Newton’s Philisophæ Naturalis Principia

Mathematica

Newton’s Life – Early Years

Born 1642 in Woolsthorpe, Lincolnshire

1661 – left home to attend Trinity College, Cambridge

Discovered contemporary scientists and a love for learning

1665 – Received his bachelor’s degree

Newton’s Life – Annus Mirabilis

Newton returned home for the plague years of 1665 – 1667

Extremely prolific time for him

Invented the calculus, experimented in light and chemistry, laid the foundations for mechanics and gravitation

Newton’s Life – Writing the Principia

1667 – Returned to Cambridge

1669 – Promoted to Lucasian professor, and turned his focus to his research

Prompted by friend and astronomer Edmund Halley, expanded his research in mechanics and gravitation

1687 – Publication of Philisophæ Naturalis Principia Mathematica

Newton’s Life - Recognition

Principia almost immediately recognized as a work of genius

“Nature, and Nature’s Laws lay hid in Night.God said, Let Newton be! And All was Light.”

Recognition by the Royal Society of London – appointed President of the Royal Society in 1703

1705 – knighted by Queen Anne, the first scientist to receive this honor

Newton’s Life - Conflict

Although a genius, Newton was also emotionally and mentally unstable

Longstanding argument with Robert Hooke

Widespread controversy and feud with Leibniz over the invention of the calculus

Newton’s Life – Beyond Science

Had many other interests, including alchemy, the Hermetic tradition, and theology

Considered a theologian in his time, but his beliefs were controversial

1693 – Newton finished with scientific exploration

Eventually became Master of the Mint and a social figure in London

Died March 20, 1727, from gallstones

Newton’s Methodology

Primary goal: uncovering the truth

Ultimate empiricist

Set the standard for experimental approach

Rules of Reasoning

Set forth in Book III of the Principia

RULE I: “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances”

RULE II: “Therefore to the same and natural effects we must, as far as possible, assign the same causes”

Rules of Reasoning (cont’d)

RULE III: “ 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”

RULE IV: “In experimental philosophy we are to look upon propositions inferred 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.”

Hypotheses non fingo

“Hypotheses non fingo” - “I do not frame hypotheses”

“Whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or of mechanical, have no place in experimental philosophy”

Advocated a scientific method based strictly on observation and induction, not prior assumptions

Hypotheses non fingo (cont’d)

Mathematical Physical Philosophical

Mathematical = Observation

Physical = induction from mathematics

Philsophical = search for the cause

Outline of Newton’s Principia

Begins with basic definitions (ex. mass, force, momentum)

His three laws of motion and corollaries follow these definitions.

Book I, The Motion of Bodies, contains his mathematical proofs, which are based in geometry, but introduced concepts of calculus.

Newton called this math “fluxions”

Section II of Book I discusses motion of a body around a fixed force center.

Section III discusses motion of bodies in conic sections.

Outline of Newton’s Principia

Newton uses Book III, The System of the World, to show how his theorems apply to natural phenomena.

Astronomical data supports Kepler’s three laws of planetary motion.

His Rules of Reasoning in Philosophy are contained in Book III.

Newton’s Definitions

Mass: “The quantity of matter is the measure of the same,

arising from its density and bulk conjunctly.”

Momentum: “The quantity of motion is the measure of the same,

arising from the velocity and quantity of matter conjunctly.”

Newton’s Definitions

Inertia: “The Innate Force of Matter, is a power of resisting,

by which every body, as much as in it lies, endeavors to persevere in its present state, whether it be of rest or of moving uniformly forward in a right line.”

Impressed force: “An impressed force is an action exerted upon a

body, in order to change its state, either of rest, or of moving uniformly forward in a right line.”

Centripetal force: “A centripetal force is that by which bodies are

drawn or impelled, or any way tend, towards a point as to a centre.”

Newton’s First Law

“Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.”

Also referred to as “law of inertia”

Differs from the concept of impetus, as inertia applies to bodies both at rest and in motion, and a body with inertia moves in a straight line path.

Newton’s Second Law

“The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force acts.”

In short: FΔt = Δp or Fα Δ pWhile F = ma famously comes from this law,

this equation is never actually stated this way in Principia.

Leonhard Euler is the first to recognize this equation as the basis for mechanics.

Can only be applied to inertial reference frames.

Newton’s Third Law

“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.”

Newton’s explanation of this law: two equal hemispheres in contact with each other at rest in empty space.

Newton’s Corollaries

Corollary I: “A body by two forces conjoined will describe the diagonal of

a parallelogram, in the same time that it would describe the sides by those forces apart.”

Corollary II: “And hence is explained the composition of any one direct

force AD, our of any two oblique forces AB and BD; and, on the contrary the resolution of any one direct force AD in two oblique forces AB and BD: which composition and resolution are abundantly confirmed in Mechanics.”

First two corollaries essentially describing vector components of forces

Parallelogram Law of Forces

This vector notation did not exist in Newton’s time, so he made use of parallelograms.

Newton’s Corollaries (cont’d)

Remaining four corollaries were primarily concerned with action-reaction forces.

Example: Corollary III explains that net forces acting on a body are the sum of the forces on that body.

Although Newton termed these ideas as corollaries, they do not follow naturally from his previous definitions and laws.

Questions?