# How did Newton know it was an inverse square law?

I passed a sign in Glasgow today** that told me I was on "Newton Street". I've passed it hundreds of times on my way to the Mitchell library, which I've been frequenting a lot as I race to finish my PhD thesis, and never noticed it before. This got me thinking about good ol' Isaac and his phenomenal contribution to our understanding of gravity. As a physicist, Newton's laws and theory of gravitation are among the main first lessons you learn at University (and even at school). Specifically, we learn that the force between two objects with masses M and m is proportional to the product of these masses and inversely proportional to the square of the distance between them, or in mathematics

Now, the more I think about this formula, the more something puzzles me. I can understand the supposition that the force is proportional to the mass of each object, it makes sense that a more massive object would provide more gravitational interaction than a smaller one. You could rationalise this from observing the solar system through telescopes since Galileo's day - larger planets have more moons and the Sun has everything in the solar system orbiting around it. And if you know it's proportional to some power of the mass, why not just assume the linear relationship (i.e no powers of 2 or 3 etc, I'm supposing this is how Newton's thinking went but the reasoning behind the linear dependence on mass may have been more subtle than this). But how did he know that the gravitational force followed an inverse square law? I think answering this question will be an interesting exercise in showing people how the scientific method works when applied to theoretical physics (I would argue that Newton's law of gravitation is the first example of pure theoretical physics that actually worked!).

Well, it turns out it seems that it's not too difficult to see where it comes from. The realisation that gravity came about from an inverse square law was through Newton doing the following:

1) supposing that gravity was universal, i.e the same law was responsible for an apple falling to Earth and for the moon orbiting the Earth and

2) knowing that he could rely on the most up-to-date observational data, which suggested that objects fell towards the Earth with an acceleration of 9.8 ms-2 and that the moon was accelerating towards the Earth at a rate of 0.00272 ms-2.  He also had to know that the moon was approximately 60 times further away form the centre of the Earth than the surface of the Earth was to the centre of the Earth.

Taking these two observations together Newton could just take the ratio of the two accelerations and observe that they are proportional to the relative distance squared

By the way it useful to note that taking a ratio like this is often a very clever idea, since it means that all the other stuff in Newton's law (which he didn't know yet!) cancels out, so he only needs to focus on the stuff he does know about (the accelerations and distances).

So this is a basic bit of theoretical physics (using your 'noggin' to work something out about how the universe works), but it is important to realise that Newton, and indeed all theoretical physicists, can't do this on their own. Remember, Newton had to rely on experimental data to feed into and check his reasoning. Theoretical physics without reference to experiment is not physics, it is just mathematics. This is important to remember, especially when you read articles in newspapers or popular science magazines about "genius theoretical physicists makes new discovery" - think Stephen Hawking, the people at CERN, Albert Einstein - they didn't, and couldn't, have done it without the most important thing in physics, experiments.

To continue this idea that theory isn't everything in physics, in a future post I will discuss how physics now has three "pillars", whereas before it had two,  theoretical, experimental and computational. Until then, Tschüss!

** Note: I wrote this aaages ago and forgot to publish it ... I'm no longer going to the Mitchell library every day.