How Newton deduced gravity obeys inverse square law | Hidden gems #2

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Both Newtonian gravity and Hooke's law admit elliptical orbits in 3D (or 2D, same thing since all solutions are planar), but is it a coincidence? Newton said no, but this connection was only discovered accidentally when he was trying to deduce why gravity obeys inverse square law from astronomical observations at the time. In fact, if you know the history between Newton and Hooke, you will find the close connection between these two forces ironic!

This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.

Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: newtonhookeduality

Sources:
Principia (Translated, first American edition): https://web.math.princeton.edu/~epryw...
Tristan Needham’s account (I used a bit of the geometry parts here): https://www.jstor.org/stable/2323768
I also used a bit of this to clarify what is going on with the geometry in the initial bits: https://journals.sagepub.com/doi/abs/...
Newton’s portrait: https://commons.wikimedia.org/wiki/Fi...
Principia title page: https://commons.wikimedia.org/wiki/Fi...
Principia’s false claim to have proved the converse:
https://www.sciencedirect.com/science...
2/3 school children don’t have access to internet: https://www.unicef.org/press-releases...
Internet access and academic results: https://ieeexplore.ieee.org/document/...

Video chapters:
00:00 Introduction
02:05 Gist of Newton's argument
03:42 Three preliminary results
07:09 Acceleration formula purely from geometry
09:49 Acceleration ratio formula
14:21 Ellipse Hooke's law
17:57 Applying acceleration ratio formula
22:06 Parabolic / hyperbolic orbits?

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