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

— Girard Desargues

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

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Girard Desargues 4

French mathematician and engineer 1591–1661

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

Girard Desargues (1591–1661) French mathematician and engineer

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

“In the field of non-Euclidean geometry, Riemann… began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
…he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom… In brief, there are no parallel lines. This … had been tried… in conjunction with the infiniteness of the straight line and had led to contradictions. However… Riemann found that he could construct another consistent non-Euclidean geometry.”

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“In philosophy, as in politics, the longest distance between two points is a straight line.”

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“The straight line is regarded as the shortest distance between two people, as if they were points.”

Theodor W. Adorno book Minima Moralia

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Heinrich Wilhelm Matthäus Olbers (1758–1840) German physician and astronomer

Sind wirklich im ganzen unendlichen Raum Sonnen vorhanden, sie mögen nun in ungefähr gleichen Abständen von einander, oder in Milchstrassen-Systeme vertheilt sein, so wird ihre Menge unendlich, und da müsste der ganze Himmel ebenso hell sein, wie die Sonne. Denn jede Linie, die ich mir von unserm Auge gezogen denken kann, wird nothwendig auf irgend einen Fixstern treffen, und also müßte uns jeder Punkt am Himmel Fixsternlicht, also Sonnenlicht zusenden.
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“As lines, so loves oblique may well
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“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.”

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“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!”