This series was motivated by a terrific article that I read in the American Mathematical Monthly about Lagrange points, which are (from Wikipedia) “points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies.” There are five such points in the Sun-Earth system, called 
The article points out a delicious historical factoid: Lagrange had a slight careless mistake in his derivation!
From the article:
Equation (d) would be just the tool to use to determine where to locate the JWST [James Webb Space Telescope, which is now in orbit about
to common denominator form is incorrect… Fortunately, at some point in the two-and-a-half centuries between Lagrange’s work and the launch of JWST, this error has been recognized and corrected.
This little historical anecdote illustrates that, despite our best efforts, even the best of us are susceptible to careless mistakes. The simplification should have been
![q' = \displaystyle \left[ 1 - \frac{1}{(m-1)^3} \right] \cdot \frac{1}{r^3}](https://s0.wp.com/latex.php?latex=q%27+%3D+%5Cdisplaystyle+%5Cleft%5B+1+-+%5Cfrac%7B1%7D%7B%28m-1%29%5E3%7D+%5Cright%5D+%5Ccdot+%5Cfrac%7B1%7D%7Br%5E3%7D&bg=ffffff&fg=000000&s=0&c=20201002)
(Parenthetically, The article also notes a clear but unintended typesetting error, as the correct but smudged exponent of 3 in the first equation became an incorrect exponent of 2 in the second.)
Mean Green Math 
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