So, there seem to be quite a few posts on the method of least squares lately, and for a reason. The method of least squares is used everywhere! Well maybe not quite, but it certainly has an illustrious history, as far as statistical methods go. It was first published in 1805 in Adrien-Marie Legendre’s

Nouvelles m´ethodes pour la d´etermination des orbites des com`etes

. But what would people do with this method over a century before computers? Well, as you’ve probably been able to guess from the title of Legendre’s book, it was used for calculating the orbits of solar system objects.

Objects orbit the sun in conic sections (such as ellipses) and the least squares method minimizes the sums of the squared distances from the center of gravity of an orbit to the location of the orbiting body. The first person to use the method in this way was a certain Carl Friedrich Gauss whom we all are familiar with. He figured out a method, after over 100 hours of computation, to predict accurately the location of the asteroid (dwarf planet) Ceres in December of 1801.

He did this using only three observations from January/February of 1801, when Ceres was first discovered. The data set was very small (only 41 days of observation) because the monk Piazzi who discovered the asteroid fell ill and the asteroid became hidden by the brightness of the sun. Gauss made his predictions using only the three observations, Kepler’s laws of planetary motion, basic algebra and trigonometry, and the method of least squares. This got him instant fame and launched his career, leading to the frustration of many a linear algebra student. Incidentally, no one knows exactly how he did the initial calculations, since he only published refined versions. He also, in his 1809 treatise

Theoria motus corporum celestium

called the method of least squares “my principle, which I have made use of since 1795” and ignored Legendre when he protested this statement.

This has a lot of information about the method of least squares, much more than this blog post.
This describes the discovery of Ceres and Gauss’s role in finding its orbit.
This is just another source I used to make this post.