Quotes about perpendicular

A collection of quotes on the topic of perpendicular, line, lining, point.

Quotes about perpendicular

“If we suppose the distance of the fixed stars from the sun to be so great that the diameter of the earth's orbit viewed from them would not subtend a sensible angle, or which amounts to the same, that their annual parallax is quite insensible; it will then follow that a line drawn from the earth in any part of its orbit to a fixed star, will always, as to sense, make the same angle with the plane of the ecliptic, and the place of the star, as seen from the earth, would be the same as seen from the sun placed in the focus of the ellipsis described by the earth in its annual revolution, which place may therefore be called its true or real place.
But if we further suppose that the velocity of the earth in its orbit bears any sensible proportion to the velocity with which light is propagated, it will thence follow that the fixed stars (though removed too far off to be subject to a parallax on account of distance) will nevertheless be liable to an aberration, or a kind of parallax, on account of the relative velocity between light and the earth in its annual motion.
For if we conceive, as before, the true place of any star to be that in which it would appear viewed from the sun, the visible place to a spectator moving along with the earth, will be always different from its true, the star perpetually appearing out of its true place more or less, according as the velocity of the earth in its orbit is greater or less; so that when the earth is in its perihelion, the star will appear farthest distant from its true place, and nearest to it when the earth is in its aphelion; and the apparent distance in the former case will be to that in the latter in the reciprocal proportion of the distances of the earth in its perihelion and its aphelion. When the earth is in any other part of its orbit, its velocity being always in the reciprocal proportion of the perpendicular let fall from the sun to the tangent of the ellipse at that point where the earth is, or in the direct proportion of the perpendicular let fall upon the same tangent from the other focus, it thence follows that the apparent distance of a star from its true place, will be always as the perpendicular let fall from the upper focus upon the tangent of the ellipse. And hence it will be found likewise, that (supposing a plane passing through the star parallel to the earth's orbit) the locus or visible place of the star on that plane will always be in the circumference of a circle, its true place being in that diameter of it which is parallel to the shorter axis of the earth's orbit, in a point that divides that diameter into two parts, bearing the same proportion to each other, as the greatest and least distances of the earth from the sun.”

James Bradley (1693–1762) English astronomer; Astronomer Royal

Miscellaneous Works and Correspondence (1832), Demonstration of the Rules relating to the Apparent Motion of the Fixed Stars upon account of the Motion of Light.

Star, Sense, Light, Distant

“Dancing Is a Perpendicular Expression of a Horizontal Desire.”

George Bernard Shaw (1856–1950) Irish playwright

attributed by George Melly in 1962 Source: Quote Investigator - Dancing Is a Perpendicular Expression of a Horizontal Desire http://quoteinvestigator.com/2016/09/11/dancing/

Funny, Dance

“Earthquakes radiate waves with periods of tenths of seconds to several minutes. Rocks behave like elastic solids at these frequencies. Elastic solids allow a variety of wave types and this makes the ground motion after an earthquake or explosion (called an event) quite complex. There are two basic types of elastic wave: one involving compression and rarefaction of the elastic material in the direction of propagation of the wave, and one involving no compression but shear of the elastic material perpendicular to its direction of propagation. These are called P and S waves respectively, for primary and secondary since the P wave travels fastest and arrives first.”

David Gubbins (1947) British university teacher

[Seismology and plate tectonics, Cambridge, UK; New York, Cambridge University Press, 1990, http://books.google.com/books?id=tZRxPzwoChIC&pg=PA4] (pp. 4–5)
Seismology and Plate Tectonics (1990)

Travel

“Can imagination act
Perpendicular to fact?
Can it be a kite that flies
Till the Earth, umbrella-wise,
Folds and drops away from sight?”

Philip José Farmer (1918–2009) American science fiction writer

"Imagination" in America Sings (1949); re-published in Pearls From Peoria (2006)

Imagination

“Elementarism has been born partly in reaction to an over-dogmatic and often narrow-minded application of NeoPlasticism [a critic on his former artist-companion Piet Mondrian, partly as its consequence but ultimately from what is primarily a radical correction of Neo-plastic ideas. Elementarism rejects the demands of pure statics which led to sterility and to the laming of creative potentialities. Instead of denying Time and Space, Elementarism acknowledge these elements to be the most elementary means for creating a new plastic expression... In contrast to the Neo-plastic [= De Stijl] manner of expression, which is restricted to two dimensions [the plane], Elementarism acknowledges a form of plastic expression in four dimensions, the realm of space-time. In opposition to the orthogonal style of plastic expression, which is 'homogeneous' with natural construction, Elementarism postulates a 'heterogeneous' contrasting, unstable manner of plastic expression based upon planes oblique in relation to the static, perpendicular axis of gravitation”

Theo van Doesburg (1883–1931) Dutch architect, painter, draughtsman and writer

Quote of van Doesburg, in van 'Painting and plastic art': Elementarism – fragment of a manifesto' Paris, December 1926 – April 1927; in De Stijl, Theo van Doesburg – series XIII, 78, 1926–27, pp. 82–87
1926 – 1931

Idea, Nature, Time

“My mantra is “stay perpendicular.” Horizontal is not as good. Half the people that came along and up with with me are either gone to another dimension or don’t remember what they had for lunch. I’m fortunate. I don’t know why. I just want to have a good time.”

Martin Landau (1928–2017) American actor and acting coach

Martin Landau: ‘Doubt Is Important’, Washington Times (December 25, 2016)

People, Time

“Hence, as the line of sight to the upper part is the longer, it makes that part look as if it were leaning back. But when the members are inclined to the front, as described above, they will seem the beholder to be plumb and perpendicular.”

Vitruvius book De architectura

Source: De architectura (The Ten Books On Architecture) (~ 15BC), Book III, Chapter V, Sec. 13

Parting

“Whether it is Juan Gris taking objects apart, Picasso replacing them with objects of his own invention, or another who replaces conical perspective by a system based on the relations between perpendiculars, all that only goes to show that Cubism was not at all born out of an authoritative theory [mot d'ordre]; that it only marked among a few painters the will to be finished with an art that never ought to have survived the condemnation pronounced upon it by Pascal.”

Jean Metzinger (1883–1956) French painter

Cubism was born

Art

“Leisure stands in a perpendicular position with respect to the working process — in just the same way as the "simple gaze" of intellectus does not consist in the "duration" (so to speak) of ratios working-out process, but instead cuts through it at the perpendicular (the ancients compared the ratio with time, the intellectus with the "always now" of eternity).”

Josef Pieper (1904–1997) German philosopher

Source: Leisure, the Basis of Culture (1948), Leisure, the Basis of Culture, p. 34

Way, Time

“He talked to her endlessly about his love of horizontals: how they, the great levels of sky and land in Lincolnshire, meant to him the eternality of the will, just as the bowed Norman arches of the church, repeating themselves, meant the dogged leaping forward of the persistent human soul, on and on, nobody knows where; in contradiction to the perpendicular lines and to the Gothic arch, which, he said, leapt up at heaven and touched the ecstasy and lost itself in the divine.”

D.H. Lawrence book Sons and Lovers

Sons and Lovers (1913)

Love, Dogs, Soul

“Light from a concave luminous body is received most powerfully at the centre. The reason for this is that, for every point of a concave body, perpendicular rays, which are stronger than others, converge in the centre. Therefore the virtues of celestial bodies are incident most powerfully in and near the centre of the world.”

John Peckham (1227–1292) Archbishop of Canterbury

Note the assumption that the heavenly sphere is concave with respect to the earth.
Perspectiva communis as quoted in J. D. North, Stars, Mind and Fate: Essays in Ancient and Mediaeval Cosmology (1989) citing D.C. Lindberg, John Pecham and the Science of Optics: Perspectiva communis (1970) p.99

World, Light

“Hitherto we have considered the apparent motion of the star about its true place, as made only in a plane parallel to the ecliptic, in which case it appears to describe a circle in that plane; but since, when we judge of the place and motion of a star, we conceive it to be in the surface of a sphere, whose centre is our eye, 'twill be necessary to reduce the motion in that plane to what it would really appear on the surface of such a sphere, or (which will be equivalent) to what it would appear on a plane touching such a sphere in the star's true place. Now in the present case, where we conceive the eye at an indefinite distance, this will be done by letting fall perpendiculars from each point of the circle on such a plane, which from the nature of the orthographic projection will form an ellipsis, whose greater axis will be equal to the diameter of that circle, and the lesser axis to the greater as the sine of the star's latitude to the radius, for this latter plane being perpendicular to a line drawn from the centre of the sphere through the star's true place, which line is inclined to the ecliptic in an angle equal to the star's latitude; the touching plane will be inclined to the plane of the ecliptic in an angle equal to the complement of the latitude. But it is a known proposition in the orthographic projection of the sphere, that any circle inclined to the plane of the projection, to which lines drawn from the eye, supposed at an infinite distance, are at right angles, is projected into an ellipsis, having its longer axis equal to its diameter, and its shorter to twice the cosine of the inclination to the plane of the projection, half the longer axis or diameter being the radius.
Such an ellipse will be formed in our present case…”

James Bradley (1693–1762) English astronomer; Astronomer Royal

Miscellaneous Works and Correspondence (1832), Demonstration of the Rules relating to the Apparent Motion of the Fixed Stars upon account of the Motion of Light.

Star, Nature, Appearance

“With regard to the diagonal, too, I am in complete agreement with you [with Theo van Doesburg ]. As soon as it appears together with straight [horizontal and vertical] lines, I believe it should be condemned... A while back I started a thing entirely in diamonds [diamond-shape] like this [his sketch in the letter of several diamond-forms]. I have to find out if it's possible: intellectually I'm inclined to say it is. There's something to be said for the idea, because perpendicular and flat lines can be seen everywhere in nature; by using a diagonal line I would be canceling that out. But I'm inclined to say that this cannot be combined with perpendicular and flat lines or with different kinds of slanting lines.”

Piet Mondrian (1872–1944) Peintre Néerlandais

quote about the growing controversy between Mondrian and Van Doesburg. concerning the use of diagonal lines
Source: quote from a letter of Mondrian to Theo van Doesburg, undated, c. May 1918; as cited in Mondrian, -The Art of Destruction, Carel Blotkamp, Reaktion Books LTD. London 2001, p. 120

Idea, Nature, Appearance

“…the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous, although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i. e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. …such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

Drawing

“Allow me to repeat what I said when you were here: deal with nature by means of the cylinder, the sphere and the cone, all placed in perspective, so that each side of an object or a plane is directed towards a central point. Lines parallel to the horizon give breadth, a section of nature, or if you prefer, of the spectacle spread before our eyes by the 'Pater Omnipotens Aeterne Deus'. Lines perpendicular to that horizon give depth. But for us men, nature has more depth than surface, hence the need to introduce in our vibrations of light, represented by reds and yellows, enough blue tints to give a feeling of air.”

Paul Cézanne (1839–1906) French painter

Quote from Cézanne's letter to Émile Bernard, 15 April 1904; as quoted in Letters of the great artists – from Blake to Pollock, Richard Friedenthal, Thames and Hudson, London, 1963, p. 180
Quotes of Paul Cezanne, after 1900

Men, Feeling, Nature, Light

“The simultaneousness of states of mind in the work of art: that is the intoxicating aim of our art... In the pictorial description of the various states of mind of a leave-taking, perpendicular lines, undulating lines and as it were worn out, clinging here and there to silhouettes of empty bodies, may well express languidness and discouragement. Confused and trepidating lines, either straight or curved, mingled with the outlined hurried gestures of people calling to one another will express a sensation of chaotic excitement. On the other hand, horizontal lines, fleeting, rapid and jerky, brutally cutting in half lost profiles of faces or crumbling and rebounding fragments of landscape, will give the tumultuous feelings of the person going away.”

Umberto Boccioni (1882–1916) Italian painter and sculptor

Source: 1912, Les exposants au public', 1912, pp. 47, 49

People, Art, Feeling, Personality

“It was the business of the Sorbonne doctors to discuss, of the pope to decide, and of a mathematician to go straight to heaven in a perpendicular line.”

Jacques Ozanam (1640–1718) French mathematician

Source: Recreations in Mathematics and Natural Philosophy, (1803), p. xv

Business

“From a careful determination of the amount of solar heat that which would fall per minute on an area of one square centimetre placed perpendicular to the radiation as it falls on Earth's surface and from a knowledge of the Earth's distance, we deduce that each square centimetre of the solar surface radiates on the average of about the rate of a nine horse-power engine.”

Gerald James Whitrow (1912–2000) British mathematician

"Why the Sun Shines" (18 July 1957)

Knowledge, Horse

“When she got to an angle of about 60 degrees, there was a sullen sort of rumbling roar as her massive boilers all left their beds and went crashing down through the bulkheads and everything that stood in their way.Up to that moment, she had stood out as clear as clear with her rows of electric lights all burning. When the boilers broke away, she was, of course, plunged into absolute darkness though her huge black outline was still perfectly distinct up against the stars and sky.Slowly she reared up on end till, at last, she was absolutely perpendicular. Then quite quietly, but quicker and quicker, she seemed just to slide away under the surface and disappear.As she vanished, everyone around me on the upturned boat, as though they could hardly believe it, just said, "She's gone."”

Charles Lightoller (1874–1952) British sailor

Interview with the BBC, 1936

Star, Light

“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".

“Treat nature in terms of the cylinder, the sphere, and the cone, the whole put into perspective so that each side of an object, or of a plane, leads towards a central point. Lines parallel to the horizon give breadth, whether a sections of nature, or, if you prefer, of the spectacle which Pater omnipotens aeterne Deus unfolds before your eyes. Lines perpendicular to this horizon give depth... Everything I am telling you [ Joachim Gasquet ] about - the sphere, the cone, cylinder, concave shadow – on mornings when I'm tired these notions of mine get me going, they stimulate me, I soon forget them once I start using my eyes.”

Paul Cézanne (1839–1906) French painter

Source: Quotes of Paul Cezanne, after 1900, Cézanne, - a Memoir with Conversations, (1897 - 1906), pp. 163-164, in: 'What he told me – I. The motif'

Nature, Forgetting

“The fact that functions may not be separated in organization does not in any way way destroy their identity as functions… The ideal of organized efficiency is not the complete segregation but the integrated correlation |of the three primary functions… The organizer must identify these functional principles, as they appear in every job, and make them the basis of his correlation… his duty is to correlate functions as such… The duty of the manager is to correlate who performs these functions… Management however represents the scalar principle… perpendicular correlations… true horizontal correlation requires other contact… The dissemination of understanding… of common purpose, of the relation of individuals to that purpose & to each other.”

James D. Mooney (1884–1957) American businessman

Source: Onward Industry!, 1931, p. 50-59, as cited in Lyndall Urwick (1937;50)

Way, Understanding, Appearance

“The discovery of Hippocrates amounted to the discovery of the fact that from the relation
(1)\frac{a}{x} = \frac{x}{y} = \frac{y}{b}it follows that(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}and if a = 2b, [then (\frac{a}{x})^3 = 2, and]a^3 = 2x^3.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations
(2)x^2 = ay, y^2 = bx, xy = ab[or equivalently…y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x} ]Doubling the Cube
the 2 solutions of Menaechmusand the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).
Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y.
The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i. e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i. e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have

\begin{cases}y^2 = b. ON = b. PM = bx\\ and\\ xy = PM. PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.
In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a.”

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

The point P where the two parabolas intersect is given by<center><math>\begin{cases}y^2 = bx\\x^2 = ay\end{cases}</math></center>whence, as before,<center><math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math></center>
Apollonius of Perga (1896)

Drawing