In the introduction to his work Von der Weltseele, however, he argues in favour of the possibility of a transmutation of species in periods incommensurable with ours.
If the number is incommensurable or consists of more than seven figures, we can take the first seven figures of it (or multiply and divide the result by any factor, and take the first seven figures of the result) and proceed as before.
This ratio, invariably denoted by 7r, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number.
If this be applied to the right-hand side of the identity m m m 2 m2 tan-=- - n n -3n-5n" it follows that the tangent of every arc commensurable with the radius is irrational, so that, as a particular case, an arc of 45 having its tangent rational, must be incommensurable with the radius; that is to say, 3r/4 is an incommensurable number."
His incommensurable and indescribable masterpiece of mingled humour, wisdom, satire, erudition, indecency, profundity, levity, imagination, realism, reflects the whole age in its mirror of hyperAristophanic farce.
Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity.
For the application of continued fractions to the problem " To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable," the reader is referred to G.
Converges to an incommensurable limit if after some finite value of n the condition an
- an verges to an incommensurable limit if after some finite value of n the condition a n ?b n +I is always satisfied, where the sign > need not always occur but must occur infinitely often.
These results were given by Lambert, and used by him to !prove that r and ir 2 are incommensurable, and also any commensurable power of e.