AN­CIENT AB­STRACTS

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gave him some black-and-white pho­to­graphs of tablets stored at the mu­seum. Ossendri­jver took no­tice of one of them, just five cen­time­tres across and five cm high. This rounded ob­ject, which he scru­ti­nized in per­son in Septem­ber 2015, proved to be a kind of Rosetta Stone.

Of­fi­cially named BH40054 by the mu­seum, and dubbed Text A by Ossendri­jver, the lit­tle tablet had mark­ings that served as a kind of ab­bre­vi­a­tion of a longer cal­cu­la­tion that looked fa­mil­iar to him. By com­par­ing Text A to the four pre­vi­ously mys­te­ri­ous tablets, he was able to de­code what was go­ing on: this was all about Jupiter. The five tablets com­puted the pre­dictable mo­tion of Jupiter rel­a­tive to the other plan­ets and the dis­tant stars.

“This tablet con­tains num­bers and com­pu­ta­tions, ad­di­tions, divi­sions, mul­ti­pli­ca­tions. It doesn’t ac­tu­ally men­tion Jupiter. It’s a highly ab­bre­vi­ated ver­sion of a more com­plete com­pu­ta­tion that I al­ready knew from five, six, seven other tablets,” he said.

Most strik­ingly, the method­ol­ogy for those com­pu­ta­tions used tech­niques that re­sem­bled the as­tro­nom­i­cal ge­om­e­try de­vel­oped in the 14th cen­tury at Ox­ford. The tablets have been au­thor­i­ta­tively dated to a pe­riod from 350 BC to 50 BC.

The peo­ple of Me­sopotamia — what is now Iraq — de­vel­oped math­e­mat­ics about 5,000 years ago. Among them were the Baby­lo­ni­ans who wrote in cuneiform script and, over time, adopted a sex­a­ges­i­mal (base 60) num­ber­ing sys­tem. Early math­e­mat­ics was es­sen­tially a form of count­ing, and the things be­ing counted were mostly sheep and the like.

Math­e­mat­ics pro­gressed, as did the shar­ing of knowl­edge in the wake of Alexan­der the Great’s con­quer­ing jour­neys across Asia. The an­cient Greek as­tronomer Aristarchus of Samos ar­gued for a he­lio­cen­tric uni­verse — one in which the Earth or­bited the sun, con­trary to what seems to be the case when one looks at the sky. That view was shared by an­other as­tronomer, pos­si­bly Greek as well, who lived in Me­sopotamia on the Ti­gris River and was known as Seleu­cus of Seleu­cia.

But Ossendri­jver said noth­ing in the newly de­coded com­pu­ta­tions sug­gests the an­cient sci­en­tist or sci­en­tists who etched the tablets un­der­stood that he­lio­cen­tric model. The cal­cu­la­tions merely de­scribe Jupiter’s mo­tion over time as it ap­pears to speed up and slow down in its jour­ney across the night sky. Those cal­cu­la­tions are done in a sur­pris­ingly ab­stract way — the same way the Ox­ford math­e­ma­ti­cians would do them a mil­len­nium and a half later.

“It’s ge­om­e­try, which is it­self old, but it’s ap­plied in a com­pletely new way, not to fields, or some­thing that lives in real space, but to some­thing that ex­ists in com­pletely ab­stract space,” Ossendri­jver said. “Any­body who stud­ies physics would be re­minded of in­te­gral cal­cu­lus.” (Which was in­vented in Europe in 1350, ac­cord­ing to his­to­ri­ans.)

“In Baby­lo­nia, be­tween 350 and 50 BC, schol­ars, or maybe one very clever guy, came up with the idea of draw­ing graphs of the ve­loc­ity of a planet against time and com­put­ing the area of this graph; of do­ing a kind of com­pu­ta­tion that seems to be thor­oughly mod­ern, that is not found un­til 1350,” he said.

Alexan­der Jones, a pro­fes­sor at New York Univer­sity’s In­sti­tute for the Study of the An­cient World, praised Ossendri­jver’s re­search, which he said shows the “revo­lu­tion­ary bril­liance of the un­known Me­sopotamian schol­ars who con­structed Babylonian math­e­mat­i­cal as­tron­omy dur­ing the se­cond half of the first mil­len­nium BC.”

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