- Source
- Archimedes. Archimède, Volume 1. Mugler, Charles, editor. Paris: Les Belles Lettres, 1970.
- Translation
- AI-generated (gemini-3.5-flash) — machine translation; may contain errors
- Languages
- 日本語 · English · 简体中文 · 한국어
- CTS URN
- urn:cts:greekLit:tlg0552.tlg002.humanitext-grc1:3#2
- Generated
- Humanitext Reader · reader.humanitext.ai · 2026-10-26
Archimedes calculates the lower bound of the ratio of the circumference to the diameter using an inscribed 96-sided polygon, proving that the circumference is greater than three and ten seventy-firsts times the diameter, thereby concluding the final estimation of pi.
Ἔστω κύκλος καὶ διάμετρος ἡ ΑΓ, ἡ δὲ ὑπὸ ΒΑΓ τρίτου ὀρθῆς·1 ἡ ΑΒ ἄρα πρὸς ΒΓ ἐλάσσονα λόγον ἔχει ἢ ὃν αταν πρὸς ψπ. Δίχα ἡ ὑπὸ ΒΑΓ τῇ ΑΗ. Ἐπεὶ οὖν ἴση ἐστὶν ἡ ὑπὸ ΒΑΗ τῇ ὑπὸ ΗΓΒ, ἀλλὰ καὶ τῇ ὑπὸ ΗΑΓ, καὶ ἡ ὑπὸ ΗΓΒ τῇ ὑπὸ ΗΑΓ ἐστὶν ἴση. Καὶ κοινὴ ἡ ὑπὸ ΑΗΓ ὀρθή· καὶ τρίτη ἄρα ἡ ὑπὸ ΗΖΓ τρίτῃ τῇ ὑπὸ ΑΓΗ ἴση. Ἰσογώνιον ἄρα τὸ ΑΗΓ τῷ ΓΗΖ τριγώνῳ· ἔστιν ἄρα, ὡς ἡ ΑΗ πρὸς ΗΓ, ἡ ΓΗ πρὸς ΗΖ καὶ ἡ ΑΓ πρὸς ΓΖ. Ἀλλʼ ὡς ἡ ΑΓ πρὸς ΓΖ, συναμφότερος ἡ ΓΑΒ2 πρὸς ΒΓ· καὶ ὡς συναμφότερος ἄρα ἡ ΒΑΓ πρὸς ΒΓ, ἡ ΑΗ πρὸς ΗΓ. Διὰ τοῦτο οὖν ἡ ΑΗ πρὸς ΗΓ ἐλάσσονα λόγον ἔχει ἤπερ βϡια πρὸς ψπ, ἡ δὲ ΑΓ πρὸς τὴν ΓΗ ἐλάσσονα ἢ ὃν γιγ U+2220΄ δ΄ πρὸς ψπ. Δίχα ἡ ὑπὸ ΓΑΗ τῇ ΑΘ· ἡ ΑΘ ἄρα διὰ τὰ αὐτὰ πρὸς τὴν ΘΓ ἐλάσσονα λόγον ἔχει ἢ ὃν εϡκδ U+2220΄ δ΄ πρὸς ψπ ἢ ὃν αωκγ πρὸς σμ· ἑκατέρα γὰρ ἑκατέρας δ ιγ3΄· ὥστε ἡ ΑΓ πρὸς τὴν ΓΘ ἢ ὃν αωλη θ ια΄ πρὸς σμ. Ἔτι δίχα ἡ ὑπὸ ΘΑΓ τῇ ΚΑ· καὶ ὁ ΑΚ πρὸς τὴν ΚΓ ἐλάσσονα λόγον ἔχει ἢ ὃν αζ πρὸς ξς· ἑκατέρα γὰρ ἑκατέρας ια μ΄. Ἡ ΑΓ ἄρα πρὸς ΚΓ ἢ ὃν αθ ϛ΄ πρὸς ξς. Ἔτι δίχα ἡ ὑπὸ ΚΑΓ τῇ ΛΑ· ἡ ΑΛ ἄρα πρὸς ΛΓ ἐλάσσονα λόγον ἔχει ἢ ὃν τὰ βις ϛ΄ πρὸς ξς, ἡ δὲ ΑΓ πρὸς ΓΛ ἐλάσσονα ἢ τὰ βιζ δ΄ πρὸς ξς. Ἀνάπαλιν4 ἄρα ἡ περίμετρος τοῦ πολυγώνου πρὸς τὴν διάμετρον μείζονα λόγον ἔχει ἤπερ ςτλς πρὸς βιζ δ΄. ἅπερ τῶν βιζ δ΄ μείζονά ἐστιν ἢ τριπλασίονα καὶ δέκα οα΄· καὶ ἡ περίμετρος ἄρα τοῦ (??)ςγώνου τοῦ ἐν τῷ κύκλῳ τῆς διαμέτρου τριπλασίων ἐστὶ καὶ μείζων ἢ ι οα΄· ὥστε καὶ ὁ κύκλος ἔτι μᾶλλον τριπλασίων ἐστὶ καὶ μείζων ἢ ι οα΄. Ἡ ἄρα τοῦ κύκλου περίμετρος τῆς διαμέτρου τριπλασίων ἐστὶ καὶ ἐλάσσονι μὲν ἢ ἑβδόμῳ μέρει, μείζονι δὲ ἢ ι οα΄ μείζων.
Let there be a circle, and the diameter ΑΓ, and the angle ΒΑΓ one-third of a right angle; therefore ΑΒ has to ΒΓ a less ratio than 1351 has to 780. Let the angle ΒΑΓ be bisected by ΑΗ. Since, then, the angle ΒΑΗ is equal to the angle ΗΓΒ, but is also equal to the angle ΗΑΓ, the angle ΗΓΒ is also equal to the angle ΗΑΓ. And the common angle ΑΗΓ is a right angle; therefore the third angle ΗΖΓ is also equal to the third angle ΑΓΗ. Therefore the triangle ΑΗΓ is equiangular with the triangle ΓΗΖ; therefore, as ΑΗ is to ΗΓ, so is ΓΗ to ΗΖ and ΑΓ to ΓΖ. But as ΑΓ is to ΓΖ, so is the sum of ΓΑ and ΑΒ to ΒΓ; and as the sum of ΒΑ and ΑΓ is to ΒΓ, so is ΑΗ to ΗΓ. Therefore on this account ΑΗ has to ΗΓ a less ratio than 2911 has to 780, while ΑΓ has to ΓΗ a less ratio than 3013 and 1/2 and 1/4 has to 780. Let the angle ΓΑΗ be bisected by ΑΘ; therefore, for the same reasons, ΑΘ has to ΘΓ a less ratio than 5924 and 1/2 and 1/4 has to 780, or than 1823 has to 240 (for each of the latter is 4/13 of each of the former); so that ΑΓ has to ΓΘ a less ratio than 1838 and 9/11 has to 240. Further, let the angle ΘΑΓ be bisected by ΚΑ; and ΑΚ has to ΚΓ a less ratio than 1007 has to 66 (for each of the latter is 11/40 of each of the former). Therefore ΑΓ has to ΚΓ a less ratio than 1009 and 1/6 has to 66. Further, let the angle ΚΑΓ be bisected by ΛΑ; therefore ΑΛ has to ΛΓ a less ratio than 2016 and 1/6 has to 66, while ΑΓ has to ΓΛ a less ratio than 2017 and 1/4 has to 66. Conversely, therefore, the perimeter of the polygon has to the diameter a greater ratio than 6336 has to 2017 and 1/4, which is greater than three times and ten seventy-firsts of 2017 and 1/4; and therefore the perimeter of the 96-sided polygon inscribed in the circle is three times the diameter and greater than ten seventy-firsts; so that the circle also is much more three times and greater than ten seventy-firsts. Therefore the perimeter of the circle is three times the diameter, and exceeds it by less than a seventh part, but by more than ten seventy-firsts.
- 3ἡ δὲ ὑπὸ ΒΑΓ τρίτου ὀρθῆς
A formulaic expression representing an angle using the preposition ὑπό. The three points defining the angle (here ΒΑΓ) follow the feminine article ἡ. The copula ἐστί is omitted.
- 3συναμφότερος ἡ ΓΑΒ
The adjective συναμφότερος (both together) is used in the feminine singular (along with the article ἡ) to express the sum of the lengths of two line segments (ΓΑ and ΑΒ) as a single noun phrase. This is a unique shorthand for addition in Greek mathematics.
- 3ἑκατέρα γὰρ ἑκατέρας δ ιγ΄
Meaning 'for each [of the latter, nominative] is 4/13 of each [of the former, genitive]'. The Greek fractional notation δ ιγ΄ represents 4/13 by combining 4 (δ) and 13 (ιγ). By multiplying both 5924 3/4 and 780 by 4/13, the ratio is simplified to the integer ratio 1823 : 240.
- 3ἀνάπαλιν
An adverb meaning 'conversely' or 'by inversion'. In mathematical contexts, it denotes the operation of inverting a ratio (from A : B < C : D to B : A > D : C), which consequently reverses the direction of the inequality.