The cartoon below, by Sidney Harris from the American Scientist magazine, illustrates one of the mysteries of science: How does a scientist create a brilliant insight? Is it an act of inspiration born out of the unconscious? Is it magic? Is it a miracle?

I’d not totally discount divine intervention in our most startling discoveries but in the main I think science moves along in rational processes from one proven idea to the next. It’s not leaps and bounds, but step by step.

A mundane example comes from the annals of detective fiction. Sherlock Holmes said that his conclusions were based on observation, finding patterns and drawing conclusions based on observations. Holmes could tell if a person were a physician by the bulge of a stethoscope in his hat, if he had gone to China by the coloring of a tattoo, or what he was thinking by the objects his attention was focused on. Dr. John Watson was always amazed at Holmes’ deductions, but said “When I hear you give your reasons the thing always appears to me to be so ridiculously simple that I could easily do it myself, though at each successive instance of your reasoning I am baffled, until you explain your process.”

And that’s it! Scientific discoveries look like magic until we fully understand the process by which those conclusions were made. Then, it all seems “ridiculously simple.”

Of course, I’ve simplified that conclusion. In reality it’s more complicated. Recently I was attracted to the work of Issac Newton and how he developed mathematical formulas that describe the motion of the planets through space. We often associate a sudden insight of inspiration with Newton as the result of a falling apple. But Newton’s insights were not born whole in an instant, as was Athena from Zeus’ forehead, but step by step. To illustrate this process, I asked a real expert to explain this for us.

First, a little background on our consultant. Dr. William Wall (Syzygy Asymptote in Second Life) was born in Canada. He received a B.Sc. from the University of Toronto, a M.Sc. from the University of British Columbia, and the Ph.D. from the University of Texas at Austin. He did his postdoctoral fellowship at NASA GSFC in Greenbelt, Maryland. He’s currently a researcher at the National Institute of Astrophysics, Optics and Electronics in Tonantzintla, Puebla, Mexico. His research involves observational studies of the interstellar medium, using radio telescopes. He occasionally gives talks to the public about astronomy.

An aside – HTML is not mathematical formula friendly, so we are using the ^ character to represent exponential indices. So 3^2, is 3 squared or 9.

Dr. Wall said, “Johannes Kepler’s laws of planetary motion and Isaac Newton’s law of universal gravitation were superb scientific accomplishments with long-reaching implications for physics, astronomy, and for comprehending our place in the universe. The former laws were published by Kepler between 1609 and 1619, and the latter published decades later by Newton in 1687. Given that the motions of the planets are governed by gravity, one might suppose that Newton conceived of his law of gravity directly from Kepler’s laws. Scientific development just flows logically like a single river from the past to the future, right?

While there certainly are logically flowing ideas in science, it’s often a confluence of separate streams of ideas that advance science, even leading to breakthroughs. Of Kepler’s three laws it is the 3rd that most closely relates to Newton’s law of universal gravitation (which I will expand upon below). Was Newton, then, inspired by Kepler’s 3rd law in formulating his law of gravitation? Such inspiration cannot be entirely ruled out, but Newton was very open to an idea that had existed for centuries before: the inverse-square law.

The inverse-square law is a mathematically compelling argument advanced by John Dumbledon of the Oxford Calculators in the 14th century. Imagine a point of light radiating isotropically (i.e., equally in all directions). Then imagine nested transparent concentric spheres centred on that point of light. The total area of each sphere is proportional to the square of its radius, and that radius is the distance from the light source. As the light rays pass through each sphere, they are spread over the larger and larger areas of the spheres, being diluted by the distance squared. Accordingly, an observer receding from the light source would perceive the light dimming according to his or her distance squared. This is a simple geometric dilution. Does this also apply to gravity?

Earlier work by Ismaël Bullialdus and correspondence from Robert Hooke suggested that the inverse-square law applies to gravity as well. Newton then made the leap that the gravity responsible for causing an apple to fall in his garden also reached the moon, maintaining it in a circular path around the Earth. Gravity was universal. But he had to test his idea.

And, to do that, he could rely on the discoveries of previous scientists. This meant not only Kepler’s laws, based on the observations of Tycho Brahe, but also the work of Eratosthenes and of Aristarchus. In 240 BC, Eratosthenes observed the shadows cast by the sun at two different locations, roughly 800 km apart, to determine the circumference of the Earth and, therefore, its radius. In 270 BC, Aristarchus used the duration of lunar eclipses (when the Moon passes through the Earth’s shadow) to determine that the Moon’s distance was 60 Earth radii. So, in Newton’s time, the distance to the Moon was known. Newton’s approach was very straightforward. If g is the known acceleration due to gravity on the Earth’s surface, then the centripetal acceleration of the Moon in its orbit around the Earth would be g/60^2, if there was inverse-square law for gravity out to the Moon’s distance. The orbital velocity of the moon, for a circular orbit, is computed from that centripetal acceleration and the orbit’s radius. That velocity for the known radius then gives the orbit’s period. Does the calculated period match the known period of 27.3 days? It did indeed! So far, the law of universal gravitation has stood up to scrutiny. But what about Kepler’s 3rd law?

A very similar argument confirms consistency of Newton’s law of universal gravitation with Kepler’s 3rd law of planetary motion. Here I’ll demonstrate such consistency for a circular orbit for simplicity. Kepler’s 3rd law, specifically, states that “the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.” This can be expressed more succinctly as a^3 ~ T^2, where a is the semi-major (radius for a circular orbit) of the orbit and T is the orbital period (time to complete one orbit). Newton’s law of universal gravitation states that “every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers,’’ or more compactly as follows: F = GMm/R^2, which is the gravitational force, F, between masses m and M orbiting each other at distance R. G is the universal gravitational constant. (Note that the gravitational force’s dependence on the masses follows naturally from the logical assumption that such a force is proportional to each of the two masses and, therefore, to their product.) Given that the centripetal force on mass m as it orbits mass M is mv^2/R, that v = 2pR/T, and that R is equivalent to a for a circular orbit, we obtain a^3/T^2 = GM/(4p^2 )

The right side of the above is constant for a given orbit and, accordingly, Newton’s law of universal gravitation is indeed consistent with Kepler’s 3rd law. This strongly supports Newton’s law.

Based on the literature (see “Suggested Reading” below), this is the sequence of reasoning that Newton likely followed. Newton’s law of universal gravitation did not immediately follow from Kepler’s laws of planetary motion, although Newton demonstrated that the latter were consistent with the former afterward. In summary, the work of Dumbledon, Bullialdus, and Hooke inspired Newton to use the inverse-square law, while the work of the ancients, Eratosthenes and Aristarchus, from observations of the sun’s shadow on the Earth and of lunar eclipses, and of the Brahe’s observations of the planets resulting in Kepler’s laws provided needed scientific tests of Newton’s new law. This was, effectively, an inspirational collaboration involving a number of people spanning many centuries.

This represents a number of disparate ideas woven into a tapestry of scientific discovery, which is how science normally works.”

To conclude. I once had a boss who, when laying out his logical processes of deduction, would preface his conclusion with the phrase, “… and as any fool can plainly see.” I must admit that often this fool did not see plainly.

**References**

- Inside the Head of a Science Cartoonist: It’s a brilliant mess, Becca Cudmore, Scientific American, January 31, 2014.
- The Complete Adventures and Memoirs of Sherlock Holmes, Arthur Conan Doyle, New York, New York: Clarkson N. Potter Inc, 1975.
- Physics – Formulas – Kepler and Newton – Orbits, AstromomyOnLine.
- Newton’s Version of Kepler’s Third Law, Jim Heath’s Starry Site, Austin Community College.
- Celestial Orbits, Part I, William Wall, The Science Circle, February 2020.

**Suggested Readings from Dr. Wall**

- Kepler’s laws of planetary motion, Wikipedia.
- Newton’s law of universal gravitation, Wikipedia.
- Gravity, Part 2: Newton, Hooke, Halley and the Three Body Problem, Dave Snyder, University Lowbrow Astronomers, University of Michigan.
- Newton’s theory of “Universal Gravitation, David P. Stern, stargaze, NASA.
- The Moon’s Distance – 1, David P. Stern, stargaze, NASA.
- Jun 19, 240 BCE: Eratosthenes Measures Circumference of Earth, National Geographic.

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