young age of 27, Newton was appointed Lucasian professor at Cambridge.
At this time Newton began serious work on the
study of optics. With the aid of a prism he discovered
that white light was composed of a spectrum of colors,
each refracting through a lens at a different angle. He
concluded then that all refracting telescopes were subject to chromatic aberrations and set to work on building a reflecting model. In 1672 he was elected as a
fellow of the prestigious ROYAL SOCIETY of London in
honor of this work. He published his text Optiks
(Optics) in 1704.
Newton also began formulating his famous laws of
motion and his laws of gravitation during the mid1660s. In 1684 eminent astronomer Edmond Halley
(1656–1742) suggested to Newton that he should
investigate the mathematics of gravitational attraction
and attempt to derive JOHANNES KEPLER’S famous
three laws. Newton apparently replied simply that he
already had many years earlier. Only after Halley’s persistent urgings did Newton agree to publish the results
in what was to become his most famous piece of work,
the 1687 piece Principia.
Newton also published in his lifetime, somewhat
delayed after completing the work, Enumeratio linearum tertii ordnis (Enumeration of lines of the third
order) in 1704, Arithmetica universalis (Universal
arithmetic), his collected works in algebra in 1707, and
Analysis per quantitatum series (Analysis by means of
various series) in 1711.
Newton resigned from his position at the University of Cambridge in 1701 to take a prestigious government position in London and participated in very little
mathematical research thereafter. In 1703 he was
elected president of the Royal Society and was reelected
to that position each year until his death on March 20,
1727. In 1705 he was knighted by Queen Anne,
becoming the first scientist to be so honored for scholarly achievement.
It is impossible to understate the influence that
Newton’s work has had on the development of all scientific research. Scholars today agree that the publication of Principia marked the beginning of the modern
scientific era.
Newton’s method In many practical situations, one is
required to find a numerical solution to an equation of
the form f(x) = 0, even if there are no clear algebraic
means for solving such an equation. (There are no general methods suitable for solving √
–
x +
+
– 5 = 0, for instance.) If the function in question is differentiable (see DIFFERENTIAL CALCULUS), then one can
employ Newton’s method to find the approximate location of a ROOT.
One begins by making an initial guess x 0 , hopefully missing the correct value of the root by just a
small amount. Suppose it turns out that the root is h
units away from x 0 (thus f(x 0 + h) = 0). Using this
value of h, we can approximate the derivative of the
function at x 0 as:
Turning this around, we see that h is approximately
–f(x 0 )/f′(x 0 ). Substituting this value for h shows that if
x 0 is our initial guess for the root, then the point:
is likely to be a much better approximation to the root.
(Geometrically, x 1 is the location at which the tangent
line to the graph y = f(x) at position x 0 crosses the xaxis.) Repeating this procedure, each time using the outcome just obtained as the next initial guess, produces a
sequence of values x 0 ,x 1 ,x 2 ,…, with
yielding successively better approximations of the root.
One can perform this procedure until a desired degree
of accuracy is obtained.
To illustrate Newton’s method, we compute √
–
2 to
four decimal places. Computing this root is equivalent
to solving the equation x
2 – 2 = 0. Setting f(x) = x
2 – 2,
we have f′(x) = 2x. Thus the sequence of approximations is given by the formula:
(Compare with HERON’S METHOD.) With x 0 = 1 as our
initial guess, we obtain the approximations:
x 1
3
2
1 5
= = .
x
x
x
x
x
x
n
n
n
n
n
n
+ =
−
− =
+
1
2
2
2
2
2
x
x
f x
f x
n
n
n
n
+ =
− ′
1
( )
( )
x x
f x
f x
1
0
0
0
=
− ′
( )
( )
′
≈
+ −
= −
f x
f x h f x
h
f x
h
( )
(
) ( )
( )
0
0
0
0
√x + 3
√x + 2
Newton’s method 353
At this time Newton began serious work on the
study of optics. With the aid of a prism he discovered
that white light was composed of a spectrum of colors,
each refracting through a lens at a different angle. He
concluded then that all refracting telescopes were subject to chromatic aberrations and set to work on building a reflecting model. In 1672 he was elected as a
fellow of the prestigious ROYAL SOCIETY of London in
honor of this work. He published his text Optiks
(Optics) in 1704.
Newton also began formulating his famous laws of
motion and his laws of gravitation during the mid1660s. In 1684 eminent astronomer Edmond Halley
(1656–1742) suggested to Newton that he should
investigate the mathematics of gravitational attraction
and attempt to derive JOHANNES KEPLER’S famous
three laws. Newton apparently replied simply that he
already had many years earlier. Only after Halley’s persistent urgings did Newton agree to publish the results
in what was to become his most famous piece of work,
the 1687 piece Principia.
Newton also published in his lifetime, somewhat
delayed after completing the work, Enumeratio linearum tertii ordnis (Enumeration of lines of the third
order) in 1704, Arithmetica universalis (Universal
arithmetic), his collected works in algebra in 1707, and
Analysis per quantitatum series (Analysis by means of
various series) in 1711.
Newton resigned from his position at the University of Cambridge in 1701 to take a prestigious government position in London and participated in very little
mathematical research thereafter. In 1703 he was
elected president of the Royal Society and was reelected
to that position each year until his death on March 20,
1727. In 1705 he was knighted by Queen Anne,
becoming the first scientist to be so honored for scholarly achievement.
It is impossible to understate the influence that
Newton’s work has had on the development of all scientific research. Scholars today agree that the publication of Principia marked the beginning of the modern
scientific era.
Newton’s method In many practical situations, one is
required to find a numerical solution to an equation of
the form f(x) = 0, even if there are no clear algebraic
means for solving such an equation. (There are no general methods suitable for solving √
–
x +
+
– 5 = 0, for instance.) If the function in question is differentiable (see DIFFERENTIAL CALCULUS), then one can
employ Newton’s method to find the approximate location of a ROOT.
One begins by making an initial guess x 0 , hopefully missing the correct value of the root by just a
small amount. Suppose it turns out that the root is h
units away from x 0 (thus f(x 0 + h) = 0). Using this
value of h, we can approximate the derivative of the
function at x 0 as:
Turning this around, we see that h is approximately
–f(x 0 )/f′(x 0 ). Substituting this value for h shows that if
x 0 is our initial guess for the root, then the point:
is likely to be a much better approximation to the root.
(Geometrically, x 1 is the location at which the tangent
line to the graph y = f(x) at position x 0 crosses the xaxis.) Repeating this procedure, each time using the outcome just obtained as the next initial guess, produces a
sequence of values x 0 ,x 1 ,x 2 ,…, with
yielding successively better approximations of the root.
One can perform this procedure until a desired degree
of accuracy is obtained.
To illustrate Newton’s method, we compute √
–
2 to
four decimal places. Computing this root is equivalent
to solving the equation x
2 – 2 = 0. Setting f(x) = x
2 – 2,
we have f′(x) = 2x. Thus the sequence of approximations is given by the formula:
(Compare with HERON’S METHOD.) With x 0 = 1 as our
initial guess, we obtain the approximations:
x 1
3
2
1 5
= = .
x
x
x
x
x
x
n
n
n
n
n
n
+ =
−
− =
+
1
2
2
2
2
2
x
x
f x
f x
n
n
n
n
+ =
− ′
1
( )
( )
x x
f x
f x
1
0
0
0
=
− ′
( )
( )
′
≈
+ −
= −
f x
f x h f x
h
f x
h
( )
(
) ( )
( )
0
0
0
0
√x + 3
√x + 2
Newton’s method 353