## Geometry   ## Reconsideration of X^3 - dx - a = 0 Based on the Cubic Equation X^3 = 15x + 4 Solved by Rafael Bombelli

Authors: Atsushi Koike

According to Pierre Wantzel’s proof of 1837 that the trisection of 60 degree is impossible, because the cubic equation of x^3 - 3x - 1 = 0 had a absence of a rational solution. And his proof reached already a consensus as a general opinion. I learned from Nobukazu Shimeno’s introductory book on complex numbers that Rafael Bombelli got a rational solution x= 4 from the cubic equation x^3 = 15x + 4 based on the Cardinal formula. x^3 = 15x + 4 is x^3 - 15x - 4 = 0, which can be further replaced by x^3 - dx - a = 0. Kentaro Yano, who introduces the trisection of angles, says that the basic equation of x^3 - 3x - 1 = 0 is x^3 - dx - a = 0. In other words, the equation that Rafael Bombelli obtained a rational number solution is the same as the equation of the trisection of the angle. On the other hand, Yano raises x^3 - 3x = 0 as an example when the angle can be divided into three equal parts. Needless to say, x^3 - 3x - 0 = 0 and x^3 - dx - a = 0. Therefore, comparing Rafael Bombelli’s solution and the equation where Yano’s angle trisection is impossible and possible, the equation for angle trisection is x = a. And I found that if x = a = 2 then the solution is obtained at all angles. This paper proves that.

### Submission history

[v1] 2019-08-04 20:39:02
[v2] 2019-08-08 09:33:02
[v3] 2019-08-12 04:48:42