Isaac Newton’s Philosophiae Naturalis Principia Mathematica, published in 1687, is routinely called the most important scientific book ever written. That claim is difficult to dispute. In a single work, Newton established the laws of motion, derived the law of universal gravitation, explained the orbits of planets and comets, accounted for the tides, and provided a mathematical framework that would dominate physics for over two centuries.
But surprisingly few people know what the Principia actually contains. The book is rarely read, even by physicists. Its reputation is so immense that it has become a monument rather than a text: everyone knows it exists, almost nobody has opened it. This is partly because Newton wrote in Latin and used geometric proofs rather than the algebraic calculus that modern readers expect. But it is also because few accessible guides to its contents exist.
The Principia is divided into three books, each with a distinct purpose. Together, they form a single argument that moves from abstract principles to the concrete structure of the solar system. Understanding what each book does, and how the three fit together, is the key to understanding what Newton achieved.
Book I: The Laws of Motion and Their Consequences
The first book establishes the foundations. Newton begins with definitions (mass, force, momentum) and then states his three laws of motion:
- First Law: A body remains at rest or in uniform straight-line motion unless acted upon by a force.
- Second Law: The change in motion of a body is proportional to the force applied and occurs in the direction of that force.
- Third Law: For every action, there is an equal and opposite reaction.
These three statements, which can be written on a single page, are the foundation of all classical mechanics. Everything that follows in the Principia is a consequence of these laws combined with the mathematics of force and motion.
After stating the laws, Newton spends the rest of Book I deriving their mathematical consequences. This is where the Principia becomes technically demanding. Newton proves, step by step, what kinds of motion result from different kinds of force. If a body is attracted toward a central point by a force that varies as the inverse square of the distance, what shape is its orbit? Newton proves that the orbit is a conic section: an ellipse, a parabola, or a hyperbola, depending on the body’s energy.
This result is the mathematical heart of the Principia. Kepler had discovered empirically that planets move in ellipses. Newton proved that elliptical orbits are the necessary consequence of an inverse-square force. He did not merely describe what happens. He explained why it happens.
Book I also contains Newton’s treatment of the two-body problem (how two masses orbit each other), the beginnings of the three-body problem (which Newton recognized as having no exact solution), and his analysis of orbital perturbations. These results would later become the foundation of celestial mechanics, the science of predicting planetary positions with extreme precision.
The entire first book is abstract. Newton does not mention gravity, the Sun, or any planet. He writes about “bodies” attracted by “centripetal forces” toward “centers.” The genius of this approach is its generality. The mathematics of Book I applies to any inverse-square force, whether gravitational, electrical, or otherwise. Newton built a mathematical engine and then, in Book III, pointed it at the sky.
Book II: Resistance and Fluids
The second book is the least famous part of the Principia, and the most puzzling to modern readers. It deals with the motion of bodies through resisting media: air, water, and other fluids. Why did Newton include an entire book on fluid resistance in a work about celestial mechanics?
The answer lies in a debate that Newton needed to win. The dominant theory of planetary motion in the late seventeenth century was not Newton’s gravity but René Descartes’s theory of vortices. Descartes proposed that the planets are carried around the Sun by vast whirlpools (tourbillons) of invisible matter that fill all of space. In this model, there is no need for a mysterious force acting at a distance. The planets move because the surrounding medium pushes them.
Newton found Descartes’s vortex theory physically implausible, but he needed to refute it mathematically. That is the purpose of Book II. By analyzing how bodies actually move through fluids, Newton demonstrated that vortex motion cannot produce the planetary orbits that Kepler had observed. Fluid resistance would slow the planets down. Vortices would not produce elliptical orbits with the correct speeds. The medium that supposedly carried the planets would interfere with their motion rather than sustain it.
Book II is also important for its contributions to fluid mechanics. Newton proposed what is now called Newtonian fluid behavior: the idea that the resistance a fluid exerts on a moving body is proportional to the body’s velocity (for slow motion) or to the square of its velocity (for faster motion). He analyzed the speed of sound in air, the motion of pendulums in resisting media, and the propagation of waves through fluids.
Some of Newton’s results in Book II were incorrect. His theory of fluid resistance contained errors that were later corrected by d’Alembert, Euler, and Navier and Stokes. But the approach was revolutionary: the idea that fluid behavior could be analyzed mathematically, using the same laws of motion that govern solid bodies, opened an entirely new field of physics.
Book III: The System of the World
The third book is where Newton turns from mathematics to physics. Its title, De Mundi Systemate (On the System of the World), announces its ambition. Newton takes the abstract mathematical results of Book I and applies them to the actual solar system. The inverse-square force that produces elliptical orbits in theory is identified with the force of gravity that pulls objects toward the Earth, the Moon toward the Earth, and the planets toward the Sun.
Newton begins Book III with a set of “Rules of Reasoning in Philosophy” that articulate principles of scientific method:
- Admit no more causes than are necessary to explain the phenomena.
- To the same effects, assign the same causes.
- Properties observed in all bodies within reach of experiment should be assumed universal.
- Propositions derived from observation should be considered true until contradicted by new observations.
These rules allowed Newton to make a breathtaking leap. The force that makes an apple fall and the force that keeps the Moon in orbit are the same force. Gravity is universal: every mass in the universe attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
With this single principle, Newton explained an astonishing range of phenomena:
- The elliptical orbits of the planets around the Sun
- The orbit of the Moon around the Earth (including its irregularities)
- The precession of the equinoxes
- The tides of the oceans (caused by the gravitational pull of the Moon and Sun)
- The oblate shape of the Earth (wider at the equator due to its rotation)
- The trajectories of comets (including a prediction of Halley’s Comet)
No previous scientific work had explained so many different phenomena with so few principles. The Principia demonstrated that the same laws govern the heavens and the Earth, unifying two domains that had been considered fundamentally different since Aristotle.
How the Three Books Fit Together
The structure of the Principia is not arbitrary. It reflects Newton’s method of scientific argument:
Book I establishes the mathematics: if an inverse-square force exists, here is what it produces. Book II eliminates the competition: Descartes’s vortices cannot explain the observations. Book III identifies the force: it is gravity, it is universal, and here is the evidence.
This three-step structure (mathematical theory, refutation of alternatives, application to nature) set the template for theoretical physics. It is the same structure used by Maxwell in his electromagnetic theory, by Einstein in his theory of relativity, and by quantum physicists in the twentieth century. Newton did not just solve the problem of planetary motion. He showed future scientists how to solve problems.
Reading the Principia Today
The Principia is not an easy book. Newton deliberately made it difficult, reportedly to avoid being “baited by little smatterers in mathematics.” The geometric proofs are dense, the Latin is technical, and the physical arguments require careful attention. Newton himself said that a reader should work through the definitions, the laws of motion, and the first three sections of Book I, then skip directly to Book III, consulting earlier propositions only as needed.
But the effort is rewarding. Reading the Principia, even selectively, reveals something that no textbook summary can capture: the experience of watching one of the greatest minds in history construct, step by step, an explanation of the universe. The logic is relentless. The ambition is staggering. The result is a work that, three centuries later, remains the foundation of our understanding of how the physical world operates.
Kronecker Wallis’s edition of the Principia preserves this structure by binding each of the three books as a separate volume within a single cover. This design echoes Newton’s own architecture: three distinct arguments that together form one unified system. For those new to Newton, the College Notebook offers a glimpse of the young mathematician before the Principia was written, showing the earliest seeds of the ideas that would change the world. And for readers interested in Newton’s other masterwork on light and color, the Opticks demonstrates a very different side of Newton: the experimentalist rather than the theorist, working with prisms and lenses rather than geometric proofs.
A Book That Still Defines Physics
The Principia was published 339 years ago, and its core results remain valid. NASA uses Newtonian mechanics to plot spacecraft trajectories. Engineers use Newton’s laws to design bridges, buildings, and vehicles. The three laws of motion are taught in every physics classroom on Earth. No other scientific work has remained in continuous use for so long.
Einstein’s general relativity replaced Newtonian gravity in extreme conditions (near black holes, at cosmological scales, at very high speeds), but for the vast majority of physical situations, Newton’s equations are both sufficient and exact. The Principia is not a historical curiosity. It is a working manual for the physical universe, as practical today as it was when Edmund Halley persuaded Newton to write it down.
No products in the cart.