卷一·第 63–73 问　球、球面诸圆与“一”

§63. What is a sphere?

第六十三问　何谓球？

It is a round body enclosed by a single surface that is not plane. Inside it is a point, its centre, such that all the straight lines drawn from that point to the surface enclosing it are equal. The generation of a sphere can be pictured as the revolution of a circle about its diameter, which stays fixed, until the circle returns to its first position.

球是由单独一个不是平面的面所包围的圆形立体。它里面有一点，就是球心，从这一点向包围它的面所引的一切直线都相等。球的生成可以设想为一个圆绕着自己固定不动的直径旋转，直到这个圆回到最初的位置。

§64. How many figures can a sphere enclose?

第六十四问　球能包住几种形体？

Of figures with equal sides and angles, all of one kind, five only, which by way of likeness are assigned to the four elements and to the heavens. Of figures composed of different kinds, they are almost without limit and beyond counting. Of the five primary figures, the first is the cube, with six square faces, which is called earthy. The second has twenty faces, equilateral triangles, and is the watery one. The third has eight faces, likewise triangles, and is the airy one. The fourth, the caltrop-shaped, has four triangular faces, and is the fiery one. The fifth has twelve pentagonal faces.

各边各角都相等、又全属同一类的形体，只有五种，人们按相似之理把它们分配给四元素与天。不同种类拼合而成的形体，那就几乎没有限度，数也数不清。五种基本形体中，第一是立方体，有六个正方形的面，称为属土的；第二有二十个面，都是等边三角形，是属水的；第三有八个面，同样是三角形，是属气的；第四是形如铁蒺藜的，有四个三角形的面，是属火的；第五有十二个五边形的面。

§65. What are great and small circles?

第六十五问　何谓大圆与小圆？

This is said only of circles on the surface of a sphere. A great circle on it is one whose plane passes through the centre of the sphere and so divides it into two halves; it has two poles, one on each side of it, from which it is equally distant. Just as circles in plane surfaces are drawn about their centres at the distance of their half-diameter, so great circles on the surface of a sphere are drawn about their poles at the distance of the side of the square inscribed in them.

这只就球面上的圆而言。球面上的大圆，是其平面穿过球心、因而把球分成两半的圆；它有两个极，两侧各一，它到两极的距离相等。正如平面上的圆以圆心为中心、以半径为距离画出，球面上的大圆则以它的极为中心、以它的内接正方形的边长为距离画出。

Small circles are those whose planes do not pass through the centre of the sphere and which divide it into two unequal parts; likewise they are unequally distant from the two poles lying on either side of them. The size of the great circles is one and the same, since it is the greatest possible on the surface of the sphere; the small circles fall short of the size of the great circle and differ among themselves, growing ever smaller.

小圆是其平面不穿过球心、把球分成不等的两部分的圆；它到两侧两极的距离也不相等。大圆的大小都是同一个，因为那已是球面上所能有的最大的圆；小圆都不及大圆之大，彼此大小也不同，越来越小。

§66. What are the properties of these circles?

第六十六问　这些圆有什么性质？

Because the great circles on a sphere are equal in size and each bisects the sphere, they necessarily cut one another, since for them to be parallel or to keep apart is altogether impossible; and each of them cuts the other into two halves at two opposite points. Whenever a great circle passes through the pole of another, it stands on it at right angles; and whenever a great circle passes through the pole of a great circle, that circle in turn passes through the pole of the first. The area of the plane of a great circle is a quarter of the area of the surface of the sphere; therefore, if its diameter is multiplied by its circumference, the result is the area of the surface of the sphere.

因为球面上的大圆大小相等，又各把球平分，所以它们必定彼此相交，要它们平行或彼此分开是完全不可能的；而且每一个都在相对的两点上把另一个分成两半。一个大圆只要穿过另一个大圆的极，就与它成直角；而一个大圆穿过另一个大圆的极时，那个大圆也反过来穿过前者的极。大圆平面的面积是球面面积的四分之一；所以用它的直径乘它的周长，所得就是球面的面积。

The small circles on a sphere, because they fall short of the size of the great circle and keep growing smaller, can be parallel to one another and to the great circles, and can be cut into two halves or into two unequal parts. Each of the small circles is drawn on the surface of the sphere not at the distance of the side of its square, as with the great circle; rather their sizes differ.

球面上的小圆，因为不及大圆之大而且越来越小，所以彼此可以平行，也可以与大圆平行；它们可以被分成两半，也可以被分成不等的两部分。每个小圆在球面上画出时，并不像大圆那样以其正方形的边长为距离，它们的大小各不相同。

§67. What are a pole and an axis?

第六十七问　何谓极与轴？

A pole is either the pole of a circle with respect to its drawing, taking the place of the centre, which has already been mentioned; or else it is a pole of the circle with respect to motion. For when a sphere moves in its place and revolves about itself, two opposite points on it must remain at rest, as on a turner's spindle, and the line joining the two points is also at rest while the sphere turns; it is called the axis.

极，或者是就画圆而言的圆之极，它占着圆心的位置，前面已经讲过；或者是就运动而言的圆之极。因为一个球在自己的位置上运动、绕自身旋转时，球上必定有相对的两点保持静止，就像旋工车床上的转轴那样；连接这两点的线在球转动时也静止不动，这条线称为轴。

§68. What is the belt of motion?

第六十八问　何谓运动带？

It is the great circle lying midway between the two poles of motion, and for that reason it is called a belt. It revolves upon itself, and its plane traces out nothing but itself, whereas all the other great circles then trace out either the whole sphere or segments of it resembling the hoop of a tambourine.

它是位于两个运动之极正中间的大圆，因此称为“带”。它在自身上旋转，它的平面除了自身之外什么也画不出；而其余一切大圆这时所画出的，要么是整个球，要么是球的若干段，形状像手鼓的鼓圈。

§69. What are the orbits?

第六十九问　何谓轨道？

They are the parallel small circles on a sphere.

就是球面上彼此平行的小圆。

§70. What are similar arcs?

第七十问　何谓相似弧？

On plane surfaces, they are the arcs of circles that lie between two straight lines drawn from the circles' centre to the circumference of the greatest of them. On the surface of a sphere, they are the parts of the parallel circles that lie between two great circles intersecting at the pole of those parallels. Similar arcs are either all of small circles, or one of them only is of a great circle. They are called similar because their ratio to their circles is one and the same, whether on the sphere or on the plane.

在平面上，相似弧是：从诸圆的圆心向其中最大一圆的圆周引两条直线，夹在这两条直线之间的各圆之弧。在球面上，相似弧是：两个相交于某组平行圆之极的大圆，夹在它们之间的那些平行圆的各段。相似弧要么全是小圆的弧，要么其中只有一段是大圆的弧。它们之所以称为相似，是因为它们与各自所属之圆的比是同一个，无论在球面上还是在平面上都是如此。

§71. What is the sector figure?

第七十一问　何谓扇形图？

It is what arises from the intersection of four great circles, each two of which meet at one point. It resembles what you get by joining the fingertips of the two middle fingers and laying the fingertips of the two forefingers on the middle joints of the middle fingers. Since the propositions of geometry are applied, especially in astronomical calculation, through arithmetic, we shall first give an account of numbers.

它是由四个大圆相交而成的图形，其中每两个交于一点。它就像这样得来的样子：把两手中指的指尖对接，再把两个食指的指尖搭在两个中指的中间关节上。因为几何的命题，尤其在天文计算中，要借算术来应用，所以我们先来讲讲数。

§72. What is one?

第七十二问　何谓一？

It is that which is marked by unity. It is the perfect, which neither increases nor decreases, nor changes at all from its state by multiplication or division. Potentially it is all numbers, and in it are all their attributes. By this property the conventional one in counted things is in the same state: it stands between the numbers that arise from its accumulation and the parts that grow ever smaller beneath it, like a balanced mean. It does not change when multiplied by its like or divided by its like, as numbers change, which increase by multiplication and decrease by division; nor as parts change, which decrease by multiplication and increase by division. One stays in its state between the two.

一就是以单一为标记的东西。它是完满的，不增也不减，乘与除都丝毫改变不了它的状态。在潜能上它就是一切数，一切数的属性都在它之中。凭着这一性质，计数之物中约定的一也处于同样的状态：它站在由它累积而生的诸数与在它之下越分越小的诸分之间，如同一个平衡的中项。它乘以同类或除以同类都不改变，不像诸数那样乘则增、除则减，也不像诸分那样乘则减、除则增。一总是在两者之间保持它的状态。

§73. How is one divided, and into how many parts?

第七十三问　一如何分，分成几份？

The true one is indivisible. The one used in sensible things is one by convention, whether it is a weight, a measure of capacity, a measure of length, or an imagined magnitude. It is impossible for the true one to admit division or to be multiplied by being divided. But the one whose unity is agreed by convention is divided, in the art of astrology, into 60 parts finer than the first; with them these units are the degrees, and they accordingly called the parts minutes ('fine ones'). This follows the custom of dividing the dirham into 60 fals and the jarīb into 60 ʿashīr. Then they divided those minutes into 60 seconds, that is, 'second time', and the second into 60 thirds, and the third into 60 fourths, and so on to the fifths, sixths, sevenths, eighths, ninths, tenths and the names of successive numbers beyond, which by nature have no end, unless the calculator chooses to stop at one of them.

真正的一不可分。用于可感之物的一是约定的一，不论它是一个重量、一个容量、一个长度，还是一个设想出来的量。真正的一不可能接受分割，也不可能因被分而变多。但这个约定为一的单位，在占星之术里被分为六十份比它更细的部分：与这些部分相对，这些单位就是度，于是人们把这些部分称为“分”，意思是“细小者”。这是沿袭了把一个迪拉姆分为六十个法尔斯、把一个贾里布分为六十个阿希尔的习惯。然后他们又把那些分各分为六十秒，“秒”的意思是“第二回”；又把一秒分为六十个三次分，一个三次分分为六十个四次分，如此下去，直到五次分、六次分、七次分、八次分、九次分、十次分，以及其后以连续诸数命名的各级，它们按本性没有尽头，除非计算者自己选定在其中某一级停下。