„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.“

Geometry as a Branch of Physics (1949)

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Howard P. Robertson Foto
Howard P. Robertson
amerikanischer Mathematiker und Physiker 1903 - 1961

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John Wallis Foto
Bernhard Riemann Foto

„Let us imagine that from any given point the system of shortest lines going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. …the square of the line-element is \sum (dx)^2 for infinitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order… an infinitesimal, therefore, of the fourth order; so that we obtain a finite quantity on dividing this by the square of the infinitesimal triangle, whose vertices are (0,0,0,…), (x1, x2, x3,…), (dx1, dx2, dx3,…). This quantity retains the same value so long as… the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is flat, i. e., when the squared line-element is reducible to \sum (dx)^2, and may therefore be regarded as the measure of the deviation of the manifoldness from flatness at the given point in the given surface-direction. Multiplied by -¾ it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. …The measure-relations of a manifoldness in which the line-element is the square root of a quadric differential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e. g., the fourth root of a quartic differential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from flatness in the squared line-element is an infinitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called flatness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them…“

—  Bernhard Riemann German mathematician 1826 - 1866

On the Hypotheses which lie at the Bases of Geometry (1873)

Andrew Marvell Foto
E. W. Hobson Foto
Archimedes Foto
Aristarchus of Samos Foto
Archimedes Foto

„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 Greek philosopher 412 - 485

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

Aristarchus of Samos Foto
Girard Desargues Foto

„When no point of a line is at a finite distance, the line itself is at an infinite distance.“

—  Girard Desargues French mathematician and engineer 1591 - 1661

Brouillion project (1639) as quoted by Harold Scott MacDonald Coxeter, Projective Geometry (1987)

Robert Grosseteste Foto

„All causes of natural effects have to be given through lines, angles and figures, for otherwise it is impossible for the reason why“

—  Robert Grosseteste English bishop and philosopher 1175 - 1253

De Lineas, Anguilis et Figuris (On Lines, Angles and Figures) as quoted in Neil Lewis, "Robert Grosseteste" http://plato.stanford.edu/entries/grosseteste/ Stanford Encyclopedia of Philosophy (2007, 2013) citing Baur, Ludwig (ed.) Die Philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln (1912) pp.59–60
Kontext: The consideration of lines, angles and figures is of the greatest utility since it is impossible for natural philosophy to be known without them... All causes of natural effects have to be given through lines, angles and figures, for otherwise it is impossible for the reason why (propter quid) to be known in them.

Margaret Cavendish Foto
Walter Scott Foto
John Dee Foto
Girard Desargues Foto

„Parallel lines have a common end point at an infinite distance.“

—  Girard Desargues French mathematician and engineer 1591 - 1661

Brouillion project (1639) as quoted by Harold Scott MacDonald Coxeter, Projective Geometry (1987)

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