From the publisher
"Principles" by I. Newton? one of the greatest works in the history of natural science. This work laid the foundations of mechanics, physics and astronomy; it formulated a program for the development of these areas of science, which remained decisive for more than a century and a half.
This publication is a facsimile reproduction of I. Newton’s book translated from Latin and with comments by Academician A. N. Krylov. The book also includes subject index, compiled by I. Newton and published in Russian for the first time.
The book is intended for a wide range of specialists in the field of natural sciences, as well as readers interested in the history of science.

Content
Preface
Isaac Newton. Mathematical principles of natural philosophy
Translator's Preface
Author's Preface to the First Edition
Author's Preface to the Second Edition
Publisher's Preface to the Second Edition
Author's Preface to the Third Edition
Definitions
Axioms or laws of motion
Book I ON THE MOTION OF BODIES
Section I. On the method of first and last relations, with the help of which the following is proved
Department P. On the determination of centripetal forces
Division III. On the motion of bodies along eccentric conic sections
Division IV. On the definition of elliptic, parabolic and hyperbolic orbits at a given focus
Section V. On finding orbits when no Focus is given
Section VI. On determining motion along given orbits
Section VII. ABOUT straight motion bodies towards or away from the center
Section VIII. On finding the orbits in which bodies rotate under the influence of any centripetal forces
Section IX. On the movement of bodies in moving orbits and on the movement of apses
Section X. On the motion of bodies on given surfaces and on the oscillatory motion of suspended bodies
Section XI. On the motion of bodies mutually attracted by centripetal forces
Section XII. On the attractive forces of spherical bodies
Section XIII. On the attraction of non-spherical bodies
Section XIV. On the movement of very small bodies under the influence of centripetal forces directed towards individual particles very big body
Translator's Note on Proposition LXVI
Book II ON THE MOTION OF BODIES
Section I. On the motion of bodies with resistance proportional to speed
Department P. On the motion of bodies with resistance proportional to the second power of speed
Division III. About the movement of bodies with resistance, partly proportional to the first power of speed, partly? second
Division IV. On the circular circulation of bodies in a resisting medium
Section V. On the density and compression of liquids and hydrostatics
Section VI. On the motion of pendulums under resistance
Section VII. On the movement of fluids and the resistance of thrown bodies
Section VIII. On motion propagating through liquids
Section IX. On the circular motion of liquids
Book III ABOUT THE WORLD SYSTEM
Rules of inference in physics
Phenomena
Offers
On the movement of the nodes of the Moon's orbit
Alphabetical subject index
Application
On the Russian translation of Isaac Newton's "Mathematical Principles of Natural Philosophy"
Name index

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Translator's Preface - page II

Publisher's Preface to the Second Edition

Definitions - page 23

Axioms or laws of motion - p.39

Book I

About the movement of bodies

Section I. On the method of first and last relations, with the help of which the following is proved - p. 57

Division II. On finding centripetal forces - p.73

Division III. On the motion of bodies along eccentric conic sections - p.91

Division IV. On the determination of elliptic, parabolic and hyperbolic orbits at a given focus - p. 106

Section V. On finding orbits when no focus is given - p.116

Section VI. On determining motion along given orbits - p. 151

Section VII. On the rectilinear motion of bodies towards or from the center - p. 160

Section VIII. On finding the orbits in which bodies rotate under the influence of any centripetal forces - p. 175

Section IX. On the movement of bodies in moving orbits and on the movement of apses - p. 184

Section X. On the motion of bodies on given surfaces and on the oscillatory motion of suspended bodies - p. 199

Section XI. On the movement of bodies mutually attracted by centripetal forces - p. 216

Section XII. On the attractive forces of spherical bodies - p.244

Section XIII. On the attraction of non-spherical bodies - p.266

Section XIV. On the movement of very small bodies under the influence of centripetal forces directed towards individual particles of a very large body - p. 280

Translator's note to sentence LXVI - p.288

Book II

About the movement of bodies

Section I. On the movement of bodies with resistance proportional to speed - p. 312

Division II. On the motion of bodies with resistance proportional to the second power of speed - p. 325

Division III. On the motion of bodies under resistance, partly proportional to the first power of speed, partly proportional to the second - p. 356

Division IV. On the circular circulation of bodies in a resisting medium - p. 369

Section V. On the density and compression of liquids and hydrostatics - p. 377

Section VI. On the movement of pendulums with resistance - p. 392

Section VII. On the movement of fluids and the resistance of thrown bodies - p.422

Section VIII. On motion propagating through liquids - p.467

Section IX. On the circular motion of liquids - p. 486

Book III

About the world system

Rules of inference in Physics - p.502

Phenomena - page 504

Offers - page 510

On the movement of the nodes of the Moon's orbit - p.572

Mathematical principles of natural philosophy

Isaac Newton

Definitions

I. The amount of matter (mass) is a measure of it, established in proportion to its density and volume.

There is four times as much air of double density in a double volume, and six times as much in a triple volume. The same applies to snow or powders when they become compacted by compression or melting. The same applies to all kinds of bodies that, for whatever reason, become denser. However, in this case I do not take into account the medium, if any, which freely penetrates into the spaces between the particles. This is the same quantity that I mean in what follows under the names body or mass. Mass is determined by the weight of the body, because it is proportional to the weight, which I found in experiments on pendulums produced in the most precise manner, as described below.

II. The quantity of motion is a measure of such, established in proportion to speed and mass.

The amount of motion of the whole is the sum of the amounts of motion of its individual parts, which means that for a mass twice as large, at equal speeds it is double, but at double speed it is quadruple.

III. The innate strength of matter is its inherent ability of resistance, by which any individual body, since it is left to itself, maintains its state of rest or uniform rectilinear motion.

This force is always proportional to the mass, and if it differs from the inertia of the mass, then only by looking at it.

From the inertia of matter it happens that every body is only with difficulty removed from its rest or movement. Therefore, the “innate force” could very sensibly be called the “force of inertia.” This force is manifested by the body only when another force applied to it produces a change in its state. The manifestation of this force can be considered in two ways: both as resistance and as pressure. As resistance - since the body resists the force acting on it, trying to maintain its state; as pressure - since the same body, with difficulty yielding to the force of the obstacle resisting it, strives to change the state of this obstacle. Resistance is usually attributed to bodies at rest, pressure - to bodies in motion. But motion and rest, when viewed in the usual way, differ only in the relation of one to the other, for what appears to be so to the simple eye is not always at rest.

IV. An applied force is an action performed on a body to change its state of rest or uniform linear motion.

Power manifests itself only in action and does not remain in the body after the action ceases. The body then continues to maintain its new state due to inertia alone. The origin of the applied force can be different: from impact, from pressure, from centripetal force (...)

Teaching

In what was stated above, it was intended to explain in what sense less well-known names are used in the following. Time, space, place and movement are generally known concepts. However, it should be noted that these concepts usually refer to what is comprehended by our senses. This is where some incorrect judgments arise, to eliminate which it is necessary to divide the above concepts into absolute and relative, true and apparent, mathematical and ordinary.

I. Absolute, true, mathematical time by itself and by its very essence, without any relation to anything external, flows uniformly and is otherwise called duration.

Relative, apparent, or ordinary time is either exact or variable, comprehended by the senses, external, accomplished through some movement, a measure of duration, used in everyday life instead of true mathematical time, such as: hour, day, month, year .

II. Absolute space, by its very essence, regardless of anything external, always remains the same and motionless.

Relative [space] is its measure or some limited moving part, which is determined by our senses by its position relative to certain bodies and which in everyday life is accepted as motionless space: for example, the extent of the spaces of underground air or aboveground, determined by their position relative to the Earth. In appearance and size, absolute and relative spaces are identical, but numerically they do not always remain the same. So, for example, if we consider the Earth to be moving, then the space of our air, which in relation to the Earth always remains the same, will constitute first one part of the absolute space, then another, depending on where the air has moved, and, therefore, the absolute space is constantly changing.

III. Place is a part of space occupied by a body and, in relation to space, can be either absolute or relative. I say part of space, and not the position of the body and not the surface enclosing it. For bodies of equal volume, the places are equal, but the surfaces, due to the dissimilarity of the shapes of the bodies, can be unequal. Position, correctly speaking, has no magnitude, and it in itself is not a place, but a property belonging to a place. The movement of the whole is the same as the totality of the movements of its parts, i.e. the movement of the whole from its place is the same as the totality of the movements of its parts from their places. Therefore the place of the whole is the same as the totality of the places of its parts, and therefore it is entirely within the whole body.

IV. Absolute motion is the movement of a body from one absolute place to another, relative motion is from relative to relative. Thus, on a ship sailing, the relative place of the body is that part of the ship in which the body is located, for example, that part of the hold that is filled with the body and which, therefore, moves with the ship. Relative rest is the presence of a body in the same area of ​​the ship or in the same part of its hold.

True peace is the presence of the body in the same part of that motionless space in which the ship moves with everything in it. Thus, if the Earth were actually at rest, then the body, which is at rest relative to the ship, would actually move with the absolute speed with which the ship is moving relative to the Earth. If the Earth itself moves, then the true absolute motion of the body can be found from the true motion of the Earth in motionless space and from the relative motions of the ship in relation to the Earth and the body in relation to the ship. (...)

Bibliography

Golin G.M., Filonovich S.R. Classics of physical science (from ancient times to the beginning of the 20th century) - M.: Vyssh. school, 1989.

History of writing

The history of the creation of this work, the most famous in the history of science along with Euclid's Elements, begins in 1682, when the passage of Halley's comet caused a rise in interest in celestial mechanics. Edmond Halley then tried to persuade Newton to publish his “general theory of motion.” Newton refused. He was generally reluctant to be distracted from his research for the painstaking task of publishing scientific works.

In August 1684, Halley came to Cambridge and told Newton that he, Wren and Hooke had discussed how to derive the ellipticity of the orbits of planets from the formula for the law of gravitation, but did not know how to approach the solution. Newton reported that he already had such a proof, and soon sent it to Halley. He immediately appreciated the significance of the result and the method, in November he visited Newton again and this time managed to persuade him to publish his discoveries.

On December 10, 1684, a historical entry appeared in the minutes of the Royal Society:

Mr. Halley... recently saw Mr. Newton in Cambridge, and he showed him an interesting treatise "De motu" [On Motion]. According to the wishes of Mr. Halley, Newton promised to send the said treatise to the Society.

The publication was supposed to be carried out with funds from the Royal Society, but at the beginning of 1686 the Society published a treatise on the history of fish that was not in demand, and thereby depleted its budget. Then Halley announced that he would bear the costs of publication himself. The Society gratefully accepted this generous offer and, as partial compensation, provided Halley with 50 free copies of a treatise on the history of fishes.

Newton's work - perhaps by analogy with the "Principles of Philosophy" ( Principia Philosophiae) Descartes - received the name “Mathematical principles of natural philosophy”, that is, on modern language, "Mathematical foundations of physics".

On April 28, 1686, the first volume of "Mathematical Principles" was presented to the Royal Society. All three volumes, after some editing by the author, were published in 1687. The circulation (about 300 copies) was sold out in 4 years - very quickly for that time. Two copies of this rare edition are kept in Russia; one of them was presented by the Royal Society during the war years (1943) to the USSR Academy of Sciences to celebrate Newton's 300th anniversary. During Newton's lifetime the book went through three editions; With each reissue, Newton made significant additions, improvements and clarifications to the text.

Summary of the work

Both the physical and mathematical level of Newton's work are incomparable with the work of his predecessors. It completely (with the exception of philosophical digressions) lacks Aristotelian or Cartesian metaphysics, with its vague reasoning and unclearly formulated, often far-fetched “first causes” of natural phenomena. Newton, for example, does not proclaim that the law of gravity operates in nature, he strictly proves this fact, based on the observed picture of the motion of the planets: from Kepler's first two laws, he deduces that the motion of the planets is controlled by a central force, and from the third law - that attraction is inversely proportional to the square of the distance.

First book

In the first chapter (chapters in the work are called departments) Newton defines the basic concepts - mass, force, inertia (“innate force of matter”), momentum, etc. The absoluteness of space and time is postulated, the measure of which does not depend on the position and speed of the observer. Based on these clearly defined concepts, the three laws of Newtonian mechanics are formulated. For the first time, general equations of motion are given, and if Aristotle’s physics argued that the speed of a body depends on driving force, then Newton makes a significant correction: not speed, but acceleration.

Further in Book I, motion in the field of an arbitrary central force is examined in detail. Newton's law of attraction is formulated (with reference to Wren, Hooke and Halley), a strict derivation of all Kepler's laws is given, and hyperbolic and parabolic orbits unknown to Kepler are also described. Newton presented Kepler's third law in a generalized form, taking into account the masses of both bodies.

Chapter X contains the theory of oscillations different types pendulums, including spherical and cycloidal. Next, the attraction of extended (no longer point-like) bodies of spherical or other shapes is examined in detail.

The methods of proof, with rare exceptions, are purely geometric; differential and integral calculus are not explicitly used (probably so as not to multiply the number of critics), although the concepts of limit (“last ratio”) and infinitesimal, with an estimate of the order of smallness, are used in many places.

Second book

Book II is actually devoted to hydromechanics, that is, the movement of bodies on Earth taking into account the resistance of the environment. For example, the oscillations of a pendulum in a resisting medium are studied. Here, in one place (Section II), Newton, as an exception, uses an analytical approach to prove several theorems and proclaims his priority in the discovery of the “method of fluxions” (differential calculus):

In letters which about ten years ago I exchanged with the very skilful mathematician Mr. Leibniz, I informed him that I had a method for determining maxima and minima, drawing tangents and solving similar questions, equally applicable to both rational and rational terms. for irrational ones, and I hid the method by rearranging the letters of the following sentence: “when given an equation containing any number of current quantities, find the fluxions and vice versa.” The most famous man answered me that he also attacked such a method and told me his method, which turned out to be barely different from mine, and then only in terms and outline of formulas.

Third book

Book 3 - world system, mainly celestial mechanics, as well as tidal theory. At the beginning of the book, Newton formulates his version of Occam's razor:

One should not accept in nature other causes than those that are true and sufficient to explain phenomena... Nature does nothing in vain, and it would be in vain for many to do what can be done by fewer. Nature is simple and does not luxury with unnecessary reasons.

In accordance with his method, Newton deduces the law of gravity from experimental data on the planets, the Moon and other satellites. To verify that gravity (weight) is proportional to mass, Newton conducted several fairly accurate experiments with pendulums.

This law is then used to describe the motion of planets. The theory of the movement of the Moon and comets and the physical causes of tides are also described in detail. A method is given for determining the mass of the planet, and the mass of the Moon is found from the height of the tides. Explained (with the help of perturbation theory) the anticipation of the equinoxes and irregularities (discrepancies) in the movement of the Moon - both known in antiquity and 7 later established (Tycho Brahe, Flamsteed).

Criticism

The publication of “Beginnings”, which laid the foundation theoretical physics, caused a huge resonance in scientific world. Along with enthusiastic responses, there were, however, sharp objections, including from famous scientists. Carthusians in Europe attacked her with fierce criticism. The three laws of mechanics did not raise any special objections; the concept of gravity was mainly criticized - a property of an incomprehensible nature, with an unclear source, which acted without a material carrier, through completely empty space. Leibniz, Huygens, Jacob Bernoulli, Cassini rejected gravity and continued to try to explain the motion of the planets by Cartesian vortices or in some other way.

From the correspondence between Leibniz and Huygens:

Leibniz: I don't understand how Newton imagines gravity or attraction. Apparently, in his opinion, this is nothing more than some inexplicable intangible quality.
Huygens: As for the reason for the tides that Newton gives, it does not satisfy me, like all his other theories based on the principle of attraction, which seems ridiculous and absurd to me.

Newton himself preferred not to speak publicly about the nature of gravity, since he had no experimental arguments in favor of the ethereal or other hypothesis, and he did not like to start empty squabbles. Newton confidently rejected the connection between gravity and magnetism suspected by a number of physicists, since the properties of these two phenomena are completely different. In personal correspondence, Newton also admitted the supernatural nature of gravity:

It is incomprehensible that inanimate gross matter could, without the mediation of something immaterial, act and influence other matter without mutual contact, as this should happen if gravity in the sense of Epicurus were essential and innate in matter. To assume that gravitation is an essential, inextricable and innate property of matter, so that a body can act on another at any distance in empty space, without the mediation of anything conveying action and force, this, in my opinion, is such an absurdity that is unthinkable for anyone who knows how to sufficiently understand philosophical subjects.

Gravity must be caused by an agent constantly acting according to certain laws. Whether, however, this agent is material or immaterial, I have left it to my readers to decide.

(From Newton's letter dated February 25, 1693 to Dr. Bentley, author of lectures on the topic "Refutation of Atheism")

Sir Isaac Newton was with me and said that he had prepared 7 pages of additions to his book on light and colors [that is, "Optics"], in a new Latin edition... He had doubts whether he could express the last question like this: " What fills the space free from bodies?” The complete truth is that he believes in the omnipresent Deity in the literal sense. Just as we feel objects when their images reach the brain, so God must feel every thing, always being present with it.

He believes that God is present in space, both free from bodies and where bodies are present. But, considering that such a formulation is too crude, he thinks of writing it like this: “What cause did the ancients attribute to gravity?” He thinks that the ancients considered God to be the cause, and not any body, for every body is already heavy in itself.

Critics also pointed out that the theory of planetary motion based on the law of gravity is insufficiently accurate, especially for the Moon and Mars. Direct measurement of the force of gravity in terrestrial conditions was carried out in 1798 by G. Cavendish using extremely sensitive torsion balances; These experiments completely confirmed Newton's theory.

Place in the history of science

Newton's book was the first work on new physics and at the same time one of the last serious works using old methods of mathematical research. All of Newton's followers already used powerful methods of mathematical analysis. Throughout the 18th century, analytical celestial mechanics developed intensively, and over time, all the mentioned discrepancies were fully explained by the mutual influence of the planets (Lagrange, Clairaut, Euler and Laplace).

From that moment until the beginning of the 20th century, all Newton's laws were considered immutable. Physicists gradually got used to long-range action, and even tried to attribute it, by analogy, to the electromagnetic field (before the advent of Maxwell's equations). The nature of gravity was revealed only with the advent of Einstein's work on General Relativity, when long-range action finally disappeared from physics.

An asteroid named after Newton's Principia

The pinnacle of Newton's scientific creativity was precisely this work, after the publication of which he largely moved away from scientific works. The greatness of the author’s plan, which subjected the system of the world to mathematical analysis, and the depth and rigor of the presentation amazed his contemporaries /2/.

Newton's preface (there is also a preface by Cotes, his student) casually sketches out the program mechanical physics: “We propose this work as the mathematical foundations of physics. The whole difficulty of physics, as will be seen, is to recognize the forces of nature from the phenomena of motion, and then to explain other phenomena using these forces (thus, in books 1 and 2, the law of action of central forces is derived from observable phenomena, and in the third, the found law is applied to the description of the world system). It would be desirable to deduce from the principles of mechanics the rest of the phenomena of nature, reasoning in a similar way, for much makes me assume that all these phenomena are determined by certain forces with which the particles of bodies, due to reasons as yet unknown, either tend to each other and interlock in regular figures, or they mutually repel and move away from each other.”

“Principles...” begin with the “Definitions” section, where definitions of the amount of matter, inertial mass, centripetal force and some others are given. This section concludes with a “Teaching,” where the definition of space, time, place, and movement is given. Next comes the section on the axioms of motion, where Newton’s famous 3 laws of mechanics, the laws of motion and the immediate consequences of them are given. Thus, we observe a certain imitation of Euclid’s “Principles...”.

Next, “Beginnings...” is divided into 3 books. The first book is devoted to the theory of gravity and movement in the field of central forces, the second - to the doctrine of environmental resistance. In the third book, Newton outlined the established laws of motion of the planets, the Moon, the satellites of Jupiter and Saturn, gave a dynamic interpretation of the laws, outlined the “method of fluxions,” and showed that the force that attracts a stone to the Earth is no different in nature from the force that keeps the Moon in orbit , and the weakening of attraction is associated only with an increase in distance.

Thanks to Newton, the Universe began to be perceived as a well-oiled clockwork mechanism. The regularity and simplicity of the basic principles that explained all observed phenomena were regarded by Newton as proof of the existence of God: “Such a most graceful conjunction of the Sun, planets and comets could not have happened except by the intention and in the power of a wise and powerful being. This one rules everything not as the soul of the world, but as the ruler of the Universe, and according to his dominion he should be called the Lord God Almighty.”