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When asked how he developed his mathematical abilities so rapidly, he
replied "by studying the masters, not their
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
character and motivation. When we teach
a child design, we
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.
of life is a ragged
Archimedes. 287-212 BC. Greek mathematician, engineer, and physicist.
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.
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
The mathematical sciences particularly
exhibit order, symmetry, and
limitation; and these are the greatest forms of the beautiful.
Stefan. 1892-1945. Polish mathematician who founded modern functional
is a person who can findanalogies
is one who can seeanalogies
and the best
can notice analogies
One can imagine that the ultimate
is one who can see 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.
taught me that without assumptions
there is no proof. Therefore, in any
examine the assumptions.
In H. Eves Return to Mathematical Circles., Boston: Prindle, Weber and
Bennis, Warren G.
b.1925. American writer, educator.
keep their eyes on the horizon, not
just on the bottom line.
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.
1891?-1963. French modernist author.
The composer opens the cage door for arithmetic,
the draftsman gives geometry its freedom.
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
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.
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
Dee, John. 1527-1608. English mathematician and
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
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.
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
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.
is the right
of all painting,
I have decided to teach
its rudiments and principles to all youngsters eager for
Course in the Art of Measurement
1879-1955. German-born American physicist and Nobel laureate.
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
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.'
Philosopher-Scientist, by Paul Arthur Schilpp, 1951.
Emerson, Ralph Waldo.
1803-1882. American essayist.
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
Escher. 1898-1972. Artist, and leading exponent of the art of
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
rotations. The result is a
´mathematical tessellation of artistic proportions.´
325 BC-265 BC. Greek geometer, author of the most important textbook of
all time, The Elements.
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,
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
that there was no royal road to geometry.'
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)
BC. Greek dramatist.
Mighty is geometry; joined with
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
The geometrical method is not so rigidly confined to
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
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
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.
Hans.1905 - 1990. German Mathematician.
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.
Galilei. 1564-1642. Italian astronomer, mathematician, and physicist.
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.
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
1601–1658. Spanish Jesuit philosopher and writer.
One must pass through the
time before arriving at the
center of opportunity.
Hardy, Godfrey H.
1877-1947. British mathematician.
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.
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
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,
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
In O. L. Dick (ed.) Brief Lives, Oxford: Oxford University
Ibn Khaldun, 1332-1406. Arab historian.
the intellect and sets one's mind right. All its
are very clear and orderly. It is hardly possible for errors to enter
because it is well arranged and orderly. Thus, the mind that constantly
applies itself to
is not likely to fall into error. In this convenient way, the person who
It has been assumed that the following statement was written upon
Plato's door: "No one who is not a
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
The Dot and the Line: A Romance in Lower
Johannes. 1571-1630. German astronomer and mathematician.
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.
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
geometry. That is how I have always
Joseph Louis. 1736-1813. French mathematician.
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
No human investigation can be called real science if it cannot be
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
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.
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
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
Benoit. 1924-. Mathematician born in Warsaw. Fractal geometer.
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
circles and squares. By the sages,
the human relations are perfectly
Napoleon Bonaparte. 1769-1821. French Emperor
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
Friedrich. 1844-1900. German classical scholar, philosopher and critic
high civilization is a
pyramid: it can stand only on a
broad base; its primary prerequisite is
a strong and soundly consolidated mediocrity.
Isaac. 1642–1727, English mathematician and natural philosopher
is the glory of geometry that from
so few principles,
fetched from without, it is able to accomplish
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
figures of the planets, the paths
of comets, and the tides of the seas first
of Alexandria. ca 290-350. Greek geometer
. . 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.
1623-1662. French mathematician, philosopher and physicist.
Nature is an infinite
sphere whose center is everywhere and
whose circumference is nowhere.
Professor of Mathematics at Santa Clara University
is a skill of the
as well as of the
Pei, I.M. Chinese
Architect, born in 1917.
trapezoid on the back of an envelope. I
drew a diagonal line
across the trapezoid and produced
two triangles. That was the
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
of which geometryaims
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.
... being perpetually
by his familiar siren, that is, by his
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
tracegeometrical figures in the ashes of
the fire, and with his finger
lines upon his body when it was anointed with oil, being in a state of
by his science.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Poincaré, Jules Henri. 1854-1912. French mathematician and physicist.
selection our mind has
itself to the conditions of the external world. It has adopted the
most advantageous to the
or, in other words, the most convenient.
is not true, it is advantageous.
Science and Method.
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)
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
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.
just as much as in poetry. Likhtenshtein
BC. Greek philosopher and mathematician
geometry in the humming of the strings.
There is music in the spacings of
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,
The oldest, shortest words - "yes' and "no"
- are those which require the most thought.
You, who wish to study great and wonderful things, who wonder about the
movement of the stars, must read these
Knowing these ideas will open
the door to all of astronomy and to certain
Bernhard. 1826-1866. German mathematician and educator.
only I had the theorems! Then I
should find the proofs
Rodin, François Auguste Rene. 1840-1917. French
sculptor noted for his renderings of the human form.
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,
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
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
or quantity of anything without changing its
Victor Hugo. 1802-1885.
French author, the most important of the Romantic authors in the French
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.
Whitehead, Alfred North. 1861-1947, British mathematician, logician and
I regret that it has been
necessary for me in this lecture to administer such a large dose of
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.
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
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.