Gauss as scientific mediator between mathematics and geodesy from the past to the present. (English) Zbl 1410.86001
Freeden, Willi (ed.) et al., Handbook of mathematical geodesy. Functional analytic and potential theoretic methods. Cham: Birkhäuser. Geosyst. Math., 1-163 (2018).
The aim of this contribution is the following (p.7): “The objective of the authors is rather to document the pioneer dimension of Gauss’s ideas, concepts, and methods in a twofold direction based on selected case examples, to demonstrate his mediation function between mathematics and geodesy firstly and secondly the historic development over the past centuries from the initial ignition by Gauss to modern characteristics and tendencies in mathematics and/or geodesy.” In the first Chapter, Gauss’s life and works are presented as an overview; the following five chapters are devoted to special topics: Chapter 2 (p. 8): From Gaussian circle problem to geosampling; in the beginning the authors explain the lattice point number theory which can be found in Gauss’s Disquisitiones arithmeticae, as it is shown, this is the basic for the development of modern sampling methods. Chapter 3 (p. 20): From Gaussian integration to geocubature; indeed, continuous fractions play a main role in Gauss’s famous method of approximate integration (1814); this treatise attracted not only the attention of the mathematicians of its time but was well recognized since then and belongs now to the repertoire of numerical methods of approximation. Chapter 4 (p. 52): From Gaussian theorem to geoidal determination; the starting point is Gauss’ potential theory (1839) and its further influence; the gravity potential, i.e. the determination of the Earth’s shape, is a main task of physical geodesy today; representative boundary value problems are defined within potential theory. Chapter 5 (p. 77): From Gaussian least squares adjustement to inverse multiscale regularization; in a historic introduction Gauss’s role in history is described; the way leads from pseudoinverse for finite-dimensional matrix equations via pseudoinverse for infinite-dimensional operator equations to multiscale solutions of inverse pseudodifferential equations. Chapter 6 (p. 142): Conclusion: Gaussian geometry and geodetic surveying; Gauss’s “Disquisitiones generales circa superficies curvas” (1828) are fundamental for this part, especially the theory of geodesics, as well as Gauss’s conformal mapping and his non-Euclidean geometry. The treatise is accompanied by References (p.151–163, 241 numbers) and many figures, which are partially presented in coulors. The article under review is not primarily a historical research paper, but it shows how Gauss’s ideas lead to modern interpretations. This kind of investigation is not so common, it is an unusual approach. Not only historians, but especially mathematicians should be interested in this kind of considerations.
For the entire collection see [Zbl 1396.86001].
For the entire collection see [Zbl 1396.86001].
Reviewer: Karin Reich (Berlin)
MSC:
| 86-03 | History of geophysics |
| 86A30 | Geodesy, mapping problems |
| 01A55 | History of mathematics in the 19th century |
| 65D30 | Numerical integration |
| 65-03 | History of numerical analysis |
Keywords:
Gauss; geometric number theory; numerical integration; integral theorems and boundary value problems; least squares adjustmentReferences:
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