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around 2000–1650 BCE

Berlin Pythagorean Theorem Papyrus

Egyptian solution to a quadratic equation

Berlin papyrus 6619 is an ancient Egyptian document dated to around 2000–1650 BCE. It contains four problems spread across two large fragments and about 50 small fragments. The most interesting of its problems suggests some knowledge of the Pythagorean theorem. On the other hand, it may represent only straightforward solution to a single quadratic equation in one unknown.

Berlin Pythagorean Theorem Papyrus

The Berlin papyrus provides the only evidence that the ancient Egyptians may have understood the Pythagorean theorem centuries before the Pythagoreans. In particular, the Berlin papyrus contains four problems, the first of which effectively states, "The area of a square of 100 is equal to that of two smaller squares. The side of one is 1/2 + 1/4 the side of the other." In modern terms, the problem suggests the simultaneous equations x² + y² = 100 and x = (3/4)y which have unique positive solution y = 8 and x = 6.

Artifact dimensions

14 cm × 8 cm, 5.5 cm × 7 cm

Artifact origin


Current artifact location

Egyptian Museum of Berlin, Berlin

Catalog number



PythagoreanTheorem timeline Babylonian Mud Wall Tablet Babylonian Inner Diagonal Tablet Babylonian Square Root of 2 Tablet Berlin Pythagorean Theorem Papyrus Plimpton 322 Vedic Fire Altar Rectangle Diagonal Vedic Fire Altar Square Diagonal Euclid's Elements Ptolemy's Quadrilateral Theorem Zhoubi Suanjing Pythagorean Theorem Proof Ibn Qurra's Pythagorean Theorem Proofs Lilāvatī of Bhāskara II

Interactive Content

Computational Explanation

Other Resources

Additional Reading

  • Clagett, M. Ancient Egyptian Science: A Source Book, Vol. 3: Ancient Egyptian Mathematics. Philadelphia, PA: American Philosophical Society, pp. 250–251, 1999.
  • Gillings, R. J. Mathematics in the Time of the Pharaohs. New York: Dover, p. 161, 1982.
  • Imhausen, A. Ägyptische Algorithmen: Eine Untersuchung zu den mittelagyptischen mathematischen Aufgabentexten. Wiesbaden, Germany: Harrassowitz Verlag, p. 50, 2003.
  • Imhausen, A. "Papyrus Berlin 6619." § 9.1.3 in Mathematics in Ancient Egypt: A Contextual History. Princeton, NJ: Princeton University Press, p. 3 and 68, 2016.
  • Miatello, L. "A Debated but Little Examined Mathematical Text: Papyrus Berlin 6619." Zeitschrift für ägyptische Sprache und Altertumskunde, Vol. 139, pp. 158–170, 2012.
  • Rossi, C. Architecture and Mathematics in Ancient Egypt. Cambridge, England: Cambridge University Press, p. 217, 2007.
  • Schack-Schackenburg, H. "Der Berliner Papyrus 6619." Zeitschrift für ägyptische Sprache und Altertumskunde, Vol. 38, pp. 135–140, 1900.
  • Schack-Schackenburg, H. "Das kleinere Fragment des Berliner Papyrus 6619." Zeitschrift für ägyptische Sprache und Altertumskunde, Vol. 40, pp. 65–66, 1902.