Position in chronology
Plimpton 322
Translation · reference
High confidenceA table of fifteen rows, each giving three numbers in sexagesimal (base-60) corresponding to Pythagorean triples — right triangles whose sides are whole numbers.
Source: Robson (2002); Mansfield & Wildberger (2017)
Translation · AI engine
read from photo1,59,0,15 | 1,59 | 2,49 | takiltum [The square of the reciprocal (held): 1;59,0,15 — the short side: 1,59 — the diagonal: 2,49 — 'that which was made to hold (i.e., the square)']
3 uncertain terms ↓
- takiltum — Akkadian verbal noun from kullum; translated variously as 'the holding-square,' 'that which was made to hold,' or 'the square (of the reciprocal).' Its precise technical meaning in this mathematical context is debated: Neugebauer & Sachs read it as referring to a squared reciprocal; Robson (2001) interprets the whole tablet as a teacher's reference list rather than a trigonometric table. The term labels the first column containing the values of the form (1/2(p/q − q/p))² + 1.
- 1.59.0.15 — Sexagesimal floating-point value; in modern notation 1;59,0,15 = 1 + 59/60 + 0/3600 + 15/216000 ≈ 1.9834. This is the squared value in column I of row 1. The precise absolute value depends on the sexagesimal place-value interpretation, which has no explicit zero-marker in the original.
- 1.59 / 2.49 — Read as integers 119 and 169 in base 10; these are the short side (b) and hypotenuse (d) of the Pythagorean triple (119, 120, 169). The long side 120 does not appear in this column arrangement, which has led to ongoing scholarly discussion about the tablet's purpose.
Reasoning ↓
Visual examination of the photograph: the tablet is a landscape-oriented clay tablet divided into ruled columns by incised vertical lines, with horizontal ruling throughout. The surface is moderately well preserved in the central and right-hand sections; the left edge shows some erosion and minor chipping. Individual wedge impressions are clearly visible across most of the surface — the characteristic horizontal, vertical, and oblique Winkelhaken wedges of Old Babylonian sexagesimal notation are legible. The top two lines (likely the header row) show denser sign clusters consistent with the column headers including the term 'takiltum' and the numerical entries. The repeated columnar structure visible in the photo aligns well with the known four-column layout of Plimpton 322 (Neugebauer & Sachs 1945). The scholar-supplied transliteration '1.59.0.15 / 1.59 / 2.49 / takiltum' corresponds to line 1 of the tablet and is consistent with what can be observed in the upper portion of the left and central columns, though independent sign-by-sign verification at this resolution is not fully possible for every wedge. The reading of takiltum as the column header 'that which was made to hold' (from kullum, 'to hold') is well established in the literature (Robson 2001, 2002; Friberg 2007), though alternative interpretations of the tablet's mathematical purpose remain debated.
Generated by claude-sonnet-4-6 · prompt 2026-05-11/v2 · May 11, 2026 · 2420 in / 834 out tokens
Why it matters
Whatever its purpose, this single tablet shows that Babylonian mathematicians, working in base-60, had an arithmetic understanding of right triangles a millennium before Pythagoras was born.
Transliteration
1.59.0.15 / 1.59 / 2.49 / takiltum (header term: 'square-of')
Scholarly note
Interpretation contested. Robson argues it's a teacher's reference table; Mansfield proposes it's a trigonometric table predating Greek trigonometry by 1500 years. Both agree the mathematics is real and sophisticated.
Attribution
Image: Columbia University, Plimpton Collection, via Wikimedia Commons. source
Translation excerpted from Robson (2002); Mansfield & Wildberger (2017).
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