Skip to content

Principia Mathematica

From the Quicksilver Metaweb.

This is an intermediate page for Principia Mathematica.

Disambiguation

The full name of Newton's 1687 work is Philosophiae Naturalis Principia Mathematica; it should not be confused with Russell and Whitehead's Principia Mathematica, published in 1910 - 1913.

Stephensonia

Cryptonomicon (p294): “ … The actual content of the business plan hews to a logical structure straight out of the Principia Mathematica. Lesser entrepreneurs purchase business-plan-writing software: packages of boilerplate text and spread sheets, craftily linked together so that you need only go through and fill in a few blanks. Avi and Beryl have written enough business plans between the two of them that they can smash them out from brute memory. … ” And Quicksilver opens with a Roger Cotes' quote from the 2nd Edition of the Principia.

Authored entries

Community entry: Philosophiae Naturalis Principia Mathematica

From Wikipedia, the free encyclopedia.

The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on July 5, 1687. Probably the most influential scientific book ever published, it contains the statement of Newton's laws of motion forming the foundation of classical mechanics as well as his law of universal gravitation. He derives Kepler's laws for the motion of the planets (which were first obtained empirically).

In formulating his physical theories, Newton had developed a field of mathematics known as calculus. However, the language of calculus was largely left out of the Principia. Instead, Newton recast the majority of his proofs as geometric arguments.

It is in the Principia that Newton expressed his famous "Hypotheses non fingo" (I feign (to assert as if true) no hypotheses). Here is the passage containing this famous remark: I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction.

The Philosophiae Naturalis Principia Mathematica is composed of three volumes. Of The Motion Of Bodies Of The Motion Of Bodies (contd.) The System Of The World

Newton's Laws

Newton's laws of motion (also called the laws of inertia) are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. These laws are fundamental to classical mechanics.

Newton first published these laws in Philosophiae Naturalis Principia Mathematica (1687) and used them to prove many results concerning the motion of physical objects. In the third volume (of the text), he showed how, combined with his Law of Universal Gravitation, the laws of motion would explain Kepler's laws of planetary motion.

Importance of Newton's laws of motion

Nature and Nature's laws lay hid in night; God said, "Let Newton be!" And there was light. -- Alexander Pope

Newton's laws of motion, together with his Law of Universal Gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena such as: the motion of spinning bodies, motion of bodies in fluids; projectiles; motion on an inclined plane; motion of a pendulum; the tides; the orbits of the Moon and the planets. The law of conservation of momentum, which Newton derived as a corollary of his second and third laws, was the first conservation law to be discovered.

Newton's laws were verified by experiment and observation for over 200 years, until 1916, when they were superseded by Einstein's theory of relativity. Newton's laws still provide a completely adequate approximation for the behaviour of objects in "everyday" situations.

Newton's First Law (Law of Inertia)

Alternative formulations: * Every object persists in its state of rest, or uniform motion (in a straight line); unless, it is compelled to change that state, by forces impressed on it. * A body remains at rest, or moves in a straight line (at a constant velocity), unless acted upon by a net outside force.

This means that a stationary object will remain stationary, and a moving object will continue to move (in a straight line and at a constant speed), unless a force acts upon it. In everyday life, the force of friction usually acts upon moving objects. Newton's law indicates that some force (gravity) must be acting upon the planets, as they do not travel in a straight line.

Newton's Second Law

Alternative formulations: * The time rate of change in momentum is proportional to the applied force and takes place in the direction of the force. * The acceleration of an object is proportional to the force acting upon it.

This is expressed by the equation:

This equation expresses that the more force an object receives, the greater its acceleration will be. The quantity m, or mass, in the above equation is the constant of proportionality, and is a characteristic of the object. This equation, therefore, indirectly defines the concept of mass.

In the equation, F = ma, a is directly measurable but F is not. The second law only has meaning if we are able to assert, in advance, the value of F. Rules for calculating force include Newton's Law of Universal Gravitation.

Taken together with Newton's Third Law of Motion, it implies the Law of Conservation of Momentum.

Newton's Third Law

Alternative formulations: * Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body. * For every action, there is an equal and opposite reaction.

If you strike an object with a force of 200 N, then the object also strikes you (with a force of 200 N). Not only does a bullet exert force upon a target; but, the target exerts equal force upon the bullet. Not only do planets accelerate toward stars; but, stars accelerate toward planets. The reaction force has the same line of action, and is of the same type and magnitude as the original force.

Newton's Law of Universal Gravitation

Newton explains, "Every object in the Universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects."

Newton eventually published his still famous law of universal gravitation in his Philosophiae Naturalis Principia Mathematica as follows:

F = \frac{G m_1 m_2}{r^2}

where: * F = gravitational force between two objects * m1 = mass of first object * m2 = mass of second object * r = distance between the objects * G = universal constant of gravitation

Strictly speaking, this law applies only to point-like objects. If the objects have spatial extent, the true force has to be found by integrating the forces between the various points.