Introduction of the Gravitational constant

Question

The constant G in Newton's law $F = G m_1m_2/r^2$ is, as far as I know, absent from Newton's work - who introduced this constant?

Answer 1

As a constant using ordinary units (e.g., metric), the gravitational constant G doesn't occur until late in the 19th century. Scientists in Newton's time and into the 18th century were quite happy to work in terms of proportionalities and ratios; the constant of proportionality never needed to be written down. For example, one does not see a gravitational constant, in any form, in Cavendish's description of his experiments to "determine the density of the Earth" (1798). This practice of not using a gravitational constant with regard to earthly matters continued throughout much of the 19th century, e.g., Pratt (1855).

One does see something much akin to the Newtonian gravitational constant in the works of Laplace (1799) and Gauss (1809). Using modern nomenclature, Newton's law of gravitation using the gravitational constant of Laplace and Gauss is $$F = k^2 \frac{m_1 m_2}{r^2}$$

The key differences between that and $F=G \frac{m_1 m_2}{r^2}$ are that the Laplace's and Gauss's $k$ is the square root of the Newtonian constant, and that the system of units is more apropos to modeling the solar system. Gauss explicitly specified his system of units: One mean solar day as the unit of time, one solar mass as the unit of mass, and one astronomical unit (the mean distance between the Earth and the Sun) as the unit of length. The Gaussian gravitational constant has a numerical value of 0.01720209895, as reported by Gauss. This value persisted as a defined constant until very recently (2012, and perhaps later).

Aside: Note that making $k$ a defined constant effectively made the astronomical unit a derived quantity, divorcing it from the size of the Earth's orbit. Keeping $k$ at the value established by Gauss was standard practice throughout the 19th and 20th centuries.

The push to measure $G$ using earthly units didn't occur until after physicists saw the value of the metric system and its predecessors, was largely driven by electromagnetism. A flurry of publications occurred late in the 19th century regarding the Newtonian gravitational constant, apparently starting with Cornu and Baille (1873). (The reason I wrote "apparently" is because I can't access that paper, and because Poynting describes that paper as "brief".) Judging a book by its cover (or a scientific paper by its title), the title "A new determination of the constant of attraction and of the mean density of the Earth" certainly does appear that Cornu and Baille made the connection between assessing the mean density of the Earth and the gravitational constant. Boys (1889) makes the connection very explicit; Boys announced he was going to use a Cavendish experiment to measure the gravitational constant as the primary goal. That this measurement also would yield an estimate of the density of the Earth was secondary.

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C.V.Boys (1889), "On the Cavendish experiment", Proc. R. Soc. London, 46, 253-268

C.V. Boys (1894), "The Newtonian constant of gravitation," Notices R. Inst. 14 353-377.

H. Cavendish (1798), "Experiments to determine the Density of the Earth," Phil. Trans. R. Soc. London, 88 469-526.

A. Cornu et B. Baille (1873), "Détermination nouvelle de la constante de l'attraction et de la densité moyenne de la Terre," Comptes Rendus lxxvi, 954-8.

Gauss (1809), "Theoria motus corporum coelestium in sectionibus conicis solem ambientium"

Laplace (1799), "Traité de mécanique céleste".

Poynting (1894), "The mean density of the Earth".

Answer 2

According to Wikipedia: "one of the first references to $G$ is in 1873, 75 years after Cavendish's work."

Newton assumed an inverse square law, as had already been proposed. Inverse square laws are usually due to spherical propagation: something is emitted in all directions from a point, so that at later times it is distributed on the surface of a sphere. This implies velocity, which Newton ignored (or assumed was too fast to mention). He could have conjectured that gravity was spherically propagated, and used $4\pi\cdot r \cdot r,$ instead of just $r\cdot r.$

G is not merely a constant of proportionality. It has units which deserve further attention. It is roughly ($c\cdot c \cdot R / M)/(4\pi),$ where $c=3\mathrm{E}8 ~\textrm{m/s}$ = speed of light (or gravity); $R=4.6\mathrm{E}26~\textrm{meters}$ = radius of visible universe; $M=3\mathrm{E}52~\textrm{kg};$ and $4\pi$ is the neglected spherical factor.