Regular derivation on The Universal Law of Gravitation?

[I am Meaningless]

Hello Everybody, I am Meaningless and I had this doubt on Newtons laws of gravitation while deriving it. My textbook stated the following derivation 9 for any two masses m₁, m₂, and radius 'r'
It stated that according to the law of product of masses,
F (Directly Proportional) m₁*m₂
And according to the inverse square law,
F (Directly Proportional) 1/r²
Now here came my doubt..:
They then said that, When both Forces (F's) were combined, we would get,
F ( Directly Proportional) m₁m₂/r²
But I thought of elaborating the ''combination'' and logically approached it. Here's what I got:
If F was directly proportional to the product of the masses, the it would be equal to the product of the masses and a proportionality constant which I took as 'k'.
Now similarly if F was directly proportional to the inverse of the square of the radius then it would be equal to the inverse of the square of the radius multiplied by a proportionality constant which I took as 'l'.
Now if I multiplied them I would get...
F² = K*L*m₁*m₂/r²
and if I rooted (square root) the entire equation on both sides I would get...
F = (SQRT)[K*L*M₁*M₂/R²]
Which did not seem to match with F = Gm₁m₂/r²
Now please tell me if:
1.There is any rule with the Proportionality that I am not aware of (or)
2.What my textbook has given is wrong
3. This was experimentally proved as an exception
4. If there is any other derivation for it
Any help will be appreciated. Thanks in advance :-)

 

[Dale]

Now if I multiplied them

This step is wrong. You need one force law which obeys both proportionalities. The product you took does not obey either.

 

[I am Meaningless]

This step is wrong. You need one force law which obeys both proportionalities. The product you took does not obey either.

Thanks a lot for the reply Dale. So can I conclude that, I can only multiply two proportionalities only if one side of both equation obeys the original proportionalities? Also could you give another derivation perhaps by using Kepler's Laws, to make my understanding clearly? If that is Impossible, then could you just give another derivation?

 

[Dale]

I can only multiply two proportionalities only if one side of both equation obeys the original proportionalities?

You can certainly multiply two proportionalities any time you feel like it. However, the square root of the resulting expression may no longer be proportional to either of the original proportionalities. The operation that you did is mathematically valid, but just doesn't have the result you require.

Also could you give another derivation perhaps by using Kepler's Laws, to make my understanding clearly? If that is Impossible, then could you just give another derivation?

I don’t think that a derivation is particularly productive. I would just take the law as a given and check to see if it matches the result of experiments. If it does, then you can use it regardless of how it is obtained.

 

[I am Meaningless]

You can certainly multiply two proportionalities any time you feel like it. However, the square root of the resulting expression may no longer be proportional to either of the original proportionalities. The operation that you did is mathematically valid, but just doesn't have the result you require.

I don’t think that a derivation is particularly productive. I would just take the law as a given and check to see if it matches the result of experiments. If it does, then you can use it regardless of how it is obtained.

Okay, Thanks a lot Dale.

 

[David Lewis]

If the units are suitably chosen, the numerical value of G = 1.