Groping Toward Linear Regression Analysis: Newton's Analysis of Hipparchus' Equinox Observations

TL;DR

This paper explores Isaac Newton's early use of regression analysis on Hipparchus' equinox data, highlighting its historical significance and its influence on modern cosmological methods.

Contribution

It uncovers Newton's pioneering application of regression techniques in astronomical data analysis, predating modern formal methods by centuries.

Findings

Newton's method closely estimated the tropical year using limited data.

Newton's approach employed the mean value, differing from contemporaries who used the median.

Regression analysis has a long history, influencing modern cosmology.

Abstract

In February 1700, Isaac Newton needed a precise tropical year to design a new universal calendar that would supersede the Gregorian one. However, 17th-Century astronomers were uncertain of the long-term variation in the inclination of the Earth's axis and were suspicious of Ptolemy's equinox observations. As a result, they produced a wide range of tropical years. Facing this problem, Newton attempted to compute the length of the year on his own, using the ancient equinox observations reported by a famous Greek astronomer Hipparchus of Rhodes, ten in number. Though Newton had a very thin sample of data, he obtained a tropical year only a few seconds longer than the correct length. The reason lies in Newton's application of a technique similar to modern regression analysis. Newton wrote down the first of the two so-called 'normal equations' known from the ordinary least-squares (OLS) method. In that procedure, Newton seems to have been the first to employ the mean (average) value of the data set, while the other leading astronomers of the era (Tycho Brahe, Galileo, and Kepler) used the median. Fifty years after Newton, in 1750, Newton's method was rediscovered and enhanced by Tobias Mayer. Remarkably, the same regression method served with distinction in the late 1920s when the founding fathers of modern cosmology, Georges Lemaitre (1927), Edwin Hubble (1929), and Willem de Sitter (1930), employed it to derive the Hubble constant.