Prior to this incident, Newton had invented something we now call the Calculus, and with it he had mathematically proven that an "inverse square law" dependence, such as gravitation on distance, MUST act as though all the mass of an object (the Earth) acts as though it is at the exact center of the Earth.
Newton was trying to think of some way of experimentally confirming what he had already calculated, of that inverse square distance dependence. In the Autumn of either 1665 or 1666, he was either 22 or 23 years old, and he was sitting out in a field near his mother's home, looking up at the Moon in the sky overhead. He believed that the Moon was orbiting the Earth because of the gravitation of the Earth. He believed that the Moon would normally have gone straight off into space, but that the Earth's gravitation caused it to "constantly fall" toward the Earth, making its path curved rather than straight. But he hadn't thought of any way to experimentally prove that!
By Newton's time, science had fairly accurately determined the radius of the Earth at just under 4,000 miles (6400 km). It was also known that the Moon orbited the Earth at an average distance of just under 240,000 miles (384,000 km), about 60 times as far from the center of the Earth as he was here on the surface. These things were known.
The Metric System would not be invented for around one hundred and fifty more years, so Newton only worked in the English system.
When an apple fell from a tree near him, it suddenly dawned on Newton that the same Earth's gravitation that must be curving the Moon's path must also have made that apple accelerate toward the Earth's center in its fall!
His calculations had shown that the acceleration should NOT depend at all on the size or mass of the object. In that, Newton had actually mathematically confirmed what Galileo had experimentally found about a hundred years earlier, where (around 1590) Galileo dropped a large cannonball and a small lead weight from the top of the Leaning Tower of Pisa, and found that they both hit the ground at the same time (both having accelerated at the same rate during the fall).
So, if the size or weight of the apple was not important, then if that apple was out at the distance of the Moon, Newton realized that it should then have the same acceleration as the Moon does, and the apple would therefore also orbit the Earth.
So now all he had to do is some fairly simple calculations!
He knew that an apple falls at "the acceleration due to gravity", about 32 feet per second per second, what we call g. And that in the first second, that apple would fall very close to 16.1 feet (193") toward the Earth.
Then, if that apple was moved to a place 60 times as far away from the center of the Earth, and if gravitation actually DID depend on an inverse square relationship, then the apple out there should fall 1/3600th as far as it did from the tree. So he multiplied 16.1 feet by 1/3600 and got an expected falling distance in one second to be 0.0535 inch.
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That meant that the Moon must "fall" 0.0535" toward the center of the Earth in every second (from an otherwise straight line). This is a REALLY small curvature (less than 1/16" over the 3,300 feet that the Moon moves every second!). But it turns out that it is still pretty easy to confirm. If you draw a really big circle that represents the orbit of the Moon, and then look at a small part of that circle, the part that the Moon moves through in one second, then simple geometry can determine that small curvature. (circle, chord, radius, etc.)
Somehow, all the historical records seem to neglect the next detail! Interestingly, in this very simple calculation, the brilliant Newton needed to convert the 4000 mile radius of the Earth to inches, and apparently Newton made a multiplication error in converting the radius of the Earth to inches! With this wrong value, there was no agreement in the results! Newton set aside this whole subject for six years!
Around then, a new calculation of the radius of the Earth had been made (by Jean Picard in 1671). Newton then decided to try the calculation again, (about six years later!) and he did it right this time, and the result was 0.0534", a virtually perfect match. The inverse square law of gravitation was therefore proven. Also proven was the fact that the mass of the object, whether apple or Moon, did not affect the acceleration results.
I guess it is understandable that no historian wanted to insult the genius Isaac Newton by mentioning this amazingly simple math error, and maybe that is why no one even today mentions the six year delay before Newton ever mentioned his experimental confirmation of his gravitational equations. As a Physics student at the University of Chicago in the late 1960s, we were informed of this actual history. Maybe the Professors wanted to let us know that even we Physics students were human, and possibly fallible, so they informed us that one of our biggest heroes, Isaac Newton, had once made a surprising math error.
Decades later, Newton calculated that there actually IS a tiny gravitational effect due to the mass of the smaller body. But it is an extremely tiny effect, for any practical sized objects, because the Earth or the Sun is so big and massive!
There is also a tiny effect due to a differential gravitational effect of the Sun, which very slightly reduces the actual value for the Moon, which even explains that 0.0001" discrepancy!
Now you know about Newton and the Apple! It really DID occur, and there is even an aspect of proof of that in the six year delay due to Newton having made a simple multiplication error!
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago