Stereometry and three dimensional number

Use in the book

After geometry's making alike in relation to plane figures, the Athenian turns to the numbers “three times increased” and like the solid nature (990d). The supplied note interprets these as products of three factors, such as cubes. A cube is a particularly clear example because its three factors are equal; the broader description of three-dimensional numbers need not be limited to that example.

Numbers that have become unlike are again made like by another craft, called stereometry by those who have encountered it. The recurrence of likeness extends the previous explanatory pattern. A new dimensional treatment makes relations intelligible without making all magnitudes numerically identical. The text compresses the operation and does not give a specific construction with a stated diagram, named instrument, and sequence of steps.

The following account of doubling reaches the solid and tangible by passing from one to eight (991a). On the edition's interpretation, doubling the side of a cube multiplies its volume by eight. This is distinct from finding a cube whose volume is twice a given cube's volume. The latter calls for a cube-root relation in modern terms. The adjacency of solids and doubling makes comparison with such problems natural, but the lexicon should not assert that the sentence explicitly states and solves “the duplication of the cube.”

The name has a consequential textual basis. Burnet prints stereometria, supported by Theon of Smyrna and a corrector of L, where A and O have geometria. The accepted text thus distinguishes the plane and solid studies by different names. The normalized headword is legitimate because the accusative στερεομετρίαν actually stands in that reading text. Its manuscript competitors belong to the apparatus, not to a second speech in which the Athenian changes the title.

The movement from arithmetic through plane and solid dimensions prepares the more general proportional claim that nature articulates form and kind. Stereometry is therefore a necessary stage in the supplied order, not a decorative addition or a digression into practical surveying. Its place in the unified curriculum matters as much as the familiar etymology of its name.

Places to read

1. 990dτοὺς τρὶς ηὐξημένους — The numbers are described as three times increased
2. 990dτῇ στερεᾷ φύσει ὁμοίους — The relevant likeness is to solid nature
3. 990dτοὺς δὲ ἀνομοίους — The contrary case consists of unlike numbers
4. 990dἑτέρᾳ τέχνῃ ὁμοιοῖ — A further craft makes them like
5. 990dστερεομετρίαν ἐκάλεσαν — The accepted name is stereometry
6. 991aεἰς τὸ στερεόν τε καὶ ἁπτὸν — The subsequent proportional sequence reaches solid and tangible magnitude
7. 991aἀφ’ ἑνὸς εἰς ὀκτὼ — Its stated numerical reach is from one to eight

Translation in this edition

The note explains both “three times increased” and the reading “stereometry,” marking the mathematical expansion as interpretation. “Three times” translates τρίς, Bekker's correction; “three” as a cardinal number would follow a different reading. The English distinguishes this study from geometry rather than silently harmonizing their titles.

Solid-number formation — αὐξάνειν (auxanein)

αὐξάνειν (auxanein)

Solid-number formation. The perfect passive participle describes numbers increased three times, interpreted through three-dimensional products such as cubes. Attested: ηὐξημένους (990d). Stereometry and three dimensional number.

Solid-dimensional treatment — ὁμοιοῦν (homoioun)

ὁμοιοῦν (homoioun)

Solid-dimensional treatment. Another craft makes unlike numbers alike, extending the geometrical language into the solid domain. Attested: ὁμοιοῖ (990d). Stereometry and three dimensional number.

Three-dimensional magnitude — στερεός (stereos)

στερεός (stereos)

Three-dimensional magnitude. The mathematical solid nature follows plane geometry; later solid and tangible names the dimensional stage reaching one to eight. Attested: στερεᾷ (990d). Stereometry and three dimensional number.

Study of solids — στερεομετρία (stereometria)

στερεομετρία (stereometria)

Study of solids. The accepted reading names the craft following plane geometry; products of three factors and cubes are explanatory interpretations. Attested: στερεομετρίαν (990d). Stereometry and three dimensional number.

Solid dimensional stage — ἁπτός (haptos)

ἁπτός (haptos)

Solid dimensional stage. The adjective accompanies solid in the sequence that reaches from one to eight; it does not make the whole number curriculum a study of bodily counting. Attested: ἁπτὸν (991a). Stereometry and three dimensional number.

Dimensional increase — τρίς (tris)

τρίς (tris)

Dimensional increase. The accepted adverb describes three-times-increased numbers, interpreted as products of three factors; it is not the cardinal three from the alternative reading. Attested: τρὶς (990d). Stereometry and three dimensional number.

Objects of solid assimilation — ἀνόμοιος (anomoios)

ἀνόμοιος (anomoios)

Objects of solid assimilation. The unlike numbers are made like by the further craft of stereometry; the operation is not spelled out as a full construction. Attested: ἀνομοίους (990d). Stereometry and three dimensional number.

Three times increased (contextual phrase)

τοὺς τρὶς ηὐξημένους (990d)

The accepted adverb is three times, not the cardinal three. The edition interprets the expression through products of three factors, such as cubes, while the sentence itself supplies no complete stereometric construction. Stereometry and three dimensional number.