Viète’s formula 525

W

ILHELM

L

EIBNIZ

(1646–1716) and the great L

EON

-

HARD

E

ULER

(1707–83) each used diagrams similar

to these to analyze arguments. In the context of

FOR

-

MAL LOGIC

Venn diagrams are often called E

ULER

DIAGRAMS

.

A single set Adivides the universal set into two

disjoint pieces, Aand its complement A′. Two subsets

Aand Bgenerally divide the universal set into four

disjoint pieces: A∩B, A′∩B, A∩B′, and A′∩B′.

(This is not the case, however, if Ais a subset of B, B

is a subset of A, or if Aand Bdo not intersect.) Three

subsets A, B, and C, in their most general arrange-

ment, divide the universal set into eight disjoint

pieces. (The diagram produced can be used to illus-

trate D

E

M

ORGAN

’

SLAWS

.)

It is not possible to draw a diagram of four over-

lapping circles to represent four general subsets A, B,

C, and Din such a way as to make the 16 disjoint

pieces of the universal set apparent. For this reason,

Venn diagrams are usually only used to illustrate rela-

tions between just two or three sets.

See also

SET THEORY

; J

OHN

V

ENN

.

Viète, François (Franciscus Vieta) (1540–1603) French

Algebra Born in Fontenay-le-Comte, France, in 1540

(his exact birth date is not known), scholar François

Viète is often referred to as “the father of algebra.” His

influential 1591 work In artem analyticam isagoge

(Introduction to the analytic arts) is noted as one of the

earliest Western texts on the topic. His practice of using

letters as symbols for unknowns represented a first step

toward the development of modern algebraic notation

and allowed him to make significant advances in the

subject. Viète also made important contributions to the

field of

TRIGONOMETRY

.

Viète was never employed as a professional mathe-

matician. After graduating from the University of

Poitiers in 1560, he began a career in legal practice, but

soon decided to change occupations by accepting a

position as a private tutor in 1564. Within this role he

pursued an interest in mathematics and soon developed

a reputation as a capable scholar. Viète occasionally

gave mathematical lectures at institutes of higher learn-

ing, and in 1589 was employed by King Henry IV of

France to decode secret messages being sent to the

enemy of the state, Philip II of Spain.

His first published work, Canon mathematicus

seu ad triangula (The mathematical canon applied to

triangles), appeared in 1579 and was intended as an

introduction to mathematical astronomy. In it Viète

used the six main trigonometric functions to solve

problems relating to plane and spherical triangles,

listed tables of values for these functions, and

explained the mathematics behind the construction of

those tables. Later, in 1593, Viète wrote two other

texts on the topic of trigonometry, Zeteticorum libri

quinque (Algebra volume five), mimicking the fifth

book in D

IOPHANTUS

’s Arithmetica, and Variorum de

rebus mathematicis responsorum (Various mathemati-

cal problems), which contains his famous formula for

πas an infinite product of radicals. He also lectured

on the classical problems of

DUPLICATING THE CUBE

and

TRISECTING AN ANGLE

.

Today Viète is best remembered for his advances in

algebra as presented in his famous 1591 text Isagoge,

which clearly demonstrated the value of manipulating

letters as symbols for both known and unknown quan-

tities as a means to solve algebraic problems. Viète also

introduced improved notation for squares, cubes, and

other powers, as well as coined the term coefficient.

With his new approach to algebra, Viète successfully

tackled a number of problems that classical scholars

had been unable to solve. Viète died in Paris, France,

on February 23, 1603.

See also

COEFFICIENT

; V

IÈTE

’

S FORMULA

.

Viète’s formula (Vieta’s formula) In 1593 French

mathematician F

RANÇOIS

V

IÈTE

discovered the follow-

ing remarkable formula relating πto an infinite prod-

uct of radicals:

His formula is established by making repeated use of

the double angle formulae from

TRIGONOMETRY

:

sin(2θ) = 2sin(θ) cos(θ)

cos(2θ) = 2 cos2(θ) – 1

Begin by writing: