Category:Euclidean Number Theory
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This category contains results about Euclidean Number Theory.
Definitions specific to this category can be found in Definitions/Euclidean Number Theory.
Euclidean number theory concerns aspects of number theory and real analysis as presented in Euclid's The Elements.
Particular attention is paid to certain properties of the irrational numbers.
Subcategories
This category has the following 6 subcategories, out of 6 total.
Pages in category "Euclidean Number Theory"
The following 168 pages are in this category, out of 168 total.
A
B
- Between two Cubes exist two Mean Proportionals
- Between two Similar Plane Numbers exists one Mean Proportional
- Between two Similar Solid Numbers exist two Mean Proportionals
- Between two Squares exists one Mean Proportional
- Binomial is Irrational
- Binomial Straight Line is Divisible into Terms Uniquely
- Binomial Straight Line is Divisible into Terms Uniquely/Lemma
C
- Classification of Irrational Straight Lines derived from Apotome
- Classification of Irrational Straight Lines derived from Apotome/Summary
- Classification of Irrational Straight Lines derived from Binomial Straight Line
- Classification of Irrational Straight Lines derived from Binomial Straight Line/Summary
- Commensurability is Transitive Relation
- Commensurability of Elements of Proportional Magnitudes
- Commensurability of Squares
- Commensurability of Squares on Proportional Straight Lines
- Commensurability of Squares on Proportional Straight Lines/Lemma
- Commensurability of Squares/Porism
- Commensurability of Sum of Commensurable Magnitudes
- Commensurable Magnitudes are Incommensurable with Same Magnitude
- Condition for Commensurability of Roots of Quadratic Equation
- Condition for Commensurability of Roots of Quadratic Equation/Lemma
- Condition for Incommensurability of Roots of Quadratic Equation
- Construction of Apotome is Unique
- Construction of Components of First Bimedial
- Construction of Components of Major
- Construction of Components of Major/Lemma
- Construction of Components of Second Bimedial
- Construction of Components of Side of Rational plus Medial Area
- Construction of Components of Side of Sum of Medial Areas
- Construction of Fifth Apotome
- Construction of Fifth Binomial Straight Line
- Construction of First Apotome
- Construction of First Apotome of Medial is Unique
- Construction of First Binomial Straight Line
- Construction of Fourth Apotome
- Construction of Fourth Binomial Straight Line
- Construction of Incommensurable Lines
- Construction of Incommensurable Lines/Lemma
- Construction of Medial Straight Lines Commensurable in Square Only containing Medial Rectangle whose Square Differences Commensurable with Greater
- Construction of Medial Straight Lines Commensurable in Square Only containing Rational Rectangle whose Square Differences Commensurable with Greater
- Construction of Minor is Unique
- Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater
- Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 1
- Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 2
- Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with Greater
- Construction of Second Apotome
- Construction of Second Apotome of Medial is Unique
- Construction of Second Binomial Straight Line
- Construction of Sequence of Numbers with Given Ratios
- Construction of Sixth Apotome
- Construction of Sixth Binomial Straight Line
- Construction of that which produces Medial Whole with Medial Area is Unique
- Construction of that which produces Medial Whole with Rational Area is Unique
- Construction of Third Apotome
- Construction of Third Binomial Straight Line
- Cube Number multiplied by Cube Number is Cube
E
F
G
I
M
- Magnitudes with Irrational Ratio are Incommensurable
- Magnitudes with Rational Ratio are Commensurable
- Magnitudes with Rational Ratio are Commensurable/Porism
- Major is Irrational
- Major Straight Line is Divisible Uniquely
- Medial Area not greater than Medial Area by Rational Area
- Medial is Irrational
- Minor is Irrational
N
- Number multiplied by Cube Number making Cube is itself Cube
- Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
- Number Squared making Cube is itself Cube
- Number whose Half is Odd is Even-Times Odd
- Numbers between which exist two Mean Proportionals are Similar Solid
- Numbers between which exists one Mean Proportional are Similar Plane
- Numbers whose Product is Square are Similar Plane Numbers
O
P
R
- Ratio of Commensurable Magnitudes
- Ratio of Products of Sides of Plane Numbers
- Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
- Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
- Root of Area contained by Rational Straight Line and Fifth Binomial
- Root of Area contained by Rational Straight Line and First Binomial
- Root of Area contained by Rational Straight Line and First Binomial/Lemma
- Root of Area contained by Rational Straight Line and Fourth Binomial
- Root of Area contained by Rational Straight Line and Second Binomial
- Root of Area contained by Rational Straight Line and Sixth Binomial
- Root of Area contained by Rational Straight Line and Third Binomial
S
- Second Apotome of Medial is Irrational
- Second Bimedial is Irrational
- Second Bimedial Straight Line is Divisible Uniquely
- Side of Area Contained by Rational Straight Line and Fifth Apotome
- Side of Area Contained by Rational Straight Line and First Apotome
- Side of Area Contained by Rational Straight Line and Fourth Apotome
- Side of Area Contained by Rational Straight Line and Second Apotome
- Side of Area Contained by Rational Straight Line and Sixth Apotome
- Side of Area Contained by Rational Straight Line and Third Apotome
- Side of Rational Plus Medial Area is Divisible Uniquely
- Side of Rational plus Medial Area is Irrational
- Side of Remaining Area from Rational Area from which Medial Area Subtracted
- Side of Sum of Medial Areas is Irrational
- Side of Sum of Two Medial Areas is Divisible Uniquely
- Similar Plane Numbers have Same Ratio as between Two Squares
- Similar Solid Numbers have Same Ratio as between Two Cubes
- Square Numbers are Similar Plane Numbers
- Square of Cube Number is Cube
- Square on Apotome applied to Rational Straight Line
- Square on Binomial Straight Line applied to Rational Straight Line
- Square on Binomial Straight Line applied to Rational Straight Line/Lemma
- Square on First Apotome of Medial Straight Line applied to Rational Straight Line
- Square on First Bimedial Straight Line applied to Rational Straight Line
- Square on Major Straight Line applied to Rational Straight Line
- Square on Medial Straight Line
- Square on Medial Straight Line/Lemma
- Square on Minor Straight Line applied to Rational Straight Line
- Square on Rational Straight Line applied to Apotome
- Square on Rational Straight Line applied to Binomial Straight Line
- Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
- Square on Second Bimedial Straight Line applied to Rational Straight Line
- Square on Side of Rational plus Medial Area applied to Rational Straight Line
- Square on Side of Sum of two Medial Area applied to Rational Straight Line
- Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
- Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
- Straight Line Commensurable with Apotome
- Straight Line Commensurable with Apotome of Medial Straight Line
- Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
- Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
- Straight Line Commensurable with Major Straight Line is Major
- Straight Line Commensurable with Medial Straight Line is Medial
- Straight Line Commensurable with Medial Straight Line is Medial/Porism
- Straight Line Commensurable with Minor Straight Line
- Straight Line Commensurable with Side of Rational plus Medial Area
- Straight Line Commensurable with Side of Sum of two Medial Areas
- Straight Line Commensurable with that which produces Medial Whole with Medial Area
- Straight Line Commensurable with that which produces Medial Whole with Rational Area
- Sum of Even Integers is Even
- Sum of Even Number of Odd Numbers is Even
- Sum of Odd Number of Odd Numbers is Odd
- Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
- Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
T
- That which produces Medial Whole with Medial Area is Irrational
- That which produces Medial Whole with Rational Area is Irrational
- Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
- Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted