Page 388 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ΔΚ ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ. ἔστι δὲ καὶ τὸ tracted (from LM), having with it the common angle
ΑΚ ἴσον τοῖς ΛΜ, ΝΞ τετραγώνοις· λοιπὸν ἄρα τὸ ΑΒ ἴσον LPM. Thus, the squares LM and NO are about the
ἐστὶ τῷ ΣΤ. τὸ δὲ ΣΤ τὸ ἀπὸ τῆς ΛΝ ἐστι τετράγωνον· τὸ same diagonal [Prop. 6.26]. Let PR be their (com-
ἄρα ἀπὸ τῆς ΛΝ τετράγωνον ἴσον ἐστὶ τῷ ΑΒ· ἡ ΛΝ ἄρα mon) diagonal, and let the (rest of the) figure have been
δύναται τὸ ΑΒ. λέγω δή, ὅτι ἡ ΛΝ ἀποτομή ἐστιν. drawn. Therefore, since the rectangle contained by AF
᾿Επεὶ γὰρ ῥητόν ἐστιν ἑκάτερον τῶν ΑΙ, ΖΚ, καί ἐστιν and FG is equal to the square EG, thus as AF is to
ἴσον τοῖς ΛΜ, ΝΞ, καὶ ἑκάτερον ἄρα τῶν ΛΜ, ΝΞ ῥητόν EG, so EG (is) to FG [Prop. 6.17]. But, as AF (is)
ἐστιν, τουτέστι τὸ ἀπὸ ἑκατέρας τῶν ΛΟ, ΟΝ· καὶ ἑκατέρα to EG, so AI (is) to EK, and as EG (is) to FG, so
ἄρα τῶν ΛΟ, ΟΝ ῥητή ἐστιν. πάλιν, ἐπεὶ μέσον ἐστὶ τὸ ΔΘ EK is to KF [Prop. 6.1]. Thus, EK is the mean pro-
καί ἐστιν ἴσον τῷ ΛΞ, μέσον ἄρα ἐστὶ καὶ τὸ ΛΞ. ἐπεὶ οὖν τὸ portional to AI and KF [Prop. 5.11]. And MN is also
μὲν ΛΞ μέσον ἐστίν, τὸ δὲ ΝΞ ῥητόν, ἀσύμμετρον ἄρα ἐστὶ the mean proportional to LM and NO, as shown before
τὸ ΛΞ τῷ ΝΞ· ὡς δὲ τὸ ΛΞ πρὸς τὸ ΝΞ, οὕτως ἐστὶν ἡ ΛΟ [Prop. 10.53 lem.]. And AI is equal to the square LM,
πρὸς τὴν ΟΝ· ἀσύμμετρος ἄρα ἐστὶν ἡ ΛΟ τῇ ΟΝ μήκει. and KF to NO. Thus, MN is also equal to EK. But, EK
καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΛΟ, ΟΝ ἄρα ῥηταί εἰσι δυνάμει is equal to DH, and MN to LO [Prop. 1.43]. Thus, DK
μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΛΝ. καὶ δύναται τὸ is equal to the gnomon UV W and NO. And AK is also
ΑΒ χωρίον· ἡ ἄρα τὸ ΑΒ χωρίον δυναμένη ἀποτομή ἐστιν. equal to (the sum of) the squares LM and NO. Thus, the
᾿Εὰν ἄρα χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τὰ ἑξῆς. remainder AB is equal to ST . And ST is the square on
LN. Thus, the square on LN is equal to AB. Thus, LN is
the square-root of AB. So, I say that LN is an apotome.
For since AI and FK are each rational (areas), and
are equal to LM and NO (respectively), thus LM and
NO—that is to say, the (squares) on each of LP and PN
(respectively)—are also each rational (areas). Thus, LP
and PN are also each rational (straight-lines). Again,
since DH is a medial (area), and is equal to LO, LO
is thus also a medial (area). Therefore, since LO is
medial, and NO rational, LO is thus incommensurable
with NO. And as LO (is) to NO, so LP is to PN
[Prop. 6.1]. LP is thus incommensurable in length with
bþ which are) commensurable in square only. Thus, LN is
PN [Prop. 10.11]. And they are both rational (straight-
lines). Thus, LP and PN are rational (straight-lines

an apotome [Prop. 10.73]. And it is the square-root of
area AB. Thus, the square-root of area AB is an apo-
tome.
Thus, if an area is contained by a rational (straight-
line), and so on . . . .

.
Proposition 92
᾿Εὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς δευτέρας, If an area is contained by a rational (straight-line) and
ἡ τὸ χωρίον δυναμένη μέσης ἀποτομή ἐστι πρώτη. a second apotome then the square-root of the area is a
first apotome of a medial (straight-line).
Α ∆ Ε Ζ Η Ν A D E F G N
Λ Ο L O
Φ V

Σ Ξ S P
Χ Π W Q
Υ U
Γ Β Θ Ι Κ C B H I K
Ρ Μ R M
Τ T

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