Page 390 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

μέσης ἀποτομή ἐστι πρώτη· καὶ δύναται τὸ ΑΒ χωρίον. and FK [Prop. 5.11]. And MN is also the mean pro-
῾Η ἄρα τὸ ΑΒ χωρίον δυναμένη μέσης ἀποτομή ἐστι portional to the squares LM and NO [Prop. 10.53 lem.].
πρώτη· ὅπερ ἔδει δεῖξαι. And AI is equal to LM, and FK to NO. Thus, MN is
also equal to EK. But, DH [is] equal to EK, and LO
equal to MN [Prop. 1.43]. Thus, the whole (of) DK is
equal to the gnomon UV W and NO. Therefore, since the
whole (of) AK is equal to LM and NO, of which DK is
equal to the gnomon UV W and NO, the remainder AB
is thus equal to T S. And T S is the (square) on LN. Thus,
the (square) on LN is equal to the area AB. LN is thus
the square-root of area AB. [So], I say that LN is the
first apotome of a medial (straight-line).
For since EK is a rational (area), and is equal to
LO, LO—that is to say, the (rectangle contained) by LP
and PN—is thus a rational (area). And NO was shown
(to be) a medial (area). Thus, LO is incommensurable
with NO. And as LO (is) to NO, so LP is to PN
[Prop. 6.1]. Thus, LP and PN are incommensurable
in length [Prop. 10.11]. LP and PN are thus medial
(straight-lines which are) commensurable in square only,
and which contain a rational (area). Thus, LN is the first
gþ of a medial (straight-line). (Which is) the very thing it
apotome of a medial (straight-line) [Prop. 10.74]. And it
is the square-root of area AB.
Thus, the square root of area AB is the first apotome

was required to show.

† There is an error in the argument here. It should just say that LP and P N are commensurable in square, rather than in square only, since LP
and P N are only shown to be incommensurable in length later on.

.
Proposition 93
᾿Εὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς τρίτης, If an area is contained by a rational (straight-line) and
ἡ τὸ χωρίον δυναμένη μέσης ἀποτομή ἐστι δευτέρα. a third apotome then the square-root of the area is a sec-
ond 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
Υ Π U Q
Χ W
Γ Β Θ Ι Κ C B H I K

Ρ Μ R M
Τ T
Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ καὶ For let the area AB have been contained by the ratio-
ἀποτομῆς τρίτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον δυ- nal (straight-line) AC and the third apotome AD. I say
ναμένη μέσης ἀποτομή ἐστι δευτέρα. that the square-root of area AB is the second apotome of
῎Εστω γὰρ τῇ ΑΔ προσαρμόζουσα ἡ ΔΗ· αἱ ΑΗ, ΗΔ a medial (straight-line).
ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ οὐδετέρα τῶν For let DG be an attachment to AD. Thus, AG and
ΑΗ, ΗΔ σύμμετρός ἐστι μήκει τῇ ἐκκειμένῃ ῥητῇ τῇ ΑΓ, ἡ GD are rational (straight-lines which are) commensu-
δὲ ὅλη ἡ ΑΗ τὴς προσαρμοζούσης τῆς ΔΗ μεῖζον δύναται rable in square only [Prop. 10.73], and neither of AG
τῷ ἀπὸ συμμέτρου ἑαυτῇ. ἐπεὶ οὖν ἡ ΑΗ τῆς ΗΔ μεῖζον and GD is commensurable in length with the (previ-

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