Principia: Deep Review Isaac Newton ## 1. Introduction and the Architecture of a Revolution Welcome to Emma’s Library. This is an original deep review and conceptual guide to one of the most formidable and transformative works in the history of human thought: Isaac Newton’s *Philosophiæ Naturalis Principia Mathematica*, or the *Mathematical Principles of Natural Philosophy*, first published in London in 1687. Please note that this episode is an original scholarly analysis and commentary, not a direct reading of the source text. Our goal is to make the conceptual architecture of this notoriously difficult book navigable and clear, explaining its internal logic, its historical context, and the profound ways in which it redefined what it means to explain the physical world. To open the *Principia* is to enter a cathedral of geometry, mechanics, and cosmology. Written in a dense, demanding Latin and structured in the style of Euclid’s *Elements*, the book was deliberately designed by Newton to be difficult. He famously admitted that he made it highly mathematical to avoid being baited by superficial critics who lacked the mathematical training to understand his arguments. Yet, beneath its forbidding exterior of geometric diagrams, lemmas, and propositions lies a unified vision of the cosmos that shattered the fractured medieval and early modern views of nature. Before Newton, the universe was divided. The earthly realm of decay, change, and straight-line motion was governed by one set of rules, while the heavenly realm of perfect, eternal, circular motion was governed by another. Newton’s *Principia* swept this duality away. With a handful of mathematical definitions, three laws of motion, and a single, universal law of gravitation, Newton bound together the falling of an apple, the trajectory of a projectile, the orbits of the planets, the behavior of the tides, and the wild, unpredictable paths of comets. In this review, we will survey all three books of the *Principia*. We will examine the definitions and axioms that lay the groundwork, trace the mathematical methods Newton used to construct his proofs, explore the dynamics of orbiting bodies in Book One, analyze his systematic destruction of rival Cartesian physics in Book Two, and marvel at the grand cosmic synthesis of Book Three. We will also examine his famous rules of reasoning, the philosophical limitations of his system, and how this masterpiece continues to shape our understanding of scientific inquiry. --- ## 2. The Problem This Book Is Trying To Solve To appreciate the magnitude of Newton’s achievement, we must understand the intellectual chaos of the seventeenth century. The scientific world was in the midst of a profound identity crisis. The ancient Aristotelian worldview, which explained the motion of objects through their inherent purposes or teleology—such as a stone falling because it desired to reach its natural place at the center of the Earth—was collapsing under the weight of new observations. Galileo Galilei had used the telescope to reveal mountains on the Moon and moons orbiting Jupiter, proving that the heavens were not made of an immutable, perfect fifth element. He had also begun to formulate a mathematical science of terrestrial motion, showing that falling bodies accelerate at a constant rate regardless of their weight. Meanwhile, Johannes Kepler, working with the precise observational data of Tycho Brahe, had discovered that the planets do not move in perfect circles, but rather in ellipses, speeding up when they are close to the Sun and slowing down when they are far away. However, these discoveries were isolated pieces of a puzzle that did not fit together. Galileo’s physics applied only to local, terrestrial motions, and he largely ignored Kepler’s elliptical orbits. Kepler’s laws were brilliant mathematical descriptions of planetary paths, but they lacked a physical explanation. Why did the planets move in ellipses? What kept them in their orbits instead of flying off into space? Kepler speculated about magnetic virtues emanating from the Sun, but he could not construct a rigorous, mathematical system to prove his ideas. In France, René Descartes had attempted to build a complete mechanical philosophy. Descartes argued that the universe is entirely filled with matter—a plenum—and that all physical changes occur through direct contact and collision. For Descartes, the planets were carried around the Sun in vast, swirling vortices of cosmic dust, much like leaves caught in a whirlpool. While Cartesian physics was highly popular because it offered an intuitive, mechanical picture of the world, it was qualitative and mathematically imprecise. It could not predict the exact positions of the planets, nor could it explain why Kepler’s laws held true. This was the crisis Newton inherited. The world was caught between the precise but unexplained mathematical descriptions of Kepler, the localized mechanics of Galileo, and the imaginative but mathematically bankrupt mechanical philosophy of Descartes. The problem Newton set out to solve was nothing less than the integration of these disparate elements into a single, mathematically rigorous, and empirically verifiable system. He needed to find a physical cause that could simultaneously account for Galilean acceleration on Earth and Keplerian orbits in the heavens, and he had to prove this cause using the absolute certainty of mathematical demonstration. --- ## 3. The Definitions and the Axioms of Motion Newton begins the *Principia* not with grand cosmological speculations, but with a series of precise definitions and axioms. This foundational section is crucial because it establishes the vocabulary of modern physics. He starts by defining "quantity of matter," which we now call mass, as the measure of matter arising from its density and bulk conjointly. This was a major conceptual breakthrough. Before Newton, weight and mass were often confused. Newton recognized that weight is a variable force that depends on an object's location, whereas mass is an intrinsic property of the body that remains constant whether the object is on the Earth, the Moon, or in deep space. He then defines "quantity of motion," which we know as momentum, as the product of an object's velocity and its mass. Following these definitions, Newton introduces his famous Scholium on Space and Time. Here, he makes a profound philosophical move by distinguishing between relative and absolute space, time, and motion. Relative space and time are what we measure with our senses and instruments—hours, miles, and positions relative to other objects. Absolute space and time, however, are the independent, unchanging backdrops of the universe. Absolute time flows equably without relation to anything external, and absolute space remains always similar and immovable. To prove that absolute motion is real and not just relative, Newton describes his famous bucket experiment. If a bucket filled with water is suspended by a strongly twisted cord and then released, the bucket begins to spin. Initially, the water remains flat because it has not yet started to spin with the bucket. Here, there is high relative motion between the bucket and the water, but no physical effect on the water's surface. As the motion is communicated to the water, it begins to spin, and its surface becomes concave, climbing up the sides of the bucket. At this point, the water and the bucket are spinning at the same speed, meaning their relative motion is zero, yet the water's surface is curved. Newton argues that this curvature is the physical effect of centrifugal force, which reveals that the water is rotating not relative to the bucket, but relative to absolute space itself. With these foundations laid, Newton presents his three Laws of Motion, which serve as the axioms of his entire physical system. The First Law, the Law of Inertia, states that every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. This overturned two thousand years of Aristotelian physics, which claimed that motion required a continuous force to sustain it. For Newton, motion is a state, not a process; it is change in motion that requires an external cause. The Second Law quantifies this change, stating that the alteration of motion is ever proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed. In modern terms, we write this as force equals mass times acceleration. This law establishes that force is not what causes velocity, but what causes acceleration—a change in speed or direction. The Third Law states that to every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and directed to contrary parts. If you press a stone with your finger, your finger is also pressed by the stone. This law is the conceptual bridge that allows Newton to construct his theory of universal gravitation, as it implies that gravity is not a one-way pull from a dominant body, but a mutual interaction between all masses in the universe. --- ## 4. The Mathematical Method of the Principia One of the greatest surprises for a modern reader opening the *Principia* is its mathematical style. We know that Newton, along with Gottfried Wilhelm Leibniz, invented the calculus—what Newton called the method of fluxions. Yet, when we look at the pages of the *Principia*, we do not find the differential and integral symbols of calculus. Instead, we find a dense thicket of Euclidean geometry, filled with triangles, circles, tangents, and ratios. Why did Newton choose to express his revolutionary physics in this ancient mathematical language? There are several reasons, both rhetorical and intellectual. First, geometry was the gold standard of mathematical certainty in the seventeenth century. Calculus was a brand-new, highly controversial tool that many mathematicians viewed with suspicion because it relied on the conceptually murky idea of infinitesimals—quantities that were infinitely small but not quite zero. If Newton had used calculus to prove his physics, his critics could have dismissed his conclusions by attacking his mathematical methods. By using geometry, he built his arguments on a foundation that no educated contemporary could dispute. However, the geometry of the *Principia* is not the static geometry of Euclid. It is a dynamic, living geometry that incorporates the concept of limits. Newton calls this the "Method of First and Last Ratios." In the opening sections of Book One, he establishes a series of mathematical lemmas that allow him to analyze curves by treating them as the limit of a series of straight lines as the intervals between them shrink to zero. Imagine a curve. Newton inscribes a series of triangles under this curve. As the width of these triangles becomes infinitely small, the sum of their areas approaches the exact area under the curve. This is, in reality, the geometric equivalent of integral calculus. Newton uses this method to calculate the continuous action of a force by breaking it down into a series of instantaneous, discrete impulses, and then finding the limit as the time between these impulses vanishes. This proof architecture is both brilliant and demanding. Newton forces his reader to visualize physical processes geometrically. A force is represented by a line segment; time is represented by an area; velocity is represented by a ratio of lengths. By translating physical dynamics into geometric relations, Newton was able to achieve a level of rigorous proof that had never before been seen in natural philosophy. He did not merely describe how things move; he demonstrated with geometric necessity why they must move in precisely that way under the influence of specific forces. --- ## 5. Book One and the Dynamics of Orbiting Bodies Book One of the *Principia*, titled *The Motion of Bodies*, is an abstract, mathematical treatise on how objects move in a vacuum under the influence of centripetal forces. A centripetal force is a "center-seeking" force—a term Newton coined to describe any force that draws or impels a body toward a central point. Newton begins by proving a proposition that is a cornerstone of his entire system: Proposition One. He demonstrates that if a body is moving in space and is continually pulled toward a single, fixed point by a centripetal force, the body will move in a single plane, and its radius vector—the line connecting the body to the center of force—will sweep out equal areas in equal times. This is an extraordinary result. It is the mathematical proof of Kepler’s Second Law of planetary motion. But whereas Kepler had discovered this law empirically by studying the orbit of Mars, Newton proved that it is a universal mathematical consequence of *any* centripetal force, regardless of its strength or nature. If an object sweeps out equal areas in equal times around a point, it must be under the influence of a force directed toward that point. Newton then turns his attention to the relationship between the shape of an orbit and the mathematical law of the force that produces it. In Proposition Eleven, he addresses the specific case of an ellipse. He proves that if a body moves in an ellipse, and the force is directed toward one of the foci of that ellipse, the strength of that force must decrease in proportion to the square of the distance between the body and the focus. This is the derivation of the famous inverse-square law. To understand the beauty of this proof, consider what Newton has accomplished. He has shown that Kepler’s first two laws of planetary motion—that planets move in ellipses with the Sun at one focus, and that they sweep out equal areas in equal times—are the direct mathematical signatures of a single, central force that decreases with the square of the distance. Book One goes on to explore other force laws, such as forces that increase directly with distance, and other orbital shapes, including parabolas and hyperbolas. Newton demonstrates that these three curves—ellipses, parabolas, and hyperbolas—are all conic sections, and that they are the only possible paths a body can take when moving under the influence of an inverse-square centripetal force. A body with low energy will fall into an elliptical orbit; a body with precisely enough energy to escape will fly away along a parabola; and a highly energetic body will trace a hyperbolic path, never to return. By the end of Book One, Newton has constructed a complete mathematical toolkit for orbital dynamics. Yet, this book remains entirely abstract. It does not mention the actual planets, the Sun, or the Earth. It is a work of pure mathematical physics, establishing what *must* happen in an idealized universe of point masses and perfect vacuums. --- ## 6. Book Two and the Physics of Resistive Media Book Two, titled *The Motion of Bodies in Resisting Mediums*, is the most mathematically complex and frequently neglected part of the *Principia*, yet it plays a vital role in Newton’s overall argument. If Book One is about motion in an idealized vacuum, Book Two is about motion in the messy, real world of fluids, air, and water. Newton’s primary motivation for writing Book Two was polemical. He needed to dismantle the dominant physical theory of his day: René Descartes’ theory of cosmic vortices. Descartes had argued that space is filled with a subtle fluid matter that swirls around the Sun, carrying the planets along with it like boats in a whirlpool. This was an attractive theory because it avoided the mysterious concept of "action at a distance"—the idea that the Sun could pull on a planet across millions of miles of empty space. To defeat Descartes, Newton had to prove that a physical vortex could not possibly produce the observed planetary orbits. He does this by systematically analyzing how bodies move through fluids and how fluids themselves behave when set in motion. He calculates the resistance experienced by spheres and cylinders moving through water and air, and he derives the laws of wave motion and the speed of sound. In the final sections of Book Two, Newton applies these hydrodynamic principles to cosmic vortices. He proves mathematically that if a vortex were responsible for carrying the planets, the fluid drag would cause the planets to lose energy and spiral into the Sun. Furthermore, he shows that a vortex cannot sustain Kepler’s Third Law, which relates a planet's distance from the Sun to its orbital period. In a vortex, the fluid at different distances must rotate at speeds that are incompatible with the precise mathematical ratios discovered by Kepler. By the time Book Two concludes, Descartes' physical system lies in ruins. Newton has demonstrated that a fluid-filled universe cannot support stable, elliptical planetary orbits. Therefore, space must be largely empty—a vacuum—and the force that holds the planets in their orbits must be capable of acting across this vast, empty void without the aid of any material medium. Having cleared the intellectual field of his greatest rival, Newton was now ready to present his own system of the world. --- ## 7. Book Three and the System of the World Book Three, titled *The System of the World*, is the climax of the *Principia*. Here, Newton transitions from abstract mathematics and fluid dynamics to the actual physical universe. He applies the mathematical theorems of Book One to the observational data of astronomy to construct his theory of Universal Gravitation. Newton begins Book Three with a series of "Phenomena"—empirical observations compiled by astronomers, including himself, Galileo, and Cassini. These phenomena include the motions of the moons of Jupiter and Saturn, which obey Kepler’s laws relative to their host planets, and the motions of the five primary planets, which obey Kepler’s laws relative to the Sun. Using these observations and the mathematical tools from Book One, Newton performs his famous "moon-test." He asks: is the force that pulls an apple to the ground the same force that keeps the Moon in its orbit around the Earth? To answer this, Newton calculates the rate at which an object falls at the surface of the Earth. He then calculates the acceleration of the Moon toward the Earth, which is determined by its orbital speed and distance. Because the Moon is sixty times further from the center of the Earth than we are from the center of the Earth, an inverse-square force means that the Earth's gravity at the distance of the Moon should be sixty squared, or thirty-six hundred times weaker than it is at the Earth's surface. When Newton compared the calculated acceleration of the Moon to the acceleration of a falling body on Earth scaled down by a factor of thirty-six hundred, the numbers matched almost perfectly. The force that pulls the apple is indeed the very same force that holds the Moon in its orbit. From this stunning realization, Newton makes a massive inductive leap. He declares that gravity is not a local property of the Earth, but a universal property of all matter. He formulates the Law of Universal Gravitation: every particle of matter attracts every other particle of matter with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In the rest of Book Three, Newton uses this single law to explain a breathtaking array of natural phenomena. He explains the shape of the Earth, showing that because our planet rotates, centrifugal force causes it to bulge at the equator and flatten at the poles, making it an oblate spheroid rather than a perfect sphere. He uses this oblate shape to explain the precession of the equinoxes—the slow, twenty-six-thousand-year wobble of the Earth's axis caused by the gravitational tug of the Sun and Moon on the equatorial bulge. Newton also solves the mystery of the tides. He shows that the tides are caused by the gravitational pull of the Moon and the Sun on the Earth's oceans. By analyzing how these forces combine, he explains why there are two high tides a day, and why the tides are exceptionally high during new and full moons, when the Sun and Moon align. Finally, Newton tames the comets. For millennia, comets were viewed with superstitious dread as atmospheric portents of disaster. Newton demonstrates that comets are physical bodies moving through deep space under the influence of the Sun's gravity. By analyzing the observations of the great comet of 1680, he shows that its path is a highly elongated ellipse—almost a parabola—confirming that comets are subject to the exact same gravitational laws as the planets. --- ## 8. The Rules of Reasoning in Philosophy In the second and third editions of the *Principia*, Newton added a crucial methodological section at the beginning of Book Three titled *Regulae Philosophandi*, or the *Rules of Reasoning in Philosophy*. These four rules are among the most important texts in the history of scientific methodology, as they define the empirical and inductive logic that would guide modern science. The First Rule is the principle of parsimony, or simplicity: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Newton famously writes that nature is pleased with simplicity, and affects not the pomp of superfluous causes. If a single cause—universal gravitation—can explain the falling of an apple, the orbits of the planets, and the tides, we must not invent separate, complicated causes for each. The Second Rule is the principle of uniformity: Therefore to the same natural effects we must, as far as possible, assign the same causes. If a stone falls in Europe and a stone falls in America, they must fall due to the same physical cause. Similarly, the light of our Sun and the light of the distant stars must be governed by the same physical principles. The Third Rule deals with induction and the properties of bodies: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever. This is the rule that allows us to generalize from local experiments to the entire universe. If every piece of matter we can touch and measure on Earth has mass, inertia, and gravitational attraction, we are justified in concluding that all matter in the universe, even in the most distant galaxies, possesses these same properties. The Fourth Rule is a defense of scientific induction against speculative hypotheses: In experimental philosophy we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions. This rule is a direct warning against the armchair physics of the Cartesians, who invented elaborate mechanical scenarios to explain the world without empirical evidence. For Newton, a theory derived from careful observation and mathematical proof must stand until it is contradicted by new empirical evidence, not by clever philosophical arguments. This methodological stance is famously summarized in the General Scholium, which Newton added to the end of the *Principia*. Addressing the criticism that he had not explained *what* gravity actually is, but had only described its mathematical behavior, Newton wrote his most famous phrase: *Hypotheses non fingo*—"I frame no hypotheses." Newton admitted that he did not know the physical cause of gravity, whether it was mechanical, spiritual, or something else. But he asserted that this did not invalidate his work. It was enough that he had proved mathematically that gravity exists, that it acts according to the inverse-square law, and that it successfully accounts for all the motions of the heavenly bodies. By separating the mathematical description of a force from the metaphysical explanation of its ultimate cause, Newton established the boundary between modern science and speculative philosophy. --- ## 9. What is Brilliant, What is Dated, and What is Dangerous Looking back at the *Principia* from our modern vantage point, we can evaluate its achievements with both admiration and critical distance. The brilliance of the *Principia* lies in its unprecedented synthesis of mathematics and observation. Newton did not just propose laws; he created a predictive engine. The proof architecture of the book—the way it moves from simple definitions to complex orbital mechanics, and then applies those mechanics to solve real-world problems like the tides and comets—became the template for all subsequent physical sciences. Newton proved that the universe is rational, mathematical, and comprehensible. He gave humanity the keys to read the language of nature. However, the *Principia* also contains elements that are dated and have been superseded by modern physics. The most significant of these is Newton’s reliance on absolute space and time. In the early twentieth century, Albert Einstein’s theories of Special and General Relativity demonstrated that space and time are not an absolute, unchanging backdrop. Instead, they are relative and dynamic, woven together into a four-dimensional fabric called spacetime, which bends and stretches in the presence of mass and energy. Furthermore, Newton’s concept of gravity as an instantaneous force acting across empty space—the very "action at a distance" that troubled his contemporaries—was replaced by Einstein’s field equations. In General Relativity, gravity is not a physical pull, but the geometric curvature of spacetime caused by mass. A planet orbits the Sun not because it is being tugged by an invisible rope, but because it is following the straightest possible path through a spacetime fabric that has been bent by the Sun's immense mass. There is also a subtle danger in the legacy of the *Principia*, often referred to as the "Newtonian Style." This is the temptation to reduce the entire world to a deterministic, mechanical clockwork. In the centuries following Newton, many thinkers attempted to apply his rigid, mathematical methods to fields where they did not fit, such as psychology, sociology, and economics. This led to a kind of reductionism that treated human beings and complex systems as mere billiard balls colliding in absolute space, ignoring the qualitative, subjective, and non-deterministic aspects of reality. Moreover, the sheer success of Newtonian mechanics created a sense of scientific hubris. By the end of the nineteenth century, some physicists believed that the work of physics was nearly complete, and that all that remained was to measure physical constants to more decimal places. This illusion was shattered by the twin revolutions of Relativity and Quantum Mechanics, which revealed that the Newtonian universe is merely an approximation—a highly accurate and useful approximation for our everyday world, but one that breaks down entirely at the scale of the very large, the very fast, and the very small. --- ## 10. How to Read the Principia Today and Who It Is For Reading the *Principia* today is a daunting task, but it is an immensely rewarding intellectual pilgrimage. If you decide to tackle this masterwork, you must approach it with patience and a clear strategy. First, do not attempt to read the *Principia* from cover to cover like a modern textbook or a novel. If you try to work through every single geometric lemma and proposition in Book One, you will likely get lost in the mathematical weeds and abandon the journey. Instead, focus on the conceptual signposts. Begin by reading the opening Definitions and the Scholium on Space and Time. These sections are written in clear, accessible prose and lay out the philosophical stakes of the book. Next, read the three Laws of Motion and the discussion that follows them. When you move into the books themselves, focus on the introductory remarks, the propositions that state major physical conclusions, and the Scholiums. In Book One, pay close attention to Proposition One, which links centripetal force to Kepler's area law, and Proposition Eleven, which links elliptical orbits to the inverse-square law. Skip the dense geometric proofs unless you are a historian of mathematics. In Book Three, read the Rules of Reasoning and the "Phenomena." Then, skip to the General Scholium at the very end of the book. The General Scholium is Newton’s grand philosophical summation, where he discusses the nature of God, the limits of physical explanation, and his famous refusal to frame hypotheses. It is one of the most beautifully written and intellectually stimulating passages in the history of literature. Who is the *Principia* for today? It is for anyone who wants to understand the origins of the modern world. It is for students of the history and philosophy of science who want to see how our concept of physical explanation was forged. It is for mathematicians and physicists who want to appreciate the raw, geometric genius of one of the greatest minds to ever live. But most of all, the *Principia* is for the curious learner who wishes to stand at the moment of transition—the precise historical pivot point where the mysterious, fragmented cosmos of the ancients was transformed into the unified, elegant, and mathematically knowable universe we inhabit today. By engaging with Newton’s masterwork, we do not just learn physics; we witness the birth of the modern mind.