It is often claimed that "Newton discovered gravity" in a flash of insight during a "miraculous year" of discovery. The alleged annus mirabilus is supposed to have occurred in 1666, when Newton was home from Cambridge University because the plague had forced its closure. Of course Newton did no such thing. The Latin word gravitas, or heaviness, was in use long before Newton's time. Indeed, people assumed that things fell because of their heaviness. Even the so-called annus mirabilus has been called into question by Newton scholars. Newton's work with the problem of the moon extended over some time. Only gradually did he become clear about all that was involved.

    This qualification of the myth of Newton's achievement is not meant to detract from the magnitude of what he accomplished, but to put it into a more accurate perspective. Newton possessed an astonishing mind. In the tiny piece of his story below you will be mightily impressed by what he did. In order to get a handle on what Newton's involvement with gravity was and its significance, read the discussion below. It makes no claim at all to be the actual route Newton followed in coming to the conclusion he did; nevertheless, creating an historical analysis can help you grasp the accomplishment with which Newton should be credited.

    William Stuckley reports that on April 15, 1726, the year before Newton died, he and Newton had dined together. "After dinner, the weather being warm, we went into the garden and drank tea, under the shade of some apple trees, only he and myself. Amidst other discourse , he told me he was in the same situation as when formerly the notion of gravitation came into his mind. It was occasioned by the fall of an apple as he sat in a contemplative mood."

     As noted above, what occurred to Newton was not the idea that there was such a thing as gravity; rather, Newton gradually realized over several years that gravity's power to make things fall might affect the moon as well as apples and other things near the surface of the earth. Henry Pemberton's account of Newton's realization makes this clear.

 As he sat alone in a garden, he fell into a speculation on the power of gravity: that as this power is not found sensibly diminished at the remotest distance from the center of the earth to which we can rise, neither at the tops of the loftiest buildings, nor even on the summits of the highest mountains, it appeared to him reasonable to conclude that this power must extend much farther than was usually thought. Why not as high as the moon, said he to himself? And if so, her motion must be influenced by it; perhaps she is retained in her orbit thereby. However, though the power of gravity is not sensibly weakened in the little change of distance at which we can place ourselves from the center of the earth, yet it is very possible that so high as the moon this power may differ much in strength from what it is here. To make an estimate what might be the degree of this dimunition, he considered with himself, that if the moon be retained in her orbit by the force of gravity, no doubt the primary planets are carried round the sun by the like power. And by comparing the periods of the several planets with their distances from the sun, he found that if any power like gravity held them in their courses, its strength must decrease in the duplicate proportion of the increase of distance.

    If the moon is to be considered in the same category as the falling apple, then the moon must be able to be seen as a falling body. Newton did regard the moon as a falling body that never completed its fall. His conjecture was that whatever makes the apple fall also makes the moon "fall." Consult the moon as a falling body and complete the exercise given there.

    Newton eventually had at his disposal a body of information, gathered from various sources, including his own work.

    1. He knew that the rate of acceleration of falling bodies at the surface of the earth was 32 ft/sec every second. In 1666 he first used the inaccurate rate published in the English translation of Galileo's Dialogues, which had appeared the year before.

    2. From Galileo he knew that the distance an accelerating body traveled equaled half the product of the acceleration times the square of the time of the fall (d = ½ at²).

    3. While he recognized Galileo's contribution of the concept of natural motion, that is, motion that would continue forever unless acted upon by an outside force, he disagreed with him that such motion occurred in a circle. For Newton (as well as for Descartes) natural or inertial motion occurred in a straight line. That is, all bodies that were in motion at a uniform velocity would continue to move at that velocity in a straight line unless their motion was interrupted by the application of an outside force. Since the moon was not traveling in a straight line something must be causing it to bend from that path. This is where the apple force came in. Perhaps whatever made apples fall also affected the moon. One can think of the apple force as a "cosmic string" attached to the moon as it twirls around the central earth.
    As implied above by his earlier confusion about two balanced forces, it is clear that in 1666 and for some years thereafter Newton was confused about inertial motion.  If there were such a force making the moon recede from the earth then it would oppose and cancel any force pulling the moon into an orbit around the earth. Should not the result of these two opposing forces be that the moon would be free to display natural motion as Newton understood it - in a straight line? But if so then it would not orbit the earth. It would take a long time, close to two decades, before Newton's thinking on all this was cleared up. So we cannot credit Newton with a moment in which suddenly all became clear.

    4. He stated for himself that when natural motion did not exist, then something must have changed that state of constant velocity into an acceleration; i.e., wherever there was an unbalanced force there must be an acceleration and whenever there was an acceleration some force must be causing it (F a and a F).

    5. He had early figured out for himself (though it was known to Huygens on the continent) that a force directed to the center of circular motion (like the one "pressing" on the moon)  would depend on several factors.  It would vary directly with the size of the object being "pressed" (mass), and inversely with the distance away from the center of motion. It varied directly with the speed with which the object moved in the circle, but as the square of the speed. Altogether, the tension F_c  mv²/r.   This result would also apply to the comparable case of twirling a rock on the end of a string, where the force would give the tension in the string.  If one thinks of the moon as being held in tension by a cosmic string, then the problem becomes one of figuring out how to use his new knowledge about the tension to solve the problem of the moon.

    6. Newton was one of the few people of his time who had actually read Kepler's work. He knew Kepler's results, though he ignored elliptical orbits for the time being because they made work much more complicated than circles. But he knew the Third Law, which related the period of a planet and the velocity with which it moved according to the proportion T²   r^3.

    Don't lose sight of Newton's conjecture: whatever makes the apple fall also affects the moon. If one thought of the apple as being pulled toward the earth, then the cosmic string was pulling the moon toward the earth. But remember, this is only a conjecture. How could Newton prove it?

       The first step is to re-express the tension in the cosmic string with data applying to a "falling" body in orbit. In general the tension was dependent on the mass of the body times the square of its velocity divided by the distance of the body from the center of the earth. Velocity equals distance divided by time. If Newton re-expressed the velocity of the body as distance over time, he could choose as distance the miles traveled in one complete orbit, and as time the so-called period of the satellite. The distance would be given by the circumference of the orbit   and the time by the period T; hence v = /T. Substituting this for v in F_c   mv²/r yields
                                            F_c  m(/T^{) 2}/r  =  ^{m() 2}/T ² x 1/r ; hence F_c kr /T²
 It's at this point that Newton realizes that the T² in the denominator above might be relevant. The earth-moon system, he concluded, would be subject to Kepler's Third Law just like the planets were. Therefore, for any system to which Kepler's Third Law applies he could substitute r³ for T² wherever he found it since they were proportional to each other. This yielded

                                                        F_c  kr/r³ or   F_c   k/r²
     The above result means that for a satellite of mass m to which Kepler's Third Law applies, the tension in the cosmic string holding it in orbit will decrease in amount according to the distance away the satellite is from the center of the earth. Specifically, the tension drops off as the inverse square of the distance from the earth's center. If, for example, the tension were an amount t at the surface of the earth it would be 1/4 t if the satellite were 2 earth radii from the center of the earth (or 1 earth radius above the earth). At 3 earth radii from the center the tension would be 1/9 t, at 4 radii 1/16 t and at 60 radii the force would be 1/3600 t. This result is known as Newton's Law of the Inverse Square.

    Now Newton is conjecturing that it was the force of this cosmic string that pulls on the moon and causes it to "fall" toward the earth into an orbit. Is there any way he can use his conjecture to calculate what we did in the exercise on the moon as a falling body - how far the moon falls in a given amount of time? He could use Galileo's formula d = ½ at² as long as he does not use a = 32 ft/sec². That value for a was for objects like apples at the surface of the earth. But Newton knows that force and acceleration are proportional to each other; hence if the force of tension is diminished as one recedes from the surface of the earth, so too must any acceleration caused by that force. Newton assumed the moon to be about 60 earth radii from the earth's center.⁽¹⁾   That means, as we saw above, that the tension in the cosmic string at the distance of the moon is 1/3600 what it is at the surface. And that in turn means that the acceleration of the moon must be 1/3600 of the rate things fall near the surface of the earth. Apples, Newton knows, fall at 32 ft/sec². So the moon (or any object at that distance away) must fall at a rate of 1/3600 x 32 ft/sec². The acceleration of the "falling" moon, a_m, = 32/3600 ft/sec².

    Using Galileo's formula d = ½ at² (where a = a_m), Newton calculated how far the moon would "fall" in one minute of time.
                                                    d = ½ (32/3600) x 60² seconds = 16 feet     This result was based on the conjecture that the force causing apples to fall was the same one that caused the moon to fall. But what if Newton's conjecture was wrong? Then his result is meaningless. It would be like calculating how much money per year you would have after taxes if you won the lotto. It's all well and good if you hit the lotto. But if you don't, then it is just an interesting exercise (and it might be almost as complicated as Newton's calculation).

    But what if there was another way to calculate how far the moon "falls" in one minute that did not depend on apples or even on forces? If we did an independent calculation of how far the moon "falls" and it came out to be 16 feet, we would be pretty sure that there was something to the first calculation. And that would mean that we could be pretty sure of the assumptions on which the first calculation was based, the assumption that made the calculation possible. That assumption, remember, was that whatever makes apples fall also makes the moon "fall."

    We can, in fact, use the approach of the engineer to find out how far the moon "falls" in one minute of time. Assume in the diagram below that C is the center of the earth and A locates the moon. AB represents how far the moon would travel in one minute if it had only natural motion. Because it is a falling body it "falls" back to D. Thus BD is the distance it falls in one minute. Our task is to find BD.

   Now, let's redraw the diagram, adding a few additional lines. We have dropped a perpendicular from D to AC, meeting it at F and we have extended the radius AC to make the diameter AE. We have also drawn FD and the line AD. Recall that our task is to find BD, the distance the moon "falls" in 1 minute.

  Using the same logic as that described two paragraphs ago, we can assume that BD = AF. If we can find the length of AF, then, we have accomplished our task.     We can find AF through geometry. In the diagram to the right the large triangle AED and the small triangle ADF are both right triangles.⁽²⁾ These two right triangles are similar since their sides are perpendicular right to right and left to left. In similar triangles correponding sides are proportional; i.e., the sides opposite the acute angle in the triangles (AD and AF) form the same ratio with the sides opposite the right angles (AE and AD). In our triangles this means

               AD/ AE = AF/AD

 Multiplying both sides by AD gives:

            AF = AD²/AE

    We know AE. It is twice the distance from the earth to the moon or twice 60 earth radii (480,000 miles). AE in feet = 480,000 x 5,280 = 2,534,400,000 feet. Since we are assuming arcAD = line AD, we can calculate AD by finding arc AD.Arc AD represents the distance traveled in one minute around the moon's orbit. We know the total orbit, and we know what portion of the total orbit one minute is. One minute out of 27.32 days is a very small fraction, but a fraction nonetheless. It is 1÷ the number of minutes in 27.32 days or 1/39,341. The total distance around the moon's orbit is 2r where r = 240,000 miles. This gives 1,507,968 miles. Arc AD, then, is

                                            1/39,341 x 1,507,968 miles = 38.33 miles or 202,386 feet

                            AF = (202,386)²/2,534,400,000 = 40,960,092,000/2,534,400,000 = 16.2 feet

    What an impressive achievement! After all this Newton has shown that the moon "falls" in one minute the distance he predicted it should if the apple force was causing its fall. One has to be willing to say that Newton's assumption about the apple force, however it works, has merit.

    In fact, the first time Newton made this calculation - as a young man of 24 - his answer varied from 16 too much to be acceptable to him. Many years later he said that in these early years he had shown the predicted value (based on his assumption about the apple force applying to the moon) agreed "pretty nearly" with the actual calculated value. This may be why he kept the result to himself and only made the whole thing public when urged to by Edmund Halley two decades later.

    In bringing this examination to a close let's look at the logic behind Newton's argument. It stands as an example of the normal logic of a scientific proof and illustrates what can and what cannot be claimed for science.

    Let p stand for Newton's assumption and q stand for the prediction based on that assumption.

    p = the same force that makes apples fall also makes the moon "fall"

    q = the moon "falls" 16 feet in one minute

    Newton's argument is:
                                                                        If p then q
                                                                        q is true
                                                                        _________
                                                                         Therefore p

    Now as impressive as Newton's demonstration is, one cannot say that it possesses logical certainty. One can think of any number of examples of this form of logical argument that are untrue. For example, let p equal the statement that I win the lotto and q be the statement that my taxes increase. The argument above then runs:

                                                    If I win the lotto then my taxes increase.
                                                    My taxes have increased.
                                                     ______________

                                                    Therefore I won the lotto

    Most scientific proof uses an argument of this structure. One conjectures that something causes a certain result and then seeks evidence that the prediction is true. Finding the evidence one concludes that the conjecture has been verified. Of course the persuasiveness of the proof depends on various factors - how exclusive the proposed cause appears, how improbable or precise the prediction is, etc. Because other factors may be involved in the production of the predicted result one can never be certain beyond all doubt that the identified cause is exclusively responsible for the result. In fact, even the formation of the conceptual and linguistic categories we use when forming the hypothesis carry with them hidden assumptions that often only become clear to later generations or people from different cultures. So the exclusiveness of the cause is never guaranteed. We have already seen examples of this and we will see more as we continue through this course.

Copyright ©1998 Frederick Gregory

1. "The mean distance of the moon from the earth ... in semidiameters of the earth is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60½, to Street, 60²/₅; and to Tycho, 56½. ... Let us assume the mean distance of 60 diameters in the syzygies, and suppose one revolution of the moon, in respect of the fixed stars, to be completed in 27 days, 7 hours, and 43 minutes, as astronomers have determined." From Isaac Newton, Mathematical Principles of Natural Philosophy, Proposition IV, Theorem IV.

2. The large triangle is a right triangle because it is inscribed in a semicircle. All angles inscribed in a circle and subtended by the diameter of the circle are right angles. The small triangle is a right triangle by construction since we dropped DF perpendicular to AE.