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.
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."
If this is the case, the apsidal angle must evidently be commensurable with -ir, and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant.
If b 2 /a 2, 3 /a 3 ..., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a commensurable number.
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.
Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number.
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.
Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.
And it occurs to no one that to admit a greatness not commensurable with the standard of right and wrong is merely to admit one's own nothingness and immeasurable meanness.
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