Finding Derivations of Newton's Law of Universal Gravitation online

Question: was Newton's law of universal gravitation found empirically (from measurements) or can it be derived mathematically? The short answer is yes -- both. The historical process is separate from what can be done now.

Empirical (historical) route

The second paragraph of the Wikipedia article on Newton's Law of Universal Gravitation says, "This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning." There's a video on YouTube describing that Newton started with

• acceleration of gravity on earth, g = 9.8 m/s^2
• distance between Earth and Moon = 60*(radius of earth)
• radius of Earth = 
• orbital period of moon = 27.32 days

Newton figured out how fast the Moon is circling the Earth using

 velocity = distance/time

where, in the case of the moon circling the Earth,

 distance the moon travels around the Earth = circumference = 2*pi*r_{orbit of Moon}

Plugging in numbers, velocity of Moon around Earth = 1022 m/s. That can then be plugged into the centripetal acceleration,

a_{centripetal} = v^2/r

How does a_{centripetal} compare to the gravitational acceleration g?

g/a_{centripetal} = 60^2

Noticing the 60 is common to the ratio and the distance between the Earth and the Moon, Newton figures that gravitation follows an inverse square law. Newton then checked this against data from observational studies of planets.

That's a big leap to F = G*(M*m)/(r^2), so are there more mathematical routes?

Mathematical Derivations

From nasa.gov

I first found NASA's classroom materials. There are some leaps that I wasn't able to follow. The same content of that page is duplicated on a related NASA page. The derivation starts with F=ma and a_{centripetal} = v^2/r. The author mentions Kepler's Third Law but says T^2 \approx R^3 (which is dimenionally inconsistent) when they mean T^2 \propto R^3. The misuse of \approx and \propto continues throughout the rest of the derivation.

velocity = distance/time

and the distance here is the circumfrence, 2*pi*r, so

period T = (2*pi*R)/v

Drop the 2*pi to get

period v \approx R/T

Square both sides and apply Kepler's Third law

T^2 \propto R^3

to get

v^2 \propto 1/R

The second source of my confusion is subscripting versus superscripting -- v_2 versus v^2.

F = (m*(v^2))/R

I tried submitting a correction to NASA's feedback page but couldn't since the Captcha is missing. :(

From a tutorial

Next I found another student-oriented page that has a derivation which is even less helpful than NASA's. The derivation presented starts from angular speed and F=mr\omega^2.

YouTube tutorial

Happily I found this helpful video from Daniel M, "Deriving Newton's Law of Universal Gravitation".