ISAAC NEWTON

1642 -1727

Isaac Newton is possibly the greatest human intellect who has ever lived. Newton's discoveries in the areas of science and mathematics have been remarkable, prolific and groundbreaking. Newton's theories are still widely studied today by both mathematics and physics students. From the mid sixteenth century to the mid to late seventeenth century there were many significant developments that took place in both the academic and social climate of Europe that set the stage for Issac Newton. Newton himself once said, "If I have been able to see further, it was only because I stood on the shoulders of giants.",[1]  recognizing the advances he made could have only come about because of the work of others. Changes in the status of the church have influenced the papacy to act in a hostile manner toward academicians. Such examples include the imprisonment of Galileo and Cardano. This causes the academic development to move away from Italy into northern parts of Europe, in particular Britain, France and Germany. Several people during this time had made some key discoveries and innovations in the scientific and mathematical realm. Before he was imprisoned Galileo began to study the motion of "falling bodies" and was able to conclude how the rate an object falls is not dependent upon its mass. A Frenchman by the name of Francois Viete makes huge strides in the development of algebra, in particular the concept of using words or symbols to stand for numbers or quantities. Rene Descartes (1596-1650) develops the Cartesian coordinate system among many other ideas. This associates geometry and algebra so that mathematicians could look at curves from an algebraic point of view creating an area of study known as analytical geometry. Descartes has a very intelligent young student by the name of Blaise Pascal (1623-1662) who is primarily interested in studying theology, but works on mathematics as a "distraction" and

makes some contributions in the area of probability. A French lawyer by the name of Pierre Fermat formulates a famous "theorem" concerning integer solutions to the equation xn+yn=zn in which he posed had no positive integer solutions other than the case when n=2. [2]

Issac Newton was born on Christmas day in 1642. He was premature and frail in the early part of his life. His father had died in October of that year leaving Isaac fatherless. When Isaac was three his mother remarried a man named Barnabus Smith who wanted nothing to do with Isaac. Smith forced Isaac to live with his grandmother and have no interaction with his mother, even though he lived within sight of her. This probably made a significant psychological impact on Isaac since he has been described as neurotic and a "loner", that he did not have friends. Later in his life he becomes obsessed with writing down and maintaining long lists of "sins" he committed such as having indecent thoughts, words or actions. As a boy he enjoys building working modals of things.[2] In 1661 Newton goes off to Trinity College (Cambridge) and finds the place in a total mess. This is the result of several years of political fighting between the Puritans led by Cromwell (who supported Oxford) and the Royalists (who supported Cambridge). The reformation of the monarchy has placed Charles II on the throne one year before Newton arrives at Trinity. The professors who are there have been appointed most for political reasons or religious views. The climate in not intellectually driven, many professors are rarely on campus spending much of their time drinking. Few give lectures or work with students.[2]

Newton first begins to study Latin and Aristolelian philosophy. He becomes discouraged and focuses on problems of a physical nature. The library is one area of the college that has been better attended to and Newton begins to read the work of Descartes Geometry. His intellectual studies flourish, he begins to study mathematics, optics, motion, heat and many other topics in

physics. In 1664 Isaac Barrow who is the Lucasian Professor of Mathematics, has recognized the talent of Isaac and promoted him to the status of scholar, which gave him funds for four years as he worked toward his master's (PhD) degree.

In the following years came Newton's "period of greatness" in which some of his most famous results coalesced. In 1665 he develops "Method of fluxions" which is what is known today as differential calculus and in 1666 he does "Inverse method of fluxions" or known today as integral calculus.[2]

In 1669 Newton is appointed to Lucasian Professor of Mathematics despite not publishing a thing because Barrow is so impressed by his actions. In 1689 at the urging of Edmund Halley, Newton publishes his Principa Mathematica. In 1689 he is appointed a member of parliament. Between 1692 and 1694 Newton undergoes a period of "derangement" that was probably associated with ingesting chemicals he was experimenting with. In 1705 Newton is knighted by Queen Anne. He dies in 1727 and is buried in Westminster Abby. We now turn our attention to two of Newton's mathematical accomplishments that were a significant development in the area of solving equations. Methods of solving equations performed by Cardano and Ferrari all relied on the method of extracting a root. This means the solution would be expressed as a combination of powers and roots. Newton thought of the idea of "closing in" on a root to "estimate" its value.  [1] website: http://turnbull.mcs.st-and.ac.uk/~history [2] Dunham William; Journey Through Genius The Great Theorems of Mathematics; Wiley (1990).

Newton was able to develop a new way to "solve" equations which today we call Newton's Method. This was not really a way to solve them in as much as it was a way to approximate (generate better estimates) solutions to an equation.

When Newton implemented his method he found he needed to multiply out binomial expressions to very large powers. In order to do this he developed a procedure or "formula" for this that today we call the Binomial Formula.

The Binomial Formula

The most common version of the Binomial Formula deals with how to expand expressions of the form (x + y)n. Where x and y are any numbers and n is an integer. this is sometimes written in the following way.

nnnnn

n

k

kknknk

nn

nnnnnnn

k

kknn

yyxnnn

yxnn

yxn

xyxyx

yxyn

nyx

nyx

nyx

nxyx

k

nyx

33221

0!!

!

133221

0

123

21

12

1

1

1321

The symbol !!

!

knk

n

k

n

is the combinatorial choose. It is sometimes written nCk.

Newton was even able to extend this formula to include fractional and negative values for n.

Pascal's Triangle

Pascal's Triangle organizes the information in the combination formula into a triangle where each is a set of a particular size n and each column are the subsets of a certain size r. Here is the table below.

0 1 2 3 4 5 6

0 1

1 1 1

2 1 2 1

3 1 3

4 1 4 6 1

5 1 10 5 1

6 1 6 20 15 6 1

nr Notice the entry in the 4th row and 2nd column is

6. Remember 4C2 = 6.

This table is built from the famous "upside down L" pattern that the entries above and to the right and directly above give the entry below.

5 + 10 = 15 3 + 1 = 4

Each row can be gotten from the previous one.5 10

15

3 1

4

This table is very useful in figuring out what (x + y)n is multiplied out.

(x + y)2 = x2 + 2xy + y2 (Look in row 2)

(x + y)3 = x3 + 3x2y + 3xy2 + y3 (Look in row 3)

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 (Look in row 4)

(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy3 + y5 (Look in row 5)

Before we explore what Newton actually did in terms of solving equations lets look at the modern view of Newton's method which is sometimes called the Newton-Raphson-Simpson method. It is important to remember this is not a method for generating and exact solution but only an approximation to a solution. The method is recursive and uses its own results to generate better estimations of a solution.

Problem: Given a differentiable function f(x) approximate a solution to f(x)=0 (i.e. a root of f(x)).

Newton’s Method

The idea for this method is to use f(x) to build another function h(x) that will generate a recursive sequence that approximates the root.

The idea here is to keep following the tangent line at a point on the graph down to the x-axis and use that for the value of x that will approximate the root. In other words h(x) represents the x-intercept of the tangent line of f(x).

x

f(x)

h(x)

f(xn)

h(xn)=xn+1

xn

root root

To get what xn is from xn+1 we write the equation of the tangent line at xn, plug in the point (xn+1,0) and solve for xn+1.

)('

)(

)('

)(

))((')(0

))((')(

1

1

1

n

nnn

nnn

n

nnnn

nnn

xf

xfxx

xxxf

xf

xxxfxf

xxxfxfy

equation of tangent at xn

substitute in (xn+1,0)

solve

this is the h(x)

The equation above gives the recursively defined sequence for xn. This is what is used for Newton’s Method. This is repeated until the answers that are generated agree in a certain number of decimal places.

1,2)( 02 xxxf

x

xxxh

2

2)(

2

n xn h(xn)

0 1 1.5

1 1.5 1.41667

2 1.41667 1.41422

3 1.41422 1.41421

2,52)( 03 xxxxf

23

52)(

2

3

x

xxxxh

n xn h(xn)

0 2 2.1

1 2.1 2.09457

2 2.09457 2.09455

3 2.09455 2.09455

Here are some examples of Newton’s Method applied.

This method has mixed in with it the influence of Raphson and Simpson. Raphson incorporated the modern idea of derivatives. Simpson saw the generalization as a recursive method and even explored equations of several variables.

The original Newton method was used to approximate solutions to polynomial equations. This seemed a useful idea at the time since a general method of solving all polynomial equations was not known, only degree 4 or less. The method that was used I will demonstrate with by applying it to the cubic equation x3-2x-5=0, but Newton saw how it could be applied to all polynomials.

Newton let f(x)=x3-2x-5 and realized that f(2)=-1 and f(3)=16 so a root must be between 2 and 3. He then Let x=2+p be the approximation for the root. Plug this back into the equation drop off any terms whose degree for p is higher than 1 and solve the linear equation (since linear equations are easiest to solve).

1.22

0110

05246128

05222

2

1021

101

101

32

3

x

p

p

pppp

pp

px

09457.2

0

052

052

561511761

1123061

1021

1123061

100061

1001123

104232

1063

1001323

10009261

10213

1021

1021

x

p

p

pppp

pp

px

Lets compare the modern Newton's Method and his original version more closely by looking at a quadratic equation. In particular the equation x2-2=0, letting f(x)=x2-2 we see the is a root between 1 and 2.

5.11

012

0221

021

1

23

21

21

2

2

x

p

p

pp

p

px

41667.1

03

023

02

1217

121

23

121

41

249

2

23

23

x

p

p

pp

p

px

Notice the answers that are generated agree exactly with the modern day Newton's method. Lets consider a general quadratic equation:

ax2+bx+c =0

Let f(x) = ax2+bx+c =0 .Lets look at what is happening from the nth step.

)('

)(

2

2

02

02

0

2

2

2

22

2

n

nn

n

nnn

n

nn

nnn

nnn

nn

n

xf

xfx

bax

cbxaxxx

bax

cbxaxp

cbxaxpbax

cbpbxappaxax

cpxbpxa

pxx