“Common to the two geometries is only the general property of one-to-one correspondence, and the rule that this correspondence determines straight lines as shortest lines as well as their relations of intersection.”

— Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

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Hans Reichenbach 41

American philosopher 1891–1953

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“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)

“A straight line can readily be drawn among each of the two series of points corresponding to the maxima and minima, thus showing that there is a simple relation between the brightness of the variables and their periods.”

Henrietta Swan Leavitt (1868–1921) astronomer

Periods of 25 Variable Stars in the Small Magellanic Cloud http://adsabs.harvard.edu/abs/1912HarCi.173....1L (1912)
Context: A remarkable relation between the brightness of these [Cepheid] variables and the length of their periods will be noticed. In H. A. 60, No.4, attention was called to the fact that the brighter variables have the longer periods, but at that time it was felt that the number was too small the drawing of general conclusions. The periods of 8 additional variables which have been determined since that time, however, conform to the same law. The relation is shown graphically in Figure 1... The two resulting curves, one for the maxima and one for the minima, are surprisingly smooth, and of remarkable form. In Figure 2, the abscissas are equal to the logarithms of the periods, and the ordinates to the corresponding magnitudes, as in Figure 1. A straight line can readily be drawn among each of the two series of points corresponding to the maxima and minima, thus showing that there is a simple relation between the brightness of the variables and their periods. The logarithm of the period increases by about 0.48 for each increase of one magnitude in brightness.

“A straight line is not the shortest distance between two points.”

Madeleine L'Engle (1918–2007) American writer

Source: A Wrinkle in Time: With Related Readings

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

Theodor W. Adorno book Minima Moralia

Nun gilt für die kürzeste Verbindung zwischen zwei Personen die Gerade, so als ob sie Punkte wären.
E. Jephcott, trans. (1974), § 20
Minima Moralia (1951)

“The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), p. 5

“A new point is determined in Euclidean Geometry exclusively in one of the three following ways:
Having given four points A, B, C, D, not all incident on the same straight line, then
(1) Whenever a point P exists which is incident both on (A, B) and on (C, D), that point is regarded as determinate.
(2) Whenever a point P exists which is incident both on the straight line (A, B) and on the circle C(D), that point is regarded as determinate.
(3) Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate.
The cardinal points of any figure determined by a Euclidean construction are always found by means of a finite number of successive applications of some or all of these rules (1), (2) and (3). Whenever one of these rules is applied it must be shown that it does not fail to determine the point. Euclid's own treatment is sometimes defective as regards this requisite.
In order to make the practical constructions which correspond to these three Euclidean modes of determination, correponding to (1) the ruler is required, corresponding to (2) both ruler and compass, and corresponding to (3) the compass only.
…it is possible to develop Euclidean Geometry with a more restricted set of postulations. For example it can be shewn that all Euclidean constructions can be carried out by means of (3) alone…”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), pp. 7-8

“But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.”

Roger Joseph Boscovich (1711–1787) Croat-Italian physicist

"Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte.
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