How Newton weighed the mass of the earth without gravitational constant $G$?

Question

I am only wondering about how Newton weighed the mass of the earth if he didn't have the value of gravitational constant $G$, so how he did it? I read about some explanations but i hadn't understand it well, can someone help me? It's for a job.

Answer 1

As a matter of fact, Newton didn't calculate the value of $G$. The first implicit measurement with accuracy within about $1$ % is attributed to Henry Cavendish in a $1798$ experiment. For a detailed look at Wikipedia. The summary of the experiment is as follows

The apparatus featured a torsion balance: a wooden rod was suspended freely from a thin wire, and a lead sphere weighing $0.73$ kg ($1.6$ pounds) hung from each end of the rod. A much larger sphere, weighing $158$ kg ($348$ pounds), was placed at each end of the torsion balance. The gravitational attraction between each larger weight and each smaller one drew the ends of the rod aside along a graduated scale. The attraction between these pairs of weights was counteracted by the restoring force from a twist in the wire, which caused the rod to move from side to side like a horizontal pendulum.

Cavendish and Michell did not conceive of their experiment as an attempt to measure G. The formulation of Newton’s law of gravitation involving the gravitational constant did not occur until the late 19th century. The experiment was originally devised to determine Earth’s density.

Michell had likely intended to move the weights by hand, but Cavendish realized that even the smallest disturbance, such as that from the difference in air temperature between the two sides of the balance, would swamp the tiny force he wanted to measure. Cavendish placed the apparatus in a sealed room designed so he could move the weights from outside. He observed the balance with a telescope. By measuring how far the rod moved from side to side and how long that motion took, Cavendish could determine the gravitational force between the larger and smaller weights. He then related that force to the larger spheres’ weight to determine Earth’s mean density as 5.48 times that of water, or, in modern units, 5.48 grams per cubic centimeter—close to the modern value of 5.51 grams per cubic centimeter.

The Cavendish experiment was significant not only for measuring Earth’s density (and thus its mass) but also for proving that Newton’s law of gravitation worked on scales much smaller than those of the solar system. Since the late 19th century, refinements of the Cavendish experiment have been used for determining $G$.

Answer 2

From the graviational law Newton had the relation $$\frac{GM}{R^2}=g \tag{1}$$ The gravitational acceleration at the surface of the earth ($g=9.8\ \text{m/s}^2$) and the radius of the earth ($R=6400\ \text{km}$) were known at Newton's time.

For the average density of the earth there was at least an imprecise estimate. It should be roughly the density of stone, $\rho=3\ \text{g/cm}^3$.

Then the mass of the earth is $$M=\rho\cdot \frac{4}{3}\pi R^3 \tag{2}$$

So you have two equations (1) and (2) for the two unknowns $G$ and $M$. And with a little math you can solve these for $G$ and $M$.