experimental-physicshistorynewtonian-gravityphysical constants
Did Newton estimate the gravitational constant $\mathrm{G}$?
In my head, he did this by comparing:
As explained here.
But did he actually take the next step and calculate what G must be to explain both accelerations? If so what value did he get?
Newton's 1st and 2nd laws weren't particularly revolutionary or surprising to anyone in the know back then. Hooke had already deduced inverse-square gravitation from Kepler's third law, so he understood the second law. He just could not prove that the bound motion in response to an inverse square attraction is an ellipse.
The source of Newton's second law was Galileo's experiments and thought experiments, especially the principle of Galilean relativity. If you believe that the world is invariant under uniform motion, as Galileo states clearly, then the velocity cannot be a physical response because it isn't invariant, only the acceleration is. Galileo established that gravity produces acceleration, and its no leap from that to the second law.
Newton's third law on the other hand was revolutionary, because it implied conservation of momentum and conservation of angular momentum, and these general principles allow Newton to solve problems. The real juicy parts of the Principia are the specific problems he solves, including the bulge of the Earth due to its rotation, which takes some thinking even now, three centuries later.
The real history of scientific developments is complex, with many people making different contributions of various magnitude. The tendency in pedagogy is to relentlessly simplify, and to credit the results to one or two people, who are sort of a handle on the era. For the early modern era, the go-to folks are Galileo and Newton. But Hooke, Kepler, Huygens, Leibniz and a host of lesser known others made crucial contributions along the way.
This is especially pernicious when you have a figure of such singular genius as Newton. Newton's actual discoveries and contributions are usually too advanced to present to beginning undergraduates, but his stature is immense, so that he is given credit for earlier more trivial results that were folklore at the time.
To repeat the answer here: Newton did not discover the second law of motion. It was well known at the time, it was used by all his contemporaries without comment and without question. The proper credit for the second law belongs almost certainly to the Italians, to Galileo and his contemporaries.
But Newton applied the second law with genius to solve the problem of inverse square motion, to find the tidal friction and precession of the equinoxes, to give the wobbly orbit of the moon (in an approximation), to find the oblateness of the Earth, and the altitude variation of the acceleration of gravity g, to give a nearly quantitative model of the propagation of sound waves, to find the isochronous property of the cycloid, and a host of other contributions which are so brilliant ad so complete in their scope, that he is justly credited as founding the modern science of physics.
But in physics classes, you aren't studying history, and the applications listed above are too advanced for a first course, and Newton did indeed state the second law, so why not just give him credit for inventing it?
Similarly, in mathematics, Newton and Leibniz are given credit for the fundamental theorem of calculus. The proper credit for the fundamental theorem of calculus is to Isaac Barrow, Newton's advisor. Leibniz does not deserve credit at all. The real meat of the calculus however is not the fundamental theorem, but the organizing principles of Taylor expansions and infinitesimal orders, with successive approximations, and differential identities applied in varied settings, like arclength problems. In this, Newton founded the field.
Leibniz gave a second set of organizing principles, based on the infinitesimal calculus of Cavalieri. Cavalieri was Galileo's contemporary in Itali, and he either revived or rediscovered the ideas originally due to Archimedes in "The Method of Mechanical Theorems" (although he might not have had access to this work, which was only definitively rediscovered in the early 20th century. One of the theorems in Archimedes reappear in Kepler's work, suggesting that perhaps the Method was available to these people in an obscure copy in some library, and only became lost at a later date. This is pure speculation on my part. Kepler might have formulated and solved the problem independently of Archimedes. It is hard to tell. The problem is the volume of a cylinder cut off by a prism, related to the problem of two cylinders intersecting at right angles). Cavalieri and Kepler hardly surpassed Archimedes, while Newton went far beyond. Leibniz gave the theory its modern form, and all the formalism of integrals, differentials, product rule, chain rule, and so on are all due to Leibniz and his infinitesimals. Leibniz was also one of the discoverers of the conservation of mechanical energy, although Huygens has his paws on it too, and I don't know the dates.
The mathematicians' early modern history is no better. Again, Newton and Leibniz are given credit for theorems they did not produce, and which were common knowledge.
This type of falsified history sometimes happens today, although the internet makes honest accounting easier. Generally, Witten gets credit for everything, whether he deserves it or not. The social phenomenon was codified by Mermin, who called it "The Matthew principle", from the biblical quote "To those that have, much will be given, and to those that have not, even the little they have will be taken away." The urge to simplify relentlessly reassigns credit to well known figures, taking credit away from lesser known figures.
The way to fight this is to simply cite correctly. This is important, because the mechanism of progress is not apparent from seeing the soup, you have to see how the soup was cooked. Future generations deserve to get the recipe, so that we won't be the only ones who can make soup.
In the summer of 2010, I had the opportunity to attend a presentation by Reiner Rummel, involved in the GOCE satellite containing a very precise gradiometer. The presentation can be found on the ESA website. It contains a table with orders of magnitude for the accelerations they could measure when still in the lab.
The gravitational acceleration in the lab in Munich they measured is $g=9.80724672 m s^{-2}$. Components (quoted literally from the aforelinked presentation), all units in $m s^{-2}$:
From the number of significant digits in their measurement, it can be seen that they are able to measure down to a $10^{-8} m s^{-2}$ precision. So it might just be possible. However, I remember an anecdote that in the lab, they could measure the metro passing by several hundred meters away. So... good luck.
Note: there are surely more authoritative sources than the one I linked, but as it's not my field of expertise, I don't know them.
Best Answer
Without knowing the mass of the Earth, calculating the gravitational constant is impossible from $g$ and the acceleration of the Moon. The best you can do is calculate the product of the gravitational constant and the Earth's mass (GM). This is why Cavendish's experiments with the gravity of lead weights was important, since the mass of the body providing the gravitational force was known. Once $G$ was calculated from this experiment, the Earth could then be weighed from using either $g$ or the Moon's acceleration (both hopefully yielding the same answer).
The suggestion in the previous paragraph that Cavendish's experiment resulting in a value for $G$ is still not quite right. While a value for $G$ could have been determined from the experiment, Cavendish only reported the specific gravity (the ratio of a density to water's density) of Earth. According to Wikipedia, the first reference in the scientific literature to the gravitational constant is in 1873--75 years after Cavendish's experiment and 186 years after Newton's Principia was first published:
Click on the link if you read French or can find a translator. Also, the symbol $f$ is used instead of $G$.
Newton's Principia can be downloaded here: https://archive.org/stream/newtonspmathema00newtrich#page/n0/mode/2up
Follow up questions copied from the comments (in case the comment-deletion strike force shows up):
Philip Wood: I'm pretty sure that Newton never wrote his law of gravitation in algebraic form, nor thought in terms of a gravitational constant. In fact the Principia looks more like geometry than algebra. Algebra was not the trusted universal tool that it is today. Even as late as the 1790s, Cavendish's lead balls experiment was described as 'weighing [finding the mass of] the Earth', rather than as determining the gravitational constant. Interestingly, Newton estimated the mean density of the Earth pretty accurately (how, I don't know) so he could have given a value for G if he'd thought algebraically
Mark H: Philip Wood is correct. Newton wrote Principia in sentences, not equations. The laws of gravity were described in two parts (quoting from a translation): "Tn two spheres mutually gravitating each towards the other, ... the weight of either sphere towards the other will be reciprocally as the square of the distance between their centres." And, "That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain." This is the full statement of the behavior of gravity. No equations or constants used.
After a quick search, I can't find who first measured $g=9.8m/s^2$. It's not a difficult measurement, but would require accurate clocks with subsecond accuracy. This is an interesting article: https://en.wikipedia.org/wiki/Standard_gravity
Actually, on page 520, Newton lists the acceleration due to gravity at Earth's surface like so: "the same body, ... falling by the impulse of the same centripetal force as before [Earth's gravity], would, in one second of time, describe 15 1/12 Paris feet." So, the value was first measured sometime between Galileo's experiments and Newton's Principia.
Newton knew that the moon was not exactly 60 earth-radii distant. He quotes a number of measurements in Principia: "The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bidliuldus, Hevelius, and Ricciolns, 59; according to Flamsted, 59 1/3; according to Tycho, 56 1/2; to Vendelin, 60; to Copernicus, 60 1/3; to Kircher, 62 1/2 (p . 391, 392, 393)." He used 60 as an average, which results in an easily calculable square, but squaring isn't a difficult calculation anyway.
The inverse square law was already being talked about by many scientists at the time, including Robert Hooke. Newton used the Moon as a confirmation of the inverse square law, not to discover it. He already knew what the answer should be if the inverse square law was true. In fact, it was the orbital laws discovered by Johannes Kepler--especially the constant ratio of the cube of the average distance from the central body and the square of the orbital period--that provided the best evidence for the inverse square law.
In "The System of the World" part of Newton's Principia, he uses astronomical data to show that gravity is a universal phenomena: the planets around the Sun, the moons around Jupiter, the moons around Saturn, and the Moon around Earth. For the last, in order to establish the ratio of forces and accelerations, you need at least two bodies. Since Earth only has one moon, he made the comparison with terrestrial acceleration.
A simple version of Kepler's Third Law to the inverse square law can be shown for circular orbits pretty easily. Define $r$ as the constant radius of the orbit, $T$ as the time period of the orbit, $v$ as the planet's velocity, $m$ as the mass of the orbiting planet, $F$ as the gravitational force, and $k$ as some constant. \begin{align} \frac{r^3}{T^2} = k &\iff r^3 = kT^2 \\ &\iff r^3 = k\left(\frac{2\pi r}{v}\right)^2 \\ &\iff r = \frac{4\pi^2k}{v^2} \\ &\iff \frac{v^2}{r} = \frac{4\pi^2k}{r^2} \\ &\iff \frac{mv^2}{r} = \frac{4\pi^2km}{r^2} \\ &\iff F = \frac{4\pi^2km}{r^2} \end{align}
The quantity $v^2/r$ is the centripetal acceleration necessary for constant speed circular motion.