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The shortest distance between two points is not a straight line- it’s a middleman. And the more middlemen, the shorter. Such is the psychology of a pretzel.

 

Ayn Rand

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

 

Theodor Adorno

 

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. 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.

 

Hans Reichenbach

 

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

 

The car is not a rabbit or a deer that jumps around in sweeping lines, but it is a man-made work of technology in need of an appropriate roadway. Rather, the car resembles a dragon fly or any other jumping animal that moves shorter distances in straight lines and then changes its direction at different points.

 

Fritz Todt

 

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!' How could you talk to a man like that?

 

Eamon de Valera