eastern tropical Pacific and Antarctica peaked
during each of the last two glacial terminations
(28), consistent with the timing of enhanced EPR
Isolating a mechanistic linkage between ridge
magmatism and glacial terminations will require
a suite of detailed proxy records from multiple
ridges that are sensitive to mantle carbon and
geothermal inputs, as well as modeling studies
of their influence in the ocean interior. The
EPR results establish the timing of hydrothermal
anomalies, an essential prerequisite for deter-
mining whether ridge magmatism can act as a
negative feedback on ice-sheet size. The data
presented here demonstrate that EPR hydro-
thermal output increased after the two largest
glacial maxima of the past 200,000 years, im-
plicating mid-ocean ridge magmatism in glacial
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ACK NOW LE DGM EN TS
We dedicate this paper to J. Dymond, whose 1981 treatise on Nazca
plate sediments made this work possible. We are also indebted
to the Oregon State University Core Repository for carefully
preserving the EPR sediment cores since they were collected in
the early 1970s. We are grateful to L. Wingate at the University of
Michigan and M. Cote at the University of Connecticut for
technical support. This work has benefited from discussions
with J. Granger, P. Vlahos, B. Fitzgerald, and M. Lyle. Data
presented here are available on the National Oceanic and
Atmospheric Administration’s Paleoclimatology Data website
was provided by the University of Michigan and the University
Materials and Methods
Figs. S1 to S11
Tables S1 to S5
14 September 2015; accepted 6 January 2016
HISTORY OF SCIENCE
Ancient Babylonian astronomers
calculated Jupiter’s position from the
area under a time-velocity graph
Theideaofcomputingabody’s displacement as an area in time-velocity space is usually traced
back to 14th -century Europe. I show that in four ancient Babylonian cuneiform tablets, Jupiter’s
displacement along the ecliptic is computed as the area of a trapez oidal figure obtained by
drawing its daily displacement ag ainst time. This interpretation is prompted by a newly
discov er ed tablet on which the same computation is presented in an equivalent arithmetical
formulation. The tablets date from 350 to 50 BCE. The trape z oid procedures offer the first
evidence for the use of geometrical methods in Bab ylonian mathematical astronomy , which was
thus far viewed as operating e xclusively with arithmetical concepts.
he so-called trapezoid procedures examined
in this paper have long puzzled historians
of Babylonian astronomy. They belong to
the corpus of Babylonian mathematical as-
tronomy, which comprises about 450 tab-
lets from Babylon and Uruk dating between 400
and 50 BCE. Approximately 340 of these tablets
are tables with computed planetary or lunar data
arranged in rows and columns (1). The remaining
110 tablets are proced ure texts with computa-
tional instructions (2), mostly aimed at comput-
ing or verifying the tables. In all of these texts the
zodiac, invented in Babylonia near the end of the
fifth century BCE (3), is used as a coordinate sys-
tem for computing celestial positions. The un-
derlying algorithms are structured as branching
chains of arithmetical operations (additions, sub-
tractions, and multiplications) that can be rep-
resented as flow charts (2). Geometrical concepts
are conspicuously absent from these texts, whereas
they are very common in the Babylonian mathe-
matical corpus (4–7). Currently four tablets, most
likely written in Babylon between 350 and 50 BCE,
are known to preserve portions of a trapezoid
procedure (8). Of the four procedures, here labeled
B to E (figs. S1 to S4), one (B) preserves a men-
tion of Jupiter and three (B, C, E) are embedded
in compendia of procedures dealing exclusively
with Jupiter. The previously unpublished text D
probably belongs to a similar compendium for
Jupiter. In spite of these indications of a connec-
tion with Jupiter, their astronomical significance
was previously not acknowledged or understood
(1, 2, 6).
A recently discovered tablet containing an un-
published procedure text, here labeled text A (Fig. 1),
sheds new light on the trapezoid procedures. Text A
most likely originates from the same period and
location (Babylon) as texts B to E (8). It contains
a nearly complete set of instructions for Jupiter’s
motion along the ecliptic in accordance with the
so-called scheme X.S
(2). Before the discovery of
text A, this scheme was too fragmentarily known
for identifying its connection with the trapezoid
procedures. Covering one complete synodic cycle,
begins with Jupiter’s heliacal rising
(first visible rising at dawn), continuing with its
first station (beginning of appa rent retrograde
motion), acronychal rising (last visible rising at
dusk), second station (end of retrograde motion),
and heliacal setting (last visible setting at dusk)
(2). Scheme X.S
and the four trapezoid procedures
are here shown to contain or imply mathematically
equivalent descriptions of Jupiter’smotionduring
the first 60 days after its first appearance. Whereas
employs a purely arithmetical ter-
minology, the trapezoid procedures operate with
482 29 JANUARY 20 16 • VOL 351 ISSUE 6272 sciencemag.org SCIENCE
Excellence Cluster TOPOI–Institute of Philosophy, Humboldt
University, Berlin, Germany.
*Corresponding author. E-mail: firstname.lastname@example.org
RESEARCH | REPORTS