Page 394 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

For since AK is rational, and is equal to the (sum of
the) squares LP and PN, the sum of the (squares) on
LP and PN is thus rational. Again, since DK is me-
dial, and DK is equal to twice the (rectangle contained)
by LP and PN, thus twice the (rectangle contained) by
LP and PN is medial. And since AI was shown (to be)
incommensurable with FK, the square on LP (is) thus
eþ in square, making the sum of the squares on them ra-
also incommensurable with the square on PN. Thus, LP
and PN are (straight-lines which are) incommensurable
tional, and twice the (rectangle contained) by them me-
dial. LN is thus the irrational (straight-line) called minor
[Prop. 10.76]. And it is the square-root of area AB.
Thus, the square-root of area AB is a minor (straight-
line). (Which is) the very thing it was required to show.

Proposition 95
.
᾿Εὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς πέμπτης, If an area is contained by a rational (straight-line) and
ἡ τὸ χωρίον δυναμένη [ἡ] μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά a fifth apotome then the square-root of the area is that
ἐστιν. (straight-line) which with a rational (area) makes a me-
dial 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 πέμπτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον δυ- tional (straight-line) AC and the fifth apotome AD. I
ναμένη [ἡ] μετὰ ῥητοῦ 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 which with a rational (area) makes a medial whole.
ΑΗ, ΗΔ ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ προ- For let DG be an attachment to AD. Thus, AG and
σαρμόζουσα ἡ ΗΔ σύμμετρός ἐστι μήκει τῇ ἐκκειμένῃ DG are rational (straight-lines which are) commensu-
ῥητῇ τῇ ΑΓ, ἡ δὲ ὅλη ἡ ΑΗ τῆς προσαρμοζούσης τῆς rable in square only [Prop. 10.73], and the attachment
ΔΗ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ. ἐὰν ἄρα GD is commensurable in length the the (previously) laid
τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΔΗ ἴσον παρὰ τὴν ΑΗ πα- down rational (straight-line) AC, and the square on the
ραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς ἀσύμμετρα αὐτὴν whole, AG, is greater than (the square on) the attach-
διελεῖ. τετμήσθω οὖν ἡ ΔΗ δίχα κατὰ τὸ Ε σημεῖον, καὶ ment, DG, by the (square) on (some straight-line) incom-
τῷ ἀπὸ τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω ἐλλεῖπον mensurable (in length) with (AG) [Def. 10.15]. Thus, if
εἴδει τετραγώνῳ καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· ἀσύμμετρος (some area), equal to the fourth part of the (square) on
ἄρα ἐστὶν ἡ ΑΖ τῇ ΖΗ μήκει. καὶ ἐπεὶ ἀσύμμετρός ἐστὶν ἡ DG, is applied to AG, falling short by a square figure,
ΑΗ τῇ ΓΑ μήκει, καί εἰσιν ἀμφότεραι ῥηταί, μέσον ἄρα ἐστὶ then it divides (AG) into (parts which are) incommensu-
τὸ ΑΚ. πάλιν, ἐπεὶ ῥητή ἐστιν ἡ ΔΗ καὶ σύμμετρος τῇ ΑΓ rable (in length) [Prop. 10.18]. Therefore, let DG have
μήκει, ῥητόν ἐστι τὸ ΔΚ. been divided in half at point E, and let (some area), equal
Συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ to the (square) on EG, have been applied to AG, falling
δὲ ΖΚ ἴσον τετράγωνον ἀφῃρήσθω τὸ ΝΞ περὶ τὴν αὐτὴν short by a square figure, and let it be the (rectangle con-
γωνίαν τὴν ὑπὸ ΛΟΜ· περὶ τὴν αὐτὴν ἄρα διάμετρόν ἐστι tained) by AF and FG. Thus, AF is incommensurable
τὰ ΛΜ, ΝΞ τετράγωνα. ἔστω αὐτῶν διάμετρος ἡ ΟΡ, καὶ in length with FG. And since AG is incommensurable

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