Page 395 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

καταγεγράφθω τὸ σχῆμα. ὁμοίως δὴ δείξομεν, ὅτι ἡ ΛΝ in length with CA, and both are rational (straight-lines),
δύναται τὸ ΑΒ χωρίον. λέγω, ὅτι ἡ ΛΝ ἡ μετὰ ῥητοῦ μέσον AK is thus a medial (area) [Prop. 10.21]. Again, since
τὸ ὅλον ποιοῦσά ἐστιν. DG is rational, and commensurable in length with AC,
᾿Επεὶ γὰρ μέσον ἐδείχθη τὸ ΑΚ καί ἐστιν ἴσον τοῖς ἀπὸ DK is a rational (area) [Prop. 10.19].
τῶν ΛΟ, ΟΝ, τὸ ἄρα συγκείμενον ἐκ τῶν ἀπὸ τῶν ΛΟ, ΟΝ Therefore, let the square LM, equal to AI, have been
μέσον ἐστίν. πάλιν, ἐπεὶ ῥητόν ἐστι τὸ ΔΚ καί ἐστιν ἴσον constructed. And let the square NO, equal to FK, (and)
τῷ δὶς ὑπὸ τῶν ΛΟ, ΟΝ, καὶ αὑτὸ ῥητόν ἐστιν. καὶ ἐπεὶ about the same angle, LPM, have been subtracted (from
ἀσύμμετρόν ἐστι τὸ ΑΙ τῷ ΖΚ, ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ NO). Thus, the squares LM and NO are about the same
ἀπὸ τῆς ΛΟ τῷ ἀπὸ τῆς ΟΝ· αἱ ΛΟ, ΟΝ ἄρα δυνάμει εἰσὶν diagonal [Prop. 6.26]. Let PR be their (common) diag-
ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ᾿ αὐτῶν onal, and let (the rest of) the figure have been drawn.
τετραγώνων μέσον, τὸ δὲ δὶς ὑπ᾿ αὐτῶν ῥητόν. ἡ λοιπὴ ἄρα So, similarly (to the previous propositions), we can show
ἡ ΛΝ ἄλογός ἐστιν ἡ καλουμένη μετὰ ῥητοῦ μέσον τὸ ὅλον that LN is the square-root of area AB. I say that LN is
ποιοῦσα· καὶ δύναται τὸ ΑΒ χωρίον. that (straight-line) which with a rational (area) makes a
῾Η τὸ ΑΒ ἄρα χωρίον δυναμένη μετὰ ῥητοῦ μέσον τὸ medial whole.
ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι. For since AK was shown (to be) a medial (area), and
is equal to (the sum of) the squares on LP and PN,
the sum of the (squares) on LP and PN is thus medial.
Again, since DK is rational, and is equal to twice the
(rectangle contained) by LP and PN, (the latter) is also
rational. And since AI is incommensurable with FK, the
(square) on LP is thus also incommensurable with the
(square) on PN. Thus, LP and PN are (straight-lines
which are) incommensurable in square, making the sum
þ the irrational (straight-line) called that which with a ra-
of the squares on them medial, and twice the (rectangle
contained) by them rational. Thus, the remainder LN is
tional (area) makes a medial whole [Prop. 10.77]. And it
is the square-root of area AB.
Thus, the square-root of area AB is that (straight-
line) which with a rational (area) makes a medial whole.
(Which is) the very thing it was required to show.

.
Proposition 96
᾿Εὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς ἕκτης, If an area is contained by a rational (straight-line) and
ἡ τὸ χωρίον δυναμένη μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά a sixth apotome then the square-root of the area is that
ἐστιν. (straight-line) which with a medial (area) makes a medial
whole.
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 ra-
ἀποτομῆς ἕκτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον δυ- tional (straight-line) AC and the sixth apotome AD. I
ναμένη [ἡ] μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν. say that the square-root of area AB is that (straight-line)
῎Εστω γὰρ τῇ ΑΔ προσαρμόζουσα ἡ ΔΗ· αἱ ἄρα ΑΗ, which with a medial (area) makes a medial whole.
ΗΔ ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ οὐδετέρα For let DG be an attachment to AD. Thus, AG and

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