The Sand-Reckoner (opening and conclusion)

The opening开篇#

There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognising that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe. Now you are aware that ‘universe’ is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account, as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premisses lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface. Now it is easy to see that this is impossible; for, since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere. We must however take Aristarchus to mean this: since we conceive the earth to be, as it were, the centre of the universe, the ratio which the earth bears to what we describe as the ‘universe’ is the same as the ratio which the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixed stars. For he adapts the proofs of his results to a hypothesis of this kind, and in particular he appears to suppose the magnitude of the sphere in which he represents the earth as moving to be equal to what we call the ‘universe.’

革隆王啊，有人认为沙的数目多到无穷；我所说的沙，不只是叙拉古和西西里其余地方的沙，还包括每一处地方的沙，无论有人居住还是无人居住。又有人虽不认为它无穷，却认为还没有哪个已被命名的数大到足以超过它的数目。显然，持这种看法的人，如果设想有一团沙，在其他方面都与大地一样大，把所有的海洋和大地上的凹陷都填满，一直填到与最高的山一样高，那他们就更加远远不会承认：能说出一个数，超过这样一团沙的数目。但我要试着用你能看懂的几何证明向你表明：在我所命名、并写在我寄给宙克西普斯的那部著作里的数中，有些不仅超过一团大小与如上所述填满的大地相等的沙的数目，而且超过一团大小与整个宇宙相等的沙的数目。你知道，大多数天文学家所说的“宇宙”，是指这样一个球：它的球心是地心，半径等于日心与地心之间的直线。这是通常的说法，你从天文学家那里听说过。但萨摩斯的阿里斯塔克斯写了一本书，提出了一些假设，从这些前提推出的结果是：宇宙比现在所说的那个要大许多倍。他的假设是：恒星和太阳不动，地球沿一个圆周绕太阳运转，太阳位于轨道的中央；而恒星天球与太阳同心，它大到这样的程度——他设想地球运行其上的那个圆，与恒星距离之比，等于球心与球面之比。不难看出，这是不可能的；因为球心没有大小，我们无法设想它与球面有任何比。不过我们应当把阿里斯塔克斯的意思理解为：既然我们把地球设想为好比宇宙的中心，那么地球与我们所说的“宇宙”之比，就等于包含他所设想的地球运行之圆的那个球与恒星天球之比。因为他的证明正是按这样一种假设来安排的；特别是，他似乎把他所设想的地球在其中运动的那个球的大小，设为等于我们所说的“宇宙”。

I say then that, even if a sphere were made up of the sand, as great as Aristarchus supposes the sphere of the fixed stars to be, I shall still prove that, of the numbers named in the Principles, some exceed in multitude the number of the sand which is equal in magnitude to the sphere referred to, provided that the following assumptions be made.

那么我说，即使有一个由沙组成的球，大到阿里斯塔克斯所设想的恒星天球那样，我仍将证明：在《原理》中命名的数里，有些超过与上述那个球同样大小的沙的数目，只要作出以下几条假定。

1. The perimeter of the earth is about 3,000,000 stadia and not greater.

1. 地球的周长约为三百万斯塔迪昂，不会更大。

It is true that some have tried, as you are of course aware, to prove that the said perimeter is about 300,000 stadia. But I go further and, putting the magnitude of the earth at ten times the size that my predecessors thought it, I suppose its perimeter to be about 3,000,000 stadia and not greater.

你当然知道，确实有人试图证明这个周长约为三十万斯塔迪昂。但我走得更远：我把地球的大小定为前人所认为的十倍，假定它的周长约为三百万斯塔迪昂，不会更大。

2. The diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth.

2. 地球的直径大于月球的直径，太阳的直径大于地球的直径。

In this assumption I follow most of the earlier astronomers.

在这条假定上，我追随了大多数早先的天文学家。

3. The diameter of the sun is about 30 times the diameter of the moon and not greater.

3. 太阳的直径约为月球直径的三十倍，不会更大。

It is true that, of the earlier astronomers, Eudoxus declared it to be about nine times as great, and Pheidias my father twelve times, while Aristarchus tried to prove that the diameter of the sun is greater than 18 times but less than 20 times the diameter of the moon. But I go even further than Aristarchus, in order that the truth of my proposition may be established beyond dispute, and I suppose the diameter of the sun to be about 30 times that of the moon and not greater.

诚然，在早先的天文学家中，欧多克索斯宣称它约为九倍，我的父亲菲迪亚斯说是十二倍，而阿里斯塔克斯则试图证明，太阳的直径大于月球直径的十八倍，小于二十倍。但我比阿里斯塔克斯走得更远，好让我的命题的真实性确立得无可争辩：我假定太阳的直径约为月球的三十倍，不会更大。

4. The diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle in the (sphere of the) universe.

4. 太阳的直径大于内接于宇宙（之球）大圆中的正千边形的一边。

I make this assumption because Aristarchus discovered that the sun appeared to be about 1/720th part of the circle of the zodiac, and I myself tried, by a method which I will now describe, to find experimentally the angle subtended by the sun and having its vertex at the eye.

我作这条假定，是因为阿里斯塔克斯发现，太阳看上去约为黄道圆的七百二十分之一；而我自己也试图用一种我现在就要描述的方法，以实验求出太阳所张的、顶点在眼睛处的那个角。

The conclusion结尾#

I conceive that these things, king Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the earth the sun and moon and the whole universe the proof will carry conviction. And it was for this reason that I thought the subject would be not inappropriate for your consideration.

革隆王啊，我想，这些事在大多数没有学过数学的人看来，会显得难以置信；但对于那些精通数学、思考过大地、太阳、月亮以及整个宇宙的距离与大小问题的人来说，这个证明将使他们信服。正因为如此，我认为这个题目呈给你考虑并非不合适。