Page:A budget of paradoxes (IA cu31924103990507).pdf/509

APPENDIX.

I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert, in the Berlin Memoirs for 1761, and has been also given in the notes to Legendre's Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.

Let a continued fraction, such as

${\displaystyle }$

be abbreviated into ${\frac {a}{b}}\!{{} \atop {+}}{\frac {c}{d}}\!{{} \atop {+}}{\frac {e}{f}}\!{{} \atop {+{\text{ }}\&{\text{c.:}}}}$ each fraction being understood as falling down to the side of the preceding sign $+$. In every such fraction we may suppose $b$, $d$, $f$, &c. positive; $a$, $c$, $e$, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if ${\frac {a}{b}}$, ${\frac {c}{d}}$, ${\frac {e}{f}}$ &c. all lie between $-1$ and $+1$, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be ${\frac {A}{B}}$, where $A$ and $B$ are integers.

First, whatever this limit may be, it lies between $-1$ and $+1$. This is obviously the case with any fraction ${\frac {p}{q+w}}$, where $w$ is between $\pm 1$: for, ${\frac {p}{q}}$, being $<1$, and $p$ and $q$ integer, cannot be brought up to $\pm$, by the value of $w$.

Hence, if we take any of the fractions

${a \over b}$,${a \over b}{{} \atop +}{c \over d}$,${a \over b}{{} \atop +}{c \over d}{{} \atop +}{e \over f}$, &c.

say ${a \over b}{{} \atop +}{c \over d}{{} \atop +}{e \over f}{{} \atop +}{g \over h}$ we have, ${\frac {g}{h}}$ being between $\pm 1$, so is ${e \over f}{{} \atop +}{g \over h}$, so therefore is ${c \over d}{{} \atop +}{e \over f}{{} \atop +}{g \over h}$; and so therefore is ${a \over b}{{} \atop +}{c \over d}{{} \atop +}{e \over f}{{} \atop +}{g \over h}$.