaders keep their eyes on the horizon, not just on the bottom line.

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.

Cibber, Colley. 1671 ­1757. English dramatist and actor-manager.

Oh! how many torments lie in the small circle of a wedding ring.

Cocteau, Jean. 1891?-1963. French modernist author.

The composer opens the cage door for arithmetic,
the draftsman gives geometry its freedom.

Cronkite, Walter. b1916. U.S. Broadcast Journalist
When Moses was alive, these pyramids were a thousand years old Here began the history of architecture. Here people learned to measure time by a calendar, to plot the stars by astronomy and chart the earth by geometry. And here they developed that most awesome of all ideas - the idea of eternity.

H.S.M. Coxeter. 1907-2003. The twentieth century's preeminent classical geometer and mathematician of polyhedra

"I’m a Platonist - a follower of Plato - who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."

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.

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?"

Cudworth, Ralph J. 1617–1688. British theologian, philosopher.

Sense is a line, the mind is a circle. Sense is like a line which is the flux of a point running out from itself, but intellect like a circle that keeps within itself.

Dee, John. 1527-1608. English mathematician and astrologer.

There is nothing which so much beautifies and adorns the soul and mind of man as does knowledge of the good arts and sciences. Many arts there are which beautify the mind of man; but of all none do more garnish and beautify it than those arts which are called mathematical, unto the knowledge of which no man can attain, without perfect knowledge and instruction of the principles, grounds, and Elements of Geometry.
The Mathematical Preface

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 judgment, which is in my mind.

These long chains of perfectly simple and easy reasoning 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.

I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery. La Geometrie.

Dürer, Albrecht. 1471-1528. German artist.

Whoever... proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.

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.

Course in the Art of Measurement

Einstein, Albert. 1879-1955. German-born American physicist and Nobel laureate.

A human being is part of a whole, called by us the Universe, a part limited in time and space. He experiences himself, his thoughts and feelings, as something separated from the rest a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest us. Our task must be to free ourselves from this prison by widening our circles of compassion to embrace all living creatures and the whole of nature in its beauty.

The pursuit of truth and beauty is a sphere of activity in which we are permitted to remain children all our lives.

To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science.

About Pythagoras Theorem Proof

'At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which --- though by no means evident --- could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in ``proving'' this theorem on the basis of the similarity of triangles ... for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time to be possible in geometry.' Albert Einstein: Philosopher-Scientist, by Paul Arthur Schilpp, 1951.

Emerson, Ralph Waldo. 1803-1882. American essayist.

A man finds room in the few square inches of the face for the traits of all his ancestors; for the expression of all his history, and his wants.

The life of man is a self-evolving circle.

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.´

Euclid. About 325 BC-265 BC. Greek geometer, author of the most important textbook of all time, The Elements.

'This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time.'  Thomas L. Heath

'Almost from the time of its writing and lasting almost to the present, The Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, The Elements may be the most translated, published, and studied of all the books produced in the Western world.' Bartel Leendert van der Waerden (1903-1996)

'Ptolemy the First  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.' Commentary on Euclid's Elements I. Proclus Diadochus. 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 three pence, since he must make gain out of what he learns.’” Euclid, Elements (ed. Thomas L. Heath)

Euripides. 480?-406 BC. Greek dramatist.

Mighty is geometry; joined with art, resistless.

Fermat, Pierre de. 1601-1665. French lawyer and mathematician. Concerning the lost proof of his "Last Theorem."

I have discovered a truly marvelous demonstration which this margin is too narrow to contain.

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.

Ford, Henry. 1863-1947. American industrialist and pioneer of the assembly-line production method.

If there is any one secret of success, it lies in the ability to get the other person's point of view and see things from that person's angle as well as from your own.

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.

Freudenthal Hans.1905 - 1990. German Mathematician.

Geometry is grasping space . . . that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it’. Freudenthal 1973: 403.

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.

Opere Il Saggiatore

Gide, Andre. 1869-1951. French critic, essayist, & novelist.

It is not always by plugging away at a difficulty and sticking to it that one overcomes it; often it is by working on the one next to it. Some things and some people have to be approached obliquely, at an angle.

Gracian, Baltasar. 1601–1658. Spanish Jesuit philosopher and writer.

One must pass through the circumference of time before arriving at the center of opportunity.

Hardy, Godfrey H. 1877-1947. British mathematician.

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.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hobbes, Thomas. 1588-1679. English philosopher

And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning. Leviathan 1651, Chapter IV, Of Speech.

For there is not one of them that begins his ratiocination from the definitions or explications of the names they are to use; which is a method that hath been used only in geometry, whose conclusions have thereby been made indisputable................ . For who is so stupid as both to mistake in geometry, and also to persist in it, when another detects his error to him?  Leviathan 1651. Chapter V, Of Reason and Science.

For I doubt not, but if it had been a thing contrary to any man's right of dominion, or to the interest of men that have dominion, that the three angles of a triangle should be equal to two angles of a square, that doctrine should have been, if not disputed, yet by the burning of all books of geometry suppressed, as far as he whom it concerned was able. Leviathan 1651. Chapter XI, Of the Difference of Manners.

Aubrey, John. 1626-1697. English antiquarian. 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", 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. Dick (ed.) Brief Lives, Oxford: Oxford University Press, 1960.

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."

Juster Norton. 1929. American architect and author.

"Once upon a time there was a sensible straight line who was hopelessly in love with a dot. 'You're the beginning and the end, the hub, the core and the quintessence,' he told her tenderly, but the frivolous dot wasn't a bit interested, for she only had eyes for a wild and unkempt squiggle who never seemed to have anything on his mind at all. All of the line's romantic dreams were in vain, until he discovered . . . angles! Now, with newfound self-expression, he can be anything he wants to be--a square, a triangle, a parallelogram. . . . And that's just the beginning!"

The Dot and the Line: A Romance in Lower Mathematics (1963).

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.

Conversation with the Sidereal Messenger (an open letter to Galileo Galilei)

Where there is matter, there is geometry.

Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.

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.

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.

Leonardo da Vinci. 1452-1519. Florentine artist, engineer, musician, and scientist.

Nothing can be found in nature that is not part of science, like continuous quantity, that is to say, geometry, which, commencing with the surfaces of bodies, is found to have its origins in lines, the boundary of these surfaces. Yet we do not remain satisfied with this, in that we know that line has its conclusion in a point, and nothing can be smaller than that which is a point. Therefore the point is the first principle of geometry, and no other thing can be found either in nature or in the human mind that can give rise to the point....  The principle of the science of painting is the point; second is the line; third is the surface; fourth is the body which is enclosed by these surfaces. And that is just what is to be represented...since in truth the scope of painting does not extend beyond the representation of the solid body or the shape of all the things that are visible.

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

While I thought I was learning how to live, I have been learning how to die.
Quoted in Des MacHale, Wisdom (London, 2002).

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

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.

Mencius. ca 372 BC - 289 BC.  Chinese philosopher and sage.

The compass and square produce perfect circles and squares. By the sages, the human relations are perfectly exhibited.

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.

Nietzsche, Friedrich. 1844-1900. German classical scholar, philosopher and critic of culture.

A high civilization is a pyramid: it can stand only on a broad base; its primary prerequisite is a strong and soundly consolidated mediocrity.

Newton, Isaac. 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.

The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.

His epitaph: Who, by vigor of mind almost divine, the motions and figures of the planets, the paths of comets, and the tides of the seas first demonstrated.

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 in constructing each. Synagoge or the Mathematical Collection, Book V.

Pascal, Blaise. 1623-1662. French mathematician, philosopher and physicist.

Nature is an infinite sphere whose center is everywhere and whose circumference is nowhere.

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.

Pei, I.M. Chinese Architect, born in 1917.

I sketched a trapezoid on the back of an envelope. I drew a diagonal line across the trapezoid and produced two triangles. That was the beginning.

In commenting on the inspiration for his design of the National Gallery's Art East Building, Washington D.C.

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.

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.

In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

Poincaré, Jules Henri. 1854-1912. French mathematician and physicist. 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.

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. 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)

Proclus Diadochus. 411 - 485.

On Euclid: According to most accounts, geometry was first discovered among the Egyptians, taking its origin from the measurement of areas. For they found it necessary by reason of the flooding of the Nile, which wiped out everybody's proper boundaries. Nor is there anything surprising in that the discovery both of this and of the other sciences should have had its origin in a practical need, since everything which is in process of becoming progresses from the imperfect to the perfect.

Pushkin, Aleksander Sergeevich. 1799-1837. Russian author.

Inspiration is needed in geometry, just as much as in poetry. 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.

Above the cloud with its shadow is the star with its light. Above all things reverence thyself.

If there be light, then there is darkness; if cold, heat; if height, depth; if solid, fluid; if hard, soft; if rough, smooth; if calm, tempest; if prosperity, adversity; if life, death.

The oldest, shortest words - "yes' and "no" - are those which require the most thought.

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

Riemann Bernhard. 1826-1866. German mathematician and educator.

If only I had the theorems! Then I should find the proofs easily enough.

Rodin, François Auguste Rene. 1840-1917. French sculptor noted for his renderings of the human form.

I have come to realize that geometry is at the bottom of sentiment or rather that each expression of sentiment is made by a movement governed by geometry. Geometry is everywhere present in nature. A woman combing her hair goes through a series of rhythmic movements that constitute a beautiful harmony. The entire rhythm of the body is governed by law.

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. 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.

Swift, Jonathan. 1667 - 1745. Anglo-Irish writer and satirist.

Gulliver's Travels (1726). About the people of Laputa (Chapter II):

Their houses are very ill built, the walls bevil, without one right angle in any apartment; and this defect arises from the contempt they bear to practical geometry, which they despise as vulgar and mechanic; those instructions they give being too refined for the intellects of their workmen, which occasions perpetual mistakes. And although they are dexterous enough upon a piece of paper, in the management of the rule, the pencil, and the divider, yet in the common actions and behavior of life, I have not seen a more clumsy, awkward, and unhandy people, nor so slow and perplexed in their conceptions upon all other subjects, except those of mathematics and music. They are very bad reasoners, and vehemently given to opposition, unless when they happen to be of the right opinion, which is seldom their case. Imagination, fancy, and invention, they are wholly strangers to, nor have any words in their language, by which those ideas can be expressed; the whole compass of their thoughts and mind being shut up within the two forementioned sciences.

Thales of Miletus. 624 BC - 547 BC. Greek philosopher and pioneer of Geometry.

Nothing is more active than thought, for it travels over the universe, and nothing is stronger than necessity for all must submit to it.

The skilful man is superior to the strong man.

Don't come to a conclusion before listening to both sides.

A small spark is enough to burn down a whole forest.

Hope is the only good that is common to all men; those who have nothing else possess hope still.

What is God? That what has nor a beginning nor an end!

What is the most difficult thing on earth? To know yourself!

What is the most easy thing on earth? To give advice to other people!

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.

Victor Hugo. 1802-1885. French author, the most important of the Romantic authors in the French language.

Mankind is not a circle with a single center but an ellipse with two focal points of which facts are one and ideas the other.

Voltaire. François Marie Arouet. 1694-1778. French philosopher and author.

There are no sects in geometry.

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.

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

Williams, Tennessee. 1911-1983. American playwright.

What is straight? A line can be straight, or a street, but the human heart, oh, no, it's curved like a road through mountains.

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.

Insights on the Shortest Distance

TSDB2P = The Shortest Distance Between 2Points


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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