Newton’s Principia

Newton’s Principia has received more attention from philosophers than any other scientific book. But from one century to the next, the causes of this and the attention’s primary emphasis have changed substantially. Philosophers have considered the Principia from the perspective of Einstein’s new theory of gravity in his theory of general relativity throughout the 20th century. What the necessity to replace Newton’s theory of gravity with Einstein’s theory of gravity indicates about the nature, scope, and boundaries of scientific knowledge has been one of the key points of contention.

Contrarily, Newton’s theory of gravity was contested throughout the majority of the 18th century, in part due to the lack of a mechanism, specifically a contact mechanism, that generated gravitational forces. Accordingly, the philosophical literature made an effort to explain and settle, in one way or another, the controversy around whether or not the Principia might be regarded as methodologically sound. Newton’s theory of gravity had been widely accepted by the 1790s among those working on orbital mechanics and physical geodesy research, making the Principia the quintessential example of science at its best.

Therefore, during the 19th century, philosophical interest in Newton’s Principia began to center on how he had accomplished this feat, partly to describe the knowledge attained and partly to pursue analogous information in other fields of study. Unfortunately, a simplification of the Principia has plagued a sizable portion of the philosophical literature throughout the three centuries. The major objective of this essay is to replace that oversimplified image with one that better captures the depth of the Principia’s approach and content.

Overview of Newton’s Principia: The Work’s Importance

No work was more influential in advancing contemporary physics and astronomy when looked at retrospectively than Newton’s Principia. It’s finding that the force keeping the planets in their orbits is similar to terrestrial gravity, putting an end to the belief that the celestial and sublunar realms require different sciences, which dates back to Aristotle. The final triumph of Newton’s theory of gravity made the identification of the basic forces of nature and their formulation in laws the main goal of physics, just as the Preface to its first edition had suggested.

A new paradigm of exact science emerged due to the theory’s success. It holds that any systematic disagreement between observation and theory—no matter how small—tells us something significant about the universe. Physical theory replaced observation as the preferred method for addressing specific questions about the world once it became clear that the theory of gravity offered a far more efficient method than observation for precisely characterizing complex orbital motions, just as Newton had suggested in the Principia in the case of the Moon’s orbit.

Following the development of Einstein’s special and general theories of relativity, the retroactive interpretation of the Principia has changed from that of the nineteenth century. In much the same way that Galileo’s and Huygens’ conclusions for motion under uniform gravity came to be regarded as holding only to high approximation in the wake of Newtonian inverse-square gravity, the Newtonian theory is now considered to hold only to high approximation in restricted conditions.

The Principia, however, was viewed as the pinnacle of empirical science in the middle of the nineteenth century, much as Euclid’s Elements had been at the start of the seventeenth century when there was no reason to believe that any confusing discrepancy between Newtonian theory and observation was ever going to emerge. The Principia has maintained its unique fundamental place in the history of physics in our post-Newtonian era because of how deeply Einsteinian theory was historically based on Newtonian science.

More remarkably, even though the Principia can no longer be regarded as an example of perfection, it is still widely regarded by physicists as an example of empirical skeptic theory because of the logical relationship between Newtonian and Einsteinian theory. Einstein demonstrated that Newtonian gravity holds as a limit-case of general relativity in the same way that Newton demonstrated (in Book 1, Section 10) that Galilean uniform gravity holds as a limit-case of universe.

Despite extravagant claims made about the Principia in the years following its publication, such as “… he seems to have exhausted his Argument, and left little to be done by those that shall succeed him”

1, the most favorable assessment of it that anyone could have substantiated during the first half of the eighteenth century would have focused more on its promise than its accomplishments. The lunar apogee’s mean speed differed by a factor of 2, undermining the idea that the Moon is held in orbit by an inverse-square force. This mismatch was one of several obvious loose ends in the theory of gravity. No one had a better understanding of the potential of the theory of gravity to answer a wide range of planetary astronomy concerns than Newton himself, which may explain why he made these loose ends impossible to discover unless by the most technically proficient, meticulous readers. When a few loose ends were tied between the late 1730s and the early 1750s, the situation underwent a significant transformation, sometimes producing extraordinary outcomes like the first fully successful descriptive explanation of the motion of the Moon in the field of astronomy.

The promise of the Principia was not only unanimously acknowledged by those engaged in empirical inquiry throughout the second half of the eighteenth century, but a significant portion of this promise was also fulfilled. This approach led to the development of what is now known as “Newtonian mechanics,” as well as the gravity-based explanations for the planets’ sometimes significant departures from Keplerian motion, Newton’s theory of gravity’s accomplishment that put a stop to all denial of it.

In many areas, the largely Cartesian worldview that had replaced the Scholastic worldview in the latter part of the seventeenth century throughout the eighteenth century was perceived as directly opposed by the Principia at that time. When Newton altered the work’s name in 1686 to Philosophiae Naturalis Principia Mathematica about Descartes’s most well-known work at the time, Principia Philosophiae, it was evident that he meant the work to be understood in this way. (The first and third words of the title of Newton’s first edition’s title page were printed in bigger font to emphasize this allusion.)

Removing the planet-carrying vortices from the celestial spheres marked Newton’s Principia’s immediate departure from the conventional worldview. Following Newton, Newtonians improved this worldview in several ways, including the idea that forces everywhere expressly act at a distance. For instance, Laplace introduced the notion of a “clockwork cosmos” late in the eighteenth century, when it was clear that the theory of gravity successfully explained intricate departures from Keplerian motion. This notion is not contained in Newton’s Principia.

Newton considered the Principia as illustrative of a new way of approaching natural philosophy in addition to seeing the theory of gravity as possibly altering orbital astronomy. The emphasis on forces was one feature of this new approach, as was stated in the Preface to the first edition:

Because identifying the forces of nature from motion-related events and then using these forces to illustrate other phenomena seems to be the entire challenge of philosophy. The broad premises in books 1 and 2 aim to achieve these goals, and in book 3, our explanation of the world’s system exemplifies these ideas. Because in book 3, we infer from celestial phenomena the gravitational forces by which things gravitate toward the Sun and the various planets, using hypotheses that were mathematically proven in books 1 and 2.

Then, by means of mathematically sound hypotheses, the movements of the planets, comets, the Moon, and the sea are inferred from these forces. If only we could apply the same thinking to other natural occurrences and derive them from mechanical principles! Because of several factors, I have a sneaking hunch that all occurrences may be influenced by some forces that drive body particles to either be attracted to one another and form regular forms or to be attracted to another and move away. Philosophers have thus far tried to test nature in vain since these forces are unknown. But I do hope that the ideas presented here will help to illuminate either this school of thought or a more accurate one. [P, 382]

A second feature of the new approach is the use of mathematical theory to cover a wide variety of different theoretical possibilities, allowing the empirical world to choose from them rather than using it to generate testable conclusions from hypotheses, as Galileo and Huygens had done. At the end of Book 1, Section 11 makes the most compelling case for this new strategy:

I use the word “attraction” here in a broad sense to refer to any effort made by bodies to approach one another, regardless of whether that effort results from the action of the bodies being drawn toward one another, acting on one another through the release of spirits, or from the action of ether, air, or any other medium — whether corporeal or incorporeal — in any way impelling the bodies floating therein toward one another.

I use the term “impulse” in the same broad meaning since, as I have stated in the definitions, this dissertation focuses on amounts and mathematical proportions rather than the types of forces and their physical characteristics. Investigating the forces and their proportions that result from any hypothesized conditions is necessary for mathematics. To determine which conditions of forces apply to each type of attractive substance, it is necessary to compare these proportions to the phenomena. Finally, it will be feasible to discuss the physical species, physical sources, and physical quantities of these forces with more assurance. [P, 588]

Thirdly, even when the mathematical theory of the species and proportions of the forces seemed to leave no other option but action at a distance, the new method was willing to put questions about the mechanism by which forces effect their changes in motion on hold. This proved to be the most contentious aspect of the method at the time. In response to its critiques, this feature was rendered explicitly polemical in the General Scholium that was inserted after the second edition. This aspect was rather implicit in the first edition.

I don’t makeup theories, but I haven’t yet been able to infer from phenomena the cause of these characteristics of gravity. Because everything that cannot be inferred from the observations is said to be a hypothesis, experimental philosophy does not allow using mechanical, occult, or metaphysical hypotheses. In this experimental philosophy, generalizations are achieved through inducing hypotheses from occurrences. This approach has led to the discovery of the laws of motion and gravity and the impenetrability, mobility, and impetus of bodies. And it is sufficient that gravity exists, behaves according to the rules we have established and is responsible for all planetary and oceanic motions. [P, 943]

The main problem Principia posed to philosophers during the eighteenth century was how to interpret a mathematical theory of forces without any mechanism other than action at a distance. However, by the century’s latter decades, there was little space for doubting if gravity operates in accordance with Newton’s principles and is sufficient to explain all motions of the celestial bodies and our waters.

Nobody could dispute the emergence of science that, at least in some regards, was so far superior to anything that had come before that it stood alone as the pinnacle of science. Philosophers were then faced with the issue of describing the specific nature and boundaries of the knowledge obtained in this discipline, followed by the methodological details of how this astounding breakthrough had been made to pave the way for other fields of study to accomplish the same.

The Newton Principia’s Historical Setting

According to a widely held belief, Newton developed his theory of gravity to explain the already well-known “laws” of orbital motion discovered by Kepler. The universality of the law of gravity ultimately led to the explanation of Keplerian motion deviations resulting from planets’ gravitational interactions. This is incorrect on several levels, the most obvious being that Kepler’s “rules” were not established prior to the Principia. In the first two decades of the seventeenth century, Kepler developed a set of principles for estimating orbital motion that dramatically improved accuracy over earlier methods.

For the motion of the Moon, Kepler’s criteria, however, did not produce comparable accuracy, and even in the case of the planets, the estimated positions were occasionally wrong by as much as a fourth of the Moon’s breadth. More significantly, by 1680, several other methods for calculating the orbits had been proposed and attained a precision that was nearly as good as Kepler’s. Newton was knowledgeable with seven alternative methods for calculating planetary orbits, all of which had nearly the same accuracy. Only two of these—and Kepler’s Jeremiah Horrocks’s—used the planets’ trajectories to find them and used Kepler’s area rule, which states that planets sweep out similar regions at about the same rates as the Sun.

The area rule was superseded by a geometric structure by Ismael Boulliau and Thomas Street after them (from whose Astronomia Carolina Newton first studied orbital astronomy). After using a point of equal angular motion oscillating about the empty focus of the ellipse earlier, Vincent Wing adopted another geometric structure in the late 1660s. Nicolaus Mercator introduced yet another geometric construction in 1676. [4] Only Horrocks and Street, who came after him, took Kepler’s 3/2 power rule seriously enough to use the periods rather than positional observations to determine the mean distances of the planets. This rule states that the periods of the planets vary as the square root of the cube of their mean distances from the Sun.

These methods adopted Kepler’s method of representing the trajectory with an ellipse. (The main historical justification for this was Kepler’s accurate forecast of Mercury’s passage across the Sun in 1631.) This does not imply that the ellipse was formed as anything other than a near approximation to the genuine orbit, which is something that may be accomplished mathematically. The known planetary orbits from that time are not very elliptical.

In all other situations, the difference between an ellipse and an eccentric circle was indistinguishable. Mercury’s minor axis is just 2% shorter than its major axis, Mars’ minor axis is only 0.4 percent shorter, and so on. By stating that “Kepler recognized the Orb to be not round but oval, and guest it to be Elliptical” in a letter to Halley in June 1686, Newton had legitimate justification for asserting his “right” to the ellipse [C, II, 436]. After reading the free copy Newton had provided him, Christiaan Huygens, the most critical reader of the first edition of the Principia, independently penned the following evaluation of its accomplishments in his notebook:

The renowned M. Newton ignored all the challenges presented by the Cartesian vortices and demonstrated that the planets are kept in their orbits by their gravitational attraction to the Sun. Additionally, the eccentrics inevitably get elliptical.

Therefore, when Newton began working on the project in 1684, it was clear that all three of Kepler’s rules—later referred to as “laws” after the Principia—were nothing more than holding to high approximation. At that time, the main question in orbital astronomy was not whether Kepler’s laws still applied but rather which of the several equally precise methods of calculating orbits should be selected.

What trajectory does a body depict while traveling under an inverse-square force directed at a central body? A question Hooke asked Newton in 1679 and Halley asked him again in 1684 is suitable given the possibility that the ellipse is simply an approximation to the genuine trajectory. Huygens’ mathematical theory of uniform circular motion, which was published in his Horologium Oscillatorium in 1673, and Kepler’s 3/2 power law were combined to provide the inverse-square component of this question: Although the squares of the periods of the planets vary as the cubes of their mean distances, the forces holding the planets in their orbits, at least to a first approximation, vary inversely with the square of the radii of their nearly circular orbits.

This is because the force in a string holding a body in a uniform circular orbit varies directly as the circle’s radius and inversely as the square of the period. Now, however, allow the orbiting body’s distance from the center to fluctuate instead of staying the same as in a circle. What trajectory would be produced if the force is always directed toward the center but changes with the inverse square of its distance? In his nine-page “De Motu Corporum in Gyrum” tract, which Newton gave to Halley in November 1684, Newton supplied the solution: an ellipse, provided the velocity is not excessive (and if it is, then instead a parabola or a hyperbola, depending on the velocity).

The crucial step in formulating this response is a generalization of uniform circular motion to the case of a motion under a “centripetal” force, which Newton coined from Huygens’s “centrifugal” force, which he used to refer to the tension in the string keeping the body in a circle; a crucial factor in this step was the discovery that a body is moving under any type of centripetal force always sweeps out equal areas in equal times concerning that center, The tract also verifies that for things orbiting in confocal ellipses driven by inverse-square centripetal pressures, Kepler’s 3/2 power law still holds.

These were impressive advancements at the time, but they represent the beginning of the environment in which Newton would later write the Principia. Newton rewrote the “De Motu” tract and added two new parts shortly after it was sent to London. This adjustment appears to have been prompted by a query regarding the impact Jupiter’s satellites’ inferred inverse-square centripetal pressures have on the Sun. Newton originally introduced two principles, which he initially referred to as “hypotheses” before redesignating them as “laws”:

Law 3: Whether a space is at rest or travels endlessly and evenly in a straight line without circular motion, the relative movements of bodies included inside it are always the same.

Law 4: The reciprocal activities of bodies do not change the shared center of gravity’s state of motion or rest. [U, 267]

The second of the two new paragraphs discuss motion in refractory media and gives readers a framework for understanding Book 2 of the Principia.

We now cite in full the first new line, referred to as the “Copernican scholium,” since it best describes what motivated Newton to conduct the further study that resulted in the five hundred-page Principia from the nine-page tract. It appears as a single, lengthy paragraph but is now divided into three parts to make commenting on it easier:

Contrary to popular belief, the entire planetary skies are either at rest or evenly moving in a straight line. Similarly, according to Law 4, the planets’ shared centers of gravity are either at rest or simultaneously moving. In either scenario, the planets’ movements relative to one another (as dictated by Law 3) proceed similarly, and their shared center of gravity is at rest relative to the entirety of space. Therefore it should be regarded as the stationary core of the entire planetary system. Thus, the Copernican system is indisputable from the start. Because if a shared center of gravity for any location of the planets is calculated, it either lies in the Sun’s body or will always be quite close to it.

The planets neither travel precisely in ellipses nor rotate again in the same orbit due to the Sun’s divergence from the center of gravity’s centripetal force, which does not always gravitate to that stationary center. Every time a planet spins, it creates a new orbit, much how the Moon moves. Each orbit depends on the total movements of all the planets and how they interact with one another. If I am not greatly misinformed, it would be beyond the scope of human intelligence to take into account so many causes of motion at once and characterize the motions by precise rules that would make calculations simple.

Leaving aside these minute details, the ellipse I previously explained will be the precise orbit representing the average of all vagaries. Anyone attempting to calculate this ellipse using trigonometry from three observations (as is typical) will be acting recklessly if they do so. As a result, there will be as many ellipses that differ from one another as there are trios of observations used.

This is because these observations will contribute to the small irregular motions neglected here and cause the ellipse to deviate somewhat from its actual magnitude and position (which ought to be the mean among all errors). Therefore, a large number of observations must be combined and assigned to a single operation that mutually modifies each observation and displays the mean ellipse in terms of location and magnitude. [U, 280]

A different aspect of the historical setting in which the Principia was written and read is highlighted in the first section. Galileo’s 1613 discovery of Venus’ phases offered irrefutable proof against the Ptolemaic system, but it was unable to establish a case for the Copernican over the Tychonic system. In the latter, the seven bodies are always in the same relative positions as in the Copernican system since Mercury, Venus, Mars, Jupiter, and Saturn circle the Sun and the Earth, respectively. One of the most talked-about topics of the seventeenth century was whether conclusive empirical evidence could be produced supporting the Copernican system over the Tychonic one.

In the first half of the century, Kepler, Galileo, and Descartes wrote significant volumes that proposed addressing this issue. Kepler and Descartes based their justifications on the physical mechanisms they had each suggested as regulating the orbital motion. However, G. D. Cassini, the most prominent observational astronomer of the century’s second half, was a Tychonist. Newton declares that the centripetal forces identified in the text of “De Motu” as governing orbital motion open the way to establishing a slightly modified form of the Copernican system in the first section of the “Copernican Scholium.” This is the technical issue at the heart of the debate between the two systems. Newton’s discovery of this line of thinking was undoubtedly a significant motivator for him to continue writing the Principia.

The second section of the “Copernican Scholium” discusses a problem in orbital astronomy that has a different role in Principia’s historical background. The question of whether the genuine movements are much more irregular and difficult than the computed motions in any of these systems was separate from the issue of which approach—or Kepler’s another—should be selected. This question was motivated in part by the complexity of the lunar orbit and the ongoing inability to characterize it accurately using the same methods as Kepler did for the planets.

Another arose from Kepler’s discovery that the real movements may entail additional oddities, as demonstrated by apparent variations in the values of orbital elements over time, which was stated in the Preface to his Rudolphine Tables and later corroborated by others. However, the most crucial factor in this matter was Descartes’ assertion that the orbits are not mathematically precise and “they are perpetually modified by the passing of the centuries” [D, 3, 34] in accordance with the changing movements of his vortices over extended periods.

In the second section of the Scholium that was cited, Newton concludes that, in contrast to the ellipse he used to respond to Hooke and Halley’s mathematical query, the real orbits are, in fact, infinitely complicated rather than ellipses. Knowledgeable readers nevertheless perceived the work as providing a negative response to the issue of whether the actual movements are mathematically flawless. This result is nowhere so strongly expressed in the published Principia.

Finally, the second and third portions make it clear that Keplerian motion is simply a rough approximation of the genuine movements and draw attention to the dangers of utilizing the orbits discovered by Kepler and others to support planetary system theories. For instance, it is not unexpected that all of the many calculational procedures were attaining equivalent accuracy, given the real movements are so complex, as they all, at most, hold only roughly.

Additionally, since the abnormalities they implied could not be easily ignored, the accuracy of the orbit calculations could not be used to refute the existence of Cartesian vortices. There might be too many theories that match the same evidence, a concern Newton had emphasized during the debate over his previous light and color studies. Even worse, from the end of the sixteenth century onward, the multiplicity of tenable hypotheses haunted the field of mathematical astronomy.

It would have been assumed that the conclusion that calculated orbits can only be rough estimates had raised the possibility that accuracy and truth were beyond the purview of mathematical astronomy. Newton’s attempt to arrive at conclusions that claim to be correct and true despite the excessive complexity of the real movements is the fundamental reason the Principia covers so much more than the “De Motu” tract.

Readers of the first edition of the Principia perceived it as addressing a number of difficulties present in the historical setting in which Newton wrote it, including Which method of determining the orbits—or Kepler’s—should be used? Was the debate over the Copernican vs. Tychonic system settled on some factual basis? Were the real motions complex and erratic when compared to the estimated movements? Is mathematical astronomy a particular field of study?

It took two hundred years for Newton’s “Copernican Scholium” to be published. Thus readers of the Principia at the time could not see how he connected these issues. Nothing, however, makes the extent to which Newton’s obsession with the issue of drawing conclusions that had a claim to being precise from facts that, in his estimation, held at best to high approximation more obvious. This worry is what led to the extended scope of the Principia. This makes the “Copernican scholium” the most instructive setting to study the Principia. The fact that it was unknown for such a long time also explains why the Principia has typically been interpreted superficially.

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