476 squaring the circle

that the numbers from √

–

2 through to √

–

17 (skipping √

–

4

= 2, √

–

9 = 3, and √

–

16 = 4) are each irrational. More

generally, one can use the

FUNDAMENTAL THEOREM OF

ARITHMETIC

to show that the square root of a positive

integer ais a

RATIONAL NUMBER

if, and only if, ais a

perfect square (that is, a= b2for some whole number b.)

H

ERON

’

S METHOD

provides a method for computing

the square root of any positive real to any prescribed

degree of accuracy.

The square-root inequality states that for any posi-

tive whole number nwe have:

An exercise in algebra establishes the validity of this

statement.

Although the statements:

√

–

ab = √

–

a√

–

b

and

are valid for all numbers aand b(with b≠0 in the sec-

ond equation), the equation √

–

a+b= √

–

a+ √

–

b, in gen-

eral, is not correct. (Substitute in the values a= 9 and

b= 16, for instance.)

See also

CUBE ROOT

/

NTH ROOT

;

IDENTITY MATRIX

;

SOLUTION BY RADICALS

;

SURD

.

squaring the circle (circle squaring, the quadrature of

the circle) One of the problems of antiquity (like

DUPLICATING THE CUBE

and

TRISECTING AN ANGLE

) of

considerable interest to the classical Greek scholars is

the task of constructing a square of the same area as a

given circle in the plane. The only tools permitted in

the construction are a compass and a straightedge (that

is, a ruler with no markings).

The problem has its origins back in the beginnings

of mathematics. Ancient Egyptian scholars of 1650

B

.

C

.

E

. describe in the R

HIND PAPYRUS

a construction of a

square of area nearly equal to that of a circle. (The con-

struction is “exact” if one works with the approximate

value 256/81 ≈3.1605 for π.) Scholar H

IPPOCRATES OF

C

HIOS

(ca. 440

B

.

C

.

E

.) was interested in the problem

and developed several methods for “squaring” a

LUNE

,

a shape made from arcs of circles, but did not succeed

in squaring the circle itself. A

RCHIMEDES OF

S

YRACUSE

(ca. 287–212

B

.

C

.

E

.) used his curve the

SPIRAL OF

A

RCHIMEDES

to square the circle, providing a solution

to the problem using tools more advanced than straight-

edge and compass alone. A

POLLONIUS OF

P

ERGA

(ca.

262–190

B

.

C

.

E

.) also solved the problem with the intro-

duction of special curves and advanced techniques. The

problem of constructing a square of the desired area

using just a straightedge and compass alone, however,

remained unsolved.

The problem garnered considerable notoriety over

the centuries that followed. Indian, Chinese, and Arab

scholars also attempted to solve the challenge, and suc-

ceeded in finding solutions to the problem if one used

various approximate values for π. The problem was stud-

ied by European scholars of the Renaissance, and it

became a popular problem of study for amateur mathe-

maticians during the 18th and 19th centuries. Novices

working on the challenge became known as “circle

squarers” and often sought fame by submitting supposed

solutions to the problem to the prestigious R

OYAL

S

OCI

-

ETY

of London. In the late 1700s, tired of being inun-

dated with large numbers of tedious and incorrect theses

on the subject, the society banned all consideration of

alleged “proofs” of squaring the circle. The French

Academy of Science followed suit soon afterward.

If we assume that the radius of the given circle is r,

then one is asked to produce a square of side-length a

so that a2= πr2. This essentially reduces the problem

to one of constructing a length √

–

πunits long using a

straightedge and a compass.

The theory of

CONSTRUCTIBLE

numbers shows that

any quantity of rational length can be constructed, and

that if two lengths l1and l2can be produced, then so

too can their sum, difference, product, and quotient,

along with the square root of each quantity. If one uses

a rational value as an approximation for π, it is not sur-

prising then that one can produce solutions to the prob-

lem using this approximate value. Moreover, if πitself is

a rational number, then one would expect it possible to

solve the problem. As no solution had ever been found,

scholars began to suspect that πis irrational.

In the mid-1600s Scottish mathematician J

AMES

G

REGORY

(1638–75), in his studies of infinite series,