-- -- -- -- -- --
1 - 1 + 2 - 3 + 2 - 5 + ...
For some recent developments in this direction the reader may consult a
paper by L. J. Rogers in the _Proceedings of the London Mathematical
Society_ (series 2, vol. 4).
_Ascending Continued Fractions._
There is another type of continued fraction called the ascending
continued fraction, the type so far discussed being called the
descending continued fraction. It is of no interest or importance,
though both Lambert and Lagrange devoted some attention to it. The
notation for this type of fraction is
b5 +
b4 + ----
a5
b3 + ---------
a4
b2 + --------------
a3
a1 + -------------------
a2
It is obviously equal to the series
b2 b3 b4 b5
a1 + -- + ---- + ------ + -------- + ...
a2 a2a3 a2a3a4 a2a3a4a5
_Historical Note._
The invention of continued fractions is ascribed generally to Pietro
Antonia Cataldi, an Italian mathematician who died in 1626. He used them
to represent square roots, but only for particular numerical examples,
and appears to have had no theory on the subject. A previous writer,
Rafaello Bombelli, had used them in his treatise on Algebra (about
1579), and it is quite possible that Cataldi may have got his ideas from
him. His chief advance on Bombelli was in his notation. They next appear
to have been used by Daniel Schwenter (1585-1636) in a _Geometrica
Practica_ published in 1618. He uses them for approximations. The
theory, however, starts with the publication in 1655 by Lord Brouncker
of the continued fraction
1 1 squared 3 squared 5 squared
-- -- -- -- as an equivalent of [pi]/4. This he is supposed
1 + 2 + 2 + 2 + ...
to have deduced, no one knows how, from Wallis' formula for
3 . 3 . 5 . 5 . 7 . 7 ...
4/[pi], viz. -------------------------
2 . 4 . 4 . 6 . 6 . 8 ...
John Wallis, discussing this fraction in his _Arithmetica Infinitorum_
(1656), gives many of the elementary properties of the convergents to
the general continued fraction, including the rule for their formation.
Huygens (_Descriptio automati planetarii_, 1703) uses the simple
continued fraction for the purpose of approximation when designing the
toothed wheels of his _Planetarium_. Nicol Saunderson (1682-1739
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