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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.

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