Page 187 - Euclid's Elements of Geometry

ST EW þ.

ELEMENTS BOOK 6

τῷ Δ [ἐπειδήπερ τὸ ΠΒ τῷ ΗΠ ὅμοιόν ἐστιν]· ὅπερ ἔδει allelogram) OB. But, OB is equal to T E, since side
ποιῆσαι. AE is equal to side EB [Prop. 6.1]. Thus, T E is also
equal to PB. Let (parallelogram) OS have been added
to both. Thus, the whole (parallelogram) T S is equal to
the gnomon V WU. But, gnomon V WU was shown (to
be) equal to C. Therefore, (parallelogram) T S is also
equal to (figure) C.
Thus, the parallelogram ST , equal to the given rec-
tilinear figure C, has been applied to the given straight-
line AB, falling short by the parallelogrammic figure QB,
kjþ
which is similar to D [inasmuch as QB is similar to GQ
[Prop. 6.24] ]. (Which is) the very thing it was required
to do.
† This proposition is a geometric solution of the quadratic equation x −α x+β = 0. Here, x is the ratio of a side of the deficit to the corresponding
2
side of figure D, α is the ratio of the length of AB to the length of that side of figure D which corresponds to the side of the deficit running along
2
AB, and β is the ratio of the areas of figures C and D. The constraint corresponds to the condition β < α /4 for the equation to have real roots.
Only the smaller root of the equation is found. The larger root can be found by a similar method. †

Proposition 29
.
Παρὰ τὴν δοθεῖσαν εὐθεῖαν τῷ δοθέντι εὐθυγράμμῳ To apply a parallelogram, equal to a given rectilin-
ἴσον παραλληλόγραμμον παραβαλεῖν ὑπερβάλλον εἴδει πα- ear figure, to a given straight-line, (the applied parallelo-
G Z to a given (parallelogram). L M K H
ραλληλογράμμῳ ὁμοίῳ τῷ δοθέντι. gram) overshooting by a parallelogrammic figure similar

F
C
D
A E B Y F D V

H X A E X B W P

O
Q
N
G
῎Εστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν Let AB be the given straight-line, and C the given
εὐθύγραμμον, ᾧ δεῖ ἴσον παρὰ τὴν ΑΒ παραβαλεῖν, τὸ Γ, rectilinear figure to which the (parallelogram) applied to
ᾧ δὲ δεῖ ὅμοιον ὑπερβάλλειν, τὸ Δ· δεῖ δὴ παρὰ τὴν ΑΒ AB is required (to be) equal, and D the (parallelogram)
εὐθεῖαν τῷ Γ εὐθυγράμμῳ ἴσον παραλληλόγραμμον παρα- to which the excess is required (to be) similar. So it is
βαλεῖν ὑπερβάλλον εἴδει παραλληλογράμμῳ ὁμοίῳ τῷ Δ. required to apply a parallelogram, equal to the given rec-
Τετμήσθω ἡ ΑΒ δίχα κατὰ τὸ Ε, καὶ ἀναγεγράθω tilinear figure C, to the given straight-line AB, overshoot-
ἀπὸ τὴς ΕΒ τῷ Δ ὅμοιον καὶ ὁμοίως κείμενον παραλ- ing by a parallelogrammic figure similar to D.
ληλόγραμμον τὸ ΒΖ, καὶ συναμφοτέροις μὲν τοῖς ΒΖ, Γ Let AB have been cut in half at (point) E [Prop. 1.10],
ἴσον, τῷ δὲ Δ ὅμοιον καὶ ὁμοίως κείμενον τὸ αὐτὸ συ- and let the parallelogram BF, (which is) similar, and
νεστάτω τὸ ΗΘ. ὁμόλογος δὲ ἔστω ἡ μὲν ΚΘ τῇ ΖΛ, ἡ δὲ similarly laid out, to D, have been described on EB
ΚΗ τῇ ΖΕ. καὶ ἐπεὶ μεῖζόν ἐστι τὸ ΗΘ τοῦ ΖΒ, μείζων ἄρα [Prop. 6.18]. And let (parallelogram) GH have been con-
ἐστὶ καὶ ἡ μὲν ΚΘ τῆς ΖΛ, ἡ δὲ ΚΗ τῇ ΖΕ. ἐκβεβλήσθωσαν structed (so as to be) both similar, and similarly laid out,
αἱ ΖΛ, ΖΕ, καὶ τῇ μὲν ΚΘ ἴση ἔστω ἡ ΖΛΜ, τῇ δὲ ΚΗ ἴση to D, and equal to the sum of BF and C [Prop. 6.25].
ἡ ΖΕΝ, καὶ συμπεπληρώσθω τὸ ΜΝ· τὸ ΜΝ ἄρα τῷ ΗΘ And let KH correspond to FL, and KG to FE. And since
ἴσον τέ ἐστι καὶ ὅμοιον. ἀλλὰ τὸ ΗΘ τῷ ΕΛ ἐστιν ὅμοιον· (parallelogram) GH is greater than (parallelogram) FB,

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