Isaac Newton

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

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Isaac Newton, 1642-1727

„ Born December 25, 1642 „ by the Julian Calendar „ or 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.

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Woolsthorpe Manor

„ Newton was slated to take over family estate and manage a farm, but was obviously too interested in books and Newton’s birthplace, Woolsthorpe study. Manor, in Lincolnshire.

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1 Trinity College, Cambridge

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

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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.

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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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2 Newton's Annus Mirabilis

„ Newton’s “miracle year,” 1666

„ During the plague, Newton returned home, thought about his new interests and made his most original insights.

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3 Major Insights of 1666

„ Light Calculus The falling apple

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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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3 Newton investigates light

„ Newton undertook to investigate using a triangular prism

„ Concluded that coloured light is simple

„ There are particles of blue light, particles of red light, etc.

„ White light is a mixture of coloured lights

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The Crucial Experiment

„ The crucial experiment--on light of a single colour

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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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4 Letter to Royal Society, 1672

A 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….

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The Calculus

„ Concerns 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/t „ In any instant, t = o, d = o „ What is 0/0?

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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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5 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.

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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. st „ But the average speed for the 1 hour is 50 km/hr; for the 2nd is 100 km/hr; and for the third is 150 km/hr.

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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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6 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.

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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 Where

„ s is the total distance fallen, st „ 4.9 is the distance the object falls in the 1 second, expressed in meters,

„ and t is the time elapsed, in seconds.

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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. rd „ Its average velocity during the 3 second of its fall is 24 meters per second.

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7 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.

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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 /∆t

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

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Working with the Galileo/Descartes equation

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

„ Now, subtract line 1. from line 4. (Equals subtracted from equals.) 2 2 2 „ 5. (s + ∆s) – s = (4.9t +9.8t∆t + 4.9[∆t ] ) - 4.9t 2 „ 6. ∆s = 9.8t∆t + 4.9(∆t )

„ 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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8 Working with the Galileo/Descartes equation, 2

2 „ 6. ∆s = 9.8t∆t + 4.9(∆t )

„ Now, divide line 6. by the time increment, ∆t . 2 „ 7. ∆s /∆t = [9.8t∆t + 4.9(∆t ) ]/∆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.

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Working with the Galileo/Descartes equation, 4

„ 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

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Working with the Galileo/Descartes equation, 5

„ 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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9 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.

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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.

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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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10 A precocious child

„ Leibniz 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.

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A university student

„ In 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.

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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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11 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.

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Christiaan Huygens

„ 1629-1695

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

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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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12 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.

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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 +

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Evaluating infinite series, 2

„ Note that any triangular number can be written as n(n+1)/2 „ Therefore 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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13 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

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Other infinite series

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

„ Newton also evaluated many infinite series.

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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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14 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.

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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.

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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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15 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.

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…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.

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