# 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