Proportion and Similarity

Elements

This flow traces the transition from Book V's theory of magnitude and ratio to Book VI's construction and measurement of similar figures.

increasing

Bearing strength in each event

decreasing

Magnitude Comparison Magnitude Comparison from "Elements". Magnitude Comparison Multiples The treatment begins by defining multiples and submultiples for magnitudes of the same kind.

Magnitude Comparison from "Elements". Magnitude Comparison Multiples The treatment begins by defining multiples and submultiples for magnitudes of the same kind.

Magnitude Comparison Multiples The treatment begins by defining multiples and submultiples for magnitudes of the same kind. This creates a rigorous language for comparison even when exact numerical measurement is unavailable. Ratio Ratio becomes a relation of magnitude rather than merely a fraction written with numbers.

Eudoxian Definition Eudoxian Definition from "Elements". Eudoxian Definition Equimultiples Four magnitudes are proportional when every comparison of equimultiples preserves the same greater equal or lesser relation.

Eudoxian Definition from "Elements". Eudoxian Definition Equimultiples Four magnitudes are proportional when every comparison of equimultiples preserves the same greater equal or lesser relation.

Eudoxian Definition Equimultiples Four magnitudes are proportional when every comparison of equimultiples preserves the same greater equal or lesser relation. The definition avoids depending on a common measure. Conceptual Precision It allows continuous magnitudes to enter a theory governed by exact logical comparison.

Ratio Operations Ratio Operations from "Elements". Ratio Operations Transformations Inversion alternation composition and division are developed as lawful operations on proportionals.

Ratio Operations from "Elements". Ratio Operations Transformations Inversion alternation composition and division are developed as lawful operations on proportionals.

Ratio Operations Transformations Inversion alternation composition and division are developed as lawful operations on proportionals. Each transformation preserves the comparison while changing its form. Generality The book thereby turns isolated geometric relations into a reusable calculus of proportional reasoning.

Similar Triangles Similar Triangles from "Elements". Similar Triangles Angle and Side Relations Equiangular triangles are shown to have proportional corresponding sides.

Similar Triangles from "Elements". Similar Triangles Angle and Side Relations Equiangular triangles are shown to have proportional corresponding sides.

Similar Triangles Angle and Side Relations Equiangular triangles are shown to have proportional corresponding sides. Conversely, proportional sides force equality of the appropriate angles under the stated conditions. Scaling Similarity becomes the mechanism by which one figure can reproduce another at a different size.

Duplicate Areas Duplicate Areas from "Elements". Duplicate Areas Linear Growth Similar triangles are related by the duplicate ratio of their homologous sides.

Duplicate Areas from "Elements". Duplicate Areas Linear Growth Similar triangles are related by the duplicate ratio of their homologous sides.

Duplicate Areas Linear Growth Similar triangles are related by the duplicate ratio of their homologous sides. Areas therefore grow as the square of a corresponding length. Polygonal Extension The same principle is extended from triangles to similar polygons, linking shape identity with quantitative scale.

at the opening of Book V

at the fifth definition of Book V

through the middle propositions of Book V

at the fourth through eighth propositions of Book VI

at the nineteenth and twentieth propositions of Book VI