Quotes about triangle

A collection of quotes on the topic of triangle, angle, other, circle.

Quotes about triangle

“Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick.”

Thales (-624–-547 BC) ancient Greek philosopher and mathematician

W. W. Rouse Ball, A Short Account of the History of Mathematics (1893, 1925)

Sun

“Music, love, death. Certainly a triangle of sorts; maybe even an eternal one.

"The only people who can see the whole picture," he murmured, "are the ones who step out of the frame." (The ground beneath her feet.)”

Salman Rushdie (1947) British Indian novelist and essayist

Love, Death, People, Music

“Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth.”

Galileo Galilei book The Assayer

From Italian: La filosofia è scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi agli occhi (io dico l'Universo), ma non si può intendere, se prima non il sapere a intender la lingua, e conoscer i caratteri ne quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro labirinto.
Other translations:
Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
The Assayer (1623), as translated by Thomas Salusbury (1661), p. 178, as quoted in The Metaphysical Foundations of Modern Science (2003) by Edwin Arthur Burtt, p. 75.
Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.
As translated in The Philosophy of the Sixteenth and Seventeenth Centuries (1966) by Richard Henry Popkin, p. 65
Il Saggiatore (1623)
Source: Galilei, Galileo. Il Saggiatore: Nel Quale Con Bilancia Efquifita E Giufta Si Ponderano Le Cofe Contenute Nellalibra Astronomica E Filosofica Di Lotario Sarsi Sigensano, Scritto in Forma Di Lettera All'Illustr. Et Rever. Mons. D. Virginio Cesarini. In Roma: G. Mascardi, 1623. Google Play. Google. Web. 22 Dec. 2015. <https://play.google.com/store/books/details?id=-U0ZAAAAYAAJ>.

Books, Mathematics, Understanding, Learning

“The general reference of the philosophical discussion is usually the triangle world: world-language-subject, the relation of the subject to the world of objects, mediated through language.”

Slavoj Žižek book The Sublime Object of Ideology

77
The Sublime Object of Ideology (1989)

World

“It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.”

Archimedes book On the Equilibrium of Planes

Book 1, Proposition 14.
On the Equilibrium of Planes

“In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.”

Archimedes book On the Equilibrium of Planes

Book 1, Proposition 13.
On the Equilibrium of Planes

“Everybody knows that the great reversed triangle of land, with its base in the north and its apex in the south, which is called India, embraces fourteen hundred thousand square miles, upon which is spread unequally a population of one hundred and eighty millions of souls. The British Crown exercises a real and despotic dominion over the larger portion of this vast country, and has a governor-general stationed at Calcutta, governors at Madras, Bombay, and in Bengal, and a lieutenant-governor at Agra.

But British India, properly so called, only embraces seven hundred thousand square miles, and a population of from one hundred to one hundred and ten millions of inhabitants. A considerable portion of India is still free from British authority; and there are certain ferocious rajahs in the interior who are absolutely independent.”

Jules Verne book Around the World in Eighty Days

Personne n'ignore que l'Inde — ce grand triangle renversé dont la base est au nord et la pointe au sud — comprend une superficie de quatorze cent mille milles carrés, sur laquelle est inégalement répandue une population de cent quatre-vingts millions d'habitants. Le gouvernement britannique exerce une domination réelle sur une certaine partie de cet immense pays. Il entretient un gouverneur général à Calcutta, des gouverneurs à Madras, à Bombay, au Bengale, et un lieutenant-gouverneur à Agra.Mais l'Inde anglaise proprement dite ne compte qu'une superficie de sept cent mille milles carrés et une population de cent à cent dix millions d'habitants. C'est assez dire qu'une notable partie du territoire échappe encore à l'autorité de la reine; et, en effet, chez certains rajahs de l'intérieur, farouches et terribles, l'indépendance indoue est encore absolue.
Source: Around the World in Eighty Days (1873), Ch. X: In Which Passepartout Is Only Too Glad to Get Off with the Loss of His Shoes

Exercises, Soul, Country

“I don't know why my brain has kept all the words to the Gilligan's Island theme song and has deleted everything about triangles.”

Jeff Foxworthy (1958) American stand-up comedian

The Tonight Show, 27 March 2007

“[L]et us assign the figures that have come into being in our theory to fire and earth and water and air. To earth let us give the cubical form; for earth is least mobile of the four and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most stable. Now of the triangles which we assumed as our starting-point that with equal sides is more stable than that with unequal; and of the surfaces composed of the two triangles the equilateral quadrangle necessarily is more stable than the equilateral triangle… Now among all these that which has the fewest bases must naturally in all respects be the most cutting and keen of all, and also the most nimble, seeing it is composed of the smallest number of similar parts… Let it be determined then… that the solid body which has taken the form of the pyramid [tetrahedron] is the element and seed of fire; and the second in order of generation let [octahedron] us say to be that of air, and the third [icosahedron] that of water. Now all these bodies we must conceive as being so small that each single body in the several kinds cannot for its smallness be seen by us at all; but when many are heaped together, their united mass is seen…”

Plato book Timaeus

Section 55e–56c, Tr. R. D. Archer-Hind, The Timaeus of Plato (1888) pp. 199-201. https://books.google.com/books?id=q2YMAAAAIAAJ&pg=PA199 Timaeus

Nature, Water, Parting

“Alec: Catarina made up the Bermuda Triangle?
Magnus: Don't be ridiculous, Alexander. That was Ragnor.”

Cassandra Clare book Lord of Shadows

Source: Lord of Shadows

“Way out in the country tonight he could smell the pumpkins ripening toward the knife and the triangle eye and the singeing candle.”

Ray Bradbury book Dandelion Wine

Source: Dandelion Wine

Way, Singing, Country

“I want to sympathize, I do, but the love triangle is just too delicious. The determined rock star and the possessive billionaire. Rawr.”

Sylvia Day (1973) American writer

Source: Reflected in You

Love, Star

“To appreciate the nature of fractals, recall Galileo's splendid manifesto that "Philosophy is written in the language of mathematics and its characters are triangles, circles and other geometric figures, without which one wanders about in a dark labyrinth." Observe that circles, ellipses, and parabolas are very smooth shapes and that a triangle has a small number of points of irregularity. Galileo was absolutely right to assert that in science those shapes are necessary. But they have turned out not to be sufficient, "merely" because most of the world is of infinitely great roughness and complexity. However, the infinite sea of complexity includes two islands: one of Euclidean simplicity, and also a second of relative simplicity in which roughness is present, but is the same at all scales.”

Benoît Mandelbrot (1924–2010) Polish-born, French and American mathematician

A Theory of Roughness (2004)

World, Mathematics, Nature, Science

“Had the acute-angled rabble been all, without exception, absolutely destitute of hope and of ambition, they might have found leaders in some of their many seditious outbreaks, so able as to render their superior numbers and strength too much even for the wisdom of the Circles. But a wise ordinance of Nature has decreed that, in proportion as the working-classes increase in intelligence, knowledge, and all virtue, in that same proportion their acute angle (which makes them physically terrible) shall increase also and approximate to the comparatively harmless angle of the Equilateral Triangle. Thus, in the most brutal and formidable of the soldier class — creatures almost on a level with women in their lack of intelligence — it is found that, as they wax in the mental ability necessary to employ their tremendous penetrating power to advantage, so do they wane in the power of penetration itself.

How admirable is this Law of Compensation! And how perfect a proof of the natural fitness and, I may almost say, the divine origin of the aristocratic constitution of the States in Flatland! By a judicious use of this Law of Nature, the Polygons and Circles are almost always able to stifle sedition in its very cradle, taking advantage of the irrepressible and boundless hopefulness of the human mind. Art also comes to the aid of Law and Order. It is generally found possible — by a little artificial compression or expansion on the part of the State physicians — to make some of the more intelligent leaders of a rebellion perfectly Regular, and to admit them at once into the privileged classes; a much larger number, who are still below the standard, allured by the prospect of being ultimately ennobled, are induced to enter the State Hospitals, where they are kept in honourable confinement for life; one or two alone of the more obstinate, foolish, and hopelessly irregular are led to execution.”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 3. Concerning the Inhabitants of Flatland

Life, Wisdom, Women, Law

“These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane… for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

Family

“The agitation for the Universal Colour Bill continued for three years; and up to the last moment of that period it seemed as though Anarchy were destined to triumph.

A whole army of Polygons, who turned out to fight as private soldiers, was utterly annihilated by a superior force of Isosceles Triangles — the Squares and Pentagons meanwhile remaining neutral. Worse than all, some of the ablest Circles fell a prey to conjugal fury. Infuriated by political animosity, the wives in many a noble household wearied their lords with prayers to give up their opposition to the Colour Bill; and some, finding their entreaties fruitless, fell on and slaughtered their innocent children and husband, perishing themselves in the act of carnage. It is recorded that during that triennial agitation no less than twenty-three Circles perished in domestic discord.

Great indeed was the peril. It seemed as though the Priests had no choice between submission and extermination; when suddenly the course of events was completely changed by one of those picturesque incidents which Statesmen ought never to neglect, often to anticipate, and sometimes perhaps to originate, because of the absurdly disproportionate power with which they appeal to the sympathies of the populace.”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 10. Of the Suppression of the Chromatic Sedition

Children, Universe, Change, Fight

“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 (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Advertisement, p.4
The Differential and Integral Calculus (1836)

Dreams, Way, Help, Drawing

“They (fans) respond mostly to what WWE trains them to respond to. An ankle lock gets over because Kurt Angle does the ankle lock and everyone submits to it. A triangle by Undertaker doesn't get over because WWE has never trained the fans to accept that as a finish because no one ever taps to it. And it was the same thing when Shamrock was in WWE.”

Bryan Alvarez (1975) Professional wrestler, editor and publisher

Quoted by Corey David LaCroix, " The Fight Network bridging MMA/wrestling gap http://web.archive.org/web/20060113150444/http://slam.canoe.ca/Slam/Wrestling/2005/11/24/1321324.html", SLAM! Wrestling, (2005-11-24)

Training

“[Relativist] Rel. There is a well-known proposition of Euclid which states that "Any two sides of a triangle are together greater than the third side." Can either of you tell me whether nowadays there is good reason to believe that this proposition is true?
[Pure Mathematician] Math. For my part, I am quite unable to say whether the proposition is true or not. I can deduce it by trustworthy reasoning from certain other propositions or axioms, which are supposed to be still more elementary. If these axioms are true, the proposition is true; if the axioms are not true, the proposition is not true universally. Whether the axioms are true or not I cannot say, and it is outside my province to consider.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Space, Time and Gravitation (1920)

Universe, Parting

“Pearl Harbor is a two-hour movie squeezed into three hours, about how on December 7, 1941, the Japanese staged a surprise attack on an American love triangle. Its centerpiece is 40 minutes of redundant special effects, surrounded by a love story of stunning banality. The film has been directed without grace, vision, or originality, and although you may walk out quoting lines of dialog, it will not be because you admire them.”

Roger Ebert (1942–2013) American film critic, author, journalist, and TV presenter

Review http://www.rogerebert.com/reviews/pearl-harbor-2001 of Pearl Harbor (25 May 2001) Reviews, One-and-a-half star reviews

Love, Quotes

“It's offense you maybe can't live with because it opens up a crack inside your thinking, and if you look down into it you see there are evil things down there, and they have little yellow eyes that don't blink, and there's a stink down there in that dark and after a while you think maybe there's a whole other universe where a square moon rises in the sky, and the stars laugh in cold voices, and some of the triangles have four sides, and some have five, and some have five raised to the fifth power of sides. In this universe there might grow roses which sing. Everything leads to everything, he would have told them if he could. Go to your church and listen to your stories about Jesus walking on the water, but if I saw a guy doing that I'd scream and scream and scream. Because it wouldn't look like a miracle to me. It would look like an offense.”

Stephen King book It

It (1986)

Guy, Star, Thinking, Universe

“The unerring signs of truth and falsehood are clear, distinct consistency and contradiction. This is also the case with revelation, in so far as that it must, in common with other truths, be free from contradiction. And just as little as miracles can prove that twice two are five or that a triangle has four angles, can a contradiction lying in the history and dogmas of Christianity be removed by any number of miracles.”

Hermann Samuel Reimarus (1694–1768) German philosopher

Source: Fragments from Reimarus: Consisting of Brief Critical Remarks on the Object of Jesus and His Disciples as Seen in the New Testament, p. 75

Truth, History, Lying

“The debate regarding which individual factor, among the three key factors producing the environmental crisis, causes more damage - the size of the human population on the planet, excessive consumption of resources or unequal/ unjust distribution of resources among countries [the wealthier countries consume much more resources than poorer countries] - is like a debate about which contributes more to a triangle, the base or the ribs of the triangle. You can not separate the three factors. If we analyze the numbers over a relatively longer time interval, we will conclude that the size of the population has a bigger impact than consumption. On the other hand, consumption and unequal distribution are also important aspects. If we do not change these three factors all at the same time, the quality of our life will change dramatically. Today humanity is delivering a serious blow to nature, but it is clear that nature will deliver the final blow.”

Paul R. Ehrlich (1932) American scientist and environmentalist

Paul Ehrlich, People should produce far fewer children, or expect the worst http://www.haaretz.co.il/1.1875624 (Dec. 2012), Haaretz

Life, Nature, Change, Time

“We have merely (!) to measure the volume V of a sphere of radius r or the sum \sigma of the angles of a triangle of measured are \delta, and from the results to compute the value of K.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“The life of the spirit may be fairly represented in diagram as a large acute-angled triangle divided horizontally into unequal parts with the narrowest segment uppermost. The lower the segment the greater it is in breadth, depth, and area.”

Wassily Kandinsky (1866–1944) Russian painter

III. The Movement of the Triangle
1910 - 1915, Concerning the Spiritual in Art, 1911

Life, Parting

“.. [by making his work 'Onement', in 1948].. from then on I had to give up any relation to nature, as seen [by himself till then]. That doesn't mean that I think my things are mathematical or removed from life. By 'nature' I mean something very specific. I think that some abstractions - for example Kandinsky's - are really nature paintings. The triangles and the spheres or circles could be bottles. They could be trees, or buildings. I think that in 'Euclydean Abyss' and 'Onement' I removed myself from nature. But I did not remove myself from life.”

Barnett Newman (1905–1970) American artist

interview, April 1965, edited for broadcasting by the BBC first published in 'The Listener', Aug. 1972; as quoted in Interviews with American Artists, by David Sylvester; Chatto & Windus, London 2001, p. 37
1960 - 1970, Interview with David Sylvester 1. Spring 1965

Life, Mathematics, Nature, Thinking

“Had some inscrutable decree of fate ordained and made it certain, with a certainty not to be disturbed, that no candidate could be returned to Parliament who would not assert the earth to be triangular, there would rise immediately a clamorous assertion of triangularity among political aspirants. The test would be innocent. Candidates have swallowed, and daily do swallow, many a worse one. As might be this doctrine of a great triangle, so is the doctrine of Home Rule. Why is a gentleman of property to be kept out in the cold by some O'Mullins because he will not mutter an unmeaning shibboleth? "Triangular? Yes, or lozenge-shaped, if you please; but, gentleman, I am the man for Tipperary."”

Anthony Trollope book The Prime Minister

Source: The Prime Minister (1876), Ch. 11

Fate, Home

“That's one of those Andromedan ships! Look how you can see through the middle. That's because it's inter-dimensional. … You hear that? The coyotes are going crazy. They always do that when the big ships fly over! No one's got the triangles on tape. This'll really freak the Greer folks out!”

James Gilliland (1952) American academic and author

James during one of the close encounters with a UFO at his Sattva Sanctuary.

Flying

“You mention the circle and I agree with your definition.... why does the circle fascinates me? It is (1) the most modern form, but asserts itself unconditionally, (2) a precise but inexhaustible variable, (3) simultaneously stable and unstable, (4) simultaneously loud and soft, (5) a single tension that caries countless tensions within it. The circle is the synthesis of the greatest oppositions. It combines the concentric and the eccentric in a single form, and in balance. Of the three primary forms (triangle, square, circle), it points most clearly to the fourth dimension.”

Wassily Kandinsky (1866–1944) Russian painter

Quote from Kandinsky's letter to Will Grohmann, c. 1926; as cited in Kandinsky, Frank Whitford, Paul Hamlyn Ltd, London 1967, p. 36
1920 - 1930

“The doctrine of Right and Wrong, is perpetually disputed, both by Pen and the Sword: Whereas the doctrine of Lines, and Figures, is not so; because men care not, in that subject what be truth, as a thing that crosses no mans ambition, profit, or lust. For I doubt not, but if it had been a thing contrary to any mans right of dominion, or to the interest of men that have dominion, That the three Angles of a Triangle, should be equall 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.”

Thomas Hobbes book Leviathan

The First Part, Chapter 11, p. 80-81
Leviathan (1651)

Men, Truth, Books

“He [Tinky Winky] is purple—the gay-pride color, and his antenna is shaped like a triangle—the gay pride symbol.”

Jerry Falwell (1933–2007) American evangelical pastor, televangelist, and conservative political commentator

"Parents Alert: Tinky Winky Comes Out of the Closet" (February 1999), National Liberty Journal, quoted in [1999-02-15, Gay Tinky Winky bad for children, BBC News, http://news.bbc.co.uk/2/hi/entertainment/276677.stm]
about Tinky Winky, a character on the children's program Teletubbies

Color

“Halfway up the slope, guarded by a group of tall, slim, cypress-trees, nestled a small strawberry-pink villa, like some exotic fruit lying in the greenery. The cypress-trees undulated gently in the breeze, as if they were busily painting the sky a still brighter blue for our arrival.
The villa was small and square, standing in its tiny garden with an air of pink-faced determination. Its shutters had been faded by the sun to a delicate creamy-green, cracked and bubbled in places. The garden, surrounded by tall fuschia hedges, had the flower beds worked in complicated geometrical patterns, marked with smooth white stones. The white cobbled paths, scarcely as wide as a rake's head, wound laboriously round beds hardly larger than a big straw hat, beds in the shape of stars, half-moons, triangles, and circles all overgrown with a shaggy tangle of flowers run wild. Roses dropped petals that seemed as big and smooth as saucers, flame-red, moon-white, glossy, and unwrinkled; marigolds like broods of shaggy suns stood watching their parent's progress through the sky. In the low growth the pansies pushed their velvety, innocent faces through the leaves, and the violets drooped sorrowfully under their heart-shaped leaves. The bougainvillaea that sprawled luxuriously over the tiny iron balcony was hung, as though for a carnival, with its lantern-shaped magenta flowers. In the darkness of the fuschia-hedge a thousand ballerina-like blooms quivered expectantly. The warm air was thick with the scent of a hundred dying flowers, and full of the gentle, soothing whisper and murmur of insects.”

Gerald Durrell book My Family and Other Animals

My Family and Other Animals (1956)

Star, Flowers, Lying, Parent

“It is inappropriate to have Tim listening to the triangle and then have a deejay come on and... go into Radiohead or something.”

David Woodard (1964) American writer, conductor and businessman

Los Angeles Times (May 9, 2001)

“I haven't really touched machinery except for a few elementary mechanisms like levers and balances. You see nature and then you try to emulate it. But, of course, when I met Mondrian [Paris, 1930] I went home and tried to paint [for a while]. The basis of everything for me is the universe. The simplest forms in the universe are the sphere and the circle. I represent them by disks and then I vary them. My whole theory about art is the disparity that exists between form, masses and movement. Even my triangles are spheres, but they are spheres of a different shape.”

Alexander Calder (1898–1976) American artist

Question, Which has influenced you more, nature or modern machinery?
1950s - 1960s, interview with Alexander Calder', (1962)

Art, Home, Nature, Universe

“I order the club sandwich all the time, but I'm not even a member, man. I don't know how I get away with it. How'd it start anyway? I like my sandwiches with three pieces of bread. So do I! Well let's form a club then. Alright, but we need more stipulations. Yes we do; instead of cutting the sandwich once, let's cut it again. Yes, four triangles, and we will position them into a circle. In the middle we will dump chips. Or potato salad. Okay. I got a question for ya, how do you feel about frilly toothpicks? I'm for 'em! Well this club is formed; spread the word on menus nationwide. I like my sandwiches with alfalfa sprouts. Well then you're not in the fuckin' club!”

Mitch Hedberg (1968–2005) American stand-up comedian

track 2, "Sandwiches"
Mitch All Together (2003)

Feeling, Question, Time

“And we must invent dynamic designs to go with them and express them in equally dynamic shapes: triangles, cones, spirals, ellipses, circles, etc.”

Giacomo Balla (1871–1958) Italian artist

(Manuscript, 1914); as quoted in Futurism, ed. Didier Ottinger; Centre Pompidou / 5 Continents Editions, Milan, 2008, p. 148
Futurist Manifesto of Men's clothing,' 1913/1914

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

Bernhard Riemann (1826–1866) German mathematician

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

Imagination, Parting, Dependency

“Mathematics and philosophy are cultivated by two different classes of men: some make them an object of pursuit, either in consequence of their situation, or through a desire to render themselves illustrious, by extending their limits; while others pursue them for mere amusement, or by a natural taste which inclines them to that branch of knowledge. It is for the latter class of mathematicians and philosophers that this work is chiefly intended j and yet, at the same time, we entertain a hope that some parts of it will prove interesting to the former. In a word, it may serve to stimulate the ardour of those who begin to study these sciences; and it is for this reason that in most elementary books the authors endeavour to simplify the questions designed for exercising beginners, by proposing them in a less abstract manner than is employed in the pure mathematics, and so as to interest and excite the reader's curiosity. Thus, for example, if it were proposed simply to divide a triangle into three, four, or five equal parts, by lines drawn from a determinate point within it, in this form the problem could be interesting to none but those really possessed of a taste for geometry. But if, instead of proposing it in this abstract manner, we should say: "A father on his death-bed bequeathed to his three sons a triangular field, to be equally divided among them: and as there is a well in the field, which must be common to the three co-heirs, and from which the lines of division must necessarily proceed, how is the field to be divided so as to fulfill the intention of the testator?"”

Jean-Étienne Montucla (1725–1799) French mathematician

This way of stating it will, no doubt, create a desire in most minds to discover the method of solving the problem; and however little taste people may possess for real science, they will be tempted to try iheir ingenuity in finding the answer to such a question at this. Source: Preface to Recreations in Mathematics and Natural Philosophy. (1803), p. ii; As cited in: Tobias George Smollett. The Critical Review: Or, Annals of Literature http://books.google.com/books?id=T8APAAAAQAAJ&pg=PA410, Volume 38, (1803), p. 410

Death, Men, People, Books

“You must show that a man is wrong before you start explaining why he is wrong. The modern method is to assume without discussion that he is wrong and then distract his attention from this (the only real issue) by busily explaining how he became so silly. In the course of the last fifteen years I have found this vice so common that I have had to invent a name for it. I call it “Bulverism”. Some day I am going to write the biography of its imaginary inventor, Ezekiel Bulver, whose destiny was determined at the age of five when he heard his mother say to his father—who had been maintaining that two sides of a triangle were together greater than a third—”Oh you say that because you are a man.” “At that moment”, E. Bulver assures us, “there flashed across my opening mind the great truth that refutation is no necessary part of argument. Assume that your opponent is wrong, and the world will be at your feet. Attempt to prove that he is wrong or (worse still) try to find out whether he is wrong or right, and the national dynamism of our age will thrust you to the wall.””

Clive Staples Lewis (1898–1963) Christian apologist, novelist, and Medievalist

That is how Bulver became one of the makers of the Twentieth Century.
"Bulverism" (1941)

Truth, Age, World, Parting

“Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. … A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw hi the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts a priori, placed into that figure and represented in it, so that, in order to know anything with certainty a priori, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept.”

Immanuel Kant book Critique of Pure Reason

Preface to the Second Edition [Tr. F. Max Müller], (New York, 1900), p. 690; as cited in: Robert Edouard Moritz, Memorabilia mathematica or, The philomath's quotation-book https://openlibrary.org/books/OL14022383M/Memorabilia_mathematica, Published 1914. p. 10 Critique of Pure Reason (1781; 1787)

People, Hope, Way, History

“New York is one of the capitals of the world and Los Angeles is a constellation of plastic, San Francisco is a lady, Boston has become Urban Renewal, Philadelphia and Baltimore and Washington wink like dull diamonds in the smog of Eastern Megalopolis, and New Orleans is unremarkable past the French Quarter. Detroit is a one-trade town, Pittsburgh has lost its golden triangle, St Louis has become the golden arch of the corporation, and nights in Kansas City close early. The oil depletion allowance makes Houston and Dallas naught but checkerboards for this sort of game. But Chicago is a great American city. Perhaps it is the last of the great American cities.”

Norman Mailer book Miami and the Siege of Chicago

Pt. 2, p. 83
Miami and the Siege of Chicago (1968)

World, Angels, Game, Past

“Nelson’s challenge was that he was outnumbered. His strategy was to risk his lead ships in order to break the coherence of his enemy’s fleet. With coherence lost, he judged, the more experienced English captains would come out on top in the ensuing melee. Good strategy almost always looks this simple and obvious and does not take a thick deck of PowerPoint slides to explain. It does not pop out of some "strategic management" tool, matrix, chart, triangle, or fill-in-the-blanks scheme. Instead, a talented leader identifi es the one or two critical issues in the situation—the pivot points that can multiply the effectiveness of effort—and then focuses and concentrates action and resources on them.”

Richard Rumelt (1942) American economist

Source: Good Strategy Bad Strategy, 2011, p. 1-2

Effort

“Well sure, my sculptures are floor pieces. Each one, like any area on the surface of the earth, supports a column of air that weighs – what is it? – 14.7 pounds per square inch. So in a sense, that might represent a column. It's not an idea, it's a sense of something you know, a demarked place. Somehow I think I always thought of it going that way, rather than an idea of a narrowing triangle going to the center of the earth… I have nothing to do with Conceptual art [in contrast to his Physical Art, as Carl Andre called his sculpture art already in 1969]]. I'm not interested in ideas. If I were interested in ideas, I'd be in a field where what we think in is ideas… I don't really know what an idea is. One thing for me is that if I can frame something in language, I would never make art out of it. I make art out of things which cannot be framed in any other way. [quote from a talk with the audience, December 1969]”

Carl Andre (1935) American artist

Source: Artists talks 1969 – 1977, p. 12

Art, Way, Quotes, Idea

“The triangles that we can measure are not large enough… to detect the curvature. Fortunately, however, we are, in a way, able to communicate with the fourth dimension. The theory of relativity has given us an insight into the structure of the real universe: …a four-dimensional structure. The study of the way in which the three space-dimensions are interwoven with the time-dimension affords a kind of outside point of view of the three-dimensional space… from this outside point of view we might be able to perceive the curvature of the three-dimensional world.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932)

World, Way, Communication, Study

“Currency competition and free banking might increase the efficiency of the financial system, and bring some small triangle welfare gains. But the key question is whether their adoption would improve macroeconomic performance. Even though Salin argues (p. 281) that ‘the best system is that which produces the least inflation’, fluctuations in output are also expensive.
Hayek states that the adoption of his proposal would end recessions. There is absolutely no reason to believe that. Nineteenth century history is evidence that free banking and currency issue, in the wrong legal and regulatory framework, can produce rather than reduce instability. The proponents of free banking and currency issue in this volume do not go much beyond a general belief in competition in justifying their views; they have certainly not explored the necessary legal and regulatory environment in any detail.”

Stanley Fischer (1943) American economist

Stanley Fischer, "Friedman versus Hayek on Private Money: Review Essay" (1986)

History, Question

“Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite.”

Ezra Pound book The Spirit of Romance

The Spirit of Romance (1910), p. 5

Mathematics, Science

“It was one to three football fields in length. It was massive, about 300 feet above the ground. It had three lights on the points of its triangle and a large red light beneath.”

Steven M. Greer (1955) American ufologist

Greer describing a close encounter he had with a UFO.
Undated
Source: [Hawley, David, Reach Out And Touch ... An Extraterrestrial, St. Paul Pioneer Press, May 8, 1993, http://nl.newsbank.com/nl-search/we/Archives?p_product=PD&s_site=twincities&p_multi=SP&p_theme=realcities&p_action=search&p_maxdocs=200&p_topdoc=1&p_text_direct-0=0EB5DCD1EE3CE7FE&p_field_direct-0=document_id&p_perpage=10&p_sort=YMD_date:D&s_trackval=GooglePM, 2007-05-13, http://nbgoku23.googlepages.com/REACHOUTANDTOUCH...ANEXTRATERRESTRIA.htm, 2007-05-13]

Light, Football

“First then I will set out the very first theorem which became known to me by means of mechanics, namely that
Any segment of a section of a right angled cone (i. e., a parabola) is four-thirds of the triangle which has the same base and equal height,
and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]…”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

Books

“As yonder tower outstretches to the earth
The dark triangle of its shade alone
When the clear day is shining on its top;
So, darkness in the pathway of man's life
Is but the shadow of God's providence,
By the great Sun of wisdom cast thereon;
And what is dark below is light in heaven.”

John Greenleaf Whittier (1807–1892) American Quaker poet and advocate of the abolition of slavery

Source: Dictionary of Burning Words of Brilliant Writers (1895), P. 282

Life, Wisdom, God, Light

“In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground”

Thomas Little Heath (1861–1940) British civil servant and academic

this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids
Achimedes (1920)

Way, Problem, Sea

“I disagree with Les. We always found good cunt at the Lyceum. Friendly cunt, clean cunt, spare cunt, jeans and knicker stuffed full of nice juicy hairy cunt, handfuls of cunt, palmful grabbing the cunt by the stem, or the root – infantile memories of cunt – backrow slides – slithery oily cunt, the cunt that breathes – the cunt that’s neatly wrapped in cotton, in silk, in nylon, that announces, that speaks or thrusts, that winks that’s squeezed in a triangle of furtive cloth backed by an arse that’s creamy, springy billowy cushiony tight, knicker lined, knicker skinned, circumscribed by flowers and cotton, by views, clinging knicker, juice ridden knicker, hot knicker, wet knicker, swelling vulva knicker, witty cunt, teeth smiling the eyes biting cunt, cultured cunt, culture vulture cunt, finger biting cunt, cunt that pours, cunt that spreads itself over your soft lips, that attacks, cunt that imagines – cunt you dream about, cunt you create as a Melba, a meringue with smooth sides – remembered from school boys’ smelly first cunt, first foreign cunt, amazing cunt – cunt that’s cruel. Cunt that protects itself and makes you want it even more cunt – cunt that smells of the air, of the earth, of bakeries, of old apples, of figs, of sweat of hands of sour yeast of fresh fish cunt. So – are we going Les? We might pick up a bit of crumpet.”

Steven Berkoff East

East (1975), Scene 17

School, Smile, Dreams, Imagination

“Translates to: for a triangle, the result of a perpendicular with the half-side is the area.”

Aryabhata (476–550) Indian mathematician-astronomer

Source: Arijit Roy “The Enigma of Creation and Destruction”, p. 27 from the Ganitapada, quoted in "The Enigma of Creation and Destruction".

“…by action on man all known force may be measured. Indeed, few men of science measured force in any other way. After once admitting that a straight line was the shortest distance between two points, no serious mathematician cared to deny anything that suited his convenience, and rejected no symbol, proved or unproveable, that helped him to accomplish work. The symbol was force, as a compass-needle or a triangle was force.”

Henry Adams (1838–1918) journalist, historian, academic, novelist

The Education of Henry Adams (1907)

Men, Way, Science, Help

“You put a disk here and then you put another disk that is a triangle at the other end and then you balance them on your finger and keep on adding. I don't use rectangles - they stop. You can use them; I have at times, but only when I want to block, to constipate movement.”

Alexander Calder (1898–1976) American artist

Question, How do you get that subtle balance in your work?
1950s - 1960s, interview with Alexander Calder', (1962)

Time

“Rapidly going over what I could recall of Jim Bursley's information about pathological curves confirmed the conjecture. The snowball curve, derived from an equilateral triangle, is a perimeter of infinite length enclosing a finite area. The angles or points of the perimeter are uncountable. An equilateral triangle projected integrally in the third dimension is a triangular pyramid of equal surfaces. The three-dimensional snowflake derived from this pyramid--hence its diamantine appearance--is a finite volume enclosed in a surface of infinite area. The convolutions of such a surface, to be gathered around its defined content extrude a number of discrete angles or points beyond all possibility of computation. The pressure on each point is infinitesimal, unmeasurably small; the total external pressure exerted on any part of the surface is an aggregate of infinitesimal values, itself infinitesimal.”

William Grey Walter (1910–1977) American-born British neuroscientist and roboticist

Source: The Curve of the Snowflake (1956), p. 126.

Appearance, Parting

“Ted, I noted, was very busy - at the pumps, at the glasses behind, the bottles below, the merrily ringing till, like a percussion-player in some modern work who dashes with confidence from xylophone to glockenspiel to triangle to wind-machine to big drum to tambourine.”

Anthony Burgess (1917–1993) English writer

Fiction, The Right to an Answer (1960)

Confidence, Business

“As long as he could whisper, he would go on as he had begun, bluntly refusing to meet his creator with the admission that the creation had taught him nothing except that the square of the hypotenuse of a right-angled triangle might for convenience be taken as equal to something else. Every man with self-respect enough to become effective, if only as a machine, has had to account to himself for himself somehow, and to invent a formula of his own for his universe, if the standard formulas failed.”

Henry Adams (1838–1918) journalist, historian, academic, novelist

The Education of Henry Adams (1907)

Universe

“The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon. the sophist, of Athens, a contemporary of Socrates, who took the first step. He inscribed a square (or, according to another account, a triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and maintained that, by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Thought this was formally incorrect, it nevertheless contained the germ of the method of exhaustion.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

“Why should the typical student be interested in those wretched triangles? …He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks.”

George Pólya (1887–1985) Hungarian mathematician

Mathematical Methods in Science (1977)

Knowledge, Time

“Recall that democratic regimes in an exchange society are at one corner of the political triangle while totalitarian regimes with the coercive society they have constructed are at another. They therefore are at opposing ends of one side of the triangle, which is a continuum ranging from Freedom to Power.”

Rudolph Rummel (1932–2014) American academic

Source: Power Kills: Democracy as a Method of Nonviolence(1997), p. 204

Freedom

“Perhaps there is nobody who would sacrifice his life for the sake of maintaining that the three angles of a triangle are together equal to two right angles, for such a truth does not demand the sacrifice of our life; but, on the other hand, there are many who have lost their lives for the sake of maintaining their religious faith. Indeed, it is truer to say that martyrs make faith than that faith makes martyrs. For faith is not the mere adherence of the intellect to an abstract principle; it is not the recognition of a theoretical truth, the process in which the will merely sets in motion our faculty of comprehension; faith is an act of the will — it is a movement of the soul towards a practical truth, towards a person, towards something that makes us not merely comprehend life, but that makes us live.”

Miguel de Unamuno (1864–1936) 19th-20th century Spanish writer and philosopher

The Tragic Sense of Life (1913), IX : Faith, Hope, and Charity

Life, Truth, Faith, Soul

“The spider-men came first, and presented her Majesty with a table full of mathematical points, lines and figures of all sorts of squares, circles, triangles, and the like; which the Empress, notwithstanding that she had a very ready wit, and quick apprehension, could not understand; but the more she endeavoured to learn, the more was she confounded”

Margaret Cavendish (1623–1673) English aristocrat, a prolific writer, and a scientist

Description of a New World, Called The Blazing World (1666)

Men, Mathematics, Understanding, Learning

“The dangers to which we are exposed from our Women must now be manifest to the meanest capacity in Spaceland. If even the angle of a respectable Triangle in the middle class is not without its dangers; if to run against a Working Man involves a gash; if collision with an officer of the military class necessitates a serious wound; if a mere touch from the vertex of a Private Soldier brings with it danger of death; — what can it be to run against a Woman, except absolute and immediate destruction? And when a Woman is invisible, or visible only as a dim sub-lustrous point, how difficult must it be, even for the most cautious, always to avoid collision!”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 4. Concerning the Women

Death, Women

“To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful… A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere… The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside… But… a being… unable to leave the surface… could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. …On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. …The spaces of zero and negative curvature are infinite, that of positive curvature is finite. …the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would… differ… by an amount too small to be appreciable… then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension…. our case with reference to three-dimensional space is exactly similar. …we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932)

Communication, Understanding, Decision, Study

“The classic instrument to measure drawn angles and to draw angles of a given measure is the protractor — essentially half a circular ring, subdivided by ray segments into 180 degrees. For reasons I was unable to find out, this instrument has recently been superseded by an isosceles right triangle — called geo-triangle, solid, transparant, made of plastic — with an angular division radiating from the midpoint of the hypotenuse to the other sides. Well, inside the triangle half a circle with the midpoint of the hypotenuse as its centre is indicated, and from the position of the degree numbers it becomes clear that it is the semicircle that really matters. One is inclined to say "an outrageously misleading instrument"…”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 363

Drawing

“When you say that if I deny, that the operations of seeing, hearing, attending, wishing, &c., can be ascribed to God, or that they exist in him in any eminent fashion, you do not know what sort of God mine is ; I suspect that you believe there is no greater perfection than such as can be explained by the aforesaid attributes. I am not astonished ; for I believe that, if a triangle could speak, it would say, in like manner, that God is eminently triangular, while a circle would say that the divine nature is eminently circular. Thus each would ascribe to God its own attributes, would assume itself to be like God, and look on everything else as ill-shaped.”

Baruch Spinoza (1632–1677) Dutch philosopher

Letter 56 (60), to Hugo Boxel (1674) http://oll.libertyfund.org/?option=com_staticxt&staticfile=show.php%3Ftitle=1711&chapter=144218&layout=html&Itemid=27 Context: When you say that if I deny, that the operations of seeing, hearing, attending, wishing, &c., can be ascribed to God, or that they exist in him in any eminent fashion, you do not know what sort of God mine is; I suspect that you believe there is no greater perfection than such as can be explained by the aforesaid attributes. I am not astonished; for I believe that, if a triangle could speak, it would say, in like manner, that God is eminently triangular, while a circle would say that the divine nature is eminently circular. Thus each would ascribe to God its own attributes, would assume itself to be like God, and look on everything else as ill-shaped. The briefness of a letter and want of time do not allow me to enter into my opinion on the divine nature, or the questions you have propounded. Besides, suggesting difficulties is not the same as producing reasons. That we do many things in the world from conjecture is true, but that our redactions are based on conjecture is false. In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. A man would perish of hunger and thirst, if he refused to eat or drink, till he had obtained positive proof that food and drink would be good for him. But in philosophic reflection this is not so. On the contrary, we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow. Again, we cannot infer that because sciences of things divine and human are full of controversies and quarrels, therefore their whole subject-matter is uncertain; for there have been many persons so enamoured of contradiction, as to turn into ridicule geometrical axioms.

God, Fashion, Nature, Perfection

“If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women.”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 4. Concerning the Women
Context: If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women. For if a Soldier is a wedge, a Woman is a needle; being, so to speak, ALL point, at least at the two extremities. Add to this the power of making herself practically invisible at will, and you will perceive that a Female, in Flatland, is a creature by no means to be trifled with.

Women

“To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle.
According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient.”

Proclus (412–485) Greek philosopher

And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application...
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

Men, Books, Problem, Parting

“If I had as clear an idea of ghosts, as I have of a triangle or a circle, I should not in the least hesitate to affirm that they had been created by God; but as the idea I possess of them is just like the ideas, which my imagination forms”

Baruch Spinoza (1632–1677) Dutch philosopher

Letter to Hugo Boxel (Oct. 1674) The Chief Works of Benedict de Spinoza (1891) Tr. R. H. M. Elwes, Vol. 2, Letter 58 (54).
Context: If I had as clear an idea of ghosts, as I have of a triangle or a circle, I should not in the least hesitate to affirm that they had been created by God; but as the idea I possess of them is just like the ideas, which my imagination forms of harpies, gryphons, hydras, &c., I cannot consider them as anything but dreams, which differ from God as totally as that which is not differs from that which is.<!--pp. 382-383

God, Imagination, Idea

“Let the letters a b c denote the three angular points of a rectilineal triangle. If the point did move continuously over the lines ab, bc, ca, that is, over the perimeter of the figure, it would be necessary for it to move at the point b in the direction ab, and also at the same point b in the direction bc. These motions being diverse, they cannot be simultaneous. There-fore, the moment of presence of the movable point at vertex b, considered as moving in the direction ab, is different from the moment of presence of the movable point at the same vertex b, considered as moving in the same direction bc. But between two moments there is time; therefore, the movable point is present at point b for some time, that is, it rests. Therefore it does not move continuously, which is contrary to the assumption. The same demonstration is valid for motion over any right lines including an assignable angle. Hence a body does not change its direction in continuous motion except by following a line no part of which is straight, that is, a curve, as Leibnitz maintained.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section III On The Principles Of The Form Of The Sensible World

Time, Parting, Change

“To me the most important thing in composition is disparity. Thus black and white are the strong colors, with a spot of red to mark the other corner of a triangle which is by no means equilateral, isosceles, or right. To vary this still further use yellow, then, later, blue. Anything suggestive of symmetry is decidedly undesirable, except possibly where an approximate symmetry is used in a detail to enhance the inequality with the general scheme.”

Alexander Calder (1898–1976) American artist

1940s, Excerpt, A Propos of Measuring a Mobile (1943)

Strong, Color

“Physical evils are in nature inseparable from animal life, they commenced existence with it, and are its concomitants through life; so that the same nature which gives being to the one, gives birth to the other also; the one is not before or after the other, but they are coexistent together, and contemporaries; and as they began existence in a necessary dependence on each other, so they terminate together in death and dissolution. This is the original order to which animal nature is subjected, as applied to every species of it. The beasts of the field, the fowls of the air, the fishes of the sea, with reptiles, and all manner of beings, which are possessed with animal life; nor is pain, sickness, or mortality any part of God's Punishment for sin. On the other hand sensual happiness is no part of the reward of virtue: to reward moral actions with a glass of wine or a shoulder of mutton, would be as inadequate, as to measure a triangle with sound, for virtue and vice pertain to the mind, and their merits or demerits have their just effects on the conscience, as has been before evinced: but animal gratifications are common to the human race indiscriminately, and also, to the beasts of the field: and physical evils as promiscuously and universally extend to the whole, so "_That there is no knowing good or evil by all that is before us, for all is vanity_."”

Ethan Allen (1738–1789) American general

It was not among the number of possibles, that animal life should be exempted from mortality: omnipotence itself could not have made it capable of eternalization [sic] and indissolubility; for the self same nature which constitutes animal life, subjects it to decay and dissolution; so that the one cannot be without the other, any more than there could be a compact number of mountains without vallies [sic], or that I could exist and not exist at the same time, or that God should effect any other contradiction in nature...

Ch. III Section IV - Of Physical Evils
Reason: The Only Oracle Of Man (1784)

Life, God, Pain, Dependency