Isaac Newton (1643-1727) built on the discoveries and insights of mainly two predecessors namely: Galileo and Kepler. But also on innovations such as Rene Descartes’s coordinate systems.

Galileo’s experiments of rolling balls on inclined planes gave insight into the mathematical relationship between gravitational force and acceleration. Decartes’s coordinate system helped make sense of the mathematical relationships.

Decartes published his treatise “Discours de la methode” in 1637. In this text he showed the usefulness of his coordinate system, which made possible the use of algebra on geometric objects. Newton, in the process of formulating his laws of motion, used Decartes’s coordinate system to develop calculus.

Galileo had been heavily involved in the study of motion during his time at the university of Pisa (1589-1592). He produced a mechanical calculator for estimating canon ball trajectories. He was also working on a publication on motion during these years named De Motu ( on motion). He never published it and moved on to other ventures such as the motions of Jovian moons which he observed in 1608. Eventually that got him into trouble with the Catholic Church, which condemned him for heresy for his heliocentric views, which he was forced to recant. He was punished with house arrest for the rest of his life, and during this time he published “Discourses on Two New Sciences” (1638) in which he laid out his earlier investigations on the motion of falling objects. This treatise contained his crucial experiment of the rolling ball on an inclined plane. Newton built on the results of this experiment through his insight of universal gravitation.

Let’s have a look at this ball rolling down an inclined plane. In the 16th century, Galileo knew that it was impossible to accurately measure objects falling straight down. His insight was to slow falling objects down by rolling balls down an inclined plane. The shallower the plane the slower the movement. He used several tools to measure time intervals. Bells, pendulums and water clocks. What he found was that for equal time intervals the distance travelled by the ball followed a constant factor times the square of the elapsed time.

In formula form:

\begin {aligned} \text{(Distance travelled)} = Constant * (Elapsed \space time)^2 \text{ \space \space Or in more familiar terms \space\space} x=C*t^2 \end{aligned}

Shortly after 1608, Galileo saw the circling moons around the planet Jupiter through the newly invented telescope and started to wonder how it was possible that these bodies of matter could circle indefinitely around the considerably more massive body of the planet without falling and crashing. The same thought applied to the Moon circling the Earth. This was proof that there is a mysterious force keeping moons circling around the more massive planets. And when Galileo saw the phases of the planet Venus, he knew for certain that it moved in front of the earth and closer to the sun in a circular path. This meant that the Earth was circling the Sun, and by extension, that all planets circled the Sun.

Around the same time, Kepler’s dogged pursuit to solve the geometry of our earth and the planets in space, culminated into his tree laws.

1: The planets move around the sun in elliptical orbits with the sun at one focal point

2:The planets sweep equal areas of the ellipse in equal time intervals

3:The orbital periods squared are equal to the cube of the ellipses semi major axes.

Kepler did not know why, but he is credited with solving the geometry of planetary motion.

The motion of a heavy lead ball swung on a string will rotate for a long time. When you increase the length of the string and keep the same circumference of rotation, you will find the motion slows down. This means less air resistance and a much longer time before the motion stops. In the 1600’s the concept of air resistance was well known. The motion of the planets that had been known to circle indefinitely since antiquity caused scientists (“natural philosophers” as they were known at the time) to consider that there was no friction resistance and thus no air in the universe to slow the rotation of the known planets and their moons down. However since Newton’s theoretical discoveries there was speculation about an inter planetary substance called “Aether” on which light waves would depend to propagate. (similar to ripples on a water surface) This was finally dispelled by the experiments of Michelson and Morley in 1887.

Newton never performed experiments on motion. His insight was theoretical. He put the experimental findings of others on a solid mathematical foundation.

The realization of friction-less motion caused Newton to come up with his first law. *Every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force*.

Now lets consider how someone in the second half 1600’s would have come up with the mathematical equation of motion through experimentation. Let’s assume we only know about Gallileo’s experiments mentioned above, as well as Decartes’s coordinate mathematics and Keplers equations. We also would have have had access to accurate pendulum clocks which were invented by Cristiaan Huygens in 1657.

Assume we have a pendulum clock that ticks exactly every second. We could release the trigger exactly at the sound of a tick and stop the motion of the cart at the second tick by moving the movable stop to the right position that synchronizes the tick of the pendulum with the snap of the collision sound when the cart hits the stop. We then read the distance travelled by the cart on the ruler fixed to the experiment table. Then repeat the procedure and have the cart stopped at the second tick of the pendulum. Repeat again for the third tick of the pendulum and so forth. The 17th century experimenter would choose the values of the weight outside the window pulling the cart forward and the weight on the cart through a bit of trial and error. The values in the image above gives reasonable results for a 3 second trial on a 5 meter long table.

We could repeat the same experiment with different weights on the cart. For instance instead of a 1 kg weight we could put a 10 kg weight on the cart and leave the 0.1 kg weight outside the window for all experiments. The results are quite different. It is obvious that for the same force the mass that is pulled has a huge effect on the speed at which the movement accelerates.

If we plot the distance travelled versus elapsed time results on Cartesian coordinate paper we can determine a relationship between distance and time elapsed.

When you see the decreasing smooth acceleration as more weight is added to the cart, the relationship between force, mass and acceleration becomes obvious. Analysis of the distance – time curves are perfect subjects for calculus, which was developed by Newton at the time. The first derivative dx/dt is the actual speed of the cart, and plots as a straight line through the origin of the coordinate system. The second derivative is the acceleration which turns out to be a constant. Based on this analysis the relationship between Force, Mass and Acceleration was discovered F = M*a.

Let’s consider this mysterious force that keeps moons circling planets and planets circling the sun. Newton’s insight was that this force has a universal aspect and that it depends on the attraction of two masses on each other. In the case of the earth and a canon ball, the mass of the canon ball is negligible in comparison. It has no influence on the earth.

To understand this mysterious force a bit better, consider a free falling object dropped from a tower. Such as the leaning tower of Pisa. (It had already been established by Galileo that different weights and sizes of objects fall at the same rate.) Suppose we have an accurate pendulum clock that ticks precisely every second. How high would a canon ball have to be lifted to fall to the ground in exactly one second. It turns out to be very close to 5 meters. That means that the acceleration of earths gravitation is close to 10 m/sec/sec.

\begin{aligned} {x=1/2*a*t^2\space or \space a=2x/t^2\space since \space x = 5 \space and \space t=1 \space it \space follows \space that \space a=10 } \end{aligned}

Now that we have established the gravitational acceleration of the earth lets consider one of Newton’s thought experiments. Imagine a very high mountain with a powerful cannon on top. Newton contended in his posthumous work “De mundi systemate” that a fast cannon ball fired in a horizontal direction (actually tangential to the earth curvature) would never touch the ground, but instead would circle the earth indefinitely.

Assume a horizontally mounted cannon on top of a 10,000 meter high mountain. Assume there is no air friction. After the cannon ball leaves the muzzle we can assume 2 motions. One horizontal the other vertical down towards the earth. We can calculate the amount of time it would take the cannon ball to fall from 10,000 meters.

\begin{aligned} x=1/2*a*t^2\space we\space know\space x=10,000\space m\space and\space a=10\ thus \space t=\sqrt{2*10000/1}=45\space seconds\end{aligned}

In the 45 seconds the horizontal (actually tangential) velocity has to cover the circumference of the earth, which is approx 40,000,000 meters. So the speed is 40,000,000 / 45 = 888,889 m/sec However the cannon ball would still hit the earth after completing that distance. It looks like we did everything right. We took into consideration the horizontal motion (not subject to gravity) and the vertical motion, which is subject to gravity. It turns out that there is an aspect we missed: the curvature of the canon ball trajectory. All curved cannon ball trajectories experience an almost mysterious force at right angles to the curved trajectory. Almost as if the cannon ball wants to straighten the trajectory out. This is due to angular acceleration, where the velocity vector changes direction while in flight. It is the same force that pushes clothes against the washing machine drum during the centrifuge spin cycle.

Lets consider newtons canon ball thought experiment where the cannon ball would circle the earth at an uniform speed.

We know the radius of the earth (roughly 6,000,000 m) and the height of the mountain (10,000 m). The radius of the trajectory would then be 6,010,000 m. The acceleration of earth gravity is approximately 10 m/sec/sec.

\begin{aligned}r=6,010,000\space a=10 \space to\space compensate \space downward \space force \space-\space v=\sqrt{r*a}=7,750\space m/sec\end{aligned}

This value for the cannon muzzle speed is very close to the actual theoretical speed required. The time required for one revolution is 2*pi*r/v or 2*3.14*6,010,000/7,750 = 4870 seconds or 81 minutes. That is close to the international space station which circles the earth in approximately 90 minutes.

In the year 1666 when Newton first contemplated gravitation he came to the realization that Kepler’s laws also must apply to our Moon. In 1610, Galileo had used the newly invented telescope to observe the planet Jupiter and found four moons rotating at varying rates around the planet. A sensation at the time that started the thinking about attractive forces between heavenly bodies. The acceleration of falling objects near the surface of the earth had already been established in the early 1600’s by Galileo at 9.8 m/sec^{2} . In Newton’s time the distance to the Moon had already been calculated fairly accurately as well as the radius of the earth. Newton was able to calculate the acceleration to keep our Moon in a circular orbit. What he found is that is was far less than the 9.8 m/sec^{2} at the earth’s surface, namely 1/3600^{th} of the 9.8 m/sec^{2}. The logic of attraction between heavenly bodies was well established based on the observations of the moon and the moons of Jupiter. So it was logical to assume that the force between them would be proportional with the mass. F is proportional to M*m. The other factor based on the calculation of the moon’s orbital acceleration would depend on the distance. Based on the ratios of the surface acceleration of the earth at 6,300,000 meters from the centre of the earth and the moons acceleration at 380,000,000 meters resulted in the proportionality relationship of 1/r^{2}.

From the diagram above we know that centrifugal acceleration is v^{2}/r where v is the velocity of our moon and r is the distance from the centre of the earth to the centre of the moon. The velocity of the moon can be found from the formula 2*pi*r/P where P is the period of one full rotation of the moon (27.4 days or 2 346 058 seconds), which is 6.28*380,000,000/2,346,058 = 1017 m/sec. The centrifugal acceleration is 1017^{2}/380,000,000=0.0027.

The ratios of gravitational acceleration at the surface of the earth and the circular orbit of our moon is: 9.8 / 0.0027 = 3629 and the inverse ratio of the distances 380,000,000/6,300,000 = 60 which is no match. However the inverse ratio of the distances squared is a very close match 144,400,000,000,000,000 / 39,690,000,000,000 = 3638. This led Newtons to his formula for universal gravitation:

\begin{aligned}F\propto \frac{M1*M2}{R^2}\end{aligned}

In it’s modern form the proportionality has been replaced with an equality. It took just over 100 years after the publication of Newton’s formula for Henry Cavendish to find the universal gravitational constant experimentally.

\begin{aligned}F=\frac{G*M1*M2}{R^2}\end{aligned}

But why, in this formula, did Newton use the product of the masses M1 and M2. Would it not have been logical to use the sum of the masses M1+M2, as in the force that the Earth exerts on the Moon plus the force the Moon exerts on the Earth, which should give the total attractive force between the two bodies?

Newton came to the now established formula through the equivalence principle in a gravitational field. That means, he knew from Galileo’s experiments on falling objects of different mass, that there was no difference in the time it took to fall from any height to the ground. If M1 is the mass of the Earth and M2 the mass of any size canon ball dropped from the tower of Pisa. Then the formula F=M2_{*}a in the Earths gravitational field is F=M2_{*}g

\begin{aligned}F=\frac{G*M1*M2}{R^2}=M2*g \space or\space g=\frac{G*M1}{R^2}\end{aligned}

Acceleration only depends on the reference mass, which in this case is the Earth. This is proof that it is the product rather than the sum of the masses. The lesser mass, in this case any of Galileo’s different weight cannon balls with mass M2, drops out of the equation.

A bit of history: By 1666 Robert Hooke and Giovanni Borelli stated that gravitation was an attractive force. And in 1670 Hooke stated that gravitation applied to all planets and moons. Hooke also stated in that year during the Gresham Lecture that the effect of gravitation decreases with distance. In 1679 Hooke stated that he thought there was an inverse square relationship between gravitation and distance in a letter to Newton.

However the inverse square law was definitively derived somewhere before 1684 by Newton. Both Robert Hooke and Isaac Newton wrote Edmund Halley that they had proven the inverse-square law of planetary motion. But only Newton offered written proof, based on Johannes Kepler’s documents which in turn were derived from the highly accurate and detailed astronomical observations by Tycho Brahe. I will dedicate a separate web page for Newton’s proof.