MadSci Network: Physics
Query:

Re: How did Newton discover the value of the universal gravitational constant?

Date: Wed Nov 11 11:08:16 1998
Posted By: Raymond Nelson, Grad student, Physics, University of New Mexico
Area of science: Physics
ID: 910234718.Ph
Message:

David:

Yours is an excellent question.  The actual value of the constant, however, 
is not really the issue.  Its numerical value is merely an artifact of the 
system of units in use and since you have cited the constant in MKS units I 
am compelled to say that this is a number with which Sir Isaac was entirely 
unfamiliar.  The MKS system of units was not in use in England at the time 
of Newton's formulation.  The Law of gravitation makes use of Newton's 
second law of motion which defines the force on an object.  Since the 
gravitation law uses previously defined quantities, it requires a constant 
to make the numbers work out right.  By the way, the precise measurement of 
the gravitational constant itself was the work of Cavendish in 1798.

Putting that aside, however, it is helpful to look at the form of the law, 
its inverse square nature.  This is the insight of Newton that deserves 
some comment.  According to Jacob Bronowski in "The Ascent of Man," Newton 
first formulated the idea of universal gravitation during the plague year 
of 1666.  In Bronowski's words: "The moon was a powerful symbol for him.  
If she follows her orbit because the earth attracts her, he reasoned, then 
the moon is like a ball (or an apple) that has been thrown very hard:  she 
is falling towards the earth, but is going so fast that she constantly 
misses it-- she keeps on going round because the earth is round."  In the 
ensuing two decades Newton published nothing about gravitation and 
seemingly failed to extend his researches into the problem of the planet's 
motions round the sun.  

In 1684 Sir Christopher Wren, Robert Hooke (of spring constant fame) and 
Edmund Halley entered into a debate on the subject. Halley travelled to 
Cambridge to ask Newton's opinion:

"After they had been sometime together, the doctor (Halley) asked him what 
he thought the curve would be that would be described by the planets, 
supposing the force of attraction towards the sun to be reciprocal to the 
square of their distance from it.  Sir Isaac replied immediately that it 
would be an ellipsis.  The doctor, struck with joy and amazement, asked him 
how he knew it.  'Why,' saith he,'I have calculated it.'  Whereupon Dr. 
Halley asked him for his calculation without any further delay.  Sir Isaac 
looked among his papers but could not find it, but he promised him to renew 
it, and then to send it to him."

The Principia was published three years later, including the calculations 
in question.  Bronowski speculates that the key problem was to understand 
the question in Section 12 of the Principia on "How does a sphere attract a 
particle?"  He had done his original calculation treating the earth and 
moon as particles.  He finally discovered that this assumption is only 
valid for azimuthally homogeneous (radial inhomogenieties are certainly 
permissible) spheres and only if the inverse square law of attraction 
holds.  The proof of this concept required the invention and manipulation 
of the calculus.  No wonder it took him so long.

So things came together nicely.  The data was already there in Kepler's and 
Brahe's figures.  The inverse square law resulted in the attraction of 
heavenly bodies as if they were point particles and in the correct periods 
and shapes of the orbits.  Newton's law appeared to be truly universal for 
230 years.  Not bad.  

For other descriptions of the process of formulation and refinement of the 
law of gravitation I recommend Bronowski's book "The Common Sense of 
Science," and Richard Feynman's "The Character of Physical Law."



Ray Nelson


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