When Newton was a young man, he attempted to mathematically verify Galileo’s conclusions that the gravitational accelerations of all bodies are absolutely equal, regardless of their unequal masses. Specifically, Newton attempted to directly compare the acceleration (fall) of the orbiting Moon to the acceleration (fall) of the apple on Earth. Newton believed that all he needed to know was the distance of the Moon from the surface of the Earth. Why? Because, as stated by Zeilik, “Newton did not need to know the mass of the apple or the mass of the moon, because [Galileo’s Leaning Tower experiment had already proved to him that] the accelerations of these bodies do not depend on their masses.”
Newton knew from Galileo’s previous work that the apple accelerates toward the Earth at about 9.8 m/s/s. He also knew that the approximate distance (radius) from the surface of the Earth to the Earth’s center was about 6,400 km. Newton then estimated the distance from the center of the Earth to the center of the Moon to be about 60 Earth radii. This turned out to be an extremely close estimate (Figure 19). From this estimate Newton was able to calculate the approximate orbital velocity of the Moon during its orbital period of 27.3 days (Figure 20).
Based on the above data, Newton arrived at the following calculation. The acceleration of the apple on Earth (9.8 m/s/s) divided by the square of the distance 602 (3,600) told Newton that the Moon should accelerate toward the Earth at the rate of about 0.27 cm/s/s (9.8 m/s/s ÷ 3,600 = 0.27 cm/s/s), or a constant distance of fall of about 0.14 cm each second (Figure 19). This calculation seemed to be approximately correct, and it appeared to validate Galileo’s law that the unequal masses of the Moon and the apple gravitationally accelerate toward Earth at the same rate. Was this really a valid direct confirmation of Galileo’s law of equal gravitational acceleration of all masses, and Newton’s conclusions based on it? Strangely enough, the answer is a resounding: NO.
Why are the above calculations not a valid direct comparison of the fall of the apple relative to the fall of the orbiting Moon (adjusted for distance), as Newton, Feynman, Zeilik, and others have suggested? The primary reason is because Newton misassumed that the Moon and the apple would accelerate toward the Earth at the same rate, 9.8 m/s/s, and he based his calculations on this false assumption. This false assumption was somewhat similar to assuming that the Sun and the apple should accelerate toward the Earth at the same rate. It was yet another example of circular reasoning, because Newton assumed that Galileo’s Leaning Tower experiment was correct.
On the contrary, beginning with the next Section T we shall empirically and conclusively demonstrate that the apple would fall toward the Earth at a much faster rate than the Moon from the same distance. We shall also demonstrate over the next 20 sections that all bodies gravitationally accelerate relative to one another at different states of velocity, depending upon their very different relative mass-ratios. For these reasons, Newton’s computations and conclusions concerning the equal gravitational acceleration of the apple and the Moon were totally incorrect. 
Another reason for Newton’s false assumption was because Newton apparently assumed that the Earth would remain stationary relative to the Moon. However, Newton’s third law of motion would require the Earth to move in a reciprocal orbit to that of the Moon. In other words, Newton’s third law of motion would cause the Earth to react and accelerate in a miniscule orbit relative to the Moon, whereas the miniscule force received from the apple would not cause the Earth to move naturally at all. We now empirically know that all of these reciprocal concepts are correct.
Another reason for Newton’s false assumption was because Newton apparently forgot about his first law of inertial resistance. Even assuming that each mass (the Moon and an apple) received the same quantity of intensive force from the Earth per unit of mass, the body with the least mass (the least inertial resistance), i.e. the apple, would act and accelerate toward the Earth much faster than the much greater inertial resistance of the Moon.
Another reason for Newton’s false assumption was because Newton apparently also forgot his second law of motion. When the same force is applied to two unequal masses, the acceleration of the smaller mass (i.e. the apple) is inversely proportional to the acceleration of the larger mass (the Moon) (Figure 9). Another reason for Newton’s false assumption was because the mass of the Moon would also pull on the tiny mass of the apple, and this would distort the motion of the apple toward the Earth the closer the Moon and the apple were relative to the Earth.
A final reason for Newton’s false assumption was because the acceleration of the Moon was only calculated for one second through a vacuum, whereas the acceleration of the apple was calculated per second per second through air. This is like comparing moving apples and stationary watermelons, for many obvious reasons.