§3#1## γ΄. Παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστὶ καὶ ἔτι ὑπερέχει ἐλάσσονι μὲν ἢ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις.
## Proposition 3 The perimeter of every circle is three times the diameter and further exceeds it by less than a seventh part of the diameter, but by more than ten seventy-firsts.
Ἔστω κύκλος καὶ διάμετρος ἡ ΑΓ καὶ κέντρον τὸ Ε καὶ ἡ ΓΛΖ ἐφαπτομένη καὶ ἡ ὑπὸ ΖΕΓ τρίτου ὀρθῆς· ἡ ΕΖ ἄρα πρὸς ΖΓ λόγον ἔχει, ὃν τς τρὸς ρνγ, ἡ δὲ ΕΓ πρὸς ΓΖ λόγον ἔχει, ὃν σξε πρὸς ρνγ, Τετμήσθω οὖν ἡ ὑπὸ ΖΕΓ δίχα τῇ ΕΗ·
Let there be a circle, and the diameter ΑΓ, and the center Ε, and the tangent ΓΛΖ, and the angle ΖΕΓ one-third of a right angle; therefore ΕΖ has to ΖΓ the ratio which 306 has to 153, while ΕΓ has to ΓΖ the ratio which 265 has to 153.
ἔστιν ἄρα, ὡς ἡ ΖΕ πρὸς ΕΓ, ἡ ΖΗ πρὸς ΗΓ.
Let, then, the angle ΖΕΓ be bisected by ΕΗ; therefore, as ΖΕ is to ΕΓ, so is ΖΗ to ΗΓ.
Ὡς ἄρα συναμφότερος ἡ ΖΕ. ΕΓ πρὸς ΖΓ, ἡ ΕΓ πρὸς ΓΗ· ὥστε ἡ ΓΕ πρὸς ΓΗ μείζονα λόγον ἔχει ἤπερ φοα πρὸς ρνγ.
Therefore, as the sum of ΖΕ and ΕΓ is to ΖΓ, so is ΕΓ to ΓΗ; so that ΓΕ has to ΓΗ a greater ratio than 571 has to 153.
Ἡ ΕΗ ἄρα πρὸς ΗΓ δυνάμει λόγον ἔχει, ὃν Μ θυν πρὸς Μ γυθ· μήκει ἄρα, ὃν φU+A7FCα η΄ πρὸς ρνγ.
Therefore ΕΗ has to ΗΓ in square the ratio which 349450 has to 23409, and in length the ratio which 591 and 1/8 has to 153.
Πάλιν δίχα ἡ ὑπὸ ΗΕΓ τῇ ΕΘ· διὰ τὰ αὐτὰ ἄρα ἡ ΕΓ πρὸς ΓΘ μείζονα λόγον ἔχει ἢ ὃν αρξβ η΄πρὸς ρνγ· ἡ ΘΕ ἄρα πρὸς ΘΓ μείζονα λόγον ἔχει ἢ ὃν αροβ η΄ πρὸς ρνγ.
Again, let the angle ΗΕΓ be bisected by ΕΘ; therefore, for the same reasons, ΕΓ has to ΓΘ a greater ratio than 1162 and 1/8 has to 153; therefore ΘΕ has to ΘΓ a greater ratio than 1172 and 1/8 has to 153.
Ἔτι δίχα ἡ ὑπὸ ΘΕΓ τῇ ΕΚ· ἡ ΕΓ ἄρα πρὸς ΓΚ μείζονα λόγον ἐχει ἢ ὃν βτλδ δ᾿ πρὸς ρνγ· ἡ ΕΚ ἄρα πρὸς ΓΚ μείζονα ἢ ὃν βτλθ δ᾿ πρὸς ρνγ.
Further, let the angle ΘΕΓ be bisected by ΕΚ; therefore ΕΓ has to ΓΚ a greater ratio than 2334 and 1/4 has to 153; therefore ΕΚ has to ΓΚ a greater ratio than 2339 and 1/4 has to 153.
Ἔτι δίχα ἡ ὑπὸ ΚΕΓ τῇ ΛΕ· ἡ ΕΓ ἄρα πρὸς ΛΓ μείζονα λόγον ἔχει ἤπερ τὰ δχογ U+2220΄ πρὸς ρνγ.
Further, let the angle ΚΕΓ be bisected by ΛΕ; therefore ΕΓ has to ΛΓ a greater ratio than 4673 and 1/2 has to 153.
Ἐπεὶ οὖν ἡ ὑπὸ ΖΕΓ τρίτου οὖσα ὀρθῆς τέτμηται τετράκις δίχα, ἡ ὑπὸ ΛΕΓ ὀρθῆς ἐστι μη΄.
Since, then, the angle ΖΕΓ, being one-third of a right angle, has been bisected four times, the angle ΛΕΓ is one ninety-sixth of a right angle.
Κείσθω οὖν αὐτῇ ἴση πρὸς τῷ Ε ἡ ὑπὸ ΓΕΜ· ἡ ἄρα ὑπὸ ΛΕΜ ὀρθῆς ἐστι κδ΄.
Let there be placed, therefore, equal to it at Ε, the angle ΓΕΜ; therefore the angle ΛΕΜ is one forty-eighth of a right angle.
Καὶ ἡ ΛΜ ἄρα εὐθεῖα τοῦ περὶ τὸν κύκλον ἐστὶ πολυγώνου πλευρὰ πλευρὰς ἔχοντος (??)ς.
And therefore the straight line ΛΜ is a side of a polygon of 96 sides circumscribed about the circle.
Ἐπεὶ οὖν ἡ ΕΓ πρὸς τὴν ΓΛ ἐδείχθη μείζονα λόγον ἔχουσα ἤπερ δχογ U+2220΄ πρὸς ρνγ, ἀλλὰ τῆς μὲν ΕΓ διπλῆ ἡ ΑΓ, τῆς δὲ ΓΛ διπλασίων ἡ ΛΜ, καὶ ἡ ΑΓ ἄρα πρὸς τὴν τοῦ (??)ς γώνου περίμετρον μείζονα λόγον ἔχει ἤπερ δχογ U+2220΄πρὸς Μ δχπη.
Since, then, ΕΓ was shown to have to ΓΛ a greater ratio than 4673 and 1/2 has to 153, but ΑΓ is double of ΕΓ, while ΛΜ is double of ΓΛ, therefore ΑΓ also has to the perimeter of the 96-sided polygon a greater ratio than 4673 and 1/2 has to 14688.
Καὶ ἐστιν τριπλασία, καὶ ὑπερέχουσιν χξζ U+2220΄, ἅπερ τῶν δχογ U+2220΄ ἐλάττονά ἐστιν ἢ τὸ ἕβδομον· ὥστε τὸ πολύγωνον τὸ περὶ τὸν κύκλον τῆς διαμέτρου ἐστὶ τριπλάσιον καὶ ἐλάττονι ἢ τῷ ἑβδόμῳ μέρει μεῖζον· ἡ τοῦ κύκλου ἄρα περίμετρος πολὺ μᾶλλον ἐλάσσων ἐστὶν ἢ τριπλασίων καὶ ἑβδόμῳ μέρει μείζων.
And it is three times, and they exceed by 667 and 1/2, which is less than a seventh of 4673 and 1/2; so that the polygon circumscribed about the circle is three times the diameter and greater by less than a seventh part; therefore the perimeter of the circle is much more less than three times and a seventh part greater.