← Archimedes

Measurement of a Circle

§3#1

Calculation of the Upper Bound for Pi using Circumscribed Polygons

Archimedes rigorously calculates the upper bound of the ratio of the circle's circumference to its diameter to be less than 3 and 1/7, by using circumscribed regular polygons up to 96 sides through successive bisections.

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§3#1

γ΄. Παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστὶ καὶ ἔτι ὑπερέχει1 ἐλάσσονι μὲν ἢ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις. 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 πρὸς ρνγ, ἡ δὲ ΕΓ πρὸς ΓΖ λόγον ἔχει, ὃν σξε πρὸς ρνγ, Τετμήσθω οὖν ἡ ὑπὸ ΖΕΓ δίχα τῇ ΕΗ· ἔστιν ἄρα, ὡς ἡ ΖΕ πρὸς ΕΓ, ἡ ΖΗ πρὸς ΗΓ. Ὡς ἄρα συναμφότερος2 ἡ ΖΕ. ΕΓ πρὸς ΖΓ, ἡ ΕΓ πρὸς ΓΗ· ὥστε ἡ ΓΕ πρὸς ΓΗ μείζονα λόγον ἔχει ἤπερ φοα πρὸς ρνγ. Ἡ ΕΗ ἄρα πρὸς ΗΓ δυνάμει3 λόγον ἔχει, ὃν Μ θυν πρὸς Μ γυθ· μήκει ἄρα, ὃν φU+A7FCα η΄ πρὸς ρνγ. 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. 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. Ἔτι δίχα ἡ ὑπὸ ΚΕΓ τῇ ΛΕ· ἡ ΕΓ ἄρα πρὸς ΛΓ μείζονα λόγον ἔχει ἤπερ τὰ δχογ U+2220΄ πρὸς ρνγ. Ἐπεὶ οὖν ἡ ὑπὸ ΖΕΓ τρίτου οὖσα ὀρθῆς τέτμηται τετράκις δίχα, ἡ ὑπὸ ΛΕΓ ὀρθῆς ἐστι μη΄. Κείσθω οὖν αὐτῇ ἴση πρὸς τῷ Ε ἡ ὑπὸ ΓΕΜ· ἡ ἄρα ὑπὸ ΛΕΜ ὀρθῆς ἐστι κδ΄. Καὶ ἡ ΛΜ ἄρα εὐθεῖα τοῦ περὶ τὸν κύκλον ἐστὶ πολυγώνου πλευρὰ πλευρὰς ἔχοντος (??)ς. 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 γώνου4 περίμετρον μείζονα λόγον ἔχει ἤπερ δχογ U+2220΄πρὸς Μ δχπη. 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.

​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. 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. Let, then, the angle ΖΕΓ be bisected by ΕΗ; therefore, as ΖΕ is to ΕΓ, so is ΖΗ to ΗΓ. 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. 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. 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. 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. 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. 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. 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. And therefore the straight line ΛΜ is a side of a polygon of 96 sides circumscribed about the circle. 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. 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.

Syntax notes4 items
  1. 3τριπλασίων ἐστὶ καὶ ἔτι ὑπερέχει

    Meaning 'is three times and further exceeds'. This is the conclusion of the proposition showing the approximation of the ratio of the circumference to the diameter ($3 \frac{10}{71} < \pi < 3 \frac{1}{7}$).

  2. 3συναμφότερος

    An adjective meaning 'both together'. Here it modifies ἡ ΖΕ. ΕΓ (the lines ΖΕ and ΕΓ), representing the sum of the two line segments ($ZE + E\Gamma$).

  3. 3δυνάμει

    In geometrical contexts, it means 'in square' or 'in power', referring to the ratio of the areas of the squares described on the lines rather than the ratio of the lengths of the lines themselves.

  4. 3(??)ς γώνου

    A damaged portion of the text, which in context corresponds to the Greek numeral ϟϛ (koppa and stigma) representing '96', referring to a 96-sided polygon.

Translated by Gemini 3.5 Flash (published 2026-05-22) · Revision history (1, latest 2026-09-21)
  • 2026-09-21 AI review — source 1
urn:cts:greekLit:tlg0552.tlg002.humanitext-grc1:3#1