Geometry and the making alike of unlike numbers

Use in the book

Geometry follows the study of numbers themselves (990c–d). The Athenian calls its familiar name ridiculous because the higher subject he is describing is not exhausted by measuring land. He redescribes it as the evident making alike of numbers not like one another by nature, with respect to the portion of plane figures. The phrase states a mathematical transformation of the terms of comparison, but it is too compressed to supply its own complete worked example.

The edition interprets it through numbers whose ratio is not the ratio of two squares, taking 1 and 2 as an example. They can be represented by squares with those areas, while the corresponding sides are incommensurable in length. In modern notation the side ratio is 1 to the square root of 2. The sides do not share a common measuring length, even though their squares have the determinate numerical ratio 1:2. This is the sense in which the plane treatment reveals an intelligible relation unavailable if one expects all corresponding lengths to be expressible as a ratio of whole numbers.

That explanation must not be turned into “geometry makes irrational numbers rational” or “incommensurable lengths become commensurable lengths.” It is precisely the respect of comparison that changes: the areas or squares can have a numerical relation where the sides do not have a common measure. The source's note invokes the mathematical discussion in the Theaetetus as context and labels the explanation interpretation. The terms “irrational” and “incommensurable” are explanatory vocabulary, not Greek words explicitly printed in this sentence of the Epinomis.

The subject is called a divine rather than human wonder to the person able to understand it. The marvel lies in the relation disclosed, not in an exemption from rational explanation. The subsequent solid-dimensional study extends the movement beyond plane figures, and the final synoptic requirement asks learners to recognize the relation of these studies to the whole mathematical and celestial order.

The local use of likeness thus differs from the earlier theological verb “liken,” apeikazein. One should not read a mathematical construction as merely an imaginative image of a god, nor assume that every occurrence of likeness throughout the dialogue names one technical operation.

Places to read

1. 990dσφόδρα γελοῖον ὄνομα γεωμετρίαν — The familiar name is sharply mocked
2. 990dτῶν οὐκ ὄντων δὲ ὁμοίων ἀλλήλοις φύσει ἀριθμῶν — The numbers are not naturally like one another
3. 990dὁμοίωσις — The positive redefinition is a making alike
4. 990dπρὸς τὴν τῶν ἐπιπέδων μοῖραν — The relevant respect is the portion of plane figures
5. 990dγεγονυῖά ἐστιν διαφανής — This making alike has become evident
6. 990dθαῦμα οὐκ ἀνθρώπινον ἀλλὰ γεγονὸς θεῖον — The intelligible relation is declared a divine wonder

Translation in this edition

“Geometry” is retained and explained through “land-measuring”; “making alike” keeps the unusual substantive rather than substituting an established modern theorem. The note's area-and-side explanation is an interpretive aid. Neither ἄλογος nor ἀσύμμετρος is attested by this passage merely because those concepts help explain it.

Disclosure of mathematical relation — διαφανής (diaphanēs)

διαφανής (diaphanēs)

Disclosure of mathematical relation. The making alike becomes evident to understanding; the term is distinct from the description of invisible middle beings being seen through. Attested: διαφανής (990d). Geometry and the making alike of unlike numbers.

Plane-number inquiry — γεωμετρία (geōmetria)

γεωμετρία (geōmetria)

Plane-number inquiry. The familiar land-measuring name is criticized and replaced by an account of making unlike numbers alike with respect to plane figures. Attested: γεωμετρίαν (990d). Geometry and the making alike of unlike numbers.

Numbers and dimensions — ὅμοιος (homoios)

ὅμοιος (homoios)

Numbers and dimensions. Numbers unlike one another by nature are considered in a plane relation, and the next numbers are likened to solid nature; likeness does not erase numerical difference. Attested: ὁμοίων; ὁμοίους (990d). Geometry and the making alike of unlike numbers.

Geometrical assimilation — ὁμοίωσις (homoiōsis)

ὁμοίωσις (homoiōsis)

Geometrical assimilation. The evident relation of naturally unlike numbers in the plane domain; the edition's interpretation uses areas whose sides are incommensurable. Attested: ὁμοίωσις (990d). Geometry and the making alike of unlike numbers.

Misleading subject-title — ὄνομα (onoma)

ὄνομα (onoma)

Misleading subject-title. Geometry's familiar name is criticized because land-measuring does not capture the higher plane-number inquiry. Attested: ὄνομα (990d). Geometry and the making alike of unlike numbers.

Two-dimensional domain — ἐπίπεδος (epipedos)

ἐπίπεδος (epipedos)

Two-dimensional domain. The substantivized plural marks the plane figures with respect to which numerical likeness becomes evident. Attested: ἐπιπέδων (990d). Geometry and the making alike of unlike numbers.

Plane domain — μοῖρα (moira)

μοῖρα (moira)

Plane domain. The portion of plane figures specifies the respect in which geometry makes unlike numbers alike. Attested: μοῖραν (990d). Geometry and the making alike of unlike numbers.

Criticism of a name — γελοῖος (geloios)

γελοῖος (geloios)

Criticism of a name. The emphatic adjective evaluates the name geometry as inadequate to the subject's higher content, not the mathematical inquiry as worthless. Attested: γελοῖον (990d). Geometry and the making alike of unlike numbers.

Divine mathematical intelligibility — θαῦμα (thauma)

θαῦμα (thauma)

Divine mathematical intelligibility. The geometrical relation is described as a divine wonder to the person capable of understanding it, not as an exemption from explanation. Attested: θαῦμα (990d). Geometry and the making alike of unlike numbers.

Likeness with respect to plane figures (contextual phrase)

πρὸς τὴν τῶν ἐπιπέδων μοῖραν (990d)

The prepositional phrase specifies the respect in which geometry's making alike takes place. In the edition's interpretation, areas can have a definite numerical ratio while their sides are incommensurable; the sides are not thereby turned into commensurable lengths. Geometry and the making alike of unlike numbers.