George Green’s Theorem

For a thirty-year old miller’s son with only one year of formal education, George Green had a strange hobby—he read papers in mathematics journals, mostly from France.  This was his escape from a dreary life running a flour mill on the outskirts of Nottingham, England, in 1823.  The tall wind mill owned by his father required 24-hour attention, with farmers depositing their grain at all hours and the mechanisms and sails needing constant upkeep.  During his one year in school when he was eight years old he had become fascinated by maths, and he had nurtured this interest after leaving school one year later, stealing away to the top floor of the mill to pore over books he scavenged, devouring and exhausting all that English mathematics had to offer.  By the time he was thirty, his father’s business had become highly successful, providing George with enough wages to become a paying member of the private Nottingham Subscription Library with access to the Transactions of the Royal Society as well to foreign journals.  This simple event changed his life and changed the larger world of mathematics.

French Analysis in England

George Green was born in Nottinghamshire, England.  No record of his birth exists, but he was baptized in 1793, which may be assumed to be the year of his birth.  His father was a baker in Nottingham, but the food riots of 1800 forced him to move outside of the city to the town of Sneinton, where he bought a house and built an industrial-scale windmill to grind flour for his business.  He prospered enough to send his eight-year old son to Robert Goodacre’s Academy located on Upper Parliament Street in Nottingham.  Green was exceptionally bright, and after one year in school he had absorbed most of what the Academy could teach him, including a smattering of Latin and Greek as well as French along with what simple math that was offered.  Once he was nine, his schooling was over, and he took up the responsibility of helping his father run the mill, which he did faithfully, though unenthusiastically, for the next 20 years.  As the milling business expanded, his father hired a mill manager that took part of the burden off George.  The manager had a daughter Jane Smith, and in 1824 she had her first child with Green.  Six more children were born to the couple over the following fifteen years, though they never married.

Without adopting any microscopic picture of how electric or magnetic fields are produced or how they are transmitted through space, Green could still derive rigorous properties that are independent of any details of the microscopic model.

            During the 20 years after leaving Goodacre’s Academy, Green never gave up learning what he could, teaching himself to read French readily as well as mastering English mathematics.  The 1700’s and early 1800’s had been a relatively stagnant period for English mathematics.  After the priority dispute between Newton and Leibniz over the invention of the calculus, English mathematics had become isolated from continental advances.  This was part snobbery, but also part handicap as the English school struggled with Newton’s awkward fluxions while the continental mathematicians worked with Leibniz’ more fruitful differential notation.  The French mathematicians in the early 1800’s were especially productive, including works by Lagrange, Laplace and Poisson.

            One block away from where Green lived stood the Free Grammar School overseen by headmaster John Topolis.  Topolis was a Cambridge graduate on a minor mission to update the teaching of mathematics in England, well aware that the advances on the continent were passing England by.  For instance, Topolis translated Laplace’s mathematically advanced Méchaniqe Celéste from French into English.  Topolis was also well aware of the work by the other French mathematicians and maintained an active scholarly output that eventually brought him back to Cambridge as Dean of Queen’s College in 1819 when Green was 26 years old.  There is no record whether Topolis and Green knew each other, but their close proximity and common interests point to a natural acquaintance.  One can speculate that Green may even have sought Topolis out, given his insatiable desire to learn more mathematics, and it is likely that Topolis would have introduced Green to the vibrant French school of mathematics.             

By the time Green joined the Nottingham Subscription Library, he must already have been well trained in basic mathematics, and membership in the library allowed him to request loans of foreign journals (sort of like Interlibrary Loan today).  With his library membership beginning in 1823, Green absorbed the latest advances in differential equations and must have begun forming a new viewpoint of the uses of mathematics in the physical sciences.  This was around the same time that he was beginning his family with Jane as well as continuing to run his fathers mill, so his mathematical hobby was relegated to the dark hours of the night.  Nonetheless, he made steady progress over the next five years as his ideas took rough shape and were refined until finally he took pen to paper, and this uneducated miller’s son began a masterpiece that would change the history of mathematics.

Essay on Mathematical Analysis of Electricity and Magnetism

By 1827 Green’s free-time hobby was about to bear fruit, and he took out a modest advertisement to announce its forthcoming publication.  Because he was an unknown, and unknown to any of the local academics (Topolis had already gone back to Cambridge), he chose vanity publishing and published out of pocket.   An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism was printed in March of 1828, and there were 51 subscribers, mostly from among the members of the Nottingham Subscription Library who bought it at 7 shillings and 6 pence per copy, probably out of curiosity or sympathy rather than interest.  Few, if any, could have recognized that Green’s little essay contained several revolutionary elements.

Fig. 1 Cover page of George Green’s Essay

            The topic of the essay was not remarkable, treating mathematical problems of electricity and magnetism, which was in vogue at that time.  As background, he had read works by Cavendish, Poisson, Arago, Laplace, Fourier, Cauchy and Thomas Young (probably Young’s Course of Lectures on Natural Philosopy and the Mechanical Arts (1807)).  He paid close attention to Laplace’s treatment of celestial mechanics and gravitation which had obvious strong analogs to electrostatics and the Coulomb force because of the common inverse square dependence. 

            One radical contribution in Green’s essay was his introduction of the potential function—one of the first uses of the concept of a potential function in mathematical physics—and he gave it its modern name.  Others had used similar constructions, such as Euler [1], D’Alembert [2], Laplace[3] and Poisson [4], but the use had been implicit rather than explicit.  Green shifted the potential function to the forefront, as a central concept from which one could derive other phenomena.  Another radical contribution from Green was his use of the divergence theorem.  This has tremendous utility, because it relates a volume integral to a surface integral.  It was one of the first examples of how measuring something over a closed surface could determine a property contained within the enclosed volume.  Gauss’ law is the most common example of this, where measuring the electric flux through a closed surface determines the amount of enclosed charge.  Lagrange in 1762 [5] and Gauss in 1813 [6] had used forms of the divergence theorem in the context of gravitation, but Green applied it to electrostatics where it has become known as Gauss’ law and is one of the four Maxwell equations.  Yet another contribution was Green’s use of linear superposition to determine the potential of a continuous charge distribution, integrating the potential of a point charge over a continuous charge distribution.  This was equivalent to defining what is today called a Green’s function, which is a common method to solve partial differential equations.

            A subtle contribution of Green’s Essay, but no less influential, was his adoption of a mathematical approach to a physics problem based on the fundamental properties of the mathematical structure rather than on any underlying physical model.  Without adopting any microscopic picture of how electric or magnetic fields are produced or how they are transmitted through space, he could still derive rigorous properties that are independent of any details of the microscopic model.  For instance, the inverse square law of both electrostatics and gravitation is a fundamental property of the divergence theorem (a mathematical theorem) in three-dimensional space.  There is no need to consider what space is composed of, such as the many differing models of the ether that were being proposed around that time.  He would apply this same fundamental mathematical approach in his later career as a Cambridge mathematician to explain the laws of reflection and refraction of light.

George Green: Cambridge Mathematician

A year after the publication of the Essay, Green’s father died a wealthy man, his milling business having become very successful.  Green inherited the family fortune, and he was finally able to leave the mill and begin devoting his energy to mathematics.  Around the same time he began working on mathematical problems with the support of Sir Edward Bromhead.  Bromhead was a Nottingham peer who had been one of the 51 subscribers to Green’s published Essay.  As a graduate of Cambridge he was friends with Herschel, Babbage and Peacock, and he recognized the mathematical genius in this self-educated miller’s son.  The two men spent two years working together on a pair of publications, after which Bromhead used his influence to open doors at Cambridge.

            In 1832, at the age of 40, George Green enrolled as an undergraduate student in Gonville and Caius College at Cambridge.  Despite his concerns over his lack of preparation, he won the first-year mathematics prize.  In 1838 he graduated as fourth wrangler only two positions behind the future famous mathematician James Joseph Sylvester (1814 – 1897).  Based on his work he was elected as a fellow of the Cambridge Philosophical Society in 1840.  Green had finally become what he had dreamed of being for his entire life—a professional mathematician.

            Green’s later papers continued the analytical dynamics trend he had established in his Essay by applying mathematical principles to the reflection and refraction of light. Cauchy had built microscopic models of the vibrating ether to explain and derive the Fresnel reflection and transmission coefficients, attempting to understand the structure of ether.  But Green developed a mathematical theory that was independent of microscopic models of the ether.  He believed that microscopic models could shift and change as newer models refined the details of older ones.  If a theory depended on the microscopic interactions among the model constituents, then it too would need to change with the times.  By developing a theory based on analytical dynamics, founded on fundamental principles such as minimization principles and geometry, then one could construct a theory that could stand the test of time, even as the microscopic understanding changed.  This approach to mathematical physics was prescient, foreshadowing the geometrization of physics in the late 1800’s that would lead ultimately to Einsteins theory of General Relativity.

Green’s Theorem and Greens Function

Green died in 1841 at the age of 49, and his Essay was mostly forgotten.  Ten years later a young William Thomson (later Lord Kelvin) was graduating from Cambridge and about to travel to Paris to meet with the leading mathematicians of the age.  As he was preparing for the trip, he stumbled across a mention of Green’s Essay but could find no copy in the Cambridge archives.  Fortunately, one of the professors had a copy that he lent Thomson.  When Thomson showed the work to Liouville and Sturm it caused a sensation, and Thomson later had the Essay republished in Crelle’s journal, finally bringing the work and Green’s name into the mainstream.

            In physics and mathematics it is common to name theorems or laws in honor of a leading figure, even if the they had little to do with the exact form of the theorem.  This sometimes has the effect of obscuring the historical origins of the theorem.  A classic example of this is the naming of Liouville’s theorem on the conservation of phase space volume after Liouville, who never knew of phase space, but who had published a small theorem in pure mathematics in 1838, unrelated to mechanics, that inspired Jacobi and later Boltzmann to derive the form of Liouville’s theorem that we use today.  The same is true of Green’s Theorem and Green’s Function.  The form of the theorem known as Green’s theorem was first presented by Cauchy [7] in 1846 and later proved by Riemann [8] in 1851.  The equation is named in honor of Green who was one of the early mathematicians to show how to relate an integral of a function over one manifold to an integral of the same function over a manifold whose dimension differed by one.  This property is a consequence of the Generalized Stokes Theorem, of which the Kelvin-Stokes Theorem, the Divergence Theorem and Green’s Theorem are special cases.

Fig. 2 Green’s theorem and its relationship with the Kelvin-Stokes theorem, the Divergence theorem and the Generalized Stokes theorem (expressed in differential forms)

            Similarly, the use of Green’s function for the solution of partial differential equations was inspired by Green’s use of the superposition of point potentials integrated over a continuous charge distribution.  The Green’s function came into more general use in the late 1800’s and entered the mainstream of physics in the mid 1900’s [9].

Fig. 3 The application of Green’s function so solve a linear operator problem, and an example applied to Poisson’s equation.

[1] L. Euler, Novi Commentarii Acad. Sci. Petropolitanae , 6 (1761)

[2] J. d’Alembert, “Opuscules mathématiques” , 1 , Paris (1761)

[3] P.S. Laplace, Hist. Acad. Sci. Paris (1782)

[4] S.D. Poisson, “Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes” Nouveau Bull. Soc. Philomathique de Paris , 3 (1813) pp. 388–392

[5] Lagrange (1762) “Nouvelles recherches sur la nature et la propagation du son” (New researches on the nature and propagation of sound), Miscellanea Taurinensia (also known as: Mélanges de Turin ), 2: 11 – 172

[6] C. F. Gauss (1813) “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,” Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378

[7] Augustin Cauchy: A. Cauchy (1846) “Sur les intégrales qui s’étendent à tous les points d’une courbe fermée” (On integrals that extend over all of the points of a closed curve), Comptes rendus, 23: 251–255.

[8] Bernhard Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Germany): Adalbert Rente, 1867

[9] Schwinger, Julian (1993). “The Greening of quantum Field Theory: George and I”: 10283. arXiv:hep-ph/9310283

Wave-Particle Duality and Hamilton’s Physics

Wave-particle duality was one of the greatest early challenges to quantum physics, partially clarified by Bohr’s Principle of Complementarity, but never easily grasped even today.  Yet long before Einstein proposed the indivisible quantum  of light (later to be called the photon by the chemist Gilbert Lewis), wave-particle duality was firmly embedded in the foundations of the classical physics of mechanics.

Light led the way to mechanics more than once in the history of physics.

 

Willebrord Snel van Royen

The Dutch physicist Willebrord Snel van Royen in 1621 derived an accurate mathematical description of the refraction of beams of light at a material interface in terms of sine functions, but he did not publish.  Fifteen years later, as Descartes was looking for an example to illustrate his new method of analytic geometry, he discovered the same law, unaware of Snel’s prior work.  In France the law is known as the Law of Descartes.  In the Netherlands (and much of the rest of the world) it is known as Snell’s Law.  Both Snell and Descartes based their work on Newton’s corpuscles of light.  The brilliant Fermat adopted corpuscles when he developed his principle of least time to explain the law of Descartes in 1662.  Yet Fermat was forced to assume that the corpuscles traveled slower in the denser material even though it was generally accepted that light should travel faster in denser media, just as sound did.  Seventy-five years later, Maupertuis continued the tradition when he developed his principle of least action and applied it to light corpuscles traveling faster through denser media, just as Descartes had prescribed.

HuygensParticle-02

The wave view of Snell’s Law (on the left). The source resides in the medium with higher speed. As the wave fronts impinge on the interface to a medium with lower speed, the wave fronts in the slower medium flatten out, causing the ray perpendicular to the wave fronts to tilt downwards. The particle view of Snell’s Law (on the right). The momentum of the particle in the second medium is larger than in the first, but the transverse components of the momentum (the x-components) are conserved, causing a tilt downwards of the particle’s direction as it crosses the interface. [i]

Maupertuis’ paper applying the principle of least action to the law of Descartes was a critical juncture in the development of dynamics.  His assumption of faster speeds in denser material was wrong, but he got the right answer because of the way he defined action for light.  Encouraged by the success of his (incorrect) theory, Maupertuis extended the principle of least action to mechanical systems, and this time used the right theory to get the right answers.  Despite Maupertuis’ misguided aspirations to become a physicist of equal stature to Newton, he was no mathematician, and he welcomed (and  somewhat appropriated) the contributions of Leonid Euler on the topic, who established the mathematical foundations for the principle of least action.  This work, in turn, attracted the attention of the Italian mathematician Lagrange, who developed a general new approach (Lagrangian mechanics) to mechanical systems that included the principle of least action as a direct consequence of his equations of motion.  This was the first time that light led the way to classical mechanics.  A hundred years after Maupertuis, it was time again for light to lead to the way to a deeper mechanics known as Hamiltonian mechanics.

Young Hamilton

William Rowland Hamilton (1805—1865) was a prodigy as a boy who knew parts of thirteen languages by the time he was thirteen years old. These were Greek, Latin, Hebrew, Syriac, Persian, Arabic, Sanskrit, Hindoostanee, Malay, French, Italian, Spanish, and German. In 1823 he entered Trinity College of Dublin University to study science. In his second and third years, he won the University’s top prizes for Greek and for mathematical physics, a run which may have extended to his fourth year—but he was offered the position of Andrew’s Professor of Astronomy at Dublin and Royal Astronomer of Ireland—not to be turned down at the early age of 21.

Hamilton1

Title of Hamilton’s first paper on his characteristic function as a new method that applied his theory from optics to the theory of mechanics, including Lagrangian mechanics as a special case.

His research into mathematical physics  concentrated on the theory of rays of light. Augustin-Jean Fresnel (1788—1827) had recently passed away, leaving behind a wave theory of light that provided a starting point for many effects in optical science, but which lacked broader generality. Hamilton developed a rigorous mathematical framework that could be applied to optical phenomena of the most general nature. This led to his theory of the Characteristic Function, based on principles of the variational calculus of Euler and Lagrange, that predicted the refraction of rays of light, like trajectories, as they passed through different media or across boundaries. In 1832 Hamilton predicted a phenomenon called conical refraction, which would cause a single ray of light entering a biaxial crystal to refract into a luminous cone.

Mathematical physics of that day typically followed experimental science. There were so many observed phenomena in so many fields that demanded explanation, that the general task of the mathematical physicist was to explain phenomena using basic principles followed by mathematical analysis. It was rare for the process to work the other way, for a theorist to predict a phenomenon never before observed. Today we take this as very normal. Einstein’s fame was primed by his prediction of the bending of light by gravity—but only after the observation of the effect by Eddington four years later was Einstein thrust onto the world stage. The same thing happened to Hamilton when his friend Humphrey Lloyd observed conical refraction, just as Hamilton had predicted. After that, Hamilton was revered as one of the most ingenious scientists of his day.

Following the success of conical refraction, Hamilton turned from optics to pursue a striking correspondence he had noted in his Characteristic Function that applied to mechanical trajectories as well as it did to rays of light. In 1834 and 1835 he published two papers On a General Method in Mechanics( I and II)[ii], in which he reworked the theory of Lagrange by beginning with the principle of varying action, which is now known as Hamilton’s Principle. Hamilton’s principle is related to Maupertuis’ principle of least action, but it was more rigorous and a more general approach to derive the Euler-Lagrange equations.  Hamilton’s Principal Function allowed the trajectories of particles to be calculated in complicated situations that were challenging for a direct solution by Lagrange’s equations.

The importance that these two papers had on the future development of physics would not be clear until 1842 when Carl Gustav Jacob Jacobi helped to interpret them and augment them, turning them into a methodology for solving dynamical problems. Today, the Hamiltonian approach to dynamics is central to all of physics, and thousands of physicists around the world mention his name every day, possibly more often than they mention Einstein’s.

[i] Reprinted from D. D. Nolte, Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford, 2018)

[ii] W. R. Hamilton, “On a general method in dynamics I,” Phil. Trans. Roy. Soc., pp. 247-308, 1834; W. R. Hamilton, “On a general method in dynamics II,” Phil. Trans. Roy. Soc., pp. 95-144, 1835.

Galileo Unbound

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: A Path Across Life, the Universe and Everything (Oxford University Press) publishes today (Sept. 26, 2018). It 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.

 

To read more: Galileo Unbound: A Path Across Life, the Universe and Everything by David D. Nolte (Oxford University Press, Sept. 26, 2018). Available at Amazon.com.

 

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) and Huygens provided a geometric proof in his Horologium Oscillatorium sive de motu pendulorum (1673) that the curve was a cycloid. 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.

HuygensIsochron

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.

[1] Galileo conceived of a 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.

 

 

 

Geometry as Motion

Nothing seems as static and as solid as geometry—there is even a subfield of geometry known as “solid geometry”. Geometric objects seem fixed in time and in space. Yet the very first algebraic description of geometry was born out of kinematic constructions of curves as René Descartes undertook the solution of an ancient Greek problem posed by Pappus of Alexandria (c. 290 – c. 350) that had remained unsolved for over a millennium. In the process, Descartes’ invented coordinate geometry.

Descartes used kinematic language in the process of drawing  curves, and he even talked about the speed of the moving point. In this sense, Descartes’ curves are trajectories.

The problem of Pappus relates to the construction of what were known as loci, or what today we call curves or functions. Loci are a smooth collection of points. For instance, the intersection of two fixed lines in a plane is a point. But if you allow one of the lines to move continuously in the plane, the intersection between the moving line and the fixed line sweeps out a continuous succession of points that describe a curve—in this case a new line. The problem posed by Pappus was to find the appropriate curve, or loci, when multiple lines are allowed to move continuously in the plane in such a way that their movements are related by given ratios. It can be shown easily in the case of two lines that the curves that are generated are other lines. As the number of lines increases to three or four lines, the loci become the conic sections: circle, ellipse, parabola and hyperbola. Pappus then asked what one would get if there were five such lines—what type of curves were these? This was the problem that attracted Descartes.

What Descartes did—the step that was so radical that it reinvented geometry—was to fix lines in position rather than merely in length. To us, in the 21st century, such an act appears so obvious as to remove any sense of awe. But by fixing a line in position, and by choosing a fixed origin on that line to which other points on the line were referenced by their distance from that origin, and other lines were referenced by their positions relative to the first line, then these distances could be viewed as unknown quantities whose solution could be sought through algebraic means. This was Descartes’ breakthrough that today is called “analytic geometry”— algebra could be used to find geometric properties.

Newton too viewed mathematical curves as living things that changed in time, which was one of the central ideas behind his fluxions—literally curves in flux.

Today, we would call the “locations” of the points their “coordinates”, and Descartes is almost universally credited with the discovery of the Cartesian coordinate system. Cartesian coordinates are the well-known grids of points, defined by the x-axis and the y-axis placed at right angles to each other, at whose intersection is the origin. Each point on the plane is defined by a pair of numbers, usually represented as (x, y). However, there are no grids or orthogonal axes in Descartes’ Géométrie, and there are no pairs of numbers defining locations of points. About the most Cartesian-like element that can be recognized in Descartes’ La Géométrie is the line of reference AB, as in Fig. 1.

Descartesgeo5

Fig. 1 The first figure in Descartes’ Géométrie that defines 3 lines that are placed in position relative to the point marked A, which is the origin. The point C is one point on the loci that is to be found such that it satisfies given relationships to the 3 lines.

 

In his radical new approach to loci, Descartes used kinematic language in the process of drawing the curves, and he even talked about the speed of the moving point. In this sense, Descartes’ curves are trajectories, time-dependent things. Important editions of Descartes’ Discourse were published in two volumes in 1659 and 1661 which were read by Newton as a student at Cambridge. Newton also viewed mathematical curves as living things that changed in time, which was one of the central ideas behind his fluxions—literally curves in flux.

 

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.

Descartes3Char

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.

 

The Oxford Scholars

 

Oxford University, and specifically Merton College, was a site of intense intellectual ferment in the middle of the Medieval Period around the time of Chaucer. A string of natural philosophers, today called the Oxford Scholars or the Oxford Calculators, began to explore early ideas of motion, taking the first bold steps beyond Aristotle. They were the first “physicists” (although that term would not be used until our own time) and laid the foundation upon which Galileo would build the first algebraic law of physics.

It is hard to imagine today what it was like doing mathematical physics in the fourteenth century. Mathematical symbolism did not exist in any form. Nothing stood for anything else, as we routinely use in algebra, and there were no equations, only proportions.

Thomas Bradwardine (1290 – 1349) was the first of the Scholars, arriving at Oxford around 1320. He came from a moderately wealthy family from Sussex on the southern coast of England not far from where the Anglo Saxon king Harold lost his kingdom and his life at the Battle of Hastings. The life of a scholar was not lucrative, so Bradwardine supported himself mainly through the royal patronage of Edward III, for whom he was chaplain and confessor during Edward’s campaigns in France, eventually becoming the Archbishop of Canterbury, although he died of the plague returning from Avignon before he could take up the position. When not campaigning or playing courtier, Bradwardine found time to develop a broad-ranging program of study that spanned from logic and theology to physics.

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Merton College, Oxford (attribution: Andrew Shiva / Wikipedia)

Bradwardine began a reanalysis of an apparent paradox that stemmed from Aristotle’s theory of motion. As anyone with experience pushing a heavy box across a floor knows, the box does not move until sufficient force is applied. Today we say that the applied force must exceed the static force of friction. However, this everyday experience is at odds with Aristotle’s theory that placed motion in inverse proportion to the resistance. In this theory, only an infinite resistance could cause zero motion, yet the box does eventually move if enough force is applied. Bradwardine sought to resolve this paradox. Within the scholastic tradition, Aristotle was always assumed to have understood the truth, even if fourteenth-century scholars could not understand it themselves. Therefore, Bradwardine constructed a mathematical “explanation” that could preserve Aristotle’s theory of proportion while still accounting for the fact that the box does not move.

It is hard to imagine today what it was like doing mathematical physics in the fourteenth century. Mathematical symbolism did not exist in any form. Nothing stood for anything else, as we routinely use in algebra, and there were no equations, only proportions. The introduction of algebra into Europe through Arabic texts was a hundred years away. Not even Euclid or Archimedes had been rediscovered by Bradwardine’s day, so all he had to work with was Pythagorean theory of ratios and positive numbers—even negative numbers did not exist—and the only mathematical tools at his disposal were logic and language. Nonetheless, armed only with these sparse tools, Bradwardine constructed a verbal argument that the proportion of impressed force to resistance must itself be as a proportionality between speeds. As awkward as this sounds in words, it is a first intuitive step towards the concept of an exponential relationship—a power law. In Bradwardine’s rule for motion, the relationships among force, resistance and speed were like compounding interest on a loan. Bradwardine’s rule is not correct physics, because the exponential function is never zero, and because motion does not grow exponentially, but it did introduce the intuitive idea that a physical process could be nonlinear (using our modern terminology), changing from small effects to large effects disproportionate to the change in the cause. Therefore, the importance of Bradwardine was more his approach than his result. He applied mathematical reasoning to a problem of kinetics and set the stage for mathematical science.

A few years after Bradwardine had devised his rule of motion, a young mathematician named William Heytesbury (1313—1373) arrived as a fellow of Merton College. In the years that they overlapped at Oxford, one can only speculate what was transmitted from the senior to the junior fellow, but by 1335 Heytesbury had constructed a theory of continuous magnitudes and their continuous changes that included motion as a subset. The concept of the continuum had been a great challenge for Aristotelian theory, leading to many paradoxes or sophisms, like Zeno’s paradox that supposedly proved the impossibility of motion. Heytesbury shrewdly recognized that the problem was the ill-defined idea of instantaneous rate of change.

Heytesbury was just as handicapped as Bradwardine in his lack of mathematical notation, but worse, he was handicapped by the Aristotelian injunction against taking ratios of unlike qualities. According to Aristotle, proportions must only be taken of like qualities, such as one linear length to another, or one mass to another. To take a proportion of a mass to a length was nonsense. Today we call it “division” (more accurately a distribution), and mass divided by length is a linear mass density. Therefore, because speed is distance divided by time, no such ratio was possible in Heytesbury’s day because distance and time are unlike qualities. Heytesbury ingeniously got around this restriction by considering the linear distances covered by two moving objects in equal times. The resulting linear distances were similar qualities and could thereby be related by proportion. The ratio of the distances become the ratio of speeds, even though speed itself could not be defined directly. This was a first step, a new tool. Using this conceit, Heytesbury was able to go much farther, to grapple with the problem of nonuniform motion and hence the more general concept of instantaneous speed.

In the language of calculus (developed by Newton and Leibniz 300 years later), instantaneous speed is a ratio of an element of length to an element of time in the limit as the elements vanish uniformly. In the language of Heytesbury (Latin), instantaneous speed is simply the average speed between two neighboring speeds (still working with ratios of distances traversed in equal times). And those neighboring speeds are similarly the averages of their neighbors, until one reaches zero speed on one end and final speed on the other. Heytesbury called this kind of motion difform as opposed to uniform motion.

A special case of difform motion was uniformly difform motion—uniform acceleration. Acceleration was completely outside the grasp of Aristotelian philosophers, even Heytesbury, but he could imagine a speed that changed uniformly. This requires that the extra distance travelled during the succeeding time unit relative to the distance travelled during the current time unit has a fixed value. He then showed, without equations, using only his math-like language, that if a form changes uniformly in time (constant rate of change) then the average value of the form over a fixed time is equal to the average of the initial and final values. This work had a tremendous importance, not only for the history of mathematics, but also for the history of physics, because when the form in question is speed, then this represents the discovery of the mean speed theorem for the case of uniform acceleration. The mean speed theorem is often attributed to Galileo, who proved the theorem as part of his law of fall, and he deserves the attribution because there is an important difference in context. Heytesbury was not a scientist nor even a natural philosopher. He was a logician interested in sophisms that arose in discussions of Aristotle. The real purpose of Heytesbury’s analysis was to show that paradoxes like that of Zeno could be resolved within the Aristotelian system. He certainly was not thinking of falling bodies, whereas Galileo was.

Not long after Heytesbury demonstrated the mean speed theorem, he was joined at Merton College by yet another young fellow, Richard Swineshead (fl. 1340-1354). Bradwardine was already gone, but his reputation survived, as well as his memory, in the person of Heytesbury, and Swineshead became another member in the tradition of the Merton Scholars. He was perhaps the most adept at mathematics of the three, and he published several monumental treatises that mathematically expanded upon both Bradwardine and Heytesbury, systematizing their results and disseminating them in published accounts that spread across scholastic Europe—all still without formulas, symbols or equations. For these works, he became known as The Calculator. By consolidating and documenting the work of the Oxford Scholars, his influence on the subsequent history of thought was considerable, as he was widely read by later mathematicians, including Leibniz, who had a copy of Heytesbury in his personal library.

( To read more about the Oxford Scholars, and their connections with members of their contemporaries in Paris, see Chapter 3 of Galileo Unbound (Oxford University Press, 2018).)