Sir Isaac Newton:
The Man, the Method, the Madness
Paper 4
History of Mathematics
December 5, 2002
Problem 11.9
Abstract
Isaac Newton is one of our leading mathematicians and
scientists of all time. This paper will take you through his
childhood, schooling, and professions. It will delve into
the controversy over the discovery of calculus and you
will learn of his disdain of conflict. The math portion will
focus on Newton’s method of approximation, its
importance, and how it is used.
Any mathematician, when asked to name history’s three or four most
influential mathematicians of all time, would probably include Isaac Newton
among the elite. What makes this fact so rare is that any physicist, when asked
to name history’s top physicists of all time, would most likely identify Newton as
one of their choices as well. Isaac Newton was born prematurely in
Woolsthorpe, England, on Christmas day in 1642 and given little chance of
survival. His illiterate father was a farmer and had died the October before Isaac
was born. Despite the complications of his birth, a harsh winter, and only a
mother to care for the delicate newborn, Newton would live to the impressive age
of 84 (Dunham Journey 160).
When Isaac was three, his mother, Hannah Ayscough Newton, remarried
a 63-year-old reverend, Barnabas Smith, who wanted a young wife but could not
handle a three-year-old child. Therefore, Newton’s mother moved in with her
new husband and left Isaac behind to be raised by his grandmother. “All
Newton’s recent biographers have seen this separation from his mother between
the ages of three and ten, as crucial in helping to form the suspicious, neurotic,
tortured personality of the adult Isaac Newton” (Fauvel et al. 11).
Newton attended local schools until the age of twelve, when he moved to
the town of Grantham where he lived with the local apothecary while he attended
grammar school where the studies were heavily focused on the learning of Latin
and Greek. Outside of school Newton mainly kept to himself, did not have many
friends, and spent most of his time constructing clever mechanical models and
conducting experiments. He built a small working windmill driven by a mouse on
a treadmill, made sundials and put them all over his quarters, and made a
wooden clock that worked by water (Eves 397). By this time, after living with the
apothecary for four years, he had returned back home to help his now twice
widowed mother to manage the farm (his originally planned profession).
Realizing Newton’s gifts and his dislike of farming, his uncle, William Ayscough,
and former schoolmaster at Grantham, Henry Stokes, convinced Newton’s
mother to send him back to Grantham to prepare him for Cambridge. Stokes
taught Newton out of a mathematical textbook that contained arithmetic through
the extraction of cube roots, surveying, elementary mensuration, plane
trigonometry, and elaborate geometric constructions. This went far beyond
anything taught in any of the universities of the period and as a result Newton
matriculated into Trinity college, the foremost college at Cambridge in the
summer of 1661 with a superior knowledge of mathematics (Swetz 483-4).
The professors at Cambridge were fairly indifferent to teaching, so Newton
was free to follow his own interests. He turned away from the Greek and Latin
that characterized the official curriculum of the times and turned toward the
exciting mathematical and scientific advances of the age (Dunham Mathematical
130).
Newton found a book on astrology at a fair, and then as a result read
Euclid’s Elements, which he found too obvious, and then read Descartes’ La
Geometrie, which he found somewhat difficult. He also read works by Oughtred,
Kepler, Viete, and Wallis. By reading mathematics Newton began doing his own
mathematical research and would make an early discovery of the generalized
binomial theorem and creation of his method of fluxions, or differential calculus
(Eves 397-8).
Newton’s work continued during the 1665-7 hiatus when Cambridge was
shut down twice from outbreaks of the plague. During this time he went back to
his family home in Woolsthorpe where and when he had his famous encounter
with the apple. According to legend, as he rested under a tree he was nearly hit
by the falling fruit. He questioned that if the Earth tugged upon an apple, “did it
not also tug upon more distant celestial bodies?” He remarked, “I began to think
of gravity extending to ye orb of the moon,” which is a pretty concise introduction
to universal gravitation (Dunham Mathematical 130).
When the plague subsided, Newton returned to Trinity College where he
became Minor Fellow in 1667 and a Major Fellow in 1668. In October of 1669, at
the age of 26, he became the second Lucasian Chair of Mathematics at
Cambridge. This post gave him security, intellectual independence, and a good
salary. Even though the position had been designed as a teaching post, the
previous chair had already turned the position into a sinecure and Newton did not
work any harder at the teaching aspects. He made only 3-10 lectures per year
for the first seventeen years to very few people and none after (Swetz 485). His
first lectures on optics were later written in a paper to the Royal Society and were
attacked by some scientists. Newton regarded the resulting arguments as so
distasteful that he vowed never to publish anything on science again. This dislike
of controversy, which bordered on the pathological, would lead to disputes of
discovery later, especially with Leibniz concerning calculus (Eves 398).
In 1687, Newton had a huge public breakthrough under the pressure of
Edmund Halley, who would later become famous for the comet that bears his
name. Newton agreed to publish his Principia Mathematica, which entailed
Newtonian mechanics in a precise, careful, and mathematical fashion. He
introduced the laws of motion and the principles of universal gravitation, and
mathematically deduced everything from tidal flows to planetary orbits. It was and
still is regarded by many as the greatest scientific book ever written. This
triumph catapulted Newton into the scientific limelight and he became a living
symbol of the “new science”. Voltaire regarded Newton as the greatest man that
ever lived and said that a genius of this proportion only comes along once in a
thousand years (Fauvel et al. 185).
After his emergence from obscurity, Newton’s life took a dramatic change.
In 1689 he served as a representative for Cambridge in Parliament. Newton
suffered a nervous breakdown in 1693, which some attributed to his common
practice of tasting the chemical compounds he used in the alchemy experiments
he often tried to do. He resigned the Lucasian chair in 1695 and left Trinity
College for London where he became warden of the Mint in 1696. In 1703 he
was elected President of the Royal Society, published The Optics in 1704, and
on April 16, 1705 had the highest honor of being knighted by Queen Anne of
England. Sir Isaac Newton died on March 20, 1727 after being ill with gout and
inflamed lungs. He was buried along with the Kings and military heroes in
Westminster Abbey. Today his statue stands at the left hand portal of the
Abbey’s great choir screen and is visible to all who enter (Dunham Journey 182).
At the time of his death, Sir Isaac Newton was a revered scientist and
mathematician, a wealthy government servant, and an English hero for his most
heroic discovery dating back to the mid-1660s when he would work late into the
night with feverish excitement. It is said that his cat grew fat from eating
Newton’s untouched dinners; but the missed meals and lost sleep resulted in his
great “method of fluxions” (Differential Calculus) and his “inverse method of
fluxions” (Integral Calculus) as well as his groundbreaking theory of colors.
Since Newton did not publish his “fluxions” however it bears Leibniz’s name of
“calculus”. Explanations as to why Newton hated publication always came back
to his personality: his distrust of others, his dislike of criticism, and his strong
desire to decline being a part of being involved in “troublesome and insignificant
disputes” (Westfall 270). Newton often got into embroiled priority controversies
with other scientists and countrymen but his most famous dispute was the one
with Leibniz over the creation of calculus. The facts are:
1. Newton had discovered his method of fluxions by the mid-1660s. He
described it in a 1669 manuscript known as De analysi and an expanded
1671 treatise called De methodis fluxionum. These circulated among a
select group of British mathematicians but were not published and hence
not widely known. Those who read them instantly recognized Newton’s
power, and one described him as “very young…but of an extraordinary
genius and proficiency.”
2. During the mid-1670s, a full decade later, Leibniz made virtually the same
discoveries. While on a diplomatic mission to London in 1676, Leibniz
saw a manuscript copy of Newton’s De analysi.
3. At about the same time, Leibniz received two letters from Isaac-what have
come to be called the epistola prior and the epistola posterior-revealing
some of Newton’s thoughts about infinite series and, much less explicitly,
about fluxions.
4. In 1684 Leibniz published the first paper on differential calculus. Nowhere
in it did he mention that he had seen manuscripts or exchanged letters
with Newton eight years earlier. In fact, nowhere did he mention Newton
at all. (Dunham Mathematical 133).
It seems that both men discovered calculus independent of each other, and
because of Newton’s secretiveness and not being willing to publish, it was
Leibniz’s 1684 paper that would make the principles of calculus known to the
public. Had Newton published his findings back in the 1660s, he would have
gotten full credit for it.
Newton does get credit and is solely famous for creating a way of
approximating solutions of equations known as Newton’s Method. The method
we use today is not precisely the one first discovered by Newton in the 1660’s, it
was modified by Joseph Raphson in 1690 and then later in 1740 by Thomas
Simpson, however, the essential idea will always be credited to Newton.
The question raised by Newton was what does a mathematician do when
confronted with an unsolvable problem such as x7 –6x5 +4x4 +x3 –17 = 0? He
found that if an exact answer was not available, you could approximate one.
After all, an accurate solution to ten decimal places would suffice for any need,
especially if the technique was fairly simple, theoretical, and could be used
repeatedly to get even more accurate estimates. These are the basic properties
of Newton’s method.
To understand the method, first consider the graph of y=f(x) in figure A.
Solving for f(x)=0 will give us the x-intercept of the function labeled C in the
figure. If we can approximate C, then we will have solved the equation f(x)=0, at
least approximately (Dunham 136).
C
Figure A
Newton’s method requires that we begin with a guess of the solution. In figure A,
we designate X n-1 as our first guess. In essence, we are saying X n-1  C, the
actual solution. Starting at X n-1 on the horizontal axis we see the corresponding
point on the curve y=f(x) is (X n-1, f(X n-1)). At this point, draw the tangent to the
curve. Recall that the slope of the tangent is the derivative of the function
X=X n-1. Symbolically, the slope of the tangent is f’(X n-1). Ideally we would
descend along f(x) until we arrive at our exact value at C, but since the exact
solution is unknown, we instead begin at point (X n-1, f(X n-1) and move down our
tangent line. The point Xn where the tangent line intersects the x-axis, although
not exactly at point C, is at least a closer approximation to C than our initial
guess was. This is the geometric representation of Newton’s method.
To determine the approximation algebraically we consider the slope of the
tangent from two different points and equate the results. As noted before, the
slope of the tangent line is given by the derivative f’(X n-1). The slope of any line
is found by the expression
Slope =
y 2  y1
x2  x1
As indicated by the diagram, the tangent goes through points (X
(Xn,0). Therefore the slope is
n-1,
f(X n-1)) and
0  f (X n -1)
f (X n -1)

X n  X n1
X n  X n -1
Equating these two expressions for slope, we solve for Xn
f (X n -1)
X n  X n -1
f (X n -1)
Xn- Xn-1= 
f ' ( X n1 )
f (X n -1)
Xn=Xn-1 
f ' ( X n1 )
In a case where Xn is not accurate enough, we simply apply the whole argument
f’(X n-1) = 
again, only this time we would start with Xn so
f (X n )
f '(X n )
This is our general formula for Newton’s Method (Dunham 138).
Xn+1=Xn 
Solve the cubic x3 –2x –5 =0 by Newton’s method for the root lying between 2
and 3 (Problem 11.9b)
f(x) = x3-2x-5
Xn+1=Xn 
=Xn 
f’(x)= 3x2-2
f (X n )
f '(X n )
( X n ) 3  2( X n )  5
3( X n ) 2  2
( X 0 ) 3  2( X 0 )  5
(2) 3  2(2)  5
X1 =X0 
=2

3( X 0 ) 2  2
3(2) 2  2
X0=2
=2 
X1=2.1
=2.1
1
10
X3=2.094551361
.061
11.23
=2.094568
X4=2.094551482
X3 =2.094568-
X2=2.094568
X2 =2.1-
X5=2.094551482
X4
.000185723
11.16164684
=2.094551361
 .000001345426
=2.09455136111.16142364
=2.094551482
X5 =2.094551482=2.094551482
5.1083  10 9
11.16143773
Solve by Newton’s Method the equation x=tanx for the root lying between 4.4 and
4.5 (Problem 11.9c)
f(x)=tanx
f’(x)=sec2x
X0=4.4
X1
X0=4.4
X1=4.107541404
X2
X2=3.639787234
X3=3.220029955
X3
X4=3.141913977
X5=3.141592654
f (X n )
f '(X n )
tan( X n )
= Xnsec 2 ( X n )
tan( X 0 )
= X0sec 2 ( X 0 )
tan( 4.4)
= 4.4sec 2 ( 4.4)
3.096323781
= 4.410.58722095
=4.107541404
1.446598116
=4.1075414043.092646112
=3.639787234
.5439605563
=3.6397872341.295893087
=3.220029955
.0785985576
=3.2200299551.006177733
=3.141913977
.0003213234761
=3.1419139771.000000103
=3.141592654
 3.27  10 11
=3.1415926541
=3.141592654
Xn+1=Xn-
X4
X6=3.141592654
X5
X6
Find 12 by Newton’s Method (Problem 11.9d)
Xn+1
12
f(X)=X2-12
f’(X)=2X
X1
X0=3
X1=3.5
X2
X2=3.464285714
X3=3.46410162
X3
X4=3.464101615
=Xn-
f (X n )
f '(X n )
=Xn-
( X n ) 2  12
2( X n )
=X0-
( X 0 ) 2  12
2( X 0 )
32  12
= 32(3)
3
= 4.46
=3.5
.25
=3.5 7
=3.464285714
=3. 464285714-
.0012755102
6.928571428
=3.46410162
X5=3.464101615
X4
X6=3.464101615
X5
3.1915  10 8
6.92820324
=3.464101615
 9.54  10 10
=3.4641016156.92820323
=3. 464101615
=3.46410162-
A special case of Newton’s Method is Heron’s Method, which is one of the
earliest methods of approximating square roots, and also one of the fastest. It
works by finding
n for any positive n (problem 11.9e). Heron’s Method states:
Let n and  be positive numbers, let d be a non-negative integer and set
h()=h(+
(1)
n

). Then:
n is between  and
n

(2) If  > 0, then h()  n
1
(  n ) 2
(3) h( )  n 
2
1  n
1
(4) If   n < 2 , then h( )  n <min( 2 d ,
)
2
10
10
(5) (Brown 3-4)
What (4) means is that if  approximates
h(d) approximates
n to d decimal places, then
n to 2d decimals. That is, each time you apply Heron’s
Method, you double the number of digits of accuracy. This method, as stated
before, is just a special case of Newton’s Method. If we let Xn+1=N() and Xn=
then Newton’s method becomes N(_=-
f (X n )
. If f(x)=x2 –n. Then f’(x)=2x
f '(X n )
and
N ( )   
f (d )
 n 1
n
  
 (  )
f ' (d )
2 2 2

Which is Heron’s approximation. (Problem 11.9f)
Today, numerical analysis is a very useful branch of mathematics that
studies the finer points of approximation procedures. Although the subject has
become very subtle and deep, its focus and trademark has always been and will
be Newton’s method. Its one of mathematics great theorems and differential
calculus’ most far-reaching applications.
Work Cited
Brown, Ezra. Square Roots From 1; 24, 51, 10 to Dan Shanks. 13 Nov 2002
<http://www.math.vt.edu/people/brown/doc/sqrts.pdf>.
Dunham, William. Journey Through Genius: The Great Theorems of
Mathematics. New York: Penguin Group, 1990.
Dunham, William. The Mathematical Universe: An Alphabetical Journey Through
the Great Proofs, Problems, and Personalities. New York: John Wiley &
Sons, Inc, 1994.
Eves, Howard. An Introduction To the History of Mathematics. Fort Worth:
Saunders College Publishing, 1990.
Fauvel, John, Raymond Flood, Michael Shortland, and Robin Wilson. Let
Newton Be. New York: Oxford University Press, 1988.
Swetz, Frank. From Five Fingers to Infinity: A Journey Through the History of
Mathematics. Illinois: Open Court, 1994.
Westfall, R.S. Never At Rest. New York: Cambridge University Press, 1980.