This subsection contains the propositions from Book 10 of Euclid's “Elements”.

- Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number
- Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm
- Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes
- Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes
- Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes
- Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable
- Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio
- Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable
- Proposition: Prop. 10.009: Commensurability of Squares
- Proposition: Prop. 10.010: Construction of Incommensurable Lines
- Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes
- Proposition: Prop. 10.012: Commensurability is Transitive Relation
- Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
- Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines
- Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes
- Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation
- Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation
- Proposition: Prop. 10.019: Product of Rational Numbers is Rational
- Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational
- Proposition: Prop. 10.021: Medial is Irrational
- Proposition: Prop. 10.022: Square on Medial Straight Line
- Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial
- Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
- Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
- Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area
- Proposition: Prop. 10.027: Construction of Components of First Bimedial
- Proposition: Prop. 10.028: Construction of Components of Second Bimedial
- Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable
- Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable
- Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I
- Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II
- Proposition: Prop. 10.033: Construction of Components of Major
- Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area
- Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas
- Proposition: Prop. 10.036: Binomial is Irrational
- Proposition: Prop. 10.037: First Bimedial is Irrational
- Proposition: Prop. 10.038: Second Bimedial is Irrational
- Proposition: Prop. 10.039: Major is Irrational
- Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational
- Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational
- Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely
- Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely
- Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely
- Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely
- Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely
- Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely
- Proposition: Prop. 10.048: Construction of First Binomial Straight Line
- Proposition: Prop. 10.049: Construction of Second Binomial Straight Line
- Proposition: Prop. 10.050: Construction of Third Binomial Straight Line
- Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line
- Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line
- Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line
- Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial
- Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial
- Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial
- Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial
- Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial
- Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial
- Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
- Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
- Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major
- Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area
- Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas
- Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
- Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
- Proposition: Prop. 10.073: Apotome is Irrational
- Proposition: Prop. 10.074: First Apotome of Medial is Irrational
- Proposition: Prop. 10.075: Second Apotome of Medial is Irrational
- Proposition: Prop. 10.076: Minor is Irrational
- Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational
- Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational
- Proposition: Prop. 10.079: Construction of Apotome is Unique
- Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique
- Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique
- Proposition: Prop. 10.082: Construction of Minor is Unique
- Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique
- Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique
- Proposition: Prop. 10.085: Construction of First Apotome
- Proposition: Prop. 10.086: Construction of Second Apotome
- Proposition: Prop. 10.087: Construction of Third Apotome
- Proposition: Prop. 10.088: Construction of Fourth Apotome
- Proposition: Prop. 10.089: Construction of Fifth Apotome
- Proposition: Prop. 10.090: Construction of Sixth Apotome
- Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome
- Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome
- Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome
- Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome
- Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome
- Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome
- Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line
- Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
- Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.103: Straight Line Commensurable with Apotome
- Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line
- Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line
- Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area
- Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area
- Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted
- Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
- Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
- Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line
- Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line
- Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome
- Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
- Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines