Historical evidence
A square, two diagonals, three numbers
YBC 7289 survives in the Yale Babylonian Collection. Its drawn square carries the number 30 and two numerical inscriptions associated with a diagonal. This is a small object with an unusually precise mathematical relationship to investigate.
In the conventional reading, one inscription gives a ratio of diagonal to side. The other gives the diagonal when the side is 30. The numbers agree to the written sexagesimal places.[YBC][CASSEL]
The surviving diagram does not tell us exactly how its scribe obtained the approximation. A successful modern method is not evidence that this particular method was used.
Historical evidence
Two signs; many places
Unit and ten signs combine to represent the digits from 1 to 59. A digit’s position changes its weight by a factor of sixty. Reading therefore involves both recognizing a cluster of signs and deciding how to interpret its place.
Old Babylonian notation did not mark the boundary between integers and fractions with our punctuation. Context helped determine scale; a modern transcription makes that interpretation explicit. Later placeholder conventions must not be projected back onto every earlier tablet.[NUMERALS]
Reconstruction
Decode the inscription
The decoder below separates each digit and assigns a power of sixty. Its wedges are schematic teaching signs. Commas, semicolons, labels and highlighted columns are modern aids.
Begin with 1;24,51,10. Change the fractional-place setting before changing any digit. The signs stay the same while the scale changes: positional notation needs an interpretation as well as a written sequence.[CASSEL]
Reconstruction · base sixty
Read the places
1 × 60^0 + 24 × 60^-1 + 51 × 60^-2 + 10 × 60^-3
Schematic signs, modern punctuation. “Empty” is an explanatory label, not an Old Babylonian zero sign. Changing the point changes our interpretation of scale.
Compare the two inscriptions
- Written ratio
- 1.4142129630
- Modern √2
- 1.4142135624
- 30 × written ratio
- 42.42638889
- Relative error
- 4.238e-7
These fixed reference readings compare the tablet’s inscriptions. Your editable number appears above. The generating algorithm is not preserved on this object.
Explore a reciprocal
1/12 = 0;5 · terminating
A finite base-sixty expansion requires only factors 2, 3 and 5 in the reduced denominator. The displayed recurring expansion is truncated after eight places.
Modern translation
From the square to a ratio
In modern Euclidean geometry, a square of side s has diagonal d with d² = 2s², so d/s = √2. That gives us a way to compare the inscription with a mathematically exact ratio.
Multiply the written approximation by 30. It gives the second line, 42;25,35. The numerical difference from 30√2 is small but nonzero. The displayed decimal approximation is a computation, not an exact equality.[CASSEL]
Modern translation
Make division a multiplication
A reciprocal turns division by a number into multiplication by its inverse. The base matters: 1/8 has the finite sexagesimal expression 0;7,30. The decoder’s reciprocal comparison lets you contrast it with 1/7, whose sexagesimal expansion recurs.
The modern explanation is a factorization condition. A reduced fraction terminates in base sixty precisely when its denominator contains only prime factors 2, 3 and 5. This explains a useful feature of the system; it does not establish why the base was originally adopted.[NUMERALS]
Open question
Calculation and the question of proof
A written result, an algorithm that could produce it, and a proof that the algorithm works answer different questions. Keep those questions separate while reading mathematical objects.
Carry that distinction into Euclid’s constructions, into the algorithms of the Nine Chapters, and into modern numerical experiments. A precise output invites explanation; its precision alone cannot supply a history of how it was obtained.[YBC][CASSEL]