圆的十七等分

       将一个圆3等分、5等分早在欧几里德时代就已经解决了。但是能否7等分、9等分、11等分，却一直没有答案。
        到了十九世纪，德国数学大师高斯证明：对奇数n，只有当它为费马素数或是不同的费马素数之积时，才能够用尺规完成n等分圆周，并亲自用圆规和直尺做出了一个正17边形。
17边形的作法：
1）作圆，过圆心作两条垂直的直径，得圆上两点P0、B：
2）作OJ=1/4OB,再作∠OJE=1/4∠OJP0, ∠FJE=45。　　
3）以FP0为直径作圆，交OB于K，以E为圆心EK 为半径作圆，交OP0于N5和N3 
4）过N5、N3分别作OB 的平行线，交圆O于P5、P3再平分P5P3得分点P4
5）P3P4 就是正17边形的一边之长，用它可在圆O上依次截得正14边形的各顶点，如上图。
高斯在18岁时发现了用直尺圆规作正17边形的方法，从此开始了他辉煌的数学生涯，成为19世纪上半叶最伟大的数学家。 

Seventeen Parts of A Circle

         Ways were found to cut a circle into three and five equal sectors early in Euclid period. However ways had not been found to cut a circle into seven, nine and eleven equal sectors.
It was not until the 19th century that the great Germany mathematician Guass proved that for an odd number n, a ruler and a compass can be used to divide a circle into n equal sectors only when n is a Fermat prime number or the product of different Fermat prime numbers. He himself make a regular seventeen-sided polygon with a compass and a ruler.
        The ways to cut a circle into 17 parts:
(1) make a circle and draw two diameters vertically with each other. Get two points in the circle P and B;
(2) make OJ=1/4OB, and make ∠OJE=1/4∠OJPO, ∠FJE=45°
(3) Draw a circle with line FPO as diameter, it intersects the line OB at K, draw a circle with E as center, line EK as radius. It intersects line OPO at N5 and N3.
(4) Draw a parallel line across N5 and N3. It intersects Circle O at P5 and P3, and then equally depart line P5P3, get the point P4;
(5) Line P3P4 is just one line of the sixteen-polygon. Using this line’s length to cut the Circle O can get every point.
Guass found the ways to draw a sixteen-polygon by using of the ruler and compass in his 18 years old. From then on he began his splendent mathematic career and become the greatest mathematician in first-half period of 19th century.