magic square 325

arctangent functions of

TRIGONOMETRY

, discovered the

G

REGORY SERIES

for π(which he used to compute an

approximation of this value correct to 11 decimal

places), and, moreover, provided accurate estimates for

the error term in truncating the series after a finite num-

ber of steps. Madhava also produced the most accurate

table of sine values of his time. The methods Madhava

used to accomplish these feats are believed to be essen-

tially the same as those developed in

CALCULUS

by G

OT

-

TFRIED

W

ILHELM

L

EIBNIZ

, S

IR

I

SAAC

N

EWTON

, and

B

ROOK

T

AYLOR

. Of course, Madhava had discovered

these techniques 300 years prior to their invention of

this subject.

Very little is known of Madhava’s life, and all of

his mathematical writings are lost. Historians have

learned of Madhava’s mathematical work through the

few astronomical texts of his that have survived, and

from the commentaries scholars following Madhava

made of his work.

It is worth mentioning that from his series expan-

sion for arctangent:

Madhava set x= 1 to obtain the familiar Gregory

series for π. It is not well known that Madhava also set

to obtain the following alternative formula

for π:

magic square A square array of numbers for which

the sum of the numbers in any row, column, or main

diagonal is the same is called a magic square. The con-

stant sum obtained is called the magic constant of the

square. Usually the numbers in a magic square are

required to be distinct, and often it is assumed that for

an n×nsquare, the specific numbers 1, 2, 3, …, n2are

used. (It is convenient to designate such a magic square

as a standard type.)

The earliest known example is the “Lho shu

square” that appears in an ancient Chinese manuscript

from the time of Emperor Yu of around 2200

B

.

C

.

E

.

Here the numbers 1 through 9 are arranged in a 3 ×3

array to produce a magic square of magic constant 15.

Up to rotations and reflections, this is the only arrange-

ment of these nine integers that produces a magic

square. (Thus we say that there is only one 3 ×3 magic

square of standard type.)

Ancient Chinese scholars, and later Arab scholars,

computed examples of standard 4 ×4, 5 ×5, and higher-

order magic squares. (The 5 ×5 magic square shown

below is attributed to Yang Hui of the 13th century, and

the 6 ×6 magic square to Chêng Ta-wei of the 16th cen-

tury.) German artist A

LBRECHT

D

ÜRER

(1471–1528)

depicted the 4 ×4 magic square below in the background

of his engraving Melancholia, and it is the believed that

this is the first introduction of a magic square to the

Western world. Famous scientist and statesman Benjamin

Franklin (1707–90) was masterful at inventing high-

order magic squares and is said to have toyed with new

squares whenever political debates became tedious.