Isaac Newton - York U

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11Math 1700 – Newton1

Isaac Newton

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


Isaac Newton, 1642-1727Born December 25, 1642

by the Julian Calendaror January 4, 1643 by the Gregorian Calendar.

From a family of yeomen farmers.

His father died some months before Newton was born (a "posthumous" child).His mother remarried and left Newton to be raised by her mother.


Woolsthorpe ManorNewton was slated to take over family estate and manage a farm, but was obviously too interested in books and study.

Newton’s birthplace, WoolsthorpeManor, in Lincolnshire.

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Trinity College, Cambridge

Sent to Trinity College, University of Cambridge in 1661. Took an ordinary BA in 1665.


Newton discovered the world of mathematics and natural philosophy

After finishing his BA, Newton planned to stay on at Cambridge for further study.

In his last years at Cambridge as an undergraduate, Newton had become very interested in the new mechanical philosophy (Descartes, etc.), in mathematics, and in Copernican astronomy.


The Plague hits England

An outbreak of the plague hit England in 1666.Cambridge closed for 18 months.Newton returned to Woolsthorpe to wait it out.

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Newton's Annus Mirabilis

Newton’s “miracle year,” 1666During the plague, Newton returned home, thought about his new interests and made his most original insights.


3 Major Insights of 1666

Light Calculus The falling apple


Light

Light one of the great mysteries.Magical nature

Connected with the Sun, with fire, with vision.

Colour thought of as a quality, e.g. blueness.Mathematical or mechanical treatment seemed impossible.

Descartes believed colours due to the spin of light particles.

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Newton investigates light

Newton undertook to investigate using a triangular prismConcluded that coloured light is simple There are particles of blue light, particles of red light, etc.White light is a mixture of coloured lights


The Crucial Experiment

The crucial experiment--on light of a single colour


Light as Particles

Particles, not waves because light rays move in straight lines.

The diagram is Newton’s illustration of what light waves would do, passing through a small opening.

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Letter to Royal Society, 1672A Letter of Mr. Isaac Newton, Professor of the Mathematicks in the University of Cambridge; containing his New Theory about Light and Colors: sent by the Author to the Publisher from Cambridge, Febr. 6. 1671/72; in order to be communicated to the R. Society.

Sir,

To perform my late promise to you I shall without further ceremony acquaint you that in the beginning of the Year 1666 (at which time I applyed my self to the grinding of Optick glasses of other figures than Spherical) I procured me a Triangular glass-Prisme, to try therewith the celebrated Phænomena of Colours.And in order thereto having darkened my chamber, and made a small hole in my window-shuts, to let in a convenient quantity of the Sun’s light, I placed my Prisme at his entrance, that it might be thereby refracted to the opposite wall. It was at first a pleasing divertisement, to view the vivid and intense colours produced thereby; but after a while applying my self to consider them more circumspectly, I became surprised to see them in an oblong form; which, according to the received laws of Refraction, I expected should have been circular….


The CalculusConcerns quantities that change over time (or space).

Example: If the speed of a falling body changes constantly, how fast is it going at any given moment?

According to Zeno, it is not going anywhere in a fixed instant.

speed = velocity = (distance travelled)/(time used) = d/tIn any instant, t = o, d = o

What is 0/0?


Average Velocity

Can be expressed as the total distance divided by the total time.Example:

A car trip from Toronto to Kingston takes 3 hours and the distance is 300 km.The average speed of the car is 300 km/3 hrs = 100 km/hr.This is the average speed, though the car may have sped up and slowed down many times.

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Instantaneous velocity

What is the car’s speed at any given moment? What does that number on the speedometer really mean?Newton’s definition:

Instantaneous Velocity = the Limit of Average Velocity as the time interval approaches zero.


Smaller and smaller time intervals

Suppose on a trip to Kingston along the 401, the car went 50km in the 1st hour, 100 km in the 2nd hour, and 150 km in the 3rd hour.The total time remains 3 hours and the total distance remains 300 km, so the average speed for the trip remains 100 km/hr.But the average speed for the 1st hour is 50 km/hr; for the 2nd is 100 km/hr; and for the third is 150 km/hr.


Still, the problem of 0/0

As the time intervals get smaller, a closer approximation to how fast the car is moving at any time is still expressible as distance divided by time.But if you get down to zero time, there is zero distance, and Zeno’s objections hold.

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Newton’s clever way to calculate the impossible

Newton found a way to manipulate an equation so that one side of it provided an answer while the other side seemed to defy common sense.For example, Galileo’s law of falling bodies, expressed as an equation in Descartes’ analytic geometry.


Deriving a meaning for 0/0

The equation for the position of a falling body near the surface of the Earth is:

s = 4.9t 2

Wheres is the total distance fallen, 4.9 is the distance the object falls in the 1st

second, expressed in meters,and t is the time elapsed, in seconds.


Calculation of the Limit of Average Velocity

The average velocity of a falling object over any interval of time during its fall is d/t, that is, distance (during that interval), divided by the time elapsed.

For example, by Galileo’s Odd-number rule, if an object falls 4.8 meters in the 1st second, in the 3rd

second it will fall 5x4.8 meters = 24 meters. Its average velocity during the 3rd second of its fall is 24 meters per second.

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But the object speeds up constantly

The smaller the time interval chosen, the closer will the average velocity be to the velocity at any moment during that interval.Suppose one could take an arbitrarily small interval of time.

Call it ∆t. Call the distance travelled during that small interval of time ∆s.


Approximating the instantaneous velocity

The average velocity of a moving object is distance divided by time.The average velocity during the arbitrarily small interval ∆t is therefore

∆s /∆tThe smaller ∆t is, the closer will be ∆s /∆t to the “real” velocity at time t.


Working with the Galileo/Descartes equation

1. s = 4.9t 2

2. s + ∆s= 4.9(t+∆t) 2

3. s + ∆s= 4.9(t 2+2t∆t + [∆t ] 2)4. s + ∆s= 4.9t 2+9.8t∆t + 4.9[∆t ] 2

Now, subtract line 1. from line 4. (Equals subtracted from equals.)

5. (s + ∆s) – s = (4.9t 2+9.8t∆t + 4.9[∆t ] 2) - 4.9t 2

6. ∆s = 9.8t∆t + 4.9(∆t ) 2

Line 6 gives a value for the increment of distance in terms of the total time, t, and the incremental time, ∆t.

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Working with the Galileo/Descartes equation, 2

6. ∆s = 9.8t∆t + 4.9(∆t ) 2

Now, divide line 6. by the time increment, ∆t .

7. ∆s /∆t = [9.8t∆t + 4.9(∆t ) 2]/∆t Which simplifies to

8. ∆s /∆t = 9.8t + 4.9∆t What happens to ∆s /∆t when ∆t (and ∆s ) go to zero?We don't know because 0/0 is not defined.



Newton’s solution:Forget about the left side of the equation:

8. ∆s /∆t = 9.8t + 4.9∆t Just look at what happens on the right side.As ∆t becomes smaller and smaller,

9. 9.8t + 4.9∆t becomes 9.8t + 4.9(0) = 9.8t



As the time increment ∆t gets closer and closer to zero,

10. ∆s /∆t gets closer to 9.8t .Since the left side of the equation must equal the right side and the left side is the velocity when ∆t goes to zero, then 9.8 t is the instantaneous velocity at time t.

Example: After 3 full seconds of free fall, the object falling will have reached the speed of 9.8 x 3 = 29.4 meters per second.

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Newton’s breakthrough

Newton’s genius in the calculus was to find a way to get around the static definitions which ruled out such calculations.He was willing to entertain the “impossible” idea of an object moving in an instant of time, and found an answer.


The Great Calculus Controversy

Curiously, the basics of the calculus were discovered independently at almost the same time by two men: Newton and Gottfried Wilhelm Leibniz.


Gottfried Wilhelm Leibniz (1646-1716)

Born in Leipzig in 1646 (three years after Newton), son of a jurist and professor at the University of Leipzig.Leibniz’s father died when Leibniz was six years old. (Remember that Newton’s father died just before Newton was born.)

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A precocious childLeibniz taught himself Latin from an illustrated copy of Livy’s history of Rome when he was eight years old.He undertook the study of Greek when he was twelve.As a result, he was given free access to his father’s library, which was otherwise kept locked away.Leibniz proceeded to educate himself by reading through his deceased father’s books.


A university studentIn the fall of 1661, Leibniz entered the University of Leipzig. He was then 15 years old.Newton, remember, entered Cambridge in the same year.Leibniz, a clear prodigy, graduated in 1663, after writing a philosophical thesis.In 1664 he earned a master’s degree in law, and was appointed to the faculty in philosophy.


A profession in law

In 1667 Leibniz obtained a doctorate in law from the University of Altdorf in Nuremberg.He went into the service of the archbishop-elector of Mainz, as a legal advisor and remained in similar work for the remainder of his life.

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Leibniz in Paris

As part of a scheme to convince Louis XIV of France to attack Egypt (instead of Germany), Leibniz persuaded the Elector of Mainz to send him to Paris.Leibniz stayed four years.

There he met Christiaan Huygens who instructed Leibniz in mathematics.


Christiaan Huygens

1629-1695Now considered primarily an astronomer, Huygens was, like Leibniz, trained in law and made significant contributions to mathematics.


Leibniz takes interest in mathematics

Leibniz had a deep interest in logical deduction and had previously written a work on the theory of permutations and combinations as it applied to logic, but otherwise was little schooled in mathematics.Huygens ignited an interest in Leibniz to master more advanced mathematics.

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Leibniz studies Descartes

Huygens had written a major work on pendular motion, Horologium Oscillatorium, and gave a copy to Leibniz. Leibniz did not have the mathematical background to read it.To educate himself in mathematics, Leibniz read the standard mathematical works of his age, including van Schooten’s two-volume edition of Descartes’ Géométrie, and the works of Blaise Pascal.


Evaluating infinite series

Leibniz took particular interest in the sums of infinite series, bragging to Huygens that he could find the sum of any series that converged.Huygens set Leibniz the task of finding the sum of the series of reciprocals of the triangular numbers:

1/1 + 1/3 + 1/6 + 1/10 + 1/15 + …


Evaluating infinite series, 2

Note that any triangular number can be written as n(n+1)/2Therefore the terms of the series of reciprocals of triangular numbers are

2/{n(n+1)}Leibniz noted that these terms could be written as the sum of two others:

2/{n(n+1)} = 2{1/n - 1/(n+1)}

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Evaluating infinite series, 3

Hence the series, 1/1 + 1/3 + 1/6 + 1/10 + 1/15 + …

Can be written as:2/(1•2) + 2/(2•3) + 2/(3•4) + 2/(4•5) + …

= 2(1-1/2) + 2(1/2-1/3) + 2(1/3-1/4) + …

= 2


Other infinite series

Leibniz found other infinite series by such manipulations, notably, the following calculation for πΠ /4 = 1–1/3+1/5–1/7+1/9–1/11+ …

Newton also evaluated many infinite series.


Leibniz invented a calculator

Leibniz was also interested in the mechanical calculation of sums and invented one of the first mechanical calculators (really an adding machine).For this invention he was inducted into the Royal Society of London.

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Leibniz’s period in Paris akin to Newton’s at Woolsthorpe

Leibniz’s four years in Paris (1672-1676) was a concentrated period of inventiveness that has comparisons to Newton’s “annus mirabilus” in 1666 when he retired to Woolsthorpe Manor during the plague.


Leibniz and Newton consider similar problems

Both Leibniz and Newton spent a great deal of time around the same time dealing with the problems of calculating the quantities that we now think of as comprising the calculus: instantaneous rates of change, areas under curves, sums of series involving infinitisimals, etc.Leibniz became aware of Newton’s work during a 2 month visit to England and later asked to be kept aprised of Newton’s mathematical work.


Both delay publication

Neither Newton nor Leibniz published their work on calculus right away. Instead their ideas were written in manuscripts and shown to a few friends.Eventually Leibniz published his writings first, in a German scientific journal (written in Latin).

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At first, collegial harmony…In the first edition of the Principia Mathematica, Netwon acknowledged that Leibniz had independently worked out some of the same solutions that Newton himself had, differing only in notation.But supporters of Newton accused Leibniz of plagiarism, and supporters of Leibniz indignantly defended him.Eventually both men were drawn in and accused each other of stealing the other’s ideas. The dispute raged for 30 years.


…and then bitter acrimony

The Newton-Leibniz calculus controversy chilled relations between England and the Continent for the next century.In sum, Newton’s presentation was much more difficult to follow than that of Leibniz.The notation adopted today of dy/dx for the

derivative and ∫ for the integral were invented by Leibniz.

Isaac Newton - York U

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