Page 379 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

(to be) impossible [Prop. 10.79]. Thus, another straight-
line cannot be (so) attached to AB.
Thus, only one straight-line, which is incommensu-
rable in square with the whole, and (together) with the
Vroi tr toi and, moreover, the (sum of the) squares on them incom-
whole makes the squares on them (added) together me-
dial, and twice the (rectangle contained) by them medial,

mensurable with the (rectangle contained) by them, can
be attached to AB. (Which is) the very thing it was re-
quired to show.

† This proposition is equivalent to Prop. 10.47, with minus signs instead of plus signs.

Definitions III
.
ιαʹ. ῾Υποκειμένης ῥητῆς καὶ ἀποτομῆς, ἐὰν μὲν ἡ ὅλη τῆς 11. Given a rational (straight-line) and an apotome, if
προσαρμοζούσης μεῖζον δύνηται τῷ ἀπὸ συμμέτρου ἑαυτῇ the square on the whole is greater than the (square on a
μήκει, καὶ ἡ ὅλη σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, straight-line) attached (to the apotome) by the (square)
καλείσθω ἀποτομὴ πρώτη. on (some straight-line) commensurable in length with
ιβʹ. ᾿Εὰν δὲ ἡ προσαρμόζουσα σύμμετρος ᾖ τῇ ἐκκειμένῃ (the whole), and the whole is commensurable in length
ῥητῇ μήκει, καὶ ἡ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται with the (previously) laid down rational (straight-line),
τῷ ἀπὸ συμμέτρου ἑαυτῇ, καλείσθω ἀποτομὴ δευτέρα. then let the (apotome) be called a first apotome.
ιγʹ. ᾿Εὰν δὲ μηδετέρα σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ 12. And if the attached (straight-line) is commen-
μήκει, ἡ δὲ ὅλη τῆς προσαρμοζούσης μεῖζον δύνηται τῷ surable in length with the (previously) laid down ra-
ἀπὸ συμμέτρου ἑαυτῇ, καλείσθω ἀποτομὴ τρίτη. tional (straight-line), and the square on the whole is
ιδʹ. Πάλιν, ἐὰν ἡ ὅλη τῆς προσαρμοζούσης μεῖζον greater than (the square on) the attached (straight-line)
δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ [μήκει], ἐὰν μὲν ἡ ὅλη by the (square) on (some straight-line) commensurable
σύμμετρος ᾖ τῇ ἐκκειμένῃ ῥητῇ μήκει, καλείσθω ἀποτομὴ (in length) with (the whole), then let the (apotome) be
τετάρτη. called a second apotome.
ιεʹ. ᾿Εὰν δὲ ἡ προσαρμόζουσα, πέμπτη. 13. And if neither of (the whole or the attached
ιϛʹ. ᾿Εὰν δὲ μηδετέρα, ἕκτη. straight-line) is commensurable in length with the (previ-
ously) laid down rational (straight-line), and the square
on the whole is greater than (the square on) the attached
(straight-line) by the (square) on (some straight-line)
commensurable (in length) with (the whole), then let the
(apotome) be called a third apotome.
14. Again, if the square on the whole is greater
than (the square on) the attached (straight-line) by the
(square) on (some straight-line) incommensurable [in
peþ (straight-line), then let the (apotome) be called a fourth
length] with (the whole), and the whole is commensu-
rable in length with the (previously) laid down rational

apotome.
15. And if the attached (straight-line is commensu-
rable), a fifth (apotome).
16. And if neither (the whole nor the attached
straight-line is commensurable), a sixth (apotome).

Proposition 85
.
Εὑρεῖν τὴν πρώτην ἀποτομήν. To find a first apotome.

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