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I know that Newton extensively used his theory of gravity to calculate many things, but how did he do so without actually knowing the value of the gravitational constant $G$? Did he estimate it somehow?

Qmechanic
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Ayden Cook
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  • I've seen on other posts that he used ratios so that the value of G dropped out.... But what exactly does that mean? – Ayden Cook Mar 31 '19 at 21:04
  • It means he used ratios so the value of $G$ dropped out. Really. Consider how Kepler's 2nd and (especially) 3rd laws are stated. Newton showed (amungst other things) that he could recover those laws. – dmckee --- ex-moderator kitten Mar 31 '19 at 21:15

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There are two contexts in which the Newton's Law of Universal Gravitation has specific consequences that Newton could work out. The first (and simpler) instance is for bodies moving near the surface of the Earth. It had been known since Galileo's time that all bodies fell at the same rate, so for calculations in this regime, all that was needed was the known value of the acceleration of gravity $g$.

The more complex case occurred in celestial mechanics. For motion in the field of the sun, the force on a body was always proportional to the product of the sun's mass $M_{\odot}$ and Newton's constant $G$. While neither of these quantities was known individually at the time, their product could be inferred from the motion of a single planet. $GM_{\odot}$ appears, for example, in the constant appearing in Kepler's Third Law; in Newton's time, the distances to various solar system bodies were reasonably well known from classical Copernican astronomy.

Buzz
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