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Euclid · Elements §10.def1.1-10.def1.4

Definitions of Commensurability and Rationality

Passage 160 of 316 · Greek

Summary

Defines commensurable and incommensurable magnitudes and straight lines (both in length and in square), and introduces the definitions of rational and irrational straight lines and squares with respect to an assigned straight line.

§10.def1.1σύμμετρα μεγέθη λέγεται τὰ τῷ αὐτῷ μέτρῳ μετρούμενα, ἀσύμμετρα δέ, ὧν μηδὲν ἐνδέχεται κοινὸν μέτρον γενέσθαι.
Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable of which no common measure can admit of coming into being.
§10.def1.2εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπʼ αὐτῶν τετράγωνα τῷ αὐτῷ χωρίῳ μετρῆται, ἀσύμμετροι δέ, ὅταν τοῖς ἀπʼ αὐτῶν τετραγώνοις μηδὲν ἐνδέχηται χωρίον κοινὸν μέτρον γενέσθαι.
Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when no area can admit of coming into being as a common measure of the squares on them.
§10.def1.3τούτων ὑποκειμένων δείκνυται, ὅτι τῇ προτεθείσῃ εὐθείᾳ ὑπάρχουσιν εὐθεῖαι πλήθει ἄπειροι σύμμετροί τε καὶ ἀσύμμετροι αἱ μὲν μήκει μόνον, αἱ δὲ καὶ δυνάμει.
With these hypotheses, it is proved that there exist, straight lines infinite in multitude, both commensurable and incommensurable with an assigned straight line, some indeed in length only, others also in square.
καλείσθω οὖν ἡ μὲν προτεθεῖσα εὐθεῖα ῥητή, καὶ αἱ ταύτῃ σύμμετροι εἴτε μήκει καὶ δυνάμει εἴτε δυνάμει μόνον ῥηταί, αἱ δὲ ταύτῃ ἀσύμμετροι ἄλογοι καλείσθωσαν.
Let then the assigned straight line be called rational, and those commensurable with it, whether in length and in square or in square only, rational, but those incommensurable with it let be called irrational.
§10.def1.4καὶ τὸ μὲν ἀπὸ τῆς προτεθείσης εὐθείας τετράγωνον ῥητόν, καὶ τὰ τούτῳ σύμμετρα ῥητά, τὰ δὲ τούτῳ ἀσύμμετρα ἄλογα καλείσθω, καὶ αἱ δυνάμεναι αὐτὰ ἄλογοι, εἰ μὲν τετράγωνα εἴη, αὐταὶ αἱ πλευραί, εἰ δὲ ἕτερά τινα εὐθύγραμμα, αἱ ἴσα αὐτοῖς τετράγωνα ἀναγράφουσαι.
And let the square on the assigned straight line be called rational, and those areas commensurable with it rational, but those incommensurable with it irrational, and the straight lines which produce them irrational—if indeed they be squares, the sides themselves, but if they be any other rectilineal figures, the straight lines which describe squares equal to them.

Notes

  1. §10.def1.2δυνάμει — The dative δυνάμει here does not mean 'potentially' or 'by power', but geometrically 'in square'. It refers to comparing lines based on the areas of the squares described on them, rather than their literal lengths.
  2. §10.def1.4δυνάμεναι — The feminine plural present participle of the verb δύναμαι, which in geometry means 'to be equivalent in square to'. Here it refers to 'the straight lines which produce those areas (αὐτά, neuter plural referring to the rational or irrational areas) in square'.

Cite this passage

Euclid, Elements §10.def1.1-10.def1.4. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:10.def1.1-10.def1.4

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