Doubling, Squaring, Cubing
Often when Newton's gravitation is a topic of conversation... you will hear that gravity varies inversely with the square of the distance. We all know how to double a line length, to square to get area, and to make a box by cubing? Right? Simple Arithmetic Geometry.
But do you really know what squaring with distance really means? I found that some persons you would think should know... don't.
Let's start with Fig-18. Any two straight lines emanating and radiating from a single point, as example P shown, spread apart the further the radiated distance. Simple right? Line PA shown, equals line PB. These lines can be any length. I drew straight line AB. Then I doubled the distance from PB and arrived at point C. Likewise PB to D. Then another straight line drawn as CD. This is called doubling. Everything is in a single plane. The result is line CD is twice the length of line AB. We double the length radiated and we get a doubling of the cross distance lines CD in relation to AB. AD to CD is a ratio of 1 to 2, in math shown as 1:2.
The line EF is created by doubling the distances of PC and PD to PE and PF. EF is twice the length of CD.
If the lines from P were radii to A, B, C, D, E, & F... and the cross lines were arcs... the above relationships of doubling remain the same.
Now looking at Fig-19, I have radiated four straight lines from a point P. And at distances similar to my above example I located squares. The circumference line, four sides, of the second square is twice that of the first. The circumference of the third square is double that of the second. Thus we are still doing the doubling of distance radiating outward from P and get doubling of respective lines.
However, The area of the second square is four times greater that the area of the first... and the area of the third square is four times greater than the second. Thus, area is NOT doubling but quadrupling. The area increases as 1 to 4 to 16 and etc... We say the area is squared because the square areas can be calculated by taking one side and multiplying by another side, squaring.
Please note, it is better to think quadrupling. If these squares were circles, the quadrupling of area relationship still remains, but the calculation of area for the circle, any regular polygon, or any similar irregular plane figures has to be done otherwise from just squaring.
In Fig-20 I have shown three dimensional spheres instead of cubes for example. The amount of volume of the spheres from the smallest to next larger increases by eight. If volume of sphere A is 1... then the volume increases outbound as 1, 8, 64... with doubling of distance as portrayed above. In your mind, imagine Fig-19 as cubic boxes. How many small boxes would you have in the second and third level cubes?
To understand the statement "gravitation varies inversely with the square of the distance", refer back to Fig-19... If gravity radiates covering the surface on the first small square ABCD with an example value of 1g... It would also cover the square surface of EFGH totally with a value of 1g spread out.
As far as anyone knows at this time of writing, there is NO weakening of gravitation per radiated distance!
There is only less gravity at a further distance per equal unit surface area!
For the area of ABCD on the surface of square EFGH there is only 1/4 of the 1g that is spread about the surface of square EFGH. Thus the further away from the emanating point the less gravitation per equal unit of area.
If there were a space craft in orbit at 103 miles from Earth, and another at 1678 miles from Earth; they would both experience the same value of Earths gravitation for one orbit each. The craft furthest away would have less gravitation acting on it for an equal short distance, as the close to Earth craft. But, if the orbits are thought of as arcs to turn, the close craft has to turn sharper, and the one further out has to turn more gradually, using less force per unit area, but both have the same gravitation for the total orbit trip of each. (Technically there are some variables as atmosphere, moon, sun, and planets... but in general, this is how it works.)
Is this what your Science, Physics or Math Instructor explained to you?
