The modern thinker often repeats the parallel axiom, 'All knowledge is experience.
"Theaetetus" by Plato
Congruence is extended in any space beyond parallel rects to all rects by two axioms depending on perpendicularity.
"The Concept of Nature" by Alfred North Whitehead
The introduction of the new axiom and of parallel lines leads to a new class of propositions.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
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Hyperbolic geometry appeared in the ﬁrst half of the 19th century, proving the independence of the parallel axiom of Euclidean geometry.
Coordinates for a new triangular tiling of the hyperbolic plane
Since pointwise deﬁnability is a strong generalization of the axiom V = HOD, it is tempting to introduce such notation as V = D or V = H D to express that a model is pointwise deﬁnable, thereby maintaining a parallel to the classical V = HOD notation while emphasizing that the deﬁnitions need no parameters.
Pointwise Definable Models of Set Theory
The axioms of ETCS and CETCS are presented in parallel and compared in Section 4.
Constructivist and Structuralist Foundations: Bishop's and Lawvere's Theories of Sets
However, the cones at e and f share the same tangent plane (because f lies in that plane), whence the intersection lines at e and f are parallel (this is part of what it means to be a tangent plane, as expressed in the cone axioms).
Using Isabelle to verify special relativity, with application to hypercomputation theory
To emphasize the parallelism to the bosonic core axioms (Appendix A.2) we keep the numbering, but we use the letter “P” (for “positive”) instead of “T”.
A positive formalism for quantum theory in the general boundary formulation
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