Ratio in mathematics

A:B generally differs from B:A: two-to-three reverses when read as three-to-two. A common positive change of unit preserves a ratio of like magnitudes; different units must first be reconciled.

Book V Definition 3 compares magnitudes of the same kind, such as length with length or area with area. That historical framework differs from dimensioned rates such as distance divided by time. Their modern operations should not automatically be read into the ancient text.

References: [1]

Definition 5 compares A:B with C:D using arbitrary positive integers m and n. The relation between mA and nB must match that between mC and nD, whether greater, equal or less. All such comparisons are required.

For two-to-three and four-to-six, both terms of the second pair are doubled, preserving every comparison. The definition does not first demand a common unit for any two lengths, so it can handle incommensurable magnitudes.

References: [2]

Requiring like magnitudes within each ratio does not require all four magnitudes of a proportion to share one kind. VI.1 equates the area ratio of equal-height triangles with the ratio of their bases: areas compare with areas and lengths with lengths.

Bases in a two-to-five ratio give areas in the same ratio when heights agree. If heights also change, the base ratio alone is insufficient. A proportion depends on its hypotheses, not merely on displaying two fraction-like expressions.

References: [3]