Robert McNees
Sir Isaac Newton's "Philosophiæ Naturalis Principia Mathematica" was published #OTD in 1687. Often referred to as just “the Principia,” it presents Newton’s laws of motion and his universal law of gravitation.
336 years old and we still teach its main results!
Robert McNees
You can thumb through one of Isaac Newton's personal copies of the Principia, courtesy of the Cambridge Digital Library.
Donated by the 5th Earl of Portsmouth in 1872, it contains many of Newton's own notes, annotations, and corrections.
Robert McNees
One of the main results contained in the Principia is Newton's "Law of Universal Gravitation."
Newton describes gravitation as producing a force between two objects "according to the quantity of solid matter which they contain and propagates on all sides to immense distances, decreasing always as the inverse square of the distances."
Some of this had been explored by earlier scientists, but this is the first time all the pieces come together in a consistent and quantitative explanation.
Robert McNees
Newton's insight gave a precise explanation of hundreds of years of astronomical observations. Despite being superseded by general relativity, it is sufficiently accurate that we used it to land people on the moon.
What does it mean, and how did he arrive at it?
Robert McNees
The first part says the gravitational force between objects is proportional to the product of their masses.
Galileo, and later Huygens, had established an important fact: Earth’s gravity causes objects to accelerate at the same rate, independent of their mass.
Newton’s 2nd law of motion, stated earlier in the Principia, equates an object’s acceleration to the force it experiences divided by its mass: a = F/m. Here, mass quantifies inertia, an object's tendency to resist changes in its motion.
Robert McNees
If gravity causes two objects with different masses to accelerate at the same rate, it follows that the gravitational force each one experiences must be proportional to its mass.
So Earth exhibits a larger gravitational force on a cannonball than on, say, an apple. But it is also more difficult to change a cannonball's motion. Drop them both and, in the absence of air resistance, they accelerate downward at the same rate.
Robert McNees
Now, Newton's 3rd law of motion says that if object A exerts a force on object B, then B exerts the same amount of force on A, but in the opposite direction. It follows that the gravitational force Earth exerts on an apple must also be proportional to the mass of the Earth.
The dependence on mass is a bit weird. In Newton's laws of motion, mass quantifies how an object resists changes to its motion. But in his law of universal gravitation it controls how much an object participates in gravity!
Robert McNees
Newton appreciated that this should not be taken for granted. He imagined an object might have an "inertial mass" m_i for his laws of motion, and a "gravitational mass" m_g for his law of gravity, and they may not be the same for objects of different composition.
If that were the case, different objects would accelerate at different rates, proportional to their ratio m_g / m_i. Newton knew, based on the work of Galileo and Huygens, that this ratio must be very nearly equal to 1.
Robert McNees
He tested it using pendulums. The period of a simple pendulum – the amount of time it takes to swing back and forth – depends only on its length and the acceleration due to gravity of the mass attached to the end.
If the inertial and gravitational properties of masses depend on their composition, this should show up in the periods of pendulums with identical lengths and masses made of different materials.
Robert McNees
As far as Newton could tell, the ratio m_g / m_i did not vary with composition. He concluded that the same mass appears in both his laws of motion and his universal law of gravitation.
Of course, this was not a very precise test! Later generations of scientists would scrutinize Newton's claim with increasing precision, most notably Baron Loránd Eötvös in a series of brilliant experiments that began in 1885.
Robert McNees
Eötvös showed that the ratio m_g / m_i differed from 1 by no more than a part in 10⁹. This phenomenal agreement between the two quantities begged an explanation, and indeed it is central to one form of the Equivalence Principle that helped Einstein formulate general relativity.
In the 1960s, Robert Dicke and his colleagues at Princeton would push the equality of inertial and gravitational mass to one part in 10¹¹.
https://www.sciencedirect.com/science/article/abs/pii/0003491664902593
Thony Christie
@mcnees The Principia Mathematica was published on 5 July 1687 is not strictly true
https://thonyc.wordpress.com/2014/07/05/published-on/
Robert McNees
This and more recent tests are so precise that they constrain the gravitational properties of *quantum mechanical* contributions to the mass of a nucleus.
An unavoidable consequence is what we call the cosmological constant problem. Quantum mechanical effects in empty spacetime should gravitate! If they didn’t, this would show up as measurable deviations in m_g/m_I for different nuclei.
Robert McNees
I am delighted by the fact that, more than 300 years after it puzzled Newton, the equality of inertial and gravitational mass tells us that we are probably missing something deep about general relativity and/or quantum mechanics.
Robert McNees
An important note!
The “publication date” of July 5, 1687 is a convenient starting point for a discussion of Principia. But as @rmathematicus explains in this blog post, it is not a publication date in the sense commonly understood today.
Rather than spoil the punchline, I encourage you to read the post:
Jimmy Havok
@mcnees And the Newton went off the rails and tried to figure out the mathematics of God.
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