之前我写了《 科普：用中学数学推导出行星椭圆轨道方程 (4） 》用中学数学推导出了行星的椭圆轨道方程。在 更早的一篇中 ，我写的【我查看了牛顿《自然哲学的数学原理》的行星轨道部分之后，发现牛顿似乎并没有导出轨道的椭圆公式，而是考虑轨道与圆接近的时候，轨道与圆轨道之间的差别。】 牛顿的《 Principia》今天很难读，因为其使用的数学工具过于简陋，没有微积分，代数都很少，基本是用欧几里得几何，数学推导非常繁琐。 今天我翻看了一本专门详细介绍牛顿的《自然哲学之数学原理》的书，名叫《Magnificent Principia》，发现我上面的说法需要纠正一下。 根据这部书，【Newton set himself the following direct problem: Given that a body moves in a conic section with the force center at the focus, what is the nature of the force involved? The inverse problem is then: given an inverse-square-law force, what is the nature of the resulting orbits? Newton gave a formula for solving the direct problem (see section 9.3), and, in this chapter, we have seen the application to conic-section orbits.】【Newton has not shown that when the force is inverse-square law, there are no other orbits than those described as conic sections. Because this is such a pinnacle in the Principia, there should be a proof. Not to give a proof suggests an error of logic: it is just not true that the solution of a problem implies the solution of its converse.】【Even people like Halley, Hooke, and Huygens did not spot this deficiency in Newton's work, but eventually there were rumblings and Johann Bernoulli recognized the problem.】 就是说，牛顿只是算了如果轨道是椭圆（这是开普勒根据观测得到的结果），那么需要的力是平方反比力，但并没有解决反过来的问题，如果力是平方反比力，轨道应该是什么。哈雷、胡克、惠更斯等人都没发现牛顿的这个缺陷，但伯努利发现了。 牛顿后来的辩解是，如果给出初速与位置，那么轨道就唯一确定了，如果某个椭圆轨道符合这个初速与位置，那轨道就是这个轨道。
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