Branches History Index

Index: Content Tree

The tree index reflects the "common-thread" structure of BoP:

Branches (2839)
- Branch: Introduction to the Axiomatic Method
- Branch: Logic (92)
  - Part: Historical Development of Logic
  - Part: Basic Concepts of Logic (18)
    - Definition: Strings (words) over an Alphabet (2)
      - Example: Examples of Strings over Alphabets (related to Definition: Strings (words) over an Alphabet)
      - Corollary: Algebraic Structure of Strings over an Alphabet (related to Definition: Strings (words) over an Alphabet) (1)
        
        Proof: (related to Corollary: Algebraic Structure of Strings over an Alphabet)
    - Definition: Language (3)
      - Definition: Concatenation of Languages
      - Example: Examples of Languages (related to Definition: Language)
      - Definition: Iteration of Languages, Kleene Star, Kleene Plus
    - Definition: Grammar (Syntax) (1)
      - Example: Examples of Syntax (related to Definition: Grammar (Syntax))
    - Definition: Formal Languages Generated From a Grammar
    - Chapter: From the Syntax to the Semantics of Formal Languages (4)
      - Axiom: Bivalence of Truth
      - Definition: Set of Truth Values (True and False)
      - Definition: Domain of Discourse
      - Definition: Interpretation of Strings of a Formal Language and Their Truth Function
    - Definition: Satisfaction Relation, Model, Tautology, Contradiction
    - Definition: Semantics of a Formal Language
    - Chapter: The Proving Machine - an Automation of Logical Reasoning (4)
      - Definition: Axioms
      - Definition: Rules of Inference
      - Definition: Logical Calculus
      - Definition: Proofs and Theorems in a Logical Calculus
    - Definition: Derivability Property
  - Part: Proof Theory (8)
    - Chapter: Putting it All Together - Syntax and Semantics of a Logical Calculus (3)
      - Definition: Soundness and Completeness of a Logical Calculus
      - Definition: Consistency and Negation-Completeness of a Logical Calculus
      - Definition: Negation of a String
    - Chapter: Classifying the Logical Calculi - Variables, Formulas, Predicates and Signatures (5)
      - Definition: Variable in a Logical Calculus (1)
        
        Example: Examples of Variables in a Logical Calculus (related to Definition: Variable in a Logical Calculus)
      - Definition: Quantifier, Bound Variables, Free Variables (1)
        
        Example: Examples of Quantifiers in a Logical Calculus (related to Definition: Quantifier, Bound Variables, Free Variables)
      - Definition: Function, Arity and Constant (1)
        
        Example: Examples of Functions in a Logical Calculus (related to Definition: Function, Arity and Constant)
      - Definition: Predicate of a Logical Calculus (1)
        
        Example: Examples of Predicates in a Logical Calculus (related to Definition: Predicate of a Logical Calculus)
      - Definition: Signature
  - Part: Propositional Logic (35)
    - Definition: Signature of Propositional Logic - PL0
    - Definition: Syntax of PL0 - Propositions as Boolean Terms
    - Definition: Interpretation of Propositions - the Law of the Excluded Middle (2)
      - Example: Examples of Propositions With a More Complex Syntax (related to Definition: Interpretation of Propositions - the Law of the Excluded Middle)
      - Definition: Paradox (1)
        
        Example: Examples of Paradoxes (related to Definition: Paradox)
    - Definition: Semantics of PL0 (15)
      - Definition: `$k$`-nary Connectives, Prime and Compound Propositions
      - Lemma: Boolean Function (2)
        
        Example: Examples of Boolean Functions (related to Lemma: Boolean Function)
        
        Proof: (related to Lemma: Boolean Function)
      - Definition: Truth Table
      - Definition: Negation
      - Definition: Conjunction (2)
        
        Corollary: Commutativity of Conjunction (related to Definition: Conjunction) (1)
        
        Proof: (related to Corollary: Commutativity of Conjunction)
        
        Proposition: Associativity of Conjunction (1)
        
        Proof: (related to Proposition: Associativity of Conjunction)
      - Definition: Disjunction (3)
        
        Corollary: Commutativity of Disjunction (related to Definition: Disjunction) (1)
        
        Proof: (related to Corollary: Commutativity of Disjunction)
        
        Proposition: Associativity of Disjunction (1)
        
        Proof: (related to Proposition: Associativity of Disjunction)
        
        Definition: Exclusive Disjunction
      - Definition: Implication (4)
        
        Lemma: Implication as a Disjunction (1)
        
        Proof: (related to Lemma: Implication as a Disjunction)
        
        Lemma: Negation of an Implication (2)
        
        Proof: (related to Lemma: Negation of an Implication)
        
        Example: Examples of Correct and Incorrect Negations of Implications (related to Lemma: Negation of an Implication)
        
        Definition: Contrapositive
      - Definition: Equivalence (1)
        
        Corollary: Commutativity of Equivalence (related to Definition: Equivalence) (1)
        
        Proof: (related to Corollary: Commutativity of Equivalence)
    - Chapter: Equivalent Transformations in Propositional Logic (5)
      - Lemma: Every Proposition Implies Itself (1)
        
        Proof: (related to Lemma: Every Proposition Implies Itself)
      - Lemma: It is true that something can be (either) true or false (1)
        
        Proof: (related to Lemma: It is true that something can be (either) true or false)
      - Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition (1)
        
        Proof: (related to Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition)
      - Lemma: De Morgan's Laws (Logic) (1)
        
        Proof: (related to Lemma: De Morgan's Laws (Logic))
      - Lemma: Distributivity of Conjunction and Disjunction (1)
        
        Proof: (related to Lemma: Distributivity of Conjunction and Disjunction)
    - Chapter: Contradictory Propositions in Propositional Logic (2)
      - Lemma: A proposition cannot be both, true and false (1)
        
        Proof: (related to Lemma: A proposition cannot be both, true and false)
      - Lemma: A proposition cannot be equivalent to its negation (1)
        
        Proof: (related to Lemma: A proposition cannot be equivalent to its negation)
    - Chapter: Normal Forms in `$PL0$` (8)
    - Definition: Boolean Algebra (1)
      - Lemma: Boolean Algebra of Propositional Logic (1)
        
        Proof: (related to Lemma: Boolean Algebra of Propositional Logic)
  - Part: PL1 - First Order Predicate Logic (3)
    - Definition: Terms in Predicate Logic
    - Definition: Atomic Formulae in Predicate Logic
    - Chapter: Peano Arithmetic
  - Part: Higher-Order Logics
  - Part: Gödel's Incompleteness Theorems
  - Part: Methods of Mathematical Proving (24)
    - Definition: Logical Arguments
    - Example: Examples of Logical Arguments (related to Part: Methods of Mathematical Proving)
    - Chapter: Invalid Logical Arguments (5)
      - Explanation: Inductive Reasoning (related to Chapter: Invalid Logical Arguments)
      - Lemma: Mixing-up the Inclusive and Exclusive Disjunction (1)
        
        Proof: (related to Lemma: Mixing-up the Inclusive and Exclusive Disjunction)
      - Lemma: Mixing-up the Sufficient and Necessary Conditions (1)
        
        Proof: (related to Lemma: Mixing-up the Sufficient and Necessary Conditions)
      - Lemma: Affirming the Consequent of an Implication (1)
        
        Proof: (related to Lemma: Affirming the Consequent of an Implication)
      - Lemma: Denying the Antecedent of an Implication (1)
        
        Proof: (related to Lemma: Denying the Antecedent of an Implication)
    - Chapter: Logical Arguments Used in Mathematical Proofs (11)
      - Explanation: Deductive Reasoning (related to Chapter: Logical Arguments Used in Mathematical Proofs)
      - Lemma: A Criterion for Valid Logical Arguments (1)
        
        Proof: (related to Lemma: A Criterion for Valid Logical Arguments)
      - Lemma: Modus Ponens (1)
        
        Proof: (related to Lemma: Modus Ponens)
      - Lemma: Modus Tollens (1)
        
        Proof: (related to Lemma: Modus Tollens)
      - Lemma: Hypothetical Syllogism (1)
        
        Proof: (related to Lemma: Hypothetical Syllogism)
      - Lemma: Disjunctive Syllogism (1)
        
        Proof: (related to Lemma: Disjunctive Syllogism)
      - Lemma: The Proving Principle by Contradiction (1)
        
        Proof: (related to Lemma: The Proving Principle by Contradiction)
      - Lemma: The Proving Principle By Contraposition, Contrapositive (1)
        
        Proof: (related to Lemma: The Proving Principle By Contraposition, Contrapositive)
      - Lemma: The Proving Principle by Complete Induction (2)
        
        Proof: By Induction (related to Lemma: The Proving Principle by Complete Induction)
        
        Proof: (related to Lemma: The Proving Principle by Complete Induction)
      - Lemma: The Proving Principle by Transfinite Induction (1)
        
        Proof: (related to Lemma: The Proving Principle by Transfinite Induction)
    - Explanation: When is something "well-defined" in mathematics? (related to Part: Methods of Mathematical Proving)
    - Explanation: Good Practices for Writing Mathematical Proofs (related to Part: Methods of Mathematical Proving)
    - Explanation: Correct Negation of Statements (related to Part: Methods of Mathematical Proving)
    - Motivation: What does not have to be proved in mathematics? (related to Part: Methods of Mathematical Proving)
    - Explanation: What does WLOG mean? (related to Part: Methods of Mathematical Proving)
    - Example: Existence and Uniqueness of Solutions (related to Part: Methods of Mathematical Proving)
  - Part: Solving Strategies and Sample Solutions to Problems in Logic
- Branch: Set Theory (153)
  - Part: Historical Development of Set Theory
  - Part: Basics about Sets (20)
    - Definition: Set, Set Element, Empty Set (1)
      - Explanation: Possibilities to Describe Sets, Venn-Diagrams, List, and Set-Builder Notations (related to Definition: Set, Set Element, Empty Set)
    - Definition: Universal Set
    - Definition: Subset and Superset
    - Definition: Power Set
    - Chapter: Set Operations (12)
      - Definition: Set Union (3)
        
        Proposition: Sets are Subsets of Their Union (1)
        
        Proof: (related to Proposition: Sets are Subsets of Their Union)
        
        Proposition: Set Union is Commutative (1)
        
        Proof: (related to Proposition: Set Union is Commutative)
        
        Proposition: Set Union is Associative (1)
        
        Proof: (related to Proposition: Set Union is Associative)
      - Definition: Set Intersection (4)
        
        Definition: Disjoint Sets
        
        Proposition: Intersection of a Set With Another Set is Subset of This Set (1)
        
        Proof: (related to Proposition: Intersection of a Set With Another Set is Subset of This Set)
        
        Proposition: Set Intersection is Commutative (1)
        
        Proof: (related to Proposition: Set Intersection is Commutative)
        
        Proposition: Set Intersection is Associative (1)
        
        Proof: (related to Proposition: Set Intersection is Associative)
      - Definition: Set Complement
      - Definition: Set Difference
      - Proposition: De Morgan's Laws (Sets) (1)
        
        Proof: (related to Proposition: De Morgan's Laws (Sets))
      - Proposition: Distributivity Laws For Sets (1)
        
        Proof: (related to Proposition: Distributivity Laws For Sets)
      - Proposition: Sets and Their Complements (1)
        
        Proof: (related to Proposition: Sets and Their Complements)
    - Definition: Index Set and Set Family
    - Definition: Mutually Disjoint Sets
    - Definition: Set Partition
    - Definition: Generalized Union of Sets
  - Motivation: Russell's Paradox (related to Branch: Set Theory)
  - Part: Zermelo-Fraenkel Set Theory (21)
    - Axiom: Zermelo-Fraenkel Axioms (21)
      - Axiom: Axiom of Existence
      - Axiom: Axiom of Empty Set (1)
        
        Corollary: Uniqueness of the Empty Set (related to Axiom: Axiom of Empty Set) (1)
        
        Proof: (related to Corollary: Uniqueness of the Empty Set)
      - Axiom: Axiom of Extensionality (1)
        
        Explanation: What does the Extensionality Principle mean? (related to Axiom: Axiom of Extensionality)
      - Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension) (8)
        
        Corollary: Justification of the Set-Builder Notation (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of the Set-Builder Notation)
        
        Corollary: Justification of Subsets and Supersets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of Subsets and Supersets)
        
        Corollary: Equality of Sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Equality of Sets)
        
        Explanation: How the Axiom of Separation Avoids Russell's Paradox (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
        
        Corollary: There is no set of all sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: There is no set of all sets)
        
        Corollary: Justification of Set Intersection (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of Set Intersection)
        
        Corollary: Justification of the Difference (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of the Difference)
        
        Corollary: Set Difference and Set Complement are the Same Concepts (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Set Difference and Set Complement are the Same Concepts)
      - Axiom: Axiom of Pairing
      - Axiom: Axiom of Union (1)
        
        Corollary: Justification of Set Union (related to Axiom: Axiom of Union) (1)
        
        Proof: (related to Corollary: Justification of Set Union)
      - Axiom: Axiom of Power Set (2)
        
        Corollary: Justification of Power Set (related to Axiom: Axiom of Power Set) (1)
        
        Proof: (related to Corollary: Justification of Power Set)
        
        Definition: Singleton
      - Axiom: Axiom of Foundation (1)
        
        Corollary: Circular References Of Self-Contained Sets Are Forbidden (related to Axiom: Axiom of Foundation) (1)
        
        Proof: By Induction (related to Corollary: Circular References Of Self-Contained Sets Are Forbidden)
      - Axiom: Axiom of Infinity (3)
        
        Definition: Inductive Set
        
        Definition: Minimal Inductive Set
        
        Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets (related to Axiom: Axiom of Infinity) (1)
        
        Proof: (related to Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets)
      - Axiom: Axiom of Replacement (Schema)
      - Axiom: Axiom of Choice
  - Part: Relations (60)
    - Definition: Ordered Pair, n-Tuple (1)
      - Proposition: Set-Theoretical Meaning of Ordered Tuples (1)
        
        Proof: By Induction (related to Proposition: Set-Theoretical Meaning of Ordered Tuples)
    - Motivation: Usage of Ordered Tuples In Other Mathematical Disciplines (related to Part: Relations)
    - Definition: Cartesian Product
    - Definition: Relation (1)
      - Example: Examples of Relations (related to Definition: Relation)
    - Chapter: Binary Relations and Their Properties (7)
      - Definition: Inverse Relation
      - Explanation: Representations of Binary Relations (related to Chapter: Binary Relations and Their Properties)
      - Definition: Total and Unique Binary Relations
      - Definition: Composition of Binary Relations (2)
        
        Lemma: Composition of Relations Preserves Their Right-Uniqueness Property (1)
        
        Proof: (related to Lemma: Composition of Relations Preserves Their Right-Uniqueness Property)
        
        Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property (1)
        
        Proof: (related to Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property)
      - Definition: Reflexive, Symmetric and Transitive Binary Relations
      - Definition: Irreflexive, Asymmetric and Antisymmetric Binary Relations
    - Definition: Equivalence Relation (7)
      - Motivation: Significance of Equivalence Relations (related to Definition: Equivalence Relation)
      - Example: Examples of Equivalence Relations (related to Definition: Equivalence Relation)
      - Proposition: The Equality of Sets Is an Equivalence Relation (1)
        
        Proof: (related to Proposition: The Equality of Sets Is an Equivalence Relation)
      - Definition: Equivalence Class
      - Definition: Quotient Set, Partition
      - Definition: Complete System of Representatives
      - Definition: Canonical Projection
    - Chapter: Functions (Maps) (25)
      - Definition: Partial and Total Maps (Functions)
      - Explanation: Some Remarks on Functions (related to Chapter: Functions (Maps))
      - Definition: Graph of a Function (1)
        
        Example: Graph of a Real-Valued Function with One Variable (related to Definition: Graph of a Function)
      - Section: Important Properties of Functions (6)
        
        Definition: Injective Function
        
        Definition: Surjective Function
        
        Definition: Bijective Function
        
        Proposition: Characterization of Bijective Functions (1)
        
        Proof: (related to Proposition: Characterization of Bijective Functions)
        
        Definition: Invertible Functions, Inverse Functions
        
        Proposition: Functions Constitute Equivalence Relations (1)
        
        Proof: (related to Proposition: Functions Constitute Equivalence Relations)
      - Section: Common Types of Functions (6)
        
        Definition: Identity Function
        
        Definition: Indicator (Characteristic) Function, Carrier
        
        Definition: Constant Function
        
        Definition: Embedding, Inclusion Map
        
        Definition: Restriction
        
        Definition: Fixed Point, Fixed Point Property
      - Lemma: Behavior of Functions with Set Operations (2)
        
        Proof: (related to Lemma: Behavior of Functions with Set Operations)
        
        Explanation: How Functions interact with Set Operations (related to Lemma: Behavior of Functions with Set Operations)
      - Lemma: Composition of Functions (7)
        
        Proof: (related to Lemma: Composition of Functions)
        
        Proposition: Composition of Functions is Associative (1)
        
        Proof: (related to Proposition: Composition of Functions is Associative)
        
        Proposition: Composition of Surjective Functions is Surjective (1)
        
        Proof: (related to Proposition: Composition of Surjective Functions is Surjective)
        
        Proposition: Composition of Injective Functions is Injective (1)
        
        Proof: (related to Proposition: Composition of Injective Functions is Injective)
        
        Proposition: Composition of Bijective Functions is Bijective (1)
        
        Proof: (related to Proposition: Composition of Bijective Functions is Bijective)
        
        Proposition: Injective, Surjective and Bijective Compositions (1)
        
        Proof: (related to Proposition: Injective, Surjective and Bijective Compositions)
        
        Proposition: The Inverse Of a Composition (1)
        
        Proof: (related to Proposition: The Inverse Of a Composition)
      - Definition: Zero of a Function
    - Chapter: Order Relations (17)
      - Definition: Preorder, Partial Order and Poset
      - Definition: Total Order and Chain
      - Definition: Comparing the Elements of Posets and Chains
      - Definition: Strict Total Order, Strictly-ordered Set
      - Lemma: Comparing the Elements of Strictly Ordered Sets (1)
        
        Proof: (related to Lemma: Comparing the Elements of Strictly Ordered Sets)
      - Explanation: Summary of Different Order Relations (related to Chapter: Order Relations)
      - Explanation: Hasse Diagram (related to Chapter: Order Relations)
      - Definition: Special Elements of Ordered Sets
      - Explanation: Notes on Special Elements of Posets (related to Chapter: Order Relations)
      - Definition: Bounded Subsets of Ordered Sets
      - Definition: Bounded Subsets of Unordered Sets
      - Lemma: Zorn's Lemma (1)
        
        Proof: (related to Lemma: Zorn's Lemma)
      - Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice (1)
        
        Proof: (related to Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice)
      - Definition: Well-order, Well-ordered Set (3)
        
        Explanation: A Note on Well-ordered Sets (related to Definition: Well-order, Well-ordered Set)
        
        Proposition: Well-ordered Sets are Chains (1)
        
        Proof: (related to Proposition: Well-ordered Sets are Chains)
        
        Proposition: Finite Chains are Well-ordered (1)
        
        Proof: (related to Proposition: Finite Chains are Well-ordered)
      - Definition: Order Embedding
  - Part: Ordinal Numbers (26)
    - Motivation: Is "Being a Set Element" ("`$\in$`") a Relation? (related to Part: Ordinal Numbers)
    - Proposition: Transitive Recursion (1)
      - Proof: (related to Proposition: Transitive Recursion)
    - Definition: Contained Relation "`$\in_X$`"
    - Proposition: Contained Relation is a Strict Order (1)
      - Proof: (related to Proposition: Contained Relation is a Strict Order)
    - Definition: Transitive Set (2)
      - Corollary: Properties of Transitive Sets (related to Definition: Transitive Set) (1)
        
        Proof: (related to Corollary: Properties of Transitive Sets)
      - Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull (1)
        
        Proof: (related to Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull)
    - Definition: Extensional Relation (3)
      - Explanation: Examples of Extensional Relations (related to Definition: Extensional Relation) (3)
        
        Proposition: The Contained Relation is Extensional (1)
        
        Proof: (related to Proposition: The Contained Relation is Extensional)
        
        Proposition: Partial Orders are Extensional (1)
        
        Proof: By Contradiction (related to Proposition: Partial Orders are Extensional)
        
        Proposition: Strict Orders are Extensional (1)
        
        Proof: By Contradiction (related to Proposition: Strict Orders are Extensional)
    - Definition: Well-founded Relation (1)
      - Example: Examples of Well-founded Relations (related to Definition: Well-founded Relation)
    - Definition: Mostowski Function and Collapse (6)
    - Definition: Ordinal Number (4)
      - Proposition: Equivalent Notions of Ordinals (1)
        
        Proof: (related to Proposition: Equivalent Notions of Ordinals)
      - Proposition: Ordinals Are Downward Closed (1)
        
        Proof: (related to Proposition: Ordinals Are Downward Closed)
      - Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals (1)
        
        Proof: (related to Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals)
      - Theorem: Trichotomy of Ordinals (1)
        
        Proof: (related to Theorem: Trichotomy of Ordinals)
    - Lemma: Properties of Ordinal Numbers (1)
      - Proof: (related to Lemma: Properties of Ordinal Numbers)
    - Lemma: Successor of Ordinal (1)
      - Proof: (related to Lemma: Successor of Ordinal)
    - Definition: Limit Ordinal
    - Explanation: Building the Successors of Ordinal Numbers (related to Part: Ordinal Numbers)
    - Motivation: Burali-Forti Paradox (related to Part: Ordinal Numbers)
    - Definition: The Class of all Ordinals `$\Omega$`
  - Part: Cardinal Numbers (23)
    - Definition: Equipotent Sets
    - Proposition: Cardinal Number (1)
      - Proof: (related to Proposition: Cardinal Number)
    - Definition: Finite Set, Infinite Set
    - Definition: Comparison of Cardinal Numbers
    - Chapter: Can Cardinals be Ordered?
    - Theorem: Schröder-Bernstein Theorem (1)
      - Proof: (related to Theorem: Schröder-Bernstein Theorem)
    - Chapter: Simple Facts Regarding Cardinals (5)
      - Proposition: Subsets of Finite Sets (1)
        
        Proof: (related to Proposition: Subsets of Finite Sets)
      - Theorem: Distinction Between Finite and Infinite Sets Using Subsets (1)
        
        Proof: (related to Theorem: Distinction Between Finite and Infinite Sets Using Subsets)
      - Proposition: More Characterizations of Finite Sets (1)
        
        Proof: (related to Proposition: More Characterizations of Finite Sets)
      - Lemma: Finite Cardinal Numbers and Set Operations (1)
        
        Proof: (related to Lemma: Finite Cardinal Numbers and Set Operations)
      - Proposition: Counting the Set's Elements Using Its Partition (1)
        
        Proof: (related to Proposition: Counting the Set's Elements Using Its Partition)
    - Motivation: Cantor's Astonishing Discoveries Regarding the Cardinals of Infinite Sets (related to Part: Cardinal Numbers)
    - Explanation: Transitive Set and Countability - Natural Numbers Have the Smallest Infinite Cardinality (related to Part: Cardinal Numbers)
    - Definition: Countable Set, Uncountable Set
    - Proposition: Union of Countably Many Countable Sets (2)
      - Corollary: Cartesian Products of Countable Sets Is Countable (related to Proposition: Union of Countably Many Countable Sets) (1)
        
        Proof: By Induction (related to Corollary: Cartesian Products of Countable Sets Is Countable)
      - Proof: (related to Proposition: Union of Countably Many Countable Sets)
    - Proposition: Cardinals of a Set and Its Power Set (1)
      - Proof: (related to Proposition: Cardinals of a Set and Its Power Set)
    - Proposition: Subset of a Countable Set is Countable (1)
      - Proof: (related to Proposition: Subset of a Countable Set is Countable)
    - Proposition: Rational Numbers are Countable (1)
      - Proof: Uncountable Set (related to Proposition: Rational Numbers are Countable)
    - Proposition: Real Numbers are Uncountable (2)
      - Proof: What does countability mean? (related to Proposition: Real Numbers are Uncountable)
      - Corollary: Irrational Numbers are Uncountable (related to Proposition: Real Numbers are Uncountable) (1)
        
        Proof: (related to Corollary: Irrational Numbers are Uncountable)
    - Proposition: Uncountable and Countable Subsets of Natural Numbers (1)
      - Proof: (related to Proposition: Uncountable and Countable Subsets of Natural Numbers)
    - Chapter: Continuum Hypothesis
  - Part: Solving Strategies and Sample Solutions to Problems in Set Theory (1)
    - Problem: To Construct a Partition of a Given Set (1)
      - Solution: (related to Problem: To Construct a Partition of a Given Set)
- Branch: Number Systems and Arithmetics (223)
  - Part: Natural Numbers (30)
    - Definition: Set-theoretic Definitions of Natural Numbers (15)
      - Proposition: Addition Of Natural Numbers (7)
        
        Proof: (related to Proposition: Addition Of Natural Numbers)
        
        Proposition: Addition Of Natural Numbers Is Associative (1)
        
        Proof: By Induction (related to Proposition: Addition Of Natural Numbers Is Associative)
        
        Proposition: Addition of Natural Numbers Is Commutative (1)
        
        Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Commutative)
        
        Proposition: Addition of Natural Numbers Is Cancellative (2)
        
        Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Cancellative)
        
        Corollary: Contraposition of Cancellative Law for Adding Natural Numbers (related to Proposition: Addition of Natural Numbers Is Cancellative) (1)
        
        Proof: (related to Corollary: Contraposition of Cancellative Law for Adding Natural Numbers)
        
        Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) (related to Proposition: Addition Of Natural Numbers) (1)
        
        Proof: (related to Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers))
        
        Proposition: Uniqueness of Natural Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Natural Zero)
      - Definition: Multiplication of Natural Numbers (6)
        
        Proposition: Multiplication of Natural Numbers Is Associative (1)
        
        Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Associative)
        
        Proposition: Multiplication of Natural Numbers is Commutative (1)
        
        Proof: By Induction (related to Proposition: Multiplication of Natural Numbers is Commutative)
        
        Proposition: Multiplication of Natural Numbers Is Cancellative (2)
        
        Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers)
        
        Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) (related to Definition: Multiplication of Natural Numbers) (1)
        
        Proof: (related to Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers))
        
        Proposition: Uniqueness Of Natural One (1)
        
        Proof: (related to Proposition: Uniqueness Of Natural One)
      - Proposition: Uniqueness Of Predecessors Of Natural Numbers (1)
        
        Proof: (related to Proposition: Uniqueness Of Predecessors Of Natural Numbers)
      - Proposition: Inequality of Natural Numbers and Their Successors (1)
        
        Proof: (related to Proposition: Inequality of Natural Numbers and Their Successors)
    - Axiom: Peano Axioms (2)
      - Definition: Set of Natural Numbers (Peano)
      - Explanation: Why do the Peano axioms define natural numbers? (related to Axiom: Peano Axioms)
    - Proposition: Algebraic Structure Of Natural Numbers Together With Addition (1)
      - Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Addition)
    - Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication)
    - Proposition: Distributivity Law For Natural Numbers (1)
      - Proof: By Induction (related to Proposition: Distributivity Law For Natural Numbers)
    - Chapter: Order (10)
      - Definition: Set-theoretic Definition of Order Relation for Natural Numbers
      - Definition: Order Relation for Natural Numbers (7)
        
        Proposition: Transitivity of the Order Relation of Natural Numbers (1)
        
        Proof: (related to Proposition: Transitivity of the Order Relation of Natural Numbers)
        
        Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities (1)
        
        Proof: (related to Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities)
        
        Proposition: Order Relation for Natural Numbers, Revised (1)
        
        Proof: (related to Proposition: Order Relation for Natural Numbers, Revised)
        
        Proposition: Every Natural Number Is Greater or Equal Zero (1)
        
        Proof: (related to Proposition: Every Natural Number Is Greater or Equal Zero)
        
        Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation (1)
        
        Proof: (related to Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation)
        
        Proposition: Comparing Natural Numbers Using the Concept of Addition (2)
        
        Proof: (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
        
        Corollary: Order Relation for Natural Numbers is Strict Total (related to Proposition: Comparing Natural Numbers Using the Concept of Addition) (1)
        
        Proof: (related to Corollary: Order Relation for Natural Numbers is Strict Total)
      - Proposition: Well-Ordering Principle of Natural Numbers (1)
        
        Proof: (related to Proposition: Well-Ordering Principle of Natural Numbers)
      - Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers (1)
        
        Proof: (related to Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers)
  - Part: Integers (25)
    - Proposition: Definition of Integers (20)
      - Proof: (related to Proposition: Definition of Integers)
      - Proposition: Addition of Integers (9)
        
        Proof: (related to Proposition: Addition of Integers)
        
        Proposition: Addition of Integers Is Associative (2)
        
        Proof: (related to Proposition: Addition of Integers Is Associative)
        
        Proof: (related to Proposition: Addition of Integers Is Associative)
        
        Proposition: Addition of Integers Is Commutative (1)
        
        Proof: (related to Proposition: Addition of Integers Is Commutative)
        
        Proposition: Addition of Integers Is Cancellative (2)
        
        Proof: (related to Proposition: Addition of Integers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Adding Integers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Integers)
        
        Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers) (1)
        
        Proof: (related to Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers))
        
        Proposition: Uniqueness of Integer Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Integer Zero)
        
        Proposition: Existence of Inverse Integers With Respect to Addition (1)
        
        Proof: (related to Proposition: Existence of Inverse Integers With Respect to Addition)
      - Definition: Subtraction of Integers
      - Proposition: Multiplication of Integers (8)
        
        Proof: (related to Proposition: Multiplication of Integers)
        
        Proposition: Multiplication of Integers Is Associative (1)
        
        Proof: (related to Proposition: Multiplication of Integers Is Associative)
        
        Proposition: Multiplication of Integers Is Commutative (1)
        
        Proof: (related to Proposition: Multiplication of Integers Is Commutative)
        
        Proposition: Multiplication of Integers Is Cancellative (2)
        
        Proof: (related to Proposition: Multiplication of Integers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Multiplying Integers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Integers)
        
        Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers) (1)
        
        Proof: (related to Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers))
        
        Proposition: Uniqueness of Integer One (1)
        
        Proof: (related to Proposition: Uniqueness of Integer One)
        
        Proposition: Multiplying Negative and Positive Integers (1)
        
        Proof: (related to Proposition: Multiplying Negative and Positive Integers)
      - Proposition: Distributivity Law For Integers (1)
        
        Proof: (related to Proposition: Distributivity Law For Integers)
    - Proposition: Algebraic Structure of Integers Together with Addition (2)
      - Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
      - Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
    - Proposition: Algebraic Structure of Integers Together with Addition and Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition and Multiplication)
    - Definition: Order Relation for Integers - Positive and Negative Integers (2)
      - Proposition: Order Relation for Integers is Strict Total (1)
        
        Proof: (related to Proposition: Order Relation for Integers is Strict Total)
      - Definition: Absolute Value of Integers
  - Part: Rational Numbers (28)
    - Proposition: Definition of Rational Numbers (22)
      - Proof: (related to Proposition: Definition of Rational Numbers)
      - Proposition: Addition Of Rational Numbers (9)
        
        Proof: (related to Proposition: Addition Of Rational Numbers)
        
        Proposition: Addition of Rational Numbers Is Associative (1)
        
        Proof: (related to Proposition: Addition of Rational Numbers Is Associative)
        
        Proposition: Addition of Rational Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Addition of Rational Numbers Is Commutative)
        
        Proposition: Addition of Rational Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Addition of Rational Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Adding Rational Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Rational Numbers)
        
        Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers) (1)
        
        Proof: (related to Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers))
        
        Proposition: Existence of Inverse Rational Numbers With Respect to Addition (2)
        
        Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Addition)
        
        Proposition: Position of Minus Sign in Rational Numbers Representations (1)
        
        Proof: (related to Proposition: Position of Minus Sign in Rational Numbers Representations)
        
        Proposition: Uniqueness of Rational Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Rational Zero)
      - Proposition: Multiplication Of Rational Numbers (10)
        
        Proof: (related to Proposition: Multiplication Of Rational Numbers)
        
        Proposition: Multiplication of Rational Numbers Is Associative (1)
        
        Proof: (related to Proposition: Multiplication of Rational Numbers Is Associative)
        
        Proposition: Multiplication Of Rational Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Multiplication Of Rational Numbers Is Commutative)
        
        Proposition: Multiplication Of Rational Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Multiplication Of Rational Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers)
        
        Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers) (1)
        
        Proof: (related to Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers))
        
        Proposition: Uniqueness Of Rational One (1)
        
        Proof: (related to Proposition: Uniqueness Of Rational One)
        
        Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication)
        
        Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication)
        
        Proposition: Multiplying Negative and Positive Rational Numbers (1)
        
        Proof: (related to Proposition: Multiplying Negative and Positive Rational Numbers)
      - Definition: Subtraction of Rational Numbers
      - Proposition: Distributivity Law For Rational Numbers (1)
        
        Proof: (related to Proposition: Distributivity Law For Rational Numbers)
    - Proposition: Algebraic Structure of Rational Numbers Together with Addition (1)
      - Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition)
    - Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication)
    - Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication (2)
      - Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
      - Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
    - Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers (1)
      - Proposition: Order Relation for Rational Numbers is Strict Total (1)
        
        Proof: (related to Proposition: Order Relation for Rational Numbers is Strict Total)
    - Definition: Absolute Value of Rational Numbers (1)
      - Corollary: The absolute value makes the set of rational numbers a metric space. (related to Definition: Absolute Value of Rational Numbers) (1)
        
        Proof: (related to Corollary: The absolute value makes the set of rational numbers a metric space.)
  - Part: Irrational Numbers (3)
    - Definition: Definition of Irrational Numbers
    - Proposition: Discovery of Irrational Numbers (2)
      - Proof: (related to Proposition: Discovery of Irrational Numbers)
      - Proof: (related to Proposition: Discovery of Irrational Numbers)
  - Part: Real Numbers (73)
    - Chapter: Real Numbers As Limits Of Rational Numbers (22)
    - Proposition: Definition of Real Numbers (39)
      - Proof: (related to Proposition: Definition of Real Numbers)
      - Proposition: Addition of Real Numbers (14)
        
        Proof: (related to Proposition: Addition of Real Numbers)
        
        Proposition: Addition Of Real Numbers Is Associative (1)
        
        Proof: (related to Proposition: Addition Of Real Numbers Is Associative)
        
        Proposition: Addition Of Real Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Addition Of Real Numbers Is Commutative)
        
        Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers) (1)
        
        Proof: (related to Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers))
        
        Proposition: Existence of Inverse Real Numbers With Respect to Addition (1)
        
        Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Addition)
        
        Proposition: Uniqueness of Real Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Real Zero)
        
        Proposition: Uniqueness of Negative Numbers (6)
        
        Proof: (related to Proposition: Uniqueness of Negative Numbers)
        
        Motivation: Motivation for the Proof of Uniqueness of Zero (related to Proposition: Uniqueness of Negative Numbers)
        
        Corollary: `\(-0=0\)` (related to Proposition: Uniqueness of Negative Numbers) (1)
        
        Proof: (related to Corollary: `\(-0=0\)`)
        
        Corollary: `\((-x)y=-(xy)\)` (related to Proposition: Uniqueness of Negative Numbers) (2)
        
        Proof: (related to Corollary: `\((-x)y=-(xy)\)`)
        
        Corollary: `\((-x)(-y)=xy\)` (related to Corollary: `\((-x)y=-(xy)\)`) (1)
        
        Proof: (related to Corollary: `\((-x)(-y)=xy\)`)
        
        Corollary: `\(-(-x)=x\)` (related to Proposition: Uniqueness of Negative Numbers) (1)
        
        Proof: (related to Corollary: `\(-(-x)=x\)`)
        
        Proposition: Addition of Real Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Addition of Real Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Adding Real Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Real Numbers)
      - Definition: Subtraction of Real Numbers
      - Proposition: Multiplication of Real Numbers (14)
        
        Proof: (related to Proposition: Multiplication of Real Numbers)
        
        Proposition: Multiplication of Real Numbers Is Associative (2)
        
        Corollary: General Associative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Associative) (1)
        
        Proof: (related to Corollary: General Associative Law of Multiplication)
        
        Proof: (related to Proposition: Multiplication of Real Numbers Is Associative)
        
        Proposition: Multiplication of Real Numbers Is Commutative (2)
        
        Proof: (related to Proposition: Multiplication of Real Numbers Is Commutative)
        
        Corollary: General Commutative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Commutative) (1)
        
        Proof: (related to Corollary: General Commutative Law of Multiplication)
        
        Proposition: Multiplication of Real Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Multiplication of Real Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers)
        
        Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers) (1)
        
        Proof: (related to Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers))
        
        Proposition: Uniqueness of Real One (1)
        
        Proof: (related to Proposition: Uniqueness of Real One)
        
        Proposition: Existence of Inverse Real Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Multiplication)
        
        Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication (3)
        
        Proof: (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
        
        Corollary: `\(1^{-1}=1\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication) (1)
        
        Proof: (related to Corollary: `\(1^{-1}=1\)`)
        
        Corollary: `\((x^{-1})^{-1}=x\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication) (1)
        
        Proof: (related to Corollary: `\((x^{-1})^{-1}=x\)`)
        
        Proposition: Multiplying Negative and Positive Real Numbers (1)
        
        Proof: (related to Proposition: Multiplying Negative and Positive Real Numbers)
      - Definition: Division of Real Numbers
      - Explanation: Why is it impossible to divide by `\(0\)`? (related to Proposition: Definition of Real Numbers)
      - Proposition: Distributivity Law For Real Numbers (3)
        
        Proof: (related to Proposition: Distributivity Law For Real Numbers)
        
        Corollary: `\(0x=0\)` (related to Proposition: Distributivity Law For Real Numbers) (2)
        
        Proof: (related to Corollary: `\(0x=0\)`)
        
        Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero. (related to Corollary: `\(0x=0\)`) (1)
        
        Proof: (related to Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero.)
      - Proposition: Unique Solvability of `\(a+x=b\)` (1)
        
        Proof: (related to Proposition: Unique Solvability of `\(a+x=b\)`)
      - Proposition: Unique Solvability of `$ax=b$` (1)
        
        Proof: (related to Proposition: Unique Solvability of `$ax=b$`)
      - Proposition: `\(-(x+y)=-x-y\)` (1)
        
        Proof: (related to Proposition: `\(-(x+y)=-x-y\)`)
      - Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)` (1)
        
        Proof: (related to Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)`)
    - Proposition: Algebraic Structure of Real Numbers Together with Addition (1)
      - Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition)
    - Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication)
    - Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication (2)
      - Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
      - Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
    - Definition: Order Relation of Real Numbers (2)
      - Corollary: Rules of Calculations with Inequalities (related to Definition: Order Relation of Real Numbers) (1)
        
        Proof: (related to Corollary: Rules of Calculations with Inequalities)
      - Proposition: Order Relation for Real Numbers is Strict and Total (1)
        
        Proof: (related to Proposition: Order Relation for Real Numbers is Strict and Total)
    - Axiom: Archimedean Axiom (5)
    - Definition: Absolute Value of Real Numbers (Modulus) (1)
      - Corollary: Properties of the Absolute Value (related to Definition: Absolute Value of Real Numbers (Modulus)) (1)
        
        Proof: (related to Corollary: Properties of the Absolute Value)
  - Part: Complex Numbers (29)
    - Chapter: Algebraic Properties of Complex Numbers (4)
    - Definition: Definition of Complex Numbers (11)
      - Definition: Addition of Complex Numbers (5)
        
        Proposition: Addition of Complex Numbers Is Associative (1)
        
        Proof: (related to Proposition: Addition of Complex Numbers Is Associative)
        
        Proposition: Addition of Complex Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Addition of Complex Numbers Is Commutative)
        
        Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers) (1)
        
        Proof: (related to Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers))
        
        Proposition: Existence of Inverse Complex Numbers With Respect to Addition (1)
        
        Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Addition)
        
        Proposition: Uniqueness of Complex Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Complex Zero)
      - Definition: Subtraction of Complex Numbers
      - Definition: Multiplication of Complex Numbers (4)
        
        Proposition: Multiplication of Complex Numbers Is Associative (1)
        
        Proof: (related to Proposition: Multiplication of Complex Numbers Is Associative)
        
        Proposition: Multiplication of Complex Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Multiplication of Complex Numbers Is Commutative)
        
        Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers) (1)
        
        Proof: (related to Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers))
        
        Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication)
      - Proposition: Distributivity Law for Complex Numbers (1)
        
        Proof: (related to Proposition: Distributivity Law for Complex Numbers)
    - Chapter: Simple Formulas Involving the Complex Numbers (5)
      - Proposition: Extracting the Real and the Imaginary Part of a Complex Number (1)
        
        Proof: (related to Proposition: Extracting the Real and the Imaginary Part of a Complex Number)
      - Proposition: Calculating with Complex Conjugates (1)
        
        Proof: (related to Proposition: Calculating with Complex Conjugates)
      - Proposition: Complex Numbers Cannot Be Ordered (1)
        
        Proof: (related to Proposition: Complex Numbers Cannot Be Ordered)
      - Proposition: Absolute Value of Complex Conjugate (1)
        
        Proof: (related to Proposition: Absolute Value of Complex Conjugate)
      - Proposition: Absolute Value of the Product of Complex Numbers (1)
        
        Proof: (related to Proposition: Absolute Value of the Product of Complex Numbers)
    - Chapter: Vector Properties of Complex Numbers (9)
      - Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)` (1)
        
        Proof: (related to Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`)
      - Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis (1)
        
        Proof: (related to Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis)
      - Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers (1)
        
        Proof: (related to Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers)
      - Definition: Complex Conjugate
      - Definition: Dot Product of Complex Numbers
      - Definition: Absolute Value of Complex Numbers
      - Definition: Argument of a Complex Number
      - Proposition: Polar Coordinates of a Complex Number (1)
        
        Proof: (related to Proposition: Polar Coordinates of a Complex Number)
      - Proposition: Multiplication of Complex Numbers Using Polar Coordinates (1)
        
        Proof: (related to Proposition: Multiplication of Complex Numbers Using Polar Coordinates)
  - Explanation: Comparison Between the Number Systems (related to Branch: Number Systems and Arithmetics)
  - Part: Solving Strategies and Sample Solutions Related to Arithmetics (34)
    - Chapter: Product Manipulation Methods (1)
      - Definition: Products
    - Chapter: Some Important Constants (3)
      - Proposition: Imaginary Unit (1)
        
        Proof: (related to Proposition: Imaginary Unit)
      - Definition: Euler's Constant
      - Definition: Number `$\pi$`
    - Chapter: Sum Manipulation Methods (22)
      - Definition: Sums
      - Proposition: Basic Rules of Manipulating Finite Sums (1)
        
        Proof: (related to Proposition: Basic Rules of Manipulating Finite Sums)
      - Proposition: Double Summation (1)
        
        Proof: (related to Proposition: Double Summation)
      - Proposition: Rule of Combining Different Sets of Indices (1)
        
        Proof: (related to Proposition: Rule of Combining Different Sets of Indices)
      - Proposition: The General Perturbation Method (1)
        
        Proof: (related to Proposition: The General Perturbation Method)
      - Proposition: Abelian Partial Summation Method (1)
        
        Proof: (related to Proposition: Abelian Partial Summation Method)
      - Section: Closed Formulas for Sums (15)
        
        Proposition: Sum of Consecutive Natural Numbers (1)
        
        Proof: By Induction (related to Proposition: Sum of Consecutive Natural Numbers)
        
        Proposition: Sum of Consecutive Odd Numbers (1)
        
        Proof: By Induction (related to Proposition: Sum of Consecutive Odd Numbers)
        
        Proposition: Sum of Squares (1)
        
        Proof: (related to Proposition: Sum of Squares)
        
        Proposition: Sum of Cube Numbers (1)
        
        Proof: (related to Proposition: Sum of Cube Numbers)
        
        Proposition: Geometric Sum (1)
        
        Proof: By Induction (related to Proposition: Geometric Sum)
        
        Proposition: Sum of Geometric Progression (1)
        
        Proof: (related to Proposition: Sum of Geometric Progression)
        
        Proposition: Sum of Arithmetic Progression (1)
        
        Proof: (related to Proposition: Sum of Arithmetic Progression)
        
        Proposition: Alternating Sum of Binomial Coefficients (1)
        
        Proof: (related to Proposition: Alternating Sum of Binomial Coefficients)
        
        Proposition: Sum of Cosines (1)
        
        Proof: (related to Proposition: Sum of Cosines)
        
        Proposition: Sum of Binomial Coefficients (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients)
        
        Proposition: Sum of Binomial Coefficients I (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients I)
        
        Proposition: Sum of Binomial Coefficients II (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients II)
        
        Proposition: Sum of Binomial Coefficients III (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients III)
        
        Proposition: Sum of Binomial Coefficients IV (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients IV)
        
        Proposition: Sum of Factorials (I) (1)
        
        Proof: By Induction (related to Proposition: Sum of Factorials (I))
      - Proposition: Product of Two Sums (Generalized Distributivity Rule) (1)
        
        Proof: (related to Proposition: Product of Two Sums (Generalized Distributivity Rule))
    - Chapter: Fractional Arithmetic (5)
      - Definition: Ratio of Two Real Numbers
      - Proposition: Equality of Two Ratios (1)
        
        Proof: (related to Proposition: Equality of Two Ratios)
      - Proposition: Sum and Difference of Two Ratios (1)
        
        Proof: (related to Proposition: Sum and Difference of Two Ratios)
      - Proposition: Product of Two Ratios (1)
        
        Proof: (related to Proposition: Product of Two Ratios)
      - Proposition: Ratio of Two Ratios (1)
        
        Proof: (related to Proposition: Ratio of Two Ratios)
    - Chapter: Sequences of Numbers (3)
      - Section: Fibonacci Numbers
      - Section: Fermat Numbers
      - Definition: Triangle Numbers
- Branch: Algebra (220)
  - Part: Algebraic Structures - Overview (54)
    - Motivation: Common Concepts Of Algebra: Substructures and Morphisms (related to Part: Algebraic Structures - Overview) (8)
      - Definition: Substructure (1)
        
        Proposition: Subset of Powers is a Submonoid (1)
        
        Proof: (related to Proposition: Subset of Powers is a Submonoid)
      - Explanation: All Types of Morphisms and Their Properties (related to Motivation: Common Concepts Of Algebra: Substructures and Morphisms)
      - Definition: Homomorphism
      - Definition: Monomorphism
      - Definition: Epimorphism
      - Definition: Isomorphism
      - Definition: Endomorphism
      - Definition: Automorphism
    - Definition: Algebraic Structure (Algebra)
    - Definition: Binary Operation
    - Chapter: Important Properties of Binary Operations (8)
      - Definition: Closure
      - Definition: Associativity (1)
        
        Corollary: General Associative Law (related to Definition: Associativity) (1)
        
        Proof: (related to Corollary: General Associative Law)
      - Definition: Commutativity (1)
        
        Corollary: General Commutative Law (related to Definition: Commutativity) (1)
        
        Proof: (related to Corollary: General Commutative Law)
      - Definition: Existence of a Neutral Element
      - Proposition: Uniqueness of the Neutral Element (1)
        
        Proof: Zero (Absorbing, Annihilating) Element, Left Zero, Right Zero (related to Proposition: Uniqueness of the Neutral Element)
      - Definition: Inverse Element
      - Proposition: Uniqueness of Inverse Elements (1)
        
        Proof: Conformity (related to Proposition: Uniqueness of Inverse Elements)
      - Definition: Cancellation Property
    - Explanation: Operation Table (related to Part: Algebraic Structures - Overview)
    - Chapter: Magmas, Semigroups, Monoids (Overview) (8)
      - Definition: Magma
      - Axiom: Axioms of Magma
      - Definition: Semigroup
      - Axiom: Axioms of Semigroup
      - Definition: Monoid
      - Axiom: Axioms of Monoid
      - Example: Examples of Magmas, Semigroups, and Monoids (related to Chapter: Magmas, Semigroups, Monoids (Overview))
      - Definition: Exponentiation in a Monoid
    - Chapter: Groups (Overview) (11)
      - Motivation: Calculations in a Group (related to Chapter: Groups (Overview)) (3)
        
        Proposition: Simple Calculations Rules in a Group (1)
        
        Proof: (related to Proposition: Simple Calculations Rules in a Group)
        
        Definition: Exponentiation in a Group (1)
        
        Corollary: Rules for Exponentiation in a Group (related to Definition: Exponentiation in a Group) (1)
        
        Proof: (related to Corollary: Rules for Exponentiation in a Group)
        
        Explanation: Calculation Rules in a Group with Additive Notation (related to Motivation: Calculations in a Group)
      - Example: Examples of Groups (related to Chapter: Groups (Overview))
      - Definition: Group
      - Axiom: Axioms of Group
      - Definition: Commutative (Abelian) Group
      - Definition: Subgroup
      - Proposition: Criteria for Subgroups (1)
        
        Proof: (related to Proposition: Criteria for Subgroups)
      - Explanation: Unions of Subgroups Are Not Subgroups (related to Chapter: Groups (Overview))
      - Proposition: Unique Solvability of `$a\ast x=b$` in Groups (1)
        
        Proof: (related to Proposition: Unique Solvability of `$a\ast x=b$` in Groups)
    - Chapter: Rings (Overview) (7)
      - Axiom: Axiom of Distributivity
      - Definition: (Unit) Ring
      - Definition: Commutative (Unit) Ring (1)
        
        Definition: Multiplicative System
      - Definition: Subring
      - Definition: Ring Homomorphism (2)
        
        Definition: Algebra over a Ring (2)
        
        Definition: Integral Element (1)
        
        Definition: Ring of Integers
        
        Definition: Integral Closure
      - Example: Examples of Ring Homomorphisms (related to Chapter: Rings (Overview))
    - Chapter: Fields (Overview) (5)
      - Example: Examples of Fields (related to Chapter: Fields (Overview))
      - Definition: Field
      - Definition: Subfield
      - Definition: Field Homomorphism
      - Proposition: In a Field, `$0$` Is Unequal `$1$` (1)
        
        Proof: (related to Proposition: In a Field, `$0$` Is Unequal `$1$`)
    - Chapter: Vector Spaces (Overview) (3)
      - Definition: Vector Space (2)
        
        Example: Examples of Vector Spaces (related to Definition: Vector Space)
        
        Proposition: Simple Consequences from the Definition of a Vector Space (1)
        
        Proof: (related to Proposition: Simple Consequences from the Definition of a Vector Space)
      - Definition: Subspace (1)
        
        Example: Trivial Subspaces, Zero Space (related to Definition: Subspace)
    - Chapter: Modules (Overview) (1)
      - Definition: Module
  - Part: Group Theory (32)
    - Definition: Generating Set of a Group
    - Definition: Group Homomorphism (6)
      - Example: Examples of Group Homomorphisms (related to Definition: Group Homomorphism)
      - Proposition: Properties of a Group Homomorphism (1)
        
        Proof: (related to Proposition: Properties of a Group Homomorphism)
      - Example: Examples of Properties of Group Homomorphisms (related to Definition: Group Homomorphism)
      - Lemma: Kernel and Image of Group Homomorphism (1)
        
        Proof: (related to Lemma: Kernel and Image of Group Homomorphism)
      - Example: Examples of Kernels and Images of Group Homomorphisms (related to Definition: Group Homomorphism)
      - Lemma: Kernel and Image of a Group Homomorphism are Subgroups (1)
        
        Proof: (related to Lemma: Kernel and Image of a Group Homomorphism are Subgroups)
    - Proposition: Additive Subgroups of Integers (1)
      - Proof: (related to Proposition: Additive Subgroups of Integers)
    - Theorem: Construction of Groups from Commutative and Cancellative Semigroups (1)
      - Proof: (related to Theorem: Construction of Groups from Commutative and Cancellative Semigroups)
    - Definition: Cyclic Group, Order of an Element
    - Example: Examples of Cyclic Groups (related to Part: Group Theory)
    - Proposition: Finite Order of an Element Equals Order Of Generated Group (1)
      - Proof: (related to Proposition: Finite Order of an Element Equals Order Of Generated Group)
    - Proposition: Group Homomorphisms with Cyclic Groups (1)
      - Proof: (related to Proposition: Group Homomorphisms with Cyclic Groups)
    - Lemma: Cyclic Groups are Abelian (1)
      - Proof: (related to Lemma: Cyclic Groups are Abelian)
    - Lemma: Subgroups of Cyclic Groups (1)
      - Proof: (related to Lemma: Subgroups of Cyclic Groups)
    - Proposition: Subgroups of Finite Cyclic Groups (1)
      - Proof: (related to Proposition: Subgroups of Finite Cyclic Groups)
    - Definition: Direct Product of Groups
    - Definition: Conjugate Elements of a Group
    - Definition: Cosets
    - Proposition: Properties of Cosets (1)
      - Proof: (related to Proposition: Properties of Cosets)
    - Lemma: Subgroups and Their Cosets are Equipotent (1)
      - Proof: (related to Lemma: Subgroups and Their Cosets are Equipotent)
    - Theorem: Order of Subgroup Divides Order of Finite Group (1)
      - Proof: (related to Theorem: Order of Subgroup Divides Order of Finite Group)
    - Theorem: Order of Cyclic Group (Fermat's Little Theorem) (1)
      - Proof: (related to Theorem: Order of Cyclic Group (Fermat's Little Theorem))
    - Chapter: Symmetry Groups
    - Definition: Normal Subgroups
    - Lemma: Factor Groups (1)
      - Proof: (related to Lemma: Factor Groups)
    - Lemma: Group Homomorphisms and Normal Subgroups (1)
      - Proof: (related to Lemma: Group Homomorphisms and Normal Subgroups)
    - Theorem: First Isomorphism Theorem for Groups (1)
      - Proof: (related to Theorem: First Isomorphism Theorem for Groups)
    - Theorem: Classification of Cyclic Groups (2)
      - Proof: (related to Theorem: Classification of Cyclic Groups)
      - Definition: Group Order
    - Theorem: Classification of Finite Groups with the Order of a Prime Number (1)
      - Proof: (related to Theorem: Classification of Finite Groups with the Order of a Prime Number)
    - Definition: Group Operation
  - Part: Algebraic Number Theory and Ring Theory (51)
    - Chapter: Divisibility in General Rings (34)
      - Definition: Zero Divisor and Integral Domain (4)
        
        Proposition: Cancellation Law (1)
        
        Proof: (related to Proposition: Cancellation Law)
        
        Definition: Zero Ring
        
        Theorem: Construction of Fields from Integral Domains (1)
        
        Proof: (related to Theorem: Construction of Fields from Integral Domains)
        
        Theorem: Finite Integral Domains are Fields (1)
        
        Proof: (related to Theorem: Finite Integral Domains are Fields)
      - Proposition: Generalization of Cancellative Multiplication of Integers (1)
        
        Proof: By Induction (related to Proposition: Generalization of Cancellative Multiplication of Integers)
      - Definition: Generalization of Divisor and Multiple
      - Definition: Generalization of the Greatest Common Divisor
      - Definition: Generalization of the Least Common Multiple
      - Definition: Unit (1)
        
        Proposition: Group of Units (1)
        
        Proof: (related to Proposition: Group of Units)
      - Definition: Associate (1)
        
        Lemma: A Criterion for Associates (1)
        
        Proof: By Induction (related to Lemma: A Criterion for Associates)
      - Definition: Irreducible, Prime
      - Definition: Factorial Ring, Generalization of Factorization
      - Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder
      - Section: Ideals (19)
        
        Definition: Ideal (3)
        
        Lemma: Greatest Common Divisor and Least Common Multiple of Ideals (1)
        
        Proof: (related to Lemma: Greatest Common Divisor and Least Common Multiple of Ideals)
        
        Definition: Divisibility of Ideals
        
        Definition: Addition of Ideals
        
        Proposition: Principal Ideals being Prime Ideals (1)
        
        Proof: By Induction (related to Proposition: Principal Ideals being Prime Ideals)
        
        Proposition: Principal Ideals being Maximal Ideals (1)
        
        Proof: By Induction (related to Proposition: Principal Ideals being Maximal Ideals)
        
        Definition: Principal Ideal (5)
        
        Proposition: Principal Ideal Generated by A Unit (1)
        
        Proof: By Induction (related to Proposition: Principal Ideal Generated by A Unit)
        
        Proposition: Criterions for Equality of Principal Ideals (1)
        
        Proof: By Induction (related to Proposition: Criterions for Equality of Principal Ideals)
        
        Lemma: Divisibility of Principal Ideals (1)
        
        Proof: (related to Lemma: Divisibility of Principal Ideals)
        
        Definition: Principal Ideal Ring
        
        Definition: Principal Ideal Domain
        
        Definition: Generating Set of an Ideal
        
        Definition: Prime Ideal (5)
        
        Definition: Spectrum of a Commutative Ring (3)
        
        Proposition: Spectrum Function of Commutative Rings (1)
        
        Proof: (related to Proposition: Spectrum Function of Commutative Rings)
        
        Proposition: Open and Closed Subsets of a Zariski Topology (1)
        
        Proof: (related to Proposition: Open and Closed Subsets of a Zariski Topology)
        
        Definition: Zariski Topology of a Commutative Ring
        
        Lemma: Fiber of Prime Ideals Under a Spectrum Function (2)
        
        Proof: (related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
        
        Lemma: Fiber of Prime Ideals (1)
        
        Proof: (related to Lemma: Fiber of Prime Ideals)
        
        Definition: Maximal Ideal (1)
        
        Lemma: Fiber of Maximal Ideals (1)
        
        Proof: (related to Lemma: Fiber of Maximal Ideals)
        
        Lemma: Prime Ideals of Multiplicative Systems in Integral Domains (1)
        
        Proof: (related to Lemma: Prime Ideals of Multiplicative Systems in Integral Domains)
        
        Theorem: Connection between Rings, Ideals, and Fields (1)
        
        Proof: (related to Theorem: Connection between Rings, Ideals, and Fields)
      - Lemma: Factor Rings, Generalization of Congruence Classes (2)
        
        Proof: (related to Lemma: Factor Rings, Generalization of Congruence Classes)
        
        Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring (1)
        
        Proof: (related to Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring)
    - Chapter: Polynomial Rings, Irreducibility, and Field Extensions (11)
      - Definition: Polynomial over a Ring, Degree, Variable (1)
        
        Definition: Reduction of an Integer Polynomial Modulo a Prime Number
      - Definition: Polynomial Ring
      - Definition: Elementary Symmetric Functions
      - Definition: Irreducible Polynomial
      - Definition: Field Extension (1)
        
        Definition: Finite Field Extension
      - Definition: Algebraic Element
      - Definition: Multiplicity of a Root of a Polynomial
      - Definition: Transcendental Element
      - Definition: Minimal Polynomial
      - Definition: Subadditive Function
      - Section: Solutions of Polynomials (1)
        
        Proposition: Quadratic Formula (1)
        
        Proof: (related to Proposition: Quadratic Formula)
    - Chapter: Algebraic and Transcendent Numbers (3)
      - Motivation: Rational Numbers and the Greatest Common Divisor (related to Chapter: Algebraic and Transcendent Numbers)
      - Definition: Continued Fractions
      - Lemma: Continuants and Convergents (1)
        
        Proof: By Induction (related to Lemma: Continuants and Convergents)
    - Theorem: Isomorphism of Rings
    - Definition: Characteristic of a Ring (2)
      - Explanation: Characteristic Zero Instead of Characteristic Infinite (related to Definition: Characteristic of a Ring)
      - Lemma: Any Positive Characteristic Is a Prime Number (1)
        
        Proof: (related to Lemma: Any Positive Characteristic Is a Prime Number)
  - Part: Finite Fields (3)
    - Definition: Finite Field
    - Definition: Characteristic of a Field
    - Definition: Prime Field
  - Part: Galois Theory
  - Part: Linear Algebra (65)
    - Chapter: Introduction to Matrices and Vectors (13)
      - Definition: Matrix, Set of Matrices over a Field
      - Definition: Transposed Matrix
      - Definition: Column Vectors and Row Vectors
      - Section: Addition of Matrices and Vectors (4)
        
        Definition: Matrix and Vector Addition
        
        Definition: Zero Matrix, Zero Vector
        
        Proposition: Abelian Group of Matrices Under Addition (2)
        
        Proof: (related to Proposition: Abelian Group of Matrices Under Addition)
        
        Corollary: Abelian Group of Vectors Under Addition (related to Proposition: Abelian Group of Matrices Under Addition) (1)
        
        Proof: (related to Corollary: Abelian Group of Vectors Under Addition)
      - Definition: Square Matrix (3)
        
        Definition: Symmetric Matrix
        
        Definition: Upper and Lower Triangular Matrix
        
        Definition: Diagonal Matrix
      - Definition: Matrix Multiplication
      - Definition: Identity Matrix
      - Definition: Invertible and Inverse Matrix
    - Chapter: Linear Equations and Systems of Linear Equations (SLEs) (14)
      - Definition: Linear Equations with many Unknowns (1)
        
        Corollary: Solutions of a Linear Equation with many Unknowns (related to Definition: Linear Equations with many Unknowns) (1)
        
        Proof: (related to Corollary: Solutions of a Linear Equation with many Unknowns)
      - Definition: Systems of Linear Equations with many Unknowns
      - Definition: Coefficient Matrix
      - Section: Solving Simple Systems of Linear Equations (5)
        
        Example: Solution to a Zero SLE (related to Section: Solving Simple Systems of Linear Equations)
        
        Example: Solution to a Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
        
        Example: Solution to a Degenerated Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
        
        Definition: Solution to an Upper Triangular SLE - Backward Substitution
        
        Definition: Solution to a Lower Triangular SLE - Forward Substitution
      - Section: Solving General Systems Of Linear Equations - Gaussian Method (4)
        
        Definition: Elementary Gaussian Operations
        
        Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE (1)
        
        Proof: (related to Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE)
        
        Definition: Gaussian Method to Solve Systems of Linear Equations, Rank of a Matrix
        
        Example: The Gaussian Method in Practice (related to Section: Solving General Systems Of Linear Equations - Gaussian Method)
      - Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs (1)
        
        Proof: (related to Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs)
      - Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations (1)
        
        Proof: By Induction (related to Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations)
    - Chapter: Vectors Revised - Vector Spaces (18)
      - Application: SLEs Revised in the Light of Vector Spaces (related to Chapter: Vectors Revised - Vector Spaces)
      - Definition: Linear Combination
      - Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector
      - Definition: Dimension of a Vector Space
      - Proposition: Quotient Space (1)
        
        Proof: (related to Proposition: Quotient Space)
      - Definition: Generating Systems
      - Definition: Basis, Coordinate System (2)
        
        Lemma: Uniqueness Lemma of a Finite Basis (1)
        
        Proof: (related to Lemma: Uniqueness Lemma of a Finite Basis)
        
        Theorem: Finite Basis Theorem (1)
        
        Proof: (related to Theorem: Finite Basis Theorem)
      - Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
      - Definition: Linear Span
      - Section: Cross Product
      - Section: Subspaces (2)
        
        Definition: Direct Sum of Vector Spaces
        
        Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace (1)
        
        Proof: (related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
      - Section: Dual Space
      - Definition: Linear Map (3)
        
        Explanation: Kernel and Image (related to Definition: Linear Map)
        
        Explanation: Matrices of Linear Maps (related to Definition: Linear Map)
        
        Definition: Multilinear Map (1)
        
        Definition: Alternating Multilinear Map
      - Definition: Bilinear Form (1)
        
        Definition: Symmetric Bilinear Form
    - Chapter: Vectors (2)
      - Definition: Dot Product, Inner Product, Scalar Product (General Field Case) (1)
        
        Corollary: Properties of a Real Scalar Product (related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case)) (1)
        
        Proof: (related to Corollary: Properties of a Real Scalar Product)
      - Definition: Dot Product, Inner Product, Scalar Product (Complex Case) (1)
        
        Proposition: Properties of a Complex Scalar Product (1)
        
        Proof: (related to Proposition: Properties of a Complex Scalar Product)
    - Chapter: Eigenvalues and Eigenvectors (2)
      - Definition: Eigenvalue
      - Definition: Eigenvector
    - Chapter: Principal Axis Transformation
    - Chapter: Jordan Normal Form
    - Chapter: Rotations and Basic Transformations
    - Chapter: Determinants (1)
      - Definition: Recursive Definition of the Determinant
    - Chapter: Affine Spaces (12)
      - Definition: Affine Subspace
      - Definition: Affine Basis, Affine Coordinate System (2)
        
        Corollary: Barycentric Coordinates, Barycenter (related to Definition: Affine Basis, Affine Coordinate System) (1)
        
        Proof: (related to Corollary: Barycentric Coordinates, Barycenter)
        
        Definition: Bounded Affine Set
      - Section: Affine Maps
      - Section: Normal Forms
      - Section: Quadrics
      - Definition: Affine Space (2)
        
        Definition: Euclidean Affine Space
        
        Definition: Unitary Affine Space
      - Definition: Affinely Dependent and Affinely Independent Points
      - Definition: Dimension of an Affine Space (1)
        
        Definition: Points, Lines, Planes, Hyperplanes
      - Definition: Convex Affine Set (2)
        
        Corollary: Intersection of Convex Affine Sets (related to Definition: Convex Affine Set) (1)
        
        Proof: (related to Corollary: Intersection of Convex Affine Sets)
        
        Definition: Convex Hull
  - Part: Constructions with Ruler and Compass
  - Part: Ordered Fields and Their Topology (11)
    - Definition: Ordered Field
    - Proposition: `$0$` Is Less Than `$1$` In Ordered Fields (1)
      - Proof: (related to Proposition: `$0$` Is Less Than `$1$` In Ordered Fields)
    - Proposition: Square of a Non-Zero Element is Positive in Ordered Fields (1)
      - Proof: (related to Proposition: Square of a Non-Zero Element is Positive in Ordered Fields)
    - Definition: Absolute Values and Non-Archimedean Absolute Values of Fields
    - Example: Examples of Absolute Values (related to Part: Ordered Fields and Their Topology)
    - Proposition: A Field with an Absolute Value is a Metric Space (1)
      - Proof: (related to Proposition: A Field with an Absolute Value is a Metric Space)
    - Definition: Dependent and Independent Absolute Values
    - Proposition: Characterization of Dependent Absolute Values (1)
      - Proof: (related to Proposition: Characterization of Dependent Absolute Values)
    - Proposition: Characterization of Non-Archimedean Absolute Values (1)
      - Proof: (related to Proposition: Characterization of Non-Archimedean Absolute Values)
    - Definition: Signum Function in An Ordered Field
    - Definition: Complete Ordered Field
  - Part: Solving Strategies and Sample Solutions to Problems in Algebra (2)
    - Chapter: Group-theoretic Problems (2)
      - Problem: Verifying Group Properties (1)
        
        Solution: (related to Problem: Verifying Group Properties)
      - Problem: Verifying Subgroup Properties (1)
        
        Solution: (related to Problem: Verifying Subgroup Properties)
- Branch: Analysis (450)
  - Part: Historical Development of Analysis (3)
  - Part: Real Analysis of One Variable and Elements of Complex Analysis (311)
    - Chapter: Basics of Real Analysis of One Variable (14)
      - Section: Real Intervals and Bounded Real Sets (11)
        
        Definition: Real Intervals
        
        Definition: Supremum, Least Upper Bound
        
        Definition: Maximum (Real Numbers)
        
        Definition: Extended Real Numbers
        
        Definition: Supremum of Extended Real Numbers
        
        Definition: Infimum, Greatest Lower Bound
        
        Definition: Minimum (Real Numbers)
        
        Definition: Infimum of Extended Real Numbers
        
        Proposition: Closed Formula for the Maximum and Minimum of Two Numbers (1)
        
        Proof: (related to Proposition: Closed Formula for the Maximum and Minimum of Two Numbers)
        
        Definition: Nested Real Intervals
        
        Proposition: Limit of Nested Real Intervals (1)
        
        Proof: (related to Proposition: Limit of Nested Real Intervals)
      - Proposition: Rational Numbers are Dense in Real Numbers (1)
        
        Proof: (related to Proposition: Rational Numbers are Dense in Real Numbers)
      - Proposition: Open Intervals Contain Uncountably Many Irrational Numbers (1)
        
        Proof: (related to Proposition: Open Intervals Contain Uncountably Many Irrational Numbers)
      - Proposition: Open Real Intervals are Uncountable (1)
        
        Proof: (related to Proposition: Open Real Intervals are Uncountable)
    - Chapter: Real-valued Sequences and Limits of Sequences and Functions (54)
      - Definition: Real Sequence
      - Definition: Convergent Real Sequence
      - Definition: Divergent Sequences (5)
        
        Example: Divergent Alternating Sequence (related to Definition: Divergent Sequences)
        
        Definition: Sequences Tending To Infinity
        
        Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero (1)
        
        Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero)
        
        Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty (1)
        
        Proof: (related to Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty)
        
        Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity (1)
        
        Proof: (related to Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity)
      - Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
      - Definition: Monotonic Sequences (1)
        
        Proposition: Real Sequences Contain Monotonic Subsequences (1)
        
        Proof: (related to Proposition: Real Sequences Contain Monotonic Subsequences)
      - Section: Theorems Regarding Limits Of Sequences (8)
        
        Proposition: Uniqueness Of the Limit of a Sequence (1)
        
        Proof: (related to Proposition: Uniqueness Of the Limit of a Sequence)
        
        Proposition: Sum of Convergent Real Sequences (1)
        
        Proof: (related to Proposition: Sum of Convergent Real Sequences)
        
        Proposition: Difference of Convergent Real Sequences (1)
        
        Proof: (related to Proposition: Difference of Convergent Real Sequences)
        
        Proposition: Product of Convegent Real Sequences (1)
        
        Proof: (related to Proposition: Product of Convegent Real Sequences)
        
        Proposition: Product of a Real Number and a Convergent Real Sequence (1)
        
        Proof: (related to Proposition: Product of a Real Number and a Convergent Real Sequence)
        
        Proposition: Quotient of Convergent Real Sequences (1)
        
        Proof: (related to Proposition: Quotient of Convergent Real Sequences)
        
        Proposition: How Convergence Preserves the Order Relation of Sequence Members (1)
        
        Proof: (related to Proposition: How Convergence Preserves the Order Relation of Sequence Members)
        
        Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members (1)
        
        Proof: (related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)
      - Definition: Limits of Real Functions
      - Section: Theorems Regarding Limits of Functions (8)
        
        Proposition: Limit of a Function is Unique If It Exists (1)
        
        Proof: (related to Proposition: Limit of a Function is Unique If It Exists)
        
        Proposition: Arithmetic of Functions with Limits - Sums (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Sums)
        
        Proposition: Arithmetic of Functions with Limits - Difference (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Difference)
        
        Proposition: Arithmetic of Functions with Limits - Division (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Division)
        
        Proposition: Preservation of Inequalities for Limits of Functions (1)
        
        Proof: (related to Proposition: Preservation of Inequalities for Limits of Functions)
        
        Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity (1)
        
        Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity)
        
        Theorem: Squeezing Theorem for Functions (1)
        
        Proof: (related to Theorem: Squeezing Theorem for Functions)
        
        Proposition: Arithmetic of Functions with Limits - Product (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Product)
      - Section: Examples of Limit Calculations (17)
        
        Proposition: Limit of 1/n (1)
        
        Proof: (related to Proposition: Limit of 1/n)
        
        Proposition: Limit of the Constant Function (1)
        
        Proof: (related to Proposition: Limit of the Constant Function)
        
        Proposition: Limit of the Identity Function (1)
        
        Proof: (related to Proposition: Limit of the Identity Function)
        
        Proposition: Limit of Nth Powers (1)
        
        Proof: By Induction (related to Proposition: Limit of Nth Powers)
        
        Proposition: Convergence Behavior of the Sequence `\((b^n)\)` (1)
        
        Proof: (related to Proposition: Convergence Behavior of the Sequence `\((b^n)\)`)
        
        Proposition: Limit of a Polynomial (1)
        
        Proof: By Induction (related to Proposition: Limit of a Polynomial)
        
        Proposition: Limit of a Rational Function (1)
        
        Proof: (related to Proposition: Limit of a Rational Function)
        
        Proposition: Limit of Nth Root of a Positive Constant (1)
        
        Proof: (related to Proposition: Limit of Nth Root of a Positive Constant)
        
        Proposition: Limit of Nth Root of N (1)
        
        Proof: (related to Proposition: Limit of Nth Root of N)
        
        Proposition: Limits of General Powers (2)
        
        Proof: (related to Proposition: Limits of General Powers)
        
        Corollary: Value of Zero to the Power of X (related to Proposition: Limits of General Powers) (1)
        
        Proof: (related to Corollary: Value of Zero to the Power of X)
        
        Proposition: Square Roots (1)
        
        Proof: (related to Proposition: Square Roots)
        
        Proposition: Limit of Exponential Growth as Compared to Polynomial Growth (1)
        
        Proof: (related to Proposition: Limit of Exponential Growth as Compared to Polynomial Growth)
        
        Proposition: Limits of Logarithm in `$[0,+\infty]$` (1)
        
        Proof: (related to Proposition: Limits of Logarithm in `$[0,+\infty]$`)
        
        Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth (1)
        
        Proof: (related to Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth)
        
        Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function (1)
        
        Proof: (related to Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function)
        
        Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function (1)
        
        Proof: (related to Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function)
      - Definition: Real Subsequence
      - Theorem: Every Bounded Real Sequence has a Convergent Subsequence (1)
        
        Proof: By Induction (related to Theorem: Every Bounded Real Sequence has a Convergent Subsequence)
      - Definition: Accumulation Point (Real Numbers) (4)
        
        Example: Examples of Accumulation Points (related to Definition: Accumulation Point (Real Numbers))
        
        Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence (1)
        
        Proof: (related to Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence)
        
        Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence (1)
        
        Proof: (related to Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence)
        
        Definition: Isolated Point (Real Numbers)
      - Definition: Asymptotical Approximation
      - Lemma: Decreasing Sequence of Suprema of Extended Real Numbers (1)
        
        Proof: (related to Lemma: Decreasing Sequence of Suprema of Extended Real Numbers)
      - Definition: Limit Superior
      - Lemma: Increasing Sequence of Infima of Extended Real Numbers (1)
        
        Proof: (related to Lemma: Increasing Sequence of Infima of Extended Real Numbers)
      - Definition: Limit Inferior
    - Chapter: Completeness of Real Numbers (7)
      - Proposition: The distance of real numbers makes real numbers a metric space. (1)
        
        Proof: (related to Proposition: The distance of real numbers makes real numbers a metric space.)
      - Definition: Real Cauchy Sequence
      - Theorem: Completeness Principle for Real Numbers (2)
        
        Proof: (related to Theorem: Completeness Principle for Real Numbers)
        
        Corollary: Non-Cauchy Sequences are Not Convergent (related to Theorem: Completeness Principle for Real Numbers) (1)
        
        Proof: (related to Corollary: Non-Cauchy Sequences are Not Convergent)
      - Theorem: Supremum Property, Infimum Property (1)
        
        Proof: (related to Theorem: Supremum Property, Infimum Property)
      - Proposition: Convergent Real Sequences Are Cauchy Sequences (1)
        
        Proof: (related to Proposition: Convergent Real Sequences Are Cauchy Sequences)
      - Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers (1)
        
        Proof: (related to Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers)
    - Chapter: Criteria for Convergence of Sequences (4)
      - Theorem: Every Bounded Monotonic Sequence Is Convergent (2)
        
        Proof: (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
        
        Application: Application of Monotonic Convergence (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
      - Proposition: Convergent Real Sequences are Bounded (1)
        
        Proof: (related to Proposition: Convergent Real Sequences are Bounded)
      - Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence (1)
        
        Proof: (related to Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence)
    - Theorem: Defining Properties of the Field of Real Numbers (1)
      - Proof: (related to Theorem: Defining Properties of the Field of Real Numbers)
    - Chapter: Useful Inequalities (21)
      - Proposition: Generalized Triangle Inequality (1)
        
        Proof: By Induction (related to Proposition: Generalized Triangle Inequality)
      - Theorem: Triangle Inequality (1)
        
        Proof: (related to Theorem: Triangle Inequality)
      - Theorem: Reverse Triangle Inequalities (1)
        
        Proof: (related to Theorem: Reverse Triangle Inequalities)
      - Definition: (Weighted) Arithmetic Mean
      - Theorem: Inequality of the Arithmetic Mean (1)
        
        Proof: (related to Theorem: Inequality of the Arithmetic Mean)
      - Theorem: Inequality of Weighted Arithmetic Mean (1)
        
        Proof: (related to Theorem: Inequality of Weighted Arithmetic Mean)
      - Theorem: Bernoulli's Inequality (1)
        
        Proof: By Induction (related to Theorem: Bernoulli's Inequality)
      - Proposition: Generalized Bernoulli's Inequality (1)
        
        Proof: By Induction (related to Proposition: Generalized Bernoulli's Inequality)
      - Proposition: Cauchy–Schwarz Inequality (1)
        
        Proof: (related to Proposition: Cauchy–Schwarz Inequality)
      - Definition: Geometric Mean
      - Theorem: Inequality Between the Geometric and the Arithmetic Mean (1)
        
        Proof: By Induction (related to Theorem: Inequality Between the Geometric and the Arithmetic Mean)
      - Lemma: Upper Bound for the Product of General Powers (1)
        
        Proof: (related to Lemma: Upper Bound for the Product of General Powers)
      - Proposition: Hölder's Inequality (1)
        
        Proof: (related to Proposition: Hölder's Inequality)
      - Proposition: Minkowski's Inequality (1)
        
        Proof: (related to Proposition: Minkowski's Inequality)
      - Proposition: Hölder's Inequality for Integral p-norms (1)
        
        Proof: (related to Proposition: Hölder's Inequality for Integral p-norms)
      - Proposition: Cauchy-Schwarz Inequality for Integral p-norms (1)
        
        Proof: (related to Proposition: Cauchy-Schwarz Inequality for Integral p-norms)
      - Proposition: Minkowski's Inequality for Integral p-norms (1)
        
        Proof: (related to Proposition: Minkowski's Inequality for Integral p-norms)
      - Proposition: Inequality between Square Numbers and Powers of `$2$` (1)
        
        Proof: By Induction (related to Proposition: Inequality between Square Numbers and Powers of `$2$`)
      - Proposition: Inequality between Powers of `$2$` and Factorials (1)
        
        Proof: By Induction (related to Proposition: Inequality between Powers of `$2$` and Factorials)
      - Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials (1)
        
        Proof: (related to Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials)
      - Proposition: Bounds for Partial Sums of Exponential Series (1)
        
        Proof: (related to Proposition: Bounds for Partial Sums of Exponential Series)
    - Chapter: Properties of Real Functions (77)
      - Definition: Continuous Functions at Single Real Numbers (9)
        
        Example: Examples of Functions Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
        
        Example: Examples of Functions Not Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
        
        Proposition: Compositions of Continuous Functions on a Whole Domain (1)
        
        Proof: (related to Proposition: Compositions of Continuous Functions on a Whole Domain)
        
        Proposition: Composition of Continuous Functions at a Single Point (1)
        
        Proof: (related to Proposition: Composition of Continuous Functions at a Single Point)
        
        Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point (1)
        
        Proof: (related to Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point)
        
        Theorem: Intermediate Value Theorem (1)
        
        Proof: (related to Theorem: Intermediate Value Theorem)
        
        Theorem: Intermediate Root Value Theorem (1)
        
        Proof: By Induction (related to Theorem: Intermediate Root Value Theorem)
        
        Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point (related to Definition: Continuous Functions at Single Real Numbers) (1)
        
        Proof: (related to Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point)
        
        Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments (related to Definition: Continuous Functions at Single Real Numbers) (1)
        
        Proof: (related to Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments)
      - Definition: Continuous Real Functions (4)
        
        Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions (1)
        
        Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions)
        
        Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain (1)
        
        Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain)
        
        Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals (1)
        
        Proof: (related to Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals)
        
        Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth (1)
        
        Proof: (related to Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth)
      - Definition: Even and Odd Complex Functions
      - Section: Differentiable Functions (15)
        
        Definition: Difference Quotient
        
        Definition: Derivative, Differentiable Functions (1)
        
        Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability
        
        Definition: Higher-Order Derivatives (1)
        
        Proposition: Generalized Product Rule (1)
        
        Proof: (related to Proposition: Generalized Product Rule)
        
        Definition: Local Extremum (3)
        
        Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum (2)
        
        Proof: (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
        
        Explanation: Local Extrema on a Closed Interval (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
        
        Proposition: Sufficient Condition for a Local Extremum (1)
        
        Proof: (related to Proposition: Sufficient Condition for a Local Extremum)
        
        Theorem: Rolle's Theorem (1)
        
        Proof: (related to Theorem: Rolle's Theorem)
        
        Proposition: Differentiable Functions and Tangent-Linear Approximation (2)
        
        Proof: (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
        
        Corollary: Differentiable Functions are Continuous (related to Proposition: Differentiable Functions and Tangent-Linear Approximation) (1)
        
        Proof: (related to Corollary: Differentiable Functions are Continuous)
        
        Proposition: Characterization of Monotonic Functions via Derivatives (1)
        
        Proof: (related to Proposition: Characterization of Monotonic Functions via Derivatives)
        
        Theorem: Darboux's Theorem (3)
        
        Proof: (related to Theorem: Darboux's Theorem)
        
        Corollary: Estimating the Growth of a Function with its Derivative (related to Theorem: Darboux's Theorem) (2)
        
        Proof: (related to Corollary: Estimating the Growth of a Function with its Derivative)
        
        Corollary: Sufficient Condition for a Function to be Constant (related to Corollary: Estimating the Growth of a Function with its Derivative) (1)
        
        Proof: (related to Corollary: Sufficient Condition for a Function to be Constant)
        
        Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule (1)
        
        Proof: (related to Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule)
        
        Proposition: Chain Rule (1)
        
        Proof: (related to Proposition: Chain Rule)
      - Definition: Continuously Differentiable Functions
      - Definition: Uniformly Continuous Functions (Real Case) (5)
        
        Proposition: Not all Continuous Functions are also Uniformly Continuous (1)
        
        Proof: (related to Proposition: Not all Continuous Functions are also Uniformly Continuous)
        
        Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions (1)
        
        Proof: (related to Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions)
        
        Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous (2)
        
        Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
        
        Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
        
        Corollary: All Uniformly Continuous Functions are Continuous (related to Definition: Uniformly Continuous Functions (Real Case)) (1)
        
        Proof: (related to Corollary: All Uniformly Continuous Functions are Continuous)
      - Definition: Periodic Functions
      - Definition: Even and Odd Functions (1)
        
        Proposition: Derivatives of Even and Odd Functions (1)
        
        Proof: (related to Proposition: Derivatives of Even and Odd Functions)
      - Section: Integrable Functions (24)
        
        Subsection: Riemann Integral (22)
        
        Proposition: Riemann Integral for Step Functions (2)
        
        Proof: (related to Proposition: Riemann Integral for Step Functions)
        
        Proposition: Linearity and Monotony of the Riemann Integral for Step Functions (1)
        
        Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral for Step Functions)
        
        Definition: Riemann-Integrable Functions (8)
        
        Proposition: Step Function on Closed Intervals are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Step Function on Closed Intervals are Riemann-Integrable)
        
        Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable)
        
        Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable)
        
        Proposition: Linearity and Monotony of the Riemann Integral (1)
        
        Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral)
        
        Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable)
        
        Proposition: Product of Riemann-integrable Functions is Riemann-integrable (1)
        
        Proof: (related to Proposition: Product of Riemann-integrable Functions is Riemann-integrable)
        
        Example: Existence of not Riemann-Integrable Functions (related to Definition: Riemann-Integrable Functions)
        
        Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions (1)
        
        Proof: (related to Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions)
        
        Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions (2)
        
        Proof: (related to Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions)
        
        Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals (1)
        
        Proof: (related to Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals)
        
        Definition: Riemann Sum With Respect to a Partition (1)
        
        Proposition: Riemann Sum Converging To the Riemann Integral (1)
        
        Proof: (related to Proposition: Riemann Sum Converging To the Riemann Integral)
        
        Theorem: Indefinite Integral, Antiderivative (2)
        
        Proof: (related to Theorem: Indefinite Integral, Antiderivative)
        
        Proposition: Antiderivatives are Uniquely Defined Up to a Constant (1)
        
        Proof: By Induction (related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
        
        Theorem: Fundamental Theorem of Calculus (1)
        
        Proof: (related to Theorem: Fundamental Theorem of Calculus)
        
        Proposition: Integrals on Adjacent Intervals (1)
        
        Proof: By Induction (related to Proposition: Integrals on Adjacent Intervals)
        
        Theorem: Integration by Substitution (1)
        
        Proof: (related to Theorem: Integration by Substitution)
        
        Theorem: Mean Value Theorem For Riemann Integrals (1)
        
        Proof: (related to Theorem: Mean Value Theorem For Riemann Integrals)
        
        Theorem: Partial Integration (1)
        
        Proof: (related to Theorem: Partial Integration)
        
        Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine (1)
        
        Proof: (related to Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine)
        
        Lemma: Trapezoid Rule (1)
        
        Proof: (related to Lemma: Trapezoid Rule)
        
        Subsection: Extended Concept of the Riemann Integral - the Improper Integral (1)
        
        Definition: Improper Integral
        
        Subsection: Riemann-Stieltjes Integral
      - Lemma: Invertible Functions on Real Intervals (2)
        
        Proof: (related to Lemma: Invertible Functions on Real Intervals)
        
        Proposition: Derivative of an Invertible Function on Real Invervals (1)
        
        Proof: (related to Proposition: Derivative of an Invertible Function on Real Invervals)
      - Definition: Convex and Concave Functions (2)
        
        Proposition: Convexity and Concaveness Test (1)
        
        Proof: (related to Proposition: Convexity and Concaveness Test)
        
        Proposition: Convex Functions on Open Intervals are Continuous (1)
        
        Proof: (related to Proposition: Convex Functions on Open Intervals are Continuous)
      - Definition: Logarithmically Convex and Concave Functions
      - Section: Uniform Convergence of Functions (8)
        
        Definition: Pointwise and Uniformly Convergent Sequences of Functions
        
        Example: Pointwise vs. Uniformly Convergent Sequences of Functions (related to Section: Uniform Convergence of Functions)
        
        Proposition: Only the Uniform Convergence Preserves Continuity (1)
        
        Proof: (related to Proposition: Only the Uniform Convergence Preserves Continuity)
        
        Definition: Supremum Norm for Functions
        
        Proposition: Supremum Norm and Uniform Convergence (1)
        
        Proof: (related to Proposition: Supremum Norm and Uniform Convergence)
        
        Proposition: Uniform Convergence Criterion of Cauchy (1)
        
        Proof: (related to Proposition: Uniform Convergence Criterion of Cauchy)
        
        Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series (1)
        
        Proof: (related to Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series)
        
        Proposition: Calculations with Uniformly Convergent Functions (1)
        
        Proof: (related to Proposition: Calculations with Uniformly Convergent Functions)
      - Definition: Monotonic Functions
      - Definition: Bounded and Unbounded Functions (2)
        
        Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals (2)
        
        Proof: (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
        
        Corollary: Continuous Real Functions on Closed Intervals are Bounded (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals) (1)
        
        Proof: (related to Corollary: Continuous Real Functions on Closed Intervals are Bounded)
    - Chapter: Types of Real Functions (86)
      - Definition: Constant Function Real Case (1)
        
        Corollary: Derivative of a Constant Function (related to Definition: Constant Function Real Case) (1)
        
        Proof: (related to Corollary: Derivative of a Constant Function)
      - Definition: Real Identity Function (1)
        
        Proposition: Identity Function is Continuous (1)
        
        Proof: (related to Proposition: Identity Function is Continuous)
      - Definition: Reciprocal Function (2)
        
        Proposition: Derivative of the Reciprocal Function (1)
        
        Proof: (related to Proposition: Derivative of the Reciprocal Function)
        
        Proposition: Integral of the Reciprocal Function (1)
        
        Proof: (related to Proposition: Integral of the Reciprocal Function)
      - Definition: Real Absolute Value Function (1)
        
        Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)` (1)
        
        Proof: (related to Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`)
      - Definition: Step Functions (2)
        
        Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval (1)
        
        Proof: (related to Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval)
        
        Definition: Order Relation for Step Functions
      - Definition: Positive and Negative Parts of a Real-Valued Function
      - Definition: Linear Function (2)
        
        Corollary: Derivative of a Linear Function `\(ax+b\)` (related to Definition: Linear Function) (2)
        
        Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
        
        Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
      - Proposition: Nth Roots of Positive Numbers (1)
        
        Proof: (related to Proposition: Nth Roots of Positive Numbers)
      - Proposition: Rational Powers of Positive Numbers (2)
        
        Proof: (related to Proposition: Rational Powers of Positive Numbers)
        
        Corollary: Limit of N-th Roots (related to Proposition: Rational Powers of Positive Numbers) (1)
        
        Proof: (related to Corollary: Limit of N-th Roots)
      - Proposition: Nth Powers (2)
        
        Proof: (related to Proposition: Nth Powers)
        
        Proposition: Derivative of the n-th Power Function (1)
        
        Proof: (related to Proposition: Derivative of the n-th Power Function)
      - Proposition: General Powers of Positive Numbers (4)
        
        Proof: (related to Proposition: General Powers of Positive Numbers)
        
        Proposition: Calculation Rules for General Powers (1)
        
        Proof: (related to Proposition: Calculation Rules for General Powers)
        
        Proposition: Derivative of General Powers of Positive Numbers (1)
        
        Proof: (related to Proposition: Derivative of General Powers of Positive Numbers)
        
        Proposition: Integral of General Powers (1)
        
        Proof: (related to Proposition: Integral of General Powers)
      - Definition: Polynomials (3)
        
        Proposition: Limits of Polynomials at Infinity (2)
        
        Proof: (related to Proposition: Limits of Polynomials at Infinity)
        
        Corollary: Real Polynomials of Odd Degree Have at Least One Real Root (related to Proposition: Limits of Polynomials at Infinity) (1)
        
        Proof: (related to Corollary: Real Polynomials of Odd Degree Have at Least One Real Root)
        
        Proposition: Eveness (Oddness) of Polynomials (1)
        
        Proof: (related to Proposition: Eveness (Oddness) of Polynomials)
      - Definition: Rational Functions (1)
        
        Proposition: Rational Functions are Continuous (1)
        
        Proof: (related to Proposition: Rational Functions are Continuous)
      - Proposition: Exponential Function (18)
        
        Proof: (related to Proposition: Exponential Function)
        
        Definition: Exponential Function of General Base (6)
        
        Proposition: Continuity of Exponential Function of General Base (1)
        
        Proof: (related to Proposition: Continuity of Exponential Function of General Base)
        
        Proposition: Functional Equation of the Exponential Function of General Base (2)
        
        Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base)
        
        Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function of General Base) (1)
        
        Proof: (related to Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property)
        
        Proposition: Exponential Function of General Base With Natural Exponents (1)
        
        Proof: By Induction (related to Proposition: Exponential Function of General Base With Natural Exponents)
        
        Proposition: Exponential Function of General Base With Integer Exponents (1)
        
        Proof: (related to Proposition: Exponential Function of General Base With Integer Exponents)
        
        Proposition: Functional Equation of the Exponential Function of General Base (Revised) (1)
        
        Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base (Revised))
        
        Proposition: Estimate for the Remainder Term of Exponential Function (1)
        
        Proof: (related to Proposition: Estimate for the Remainder Term of Exponential Function)
        
        Proposition: Functional Equation of the Exponential Function (6)
        
        Proof: (related to Proposition: Functional Equation of the Exponential Function)
        
        Corollary: Reciprocity of Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function) (2)
        
        Proof: (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
        
        Corollary: (Real) Exponential Function Is Always Positive (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property) (1)
        
        Proof: (related to Corollary: (Real) Exponential Function Is Always Positive)
        
        Corollary: Exponential Function Is Strictly Monotonically Increasing (related to Proposition: Functional Equation of the Exponential Function) (1)
        
        Proof: (related to Corollary: Exponential Function Is Strictly Monotonically Increasing)
        
        Corollary: Exponential Function Is Non-Negative (Real Case) (related to Proposition: Functional Equation of the Exponential Function) (1)
        
        Proof: (related to Corollary: Exponential Function Is Non-Negative (Real Case))
        
        Corollary: Exponential Function and the Euler Constant (related to Proposition: Functional Equation of the Exponential Function) (1)
        
        Proof: (related to Corollary: Exponential Function and the Euler Constant)
        
        Proposition: Continuity of Exponential Function (1)
        
        Proof: (related to Proposition: Continuity of Exponential Function)
        
        Proposition: `\(\exp(0)=1\)` (1)
        
        Proof: (related to Proposition: `\(\exp(0)=1\)`)
        
        Proposition: Derivative of the Exponential Function (1)
        
        Proof: (related to Proposition: Derivative of the Exponential Function)
        
        Proposition: Integral of the Exponential Function (1)
        
        Proof: (related to Proposition: Integral of the Exponential Function)
      - Section: Logarithms (5)
        
        Proposition: Natural Logarithm (4)
        
        Proof: (related to Proposition: Natural Logarithm)
        
        Proposition: Functional Equation of the Natural Logarithm (1)
        
        Proof: (related to Proposition: Functional Equation of the Natural Logarithm)
        
        Proposition: Derivative of the Natural Logarithm (1)
        
        Proof: (related to Proposition: Derivative of the Natural Logarithm)
        
        Proposition: Integral of the Natural Logarithm (1)
        
        Proof: (related to Proposition: Integral of the Natural Logarithm)
        
        Proposition: Logarithm to a General Base (1)
        
        Proof: (related to Proposition: Logarithm to a General Base)
      - Section: Trigonometric Functions (30)
        
        Definition: Cosine of a Real Variable (5)
        
        Proposition: Eveness of the Cosine of a Real Variable (1)
        
        Proof: (related to Proposition: Eveness of the Cosine of a Real Variable)
        
        Proposition: Zero of Cosine (1)
        
        Proof: (related to Proposition: Zero of Cosine)
        
        Proposition: Derivative of Cosine (1)
        
        Proof: (related to Proposition: Derivative of Cosine)
        
        Proposition: Integral of Cosine (1)
        
        Proof: (related to Proposition: Integral of Cosine)
        
        Corollary: Representing Real Cosine by Complex Exponential Function (related to Definition: Cosine of a Real Variable) (1)
        
        Proof: (related to Corollary: Representing Real Cosine by Complex Exponential Function)
        
        Definition: Sine of a Real Variable (4)
        
        Proposition: Oddness of the Sine of a Real Variable (1)
        
        Proof: (related to Proposition: Oddness of the Sine of a Real Variable)
        
        Proposition: Derivative of Sine (1)
        
        Proof: (related to Proposition: Derivative of Sine)
        
        Proposition: Integral of Sine (1)
        
        Proof: (related to Proposition: Integral of Sine)
        
        Corollary: Representing Real Sine by Complex Exponential Function (related to Definition: Sine of a Real Variable) (1)
        
        Proof: (related to Corollary: Representing Real Sine by Complex Exponential Function)
        
        Definition: Tangent of a Real Variable (2)
        
        Proposition: Additivity Theorem of Tangent (1)
        
        Proof: (related to Proposition: Additivity Theorem of Tangent)
        
        Proposition: Derivative of Tangent (1)
        
        Proof: (related to Proposition: Derivative of Tangent)
        
        Proposition: Continuity of Cosine and Sine (1)
        
        Proof: (related to Proposition: Continuity of Cosine and Sine)
        
        Proposition: Pythagorean Identity (1)
        
        Proof: (related to Proposition: Pythagorean Identity)
        
        Proposition: Additivity Theorems of Cosine and Sine (1)
        
        Proof: (related to Proposition: Additivity Theorems of Cosine and Sine)
        
        Proposition: Infinite Series for Cosine and Sine (2)
        
        Proof: (related to Proposition: Infinite Series for Cosine and Sine)
        
        Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine (1)
        
        Proof: (related to Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine)
        
        Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function (6)
        
        Proof: (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
        
        Corollary: Cosine and Sine are Periodic Functions (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Cosine and Sine are Periodic Functions)
        
        Corollary: Negative Cosine and Sine vs Shifting the Argument (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Negative Cosine and Sine vs Shifting the Argument)
        
        Corollary: Arguments for which Cosine and Sine are Equal to Each Other (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Arguments for which Cosine and Sine are Equal to Each Other)
        
        Corollary: All Zeros of Cosine and Sine (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: All Zeros of Cosine and Sine)
        
        Corollary: More Insight to Euler's Identity (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: More Insight to Euler's Identity)
        
        Proposition: Inverse Cosine of a Real Variable (1)
        
        Proof: (related to Proposition: Inverse Cosine of a Real Variable)
        
        Proposition: Inverse Sine of a Real Variable (3)
        
        Proof: (related to Proposition: Inverse Sine of a Real Variable)
        
        Proposition: Derivative of the Inverse Sine (1)
        
        Proof: (related to Proposition: Derivative of the Inverse Sine)
        
        Proposition: Integral of Inverse Sine (1)
        
        Proof: (related to Proposition: Integral of Inverse Sine)
        
        Proposition: Inverse Tangent of a Real Variable (4)
        
        Proof: (related to Proposition: Inverse Tangent of a Real Variable)
        
        Proposition: Inverse Tangent and Complex Exponential Function (1)
        
        Proof: (related to Proposition: Inverse Tangent and Complex Exponential Function)
        
        Proposition: Derivative of the Inverse Tangent (1)
        
        Proof: (related to Proposition: Derivative of the Inverse Tangent)
        
        Proposition: Integral of the Inverse Tangent (1)
        
        Proof: (related to Proposition: Integral of the Inverse Tangent)
      - Section: Cyclometric Functions
      - Section: Hyporbolic Functions (3)
        
        Definition: Hyperbolic Cosine (1)
        
        Proposition: Sum of Arguments of Hyperbolic Cosine (1)
        
        Proof: (related to Proposition: Sum of Arguments of Hyperbolic Cosine)
        
        Definition: Hyperbolic Sine (1)
        
        Proposition: Sum of Arguments of Hyperbolic Sine (1)
        
        Proof: (related to Proposition: Sum of Arguments of Hyperbolic Sine)
        
        Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine (1)
        
        Proof: (related to Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine)
      - Section: Inverse Hyperbolic Functions (2)
        
        Proposition: Inverse Hyperbolic Sine (1)
        
        Proof: (related to Proposition: Inverse Hyperbolic Sine)
        
        Proposition: Inverse Hyperbolic Cosine (1)
        
        Proof: (related to Proposition: Inverse Hyperbolic Cosine)
      - Section: Mixed Functions
      - Proposition: Gamma Function (2)
        
        Proposition: Gamma Function Interpolates the Factorial (1)
        
        Proof: (related to Proposition: Gamma Function Interpolates the Factorial)
        
        Proof: (related to Proposition: Gamma Function)
      - Example: Examples of Real Functions, Whose Graphs Cannot be Plotted (related to Chapter: Types of Real Functions)
    - Chapter: Infinite Series - Overview (40)
      - Definition: Convergent Complex Series (1)
        
        Proposition: A General Criterion for the Convergence of Infinite Complex Series (1)
        
        Proof: (related to Proposition: A General Criterion for the Convergence of Infinite Complex Series)
      - Definition: Infinite Series, Partial Sums
      - Definition: Convergent Real Series (4)
        
        Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent (1)
        
        Proof: (related to Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent)
        
        Proposition: Sum of Convergent Real Series (1)
        
        Proof: (related to Proposition: Sum of Convergent Real Series)
        
        Proposition: Difference of Convergent Real Series (1)
        
        Proof: (related to Proposition: Difference of Convergent Real Series)
        
        Proposition: Product of a Real Number and a Convergent Real Series (1)
        
        Proof: (related to Proposition: Product of a Real Number and a Convergent Real Series)
      - Definition: Absolutely Convergent Series (2)
        
        Proposition: Convergence Behaviour of Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Convergence Behaviour of Absolutely Convergent Series)
        
        Proposition: Cauchy Product of Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Series)
      - Definition: Complex Infinite Series
      - Definition: Absolutely Convergent Complex Series (3)
        
        Proposition: Direct Comparison Test For Absolutely Convergent Complex Series (1)
        
        Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Complex Series)
        
        Proposition: Ratio Test For Absolutely Convergent Complex Series (1)
        
        Proof: By Induction (related to Proposition: Ratio Test For Absolutely Convergent Complex Series)
        
        Proposition: Cauchy Product of Absolutely Convergent Complex Series (1)
        
        Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Complex Series)
      - Definition: Divergent Series
      - Section: Convergence and Divergence Criteria for Real Series (20)
        
        Proposition: Monotony Criterion (2)
        
        Corollary: Monotony Criterion for Absolute Series (related to Proposition: Monotony Criterion) (1)
        
        Proof: (related to Corollary: Monotony Criterion for Absolute Series)
        
        Proof: (related to Proposition: Monotony Criterion)
        
        Proposition: Leibniz Criterion for Alternating Series (1)
        
        Proof: (related to Proposition: Leibniz Criterion for Alternating Series)
        
        Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero (1)
        
        Proof: (related to Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero)
        
        Proposition: Cauchy Condensation Criterion (2)
        
        Example: Applications of the Cauchy Condensation Criterion (related to Proposition: Cauchy Condensation Criterion)
        
        Proof: (related to Proposition: Cauchy Condensation Criterion)
        
        Proposition: Direct Comparison Test For Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Series)
        
        Proposition: Direct Comparison Test For Divergent Series (1)
        
        Proof: (related to Proposition: Direct Comparison Test For Divergent Series)
        
        Proposition: Limit Comparizon Test (1)
        
        Proof: (related to Proposition: Limit Comparizon Test)
        
        Proposition: Root Test (1)
        
        Proof: (related to Proposition: Root Test)
        
        Proposition: Ratio Test (1)
        
        Proof: By Induction (related to Proposition: Ratio Test)
        
        Proposition: Limit Test for Roots or Ratios (1)
        
        Proof: (related to Proposition: Limit Test for Roots or Ratios)
        
        Lemma: Convergence Test for Telescoping Series (1)
        
        Proof: (related to Lemma: Convergence Test for Telescoping Series)
        
        Proposition: Raabe's Test (1)
        
        Proof: (related to Proposition: Raabe's Test)
        
        Subsection: Convergence Criteria for Infinite Series Involving Products (4)
        
        Lemma: Abel's Lemma for Testing Convergence (1)
        
        Proof: (related to Lemma: Abel's Lemma for Testing Convergence)
        
        Proposition: Cauchy-Schwarz Test (1)
        
        Proof: (related to Proposition: Cauchy-Schwarz Test)
        
        Proposition: Abel's Test (1)
        
        Proof: (related to Proposition: Abel's Test)
        
        Proposition: Dirichlet's Test (1)
        
        Proof: (related to Proposition: Dirichlet's Test)
        
        Proposition: Integral Test for Convergence (1)
        
        Proof: (related to Proposition: Integral Test for Convergence)
        
        Proposition: Cauchy Criterion (1)
        
        Proof: (related to Proposition: Cauchy Criterion)
      - Definition: Rearrangement of Infinite Series (2)
        
        Proposition: Rearrangement of Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Rearrangement of Absolutely Convergent Series)
        
        Proposition: Rearrangement of Convergent Series (1)
        
        Proof: (related to Proposition: Rearrangement of Convergent Series)
      - Section: Closed Formulas for Infinite Real Series (1)
        
        Proposition: Infinite Geometric Series (1)
        
        Proof: (related to Proposition: Infinite Geometric Series)
      - Definition: `\(b\)`-Adic Fractions (4)
        
        Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences (1)
        
        Proof: (related to Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences)
        
        Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions (2)
        
        Proof: (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
        
        Corollary: Real Numbers Can Be Approximated by Rational Numbers (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions) (1)
        
        Proof: (related to Corollary: Real Numbers Can Be Approximated by Rational Numbers)
        
        Definition: Decimal Representation of Real Numbers
    - Chapter: Infinite Products - Overview
    - Chapter: Representation of Functions as Taylor Series (4)
      - Proposition: Approximation of Functions by Taylor's Formula (1)
        
        Proof: (related to Proposition: Approximation of Functions by Taylor's Formula)
      - Proposition: Taylor's Formula with Remainder Term of Lagrange (1)
        
        Proof: (related to Proposition: Taylor's Formula with Remainder Term of Lagrange)
      - Theorem: Taylor's Formula (2)
        
        Corollary: Taylor's Formula for Polynomials (related to Theorem: Taylor's Formula) (1)
        
        Proof: (related to Corollary: Taylor's Formula for Polynomials)
        
        Proof: By Induction (related to Theorem: Taylor's Formula)
    - Chapter: Power Series Introduction
    - Chapter: Fourier Series
  - Part: Real Analysis of Multiple Variables (17)
    - Chapter: Differentiability (6)
      - Section: Implicit Functions
      - Definition: Directional Derivative (1)
        
        Definition: Higher Order Directional Derivative
      - Section: Extremal Values with Side Conditions
      - Definition: `\(n\)` times Continuously Differentiable Functions
      - Definition: Totally Differentiable Functions, Total Derivative
      - Definition: Derivative of an n-Dimensional Curve
    - Chapter: Integrability (4)
      - Section: Path Integrals
      - Section: Multiple Riemann Integrals
      - Section: Multiple Lebesgue Integrals
      - Section: Jordan Measurability
    - Chapter: Fixed Point Theory (3)
      - Section: Brouwer's Theorem
      - Section: Schauder's Theorem
      - Section: Kakutani's Theorem
    - Definition: Curves In the Multidimensional Space `\(\mathbb R^n\)` (2)
      - Definition: Jordan Arc (Simple Curve)
      - Definition: Closed Curve, Open Curve
    - Definition: Generalized Polynomial Function
    - Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm (1)
      - Proof: (related to Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm)
  - Part: Complex Analysis (68)
    - Chapter: Topological Aspects (25)
      - Definition: Open and Closed Discs
      - Definition: Closed and Open Regions of the Complex Plane
      - Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
      - Proposition: The distance of complex numbers makes complex numbers a metric space. (1)
        
        Proof: (related to Proposition: The distance of complex numbers makes complex numbers a metric space.)
      - Proposition: Complex Convergent Sequences are Bounded (1)
        
        Proof: (related to Proposition: Complex Convergent Sequences are Bounded)
      - Definition: Continuous Functions at Single Complex Numbers
      - Proposition: Convergent Complex Sequences Are Cauchy Sequences (1)
        
        Proof: (related to Proposition: Convergent Complex Sequences Are Cauchy Sequences)
      - Definition: Complex Sequence
      - Definition: Bounded Complex Sequences (1)
        
        Proposition: Convergent Complex Sequences Are Bounded (1)
        
        Proof: (related to Proposition: Convergent Complex Sequences Are Bounded)
      - Definition: Convergent Complex Sequence (5)
        
        Proposition: Quotient of Convergent Complex Sequences (1)
        
        Proof: (related to Proposition: Quotient of Convergent Complex Sequences)
        
        Proposition: Sum of Convergent Complex Sequences (1)
        
        Proof: (related to Proposition: Sum of Convergent Complex Sequences)
        
        Proposition: Difference of Convergent Complex Sequences (1)
        
        Proof: (related to Proposition: Difference of Convergent Complex Sequences)
        
        Proposition: Product of Convegent Complex Sequences (1)
        
        Proof: (related to Proposition: Product of Convegent Complex Sequences)
        
        Proposition: Product of a Complex Number and a Convergent Complex Sequence (1)
        
        Proof: (related to Proposition: Product of a Complex Number and a Convergent Complex Sequence)
      - Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences (2)
        
        Proof: (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
        
        Corollary: Convergence of Complex Conjugate Sequence (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences) (1)
        
        Proof: (related to Corollary: Convergence of Complex Conjugate Sequence)
      - Definition: Limits of Complex Functions
      - Definition: Complex Cauchy Sequence
      - Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences (1)
        
        Proof: (related to Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences)
      - Theorem: Completeness Principle for Complex Numbers (1)
        
        Proof: (related to Theorem: Completeness Principle for Complex Numbers)
      - Definition: Accumulation Points (Complex Numbers)
      - Section: Paths
      - Section: Homology and Winding Numbers
      - Section: Homotopy
      - Definition: Bounded Complex Sets
    - Chapter: Geometric Aspects (6)
      - Section: Moebius Transformations (3)
        
        Subsection: Inversion
        
        Subsection: Fixed Points
        
        Subsection: Cross-Ratio
      - Section: Riemann's Plane and Sphere
      - Section: Isogonality
      - Section: Analytic Continuation
    - Chapter: Properties of Complex Functions (9)
      - Definition: Continuous Complex Functions
      - Section: Differentiable Complex Functions
      - Section: Periodic Complex Functions
      - Section: Invertible Complex Functions
      - Section: Integrable Complex Functions (2)
        
        Subsection: Indefinite Integral
        
        Subsection: Path Integrals
      - Section: Holomorphic Functions (3)
        
        Subsection: Criteria for Holomorphic Functions
        
        Subsection: Properties
        
        Subsection: Harmonic Functions
    - Chapter: Types of Complex Functions (20)
      - Proposition: Complex Exponential Function (7)
        
        Proof: (related to Proposition: Complex Exponential Function)
        
        Proposition: Estimate for the Remainder Term of Complex Exponential Function (1)
        
        Proof: (related to Proposition: Estimate for the Remainder Term of Complex Exponential Function)
        
        Proposition: Functional Equation of the Complex Exponential Function (2)
        
        Proof: (related to Proposition: Functional Equation of the Complex Exponential Function)
        
        Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property)
        
        Proposition: `\(\exp(0)=1\)` (Complex Case) (1)
        
        Proof: (related to Proposition: `\(\exp(0)=1\)` (Complex Case))
        
        Proposition: Continuity of Complex Exponential Function (1)
        
        Proof: (related to Proposition: Continuity of Complex Exponential Function)
        
        Proposition: Complex Conjugate of Complex Exponential Function (1)
        
        Proof: (related to Proposition: Complex Conjugate of Complex Exponential Function)
      - Lemma: Unit Circle (1)
        
        Proof: (related to Lemma: Unit Circle)
      - Proposition: Euler's Formula (1)
        
        Proof: (related to Proposition: Euler's Formula)
      - Proposition: n-th Roots of Unity (1)
        
        Proof: (related to Proposition: n-th Roots of Unity)
      - Theorem: De Moivre's Identity, Complex Powers (1)
        
        Proof: By Induction (related to Theorem: De Moivre's Identity, Complex Powers)
      - Section: Complex Trigonometric Functions
      - Definition: Complex Polynomials
      - Section: Complex Rational Functions
      - Section: Complex Logarithms
      - Section: Complex Cyclometric Functions
      - Section: Complex Hyporbolic Functions
      - Section: Complex Inverse Hyperbolic Functions
      - Section: Complex Mixed Functions
      - Lemma: Euler's Identity (1)
        
        Proof: (related to Lemma: Euler's Identity)
    - Chapter: Laurant Series
    - Chapter: Isolated Singularities
    - Chapter: Meromorphic Functions
    - Chapter: Residue Theorem
    - Chapter: Discrete Fourier Transform (DFT) (4)
      - Definition: Even Complex Sequence
      - Definition: Odd Complex Sequence
      - Definition: n-Periodical Complex Sequence
      - Lemma: Sum of Roots Of Unity in Complete Residue Systems (1)
        
        Proof: (related to Lemma: Sum of Roots Of Unity in Complete Residue Systems)
  - Part: Differential Equations (14)
    - Chapter: Classification of Differential Equations (5)
      - Definition: First-Order Ordinary Differential Equation (ODE) (1)
        
        Proposition: Differential Equation of the Exponential Function (1)
        
        Proof: (related to Proposition: Differential Equation of the Exponential Function)
      - Section: Linear vs. Non-Linear DE
      - Section: Order of DE
      - Section: Systems of DE
      - Proposition: Legendre Polynomials and Legendre Differential Equations (1)
        
        Proof: (related to Proposition: Legendre Polynomials and Legendre Differential Equations)
    - Chapter: Methods to Solve Ordinary DE (7)
      - Section: Linear 1. Order DE
      - Section: Linear 2. Order DE
      - Section: Linear n-th Order DE
      - Section: Linear Systems of 1. Order DE
      - Section: Non-Linear DE
      - Section: Boundary Value Problems
      - Definition: Solution of Ordinary DE
    - Chapter: Methods to Solve Partial DE (2)
      - Section: Linear 1. Order DE
      - Section: Linear 2. Order DE
  - Part: Linear Integral Equations (4)
    - Chapter: Fredholm Integral Equations
    - Chapter: Volterra Integral Equation
    - Chapter: Liouville-Neumann Series
    - Chapter: Abel Integral Equations
  - Part: Functional Analysis (29)
    - Theorem: Nested Closed Subset Theorem (1)
      - Proof: (related to Theorem: Nested Closed Subset Theorem)
    - Chapter: Banach Spaces and Banach Algebra (2)
      - Definition: Complete Metric Space
      - Definition: Banach Space
    - Chapter: Hilbert Spaces (1)
      - Definition: Hilbert Space
    - Chapter: Linear Operators and Linear Functionals
    - Chapter: Non-linear Operator
    - Chapter: Lebesgue Measure and Lebesgue Integral
    - Chapter: Simple Connectivity
    - Chapter: The `$C(X)$` Algebra
    - Chapter: Compact Sets (13)
      - Definition: Heine-Borel Property Defines Compact Subsets (1)
        
        Explanation: Explanation of the Heine-Borel Property (related to Definition: Heine-Borel Property Defines Compact Subsets)
      - Proposition: Image of a Compact Set Under a Continuous Function (1)
        
        Proof: (related to Proposition: Image of a Compact Set Under a Continuous Function)
      - Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space (1)
        
        Proof: (related to Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space)
      - Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space (1)
        
        Proof: (related to Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space)
      - Proposition: Closed n-Dimensional Cuboids Are Compact (2)
        
        Proof: (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
        
        Corollary: Closed Real Intervals Are Compact (related to Proposition: Closed n-Dimensional Cuboids Are Compact) (1)
        
        Proof: (related to Corollary: Closed Real Intervals Are Compact)
      - Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed (1)
        
        Proof: (related to Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed)
      - Proposition: Convergent Sequences are Bounded (1)
        
        Proof: (related to Proposition: Convergent Sequences are Bounded)
      - Theorem: Heine-Borel Theorem (1)
        
        Proof: (related to Theorem: Heine-Borel Theorem)
      - Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum (1)
        
        Proof: (related to Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum)
      - Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains (2)
        
        Proof: (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
        
        Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains) (1)
        
        Proof: (related to Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded)
      - Proposition: Closed Subsets of Compact Sets are Compact (1)
        
        Proof: (related to Proposition: Closed Subsets of Compact Sets are Compact)
    - Definition: Sigma-Algebra (5)
      - Definition: Measurable Set
      - Definition: Measureable Function
      - Definition: Measure (1)
        
        Definition: Finite and Sigma-Finite Measure
      - Definition: Pre-measure (1)
        
        Definition: Finite and Sigma-Finite Pre-measure
      - Definition: Measurable Space
    - Definition: Ring of Sets (measure-theoretic definition)
    - Definition: Functional (1)
      - Definition: Functional Equation
  - Part: Calculus of Variations
  - Part: Vector Analysis (3)
    - Definition: Vector Field
    - Chapter: Differential Operations on Vector Fields
    - Chapter: Integral Operations on Vector Fields
- Branch: Topology (86)
  - Part: Historical Development of Topology
  - Part: The Basics Set-theoretic Topology (63)
    - Chapter: Basic Topological Concepts (13)
      - Definition: Topological Space, Topology
      - Proposition: Clopen Sets and Boundaries (1)
        
        Proof: (related to Proposition: Clopen Sets and Boundaries)
      - Definition: Discrete and Indiscrete Topology
      - Definition: Ordering of Topologies
      - Definition: Open, Closed, Clopen
      - Proposition: Alternative Characterization of Topological Spaces (1)
        
        Proof: (related to Proposition: Alternative Characterization of Topological Spaces)
      - Definition: Neighborhood
      - Proposition: Properties of the Set of All Neighborhoods of a Point (1)
        
        Proof: (related to Proposition: Properties of the Set of All Neighborhoods of a Point)
      - Proposition: A Necessary Condition of a Neighborhood to be Open (1)
        
        Proof: (related to Proposition: A Necessary Condition of a Neighborhood to be Open)
      - Definition: Boundary Points, Closures, Interiors, and Exteriors
      - Proposition: How the Boundary Changes the Property of a Set of Being Open (1)
        
        Proof: (related to Proposition: How the Boundary Changes the Property of a Set of Being Open)
      - Definition: Regular Open, Regular Closed
      - Explanation: Fourteen Sets Formed By Closure, Interior and Complement Operations (related to Chapter: Basic Topological Concepts)
    - Chapter: Construction of Topologies (6)
      - Definition: Topological Subspaces and Subspace Topologies
      - Definition: Hereditary and Weakly Hereditary Properties
      - Proposition: Construction of Topological Spaces Using a Subbasis (1)
        
        Proof: (related to Proposition: Construction of Topological Spaces Using a Subbasis)
      - Definition: Topological Sum, Disjoint Union
      - Definition: Topological Product, Product Topology
      - Definition: Subbasis and Basis of Topology
    - Chapter: Sequences and Limits (12)
      - Definition: Isolated, Adherent, Limit, `$\omega$`-Accumulation and Condensation Points
      - Definition: Sequence
      - Definition: Carrier Set
      - Definition: Subsequence
      - Definition: Limits and Accumulation Points of Sequences
      - Example: Examples of Convergent Sequences in Topological Spaces (related to Chapter: Sequences and Limits)
      - Definition: Derived, Dense-in-itself, and Perfect Sets
      - Proposition: Perfect Sets vs. Derived Sets (1)
        
        Proof: (related to Proposition: Perfect Sets vs. Derived Sets)
      - Definition: Comparison of Filters, Finer and Coarser Filters
      - Axiom: Filter
      - Proposition: Filter Base (1)
        
        Proof: (related to Proposition: Filter Base)
      - Definition: Ultrafilter
    - Chapter: Density and Countability (2)
      - Definition: Dense Sets, Nowhere Dense Sets
      - Definition: First and Second Category Sets
    - Chapter: Continuity (9)
      - Definition: Continuous Function
      - Example: Example of Continuous Functions in Topological Spaces (related to Chapter: Continuity)
      - Proposition: Equivalent Notions of Continuous Functions (1)
        
        Proof: (related to Proposition: Equivalent Notions of Continuous Functions)
      - Proposition: Continuity of Compositions of Functions (1)
        
        Proof: (related to Proposition: Continuity of Compositions of Functions)
      - Definition: Open and Closed Functions
      - Proposition: Bijective Open Functions (1)
        
        Proof: (related to Proposition: Bijective Open Functions)
      - Definition: Homeomorphism, Homeomorphic Spaces
      - Proposition: Equivalent Notions of Homeomorphisms (1)
        
        Proof: (related to Proposition: Equivalent Notions of Homeomorphisms)
      - Definition: Topological, Continuous, Open, and Closed Invariants
    - Definition: Limit of a Function
    - Definition: Convergent Sequences and Limits (1)
      - Lemma: Criteria for Convergent Sequences (1)
        
        Proof: (related to Lemma: Criteria for Convergent Sequences)
    - Proposition: Uniqueness of the Limit of a Sequence (1)
      - Proof: (related to Proposition: Uniqueness of the Limit of a Sequence)
    - Definition: Cauchy Sequence (1)
      - Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces) (1)
        
        Proof: (related to Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces))
    - Theorem: Theorem of Bolzano-Weierstrass (1)
      - Proof: (related to Theorem: Theorem of Bolzano-Weierstrass)
    - Definition: Norm, Normed Vector Space (3)
      - Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm (2)
        
        Proof: (related to Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm)
        
        Proposition: Maximum Norm as a Limit of p-Norms (1)
        
        Proof: (related to Proposition: Maximum Norm as a Limit of p-Norms)
      - Proposition: Integral p-Norm (1)
        
        Proof: (related to Proposition: Integral p-Norm)
    - Definition: Isometry (1)
      - Proposition: Isometry is Injective (1)
        
        Proof: (related to Proposition: Isometry is Injective)
    - Definition: Bounded Sequence
    - Definition: Bounded Subset of a Metric Space
    - Proposition: Metric Spaces are Hausdorff Spaces (1)
      - Proof: (related to Proposition: Metric Spaces are Hausdorff Spaces)
    - Proposition: Distance in Normed Vector Spaces (1)
      - Proof: (related to Proposition: Distance in Normed Vector Spaces)
    - Definition: Open Cover
    - Definition: Continuous Functions in Metric Spaces (6)
      - Definition: Pointwise and Uniform Convergence (2)
        
        Example: A pointwise convergent sequence of functions, which is not uniformly convergent (related to Definition: Pointwise and Uniform Convergence)
        
        Corollary: Every uniformly convergent sequence of functions is pointwise convergent. (related to Definition: Pointwise and Uniform Convergence) (1)
        
        Proof: (related to Corollary: Every uniformly convergent sequence of functions is pointwise convergent.)
      - Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity (4)
        
        Proof: (related to Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity)
        
        Definition: Modulus of Continuity of a Continuous Function (3)
        
        Proposition: Modulus of Continuity is Subadditive (1)
        
        Proof: (related to Proposition: Modulus of Continuity is Subadditive)
        
        Proposition: Modulus of Continuity is Monotonically Increasing (1)
        
        Proof: (related to Proposition: Modulus of Continuity is Monotonically Increasing)
        
        Proposition: Modulus of Continuity is Continuous (1)
        
        Proof: (related to Proposition: Modulus of Continuity is Continuous)
    - Definition: Uniformly Continuous Functions (General Metric Spaces Case) (1)
      - Theorem: Continuous Functions on Compact Domains are Uniformly Continuous (1)
        
        Proof: (related to Theorem: Continuous Functions on Compact Domains are Uniformly Continuous)
  - Part: Separation Of Topological Spaces (5)
    - Axiom: Separation Axioms
    - Proposition: Characterization of `$T_1$` Spaces (1)
      - Proof: (related to Proposition: Characterization of `$T_1$` Spaces)
    - Proposition: Inheritance of the `$T_1$` Property (1)
      - Proof: (related to Proposition: Inheritance of the `$T_1$` Property)
    - Proposition: Characterization of `$T_2$` Spaces (1)
      - Proof: (related to Proposition: Characterization of `$T_2$` Spaces)
    - Proposition: Inheritance of the `$T_2$` Property (1)
      - Proof: (related to Proposition: Inheritance of the `$T_2$` Property)
  - Part: Metric Spaces (7)
    - Definition: Metric (Distance) (1)
      - Corollary: Every Distance Is Positive Definite (related to Definition: Metric (Distance)) (1)
        
        Proof: (related to Corollary: Every Distance Is Positive Definite)
    - Definition: Metric Space
    - Definition: Open Sets in Metric Spaces (2)
      - Lemma: Characterization of Closed Sets by Limits of Sequences (1)
        
        Proof: (related to Lemma: Characterization of Closed Sets by Limits of Sequences)
      - Definition: Open Function, Closed Function
    - Definition: Open Ball, Neighborhood
    - Definition: Diameter In Metric Spaces
    - Proposition: Metric Spaces and Empty Sets are Clopen (1)
      - Proof: (related to Proposition: Metric Spaces and Empty Sets are Clopen)
  - Part: Homotopy (9)
    - Definition: Manifold (6)
      - Definition: Differentiable Manifold, Atlas
      - Definition: `\(C^n\)` Differentiable Function
      - Definition: Tangent Bundle
      - Definition: Cotangent Bundle
      - Definition: Section over a Base Space
      - Definition: Differential Form of Degree k
    - Definition: Topological Chart
    - Definition: Transition Map
    - Definition: `\(C^{n}\)`-Diffeomorphism
  - Part: Algebraic Topology (1)
    - Definition: Simplex
- Branch: Geometry (700)
  - Part: Historical Development of Geometry
  - Chapter: Euclid's “Elements” (670)
    - Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines (106)
      - Subsection: Definitions from Book 1 (28)
        
        Definition: 1.01: Point (1)
        
        Explanation: How a point is different from a solid, a surface and a line? (related to Definition: 1.01: Point)
        
        Definition: 1.02: Line, Curve (1)
        
        Explanation: How a line is different from a solid and a surface? (related to Definition: 1.02: Line, Curve)
        
        Definition: 1.03: Intersections of Lines
        
        Definition: 1.04: Straight Line, Segment and Ray (1)
        
        Definition: Collinear Points, Segments, Rays
        
        Definition: 1.05: Surface
        
        Definition: 1.06: Intersections of Surfaces
        
        Definition: 1.07: Plane
        
        Definition: 1.08: Plane Angle
        
        Definition: 1.09: Angle, Rectilinear, Vertex, Legs (3)
        
        Definition: Exterior, Interior, Alternate and Corresponding Angles
        
        Definition: Sum of Angles
        
        Definition: Supplemental Angles
        
        Definition: 1.10: Right Angle, Perpendicular Straight Lines (1)
        
        Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines) (1)
        
        Proof: By Euclid (related to Corollary: The supplemental angle of a right angle is another right angle.)
        
        Definition: 1.11: Obtuse Angle
        
        Definition: 1.12: Acute Angle
        
        Definition: 1.13: Boundary
        
        Definition: 1.14: Plane Figure
        
        Definition: 1.15: Circle, Circumference, Radius
        
        Definition: 1.16: Center of the Circle (1)
        
        Definition: Concentric Circles
        
        Definition: 1.17: Diameter of the Circle
        
        Definition: 1.18: Semicircle
        
        Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure (4)
        
        Definition: Triangle
        
        Definition: Pentagon
        
        Definition: Hexagon
        
        Definition: Decagon
        
        Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle (1)
        
        Definition: Altitude of a Triangle
        
        Definition: 1.21: Right Triangle, Obtuse Triangle, Acute Triangle
        
        Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium (1)
        
        Definition: Diagonal
        
        Definition: 1.23: Parallel Straight Lines
      - Subsection: Axioms from Book 1 (5)
        
        Axiom: 1.1: Straight Line Determined by Two Distinct Points (1)
        
        Explanation: Why did Euclid postulate the axiom of straight line determined by two points? (related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
        
        Axiom: 1.2: Segment Extension
        
        Axiom: 1.3: Circle Determined by its Center and its Radius
        
        Axiom: 1.4: Equality of all Right Angles
        
        Axiom: 1.5: Parallel Postulate
      - Subsection: Common Notions (all Books) (5)
        
        Explanation: 1.1: Equality is an Equivalence Relation (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.2: Adding Equations Preserves Equality (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.3: Subtracting Equations Preserves Equality (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.4: Congruent Figures (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.5: Comparing the Size of Sets and Their Subsets (related to Subsection: Common Notions (all Books))
      - Subsection: Propositions from Book 1 (68)
        
        Proposition: 1.01: Constructing an Equilateral Triangle (1)
        
        Proof: By Euclid (related to Proposition: 1.01: Constructing an Equilateral Triangle)
        
        Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment (1)
        
        Proof: By Euclid (related to Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment)
        
        Proposition: 1.03: Cutting a Segment at a Given Size (1)
        
        Proof: By Euclid (related to Proposition: 1.03: Cutting a Segment at a Given Size)
        
        Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle (1)
        
        Proof: By Euclid (related to Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle)
        
        Proposition: 1.05: Isosceles Triangles I (2)
        
        Proof: By Euclid (related to Proposition: 1.05: Isosceles Triangles I)
        
        Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I) (1)
        
        Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
        
        Proposition: 1.06: Isosceles Triagles II (2)
        
        Proof: By Euclid (related to Proposition: 1.06: Isosceles Triagles II)
        
        Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II) (1)
        
        Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
        
        Proposition: 1.07: Uniqueness of Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.07: Uniqueness of Triangles)
        
        Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles)
        
        Proposition: 1.09: Bisecting an Angle (1)
        
        Proof: By Euclid (related to Proposition: 1.09: Bisecting an Angle)
        
        Proposition: 1.10: Bisecting a Segment (1)
        
        Proof: By Euclid (related to Proposition: 1.10: Bisecting a Segment)
        
        Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line (1)
        
        Proof: (Euclid) (related to Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line)
        
        Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line (1)
        
        Proof: (Euclid) (related to Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line)
        
        Proposition: 1.13: Angles at Intersections of Straight Lines (3)
        
        Proof: By Euclid (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
        
        Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines) (1)
        
        Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
        
        Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines) (1)
        
        Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
        
        Proposition: 1.14: Combining Rays to Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 1.14: Combining Rays to Straight Lines)
        
        Proposition: 1.15: Opposite Angles on Intersecting Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 1.15: Opposite Angles on Intersecting Straight Lines)
        
        Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles (2)
        
        Proof: By Euclid (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
        
        Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles) (1)
        
        Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
        
        Proposition: 1.17: The Sum of Two Angles of a Triangle (1)
        
        Proof: By Euclid (related to Proposition: 1.17: The Sum of Two Angles of a Triangle)
        
        Proposition: 1.18: Angles and Sides in a Triangle I (1)
        
        Proof: (Euclid) (related to Proposition: 1.18: Angles and Sides in a Triangle I)
        
        Proposition: 1.19: Angles and Sides in a Triangle II (1)
        
        Proof: (Euclid) (related to Proposition: 1.19: Angles and Sides in a Triangle II)
        
        Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality) (1)
        
        Proof: By Euclid (related to Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality))
        
        Proposition: 1.21: Triangles within Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.21: Triangles within Triangles)
        
        Proposition: 1.22: Construction of Triangles From Arbitrary Segments (1)
        
        Proof: By Euclid (related to Proposition: 1.22: Construction of Triangles From Arbitrary Segments)
        
        Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle (1)
        
        Proof: By Euclid (related to Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle)
        
        Proposition: 1.24: Angles and Sides in a Triangle III (1)
        
        Proof: (Euclid) (related to Proposition: 1.24: Angles and Sides in a Triangle III)
        
        Proposition: 1.25: Angles and Sides in a Triangle IV (2)
        
        Proof: (Euclid) (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
        
        Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV) (1)
        
        Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
        
        Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles)
        
        Proposition: 1.27: Parallel Lines I (1)
        
        Proof: By Euclid (related to Proposition: 1.27: Parallel Lines I)
        
        Proposition: 1.28: Parallel Lines II (1)
        
        Proof: By Euclid (related to Proposition: 1.28: Parallel Lines II)
        
        Proposition: 1.29: Parallel Lines III (2)
        
        Proof: By Euclid (related to Proposition: 1.29: Parallel Lines III)
        
        Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III) (1)
        
        Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
        
        Proposition: 1.30: Transitivity of Parallel Lines (1)
        
        Proof: By Euclid (related to Proposition: 1.30: Transitivity of Parallel Lines)
        
        Proposition: 1.31: Constructing a Parallel Line from a Line and a Point (1)
        
        Proof: (Euclid) (related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)
        
        Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle (6)
        
        Proof: By Euclid (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
        
        Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
        
        Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Angles of Right Triangle)
        
        Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
        
        Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
        
        Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Similar Triangles)
        
        Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram (1)
        
        Proof: By Euclid (related to Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram)
        
        Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms (8)
        
        Proof: By Euclid (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
        
        Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
        
        Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
        
        Definition: Parallelogram - Defining Property III
        
        Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Rhombus as a Special Case of a Parallelogram)
        
        Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
        
        Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
        
        Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
        
        Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels (1)
        
        Proof: By Euclid (related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
        
        Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels (1)
        
        Proof: By Euclid (related to Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels)
        
        Proposition: 1.37: Triangles of Equal Area I (1)
        
        Proof: By Euclid (related to Proposition: 1.37: Triangles of Equal Area I)
        
        Proposition: 1.38: Triangles of Equal Area II (1)
        
        Proof: By Euclid (related to Proposition: 1.38: Triangles of Equal Area II)
        
        Proposition: 1.39: Triangles of Equal Area III (1)
        
        Proof: By Euclid (related to Proposition: 1.39: Triangles of Equal Area III)
        
        Proposition: 1.40: Triangles of Equal Area IV (1)
        
        Proof: By Euclid (related to Proposition: 1.40: Triangles of Equal Area IV)
        
        Proposition: 1.41: Parallelograms and Triagles (1)
        
        Proof: By Euclid (related to Proposition: 1.41: Parallelograms and Triagles)
        
        Proposition: 1.42: Construction of Parallelograms I (1)
        
        Proof: By Euclid (related to Proposition: 1.42: Construction of Parallelograms I)
        
        Proposition: 1.43: Complementary Segments of Parallelograms (1)
        
        Proof: (Euclid) (related to Proposition: 1.43: Complementary Segments of Parallelograms)
        
        Proposition: 1.44: Construction of Parallelograms II (1)
        
        Proof: By Euclid (related to Proposition: 1.44: Construction of Parallelograms II)
        
        Proposition: 1.45: Construction of Parallelograms III (1)
        
        Proof: By Euclid (related to Proposition: 1.45: Construction of Parallelograms III)
        
        Proposition: 1.46: Construction of a Square on a Given Segment (2)
        
        Proof: By Euclid (related to Proposition: 1.46: Construction of a Square on a Given Segment)
        
        Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment) (1)
        
        Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
        
        Proposition: 1.47: Pythagorean Theorem (1)
        
        Proof: By Euclid (related to Proposition: 1.47: Pythagorean Theorem)
        
        Proposition: 1.48: The Converse of the Pythagorean Theorem (1)
        
        Proof: By Euclid (related to Proposition: 1.48: The Converse of the Pythagorean Theorem)
    - Section: Book 02: Fundamentals of Geometric Algebra (16)
      - Subsection: Definitions from Book 2 (2)
        
        Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides (1)
        
        Definition: Point of Division, Point of External Division
        
        Definition: 2.2: Gnomon
      - Subsection: Propositions from Book 2 (14)
        
        Proposition: 2.01: Summing Areas or Rectangles (1)
        
        Proof: By Euclid (related to Proposition: 2.01: Summing Areas or Rectangles)
        
        Proposition: 2.02: Square is Sum of Two Rectangles (1)
        
        Proof: By Euclid (related to Proposition: 2.02: Square is Sum of Two Rectangles)
        
        Proposition: 2.03: Rectangle is Sum of Square and Rectangle (1)
        
        Proof: By Euclid (related to Proposition: 2.03: Rectangle is Sum of Square and Rectangle)
        
        Proposition: 2.04: Square of Sum (1)
        
        Proof: By Euclid (related to Proposition: 2.04: Square of Sum)
        
        Proposition: 2.05: Rectangle is Difference of Two Squares (1)
        
        Proof: By Euclid (related to Proposition: 2.05: Rectangle is Difference of Two Squares)
        
        Proposition: 2.06: Square of Sum with One Halved Summand (1)
        
        Proof: By Euclid (related to Proposition: 2.06: Square of Sum with One Halved Summand)
        
        Proposition: 2.07: Sum of Squares (1)
        
        Proof: By Euclid (related to Proposition: 2.07: Sum of Squares)
        
        Proposition: 2.08: Square of Sum with One Doubled Summand (1)
        
        Proof: By Euclid (related to Proposition: 2.08: Square of Sum with One Doubled Summand)
        
        Proposition: 2.09: Sum of Squares of Sum and Difference (1)
        
        Proof: By Euclid (related to Proposition: 2.09: Sum of Squares of Sum and Difference)
        
        Proposition: 2.10: Sum of Squares (Half) (1)
        
        Proof: By Euclid (related to Proposition: 2.10: Sum of Squares (Half))
        
        Proposition: 2.11: Constructing the Golden Ratio of a Segment (1)
        
        Proof: By Euclid (related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)
        
        Proposition: 2.12: Law of Cosines (for Obtuse Angles) (1)
        
        Proof: By Euclid (related to Proposition: 2.12: Law of Cosines (for Obtuse Angles))
        
        Proposition: 2.13: Law of Cosines (for Acute Angles) (1)
        
        Proof: By Euclid (related to Proposition: 2.13: Law of Cosines (for Acute Angles))
        
        Proposition: 2.14: Constructing a Square from a Rectilinear Figure (1)
        
        Proof: By Euclid (related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)
    - Section: Book 03: Fundamentals of Plane Geometry Involving Circles (50)
      - Subsection: Definitions from Book 3 (11)
        
        Definition: 3.01: Congruent Circles
        
        Definition: 3.02: Tangent to the Circle, Straight Line Touching The Circle
        
        Definition: 3.03: Circles Touching One Another
        
        Definition: 3.04: Chords Equally Far From the Center of a Circle
        
        Definition: 3.05: Chords Being Further from the Center of a Circle
        
        Definition: 3.06: Segment of a Circle, Arc
        
        Definition: 3.07: Angle of a Segment
        
        Definition: 3.08: Angle in the Segment (Inscribed Angle)
        
        Definition: 3.09: Angle Standing Upon An Arc
        
        Definition: 3.10: Circular Sector, Central Angle
        
        Definition: 3.11: Similar Circular Segments
      - Subsection: Propositions from Book 3 (39)
        
        Proposition: 3.01: Finding the Center of a given Circle (2)
        
        Proof: By Euclid (related to Proposition: 3.01: Finding the Center of a given Circle)
        
        Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle) (1)
        
        Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
        
        Proposition: 3.02: Chord Lies Inside its Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.02: Chord Lies Inside its Circle)
        
        Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector (1)
        
        Proof: By Euclid (related to Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector)
        
        Proposition: 3.04: Chords do not Bisect Each Other (1)
        
        Proof: By Euclid (related to Proposition: 3.04: Chords do not Bisect Each Other)
        
        Proposition: 3.05: Intersecting Circles have Different Centers (1)
        
        Proof: By Euclid (related to Proposition: 3.05: Intersecting Circles have Different Centers)
        
        Proposition: 3.06: Touching Circles have Different Centers (1)
        
        Proof: By Euclid (related to Proposition: 3.06: Touching Circles have Different Centers)
        
        Proposition: 3.07: Relative Lengths of Lines Inside Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.07: Relative Lengths of Lines Inside Circle)
        
        Proposition: 3.08: Relative Lengths of Lines Outside Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.08: Relative Lengths of Lines Outside Circle)
        
        Proposition: 3.09: Condition for Point to be Center of Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.09: Condition for Point to be Center of Circle)
        
        Proposition: 3.10: Two Circles have at most Two Points of Intersection (1)
        
        Proof: By Euclid (related to Proposition: 3.10: Two Circles have at most Two Points of Intersection)
        
        Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally (1)
        
        Proof: By Euclid (related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
        
        Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally (1)
        
        Proof: By Euclid (related to Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally)
        
        Proposition: 3.13: Circles Touch at One Point at Most (1)
        
        Proof: By Euclid (related to Proposition: 3.13: Circles Touch at One Point at Most)
        
        Proposition: 3.14: Equal Chords in Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.14: Equal Chords in Circle)
        
        Proposition: 3.15: Relative Lengths of Chords of Circles (1)
        
        Proof: By Euclid (related to Proposition: 3.15: Relative Lengths of Chords of Circles)
        
        Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (2)
        
        Proof: By Euclid (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
        
        Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle) (1)
        
        Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
        
        Proposition: 3.17: Construction of Tangent from Point to Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.17: Construction of Tangent from Point to Circle)
        
        Proposition: 3.18: Radius at Right Angle to Tangent (1)
        
        Proof: By Euclid (related to Proposition: 3.18: Radius at Right Angle to Tangent)
        
        Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center (1)
        
        Proof: By Euclid (related to Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center)
        
        Proposition: 3.20: Inscribed Angle Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.20: Inscribed Angle Theorem)
        
        Proposition: 3.21: Angles in Same Segment of Circle are Equal (1)
        
        Proof: By Euclid (related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
        
        Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral (1)
        
        Proof: By Euclid (related to Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral)
        
        Proposition: 3.23: Segment on Given Base Unique (1)
        
        Proof: By Euclid (related to Proposition: 3.23: Segment on Given Base Unique)
        
        Proposition: 3.24: Similar Segments on Equal Bases are Equal (1)
        
        Proof: By Euclid (related to Proposition: 3.24: Similar Segments on Equal Bases are Equal)
        
        Proposition: 3.25: Construction of Circle from Segment (1)
        
        Proof: By Euclid (related to Proposition: 3.25: Construction of Circle from Segment)
        
        Proposition: 3.26: Equal Angles and Arcs in Equal Circles (1)
        
        Proof: By Euclid (related to Proposition: 3.26: Equal Angles and Arcs in Equal Circles)
        
        Proposition: 3.27: Angles on Equal Arcs are Equal (1)
        
        Proof: By Euclid (related to Proposition: 3.27: Angles on Equal Arcs are Equal)
        
        Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles (1)
        
        Proof: By Euclid (related to Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles)
        
        Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
        
        Proposition: 3.30: Bisection of Arc (1)
        
        Proof: By Euclid (related to Proposition: 3.30: Bisection of Arc)
        
        Proposition: 3.31: Relative Sizes of Angles in Segments (1)
        
        Proof: By Euclid (related to Proposition: 3.31: Relative Sizes of Angles in Segments)
        
        Proposition: 3.32: Angles made by Chord with Tangent (1)
        
        Proof: By Euclid (related to Proposition: 3.32: Angles made by Chord with Tangent)
        
        Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle (1)
        
        Proof: By Euclid (related to Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle)
        
        Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle (1)
        
        Proof: By Euclid (related to Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle)
        
        Proposition: 3.35: Intersecting Chord Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.35: Intersecting Chord Theorem)
        
        Proposition: 3.36: Tangent Secant Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.36: Tangent Secant Theorem)
        
        Proposition: 3.37: Converse of Tangent Secant Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.37: Converse of Tangent Secant Theorem)
    - Section: Book 04: Circles: Inscription and Circumscription (24)
      - Subsection: Definitions from Book 4 (7)
        
        Definition: 4.1: Rectilinear Figure Inscribed in Another Rectilinear Figure
        
        Definition: 4.2: Rectilinear Figure Circumscribed about Another Rectilinear Figure
        
        Definition: 4.3: Inscribing Rectilinear Figures in Circles
        
        Definition: 4.4: Circumscribing Rectilinear Figures about Circles
        
        Definition: 4.7: Chord and Secant
        
        Definition: 4.6: Circumscribing Circles about Rectilinear Figures
        
        Definition: 4.5: Inscribing Circles in Rectilinear Figures
      - Subsection: Propositions from Book 4 (17)
        
        Proposition: 4.01: Fitting Chord Into Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.01: Fitting Chord Into Circle)
        
        Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles (1)
        
        Proof: By Euclid (related to Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles)
        
        Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles (1)
        
        Proof: By Euclid (related to Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles)
        
        Proposition: 4.04: Inscribing Circle in Triangle (1)
        
        Proof: By Euclid (related to Proposition: 4.04: Inscribing Circle in Triangle)
        
        Proposition: 4.05: Circumscribing Circle about Triangle (1)
        
        Proof: By Euclid (related to Proposition: 4.05: Circumscribing Circle about Triangle)
        
        Proposition: 4.06: Inscribing Square in Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.06: Inscribing Square in Circle)
        
        Proposition: 4.07: Circumscribing Square about Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.07: Circumscribing Square about Circle)
        
        Proposition: 4.08: Inscribing Circle in Square (1)
        
        Proof: By Euclid (related to Proposition: 4.08: Inscribing Circle in Square)
        
        Proposition: 4.09: Circumscribing Circle about Square (1)
        
        Proof: By Euclid (related to Proposition: 4.09: Circumscribing Circle about Square)
        
        Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex (1)
        
        Proof: By Euclid (related to Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex)
        
        Proposition: 4.11: Inscribing Regular Pentagon in Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.11: Inscribing Regular Pentagon in Circle)
        
        Proposition: 4.12: Circumscribing Regular Pentagon about Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.12: Circumscribing Regular Pentagon about Circle)
        
        Proposition: 4.13: Inscribing Circle in Regular Pentagon (1)
        
        Proof: By Euclid (related to Proposition: 4.13: Inscribing Circle in Regular Pentagon)
        
        Proposition: 4.14: Circumscribing Circle about Regular Pentagon (1)
        
        Proof: By Euclid (related to Proposition: 4.14: Circumscribing Circle about Regular Pentagon)
        
        Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (2)
        
        Proof: By Euclid (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
        
        Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle) (1)
        
        Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
        
        Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle)
    - Section: Book 05: Proportion (45)
      - Subsection: Definitions from Book 5 (18)
        
        Definition: 5.01: Magnitude is Aliquot Part
        
        Definition: 5.02: Multiple of a Real Number
        
        Definition: 5.03: Ratio
        
        Definition: 5.04: Having a Ratio
        
        Definition: 5.05: Having the Same Ratio
        
        Definition: 5.06: Proportional Magnitudes (1)
        
        Definition: Geometric Progression, Continued Proportion
        
        Definition: 5.07: Having a Greater Ratio
        
        Definition: 5.08: Proportion in Three Terms
        
        Definition: 5.09: Squared Ratio
        
        Definition: 5.10: Cubed Ratio
        
        Definition: 5.11: Corresponding Magnitudes
        
        Definition: 5.12: Alternate Ratio
        
        Definition: 5.13: Inverse Ratio
        
        Definition: 5.14: Composition of a Ratio
        
        Definition: 5.15: Separation of a Ratio
        
        Definition: 5.16: Conversion of a Ratio
        
        Definition: 5.17: Ratio ex Aequali
        
        Definition: 5.18: Perturbed Proportion
      - Subsection: Propositions from Book 5 (27)
        
        Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition (1)
        
        Proof: By Euclid (related to Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition)
        
        Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition (1)
        
        Proof: By Euclid (related to Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition)
        
        Proposition: 5.03: Multiplication of Numbers is Associative (1)
        
        Proof: By Euclid (related to Proposition: 5.03: Multiplication of Numbers is Associative)
        
        Proposition: 5.04: Multiples of Terms in Equal Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.04: Multiples of Terms in Equal Ratios)
        
        Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction (1)
        
        Proof: By Euclid (related to Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction)
        
        Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction (1)
        
        Proof: By Euclid (related to Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction)
        
        Proposition: 5.07: Ratios of Equal Magnitudes (2)
        
        Proof: By Euclid (related to Proposition: 5.07: Ratios of Equal Magnitudes)
        
        Corollary: 5.07: Ratios of Equal Magnitudes (related to Proposition: 5.07: Ratios of Equal Magnitudes) (1)
        
        Proof: By Euclid (related to Corollary: 5.07: Ratios of Equal Magnitudes)
        
        Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes (1)
        
        Proof: By Euclid (related to Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes)
        
        Proposition: 5.09: Magnitudes with Same Ratios are Equal (1)
        
        Proof: By Euclid (related to Proposition: 5.09: Magnitudes with Same Ratios are Equal)
        
        Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios)
        
        Proposition: 5.11: Equality of Ratios is Transitive (1)
        
        Proof: By Euclid (related to Proposition: 5.11: Equality of Ratios is Transitive)
        
        Proposition: 5.12: Sum of Components of Equal Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.12: Sum of Components of Equal Ratios)
        
        Proposition: 5.13: Relative Sizes of Proportional Magnitudes (1)
        
        Proof: By Euclid (related to Proposition: 5.13: Relative Sizes of Proportional Magnitudes)
        
        Proposition: 5.14: Relative Sizes of Components of Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.14: Relative Sizes of Components of Ratios)
        
        Proposition: 5.15: Ratio Equals its Multiples (1)
        
        Proof: By Euclid (related to Proposition: 5.15: Ratio Equals its Multiples)
        
        Proposition: 5.16: Proportional Magnitudes are Proportional Alternately (1)
        
        Proof: By Euclid (related to Proposition: 5.16: Proportional Magnitudes are Proportional Alternately)
        
        Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated (1)
        
        Proof: By Euclid (related to Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated)
        
        Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded (1)
        
        Proof: By Euclid (related to Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded)
        
        Proposition: 5.19: Proportional Magnitudes have Proportional Remainders (2)
        
        Proof: By Euclid (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
        
        Corollary: 5.19: Proportional Magnitudes have Proportional Remainders (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders) (1)
        
        Proof: By Euclid (related to Corollary: 5.19: Proportional Magnitudes have Proportional Remainders)
        
        Proposition: 5.20: Relative Sizes of Successive Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.20: Relative Sizes of Successive Ratios)
        
        Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion)
        
        Proposition: 5.22: Equality of Ratios Ex Aequali (1)
        
        Proof: By Euclid (related to Proposition: 5.22: Equality of Ratios Ex Aequali)
        
        Proposition: 5.23: Equality of Ratios in Perturbed Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.23: Equality of Ratios in Perturbed Proportion)
        
        Proposition: 5.24: Sum of Antecedents of Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.24: Sum of Antecedents of Proportion)
        
        Proposition: 5.25: Sum of Antecedent and Consequent of Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.25: Sum of Antecedent and Consequent of Proportion)
    - Section: Book 06: Similar Figures (39)
      - Subsection: Definitions from Book 6 (3)
        
        Definition: 6.01: Similar Rectilinear Figures
        
        Definition: 6.02: Cut in Extreme and Mean Ratio
        
        Definition: 6.03: Height of a Figure
      - Subsection: Propositions from Book 6 (36)
        
        Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base (1)
        
        Proof: By Euclid (related to Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base)
        
        Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally (1)
        
        Proof: By Euclid (related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
        
        Proposition: 6.03: Angle Bisector Theorem (1)
        
        Proof: By Euclid (related to Proposition: 6.03: Angle Bisector Theorem)
        
        Proposition: 6.04: Equiangular Triangles are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.04: Equiangular Triangles are Similar)
        
        Proposition: 6.05: Triangles with Proportional Sides are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.05: Triangles with Proportional Sides are Similar)
        
        Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar)
        
        Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar)
        
        Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles (2)
        
        Proof: By Euclid (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
        
        Corollary: 6.08: Geometric Mean Theorem (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles) (1)
        
        Proof: By Euclid (related to Corollary: 6.08: Geometric Mean Theorem)
        
        Proposition: 6.09: Construction of Part of Line (1)
        
        Proof: By Euclid (related to Proposition: 6.09: Construction of Part of Line)
        
        Proposition: 6.10: Construction of Similarly Cut Straight Line (1)
        
        Proof: By Euclid (related to Proposition: 6.10: Construction of Similarly Cut Straight Line)
        
        Proposition: 6.11: Construction of Segment in Squared Ratio (1)
        
        Proof: By Euclid (related to Proposition: 6.11: Construction of Segment in Squared Ratio)
        
        Proposition: 6.12: Construction of Fourth Proportional Straight Line (1)
        
        Proof: By Euclid (related to Proposition: 6.12: Construction of Fourth Proportional Straight Line)
        
        Proposition: 6.13: Construction of Mean Proportional (1)
        
        Proof: By Euclid (related to Proposition: 6.13: Construction of Mean Proportional)
        
        Proposition: 6.14: Characterization of Congruent Parallelograms (1)
        
        Proof: By Euclid (related to Proposition: 6.14: Characterization of Congruent Parallelograms)
        
        Proposition: 6.15: Characterization of Congruent Triangles (1)
        
        Proof: By Euclid (related to Proposition: 6.15: Characterization of Congruent Triangles)
        
        Proposition: 6.16: Rectangles Contained by Proportional Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 6.16: Rectangles Contained by Proportional Straight Lines)
        
        Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)
        
        Proposition: 6.18: Construction of Similar Polygon (1)
        
        Proof: By Euclid (related to Proposition: 6.18: Construction of Similar Polygon)
        
        Proposition: 6.19: Ratio of Areas of Similar Triangles (2)
        
        Proof: By Euclid (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
        
        Corollary: 6.19: Ratio of Areas of Similar Triangles (related to Proposition: 6.19: Ratio of Areas of Similar Triangles) (1)
        
        Proof: By Euclid (related to Corollary: 6.19: Ratio of Areas of Similar Triangles)
        
        Proposition: 6.20: Similar Polygons are Composed of Similar Triangles (2)
        
        Proof: By Euclid (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
        
        Corollary: 6.20: Similar Polygons are Composed of Similar Triangles (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles) (1)
        
        Proof: By Euclid (related to Corollary: 6.20: Similar Polygons are Composed of Similar Triangles)
        
        Proposition: 6.21: Similarity of Polygons is Transitive (1)
        
        Proof: By Euclid (related to Proposition: 6.21: Similarity of Polygons is Transitive)
        
        Proposition: 6.22: Similar Figures on Proportional Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)
        
        Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms (1)
        
        Proof: By Euclid (related to Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms)
        
        Proposition: 6.24: Parallelograms About Diameter are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.24: Parallelograms About Diameter are Similar)
        
        Proposition: 6.25: Construction of Figure Similar to One and Equal to Another (1)
        
        Proof: By Euclid (related to Proposition: 6.25: Construction of Figure Similar to One and Equal to Another)
        
        Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter (1)
        
        Proof: By Euclid (related to Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter)
        
        Proposition: 6.27: Similar Parallelogram on Half a Straight Line (1)
        
        Proof: By Euclid (related to Proposition: 6.27: Similar Parallelogram on Half a Straight Line)
        
        Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram (1)
        
        Proof: By Euclid (related to Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram)
        
        Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram (1)
        
        Proof: By Euclid (related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)
        
        Proposition: 6.30: Construction of the Inverse Golden Section (1)
        
        Proof: By Euclid (related to Proposition: 6.30: Construction of the Inverse Golden Section)
        
        Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle (1)
        
        Proof: By Euclid (related to Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle)
        
        Proposition: 6.32: Triangles with Two Sides Parallel and Equal (1)
        
        Proof: By Euclid (related to Proposition: 6.32: Triangles with Two Sides Parallel and Equal)
        
        Proposition: 6.33: Angles in Circles have Same Ratio as Arcs (1)
        
        Proof: By Euclid (related to Proposition: 6.33: Angles in Circles have Same Ratio as Arcs)
    - Section: Book 07: Elementary Number Theory (62)
      - Subsection: Definitions from Book 7 (22)
        
        Definition: 7.01: Unit
        
        Definition: 7.02: Number
        
        Definition: 7.03: Proper Divisor
        
        Definition: 7.04: Aliquant Part, a Number Being Not a Divisor of Another Number
        
        Definition: 7.05: Multiple, Number Multiplying another Number
        
        Definition: 7.06: Even Number
        
        Definition: 7.07: Odd Number
        
        Definition: 7.08: Even-Times-Even Number
        
        Definition: 7.09: Even-Times-Odd Number
        
        Definition: 7.10: Odd-Times-Odd Number
        
        Definition: 7.11: Prime Number
        
        Definition: 7.12: Co-prime (Relatively Prime) Numbers
        
        Definition: 7.13: Composite Number
        
        Definition: 7.14: Not Co-prime Numbers
        
        Definition: 7.15: Multiplication of Numbers
        
        Definition: 7.16: Rectangular Number, Plane Number
        
        Definition: 7.17: Cuboidal Number, Solid Number
        
        Definition: 7.18: Square Number
        
        Definition: 7.19: Cubic Number, Cube Number
        
        Definition: 7.20: Proportional Numbers
        
        Definition: 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers
        
        Definition: 7.22: Perfect Number
      - Subsection: Propositions from Book 7 (40)
        
        Proposition: 7.01: Sufficient Condition for Coprimality (1)
        
        Proof: By Euclid (related to Proposition: 7.01: Sufficient Condition for Coprimality)
        
        Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm (2)
        
        Proof: By Euclid (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
        
        Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm) (1)
        
        Proof: By Euclid (related to Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor)
        
        Proposition: 7.03: Greatest Common Divisor of Three Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)
        
        Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers)
        
        Proposition: 7.05: Divisors Obey Distributive Law (Sum) (1)
        
        Proof: By Euclid (related to Proposition: 7.05: Divisors Obey Distributive Law (Sum))
        
        Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum) (1)
        
        Proof: By Euclid (related to Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum))
        
        Proposition: 7.07: Divisors Obey Distributive Law (Difference) (1)
        
        Proof: By Euclid (related to Proposition: 7.07: Divisors Obey Distributive Law (Difference))
        
        Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference) (1)
        
        Proof: By Euclid (related to Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference))
        
        Proposition: 7.09: Alternate Ratios of Equal Fractions (1)
        
        Proof: By Euclid (related to Proposition: 7.09: Alternate Ratios of Equal Fractions)
        
        Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions (1)
        
        Proof: By Euclid (related to Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions)
        
        Proposition: 7.11: Proportional Numbers have Proportional Differences (1)
        
        Proof: By Euclid (related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
        
        Proposition: 7.12: Ratios of Numbers is Distributive over Addition (1)
        
        Proof: By Euclid (related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
        
        Proposition: 7.13: Proportional Numbers are Proportional Alternately (1)
        
        Proof: By Euclid (related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
        
        Proposition: 7.14: Proportion of Numbers is Transitive (1)
        
        Proof: By Euclid (related to Proposition: 7.14: Proportion of Numbers is Transitive)
        
        Proposition: 7.15: Alternate Ratios of Multiples (1)
        
        Proof: By Euclid (related to Proposition: 7.15: Alternate Ratios of Multiples)
        
        Proposition: 7.16: Natural Number Multiplication is Commutative (1)
        
        Proof: By Euclid (related to Proposition: 7.16: Natural Number Multiplication is Commutative)
        
        Proposition: 7.17: Multiples of Ratios of Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.17: Multiples of Ratios of Numbers)
        
        Proposition: 7.18: Ratios of Multiples of Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.18: Ratios of Multiples of Numbers)
        
        Proposition: 7.19: Relation of Ratios to Products (1)
        
        Proof: By Euclid (related to Proposition: 7.19: Relation of Ratios to Products)
        
        Proposition: 7.20: Ratios of Fractions in Lowest Terms (1)
        
        Proof: By Euclid (related to Proposition: 7.20: Ratios of Fractions in Lowest Terms)
        
        Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms (1)
        
        Proof: By Euclid (related to Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms)
        
        Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime)
        
        Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other (1)
        
        Proof: By Euclid (related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
        
        Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole (1)
        
        Proof: By Euclid (related to Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole)
        
        Proposition: 7.25: Square of Co-prime Number is Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.25: Square of Co-prime Number is Co-prime)
        
        Proposition: 7.26: Product of Co-prime Pairs is Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.26: Product of Co-prime Pairs is Co-prime)
        
        Proposition: 7.27: Powers of Co-prime Numbers are Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.27: Powers of Co-prime Numbers are Co-prime)
        
        Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both (1)
        
        Proof: By Euclid (related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)
        
        Proposition: 7.29: Prime not Divisor implies Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.29: Prime not Divisor implies Co-prime)
        
        Proposition: 7.30: Euclidean Lemma (1)
        
        Proof: By Euclid (related to Proposition: 7.30: Euclidean Lemma)
        
        Proposition: 7.31: Existence of Prime Divisors (1)
        
        Proof: By Euclid (related to Proposition: 7.31: Existence of Prime Divisors)
        
        Proposition: 7.32: Natural Number is Prime or has Prime Factor (1)
        
        Proof: By Euclid (related to Proposition: 7.32: Natural Number is Prime or has Prime Factor)
        
        Proposition: 7.33: Least Ratio of Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.33: Least Ratio of Numbers)
        
        Proposition: 7.34: Existence of Least Common Multiple (1)
        
        Proof: By Euclid (related to Proposition: 7.34: Existence of Least Common Multiple)
        
        Proposition: 7.35: Least Common Multiple Divides Common Multiple (1)
        
        Proof: By Euclid (related to Proposition: 7.35: Least Common Multiple Divides Common Multiple)
        
        Proposition: 7.36: Least Common Multiple of Three Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.36: Least Common Multiple of Three Numbers)
        
        Proposition: 7.37: Integer Divided by Divisor is Integer (1)
        
        Proof: By Euclid (related to Proposition: 7.37: Integer Divided by Divisor is Integer)
        
        Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer (1)
        
        Proof: By Euclid (related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
        
        Proposition: 7.39: Least Number with Three Given Fractions (1)
        
        Proof: By Euclid (related to Proposition: 7.39: Least Number with Three Given Fractions)
    - Section: Book 08: Continued Proportion (28)
      - Subsection: Propositions from Book 8 (28)
        
        Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms (1)
        
        Proof: By Euclid (related to Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms)
        
        Proposition: 8.02: Construction of Geometric Progression in Lowest Terms (2)
        
        Proof: By Euclid (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
        
        Corollary: 8.02: Construction of Geometric Progression in Lowest Terms (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms) (1)
        
        Proof: By Euclid (related to Corollary: 8.02: Construction of Geometric Progression in Lowest Terms)
        
        Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes (1)
        
        Proof: By Euclid (related to Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes)
        
        Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios (1)
        
        Proof: By Euclid (related to Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios)
        
        Proposition: 8.05: Ratio of Products of Sides of Plane Numbers (1)
        
        Proof: By Euclid (related to Proposition: 8.05: Ratio of Products of Sides of Plane Numbers)
        
        Proposition: 8.06: First Element of Geometric Progression not dividing Second (1)
        
        Proof: By Euclid (related to Proposition: 8.06: First Element of Geometric Progression not dividing Second)
        
        Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second (1)
        
        Proof: By Euclid (related to Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second)
        
        Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements (1)
        
        Proof: By Euclid (related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
        
        Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers)
        
        Proposition: Prop. 8.10: Product of Geometric Progressions from One (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.10: Product of Geometric Progressions from One)
        
        Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional)
        
        Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals)
        
        Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression)
        
        Proposition: Prop. 8.14: Number divides Number iff Square divides Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.14: Number divides Number iff Square divides Square)
        
        Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube)
        
        Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
        
        Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
        
        Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional)
        
        Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals)
        
        Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane)
        
        Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid)
        
        Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)
        
        Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube)
        
        Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)
        
        Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube)
        
        Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares)
        
        Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes)
    - Section: Book 09: Applications of Number Theory (37)
      - Subsection: Propositions from Book 9 (37)
        
        Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square)
        
        Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers)
        
        Proposition: Prop. 9.03: Square of Cube Number is Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.03: Square of Cube Number is Cube)
        
        Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube)
        
        Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube)
        
        Proposition: Prop. 9.06: Number Squared making Cube is itself Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.06: Number Squared making Cube is itself Cube)
        
        Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number)
        
        Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number)
        
        Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number)
        
        Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number)
        
        Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements (2)
        
        Proof: By Euclid (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
        
        Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements) (1)
        
        Proof: By Euclid (related to Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements)
        
        Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime)
        
        Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime)
        
        Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic (1)
        
        Proof: By Euclid (related to Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic)
        
        Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element)
        
        Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional)
        
        Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second)
        
        Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers)
        
        Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers)
        
        Proposition: Prop. 9.20: Infinite Number of Primes (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.20: Infinite Number of Primes)
        
        Proposition: Prop. 9.21: Sum of Even Numbers is Even (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.21: Sum of Even Numbers is Even)
        
        Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even)
        
        Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd)
        
        Proposition: Prop. 9.24: Even Number minus Even Number is Even (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.24: Even Number minus Even Number is Even)
        
        Proposition: Prop. 9.25: Even Number minus Odd Number is Odd (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
        
        Proposition: Prop. 9.26: Odd Number minus Odd Number is Even (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.26: Odd Number minus Odd Number is Even)
        
        Proposition: Prop. 9.27: Odd Number minus Even Number is Odd (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.27: Odd Number minus Even Number is Odd)
        
        Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even)
        
        Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd)
        
        Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)
        
        Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)
        
        Proposition: Prop. 9.32: Power of Two is Even-Times Even Only (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.32: Power of Two is Even-Times Even Only)
        
        Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd)
        
        Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd (1)
        
        Proof: By Euclid (related to Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd)
        
        Proposition: 9.35: Sum of Geometric Progression (1)
        
        Proof: By Euclid (related to Proposition: 9.35: Sum of Geometric Progression)
        
        Proposition: 9.36: Theorem of Even Perfect Numbers (First Part) (1)
        
        Proof: By Euclid (related to Proposition: 9.36: Theorem of Even Perfect Numbers (First Part))
    - Section: Book 10: Incommensurable Magnitudes (147)
    - Section: Book 11: Elementary Stereometry (70)
      - Subsection: Definitions from Book 11 (28)
        
        Definition: Def. 11.01: Solid Figures, Three-Dimensional Polyhedra
        
        Definition: Def. 11.02: Surface of a Solid Figure
        
        Definition: Def. 11.03: Straight Line at Right Angles To a Plane
        
        Definition: Def. 11.04: Plane at Right Angles to a Plane
        
        Definition: Def. 11.05: Inclination of a Straight Line to a Plane
        
        Definition: Def. 11.06: Inclination of a Plane to a Plane
        
        Definition: Def. 11.07: Similarly Inclined Planes
        
        Definition: Def. 11.08: Parallel Planes
        
        Definition: Def. 11.09: Similar Solid Figures
        
        Definition: Def. 11.10: Equal Solid Figures
        
        Definition: Def. 11.11: Solid Angle
        
        Definition: Def. 11.12: Pyramid, Tetrahedron
        
        Definition: Def. 11.13: Prism, Parallelepiped
        
        Definition: Def. 11.14: Sphere
        
        Definition: Def. 11.15: Axis of a Sphere
        
        Definition: Def. 11.16: Center of a Sphere
        
        Definition: Def. 11.17: Diameter of a Sphere
        
        Definition: Def. 11.18: Cone
        
        Definition: Def. 11.19: Axis of a Cone
        
        Definition: Def. 11.20: Base of a Cone
        
        Definition: Def. 11.21: Cylinder
        
        Definition: Def. 11.22: Axis of a Cylinder
        
        Definition: Def. 11.23: Bases of a Cylinder
        
        Definition: Def. 11.24: Similar Cones, Similar Cylinders
        
        Definition: Def. 11.25: Cube
        
        Definition: Def. 11.26: Octahedron
        
        Definition: Def. 11.27: Icosahedron
        
        Definition: Def. 11.28: Dodecahedron
      - Subsection: Propositions from Book 11 (42)
        
        Proposition: Prop. 11.01: Straight Line cannot be in Two Planes (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.01: Straight Line cannot be in Two Planes)
        
        Proposition: 11.02: Two Intersecting Straight Lines are in One Plane (1)
        
        Proof: By Euclid (related to Proposition: 11.02: Two Intersecting Straight Lines are in One Plane)
        
        Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line)
        
        Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane)
        
        Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane)
        
        Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel)
        
        Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane)
        
        Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane)
        
        Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other)
        
        Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles)
        
        Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane)
        
        Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane)
        
        Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique)
        
        Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel)
        
        Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel)
        
        Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel)
        
        Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes)
        
        Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane)
        
        Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane)
        
        Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle)
        
        Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles)
        
        Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle)
        
        Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles (2)
        
        Proof: By Euclid (related to Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles)
        
        Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares (1)
        
        Proof: By Euclid (related to Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares)
        
        Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms)
        
        Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes)
        
        Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle)
        
        Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped)
        
        Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected)
        
        Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume)
        
        Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume)
        
        Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume)
        
        Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases)
        
        Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides (2)
        
        Proof: By Euclid (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
        
        Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides) (1)
        
        Proof: By Euclid (related to Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
        
        Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights)
        
        Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles (2)
        
        Proof: By Euclid (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
        
        Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles) (1)
        
        Proof: By Euclid (related to Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
        
        Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme)
        
        Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional)
        
        Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube)
        
        Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base (1)
        
        Proof: By Euclid (related to Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base)
    - Section: Book 12: Proportional Stereometry (23)
      - Subsection: Propositions from Book 12 (23)
        
        Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters)
        
        Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters (2)
        
        Proof: By Euclid (related to Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters)
        
        Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters (1)
        
        Proof: By Euclid (related to Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters)
        
        Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms)
        
        Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms (2)
        
        Proof: By Euclid (related to Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
        
        Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms (1)
        
        Proof: By Euclid (related to Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
        
        Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases)
        
        Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases)
        
        Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra (2)
        
        Proof: By Euclid (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
        
        Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra) (1)
        
        Proof: By Euclid (related to Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
        
        Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides (2)
        
        Proof: By Euclid (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
        
        Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides) (1)
        
        Proof: By Euclid (related to Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
        
        Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights)
        
        Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height)
        
        Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases)
        
        Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases)
        
        Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis)
        
        Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights)
        
        Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights)
        
        Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles)
        
        Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres (2)
        
        Proof: By Euclid (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
        
        Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres) (1)
        
        Proof: By Euclid (related to Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
        
        Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters (1)
        
        Proof: By Euclid (related to Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters)
    - Section: Book 13: Platonic Solids (23)
      - Subsection: Propositions from Book 13 (23)
        
        Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
        
        Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio (2)
        
        Proof: By Euclid (related to Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
        
        Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio (1)
        
        Proof: By Euclid (related to Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
        
        Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
        
        Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
        
        Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment)
        
        Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome)
        
        Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal)
        
        Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio)
        
        Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio)
        
        Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa)
        
        Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor)
        
        Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle)
        
        Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere (2)
        
        Proof: By Euclid (related to Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere)
        
        Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere (1)
        
        Proof: By Euclid (related to Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere)
        
        Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere)
        
        Proposition: Prop. 13.15: Construction of Cube within Given Sphere (1)
        
        Proof: By Euclid (related to Proposition: Prop. 13.15: Construction of Cube within Given Sphere)
        
        Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere (2)
        
        Proof: By Euclid (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
        
        Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere) (1)
        
        Proof: By Euclid (related to Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere)
        
        Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere (2)
        
        Proof: By Euclid (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
        
        Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere) (1)
        
        Proof: By Euclid (related to Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere)
        
        Proposition: Prop. 13.18: There are only Five Platonic Solids (2)
        
        Proof: By Euclid (related to Proposition: Prop. 13.18: There are only Five Platonic Solids)
        
        Lemma: Lem. 13.18: Angle of the Pentagon (1)
        
        Proof: By Euclid (related to Lemma: Lem. 13.18: Angle of the Pentagon)
  - Part: Euclidean Geometry (12)
    - Chapter: Hilbert's Axiomatic System (8)
      - Section: Axioms of Connection and Their Consequences (7)
        
        Definition: Points, Straight Lines, and Planes
        
        Definition: "Lies on" Relation
        
        Axiom: Axioms of Connection
        
        Proposition: Common Points of Two Distinct Straight Lines in a Plane (1)
        
        Proof: (related to Proposition: Common Points of Two Distinct Straight Lines in a Plane)
        
        Proposition: Common Points of a Plane and a Straight Line Not in the Plane (1)
        
        Proof: (related to Proposition: Common Points of a Plane and a Straight Line Not in the Plane)
        
        Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line (1)
        
        Proof: (related to Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line)
        
        Proposition: Plane Determined by two Crossing Straight Lines (1)
        
        Proof: (related to Proposition: Plane Determined by two Crossing Straight Lines)
      - Section: Axioms of Order and Their Consequences (1)
        
        Axiom: "Between" Relation, Axioms of Order
    - Definition: Euclidean Movement - Isometry
    - Definition: Congruence
    - Definition: Similarity
    - Definition: Ellipse
  - Part: Analytic Geometry (7)
    - Definition: Points in a Coordinate System - Number Spaces
    - Definition: Points vs. Vectors in a Number Space
    - Definition: Describing a Straight Line Using Two Vectors
    - Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors (1)
      - Proof: (related to Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors)
    - Theorem: Existence and Uniqueness of a Straight Line Through Two Points (1)
      - Proof: (related to Theorem: Existence and Uniqueness of a Straight Line Through Two Points)
    - Proposition: Presentation of a Straight Line in a Plane as a Linear Equation (1)
      - Proof: (related to Proposition: Presentation of a Straight Line in a Plane as a Linear Equation)
    - Definition: Hyperplane of a Number Space (1)
      - Explanation: Geometrical Interpretation of Hyperplanes (related to Definition: Hyperplane of a Number Space)
  - Part: Projective Geometry (7)
    - Chapter: Basic Geometric Concepts in Projective Geometry (6)
      - Definition: Incidence
      - Definition: Collinear Points
      - Definition: Concurrent Straight Lines
      - Definition: Coplanar Points and Straight Lines
      - Definition: Projectivities, Ranges and Pencils
      - Definition: Perspectivities
    - Chapter: An Introduction to the Principle of Duality in Projective Geometry
  - Part: Hyperbolic Geometry
  - Part: Elliptic Geometry
  - Part: Differential Geometry
- Branch: Combinatorics and Discrete Mathematics (57)
  - Part: Historical Development of Combinatorics
  - Part: Set-theoretic Prerequisites Needed For Combinatorics (8)
    - Proposition: Indicator Function and Set Operations (1)
      - Proof: (related to Proposition: Indicator Function and Set Operations)
    - Proposition: Fundamental Counting Principle (1)
      - Proof: (related to Proposition: Fundamental Counting Principle)
    - Explanation: Connection Between the Cartesian Product and Functions Mapping Finite Sets into a Non-empty Set (related to Part: Set-theoretic Prerequisites Needed For Combinatorics) (1)
      - Proposition: Number of Ordered n-Tuples in a Set (1)
        
        Proof: (related to Proposition: Number of Ordered n-Tuples in a Set)
    - Explanation: Indicator Function and Complete Induction (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
    - Theorem: Inclusion-Exclusion Principle (Sylvester's Formula) (1)
      - Proof: (related to Theorem: Inclusion-Exclusion Principle (Sylvester's Formula))
    - Explanation: Connection Between the Power Set and Functions Mapping Sets into a Set with Two Elements (related to Part: Set-theoretic Prerequisites Needed For Combinatorics) (2)
      - Proposition: Number of Subsets of a Finite Set (2)
        
        Proof: By Induction (related to Proposition: Number of Subsets of a Finite Set)
        
        Proof: (related to Proposition: Number of Subsets of a Finite Set)
    - Proposition: Recursively Defined Arithmetic Functions, Recursion (1)
      - Proof: (related to Proposition: Recursively Defined Arithmetic Functions, Recursion)
  - Part: Cycles, Permutations, Combinations and Variations (10)
    - Definition: Cycles
    - Definition: Permutations
    - Theorem: Approximation of Factorials Using the Stirling Formula (1)
      - Proof: (related to Theorem: Approximation of Factorials Using the Stirling Formula)
    - Definition: Combinations (6)
      - Definition: Binomial Coefficients
      - Proposition: Closed Formula For Binomial Coefficients (1)
        
        Proof: (related to Proposition: Closed Formula For Binomial Coefficients)
      - Proposition: Recursive Formula for Binomial Coefficients (1)
        
        Proof: (related to Proposition: Recursive Formula for Binomial Coefficients)
      - Explanation: Pascal's Triangle (Triangle of Binomial Coefficients) (related to Definition: Combinations) (1)
        
        Explanation: Properties of the Pascal's Triangle (related to Explanation: Pascal's Triangle (Triangle of Binomial Coefficients))
      - Theorem: Binomial Theorem (1)
        
        Proof: By Induction (related to Theorem: Binomial Theorem)
      - Proposition: Simple Binomial Identities (1)
        
        Proof: (related to Proposition: Simple Binomial Identities)
    - Proposition: Factorial (1)
      - Proof: (related to Proposition: Factorial)
  - Part: Stirling Numbers (11)
    - Definition: Stirling Numbers of First and Second Kind
    - Proposition: Recursive Formula for the Stirling Numbers of the First Kind (1)
      - Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the First Kind)
    - Chapter: Triangle of the Stirling Numbers of the First Kind
    - Chapter: Triangle of the Stirling Numbers of the Second Kind
    - Explanation: Combinatorial Interpretation of Stirling Numbers of the First Kind (related to Part: Stirling Numbers)
    - Explanation: Combinatorial Interpretation of Stirling Numbers of the Second Kind (related to Part: Stirling Numbers)
    - Proposition: Factorials and Stirling Numbers of the First Kind (1)
      - Proof: (related to Proposition: Factorials and Stirling Numbers of the First Kind)
    - Proposition: Inversion Formulas For Stirling Numbers (1)
      - Proof: (related to Proposition: Inversion Formulas For Stirling Numbers)
    - Lemma: Stirling Numbers and Rising Factorial Powers (1)
      - Proof: (related to Lemma: Stirling Numbers and Rising Factorial Powers)
    - Proposition: Recursive Formula for the Stirling Numbers of the Second Kind (1)
      - Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the Second Kind)
    - Proposition: Comparison between the Stirling numbers of the First and Second Kind (1)
      - Proof: (related to Proposition: Comparison between the Stirling numbers of the First and Second Kind)
  - Proposition: Number of Relations on a Finite Set (1)
    - Proof: (related to Proposition: Number of Relations on a Finite Set)
  - Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters (1)
    - Proof: (related to Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters)
  - Part: Solving Strategies and Sample Solutions to Problems in Combinatorics (6)
    - Chapter: Applications of the Fundamental Theorem of the Difference Calculus (5)
      - Problem: Sums of Powers of Two (1)
        
        Solution: (related to Problem: Sums of Powers of Two)
      - Problem: Sums of Falling Factorial Powers (1)
        
        Solution: (related to Problem: Sums of Falling Factorial Powers)
      - Problem: Calculating an Infinite Sum (1)
        
        Solution: (related to Problem: Calculating an Infinite Sum)
      - Problem: Sum of Consecutive Positive Integers (1)
        
        Solution: (related to Problem: Sum of Consecutive Positive Integers)
      - Problem: Sum of Consecutive Squares (1)
        
        Solution: (related to Problem: Sum of Consecutive Squares)
    - Problem: Interpolating Numbers With a Polynomial (1)
      - Solution: (related to Problem: Interpolating Numbers With a Polynomial)
  - Part: Discrete Calculus and Difference Equations (16)
    - Definition: Difference Operator
    - Proposition: Basic Calculations Involving the Difference Operator (1)
      - Proof: (related to Proposition: Basic Calculations Involving the Difference Operator)
    - Proposition: Nth Difference Operator (1)
      - Proof: By Induction (related to Proposition: Nth Difference Operator)
    - Proposition: Difference Operator of Powers (1)
      - Proof: (related to Proposition: Difference Operator of Powers)
    - Definition: Factorial Polynomials (2)
      - Proposition: Factorial Polynomials have a Unique Representation (1)
        
        Proof: (related to Proposition: Factorial Polynomials have a Unique Representation)
      - Proposition: Factorial Polynomials vs. Polynomials (1)
        
        Proof: By Induction (related to Proposition: Factorial Polynomials vs. Polynomials)
    - Definition: Falling and Rising Factorial Powers of Functions
    - Example: Difference Operators of Some Functions (related to Part: Discrete Calculus and Difference Equations)
    - Theorem: Taylor's Formula Using the Difference Operator (1)
      - Proof: (related to Theorem: Taylor's Formula Using the Difference Operator)
    - Definition: Indefinite Sum, Antidifference (1)
      - Proposition: Antidifferences are Unique Up to a Periodic Constant (1)
        
        Proof: (related to Proposition: Antidifferences are Unique Up to a Periodic Constant)
    - Proposition: Basic Calculations Involving Indefinite Sums (1)
      - Proof: (related to Proposition: Basic Calculations Involving Indefinite Sums)
    - Proposition: Antidifferences of Some Functions (1)
      - Proof: (related to Proposition: Antidifferences of Some Functions)
    - Definition: Falling And Rising Factorial Powers (2)
      - Motivation: Formulae of Negative Factorial Powers Explained (related to Definition: Falling And Rising Factorial Powers)
      - Corollary: Reciprocity Law of Falling And Rising Factorial Powers (related to Definition: Falling And Rising Factorial Powers) (1)
        
        Proof: (related to Corollary: Reciprocity Law of Falling And Rising Factorial Powers)
    - Proposition: Difference Operator of Falling Factorial Powers (1)
      - Proof: (related to Proposition: Difference Operator of Falling Factorial Powers)
    - Theorem: Fundamental Theorem of the Difference Calculus (1)
      - Proof: (related to Theorem: Fundamental Theorem of the Difference Calculus)
  - Proposition: Multinomial Coefficient (3)
    - Proof: (related to Proposition: Multinomial Coefficient)
    - Example: Examples of Multinomial Coefficients (related to Proposition: Multinomial Coefficient)
    - Theorem: Multinomial Theorem (1)
      - Proof: By Induction (related to Theorem: Multinomial Theorem)
- Branch: Probability Theory and Statistics (53)
  - Part: Randomness and Probability Calculus (27)
    - Chapter: Laplace Probability (2)
      - Definition: Laplace Experiments and Elementary Events (2)
        
        Example: Fair Dice (related to Definition: Laplace Experiments and Elementary Events)
        
        Corollary: Probability of Laplace Experiments (related to Definition: Laplace Experiments and Elementary Events) (1)
        
        Proof: (related to Corollary: Probability of Laplace Experiments)
    - Chapter: Random Sample Models
    - Definition: Independent Events (4)
      - Proposition: Characterization of Independent Events (1)
        
        Proof: (related to Proposition: Characterization of Independent Events)
      - Proposition: Characterization of Independent Events II (1)
        
        Proof: (related to Proposition: Characterization of Independent Events II)
      - Definition: Mutually Independent Events (1)
        
        Proposition: Replacing Mutually Independent Events by Their Complements (1)
        
        Proof: (related to Proposition: Replacing Mutually Independent Events by Their Complements)
      - Definition: Pairwise Independent Events (1)
        
        Explanation: Mutually Exclusive vs. Pairwise Independent Events (related to Definition: Pairwise Independent Events)
    - Definition: Conditional Probability
    - Theorem: Law of Total Probability (2)
      - Proof: (related to Theorem: Law of Total Probability)
      - Problem: Broken Items in the Box (1)
        
        Solution: Application of the Law of Total Probability (related to Problem: Broken Items in the Box)
    - Theorem: Bayes' Theorem (1)
      - Proof: (related to Theorem: Bayes' Theorem)
    - Definition: Random Experiments and Random Events (8)
      - Definition: Certain and Impossible Event
      - Explanation: How to Interpret Events Which Are Constructed From Other Events Doing Set Operations? (related to Definition: Random Experiments and Random Events)
      - Proposition: Probability of the Complement Event (2)
        
        Proof: (related to Proposition: Probability of the Complement Event)
        
        Corollary: Probability of the Impossible Event (related to Proposition: Probability of the Complement Event) (1)
        
        Proof: (related to Corollary: Probability of the Impossible Event)
      - Proposition: Probability of Included Event (1)
        
        Proof: (related to Proposition: Probability of Included Event)
      - Proposition: Probability of Event Difference (1)
        
        Proof: (related to Proposition: Probability of Event Difference)
      - Proposition: Probability of Event Union (1)
        
        Proof: (related to Proposition: Probability of Event Union)
      - Proposition: Probability of Joint Events (1)
        
        Proof: (related to Proposition: Probability of Joint Events)
    - Definition: Probability and its Axioms (3)
      - Axiom: Probability of the Certain Event
      - Axiom: Probability as a Non-Negative Number
      - Axiom: Addition of the Probability of Mutually Exclusive Events
    - Definition: Mutually Exclusive and Collectively Exhaustive Events
    - Definition: Geometric Probability
    - Proposition: Urn Model Without Replacement (2)
      - Proof: (related to Proposition: Urn Model Without Replacement)
      - Problem: Broken Items in the Box II (1)
        
        Solution: Application of the Urn Model Without Replacement (related to Problem: Broken Items in the Box II)
    - Proposition: Urn Model With Replacement (1)
      - Proof: (related to Proposition: Urn Model With Replacement)
  - Part: Laws of Large Numbers (4)
    - Chapter: Chebyshev's Lemma
    - Chapter: Weak Law of Large Numbers
    - Chapter: Strong Law of Large Numbers
    - Theorem: Theorem of Large Numbers for Relative Frequencies (1)
      - Proof: (related to Theorem: Theorem of Large Numbers for Relative Frequencies)
  - Part: Discrete Distributions (8)
    - Definition: Discrete Random Variables
    - Chapter: Uniform Distribution
    - Proposition: Geometric Distribution (1)
      - Proof: (related to Proposition: Geometric Distribution)
    - Proposition: Binomial Distribution (2)
      - Proof: (related to Proposition: Binomial Distribution)
      - Definition: Bernoulli Experiment
    - Chapter: Hypergeometric Distribution
    - Chapter: Poisson Distribution
    - Proposition: Multinomial Distribution (1)
      - Proof: (related to Proposition: Multinomial Distribution)
  - Part: Continuous Distributions (6)
    - Definition: Continuous Random Variables
    - Chapter: Normal Distribution
    - Chapter: Logarithmic Normal Distribution
    - Chapter: Student Distribution
    - Chapter: The `$\chi^2$` Distribution
    - Chapter: Fisher Distribution
  - Part: Multidimensional Random Variables
  - Part: Markov Chains (2)
    - Chapter: Discrete Time Markov Chains
    - Chapter: Continuous Time Markov Chains
  - Part: Estimation Theory
  - Definition: Random Variable, Realization, Population and Sample (2)
    - Explanation: Why is a random variable neither random, nor variable? (related to Definition: Random Variable, Realization, Population and Sample)
    - Definition: Relative and Absolute Frequency
  - Definition: Probability Distribution (1)
    - Proposition: Monotonically Increasing Property of Probability Distributions (1)
      - Proof: (related to Proposition: Monotonically Increasing Property of Probability Distributions)
  - Definition: Probability Mass Function
- Branch: Number Theory (130)
  - Part: Historical Development of Number Theory
  - Part: Elementary Number Theory (106)
    - Chapter: Divisibility (20)
      - Definition: Divisor, Complementary Divisor, Multiple (1)
        
        Example: Divisibility Examples (related to Definition: Divisor, Complementary Divisor, Multiple)
      - Proposition: Sign of Divisors of Integers (1)
        
        Proof: (related to Proposition: Sign of Divisors of Integers)
      - Proposition: Finite Number of Divisors (1)
        
        Proof: (related to Proposition: Finite Number of Divisors)
      - Definition: Divisor-Closed Sets
      - Proposition: Divisibility Laws (1)
        
        Proof: (related to Proposition: Divisibility Laws)
      - Definition: Sets of Integers Co-Prime To a Given Integer (1)
        
        Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed (1)
        
        Proof: (related to Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed)
      - Definition: Even and Odd Numbers (4)
        
        Proposition: Every Integer Is Either Even or Odd (1)
        
        Proof: (related to Proposition: Every Integer Is Either Even or Odd)
        
        Proposition: Product of Two Even Numbers (1)
        
        Proof: (related to Proposition: Product of Two Even Numbers)
        
        Proposition: Product of Two Odd Numbers (1)
        
        Proof: (related to Proposition: Product of Two Odd Numbers)
        
        Proposition: Product of an Even and an Odd Number (1)
        
        Proof: (related to Proposition: Product of an Even and an Odd Number)
      - Lemma: Division with Quotient and Remainder (1)
        
        Proof: (related to Lemma: Division with Quotient and Remainder)
      - Proposition: Least Common Multiple (1)
        
        Proof: (related to Proposition: Least Common Multiple)
      - Proposition: Greatest Common Divisor (1)
        
        Proof: Commutativity of the Greatest Common Divisor (related to Proposition: Greatest Common Divisor)
      - Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple (1)
        
        Proof: (related to Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple)
      - Definition: Co-prime Numbers (4)
        
        Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor (1)
        
        Proof: (related to Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor)
        
        Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers (1)
        
        Proof: (related to Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers)
        
        Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor (2)
        
        Proof: (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor)
        
        Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor) (1)
        
        Proof: (related to Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor)
      - Proposition: Greatest Common Divisor of More Than Two Numbers (1)
        
        Proof: (related to Proposition: Greatest Common Divisor of More Than Two Numbers)
      - Proposition: Least Common Multiple of More Than Two Numbers (1)
        
        Proof: (related to Proposition: Least Common Multiple of More Than Two Numbers)
    - Chapter: Elementary Results About Prime Numbers (22)
      - Definition: Prime Numbers (9)
        
        Proposition: Existence of Prime Divisors (2)
        
        Proof: (related to Proposition: Existence of Prime Divisors)
        
        Corollary: Primality of the Smallest Non-Trivial Divisor (related to Proposition: Existence of Prime Divisors) (1)
        
        Proof: (related to Corollary: Primality of the Smallest Non-Trivial Divisor)
        
        Proposition: Natural Numbers and Products of Prime Numbers (1)
        
        Proof: By Induction (related to Proposition: Natural Numbers and Products of Prime Numbers)
        
        Definition: Composite Number
        
        Proposition: Co-prime Primes (1)
        
        Proof: (related to Proposition: Co-prime Primes)
        
        Lemma: Generalized Euclidean Lemma (2)
        
        Proof: (related to Lemma: Generalized Euclidean Lemma)
        
        Corollary: Prime Dividing Product of Primes Implies Prime Divisor (related to Lemma: Generalized Euclidean Lemma) (1)
        
        Proof: (related to Corollary: Prime Dividing Product of Primes Implies Prime Divisor)
        
        Proposition: Greatest Common Divisors Of Integers and Prime Numbers (1)
        
        Proof: (related to Proposition: Greatest Common Divisors Of Integers and Prime Numbers)
        
        Definition: Perfect Square
      - Theorem: Infinite Set of Prime Numbers (3)
        
        Proof: (related to Theorem: Infinite Set of Prime Numbers)
        
        Proof: (related to Theorem: Infinite Set of Prime Numbers)
        
        Proof: (related to Theorem: Infinite Set of Prime Numbers)
      - Theorem: Fundamental Theorem of Arithmetic (1)
        
        Proof: (related to Theorem: Fundamental Theorem of Arithmetic)
      - Definition: Canonical Representation of Natural Numbers, Factorization
      - Definition: Subsets of Prime Numbers Not Dividing a Natural Number
      - Definition: Canonical Representation of Positive Rational Numbers
      - Definition: Floor and Ceiling Functions (4)
        
        Proposition: Properties of Floors and Ceilings (1)
        
        Proof: (related to Proposition: Properties of Floors and Ceilings)
        
        Proposition: Floor Function and Division with Quotient and Remainder (1)
        
        Proof: (related to Proposition: Floor Function and Division with Quotient and Remainder)
        
        Lemma: Sums of Floors (1)
        
        Proof: (related to Lemma: Sums of Floors)
        
        Lemma: Reciprocity Law for Floor Functions (1)
        
        Proof: (related to Lemma: Reciprocity Law for Floor Functions)
      - Proposition: Number of Multiples of a Given Number Less Than Another Number (1)
        
        Proof: (related to Proposition: Number of Multiples of a Given Number Less Than Another Number)
      - Proposition: Factorization of Greatest Common Divisor and Least Common Multiple (1)
        
        Proof: (related to Proposition: Factorization of Greatest Common Divisor and Least Common Multiple)
    - Chapter: Arithmetic Functions (20)
      - Definition: Arithmetic Function
      - Example: Examples of Important Arithmetic Functions (related to Chapter: Arithmetic Functions) (5)
        
        Definition: Prime-Counting Function
        
        Definition: Number of Divisors
        
        Definition: Möbius Function, Square-free
        
        Definition: Euler function
        
        Definition: Von Mangoldt Function
      - Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors (1)
        
        Proof: (related to Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors)
      - Theorem: Number of Multiples of a Prime Number Less Than Factorial (1)
        
        Proof: (related to Theorem: Number of Multiples of a Prime Number Less Than Factorial)
      - Proposition: Calculating the Number of Positive Divisors (1)
        
        Proof: (related to Proposition: Calculating the Number of Positive Divisors)
      - Definition: Multiplicative Functions (2)
        
        Corollary: Simple Conclusions For Multiplicative Functions (related to Definition: Multiplicative Functions) (1)
        
        Proof: (related to Corollary: Simple Conclusions For Multiplicative Functions)
        
        Example: Examples of Multiplicative Functions (related to Definition: Multiplicative Functions)
      - Section: Some Properties of the Möbius Function (6)
        
        Proposition: Sum of Möbius Function Over Divisors (1)
        
        Proof: (related to Proposition: Sum of Möbius Function Over Divisors)
        
        Lemma: Sum of Möbius Function Over Divisors With Division (1)
        
        Proof: 624 BC (related to Lemma: Sum of Möbius Function Over Divisors With Division)
        
        Lemma: Möbius and Floor Functions Combined (2)
        
        Proof: (related to Lemma: Möbius and Floor Functions Combined)
        
        Explanation: Changing the Index Of Sums in Number Theory (related to Lemma: Möbius and Floor Functions Combined)
        
        Lemma: Upper Bound of Harmonic Series Times Möbius Function (1)
        
        Proof: (related to Lemma: Upper Bound of Harmonic Series Times Möbius Function)
        
        Theorem: Möbius Inversion Formula (1)
        
        Proof: (related to Theorem: Möbius Inversion Formula)
      - Section: An Application of the Möbius Inversion Formula (2)
        
        Proposition: Sum of Euler Function (1)
        
        Proof: 1620 BC (related to Proposition: Sum of Euler Function)
        
        Proposition: Explicit Formula for the Euler Function (1)
        
        Proof: 800 BC (related to Proposition: Explicit Formula for the Euler Function)
      - Section: Dirichlet Convolution (1)
        
        Theorem: Commutative Group of Multiplicative Functions (1)
        
        Proof: (related to Theorem: Commutative Group of Multiplicative Functions)
    - Chapter: Congruence Classes and Modular Arithmetic (22)
      - Definition: Congruent, Residue (8)
        
        Proposition: Congruence Classes (1)
        
        Proof: (related to Proposition: Congruence Classes)
        
        Proposition: Congruences and Division with Quotient and Remainder (1)
        
        Proof: (related to Proposition: Congruences and Division with Quotient and Remainder)
        
        Explanation: Explanation of Congruence Classes (related to Definition: Congruent, Residue)
        
        Proposition: Connection between Quotient, Remainder, Modulo and Floor Function (1)
        
        Proof: (related to Proposition: Connection between Quotient, Remainder, Modulo and Floor Function)
        
        Lemma: Coprimality and Congruence Classes (1)
        
        Proof: (related to Lemma: Coprimality and Congruence Classes)
        
        Proposition: Multiplication of Congruences with a Positive Factor (1)
        
        Proof: (related to Proposition: Multiplication of Congruences with a Positive Factor)
        
        Proposition: Congruence Modulo a Divisor (1)
        
        Proof: (related to Proposition: Congruence Modulo a Divisor)
        
        Definition: Modulo Operation for Real Numbers
      - Section: Algebraic Structures of Complete Residue Systems (6)
        
        Definition: Complete Residue System
        
        Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$` (2)
        
        Proof: (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`)
        
        Corollary: Sums, Products, and Powers Of Congruences (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`) (1)
        
        Proof: (related to Corollary: Sums, Products, and Powers Of Congruences)
        
        Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$` (1)
        
        Proof: (related to Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$`)
        
        Proposition: Cancellation of Congruences with General Factor (1)
        
        Proof: (related to Proposition: Cancellation of Congruences with General Factor)
        
        Proposition: Creation of Complete Residue Systems From Others (1)
        
        Proof: (related to Proposition: Creation of Complete Residue Systems From Others)
      - Section: Algebraic Structures of Reduced Residue Systems (8)
        
        Definition: Reduced Residue System
        
        Proposition: Creation of Reduced Residue Systems From Others (1)
        
        Proof: (related to Proposition: Creation of Reduced Residue Systems From Others)
        
        Proposition: Existence and Number of Solutions of Congruence With One Variable (1)
        
        Proof: By Induction (related to Proposition: Existence and Number of Solutions of Congruence With One Variable)
        
        Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$` (1)
        
        Proof: (related to Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$`)
        
        Theorem: Euler-Fermat Theorem (1)
        
        Proof: (related to Theorem: Euler-Fermat Theorem)
        
        Proposition: A Necessary Condition for an Integer to be Prime (1)
        
        Proof: (related to Proposition: A Necessary Condition for an Integer to be Prime)
        
        Proposition: Wilson's Condition for an Integer to be Prime (1)
        
        Proof: By Induction (related to Proposition: Wilson's Condition for an Integer to be Prime)
        
        Proposition: Complete and Reduced Residue Systems (Revised) (1)
        
        Proof: (related to Proposition: Complete and Reduced Residue Systems (Revised))
    - Chapter: Solving Diophantine Equations (9)
      - Definition: Diophantine Equations
      - Proposition: Diophantine Equations of Congruences (2)
        
        Proof: (related to Proposition: Diophantine Equations of Congruences)
        
        Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions (related to Proposition: Diophantine Equations of Congruences) (1)
        
        Proof: (related to Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions)
      - Proposition: Existence of Solutions of an LDE With More Variables (1)
        
        Proof: By Induction (related to Proposition: Existence of Solutions of an LDE With More Variables)
      - Proposition: All Solutions Given a Solution of an LDE With Two Variables (1)
        
        Proof: (related to Proposition: All Solutions Given a Solution of an LDE With Two Variables)
      - Theorem: Chinese Remainder Theorem (1)
        
        Proof: By Induction (related to Theorem: Chinese Remainder Theorem)
      - Proposition: A Linear Term for 1 Using Two Co-prime Coefficients (1)
        
        Proof: (related to Proposition: A Linear Term for 1 Using Two Co-prime Coefficients)
      - Proposition: Counting the Solutions of Diophantine Equations of Congruences (1)
        
        Proof: (related to Proposition: Counting the Solutions of Diophantine Equations of Congruences)
      - Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number (1)
        
        Proof: (related to Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number)
    - Chapter: Quadratic Residues (13)
      - Definition: Quadratic Residue, Quadratic Nonresidue
      - Definition: Legendre Symbol
      - Proposition: Legendre Symbols of Equal Residues (1)
        
        Proof: (related to Proposition: Legendre Symbols of Equal Residues)
      - Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime (1)
        
        Proof: (related to Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime)
      - Proposition: Euler's Criterion For Quadratic Residues (1)
        
        Proof: (related to Proposition: Euler's Criterion For Quadratic Residues)
      - Proposition: Multiplicativity of the Legendre Symbol (1)
        
        Proof: (related to Proposition: Multiplicativity of the Legendre Symbol)
      - Section: Calculating Legendre Symbols (4)
        
        Theorem: Quadratic Reciprocity Law (1)
        
        Proof: (related to Theorem: Quadratic Reciprocity Law)
        
        Theorem: First Supplementary Law to the Quadratic Reciprocity Law (1)
        
        Proof: (related to Theorem: First Supplementary Law to the Quadratic Reciprocity Law)
        
        Lemma: Gaussian Lemma (Number Theory) (1)
        
        Proof: (related to Lemma: Gaussian Lemma (Number Theory))
        
        Theorem: Second Supplementary Law to the Quadratic Reciprocity Law (1)
        
        Proof: (related to Theorem: Second Supplementary Law to the Quadratic Reciprocity Law)
      - Section: Generalizations of the Legendre symbol - Jacobi and Kronecker Symbols (3)
        
        Definition: Jacobi Symbol
        
        Theorem: Properties of the Jacobi Symbol (1)
        
        Proof: (related to Theorem: Properties of the Jacobi Symbol)
        
        Explanation: Jacobi Symbol vs. Solvability of Quadratic Congruences (related to Section: Generalizations of the Legendre symbol - Jacobi and Kronecker Symbols)
  - Part: Algebraic Number Theory (Link)
  - Part: Analytic Number Theory (9)
    - Chapter: Prime Number Theorem
    - Chapter: Prime Number Theorem for Arithmetic Progressions
    - Definition: Twin Prime Numbers
    - Definition: Harmonic Series (2)
      - Proposition: Divergence of Harmonic Series (1)
        
        Proof: (related to Proposition: Divergence of Harmonic Series)
      - Proposition: Convergence of Alternating Harmonic Series (1)
        
        Proof: (related to Proposition: Convergence of Alternating Harmonic Series)
    - Chapter: Zeta Functions
    - Chapter: Sieve Methods (3)
      - Definition: Sieve of Eratosthenes
      - Lemma: Sieve for Twin Primes (1)
        
        Proof: (related to Lemma: Sieve for Twin Primes)
      - Definition: Sieve, Sieve Problem
  - Part: Additive Number Theory (7)
    - Definition: Sum of Divisors
    - Proposition: Calculating the Sum of Divisors (1)
      - Proof: (related to Proposition: Calculating the Sum of Divisors)
    - Definition: Perfect Number
    - Proposition: Even Perfect Numbers (2)
      - Proof: (related to Proposition: Even Perfect Numbers)
      - Proposition: Numbers Being the Product of Their Divisors (1)
        
        Proof: (related to Proposition: Numbers Being the Product of Their Divisors)
    - Chapter: Goldbach Conjecture
    - Theorem: Fermat's Last Theorem (1)
      - Proof: (related to Theorem: Fermat's Last Theorem)
  - Part: Some Unsolved Number-Theoretic Problems (4)
    - Problem: Even Perfect Numbers Problem
    - Problem: Odd Perfect Numbers Problem
    - Problem: Are there infinitely many primorial primes?
    - Problem: Twin Prime Problem
  - Part: Solving Strategies and Sample Solutions Related to Number Theory (2)
    - Problem: Calculating Quadratic Residues (1)
      - Solution: (related to Problem: Calculating Quadratic Residues)
    - Problem: Applications of the Jacobi Symbol (1)
      - Solution: (related to Problem: Applications of the Jacobi Symbol)
- Branch: Graph Theory (91)
  - Part: Basics Concepts in Graph Theory (22)
    - Chapter: Graphs (13)
      - Definition: Undirected Graph, Vertices, Edges, Simple Graph (6)
        
        Definition: Order of a Graph (1)
        
        Definition: Finite and Infinite Graphs
        
        Definition: Size of a Graph (1)
        
        Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices (1)
        
        Proof: (related to Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices)
        
        Definition: Complete Graph
        
        Definition: Cycle Graph
        
        Definition: Null Graph
        
        Definition: Complement Graph
      - Definition: Vertex Degrees for Undirected Graphs (2)
        
        Definition: Degree Sequence
        
        Definition: Regular Graph
      - Definition: Leaf
      - Definition: Subgraphs and Supergraphs; Induced Subgraph (1)
        
        Definition: Spanning Subgraph
      - Definition: Incidence, Adjacency, Neighbours
      - Lemma: Handshaking Lemma for Finite Graphs (2)
        
        Proof: (related to Lemma: Handshaking Lemma for Finite Graphs)
        
        Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs (related to Lemma: Handshaking Lemma for Finite Graphs) (1)
        
        Proof: (related to Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs)
    - Chapter: Digraphs (6)
    - Chapter: Contraction and Minors
    - Chapter: Graph Representations (2)
      - Definition: Adjacency Matrix (1)
        
        Example: Examples of Adjacency Matrices (related to Definition: Adjacency Matrix)
      - Definition: Adjacency List Representation
  - Part: Paths, Cycles and Connectivity (22)
    - Definition: Walks, Trails, and Paths
    - Definition: Closed Walks, Closed Trails, and Cycles (2)
      - Definition: Girth and Circumference
      - Definition: Cyclic, Acyclic Graph
    - Chapter: Bipartite Graphs (2)
      - Definition: Bipartite Graph (1)
        
        Definition: Complete Bipartite Graph
      - Theorem: Characterization of Bipartite Graphs (1)
        
        Proof: (related to Theorem: Characterization of Bipartite Graphs)
    - Definition: Hamiltonian Cycle (3)
      - Definition: Hamiltonian Graph (3)
        
        Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph (1)
        
        Proof: (related to Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph)
        
        Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree (1)
        
        Proof: (related to Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree)
        
        Theorem: Characterization of Planar Hamiltonian Graphs (1)
        
        Proof: (related to Theorem: Characterization of Planar Hamiltonian Graphs)
    - Definition: Eulerian Tour (2)
      - Definition: Eulerian Graph (2)
        
        Lemma: Splitting a Graph with Even Degree Vertices into Cycles (1)
        
        Proof: (related to Lemma: Splitting a Graph with Even Degree Vertices into Cycles)
        
        Theorem: Characterization of Eulerian Graphs (1)
        
        Proof: (related to Theorem: Characterization of Eulerian Graphs)
    - Definition: Connected and Disconnected Graphs, Bridges and Cutvertices (1)
      - Proposition: Characterization of Cutvertices (1)
        
        Proof: (related to Proposition: Characterization of Cutvertices)
    - Definition: Weakly and Strongly Connected Digraphs
    - Algorithm: Get All Components of a Graph (2)
      - Proof: (related to Algorithm: Get All Components of a Graph)
      - Proof: (related to Algorithm: Get All Components of a Graph)
    - Definition: Connected Vertices
    - Definition: Biconnected Graphs, `\(k\)`-Connected Graphs
    - Definition: Semi-Eulerian Tour, Open Trail (1)
      - Definition: Semi-Eulerian Graph (1)
        
        Theorem: Characterization of Semi-Eulerian Graphs (1)
        
        Proof: (related to Theorem: Characterization of Semi-Eulerian Graphs)
    - Definition: Semi-Hamiltonian Path (1)
      - Definition: Semi-Hamiltonian Graph
    - Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph (1)
      - Proof: (related to Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph)
    - Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex (2)
      - Proof: (related to Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex)
      - Proof: By Induction (related to Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex)
    - Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph (1)
      - Proof: (related to Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph)
  - Definition: Trees and Forests (6)
    - Proposition: Equivalent Definitions of Trees (1)
      - Proof: By Induction (related to Proposition: Equivalent Definitions of Trees)
    - Definition: Root, Degree of a Tree, Subtree, Height
    - Lemma: Lower Bound of Leaves in a Tree (1)
      - Proof: (related to Lemma: Lower Bound of Leaves in a Tree)
    - Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree (1)
      - Proof: By Induction (related to Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree)
    - Definition: Spanning Tree (1)
      - Theorem: Number of Labeled Spanning Trees (1)
        
        Proof: (related to Theorem: Number of Labeled Spanning Trees)
    - Definition: Graph Decomposable Into `\(k\)` Trees (1)
      - Definition: Minimal Tree Decomposability (1)
        
        Corollary: Bounds for the Minimal Tree Decomposability (related to Definition: Minimal Tree Decomposability) (1)
        
        Proof: (related to Corollary: Bounds for the Minimal Tree Decomposability)
  - Part: Planar Graphs (20)
    - Definition: Planar Drawing (Embedding)
    - Definition: Planar Graph
    - Definition: Face, Infinite Face
    - Definition: Face Degree
    - Lemma: Handshaking Lemma for Planar Graphs (1)
      - Proof: (related to Lemma: Handshaking Lemma for Planar Graphs)
    - Theorem: Euler Characteristic for Planar Graphs (1)
      - Proof: (related to Theorem: Euler Characteristic for Planar Graphs)
    - Definition: Dual Planar Graph (1)
      - Corollary: Number of Vertices, Edges, and Faces of a Dual Graph (related to Definition: Dual Planar Graph) (1)
        
        Proof: (related to Corollary: Number of Vertices, Edges, and Faces of a Dual Graph)
    - Chapter: Conditions for Planarity and Planarity Testing (6)
      - Proposition: A Necessary Condition for a Graph to be Planar (1)
        
        Proof: (related to Proposition: A Necessary Condition for a Graph to be Planar)
      - Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (1)
        
        Proof: (related to Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar)
      - Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II) (1)
        
        Proof: (related to Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II))
      - Explanation: Examples of Dual Statements for Planar Graphs (related to Chapter: Conditions for Planarity and Planarity Testing)
      - Proposition: Relationship Between Planarity and Connectivity of Graphs (1)
        
        Proof: (related to Proposition: Relationship Between Planarity and Connectivity of Graphs)
      - Proposition: Relationship Between Planarity and Biconnectivity of Graphs (1)
        
        Proof: (related to Proposition: Relationship Between Planarity and Biconnectivity of Graphs)
    - Definition: Pieces of a Graph With Respect to A Cycle
    - Definition: Separating and Non-Separating Cycles
    - Definition: Interlacing Pieces with Respect to a Cycle, Interlacement Graph
    - Definition: Subdivision of a Graph (1)
      - Corollary: Planarity of Subdivisions (related to Definition: Subdivision of a Graph) (1)
        
        Proof: (related to Corollary: Planarity of Subdivisions)
    - Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle? (1)
      - Proof: (related to Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?)
    - Theorem: Characterization of Biconnected Planar Graphs (1)
      - Proof: (related to Theorem: Characterization of Biconnected Planar Graphs)
    - Theorem: Characterization of Planar Graphs (1)
      - Proof: (related to Theorem: Characterization of Planar Graphs)
  - Part: Vertex Colorings and Decompositions (8)
    - Theorem: Brooks' Theorem (1)
      - Proof: (related to Theorem: Brooks' Theorem)
    - Definition: Chromatic Number and `$k$`-Coloring of a Graph
    - Proposition: Chromatic Number and Maximum Vertex Degree (1)
      - Proof: By Induction (related to Proposition: Chromatic Number and Maximum Vertex Degree)
    - Chapter: Colorings of Planar Graphs (5)
      - Theorem: Six Color Theorem for Planar Graphs (1)
        
        Proof: (related to Theorem: Six Color Theorem for Planar Graphs)
      - Theorem: Five Color Theorem for Planar Graphs (1)
        
        Proof: (related to Theorem: Five Color Theorem for Planar Graphs)
      - Theorem: Four Color Theorem for Planar Graphs (1)
        
        Proof: (related to Theorem: Four Color Theorem for Planar Graphs)
      - Lemma: Coloring of Trees (1)
        
        Proof: (related to Lemma: Coloring of Trees)
      - Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph (1)
        
        Proof: (related to Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph)
  - Part: Transitive Hull and Irreducible Kernels
  - Part: Graph Transformations (5)
    - Chapter: Isomorphism (Graphs) (2)
      - Definition: Isomorphic Undirected Graphs
      - Definition: Isomorphic Digraphs
    - Chapter: Reduction
    - Chapter: Similarity of Graphs
    - Definition: Suppressing Vertices, Suppressed Multigraph
  - Part: Flows
  - Part: Matchings
  - Part: Network Design and Routing
  - Part: Infinite Graphs
  - Part: Extremal Problems in Graph Theory
  - Part: Solving Strategies and Sample Solutions to Problems in Graph Theory (2)
    - Problem: Checking if `$K_{3,3}$` is Planar (1)
      - Solution: (related to Problem: Checking if `$K_{3,3}$` is Planar)
    - Problem: Checking if `$K_5$` is Planar (1)
      - Solution: (related to Problem: Checking if `$K_5$` is Planar)
- Branch: Knot Theory (5)
  - Definition: Knot Diagram, Classical Crossing, Virtual Crossing
  - Definition: Reidemeister Moves, Planar Isotopy Moves, Diagrammatic Moves
  - Definition: Knot
  - Definition: Unknot
  - Proposition: Equivalent Knot Diagrams (1)
    - Proof: (related to Proposition: Equivalent Knot Diagrams)
- Branch: Game Theory
- Branch: Theoretical Physics (17)
  - Part: Classical Physics (5)
    - Definition: Average Velocity
    - Definition: Instantaneous Velocity
    - Theorem: First Law of Planetary Motion (1)
      - Proof: (related to Theorem: First Law of Planetary Motion)
    - Theorem: Second Law of Planetary Motion (1)
      - Proof: (related to Theorem: Second Law of Planetary Motion)
    - Theorem: Third Law of Planetary Motion (1)
      - Proof: (related to Theorem: Third Law of Planetary Motion)
  - Part: Special Relativity (9)
    - Motivation: Michelson-Morley Experiment (related to Part: Special Relativity)
    - Motivation: Measurement of the Speed of Light - First Experimental Proof that the Speed of Light is Finite (related to Part: Special Relativity)
    - Axiom: Principle of Relativity (1)
      - Definition: Second
    - Axiom: Principle of the Constancy of Light Speed (1)
      - Definition: Meter
    - Definition: Frame of Reference (1)
      - Definition: Inertial and Noninertial Frames of Reference
    - Definition: Spacetime Diagram
    - Proposition: Construction of a Light Clock (1)
      - Proof: (related to Proposition: Construction of a Light Clock)
    - Proposition: Time Dilation, Lorentz Factor (2)
      - Proof: (related to Proposition: Time Dilation, Lorentz Factor)
      - Explanation: Einstein's Interpretation of the Time Dilation (related to Proposition: Time Dilation, Lorentz Factor)
  - Part: General Relativity
  - Part: Quantum Mechanics (1)
    - Chapter: Quantum Computation
  - Part: Quantum Field Theory
- Branch: Theoretical Computer Science (113)
  - Part: Formal Languages (21)
    - Definition: Equivalent Grammars
    - Definition: Abstract Syntax Tree
    - Definition: Ambiguous and Unambiguous Grammars
    - Example: Examples of ASTs (related to Part: Formal Languages)
    - Chapter: Classification of Formal Languages (5)
      - Definition: Type-3 (Linear) Grammars and Regular Languages
      - Definition: Type-2 (cf) Grammars and Context-free Languages
      - Definition: Type-1 (cs) Grammars and Context-sensitive Languages
      - Definition: Type-0 (Phrase Structure) Grammars and Recursively Enumerable Languages
      - Explanation: Chomsky's Hierarchy of Languages (related to Chapter: Classification of Formal Languages)
    - Chapter: Finite Automata (Finite Sequential Machines) (12)
  - Part: Computability (23)
    - Definition: Computational Problem, Solution
    - Example: Examples of Computational Problems (related to Part: Computability)
    - Definition: Algorithm (Effective Procedure)
    - Definition: `$\mathcal P$`-Computable and `$\mathcal P$`-Decidable Problems
    - Chapter: Models of Computation (16)
      - Section: Turing Machine
      - Definition: Unit-Cost Random Access Machine (15)
        
        Theorem: Simulating LOOP Programs Using WHILE Programs (1)
        
        Proof: (related to Theorem: Simulating LOOP Programs Using WHILE Programs)
        
        Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa) (1)
        
        Proof: (related to Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa))
        
        Definition: LOOP Command, LOOP Program (8)
        
        Example: Simulating More Complex Commands Using Basic `\(L O O P\)` Commands (related to Definition: LOOP Command, LOOP Program) (8)
        
        Algorithm: Setting the value of a register to the value of another register plus or minus a constant
        
        Algorithm: Conditional execution of `\(L O O P\)` programs - IF-THEN-ELSE construct
        
        Algorithm: Conditional execution of `\(L O O P\)` programs - IF-THEN construct
        
        Algorithm: Addition (or Subtraction) of Two Registers
        
        Algorithm: Multiplication of Two Registers
        
        Algorithm: No-Operation Command (NOP)
        
        Algorithm: Division with Quotient and Remainder (Unit-cost Random Access Machine)
        
        Algorithm: Assignment of a Constant `\(c\)` to the Register `\(r_i\)` with a `\(L O O P\)`-Program
        
        Definition: WHILE Command, WHILE Program
        
        Definition: GOTO Command, GOTO Program, Index
        
        Definition: LOOP-Computable Functions (1)
        
        Lemma: LOOP-Computable Functions are Total (1)
        
        Proof: (related to Lemma: LOOP-Computable Functions are Total)
        
        Definition: WHILE-Computable Functions (1)
        
        Corollary: The set of WHILE-computable functions is included in the set of partially WHILE-computable functions (related to Definition: WHILE-Computable Functions) (1)
        
        Proof: (related to Corollary: The set of WHILE-computable functions is included in the set of partially WHILE-computable functions)
        
        Definition: GOTO-Computable Functions
    - Chapter: Correctness
    - Chapter: Decidability
    - Chapter: Entropy
  - Part: Computational Complexity Theory (8)
    - Section: Asymptotic Notation (2)
      - Definition: Big O Notation
      - Proposition: Calculation Rules for the Big O Notation (1)
        
        Proof: (related to Proposition: Calculation Rules for the Big O Notation)
    - Section: Worst Case
    - Section: Average Case
    - Section: Uniform Complexity Measure and Bit Complexity
    - Section: Polynomial Time
    - Section: NP-Completeness (1)
      - Problem: TSP - The Traveling Salesman Problem
    - Section: Probabilistic Complexity
  - Part: Optimization Methods (7)
    - Chapter: Genetic Programming
    - Chapter: Dynamic Programming
    - Chapter: Fuzzy Logic
    - Chapter: Neural Networks
    - Chapter: Greedy Algorithms
    - Chapter: Randomized Rounding
    - Chapter: Simulated Annealing
  - Part: Data Structures (9)
    - Chapter: Linked Lists (1)
      - Definition: Linked List, List Nodes
    - Chapter: Trees
    - Chapter: Sets
    - Chapter: Stacks
    - Chapter: Queues
    - Chapter: Strings
    - Chapter: Files
    - Chapter: Graphs (2)
      - Algorithm: Graph (Python)
      - Algorithm: getSubgraph
  - Part: Basic Algorithms (15)
    - Chapter: Searching (1)
      - Algorithm: Sequential Search
    - Chapter: Hashing
    - Chapter: Sorting
    - Chapter: String Matching
    - Chapter: External Sorting
    - Chapter: Mixing
    - Chapter: Data Compression
    - Chapter: Branch and Bound
    - Chapter: And / Or Trees
    - Chapter: Min-Max and Alpha-Beta
    - Chapter: Satisfaction Problem Solving
    - Chapter: Combinatorial Algorithms
    - Chapter: Graph-Theoretic Algorithms
    - Chapter: Network Paths
    - Chapter: Network Flows
  - Part: Semi-numerical Algorithms (15)
    - Chapter: Positional Notation Systems
    - Chapter: Roman Numbers (2)
      - Algorithm: Converting Decimal Numbers to Roman Numbers
      - Algorithm: Converting Roman Numbers to Decimal Numbers
    - Algorithm: Complex Numbers
    - Chapter: Pseudo-Random Numbers
    - Chapter: Multiplication
    - Chapter: Divisibility and Modular Arithmetic (5)
      - Algorithm: Greatest Common Divisor (Python) (1)
        
        Proof: (related to Algorithm: Greatest Common Divisor (Python))
      - Algorithm: Extended Greatest Common Divisor (Python) (1)
        
        Proof: (related to Algorithm: Extended Greatest Common Divisor (Python))
      - Algorithm: Calculation of Inverses Modulo a Number (Python) (1)
        
        Proof: (related to Algorithm: Calculation of Inverses Modulo a Number (Python))
      - Algorithm: Continued Fraction (Python) (1)
        
        Proof: (related to Algorithm: Continued Fraction (Python))
      - Algorithm: Jacobi Symbol (Python) (1)
        
        Proof: (related to Algorithm: Jacobi Symbol (Python))
    - Chapter: Cryptography
    - Chapter: Calendar
    - Algorithm: Horner Scheme (2)
      - Proof: (related to Algorithm: Horner Scheme)
      - Proof: (related to Algorithm: Horner Scheme)
  - Part: Numerical Algorithms (15)
    - Chapter: Linear Programming
    - Chapter: Infinite Series Approximations
    - Chapter: Calculation of Roots
    - Chapter: Interpolation
    - Chapter: Extrapolation
    - Chapter: Differentiation
    - Chapter: Integration
    - Chapter: Linear Equation Systems (2)
      - Section: Elimination
      - Section: Iterative Methods
    - Chapter: Non-Linear Equation Systems (1)
      - Section: Iterative Methods
    - Chapter: Orthogonalisation
    - Chapter: Eigenvalues and Eigenvectors
    - Chapter: Integration of Starting Value Problems (Ordinary DE) (2)
      - Section: One-Step Methods
      - Section: Multi-Step Methods
    - Chapter: Error Analysis
- Branch: Riddles, Puzzles and Brain-Teasers (447)
  - Part: Tricks for Mental Maths (3)
  - Part: Jokes (5)
    - Example: Analytical Thinking Joke (related to Part: Jokes)
    - Example: Sexy Parabola (Joke) (related to Part: Jokes)
    - Example: Is Reality Real? (related to Part: Jokes)
    - Example: What is time? (related to Part: Jokes)
    - Example: Pizza Lovers (related to Part: Jokes)
  - Part: Oddities and Curiosities (3)
  - Part: Forgotten Reckoning Methods
  - Part: Dudeney's Amusements in Mathematics (435)
    - Chapter: Preface
    - Chapter: Arithmetical and Algebraic Problems (141)
      - Section: Money Puzzles (39)
        
        Problem: A Postoffice Perplexity (1)
        
        Solution: (related to Problem: A Postoffice Perplexity)
        
        Problem: Youthful Precocity (1)
        
        Solution: (related to Problem: Youthful Precocity)
        
        Problem: At a Cattle Market (1)
        
        Solution: (related to Problem: At a Cattle Market)
        
        Problem: The Beanfeast Puzzle (1)
        
        Solution: (related to Problem: The Beanfeast Puzzle)
        
        Problem: A Queer Coincidence (1)
        
        Solution: (related to Problem: A Queer Coincidence)
        
        Problem: A Charitable Bequest (1)
        
        Solution: (related to Problem: A Charitable Bequest)
        
        Problem: The Widow's Legacy (1)
        
        Solution: (related to Problem: The Widow's Legacy)
        
        Problem: Indiscriminate Charity (1)
        
        Solution: (related to Problem: Indiscriminate Charity)
        
        Problem: The Two Aeroplanes (1)
        
        Solution: (related to Problem: The Two Aeroplanes)
        
        Problem: Buying Presents (1)
        
        Solution: (related to Problem: Buying Presents)
        
        Problem: The Cyclists' Feast (1)
        
        Solution: (related to Problem: The Cyclists' Feast)
        
        Problem: A Queer Thing in Money (1)
        
        Solution: (related to Problem: A Queer Thing in Money)
        
        Problem: A New Money Puzzle (1)
        
        Solution: (related to Problem: A New Money Puzzle)
        
        Problem: Square Money (1)
        
        Solution: (related to Problem: Square Money)
        
        Problem: Pocket Money (1)
        
        Solution: (related to Problem: Pocket Money)
        
        Problem: The Millionaire's Perplexity (1)
        
        Solution: (related to Problem: The Millionaire's Perplexity)
        
        Problem: The Puzzling Money-Boxes (1)
        
        Solution: (related to Problem: The Puzzling Money-Boxes)
        
        Problem: The Market Women (1)
        
        Solution: (related to Problem: The Market Women)
        
        Problem: The new Year's Eve Suppers (1)
        
        Solution: (related to Problem: The new Year's Eve Suppers)
        
        Problem: Beef and Sausages (1)
        
        Solution: (related to Problem: Beef and Sausages)
        
        Problem: A Deal in Apples (1)
        
        Solution: (related to Problem: A Deal in Apples)
        
        Problem: A Deal in Eggs (1)
        
        Solution: (related to Problem: A Deal in Eggs)
        
        Problem: The Christmas-Boxes (1)
        
        Solution: (related to Problem: The Christmas-Boxes)
        
        Problem: A Shopping Perplexity (1)
        
        Solution: (related to Problem: A Shopping Perplexity)
        
        Problem: Awkward Money (1)
        
        Solution: (related to Problem: Awkward Money)
        
        Problem: The Junior Clerk's Puzzle (1)
        
        Solution: (related to Problem: The Junior Clerk's Puzzle)
        
        Problem: Giving Change (1)
        
        Solution: (related to Problem: Giving Change)
        
        Problem: Defective Observation (1)
        
        Solution: (related to Problem: Defective Observation)
        
        Problem: The Broken Coins (1)
        
        Solution: (related to Problem: The Broken Coins)
        
        Problem: Two Questions in Probabilities (1)
        
        Solution: (related to Problem: Two Questions in Probabilities)
        
        Problem: Domestic Economy (1)
        
        Solution: (related to Problem: Domestic Economy)
        
        Problem: The Excursion Ticket Puzzle (1)
        
        Solution: (related to Problem: The Excursion Ticket Puzzle)
        
        Problem: A Puzzle in Reversals (1)
        
        Solution: (related to Problem: A Puzzle in Reversals)
        
        Problem: The Grocer and Draper (1)
        
        Solution: (related to Problem: The Grocer and Draper)
        
        Problem: Judkins's Cattle (1)
        
        Solution: (related to Problem: Judkins's Cattle)
        
        Problem: Buying Apples (1)
        
        Solution: (related to Problem: Buying Apples)
        
        Problem: Buying Chestnuts (1)
        
        Solution: (related to Problem: Buying Chestnuts)
        
        Problem: The Bicylce Thief (1)
        
        Solution: (related to Problem: The Bicylce Thief)
        
        Problem: The Costermonger's Puzzle (1)
        
        Solution: (related to Problem: The Costermonger's Puzzle)
      - Section: Age and Kinship Puzzles (17)
        
        Problem: Mamma's Age (1)
        
        Solution: (related to Problem: Mamma's Age)
        
        Problem: Their Ages (1)
        
        Solution: (related to Problem: Their Ages)
        
        Problem: The Family Ages (1)
        
        Solution: (related to Problem: The Family Ages)
        
        Problem: Mrs. Timpkin's Age (1)
        
        Solution: (related to Problem: Mrs. Timpkin's Age)
        
        Problem: A Census Puzzle (1)
        
        Solution: (related to Problem: A Census Puzzle)
        
        Problem: Mother and Daughter (1)
        
        Solution: (related to Problem: Mother and Daughter)
        
        Problem: Mary and Marmaduke (1)
        
        Solution: (related to Problem: Mary and Marmaduke)
        
        Problem: Rover's Age (1)
        
        Solution: (related to Problem: Rover's Age)
        
        Problem: Concerning Tommy's Age (1)
        
        Solution: (related to Problem: Concerning Tommy's Age)
        
        Problem: Next-door Neighbors (1)
        
        Solution: (related to Problem: Next-door Neighbors)
        
        Problem: The Bag of Nuts (1)
        
        Solution: (related to Problem: The Bag of Nuts)
        
        Problem: How Old Was Mary? (1)
        
        Solution: (related to Problem: How Old Was Mary?)
        
        Problem: Queer Relationships (1)
        
        Solution: (related to Problem: Queer Relationships)
        
        Problem: Heard on the Tube Railway (1)
        
        Solution: (related to Problem: Heard on the Tube Railway)
        
        Problem: A Family Party (1)
        
        Solution: (related to Problem: A Family Party)
        
        Problem: A Mixed Pedigree (1)
        
        Solution: (related to Problem: A Mixed Pedigree)
        
        Problem: Wilson's Poser (1)
        
        Solution: (related to Problem: Wilson's Poser)
      - Section: Clock Puzzles (10)
        
        Problem: What Was the Time? (1)
        
        Solution: (related to Problem: What Was the Time?)
        
        Problem: A Time Puzzle (1)
        
        Solution: (related to Problem: A Time Puzzle)
        
        Problem: A Puzzling Watch (1)
        
        Solution: (related to Problem: A Puzzling Watch)
        
        Problem: The Wapshaw's Wharf Mystery (1)
        
        Solution: (related to Problem: The Wapshaw's Wharf Mystery)
        
        Problem: Changing Places (1)
        
        Solution: (related to Problem: Changing Places)
        
        Problem: The Club Clock (1)
        
        Solution: (related to Problem: The Club Clock)
        
        Problem: The Stop-Watch (1)
        
        Solution: (related to Problem: The Stop-Watch)
        
        Problem: The Three Clocks (1)
        
        Solution: (related to Problem: The Three Clocks)
        
        Problem: The Railway Station Clock (1)
        
        Solution: (related to Problem: The Railway Station Clock)
        
        Problem: The Village Simpleton (1)
        
        Solution: (related to Problem: The Village Simpleton)
      - Section: Locomotion and Speed Puzzles (9)
        
        Problem: Average Speed (1)
        
        Solution: (related to Problem: Average Speed)
        
        Problem: The Two Trains (1)
        
        Solution: (related to Problem: The Two Trains)
        
        Problem: The Three Villages (1)
        
        Solution: (related to Problem: The Three Villages)
        
        Problem: Drawing Her Pension (1)
        
        Solution: (related to Problem: Drawing Her Pension)
        
        Problem: Sir Edwyn De Tudor (1)
        
        Solution: (related to Problem: Sir Edwyn De Tudor)
        
        Problem: The Hydroplane Question (1)
        
        Solution: (related to Problem: The Hydroplane Question)
        
        Problem: Donkey Riding (1)
        
        Solution: (related to Problem: Donkey Riding)
        
        Problem: The Basket Of Potatoes (1)
        
        Solution: (related to Problem: The Basket Of Potatoes)
        
        Problem: The Passenger's Fare (1)
        
        Solution: (related to Problem: The Passenger's Fare)
      - Section: Digital Puzzles (21)
        
        Problem: The Barrel of Beer (1)
        
        Solution: (related to Problem: The Barrel of Beer)
        
        Problem: Digits and Squares (1)
        
        Solution: (related to Problem: Digits and Squares)
        
        Problem: Odd And Even Digits (1)
        
        Solution: (related to Problem: Odd And Even Digits)
        
        Problem: The Lockers Puzzle (1)
        
        Solution: (related to Problem: The Lockers Puzzle)
        
        Problem: The Three Groups (1)
        
        Solution: (related to Problem: The Three Groups)
        
        Problem: The Nine Counters (1)
        
        Solution: (related to Problem: The Nine Counters)
        
        Problem: The Ten Counters (1)
        
        Solution: (related to Problem: The Ten Counters)
        
        Problem: Digital Multiplication (1)
        
        Solution: (related to Problem: Digital Multiplication)
        
        Problem: The Pierrot's Puzzle (1)
        
        Solution: (related to Problem: The Pierrot's Puzzle)
        
        Problem: The Cab Numbers (1)
        
        Solution: (related to Problem: The Cab Numbers)
        
        Problem: Queer Multiplication (1)
        
        Solution: (related to Problem: Queer Multiplication)
        
        Problem: The Number Checks Puzzle (1)
        
        Solution: (related to Problem: The Number Checks Puzzle)
        
        Problem: Digital Division (1)
        
        Solution: (related to Problem: Digital Division)
        
        Problem: Adding The Digits (1)
        
        Solution: (related to Problem: Adding The Digits)
        
        Problem: The Century Puzzle (1)
        
        Solution: (related to Problem: The Century Puzzle)
        
        Problem: More Mixed Fractions (1)
        
        Solution: (related to Problem: More Mixed Fractions)
        
        Problem: Digital Square Numbers (1)
        
        Solution: (related to Problem: Digital Square Numbers)
        
        Problem: The Mystic Eleven (1)
        
        Solution: (related to Problem: The Mystic Eleven)
        
        Problem: The Digital Century (1)
        
        Solution: (related to Problem: The Digital Century)
        
        Problem: The Four Sevens (1)
        
        Solution: (related to Problem: The Four Sevens)
        
        Problem: The Dice Numbers (1)
        
        Solution: (related to Problem: The Dice Numbers)
      - Section: Various Arithmetical and Algebraic Problems (45)
        
        Problem: The Spot on the Table (1)
        
        Solution: (related to Problem: The Spot on the Table)
        
        Problem: Academic Courtesies (1)
        
        Solution: (related to Problem: Academic Courtesies)
        
        Problem: The Thirty-Three Pearls (1)
        
        Solution: (related to Problem: The Thirty-Three Pearls)
        
        Problem: The Laborer's Puzzle (1)
        
        Solution: (related to Problem: The Laborer's Puzzle)
        
        Problem: The Trusses of Hay (1)
        
        Solution: (related to Problem: The Trusses of Hay)
        
        Problem: Mr. Gubbins in a Fog (1)
        
        Solution: (related to Problem: Mr. Gubbins in a Fog)
        
        Problem: Painting the Lamp-Posts (1)
        
        Solution: (related to Problem: Painting the Lamp-Posts)
        
        Problem: Catching the Thief (1)
        
        Solution: (related to Problem: Catching the Thief)
        
        Problem: The Parish Council Election (1)
        
        Solution: (related to Problem: The Parish Council Election)
        
        Problem: The Muddletown Election (1)
        
        Solution: (related to Problem: The Muddletown Election)
        
        Problem: The Suffragists' Meeting (1)
        
        Solution: (related to Problem: The Suffragists' Meeting)
        
        Problem: The Leap-Year Ladies (1)
        
        Solution: (related to Problem: The Leap-Year Ladies)
        
        Problem: The Great Scramble (1)
        
        Solution: (related to Problem: The Great Scramble)
        
        Problem: The Abbot's Puzzle (1)
        
        Solution: (related to Problem: The Abbot's Puzzle)
        
        Problem: Reeping the Corn (1)
        
        Solution: (related to Problem: Reeping the Corn)
        
        Problem: A Puzzling Legacy (1)
        
        Solution: (related to Problem: A Puzzling Legacy)
        
        Problem: The Torn Number (1)
        
        Solution: (related to Problem: The Torn Number)
        
        Problem: Curious Numbers (1)
        
        Solution: (related to Problem: Curious Numbers)
        
        Problem: A Printer's Error (1)
        
        Solution: (related to Problem: A Printer's Error)
        
        Problem: The Converted Miser (1)
        
        Solution: (related to Problem: The Converted Miser)
        
        Problem: A Fence Problem (1)
        
        Solution: (related to Problem: A Fence Problem)
        
        Problem: Circling the Squares (1)
        
        Solution: (related to Problem: Circling the Squares)
        
        Problem: Rackbrane's Little Loss (1)
        
        Solution: (related to Problem: Rackbrane's Little Loss)
        
        Problem: The Farmer and His Sheep (1)
        
        Solution: (related to Problem: The Farmer and His Sheep)
        
        Problem: Heads or Tails (1)
        
        Solution: (related to Problem: Heads or Tails)
        
        Problem: The See-saw Puzzle (1)
        
        Solution: (related to Problem: The See-saw Puzzle)
        
        Problem: A Legal Difficulty (1)
        
        Solution: (related to Problem: A Legal Difficulty)
        
        Problem: A Question of Definition (1)
        
        Solution: (related to Problem: A Question of Definition)
        
        Problem: The Miners' Holiday (1)
        
        Solution: (related to Problem: The Miners' Holiday)
        
        Problem: Simple Multiplication (1)
        
        Solution: (related to Problem: Simple Multiplication)
        
        Problem: Simple Division (1)
        
        Solution: (related to Problem: Simple Division)
        
        Problem: A Problem in Squares (1)
        
        Solution: (related to Problem: A Problem in Squares)
        
        Problem: The Battle of Hastings (1)
        
        Solution: (related to Problem: The Battle of Hastings)
        
        Problem: The Sculptor's Problem (1)
        
        Solution: (related to Problem: The Sculptor's Problem)
        
        Problem: The Spanish Miser (1)
        
        Solution: (related to Problem: The Spanish Miser)
        
        Problem: The Nine Treasure Boxes (1)
        
        Solution: (related to Problem: The Nine Treasure Boxes)
        
        Problem: The Five Brigands (1)
        
        Solution: (related to Problem: The Five Brigands)
        
        Problem: The Banker's Puzzle (1)
        
        Solution: (related to Problem: The Banker's Puzzle)
        
        Problem: The Stonemason's Problem (1)
        
        Solution: (related to Problem: The Stonemason's Problem)
        
        Problem: The Sultan's Army (1)
        
        Solution: (related to Problem: The Sultan's Army)
        
        Problem: A Study in Thrift (1)
        
        Solution: (related to Problem: A Study in Thrift)
        
        Problem: The Artilleryman's Dilemma (1)
        
        Solution: (related to Problem: The Artilleryman's Dilemma)
        
        Problem: The Dutchmen's Wives (1)
        
        Solution: (related to Problem: The Dutchmen's Wives)
        
        Problem: Find Ada's Surname (1)
        
        Solution: (related to Problem: Find Ada's Surname)
        
        Problem: Saturday Marketing (1)
        
        Solution: (related to Problem: Saturday Marketing)
    - Chapter: Geometrical Problems (65)
      - Section: Dissection Puzzles
      - Section: Greek Cross Puzzles (4)
        
        Problem: The Silk Patchwork (1)
        
        Solution: (related to Problem: The Silk Patchwork)
        
        Problem: Two Crosses From One (1)
        
        Solution: (related to Problem: Two Crosses From One)
        
        Problem: The Cross and the Triangle (1)
        
        Solution: (related to Problem: The Cross and the Triangle)
        
        Problem: The Folded Cross (1)
        
        Solution: (related to Problem: The Folded Cross)
      - Section: Various Dissection Puzzles (24)
        
        Problem: An Easy Dissection Puzzle (1)
        
        Solution: (related to Problem: An Easy Dissection Puzzle)
        
        Problem: An Easy Square Puzzle (1)
        
        Solution: (related to Problem: An Easy Square Puzzle)
        
        Problem: The Bun Puzzle (1)
        
        Solution: (related to Problem: The Bun Puzzle)
        
        Problem: The Chocolate Squares (1)
        
        Solution: (related to Problem: The Chocolate Squares)
        
        Problem: Dissecting a Mittre (1)
        
        Solution: (related to Problem: Dissecting a Mittre)
        
        Problem: The Joiner's Problem (1)
        
        Solution: (related to Problem: The Joiner's Problem)
        
        Problem: Another Joiner's Problem (1)
        
        Solution: (related to Problem: Another Joiner's Problem)
        
        Problem: A Cutting-out Puzzle (1)
        
        Solution: (related to Problem: A Cutting-out Puzzle)
        
        Problem: Mrs. Hobson's Hearthrug (1)
        
        Solution: (related to Problem: Mrs. Hobson's Hearthrug)
        
        Problem: The Pentagon and Square (1)
        
        Solution: (related to Problem: The Pentagon and Square)
        
        Problem: The Dissected Triangle (1)
        
        Solution: (related to Problem: The Dissected Triangle)
        
        Problem: The Table-top and Stools (1)
        
        Solution: (related to Problem: The Table-top and Stools)
        
        Problem: The Great Monad (1)
        
        Solution: (related to Problem: The Great Monad)
        
        Problem: The Square of Veneer (1)
        
        Solution: (related to Problem: The Square of Veneer)
        
        Problem: The Two Horseshoes (1)
        
        Solution: (related to Problem: The Two Horseshoes)
        
        Problem: The Betsy Ross Puzzle (1)
        
        Solution: (related to Problem: The Betsy Ross Puzzle)
        
        Problem: The Cardboard Chain (1)
        
        Solution: (related to Problem: The Cardboard Chain)
        
        Example: The Paper Box (related to Section: Various Dissection Puzzles)
        
        Problem: The Potato Puzzle (1)
        
        Solution: (related to Problem: The Potato Puzzle)
        
        Problem: The Seven Pigs (1)
        
        Solution: (related to Problem: The Seven Pigs)
        
        Problem: The Landowner's Fences (1)
        
        Solution: (related to Problem: The Landowner's Fences)
        
        Problem: The Wizard's Cats (1)
        
        Solution: (related to Problem: The Wizard's Cats)
        
        Problem: The Christmas Pudding (1)
        
        Solution: (related to Problem: The Christmas Pudding)
        
        Problem: A Tangram Paradox (1)
        
        Solution: (related to Problem: A Tangram Paradox)
      - Section: Patchwork Puzzles (8)
        
        Problem: The Cushion Covers (1)
        
        Solution: (related to Problem: The Cushion Covers)
        
        Problem: He Banner Puzzle (1)
        
        Solution: (related to Problem: He Banner Puzzle)
        
        Problem: Mrs. Smiley's Christmas Present (1)
        
        Solution: (related to Problem: Mrs. Smiley's Christmas Present)
        
        Problem: Mrs. Perkins's Quilt (1)
        
        Solution: (related to Problem: Mrs. Perkins's Quilt)
        
        Problem: The Squares Of Brocade (1)
        
        Solution: (related to Problem: The Squares Of Brocade)
        
        Problem: Another Patchwork Puzzle (1)
        
        Solution: (related to Problem: Another Patchwork Puzzle)
        
        Problem: Linoleum Cutting (1)
        
        Solution: (related to Problem: Linoleum Cutting)
        
        Problem: Another Linoleum Puzzle (1)
        
        Solution: (related to Problem: Another Linoleum Puzzle)
      - Section: Various Geometrical Puzzles (28)
        
        Problem: The Cardboard Box (1)
        
        Solution: (related to Problem: The Cardboard Box)
        
        Problem: Stealing The Bell-ropes (1)
        
        Solution: (related to Problem: Stealing The Bell-ropes)
        
        Problem: The Four Sons (1)
        
        Solution: (related to Problem: The Four Sons)
        
        Problem: The Three Railway Stations (1)
        
        Solution: (related to Problem: The Three Railway Stations)
        
        Problem: The Garden Puzzle (1)
        
        Solution: (related to Problem: The Garden Puzzle)
        
        Problem: Drawing A Spiral (1)
        
        Solution: (related to Problem: Drawing A Spiral)
        
        Problem: How To Draw An Oval (1)
        
        Solution: (related to Problem: How To Draw An Oval)
        
        Problem: St. George's Banner (1)
        
        Solution: (related to Problem: St. George's Banner)
        
        Problem: The Clothes Line Puzzle (1)
        
        Solution: (related to Problem: The Clothes Line Puzzle)
        
        Problem: The Milkmaid Puzzle (1)
        
        Solution: (related to Problem: The Milkmaid Puzzle)
        
        Problem: The Ball Problem (1)
        
        Solution: (related to Problem: The Ball Problem)
        
        Problem: The Yorkshire Estates (1)
        
        Solution: (related to Problem: The Yorkshire Estates)
        
        Problem: Farmer Wurzel's Estate (1)
        
        Solution: (related to Problem: Farmer Wurzel's Estate)
        
        Problem: The Crescent Puzzle (1)
        
        Solution: (related to Problem: The Crescent Puzzle)
        
        Problem: The Puzzle Wall (1)
        
        Solution: (related to Problem: The Puzzle Wall)
        
        Problem: The Sheep-fold (1)
        
        Solution: (related to Problem: The Sheep-fold)
        
        Problem: The Garden Walls (1)
        
        Solution: (related to Problem: The Garden Walls)
        
        Problem: Lady Belinda's Garden (1)
        
        Solution: (related to Problem: Lady Belinda's Garden)
        
        Problem: The Tethered Goat (1)
        
        Solution: (related to Problem: The Tethered Goat)
        
        Problem: The Compasses Puzzle (1)
        
        Solution: (related to Problem: The Compasses Puzzle)
        
        Problem: The Eight Sticks (1)
        
        Solution: (related to Problem: The Eight Sticks)
        
        Problem: Papa's Puzzle (1)
        
        Solution: (related to Problem: Papa's Puzzle)
        
        Problem: A Kite-flying Puzzle (1)
        
        Solution: (related to Problem: A Kite-flying Puzzle)
        
        Problem: How To Make Cisterns (1)
        
        Solution: (related to Problem: How To Make Cisterns)
        
        Problem: The Cone Puzzle (1)
        
        Solution: (related to Problem: The Cone Puzzle)
        
        Problem: Concerning Wheels (1)
        
        Solution: (related to Problem: Concerning Wheels)
        
        Problem: A New Match Puzzle (1)
        
        Solution: (related to Problem: A New Match Puzzle)
        
        Problem: The Six Sheep-pens (1)
        
        Solution: (related to Problem: The Six Sheep-pens)
    - Chapter: Points and Lines Problems (8)
      - Problem: The King And The Castles (1)
        
        Solution: (related to Problem: The King And The Castles)
      - Problem: Cherries And Plums (1)
        
        Solution: (related to Problem: Cherries And Plums)
      - Problem: A Plantation Puzzle (1)
        
        Solution: (related to Problem: A Plantation Puzzle)
      - Problem: The Twenty-one Trees (1)
        
        Solution: (related to Problem: The Twenty-one Trees)
      - Problem: The Ten Coins (1)
        
        Solution: (related to Problem: The Ten Coins)
      - Problem: The Twelve Mince-pies (1)
        
        Solution: (related to Problem: The Twelve Mince-pies)
      - Problem: The Burmese Plantation (1)
        
        Solution: (related to Problem: The Burmese Plantation)
      - Problem: Turks And Russians (1)
        
        Solution: (related to Problem: Turks And Russians)
    - Chapter: Moving Counter Problems (26)
      - Problem: The Six Frogs (2)
        
        Solution: The Six Frogs - Solved (related to Problem: The Six Frogs)
        
        Solution: (related to Problem: The Six Frogs)
      - Problem: The Grasshopper Puzzle (1)
        
        Solution: (related to Problem: The Grasshopper Puzzle)
      - Problem: The Educated Frogs (1)
        
        Solution: (related to Problem: The Educated Frogs)
      - Problem: The Twickenham Puzzle (1)
        
        Solution: (related to Problem: The Twickenham Puzzle)
      - Problem: The Victoria Cross Puzzle (1)
        
        Solution: (related to Problem: The Victoria Cross Puzzle)
      - Problem: The Letter Block Puzzle (1)
        
        Solution: (related to Problem: The Letter Block Puzzle)
      - Problem: A Lodging-house Difficulty (1)
        
        Solution: (related to Problem: A Lodging-house Difficulty)
      - Problem: The Eight Engines (1)
        
        Solution: (related to Problem: The Eight Engines)
      - Problem: A Railway Puzzle (1)
        
        Solution: (related to Problem: A Railway Puzzle)
      - Problem: A Railway Muddle (1)
        
        Solution: (related to Problem: A Railway Muddle)
      - Problem: The Motor-garage Puzzle (1)
        
        Solution: (related to Problem: The Motor-garage Puzzle)
      - Problem: The Ten Prisoners (1)
        
        Solution: (related to Problem: The Ten Prisoners)
      - Problem: Round The Coast (1)
        
        Solution: (related to Problem: Round The Coast)
      - Problem: Central Solitaire (1)
        
        Solution: (related to Problem: Central Solitaire)
      - Problem: The Ten Apples (1)
        
        Solution: (related to Problem: The Ten Apples)
      - Problem: The Twelve Pennies (1)
        
        Solution: (related to Problem: The Twelve Pennies)
      - Problem: The Nine Almonds (1)
        
        Solution: (related to Problem: The Nine Almonds)
      - Problem: Plates And Coins (1)
        
        Solution: (related to Problem: Plates And Coins)
      - Problem: Catching The Mice (1)
        
        Solution: (related to Problem: Catching The Mice)
      - Problem: The Eccentric Cheesemonger (1)
        
        Solution: (related to Problem: The Eccentric Cheesemonger)
      - Problem: The Exchange Puzzle (1)
        
        Solution: (related to Problem: The Exchange Puzzle)
      - Problem: Torpedo Practice (1)
        
        Solution: (related to Problem: Torpedo Practice)
      - Problem: The Hat Puzzle (1)
        
        Solution: (related to Problem: The Hat Puzzle)
      - Problem: Boys And Girls (1)
        
        Solution: (related to Problem: Boys And Girls)
      - Problem: Arranging The Jampots (1)
        
        Solution: (related to Problem: Arranging The Jampots)
    - Chapter: Unicursal and Route Problems (23)
      - Problem: A Juvenile Puzzle (1)
        
        Solution: (related to Problem: A Juvenile Puzzle)
      - Problem: The Union Jack (1)
        
        Solution: (related to Problem: The Union Jack)
      - Problem: The Dissected Circle (1)
        
        Solution: (related to Problem: The Dissected Circle)
      - Problem: The Tube Inspector's Puzzle (1)
        
        Solution: (related to Problem: The Tube Inspector's Puzzle)
      - Problem: Visiting The Towns (1)
        
        Solution: (related to Problem: Visiting The Towns)
      - Problem: The Fifteen Turnings (1)
        
        Solution: (related to Problem: The Fifteen Turnings)
      - Problem: The Fly On The Octahedron (1)
        
        Solution: (related to Problem: The Fly On The Octahedron)
      - Problem: The Icosahedron Puzzle (1)
        
        Solution: (related to Problem: The Icosahedron Puzzle)
      - Problem: Inspecting A Mine (1)
        
        Solution: (related to Problem: Inspecting A Mine)
      - Problem: The Cyclists' Tour (1)
        
        Solution: (related to Problem: The Cyclists' Tour)
      - Problem: The Sailor's Puzzle (1)
        
        Solution: (related to Problem: The Sailor's Puzzle)
      - Problem: The Grand Tour (1)
        
        Solution: (related to Problem: The Grand Tour)
      - Problem: Water, Gas, And Electricity (1)
        
        Solution: (related to Problem: Water, Gas, And Electricity)
      - Problem: A Puzzle For Motorists (1)
        
        Solution: (related to Problem: A Puzzle For Motorists)
      - Problem: A Bank Holiday Puzzle (1)
        
        Solution: (related to Problem: A Bank Holiday Puzzle)
      - Problem: The Motor-car Tour (1)
        
        Solution: (related to Problem: The Motor-car Tour)
      - Problem: The Level Puzzle (1)
        
        Solution: (related to Problem: The Level Puzzle)
      - Problem: The Diamond Puzzle (1)
        
        Solution: (related to Problem: The Diamond Puzzle)
      - Problem: The Deified Puzzle (1)
        
        Solution: (related to Problem: The Deified Puzzle)
      - Problem: The Voters' Puzzle (1)
        
        Solution: (related to Problem: The Voters' Puzzle)
      - Problem: Hannah's Puzzle (1)
        
        Solution: (related to Problem: Hannah's Puzzle)
      - Problem: The Honeycomb Puzzle (1)
        
        Solution: (related to Problem: The Honeycomb Puzzle)
      - Problem: The Monk And The Bridges (1)
        
        Solution: (related to Problem: The Monk And The Bridges)
    - Chapter: Combination and Group Problems (26)
      - Problem: Those Fifteen Sheep (1)
        
        Solution: (related to Problem: Those Fifteen Sheep)
      - Problem: King Arthur's Knights (1)
        
        Solution: (related to Problem: King Arthur's Knights)
      - Problem: The City Luncheons (1)
        
        Solution: (related to Problem: The City Luncheons)
      - Problem: A Puzzle for Card-players (1)
        
        Solution: (related to Problem: A Puzzle for Card-players)
      - Problem: A Tennis Tournament (1)
        
        Solution: (related to Problem: A Tennis Tournament)
      - Problem: The Wrong Hats (1)
        
        Solution: (related to Problem: The Wrong Hats)
      - Problem: The Peal Of Bells (1)
        
        Solution: (related to Problem: The Peal Of Bells)
      - Problem: Three Men In A Boat (1)
        
        Solution: (related to Problem: Three Men In A Boat)
      - Problem: The Glass Balls (1)
        
        Solution: (related to Problem: The Glass Balls)
      - Problem: Fifteen Letter Puzzle (1)
        
        Solution: (related to Problem: Fifteen Letter Puzzle)
      - Problem: The Nine Schoolboys (1)
        
        Solution: (related to Problem: The Nine Schoolboys)
      - Problem: The Round Table (1)
        
        Solution: (related to Problem: The Round Table)
      - Problem: The Mouse-trap Puzzle (1)
        
        Solution: (related to Problem: The Mouse-trap Puzzle)
      - Problem: The Sixteen Sheep (1)
        
        Solution: (related to Problem: The Sixteen Sheep)
      - Problem: The Eight Villas (1)
        
        Solution: (related to Problem: The Eight Villas)
      - Problem: Counter Crosses (1)
        
        Solution: (related to Problem: Counter Crosses)
      - Problem: A Dormitory Puzzle (1)
        
        Solution: (related to Problem: A Dormitory Puzzle)
      - Problem: The Barrels Of Balsam (1)
        
        Solution: (related to Problem: The Barrels Of Balsam)
      - Problem: Building The Tetrahedron (1)
        
        Solution: (related to Problem: Building The Tetrahedron)
      - Problem: Painting A Pyramid (1)
        
        Solution: (related to Problem: Painting A Pyramid)
      - Problem: The Antiquary's Chain (1)
        
        Solution: (related to Problem: The Antiquary's Chain)
      - Problem: The Fifteen Dominoes (1)
        
        Solution: (related to Problem: The Fifteen Dominoes)
      - Problem: The Cross Target (1)
        
        Solution: (related to Problem: The Cross Target)
      - Problem: The Four Postage Stamps (1)
        
        Solution: (related to Problem: The Four Postage Stamps)
      - Problem: Painting The Die (1)
        
        Solution: (related to Problem: Painting The Die)
      - Problem: An Acrostic Puzzle (1)
        
        Solution: (related to Problem: An Acrostic Puzzle)
    - Chapter: Chessboard Problems (75)
      - Section: The Chessboard (7)
        
        Problem: Chequered Board Divisions (1)
        
        Solution: (related to Problem: Chequered Board Divisions)
        
        Problem: Lions And Crowns (1)
        
        Solution: (related to Problem: Lions And Crowns)
        
        Problem: Boards With An Odd Number Of Squares (1)
        
        Solution: (related to Problem: Boards With An Odd Number Of Squares)
        
        Problem: The Grand Lama's Problem (1)
        
        Solution: (related to Problem: The Grand Lama's Problem)
        
        Problem: The Abbot's Window (1)
        
        Solution: (related to Problem: The Abbot's Window)
        
        Problem: The Chinese Chessboard (1)
        
        Solution: (related to Problem: The Chinese Chessboard)
        
        Problem: The Chessboard Sentence (1)
        
        Solution: (related to Problem: The Chessboard Sentence)
      - Section: Statical Chess Puzzles (25)
        
        Problem: The Eight Rooks (1)
        
        Solution: (related to Problem: The Eight Rooks)
        
        Problem: The Four Lions (1)
        
        Solution: (related to Problem: The Four Lions)
        
        Problem: Bishops — unguarded (1)
        
        Solution: (related to Problem: Bishops — unguarded)
        
        Problem: Bishops — guarded (1)
        
        Solution: (related to Problem: Bishops — guarded)
        
        Problem: Bishops in Convocation (1)
        
        Solution: (related to Problem: Bishops in Convocation)
        
        Problem: The Eight Queens (1)
        
        Solution: (related to Problem: The Eight Queens)
        
        Problem: The Eight Stars (1)
        
        Solution: (related to Problem: The Eight Stars)
        
        Problem: A Problem in Mosaics (1)
        
        Solution: (related to Problem: A Problem in Mosaics)
        
        Problem: Under the Veil (1)
        
        Solution: (related to Problem: Under the Veil)
        
        Problem: Bachet's Square (1)
        
        Solution: (related to Problem: Bachet's Square)
        
        Problem: The Thirty-six Letter Blocks (1)
        
        Solution: (related to Problem: The Thirty-six Letter Blocks)
        
        Problem: The Crowded Chessboard (1)
        
        Solution: (related to Problem: The Crowded Chessboard)
        
        Problem: The Coloured Counters (1)
        
        Solution: (related to Problem: The Coloured Counters)
        
        Problem: The Gentle Art Of Stamp-licking (1)
        
        Solution: (related to Problem: The Gentle Art Of Stamp-licking)
        
        Problem: The Forty-nine Counters (1)
        
        Solution: (related to Problem: The Forty-nine Counters)
        
        Problem: The Three Sheep (1)
        
        Solution: (related to Problem: The Three Sheep)
        
        Problem: The Five Dogs Puzzle (1)
        
        Solution: (related to Problem: The Five Dogs Puzzle)
        
        Problem: The Five Crescents of Byzantium (1)
        
        Solution: (related to Problem: The Five Crescents of Byzantium)
        
        Problem: Queens And Bishop Puzzle (1)
        
        Solution: (related to Problem: Queens And Bishop Puzzle)
        
        Problem: The Southern Cross (1)
        
        Solution: (related to Problem: The Southern Cross)
        
        Problem: The Hat-peg Puzzle (1)
        
        Solution: (related to Problem: The Hat-peg Puzzle)
        
        Problem: The Amazons (1)
        
        Solution: (related to Problem: The Amazons)
        
        Problem: A Puzzle With Pawns (1)
        
        Solution: (related to Problem: A Puzzle With Pawns)
        
        Problem: Lion-hunting (1)
        
        Solution: (related to Problem: Lion-hunting)
        
        Problem: The Knight-guards (1)
        
        Solution: (related to Problem: The Knight-guards)
      - Section: The Guarded Chessboard
      - Section: Dynamical Chess Puzzles (26)
        
        Problem: The Rook's Tour (1)
        
        Solution: (related to Problem: The Rook's Tour)
        
        Problem: The Rook's Journey (1)
        
        Solution: (related to Problem: The Rook's Journey)
        
        Problem: The Languishing Maiden (1)
        
        Solution: (related to Problem: The Languishing Maiden)
        
        Problem: A Dungeon Puzzle (1)
        
        Solution: (related to Problem: A Dungeon Puzzle)
        
        Problem: The Lion And The Man (1)
        
        Solution: (related to Problem: The Lion And The Man)
        
        Problem: An Episcopal Visitation (1)
        
        Solution: (related to Problem: An Episcopal Visitation)
        
        Problem: A New Counter Puzzle (1)
        
        Solution: (related to Problem: A New Counter Puzzle)
        
        Problem: A New Bishop's Puzzle (1)
        
        Solution: (related to Problem: A New Bishop's Puzzle)
        
        Problem: The Queen's Tour (1)
        
        Solution: (related to Problem: The Queen's Tour)
        
        Problem: The Star Puzzle (1)
        
        Solution: (related to Problem: The Star Puzzle)
        
        Problem: The Yacht Race (1)
        
        Solution: (related to Problem: The Yacht Race)
        
        Problem: The Scientific Skater (1)
        
        Solution: (related to Problem: The Scientific Skater)
        
        Problem: The Forty-nine Stars (1)
        
        Solution: (related to Problem: The Forty-nine Stars)
        
        Problem: The Queen's Journey (1)
        
        Solution: (related to Problem: The Queen's Journey)
        
        Problem: St. George And The Dragon (1)
        
        Solution: (related to Problem: St. George And The Dragon)
        
        Problem: Farmer Lawrence's Cornfields (1)
        
        Solution: (related to Problem: Farmer Lawrence's Cornfields)
        
        Problem: The Greyhound Puzzle (1)
        
        Solution: (related to Problem: The Greyhound Puzzle)
        
        Problem: The Four Kangaroos (1)
        
        Solution: (related to Problem: The Four Kangaroos)
        
        Problem: The Board In Compartments (1)
        
        Solution: (related to Problem: The Board In Compartments)
        
        Problem: The Four Knights' Tours (1)
        
        Solution: (related to Problem: The Four Knights' Tours)
        
        Problem: The Cubic Knight's Tour (1)
        
        Solution: (related to Problem: The Cubic Knight's Tour)
        
        Problem: The Four Frogs (1)
        
        Solution: (related to Problem: The Four Frogs)
        
        Problem: The Mandarin's Puzzle (1)
        
        Solution: (related to Problem: The Mandarin's Puzzle)
        
        Problem: Exercise For Prisoners (1)
        
        Solution: (related to Problem: Exercise For Prisoners)
        
        Problem: The Kennel Puzzle (1)
        
        Solution: (related to Problem: The Kennel Puzzle)
        
        Problem: The Two Pawns (1)
        
        Solution: (related to Problem: The Two Pawns)
      - Section: Various Chess Puzzles (16)
        
        Problem: Setting The Board (1)
        
        Solution: (related to Problem: Setting The Board)
        
        Problem: Counting The Rectangles (1)
        
        Solution: (related to Problem: Counting The Rectangles)
        
        Problem: The Rookery (1)
        
        Solution: (related to Problem: The Rookery)
        
        Problem: Stalemate (1)
        
        Solution: (related to Problem: Stalemate)
        
        Problem: The Forsaken King (1)
        
        Solution: (related to Problem: The Forsaken King)
        
        Problem: The Crusader (1)
        
        Solution: (related to Problem: The Crusader)
        
        Problem: Immovable Pawns (1)
        
        Solution: (related to Problem: Immovable Pawns)
        
        Problem: Thirty-six Mates (1)
        
        Solution: (related to Problem: Thirty-six Mates)
        
        Problem: An Amazing Dilemma (1)
        
        Solution: (related to Problem: An Amazing Dilemma)
        
        Problem: Checkmate! (1)
        
        Solution: (related to Problem: Checkmate!)
        
        Problem: Queer Chess (1)
        
        Solution: (related to Problem: Queer Chess)
        
        Problem: Ancient Chinese Puzzle (1)
        
        Solution: (related to Problem: Ancient Chinese Puzzle)
        
        Problem: The Six Pawns (1)
        
        Solution: (related to Problem: The Six Pawns)
        
        Problem: Counter Solitaire (1)
        
        Solution: (related to Problem: Counter Solitaire)
        
        Problem: Chessboard Solitaire (1)
        
        Solution: (related to Problem: Chessboard Solitaire)
        
        Problem: The Monstrosity (1)
        
        Solution: (related to Problem: The Monstrosity)
    - Chapter: Measuring, Weighing, and Packing Problems (11)
      - Problem: The Wassail Bowl (1)
        
        Solution: (related to Problem: The Wassail Bowl)
      - Problem: The Doctor's Query (1)
        
        Solution: (related to Problem: The Doctor's Query)
      - Problem: The Barrel Puzzle (1)
        
        Solution: (related to Problem: The Barrel Puzzle)
      - Problem: New Measuring Puzzle (1)
        
        Solution: (related to Problem: New Measuring Puzzle)
      - Problem: The Honest Dairyman (1)
        
        Solution: (related to Problem: The Honest Dairyman)
      - Problem: Wine And Water (1)
        
        Solution: (related to Problem: Wine And Water)
      - Problem: The Keg Of Wine (1)
        
        Solution: (related to Problem: The Keg Of Wine)
      - Problem: Mixing The Tea (1)
        
        Solution: (related to Problem: Mixing The Tea)
      - Problem: A Packing Puzzle (1)
        
        Solution: (related to Problem: A Packing Puzzle)
      - Problem: Gold Packing In Russia (1)
        
        Solution: (related to Problem: Gold Packing In Russia)
      - Problem: The Barrels Of Honey (1)
        
        Solution: (related to Problem: The Barrels Of Honey)
    - Chapter: Crossing River Problems (5)
      - Problem: Crossing The Stream (1)
        
        Solution: (related to Problem: Crossing The Stream)
      - Problem: Crossing The River Axe (1)
        
        Solution: (related to Problem: Crossing The River Axe)
      - Problem: Five Jealous Husbands (1)
        
        Solution: (related to Problem: Five Jealous Husbands)
      - Problem: The Four Elopements (1)
        
        Solution: (related to Problem: The Four Elopements)
      - Problem: Stealing The Castle Treasure (1)
        
        Solution: (related to Problem: Stealing The Castle Treasure)
    - Chapter: Problems Concerning Games (14)
      - Problem: Dominoes In Progression (1)
        
        Solution: (related to Problem: Dominoes In Progression)
      - Problem: The Five Dominoes (1)
        
        Solution: (related to Problem: The Five Dominoes)
      - Problem: The Domino Frame Puzzle (1)
        
        Solution: (related to Problem: The Domino Frame Puzzle)
      - Problem: The Card Frame Puzzle (1)
        
        Solution: (related to Problem: The Card Frame Puzzle)
      - Problem: The Cross Of Cards (1)
        
        Solution: (related to Problem: The Cross Of Cards)
      - Problem: The "T" Card Puzzle (1)
        
        Solution: (related to Problem: The "T" Card Puzzle)
      - Problem: Card Triangles (1)
        
        Solution: (related to Problem: Card Triangles)
      - Problem: "Strand" Patience (1)
        
        Solution: (related to Problem: "Strand" Patience)
      - Problem: A Trick With Dice (1)
        
        Solution: (related to Problem: A Trick With Dice)
      - Problem: The Village Cricket Match (1)
        
        Solution: (related to Problem: The Village Cricket Match)
      - Problem: Slow Cricket (1)
        
        Solution: (related to Problem: Slow Cricket)
      - Problem: The Football Players (1)
        
        Solution: (related to Problem: The Football Players)
      - Problem: The Horse-race Puzzle (1)
        
        Solution: (related to Problem: The Horse-race Puzzle)
      - Problem: The Motor-car Race (1)
        
        Solution: (related to Problem: The Motor-car Race)
    - Chapter: Puzzle Games (7)
      - Problem: The Pebble Game (1)
        
        Solution: (related to Problem: The Pebble Game)
      - Problem: The Two Rooks (1)
        
        Solution: (related to Problem: The Two Rooks)
      - Problem: Puss In The Corner (1)
        
        Solution: (related to Problem: Puss In The Corner)
      - Problem: A War Puzzle Game (1)
        
        Solution: (related to Problem: A War Puzzle Game)
      - Problem: A Match Mystery (1)
        
        Solution: (related to Problem: A Match Mystery)
      - Problem: The Montenegrin Dice Game (1)
        
        Solution: (related to Problem: The Montenegrin Dice Game)
      - Problem: The Cigar Puzzle (1)
        
        Solution: (related to Problem: The Cigar Puzzle)
    - Chapter: Magic Square Problems (14)
      - Problem: The Troublesome Eight (1)
        
        Solution: (related to Problem: The Troublesome Eight)
      - Problem: The Magic Strips (1)
        
        Solution: (related to Problem: The Magic Strips)
      - Problem: Eight Jolly Gaol Birds (1)
        
        Solution: (related to Problem: Eight Jolly Gaol Birds)
      - Problem: Nine Jolly Gaol Birds (1)
        
        Solution: (related to Problem: Nine Jolly Gaol Birds)
      - Problem: The Spanish Dungeon (1)
        
        Solution: (related to Problem: The Spanish Dungeon)
      - Problem: The Siberian Dungeons (1)
        
        Solution: (related to Problem: The Siberian Dungeons)
      - Problem: Card Magic Squares (1)
        
        Solution: (related to Problem: Card Magic Squares)
      - Problem: The Eighteen Dominoes (1)
        
        Solution: (related to Problem: The Eighteen Dominoes)
      - Section: Subtracting, Multiplying, and Dividing Magics (2)
        
        Problem: Two New Magic Squares (1)
        
        Solution: (related to Problem: Two New Magic Squares)
        
        Problem: Magic Squares Of Two Degrees (1)
        
        Solution: (related to Problem: Magic Squares Of Two Degrees)
      - Section: Magic Squares of Primes (4)
        
        Problem: The Baskets Of Plums (1)
        
        Solution: (related to Problem: The Baskets Of Plums)
        
        Problem: The Mandarin's "T" Puzzle (1)
        
        Solution: (related to Problem: The Mandarin's "T" Puzzle)
        
        Problem: A Magic Square Of Composites (1)
        
        Solution: (related to Problem: A Magic Square Of Composites)
        
        Problem: The Magic Knight's Tour (1)
        
        Solution: (related to Problem: The Magic Knight's Tour)
    - Chapter: Mazes and How to Thread Them
    - Chapter: The Paradox Party (1)
      - Problem: A Chessboard Fallacy (1)
        
        Solution: (related to Problem: A Chessboard Fallacy)
    - Chapter: Unclassified Problems (17)
      - Problem: Who Was First? (1)
        
        Solution: (related to Problem: Who Was First?)
      - Problem: A Wonderful Village (1)
        
        Solution: (related to Problem: A Wonderful Village)
      - Problem: A Calendar Puzzle (1)
        
        Solution: (related to Problem: A Calendar Puzzle)
      - Problem: The Tiring Irons (1)
        
        Solution: (related to Problem: The Tiring Irons)
      - Problem: Such A Getting Upstairs (1)
        
        Solution: (related to Problem: Such A Getting Upstairs)
      - Problem: The Five Pennies (1)
        
        Solution: (related to Problem: The Five Pennies)
      - Problem: The Industrious Bookworm (1)
        
        Solution: (related to Problem: The Industrious Bookworm)
      - Problem: A Chain Puzzle (1)
        
        Solution: (related to Problem: A Chain Puzzle)
      - Problem: The Sabbath Puzzle (1)
        
        Solution: (related to Problem: The Sabbath Puzzle)
      - Problem: The Ruby Brooch (1)
        
        Solution: (related to Problem: The Ruby Brooch)
      - Problem: The Dovetailed Block (1)
        
        Solution: (related to Problem: The Dovetailed Block)
      - Problem: Jack And The Beanstalk (1)
        
        Solution: (related to Problem: Jack And The Beanstalk)
      - Problem: The Hymn-board Poser (1)
        
        Solution: (related to Problem: The Hymn-board Poser)
      - Problem: Pheasant-shooting (1)
        
        Solution: (related to Problem: Pheasant-shooting)
      - Problem: The Gardener And The Cook (1)
        
        Solution: (related to Problem: The Gardener And The Cook)
      - Problem: Placing Halfpennies (1)
        
        Solution: (related to Problem: Placing Halfpennies)
      - Problem: Find The Man's Wife (1)
        
        Solution: (related to Problem: Find The Man's Wife)
History (3154)
- Epoch: Prehistory (2)
  - Topic: to 22 000 BC: Ishango Bone
  - Topic: about 4000 BC: Clay Accounting Tokens
- Epoch: Ancient World (from 4000 BC to 1 BC) (62)
  - Person: Ahmes
  - Person: Baudhayana
  - Person: Manava
  - Person: Thales Of Miletus
  - Person: Anaximander Of Miletus
  - Person: Apastamba
  - Person: Pythagoras Of Samos
  - Person: Panini
  - Person: Anaxagoras Of Clazomenae
  - Person: Empedocles Of Acragas
  - Person: Oenopides Of Chios
  - Person: Zeno Of Elea
  - Person: Antiphon
  - Person: Leucippus Of Miletus
  - Person: Hippocrates Of Chios
  - Person: Theodorus Of Cyrene
  - Person: Democritus Of Abdera
  - Person: Hippias Of Elis
  - Person: Bryson Of Heraclea
  - Person: Archytas Of Tarentum
  - Person: Plato
  - Person: Theaetetus Of Athens
  - Person: Eudoxus Of Cnidus
  - Person: Gan De
  - Person: Thymaridas Of Paros
  - Person: Xenocrates Of Chalcedon
  - Person: Dinostratus
  - Person: Heraclides Of Pontus
  - Person: Aristotle
  - Person: Menaechmus
  - Person: Aristaeus the Elder
  - Person: Callippus Of Cyzicus
  - Person: Autolycus Of Pitane
  - Person: Eudemus Of Rhodes
  - Person: Euclid Of Alexandria
  - Person: Aristarchus Of Samos
  - Person: Archimedes Of Syracuse
  - Person: Chrysippus Of Soli
  - Person: Conon Of Samos
  - Person: Nicomedes
  - Person: Philon Of Byzantium
  - Person: Eratosthenes Of Cyrene
  - Person: Apollonius Of Perga
  - Person: Dionysodorus
  - Person: Diocles Of Carystus
  - Person: Katyayana
  - Person: Zenodorus
  - Person: Hipparchus Of Rhodes
  - Person: Hypsicles Of Alexandria
  - Person: Perseus
  - Person: Theodosius Of Bithynia
  - Person: Zeno Of Sidon
  - Person: Posidonius Of Rhodes
  - Person: Hong, Luoxia
  - Person: Pollio, Marcus Vitruvius
  - Person: Geminus
  - Person: Hippasus of Metapontum
  - Topic: Egyptian Mathematics
  - Topic: Babylonian Mathematics
  - Topic: Ancient Chinese Mathematics
  - Topic: Indian Mathematics
  - Topic: Sumerian Mathematics
- Epoch: Late Ancient World (from 1 AD to 499 AD) (32)
  - Person: Cleomedes
  - Person: Heron Of Alexandria
  - Person: Nicomachus Of Gerasa
  - Person: Menelaus Of Alexandria
  - Person: Theon Of Smyrna
  - Person: Heng, Zhang
  - Person: Ptolemy, Claudius
  - Person: Yavanesvara
  - Person: Hong (2), Liu
  - Person: Yue, Xu
  - Person: Diophantus Of Alexandria
  - Person: Hui, Liu
  - Person: Malchus, Porphyry
  - Person: Sporus Of Nicaea
  - Person: Pappus Of Alexandria
  - Person: Pandrosion
  - Person: Serenus
  - Person: Theon Of Alexandria
  - Person: Hypatia Of Alexandria
  - Person: Zi, Sun
  - Person: Yang, Xiahou
  - Person: Diadochus, Proclus
  - Person: Domninus Of Larissa
  - Person: Chongzhi, Zu
  - Person: Qiujian, Zhang
  - Person: Marinus Of Neapolis
  - Person: Geng, Zu
  - Person: Anthemius Of Tralles
  - Person: Aryabhata The Elder
  - Person: Boethius, Anicius Manlius Severinus
  - Person: Eutocius Of Ascalon
  - Person: Simplicius
- Epoch: Early Middle Ages (47)
  - Person: Yativrsabha
  - Person: Varahamihira
  - Person: Xiaotong, Wang
  - Person: Brahmagupta
  - Person: Bhaskara
  - Person: Chunfeng, Li
  - Person: Lalla
  - Person: Alcuin
  - Person: Al-Khwarizmi, Abu Ja&amp;#x27;far Muhammad ibn Musa
  - Person: Al-Jawhari, al-Abbas ibn Said
  - Person: Musa, Jafar Muhammad Banu
  - Person: Banu Musa Brothers
  - Person: Govindasvami
  - Person: Mahāvīra
  - Person: Al-Kindi, Abu Yusuf Yaqub ibn Ishaq al-Sabbah
  - Person: Musa (2), Ahmad Banu
  - Person: Hunayn Ibn Ishaq
  - Person: Al-Hasan Banu Musa
  - Person: Al-Mahani, Abu Abd Allah Muhammad ibn Isa
  - Person: Prthudakasvami
  - Person: Ibn Yusuf, Ahmed
  - Person: Thabit Ibn Qurra, Al-Sabi
  - Person: Sankara, Narayana
  - Person: Abu Kamil, ibn Aslam Shuja
  - Person: Al-Battani, Abu Abdallah Mohammad ibn Jabir
  - Person: Sridhara
  - Person: Al-Nayrizi, Abu&amp;#x27;l Abbas al-Fadl ibn Hatim
  - Person: Sinan, Abu ibn Thabit ibn Qurra
  - Person: Al-Khazin, Abu Jafar Muhammad ibn Al-Hasan
  - Person: Ibrahim Ibn Sinan
  - Person: Al-Uqlidisi, Abu&amp;#x27;l Hasan Ahmad ibn Ibrahim
  - Person: Aryabhata The Ii
  - Person: Mohammad Abu&amp;#x27;L-Wafa, Al-Buzjani
  - Person: Al-Khujandi, Abu Mahmud Hamid ibn al-Khidr
  - Person: Al-Quhi, Abu Sahl Waijan ibn Rustam
  - Person: Vijayanandi
  - Person: Al-Sijzi, Abu Said Ahmad ibn Muhammad
  - Person: Gerbert Of Aurillac
  - Person: Ibn Yunus, Abu&amp;#x27;l-Hasan Ali Ibn Abd al-Rahman
  - Person: Al-Karaji, Abu Bekr ibn Muhammad ibn Al-Husayn
  - Person: Alhazen, Abu Ali Al-Hasan ibn Al-Haytham
  - Person: Mansur, Abu Nasr ibn Ali ibn Iraq
  - Person: Kushyar Ibn Labban
  - Person: Al-Biruni, Abu Arrayhan Muhammad ibn Ahmad
  - Person: Al-Baghdadi, Abu Mansur ibn Tahir
  - Person: Avicenna, Abu Ali al-Husain ibn Abdallah ibn Sina
  - Person: Al-Jayyani, Abu Abd Allah Muhammad ibn Muadh
- Epoch: 11th Century (12)
  - Person: Al-Nasawi
  - Person: Xian, Jia
  - Person: Hermann Of Reichenau
  - Person: Sripati
  - Person: Al-Zarqali
  - Person: Kua, Shen
  - Person: Khayyam, Omar
  - Person: Brahmadeva
  - Person: Ha-Nasi, Abraham bar Hiyya
  - Person: Adelard Of Bath
  - Person: Hemchandra, Acharya
  - Person: Abraham Ibn Ezra, ben Meir
- Epoch: 12th Century (11)
  - Person: Jabir Ibn Aflah
  - Person: Bhaskara (2)
  - Person: Gherard Of Cremona
  - Person: Al-Samawal
  - Person: Al-Tusi, Sharaf al-Din
  - Person: Reinher Of Paderborn
  - Person: Grosseteste, Robert
  - Person: Fibonacci, Leonardo Pisano
  - Person: Scot, Michael
  - Person: Zhi, Li
  - Person: De Sacrobosco, Johannes
- Epoch: 13th Century (19)
  - Person: Magnus, Saint Albertus
  - Person: Al-Tusi (2), Nasir al-Din
  - Person: Jiushao, Qin
  - Person: Bacon, Roger
  - Person: Al-Maghribi, Muhyi l&amp;#x27;din
  - Person: Campanus Of Novara
  - Person: Nemorarius, Jordanus
  - Person: Shoujing, Guo
  - Person: Llull, Ramon
  - Person: Ibn Tibbon, Jacob ben Machir
  - Person: Hui (2), Yang
  - Person: Al-Samarqandi, Shams al-Din
  - Person: Al-Banna
  - Person: Al-Farisi
  - Person: Shijie, Zhu
  - Person: Youqin, Zhao
  - Person: Gerson, Rabbi Levi Ben
  - Person: William Of Ockham
  - Person: Bradwardine, Thomas
- Epoch: 14th Century (11)
  - Person: Albert Of Saxony
  - Person: Al-Khalili
  - Person: Oresme, Nicole
  - Person: Suri, Mahendra
  - Person: Pandit, Narayana
  - Person: Madhava Of Sangamagramma
  - Person: Al-Rumi, Qadi Zada
  - Person: Paramesvara
  - Person: Brunelleschi, Filippo
  - Person: Al-Kashi
  - Person: Beg, Ulugh
- Epoch: 15th Century (30)
  - Person: Al-Umawi
  - Person: Nicholas Of Cusa
  - Person: Qushji, Ali
  - Person: Alberti, Leone Battista
  - Person: Al Qalasadi, Abu&amp;#x27;l Hasan ibn Ali
  - Person: Della Francesca, Piero
  - Person: Peurbach, Georg
  - Person: Borgi, Piero
  - Person: Regiomontanus, Johann Müller
  - Person: Somayaji, Nilakantha
  - Person: Chuquet, Nicolas
  - Person: Pacioli, Luca
  - Person: Da Vinci, Leonardo
  - Person: Widman, Johannes
  - Person: Ferro, del Scipione
  - Person: Werner, Johann
  - Person: Maior, John
  - Person: Roche, Estienne de La
  - Person: De Bouvelles, Charles
  - Person: Dürer, Albrecht
  - Person: Copernicus, Nicolaus
  - Person: Tunstall, Cuthbert
  - Person: De Ortega, Juan
  - Person: Lax, Gaspar
  - Person: Stifel, Michael
  - Person: Ries, Adam
  - Person: Fine, Oronce
  - Person: Maurolico, Francesco
  - Person: Apianus, Petrus
  - Person: Rudolff, Christoff
- Epoch: 16th Century (85)
  - Person: Jyesthadeva
  - Person: Tartaglia, Niccolò
  - Person: Cardan, Girolamo
  - Person: Salaciense, Pedro Nunes
  - Person: Commandino, Frederico
  - Person: Frisius, Regnier Gemma
  - Person: Recorde, Robert
  - Person: Reinhold, Erasmus
  - Person: Mercator, Gerard
  - Person: Rheticus, Georg Joachim von Lauchen
  - Person: Ramus, Peter
  - Person: Ferrari, Lodovico
  - Person: Bombelli, Rafael
  - Person: Dee, John
  - Person: Benedetti, Giovanni Battista
  - Person: Dawei, Cheng
  - Person: Della Porta, Giambattista
  - Person: Danti, Egnatio Pellegrino Rainaldi
  - Person: Barocius, Franciscus
  - Person: Clavius, Christopher
  - Person: Allen, Thomas
  - Person: Van Ceulen, Ludolph
  - Person: Viète, François
  - Person: Monte, del Guidobaldo Marchese
  - Person: Brahe, Tycho
  - Person: Digges, Thomas
  - Person: Wittich, Paul
  - Person: Al-Amili, Baha&amp;#x27; al-Din
  - Person: Bruno, Giordano
  - Person: Cataldi, Pietro Antonio
  - Person: Stevin, Simon
  - Person: Savile, Sir Henry
  - Person: Mästlin, Michael
  - Person: Napier, John
  - Person: Bürgi, Jost
  - Person: Ricci, Matteo
  - Person: Valerio, Luca
  - Person: Delgado, João
  - Person: Lavanha, João Baptista
  - Person: Magini, Giovanni Antonio
  - Person: Harriot, Thomas
  - Person: Briggs, Henry
  - Person: Fincke, Thomas
  - Person: Van Lansberge, Johan Philip
  - Person: Liddel, Duncan
  - Person: Pitiscus, Bartholomeo
  - Person: Van Roomen, Adriaan
  - Person: Guangqi, Xu
  - Person: Galilei, Galileo
  - Person: Biancani, Giuseppe
  - Person: Ghetaldi, Marino
  - Person: Kepler, Johannes
  - Person: Mayr, Simon
  - Person: Scheiner, Christoph
  - Person: Oughtred, William
  - Person: Guldin, Paul
  - Person: Castelli, Benedetto Antonio
  - Person: Faulhaber, Johann
  - Person: Hérigone, Pierre
  - Person: Snell, Willebrord van Royen
  - Person: Bachet, Claude Gaspar
  - Person: Gunter, Edmund
  - Person: Morin, Jean-Baptiste
  - Person: Gregory Of Saint-Vincent
  - Person: Vernier, Pierre
  - Person: Mydorge, Claude
  - Person: Turner, Peter
  - Person: Jungius, Joachim
  - Person: Bramer, Benjamin
  - Person: Hobbes, Thomas
  - Person: Mersenne, Marin
  - Person: Pascal, Étienne
  - Person: Beaugrand, Jean
  - Person: Desargues, Girard
  - Person: Gassendi, Pierre
  - Person: Schickard, Wilhelm
  - Person: Petit, Pierre
  - Person: Girard, Albert
  - Person: Descartes, René
  - Person: Golius, Jacobus
  - Person: Gellibrand, Henry
  - Person: Faille, Jean Charles de La
  - Person: Cavalieri, Bonaventura Francesco
  - Person: Hardy, Claude
  - Person: Riccioli, Giovanni Battista
- Epoch: 17th Century (145)
  - Person: De Carcavi, Pierre
  - Person: Delamain, Richard
  - Person: Vlacq, Adriaan
  - Person: De Beaune, Florimond
  - Person: De Fermat, Pierre
  - Person: De Billy, Jacques
  - Person: Greaves, John
  - Person: De Roberval, Gilles Personne
  - Person: De Bessy, Bernard Frénicle
  - Person: Boulliau, Ismael
  - Person: Caramuel, Juan
  - Person: Cunitz, Maria
  - Person: Borelli, Giovanni Alfonso
  - Person: Fabri, Honoré
  - Person: Torricelli, Evangelista
  - Person: Tenneur, Jacques Alexandre Le
  - Person: Stampioen, Jan Jansz de Jonge
  - Person: Hevelius, Johannes
  - Person: Pell, John
  - Person: Arnauld, Antoine
  - Person: Tacquet, Andrea
  - Person: More, Henry
  - Person: Wilkins, John
  - Person: Van Schooten, Frans
  - Person: Kamalakara
  - Person: Wallis, John
  - Person: Ward, Seth
  - Person: Grimaldi, Francesco Maria
  - Person: Horrocks, Jeremiah
  - Person: Mouton, Gabriel
  - Person: Mylon, Claude
  - Person: Ricci (2), Michelangelo
  - Person: Brouncker, William
  - Person: Mercator (2), Nicolaus
  - Person: Picard, Jean
  - Person: Dechales, Claude François Milliet
  - Person: Auzout, Adrien
  - Person: Rahn, Johann Heinrich
  - Person: De Sluze, René François Walter
  - Person: Viviani, Vincenzo
  - Person: Angeli, Stefano degli
  - Person: Pascal (2), Blaise
  - Person: Verbiest, Ferdinand
  - Person: Collins, John
  - Person: Guarini, Guarino
  - Person: Apáczai, János
  - Person: Bartholin, Erasmus
  - Person: Cassini, Giovanni Domenico
  - Person: De Witt, Jan
  - Person: Mengoli, Pietro
  - Person: Boyle, Robert
  - Person: Moore, Jonas
  - Person: Hudde, Johann van Waveren
  - Person: Huygens, Christiaan
  - Person: Barrow, Isaac
  - Person: Richer, Jean
  - Person: Cocker, Edward
  - Person: Wren, Sir Christopher
  - Person: Coley, Henry
  - Person: Wending, Mei
  - Person: Van Heuraet, Hendrik
  - Person: Hooke, Robert
  - Person: Heins, Valentin
  - Person: Neile, William
  - Person: Gregory (2), James
  - Person: Malebranche, Nicolas
  - Person: Hire, Philippe de La
  - Person: Lamy, Bernard
  - Person: Mohr, Georg
  - Person: Ozanam, Jacques
  - Person: Seki, Takakazu Shinsuke
  - Person: Newton, Sir Isaac
  - Person: Meissner, Heinrich
  - Person: Gongora, Siguenza y
  - Person: Seok-Jeong, Choi
  - Person: Flamsteed, John
  - Person: Von Leibniz, Gottfried Wilhelm
  - Person: Ceva, Giovanni
  - Person: Koopman, Elisabetha
  - Person: Papin, Denis
  - Person: Aldrich, Henry
  - Person: Ceva (2), Tommaso
  - Person: Von Tschirnhaus, Ehrenfried Walter
  - Person: Fèvre, Jean Le
  - Person: Rolle, Michel
  - Person: Sharp, Abraham
  - Person: Varignon, Pierre
  - Person: Bernoulli, Jacob
  - Person: Halley, Edmond
  - Person: Molyneux, William
  - Person: Reyneau, Charles René
  - Person: De Fontenelle, Bernard le Bovier
  - Person: Gregory (3), David
  - Person: Saurin, Joseph
  - Person: De Lagny, Thomas Fantet
  - Person: De L&amp;#x27;Hôpital, Guillaume
  - Person: Carré, Louis
  - Person: Craig, John
  - Person: Katahiro, Takebe
  - Person: Arbuthnot, John
  - Person: Bernoulli (2), Johann
  - Person: De Moivre, Abraham
  - Person: Saccheri, Giovanni Girolamo
  - Person: Whiston, William
  - Person: Raphson, Joseph
  - Person: Magnitsky, Leonty Filippovich
  - Person: Kirch, Maria Margarethe Winckelmann
  - Person: Grandi, Luigi Guido
  - Person: Keill, John
  - Person: Manfredi, Eustachio
  - Person: Clarke, Samuel
  - Person: Jones, William
  - Person: Riccati, Jacopo Francesco
  - Person: Cassini (2), Jacques
  - Person: Doppelmayr, Johann Gabriel
  - Person: De Molières, Joseph Privat
  - Person: Hayes, Charles
  - Person: Hermann, Jakob
  - Person: De Montmort, Pierre Rémond
  - Person: Colson, John
  - Person: Machin, John
  - Person: Manfredi (2), Gabriele
  - Person: Juecheng, Mei
  - Person: Cotes, Roger
  - Person: Fagnano, Giulio
  - Person: Hadley, John
  - Person: Saunderson, Nicholas
  - Person: Poleni, Giovanni
  - Person: Berkeley, George
  - Person: Taylor, Brook
  - Person: Bernoulli (3), Nicolaus
  - Person: Simson, Robert
  - Person: Castel, Louis Bertrand
  - Person: &amp;#x27;sGravesande, Willem Jacob
  - Person: Molyneux (2), Samuel
  - Person: Goldbach, Christian
  - Person: Jagannatha
  - Person: Stirling, James
  - Person: Bradley, James
  - Person: Bernoulli (4), Nicolaus
  - Person: Rampinelli, Ramiro
  - Person: Bouguer, Pierre
  - Person: Maclaurin, Colin
  - Person: De Maupertuis, Pierre Louis Moreau
  - Person: Camus, Charles Étienne Louis
- Epoch: 18th Century (252)
  - Person: Paman, Roger
  - Person: Bernoulli (5), Daniel
  - Person: Bliss, Nathaniel
  - Person: Braikenridge, William
  - Person: Stone, Edmund
  - Person: Celsius, Anders
  - Person: Emerson, William
  - Person: Condamine, Charles Marie de La
  - Person: Bayes, Thomas
  - Person: Calandrini, Jean-Louis
  - Person: Deparcieux, Antoine
  - Person: Hallerstein, Augustin
  - Person: Castillon, Johann Francesco Melchiore Salvemini
  - Person: Cramer, Gabriel
  - Person: Bertins, Alexis Fontaine des
  - Person: Von Segner, Johann Andreas
  - Person: Du Châtelet, Émilie
  - Person: Franklin, Benjamin
  - Person: Robins, Benjamin
  - Person: De Buffon, Georges Louis Leclerc Comte
  - Person: Euler, Leonhard
  - Person: Riccati (2), Vincenzo
  - Person: Riccati (3), Giordano
  - Person: Bernoulli (6), Johann
  - Person: Fuller, Thomas
  - Person: Simpson, Thomas
  - Person: Bassi, Laura Maria Catarina
  - Person: Boscovich, Ruggero Giuseppe
  - Person: Jacquier, François
  - Person: König, Samuel
  - Person: Clairaut, Alexis Claude
  - Person: De Lacaille, Nicolas-Louis
  - Person: Juan, Jorge
  - Person: De Thury, César-François Cassini
  - Person: Wilson, Alexander
  - Person: Fagnano (2), Giovanni
  - Person: D&amp;#x27;Alembert, Jean
  - Person: Stewart, Matthew
  - Person: Agnesi, Maria Gaëtana
  - Person: Hatvani, István
  - Person: Kästner, Abraham
  - Person: Landen, John
  - Person: Casamayor, María Andresa
  - Person: Lepaute, Nicole-Reine Etable de Labrière
  - Person: Mayer, Tobias
  - Person: Price, Richard
  - Person: Aepinus, Franz Maria Ulrich Theodosius
  - Person: D&amp;#x27;Arcy, Patrick
  - Person: Montucla, Jean Étienne
  - Person: Hutton, James
  - Person: Frisi, Paolo
  - Person: Lambert, Johann Heinrich
  - Person: De Bougainville, Louis Antoine
  - Person: Bézout, Étienne
  - Person: Bossut, Charles
  - Person: Banneker, Benjamin
  - Person: Malfatti, Gian Francesco
  - Person: Maseres, Francis
  - Person: Ajima, Chokuyen Naonobu
  - Person: Karsten, Wenceslaus Johann Gustav
  - Person: De Lalande, Joseph-Jérôme Lefrançais
  - Person: Maskelyne, Nevil
  - Person: Mutis, José
  - Person: Rittenhouse, David
  - Person: De Borda, Jean Charles
  - Person: Du Séjour, Achille Pierre Dionis
  - Person: Da Rocha, Monteiro
  - Person: Lorgna, Antonio Maria
  - Person: Ramsden, Jesse
  - Person: Vandermonde, Alexandre-Théophile
  - Person: Bring, Erland Samuel
  - Person: De Coulomb, Charles Augustin
  - Person: Lagrange, Joseph-Louis
  - Person: Tetens, Johannes Nikolaus
  - Person: Waring, Edward
  - Person: Hutton (2), Charles
  - Person: Vilant, Nicolas
  - Person: Herschel, William
  - Person: Klügel, Georg
  - Person: Giannini, Pietro
  - Person: Lexell, Anders Johan
  - Person: Hindenburg, Carl Friedrich
  - Person: Wilson (2), John
  - Person: De Condorcet, Marquis
  - Person: Bernoulli (7), Johann
  - Person: Da Cunha, José Anastácio
  - Person: Méchain, Pierre
  - Person: Atwood, George
  - Person: Buée, Adrien Quentin
  - Person: Wessel, Caspar
  - Person: Monge, Gaspard
  - Person: Piazzi, Giuseppe
  - Person: Trail, William
  - Person: Yasuaki, Aida
  - Person: Cassini (3), Dominique
  - Person: Playfair, John
  - Person: De Tinseau, D&amp;#x27;Amondans Charles
  - Person: Delambre, Jean Baptiste Joseph
  - Person: Hellins, John
  - Person: Laplace, Pierre-Simon
  - Person: Vince, Samuel
  - Person: Glenie, James
  - Person: Herschel (2), Caroline
  - Person: Lhuilier, Simon Antoine Jean
  - Person: Mascheroni, Lorenzo
  - Person: Morgan, William
  - Person: Bonnycastle, John
  - Person: Legendre, Adrien-Marie
  - Person: Carnot, Lazare Nicolas Marguérite
  - Person: Fergola, Nicola
  - Person: Stewart (2), Dugald
  - Person: Troughton, Edward
  - Person: Canard, Nicolas-François
  - Person: Von Vega, Georg Freiherr
  - Person: De Prony, Gaspard Clair François Marie Riche
  - Person: Fuss, Nicolaus
  - Person: Parseval, Marc-Antoine
  - Person: Gerstner, František Josef
  - Person: West, John
  - Person: Arbogast, Louis François Antoine
  - Person: Bernoulli (8), Jacob
  - Person: Paoli, Pietro
  - Person: Kramp, Christian
  - Person: De Boislaurent, Ferdinand François Désiré Budan
  - Person: Yuan, Ruan
  - Person: Girard (2), Pierre-Simon
  - Person: Ivory, James
  - Person: Lacroix, Sylvestre François
  - Person: Osipovsky, Timofei Fedorovic
  - Person: Pfaff, Johann Friedrich
  - Person: Ruffini, Paolo
  - Person: Brinkley, John Mortimer
  - Person: Farey, John
  - Person: Leslie, John
  - Person: Bouvard, Alexis
  - Person: Argand, Jean Robert
  - Person: Fourier, Jean Baptiste Joseph
  - Person: Français, François
  - Person: Harlay, Marie-Jeanne Amélie
  - Person: Rui, Li
  - Person: Marescotti, Pietro Abbati
  - Person: Servois, François Joseph
  - Person: Wallace, William
  - Person: Bartels, Johann Christian Martin
  - Person: Giordano, Annibale
  - Person: Hachette, Jean Nicolas Pierre
  - Person: Puissant, Louis
  - Person: Gergonne, Joseph Diaz
  - Person: Reynaud, Antoine-André-Louis
  - Person: Haldane, Robert
  - Person: Lloyd, Bartholomew
  - Person: Bowditch, Nathaniel
  - Person: Burckhardt, Johann Karl
  - Person: Francoeur, Louis Benjamin
  - Person: Woodhouse, Robert
  - Person: Young, Thomas
  - Person: Biot, Jean Baptiste
  - Person: Gregory (4), Olinthus
  - Person: Mollweide, Karl Brandan
  - Person: Adrain, Robert
  - Person: Ampère, André-Marie
  - Person: Bolyai, Farkas
  - Person: Français (2), Jacques
  - Person: Malus, Étienne Louis
  - Person: Barlow, Peter
  - Person: Germain, Marie-Sophie
  - Person: Brisson, Barnabé
  - Person: Gauss, Johann Carl Friedrich
  - Person: Hecht, Daniel Friedrich
  - Person: Poinsot, Louis
  - Person: Spence, William
  - Person: De Wronski, Josef-Maria Hoëné
  - Person: Gompertz, Benjamin
  - Person: Crelle August Leopold
  - Person: Somerville, Mary Fairfax Greig
  - Person: Bidone, Giorgio
  - Person: Bolzano, Bernard Placidus Johann Nepomuk
  - Person: Plana, Giovanni Antonio Amedeo
  - Person: Poisson, Siméon Denis
  - Person: Flauti, Vincenzo
  - Person: Brianchon, Charles Julien
  - Person: Mathieu, Claude-Louis
  - Person: Bessel, Friedrich Wilhelm
  - Person: Dupin, Pierre Charles François
  - Person: Navier, Claude Louis Marie Henri
  - Person: Arago, Dominique François Jean
  - Person: Binet, Jacques Philippe Marie
  - Person: Bland, Miles
  - Person: Horner, William George
  - Person: Nicollet, Jean-Nicolas
  - Person: Thomson, James
  - Person: Walsh, John
  - Person: Cruickshank, John
  - Person: De Fourcy, Louis Lefébure
  - Person: Davies, Griffith
  - Person: Dirksen, Enno Heeren
  - Person: Fallows, Fearon
  - Person: Fresnel, Augustin Jean
  - Person: Hamilton, William
  - Person: Poncelet, Jean Victor
  - Person: Bordoni, Antonio Maria
  - Person: Cauchy, Augustin Louis
  - Person: Ohm, Georg Simon
  - Person: Bélanger, Jean-Baptiste
  - Person: Champollion, Jean-François
  - Person: Möbius, August
  - Person: Strong, Theodore
  - Person: Terrot, Charles Hughes
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  - Person: Vladimirov, Vasilii Sergeevich
  - Person: Wilkins (2), J. Ernest
  - Person: Yamabe, Hidehiko
  - Person: Andreotti, Aldo
  - Person: Ashour, Attia
  - Person: Aubert, Karl
  - Person: Backus, John Warner
  - Person: Bartik, Betty Jean Jennings
  - Person: Cohn-Vossen (2), Stefan Emmanuilovich
  - Person: Davies (3), Donald
  - Person: Dixmier, Jacques
  - Person: Dynkin, Eugene Borisovich
  - Person: Everitt, Norrie
  - Person: Granville, Evelyn Boyd
  - Person: Horváth, János
  - Person: Høyland, Arnljot
  - Person: Karlin, Samuel
  - Person: Kiefer (2), Jack Carl
  - Person: Kilmister, Clive William
  - Person: Klingenberg, Wilhelm Paul Albert
  - Person: Krasovskii, Nikolai Nikolaevich
  - Person: Cam, Lucien Marie Le
  - Person: Lighthill, Michael James
  - Person: Mandelbrot, Benoit
  - Person: Popoviciu (2), Elena Moldovan
  - Person: Reiner, Irving
  - Person: Reizins, Linards Eduardovich
  - Person: Rudin (2), Mary Ellen
  - Person: Singer, Isadore Manuel
  - Person: Steinfeld, Ottó
  - Person: Berdichevsky, Cecilia
  - Person: Dahlquist, Germund
  - Person: Divinsky, Nathan Joseph Harry
  - Person: Gomide, Elza Furtado
  - Person: Graham (2), Eugene
  - Person: Gupta, Shanti Swarup
  - Person: Hoehnke, Hans-Jürgen
  - Person: Kadison, Richard
  - Person: Kruskal (2), Martin
  - Person: Lehto, Olli Erkki
  - Person: Marchuk, Guri Ivanovich
  - Person: Moreno, Adelina Gutiérrez
  - Person: Nirenberg, Louis
  - Person: Oleinik, Olga Arsen&amp;#x27;evna
  - Person: Pople, John Anthony
  - Person: Potts, Renfrey Burnard
  - Person: Preston, Gordon Bamford
  - Person: Prodi, Giovanni
  - Person: Roth (2), Klaus
  - Person: Rund, Hanno
  - Person: Stewartson, Keith
  - Person: Tate, John Torrence
  - Person: Williams (3), Lloyd
  - Person: Wright (3), Fred
  - Person: Zeeman, Erik Christopher
  - Person: Aitchison, John
  - Person: Auslander, Maurice
  - Person: Babuska, Ivo Milan
  - Person: Berge, Claude Jacques Roger
  - Person: Bremermann, Hans-Joachim
  - Person: Clunie, James Gourlay
  - Person: Dye, Henry Abel
  - Person: Eells, James
  - Person: Fiedler (3), Miroslav
  - Person: Green (2), J. A.
  - Person: De Guber, Rebeca Cherep
  - Person: Hájek, Jaroslav
  - Person: Haselgrove, Colin Brian
  - Person: Hayman, Walter Kurt
  - Person: Hönig, Chaim Samuel
  - Person: Ikeda, Gündüz
  - Person: Kahane, Jean-Pierre
  - Person: Kahn, Franz
  - Person: Karp, Carol Ruth
  - Person: Kemeny, John
  - Person: Lax (3), Peter
  - Person: Leech, John
  - Person: Mouján, Magdalena
  - Person: Nijenhuis, Albert
  - Person: Nygaard, Kristen
  - Person: Pawlak, Zdzisław
  - Person: Serre, Jean-Pierre Albert Achille
  - Person: Serrin, James Burton
  - Person: Shepherdson, John Cedric
  - Person: Spence (2), David
  - Person: Springer, Tonny Albert
  - Person: Suzuki, Michio
  - Person: Tims, Sydney Rex
  - Person: Vinti, Calogero
  - Person: Bharucha-Reid, Albert
  - Person: Birman, Joan Sylvia Lyttle
  - Person: Britton, John Leslie
  - Person: Browder, Felix
  - Person: Dajono, Slamet
  - Person: Frank (2), Marguerite Straus
  - Person: Griffiths (2), Brian
  - Person: Helgason, Sigurður
  - Person: Hirzebruch, Friedrich Ernst Peter
  - Person: Horn, Friedrich Josef Maria
  - Person: Lang (2), Serge
  - Person: Leopoldt, Heinrich-Wolfgang
  - Person: Levin (2), Frank
  - Person: De Lyra, Carlos Benjamin
  - Person: Mackay (2), James Peter Hymers
  - Person: Maurer, Gyula Iulius
  - Person: Moiseiwitsch, Benjamin
  - Person: Moser (2), William
  - Person: Nagata, Masayoshi
  - Person: Postnikov, Mikhail Mikhailovich
  - Person: Sachs, Horst
  - Person: Shabazz, Abdulalim
  - Person: Shephard, Geoffrey
  - Person: Shields, Allen Lowell
  - Person: Skopin, Alexander Ivanovich
  - Person: Stancu, Dimitrie D.
  - Person: Taniyama, Yutaka
  - Person: Tụy, Hoàng
  - Person: Wussing, Hans
  - Person: Adamson, Iain Thomas Arthur Carpenter
  - Person: Auslander (2), Louis
  - Person: Bauer (2), Heinz
  - Person: Bell (4), Charles
  - Person: Bell (3), John
  - Person: Keller (3), Joseph Bishop
  - Person: Butzer, Paul Leo
  - Person: Carleson, Lennart Axel Edvard
  - Person: Costley, Charles
  - Person: Davis, Martin David
  - Person: De Giorgi, Ennio
  - Person: Fennema, Elizabeth
  - Person: Gohberg, Israel
  - Person: Greenspan, Donald
  - Person: Grothendieck, Alexander
  - Person: Gruenberg, Karl Walter
  - Person: Higman (2), Donald
  - Person: Temple, George Frederick James
  - Person: Kruskal (3), Joseph
  - Person: Lions, Jacques-Louis
  - Person: Malgrange, Bernard
  - Person: Milner, Eric
  - Person: Moser (3), Jürgen
  - Person: Nash, John Forbes
  - Person: Okonjo, Chukwuka
  - Person: Pask, Andrew Gordon Speedie
  - Person: Ribenboim, Paulo
  - Person: Routledge, Norman Arthur
  - Person: Sato, Mikio
  - Person: Valdivia, Manuel
  - Person: Varga (2), Richard Steven
  - Person: Walter, Marion Ilse
  - Person: Atiyah, Michael Francis
  - Person: Benjamin, Thomas Brooke
  - Person: Choquet-Bruhat, Yvonne Suzanne Marie-Louise
  - Person: Cheney, Ward
  - Person: Grünbaum, Branko
  - Person: Harper, Laurence Raymond
  - Person: Hoe, Jock
  - Person: Kennedy, Patrick Brendan
  - Person: Kertész, Andor
  - Person: Kiss, Elemér
  - Person: Kostrikin, Aleksei Ivanovich
  - Person: Lima, Elon
  - Person: Lumer, Günter
  - Person: Mcleod, John Bryce
  - Person: Munn, Walter Douglas
  - Person: Petryshyn, Volodymyr
  - Person: Piatetski-Shapiro, Ilya Iosifovich
  - Person: Prokhorov, Yurii Vasilevich
  - Person: Puri, Madan Lal
  - Person: Rayner, Margaret
  - Person: Spencer (2), Anthony
  - Person: Spinadel, Vera
  - Person: Abhyankar, Shreeram Shankar
  - Person: Adams (4), Frank
  - Person: Alling, Norman Larrabee
  - Person: Curle, Newby
  - Person: Dijkstra, Edsger Wybe
  - Person: Escalante, Jaime
  - Person: Ezeilo, James
  - Person: Feit, Walter
  - Person: Foulis, David James
  - Person: Goldberg, Anatolii Asirovich
  - Person: Grauert, Hans
  - Person: Kalman, Rudolf Emil
  - Person: Kellogg (2), Bruce
  - Person: Kelly (2), Gregory Maxwell
  - Person: Mohamed, Ismail
  - Person: Morton (2), Keith William
  - Person: Polkinghorne, John Charlton
  - Person: Schäffer, Juan Jorge
  - Person: Schwartz (3), Jacob T.
  - Person: Selten, Reinhard
  - Person: Shimura, Goro
  - Person: Skorokhod, Anatolii Volodymyrovych
  - Person: Smale, Stephen
  - Person: Sós, Vera
  - Person: Suzuki (2), Satoshi
  - Person: Tits, Jacques
  - Person: Urbanik, Kazimierz
  - Person: Yuan (2), Wang
  - Person: Adian, Sergei Ivanovich
  - Person: Borok, Valentina Mikhailovna
  - Person: Hajnal, András
  - Person: Hironaka, Heisuke
  - Person: Hörmander, Lars
  - Person: Kluvánek, Igor
  - Person: Knapowski, Stanislaw
  - Person: Mercer (2), Alan
  - Person: Milnor, John Willard
  - Person: Murray, James Dickson
  - Person: Olech, Czeslaw
  - Person: Penrose, Roger
  - Person: Pless, Vera
  - Person: Stein (2), Elias Menachem
  - Person: Thompson (2), Robert C
  - Person: Vitushkin, Anatoli Georgievich
  - Person: Kreindler, Eléna
  - Person: Wilf, Herbert Saul
  - Person: Alele-Williams, Grace
  - Person: Allotey, Francis
  - Person: Appel, Kenneth Ira
  - Person: Audin, Maurice
  - Person: Bass, Hyman
  - Person: De Bourcia, Louis de Branges
  - Person: Cartier, Pierre Emile Jean
  - Person: Golub, Gene Howard
  - Person: Honda, Taira
  - Person: Lewis (2), John
  - Person: Loibel, Gilberto
  - Person: Malone-Mayes, Vivienne
  - Person: Nash-Williams, Crispin
  - Person: Neubüser, Joachim
  - Person: Neveu, Jacques
  - Person: Ringrose, John Robert
  - Person: Rota, Gian-Carlo
  - Person: Segel, Lee
  - Person: Solitar, Donald Moiseyevitch
  - Person: Świerczkowski, Stanisław
  - Person: Țarină, Marian
  - Person: Thompson (3), John
  - Person: Van Lint, Jacobus Hendricus
  - Person: Vernon, Siobhan O&amp;#x27;Shea
  - Person: Warner, Mary Wynne
  - Person: Whiteside, Derek Thomas
  - Person: Almgren, Frederick Justin
  - Person: Askey, Richard
  - Person: Baumslag, Gilbert
  - Person: Carmeli, Moshe (Ehezkel)
  - Person: Jingrun, Chen
  - Person: Falconer, Etta Zuber
  - Person: Foster (2), Dorothy
  - Person: Greenspan (2), Harvey
  - Person: Leslie (2), Joshua
  - Person: Lesokhin, Mikhail Moiseevich
  - Person: O&amp;#x27;Raifeartaigh, Lochlainn
  - Person: Pier, Jean-Paul
  - Person: Rigby, John Frankland
  - Person: Roach, Gary
  - Person: Schmidt (5), Wolfgang
  - Person: Vanstone, Ray
  - Person: Artin (2), Michael
  - Person: Browder (2), William
  - Person: Cohen (2), Jacob Willem
  - Person: Faddeev (2), Ludwig
  - Person: Glimm, James
  - Person: Kerr (2), Roy
  - Person: Meyer (2), Paul-André
  - Person: Ornstein, Donald Samuel
  - Person: Ostrovskii, Iossif Vladimirovich
  - Person: Parry, William
  - Person: Sherif, Soraya
  - Person: Tiit, Ene-Margit
  - Person: Wiegold, James
  - Person: Bourbaki, Nicolas
  - Person: Chuaqui, Rolando
  - Person: Furstenberg, Hillel
  - Person: Graham (3), Thomas Smith
  - Person: Grigelionis, Bronius
  - Person: Hay, Louise Schmir
  - Person: Hewitt (2), Gloria
  - Person: Hildebrant, John A
  - Person: Kuczma, Marek
  - Person: Kumano-Go, Hitoshi
  - Person: Olunloyo, Victor
  - Person: Sinai, Yakov Grigorevich
  - Person: Srinivasan, Bhama
  - Person: Stallings, John Robert
  - Person: Tverberg, Helge
  - Person: Allan, Graham
  - Person: Anosov, Dmitrii Viktorovich
  - Person: Burton, Leone Minna Gold
  - Person: Cofman, Judita
  - Person: Hammer, Peter Ladislaw
  - Person: Howie, John Mackintosh
  - Person: Iribarren, GonzaloPérez
  - Person: Keisler, Jerome
  - Person: Kim, Ki Hang
  - Person: Kirillov, Alexandre Aleksandrovich
  - Person: Kovács, László
  - Person: Langlands, Robert Phelan
  - Person: Lindenstrauss, Joram
  - Person: May (2), Robert
  - Person: Schelp, Richard Herbert
  - Person: Sharkovsky, Oleksandr Mikolaiovich
  - Person: Strassen, Volker
  - Person: Sultangazin, Umirzak
  - Person: Tichy, Pavel
  - Person: Velez-Rodriguez, Argelia
  - Person: Eckert (3), Wallace John
  - Person: Arnold, Vladimir Igorevich
  - Person: Boersma, Johannes
  - Person: Book, Ronald Vernon
  - Person: Conway (2), John Horton
  - Person: Flato, Moshé
  - Person: Fowler (2), David
  - Person: Hayes (3), David
  - Person: Iyahen, Sunday
  - Person: Johnson (5), Barry
  - Person: Khruslov, Evgenii Yakovlevich
  - Person: Knopfmacher, John
  - Person: Manin, Yuri Ivanovich
  - Person: Mazur (2), Barry
  - Person: Maz&amp;#x27;Ya, Vladimir
  - Person: Mochizuki, Horace Yomishi
  - Person: Mumford, David Bryant
  - Person: Pastur, Leonid Andreevich
  - Person: Planchart, Enrique
  - Person: Sims, Charles
  - Person: Akyeampong, Daniel
  - Person: Andrews, George W. Eyre
  - Person: Animalu, Alexander
  - Person: Black (2), Fischer Sheffey
  - Person: Carlson, Donald Earle
  - Person: Craik, Alex
  - Person: Griffiths (3), Phillip Augustus
  - Person: Gromoll, Detlef
  - Person: Kirby, Robion Cromwell
  - Person: Knuth, Donald Ervin
  - Person: Meinhardt, Hans
  - Person: Novikov (2), Sergei
  - Person: Ramanujam, Chidambaram Padmanabhan
  - Person: Ratner, Marina
  - Person: Samoilenko, Anatoly Mykhailovych
  - Person: Subbotovskaya, Bella Abramovna
  - Person: Tanaka, Hideo
  - Person: Baker (3), Alan
  - Person: Calderón (2), Calixto
  - Person: Cercignani, Carlo
  - Person: Gray (4), Marion Cameron
  - Person: Hartley, Brian
  - Person: Kingman, John Frank Charles
  - Person: Moalla, Fatma
  - Person: Pahlings, Herbert
  - Person: Schattschneider, Doris
  - Person: Senechal, Marjorie
  - Person: Sleeman, Brian
  - Person: Swart, Hendrika
  - Person: Tapia, Richard Alfred
  - Person: Wschebor-Wonsever, Mario Israel
  - Person: Bombieri, Enrico
  - Person: Keen, Linda Goldway
  - Person: Palis Jr, Jacob
  - Person: Quillen, Daniel Gray
  - Person: Sadosky (2), Cora
  - Person: Szafraniec, Franciszek Hugon
  - Person: Szemerédi, Endre
  - Person: Varadhan, Sathamangalam Ranga Iyengar Srinivasa
  - Person: Day, Richard Alan
  - Person: Filep, László
  - Person: Grattan-Guinness, Ivor
  - Person: Kuku, Aderemi
  - Person: Okikiolu, George Olatokunbo
  - Person: Stefan (2), Peter
  - Person: Sullivan, Dennis Parnell
  - Person: Szász (2), Domokos
  - Person: Waller, Derek Arthur
  - Person: Aubin, Thierry Émilien Flavien
  - Person: Blum, Lenore
  - Person: Brown (4), Gavin
  - Person: Crighton, David George
  - Person: Fedorchuk, Vitaly Vitalievich
  - Person: Hawking, Stephen William
  - Person: Keast, Patrick
  - Person: Lupas, Alexandru Ioan
  - Person: Obada, Abdel-Shafy
  - Person: Pedley, Timothy
  - Person: Rallis, Stephen James
  - Person: Trudinger, Neil Sidney
  - Person: Uhlenbeck (2), Karen
  - Person: Bollobás, Béla
  - Person: Gromov, Mikhael Leonidovich
  - Person: Nelson, Evelyn Merle Roden
  - Person: Orszag, Steven Alan
  - Person: Robertson (2), Edmund
  - Person: Deligne, Pierre René
  - Person: Feigenbaum, Mitchell Jay
  - Person: Kuperberg, Krystyna Maria Trybulec
  - Person: Millington, Margaret Hilary Ashworth
  - Person: Trahtman, Avraham Naumovich
  - Person: Adleman, Leonard
  - Person: Coates, John Henry
  - Person: Diaconis, Persi Warren
  - Person: Guidy-Wandja, Joséphine
  - Person: Knorr, Wilbur Richard
  - Person: Mcduff, Margaret Dusa Waddington
  - Person: O&amp;amp;amp;amp;#x27;Connor, John
  - Person: Patodi, Vijay Kumar
  - Person: Saitoti, George
  - Person: Shelah, Saharon
  - Person: Sigmund, Karl
  - Person: Simon, Leon Melvyn
  - Person: Bratteli, Ola
  - Person: Davies (4), Paul
  - Person: Melo, Welingtonde
  - Person: Hoyles, Celia
  - Person: Iwanik, Anzelm
  - Person: Kalton, Nigel John
  - Person: Loday, Jean-Louis
  - Person: Lusztig, George
  - Person: Margulis, Gregori Aleksandrovic
  - Person: Thurston, William Paul
  - Person: Arino, Ovide
  - Person: Benkart, Georgia
  - Person: Bowen, Robert Edward
  - Person: Connes, Alain
  - Person: Eisenbud, David
  - Person: Huhn, András P
  - Person: Jorgensen, Palle
  - Person: Matiyasevich, Yuri Vladimirovich
  - Person: Rothblum, Uriel George
  - Person: Caffarelli, Luis
  - Person: Carr, John
  - Person: Chang, Sun-Yung Alice
  - Person: Flajolet, Philippe
  - Person: Kintala, Chandra Mohan Rao
  - Person: Lovász, Laszlo
  - Person: Mañé, Ricardo
  - Person: Murphy (2), Gerard
  - Person: Praeger, Cheryl Elisabeth
  - Person: Chung, Fan Rong K. Graham
  - Person: Fefferman, Charles Louis
  - Person: Grunewald, Fritz Alfred Joachim
  - Person: Harvey, Josephat Martin (Fanyana Mutyambizi)
  - Person: Haxhibeqiri, Qamil
  - Person: Rudolph, Daniel Jay
  - Person: Valiant, Leslie Gabriel
  - Person: Yau, Shing-Tung
  - Person: Bolibrukh, Andrei Andreevich
  - Person: Broomhead, David S
  - Person: Gorbunov, Viktor Aleksandrovich
  - Person: Ismail, Gamal
  - Person: Jacimovic, Milojica
  - Person: Pisier, Jean Georges Gilles
  - Person: Schoen, Richard Melvin
  - Person: Freedman, Michael Hartley
  - Person: Mori, Shigefumi
  - Person: Stromme, Stein Arild
  - Person: Ugbebor, Olabisi
  - Person: Witten, Edward
  - Person: Cafaro, Emilio
  - Person: Jones (4), Vaughan
  - Person: Thomason, Robert Wayne
  - Person: Young (7), Lai-Sang
  - Person: Balogh, Zoltán Tibor
  - Person: Wiles, Andrew John
  - Person: Bourgain, Jean
  - Person: Daubechies, Ingrid
  - Person: Dimovski, Dončo
  - Person: Drinfeld, Vladimir Gershonovich
  - Person: Escobar, José
  - Person: Faltings, Gerd
  - Person: Krejčí, Pavel
  - Person: Masanja, Verdiana
  - Person: Mushtaq, Qaiser
  - Person: Perrin-Riou, Bernadette
  - Person: Zelmanov, Efim Isaakovich
  - Person: Floer, Andreas
  - Person: Lions (2), Pierre-Louis
  - Person: Soueif, Laila
  - Person: Donaldson, Simon Kirwan
  - Person: Yoccoz, Jean-Christophe
  - Person: Bahouri, Hajer
  - Person: Mcmullen, Curtis Tracy
  - Person: Seress, Akos
  - Person: Thompson (4), Abigail
  - Person: Borcherds, Richard Ewen
  - Person: Franca, Leopoldo
  - Person: Hau, Lene Vestergaard
  - Person: Maini, Philip
  - Person: Perthame, Benoît
  - Person: Heinonen, Juha
  - Person: Khaketla, &amp;#x27;Mamphono
  - Person: Olubummo (2), Yewande
  - Person: Taleb, Nassim
  - Person: Schramm, Oded
  - Person: Agwu, Nkechi
  - Person: Motwani, Rajeev
  - Person: Taylor (6), Richard
  - Person: Gowers, William Timothy
  - Person: Dubickas, Artūras
  - Person: Kontsevich, Maxim Lvovich
  - Person: Nessyahu, Haim
  - Person: Wu (2), Sijue
  - Person: Okikiolu (2), Katherine
  - Person: Smith (4), Karen
  - Person: Jitomirskaya, Svetlana
  - Person: Lafforgue, Laurent
  - Person: Perelman, Grigori Yakovlevich
  - Person: Voevodsky, Vladimir Aleksandrovich
  - Person: Michler, Ruth
  - Person: Ouedraogo, Marie Françoise
  - Person: Werner (2), Wendelin
  - Person: Cariolaro, David
  - Person: Okounkov, Andrei Yuryevich
  - Person: Goins, Edray Herber
  - Person: Washington, Talitha
  - Person: Tao, Terence Chi-Shen
  - Person: Mirzakhani, Maryam
  - Person: Ávila, Artur
  - Person: Gashi, Qëndrim
  - Person: Viazovska, Maryna
- Epoch: 21th Century
Index (8)
- Index: Content Tree
- Index: By Building Blocks
- Index: By Contributors. Thank you!
- Index: Symbolic Notation
- Index: Widgets
- Index: Person
- Index: By Site Issues
- Index: Keyword

Thank you to the contributors under CC BY-SA 4.0!

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