# Bisecting the trapezoid: tracing the origins of a Babylonian computation of Jupiter’s motion

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## Abstract

Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure representing the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics (1900–1700 BCE). Some implications of the use of geometric methods by Babylonian astronomers are also explored.

## Notes

### Acknowledgements

The Trustees of the British Museum are acknowledged for permission to study and publish the tablets kept in the British Museum. Christopher B.F. Walker is acknowledged for making available his catalogue of astronomical fragments in the Babylon collection of the British Museum.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## References

- Aaboe, A. 1964.
*Episodes from the early history of mathematics*. New York: Random House.zbMATHGoogle Scholar - Brack-Bernsen, L., and O. Schmidt. 1990. Bisectable Trapezia in Babylonian Mathematics.
*Centaurus*33: 1–38.MathSciNetCrossRefzbMATHGoogle Scholar - Britton, J.P. 2010. Studies in Babylonian Lunar Theory: Part III. The Introduction of the Uniform Zodiac.
*AHES*64: 617–663.MathSciNetCrossRefzbMATHGoogle Scholar - Bruins, E.M., and M. Rutten. 1961.
*Textes Mathématiques de Suse. Mémoires de la Mission Archéologique en Iran 34*. Paris: Paul Geuthner.zbMATHGoogle Scholar - Clagett, M. 1968.
*Nicole Oresme and the Medieval Geometry of Qualities and Motions. A Treatise on the Uniformity and Difformity of Intensities known as Tractatus de configurationibus qualitatum et motuum*. The University of Wisconsin Press: Madison and London.Google Scholar - Edzard, D.O. 2003.
*Sumerian Grammar, Handbuch des Orients*. Leiden: Brill.CrossRefGoogle Scholar - Frank, C. 1928.
*Strassburger Keilschrifttexte in sumerischer und babylonischer Sprache*. Berlin: De Gruyter.zbMATHGoogle Scholar - Friberg, J. 1990. Mathematik. In
*Reallexikon der Assyriologie*, ed. D.O. Edzard, 531–585. Berlin: De Gruyter.Google Scholar - Friberg, J. 1997. Seeds and Reeds Continued. Another Metro-Mathematical Topic Text from Late Babylonian Uruk.
*Bagdhader Mitteilungen*28: 251–365.Google Scholar - Friberg, J. 2000. Mathematics at Ur in the Old Babylonian period.
*Revue d’Assyriologie et d’archéologie orientale*94: 97–188.Google Scholar - Friberg, J. 2005.
*Unexpected Links between Egyptian and Babylonian Mathematics*. Singapore: World Scientific.CrossRefzbMATHGoogle Scholar - Friberg, J. 2007a.
*A Remarkable Collection of Babylonian Mathematical Texts*. New York: Springer.CrossRefzbMATHGoogle Scholar - Friberg, J. 2007b.
*Amazing Traces of a Babylonian Origin in Greek Mathematics*. Singapore: World Scientific.CrossRefzbMATHGoogle Scholar - Friberg, J., and F. Al-Rawi. 2016.
*New Mathematical Cuneiform Texts*. New York: Springer.CrossRefzbMATHGoogle Scholar - Friberg, J., and A. George. 2010. Six More Mathematical Cuneiform Texts in the Schøyen Collection. In
*Essays and Texts in Honour of Martin Schøyen. Papyri Graecae Schøyen*II, ed. D. Minutoli and R. Pintaudi, 123–195. Florence: Gonnelli.Google Scholar - Gandz, S. 1948. Studies in Babylonian Mathematics I: Indeterminate Analysis in Babylonian Mathematics.
*Osiris*8: 12–40.MathSciNetCrossRefzbMATHGoogle Scholar - Høyrup, J. 1996. Changing trends in the historiography of Mesopotamian mathematics: an insider’s view.
*History of Science*34: 1–32.MathSciNetCrossRefGoogle Scholar - Høyrup, J. 2002.
*Lengths, Widths, Surfaces. A Portrait of Old-Babylonian Algebra and its Kin*. New York: Springer.CrossRefzbMATHGoogle Scholar - Høyrup, J. 2017. What is “Geometric Algebra”, and What Has it Been in Historiography?
*AIMS Mathematics*2 (1): 128–160. https://doi.org/10.3934/Math.2017.1.128.CrossRefGoogle Scholar - Huber, P.J. 1955. Zu einem mathematischen Keilschrifttext (VAT 8512).
*Isis*46: 104–106.MathSciNetCrossRefzbMATHGoogle Scholar - Huber, P.J. 1957. Zur täglichen Bewegung des Jupiter nach babylonischen Texten.
*Zeitschrift für Assyriologie*52: 265–303.CrossRefGoogle Scholar - Imhausen, A., and P. Pommerening (eds.). 2016.
*Translating Writings of Early Scholars in the Ancient Near East, Egypt*. Methodological Aspects with Examples, De Gruyter: Greece and Rome.Google Scholar - Jones, A. 1999. Astronomical Papyri from Oxyrhynchus, Memoirs of the American Philosophical Society 233 (Philadelphia)Google Scholar
- Muroi, K. 2001. Inheritance Problems in the Susa Mathematical Text No. 26.
*Historia Scientiarum*10: 226–234.MathSciNetzbMATHGoogle Scholar - Neugebauer, O. 1935–1937.
*Mathematische Keilschrifttexte*, Vols. I–III . Berlin: Springer.Google Scholar - Neugebauer, O. 1955.
*Astronomical Cuneiform Texts*. London: Humphries.CrossRefzbMATHGoogle Scholar - Neugebauer, O. 1975.
*A History of Ancient Mathematical Astronomy*. New York: Springer.CrossRefzbMATHGoogle Scholar - Neugebauer, O. 1988. A Babylonian Lunar Ephemeris from Roman Egypt, In
*A Scientific Humanist: Studies in Memory of Abraham Sachs. Occasional publications of the Samuel Noah Kramer Fund 9*ed. Leichty, E., Ellis M., and Gerardi P., 301–304. Philadelphia: University Museum.Google Scholar - Neugebauer, O., and A. Sachs. 1945.
*(Reprint 1986), Mathematical Cuneiform Texts*. New Haven: American Oriental Society.Google Scholar - Ossendrijver, M. 2012.
*Babylonian Mathematical Astronomy. Procedure Texts*. New York: Springer.CrossRefzbMATHGoogle Scholar - Ossendrijver, M. 2016. Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph.
*Science*351: 482–484. Supplementary Materials 1–19.MathSciNetCrossRefzbMATHGoogle Scholar - Ossendrijver, M. 2017. New results on a Babylonian scheme for Jupiter’s motion along the zodiac.
*Journal for Near Eastern Studies*76: 231–247.CrossRefGoogle Scholar - Ossendrijver, M. 2018a. The Trapezoid Procedures: Area Computations in Babylonian Astronomy In K. Chemla (ed.).
*Proceedings of the SAW Conference*, Paris, forthcoming.Google Scholar - Ossendrijver, M. 2018b. Scholarly Mathematics in the Rēš Temple. In C. Proust and J. Steele (eds.),
*Scholars and Scholarship in Late Babylonian Uruk*(Brill, forthcoming).Google Scholar - Pedersen, O. 1974.
*Early Physics and Astronomy. A Historical Introduction*. Cambridge: Cambridge University Press.Google Scholar - Powell, M.A. 1987–1990. Masse und Gewichte. In D. O. Edzard (ed.),
*Reallexikon der Assyriologie und Vorderasiatischen Archäologie*. Siebter Band. Libanukšabaš– Medizin (Berlin—New York: De Gruyter), 457–517.Google Scholar - Proust, C. 2012. Problèmes de partage: des cadastres à l’arithmétique. website CultureMATH. http://culturemath.ens.fr/histoire%20des%20maths/htm/Proust12/problemes-de-partage.html.
- Robson, E. 1999.
*Mesopotamian Mathematics 2000–1600 BC. Technical Constants in Bureaucracy and Education*. Oxford: Clarendon Press.zbMATHGoogle Scholar - Robson, E. 2007. Mesopotamian Mathematics. In Katz, V. (ed.),
*The Mathematics of Egypt, Mesopotamia, China, India, and Islam*. A Sourcebook (Princeton University Press), 57–186.Google Scholar - Robson, E. 2008.
*Mathematics in Ancient Iraq. A Social History*. Princeton: Princeton University Press.zbMATHGoogle Scholar - Sylla, E.D. 1982. The Oxford calculators. In N. Kretzmann, A. Kenny, J. Pinborg (eds.), The Cambridge History of Later Medieval Philosophy, 540–563.Google Scholar
- Thureau-Dangin, F. 1934. Une nouvelle tablette mathématique de Warka.
*Revue d’Assyriologie*31: 61–69.zbMATHGoogle Scholar - Thureau-Dangin, F. 1938.
*Textes Mathématiques Babyloniens*. Leiden: Brill.zbMATHGoogle Scholar - Vaiman, A.A. 1961. Shumero-Vavilonskaya matematika III–I tysyacheletiya do n. e. (Izdatel’stvo Vostochnoy Literatury, Moscow).Google Scholar
- van der Waerden, B.L. 1966.
*Erwachende Wissenschaft I. Ägyptische, babylonische und griechische Mathematik*. Basel–Stuttgart: Birkhäuser.zbMATHGoogle Scholar - Vogel, K. 1958. Ist die babylonische Mathematik sumerisch oder akkadisch?
*Mathematische Nachrichten*18: 377–382.MathSciNetCrossRefzbMATHGoogle Scholar - Vogel, K. 1959.
*Vorgriechische Mathematik II. Die Mathematik der Babylonier*. Hannover: H. Schroedel.zbMATHGoogle Scholar