Galileo’s Moons in the History of Science

When Galileo trained his crude telescope on the planet Jupiter, hanging above the horizon in 1610, and observed moons orbiting a planet other than Earth, it created a quake whose waves have rippled down through the centuries to today.  Never had such hard evidence been found that supported the Copernican idea of non-Earth-centric orbits, freeing astronomy and cosmology from a thousand years of error that shaded how people thought.

The Earth, after all, was not the center of the Universe.

Galileo’s moons: the Galilean Moons—Io, Europa, Ganymede, and Callisto—have drawn our eyes skyward now for over 400 years.  They have been the crucible for numerous scientific discoveries, serving as a test bed for new ideas and new techniques, from the problem of longitude to the speed of light, from the birth of astronomical interferometry to the beginnings of exobiology.  Here is a short history of Galileo’s Moons in the history of physics.

Galileo (1610): Celestial Orbits

In late 1609, Galileo (1564 – 1642) received an unwelcome guest to his home in Padua—his mother.  She was not happy with his mistress, and she was not happy with his chosen profession, but she was happy to tell him so.  By the time she left in early January 1610, he was yearning for something to take his mind off his aggravations, and he happened to point his new 20x telescope in the direction of the planet Jupiter hanging above the horizon [1].  Jupiter appeared as a bright circular spot, but nearby were three little stars all in line with the planet.  The alignment caught his attention, and when he looked again the next night, the position of the stars had shifted.  On successive nights he saw them shift again, sometimes disappearing into Jupiter’s bright disk.  Several days later he realized that there was a fourth little star that was also behaving the same way.  At first confused, he had a flash of insight—the little stars were orbiting the planet.  He quickly understood that just as the Moon orbited the Earth, these new “Medicean Planets” were orbiting Jupiter.  In March 1610, Galileo published his findings in Siderius Nuncius (The Starry Messenger). 

Page from Galileo’s Starry Messenger showing the positions of the moon of Jupiter

It is rare in the history of science for there not to be a dispute over priority of discovery.  Therefore, by an odd chance of fate, on the same nights that Galileo was observing the moons of Jupiter with his telescope from Padua, the German astronomer Simon Marius (1573 – 1625) also was observing them through a telescope of his own from Bavaria.  It took Marius four years to publish his observations, long after Galileo’s Siderius had become a “best seller”, but Marius took the opportunity to claim priority.  When Galileo first learned of this, he called Marius “a poisonous reptile” and “an enemy of all mankind.”  But harsh words don’t settle disputes, and the conflicting claims of both astronomers stood until the early 1900’s when a scientific enquiry looked at the hard evidence.  By that same odd chance of fate that had compelled both men to look in the same direction around the same time, the first notes by Marius in his notebooks were dated to a single day after the first notes by Galileo!  Galileo’s priority survived, but Marius may have had the last laugh.  The eternal names of the “Galilean” moons—Io, Europe, Ganymede and Callisto—were given to them by Marius.

Picard and Cassini (1671):  Longitude

The 1600’s were the Age of Commerce for the European nations who relied almost exclusively on ships and navigation.  While latitude (North-South) was easily determined by measuring the highest angle of the sun above the southern horizon, longitude (East-West) relied on clocks which were notoriously inaccurate, especially at sea. 

The Problem of Determining Longitude at Sea is the subject of Dava Sobel’s thrilling book Longitude (Walker, 1995) [2] where she reintroduced the world to what was once the greatest scientific problem of the day.  Because almost all commerce was by ships, the determination of longitude at sea was sometimes the difference between arriving safely in port with a cargo or being shipwrecked.  Galileo knew this, and later in his life he made a proposal to the King of Spain to fund a scheme to use the timings of the eclipses of his moons around Jupiter to serve as a “celestial clock” for ships at sea.  Galileo’s grant proposal went unfunded, but the possibility of using the timings of Jupiter’s moons for geodesy remained an open possibility, one which the King of France took advantage of fifty years later.

In 1671 the newly founded Academie des Sciences in Paris funded an expedition to the site of Tycho Brahe’s Uranibourg Observatory in Hven, Denmark, to measure the time of the eclipses of the Galilean moons observed there to be compared the time of the eclipses observed in Paris by Giovanni Cassini (1625 – 1712).  When the leader of the expedition, Jean Picard (1620 – 1682), arrived in Denmark, he engaged the services of a local astronomer, Ole Rømer (1644 – 1710) to help with the observations of over 100 eclipses of the Galilean moon Io by the planet Jupiter.  After the expedition returned to France, Cassini and Rømer calculated the time differences between the observations in Paris and Hven and concluded that Galileo had been correct.  Unfortunately, observing eclipses of the tiny moon from the deck of a ship turned out not to be practical, so this was not the long-sought solution to the problem of longitude, but it contributed to the early science of astrometry (the metrical cousin of astronomy).  It also had an unexpected side effect that forever changed the science of light.

Ole Rømer (1676): The Speed of Light

Although the differences calculated by Cassini and Rømer between the times of the eclipses of the moon Io between Paris and Hven were small, on top of these differences was superposed a surprisingly large effect that was shared by both observations.  This was a systematic shift in the time of eclipse that grew to a maximum value of 22 minutes half a year after the closest approach of the Earth to Jupiter and then decreased back to the original time after a full year had passed and the Earth and Jupiter were again at their closest approach.  At first Cassini thought the effect might be caused by a finite speed to light, but he backed away from this conclusion because Galileo had shown that the speed of light was unmeasurably fast, and Cassini did not want to gainsay the old master.

Ole Rømer

Rømer, on the other hand, was less in awe of Galileo’s shadow, and he persisted in his calculations and concluded that the 22 minute shift was caused by the longer distance light had to travel when the Earth was farthest away from Jupiter relative to when it was closest.  He presented his results before the Academie in December 1676 where he announced that the speed of light, though very large, was in fact finite.  Unfortnately, Rømer did not have the dimensions of the solar system at his disposal to calculate an actual value for the speed of light, but the Dutch mathematician Huygens did.

When Huygens read the proceedings of the Academie in which Rømer had presented his findings, he took what he knew of the radius of Earth’s orbit and the distance to Jupiter and made the first calculation of the speed of light.  He found a value of 220,000 km/second (kilometers did not exist yet, but this is the equivalent of what he calculated).  This value is 26 percent smaller than the true value, but it was the first time a number was given to the finite speed of light—based fundamentally on the Galilean moons. For a popular account of the story of Picard and Rømer and Huygens and the speed of light, see Ref. [3].

Michelson (1891): Astronomical Interferometry

Albert Michelson (1852 – 1931) was the first American to win the Nobel Prize in Physics.  He received the award in 1907 for his work to replace the standard meter, based on a bar of metal housed in Paris, with the much more fundamental wavelength of red light emitted by Cadmium atoms.  His work in Paris came on the heels of a new and surprising demonstration of the use of interferometry to measure the size of astronomical objects.

Albert Michelson

The wavelength of light (a millionth of a meter) seems ill-matched to measuring the size of astronomical objects (thousands of meters) that are so far from Earth (billions of meters).  But this is where optical interferometry becomes so important.  Michelson realized that light from a distant object, like a Galilean moon of Jupiter, would retain some partial coherence that could be measured using optical interferometry.  Furthermore, by measuring how the interference depended on the separation of slits placed on the front of a telescope, it would be possible to determine the size of the astronomical object.

From left to right: Walter Adams, Albert Michelson, Walther Mayer, Albert Einstein, Max Ferrand, and Robert Milliken. Photo taken at Caltech.

In 1891, Michelson traveled to California where the Lick Observatory was poised high above the fog and dust of agricultural San Jose (a hundred years before San Jose became the capitol of high-tech Silicon Valley).  Working with the observatory staff, he was able to make several key observations of the Galilean moons of Jupiter.  These were just close enough that their sizes could be estimated (just barely) from conventional telescopes.  Michelson found from his calculations of the interference effects that the sizes of the moons matched the conventional sizes to within reasonable error.  This was the first demonstration of astronomical interferometry which has burgeoned into a huge sub-discipline of astronomy today—based originally on the Galilean moons [4].

Pioneer (1973 – 1974): The First Tour

Pioneer 10 was launched on March 3, 1972 and made its closest approach to Jupiter on Dec. 3, 1973. Pioneer 11 was launched on April 5, 1973 and made its closest approach to Jupiter on Dec. 3, 1974 and later was the first spacecraft to fly by Saturn. The Pioneer spacecrafts were the first to leave the solar system (there have now been 5 that have left, or will leave, the solar system). The cameras on the Pioneers were single-pixel instruments that made line-scans as the spacecraft rotated. The point light detector was a Bendix Channeltron photomultiplier detector, which was a vacuum tube device (yes vacuum tube!) operating at a single-photon detection efficiency of around 10%. At the time of the system design, this was a state-of-the-art photon detector. The line scanning was sufficient to produce dramatic photographs (after extensive processing) of the giant planets. The much smaller moons were seen with low resolution, but were still the first close-ups ever to be made of Galileo’s moons.

Voyager (1979): The Grand Tour

Voyager 1 was launched on Sept. 5, 1977 and Voyager 2 was launched on August 20, 1977. Although Voyager 1 was launched second, it was the first to reach Jupiter with closest approach on March 5, 1979. Voyager 2 made its closest approach to Jupiter on July 9, 1979.

In the Fall of 1979, I had the good fortune to be an undergraduate at Cornell University when Carl Sagan gave an evening public lecture on the Voyager fly-bys, revealing for the first time the amazing photographs of not only Jupiter but of the Galilean Moons. Sitting in the audience listening to Sagan, a grand master of scientific story telling, made you feel like you were a part of history. I have never been so convinced of the beauty and power of science and technology as I was sitting in the audience that evening.

The camera technology on the Voyagers was a giant leap forward compared to the Pioneer spacecraft. The Voyagers used cathode ray vidicon cameras, like those used in television cameras of the day, with high-resolution imaging capabilities. The images were spectacular, displaying alien worlds in high-def for the first time in human history: volcanos and lava flows on the moon of Io; planet-long cracks in the ice-covered surface of Europa; Callisto’s pock-marked surface; Ganymede’s eerie colors.

The Voyager’s discoveries concerning the Galilean Moons were literally out of this world. Io was discovered to be a molten planet, its interior liquified by tidal-force heating from its nearness to Jupiter, spewing out sulfur lava onto a yellowed terrain pockmarked by hundreds of volcanoes, sporting mountains higher than Mt. Everest. Europa, by contrast, was discovered to have a vast flat surface of frozen ice, containing no craters nor mountains, yet fractured by planet-scale ruptures stained tan (for unknown reasons) against the white ice. Ganymede, the largest moon in the solar system, is a small planet, larger than Mercury. The Voyagers revealed that it had a blotchy surface with dark cratered patches interspersed with light smoother patches. Callisto, again by contrast, was found to be the most heavily cratered moon in the solar system, with its surface pocked by countless craters.

Galileo (1995): First in Orbit

The first mission to orbit Jupiter was the Galileo spacecraft that was launched, not from the Earth, but from Earth orbit after being delivered there by the Space Shuttle Atlantis on Oct. 18, 1989. Galileo arrived at Jupiter on Dec. 7, 1995 and was inserted into a highly elliptical orbit that became successively less eccentric on each pass. It orbited Jupiter for 8 years before it was purposely crashed into the planet (to prevent it from accidentally contaminating Europa that may support some form of life).

Galileo made many close passes to the Galilean Moons, providing exquisite images of the moon surfaces while its other instruments made scientific measurements of mass and composition. This was the first true extended study of Galileo’s Moons, establishing the likely internal structures, including the liquid water ocean lying below the frozen surface of Europa. As the largest body of liquid water outside the Earth, it has been suggested that some form of life could have evolved there (or possibly been seeded by meteor ejecta from Earth).

Juno (2016): Still Flying

The Juno spacecraft was launched from Cape Canaveral on Aug. 5, 2011 and entered a Jupiter polar orbit on July 5, 2016. The mission has been producing high-resolution studies of the planet. The mission was extended in 2021 to last to 2025 to include several close fly-bys of the Galilean Moons, especially Europa, which will be the object of several upcoming missions because of the possibility for the planet to support evolved life. These future missions include NASA’s Europa Clipper Mission, the ESA’s Jupiter Icy Moons Explorer, and the Io Volcano Observer.

Epilog (2060): Colonization of Callisto

In 2003, NASA identified the moon Callisto as the proposed site of a manned base for the exploration of the outer solar system. It would be the next most distant human base to be established after Mars, with a possible start date by the mid-point of this century. Callisto was chosen because it is has a low radiation level (being the farthest from Jupiter of the large moons) and is geologically stable. It also has a composition that could be mined to manufacture rocket fuel. The base would be a short-term way-station (crews would stay for no longer than a month) for refueling before launching and using a gravity assist from Jupiter to sling-shot spaceships to the outer planets.

[1] See Chapter 2, A New Scientist: Introducing Galileo, in David D. Nolte, Galileo Unbound (Oxford University Press, 2018).

[2] Dava Sobel, Longitude: The True Story of a Lone Genius who Solved the Greatest Scientific Problem of his Time (Walker, 1995)

[3] See Chap. 1, Thomas Young Polymath: The Law of Interference, in David D. Nolte, Interference: The History of Optical Interferometry and the Scientists who Tamed Light (Oxford University Press, 2023)

[4] See Chapter 5, Stellar Interference: Measuring the Stars, in David D. Nolte, Interference: The History of Optical Interferometry and the Scientists who Tamed Light (Oxford University Press, 2023).

Galileo Unbound

Book Outline Topics

  • Chapter 1: Flight of the Swallows
    • Introduction to motion and trajectories
  • Chapter 2: A New Scientist
    • Galileo’s Biography
  • Chapter 3: Galileo’s Trajectory
    • His study of the science of motion
    • Publication of Two New Sciences
  • Chapter 4: On the Shoulders of Giants
    • Newton’s Principia
    • The Principle of Least Action: Maupertuis, Euler, and Voltaire
    • Lagrange and his new dynamics
  • Chapter 5: Geometry on my Mind
    • Differential geometry of Gauss and Riemann
    • Vector spaces rom Grassmann to Hilbert
    • Fractals: Cantor, Weierstrass, Hausdorff
  • Chapter 6: The Tangled Tale of Phase Space
    • Liouville and Jacobi
    • Entropy and Chaos: Clausius, Boltzmann and Poincare
    • Phase Space: Gibbs and Ehrenfest
  • Chapter 7: The Lens of Gravity
    • Einstein and the warping of light
    • Black Holes: Schwarzschild’s radius
    • Oppenheimer versus Wheeler
    • The Golden Age of General Relativity
  • Chapter 8: On the Quantum Footpath
    • Heisenberg’s matrix mechanics
    • Schrödinger’s wave mechanics
    • Bohr’s complementarity
    • Einstein and entanglement
    • Feynman and the path-integral formulation of quantum
  • Chapter 9: From Butterflies to Hurricanes
    • KAM theory of stability of the solar system
    • Steven Smale’s horseshoe
    • Lorenz’ butterfly: strange attractor
    • Feigenbaum and chaos
  • Chapter 10: Darwin in the Clockworks
    • Charles Darwin and the origin of species
    • Fibonnacci’s bees
    • Economic dynamics
    • Mendel and the landscapes of life
    • Evolutionary dynamics
    • Linus Pauling’s molecular clock and Dawkins meme
  • Chapter 11: The Measure of Life
    • Huygens, von Helmholtz and Rayleigh oscillators
    • Neurodynamics
    • Euler and the Seven Bridges of Königsberg
    • Network theory: Strogatz and Barabasi

In June of 1633 Galileo was found guilty of heresy and sentenced to house arrest for what remained of his life. He was a renaissance Prometheus, bound for giving knowledge to humanity. With little to do, and allowed few visitors, he at last had the uninterrupted time to finish his life’s labor. When Two New Sciences was published in 1638, it contained the seeds of the science of motion that would mature into a grand and abstract vision that permeates all science today. In this way, Galileo was unbound, not by Hercules, but by his own hand as he penned the introduction to his work:

. . . what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote corners.

            Galileo Galilei (1638) Two New Sciences

Galileo Unbound (Oxford University Press, 2018) explores the continuous thread from Galileo’s discovery of the parabolic trajectory to modern dynamics and complex systems. It is a history of expanding dimension and increasing abstraction, until today we speak of entangled quantum particles moving among many worlds, and we envision our lives as trajectories through spaces of thousands of dimensions. Remarkably, common themes persist that predict the evolution of species as readily as the orbits of planets. Galileo laid the foundation upon which Newton built a theory of dynamics that could capture the trajectory of the moon through space using the same physics that controlled the flight of a cannon ball. Late in the nineteenth-century, concepts of motion expanded into multiple dimensions, and in the 20th century geometry became the cause of motion rather than the result when Einstein envisioned the fabric of space-time warped by mass and energy, causing light rays to bend past the Sun. Possibly more radical was Feynman’s dilemma of quantum particles taking all paths at once—setting the stage for the modern fields of quantum field theory and quantum computing. Yet as concepts of motion have evolved, one thing has remained constant—the need to track ever more complex changes and to capture their essence—to find patterns in the chaos as we try to predict and control our world. Today’s ideas of motion go far beyond the parabolic trajectory, but even Galileo might recognize the common thread that winds through all these motions, drawing them together into a unified view that gives us the power to see, at least a little, through the mists shrouding the future.

Huygens’ Tautochrone

In February of 1662, Pierre de Fermat wrote a paper Synthesis ad refractiones that explained Descartes-Snell’s Law of light refraction by finding the least time it took for light to travel between two points. This famous approach is now known as Fermat’s principle, and it motivated other searches for minimum principles. A few years earlier, in 1656, Christiaan Huygens had invented the pendulum clock [1], and he began a ten-year study of the physics of the pendulum. He was well aware that the pendulum clock does not keep exact time—as the pendulum swings wider, the period of oscillation slows down. He began to search for a path of the pendular mass that would keep the period the same (and make pendulum clocks more accurate), and he discovered a trajectory along which a mass would arrive at the same position in the same time no matter where it was released on the curve. That such a curve could exist was truly remarkable, and it promised to make highly accurate time pieces.

It made minimization problems a familiar part of physics—they became part of the mindset, leading ultimately to the principle of least action.

This curve is known as a tautochrone (literally: same or equal time in Greek) and Huygens provided a geometric proof in his Horologium Oscillatorium sive de motu pendulorum (1673) that the curve was a cycloid [2]. A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls without slipping along a straight line. Huygens invented such a pendulum in which the mass executed a cycloid curve. It was a mass on a flexible yet inelastic string that partially wrapped itself around a solid bumper on each half swing. In principle, whether the pendulum swung gently, or through large displacements, the time would be the same. Unfortunately, friction along the contact of the string with the bumper prevented the pendulum from achieving this goal, and the tautochronic pendulum did not catch on.


Fig. 1 Huygens’ isochronous pendulum.  The time it takes the pendulum bob to follow the cycloid arc is independent of the pendulum’s amplitude, unlike for the circular arc, as the pendulum slows down for larger excursions.

The solution of the tautochrone curve of equal time led naturally to a search for the curve of least time, known as the brachistochrone curve for a particle subject to gravity, like a bead sliding on a frictionless wire between two points. Johann Bernoulli published a challenge to find the brachistochrone in 1696 in the scientific journal Acta Eruditorum that had been founded in 1682 by Leibniz in Germany in collaboration with Otto Mencke. Leibniz envisioned the journal to be a place where new ideas in the natural sciences and mathematics could be published and disseminated rapidly, and it included letters and commentaries, acting as a communication hub to help establish a community of scholars across Europe. In reality, it was the continental response to the Proceedings of the Royal Society in England.  Naturally, the Acta and the Proceedings would later take partisan sides in the priority dispute between Leibniz and Newton for the development of the calculus.

When Bernoulli published his brachistochrone challenge in the June issue of 1696, it was read immediately by the leading mathematicians of the day, many of whom took up the challenge and replied. The problem was solved and published in the May 1697 issue of the Acta by no less than five correspondents, including Johann Bernoulli, Jakob Bernoulli (Johann’s brother), Isaac Newton, Gottfried Leibniz and Ehrenfried Walther von Tschirnhaus. Each of them varied in their approaches, but all found the same solution. Johann and Jakob each considered the problem as the path of a light beam in a medium whose speed varied with depth. Just as in the tautochrone, the solved curve was a cycloid. The path of fastest time always started with a vertical path that allowed the fastest acceleration, and the point of greatest depth always was at the point of greatest horizontal speed.

The brachistrochrone problem led to the invention of the variational calculus, with first steps by Jakob Bernoulli and later more rigorous approaches by Euler.  However, its real importance is that it made minimization problems a familiar part of physics—they became part of the mindset, leading ultimately to the principle of least action.

Mathematical Description of the Tautochrone Curve

If the length of the pendulum is L, the position coordinates (relative to the cusp of the cycloid) are

where the angle θ is between the straight section of the string and the vertical and depends on time as

for a maximum angle θ0. The angular frequency of the isochronous pendulum is given by the standard expression for a small-amplitude pendulum as

For small-angle oscillations, the tautochrone and the conventional pendulum have the same periods and amplitudes. The difference is that the tautochrone has that same period regardless of the amplitude, so no approximation is needed.

Huygens’ proof of the tautochrone curve was made geometrically without the use of calculus, requiring almost 18 pages and 16 figures.

The modern proof (due to Euler) takes only a few lines of calculus. All that is needed is to recognize that the equation of motion of the pendulum bob along its path of length s should be that of a simple-harmonic oscillator

because a simple harmonic oscillator always oscillates at the same frequency regardless of amplitude. Of course, a simple pendulum is not a simple harmonic oscillator because it becomes nonlinear and slows down for large amplitudes. The right-hand-side is the standard force on the bob given by the gradient of the potential energy along the curve

yielding the simple differential equation

which is integrated to give a quadratic dependence of height on path length

To get an equation for the curve itself, plug this back in to the differential equation and square it to give

Euler recognized this as the differential equation of a cycloid


[1] Galileo conceived the pendulum clock in 1641, and his son Vincenzo started construction, but it was never finished.  Huygens submitted and received a patent in 1657 for a practical escape mechanism on pendulum clocks that is still used today.

[2] Huygens, Christiaan; Blackwell,, Richard J., trans. (1986). Horologium Oscillatorium (The Pendulum Clock, or Geometrical demonstrations concerning the motion of pendula as applied to clocks). Ames, Iowa: Iowa State University Press

Descartes’ Odd Geometry

Rene Descartes was an unlikely candidate to revolutionize geometry. He began his career as a mercenary soldier, his mind wrapped around things like war and women, which are far from problems of existence and geometry. Descartes’ strange conversion from a life of action to a life of mind occurred on the night of November 10-11 in 1619 while he was bivouacked in an army encampment in Bavaria as a mercenary early in the Thirty Years’ War (1618—1648). On that night, Descartes dreamed that exact rational thought, even mathematical method, could be applied to problems of philosophy. This became his life’s work, and because he was a man of exceptional talent, he succeeded in exceptional ways.

Even Descartes’ footnotes were capable of launching new fields of thought.

Descartes left his mercenary employment and established himself in the free-thinking republic of the Netherlands which was ending the long process of casting off the yolk of Spanish rule towards the end of the Eighty Years War (1568—1648). In 1623, he settled in The Hague, a part of the republic that had been free of Spanish troops for many years, and after a brief absence (during which he witnessed the Siege of Rochelle by Cardinal Richelieu), he returned to the Netherlands in 1628, at the age of 32. He remained in the Netherlands, moving often, taking classes or teaching classes at the Universities of Leiden and Utrecht until 1649, when he was enticed away by Queen Christina of Sweden to colder climes and ultimately to his death.


Descartes’ original curve (AC), constructed on non-orthogonal (oblique) x and y coordinates (La Géométrie, 1637)

Descartes’ years in the Netherlands were the most productive epoch of his life as he created his philosophy and pursued his dream’s promise. He embarked on an ambitious project to codify his rational philosophy to gain a full understanding of natural philosophy. He called this work Treatise on the World, known in short as Le Monde, and it quite naturally adopted Copernicus’ heliocentric view of the solar system, which by that time had become widely accepted in learned circles even before Galileo’s publication in 1632 of his Dialogue Concerning the Two Chief World Systems. However, when Galileo was convicted in 1633 of suspicion of heresy (See Galileo Unbound, Oxford University Press, 2018), Descartes abruptly abandoned his plans to publish Le Monde, despite being in the Netherlands where he was well beyond the reach of the Church. It was, after all, the Dutch publisher Elzevir who published Galileo’s last work on the Two Sciences in 1638 when no Italian publishers would touch it. However, Descartes remained a devout Catholic throughout his life and had no desire to oppose its wishes. Despite this setback, Descartes continued to work on less controversial parts of his project, and in 1637 he published three essays preceded by a short introduction.

The introduction was called the Discourse on the Method (which contained his famous cogito ergo sum), and the three essays were La Dioptrique on optics, Les Météores on atmosphere and weather and finally La Géométrie on geometry in which he solved a problem posed by Pappus of Alexandria in the fourth century AD. Descartes sought to find a fundamental set of proven truths that would serve as the elements one could use in a deductive method to derive higher-level constructs. It was partially as an exercise in deductive reasoning that he sought to solve the classical mathematics problem posed by Pappus. La Géométrie was published as an essay following the much loftier Discourse, so even Descartes’ footnotes were capable of launching new fields of thought. The new field is called analytical geometry, also known as Cartesian or coordinate geometry, in which algebra is applied to geometric problems. Today, coordinates and functions are such natural elements of mathematics and physics, that it is odd to think that they emerged as demonstrations of abstract philosophy.

Bibliography:  R. Descartes, D. E. Smith, and M. L. Latham, The geometry of René Descartes. Chicago: Open Court Pub. Co., 1925.


A Wealth of Motions: Six Generations in the History of the Physics of Motion

SixGenerations3 Since Galileo launched his trajectory, there have been six broad generations that have traced the continuing development of concepts of motion. These are: 1) Universal Motion; 2) Phase Space; 3) Space-Time; 4) Geometric Dynamics; 5) Quantum Coherence; and 6) Complex Systems. These six generations were not all sequential, many evolving in parallel over the centuries, borrowing from each other, and there surely are other ways one could divide up the story of dynamics. But these six generations capture the grand concepts and the crucial paradigm shifts that are Galileo’s legacy, taking us from Galileo’s trajectory to the broad expanses across which physicists practice physics today. Universal Motion emerged as a new concept when Isaac Newton proposed his theory of universal gravitation by which the force that causes apples to drop from trees is the same force that keeps the Moon in motion around the Earth, and the Earth in motion around the Sun. This was a bold step because even in Newton’s day, some still believed that celestial objects obeyed different laws. For instance, it was only through the work of Edmund Halley, a contemporary and friend of Newton’s, that comets were understood to travel in elliptical orbits obeying the same laws as the planets. Universal Motion included ideas of momentum from the start, while concepts of energy and potential, which fill out this first generation, took nearly a century to develop in the hands of many others, like Leibniz and Euler and the Bernoullis. This first generation was concluded by the masterwork of the Italian-French mathematician Joseph-Louis Lagrange, who also planted the seed of the second generation. The second generation, culminating in the powerful and useful Phase Space, also took more than a century to mature. It began when Lagrange divorced dynamics from geometry, establishing generalized coordinates as surrogates to directions in space. Ironically, by discarding geometry, Lagrange laid the foundation for generalized spaces, because generalized coordinates could be anything, coming in any units and in any number, each coordinate having its companion velocity, doubling the dimension for every freedom. The Austrian physicist Ludwig Boltzmann expanded the number of dimensions to the scale of Avogadro’s number of particles, and he discovered the conservation of phase space volume, an invariance of phase space that stays the same even as 1023 atoms (Avogadro’s number) in ideal gases follow their random trajectories. The idea of phase space set the stage for statistical mechanics and for a new probabilistic viewpoint of mechanics that would extend into chaotic motions. The French mathematician Henri Poincaré got a glimpse of chaotic motion in 1890 as he rushed to correct an embarrassing mistake in his manuscript that had just won a major international prize. The mistake was mathematical, but the consequences were profoundly physical, beginning the long road to a theory of chaos that simmered, without boiling, for nearly seventy years until computers became common lab equipment. Edward Lorenz of MIT, working on models of the atmosphere in the late 1960s, used one of the earliest scientific computers to expose the beauty and the complexity of chaotic systems. He discovered that the computer simulations were exponentially sensitive to the initial conditions, and the joke became that a butterfly flapping its wings in China could cause hurricanes in the Atlantic. In his computer simulations, Lorenz discovered what today is known as the Lorenz butterfly, an example of something called a “strange attractor”. But the term chaos is a bit of a misnomer, because chaos theory is primarily about finding what things are shared in common, or are invariant, among seemingly random-acting systems. The third generation in concepts of motion, Space-Time, is indelibly linked with Einstein’s special theory of relativity, but Einstein was not its originator. Space-time was the brain child of the gifted but short-lived Prussian mathematician Hermann Minkowski, who had been attracted from Königsberg to the mathematical powerhouse at the University in Göttingen, Germany around the turn of the 20th Century by David Hilbert. Minkowski was an expert in invariant theory, and when Einstein published his special theory of relativity in 1905 to explain the Lorentz transformations, Minkowski recognized a subtle structure buried inside the theory. This structure was related to Riemann’s metric theory of geometry, but it had the radical feature that time appeared as one of the geometric dimensions. This was a drastic departure from all former theories of motion that had always separated space and time: trajectories had been points in space that traced out a continuous curve as a function of time. But in Minkowski’s mind, trajectories were invariant curves, and although their mathematical representation changed with changing point of view (relative motion of observers), the trajectories existed in a separate unchanging reality, not mere functions of time, but eternal. He called these trajectories world lines. They were static structures in a geometry that is today called Minkowski space. Einstein at first was highly antagonistic to this new view, but he relented, and later he so completely adopted space-time in his general theory that today Minkowski is almost forgotten, his echo heard softly in expressions of the Minkowski metric that is the background to Einstein’s warped geometry that bends light and captures errant space craft. The fourth generation in the development of concepts of motion, Geometric Dynamics, began when an ambitious French physicist with delusions of grandeur, the historically ambiguous Pierre Louis Maupertuis, returned from a scientific boondoggle to Lapland where he measured the flatness of the Earth in defense of Newtonian physics over Cartesian. Skyrocketed to fame by the success of the expedition, he began his second act by proposing the Principle of Least Action, a principle by which all motion seeks to be most efficient by taking a geometric path that minimizes a physical quantity called action. In this principle, Maupertuis saw both a universal law that could explain all of physical motion, as well as a path for himself to gain eternal fame in the company of Galileo and Newton. Unfortunately, his high hopes were dashed through personal conceit and nasty intrigue, and most physicists today don’t even recognize his name. But the idea of least action struck a deep chord that reverberates throughout physics. It is the first and fundamental example of a minimum principle, of which there are many. For instance, minimum potential energy identifies points of system equilibrium, and paths of minimum distances are geodesic paths. In dynamics, minimization of the difference between potential and kinetic energies identifies the dynamical paths of trajectories, and minimization of distance through space-time warped by mass and energy density identifies the paths of falling objects. Maupertuis’ fundamentally important idea was picked up by Euler and Lagrange, expanding it through the language of differential geometry. This was the language of Bernhard Riemann, a gifted and shy German mathematician whose mathematical language was adopted by physicists to describe motion as a geodesic, the shortest path like a great-circle route on the Earth, in an abstract dynamical space defined by kinetic energy and potentials. In this view, it is the geometry of the abstract dynamical space that imposes Galileo’s simple parabolic form on freely falling objects. Einstein took this viewpoint farther than any before him, showing how mass and energy warped space and how free objects near gravitating bodies move along geodesic curves defined by the shape of space. This brought trajectories to a new level of abstraction, as space itself became the cause of motion. Prior to general relativity, motion occurred in space. Afterwards, motion was caused by space. In this sense, gravity is not a force, but is like a path down which everything falls. The fifth generation of concepts of motion, Quantum Coherence, increased abstraction yet again in the comprehension of trajectories, ushering in difficult concepts like wave-particle duality and quantum interference. Quantum interference underlies many of the counter-intuitive properties of quantum systems, including the possibility for quantum systems to be in two or more states at the same time, and for quantum computers to crack unbreakable codes. But this new perspective came with a cost, introducing fundamental uncertainties that are locked in a battle of trade-offs as one measurement becomes more certain and others becomes more uncertain. Einstein distrusted Heisenberg’s uncertainty principle, not that he disagreed with its veracity, but he felt it was more a statement of ignorance than a statement of fundamental unknowability. In support of Einstein, Schrödinger devised a thought experiment that was meant to be a reduction to absurdity in which a cat is placed in a box with a vial of poison that would be broken if a quantum particle decays. The cruel fate of Schrödinger’s cat, who might or might not be poisoned, hinges on whether or not someone opens the lid and looks inside. Once the box is opened, there is one world in which the cat is alive and another world in which the cat is dead. These two worlds spring into existence when the box is opened—a bizarre state of affairs from the point of view of a pragmatist. This is where Richard Feynman jumped into the fray and redefined the idea of a trajectory in a radically new way by showing that a quantum trajectory is not a single path, like Galileo’s parabola, but the combined effect of the quantum particle taking all possible paths simultaneously. Feynman established this new view of quantum trajectories in his thesis dissertation under the direction of John Archibald Wheeler at Princeton. By adapting Maupertuis’ Principle of Least Action to quantum mechanics, Feynman showed how every particle takes every possible path—simultaneously—every path interfering in such as way that only the path with the most constructive interference is observed. In the quantum view, the deterministic trajectory of the cannon ball evaporates into a cloud of probable trajectories. In our current complex times, the sixth generation in the evolution of concepts of motion explores Complex Systems. Lorenz’s Butterfly has more to it than butterflies, because Life is the greatest complex system of our experience and our existence. We are the end result of a cascade of self-organizing events that began half a billion years after Earth coalesced out of the nebula, leading to the emergence of consciousness only about 100,000 years ago—a fact that lets us sit here now and wonder about it all. That we are conscious is perhaps no accident. Once the first amino acids coagulated in a muddy pool, we have been marching steadily uphill, up a high mountain peak in a fitness landscape. Every advantage a species gained over its environment and over its competitors exerted a type of pressure on all the other species in the ecosystem that caused them to gain their own advantage. The modern field of evolutionary dynamics spans a wide range of scales across a wide range of abstractions. It treats genes and mutations on DNA in much the same way it treats the slow drift of languages and the emergence of new dialects. It treats games and social interactions the same way it does the evolution of cancer. Evolutionary dynamics is the direct descendant of chaos theory that turned butterflies into hurricanes, but the topics it treats are special to us as evolved species, and as potential victims of disease. The theory has evolved its own visualizations, such as the branches in the tree of life and the high mountain tops in fitness landscapes separated by deep valleys. Evolutionary dynamics draws, in a fundamental way, on dynamic processes in high dimensions, without which it would be impossible to explain how something as complex as human beings could have arisen from random mutations. These six generations in the development of dynamics are not likely to stop, and future generations may arise as physicists pursue the eternal quest for the truth behind the structure of reality.

Read all the stories behind the history of the physics of motion, in Galileo Unbound from Oxford Press: