Verification and estimation of uncertainties of Tobias Mayer's 18th century astronomical observations

Abstract

Perceiving the uncertainty of the measurement has been changing over the past centuries, reflecting the advancement in the experimental techniques, the urge for reliable and reproducible measurement methodology, and development of mathematical data processing and evaluating algorithms. From the historical perspective, the concepts of considering the measurement uncertainty were firstly introduced with geographic and cartographic measurements. In this context, the works of Tobias Mayer on lunar landscape measurements are widely highlighted which, at that time, presented innovative approaches in data processing with the method of averages and pioneeringly addressed the issue of measurement error. In this study, we analyze in details the Mayer's set of 27 non-linear equations with 3 unknown parameters and discuss the effect of Mayer's linearization and subsequent mathematical procedures on the accuracy of the parameter values in contrast with the results from rigorous treatment of non-linear regression model involving the least-square method. In particular, we compare the values of the unknown parameters and their uncertainties in several variants in the linearized and nonlinearized model, providing monitoring of a small deviation of the Mayer's linearization. The results, presented here, show that despite the conceptual and computational simplification of the Mayer's method, such an approach to data processing can be exploited, with an acceptable level of accuracy, in several practical situations even today.