“[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

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

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John Wallis 34

English mathematician 1616–1703

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

Geometry as a Branch of Physics (1949)

“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

“The disharmoniousness (one might say, the negative rhythm) of the individual forms was that which primarily drew me, attracted me, during the period to which this watercolor belongs. The so-called rhythmic always comes on its own because in general the person himself is rhythmically built. Thus at least on the surface, the rhythmic is innate in people. Children, 'primitive' peoples, and laymen draw rhythmically..
In that period my soul was especially enchanted by the not-fitting-together of drawn and painterly form. Line serves the plane in that the former bounds the latter. And it makes my heart race in those cases when the independent plane springs over the confining line: line and plane are not in tune! It was this that produced a strong inner emotion in me, the inner 'ah!”

Wassily Kandinsky (1866–1944) Russian painter

2 quotes from Kandinsky's letter to Hans Arp, November 1912; in Friedel, Wassily Kandinsky, p. 489; as cited in Negative Rhythm: Intersections Between Arp, Kandinsky, Münter, and Taeuber, Bibiana K. Obler (including transl. - Yale University Press, 2014
Kandinsky was trying to explain to Arp his state of mind when he made his sketch for 'Improvisation with Horses' https://upload.wikimedia.org/wikipedia/commons/c/ce/Wassily_Kandinsky_Cossacks_or_Cosaques_1910%E2%80%931.jpg, 1911, a watercolor belonging to Arp. Kandinsky had told Arp that he could have one of his pictures included in the 'Moderne Bund' (second) exhibition in Zurich, 1912, and this was the one Arp selected
1910 - 1915

“Lines should not be drawn simply for the sake of drawing lines”

Felix Frankfurter (1882–1965) American judge

Dissenting in Pearce v. Commissioner of Internal Revenue, 315 U.S. 543, 558 (1942).
Judicial opinions
Context: The line must follow some direction of policy, whether rooted in logic or experience. Lines should not be drawn simply for the sake of drawing lines.

“A work based only on a line concept is scarcely more than a illustration; it fails to achieve pictorial structure. Pictorial structure is based on a plane concept. The line originates in the meeting of two planes … we can lose ourselves in a multitude of lines, if through them we lose our senses for the planes.”

Hans Hofmann (1880–1966) American artist

'Terms' p. 71
Search for the Real and Other Essays (1948)

“The colour line must go; the line will be drawn at personal merit.”

John Ireland (bishop) (1838–1918) Catholic bishop

Sermon at St. Augustine Catholic Church (1890)

“For as positio refers to a line, quadratum to the surface, and cubum to a solid body, it would be very foolish for us to go beyond this point. Nature does no permit it.”

Girolamo Cardano (1501–1576) Italian Renaissance mathematician, physician, astrologer

The Great Rules of Algebra (1968)
Context: Although a long series of rules might be added and a long discourse given about them, we conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio refers to a line, quadratum to the surface, and cubum to a solid body, it would be very foolish for us to go beyond this point. Nature does no permit it.

“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.

“It requires courage to make a frontal attack on nature through the broad planes and the large lines and it is cowardly to do it by the facets and details. It is a battle.”

Edgar Degas (1834–1917) French artist

posthumous quotes, The Shop-Talk of Edgar Degas', (1961)

“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)

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John Wallis 34

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