“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

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

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Paul Cézanne62

French painter 1839–1906

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“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'

“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)

“[W]hereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids;) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension.
A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-plane? This is a Monster in Nature, and less possible than a Chimera or a Centaure. For Length, Breadth and Thickness, take up the whole of Space.”

John Wallis (1616–1703) English mathematician

Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three.
Treatise of Algebra (1685)

“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.

“Perspective is nothing more than a rational demonstration applied to the consideration of how objects in front of the eye transmit their image to it, by means of a pyramid of lines. The Pyramid is the name I apply to the lines which, starting from the surface and edges of each object, converge from a distance and meet in a single point.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), II Linear Perspective

“… spread on a plane the surface of a sphere in such a way that the positions of all places shall correspond on all sides with each other both in so far as true direction and distance are concerned and as concerns true longitudes and latitudes.”

Gerardus Mercator (1512–1594) cartographer, philosopher and mathematician

Legend on 1569 map

“Always lines, never forms. Where do they find these lines in Nature? Personally I see only forms that are lit up and forms that are not, planes that advance and planes that recede, relief and depth. My eye never sees outlines or particular features or details… …My brush should not see better than I do.”

Francisco De Goya (1746–1828) Spanish painter and printmaker (1746–1828)

Goya, in a recall of an overheard conversation
conversation of c. 1808, in the earliest biography of Goya: Goya, by Laurent Matheron, Schulz et Thuillié, Paris 1858; as quoted by Robert Hughes, in: Goya. Borzoi Book - Alfred Knopf, New York, 2003, p. 176
probably not accurate word for word, but according to Robert Hughes it rings true in all essentials, of the old Goya, in exile
1800s

“…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)

“It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. …the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection… analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". …If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels.”

Hans Reichenbach (1891–1953) American philosopher

The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface.
The Philosophy of Space and Time (1928, tr. 1957)

“We speak of concrete and not abstract painting because nothing is more concrete, more real then a line, a colour, a surface. A woman, a tree, a cow; are these concrete elements in a painting? No. A woman, a tree and a cow are concrete only in nature; in painting they are abstract, illusionistic, vague and speculative. However, a plane is a plane, a line is a line and no more or no less than that. 'Concrete paintin.”

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

Spirit has arrived at the age of maturity...
Quote in 'Comments on the basic of concrete painting', Paris, January 1930, in 'Art Concret', April 1930, pp. 2–4
1926 – 1931

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