F.W. Bessel (1825): The calculation of longitude and latitude from geodesic measurements
Abstract
Issue No. 86 (1825 October) of the Astronomische Nachrichten was largely devoted to a single paper by F. W. Bessel on the solution of the direct geodesic problem (see the first sentences of the paper). For the most part, the paper stands on its own and needs little introduction. However, a few words are in order to place this paper in its historical context.
First of all, it should be no surprise that a paper on this subject appeared in an astronomical journal. At the time, the disciplines of astronomy, navigation, and surveying were inextricably linked  the methods and, in many cases, the practitioners (in particular, Bessel) were the same. Prior to Bessel's paper, the solution of the geodesic problem had been the subject of several studies by Clairaut, Euler, du Séjour, Legendre, Oriani, and others. The interest in the subject was twofold. It combined several new fields of mathematics: the calculus of variations, the theory of elliptic functions, and the differential geometry of curved surfaces. It also addressed very practical needs: the determination of the figure of the earth, the requirements of large scale surveys, and the construction of map projections. With the papers of Legendre and of Oriani in 1806, the framework for the mathematical solution for an ellipsoid of revolution had been established.
However, Bessel was firmly in the practical camp; he carried out the East Prussian survey that connected the West European and Russian triangulation networks and later he made the first accurate estimate of the figure of the Earth, the ``Bessel ellipsoid''. He lays out his goal for this paper in its first section: to simplify the numerical solution of the geodesic problem. In Sects. \ref{sec2}\ref{sec4}, Bessel gives a clear and concise summary of the previous work on the problem. In the remaining sections, he develops series for the distance and longitude integrals and constructs the tables which allow geodesics to be calculated to an accuracy of about 3 cm over distances in excess of 1000 km (and the method remains accurate for geodesics that encircle the Earth). Despite the use of logarithms, Bessel's numerical methods are surprisingly uptodate: he writes out his series in a form that allows them to be extended to any order and he carries out a rather detailed analysis of the numerical errors.
Bessel's derivation and tables were extensively used throughout the nineteenth century and many twentieth century works continued to refer to ``Bessel's method''. However, over time, the attributions to Bessel have become diluted as authors cite more recent works. This trend accelerated with the introduction of electronic calculators when Bessel's algorithms were thought to be too complex and simpler less accurate ones were substituted (these approximate algorithms are still in widespread use). However, now that floatingpoint hardware is fast and accurate, it is these later algorithms that often seem outdated, while Bessel's are easily adapted for implementation on modern computers.
 Publication:

Astronomische Nachrichten
 Pub Date:
 August 2010
 DOI:
 10.1002/asna.201011352
 arXiv:
 arXiv:0908.1824
 Bibcode:
 2010AN....331..852K
 Keywords:

 history and philosophy of astronomy;
 Physics  Computational Physics;
 Physics  Geophysics;
 Physics  History and Philosophy of Physics
 EPrint:
 11 pages, including 1 figure and 4 pages of tables. Version 2 and 3 fix some minor errors. This translation was edited by Charles F. F. Karney and Rodney E. Deakin. A transcription of the original paper is available at arXiv:0908.1823 . For links to other 18th and 19th century papers on geodesics, see http://geographiclib.sourceforge.net/geodesicpapers/biblio.html