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around 1900–1600 BCE

Babylonian Inner Diagonal Tablet

The Pythagorean theorem in three dimensions

This Babylonian cuneiform tablet fragment, dated to around 1900–1600 BCE, consists of the upper-left quarter of a full tablet. The presence of a detailed summary of its contents on the surviving portion indicates it originally contained 16 different geometric exercises concerning circles, diagonals, chords and a city gate.

Babylonian Inner Diagonal Tablet

One of the most interesting of the tablet's surviving problems concerns finding the "inner diagonal" drawn from the inner-left bottom corner to the outer-right top corner of a cuboid. This involves finding a solution in integers to the three-dimensional generalization of the Pythagorean theorem more than a millennium before Pythagoras lived.

Artifact dimensions

11.5 cm × 6.4 cm × 2.2 cm

Original artifact location

Uruk (historical name), Al-Muthannia, Iraq (current name)

Current artifact location

Oslo, Norway

Catalog number

MS 3049

Timeline

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

  • Friberg, J. "MS 3049, § 5. The Inner Diagonal of a Rectangular Gate in a Wall" §11.1 d in A Remarkable Collection of Babylonian Mathematical Texts: Manuscripts in the Schøyen Collection: Cuneiform Texts I. New York: Springer, pp. 301–304 and 495, 2007.