Page 397 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

rable in square, making the sum of the squares on them
medial, and twice the (rectangle contained) by medial,
zþ them. Thus, LN is the irrational (straight-line) called
and, furthermore, the (sum of the) squares on them in-
commensurable with twice the (rectangle contained) by

that which with a medial (area) makes a medial whole
[Prop. 10.78]. And it is the square-root of area AB.
Thus, the square-root of area (AB) is that (straight-
line) which with a medial (area) makes a medial whole.
(Which is) the very thing it was required to show.
A
B H
.
Proposition 97
Τὸ ἀπὸ ἀποτομῆς παρὰ ῥητὴν παραβαλλόμενον πλάτος The (square) on an apotome, applied to a rational
G Z C A F B N K M
ποιεῖ ἀποτομὴν πρώτην. (straight-line), produces a first apotome as breadth.
G

D E X

D
L
O
H
E
῎Εστω ἀποτομὴ ὴ ΑΒ, ῥητὴ δὲ ἡ ΓΔ, καὶ τῷ ἀπὸ τῆς Let AB be an apotome, and CD a rational (straight-
ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω τὸ ΓΕ πλάτος ποιοῦν line). And let CE, equal to the (square) on AB, have
τὴν ΓΖ· λέγω, ὅτι ἡ ΓΖ ἀποτομή ἐστι πρώτη. been applied to CD, producing CF as breadth. I say that
῎Εστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα ΑΗ, CF is a first apotome.
ΗΒ ῥηταί εἰσι δυνάμει μόνον σύμμετροι. καὶ τῷ μὲν ἀπὸ For let BG be an attachment to AB. Thus, AG and
τῆς ΑΗ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω τὸ ΓΘ, τῷ δὲ GB are rational (straight-lines which are) commensu-
ἀπὸ τῆς ΒΗ τὸ ΚΛ. ὅλον ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς ἀπὸ rable in square only [Prop. 10.73]. And let CH, equal
τῶν ΑΗ, ΗΒ· ὧν τὸ ΓΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ· λοιπὸν to the (square) on AG, and KL, (equal) to the (square)
ἄρα τὸ ΖΛ ἴσον ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ. τετμήσθω on BG, have been applied to CD. Thus, the whole of
ἡ ΖΜ δίχα κατὰ τὸ Ν σημεῖον, καὶ ἤχθω διὰ τοῦ Ν τῇ ΓΔ CL is equal to the (sum of the squares) on AG and GB,
παράλληλος ἡ ΝΞ· ἑκάτερον ἄρα τῶν ΖΞ, ΛΝ ἴσον ἐστὶ τῷ of which CE is equal to the (square) on AB. The re-
ὑπὸ τῶν ΑΗ, ΗΒ. καὶ ἐπεὶ τὰ ἀπὸ τῶν ΑΗ, ΗΒ ῥητά ἐστιν, mainder FL is thus equal to twice the (rectangle con-
καί ἐστι τοῖς ἀπὸ τῶν ΑΗ, ΗΒ ἴσον τὸ ΔΜ, ῥητὸν ἄρα tained) by AG and GB [Prop. 2.7]. Let FM have been
ἐστὶ τὸ ΔΜ. καὶ παρὰ ῥητὴν τὴν ΓΔ παραβέβληται πλάτος cut in half at point N. And let NO have been drawn
ποιοῦν τὴν ΓΜ· ῥητὴ ἄρα ἐστὶν ἡ ΓΜ καὶ σύμμετρος τῇ ΓΔ through N, parallel to CD. Thus, FO and LN are each
μήκει. πάλιν, ἐπεὶ μέσον ἐστὶ τὸ δὶς ὑπὸ τῶν ΑΗ, ΗΒ, καὶ equal to the (rectangle contained) by AG and GB. And
τῷ δὶς ὑπὸ τῶν ΑΗ, ΗΒ ἴσον τὸ ΖΛ, μέσον ἄρα τὸ ΖΛ. καὶ since the (sum of the squares) on AG and GB is rational,
παρὰ ῥητὴν τὴν ΓΔ παράκειται πλάτος ποιοῦν τὴν ΖΜ· ῥητὴ and DM is equal to the (sum of the squares) on AG and
ἄρα ἐστὶν ἡ ΖΜ καὶ ἀσύμμετρος τῇ ΓΔ μήκει. καὶ ἐπεὶ τὰ GB, DM is thus rational. And it has been applied to the
μὲν ἀπὸ τῶν ΑΗ, ΗΒ ῥητά ἐστιν, τὸ δὲ δὶς ὑπὸ τῶν ΑΗ, rational (straight-line) CD, producing CM as breadth.
ΗΒ μέσον, ἀσύμμετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΗ, ΗΒ τῷ δὶς Thus, CM is rational, and commensurable in length with
ὑπὸ τῶν ΑΗ, ΗΒ. καὶ τοῖς μὲν ἀπὸ τῶν ΑΗ, ΗΒ ἴσον ἐστὶ CD [Prop. 10.20]. Again, since twice the (rectangle con-
τὸ ΓΛ, τῷ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ τὸ ΖΛ· ἀσύμμετρον ἄρα tained) by AG and GB is medial, and FL (is) equal to
ἐστὶ τὸ ΔΜ τῷ ΖΛ. ὡς δὲ τὸ ΔΜ πρὸς τὸ ΖΛ, οὕτως ἐστὶν twice the (rectangle contained) by AG and GB, FL (is)
ἡ ΓΜ πρὸς τὴν ΖΜ. ἀσύμμετρος ἄρα ἐστὶν ἡ ΓΜ τῇ ΖΜ thus a medial (area). And it is applied to the rational
μήκει. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ἄρα ΓΜ, ΜΖ ῥηταί εἰσι (straight-line) CD, producing FM as breadth. FM is

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