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Application of Bošković Geometric Adjustment Method on Nine Meridian Degrees

Martina TRIPLAT HORVAT, Dražen TUTIĆ, Miljenko LAPAINE, University of Zagreb Faculty of Geodesy, Kačićeva 26, 10000 Zagreb, Croatia

Josip Ruđer Bošković (Dubrovnik, 18th May 1711 – Milan, 13th February 1787) began to publish theses of Earth’s shape and size as a young scientist. These issues were a major scientific problem of the 18th century. To accurately determine the shape of the Earth, in his first attempt to find ellipticity, Bošković compared five arc lengths of one meridian degree. Those were measurements of meridian degrees, carried out in Quito in South America, the Cape of Good Hope in South Africa, Paris in France, the Finland province of Lapland, and his own, carried out in Rome in Italy. In his second attempt to determine ellipticity, he compared nine arc lengths of one meridian degree.

Whereas astronomic and geodetic measurements are liable to errors caused from various sources, Bošković was aware the causes of errors can not be fully eliminated during the construction of instruments and measurements. When comparing five, then nine degrees of meridian, Bošković could not determine an ellipsoid which would be consistent with all measurements. Hence, he decided to determine corrections which would fix all degrees in order to get a better estimate of true values.

Bošković sets the following three conditions on the data on lengths of meridian degrees:

  1. The differences of the meridian degrees are proportional to the differences of the versed sines of double latitudes
  2. The sum of positive corrections is equal to the sum of the negative ones (by their absolute values) and
  3. The absolute sum of all corrections, positive as well as negative, is the least possible one, in the case in which the first two conditions are fulfilled.

In all his works Bošković provided geometric description of solutions for the mentioned conditions. In the paper, we describe in detail the example with nine meridian degrees. Data were taken from Bošković’s original book. The geometric solution, described by Bošković himself, is not easy to understand at first, as it is noted by other authors who also studied the Bošković method. Nowadays, software for interactive geometry enables to define and visualize his method analytically in a way which provides better understanding. GeoGebra was used for this purpose which is free mathematics software which connects geometry, algebra, statistics and calculus in one easy-to-use package.

Keywords: Josip Ruđer Bošković, geometric adjustment method, GeoGebra

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