The Method: the Letter to Eratosthenes

Now these theorems differ in character from those communicated before; for we compared the figures then in question, conoids and spheroids and segments of them, in respect of size, with figures of cones and cylinders: but none of those figures have yet been found to be equal to a solid figure bounded by planes; whereas each of the present figures bounded by two planes and surfaces of cylinders is found to be equal to one of the solid figures which are bounded by planes. The proofs then of these theorems I have written in this book and now send to you. Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer of mathematical inquiry, I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely that the cone is a third part of the cylinder, and the pyramid of the prism, having the same base and equal height, we should give no small share of the credit to Democritus who was the first to make the assertion with regard to the said figure though he did not prove it. I am myself in the position of having first made the discovery of the theorem now to be published by the method indicated, and I deem it necessary to expound the method partly because I have already spoken of it and I do not want to be thought to have uttered vain words, but equally because I am persuaded that it will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.

这些定理与以前所寄的那些性质不同；因为那时我们是把所讨论的图形——劈锥曲面体、旋转椭球体以及它们的截段——在大小上同圆锥和圆柱这类图形相比较；但那些图形，还没有一个被发现等于某个由平面围成的立体；而现在这些由两个平面和圆柱曲面围成的图形，每一个都被发现等于某个由平面围成的立体。这些定理的证明，我已写在这本书里，现在寄给你。此外，正如我所说，我看你是一位热心的学者，在哲学上卓有声望，又爱好数学的探究，所以我认为应当在同一本书里为你写出并详细说明一种方法的特点；借助这种方法，你将能够起步，用力学来研究数学中的某些问题。我相信，这种做法对于定理本身的证明也同样有用；因为有些事情是先通过力学方法向我显明的，虽然后来还得用几何来证明，因为用这种方法所作的研究并不构成真正的证明。但是当然，如果我们先已用这种方法对问题有了一些认识，再去补出证明，要比毫无认识地去寻找证明容易得多。正因为如此，对于那些由欧多克索斯首先找到证明的定理——即圆锥是同底等高圆柱的三分之一，棱锥是同底等高棱柱的三分之一——我们应当把不小的一份功劳归于德谟克利特，因为是他首先对这一图形作出了断言，虽然他没有证明。至于我自己，现在要发表的这条定理，我也是先用上述方法发现的；我认为有必要把这种方法讲解出来，一方面是因为我以前已经提到过它，不想被人认为说了空话；另一方面同样是因为我相信，它对数学将大有裨益：因为我预料，我的同时代人或者我的后来者中，会有人借助这种方法，一旦它确立起来，再发现另一些我还没有想到的定理。

Any segment of a section of a right-angled cone (i.e. a parabola) is four-thirds of the triangle which has the same base and equal height,

直角圆锥截线（即抛物线）的任一弓形，等于同底等高三角形的三分之四；