A fragment of clay in the British Museum, written somewhere between 350 and 50 BC and found near the Esagila temple in Babylon, carries a calculation of how far Jupiter moves across the sky. The method is what makes it notable. The scribe works out the distance by finding the area of a trapezoid.
BM 34757 alongside a modern reconstruction of the procedure it describes. Jupiter's daily motion falls from 12 minutes of arc to 9 minutes 30 seconds over sixty days, and the area of the resulting trapezoid, 10 degrees 45 minutes, is the distance travelled. The dotted line marks the bisection that gives the halfway point. No such drawing appears on the tablet itself. Credit: Trustees of the British Museum and Mathieu Ossendrijver
The tablet is catalogued as BM 34757 and designated Text B in the study that made it famous, published in Science in January 2016 by Mathieu Ossendrijver, an astrophysicist turned historian of ancient science at Humboldt University in Berlin.
The area of the trapezoid is the distance the planet travelled
Jupiter does not move at a steady rate against the stars. Its apparent daily displacement changes as it goes.
The Babylonian procedure takes that changing daily displacement as one dimension and time as the other. Plotted against each other, they describe a trapezoid, and the area of that trapezoid is the total distance the planet has covered. The tablets state this outright, giving Jupiter’s travel over the sixty days following its first appearance as 10 degrees 45 minutes.
Then they go further. The trapezoid is divided into two smaller trapezoids of equal area, which gives the moment at which Jupiter has covered half the distance. That is a harder problem than the first one.
Four tablets sat unexplained until a colleague sent a photograph
Ossendrijver had been working through Babylonian mathematical astronomy for years, and had examined around 110 tablets. Geometry was conspicuously missing from them. The procedures are chains of arithmetic, additions and subtractions and multiplications, which can be laid out as flow charts.
Four tablets with trapezoid procedures were known, and nobody could say what they were computing. The answer arrived when Hermann Hunger in Vienna sent him a photograph of an undeciphered tablet in the British Museum, BM 40054, which set out a full cycle of Jupiter’s motion. With that in hand, the other four became legible.
Historians had placed the invention of this method in fourteenth-century Europe
The reason this reached the front pages is the comparison it invites.
Computing distance as the area under a velocity against time figure is credited to fourteenth-century Europe. The Oxford Calculators at Merton College formulated the mean speed theorem, and Nicole Oresme in Paris proved it graphically, using a trapezoid of width t and heights u and v.
Ossendrijver’s position is that the Babylonian procedures are concrete instances of the same computation, carried out at least fourteen centuries earlier.
Not everyone reads it that way
Francesca Rochberg of Berkeley called the connection Ossendrijver drew an exciting one.
James Evans, a physicist and historian at the University of Puget Sound, is more cautious. His objection is concrete. There are no drawings on these tablets. Babylonian mathematical tablets dealing with land measurement do carry geometrical figures, and these do not. He doubts the scribes would have understood what they were doing in the way Oresme did, though he allows they had a working grasp of the mean speed relation.
That distinction is worth holding onto. A procedure that produces the right answer is not the same as a graphical method for representing motion, and the tablets give us the procedure without telling us how the scribes pictured it.
Ossendrijver himself has continued to revise the reading. In a 2018 paper he replaced one of his own 2016 interpretations of a key term, taking it as a length rather than as an astronomical station.
There are no limits and no infinitesimals on these tablets
Popular accounts of these tablets frequently describe them as the earliest known integral calculus. That goes too far.
What is on the tablets is the area of a trapezoid, worked out by hand. There are no limits, no infinitesimals, no summing of arbitrarily many pieces. Oresme’s fourteenth-century work is sometimes described as an early form of integration, so the honest version of the claim is that the Babylonians anticipated a precursor of calculus rather than calculus itself.
That is still a considerable thing. It is a piece of abstract mathematics, applied to a quantity that has no physical extent, in a tradition that historians had until recently thought was purely arithmetical.
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Sources: Ossendrijver, M. (2016). “Ancient Babylonian Astronomers Calculated Jupiter’s Position From the Area Under a Time-Velocity Graph.” Science 351, 482 to 484. Ossendrijver, M. (2018). “Bisecting the Trapezoid.” Archive for History of Exact Sciences 72, 145 to 189. Comment from Francesca Rochberg and James Evans reported by Physics World, January 2016.
I find it curious that there isn't more information about Sumaria. It is all hidden in Museums around the world. I think the reason is that it disputes Darwinism and incidently, Avraham came from Ur in Sumaria.
The Greeks had regular communication with the nations of western Asia, as well as with Egypt. knowledge gleaned from conversations with Assyrians and Babylonians could well have included scientific and mathematical knowledge that the older civilizations had determined.