Proportion proportion, that relation of magnitudes by which one is measured against another, is a fundamental principle in geometry and number. Let a line be divided into segments such that the whole is to the greater part as the greater part is to the lesser. This is the proportion known as the extreme and mean ratio. It is observed that when four magnitudes are in proportion, the first is to the second as the third is to the fourth. Let the first be A, the second B, the third C, and the fourth D. If A is to B as C is to D, then the ratio of A to B is equal to the ratio of C to D. This equality of ratios constitutes proportion. It is demonstrated that if two ratios are equal, their terms may be rearranged in various ways without disturbing the equality. For instance, if A is to B as C is to D, then alternately A is to C as B is to D. Likewise, inversely, B is to A as D is to C. These transformations follow from the definition of proportion and the properties of equal ratios. The terms may be compounded: if A is to B as C is to D, and B is to E as D is to F, then A is to E as C is to F. This is the principle of ex aequali. Continued proportion occurs when three or more magnitudes are such that the first is to the second as the second is to the third, and so forth. Let three magnitudes be A, B, and C. If A is to B as B is to C, then B is the mean proportional between A and C. The square on B is equal to the rectangle contained by A and C. This relationship is foundational in the theory of similar figures. In similar triangles, the sides about the equal angles are proportional. Let triangle ABC and triangle DEF be similar. Then AB is to DE as BC is to EF as AC is to DF. The angles correspond, and the sides are in proportion. Proportion is not confined to lines. It extends to areas and solids. If two plane figures are similar, the ratio of their areas is the square of the ratio of their corresponding sides. If two solids are similar, the ratio of their volumes is the cube of the ratio of their corresponding sides. Let two cubes be constructed, one with side A, the other with side B. The volume of the first is to the volume of the second as A cubed is to B cubed. This holds universally for all similar figures, whether bounded by straight lines or curved boundaries. A proportion may be extended to more than four terms. When magnitudes are in continued proportion, the first is said to have to the last the ratio compounded of the ratios of the intermediate terms. Let five magnitudes be A, B, C, D, E. If A is to B as B is to C as C is to D as D is to E, then A is to E as the ratio compounded of A to B, B to C, C to D, and D to E. This compound ratio is determined by successive multiplication of the terms. It is further observed that when magnitudes are proportional, the greater is to the greater as the lesser is to the lesser. If A is greater than B, and C is greater than D, and A is to B as C is to D, then A minus B is to B as C minus D is to D. This is the principle of separation of ratios. Similarly, if magnitudes are proportional and one pair is increased by a common quantity, their ratios remain unchanged under addition, provided the addition is proportional to the original terms. Proportion, when applied to numbers, retains its geometric character. Let numbers be expressed as lines, and their ratios as the ratios of those lines. The numerical proportion of 2 to 4 is the same as that of 3 to 6, because the former is half the latter, and so too the latter. This equality of ratios holds whether the quantities are commensurable or incommensurable. The irrational, such as the diagonal of a square to its side, may enter into proportion as readily as the rational, so long as the relation remains consistent. It is possible for magnitudes to be proportional in one respect and not in another. Two figures may have equal angles and yet not have proportional sides. Two numbers may be in the same ratio as two lines, yet differ in kind. Proportion depends not on similarity of nature, but on equality of relation. What must be true of magnitudes if three of them are proportional, and a fourth is added such that the whole is to the part as the part is to the remainder? [role=marginalia, type=clarification, author="a.husserl", status="adjunct", year="2026", length="38", targets="entry:proportion", scope="local"] Proportion is not merely numerical equivalence but a structural intentionality—a lived harmony between magnitudes as they disclose themselves in pure intuition. The ratio’s equality reveals not calculation, but the eidetic unity of consciousness ordering being through pure relation. [role=marginalia, type=objection, author="a.simon", status="adjunct", year="2026", length="46", targets="entry:proportion", scope="local"] Proportion, as here defined, presumes magnitudes are commensurable—yet Greek thinkers themselves grappled with incommensurables. Euclid’s definition avoids arithmetic ratios, precisely to accommodate irrational relations. To equate proportion with equality of numerical ratios risks anachronism; it conflates measure with relation, obscuring the deeper geometric ontology at play. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:proportion", scope="local"]