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Viète’s construction

For approximation of the value of π Viète used the antique idea already used by Archimedes. He compared the area of an inscribed regular typeset structure-gon to the area of the inscribed regular typeset structure-gon.  

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The area of the typeset structure-gon is

A(n) = n - times the area ΔAOB = 1/2 n r^2 sin (2 β) = n r^2 sin β cos β ,

and similarly

A(2 n) = n r^2 sin β .

Comparing we get

A(n)/A(2 n) = cos β ,

and then

A(n)/A(4 n) = A(n)/A(2 n) . A(2 n)/A(4 n) = cos β cos(β/2) .

Consequently

A(n)/A(2^k n) = A(n)/A(2 n) . A(2 n)/A(4 n) . ··· . A(2^(k - 1) n)/A(2^r n) = cos β cos(β/2) ··· cos(β/2^k) .

For typeset structure we get1

lim _ (k -> ∞) A(2^k n) = π r^2,

i.e.

π = (1/2 n sin(2 β))/(cos β cos(β/2) cos (β/2^2) cos(β/2^3) ···) .

Viète now started with typeset structure, in which case typeset structure and typeset structure. Since typeset structure, we get

π = 2/(1/2^(1/2) (1/2 + 1/2 1/2^(1/2))^(1/2) (1/2 + 1/2 (1/2 + 1/2 1/2^(1/2))^(1/2))^(1/2) ···) .(1)

To approximate typeset structure using  (1) go to .

Note that Euler proved  using another approach a generalization of  Viète’s formula in the form

π = (sin (2 π)/n)/(cos π/n    cos π/(2 n) cos π/(2^2 n) ···),

which for typeset structure yields Viète’s one.

Notes

1 Viète did not worried about the convergence, because this notion was completely unknown in his time. Although the formula is not very suitable for computation,  for approximation purposes we can take typeset structure sufficiently large. That the Viète’s formula is really convergent was proved by F.Rudio (Zeitschrift für Mathematik und Physik 36 (1891), 139-140).

References

[1]  Viète, F. (1593). Uriorum de rebus mathematicis responsorum. Liber VII.  Reprinted in New York: Georg Olms, pp. 398-400 and 436-446, 1970. .

Cite this web-page as:

Štefan Porubský: Viète construction.

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