“Neither the circle without the line, nor the line without the point, can be artificially produced. It is, therefore, by virtue of the point and the Monad that all things commence to emerge in principle.
That which is affected at the periphery, however large it may be, cannot in any way lack the support of the central point.”

Theorem II
Monas Hieroglyphica (1564)

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John Dee10

English mathematican, astrologer and antiquary 1527–1608

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“Therefore, the central point which we see in the centre of the hieroglyphic Monad produces the Earth, round which the Sun, the Moon, and the other planets follow their respective paths. The Sun has the supreme dignity, and we represent him by a circle having a visible centre.”

John Dee (1527–1608) English mathematican, astrologer and antiquary

Theorem III
Monas Hieroglyphica (1564)

“The line has in itself neither matter nor substance and may rather be called an imaginary idea than a real object; and this being its nature it occupies no space. Therefore an infinite number of lines may be conceived of as intersecting each other at a point, which has no dimensions and is only of the thickness (if thickness it may be called) of one single line.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

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

“The geometric line is an invisible thing. It is the track made by the moving point; that is, its product. It is created by movement – specifically through the destruction of the intense self-contained repose of the point. Here, the leap out of the static to the dynamic occurs. […] The forces coming from without which transform the point into a line, can be very diverse. The variation in lines depends upon the number of these forces and upon their combinations.”

Wassily Kandinsky (1866–1944) Russian painter

1920 - 1930, Point and line to plane, 1926

“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)

“For this, to draw a right line from every point, to every point, follows the definition, which says, that a line is the flux of a point, and a right line an indeclinable and inflexible flow.”

Proclus (412–485) Greek philosopher

Book III. Concerning Petitions and Axioms.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i. e., in the following principle:
"If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions."
…every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed.”

Richard Dedekind (1831–1916) German mathematician

Stetigkeit und irrationale Zahlen (1872)

“In s (in which 180° = \pi [radians]). Further, each full line (great circle) is of finite length 2 \pi R, and any two full lines meet in two points—there are no parallels!”

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

Geometry as a Branch of Physics (1949)

“Every measurable thing except numbers is imagined in the manner of a continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined. For in them… measure or ratio is initially found… Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e. g., a quality… And since the quantity or ratio of lines is better known and is more readily conceived by us—nay the line is in the first species of continua, therefore such intensity ought to be imagined by lines… Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally.”

Nicole Oresme (1323–1382) French philosopher

Tractatus de Configurationibus et Qualitatibus et Motuum (c. 1350)

“…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)

“How could one argue with a man who was always drawing lines and circles to explain the position; who, one day, drew a diagram [here Michael illustrated with pen and paper] saying 'take a point A, draw a straight line to point B, now three-fourths of the way up the line take a point C. The straight line AB is the road to the Republic; C is where we have got to along the road, we canot move any further along the straight road to our goal B; take a point out there, D [off the line AB]. Now if we bend the line a bit from C to D then we can bend it a little further, to another point E and if we can bend it to CE that will get us around Cathal Brugha which is what we want!”

Michael Collins (Irish leader) (1890–1922) Irish revolutionary leader

How could you talk to a man like that?
Referring to Eamon de Valera in conversation with Michael Hayes, at the debates over the Anglo-Irish Treaty in 1921
Michael Hayes Papers, P53/299, UCDA
Quoted in Doherty, Gabriel and Keogh, Dermot (2006). Michael Collins and the Making of the Irish State. Mercier Press, p. 153.

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