# „The trisection of an angle was effected by means of a curve discovered by Hippias of Elis, the sophist, a contemporary of Hippocrates as well as of Democritus and Socrates. The curve was called the quadratrix because it also served (in the hands, as we are told, of Dinostratus, brother of Menæchmus, and of Nicomedes) for squaring the circle. It was theoretically constructed as the locus of the point of intersection of two straight lines moving at uniform speeds and in the same time, one motion being angular and the other rectilinear.“

p, 125
Achimedes (1920)

Obtido da Wikiquote. Última atualização 3 de Junho de 2021. História
##### Thomas Little Heath46
British civil servant and academic 1861 - 1940

### „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 German philosopher 1724 - 1804

Section III On The Principles Of The Form Of The Sensible World

### „Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, (α) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x2=ay, y2=bx, and (β) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x2=ay, and xy=ab respectively. It would appear that it was in the effort to solve this problem that Menæchmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menæchmus."“

—  Thomas Little Heath British civil servant and academic 1861 - 1940

Achimedes (1920)

### „The great Cartesian invention had its roots in those famous problems of antiquity which originated in the days of Plato. In endeavoring to solve the problems of the trisection of an angle, of the duplication of the cube and of the squaring of the circle, the ruler and compass having failed them, the Greek geometers sought new curves. They stumbled on the conic sections…There we find the nucleus of the method which Descartes later erected into a principle. Thus Apollonius referred the parabola to its axis and principal tangent, and showed that the semichord was the mean propotional between the latus rectum and the height of the segment. Today we express this relation by x2 = Ly, calling the height the ordinate (y) and the semichord the abscissa (x); the latus rectum being… L. …the Greeks named these curves and many others… loci… Thus the ellipse was the locus of a point the sum of the distances of which from two fixed points was constant. Such a description was a rhetorical equation of the curve…“

—  Tobias Dantzig American mathematician 1884 - 1956

Number: The Language of Science (1930)

### „Time, so to say, runs at right angles to the page at each point on the curve.“

—  Joan Robinson English economist 1903 - 1983

Fonte: Economic Heresies (1971), Chapter VII, The Theory of the Firm, p. 104

### „By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded pari passu with the completion of the body of the Elements.“

—  Thomas Little Heath British civil servant and academic 1861 - 1940

p, 125
Achimedes (1920)

### „Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth… No force of gravitation is… needed to impress curvature on world-lines; the curvature is inherent in the space…“

—  James Jeans British mathematician and astronomer 1877 - 1946

The Growth of Physical Science (1947)

### „Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points.“

—  Stephen Hawking, livro A Brief History of Time

Fonte: A Brief History of Time (1988), Ch. 2

### „I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely.“

—  René Descartes, livro La Géométrie

Second Book
La Géométrie (1637)

### „The straight line belongs to Man. The curved line belongs to God.“

—  Antoni Gaudí Catalan architect 1852 - 1926

The real author seems to be Pierre Albert-Birot https://books.google.com/books?id=3Ul51CwjUOcC&pg=PA290&dq=%22the+curved+line+that+belongs+let%27s+say+to+God+and+the+straight+line+that+belongs+to+man%22&hl=de&sa=X&redir_esc=y#v=onepage&q=%22the%20curved%20line%20that%20belongs%20let%27s%20say%20to%20God%20and%20the%20straight%20line%20that%20belongs%20to%20man%22&f=false.
Attributed

### „But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.“

—  Roger Joseph Boscovich Croat-Italian physicist 1711 - 1787

"Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte.
"Transient pressure analysis in composite reservoirs" (1982) by Raymond W. K. Tang and William E. Brigham.
"Non-Newtonian Calculus" (1972) by Michael Grossman and Robert Katz.

### „A Curve does not exist in its full power until contrasted with a straight line.“

—  Robert Henri American painter 1865 - 1929

### „…the relation of betweenness on the torus is undetermined for curves that cannot be contracted to a point [e. g., circles around a doughnut hole], i. e., for three of such curves it is not uniquely determined which of them lies between the other two... This indeterminateness… has the consequence that such a curve [alone] does not divide the surface of the torus into two separate domains; between points to the "right" and to the "left" of the line.“

—  Hans Reichenbach American philosopher 1891 - 1953

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

### „Common to the two geometries is only the general property of one-to-one correspondence, and the rule that this correspondence determines straight lines as shortest lines as well as their relations of intersection.“

—  Hans Reichenbach American philosopher 1891 - 1953

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

### „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.“

—  Tennessee Williams, A Streetcar Named Desire

Fonte: A Streetcar Named Desire

### „It is not the right angle that attracts me, nor the straight line, hard and inflexible, created by man. What attracts me is the free and sensual curve — the curve that I find in the mountains of my country, in the sinuous course of its rivers, in the body of the beloved woman.“

—  Oscar Niemeyer Brazilian architect 1907 - 2012

As quoted in Plans, Sections and Elevations : Key Buildings of the Twentieth Century (2004) by Richard Weston
Variant translations:
It is not the right angle that attracts me,
Nor the hard, inflexible straight line, man-made.
What attracts me are free and sensual curves.
The curves in my country’s mountains,
In the sinuous flow of its rivers,
In the beloved woman’s body.
As quoted in "Architect of Optimism" http://www.ft.com/cms/s/0/db740a7a-e897-11db-b2c3-000b5df10621.html?nclick_check=1, Angel Gurria-Quintana, Financial Times (2007-04-13)
It is not the right angle that attracts me.
nor the straight line, tough, inflexible,
created by man.
what attracts me is the free, sensual curve.
the curve I find in the mountains of my country,
in the sinuous course of its rivers,
in the waves of the sea,
in the clouds of the sky,
in the body of the favourite woman.
Of curves is made all the universe.
As quoted on a Photo page on the Museum of Contemporary Art over Baia da Guanabara http://app.tabblo.com/studio/stories/view/122423/?nextnav=favs&navuser=1

### „I deliberately disregarded the right angle and rationalist architecture designed with ruler and square to boldly enter the world of curves and straight lines offered by reinforced concrete.… This deliberate protest arose from the environment in which I lived, with its white beaches, its huge mountains, its old baroque churches, and the beautiful suntanned women.“

—  Oscar Niemeyer Brazilian architect 1907 - 2012

The Curves of Time: The Memoirs of Oscar Niemeyer (2000), p. 62.

### „I ask at what part of its curved motion the moving cause will leave the thing moved and moveable.“

—  Leonardo Da Vinci Italian Renaissance polymath 1452 - 1519

XXI Letters. Personal Records. Dated Notes.

### „A smile is a curve that sets everything straight. “

—  Phyllis Diller American actress and stand-up comedianne 1917 - 2012

### „Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.“

—  Aristarchus of Samos ancient Greek astronomer and mathematician

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)

### „There will always be people who are ahead of the curve, and people who are behind the curve. But knowledge moves the curve.“

—  Bill James American baseball writer and statistician 1949

The Mind of Bill James, 2006, p. 191