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.
Morris KlineRelated topics
attention axiom begin calling comes concept condition conjunction construct end field found geometry illustrated led length line once parallel straight try yet Riemann reachesRelated quotes
The concept of peace is easy to grasp; that of international security is more complex, for a pattern of contradictions has arisen here as well. As major nuclear Powers have begun to negotiate arms reduction agreements, the proliferation of weapons of mass destruction threatens to increase and conventional arms continue to be amassed in many parts of the world. As racism becomes recognized for the destructive force it is and as apartheid is being dismantled, new racial tensions are rising and finding expression in violence. Technological advances are altering the nature and the expectation of life all over the globe. The revolution in communications has united the world in awareness, in aspiration and in greater solidarity against injustice. But progress also brings new risks for stability: ecological damage, disruption of family and community life, greater intrusion into the lives and rights of individuals.
Boutros Boutros-Ghali
I shall never forget the impression which our first landing on the beach of California make upon me. The sun had just gone down; it was getting dusky; the damp night wind was beginning to blow, and the heavy swell of the Pacific was setting in, and breaking in loud and high "combers" on the beach... we put our oars in the boat, and, leaving one to watch it, walked about the beach to see what we could of the place. The beach is nearly a mile in length between the two points, and of smooth sand... It was growing dark, so that we could just distinguish the dim outlines of the two vessels in the offing; and the great seas were rolling in in regular lines, growing larger and larger as they approached the shore, and hanging over the beach upon which they were to break, when their tops would curl over and turn white with foam, and, being at one extreme of the line, break rapidly to the other, as a child's long card house falls when a card is knocked down at one end.
Richard Henry Dana, Jr.
I have always held the old-fashioned opinion that the primary object of a work of fiction should be to tell a story; and I have never believed that the novelist who properly performed this first condition of his art, was in danger, on that account, of neglecting the delineation of character - for this plain reason, that the effect produced, by any narrative of events is essentially dependent, not on the events themselves, but on the human interest which is directly connected with them. It may be possible in novel-writing to present characters successfully without telling a story; but it is not possible to tell a story successfully without presenting characters: their existence, as recognizable realities, being the sole condition on which the story can be effectively told. The only narrative which can hope to lay a strong hold on the attention of readers is a narrative which interests them about men and women - for the perfectly obvious reasons that they are men and women themselves.
Wilkie Collins
It was a long way down. Before she had gone very far the curious dizziness she had known before came over her again, a dizziness not entirely induced by the spirals she whirled around, but a deeper, atomic unsteadiness as if not only she but also the substances around her were shifting. There was something queer about the angles of those curves. She was no scholar in geometry or aught else, but she felt intuitively that the bend and slant of the way she went were somehow outside any other angles or bends she had ever known. They led into the unknown and the dark, but it seemed to her obscurely that they led into deeper darkness and mystery than the merely physical, as if, though she could not put it clearly even into thoughts, the peculiar and exact lines of the tunnel had been carefully angled to lead through poly-dimensional space as well as through the underground-perhaps through time, too.
C. L. Moore
The Divine Existence (Daseyn),-his Existence, I say, which, according to the distinction already laid down, is his (Manifestation and Revelation of himself-is absolute- only through itself, and of necessity, LIGHT:-namely, y the inward and spiritual Light. This Light, left to itself, separates and divides itself into an infinite multiplicity of individual rays; and in this way, in these individual rays, becomes estranged from itself and its | original source. But this same Light may also again concentrate itself from out this separation, and conceive and comprehend itself as One, as that which it is in itself,-the Existence and Revelation of God; remaining indeed, even in this conception, that which it is in its form,-Light; but yet in this condition, and even by means of this very condition, announcing it-to self as having no real Being in itself, but as only the, Existence and Self-Manifestation of God.
Johann Gottlieb Fichte