By ascertaining an approxiamte value of G , perhaps? After that, you know M_earth, and already knowing Earth’s geometry, one arrives at average density rho.
It's also a notoriously difficult experiment to perform. When I did it at university, the value of G I got was out by an order of magnitude - and that was considered a good result!
Oh so the earth density is merely the motivation for the experiment? I read it as the earth mass actually being used somewhere in the formulas within the setup itself which was what confused me.
He uses his experiment to calculate G based only on the test masses and spring and then the _result_ of the calculation was just used as a final step to calculate the mass of the earth, and then from that the density?
Y’know, I went and looked at the current Wikipedia page for the Cavendish experiment. There’s apparently more nuance than my simple outline offers, particularly as G wasn’t treated as an isolated quantity till decades after his experiment.
Several years back when I first learned about the Cavendish experiment, I was indeed surprised that he was able to measure anything related to gravity without using a planetary-scale mass as one of experimental masses.
Clever experiments to measure tiny values are probably my favorite kind of science. Millikan found the charge of an electron by looking for a common integer multiple of charge in tiny oil droplets. Or LIGO measures gravitational waves by watching the interference pattern generated over time by two light beams taking different paths.
F_gravitational = G m1 m2 /r^2
g = G Mass_earth / r_earth^2
Mass_earth = r_earth^2 * g/G
Density_earth = r_earth^2 * g/G / V_earth
Density_earth = 3*g / (4*Pi*G*r_earth)
Prior to Cavendish we already new g and r_earth, just missing G.