pr ecedes the solution procedure: “ Concerning

this 10;45, you see when it is halved. ” The time

in which Jupiter reaches l

c

,s a y t

c

,i st h e nc o m -

puted by the following geometrical method: The

trapezoid for days 0 to 60 is divided into two

smaller trapezoids of equal area (Fig. 3). In order

t oa c h i e v et h i s ,t h eB a b y l o n i a na s t r o n o m e r sa p -

plied a partitio n procedur e that is w ell-attested in

Old Babylonian (2000 to 18 00 BCE) mathemati cs

( 5 , 6 ). In modern terms, it can be formulated as

follows: If v

0

and v

60

are th e parallel sides of a

trapezoid, then the inter mediate parallel that

divides i t i nto two trapezoids of e qual area has a

height v

c

=[ ( v

0

2

+ v

60

2

)/2]

1/2

. In the present case,

v

c

deno tes Jupi ter ’ s daily di splace ment w hen it

is at the “ crossing. ” This expression follows from

equating the areas of the partial trapezoids, S

1

=

t

c

• ( v

0

+ v

c

)/2 = S

2

= t

2

• ( v

c

+ v

60

)/2, where t

c

and t

2

ar e the width s of these trapez oids, and using t

c

=

t • ( v

0

– v

c

)/( v

0

– v

60

), wher e t=t

c

+t

2

is the

widt h of th e original trapez oid ( 6 , 10 ). Insertin g

v

0

=0 ; 1 2 ° / d , v

60

=0 ; 9 , 3 0 ° / d ,a n d t =1 , 0d ,w eo b -

tain v

c

=[ ( 0 ; 2 , 2 4+0 ; 1 , 3 0 , 1 5 ) / 2 ]

1/2

= (0;1,57,7,30)

1/2

=

0;10,49,2 0,44,58,...° /d, t

c

= 28;15,42,0,48,... d, and

t

2

= 3 1;44,17,59,12,...d. The computatio n of v

c

is

partly preserve d in text D up to the addition

0;2,24 + 0;1,30,15 ( 8 ). In text B, the related quan-

tity u

2

=( v

0

2

– v

60

2

)/2 = (0;2,24 – 0;1,30,15)/2 =

0;0,26,52,30 is computed. This was most likely

followed by another step in which v

c

was com-

puted using v

c

2

= v

0

2

– u

2

. Whereas all known

Old Babylonian examples of the partition algo-

rithm c oncern trapezoids for which v

c

, v

0

,a n d

v

60

are terminating sexagesimal numbers ( 6 ), the

present solution does not terminate in the sex-

agesimal system. Hence, texts B to E can only

have offered rounded results for v

c

and t

c

.N o t h i n g

remains of this in texts B to D, but text E partly

preserves a computation involving 0;10,50, whi ch

is, most plausibly, an approximation of v

c

.T h i s

interpretation is confirmed by the fact that text

E also mentions the value t

c

= 28 d and, very

likely, t

2

= 32 d, both in exact agreement with

t

c

=6 0 • ( v

0

– v

c

)/( v

0

– v

60

) and t

2

=6 0 – t

c

if one

approximates v

c

= 0;10,50°/d. By rounding v

c

,

only an approximately equal partition of the trap-

ezoid is achieved.

Also partly preserved in text E is a computa-

tion of the area of the second partial trapezoid,

using the same method as before, leading to S

2

=

t

2

• ( v

c

+ v

60

)/2, whe re t

2

=3 2d a y s , v

c

=0 ; 1 0 , 5 0 ° / d ,

and v

60

=0 ; 9 , 3 0 ° / d .T h ev a l u eo f S

2

is broken

away but can be res tored as 5;25, 20° . The proba b le

purpose of this computation was to verify the

solution for v

c

,a si sd o n ei nt h eO l dB a b y l o n i a n

mathematical text UET 5, 858 ( 5 , 11 ). The anal-

ogous computation of the area of the first par-

tial trapezoid, which c an be reconstructed a s S

1

=

t

c

• ( v

0

+ v

c

)/2 = 5;19,40° , is not preserved. Neither

of these values equals 5;22,30° = S /2 as the y

ide all y should (Fig . 3), a direct consequence of the

rounding of v

c

to 0;10,50°/d. At most two more

lines are partl y preserved in texts B, D, and E, but

they are too fragmentary for an interpretation.

The evidence presented here demonstrates

that Babylonian astronomers construed Jupiter ’ s

displacement along the ecliptic during the f irst

60 days after its first appe arance as the area of a

trapezoid in time-velocity sp ace. More over , they

computed the time when Jupiter covers half this

distance by partitioni ng the trapezoid into two

smaller ones of ideally equal area. These compu-

tations predate the use of similar techniques by

medieval European scholars by at le ast 14 cen-

turies. The “ Oxford calculat ors ” of the 14 th cen-

tury CE, who were ce ntered at Merton Colle ge,

Oxford, a re credited with formulating the “ Mer-

tonian mean speed theorem ” for the distance

traveled by a uniformly accelerating body, cor-

responding to the modern formula s = t • ( v

0

+

v

1

)/2, where v

0

and v

1

are the initial and final

velocities ( 12 , 13 ). In the same century Nicole

Oresme, in Paris, devised graphical methods that

enabled him to prove this relation by computing

s as the area of a tra pezoid of width t an d he ig hts

v

0

and v

1

( 12 ). Part I of the Babylonian trapezoid

procedures can be viewed as a concrete example

of the same computation. They also show that

Babylonian astronomers did, at least oc casionally,

use g eometrical methods for computin g pla netary

positions. Ancient Greek astronomers such as

Aristarchus of Samos, Hipparchus, and Claudius

Ptolemy also used geometrical methods ( 12 ),

while arithmetical methods are attested in the

Antikythe ra mechanism ( 14 )a n di nG r e c o - R o m a n

astronomical papyri from Egypt ( 15 ). However ,

the Babylonian trapezoid procedures are geo-

metr ical in a differen t sense than the methods

of the mentione d Greek astronom ers, si nce the

geom etri cal fig ure s desc ri be con fig ura ti ons not

in physical space but in an abstract mathemat-

ical space de fi n ed b y ti me a nd v el oc it y (d ail y

displacemen t).

REFERENCES AND NOTES

1. O. Neugebau er , Ast rono mic al Cune ifo rm Text s (Lun d Hump hrie s,

London, 1955).

2. M. Ossendrijver , Babylonian Mathematical Astronomy:

Procedure Texts (Springer , New York, 2012).

3. J. P. Britto n, Arch. Hist. Exact Sci. 64 ,6 1 7 – 663

(2010).

4. J . Hø yr u p, Lengths, Width s, Surfaces. A Portrait of Old

Babylonian Algebra and Its Kin (Springer , New York,

2002).

5. A. A. Vaiman, Shumero-Vavilonskaya matematika III-I

tysyacheletiya do n. e. (Izdatel ’ stvo Vostochnoy Literatury,

Moscow , 1961).

6. J. Friberg, in Reallexikon der Assyriologie , D. O. Edzard, Ed.

(De Gruyter, Berlin, 1990), vol. 7, pp. 561 – 563.

7. J. Friberg, A Remarkable Collection of Babylonian Mathematical

Texts. Manuscripts in the Schøyen Collection: Cuneiform

Texts I (Springer , New York, 2007).

8. Materials and methods are available as supplementary

materials on Science Online.

9. P. J. Huber, Z. Assyriol. 52 , 265 – 303 (1957).

10. O. Neugebauer , Mathematische Keilschrifttexte, I – III (Springer ,

Berlin, 1935 – 1937).

11. J. Friberg, Rev. Assyriol. Archeol. Orient. 94 ,9 7 – 188

(2000).

12. O. Ped er se n, Early Physics and Astronom y. A Historical

Introduction (Cambridge Univ. Press, Cambridge, 1974).

13. E. D. Sylla, in The Cambridge History of Later Medieval

Philosophy , N. Kretzmann, A. Kenny, J. Pinborg, Eds.

(Cambridge Univ. Press, Cambridge, 1982), pp. 540 – 563.

14. T. Freeth, A. Jones, J. M. Steele, Y. Bitsakis, Nature 454 ,

614 – 617 (2008).

15. A. Jones, Astronomical Papyri from Oxyrhynchus (American

Philosophical Society, Philadelphia, 1999).

AC KN OW LE D GM E NT S

The Trustees of the British Museum (London) are thanked for

permission to photograph, study, and publish the tablets. Work

was supported by the Excellence Cluster TOPOI, “ The Formation

and Transformation of Space and Knowledge in Ancient Cultures ”

(Deutsche Forschungsgemeinschaft grant EXC 264), Berlin.

Photographs, transliterations, and translations of the relevant parts

of the tablets are included in the supplementary materials. The

tablets are accessible in the Middle Eastern Department of the

British Museum under the registration numbers BM 40054

(text A), BM 36801, BM 41043, BM 34757 (text B), BM

34081+34622+34846+42816+45851+46135 (text C), BM 35915

(text D), and BM 82824+99697 +99742 ( text E). H. Hunger (Vienna)

is ac know led ged for provi din g an unp ubli shed phot ogra ph of

BM 40054.

SUPPLEMENT ARY MA TERIALS

www .sciencemag.org/content/351/6272/482/suppl/DC1

Materials and Methods

Figs. S1 to S4

References ( 16 – 21 )

4 November 2015; accepted 23 December 2015

10.1126/science.aad8 085

484 29 JANUARY 2016 • VOL 351 ISSUE 6272 sciencemag.org SCIE NCE

Fig. 3. P arti tion ing the tr ape z oid f or

day s 0 to 60 . The time at which

Jupiter reaches the “ crossing, ” t

c

,

where it has cover ed the distance

5;22,30° = 10;45°/2, is computed

ge ome tri cal ly by div idi ng the tr ape z oid

for days 0 to 60 into two small er

trapez oids of equal area. In text E, v

c

is

rounded to 0;10,50°/ d, resulting in t

c

=

28 d, S

1

=5 ; 1 9 , 4 0 ° , t

2

=3 2d ,a n d S

2

=

5;25,20° .

0;12

0;10

0;8

0;6

0;4

0;2

=0;12

=0;10,49,20,...

=0;9,30 v

v

v

0

0 =28;15,42,... 1,0 t

oo

0

60

c

c

time [days]

v [

o

/day]

5;22,30 5;22,30