Chapter 4

The shape of the heaven is of necessity spherical¹; for that is the shape most appropriate to its substance and also by nature primary.

First, let us consider generally which shape is primary among planes and solids alike. Every plane figure must be either rectilinear or curvilinear. Now the rectilinear is bounded by more than one line, the curvilinear by one only. But since in any kind the one is naturally prior to the many and the simple to the complex, the circle will be the first of plane figures. Again, if by complete, as previously² defined, we mean a thing outside which no part of itself can be found, and if addition is always possible to the straight line but never to the circular, clearly the line which embraces the circle is complete. If then the complete is prior to the incomplete, it follows on this ground also that the circle is primary among figures. And the sphere holds the same position among solids. For it alone is embraced by a single surface, while rectilinear solids have several. The sphere is among solids what the circle is among plane figures. Further, those who divide bodies into planes and generate them out of planes seem to bear witness to the truth of this. Alone among solids they leave the sphere undivided, as not possessing more than one surface: for the division into surfaces is not just dividing a whole by cutting it into its parts, but division of another fashion into parts different in form. It is clear, then, that the sphere is first of solid figures.

If, again, one orders figures according to their numbers, it is most natural to arrange them in this way. The circle corresponds to the number one, the triangle, being the sum of two right angles, to the number two. But if one is assigned to the triangle, the circle will not be a figure at all.

Now the first figure belongs to the first body, and the first body is that at the farthest circumference³. It follows that the body which revolves⁴ with a circular movement must be spherical. The same then will be true of the body continuous with it: for that which is continuous with the spherical is spherical. The same again holds of the bodies between these and the centre. Bodies which are bounded by the spherical and in contact with it must be, as wholes, spherical; and the bodies below the sphere of the planets are contiguous with the sphere above them. The sphere then will be spherical throughout; for every body within it is contiguous and continuous with spheres.

Again, since the whole revolves, palpably and by assumption, in a circle, and since it has been shown that outside the farthest circumference there is neither void nor place, from these grounds also it will follow necessarily that the heaven is spherical. For if it is to be rectilinear in shape, it will follow that there is place and body and void without it. For a rectilinear figure as it revolves never continues in the same room, but where formerly⁵ was body, is now none, and where now is none, body will be in a moment because of the projection⁶ at the corners. Similarly, if the world had some other figure with unequal radii⁷, if, for instance, it were lentiform, or oviform, in every case we should have to admit space and void outside the moving body, because the whole body would not always occupy the same room.

Again, if the motion of the heaven is the measure of all movements whatever in virtue⁸ of being alone continuous and regular and eternal, and if, in each kind, the measure is the minimum, and the minimum movement is the swiftest, then, clearly, the movement of the heaven must be the swiftest of all movements. Now of lines which return upon themselves the line which bounds the circle is the shortest; and that movement is the swiftest which follows the shortest line. Therefore, if the heaven moves in a circle and moves more swiftly than anything else, it must necessarily be spherical.

Corroborative⁹ evidence may be drawn¹⁰ from the bodies whose position is about the centre. If earth is enclosed by water, water by air, air by fire, and these similarly by the upper bodies-which while not continuous are yet contiguous with them-and if the surface of water is spherical, and that which is continuous with or embraces the spherical must itself be spherical, then on these grounds also it is clear that the heavens are spherical. But the surface of water is seen to be spherical if we take as our starting-point the fact that water naturally tends to collect in a hollow place-’hollow’ meaning ‘nearer the centre’. Draw from the centre the lines AB, AC, and let their extremities¹¹ be joined by the straight line BC. The line AD, drawn to the base of the triangle, will be shorter than either of the radii. Therefore the place in which it terminates will be a hollow place. The water then will collect there until equality is established, that is until the line AE is equal to the two radii. Thus water forces its way to the ends of the radii, and there only will it rest: but the line which connects the extremities of the radii is circular: therefore the surface of the water BEC is spherical.

It is plain from the foregoing that the universe is spherical. It is plain, further, that it is turned (so to speak) with a finish which no manufactured thing nor anything else within the range of our observation can even approach. For the matter of which these are composed does not admit of anything like the same regularity¹² and finish as the substance of the enveloping¹³ body; since with each step away from earth the matter manifestly becomes finer in the same proportion as water is finer than earth.

1 spherical   

adj.球形的；球面的

参考例句：

• The Earth is a nearly spherical planet.地球是一个近似球体的行星。
• Many engineers shy away from spherical projection methods.许多工程师对球面投影法有畏难情绪。

2 previously   

adv.以前，先前（地）

参考例句：

• The bicycle tyre blew out at a previously damaged point.自行车胎在以前损坏过的地方又爆开了。
• Let me digress for a moment and explain what had happened previously.让我岔开一会儿，解释原先发生了什么。

3 circumference   

n.圆周，周长，圆周线

参考例句：

• It's a mile round the circumference of the field.运动场周长一英里。
• The diameter and the circumference of a circle correlate.圆的直径与圆周有相互关系。

4 revolves   

v.（使）旋转( revolve的第三人称单数 )；细想

参考例句：

• The earth revolves both round the sun and on its own axis. 地球既公转又自转。 来自《现代汉英综合大词典》
• Thus a wheel revolves on its axle. 于是，轮子在轴上旋转。 来自《简明英汉词典》

5 formerly   

adv.从前，以前

参考例句：

• We now enjoy these comforts of which formerly we had only heard.我们现在享受到了过去只是听说过的那些舒适条件。
• This boat was formerly used on the rivers of China.这船从前航行在中国内河里。

6 projection   

n.发射，计划，突出部分

参考例句：

• Projection takes place with a minimum of awareness or conscious control.投射在最少的知觉或意识控制下发生。
• The projection of increases in number of house-holds is correct.对户数增加的推算是正确的。

7 radii   

n.半径；半径（距离）( radius的名词复数 )；用半径度量的圆形面积；半径范围；桡骨

参考例句：

• Hence, the damage radii can not be determined from overpressure alone. 因此，破坏半径不能单单由超压力大小来决定。 来自辞典例句
• It is now necessary to introduce a sign convention for radii of curvature. 现在必须介绍曲率半径的正负号规则。 来自辞典例句

8 virtue   

n.德行，美德；贞操；优点；功效，效力

参考例句：

• He was considered to be a paragon of virtue.他被认为是品德尽善尽美的典范。
• You need to decorate your mind with virtue.你应该用德行美化心灵。

9 corroborative   

adj.确证（性）的，确凿的

参考例句：

• Is there any corroborative evidence for this theory? 是否有进一步说明问题的论据来支持这个理论？
• They convicted the wrong man on the basis of a signed confession with no corroborative evidence. 凭一张有签名的认罪书而没有确凿的佐证，他们就错误地判了那人有罪。 来自《简明英汉词典》

10 drawn   

v.拖，拉，拔出；adj.憔悴的，紧张的

参考例句：

• All the characters in the story are drawn from life.故事中的所有人物都取材于生活。
• Her gaze was drawn irresistibly to the scene outside.她的目光禁不住被外面的风景所吸引。

11 extremities   

n.端点( extremity的名词复数 )；尽头；手和足；极窘迫的境地

参考例句：

• She was most noticeable, I thought, in respect of her extremities. 我觉得她那副穷极可怜的样子实在太惹人注目。 来自辞典例句
• Winters may be quite cool at the northwestern extremities. 西北边区的冬天也可能会相当凉。 来自辞典例句

12 regularity   

n.规律性，规则性；匀称，整齐

参考例句：

• The idea is to maintain the regularity of the heartbeat.问题就是要维持心跳的规律性。
• He exercised with a regularity that amazed us.他锻炼的规律程度令我们非常惊讶。

13 enveloping   

v.包围，笼罩，包住( envelop的现在分词 )

参考例句：

• Always the eyes watching you and the voice enveloping you. 那眼睛总是死死盯着你，那声音总是紧紧围着你。 来自英汉文学
• The only barrier was a mosquito net, enveloping the entire bed. 唯一的障碍是那顶蚊帐罩住整个床。 来自辞典例句