Department of History
University of California, Irvine
Instructor:    Dr. Barbara J. Becker

Lecture 11.  Mathematics and Motion.


New Philosophers, New Philosophy:  New Physics
Before the Copernican system could be accepted as more than a mathematical convenience,
a new physics would have to be developed to explain the "natural" motions of things.

In the old Aristotelian physics:

  • There is a distinction between terrestrial and celestial physics.
  • Every terrestrial body has a rightful place and must move naturally toward it.
  • The material composition of a terrestrial body determines where that rightful place is.

What should a new "Copernican" physics be like?

  • If earth is not the center of the universe, or if there are multiple centers--then what?  How can the natural motion of terrestrial bodies be explained??
  • If earth is a celestial body, does it still make sense to distinguish between terrestrial physics and celestial physics?
Pre-paradigm phase
  • no clear guidelines
  • more open discussion, argument, and controversy
  • more questions about validity of research questions, methods, results and interpretation
  • much interest and enthusiasm--a sense of mission
Maybe all motion is forced motion of one sort or another.

Each star in René Descartes' universe is surrounded by a swirling vortex of matter.  Planets do not possess their own independent motion, but are simply carried around their sun with the rest of the vortex matter.  In this way, Descartes could agree with the general structure of the Copernican system and still argue that the earth does not move relative to its surroundings.

René Descartes  (1596-1650)

As a young boy, Descartes received an excellent education.  His lessons came from textbooks based on the writings of the ancient Greeks.  Young René quickly mastered the mathematics of Euclid, the physics of Aristotle, and the anatomy of Galen.  The more he learned, the more he wanted to know.  The more he knew, the more questions he asked.  He wondered, for example, how Aristotle could be so sure that the earth was at rest in the middle of the universe while the mathematician Copernicus was equally certain that the earth was in motion around the sun.  Aristotle had reached his conclusions by relying on his senses.  Copernicus had reached his conclusions by relying on mathematics. 

Who was right?

While looking for answers to such puzzling questions, Descartes became distrustful of the knowledge he had learned in school.  He gave up reading all books and set out on his own personal search for truth.  His search was adventurous and exhausting.  For several years, he served as a volunteer soldier and travelled throughout Europe.  But at the age of 25, he gave up the soldier's life and settled in Holland where he devoted himself to the study of mathematics and philosophy.

René Descartes made many important contributions to mathematics and science.  He derived a mathematical expression to describe the bending (refraction) of light as it travels through different transparent materials.  He introduced the use of exponents to mathematics.  Anyone who has ever plotted points on a graph using an algebraic formula owes Descartes a debt of gratitude.

Perhaps his greatest achievement was his creation of an entirely new method of scientific investigation, a method which he proposed as a replacement for that of the ancients.

Descartes was convinced that humans can gain certain knowledge of the world and how it works, but only if they build that knowledge on a firm foundation of simple and indisputable truths.  Ordinarily, Descartes argued, our minds are cluttered by the distracting and deceptive information that bombards our senses every day.  That is why these simple and indisputable truths have remained inaccessible to our understanding.

Putting his own method to work, Descartes cleared his mind of all preconceptions.  Starting from scratch, he constructed an entirely new system of the world--a world composed only of matter and motion.

The complexity of the observable world is only an illusion, he claimed.  In reality, everything we humans sense is the product of innumerable collisions between extremely small particles of matter.  These collisions are controlled by two basic ideas:

  • The total quantity of motion in the universe is always constant.
  • Collisions are like contests whose outcomes are determined by the size and velocity of the contestants.
Descartes' new method and the system of the world he envisioned inspired the thoughts and work of those who lived after him, most notably Christiaan Huygens, Robert Hooke, and Isaac Newton.

Christiaan Huyghens was only about 15 years old when he first encountered the ideas of Descartes.  He was enthused by much of what he read.  Still, there were times when the French philosopher's words just didn't seem to make any sense at all.  For a while, the young scholar blamed his inability to fully understand Descartes' system of the world on his own ignorance.  But gradually, he began to realize that the problem lay in Descartes, not in himself.  Huygens became determined to correct Descartes' errors.

Huyghens did not share Descartes' extreme skepticism about the value of experimental results.  Human senses may be faulty, he admitted, but they are not totally unreliable. 

Christiaan Huyghens (1625-1695)

With adequate care and repetition of controlled scenarios, Huyghens was confident the limitations inherent in human sensory perception could be reduced.  He knew from experience that human senses could be artificially extended through the use of instruments.

At the age of 26, Huyghens designed and built an excellent telescope with which he discovered the ring of Saturn as well as its moon Titan, the first moon of that planet to be identified.  At the age of 27, he designed and built an accurate clock.  By introducing this new instrument into the investigator's toolkit, Huyghens revolutionized the study of moving bodies.  His pendulum-driven timepiece allowed him to make more precise measures in free-fall experiments than any of his predecessors.

For Huyghens, using experimentation hand-in-hand with reason proved a winning combination throughout his life.  His methodical study of colliding bodies led him to reject Descartes' ideas on collision and propose a very different set of rules governing their behavior.

What Happens in Simple Collisions?
Collision I
Two equal bodies move toward each other at equal speeds.

They will be reflected and continue to move at equal speeds.

They will be reflected and continue to move at equal speeds.

Collision II
Two equal bodies move toward one other at different speeds.

The slower one will be reflected.  Both bodies will continue to move in the direction of the faster one with a speed that is the average of the two.

They will move with their speeds reciprocally exchanged.

Collision III
One body moves toward another equal body that is at rest.

The moving body will give some of its speed to the one at rest, and will be reflected back with the larger share of that speed.

The moving body will stop while the one that had been at rest will move forward with the same speed as the body which struck it.

Collision IV
One body moves toward a slightly bigger body that is at rest.

No matter how fast the moving body is going, the body at rest will not be moved at all while the one in motion will be reflected.

No matter how slowly the moving body is going, the body at rest will be moved.

Why Do Things Fall?
According to René Descartes (ca. 1630):
According to Isaac Newton (ca. 1664-1675):

Isaac Newton  (1642-1727)

First of the Age of Reason?
Last of the Magicians?

A calculus fit to compute on,
White light, and a head to drop fruit on,
A mind to absorb it,
And soar into orbit--
That's all that it takes to be Newton.

       --Gina Berkeley

Nature and Nature's laws lay hid in night:
God said, "Let Newton be!" and all was light.

       --Alexander Pope (1688-1744)

Events in the Life of Isaac Newton
born at Grantham
enrolled at Trinity College, Cambridge
Anni Mirabiles (Remarkable Years)
  • optics
  • calculus
  • gravitation (mathematical description can be extended to include motion of moon)
"New Theory about Light and Colors" published in Philosophical Transactions
Edmund Halley visited Isaac Newton to discuss the question of why planetary orbits are elliptical
composed Principia
Philosophiae Naturalis Principia Mathematica [Mathematical Principles of Natural Philosophy] published
appointed Warden of the Mint
appointed Master of the Mint
elected to Parliament
elected President of the Royal Society
published Opticks
published second edition of Principia
published third edition of Principia
died at London; buried in Westminster Abbey

Young Newton's Guides to Conducting Experiments

Francis Bacon, Novum Organum (1620):

Suppose you are curious about the true nature of some phenomenon.  Take heat, for example.  How would you begin to investigate it?

In my opinion, the best way to start is by making a list of all the things you can think of that have heat.

The items on your list can be very different from one another in their other properties just so long as they all have the property of heat in common.

It is important to make the list without any advanced notions about what might be "right" or "wrong" to include.  Let those decisions come later.

It is better to include too many things than to leave an important clue off the list.

Example:  Francis Bacon's List of Things that Have Heat
  • The rays of the sun.
  • Comets.
  • Falling stars.
  • Rainbows, haloes, sun pillars, etc.
  • Lightning.
  • Eruptions of flames from the cavities of mountains.
  • Flame of every kind.
  • Natural hot springs.
  • Damp hot weather at any time of the year.
  • All shaggy substances (wool, animal skins, feathers).
  • All bodies placed near fire for any time.
  • Sparks arising from striking flint against steel.
  • All bodies rubbed violently.
  • Iron, when first dissolved by acids in a glass.
  • The bodies of animals (except insects).
  • Horse dung, and other animal excrement, when fresh.
  • Strong sulfuric acid will burn a hole in a piece of fabric.
  • Alcohol burns the mouth.
  • Some plants are warm to the tongue and have an almost burning taste when chewed.
  • Strong vinegar and all acids cause a pain almost like that produced by heat when they come into contact with any part of the body not covered by skin.
  • Severe and intense cold produces a sensation of burning.

René Descartes, Rules for Philosophizing (c. 1628):
  • Direct the mind to forming true & sound judgments about whatever comes before it....
  • Attend only to those objects of which our minds seem capable of having certain & indubitable cognition....
  • We need a method if we are to investigate the truth of things.
  • The method consists entirely in--
    • ordering and arranging objects on which we must concentrate;
    • reducing complicated & obscure propositions step by step to simpler ones....
  • Make use of all the aids which intellect, imagination, sense-perception, & memory afford in order to--
    • intuit simple propositions distinctly;
    • combine matters under investigation with what we already know; &
    • find out what things should be compared with each other so that we make the most thorough use of all our human powers.

Go to:
  • The Regulæ [Rules for the Direction of the Mind] (1628) by René Descartes (1596-1650);
    • Philosophia Naturalis Principia Mathematica [Mathematical Principles of Natural Philosophy] (1689);
    • Opticks (1704); and
Weekly Readings
Lecture Notes