Irrationality, Transcendence and the Circle-Squaring Problem by Eduardo Dorrego López & Elías Fuentes Guillén

Author:Eduardo Dorrego López & Elías Fuentes Guillén
Language: eng
Format: epub
ISBN: 9783031243639
Publisher: Springer International Publishing

which also converges more markedly than any geometric series and has an irrational sum.

§. 15.

Since, therefore, the tangent of every rational arc is irrational, then, conversely, the arc of every rational tangent is irrational. For, if one were to assume the arc to be rational, then, contrary to the assumption, the tangent would be irrational by virtue of what was initially proven.

§. 16.

In the trigonometric tables we have a single rational tangent, namely that of 45 deg., which is equal to the radius and, therefore, . For this reason, the arc of 45 deg. is irrational and likewise irrational, consequently, are the arcs of 90, 180 and 360 deg., or in other words, these arcs have no rational ratio to the radius of the circle.

§. 17.

From what has been said so far it is clear that no arc can at the same time have a rational ratio to the radius and to its tangent. There are, however, innumerable ways in which an arc can have a rational ratio to its tangent. But it can also be shown that, in all such cases, both the arc and its tangent are incommensurable with the radius. Because in the first place, by virtue of what has already been proven, it is not possible for both to have at the same time a rational ratio with respect to the radius. Let it therefore be assumed that only the tangent or the arc is rational. In the first case, the tangent would have to be commensurable with both the radius and the arc. And, thus, the arc would also be commensurable with the radius, since the sum or the difference of two rational ratios is also rational. In the other case, the arc would be commensurable with the tangent as well as with the radius, and thus the tangent would also have a rational ratio to the radius. Now, since, by virtue of what has been proven above, the radius, the arc and the tangent are not all at the same time commensurable, then both of the above cases are invalidated. Accordingly, if the arc and the tangent have a rational ratio to each other, then both are incommensurable with the radius.

§. 18.

I will end by briefly addressing two cases which present some plausibility with regard to the quadrature of the circle. The first is the following proposition: if one describes an arbitrary regular or irregular polygon around a circle, so that each side of the former touches the circle, then the perimeter of the polygon will thereby stand in the same relation to its content as does the circumference of the circle to its own content. I omit the proof, because it is very easy. The other case is a phaenomenon which occurs in the following way: if one divides 1 by 0, 7853981634..., as a fourth part of the Ludolphian numbers, it occurs 1 time and subtracts 0, 2146018366.... If one further divides 0, 7853981634..., which was previously the divisor, by this remainder, then it occurs 3 times and subtracts 0, 1415926536.

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