• Wilhelm Weber Derives his Electric Force Law

  In this post, based on Weber’s First Memoir, I wrote about the development of his general fundamental law, which combined the electrostatics of Charles-Augustin de Coulomb (1736-1806), the electrostatics of André-Marie Ampère (1775-1836), and the induction of Michael Faraday (1791-1867).

In his 2010 book The German Genius,1 Peter Watson summarizes the work of Weber with two facts: 1) as the co-inventor with Carl Gauss (1777-1855) of the telegraph and 2) as the teacher of physics for Bernhard Riemann (1826-1866). There is simply no mention of Weber’s groundbreaking work in electricity, neither at the experimental level, nor the theoretical; today, Weber’s work is simply unknown. Compare Watson’s assessment with that of Ferdinand Rosenberger (1845-1899) in the 1880s in his Die Geschichte der Physik2 (The History of Physics):

Weber’s law of electric force was a revolutionary achievement of the first order, one that foreshadowed far-reaching consequences. Since Newton, all forces had ultimately been traced back to fundamental properties of matter, whose effects were entirely independent of the state of motion of the matter. According to Weber, however, this fundamental effect is modified by motion, and not only velocity but even acceleration introduces new forces independent of the properties of matter at rest. Newton’s unified view of force—which had already been believed to be firmly established for the entire field of physics—was thus once again called into question by that old revolutionary element, electricity, and a tendency toward the old notions of motion as the actual force-producing element seemed unmistakable in this. However, at that time, physicists were still largely reluctant to delve more deeply into questions of such a fundamental nature even in this field, and so no further objections were raised at that time against the new view of the action of electric forces. Weber, on the other hand, was compelled to grapple with the work of other physicists who had derived the phenomena of galvanic induction primarily from an entirely different basis, yet whose results corresponded to the observations. [Volume 3, pp.507-508, Rosenberger’s emphasis]

In this post, I will give an overview of Weber’s achievements after he published his First Memoir in 1846, in which he presented his electric force law. This summary will help give an idea of the “far-reaching consequences” mentioned by Rosenberger. Since I will be covering a period of close to four decades, each achievement can only be briefly presented. Future posts will focus on specific topics pertaining to Weber.

What is striking when reading Weber’s memoirs is the clear materialist approach that he took, a completely different overall method compared to that of many of his contemporaries. Most of his memoirs begin with a detailed description of the instruments that he designed and had made by the best instrument-makers in Leipzig, Göttingen or elsewhere, followed by an extensive presentation of the experiments undertaken using said instruments. The key conclusions from those experiments then serve as the basis for the derivation of mathematical laws. From the latter, he then considers what might be happening at a microscopic or molecular level, ultimately leading to his model of an electric atom in his Eighth Memoir, published posthumously in 1894.

For the elaboration of this post, I will be referring to Volumes II-IV of Wilhem’s Weber Main Works on Electrodynamics, edited by André Assis.345

1852a: Second Memoir, relating specially to Measures of Resistance

When Weber began work on this memoir, the resistance law of Georg Simon Ohm (1789-1854), published in 1827, was well known. However, what was being measured, and the underlying physical processes, were unknown. André Assis writes:

Weber showed in the present work that in order to have a steady current flowing along this resistive wire, it was necessary to have a distribution of charges spread along the surface of the conductor. This distribution of surface charges would generate a constant electric force acting on any charge of the conductor, like a stationary ion at rest in the lattice or a mobile charge of the wire (as the modern conduction electron). This distribution of surface charges was maintained by the battery. It was constant in time for steady currents, but its magnitude varied along the length of the wire. In this case of a straight wire, Weber showed that the surface charge density σ increased linearly as a function of the longitudinal coordinate z. [Volume II, pp.287-288, my emphasis]

In his First Memoir, Weber had, as had Ampère, supposed that a current-carrying wire consisted of two electrical fluids flowing in opposite directions. In studying the causes of electric resistance, Weber’s conclusion was that having these two fluids flowing around each other would be problematic:

One then further asks, from whence originate the forces that preserve the equilibrium of the continuing electromotive forces for a steady, uniform current, and in that way prevent further acceleration of the electric fluid in its motion? Are those forces purely-electric forces or are they forces that the ponderable parts exert upon the electric fluids that flow through them? If we assume, as we have always done, that there are two electric fluids in the galvanic current that simultaneously flow through that conductor in opposite directions, then we are very close to looking for a cause of the resistance to the motion of that fluid in the fluid that opposes it. Namely, the positive and negative fluid will combine into a neutral mixture at the moment that they meet, and no matter how easy that it might be to also separate that neutral combination, such a new separation could result only from a new electromotive force, and not as a consequence of a persistence of the motions that the two fluids possessed before their combination, because they must be considered as cancelling each other out as a result of their meeting and combination. That implies that whereas each fluid must be ascribed a persistent motion by itself, the motions of both fluids together will not persist in the double current. [Volume II, pp.376-377, my emphasis]

Weber’s solution was that in fact one of the fluids would in fact not be moving, or barely, and the other would flow around the first. This was, in fact, the first proposal of an electric atom:

At a, one presently finds a mobile negative electric mass upon which the neighboring positive mass at A acts strongly enough that the action of the more distant masses B, C, ... can be neglected. The masses at A and a act upon each other with a force that depends upon their magnitudes, relative velocity, and the change in it. Meanwhile, for the sake of simplicity, one might assume that the correction to the electrostatic force (which depends upon the masses and the distance) from the relative velocity and its change proves to be so small in comparison to the latter force that it can likewise be neglected. With those assumptions, it will follow that when no other force acts upon the mass at a, that mass must obey the laws of motion in central forces, which are inversely proportional to the square of the distance. As a result, the mass at a will describe, e.g., an elliptical orbit around A, according to Kepler’s laws. However, a perturbation of that motion of the mass in question will occur at A as long as an electromotive force parallel to the line AB with constant intensity acts upon the mass considered, in addition to the central force. The elements of the elliptical motion up to now would then be changed continually, and the orbit that is described by the mass considered would go over to a spiral curve in that way, in which the mass considered would ultimately be led so far from A that it would leave the sphere of action of A and arrive in the sphere of action of B, and so on, until it had described a number of spiral windings around B that also led it so far from B that it left the sphere of action of B and entered the sphere of action of C. In that way, an electromotive force can then cause the negative electricity to flow away in the direction ABC, which the positive masses at A, B, C take no part in. [Volume II, p.378, my emphasis]

1852b: Third Memoir, relating specially to Diamagnetism

Diamagnetism is a phenomenon in which a substance is repelled by both the north and south poles of a magnet. The first substance for which this phenomenon was discovered was for bismuth.

The First Memoir brought together electrostatics, electrodynamics and magnetism. The Third Memoir added diamagnetism, as a phenomenon of induced currents. The key conclusion of this memoir is that the theory of two magnetic fluids is clearly rejected. Combined with the results of the Second Memoir, the theory of two electric fluids, one of which is essentially stationary, becomes the preferred one.

From the compilation and consideration of the possible cases above, through which a state of a material can occur representable by an ideal distribution of magnetic fluids, we found only one [case] compatible with the fundamental phenomena during the emergence of diamagnetism. It follows from this, that one can explain the emergence of a diamagnetic state of a material only if one assume that this case really exists. Namely the case where the diamagnetic state emerges due to the induced forces which acted on the material and the electric fluids in the material which move without resistance on circular orbits. Therefore one assumes that a bar of bismuth consists of molecules which contain closed orbits (or canals), in which the electric fluids can move without resistance, while in all other orbits they can only move after overcoming a resistance proportional to its velocity. The occurrence of a pure diamagnetism not intermingled with magnetism also requires, that the molecules together with those orbits or canals cannot be rotated. Otherwise rotating molecular currents would emerge leading to a magnetic state, if during the rotation their intensity does not change, as proved by Ampère. [Volume III, pp.58-59, my emphasis]

Weber’s conclusion is even more categorical: the theory of magnetic fluids has been disproven.

The diamagnetic phenomena discovered by Faraday decide between these two theories in the same way as the phenomena of interference decided between the emission and wave theory in optics. This is the most essential and important meaning associated to this discovery. Thanks to the discovery of diamagnetism the hypothesis of electric molecular currents in the interior of materials gets affirmed and the hypothesis of magnetic fluids in the interior of materials gets disproved. [Volume III, p.68, my emphasis]

1857, with Kohlrausch: Fourth Memoir, specially Attributing Mechanical Units to Measures of Current Intensity

The Fourth Memoir was written jointly with Rudolph Kohlrausch (1809-1858).

With the first three memoirs, all of the studied phenomena could be given equations. However, in most of these equations, a constant c appeared. It turned out to be a velocity:

From the fundamental law of electrical action that was established in the first treatise on Electrodynamic Measurements, which encompassed electrostatics, electrodynamics, and induction, the force that an amount of electricity e exerts upon an amount of electricity e′ at a distance of r with a relative velocity of dr/dt and an acceleration of d²r/dt² is expressed by:

That force splits into two parts, the first of which, which is equal to ee′/r², can be called the electrostatic force, and the second of which, which is equal to

−(ee′/c²r²)(dr²/dt² −2rd²r/dt²),

can be called the electrodynamic force. The ratio of those two forces is determined from the constant c. c means the value of relative velocity (assumed uniform) at which the electrostatic force would cancel the electrodynamic force. [Volume III, p.178, my emphasis]

It turns out that the value of c is 439 450 · 10⁶ mm, or 439.450 · 10⁶ m. They write:

From this determination of the constant c, one then sees that two electrical masses must move with a very large velocity with respect to each other if the electrodynamic force were to cancel the electrostatic one, namely, with a velocity of 439 million meters or 59 320 miles per second, which exceeds the speed of light significantly. [Volume III, p.179]

1857, Kirchhoff: On the Motion of Electricity in Wires

Gustav Kirchhoff (1824-1887) was the first to formulate, in 1857, telegraph equations, describing how electricity flows in a long, thin wire, by presupposing Weber’s electrodynamics. This is 19 years before 1876, when Oliver Heaviside (1850-1925) formulated his own telegraph equations.

In this paper, Kirchhoff discovered that an electric wave propagates through a wire at the speed of light:

The velocity of propagation of an electric wave is here found to be = c/√2, hence it is independent of the cross section, of the conductivity of the wire, also, finally, of the density of the electricity: its value is 41 950 German miles in a second, hence very nearly equal to the velocity of light in vacuo. [Volume III, p.214, my emphasis]

So Kirchhoff computed the speed of light in 1857, eight years before James Clerk Maxwell (1831-1879) did so in 1865.

1864: Fifth Memoir, relating specially to Electric Oscillations

According to Johann Christian Poggendorff (1796-1877), publisher of the Annalen, Weber had shown him in 1857 work of a similar—but more general—nature to that of Kirchhoff, but that Weber wanted to undertake a number of experiments to support his work. Publication of the Fifth Memoir only took place in 1864 because of the untimely death in 1858 of Kohlrausch, who had been working with Weber on this topic.

1871: Sixth Memoir, relating specially to the Principle of Conservation of Energy

In the following post, I will focus on the bitter opposition to Weber’s work by Hermann Helmholtz (1821-1894, later von Helmholtz) throughout the latter’s life, as well as by William Thomson (1824-1907, later Baron Kelvin). One of the key arguments put forward against Weber is that his fundamental law of electricity violated the principle of conservation of energy, because his law depended not only on the position of electric masses, but also on their relative velocity.

When Helmholtz formulated in 1847 his principle of conservation of force, later renamed energy, he had already criticized Weber’s law, published in 1846. What is interesting is that in the paper summarizing his First Memoir, published by Weber in 1848, the latter had already formulated the equation for a potential between two electrical masses, hence yielding the definition of a potential energy. Hence, Weber had already implicitly replied to Helmholtz.

In the Sixth Memoir, 23 years later, Weber demonstrated without a doubt, as would agree people such as Maxwell, that the former’s law did respect the principle of conservation of energy.

A comment in the conclusions gives an indication of the research that Weber would undertake in his latter years:

The theory of molecular motions requires in any case a special development, which as yet is wanting throughout. But as long as such a theory remains excluded from mechanical investigations, any doubts as to physical admissibility in relation to molecular motions are without foundation. [Volume IV, p.111, my emphasis]

This would be the focus of much of the work published posthumously in his Eighth Memoir.

1878: Seventh Memoir, relating specially to the Energy of Interaction

In the Seventh Memoir, Weber refocused on the conservation of energy, but by looking at things differently. He would derive the fundamental law, starting from the principle of the conservation of energy:

If by this, Helmholtz was entitled to formulate the principle of energy tentatively in such a way that my fundamental law, condemned by him, is in contradiction with it, then obviously the opposite is equally legitimate, namely, to formulate the same principle tentatively in such a way that it is not only in agreement with that fundamental law, but that the latter even results as a necessary consequence of the principle by proving that all electrodynamic laws, to which that fundamental law belongs, can be derived via the tentative principle from the electrostatic laws. To try it this way around is the intention of the present treatise whereby, instead of beginning with the general fundamental law of electrical interaction established in the first treatise which simultaneously embraced electrostatics and electrodynamics, the principle of the conservation of energy will be the starting point, from which then, in combination with the fundamental law of static interaction between two particles, we shall deduce, firstly, that fundamental law of electrical interaction and, secondly, the existence of an energy function for every pair of particles from which follows the validity of the ordinary energy principle, as it was spelled out by Neumann. [Volume IV, p.125, my emphasis]

In this memoir, Weber distinguishes between the energy of motion and the energy of interaction. In the equation for the latter appears a constant ρ, which corresponds to a distance. Two electric particles e and e' can never be at a distance ρ from each other. If they are of the same charge, and are further away than ρ from each other, they repel each other. If, on the other hand, they are less than ρ from each other, they attract each other!

The expression for the electric force in this form can now serve as a better overview of the forces that each of the two particles exert upon one another, but one has to bear in mind that in this form the forces cannot be composed according to the parallelogram law. One can see that for positive values of the product ee′ this force would be infinitely large, not only for r = 0, but also for r = ρ; but one can also see that in reality the case r = ρ will never happen, because, no matter how large the relative velocity might be at any distance that differs from ρ by a finite value, then r will never be equal to ρ.

Since ee′ is positive, the force for r > ρ is repulsive, and it increases to infinity while r decreases towards ρ, from which it is clear that the approximate velocity, which although very large is not yet infinite, must be cancelled by this repulsive force increasing to infinity, before r reaches ρ, and that then r immediately starts increasing again. It follows from this that r never equals ρ, but that the two particles necessarily have to stay apart at a distance larger than ρ.

For r < ρ the force is attractive and it increases with r to a value which approximates the limit ρ, to infinity, from which it is clear that the distant velocity, which although is very large is not yet infinite, must be neutralized by the attracting force increasing to infinity, before r reaches ρ, and that then r immediately starts decreasing again. Hence in this case the two particles always will stay apart by a distance smaller than ρ. [Volume IV, p.139, my emphasis]

Weber did not have the means to calculate the value of ρ. Today we can, and it turns out to be in the order of 10^-15 m, which is the order of the diameter of a hydrogen atom’s nucleus! This means that when charges of the same sign are within that distance ρ, they attract each other. There is no need for additional forces to hold the nucleus together!

1894, posthumous: Eighth Memoir, relating specially to the Connection of the Fundamental Law of Electricity with the Law of Gravitation

This memoir, thought to be written—but not completed—by Wilhelm Weber in the 1880s, was edited by his nephew Heinrich Friedrich Weber (1843-1912).

In the introduction to the text, Weber writes about the different parts of physics, which can only be understood together if we study the interactions of components of ponderable masses, and if we understand light, heat, electricity and magnetism as phenomena resulting from the interaction of electrical masses.

A distinction has been made between the ponderable bodies in space in solid, liquid, and gaseous aggregate states, and imponderable bodies, in particular light, heat material, two electric and two magnetic substances and, accordingly, all of physics has been categorized into the Mechanics of ponderable solid, liquid, and gaseous bodies, and the Theories of Light, of Heat, of Electricity, and of Magnetism.

In the course of the development of these theories, however, the theory of magnetism along with its extension into the theory of diamagnetism, has been entirely absorbed by the theory of electricity, so that the assumption of two special magnetic materials has been discarded. Likewise, the differentiation of a light ether for light radiation from a heat ether for heat radiation, has also been dropped; an ether, however, for light and heat radiation remains, and a heat material as a conductor for heat in the interior of ponderable bodies, in particular metallic heat conductors.

It is interesting to pursue this present development of physical research further and indeed, one observes:

Firstly, that the theory of magnetism can only be absorbed by the theory of electricity under the assumption of movable parts in the interior of all magnetic and magnetizable bodies, i.e., positive electric molecules, which form molecular currents around the negatively electrically charged ponderable molecules, in the interior of all magnetizable bodies. [Here, Weber had the positive masses moving around the negative masses, but at that time there was no way to determine which revolved around which.]

Secondly, by the further consideration, that the theory of galvanism and of heat, in order likewise to be absorbed by the theory of electricity, must also presuppose movable parts in the interior of all galvanic conductors and heat conductors; that, however, these need by no means be different parts which produce in the interior of ponderable bodies, magnetism, and others which produce the motion of galvanism, and still others, whose movements produce heat; rather that the same parts, according to the difference of their motions, can produce magnetism, galvanism, and heat, sometimes together, sometimes separately, and that these moving parts in the interior of ponderable bodies are molecules of one electricity, which ought to be called positive electricity.

Thirdly, it is to be considered, that the movements of these positive electric molecules around the negatively electrically charged ponderable molecules of bodies, either form closed orbits, or spiral orbits differing only slightly from circular orbits with periodically increasing and attenuating diameter, or spiral-shaped orbits with continuously increasing diameter, whereby they ultimately pass over into a ballistic trajectory, thus effecting the transfer of this electric molecule from one ponderable molecule to another neighboring ponderable molecule, whereupon in part heat conduction, and in part galvanic currents in metallic conductors, are based.

Fourthly, and finally, it is furthermore to be considered, that by means of magnetic or electrodynamic induction from the outside, circular currents around the ponderable molecules of a body may be excited, or circular currents already in existence may be enhanced, weakened, or changed in direction.

It is self evident, that all metallic heat- and electricity- conductors belong to the class of ponderable bodies, around whose molecules positive electric molecules move in circular orbits, albeit with increasing diameters, which make a transition into a ballistic trajectory and thus a transition from those ponderable molecules about which they revolve, to neighboring ponderable molecules; that, on the other hand, all transparent bodies, such as glass and crystals, belong to the class of ponderable bodies about whose molecules positive electric molecules indeed move, but only in tighter circles without transition into a ballistic trajectory (which, therefore, are neither conductors of heat nor electricity), while the greater remaining part of the body consists of the space (like the cosmic space) between the ponderable molecules, with positive electric molecules forming the light ether, and characterized by ballistic or wave motion. [Volume IV, pp.174-175]

In the memoir, Weber further develops his electric atom, and intended to write further, as can be seen in a final note provided by the editor Heinrich Weber:

In addition to the Sections presented here, the original manuscript of Wilhelm Weber also included four other Sections, with the following titles:

1. The theory of the reflection and scattering of electric rays is not applicable for the foundation of a theory of dynamic media.

2. The theory of the light-ether in space as a static medium.

3. On the wave-theory of so-called dynamic media.

4. Laws of repulsion and scattering of gas molecules in ballistic trajectories according to the dynamic theory of gases upon their collision, on the hypothesis, that gas molecules are compounds of positive and negative electric molecules. […]

Wilhelm Weber, however, later edited these Sections out of the discussion, for which reason they are not published here. [Volume IV, p.210]

Concluding Thoughts

This cursory overview of forty years of memoirs on electrodynamics written by Wilhelm Weber should give an idea of just how far-reaching were his thoughts and research. Yet, today, he is, as I mentioned in the introduction to this post, barely known: simply as a fine experimentalist who worked with Gauss and who taught physics to Riemann.

As we will see in the following post, Weber’s work was not superseded by better research, but by a deliberate process of erasure.

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Je suis heureux d’annoncer que mes traductions en français et en espagnol de La méthode illustrée d’Archimède : utilisation de la loi du levier pour calculer les aires, les volumes et les centres de gravité, d’André Koch Torres Assis et Ceno Pietro Magnaghi, ont été publiées par Apeiron à Montréal. Ces ouvrages peuvent être commandés sur Amazon ou téléchargés depuis le site web d’André Assis.

Me complace anunciar que mis traducciones al francés y al castellano de El método ilustrado de Arquímedes: uso de la ley de la palanca para calcular áreas, volúmenes y centros de gravedad, de André Koch Torres Assis y Ceno Pietro Magnaghi, han sido publicadas por Apeiron en Montreal. Los libros se pueden comprar en Amazon o descargar de la página web de André Assis.

• PDF: https://www.ifi.unicamp.br/~assis/The-Illustrated-Method-of-Archimedes-in-French.pdf

• Amazon: https://www.amazon.com/dp/1987980557

• PDF: https://www.ifi.unicamp.br/~assis/The-Illustrated-Method-of-Archimedes-in-Spanish.pdf

• Amazon: https://www.amazon.com/dp/1987980573

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• Two Productive Research Programs in the Early 19th Century

• How Two Productive Research Programs Were Consciously Thrown Out

• The Old Productive Research Programs Refuse to Die

• English Translations of Coulomb, Ampère and Weber

• André Koch Torres Assis Publishes the Fifth Volume of Wilhelm Eduard Weber's Works

In this post, I will focus on a key paper published by Weber in 1846, available in Volume II of Wilhem’s Weber Main Works on Electrodynamics, entitled Weber’s Fundamental Force and the Unification of the Laws of Coulomb, Ampère and Faraday:1

• Electrodynamic Measurements, First Memoir, relating specially to a General Fundamental Law of Electric Action (Chapter 5, pp.33-203).

In this paper, Weber developed his electric force law, after demonstrating experimentally the validity of André-Marie Ampère’s (1775–1836) law and of Michael Faraday’s (1791-1867) induction. Weber’s law would remain the basis for all of his subsequent research. For more about Ampère’s law, see my post The Development of André-Marie Ampère’s Electric Force Law.

Weber, with Gauss, Studies the Earth’s Magnetism and Invents the Telegraph

When Weber started to work on validating Ampère’s work in 1845, he was already an accomplished researcher. At the precocious age of 21, in 1825, he and his brother Ernst Heinrich (1795-1878) published a 575-page monograph Wellenlehre auf Experimente gegründet (Wave Theory based on Experiments).2

Weber was appointed Professor of Physics at the University of Göttingen in 1831, after having impressed both Alexander von Humboldt (1769-1859) and Carl Gauss (1777-1855) in 1828 at a meeting of the Gesellschaft Deutscher Naturforscher und Arzte (Society of German Naturalists and Physicians) in Berlin. Weber would remain in Göttingen until 1843, despite losing his position in 1837 as one of the Göttingen Seven, who had opposed the revocation of the Hannover constitution by the new king Ernest Augustus (1771-1851), the uncle of Queen Victoria (1819-1901) of Great Britain.

During the period 1831-1843, Weber and Gauss collaborated extensively. Upon a suggestion of Humboldt, they became the center of a pan-European network of researchers measuring Terrestrial magnetism in multiple sites around the world. To support their own research, in 1833, they invented the telegraph, drawing a 3km-long cable across the rooftops of Göttingen between Weber’s university laboratory and Gauss’s university astronomical observatory.

I intend to write more extensively about Weber’s earlier work, along with his collaboration with Gauss, in a future series of posts.

Weber Experimentally Verifies Ampère’s Law

In 1843, Weber was appointed Professor of Physics at the University of Leipzig, taking over the position previously held by Gustav Fechner (1801-1887). These two would continue to collaborate, even though Fechner was losing his sight.

In 1845, Weber and Gauss exchanged some letters [Volume II, Chapter 2], in which Weber stated his intention to study the work of Ampère. In so doing, Weber expressed his skepticism about one of the consequences of Ampère’s force law between two current elements, namely that two collinear current elements, with the currents pointing in the same direction, would repel each other. It must be stated that this skepticism is still alive today, as many (most?) of today’s researchers deny the existence of longitudinal electrical forces.

Ampère had demonstrated the validity of this prediction with his bridge experiment. But Weber felt that the experimental apparatuses used by Ampère were not sufficiently precise to guarantee the validity of the latter’s work. In particular, when examining the equilibria cases from which Ampère derived his force law in his Théorie, it was unclear that the experimentally verified equilibria were actually such, or whether they were the result of friction forces that could not be overcome by weaker electric forces. Gauss’s response was that Ampère’s law was a whole, and it would be inappropriate to simply throw out part of it without validation.

The first part of the 1846 paper by Weber focuses on the design and construction of precise instruments, including his electrodynamometer, built by Moritz Meyerstein (1808-1882) in 1841 in Göttingen. These instruments,

designed for electromagnetic measurement of the intensity of currents and electrodynamic measurement of the interaction of two portions of the circuit, [p.52]

were used to undertake detailed measurements of various phenomena and then to compare these measurements with the relevant predictions using Ampère’s law. After conducting these experiments, Weber’s conclusion was unequivocal:

This complete agreement between the values calculated according to Ampère’s formula and the observed values (namely, the differences never exceed the possible amount contributed by unavoidable observational error) is, under such diverse conditions, a full proof of the truth of Ampère’s law. [p.84, my emphasis]

Weber Studies Faraday’s Induction Experimentally

Weber then moved on to studying induction, first discovered by Faraday in late 1831, in which a current can be induced in a wire A by moving a magnet or another current-carrying wire B towards A. Using the same instruments as used for validating Ampère’s law, Weber conducted experiments similar to those of Faraday, in order to produce numerical results. Faraday’s work had been studied from a qualitative point of view by other researchers, notably by Leopoldo Nobili (1784-1835) and Heinrich Friedrich Emil Lenz (1804-1865). Weber’s intent was to produce quantitative results, which were needed in order to test the electric force law that he was developing.

Weber Derives His Electric Force Law Starting from Three Facts

After undertaking his experiments, Weber moved on to the mathematical work necessary for the creation of an electric force law. The derivations can be easily followed, starting from the certain basic principles, which he stated explicitly:

In order to obtain for this investigation the most reliable possible guideline based on experience, the foundation will be three special facts, which are in part based indirectly on observation, in part contained directly in Ampère’s law, which is confirmed by all measurements.

The first fact is, that two current elements lying in a straight line which coincides with their direction, repel or attract each other, according to whether the electricity flows through them in the same or opposite way.

The second fact is, that two parallel current elements, which form right angles with a line connecting them, attract or repel each other, according to whether the electricity flows through them in the same or opposite way.

The third fact is, that a current element, which lies together with a wire element in a straight line coinciding with the directions of both elements, induces a like- or opposite-directed current in the wire element, according to whether the intensity of its own current decreases or increases. [pp.130-131]

Note that the first fact is precisely about the fact that collinear current elements mutually repel or attract each other, depending on the relative directions of the currents. This was precisely the topic of Weber’s skepticism when he was exchanging correspondence with Gauss. In private communication, André Assis told me that he believes that Weber was convinced by Gauss that Ampère’s law was correct in all aspects.

These three facts are, of course, not directly given through experience, because the effect of one element on another can not be directly observed; yet they are so closely connected with directly observed facts, that they have almost the same validity as the latter. The first two facts were already comprehended under Ampère’s law; the third was added by Faraday’s discovery.

The three adduced facts are considered as electrical, viz., we consider the indicated forces as actions of electrical masses on each other. The electrical law of this interaction is still unknown, however; for, even if the first two facts are comprehended under Ampère’s law, nevertheless, even apart from the third fact, which is not comprehended by it, Ampère’s law is itself, in the strict sense, no electrical law, because it identifies no electrical force, which an electrical mass exerts on the other. Ampère’s law merely provides a way to identify a force acting on the ponderable mass of the conductor. Ampère did not deal with the electrical forces which the electrical fluids flowing through the conductor exert on one another, though he repeatedly expressed the hope that it would be possible to explain the reciprocal effect of the ponderable conductors identified by his law, in terms of the interactions of the electric fluids contained in them. [p.131, my emphasis]

What Weber wrote above is crucial. Ampère’s law described the interactions between bodies through which passed electrical currents. Weber’s intent was to develop a law describing the interactions between what he called electrical masses, the material entities that moved to create an electrical current.

If we now direct our attention to the electrical fluids in the two current elements themselves, we have in them like amounts of positive and negative electricity, which, in each element, are in motion in an opposing fashion. This simultaneous opposite motion of positive and negative electricity, as we are accustomed to assume it in all parts of a linear conducting wire, admittedly can not exist in reality, yet can be viewed for our purposes as an ideal motion, which, in the cases we are considering, where it is simply a matter of actions at a distance, represents the actually occurring motions in relation to all the actions to be taken into account, and thereby has the advantage, of subjecting itself better to calculation. The actually occurring lateral motion through which the particles encountering each other in the conducting wire (which latter forms no mathematical line) avoid each other, must be considered as without influence on the actions at a distance, hence it seems permissible for our purpose, to adhere to the foregoing simple view of the matter…. [p.131]

As did Ampère previously, Weber in 1846 was working with the idea of an electric current consisting of two electrical fluids moving in opposite directions. This is an important point, because he would be criticized in 1867 for this assumption by William Thomson (Baron Kelvin, 1824-1907) and Peter Guthrie Tait (1831-1901) in the first edition of their Treatise on Natural Philosophy. But already in 1852, in his Second Memoir, Weber would change his mind, and fix the positive electric masses and allow the negative electric masses to flow around the positive ones, thereby creating the very first model of an electric atom, six decades before Ernest Rutherford (1871-1937) announced the discovery of the atomic nucleus.

We have, then, in the two current elements we are considering, four interactions of electrical masses to consider, two repulsive, between the two positive and between the two negative masses in the current elements, and two attractive, between the positive mass in the first and the negative mass in the second, and between the negative mass in the first and the positive mass in the second. [p.131]

So, in order to develop his force law, Weber started with Coulomb’s law for electrostatics:

and then considered what needed to be added to produce a law consistent with those three facts that he gave above. After a lengthy discussion, he concluded that the relative motion of the electrical masses would have to be taken into account.

This presupposes, as shown by the first two facts stated above, that the resultant of those four interactions of the electrical masses contained in the two current elements under consideration, which, according to the electrostatic laws, ought to be zero, departs more from zero, the greater the velocity, with which the electrical masses flow through both current elements, that is, the greater the current intensities.

From this it follows, therefore, that the electrostatic laws have no unconditional application to electrical masses which are in motion with respect to one another, but on the contrary, they merely provide for the forces, which these masses reciprocally exert upon each other, a limiting value, to which the true value of these forces approximates more closely, the slighter the reciprocal motions of the masses, and from which, on the contrary, the true value is more divergent, the greater the reciprocal motions. To the values, which the electrostatic laws give for the force exerted by two electrical masses upon one another, must thus be added a complement dependent upon their reciprocal motion, if this force is to be correctly determined, not simply for the case of mutual rest and equilibrium, but universally, including any arbitrary motion of the two masses with respect to one another. This complement, which would confer upon the electrostatic laws a more general applicability than they presently possess, will now be sought. [pp.132-133]

Considering the first fact, about the mutual repulsion or attraction of collinear current elements, Weber concluded something remarkable:

that the electrical masses, which have an opposite motion, act upon one another more weakly, than those which are moved in the same direction.

For, 1) if the direction of the current is the same in the two elements, then repulsion occurs, consequently the attractive force of the unlike masses must be weaker than the repulsive forces of the like masses. In this case, however, it is the unlike masses, which are in opposite motion. If, however, 2) the direction of the current in the two elements is opposite, then attraction occurs; consequently the repulsive forces of the like masses must be weaker than the attractive forces of the unlike masses. In this case, however, it is the like masses, which are put into opposite motion. In both cases it is thus the masses in opposite motion, which act more weakly upon one another, confirming the statement above. [p.133]

And this conclusion leads directly to an addition to Coulomb’s law:

The first fact, to which the statement above was referred, further permits the following, more precise, determination to be added,

that two electrical masses (repulsive or attractive, according to whether they are like or unlike) act more weakly upon one another, the greater the square of their relative velocity.

The relative velocity of two electrical masses can, if r denotes the distance between the two masses, be expressed as dr/dt, and is positive or negative, according to whether the two masses are withdrawing from or approaching one another; since, however, this difference between approach and withdrawal, or, in short, the difference of the sign for dr/dt, has no influence upon the magnitude of the force, it was necessary in the just-stated rule to introduce, instead of the relative velocity itself, its square. [p.133]

This leads to the following force law, where a is some constant:

Weber then moved on to the second fact, which refers to the mutual attraction or repulsion of current elements in parallel wires. In this case, there will always be a moment in which the relative velocity is zero, yet the attraction or repulsion is real; therefore it will be the relative acceleration which will be relevant:

If we now proceed to the second fact stated above, it is clear that the supplement to the electrostatic law just provided will no longer suffice here, because for all cases included under this second fact, it yields the value of the relative velocity of the electrical masses dr/dt = 0. That is to say, if we follow two electrical particles in their paths, the result is that their relative distance decreases up to the moment in question, and from then on increases again, and therefore, at the moment in question itself, neither increase nor decrease in the distance takes place; consequently, for all these cases, the electrostatic law itself, would be brought into application in order to determine the four interactions of the electric masses in both current elements, without applying a supplement to the law, according to which the two current elements ought to have no effect at all upon one another, which is not the case.

It is easily proven, however, that for this second class of cases, where the value of the relative velocity dr/dt disappears, the value of the relative acceleration d²r/dt² stands out all the more significantly, while for the first class, where the latter value d²r/dt² disappears, the first dr/dt stood out all the more significantly. [p.135]

This leads to the following force law, where a and b are constants:

Now, working from Ampère’s law, Weber was able to deduce that b = 2ra², yielding the following force law:

So what about the third fact? Well, it turned out that it could be used as a test of the validity of the derived force law:

Thus we are led by this third fact, as well, to add to the electrical forces determined by the electrostatic law, a supplement contingent upon their motion, and the question is merely, whether this justifies exactly the same supplement, as that which was established on the basis of the first two facts. This third fact therefore yields a criterion for testing the results already obtained, and is especially suited to their rejection or their firmer substantiation. [p.139]

And the answer is clear:

From this it follows, that this third fact confirms the result derived from the first two, in that the same complement of the electrostatic law into a general law, which served to explain the first two facts, also suffices to explain the third. [p.141]

After developing his force law starting from the first two facts, Weber began to explore the implications of the law. First, he demonstrated that the same law could also be derived directly from Ampère’s law. Second, he derived the laws of induction, and demonstrated that these were consistent with the detailed experimentation that he had previously undertaken.

Weber Discusses the Philosophical Implications of His Law

When carefully reading Weber’s paper, it becomes clear that he was very much aware of the philosophical implications of his work. Below, I will focus on three points.

First, as will be seen in future posts, Weber was the target of vicious attacks from Hermann von Helmholtz (1821-1894) and William Thomson. One of the key points being argued by Helmholtz, starting in 1847, and right to his death bed in 1894, was that Weber’s law violated the principle of conservation of energy, because the law depended on time and relative velocity between charges, not just distance.

It is interesting that Weber himself preëmpted—in 1846—discussion on this topic of his law depending on time and velocity:

The law of the dependence of forces upon given physical relationships is called the fundamental law of physics, and, in accordance with the goals of physics, it is not supposed to provide an explanation of the forces based on their true causes, but only a clearly demonstrated and useful general method for quantitative determination of forces, according to the fundamental measures established in physics for space and time. Hence, from the standpoint of physics, one can not take offense at the fact that a force is made into a function of a relationship dependent on time, any more than one can take offense at the fact that it is made into a function of distance, because a relationship dependent on time is just as measurable a magnitude as a distance; therefore, in virtue of their nature, both are suited to more rigorous quantitative determination, even if it is not appropriate to seek in them the inherent reason for a force.

At most, accordingly, against the introduction of a time-dependent relationship in the general expression for a force, the analogy with another fundamental law of physics, e.g. with the law of gravitation, may be asserted, where this time-dependent relationship does not occur. Yet such an analogy can only be viewed as binding, when it offers ways and means to achieve the goal; where the analogy with known cases does not suffice, in the nature of the case new paths must be sought.

If, therefore, the introduction of such time-dependent relationships in the general expression for a force cannot be rejected in general, then all the less so, if those relationships are an essential part of the complete determination of the existing condition of masses acting upon one another, since in any case the force, which two masses exert upon one another, since it does not always remain the same, must be thought of subject to the condition existing at the time. Complete determination of the present condition of two masses, however, essentially involves, in addition to the determination of their relative position by means of their mutual distance r, the determination of their relative movement by means of their relative velocity dr/dt. For, according to the principle of inertia, one has no choice but to calculate the velocity of a body essentially in its present condition, because the reason for the inertia lies, according to that principle, in the body itself, and consequently the persistence in different motion must correspond to different internal conditions of the body, which, themselves inaccessible to our observation, can only be distinguished by means of their effects emerging over time. [pp.142-143, my emphasis]

It should also be noted that already in 1848, in a shorter version of this First Memoir, Weber would introduce a potential from which could be derived his force law, showing that the criticism from Helmholtz was invalid.

Second, while deriving his force law, Weber presupposed that the negative and positive electrical masses had the same effective mass. However, he already considered the possibility that this might not actually be true:

By a galvanic current, as opposed to other electrical motions not comprised under this name, should be understood a motion of the electricity in a closed conductor, such that the same amounts of positive and negative electricity flow through all its cross-sections simultaneously in the opposite directions. This equality of the positive and negative electricity flowing through does not necessarily presuppose the equality of the moving positive and negative masses, which was previously assumed, but rather, it can exist even when the latter are of unequal magnitudes, if the larger mass flows slower, the smaller one faster. In a galvanic current of the latter kind, new electromotive forces arise from the interaction of the elements, by means of which forces the unequal relationship of the original electromotive forces can be equalized. For as soon as the positive amount of electricity in an element is not equal to the negative, viz., as soon as the element, because of an excess of one electricity, is charged with free electricity, this free electricity itself, in accordance with the laws of the excitation of electricity by means of separation, becomes a source of electromotive forces for all other elements, which, through intensifying that charge, can be increased such that, added to the original electromotive forces, they become proportional to the resistance in all elements, for which, in the galvanic circuits with which we are familiar, a very low degree of electrical charge suffices. [pp.192-193]

Weber would further this idea, ultimately developing his planetary model of an electric atom, first in 1852 in his Second Memoir, then in his work of the 1870s and the 1880s, posthumously published in his Eighth Memoir in 1894.

Third, Weber, like Ampère, did consider the possibility of a medium, which could play either the rôle of subtly influencing the interactions between charged particles, in ways which up to that time could not be detected experimentally, or as the mechanism through which the action-at-distance laws were physically realized:

Another still undecided question is, however, whether the knowledge of the transmitting medium, even if it is not necessary for the determination of forces, would nevertheless be useful. That is, the general rule for determination of forces could perhaps be expressed still more simply, when the transmitting medium were taken into consideration, than was otherwise possible in the fundamental electrical law presented here. However, investigation of the transmitting medium, which perhaps would elucidate many other things as well, is itself necessary in order to decide this question.

The idea of the existence of such a transmitting medium is already found in the idea of the all-pervasive neutral electrical fluid, and even if this neutral fluid, apart from conductors, has up to now almost entirely evaded the physicists’ observations, nevertheless there is now hope that we can succeed in gaining more direct elucidation of this all-pervasive fluid in several new ways. Perhaps in other bodies, apart from conductors, no currents appear, but only vibrations, which can be observed more precisely for the first time with the methods discussed…. Further, I need only recall Faraday’s latest discovery of the influence of electrical currents on light vibrations, which make it not improbable, that the all-pervasive neutral electrical medium is itself that all-pervasive ether, which creates and propagates light vibrations, or that at least the two are so intimately interconnected, that observations of light vibrations may be able to explain the behavior of the neutral electrical medium.

Ampère has already called attention to the possibility of an indirect action of electrical masses on each other, as cited in the introduction …,

namely, according to which, the electrodynamic phenomena

would be ascribed

to the motions communicated to the ether by electrical currents.

Ampère himself, however, pronounced the examination of this possibility an extraordinarily difficult investigation, which he would have no time to undertake. [pp.202-203]

In my next post, I will examine some of the key findings made by Weber and his fellow researchers, based on the groundbreaking results presented in this First Memoir.

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• In Three Remarkable Weeks, André-Marie Ampère Discovered the Fundamental Principles of Electrodynamics

  After hearing of Hans Christian Ørsted’s (1777-1851) experiment demonstrating that a current-carrying wire deviates a magnetized needle, Ampère intuited that magnetism is a phenomenon resulting from the interaction of electric circuits. He proceeded to develop a number of experiments, ultimately demonstrating that two parallel current-carrying wires are mutually attracted if their currents are in the same direction, and mutually repelled if their currents are in opposite directions.

• The Fruitful Collaboration of André-Marie Ampère and Augustin-Jean Fresnel

  Ampère and Fresnel (1788-1827) discussed physics extensively for several years. It turns out that Fresnel had told Ampère that his idea of macroscopic electric currents could not explain magnetic phenomena, since macroscopic currents have a tendency to heat up the objects through which they pass. Instead, Fresnel suggested that microscopic currents, now called molecular currents, should suffice. Meanwhile, it was Ampère who suggested to Fresnel that light travels as a transverse wave through the aether.

• The Development of André-Marie Ampère’s Electric Force Law

  Once Ampère understood that the key to magnetism was the interaction of electric currents, and their mutual attraction and repulsion, he began to work in October 1820 on a force law for electric currents. This was defined as a force between current elements, i.e., infinitesimal components of currents, acting along the straight line between the two elements, taking into account the angles between the current element directions and that line, with an inverse-square relation with respect to the distance therebetween. The force law was completed in June 1822.

In this post, I will present the opposition to Ampère’s work that came up, not only during the period 1820-1826 when he was working on electrodynamics, but also long after he passed away. In particular, I will argue that Ampère’s work was “repackaged” to be compatible with later theories, often into the exact opposite of what he had proposed when he was alive. As in my first post, my primary source of information is the book by A.K.T. Assis and J.P.M.C. Chaib, Ampère’s electrodynamics: analysis of the meaning and evolution of Ampère’s force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience,1 which was translated by the authors themselves from the original Portuguese into English.

Initial Opposition: The Refusal to Accept that Magnetism is Caused by the Interaction of Electric Currents

The initial opposition to Ampère’s work on electricity and magnetism arose as he was conducting his very first experiments, from other researchers themselves conducting experiments in these domains: Ørsted, the pair Jean-Baptiste Biot (1774-1862) and Félix Savart (1791-1841), and Michael Faraday (1791-1867). Although their approaches differed, they all concurred, in opposition to Ampère, that magnetism could not be explained simply as a result of the interaction of electric currents.

That is, to Ampère magnetism could be explained as the result of the interaction of electric currents. Ørsted, Biot, Savart and Faraday, on the other hand, believed in the existence of magnetic poles and tried to explain the experiments conducted by Ørsted and others through the interaction between magnetic poles (Biot, Savart and several others), through the interaction between a charge and a current element (Ørsted), or through the interaction between a magnetic pole and a current element (Faraday).

As I wrote in my first post, Ampère soon wrote after hearing of Ørsted’s experiment:

I reflected that since the order in which two facts are discovered in no way affects any conclusions which can be drawn from the analogies they present, it might, before we knew that a magnetized needle points constantly from South to North, have first been known that a magnetized needle has the property of being influenced by an electric current into a position perpendicular to the current… [1820, p.55, my emphasis]

Clearly, this conclusion of Ampère’s was not accepted by many—perhaps most—of his colleagues. So his opponents came up with various artifices to explain both Ørsted’s experiment, as well as some of Ampère’s discoveries.

1. Ørsted’s understanding changed over time, but the theme of electricity flowing around wires in circles or spirals recurred, as did the physical existence of magnetic poles. In his original 1820 paper, he wrote:

All the effects on the north pole above-mentioned are easily understood by supposing that negative electricity moves in a spiral line bent towards the right, and propels the north pole, but does not act on the south pole. The effects on the south pole are explained in a similar manner, if we ascribe to positive electricity a contrary motion and power of acting on the south pole, but not upon the north. [p.202, my emphasis]

In 1821, he wrote:

When opposite electrical powers meet under circumstances which offer resistance, they are subjected to a new form of action, and in this state they act upon the magnetic needle in such a manner that positive electricity repels the south, and attracts the north pole of the compass: and negative electricity repels the north, and attracts the south pole; but the direction followed by the electrical powers in this state is not a right line, but a spiral one, turning from left hand to the right. [p.203, my emphasis]

According to Assis and Chaib, Ørsted did not accept “The supposition of a direct interaction between two current-carrying conductors [demonstrated by Ampère], without the intermediation of a flux of charges circulating around the wire.” [p.204].

2. Biot and Savart proposed, on the other hand, that somehow magnetism was primary over electricity. They suggested that when a current flowed through a wire, it became magnetic. In 1820, they wrote:

By these procedures MM. Biot and Savart arrived at the following result which rigorously represents the action experienced by a molecule of austral or boreal magnetism when placed at some distance from a fine and indefinite cylindrical wire which is made magnetic by voltaic current. Drawing a perpendicular to the axis of the wire from the point where the magnetic molecule resides, the force influencing the molecule is perpendicular to this line and to the axis of the wire. [p.207, my emphasis]

Similarly, in 1821, Biot wrote:

However, what nobody had supposed, before M. Oersted made the observation, is that the electric current has another power. When it flows along metal bodies of any nature, it gives them momentarily a magnetic virtue; they are then able to attract soft and non magnetized iron. When one presents to them a steel magnetized needle, they attract one of its poles and repel the other, but only according to the parts of their surfaces which it faces. At last, what completes the characteristic of a magnetic action, they do not act at all on silver or copper needles, but act only on substances which are capable of magnetization. [p.207, my emphasis]

Assis and Chaib write:

According to Biot and Savart, the interaction between current-carrying wires [discovered by Ampère] was a secondary effect arising from something more basic, namely, the magnetization of these wires due to the passage of the electric current. Accordingly, the interaction between two current-carrying wires would be due to the supposed magnetic poles spread over the cross section of one wire interacting with the magnetic poles spread over the cross section of the second wire. [p.208, my emphasis]

Furthermore, Biot and Savart opposed Ampère’s force law, developed in the Newtonian tradition, by claiming to be the only true heirs of Newton, rejecting the force law because it took into account the directions of the current elements. All previous Newtonian laws were between point-like objects. Assis and Chaib write:

Biot and Savart rejected not only Ampère’s conception of a force between current elements depending on the spatial orientations of these elements, but also the more basic supposition of the existence of current elements which were spatially orientated. [p.208, my emphasis]

3. Faraday, like Biot and Savart, supposed that magnetism was primary over electricity. However, for him Ampère’s experiments did not demonstrate any attraction or repulsion between current-carrying wires, but, rather, were the result of compound actions derived from revolving entities:

Several important conclusions flow from these facts; such as that there is no attraction between the wire and either pole of a magnet; that the wire ought to revolve round a magnetic pole, and a magnetic pole round the wire; that both attraction and repulsion of connecting wires, and probably magnets, are compound actions; [p.211, my emphasis]

Furthermore, Faraday even denied that an electric current in a wire consisted of the movement of fluids or particles within said wire:

Those who consider electricity as a fluid, or as two fluids, conceive that a current or currents of electricity are passing through the wire during the whole time it forms the connection between the poles of an active [voltaic] apparatus. There are many arguments in favour of the materiality of electricity, and but few against it; but still it is only a supposition; and it will be as well to remember, while pursuing the subject of electro-magnetism, that we have no proof of the materiality of electricity, or of the existence of any current through the wire. [pp.212-213, my emphasis]

This last sentence demonstrates the anti-materialist bent taken by Faraday, which would be taken up as well by James Clerk Maxwell (1831-1879) and William Thomson (Baron Kelvin, 1824-1907). Here is what Maxwell wrote five decades later:2

It appears to me, however, that while we derive great advantage from the recognition of the many analogies between the electric current and a current of material fluid, we must carefully avoid making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as measured in feet per second. [Maxwell, §574, p.218, my emphasis]

This idea had previously been expressed by Thomson and Peter Guthrie Tait (1831-1901), who wrote in the first edition (1867) of their Treatise on Natural Philosophy,3 while attacking Wilhelm Eduard Weber (1804-1891), that:

it is impossible to conceive that the theory of two electrical fluids can be true [Thomson and Tait, §385, p.312, my emphasis]

It is important to remember that Ampère, despite teaching in two prestigious institutions, the École Polytechnique and the Collège de France, never had full institutional support, unlike Biot, who became director of the Bureau des Longitudes, or Faraday, who became director of the Royal Institution. Ampère literally had to pay out of his own pocket for the creation of the instruments necessary for his experiments. This imbalance meant that the opposition put up against Ampère would have lasting repercussions.

Ampère’s Rebuttal: Newton’s Third Law is Fundamental

Ampère carefully responded to all of these attacks. His most important argument was that his work was consistent with the work of Newton, and in particular that only his approach respected Newton’s Third Law, which states that for every action, there is an equal and opposite reaction. As I quoted in a previous post Descartes’s Vortices Keep Resurfacing, Ampère wrote in 1823 to Paul Erman (1764-1851):

According to a passage in your letter, I believe there is a certain obscurity in Germany as regards this aspect of my theory. By seeing the planets revolving around the Sun, Cartesian physics imagines that they are pushed in the sense in which they move by vortices revolving in the same sense; when it [the Cartesian physics] sees a pole of a magnet carried to the right and the other [pole carried] to the left of a conducting wire, it supposes a vortex independent of the wire. Newtonian physics explains all celestial phenomena by an attraction directed along the straight line connecting the two interacting particles and the motion is a complicated result of this attraction. As regards the new [electromagnetic and electrodynamic] phenomena I make what Newton has done for celestial motions, [namely,] I explain them by attractive and repulsive forces. And the direction of the electric current, which follows a straight line when the conductor is rectilinear, is only introduced in the calculation due to the fact that the mathematical value of the exerted forces depends, as you can see in my Recueil, on the angles formed by this direction with the direction of another current and with the direction of the straight line connecting the particles which are in mutual action. [p.255, my emphasis]

Ampère referred explicitly to the Third Law in his Théorie:

The physicists who adopted this opinion are then forced to admit a really elementary action, consisting in two equal forces acting in opposite senses along parallel straight lines, forming in this way a primitive couple, which cannot be reduced in terms of forces for which the action and reaction are opposite along the same straight line. I always considered this hypothesis of the primitive couples as absolutely contrary to the first laws of mechanics, among which one should count, with Newton, the equality of action and reaction in opposite senses along the same straight line. [p.261, my emphasis]

But Ampère did not just presuppose the Third Law; he even proved it in one of the latter parts of his Théorie. He wrote:

one has a direct proof of the principle which rests a part of the explanations that I have given, namely: that the portions of currents which traverse the magnet do not act in any manner on it, because the forces which would result from their action on the currents proper to the magnet, or on those that one calls the magnetic molecules, by occurring between the particles of the same rigid body, are necessarily destroyed by an equal and opposite reaction.

I confess that this experimental proof of a principle which is nothing else but a necessary consequence of the first laws of mechanics, appears to me completely useless, as it should have been clear to all the physicists who considered this principle one of the foundations of science. I would not have made this observation, if it had not been assumed [by others] that the mutual action of one element of a conducting wire and of a magnetic molecule, consisted in a primitive couple composed of two forces equal and parallel without being directly opposed, by virtue of which a portion of current which is located inside a magnet might move it; [this] supposition is contrary to the principle which is being discussed here, and is denied by the previous experiment from which there is no action exerted on the magnet by the portions of currents which traverse it when it is not covered by an insulating envelope, since the movement which takes place in this case stays the same if one prevents the currents from traversing the magnet, by enclosing it in this [insulating] envelope. [p.448, my emphasis]

Grassmann’s Ideological Opposition to Ampère’s Work

Ampère’s defense of the Third Law, and his proof thereof in his Théorie, is crucial, for in 1845, Hermann Günther Grassmann (1809-1877) published a paper in Poggendorff’s Annalen, now known as Annalen der Physik, entitled Neue Theorie der Electrodynamik4 [New theory of electrodynamics]. Therein, Grassmann attacks Ampère purely on ideological grounds:

When I submitted the explanation offered by Ampère for the interaction of two infinitely small current-sections on one another to a more exacting analysis, this explanation seemed to me a highly improbable one; […]

(2) Ampère was obliged, therefore, in order to obtain his formula, to use an arbitrary assumption together with the experimental results. The assumption used for this purpose is, at first glance, very simple and natural, consisting in the supposition that two infinitely small circuit elements exert forces on each other along the straight line connecting their midpoints, either of attraction or repulsion […]

3) The complicated form of this formula arouses suspicion, and the suspicion is heightened when an attempt is made to apply it. […] This is such an unlikely result, that the principle from which it is derived must come under the gravest suspicion and with it the supposition that the force in question must show an analogy with all other forces. [pp.219-220, my emphasis]

A “highly improbable explanation”, based on “an arbitrary assumption”, presented in “a complicated form” which “arouses suspicion”! Hard to be more negative!

And what did Grassmann propose instead? A force law in which the force exerted by current element I ds on current element I' ds' is not equal to the force exerted by current element I' ds' on current element I ds! Simply unbelievable!

In the words of Chaib5:

Grassmann, then, proposes the intrinsic torque between current-carrying elements and states: “It will therefore be my task to derive the new explanation, and that of an experienced physicist to test it experimentally.” Thus, he employed a principle “against arbitrariness” to disqualify Ampère’s force without a phenomenon to support his choice and without an epistemological discussion explaining why introducing intrinsic torque is “less arbitrary.” To this day, no one has done so. [Chaib, p.22, my emphasis]

The Repackaging of Ampère by Maxwell and Oliver Heaviside

Among those who opposed Ampère, some offered high praise at different times in their careers. For example, in 1831, after his discovery of induction, Faraday wrote in a letter to Phillips, dated 29 November 1831:

Then I found that magnets would induce just like voltaic currents, and by bringing helices and wires and jackets up to the poles of magnets, electrical currents were produced in them; these currents being able to deflect the galvanometer, or to make, by means of the helix, magnetic needles, or in one case even to give a spark. Hence the evolution of electricity from magnetism. The currents were not permanent. They ceased the moment the wires ceased to approach the magnet, because the new and apparently quiescent state was assumed, just as in the case of the induction of currents. But when the magnet was removed, and its induction therefore ceased, the return currents appeared as before. These two kinds of induction I have distinguished by the terms volta-electric and magneto-electric induction. Their identity of action and results is, I think, a very powerful proof of M. Ampère’s theory of magnetism. [p.213, my emphasis]

Similarly, Maxwell highly praised Ampère:

The experimental investigation by which Ampère established the laws of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the ‘Newton of electricity.’ It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics. [Maxwell, §528, p.175, my emphasis]

But immediately after this praise, Maxwell wrote:

The method of Ampère, however, though cast into an inductive form, does not allow us to trace the formation of the ideas which guided it. We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shewn us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it. [Maxwell, §528, pp.175-176, my emphasis]

So, right after calling Ampère the “Newton of electricity”, Maxwell leads us to believe that Ampère was hiding something from us.

Arguably, it is Maxwell who is hiding something from us. First, as Assis and Chaib amply demonstrated in Part II (Origins and Evolution of Ampère’s Force between Current Elements) and Part III (The Last Period of Ampère’s Electrodynamic Researches), covering some 146 pages, of their book, Ampère clearly presented the development of his ideas, and the clarifications undertaken in his experiments. Second, as I wrote in my post on the development of the force law, the Théorie was organized as a deductive work, starting from four equilibrium cases, which had already been demonstrated through a series of experiments.

In his paper “On Action at a Distance”,6 Maxwell offered another backhanded compliment to Ampère, discussing Ampère’s work after Ørsted announced his experimental results.

But Ampère, by a combination of mathematical skill with experimental ingenuity, first proved that two electric currents act on one another, and then analysed this action into the resultant of a system of push-and-pull forces between the elementary parts of these currents.

The formula of Ampère, however, is of extreme complexity, as compared with Newton’s law of gravitation, and many attempts have been made to resolve it into something of greater apparent simplicity. [Maxwell, p.317, my emphasis]

[From my post Michael Faraday and James Clerk Maxwell on Action-at-a-Distance, in which I demonstrated that each of Faraday and Maxwell actually accepted action-at-a-distance, simply moving it to a finer structure of matter.]

Once again, as did Grassmann, Maxwell claimed that Ampère’s law is complicated….

So the repackaging of Ampère began with Maxwell’s rhetoric of praising Ampère while subtly throwing jabs in his direction.

But that is not all, for in 1855, Maxwell introduced what is today called “Ampère’s circuital law”, which states that

The line integral of a magnetic field around any closed loop is equal to the permeability of free space (μ₀) multiplied by the total electric current enclosed by that loop.

But this has nothing to do with Ampère. The very notion of magnetic field was introduced by Faraday in 1845, several years after Ampère’s death!

So where did this substitution come from? Here is where Heaviside (1850-1925) steps in. He wrote in 1888:7

It has been stated, on no less authority than that of the great Maxwell [§528] that Ampère’s law of force between a pair of current-elements is the cardinal formula of electrodynamics. If so, should we not be always using it? Do we ever use it? Did Maxwell, in his treatise? Surely there is some mistake. I do not in the least mean to rob Ampère of the credit of being the father of electrodynamics; I would only transfer the name of cardinal formula to another due to him, expressing the mechanical force on an element of a conductor supporting current in any magnetic field; the vector product of current and induction. There is something real about it; it is not like his force between a pair of unclosed elements; it is fundamental; and, as everybody knows, it is in continual use, either actually or virtually (through electromotive force) both by theorists and practicians. [Heaviside, pp.501-502, my emphasis]

Since Ampère and Faraday had demonstrated mutual respect for each other when they were working in the 1820s, and given the praise of Ampère offered by Faraday in 1831 after the latter’s discovery of induction, it was not possible to erase Ampère, as would be done for Wilhelm Weber. Instead, Ampère was repackaged so that both his method and his results could be safely hidden, by pretending that his work had been superseded, in order to avoid that later generations actually read his papers.

In coming posts, I will be writing several posts about the work of Wilhelm Weber, and of its erasure, organized by William Thomson and Hermann von Helmholtz.

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• In Three Remarkable Weeks, André-Marie Ampère Discovered the Fundamental Principles of Electrodynamics

  After hearing of Hans Christian Ørsted’s (1777-1851) experiment demonstrating that a current-carrying wire deviates a magnetized needle, Ampère intuited that magnetism is a phenomenon resulting from the interaction of electric circuits. He proceeded to develop a number of experiments, ultimately demonstrating that two parallel current-carrying wires are mutually attracted if their currents are in the same direction, and mutually repelled if their currents are in opposite directions.

• The Fruitful Collaboration of André-Marie Ampère and Augustin-Jean Fresnel

  Ampère and Fresnel (1788-1827) discussed physics extensively for several years. It turns out that Fresnel had told Ampère that his idea of macroscopic electric currents could not explain magnetic phenomena, since macroscopic currents have a tendency to heat up the objects through which they pass. Instead, Fresnel suggested that microscopic currents, now called molecular currents, should suffice.

In this post, I will give an overview of the steps leading to the development and publication of Ampère’s force law. It is of course, important to restate that the force law should not be confused with the so-called “Ampère’s circuital law”, proposed by James Clerk Maxwell (1831-1879) in 1855. As in my first post, my primary source of information is the book by A.K.T. Assis and J.P.M.C. Chaib, Ampère’s electrodynamics: analysis of the meaning and evolution of Ampère’s force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience,1 which was translated by the authors themselves from the original Portuguese into English.

I first wrote about the force law in Two Productive Research Programs in the Early 19th Century, the first part of a three-part series on the replacement of Continental physics in the 19th century. Therein, I presented the force law as being a part of a series of laws—with similar structures—developed successively by Isaac Newton (1643-1727, gravitation), Charles-Augustin de Coulomb (1736-1806, electrostatics), Ampère (electrodynamics) and Wilhelm Eduard Weber (1804-1891, electric charges in motion).

The First Attempts at a Force Law (October 1820 - January 1821)

Following the initial three weeks of my first post, Ampère continued with his experiments. By early October 1820, he had been able to demonstrate the following phenomena:

1. Forces of attraction and repulsion between a rectilinear wire carrying a steady current and a magnet.

2. Attraction and repulsion between planar spirals carrying steady currents.

3. Attraction and repulsion between straight conductors carrying steady currents.

4. Torque exerted by a current-carrying loop and acting on another current-carrying loop.

5. Torque exerted by the Earth on a current-carrying loop.

6. He discovered that an electric current also flows inside a battery, and obtained the direction of this current. These were important results in order to establish the concept of a closed circuit or the concept of a current following a closed path.

7. He reproduced electrodynamically the torques and forces between interacting magnets. To this end he replaced the magnets by planar spirals or by helices coiled around glass tubes.

8. He reproduced the directive action of the Earth, that is, the orientation of a magnetic needle along the magnetic meridian due to the magnetic influence of the Earth. To this end he replaced the compass needle by a large circular loop carrying a steady current.

9. He reproduced Ørsted’s original experiment electrodynamically. To this end he replaced Ørsted’s magnetic needle by a mobile loop carrying a steady current. [p.72]

Once this initial experimentation was completed, Ampère had a qualitative understanding of how electrical currents in wires interacted. It seems that in the second of week of October 1820 [p.74], he started to look at the quantitative aspects. Ampère’s basic assumptions were as follows:

(A) The force between two finite conductors could be obtained by the integration over each circuit of a supposed infinitesimal force between two current elements.

(B) This infinitesimal force was assumed to act along the straight line connecting the centers of these current elements.

(C) This force should comply with Newton’s action and reaction law.

(D) He also initially assumed, in analogy with the other known forces of gravitation, electrostatics and magnetostatics, that this force should vary with the inverse square of the distance between the current elements.

(E) He supposed that this force should be proportional to the product of the intensities of both current elements and also to the product of their infinitesimal lengths, although initially he was not yet clear about the distinction between current intensity and the length of the current element. [p.73]

One of the surprises encountered by Ampère was when he allowed current to pass through two heliacal wires in a parallel arrangement, as seen in the following diagram:

Since the currents of the closest parts of the AB and CD helices had opposite directions, Ampère expected the helices to repel each other; in fact, they attracted one another. This led him to understand that a heliacal current has two components, as seen in this diagram:

Ampère came up with his “addition law”, also known as the “law of sinuous currents”. According to this law, the interaction between those two helices is the sum of the interactions of:

1. the rectilinear current of AB and the rectilinear current of CD;

2. the rectilinear current of AB and the azimuthal current of CD;

3. the azimuthal current of AB and the rectilinear current of CD; and

4. the azimuthal current of AB and the azimuthal current of CD;

and that each of these interactions can be analyzed separately. In the case of this particular experiment, the first interaction dominated, hence the mutual attraction.

When analyzing the interaction between two current elements ds and ds', Ampère worked with three angles:

• α, the angle between the first current element ds and the straight line connecting the two current elements;

• β, the angle between the second current element ds' and the straight line connecting the two current elements; and

• γ, the angle between two planes; for each current element, the corresponding plane has as direction vectors the extrapolation of the direction of the current element and the straight line connecting the two elements.

By December 1820, Ampère had developed a general formula, which, assuming that the intensities of the current elements are g and h, gives:

with n and m unknown natural numbers. The cos α cos β component is called the collinear component. Ampère suspected that k = n/m = 0, but it would be only shown to be -1/2 in June 1822.

Ampère stopped this avenue of research in January 1821, for reasons of illness and fatigue. Furthermore, his assumption that k = 0 led to inconsistencies with respect to experimental results.

The Finalization of the Force Law (October 1821 - June 1822)

In October 1821, Ampère was informed by Michael Faraday (1791-1867) of the development of what is now considered to be the first electric motor.

What followed is equivalent to what followed the announcement of Ørsted’s experiment with the compass-deviating current-carrying wire. Ampère got to work, and devised his own experiments, inspired by Faraday’s motor, all based on the concept of continuous rotation.

• Ampère reproduced Faraday’s motor, using a horseshoe conductor turning around its axis in acidulated water.

  

• Ampère was the first scientist to obtain continuous rotation of a conductor utilizing only Terrestrial magnetism, without the presence of any other magnet.

  By removing the magnets and replacing the mobile conductor by another conductor represented in figure …, M. Ampère observed this conductor, under the action of the Earth, turn slowly and constantly, from East to West through the South. When the austral [South] pole of a magnet was placed below the conductor and close to it, the conductor is made to turn in the opposite direction; returning to its initial motion by removing the magnet. [p.122]

• Ampère was the first scientist to obtain a continuous rotation of a magnet around its axis, a phenomenon Faraday considered improbable. Faraday had written:

  Having succeeded thus far, I endeavoured to make a wire and a magnet revolve on their own axis by preventing the rotation in a circle round them, but have not been able to get the slightest indications that such can be the case; nor does it, on consideration, appear probable. [p.123]

  Here is Ampère’s instrument:

  

  Ampère wrote:

  According to what came before [Ampère had reproduced Faraday’s experiment of continuous rotation], the motion of translation of the magnet takes place whenever it is only traversed by currents entering it from one side and leaving from the other [side]; however, by making all [currents] penetrating the magnet, or making all of them leaving from it, only a motion of rotation of the magnet around itself will result. In order to produce this motion, a motion which I was the first to obtain, mercury is placed on the upper cavity of the cylindrical bar cc′ (figure 8 [our figure 7.8; this cylindrical bar is magnetized, with its North pole in the upper extremity and its South pole in the lower extremity, the magnet kept floating vertically with the help of the counterweight P]), and the wire Z is immersed in it [that is, the wire Z of Ampère’s original figure, our figure 7.8, is immersed in the mercury filled upper cavity of the magnet]. In this configuration all currents diverge from the axis of the magnet towards the copper ring [that is, towards the copper ring HIG of Ampère’s original figure 7, our figure 7.8]. [p.124]

• Finally, Ampère devised an instrument producing continuous rotation utilizing only current-carrying conductors, with no magnets or terrestrial magnetism. This parallels the work he undertook with parallel current-carrying wires, simulating magnetic attraction and repulsion with no magnets, as described in my first post.

It is at this point that things got interesting. Ampère attempted to create continuous rotation using astatic coils, which are not subject to Terrestrial magnetism. The word “astatic” comes from the Greek άστατος (astatos = unstable). However, once the instrument was built, it did not produce his anticipated results:

When everything is so arranged, the mobile conductor … does not turn continuously, as happened with the conductor of figure …, it does not acquire any motion or then it oscillates around a position of stable equilibrium. [p.132]

It turned out that his erroneous prediction followed from his assumption that the value of the constant k = n/m in his force law was zero. Taking as given the equilibrium that he found in this new experiment, he was able to deduce that k = -1/2, leaving the following force law, presented to the Académie on 10 June 1822:

The Coining of the Words Electrostatic and Electrodynamic

It was during this second period of work on electricity that Ampère coined the words “electrostatic” and “electrodynamic”:

The word electromagnetic, given to the phenomena produced by the conducting wire of Volta’s pile, could only describe these phenomena conveniently at the time in which there were known only those phenomena discovered by M. Œrsted between an electric current and a magnet. I believe that I should utilize the denomination electrodynamic, in order to combine in a single word all those phenomena and, especially, to designate those phenomena which I discovered between two voltaic conductors. This name expresses the characteristic property of these phenomena, namely, to be produced by electricity in motion; while the attractions and repulsions known for a long time are the electrostatic phenomena produced by the unequal distribution of electricity at rest in the bodies in which these phenomena are observed. [1822, p.25]

The Théorie (1826)

The final presentation of Ampère’s electric force law would be in his Théorie mathématique des phénomènes électro-dynamiques, uniquement déduite de l’expérience (Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience), which appeared two hundred years ago, in 1826.

The presentation of the force law in the Théorie is not done inductively, but, rather, deductively, taking as starting points four separate equilibrium cases. I intend to write about how the Théorie is structured, when I start to focus on different methodologies for the development of scientific theories, but this will not be in the immediate future. Nevertheless, one can immediately see certain key advantages of working with equilibrium situations: since nothing is in motion, the resulting theories do not depend on measurements of uncertain precision, nor on factors such as friction.

In the next post, I will focus on the fact that, notwithstanding the brilliance of Ampère’s work, both experimental and theoretical, it has not only been opposed by important researchers, but even “repackaged” so that today’s readers are not actually aware of what he actually undertook.

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Copies of this publication were sent to researchers across Europe, and translated into French, English, German and other languages. François Arago (1786-1853) presented Ørsted’s work in Paris to the Académie royale des Sciences on 4 September 1820. Due to generalized disbelief, he repeated this experiment in front of the members of the Académie on 11 September 1820. Among the audience was the mathematician André-Marie Ampère (1775-1836), who, despite already being in his mid-forties, set out to develop an entire new discipline of physics, electrodynamics, a term which he himself would coin in 1822.

In this post, I will present the overview of the phenomenal achievements attained by Ampère during the first three weeks. The theory would only be completed in 1826, with the publication of his masterpiece Théorie mathématique des phénomènes électro-dynamiques, uniquement déduite de l’expérience (Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience), whose development will be presented in upcoming posts. However, the broad lines of that work were already present only three weeks after Ørsted’s paper had been read to the Académie. Two hundred years later, Ampère’s achievements remain unchallenged.

As source material for this post, I will be using Chapter 3 (Ampère’s Initial Experiments) of the book by A.K.T. Assis and J.P.M.C. Chaib, Ampère’s electrodynamics: analysis of the meaning and evolution of Ampère’s force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience,1 which was translated by the authors themselves from the original Portuguese into English. I have also been inspired by the lengthy eulogy2 written by Arago, Ampère’s friend and supporter.

The Initial Insight: Magnetism is an Epiphenomenon of Electric Currents

When Ampère first viewed the experiment, his conclusions were very different from those of his colleagues. Instead of trying to understand the experiment as the result of some interaction between magnetism and electricity, he proposed that everything known up to that point be ignored, and to start from the results of the experiment as initial observations. He stated that the natural conclusion would be that magnetism is a phenomenon caused by electric currents, and that inside a magnet, there must be electric currents provoking the magnetic properties; this conclusion applied equally to the Earth, known to be a magnet since at least William Gilbert (1544-1603); see my summary post https://johnplaice.substack.com/p/my-writings-on-william-gilbert.

When first I wanted to find the causes of the new phenomena discovered by M. Œrsted, I reflected that since the order in which two facts are discovered in no way affects any conclusions which can be drawn from the analogies they present, it might, before we knew that a magnetized needle points constantly from South to North, have first been known that a magnetized needle has the property of being influenced by an electric current into a position perpendicular to the current, in such a way that the austral pole of the magnet is carried to the left of the current, and it could then have subsequently been discovered that the extremity of the needle which is carried to the left of the current points constantly towards the North: would not the simplest idea, and the one which would immediately occur to anyone who wanted to explain the constant direction from South to North, be to postulate an electric current in the Earth in a direction such that the North would be to the left of a man who, lying on its surface facing the needle, received this current in the direction from his feet to his head, and to draw the conclusion that it takes place from East to West in a direction perpendicular to the magnetic meridian? [1820, p.55, my emphasis]

This conclusion was reached almost immediately after Arago’s presentation of Ørsted’s experiment to the Académie.

The Astatic Magnetic Needle

But Ampère went way beyond just speculating. He devised new instruments, which according to Assis, were built by a very young engineer named Hippolyte Pixii (1808-1835) —in 1820, Pixii was only 12 years old! Using these instruments, Ampère devised a number of experiments to test and validate his intentions.

On 18 September 1820, his first instrument presented to the Académie was the astatic magnetic needle, which was designed so that its alignment would be unaffected by Terrestrial magnetism. In other words, whatever the alignment of the astatic needle, it would not realign itself in a North-South (or any other) direction if only subject to the Earth’s magnetism.

Why was this important? Ørsted had argued that the electric current in a wire would deflect a magnetized needle by 45º from its normal North-South alignment. Ampère, on the other hand, argued that the deflection should be 90º, but that an observed 45º could be explained by the fact that the magnetized needle was being simultaneously affected by both the current-carrying wire and by the electric currents in the Earth. With the astatic needle, Terrestrial magnetism could be safely ignored. Lo and behold, when a current-carrying wire was actually placed next to an astatic needle, the latter was deflected 90º.

Then, when a galvanic current is close [to an astatic needle], its directive action will be the only one affecting the needle, and experiment shows that the needle always becomes exactly perpendicular to the direction of the current. [p.58]

The Mutual Attraction or Repulsion of Parallel Electric Currents

So Ampère’s proposal of electric currents as the source of magnetism seemed to be explain the torque forcing magnets to turn. Could his proposal also explain how properly-aligned magnets can mutually attract or repel each other? To address this question, he devised a number of experiments, in which coiled current-carrying wires could simulate magnets, ultimately leading to the conclusion that parallel electric currents mutually attract or repel each other!

The first experiment consisted of demonstrating that a magnetized needle is attracted or repelled by a current-carrying wire.

For the further experiments, Ampère made a momentous discovery, namely that an electric current in a wire attached to the two ends of a battery passes not only through the wire, but also through the battery itself:

At the meetings of the French Academy which took place on 18 and 25 September 1820, Ampère described another very important experiment…. He utilized his galvanoscope…. With this experiment Ampère concluded that in Volta’s original experiment in which the terminals of a battery were connected to a conducting wire, the electric current flowed in a closed circuit. That is, it flows not only along the metal wire, but also inside the battery. In order to reach this conclusion, he placed his galvanoscope not only above or below the wire, as Ørsted had done, but also above the trough battery. When a current was flowing in this circuit, he observed the deflection of the magnetized needle of his galvanoscope placed above the battery. This fact indicated that the current was also flowing inside the battery. He could also ascertain the sense of the current inside the battery by the direction of the deflection of the magnetic needle. He concluded that the current followed a closed path flowing always in the same sense (clockwise, for instance) along the whole circuit composed of the battery and connecting metal wire. [p.60, my emphasis]

He then devised a wire coiled into a spiral that could be used to simulate a magnet:

With this wire connected to a battery, he could simulate a magnet, and test to see if a current-carrying wire could attract or repel a magnet.

Ampère wrote:

Depending on the direction of the current, such a spiral is greatly attracted or repelled by the pole of a magnet which is presented with its axis perpendicular to the plane of the spiral, according as the current of the spiral and of the pole of the magnet flow in the same or opposite directions. [p.64]

So the next step was to replace the second magnet with another spiral current-carrying wire. The experiment looked like this.

Ampère wrote:

In replacing the magnet by another spiral with its current in the same direction, the same attractions and repulsions occur. It is in this way that I discovered that two electric currents attract each other when they flow in the same direction and repel each other in the other case. [p.62, my emphasis]

Ampère’s Article in the Journal de Physique

The French translation of Ørsted’s 4-page article was published in the July 1820 issue, volume XCI, of the Journal de physique, de chimie, d’histoire naturelle et des arts (Journal of Physics, Chemistry, Natural History and the Arts), whose latest entry was dated 29 September 1820. It is interesting to see the last three articles in this issue:

• pp.72-76: M. J.C. Œrsted.

  Expériences sur un effet que le courant de la Pile excite dans l’Aiguille aimantée

  (Experiments on an effect produced by the current from a battery in a magnetised needle).

• pp.76-78: M. Ampère.

  Conclusions d’un Mémoire sur l’Action mutuelle de deux courans électriques, sur celle qui existe entre un courant électrique et un aimant, et celle de deux aimans l’un sur l’autre; lu à l’Académie royale des Sciences le 25 septembre 1820

  (Conclusions from a Thesis on the Mutual Action of Two Electric Currents, on the interaction between an electric current and a magnet, and on the interaction between two magnets; read to the Royal Academy of Sciences on 25 September 1820).

• pp.78-80: M. J.C. Œrsted.

  Nouvelles expériences électro-magnétiques communiquées dans une lettre reçue à Paris, le 29 septembre 1820

  (New electromagnetic experiments reported in a letter received in Paris on 29 September 1820).

The first of the three is Ørsted’s announcement of his famous experiment, and the third is the announcement of further experiments he undertook.

Sandwiched between Ørsted’s two papers is a two-page paper by Ampère stating his initial conclusions. Below is an English-language translation thereof, with my emphasis:

Conclusions from a Thesis on the Mutual Action of Two Electric Currents, on the interaction between an electric current and a magnet, and on the interaction between two magnets; read to the Royal Academy of Sciences on 25 September 1820;

By Mr Ampère.

1. Two electric currents attract one another when they flow in parallel in the same direction; they repel one another when they flow in parallel in opposite directions.

2. It follows that when the metal wires through which they flow can only rotate in parallel planes, each of the two currents tends to bring the other into a position where it is parallel to it and directed in the same direction.

3. These attractions and repulsions are entirely different from ordinary electrical attractions and repulsions.

4. All the phenomena arising from the mutual interaction of an electric current and a magnet, discovered by Mr Œrsted, which I analysed and reduced to two general principles in a previous paper, read to the Academy on 18 September 1820, fall within the law of attraction and repulsion between two electric currents, as it has just been stated, assuming that a magnet is merely a collection of electric currents produced by the interaction of the particles of steel with one another—an interaction analogous to that of the elements in a Voltaic battery—and that these currents move in planes perpendicular to the line joining the two poles of the magnet.

5. When the magnet is in the position it tends to assume due to the action of the Earth, these currents are directed in the opposite direction to that of the apparent motion of the Sun; so that when the magnet is placed in the opposite position, so that those of its poles which face the Earth’s poles are of the same kind as them, the same currents are found to be in the direction of the apparent motion of the Sun.

6. The well-known phenomena observed when two magnets interact with one another are governed by the same law.

7. The same applies to the action exerted by the Earth on a magnet, assuming that electric currents exist in planes perpendicular to the direction of the magnetic needle, and that these currents move from east to west.

8°. There is nothing more at one pole of a magnet than at the other; the only difference between them is that one lies to the left and the other to the right of the electric currents that impart magnetic properties to the steel.

9. When Volta had demonstrated that the two kinds of electricity, positive and negative, at the two ends of the battery attracted and repelled one another according to the same laws as the two kinds of electricity produced by the methods known before him, he had not thereby fully demonstrated the identity of the fluids set in motion by the battery and by friction; but this identity was established as fully as a physical truth can be, when he showed that two bodies, one of which had been electrified by the contact of metals and the other by friction, acted upon one another, in all circumstances, as if they had both been electrified by the pile or by the ordinary electric machine. The same kind of evidence is found here regarding the identity of the attractions and repulsions of electric currents and magnets. I have just demonstrated to the Academy the mutual action of two currents; the phenomena previously known concerning that of two magnets fall under the same law; starting from this similarity, one would merely prove that electric and magnetic fluids are subject to the same laws, as has long been accepted, and the only change to be made to the ordinary theory of magnetisation would be to admit that magnetic attractions and repulsions must not be equated with those resulting from electric voltage, but with those I have observed between two currents. Mr Œrsted’s experiments, in which an electric current produces the same effects on a magnet, further prove that it is the same fluids that are at work in both cases.

The nine points that Ampère made in his short paper laid the basis for the discipline that he would two years later call electrodynamics. And these points were published in the very same issue announcing for the first time in French the results of Ørsted’s experiments. From the first announcement on 4 September 1820 of Ørsted’s first experiment in front of the Académie royale des Sciences to Ampère’s summary dated 25 September 1820 was a mere three weeks!

In the following posts, I will write about Ampère’s development of his force law (not to be confused with the so-called “Ampère’s circuital law”, proposed by James Clerk Maxwell (1831-1879) in 1855). I will also focus on the opposition to Ampère’s ideas, and the debates in which he participated.

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Me complace anunciar que mis traducciones al castellano de los dos volúmenes de Los fundamentos experimentales e históricos de la electricidad, de André Koch Torres Assis, han sido publicadas por Apeiron en Montreal. Los libros se pueden comprar en Amazon o descargar de la página web de André Assis.

• PDF: https://www.ifi.unicamp.br/~assis/Electricidad-Vol-1.pdf

• Amazon: https://www.amazon.com/dp/1987980514

• PDF: https://www.ifi.unicamp.br/~assis/Electricidad-Vol-2.pdf

• Amazon: https://www.amazon.com/dp/1987980530

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• Two Productive Research Programs in the Early 19th Century

• How Two Productive Research Programs Were Consciously Thrown Out

• The Old Productive Research Programs Refuse to Die

When I first wrote those posts, I was working mostly from intuition. Now it is time to move on to the details, and focus on the contributions made by a number of researchers, ultimately leading to the Belle Époque (1871-1914), arguably the height of European civilization, before its collective suicide commonly known as WWI.

The first researcher that I will focus on is Charles-Augustin de Coulomb (1736–1806), with whom we associate Coulomb’s electrostatic and magnetic laws, whose modern forms are presented below.

Each of these laws follows the principle of Isaac Newton (1643-1727) in his law of universal gravitation, namely that there is a force between two objects, along the straight line between the two, and whose intensity falls off with the inverse square of the distance separating the two. What changes is that gravity is uniquely attractive, while for each of the electrostatic and the magnetic laws, the force can be attractive or repulsive.

In this post, I wish to focus on three basic principles which guided Coulomb’s research. For this, I will be referring to Coulomb’s Memoirs on Torsion, Electricity and Magnetism Translated Into English,1 edited and commented by Andre Koch Torres Assis and Louis L. Bucciarelli. This book was presented in one of my earliest posts: English Translations of Coulomb, Ampère and Weber: Electricity and Magnetism Without Fields. I will be focusing on Section 4.4 (Magnetic Researches, pp.42-47).

For these principles, three of Coulomb’s works are relevant:

• 1777: Investigations of the Best Method of Making Magnetized Needles, of Suspending Them, of Ensuring that They are in the True Magnetic Meridian; Finally, of Explaining Their Regular Diurnal Variations (Chapter 5)

• 1793: Seventh Memoir on Electricity and Magnetism: On Magnetism (Chapter 26)

• 1799: Theoretical and Experimental Determination of the Forces that Bring Different Magnetic Needles to Saturation Back to Their Magnetic Meridian (Chapter 28)

Principle 1: Opposition to Cartesian Vortices

In an earlier post, Descartes's Vortices Keep Resurfacing, I wrote that the vortices of René Descartes (1596-1650) keep resurfacing:

So clearly the vortex did not appear with Descartes. Now what happened after the decline of Cartesianism in the eighteenth century? Well, the vortex reappeared in the nineteenth century with the study of magnetism.

At the time, I did not know that this use of the vortex to explain magnetism was a recurring theme from the 17th century, starting from Descartes himself, all the way through the 18th century, with the likes of Leonhard Euler (1707-1783); Étienne François Dutour de Salvert (1711-1789); and Daniel Bernoulli (1700-1782) and Jean Bernoulli, in the latter case unclear whether Jean I (1667-1748) or Jean II (1710-1790); all writing treatises supporting the vortex idea.

Coulomb, like many French researchers in the second half of the 18th century, supported the principles of Newton, and explicitly opposed the Cartesian vortices:

Therefore it seems that it follows from experiment, that it is not vortices which produce the different magnetic phenomena; and that, to explain them, we must necessarily resort to attractive and repulsive forces of the nature of those which we are not obliged to make use of to explain the weight of bodies and celestial physics. [1777, p.60, my emphasis]

Principle 2: There Are Two Kinds of Magnetic Fluids

Supporters of the Newtonian tradition argued as to whether there was one magnetic fluid or there were two different magnetic fluids, one boreal (North) and the other austral (South). This debate mirrored the debate for electricity, during which Benjamin Franklin (1706-1790) supported the single-fluid model. For magnetism, the most prominent supporter of the single-fluid model was Franz Aepinus (1724-1802).

Coulomb was a supporter of the two-fluid model. He could not see how there could possibly be repulsion of the south poles of two magnets, based on a mutual lack of the hypothetical single fluid.

It follows from these experiments that, whatever the cause of magnetic phenomena, all these phenomena could be explained and subjected to calculation, assuming in the steel laminae or in their molecules, two magnetic fluids, the parts of each fluid repelling each other in direct proportion to their density, and in inverse proportion to the square of their distance, and attracting the molecules of the other fluid in the same ratio; so that each lamina of iron and steel contains in each molecule, before being magnetized, a sufficient quantity of both fluids to saturate or balance each other, that the two fluids thus combined no longer exert any action on each other. [1799, p.499, my emphasis]

Principle 3: Molecular Magnetic Polarization

If we accept the two-fluid model, then we are left with the question of how these two fluids are spread through a magnet. Suppose that, say, in a bar magnet, all of the boreal fluid is around the North pole and all of the austral fluid is around the South pole. Should the bar magnet be broken in two half-way down its length, we get two magnets. But this should not be possible should that supposition have held to be true.

For Coulomb, therefore, already in 1777, he was considering the possibility of polarizing at each point of a magnet:

Thus, each point of a magnet or of a magnetized bar can be regarded as the pole of a tiny magnet… [1777, p.72]

In his Seventh and last Memoir, he clarified this intuition with this image and the following text:

I believe that we could reconcile the result of the experiments with the calculation, by making some changes to the hypotheses; here is one which appears to be able to explain all the magnetic phenomena of which the preceding tests have given precise measurements. It consists in supposing, in the system of Mr. Œpinus, that the magnetic fluid is enclosed in each molecule or integral part of the magnet or of the steel; that the fluid can be transported from one end of this molecule to the other, which gives each molecule two poles; but that this fluid cannot pass from one molecule to another. Thus, for example, if a magnetized needle were of a very small diameter, or if, Figure 7, each molecule could be regarded as a small needle whose North extremity would be united to the South extremity of the needle which precedes it, only the two extremities n and s of this needle would give signs of magnetism; because it would only be at the two extremities where one of the poles of the molecules would not be in contact with the opposite pole of another molecule. [1793, pp. 479-480]

The focus on providing molecular explanations for physical phenomena that was taken by Coulomb would be very productive. As we will see in future posts, this idea would be extensively developed by André-Marie Ampère (1775-1836) and by Wilhelm Eduard Weber (1804-1891), who would both work with molecular electric currents. And their opponents would fight bitterly against these materialist ideas.

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I have written a few posts touching on some of these subjects, and will continue to do so. For the time being, in this summary post, I have gathered my existing posts on these questions in order to make them more accessible, both for my readers and myself. In addition, the following two collections contain posts related to this topic:

• My Writings About Action-at-a-Distance

• My Writings on the Infinite Universe

The atomists

Most people today are familiar with the idea of atomism, namely that the material world is made up of atoms, themselves separated by empty space.

• Lucretius's Of the Nature of Things: Atoms and the Void

  Lucretius (died 55BC), who wrote De rerum natura (Of the Nature of Things), is known as one of the most important atomists of the ancient Western world.

• Galileo Galilei Supports Atomism, Part I

  Galileo Galilei (1564-1642) was one of the early supporters of atomism during the Scientific Revolution of the 17th century, writing in support thereof in Il Saggiatore (The Assayer).

• In 1651 the Jesuit Order Banned the Teaching of Modern Science

  The teaching of atomism, along with many other topics, was literally banned by the Jesuit Order in 1651.

The aetherists

Those who oppose atomism are called plenists; they believe that the Universe is a plenum, meaning that all of space is occupied by matter, typically referred to as the aether, since this matter is understood to be very light.

• Descartes's Vortices Keep Resurfacing

  René Descartes (1596-1650) proposed an aether composed of three sets of successively finer corpuscles, which together occupy all of space, and whose primary motion is circular.

• Christiaan Huygens Proposes an Infinite Series of Levels to the Universe

  Christiaan Huygens (1629-1695) , in his Traité de la Lumière (Treatise on Light), in which he first presented the wave theory of light, proposed an aether composed of an infinite series of successively finer particles.

• A Universe Infinite in Scale with No Action-at-a-Distance

  Tom Van Flandern (1940-2009) presented a theory of the Universe with an infinite series of successively finer particles, similar to that of Huygens, the limit of which fills all of space.

The Monads

The monads, first proposed by Gottfried Wilhelm Leibniz (1646-1716) in his Théodicée (Theodicy), evolved through the writings of successive researchers.

• Leibniz’s Monads: Infinitesimals of Consciousness

  Leibniz, in the Théodicée, was wrestling with the question of evil in a Universe which was “the best of all possible worlds”. For him, the ultimate entities in the Universe were monads, which were ideal, not physical.

• Christian Wolff’s Elements Are Not Identical to Leibniz’s Monads

  Christian Wolff (1679-1754) proposed that the physical world was made up of corpuscles, themselves made up of elements, which resembled monads in some ways, but which were physical—rather than ideal—, but with no extent.

• Mikhail Lomonosov Adapts Christian Wolff's Elements to Chemistry

  Mikhail Vasilyevich Lomonosov (Михаил Васильевич Ломоносов, 1711-1765), was a student of Wolff’s who made important contributions to chemistry. Lomonosov used the same terms as Wolff, but defined the elements as having extent.

The Point Atoms

Point atoms are not material atoms but, rather, simply points, separated by empty space, with position and inertia but no extent, and are subject to external forces. They are well suited as the basis for the technical discussion of a plenum.

• Galileo Galilei Supports Atomism, Part II

  In his last work, the Discorsi (Discources), Galileo moved away from the material atoms that he had supported in Il Saggiatore, and wrote in support of point atoms.

• The Point-atoms of Ruđer Josip Bošković: The Key to the Fields of Michael Faraday

  The most important proponent of point atoms was the Jesuit Roger Joseph Boscovich (Ruđer Josip Bošković, 1711-1787), who had a profound influence on Michael Faraday (1791-1867) and William Thomson (Baron Kelvin, 1824-1907).

• How Two Productive Research Programs Were Consciously Thrown Out

  In this second post of a three-part series on the replacement of Continental science by British empirical science in the second half of the 19th century, I refer to writings of Faraday and Thomson in support of Boscovich.

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• Leibniz's Monads: Infinitesimals of Consciousness

• Christian Wolff's Elements Are Not Identical to Leibniz's Monads

The objective of this series was to first understand, in my own terms, what are exactly Gottfried Wilhelm Leibniz’s (1646-1716) monads, and then to follow their evolution through the thinking of successive researchers.

After reading several of Leibniz’s works on this question, my conclusion was that his monads could be considered to be “infinitesimals of consciousness”, i.e., they were ideal concepts, which could help us understand the basis for life and consciousness, but not the material world. In fact, this topic was the subject of extensive correspondence—with no result—with the Jesuit Bartholomaeus Des Basses, who translated Leibniz’s Theodicy from French into Latin. The Theodicy covered such difficult topics as the existence of evil in Leibniz’s “best of all possible worlds”. For my take on the latter, see my post The Best of All Possible Worlds.

Christian Wolff (1679-1754), the most important German philosopher after Leibniz and before Immanuel Kant (1724-1804), moved away from Leibniz’s monads, and came up with his corpuscles, made up of elements. When I looked at the elements, I found them to be “infinitesimals of physicality”, i.e., they tried to explain the basis for the material world. But since they were infinitesimals, they did not have any extent. And, just as Leibniz could not use his monads to explain the material world, Wolff had to hand-wave to try to explain how real objects, which all have extent, could ultimately be made up from his elements, which had none.

One of Wolff’s students in Marburg was Mikhail Vasilyevich Lomonosov (Михайл Васильевич Ломоносов, 1711-1765), the first Russian-born to be a member of the Saint Petersburg Academy of Sciences, and the founder of Moscow State University, now named after him. Lomonosov’s primary scientific interest was chemistry, and he was the first to propose that there is what we now call “conservation of mass” during combustion, although at the time he still believed in the phlogiston theory.

The most readily-available source in English on Lomonosov is the book Russia’s Lomonosov: Chemist, Courtier, Physicist, Poet,1 written by the chemist Boris Nikolaevich Menshutkin (Борис Николаевич Меншуткин, 1874-1938). Lomonosov’s collected works, which cover a wide variety of subjects, including a Russian grammar and poetry, consist of nine volumes, the first four covering scientific subjects. I will be looking at two articles appearing in Volume 12, both referred to by Menshutkin:

• Elementa chimiae mathematicae (Elements of mathematical chemistry, 1741, pp.65-83)

• De particulis physicis insensibilibus corpora naturalia constituentibus, in quibus qualitatum particularium ratio sufficiens continetur (Of the imperceptible physical particles that constitute natural bodies, in which a sufficient account of the particular qualities is contained, pp.279-313)

In the collected works, these articles appear as side-by-side Latin-Russian bilingual editions. I chose to work with the Latin, as I find it easier to read than the Russian.

In the Elements of mathematical chemistry, Lomonosov uses the same terminology as does Wolff:3

38) An element is a part of a body which consists of no other smaller bodies that are different from itself.

39) A corpuscle is a collection of elements into one mass. [my emphasis]

From these definitions, he moved on to define further concepts, such as homogeneous, heterogeneous and mixed bodies. But what is clear is that he had in mind preparing the groundwork, not for general philosophical arguments but, rather, for discussing Chemistry:4

Since in Chemistry things must be demonstrated, they are demonstrated from distinct notions of the thing itself. A distinct notion must be sought in the enumeration of the notes, that is, in the parts of the whole, and thus the parts of the mixture must be known, which since they cannot be known in any way better than if they were viewed separately, which since they are so minute cannot be distinguished in the mixture, and therefore in order to know the mixture they must be separated. But separation supposes that the parts change place and are thus moved. Therefore, to know and demonstrate the truths of Chemistry and Mechanics, there is need of knowledge. [my emphasis]

In the second work, it becomes clear that Lomonosov is focussing on the actual, real, material world, and he clearly differentiates between mathematics and the physical world, and that mathematical and physical division should not be conflated:5

Bodies can be divided into the smallest parts because of the matter of which they are composed. This division can be considered in two ways, namely mathematically and physically. I say that a body is mathematically divided when, in its given extension, parts are assigned at will by calculation. Physically a body is divided when its parts are actually separated from one another. But since the purely mathematical division is determined by arbitrary choice and cannot be demonstrated, that the mathematically assigned parts exist in bodies that are actually separable from one another; for this reason, we will be less concerned with it and will attempt to investigate only the physical division of bodies; nevertheless, where it is permissible, we will mathematically determine the universal qualities of the physical parts. [his emphasis]

The essence of Lomonosov’s approach can be understood by focussing on the passages that he himself emphasized:6

• Physically, bodies are divided into particles of astonishing fineness.

• [T]here are minute, insensible, physically separable particles in bodies.

• The individual particles of physical bodies are extended.

• Individual physical particles are impenetrable, endowed with the force of inertia, capable of motion and rest.

• Therefore, since physical particles, even insensible ones, are extended, impenetrable, endowed with the force of inertia, capable of motion and rest, they are consequently bodies, and therefore all that proceeds from the universal qualities in them can be explained by mechanical rules.

In other words, Lomonosov is stating that studying chemistry must be undertaken by using all of the knowledge we have from mechanics; we have clearly left the world of alchemy. Furthermore, his ultimate particles are physical, and extended. The inherent philosophical problems inherent in the work of Leibniz and Wolff in trying to explain the physical world have been left behind.

The ideas of atomism were always most readily accepted by those—such as Lomonosov—working in chemistry. As we will see in posts over the coming months, there was strong resistance throughout the 19th century to the idea of physical atoms among physicists.

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So I think that with this cursory look at the monads, I can conclude that the late Leibniz tends towards idealism, and considers the material world to be a phenomenon of the ideal world, made up of an eternal, infinite set of living, conscious monads. Given that these monads have no extent, and given that Leibniz developed the calculus through the use of infinitesimals, I think it is appropriate to consider the monads to be infinitesimals of consciousness. It will be interesting to see how Leibniz’s monads influenced the thoughts of successive researchers. [my emphasis]

The most important immediate successor for Leibniz was Christian Wolff (1679-1754). So I set myself to figure out what Wolff’s elements correspond to. It was not obvious, but after reading a number of papers, I realized that the foremost expert on Wolff’s metaphysics was Jean École, the principal editor of Wolff’s collected works. After much searching, I was able to get a copy of École’s La métaphysique de Christian Wolff1 [The Metaphysics of Christian Wolff] from Celler Versandantiquariat in Eicklingen, near Hannover.

Much to my delight, I found that there was therein a inscription for the publishers signed by Jean École himself.

À Monsieur et Madame Olms, en hommage amical et reconnaissant, cet ouvrage qui est le fruit de trente années de travail et le témoignage d’une amitié déjà aussi longue. Jean École.

To Mr and Mrs Olms, as a token of my friendship and gratitude, I dedicate this book, which is the fruit of thirty years’ labour and a testament to a friendship that has already lasted just as long. Jean École.

And now I can start thinking about the text. Wolff, like Leibniz, wrote about a huge variety of topics. In this post, I will only focus on his elements, or simple substances. In École’s book, these are covered in Part IV: The theory of the world in general, Chapter III: The components of the world, Section II: The elements [pp.238-241]. The quoted passages are translated into English from a mixture of École’s French and Wolff’s Latin, with the originals in footnotes.

An element is defined to be a simple substance, which cannot be further subdivided.

1. Wolff refers to these simple substances as ‘elements’, defining the term as follows: ‘the internal principle of bodies, which cannot be reduced to another, nor to the first’, to which he adds the following commentary: ‘We understand here the principle of the existence of bodies, which contains within itself the reason for their possibility…, or from which it is understood why bodies are possible… It is called irreducible because it does not depend on any prior principle’.2

The elements do not have extent:

3. Owing to their simplicity, these elements are not extended; they have neither configuration nor magnitude, do not occupy any space, are not subject to internal motion, and are indivisible.3

They elements are not the atoms either of Epicurus or Leucippus. [I have previously written about the ancient Atomists in my post Lucretius's Of the Nature of Things: Atoms and the Void]:

4. Nor, therefore, can they be confused with Epicurus’ atoms, which, whilst being indivisible due to their hardness, are nevertheless small bodies endowed with a configuration by which they differ from one another; nor with those of Lucretius, who attributes to them a massive simplicity excluding any composition of parts, but does not deny that they form a homogeneous and coherent whole of irreducible particles.4

So Wolff’s elements, like Leibniz’s monads, can be considered to be ‘Atoms of nature’, using Leibniz’s term, as opposed to the material atoms of Epicurus or Leucippus:

If, therefore, one wishes nevertheless to call these elements ‘atoms’, it must be made clear that they are what Leibniz termed ‘atoms of nature’, in order to distinguish them from the material atoms of the two preceding authors.5

But it would be a mistake—as was made by Johann Joachim Lange [1670-1744]—to equate Wolff’s elements and Leibniz’s monads, because the elements are physical, not metaphysical, endowed with their own internal vis activa [living force, now understood as kinetic energy]:

5. Now, it is well known that Leibniz used this expression to refer to his monads, which he regarded as ‘the elements of things’. But the identity of the terms must not be misleading. Wolff, in fact, does not subscribe to monadism. He states quite unequivocally: ‘I leave it to Leibniz to deal with his doctrine of monads… For it matters little to me whether Leibnizian monads are held in high esteem or condemned and rejected’, and he returns to this point on several occasions. Whilst his elements share with Leibnizian monads the characteristic of being perfectly simple substances, they are not, like them, ‘metaphysical points … [which] possess something vital and a kind of perception’, which, moreover, are one and the same as the vis activa primitiva, but rather physical units endowed with a vis activa that is itself physical, with which they are not identical and which are not capable of perception; hence, they could be called physical atoms.6 [my emphasis]

Finally, the elements are not the points of Zeno of Elea [c.490-c.430 BC], since the elements are all different from one another:

6. Wolff’s elements differ not only from the material atoms of the Atomists and Leibniz’s monads, but also from Zenonian points. For whilst the latter are simple entities, they possess no intrinsic determination other than the absence of parts and are mathematical points, all alike. However, apart from regarding them as physical points, Wolff attributes to his elements, amongst other properties, that of being all dissimilar.7

In my post about Leibniz’s monads, I wrote that Leibniz’s understanding of the relation between his monads and the material world was never fully clarified, even leading to an “extensive correspondence [over 500 pages] that took place between Leibniz and the Jesuit Bartholomaeus Des Basses, who translated the Theodicy from French into Latin.”

Similarly, for Wolff, the relationship between his elements with no extent and material bodies that do have extent is not clear:

2. As regards extension, the elements certainly do not possess it, as we have said. But we know that extension consists of the union of different coexisting beings, that is to say, ‘existing independently of one another’. Now, whilst this is not the case with Zenonian points, the elements that unite to form bodies are all dissimilar; otherwise, one could not explain why some constitute one body and others another; furthermore, they possess within themselves the reason for their union; otherwise, this would be a ‘pure chance’; and from this union, therefore, both the extension and the continuity of their aggregates—which are bodies—can arise. In short, extension results from that which is not extended. This may seem paradoxical, but, according to Wolff, it presents no greater difficulty than the emergence of the multitude from unity.8 [my emphasis]

In order for bodies to be capable of doing anything, their elements need to possess internal forces and powers:

3. If the elements are not extended, they must, on the other hand, possess a force of inertia and an active force, a passive power and an active power; otherwise, one could not explain those powers which must be assumed to exist in bodies in order to account for everything that takes place within them. Wolff establishes this for the force of inertia, the active force and the active power, but does not bother to do so for the active power, no doubt because it is evident that it is the primary condition of action, just as the passive power is that of passion.9 [my emphasis]

As for the forces within bodies, these arise as phenomena from the forces within their elements:

However, because we do not perceive clearly how extension arises from them, nor how the forces of bodies result from their force of inertia and their motive force, we perceive both only in a confused manner. And this is why Wolff describes them as phenomena, specifying that by this he means—unlike the Idealists (and he is no doubt referring to Berkeley)—‘that which merely appears to exist has no reality outside the mind’, but, following Goclenius, ‘everything that presents itself to the senses is perceived in a confused manner’. In other words, matter and the motive force of bodies are, for him, what Leibniz called ‘well-founded phenomena’, and he claims that Leibniz regarded them as such.10 [my emphasis]

Because of the forces within the elements, their internal state is constantly changing:

1. Because the active force of the elements, like any force of this kind, consists in a continual tendency towards change, and since nothing within them opposes this tendency owing to their simplicity, their internal state is constantly changing. Now, this change takes place in such a way that the present state of an element is always the cause of the next, and that its successive states form a series within which they are all mutually dependent in terms of their existence; from which it follows that, from the present state of an element, one can infer what its future state will be and what its past state was. Because all elements are dissimilar, the sequence of changes in the state of one element differs from that of another, and the state of one element is at all times different from that of another.11 [my emphasis]

Finally, the elements never exist separately, but combine for form bodies:

2. Elements do not exist in a separate state; they unite to form bodies. Because they possess within themselves the reason for their coexistence, they are all linked to one another within these bodies, and every state of each element stands in relation to those of the other elements that coexist with it within the same body.12 [my emphasis]

From the above discussion, it seems to me that the physical world is of more importance to Wolff than it is to Leibniz. Yet, Wolff’s elements have no extent. I had described Leibniz’s monads as “infinitesimals of consciousness”. For Wolff’s elements, I think the term “infinitesimals of physicality” is appropriate. Each has focused on an integral part of reality, but neither has provided an overarching whole.

In addition to being an important philosopher, Wolff was also the teacher of the Russian polymath Mikhail Vasilyevich Lomonosov (Михаил Васильевич Ломоносов, 1711-1765), founder of Moscow State University, who first proposed a law of mass conservation during combustion. I will soon be looking at the different ideas of atomism of the early chemists, including Robert Boyle (1627-1691), Lomonosov and Antoine Lavoisier (1743-1794).

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Historical posts

• Geoheliocentrism: The Dismissed Theory

  This post focuses on Heraclides Ponticus (c.390 BCE - c.310 BCE), who supported the idea of the diurnal motion of the Earth, and clearly demonstrated that Mercury and Venus revolve around the Sun. At the beginning of the 20th century, John Louis Emil Dreyer (1852–1926) and Pierre Duhem (1861-1916) argued whether Heraclides was a geoheliocentrist (Dreyer) or a heliocentrist (Duhem). My impression is that Duhem “promoted” Heraclides to the status of a heliocentrist, because heliocentrism was considered to be “superior”.

• The Geoheliocentric Systems of Nilakantha Somayaji and Tycho Brahe

  In this post, I focus on the origins of Tycho Brahe’s (1546-1601) geoheliocentric model, published in 1588, which came to him “as if unexpectedly”. This model is identical to that published in 1501 by Nilakantha Somayaji (1444-1544). I give a summary of Nilakantha’s model and explain how it could have arrived in Tycho’s hands.

  

• Johannes Kepler “Disappears” Tycho Brahe’s Principal Critique of Copernicanism

  During the 17th century, Tycho Brahe’s model was widely accepted by astronomers. Nevertheless, the model was pilloried by Christiaan Huygens (1629-1695), who mocked the lack of diurnal motion therein, despite the fact that Tycho’s successor, Christen Sørensen Longomontanus (1562-1647), had added it.

  More importantly, Huygens completely ignored Tycho’s principal criticism of the Copernican model, namely that because of the lack of any visible parallax for the fixed stars as the Earth revolved around the Sun, those same stars would have to be incredibly far away. This problem was “fixed” by Johannes Kepler (1571-1630), who simply declared that “the fixed stars can be represented as mere points” on the outer crystalline sphere.

• Benedetto Castelli and Galileo Galilei Were the First to Observe a Binary Star System

  The first known observation of a binary star system was made by Benedetto Castelli (1577-1643) and Galileo Galilei (1564-1642), who observed the Mizar system in 1617. However, they never published their results, which were only brought to light in 1949 by Umberto Fedele. Likely Castelli and Galilei did not publish this momentous discovery because they understood that it would be contrary to the Copernican model that they were supporting.

Posts about the TYCHOS

In Simon Shack’s TYCHOS system1, Mars and the Sun form a binary system, rotating around a barycentre close to the Earth, and the Earth itself follows a slow 1.6 km/h orbit with a period of 25'344 years. The Tychosium simulator (https://ts.tychos.space), developed by Patrik Holmqvist, allows an exploration of our stellar system from the TYCHOS perspective, allowing detailed predictions and retrodictions.

• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

  In the TYCHOS system, the true mean synodic period (TMSP) for the Moon is 29.22 days, and all of the major components of our stellar system have periods that are simple multiples of the TMSP. Our system can therefore be viewed as some kind of complex clock, with the Moon providing the rhythm for the entire system.

• Are the Mayan Long and Short Periods for Mars Correct?

  The Mayan astronomers kept track of the empirical sidereal intervals (ESI) for Mars, which are “the number of days elapsed between consecutive passages of Mars through a given celestial longitude while in prograde motion”. They found that there would be a sequence of 7 or 8 long ESIs, just over 700 days long, followed by a short ESI, around 546 days long. For the 8th-century data found in the Mayan Dresden Codex, the TYCHOS provides identical retrodictions for that time. The explanation is that every 8th or 9th interval, Mars’s retrograde loop is skipped, hence the short ESI.

• Hamlet’s Mill Advances at 1.6 km/h

  In their 1969 book Hamlet’s Mill: An essay on myth and the frame of time2, Giorgio de Santillana (1902-1974) and Hertha von Dechend (1915-2001) wrote that the precession of the equinoxes can be found in the myths of humanity world-wide, demonstrating an understanding of this phenomenon going back to 5000 BCE, when the Sun was in Gemini. The TYCHOS provides a simple explanation: during the slow 25'344-year orbit of the Earth around our stellar system’s barycentre, the Northern hemisphere “leans out” from the orbit; over time, the polar stars for the Earth change. For this reason, this orbit is called the Polaris-Vega-Polaris (PVP) orbit.

• The Great Inequality of Jupiter and Saturn (four parts):

  Soon after Kepler published his 1627 Rudolphine Tables, it became clear even to him that the orbits of Jupiter and Saturn were not moving as had been computed. Seemingly, in the span of a single human lifetime, it was observed that the orbits of Jupiter and Saturn had sped up or slowed down. This was one of the most important subjects of research of the greatest astronomers of the 17th and 18th centuries, but also of the greatest mathematicians of the 18th century. I wrote a four-part post on this topic.

  Part 1: The English Astronomers

  The English astronomers of the 17th century attempted, without much success, to reconcile the observed movements of Jupiter and Saturn with Kepler’s tables. Ultimately, Edmond Halley (1656-1742) printed in 1719—but only published posthumously in 1752—revised astronomical tables, with “an anomalous acceleration in the mean motion of Jupiter and an anomalous deceleration in the mean motion of Saturn”.

  Part 2: The French Astronomers

  This problem was picked up by Maraldi I (Giacomo Filippo Maraldi, 1665-1729), who first began looking at this problem in 1704. In his 1718 paper, he stated clearly that, since ancient times, Jupiter has been accelerating and Saturn decelerating, but not uniformly. In 1746, both Cassini II (Jacques Cassini, 1677-1756) and Pierre Charles Le Monnier (1715-1799) published papers attempting to explain the anomalies. After the reading of Le Monnier’s papers, the Académie Royale announced a prize for a mathematician resolving the anomaly.

  Part 3: The Mathematicians

  The resolution of the Great Inequality from the perspective of Newtonian gravitation was taken up by the foremost mathematicians of the 18th century, including Leonhard Euler (1707-1783), Joseph-Louis Lagrange (1736-1813) and Pierre-Simon Laplace (1749-1827). Over a period of several decades, new mathematical techniques needed to be developed, ultimately providing an infrastructure—published in 1785 by Laplace—allowing the known anomalies in the orbit of the Moon, as well as those of Jupiter, Saturn and their moons, to be solved, leading to the publication of much more accurate astronomical tables. However, the solution is a set of mathematical equations that is as complicated as the epicycle-and-deferent Ptolemaic system that Copernicanism supposedly replaced.

  Part 4: In the TYCHOS, There Is No Inequality

  In the TYCHOS, the Great Inequality can be easily explained through the Earth’s PVP orbit: the inequality “is nothing more than an illusion of perspective”.

  In other words, a problem that vexed Europe’s leading astronomers and mathematicians for most of the 17th and 18th centuries can be explained using the most basic mathematics.

  The slow advance along the PVP orbit also explains a number of the anomalies that arise by presupposing the Copernican model: variations in the Moon’s orbit, the analemma, stellar aberration, Earth’s nutation, and so on. For each of these, simply taking into account the motion of the Earth along the PVP changes our perspective and makes these anomalies disappear.

• Accurately Retrodicting 3000 Years of Passages of Halley’s Comet

  Shack’s longest chapter is about Halley’s comet, henceforth simply written as Halley. For him, Halley has a fixed orbital period of 75.7 years and its motion is identical to that of other celestial bodies, such as Jupiter and Saturn, in our stellar system, namely a circular orbit around a point which itself follows the circumference of a second circle. The difference for Halley is that the center of the second circle is far from that of the Sun-Earth-Mars nucleus of our system. Shack has been able to match historical observations of Halley going back more than three millennia.

• Can a Physics Be Developed for the TYCHOS?

  Here I speculate about various ideas which could contribute to a physical understanding of the geometrical ideas of the TYCHOS. I claim that viewing the entire system through the lens of electricity might be the key, and that viewing our stellar system as a giant atom, with the Sun-Earth-Mars system—along with their respective moons Mercury, Venus, Moon, Deimos, Phobos—as the nucleus might well prove useful. I also point out two very interesting phenomena, which might guide this research: the major celestial bodies of our system all move at approximately 30km/s, and binary systems appear at multiple levels of the Universe.

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet’s Mill Advances at 1.6 km/h

• The Great Inequality of Jupiter and Saturn (four parts):

  Part 1: The English Astronomers

  Part 2: The French Astronomers

  Part 3: The Mathematicians

  Part 4: In the TYCHOS, There Is No Inequality

• Accurately Retrodicting 3000 Years of Passages of Halley’s Comet

As we have seen in the previous posts, the TYCHOS defines a remarkably simple geometry describing the motions of the celestial bodies in our stellar system. There is one problem, however: there is no associated physics, explaining why these bodies move as they do, following their trajectories of circles on circles.

This is a valid criticism, which I intend to address in this post. However, I think two points need to be made about mainstream astronomy.

First, it should be remembered just how long it took to develop the Copernican system into a coherent scientific theory, based on the principle of gravitation, providing accurate predictions:

• 1543: De revolutionibus (Nicolaus Copernicus, 1473-1543)

• 1619: Harmonices mundi (Johannes Kepler, 1571-1630)

• 1687: Principia, first edition (Isaac Newton, 1643-1727)

• 1785: Solution of Great Inequality (Pierre-Simon Laplace, 1749-1827)

• 1798-1825: Traité de mécanique céleste (Laplace)

Second, it is not so clear to me that today’s mainstream astronomy is itself actually based on a clear physics. Consider, for example, the following statement in Wikipedia about lunar theory:2

The analysts of the mid-18th century expressed the perturbations of the Moon’s position in longitude using about 25-30 trigonometrical terms. However, work in the nineteenth and twentieth century led to very different formulations of the theory so these terms are no longer current. The number of terms needed to express the Moon’s position with the accuracy sought at the beginning of the twentieth century was over 1400; and the number of terms needed to emulate the accuracy of modern numerical integrations based on laser-ranging observations is in the tens of thousands: there is no limit to the increase in number of terms needed as requirements of accuracy increase. [my emphasis]

So, for the Moon alone, tens of thousands of parameters are needed to accurately describe its motion. Is there a physical explanation for each of these parameters? The answer is a clear no, they are calculated using large computers using statistical analysis. They are the result of curve-fitting. And how many parameters are needed for the motions of the other major celestial bodies in our stellar system? We are a long way from the geometrical arguments used by Newton when writing his Principia.

So, yes, a physics for the TYCHOS would be welcome, and a search therefore is a completely valid exercise. However, it might take time. What does give us hope is that the Tychosium simulator developed by Patrik Holmqvist gives reliable predictive capability using only a few kB of initial data, while the best standard simulators need GB of initial data.

In the meantime, here are my thoughts on this question.

First, it seems to me that a proper physics for our stellar system—and others as well—should presuppose electricity as the dominant force of the universe. Here are some of my initial thoughts on this topic:

• Kristian Birkeland Theorized that the Universe is Electric

• The Self-Similar Nature of Cosmic Plasma

Second, if we accept the idea that the universe is self-similar at different scales, then one possible research direction is to consider our stellar system to be akin to an atom, in which the Sun-Earth-Mars core, along with the satellites Venus, Mercury, Moon, Phobos and Deimos, together can be considered to be equivalent to the nucleus of an atom.

For me, the planetary model of the atom of Wilhelm Eduard Weber (1804-1891) holds promise. I first presented this model in my post How Does a Positively-Charged Body or an Atomic Nucleus Hold Together?, in the section entitled Does Weber’s Electric Force Law Provide a Solution?

In Weber’s atom, the charges inside the nucleus are held together by his force law: when identical charges are sufficiently close together (approximately 10^-15 m, which is the typical diameter of an atomic nucleus), and with non-zero relative velocity and relative acceleration, then they will attract each other. Is something similar holding together the components of the core of our stellar system?

Third, when I discussed with Simon Shack recently, he pointed out to me that binary systems also seem to occur at many different scales. Consider, for example, this image of neighboring spiral galaxy M31, located in the constellation Andromeda.34

If this is correct, then a proper physics would cover not only the Sun-Mars binary system, but also how binary systems develop in general.

Finally, Shack and Holmqvist have recently discovered that the speeds of a number of celestial bodies in our stellar system are all very similar:5

In summary, here are the respective (approximate) translational speeds of the Sun, Mars, Halley’s comet, Jupiter, Saturn, Uranus and Neptune—according to the TYCHOS model:

• Sun: ≈107'226 km/h or 29.79 km/s

• Mars: ≈107'192 km/h or 29.78 km/s

• Halley: ≈107'226 km/h or 29.79 km/s

• Jupiter: ≈112'465 km/h or 31.24 km/s

• Saturn: ≈109'953 km/h or 30.54 km/s

• Uranus: ≈109'076 km/h or 30.30 km/s

• Neptune: ≈108'988 km/h or 30.28 km/s

Needless to say, these (slightly different) speed values are probably not quite exact. Thus, it is not entirely inconceivable to think that they may possibly all be identical. [Shack’s emphasis]

Once again, a coherent physics for the TYCHOS would be able to explain why these numbers are so similar, or even possibly identical. This could be very interesting.

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet’s Mill Advances at 1.6 km/h

• The Great Inequality of Jupiter and Saturn (four parts):

  Part 1: The English Astronomers

  Part 2: The French Astronomers

  Part 3: The Mathematicians

  Part 4: In the TYCHOS, There Is No Inequality

When Isaac Newton (1643-1727) was developing his theory of gravitation, predicting the elliptical trajectories of comets was a daunting task. His colleague, Edmond Halley (1656-1742), used this new theory in 1705 in the Synopsis of the Astronomy of Comets to predict the return of a comet in 1758. This comet, the best known of all, is today known as Halley’s comet.

The Wikipedia entry for Halley’s comet2 reads as follows:

Halley's orbital period has varied between 74 and 80 years since 240 BC. Its orbit around the Sun is highly elliptical, with an orbital eccentricity of 0.967 (with 0 being a circle and 1 being a parabolic trajectory). The perihelion, the point in the comet's orbit when it is nearest the Sun, is 0.59 au (88 million km). This is between the orbits of Mercury and Venus. Its aphelion, or farthest distance from the Sun, is 35 au (5.2 billion km), roughly the orbital distance of Pluto. [18 July 2026, my emphasis]

So mainstream astronomy tells us that Halley’s comet has a highly elliptical orbit with a substantial variance in its orbital period.

In his book about the TYCHOS, Shack’s longest chapter is about Halley’s comet, henceforth simply written as Halley. And Shack’s understanding of Halley is completely different from that of mainstream astronomy. For him, Halley has a fixed orbital period of 75.7 years and its motion is identical to that of other celestial bodies, such as Jupiter and Saturn, in our stellar system, namely a circular orbit around a point which itself follows the circumference of a second circle. The difference for Halley is that the center of the second circle is far from that of the Sun-Earth-Mars nucleus of our system.

What is not visible from that first diagram is that Halley, Jupiter and Saturn are all moving with approximately the same velocity. Furthermore, Halley’s loops are of the same size as the Sun’s, as can be seen in the next diagram.

Because of the looping trochoidal trajectory of Halley, when it approaches the nucleus of our system, in successive years there can be two, three or even four transits. For example, in the latest approach, which officially took place in 1986, there was a transit in 1985 to the north and another in 1986 to the south. I will use the term “transit set” for the set of transits corresponding to a specific 75.7-year cycle.

Given the completely different trajectories for Halley in standard astronomy and in the TYCHOS model, the retrodiction of Halley transits gives completely different results, as can be seen in the following two tables:

It is not surprising that most of Shack’s chapter about Halley focuses on comparing historical sightings of comets and comparing them with the TYCHOS historical reconstruction of Halley’s transits. Two kinds of surprise show up:

• A number of comets, appearing only once in the historical record, but within a year of Halley’s official passage, and named after the first person to see them in the sky, turn out to be unexpected transits of Halley.

• Some of the recorded transits of comets, sometimes named after Halley, were in fact transits of the asteroid Eros.

This chapter is a fascinating read, with the citation of contemporary observers. The seeming concordance with the TYCHOS retrodictions is impressive.

After reading the chapter, I decided to write this post, and started to do some research of my own. I quickly realized that there are literally hundreds of articles in the literature about Halley, coming from all sorts of perspectives. I was relieved when I came across the work of the late Italian astronomer Paolo Maffei (1926-2009), who published in 1984 La cometa di Halley: dal passato al presente [Halley’s Comet: From the Past to the Present]. I was not able to find a copy of the Italian original, but did find a copy of the French translation La comète de Halley: Une révolution scientifique.3 It is a monumental 504-page tome, containing discussions of many of the historical Chinese sightings.

I sent this copy to Shack, and he recounted to me that he ran with it. He came across in Maffei’s book an account of a historical sighting from China dated to 12 BC, which as we can see from the above table, does not correspond to a Halley transit. Using the Tychosium simulator, Shack was able to determine that it was another passing of Eros!4

Shack continues to work on historical transits of Halley, and has recently written a short paper5 thereon. Therein, he covers 33 transit sets of Halley, for which he was able to accurately match Tychosium retrodictions with more than 70 historical sightings. Here is what he writes for the first two transit sets, which correspond to historical sightings that took place before the dates recorded in the above tables.

1. 1194-1193 BC. Source: “Comet Lore”, by Edwin Emerson (1910): “In the year B. C. 1194, we are told by Hyginus that ‘On the fall of Troy, one of the Pleiades group of stars rushed along the Heavens toward the Arctic pole, where the star remained visible with disheveled hair, to which the name of Comet is applied’.” In the Tychosium, Halley’s comet transits at 0.45 AU from the Earth, right in front of the Pleiades (at RA 3h47m) on -1194-07-09 BC. It then returns the following year at a mere 0.08 AU (!) from Earth on -1193-05-05 (i.e. 1194 BC/historical).

2. 362-361 BC. Source: “Broom Star, China”/ quantumfuturegroup.org: “China, a broom star comet appeared in the west in 362 BC”. Source: ChatGPT: “Chinese records report: ‘In the 6th year of King Xian of Zhou (363 BC), a broom star appeared in the western sky.’ The following year (7th year, 362 BC) a comet was seen in the east. Many historians think these are two appearances of the same long-lived comet seen on successive returns or during a very long visibility interval, depending on record interpretation. The 362 BC comet is one of the earliest clearly datable comet records in Chinese history. It does not correspond to a known periodic comet like Halley’s — Halley’s known perihelion dates around then are c. 466 BC and 240 BC, skipping 362 BC”. In the Tychosium however, Halley transits at 0.24 AU on -362-08-24 BC (appearing “in the west” as viewed from China) and the following year at 0.37 AU on -361-06-10 BC (appearing “in the east” as viewed from China). Boom!  [Shack’s emphasis]

I find the combined work of Simon Shack and Patrik Holmqvist, who programmed the Tychosium simulator, gives a detailed theory of our stellar system that produces detailed predictions and retrodictions of the transits of Halley, with a precision far greater than anything that mainstream astronomy has produced.

The work deserves serious analysis and wider renown.

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet's Mill Advances at 1.6 km/h

In this post, we are focusing on the Great Inequality, in which seemingly, in the span of a single human lifetime, it was observed that the orbits of Jupiter and Saturn had sped up or slowed down. The first three parts of this post were:

• The Great Inequality of Jupiter and Saturn, Part 1: The English Astronomers

  This problem was first noted by Johannes Kepler (1571-1630) after the publishing in 1627 of his Rudolphine Tables, and the English astronomers of the 17th century attempted, without much success, to reconcile the observed movements of Jupiter and Saturn with Kepler’s tables. Ultimately, Edmond Halley (1656-1742) printed in 1719—but only published posthumously in 1752—revised astronomical tables, with “an anomalous acceleration in the mean motion of Jupiter and an anomalous deceleration in the mean motion of Saturn”.

• The Great Inequality of Jupiter and Saturn, Part 2: The French Astronomers

  This problem was picked up by Maraldi I (Giacomo Filippo Maraldi, 1665-1729), who first began looking at this problem in 1704. In his 1718 paper, he stated clearly that, since ancient times, Jupiter has been accelerating and Saturn decelerating, but not uniformly. In 1746, both Cassini II (Jacques Cassini, 1677-1756) and Pierre Charles Le Monnier (1715-1799) published papers attempting to explain the anomalies. After the reading of Le Monnier’s papers, the Académie Royale announced a prize for a mathematician resolving the anomaly.

• The Great Inequality of Jupiter and Saturn, Part 3: The Mathematicians

  The resolution of the Great Inequality from the perspective of Newtonian gravitation was taken up by the foremost mathematicians of the 18th century, including Leonhard Euler (1707-1783), Joseph-Louis Lagrange (1736-1813) and Pierre-Simon Laplace (1749-1827). Over a period of several decades, new mathematical techniques needed to be developed, ultimately providing an infrastructure—published in 1785 by Laplace—allowing the known anomalies in the orbit of the Moon, as well as those of Jupiter, Saturn and their moons, to be solved, leading to the publication of much more accurate astronomical tables. However, the solution is a set of mathematical equations that is as complicated as the epicycle-and-deferent Ptolemaic system that Copernicanism supposedly replaced.

In this fourth and final part of the post, we leave the Copernican model, as mathematized by Laplace, and go back to Simon Shack’s neo-Tychonian model, the TYCHOS. In this model, the Sun and Mars form a binary system, rotating around the Earth, which is close to—but not exactly—the barycentre of the system. The Earth plods along at a pace of 1.6 km/h (one mile an hour), in a slow orbit—called the PVP orbit—of 25'344 years, which explains, as was written in my post Hamlet’s Mill Advances at 1.6 km/h, the precession of the equinoxes.

So what does this have to do with the Great Inequality? The answer is simple:

In the TYCHOS, the Great Inequality Does Not Exist!

The published orbital periods of Jupiter and Saturn are respectively 11.862 and 29.4571 years, i.e., close to 12 and 30 years. This means that after 60 years, Jupiter will have completed just over 5 orbits and Saturn just over 2 orbits. Meanwhile, the Earth will have plodded along its slow orbit, advancing some 840 thousand kilometres. Earth’s advance during this period will have changed our perspective of the orbits of Jupiter and Saturn. And, as can be seen in the diagram below, depending on the current position of the Earth along the PVP orbit, we will have gotten the impression that Jupiter was accelerating and Saturn decelerating, or the reverse.

Shack’s explanation of the diagram is given below:

As viewed from Earth, Jupiter and Saturn will appear to alternately “accelerate and decelerate” depending on the timeframe chosen to measure their “60-year” conjunctions. This, due to Earth’s 1.6 km/h orbital motion. Two successive Jupiter-Saturn “60-year” conjunctions: in (a) the “upper” and (b) the “lower” quadrant of our celestial sphere. The so-called “Great Inequality” is nothing more than an illusion of perspective. [p.156, my emphasis]

In other words, a problem that vexed Europe’s leading astronomers and mathematicians for most of the 17th and 18th centuries can be explained using the most basic mathematics, which has been programmed into the Tychosium simulator (https://ts.tychos.space).

The slow advance along the PVP orbit explains a number of the anomalies that arise by presupposing the Copernican model: variations in the Moon’s orbit, the analemma, stellar aberration, Earth’s nutation, and so on. For each of these, simply taking into account the motion of the Earth along the PVP changes our perspective and makes these anomalies disappear.

It should be said that it took me months to prepare the three previous parts of this post, perusing through a century of articles published by the Académie Royale des Sciences, and working through Curtis Wilson’s lengthy article2 on the topic. As for this (fourth) part of the post, it took me two hours to prepare. The simplicity of the TYCHOS model is impressive.

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet's Mill Advances at 1.6 km/h

In this post, we are focusing on the Great Inequality, in which seemingly, in the span of a single human lifetime, it was observed that the orbits of Jupiter and Saturn had sped up or slowed down. The first two parts of this post were:

• The Great Inequality of Jupiter and Saturn, Part 1: The English Astronomers:

  This problem was first noted by Johannes Kepler (1571-1630) after the publishing in 1627 of his Rudolphine Tables, and the English astronomers of the 17th century attempted, without much success, to reconcile the observed movements of Jupiter and Saturn with Kepler’s tables. Ultimately, Edmond Halley (1656-1742) printed in 1719—but only published posthumously in 1752—revised astronomical tables, with “an anomalous acceleration in the mean motion of Jupiter and an anomalous deceleration in the mean motion of Saturn”.

• The Great Inequality of Jupiter and Saturn, Part 2: The French Astronomers:

  This problem was picked up by Maraldi I (Giacomo Filippo Maraldi, 1665-1729), who first began looking at this problem in 1704. In his 1718 paper, he stated clearly that, since ancient times, Jupiter has been accelerating and Saturn decelerating, but not uniformly. In 1746, both Cassini II (Jacques Cassini, 1677-1756) and Pierre Charles Le Monnier (1715-1799) published papers attempting to explain the anomalies. After the reading of Le Monnier’s papers, the Académie Royale announced a prize for a mathematician resolving the anomaly.

The man associated with the mathematical resolution of the Great Inequality is Pierre-Simon Laplace (1749-1827), who, after making a presentation to the Académie Royale in November 1785, submitted three papers234 for the 1784 (published in 1787), 1785 and 1786 (published in 1788) proceedings. These would be the last proceedings of the Académie Royale, which was replaced following the French Revolution.

This third part will focus on key developments leading to Laplace’s achievement, achieved approximately one century after Isaac Newton (1643-1727) published his Principia in 1687. As I did for the first part, for historical perspective, I will be using as guide the very lengthy paper by Curtis Wilson (1921-2012), “The great inequality of Jupiter and Saturn: from Kepler to Laplace.”5

It should be kept in mind that gravity was not explained in Newton’s Principia using the calculus, but, rather, geometry. It would take several decades before gravity was understood in terms of the calculus. For an introduction to this topic, see my post The Work of Leonhard Euler and the Bernoulli Family Ensured the Success of Newtonian Mechanics.

The three main actors in this development were arguably the three most important mathematicians of the 18th century: Leonhard Euler (1707-1783), Joseph-Louis Lagrange (1736-1813) and Laplace. Other contributors were Alexis Claude Clairaut (1713-1765) and Jean le Rond d’Alembert (1717-1783).

It is clear that I can only give a very high-level overview of the developments, since I am covering several decades worth of lengthy publications, each including pages and pages of detailed derivations. And Substack posts are limited in length.

Before gravitation could be presented in analytical terms, some basic analytical concepts had to be developed. This work would fall on Euler, who introduced the trigonometric series in 1729. These are of the form

a₀/2 + ∑_n=1..∞ (aₙ cos nθ + bₙ sin nθ)

where the aₙ and bₙ are constants. According to Wilson,

Without the extensive employment of such series, the celestial mechanics of Lagrange and Laplace would be inconceivable. [Wilson, pp.89-90]

With the introduction of these series, physics and mathematics would take a new turn:

The specific problems in the context of which these symbolic conceptions arose are left behind, and a new and apparently autonomous field of study emerges: the transformations and interrelations of algebraic and transcendental functions. It is within this field of study that the conception of the trigonometric series arises, as a means of approximating some other function, defined mathematically or physically. [Wilson, p.90, my emphasis]

In 1747, Euler made two significant steps towards the problem of resolving the motions of the planets. First, on 8 June 1747, he presented to the Berlin Academy the differential equations for motion. Second, three days later, he finished his submission for the 1748 prize—which he won—the Académie Royale had offered for resolving the motions of Jupiter and Saturn. Although he won both that prize and that of 1752, his results are unsatisfactory, as he had computed an acceleration for both Jupiter’s and Saturn’s orbits. This was around the same time as the discussions around the Principle of Least Action; see my post Pierre Louis Maupertuis and the Principle of Least Action.

Around the same time, and independently, Clairaut and d’Alembert also developed differential equations of motion. Without such equations, Newton’s gravity could never have been studied from an analytical point of view.

The problems to be solved were not simply mathematical, as not all agreed on the underlying physical principles. Euler, for example, was not a fully convinced Newtonian:

Up to 1750 Euler remained fundamentally sceptical as to the correctness of the inverse-square law of universal gravitation, and vigorously defended the view that all forces arise from the impenetrability of matter, and are thus forces of contact. Only with Clairaut’s proof that the inverse-square law could account for the motion of the lunar apogee did Euler come to agree that all further work on planetary theory must assume this law to be exact. And even after so agreeing, Euler remained convinced that the aether responsible for transmitting light through the interplanetary spaces must introduce frictional forces, and so cause the planets to fall gradually toward the Sun, with accelerating mean motions. The inverse-square law might be necessary for the explanation of the phenomena, but Euler did not believe it to be sufficient. Euler was fortified in this conviction by the phenomenon of the secular acceleration of the Moon, which until 1787 seemed to be inexplicable otherwise than by the assumption of frictional forces. But if the inverse-square law were fundamentally insufficient, then so also were the Keplerian rules that followed from it in the solution of the two-body problem. [Wilson, pp.74-75, my emphasis]

One could argue that Euler was channelling Paolo Sarpi (1552-1623) and Galileo Galilei (1564-1642). See one of my very first posts, Paolo Sarpi on Local Motion. In addition, Euler felt that the aether was making the celestial bodies slow down:

In his Opuscula varii argumenti of 1746, Euler had shown that a resisting aether of very low density would cause a planet or satellite to fall with extreme slowness toward the central body, and hence to have an ever decreasing period, or ever increasing mean angular motion about the central body. During these years as later, it was Euler’s conviction that light was produced by waves in an aether, and that the matter of this aether could not fail to have inertia and therefore to exercise resistance against any body moving through it. [Wilson, p.135, my emphasis]

Nor was Laplace consistently Newtonian:

Laplace, who appears to have shared neither Euler’s theological convictions nor his belief in a luminiferous aether, in 1774 put forward a modification of Newton’s theory, involving a finite speed of transmission of the gravitational force, and a difference in the resulting action depending on the speed and direction of motion of the body acted on; and these assumptions led to the same result as did the assumption of a resisting medium, namely that the mean motions of planets and satellites would undergo slow, secular accelerations. The apparent deceleration of Saturn, on the other hand, could not be accounted for in the same way, since it implied an increase in what we would now call potential energy; here the best Laplace could do was to invoke a hypothetical interaction between Saturn and one or more comets, as possibly having produced the effect. He was still putting forward these hypotheses as probable a decade later, in 1784. [Wilson, p.21, my emphasis]

Back to the mathematics. It turned out that the differential equations of motion introduced by Euler—when applied to the motion of a planet perturbing the motion of another planet—referred to the inverse cube of the distance between the perturbing planet and the perturbed planet. The result was therefore an irrational expression, impossible to integrate except by integration. Euler resolved the situation by introducing trigonometric series—easy to integrate, term by term—to approximate that expression. This still led to the task of computing the initial coefficients of the terms of the series; the initial work was done by Euler, with successive refinements, leading to faster convergence of the series, by Lagrange and Laplace.

But there was a problem. The solution of the differential equations for the radius vector—the distance from the Sun to a planet—turned out to contain terms proportional to the time. These terms were called at the time «arcs de cercles» [“arcs of circles”]. What does this mean? Occurrences of t—for “time”—occurred independently, and not as part of arguments of the sine or cosine functions. There were two possibilities: the first was that these occurrences were artifacts of the approximation process, and the second that the solar system was unstable. Should the second case apply, this would mean that the Great Inequality were not some long-term oscillatory process.

Laplace found a workaround in which the «arcs de cercles» could be avoided:

Laplace will write again, seeking a formal generality, on the subject of eliminating “arcs de cercle” from the solutions of differential equations, when these equations are themselves free of such “arcs”. And Lagrange will respond by showing that Laplace’s procedure is subject to certain limitations, and by presenting a different procedure that avoids these limitations and is also free of certain features of Laplace’s memoir that Lagrange finds “metaphysically unsatisfying”. Lagrange’s metaphysical discontent with the Laplacian techniques presumably stems here, as it stems elsewhere, from his dislike of any hint of the infinitesimal. [Wilson, pp.129-130, my emphasis]

The highlighted sentence reflects a distrust of any kind of indivisible or infinitesimal in the world of mathematics, since their introduction in Western Europe during the 17th century. For more on this topic, including Lagrange’s position, see my post The War Against the Infinitesimals.

Lagrange and Laplace each wrote several papers in which the Great Inequality appears, and each, with mutual respect, carefully read the work of the other, and adjusted his work accordingly. But for neither was this topic the main focus. Compare the titles of their first papers:

• Lagrange: «Solution de différents problèmes de calcul intégral»6 [Solution of several problems of the integral calculus], first published in 1762-1765 in Miscellanea Taurinensia, the Turin-based journal created by Lagrange with some colleagues;

• Laplace: «Sur le principe de la gravitation universelle et sur les inégalités séculaires des planètes qui en dépendent»7 [On the principle of universal gravitation and the secular inequalities of the planets that depend upon it], dated 1773 and published in 1776.

As we can see, Lagrange was focusing on the development of the integral calculus, and only the last section, 60 of 200 pages, targeted a particular use case: «Application de la solution précédente à la théorie de Jupiter and de Saturne» [Application of the previous solution to the theory of Jupiter and Saturn]. Laplace, on the other hand, was focusing on the development of Newton’s theory of universal gravitation. It seems to me that Lagrange was focusing on mathematical elegance, while Laplace was acting “like an engineer”, looking for a solution “that worked”.

Here is the overall assumption under which they were working:

In the view of planetary motion that guides the work of Laplace and Lagrange throughout the 1770’s and 1780’s, the fundamental motion of each planet is the Keplerian motion, following the elliptical orbit in accordance with Kepler’s areal rule. But owing to their mutual action, the planets are subject to inequalities that perturb the Keplerian motion, and these inequalities are seen as of two kinds. Some of them, labelled “periodic”, depend on the positions of the planets, either with respect to each other or with respect to their aphelia; these inequalities prove to be small in relation to the equation of center, and in general, reestablish themselves within relatively short periods, usually not longer than a small number of years. In treating of these inequalities, Laplace views the orbital elements as fixed and unchanging, and the planet as merely executing an oscillatory dance of small ambit about a mean position which moves forward in strict accordance with the Keplerian rules. The other inequalities, called “secular”, alter the elements of the orbits by nearly insensible nuances at each revolution, but, accumulating unceasingly, they end by changing entirely, over the course of centuries, the nature and position of the orbits. But even the secular variations, as Lagrange first argues and Laplace comes to accept as established doctrine, are periodic; their periods, however, are very long, measured in tens of thousands of years rather than a few years or decades, and their formulas do not involve the positions of the planets. [Wilson, pp.166-167, my emphasis]

But the inequalities all affect each other, and there are constraints that needed to be respected:

In accordance with this formula, the secular variations of the eccentricities and inclinations that Lagrange and Laplace have educed imply some exchange of angular momentum between the planets, but the gains and losses of angular momentum deriving from these variations turn out to balance one another, without the mean solar distances being assumed to change. But if one does suppose an exchange of angular momentum involving change in the mean solar distances and not the eccentricities or inclinations, then the relation between the changes in mean motion derivable from this assumption… will be contradicted by the observations. Indeed, there can be no exchange of angular momentum involving a change in the mean solar distances, without changes in eccentricity or inclination or both being involved as well. This is because a second requirement has to be met, the conservation of forces vives or, as we now say, energy, and this puts a restriction on the ways in which angular momentum can be exchanged. Laplace in 1774 gives no indication of being aware of the necessity of premising conservation of forces vives as well as angular momentum. We believe it is from Lagrange that he will learn to take energy conservation into account. [Wilson, pp.167-168, my emphasis]

So by late 1774 or early 1775, significant progress had taken place, but Laplace was still convinced that gravitation was not the cause of the Great Inequality. During the next decade, Laplace would focus on other problems, such as the tides:

In the decade from the mid-1770’s to the mid-1780’s, Laplace, insofar as he dealt with questions of gravitational action, focussed on problems such as those of the tides and the precession of the equinoxes, that depend on the extended mass and non-spherical shape of planetary bodies. Having concluded that the apparent acceleration of Jupiter and the apparent deceleration of Saturn were not caused by gravitation, he had hypothesized tentatively that the acceleration of Jupiter could be due to the finite speed of propagation of the gravitational force, and that the deceleration of Saturn must be due to the action of comets. It would appear that he had decided that further investigation of the anomalies of these planets, at this time, was unlikely to be profitable. [Wilson, p.71, my emphasis]

Lagrange, on the other hand, kept at it, and ultimately introduced what Laplace would call the “perturbing function”.

Lagrange in these years, by contrast, continued to engage in systematic development of the general theory of planetary perturbations. He introduced at least three ideas that would prove of the greatest importance in Laplace’s final resolution of the anomalies. He defined the perturbing function, which was to become an essential feature of Laplace’s procedure for calculating higher-order perturbations. The perturbing function is a potential function; Lagrange uses it in the derivation of the various integrals of the equations of motion, including the energy or force vive integral. This integral, which had not entered into Laplace’s earlier work, will play a significant role in his drawing the conclusion contrary to his earlier view, that the anomalies in the motions of Jupiter and Saturn can be caused by gravitational action. Finally, it will be Lagrange who first remarks on the fact that near-commensurability of the mean motions of the planets implies long-term and potentially sizable inequalities—a suggestion that [Wilson] believe[d] led to Laplace’s renewed attack on the anomalies of Jupiter and Saturn in 1785. [Wilson, p.71, my emphasis]

[A potential function is a function whose value, scalar or vector, varies with the position in space.]

Among Lagrange’s successive works, reading Wilson’s paper, these two stand out:

• «Remarques générales sur le mouvement de plusieurs corps qui s’attirent mutuellement en raison inverse des carrés des distances»8 [General remarks on the motion of several bodies which attract one another in inverse proportion to the squares of the distances between them], published in 1779.

  In this memoir Lagrange shows how the perturbing function simplifies the derivation of the integrals for the center of gravity, angular momentum, and force vive of a system of gravitating bodies. [Wilson, p.206]

• «Théorie des variations périodiques (Première partie contenant les formules générales de ces variations)» [Theory of Periodic Variations (Part One: containing the general formulae for these variations)], published in 1785.

  In this memoir Lagrange mentioned, in passing, the circumstances under which a periodic inequality, even though proportional to the eccentricity or inclination, might prove to be sizable. The context was not the same as the one in which Laplace would later apply the idea so successfully. Nevertheless, in the total absence of other evidence as to why Laplace returned to the problem of Jupiter and Saturn in 1785, after a lapse of ten years, and at the same time relinquished the earlier hypotheses he had entertained on this subject, we are inclined to think Lagrange’s publication of 1785 on perturbations had a good deal to do with it; and the passage therein pointing out that inequalities otherwise expected to be small can become large because of a near-commensurability of the mean motions of perturbing and perturbed planets is the most relevant of all to Laplace’s discovery. It may also be worth mentioning that in the same memoir Lagrange derived the very formula that Laplace would later use in the actual calculation of the great inequality; however, Lagrange’s derivation differs from the one that Laplace will later give. [Wilson, p.215, my emphasis]

And this brings us to Laplace’s «Théorie de Jupiter et de Saturne» [Theory of Jupiter and Saturn], reference 3. Wilson provides a quick overview of the contributions. First, at the theoretical level, Laplace computed expressions involving higher powers of the basic parameters:

Of the new Laplacian procedures, the most important on the theoretical side concerned the calculation of perturbations proportional to the powers and products of the second and higher dimensions of the orbital eccentricities and inclinations. No planetary perturbations of this type had previously been calculated; the task was an exceedingly laborious one, most terms of this order were negligible, and previously there had been no procedure for determining in advance whether the calculation of a given perturbational term was likely to yield an empirically measurable result. Laplace, following a clue that he had undoubtedly gleaned from Lagrange, developed a rule for the purpose; and this rule became the basis of subsequent practice. Since many of the inequalities discovered by its means were quite as large as and even larger than the lower-order terms previously calculated, the result was a sharp increment in the accuracy of planetary prediction. [Wilson, p.23]

Second, in order to determine the constants in the expressions, he used an averaging technique based on actual observations made by astronomers:

A second important technique introduced into general use through Laplace’s example in the “Théorie de Jupiter et de Saturne” was the employment of equations of condition in the correction of orbital elements, and in the evaluation of coefficients of perturbational terms. The method is used to bring large numbers of observations to bear on the determination of constants so related to one another that the value given to one of them influences the values given to the others. It had been invented by Euler, and used successfully by Tobias Mayer and by Roger Boscovich. But it was Laplace in 1786 who first obtained general recognition for the method, with his application of it to the determination of the orbital elements of Jupiter and Saturn. Delambre, imitating Laplace’s example, made use of it in determining the constants of his solar theory, and in the refinement of the empirical constants of Laplace’s theories of Jupiter and Saturn, of the satellites of Jupiter, and of the planet Uranus. With the evident superiority of the resulting tables over all earlier tables, the employment of statistical procedures in astronomy became de rigueur. [Wilson, pp.23-24]

The techniques used in the «Théorie de Jupiter et de Saturne» turned out to be successful in resolving a number of issues, and astronomers successfully used Laplace’s ideas:

Laplace was an opportunist in mathematical astronomy—successfully, importantly so. He became the ‘organizer of victory’ for the science that had earlier been called “physical astronomy”, and which he in 1799 was to re-name “celestial mechanics”. His “Théorie de Jupiter et de Saturne” is the first in a series of victories leading beyond the earlier stalemate to a new level of precision in the agreement of planetary tables and observations. [Wilson, pp.23-24]

Still, for me, this success raises some issues. When we learned about the Ptolemaic and the Copernican systems, we were taught to mock the elaborate system of epicycles and deferents. Yet, if we look at the perturbation theory introduced by Newton, then successively developed by Euler, Lagrange, Laplace and others, aren’t we simply looking at another form of epicycles?

With the introduction of trigonometric series the mathematics became in a sense Ptolemaic, epicyclic, and therewith opaque to the force relations, the consequences of which it was designed to express. [Wilson, p.25]

In the next post, we will see how the motions of Jupiter and Saturn are understood in the TYCHOS system.

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet's Mill Advances at 1.6 km/h

In this post, we will focus on the Great Inequality, in which seemingly, in the span of a single human lifetime, it was observed that the orbits of Jupiter and Saturn had sped up or slowed down.

In The Great Inequality of Jupiter and Saturn, Part 1: The English Astronomers, I wrote that this problem was first noted by Johannes Kepler (1571-1630) after the publishing in 1627 of his Rudolphine Tables, and that the English astronomers of the 17th century attempted, without much success, to reconcile the observed movements of Jupiter and Saturn with Kepler’s tables. Ultimately, Edmond Halley (1656-1742) printed in 1719—but only published posthumously in 1752—revised astronomical tables, with “an anomalous acceleration in the mean motion of Jupiter and an anomalous deceleration in the mean motion of Saturn”.

The French Astronomers

Focus on the Great Inequality moved across the English Channel in the early 18th century. Key players in French astronomy at the time were the Cassinis and the Maraldis, who originally hailed from Perinaldo in Liguria:

• Giovanni Domenico Cassini (1625-1712), known as Cassini I.

• Giacomo Filippo Maraldi (1665-1729), nephew of Cassini I, known as Maraldi I.

• Jacques Cassini (1677-1756), son of Cassini I, known as Cassini II.

• Giovanni Domenico Maraldi (1709-1788), nephew of Maraldi I, known as Maraldi II.

• César-François Cassini de Thury (1714-1784), son of Cassini II, known as Cassini III.

• Jean-Dominique, comte de Cassini (1748-1845), son of Cassini III, known as Cassini IV.

It is important to remember that the French and English scientific environments were quite different. For example, as I wrote in my post Johannes Kepler “Disappears” Tycho Brahe’s Principal Critique of Copernicanism, Cassini I and Ole Christensen Rømer (1644-1710), both working in Paris, were, in the words of Cristiaan Huygens (1629-1695), complete dunces, “whose Dulness can’t comprehend it” («d’une intelligence tardive»), as they did not believe the Copernican system to be true. And even after Cassini I, when the Copernican system was accepted in France, there were still holdouts, such as Cassini II when he was young, supporting the idea of Cartesian vortices—as opposed to Newtonian gravity—as a cause of the motion of the planets. The vortices had also been supported by Huygens, as I wrote in my post Christiaan Huygens Accepts the Inverse-Square Law But Not Action-at-a-Distance. For more on vortices, see my post Descartes’s Vortices Keep Resurfacing.

I perused through the proceedings of the Académie Royale des Sciences, which were published during the years 1733-1788, for the years 1666-1786. After the French Revolution, the Académie Royale was shut and replaced by a series of other institutions, ultimately leading to the creation of the modern Académie des Sciences. Each of the proceedings of the Académie Royale, sometimes published several years after the nominal year, consisted of two parts. The Histoire part consisted of summary papers with no attributed author, each of which reported on the importance of one or more authored papers appearing in the second part. The Mémoires part consisted of authored papers. Below are the papers that I found that are relevant to the Great Inequality.

• Maraldi I: 1704 paper on Saturn23

• Maraldi I: 1706 paper on Jupiter45

• Maraldi I: 1718 paper on Jupiter67

• Cassini II: 1728 paper on Saturn89

• Cassini II: 1743 paper on Mars, Saturn and Jupiter1011

• Cassini II: 1746 paper on Saturn and Jupiter1213

• Pierre Charles Le Monnier (1715-1799): two 1746 papers1415

Maraldi I’s 1704 and 1706 papers presented attempts to reconcile observations of the motions of Jupiter and Saturn over a period of several decades, initially by Cassini I, then jointly with Maraldi I, with the predictions in the Keplerian tables.

The 1704 paper, along with the accompanying Histoire paper [reference 2], outlines the difficulty of constructing astronomical tables. Initially, a mean or average motion for a planet’s entire orbit around the Sun was computed. Then, given an epoch, i.e., a designated point in time for which the observations were certain, the tables for the entire orbit could be computed, by supposing that for about half of the orbit, the calculated motion was computed by adding to the mean motion, while for the other half of the orbit, the calculated motion was computed by subtracting from the mean motion. From there, the aphelion (furthest point from the Sun), perihelion (closest to the Sun), apogee (furthest from the Earth), perigee (closest to the Earth) and nodes (crossings of the ecliptic) of the orbit could be determined.

Any error in observations of a planet’s position in the sky, or subsequent calculations of its orbit, could lead to significant errors in the tables, for the entire orbit of that planet. Therefore, trying to determine if there were anomalies in the Keplerian trajectory of a planet was no easy task, as it have been that the predictions as appearing in a set of tables were simply the result of errors in observations and calculations, as opposed to the planet not moving according to Keplerian principles.

In the 1704 paper, Maraldi I proposed some adjustments to the orbit of Saturn, and in the 1706 paper, to that of Jupiter.

To establish the hypotheses concerning Saturn, we first calculated a large number of observations made during the planet’s opposition to the Sun at various points along the planet’s orbit, and we compared these observations with Kepler’s Tables. This comparison revealed that in the years 1672 and 1673, these Tables gave Saturn’s position as being further advanced in the Zodiac than the observations by 20 to 21 minutes; that in the years 1686 and 1687, the difference between the observations and the Tables there was a difference of only 10 to 12 minutes, with the Tables being more advanced; and that finally, in the years 1700, 1701 and 1702, this difference was approximately 21 minutes, as it had been thirty years earlier. [1704, pp.314-315]

This has shown us that we must increase the greatest equation of Saturn determined by Kepler—which he sets at 6° 31' 30''—by half of this difference, which is 5 minutes, making it 6° 36' 30'', which is within one minute of that determined by Mr Bouillaud. In addition to this correction, a second must be made to the Epoch, by subtracting 16 minutes from the mean motion of 1607, which Kepler gives as 6 28° 3' 52'', and setting it at 6 27° 47' 52''. [1704, pp.315-316]

To determine the greatest inequality of Jupiter, we compared the calculation derived from the same Tables with the observations made near the mean distances, & we found that to properly represent these different observations, the greatest equation of Jupiter had to be set at times to 5° 34' 55'', sometimes 5° 35' 25'' and sometimes 5° 35' 15'', on which we have settled, as it is the mean between the extremes with which it agrees to the minute; thus, in accordance with this determination, it will be necessary to increase by three and a half minutes the greatest equation of Jupiter determined by Kepler, and by one minute that determined by Mr Boüillaud. [1706, pp.62-63]

But it would be in the 1718 paper that Maraldi I proposed that there was some serious problem to be resolved. It was known that seven orbits of Jupiter take place in just over 83 years. Maraldi I compared the lengths of seven-Jupiter-orbit periods of his time with that of Tycho Brahe and also that of medieval, Ptolemaic and Babylonian astronomers. His conclusion was that these periods were not of the same length, and apparently the orbit of Jupiter had accelerated, although not uniformly, over a period of some two millennia, and that the greatest acceleration had taken place in the most recent centuries:

It is clear from all that we have just reported, 1. That the four earlier observations, when compared with one another in the order in which they were made, all yield a smaller value for the motion than that obtained by comparing the earlier observations with the modern ones, and even smaller than that found by comparing the modern observations with one another. 2. That the difference in motion resulting from the comparison of one ancient observation with two others made by Walterus within an interval of eight months is not what it ought to be, which shows how little reliance can be placed on it. 3. That this acceleration, as indicated by the ancient observations, amounts to less than one minute in 83 years, whereas from the modern observations it amounts to at least 9 to 10 minutes over a similar period. Thus, although this acceleration appears certain when observations made over the past century are compared with one another, it is not so when based on the ancient observations. The small differences found therein may be attributed partly to the hypotheses one is obliged to employ to reduce the true motions to their mean values, and partly to certain minor errors to which the observations are subject, chiefly the older ones which lack the degree of precision required by this research, having been made by the naked eye, and being, moreover, too few in number. [1718, pp.326-327]

As for Saturn’s orbit, referring to his 1704 paper, he concluded that it has been decelerating:

Thus, if one were to accurately represent the most precise ancient and modern observations of Saturn, one would have to assume that the average motion of this planet has been slower in recent times than it was in the past, contrary to what appears to follow from the observations of Jupiter that we have just compared. [1718, p.327]

The accompanying Histoire paper [reference 6] concludes that there is clearly a problem to be resolved:

Mr Maraldi even admits that, by investigating in the same way he had previously done whether there is any variation in Saturn’s motion, he found that it would be opposite to that of Jupiter; that is to say, that Saturn’s motion would be retarded; yet if there were any slow and gradual variation in the motion of a planet, it seems quite likely that not only would there be such a variation for all of them, but that it would be of the same kind for all, that is to say, accelerated or delayed, unless, however, the acceleration of some caused the delay of others, or vice versa, which would be quite possible, since the total quantity of motion would always have to remain the same. Be that as it may, a mere suspicion on this matter is quite enough for the present, and had physics not made such great progress, one would not have the audacity to entertain it. [Histoire, 1718, pp.69-70]

The articles of Cassini II and Le Monnier continued on from the work of Maraldi I. They consisted of very detailed discussions of astronomical observations of the motions of both Jupiter and Saturn, made both in ancient and modern times, and confirmed what Maraldi I had observed: over the centuries, the orbit of Jupiter had accelerated and that of Saturn had decelerated.

What differentiates the work of Cassini II and Le Monnier from that of Maraldi I is their attempts to provide physical explanations for these anomalies in the orbits of Jupiter and Saturn.

For Cassini II, what is fascinating is how he initially supported Descartes’s vortices and then later accepted the Newtonian model of gravity. Compare the following passages from 1723:

It therefore remains to be examined whether some other cause can be attributed to them which is subject to some law.

It is known that some modern physicists have supposed that the planets may undergo some alteration in their motion due to the various positions they occupy in relation to one another. This hypothesis in no way contradicts the most generally accepted principles of physics; for since everything is full, the vortices of these planets cannot approach or move away from one another without the bodies contained within them receiving some impression from the combination of these movements. But the question is whether these effects are sufficiently noticeable for us to perceive them. [1723, p.86, my emphasis]

and from 1746:

However, as one might be reluctant to accept this hypothesis if it were not supported by arguments that had at least some semblance of plausibility, I have examined whether this variation in the motion of these planets could be explained by the laws of gravity, which, as Mr Newton has shown, are consistent with the motions of the planets, whatever the cause of that gravity may be. [1746, pp.474-475, my emphasis]

As for Le Monnier’s article, its most lasting legacy is a footnote found at the bottom of its first page, in which it was announced that after he presented the results of this article to the Académie Royale, it was decided that the subject for the next Académie prize would be the theory of Saturn and Jupiter.

Henceforth, it would be the mathematicians who would take up the baton in trying to resolve the Great Inequality. That will be the topic of the third part of this post.

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet's Mill Advances at 1.6 km/h

In this post, we will focus on the Great Inequality, in which seemingly, in the span of a single human lifetime, it was observed that the orbits of Jupiter and Saturn had sped up or slowed down. This was one of the most important subjects of research of the greatest astronomers of the 17th and 18th centuries, but also of the greatest mathematicians of the 18th century, including Leonhard Euler (1707-1783), Alexis Claude Clairaut (1713-1765), Jean le Rond d’Alembert (1717-1783), Joseph-Louis Lagrange (1736-1813) and Pierre-Simon Laplace (1749-1827).

For the historical perspective part of this post, I will be using as guide the very lengthy paper by Curtis Wilson (1921-2012), “The great inequality of Jupiter and Saturn: from Kepler to Laplace.”2 This paper follows on from two other lengthy papers by Wilson, “Perturbations and Solar Tables from [Abbé Nicolas-Louis de] Lacaille [1713-1762] to [Jean-Baptiste Joseph] Delambre [1749-1822]: the Rapprochement of Observation and Theory”.34 We have previously come across Wilson’s name in the post on Mars, Are the Mayan Long and Short Periods for Mars Correct? He was the first to write about the fact that Johannes Kepler’s (1571-1630) calculations about the motion Mars could not possibly have been derived from previous observations by Tycho Brahe (1546-1601).5

Kepler and the Rudolphine Tables

In 1627, Kepler published his Rudolphine Tables, consisting of a star catalog and planetary tables, allowing one to locate, as time passed, the moon and the planets. These tables would be the primary reference for the planets for more than a century. As astronomers made more and more accurate observations, using more and more precise tools, and as mathematicians adjusted astronomical theories, they could refer to Kepler’s tables as baseline to express a difference in observations or predictions. At the same time, work in libraries and careful reading of old texts also allowed retrodiction and comparison with Babylonian and Ptolemaic observations.

It must be remembered that Kepler’s publishing of the Rudolphine Tables took place some six decades before Isaac Newton (1643-1727) published the first edition of the Principia and some twelve decades before Euler transformed the Newtonian system into a useful mathematical model based on analysis. This extended period of time often gets collapsed into a handful of paragraphs in a school or university textbook.

However, one of the major problems to be resolved was that neither Jupiter nor Saturn seemed to be behaving correctly, according to the Keplerian model. In fact, this was first noticed by Kepler himself, who argued in a June 1625 letter to Matthias Bernegger (1582-1640) for the need for “secular equations” to take into account long-term changes in the positions of Saturn, Jupiter and Mars:

... In the meantime I in Maestlin’s [1550-1631] library at Tübingen have convoked a council of all mathematicians who have flourished during the past 2000 years, or who will exist in the future: it is easier for them to be present—and, let me add, to agree—than for those now living. The topics of deliberation of greatest moment are these: 1. I exhibit the form of my tables. 2. The mean motions are no longer mean, since I have learned from the observations of Regiomontanus [1436-1476] and Waltherus [1430-1504] that all the positions of Saturn in particular, if not also those of Jupiter and Mars, everywhere in their eccentric circles, were then more advanced or less advanced than would be implied by mean motions running uniformly from Ptolemy to Tycho.

And thus there is need for a secular equation, of exactly what kind cannot be discovered for several centuries. 3. Meanwhile, supporting myself on the mean motions hitherto accepted as settled, and lacking the secular equation, I find back at the beginning of the world, on a certain special day, all the epochs of all the planets disposed as if in a cross, or in places at right angles as seen from the center of the Sun; thus the hope is that, a minimal secular equation being introduced, all those places will agree in the four cardinal points ... [Kepler, Gesammelte Werke, XVIII, pp.237-238 (Munich: C.H. Beck, 1959), as translated and quoted in Wilson, pp.36-37, my emphasis]

The English Astronomers

The process leading from Kepler’s laws towards Newton’s law of universal gravitation required the support of further data provided by English astronomers. The first of these was Jeremiah Horrocks (1618?-1641). Despite his very short life, he made some remarkable observations, including the first observation of a passage of Venus in front of the Sun.6 He also substantially reduced the solar parallax compared to what it had been assumed to be by previous astronomers. His works were gathered by John Wallis (1616-1703), his former classmate from Emmanuel College (Cambridge), and includes extensive correspondence with William Crabtree (1610-1644), along with additional material by Crabtree and John Flamsteed (1646-1719).7

During the last four years of his life, Horrocks repeatedly expressed the difficulties in correlating the observed motions of Jupiter and Saturn with Kepler’s tables. Here is Wilson writing about Horrocks’s first mention of the problem:

HORROCKS first obtained a copy of the Rudolphine Tables in May, 1637. Already in a letter to CRABTREE of 3 June 1637 he is proposing to correct the elements of Saturn by adding 2° to the position of the aphelion, and subtracting 4' from the motus medius, or position as deduced from the tables of mean motion.

“Jupiter causes me annoyance ... [From observation] it seems that both its aphelion and its mean motion are notably more rapid than Kepler would have them. Your observations of 2 and 17 December 1635, and 25 January and 25 February 1636, make Jupiter 14' or 15' more advanced than Kepler does, and the observation of 8 March at least 10' more advanced. Kepler thinks that the inequalities in the motions of the stars are irregular on account of uncertain physical causes, and promises to write a book in which this will be demonstrated. If this is so, what, may I ask, is to be done? Certainly the motions of Jupiter almost incline me to this opinion ... See the preface to the Rudolphine Tables, pages 7,34,118.” [Opera Posthuma, p.287, quoted in Wilson, p.39]

And here is Horrocks’s last writing on Jupiter and Saturn, in 1640:

In Saturn and Jupiter Kepler’s tables of equations [of center] are exact enough; only in Jupiter 30' is to be added to the aphelion, 5' to the equal motion for the beginning of the year 1640, and 6' for the beginning of the year 1650. For it is about 1' faster than the Rudolphine [numbers] make it in this century. Whether it will continue thus or not, I do not at all know. For between 1490 and 1590 it was sensibly faster than it is now. [Opera Posthuma, p.334, quoted in Wilson, p.41]

If the attention Horrocks paid to Jupiter and Saturn was a mere four years, this was very different for Flamsteed, whose extensive correspondence with Newton and other researchers on this question, which was published as part of The Correspondence of Isaac Newton8910 and in Francis Baily’s 1835 biography of Flamsteed,11 spans over a period of over four decades. It became clear over time that the fluctuations of Jupiter’s and Saturn’s orbits were interlinked. For example, here is part of a letter dated 11 January 1698/99 (O.S.) from Flamsteed to William Bossley (fl. 1695–1710):

I gave Mr Newton, about 4 years ago, the same places of Saturn and Jupiter I gave you on our first acquaintance. He tells me he had tried to answer those of Saturn, and found he could do it nearly by only liberating [!] his aphelion: but that it liberates [!] very oddly: and that Kepler’s eccentricity was well. In some discourse I had with him lately, he added further that the greatest fault in the superiors was when they approached each other nearest; and that, when they were more remote, each pursued his own way with his primary inequality little disturbed. You will hereby understand that he accounts the inequality to be restored every revolution of Saturn to Jupiter, which they would precisely if their distances of the one planet from the other were exactly the same in every conjunction: but these changing, and not being restored but in 59 years very nearly, these inequalities are only restored at that period: and at all other times, they are less according as the planets are more remote from each other.

And I doubt not but Saturn moves swiftest in that revolution in which the sum of all his distances from Jupiter is least; and Jupiter, on the contrary, slowest. And again Saturn is slowest in that revolution wherein all his distances from Jupiter make the biggest sum, and Jupiter then swiftest: and that this is the reason why the motions of these two planets are found swifter in some ages, or parts of an age, than in others. I give you these thoughts to assist you: in the mean time you may try whether your own way will do; and you may see whether these same thoughts are agreeable to what you find by experience on comparing the observed places with the calculated. [Bailey, pp.169-170, quoted in Wilson, p.47, my emphasis]

After Flamsteed’s failure to resolve the problem, the task would be taken up by the second Astronomer Royal, Edmond Halley (1656-1742), who printed in 1719 a set of revised tables, which were published posthumously in 1752.12

Wilson writes:

These tables, the most highly respected at the time of their publication, and still in use in the 1780’s, were purely Keplerian in principle, except for HALLEY’S inclusion in his tables of an anomalous acceleration in the mean motion of Jupiter and an anomalous deceleration in the mean motion of Saturn. [Wilson, p.16]

Nevertheless, Halley himself recognized that there were still problems, and that Newtonian Geometry might not suffice in explaining the inequality:

You see also that our Tables are made to undergo an examination rigid enough. And yet be not surprized that in Jupiter the error sometimes arises to eight minutes, and sometimes in Saturn to ten: For indeed it cannot be otherwise without assuming new hypotheses not yet sufficiently known and proved by the Touchstone of the Heavens. Jupiter indeed in his motion from the opposition in the year 1677 to that of the year 1689, according to undoubted Observations, was found slower by twelve whole minutes [of arc] than he was in the preceding or subsequent Revolution. Also Saturn’s Period, between the years 1668 and 1698, was shorter than this mean Revolution by almost a whole week; and another Period of his compleated between the years 1689 and 1719, was longer than the mean Revolution by about as many days; so that their difference in duration was about thirteen days. But whether this will be so for the time to come must be left to the Observation of Posterity.

It is more than probable that this is owing to the mutual actions of the greater Planets upon one another, disturbing the centripetal forces of the Sun. For what happened in the year 1683 seemed no slight argument to prove this; For then there was a conjunction of Jupiter and Saturn in those parts of their orbits, where, on account of the situation of the Apses, the Planets approached very near to each other; their joint forces then urged Saturn towards the Sun, and on the contrary Jupiter from the Sun; wherefore Jupiter, having his proper velocity increased, and the centripetal force of the Sun being decreased, must have run out into a greater Orbit, and would require a longer time for a Revolution: while Saturn, at the same time, having his proper velocity diminished, and being urged with a greater force towards the Sun, must have been forced into a less Orbit, and so revolve in a less time. If the same happen again and again, when Jupiter and Saturn are in conjunction in Leo, we may justly hope that the errors we find in their motions, as they are owing only to the joint efficacy of the three Centers, at length may be removed by the Newtonian Geometry. But if not, or if the Periods should prove longer where they are now shortest, on the contrary, there must be some extrinsic cause, of which we are now ignorant. [pp.52-53, my emphasis]

So, for the first eight decades after Kepler’s publishing of the Rudolphine Tables, the study and attempted resolution of the Great Inequality took place in England. During the 18th century, it would be for the French astronomers and mathematicians to take up this task. This will be the subject of the next two parts.

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Je suis heureux d’annoncer que mes traductions en français des deux volumes de The Experimental and Historical Foundations of Electricity, d’André Koch Torres Assis, ont été publiées par Apeiron à Montréal. Ces ouvrages peuvent être commandés sur Amazon ou téléchargés depuis le site web d’André Assis.

• PDF: https://www.ifi.unicamp.br/~assis/Electricite-Vol-1.pdf

• Amazon: https://www.amazon.com/dp/1987980476

• PDF: https://www.ifi.unicamp.br/~assis/Electricite-Vol-2.pdf

• Amazon: https://www.amazon.com/dp/1987980492

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• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

In this post we will focus on the precession of the equinoxes, which corresponds to the long-term movement of the equinoxes westward along the ecliptic relative to the fixed stars, in the opposite direction of the Sun along the ecliptic. As a reminder, the ecliptic is the “great circle on the celestial sphere representing the sun’s apparent path during the year, so called because lunar and solar eclipses can only occur when the moon crosses it.” (Oxford)

As a result of this precession, the polar stars change over time. For example, the star approximately “above” the North Pole today is Polaris (α Ursae Minoris). Around 1500 BCE, it was Kochab (β Ursae Minoris). and between 3000 and 1500 BCE, it was Thuban (α Draconis).

Ptolemy: Hipparchus Discovered the Precession

One of the most important documents of ancient astronomy is Ptolemy’s Almagest.2 Therein, Ptolemy (c.100 – 160s/170) states that Hipparchus (c.190 – c.120 BCE) discovered the precession of the equinoxes, and confirms the same through his own calculations:

2. {That the sphere of the fixed stars, too, performs a rearward motion along the ecliptic}

From these considerations, and others like these, we can be assured that absolutely all the so-called fixed stars maintain one and the same position relative [to each other], and share one and the same motion. But the sphere of the fixed stars also performs a motion of its own in the opposite direction to the revolution of the universe, that is, [the motion of] the great circle through both poles, that of the equator and that of the ecliptic. We can see this mainly from the fact that the same stars do not maintain the same distances with respect to the solsticial and equinoctial points in our times as they had in former times: rather, the distance [of a given star] towards the rear with respect to [one of] those same points is found to be greater in proportion as the time [of observation] is later.

For Hipparchus too, in his work ‘On the displacement of the solsticial and equinoctial points’, adducing lunar eclipses from among those accurately observed by himself, and from those observed earlier by Timocharis, computes that the distance by which Spica is in advance of the autumnal [equinoctial] point is about 6° in his own time, but was about 8° in Timocharis’ time. For his final conclusion is expressed as follows: ‘If, then, Spica, for example, was formerly 8°, in zodiacal longitude, in advance of the autumnal [equinoctial] point, but is now 6° in advance’, and so forth. Furthermore he shows that in the case of almost all the other fixed stars for which he carried out the comparison, the rearward motion was of the same amount. And we also, comparing the distances of fixed stars from the solsticial and equinoctial points as they appear in our time with those observed and recorded by Hipparchus, find that their motion towards the rear with respect to the ecliptic is, proportionally, similar to the above amount. We conducted this type of investigation by means of the instrument which we constructed previously … for the observations of individual moon-sun distances. [In this case] we set one of the astrolabe rings to the apparent position of the moon (computed for the moment of observation), then adjusted the other astrolabe ring to align it with the star being sighted, so that both moon and star would be sighted simultaneously in the proper positions. Thus we obtained the position of every one of the bright stars from its distance from the moon. [H12-H13, pp.327-328, my emphasis]

Ptolemy dates the discovery to “the 50th year of the Third Kallippic Cycle”, which works out to -128/7, or 129/8 BCE:

But in the 50th year of the Third Kallippic Cycle [-128/7], as Hipparchus records from his own observations, [that star] had a distance to the rear of the summer solstice of 29 5/6°. Therefore the star on the heart of Leo has moved 25° towards the rear along the ecliptic in the 265 or so years from the observation of Hipparchus to the beginning [of the reign] of Antoninus [137/8], which was when we made the majority of our observations of the positions of the fixed stars. From this we find that 1° rearward motion takes place in approximately 100 years, as Hipparchus too seems to have suspected, according to the following quotation from his work ‘On the length of the year’: ‘For if the solstices and equinoxes were moving, from that cause, not less than 1/100th of a degree in advance [i.e. in the reverse order] of the signs, in the 300 years they should have moved not less than 3°’. [H15-H16, p.328]

Hamlet’s Mill: Knowledge of the Precession Is Ancient

But it is probable that knowledge of the precession of the equinoxes is much older than the work of Hipparchus. This idea was developed by Giorgio de Santillana (1902-1974) and Hertha von Dechend (1915-2001) in their remarkable 1969 book, Hamlet’s Mill: An essay on myth and the frame of time.3

Hamlet’s Mill is not an easy read. It is written in an inductive manner, starting from the Shakespeare play of that name, and moving on to tales from around the world, of which Hamlet is a latecomer, which share many aspects of that story.

With suggestive insights from other continents, it is time to take a fresh look at Shakespeare’s Gentle Prince, a cultivated, searching intellectual, the glass of fashion and the mold of form in the Danish Court, who was known once upon a time as a personage of no ordinary power, of universal position, and, in the North, as the owner of a formidable mill. [p.86, my emphasis]

Hamlet is directly inspired from the legend of Amleth (Old Norse: Amlóði; Latinized as Amlethus), as written by Saxo Grammaticus (c.1150 – c.1220). Therein appears a theme around a mill:

T’is said, sang Snaebjorn, that far out, off yonder ness, the Nine Maids of the Island Mill stir amain the host-cruel skerry-quern — they who in ages past ground Hamlet’s meal. The good chieftain furrows the hull’s lair with his ship’s beaked prow. Here the sea is called Amlodhi’s Mill. [p.87, my emphasis]

De Santillana and von Dechend continue:

The Mill is thus not only very great and ancient, but it must also be central to the original Hamlet story. It reappears in the Skaldskaparmal, where Snorri explains why a kenning for gold is “Frodhi’s meal.” Frodhi appears in the chronicles, but his name is really an alias of Freyr, one of the great Vanir or Titans of Norse myth. [p.87, my emphasis]

Through the analysis of numerous tales and myths from around the world, de Santillana and von Dechend conclude that the underlying themes are not terrestrial, but cosmic. At the core of those themes is the precession of the equinoxes. The mill is the nightly spinning of the heavens, attached to a pole or peg. But every couple of thousand years or so, the pole or peg is broken, a sort of End of Times arises, and a new pole or peg needs to be attached to the mill, for a New Age to begin. Furthermore, the older the variant of the story, the clearer it becomes that the themes are cosmic.

The historical national epic of Iran is the Book of Kings, i.e., the Shahnama [also written Shahnameh]. It was written around 1010 by Firdausi [Abolqasem Mansour bin Hassan Tusi (940–1025), also written Ferdowsi]. Therein, the hero, corresponding to Amlodhi, is named Kai Khusrau. At some point he says:

The whole world is my kingdom, all is mine
From Pisces downward to the Bull's head.

This is how de Santillana and von Dechend explain that passage:

He designates that section of the zodiac comprised between Pisces and Aldebaran, the thirty degrees which cover the constellation Aries. It means that his reign is not only of heaven, it is essentially of Time. The dimension of heaven is Time. [p.41, my emphasis]

Note that Kai Khusrau does not reign forever, but only for the duration of one constellation. If we use the estimate of 26'000 years from Wikipedia for the full cycle of Axial precession,4 then that means that Kai Khusrau’s reign is around 2166 years.

According to de Santillana and von Dechend, knowledge of the precession of the equinoxes goes back to 5000 BCE!

The sun’s position among the constellations at the vernal equinox was the pointer that indicated the “hours” of the precessional cycle — very long hours indeed, the equinoctial sun occupying each zodiacal constellation for about 2,200 years. The constellation that rose in the east just before the sun (that is, rose heliacally) marked the “place” where the sun rested. At this time it was known as the sun’s “carrier,” and as the main “pillar” of the sky, the vernal equinox being recognized as the fiducial point of the “system,” determining the first degree of the sun’s yearly circle, and the first day of the year. (When we say, it was “recognized,” we mean that it was spelled “carrier” or “pillar,” and the like: it must be kept in mind that we are dealing with a specific terminology, and not with vague and primitively rude “beliefs.”) At Time Zero (say, 5000 B.C. — there are reasons for this approximate date), the sun was in Gemini; it moved ever so slowly from Gemini into Taurus, then Aries, then Pisces, which it still occupies and will for some centuries more. The advent of Christ the Fish marks our age. It was hailed by Virgil, shortly before Anno Domini: “a new great order of centuries is now being born ...” which earned Virgil the strange title of prophet of Christianity. The preceding age, that of Aries, had been heralded by Moses coming down from Mount Sinai as “two-horned,” that is, crowned with the Ram’s horns, while his flock disobediently insisted upon dancing around the “Golden Calf” that was, rather, a “Golden Bull,” Taurus.

Thus, the revolving heavens gave the key, the events of our globe receding into insignificance. Attention was focused on the supernal presences, away from the phenomenal chaos around us. What moved in heaven of its own motion, the planets in their weeks and years, took on ever more awesome dignity. They were the Persons of True Becoming. The zodiac was where things really happened, for the planets, the true inhabitants, knew what they were doing, and mankind was only passive to their behest. [pp.59-60]

So, although we cannot name those who discovered the precession of the equinoxes, it is surely very ancient knowledge.

Copernicus’s Third Motion

Fast forward to Nicolaus Copernicus (1473-1543), to understand how the precession of the equinoxes is currently understood. Copernicus’s model of our stellar system includes three kinds of motion (Google AI Overview):

• Annual Revolution: The Earth travels in a circular orbit around the Sun once per year. This movement, combined with the Earth’s speed relative to other planets, explains the apparent retrograde motion of the planets.

• Daily Rotation (Diurnal Motion): The Earth spins on its axis once every 24 hours, accounting for the apparent daily rise and set of the Sun and stars.

• Third Motion (Annual Tilting): Known as motus declinationis, this motion involves the Earth’s axis maintaining a relatively constant angle, which contributes to the changing seasons and explains the precession of the equinoxes.

The first two motions are not difficult to imagine. The third motion, on the other hand, is variously explained as the Earth acting like a gyroscope, or some kind of spinning top with a wobble. Although it can be described mathematically, it is much harder to imagine an underlying physics.

Of course, today the first motion is not described as being circular. Rather, we use the elliptical orbits and the three laws of Johannes Kepler (1571-1630), and the law of universal gravitation of Isaac Newton (1643-1727). See my summary post My writings on Johannes Kepler, as well as my post Isaac Newton Derives Kepler's Ellipses from the Law of Universal Gravitation.

The TYCHOS Snail

So now it’s time to examine the TYCHOS. The three Copernican motions have equivalents in the TYCHOS:

• Tycho Brahe (1546-1601): Binary Sun/Mars system, with Earth close to the center of this system. This is described in detail in the previous post.

• Christen Sørensen Longomontanus (1562-1647): Daily rotation. This is the same as for Copernicus.

• Simon Shack: The Earth moves in a slow orbit whose period is 25'344 years.

The third part is the real innovation of the TYCHOS system. Shack gives the motivation:

Assuming that, contrary to Copernican dogma, Earth does not wobble around its polar axis, but moves clockwise in a local orbit under the pole stars, the same effect would be produced. It may at first seem highly unorthodox to assign a local orbit to Earth, but is it really? After all, every single celestial body in our skies is known to move in a local orbit of its own. [p.91, my emphasis]

The key question, of course, is the exact nature of the orbit of the Earth’s motion. What is the diameter of this orbit, and how fast does the Earth move along this orbit? These questions can be answered with the following diagram.

In the top right corner, we see the north-star ring in a 25'344 year cycle. Currently our North star is Polaris, with an 89º declination, i.e., Polaris is 1º away from being directly over the North Pole. In some 11'500 years, our North star will be Vega, with an 86º declination. The north-star ring subtends about 44º of our celestial sphere. The 44º corresponds to 2h56min of RA (Right Ascension), which the Sun, in its voyage around the Earth, covers in 44 days (1056 hours).

Now what Shack did is to imagine a “solar polar orbit”, whose center is that of the Earth’s orbit, and whose diameter is such that the chord for 44º is the same length as that of the arc of the Sun’s normal orbit for 44º, as can be seen in the main part of the diagram. That length is taken as the diameter of the Earth’s orbit, called PVP, which stands for Polaris-Vegas-Polaris. [p.92]

The diameter of the PVP orbit can be estimated with a simple calculation. Assuming the Sun travels at 107226 km/h and covers the distance between Polaris and Vega in 44 days (1056 hours), we would have:

107226 × 1056 = 113 230 656 (km)

So, according to Shack’s calculations, the diameter of the PVP orbit is in the order of 113.2 Gm; furthermore, the period of the PVP orbit is 25'344 years. Thus it becomes possible to compute the Earth’s orbital speed [p.94]:

• 1 year = 365.25 days = 8766 hours

• 1 TGY = 25344 solar years × 8766 hours = 222 165 504 hours

• The PVP orbit’s circumference = 355 724 597 km

• Orbital speed of Earth ≈ 1.601169 km/h

So, in the TYCHOS, the Earth moves along at a true snail’s pace, approximately 1.6 km/h [one mile an hour]. Hence my title,

Hamlet’s Mill Advances at 1.6 km/h.

So what happens with the Earth’s tilt, as it moves along this slow orbit? According to Shack, this is quite simple. The northern hemisphere of the Earth is much heavier than the southern hemisphere. In the Wikipedia entry for “Hemispheres of Earth”,5 the northern hemisphere “contains approximately 68% of Earth's landmass”. He writes:

It would thus seem intuitively logical, even to devout Newtonian advocates, that Earth’s heavier hemisphere would hang ‘outwards’ as our planet goes around its Polaris-Vega-Polaris (PVP) orbit. [p.73]

This is illustrated in the following diagram.

Now it must be said that when first reading the manner in which the PVP orbit has been calculated, it is not clear how sound is the reasoning. One possible indicator that the reasoning is sound is that, according to these calculations, when Mars is closest to Earth, it passes close to the center of the PVP orbit! Shack writes:

Mars can transit as close as 0.373 AU from Earth (as it did in 2003). However, … the mean figure of its closest passages is about 0.379 AU [= 56.6 Mkm].

I’d like to state, for the record, that I only realized this astounding fact long after I had estimated the diameter of the PVP orbit (113.2 Mkm, or 2 × 56.6 Mkm). Needless to say, it lends considerable support to the proposed diameter of the PVP orbit—unless you are willing to chalk it all up to sheer coincidence. [p.96, my emphasis]

So what does this mean? He writes:

So, if Mars regularly transits in the middle of the PVP orbit, what long-term implications would this have under the TYCHOS paradigm? Well, as Patrik Holmqvist and I proceeded to fine-tune the Tychosium simulator, we were obviously curious to see how Mars would ‘behave’ over a full Great Year of 25344 solar years. The result of this test is illustrated in [the image below] which was put together by simply superimposing 4 screenshots from the Tychosium simulator, each one of them separated by 6336 years. All in all, the TYCHOS model reveals the breathtaking beauty and geometric harmony of our Solar System—in an even more spectacular manner than Johannes Kepler ever envisioned in his dreamy “Harmonices Mundi” treatise.

Note how even the highly eccentric orbit of Mercury maintains its geometric relationship with the Sun throughout the TGY of 25344 years. [p.97]

In the diagram below, the green dot is the center of our stellar system, the turquoise circle is the Earth’s PVP orbit, the red circles are orbits of Mars, the yellow circles are orbits of the Sun, and around the Sun are the circles for the orbits of Venus and Mercury.

If the TYCHOS is correct, we get a remarkably simple explanation of the precession of the equinoxes, a phenomenon that has fascinated our ancestors for at least 7000 years: the Earth is following a circular orbit, called the PVP orbit, at a snail’s pace of 1.6 km/h, with the North Pole always pointing outwards from this orbit.

For me, this is a fascinating exploration of our stellar system. Our next stop will be the Great Inequality of Jupiter and Saturn.

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