Title: Theory / Unimportant / Math / Geometry/ / Non-Euclidean - Sommerville;
Tags: fromImplication toBinaryRelation typToBinary

Name: Axiom of connection
Definition: Axioms connecting point, line and plane
Page: 27
Chapter: 2
Section: 1
Kind: Def
Name2: Axiom of classification

Name: Axiom of order
Definition: Axioms explaining the idea of «Between»
Page: 27
Chapter: 2
Section: 1
Kind: Def

Question: Vertically opposite angles are
Answer: equal
Page: 28
Chapter: 2
Section: 1
Kind: Def

Question: The sum of the four angles made by two intersecting lines is
Answer: an absolute constant.
Page: 28
Chapter: 2
Section: 1
Kind: Def

Name: Right angle
Definition: Quarter of the sum of the four angles of two intersecting lines.
Page: 28
Chapter: 2
Section: 1
Kind: Def

Question: The number of perdendicular to \(L\) through \(P\) is
Answer: exactly one
Page: 28
Chapter: 2
Section: 1
Variables: \(P\) a pont and \(L\) a line
Kind: Def

Name: A flatten angle
Definition: Two right angle
Page: 28
Chapter: 2
Section: 1
Notation: \(\pi\)
Kind: Def

Smaller0: \(L\) cuts one side of a triangle
Greater0: either \(L\) passes through a vertex, or it cut one of the other sides
Page: 29
Chapter: 2
Section: 1
Variables: \(T\) a triangle,
\(L\) a straight line
Name: Pasch's Axiom

Name: The parallel to \(NA\) through \(O\)
Definition: The limit of the line \(OP\) where \(P\) tends in the direction \(\vec{AN}\) on \(l\), if this limit exists.
Page: 29
Chapter: 2
Section: 1
Variables: \(l\) a line,
\(O\not\in l\) a point,
\(N\in l\) such that \(ON\bot l\),
\(A\in L\) distinct from \(N\)

Page: 29, 32
Definition: -\(AA'\) and \(BB'\) lie in the same plane,
-\(AA'\) is disjoint from \(BB'\)
-every ray drawn through \(A\) within the angle \(BAA'\) meet the ray \(BB'\)
Chapter: 2
Section: 1
Variables: \(A,A',B,B'\) points
Notation: \(AA'\parallel BB'\)
Name: \(AA'\) is parallel to \(BB'\)
Index: , (2)
Transitive: Equivalence relation

Question: The angles \(NOL\) and \(NOM\) are
Answer: equal
Page: 29-30
Chapter: 2
Section: 1
Variables: \(B,N,A\) in this order
\(O\not\in ABl\) a point with \(ON\bot BA\),
\(L,M\) such that \(OL||NA\) and \(OM||NB\)

Question: The angle \(NOL\) is a function of
Answer: the distance \(NO\)
Page: 29
Chapter: 2
Section: 1
Variables: \(l\) a line,
\(O\not\in l\) a point,
\(N,A\in L\) distinct such that \(ON\bot l\),
\(L\) such that \(OL||NA\)

Name: The angle of parallelism
Definition: The function associating to a distance \(p\) the angle of \(NOL\), assuming that \(NO\bot NA\) and \(NA||OL\).
Page: 30
Chapter: 2
Section: 1
Context: Geometry
Notation: \(\Pi\)
Name2: The parallel-angle

Smaller0: \(R\) cuts \(BB'\)
Greater0: \(R\) is within the angle \(BPA'\)
Page: 31
Chapter: 2
Section: 1
Variables: \(AA'||BB'\),
\(P\in AA'\),
\(R\) a ray through \(P\)
Smaller to greater: iff

Name: Five groups of axioms
Definition: connection
Page: 27
Chapter: 2
Section: 1
Context: Geometry
Kind: Def
Definition2: order
Conjdef: All of
Definition3: congruence
Definition4: parallels
Definition5: continuity