A long time ago I saw something similar to the following diagram attributed to a very young Albert Einstein which neatly and intuitively (I think) proves Pythagoras' theorem without any maths if you understand the principles behind it:
I have long been wondering about the maths behind there being no gravity inside a hollow sphere. I have drawn the following diagram in the hopes that it can intuitively explain the phenomenon a la Einstein above:
The lengths of the red intervals are the same between the two concentric circles, as are the lengths of the green intervals (their lengths inside the inner circle are unimportant). Between the red and green intervals on one side, there is less mass (between about 1 o'clock and 2 o'clock) but it is closer. On the other side there is more mass (between about 7:30 and 9 o'clock) but it is further away. I think - but this is not the bit I'm sure about - that the two forces of just these opposite segments exactly counteract each other.
The point being measured is at the intersection of the red and green intervals, and if it's true that there is no nett force on it, it's true for all other points inside the inner circle or sphere by the same reasoning.
Is my reasoning mathematically correct?
I apologise for the images. If someone suggests an easy-to-use tool that can generate nicer images I will replace them.