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引用次数: 0
摘要
本文全面介绍了Brianchon和Poncelet关于在四个给定条件下的等边双曲线的联合回忆录,重点阐述了其中最重要的定理,以及“九点圆”的确定。我们还讨论了这个当时非常罕见的合作工作例子的起源,以及寻找通过m个点并与给定位置的n条直线相切的圆锥截面中心所描述的轨迹的性质的一般挑战 + n = 4,这是在他们的工作结束时提出的。在m的情况下 = 4,即当圆锥截面必须穿过四边形的顶点时,中心轨迹是分别穿过四边形相对边和两条对角线的交点的另一个圆锥截面,并且,正如Gergonne不久后分析所示,穿过与四边形连接的其他重要点;这条曲线后来被命名为“九点二次曲线”,是上述圆的自然推广。
Brianchon and Poncelet’s joint memoir, the nine-point circle, and beyond
In this paper, we give a thorough account of Brianchon and Poncelet’s joint memoir on equilateral hyperbolas subject to four given conditions, focusing on the most significant theorems expounded therein, and the determination of the “nine-point circle”. We also discuss about the origin of this very rare example of collaborative work for the time, and the general challenge of finding the nature of the loci described by the centres of the conic sections required to pass through m points and to be tangent to n straight lines given in position, m + n = 4, which was posed at the end of their work. In the case m = 4, i.e. when the conic sections have to pass through the vertices of a quadrilateral, the locus of centres is another conic section passing through the intersection points of the opposite sides and the two diagonals of the quadrilateral, respectively, and, as Gergonne showed analytically shortly after, through other significant points connected with the quadrilateral; this curve was later given the name of the “nine-point conic”, being a natural generalization of the above mentioned circle.
期刊介绍:
The Archive for History of Exact Sciences casts light upon the conceptual groundwork of the sciences by analyzing the historical course of rigorous quantitative thought and the precise theory of nature in the fields of mathematics, physics, technical chemistry, computer science, astronomy, and the biological sciences, embracing as well their connections to experiment. This journal nourishes historical research meeting the standards of the mathematical sciences. Its aim is to give rapid and full publication to writings of exceptional depth, scope, and permanence.