I have long been fascinated by Newton’s proof of Kepler’s Equal Areas Law and wanted to write about it. Of course, others have as well, but I wanted to emphasize an aspect of the proof that supported my philosophy of mathematics.
Before I get to Newton, however, I wanted to discuss how Kepler himself justified this law, since his approach has a number of fascinating historical aspects to it. I have previously discussed Kepler’s ellipse and in the process of doing that research, I came across a number of articles about how Kepler arrived at his equal areas law. One notable result is that even though now we call the idea that a planet orbits the Sun in an elliptical path with the Sun at one focus, Kepler’s First Law, and the idea that the line from the Sun to the planet sweeps out equal areas in equal times, Kepler’s Second Law, Kepler actually discovered these laws in reverse order. See Kepler’s Equal Areas Law
I had been exploring how Kepler originally discovered his first two laws and became fascinated by what he did in his Astronomia Nova (1609), as presented by a number of researchers. Among the writers was A. E. L. Davis. She mentioned that the characterization of the ellipse that Kepler was using was the idea of a “compressed circle,” that is, a circle all of whose points were shrunk vertically by a constant amount towards a fixed diameter of the circle. I did not recall ever hearing this idea before and tried to track down its origin together with a proof — futilely, Davis’s references notwithstanding. I then tried to prove it myself. It was easy to do with analytic geometry. But in the spirit of the Kepler era (before the advent of Fermat’s and Descartes’s beginnings at fusing algebra and geometry) I tried to prove it solely within Euclid’s plane geometry. Some critical steps seemed to come from the great work of Apollonius of Perga (262-190 BC) on Conics. But for me a final elegant proof was not evident until 1822 when Dandelin employed his inscribed spheres. See 
This is another problem from the 
This is a mildly pointless 2015 article about Leonardo Da Vinci’s famous drawing of the Vitruvian Man spread-eagled and inscribed in a circle and a square. I started wondering about the positions and whether they over-determined the circle and square. What hidden constraints were being assumed? One assumption turned out to be famous, namely, that the height of a man equaled the distance between his finger tips when he holds his arms straight out to either side of his body. I had been told this in childhood, and I never knew where it came from. Also, I don’t think it is true in every case (what about women?), though it does appear to be close (and is true in my case). See the