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Page:Carroll - Notes by an Oxford Chiel.djvu/19

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9
THE EVALUATION OF π.

I. Rationalisation.


The peculiarity of this process consists in its affecting all quantities alike with a negative sign.

To apply it, let H = High Church, and L= Low Church, then the geometric mean= √HL: call this 'B' (Broad Church).

∴ HL = B².

Also let x and y represent unknown quantities.

The process now requires the breaking up of U into its partial factions, and the introduction of certain combinations. Of the two principal factions thus formed, that corresponding with P presented no further difficulty, but it appeared hopeless to rationalise the other.

A 'reductio ad absurdum' was therefore attempted, and it was asked 'why should π not be evaluated?' The great difficulty now was, to discover y.

Several ingenious substitutions and transformations were then resorted to, with a view to simplifying the equation, and it was at one time asserted, though never actually proved,