<s>
In	O
mathematics	O
,	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
is	O
the	O
following	O
infinite	O
product	O
of	O
nested	O
radicals	O
representing	O
twice	O
the	O
reciprocal	O
of	O
the	O
mathematical	O
constant	O
:	O
</s>
<s>
As	O
the	O
first	O
formula	O
of	O
European	O
mathematics	O
to	O
represent	O
an	O
infinite	O
process	O
,	O
it	O
can	O
be	O
given	O
a	O
rigorous	O
meaning	O
as	O
a	O
limit	B-Algorithm
expression	O
,	O
and	O
marks	O
the	O
beginning	O
of	O
mathematical	O
analysis	O
.	O
</s>
<s>
It	O
has	O
linear	B-Architecture
convergence	I-Architecture
,	O
and	O
can	O
be	O
used	O
for	O
calculations	O
of	O
,	O
but	O
other	O
methods	O
before	O
and	O
since	O
have	O
led	O
to	O
greater	O
accuracy	O
.	O
</s>
<s>
The	O
formula	O
can	O
be	O
derived	O
as	O
a	O
telescoping	O
product	O
of	O
either	O
the	O
areas	O
or	O
perimeters	B-Application
of	O
nested	O
polygons	B-General_Concept
converging	O
to	O
a	O
circle	O
.	O
</s>
<s>
Alternatively	O
,	O
repeated	O
use	O
of	O
the	O
half-angle	O
formula	O
from	O
trigonometry	O
leads	O
to	O
a	O
generalized	O
formula	O
,	O
discovered	O
by	O
Leonhard	O
Euler	O
,	O
that	O
has	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
as	O
a	O
special	O
case	O
.	O
</s>
<s>
As	O
the	O
first	O
representation	O
in	O
European	O
mathematics	O
of	O
a	O
number	O
as	O
the	O
result	O
of	O
an	O
infinite	O
process	O
rather	O
than	O
of	O
a	O
finite	O
calculation	O
,	O
Eli	O
Maor	O
highlights	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
as	O
marking	O
the	O
beginning	O
of	O
mathematical	O
analysis	O
and	O
Jonathan	O
Borwein	O
calls	O
its	O
appearance	O
"	O
the	O
dawn	O
of	O
modern	O
mathematics	O
"	O
.	O
</s>
<s>
Beyond	O
its	O
mathematical	O
and	O
historical	O
significance	O
,	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
can	O
be	O
used	O
to	O
explain	O
the	O
different	O
speeds	O
of	O
waves	O
of	O
different	O
frequencies	O
in	O
an	O
infinite	O
chain	O
of	O
springs	O
and	O
masses	O
,	O
and	O
the	O
appearance	O
of	O
in	O
the	O
limiting	O
behavior	O
of	O
these	O
speeds	O
.	O
</s>
<s>
Additionally	O
,	O
a	O
derivation	O
of	O
this	O
formula	O
as	O
a	O
product	O
of	O
integrals	O
involving	O
the	O
Rademacher	B-Algorithm
system	I-Algorithm
,	O
equal	O
to	O
the	O
integral	O
of	O
products	O
of	O
the	O
same	O
functions	O
,	O
provides	O
a	O
motivating	O
example	O
for	O
the	O
concept	O
of	O
statistical	O
independence	O
.	O
</s>
<s>
For	O
each	O
choice	O
of	O
,	O
the	O
expression	O
in	O
the	O
limit	B-Algorithm
is	O
a	O
finite	O
product	O
,	O
and	O
as	O
gets	O
arbitrarily	O
large	O
these	O
finite	O
products	O
have	O
values	O
that	O
approach	O
the	O
value	O
of	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
arbitrarily	O
closely	O
.	O
</s>
<s>
Viète	O
did	O
his	O
work	O
long	O
before	O
the	O
concepts	O
of	O
limits	O
and	O
rigorous	O
proofs	O
of	O
convergence	B-Algorithm
were	O
developed	O
in	O
mathematics	O
;	O
the	O
first	O
proof	O
that	O
this	O
limit	B-Algorithm
exists	O
was	O
not	O
given	O
until	O
the	O
work	O
of	O
Ferdinand	O
Rudio	O
in	O
1891	O
.	O
</s>
<s>
The	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
limit	B-Algorithm
governs	O
the	O
number	O
of	O
terms	O
of	O
the	O
expression	O
needed	O
to	O
achieve	O
a	O
given	O
number	O
of	O
digits	O
of	O
accuracy	O
.	O
</s>
<s>
In	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
,	O
the	O
numbers	O
of	O
terms	O
and	O
digits	O
are	O
proportional	O
to	O
each	O
other	O
:	O
the	O
product	O
of	O
the	O
first	O
terms	O
in	O
the	O
limit	B-Algorithm
gives	O
an	O
expression	O
for	O
that	O
is	O
accurate	O
to	O
approximately	O
digits	O
.	O
</s>
<s>
This	O
convergence	B-Architecture
rate	I-Architecture
compares	O
very	O
favorably	O
with	O
the	O
Wallis	B-Algorithm
product	I-Algorithm
,	O
a	O
later	O
infinite	O
product	O
formula	O
for	O
.	O
</s>
<s>
Although	O
Viète	O
himself	O
used	O
his	O
formula	O
to	O
calculate	O
only	O
with	O
nine-digit	O
accuracy	O
,	O
an	O
accelerated	B-Algorithm
version	O
of	O
his	O
formula	O
has	O
been	O
used	O
to	O
calculate	O
to	O
hundreds	O
of	O
thousands	O
of	O
digits	O
.	O
</s>
<s>
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
may	O
be	O
obtained	O
as	O
a	O
special	O
case	O
of	O
a	O
formula	O
for	O
the	O
sinc	B-Algorithm
function	I-Algorithm
that	O
has	O
often	O
been	O
attributed	O
to	O
Leonhard	O
Euler	O
,	O
more	O
than	O
a	O
century	O
later	O
:	O
</s>
<s>
gives	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
derive	O
from	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
a	O
related	O
formula	O
for	O
that	O
still	O
involves	O
nested	O
square	O
roots	O
of	O
two	O
,	O
but	O
uses	O
only	O
one	O
multiplication	O
:	O
</s>
<s>
Viète	O
obtained	O
his	O
formula	O
by	O
comparing	O
the	O
areas	O
of	O
regular	O
polygons	B-General_Concept
with	O
and	O
sides	O
inscribed	O
in	O
a	O
circle	O
.	O
</s>
<s>
The	O
first	O
term	O
in	O
the	O
product	O
,	O
,	O
is	O
the	O
ratio	O
of	O
areas	O
of	O
a	O
square	O
and	O
an	O
octagon	B-Application
,	O
the	O
second	O
term	O
is	O
the	O
ratio	O
of	O
areas	O
of	O
an	O
octagon	B-Application
and	O
a	O
hexadecagon	O
,	O
etc	O
.	O
</s>
<s>
Thus	O
,	O
the	O
product	O
telescopes	O
to	O
give	O
the	O
ratio	O
of	O
areas	O
of	O
a	O
square	O
(	O
the	O
initial	O
polygon	B-General_Concept
in	O
the	O
sequence	O
)	O
to	O
a	O
circle	O
(	O
the	O
limiting	O
case	O
of	O
a	O
-gon	O
)	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
terms	O
in	O
the	O
product	O
may	O
be	O
instead	O
interpreted	O
as	O
ratios	O
of	O
perimeters	B-Application
of	O
the	O
same	O
sequence	O
of	O
polygons	B-General_Concept
,	O
starting	O
with	O
the	O
ratio	O
of	O
perimeters	B-Application
of	O
a	O
digon	O
(	O
the	O
diameter	O
of	O
the	O
circle	O
,	O
counted	O
twice	O
)	O
and	O
a	O
square	O
,	O
the	O
ratio	O
of	O
perimeters	B-Application
of	O
a	O
square	O
and	O
an	O
octagon	B-Application
,	O
etc	O
.	O
</s>
<s>
leads	O
to	O
a	O
proof	B-Algorithm
by	I-Algorithm
mathematical	I-Algorithm
induction	I-Algorithm
that	O
,	O
for	O
all	O
positive	O
integers	O
,	O
</s>
<s>
The	O
term	O
goes	O
to	O
in	O
the	O
limit	B-Algorithm
as	O
goes	O
to	O
infinity	O
,	O
from	O
which	O
Euler	O
's	O
formula	O
follows	O
.	O
</s>
<s>
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
may	O
be	O
obtained	O
from	O
this	O
formula	O
by	O
the	O
substitution	O
.	O
</s>
