Gravitation demonstrated by the discoveries of Galileo and Newton

Galileo, the restorer of reason in Italy, discovered the important proposition that heavy bodies falling to Earth (disregarding the slight resistance of the air) have an accelerated motion in a proportion that I will try to make clear. A body left to itself from the top of a tower travels, in the first second, a distance that was found to be 15 Paris feet, according to the discoveries of Huygens, a mathematical inventor. Before Galileo, it was believed that this body, in two seconds, would have traveled only twice that distance, and would thus travel 150 feet in ten seconds, and 900 feet in one minute: this was the general opinion, and even quite plausible to those who don’t look closely; however, it is true that in one minute this body would have traveled a path of 54,000 feet, and 216,000 feet in two minutes.

Here’s how this progress, which at first surprises the imagination, operates necessarily and with simplicity. A body is precipitated by its own weight: this force, whatever it is, that causes it to descend 15 feet in the first second acts equally at every instant, because, nothing having changed, it must always be the same: thus, in the second second, the body will have the force it acquired at every instant of the first second, and the force it experiences every instant of the second. Now, by the force that animated it in the first second, it traveled 15 feet; it therefore still has this force when it descends in the second second. In addition, it has the force of another 15 feet that it acquired as it descended in that first second: that makes 30; it must, since nothing has changed, in the time of this second second, still have the force to travel 15 feet: that makes 45; for the same reason, the body will travel 75 feet in the third second, and so on.

From this it follows:

1. That the moving object acquires in infinitely small equal times infinitely small degrees of speed, which accelerate its movement towards the center of the earth, as long as it does not encounter resistance;

2. That the speeds it acquires are proportional to the times it takes to descend;

3. That the spaces it travels are proportional to the squares of these times or these speeds;

4. That the progression of the spaces traveled by this moving object is like the odd numbers 1, 3, 5, 7. This necessary knowledge of this phenomenon that happens around us at all instants will be made perceptible even to those who might at first be a little confused by all these relationships: it only takes a little attention by looking at this small table, which each reader can expand as they wish. Time the object falls Spaces traveled in each interval Spaces traveled are as the squares of the times Odd numbers marking the progression of movement and spaces traveled

1st Second, one unit of speed. The body descends 15 feet. The square of one is one; the body travels 15 feet. 1 times 15. 2nd Second, 2 units of speed. The body travels 45 feet. The square of 2 seconds or 2 units of speed is 4: 4 times 15 is 60: so the body has traveled 60 feet; that is, 15 in the first second, and 45 in the second. 3 times 15; thus the progression is from 1 to 3 in this second.

3rd Second, 3 units of speed. The body travels 75 feet. The square of 3 seconds is 9; now, 9 times 15 is 135: so the body has traveled 135 feet in the 3 seconds. 5 times 15 feet; thus the progression is visibly according to the odd numbers 1, 3, 5, etc.

It is clear that the power which always acts equally at every instant, and which loses none of its force, must thus increase its effect, until some other force comes to oppose it.

From this small table, a glance will demonstrate that at the end of one minute, the moving object will have traveled 54,000 feet, for 3,600 is the square of sixty seconds: now, 15 multiplied by the square of 60, which is 3,600, gives 54,000. From this beautiful discovery by Galileo, a new question arose. People asked: “Will a body always descend about 15 feet in the first second, no matter where in the universe it is placed?” We see that the fall of bodies accelerates as they fall back on our globe: they all obviously tend, as they fall back, towards the center of this globe; is there not some power that attracts them towards this center? And does not this power increase its force as this center is closer? Copernicus had already had some faint glimmer of this idea. Kepler had embraced it, but without a method. Chancellor Bacon formally stated that it is probable that there is an attraction of bodies to the center of the earth, and from this center to the bodies. In his excellent book Novum scientiarum Organum, he proposed that experiments be done with pendulums on the highest towers and at the greatest depths: “For,” he said, “if the same pendulums make more rapid vibrations at the bottom of a well than on a tower, we must conclude that gravity, which is the principle of these vibrations, will be much stronger at the center of the earth, which this well is closer to.” He also tried to make objects descend from different heights, and to observe if they would descend less than fifteen feet in the first second; but no variation ever appeared in these experiments, as the heights or depths where they were done were too small. So, we remained in uncertainty, and the idea of this force acting from the center of the earth remained a vague suspicion. Descartes was aware of it: he even speaks of it when dealing with gravity; but the experiments that were to shed light on this great question were still lacking. The system of vortices carried away this sublime and vast genius: in creating his universe, he wanted to give the direction of everything to his subtle matter: he made it the dispenser of all movement and all gravity; little by little Europe adopted his system, despite the protests of Gassendi, who was less followed because he was less bold.

One day, in the year 1666, Newton, retired in the countryside, and seeing fruit fall from a tree, as his niece (Mrs. Conduit) told me, let himself fall into a deep meditation on the cause that thus drags all bodies along in a line which, if extended, would pass approximately through the center of the earth.

He asked himself: “What is this force that cannot come from all these imaginary vortices, which have been proven to be so false? It acts on all bodies in proportion to their masses, and not their surfaces; it would act on the fruit that just fell from this tree, even if it were raised three thousand fathoms, or ten thousand. If this is the case, this force must act from the moon’s position all the way to the center of the earth; if so, this power, whatever it may be, may therefore be the same one that makes the planets tend towards the sun, and that makes Jupiter’s satellites gravitate on Jupiter. Now it is demonstrated, by all the deductions drawn from Kepler’s laws, that all these secondary planets gravitate towards the center of their orbits, all the more so the closer they are to it, and all the less so the further they are from it, that is to say, reciprocally according to the square of their distances.”

A body placed where the moon is, which circulates around the earth, and a body placed near the earth, must therefore both weigh on the earth precisely according to this law.

Therefore, to be sure if it is the same cause that retains the planets in their orbits and that makes heavy bodies fall here, we only need measurements, we only need to examine what space a heavy body travels when falling on earth, in a given time, and what space a body placed in the moon’s region would travel in a given time. The moon itself is this body which can be considered as actually falling from its highest point of the meridian. But this is not a hypothesis that one adjusts as one can to a system; it is not a calculation where one should be content with approximation. We must begin by knowing exactly the distance of the moon from the earth, and, to know it, it is necessary to have the measure of our globe. This is how Newton reasoned; but for the measure of the earth, he stuck to the faulty estimate of the pilots, who counted sixty English miles, that is to say twenty French leagues, for one degree of latitude, whereas it was necessary to count seventy miles. There was, to tell the truth, a more accurate measure of the earth. Norvood, an English mathematician, had, in 1636, measured a degree of the meridian quite accurately; he had found it, as it should be, to be about seventy miles. But this operation, done thirty years earlier, was unknown to Newton. The civil wars which had afflicted England, always as fatal to the sciences as to the State, had buried in oblivion the only accurate measure of the earth that existed, and people stuck to this vague estimate of the pilots. By this account, the moon was too close to the earth, and the proportions sought by Newton were not found with accuracy. He did not believe that he was allowed to supply anything, and to accommodate nature to his ideas; he wanted to accommodate his ideas to nature: he therefore abandoned this beautiful discovery, which the analogy with the other stars made so plausible, and which lacked so little to be demonstrated; a very rare good faith, and which alone should give great weight to his opinions. Finally, based on more accurate measurements taken in France several times, and of which we will speak, he found the demonstration of his theory. The degree of the earth was valued at twenty-five of our leagues, the moon was found to be sixty half-diameters from the earth, and Newton thus resumed the thread of his demonstration. Gravity on our globe is in inverse proportion to the squares of the distances of heavy bodies from the center of the earth; that is to say that a body that weighs one hundred pounds at one diameter of the earth will weigh only one pound if it is ten diameters away. The force that causes gravity does not depend on the vortices of subtle matter, whose existence is proven false. This force, whatever it may be, acts on all bodies, not according to their surfaces, but according to their masses. If it acts at one distance, it must act at all distances; if it acts in inverse proportion to the square of these distances, it must always act according to this proportion on known bodies, when they are not at the point of contact, I mean as close as it is possible to be without being united.

If, according to this proportion, this force makes a body travel 54,000 feet in 60 seconds on our globe, a body which will be about sixty radii from the center of the earth must, in 60 seconds, fall only 15 Paris feet or about.

The moon, in its average movement, is distant from the center of the earth by about sixty radii of the globe of the earth: now, by the measurements taken in France, we know how many feet the orbit that the moon describes contains; we know from this that in its average movement it describes 187,001 Paris feet in one minute.

The moon, in its average movement, fell from A to B (figure 48): it therefore obeyed the projectile force that pushes it in the tangent A C, and the force that would make it descend along the line A D, equal to B C; take away the force that directs it from A to C, there will remain a force that can be evaluated by the line C B: this line C B is equal to the line A D; but it is demonstrated that the curve A B, worth 187,961 feet, the line A D or C B will be worth only fifteen: therefore, whether the moon fell in A or in D, it is the same thing here, it would have traveled 15 feet in one minute from C to B; therefore it would have traveled 15 feet also from A to D in one minute. But, by traveling this space in one minute, it travels precisely 3,600 times less of a path than a moving object would here on earth; 3,600 is exactly the square of its distance: therefore the gravitation that acts thus on all bodies also acts between the earth and the moon precisely in this ratio of the inverse reason of the square of the distances.

But if this power that animates bodies directs the moon in its orbit, it must also direct the earth in its own, and the effect it operates on the planet of the moon, it must operate it on the planet of the earth, for this power is everywhere the same; all the other planets must be subject to it: the sun must also experience its law, and if there is no movement of the planets with respect to each other which is not the necessary effect of this power, we must then admit that all of nature demonstrates it. This is what we are going to observe more fully[2].