√

x+ 3

√

x+ 2

Newton’s method 353

x

5

young age of 27, Newton was appointed Lucasian pro-

fessor 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 sub-

ject to chromatic aberrations and set to work on build-

ing a reflecting model. In 1672 he was elected as a

fellow of the prestigious R

OYAL

S

OCIETY

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 mid-

1660s. In 1684 eminent astronomer Edmond Halley

(1656–1742) suggested to Newton that he should

investigate the mathematics of gravitational attraction

and attempt to derive J

OHANNES

K

EPLER

’

S

famous

three laws. Newton apparently replied simply that he

already had many years earlier. Only after Halley’s per-

sistent 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 lin-

earum 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 Univer-

sity of Cambridge in 1701 to take a prestigious govern-

ment 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 schol-

arly achievement.

It is impossible to understate the influence that

Newton’s work has had on the development of all sci-

entific research. Scholars today agree that the publica-

tion 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 gen-

eral methods suitable for solving √

–

x+ +

– 5 = 0, for instance.) If the function in question is differ-

entiable (see

DIFFERENTIAL CALCULUS

), then one can

employ Newton’s method to find the approximate loca-

tion of a

ROOT

.

One begins by making an initial guess x0, hope-

fully missing the correct value of the root by just a

small amount. Suppose it turns out that the root is h

units away from x0(thus f(x0+ h) = 0). Using this

value of h, we can approximate the derivative of the

function at x0as:

Turning this around, we see that his approximately

–f(x0)/f′(x0). Substituting this value for hshows that if

x0is our initial guess for the root, then the point:

is likely to be a much better approximation to the root.

(Geometrically, x1is the location at which the tangent

line to the graph y= f(x) at position x0crosses the x-

axis.) Repeating this procedure, each time using the out-

come just obtained as the next initial guess, produces a

sequence of values x0,x1,x2,…, 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 x2– 2 = 0. Setting f(x) = x2– 2,

we have f′(x) = 2x. Thus the sequence of approxima-

tions is given by the formula:

(Compare with H

ERON

’

S METHOD

.) With x0= 1 as our

initial guess, we obtain the approximations: