卷一·第 47–62 问　比例诸法、杆秤、立体与圆锥截线

§47. What is the conversion of a ratio?

第四十七问　何谓转比？

It is the ratio of the first to its excess over the second, as the ratio of the third to its excess over the fourth. In our example, if the ratio is inverted so that the second becomes first, it is the ratio of five times; and when it is then converted, as we have said, it is the ratio of five quarters [corrected: MS has 'five times'].

它是第一比它超出第二之数，如同第三比它超出第四之数。在我们的例子里，若把比反过来，使第二成为第一，它就是五倍的比；然后再照我们所说的转过来，就是四分之五的比〔抄本作「五倍」，据算改正〕。

§48. What is ordered equality of ratio?

第四十八问　何谓顺序等比？

If the ratio of the first to the second is as the ratio of the third to the fourth, and the ratio of the second to the fifth as the ratio of the fourth to the sixth, and so on for however many there are, then the extremes are proportional by equality of ratio: that is, the ratio of the first to the fifth is as the ratio of the third to the sixth. In our example the ratio of the first to the second is that of a fifth, and the ratio of the second to the fifth is that of a quarter, so the ratio of the first to the fifth is that of a fifth of a quarter, as is the ratio of the third to the sixth; and so it is [corrected: MS has 'the ratio of the second to the fifth is that of a fifth of a quarter, as the ratio of the third to the sixth'].

若第一比第二如同第三比第四，第二比第五如同第四比第六，依此类推，不论有多少项，那么两端的项按等比成比例：就是说，第一比第五如同第三比第六。在我们的例子里，第一比第二是五分之一的比，第二比第五是四分之一的比，所以第一比第五是四分之一的五分之一的比，第三比第六也是如此；事情正是这样〔抄本作「第二比第五是四分之一的五分之一的比，如同第三比第六」，据算改正〕。

§49. What is perturbed equality of ratio?

第四十九问　何谓错序等比？

If the ratio of the first to the second is as the ratio of the fourth to the sixth, and the ratio of the second to the fifth as the ratio of the third to the fourth, then, when the middle terms are dropped, the extremes remain proportional: that is, the ratio of the first to the fifth is as the ratio of the third to the sixth. This is called perturbed equality. In our example the ratio of the first to the second is that of a fifth, and the ratio of the fourth to the sixth likewise; the ratio of the second to the fifth is that of a quarter, and the ratio of the third to the fourth likewise. So the ratio of the first to the fifth, which is that of a quarter of a fifth, is as the ratio of the third to the sixth.

若第一比第二如同第四比第六，第二比第五如同第三比第四，那么去掉中间各项，两端的项仍成比例：就是说，第一比第五如同第三比第六。这叫作错序等比。在我们的例子里，第一比第二是五分之一的比，第四比第六也是如此；第二比第五是四分之一的比，第三比第四也是如此。所以第一比第五，即五分之一的四分之一的比，如同第三比第六。

§50. What is a ratio duplicated by repetition?

第五十问　何谓重复而成的二重比？

If magnitudes follow one another in succession, and the ratio of the first of them to the second is as the ratio of the second to the third, and as that of the third to the fourth, and as that of the fourth to the fifth, and so on as far as they go, then the ratio of the first to the third is the ratio of the first to the second duplicated by repetition, and to the fourth it is that ratio triplicated by repetition, and the rest follow by analogy. Do you not see that if the ratio between these magnitudes is, say, a half, the first is half the second and half of half the third, so that the half is repeated twice; and the first is half of half of half the fourth, the half being repeated three times; and at the fifth it is repeated four times? It is the same if you take the ratio between them to be other than a half, such as a third or a quarter, or any of the other parts and multiples.

若诸量依次相续，其中第一比第二，如同第二比第三，如同第三比第四，如同第四比第五，依此直到最后，那么第一比第三，就是第一比第二的比重复而成的二重比；第一比第四，就是这个比重复而成的三重比；其余依此类推。你难道看不出：比如这些量之间的比是一半，第一就是第二的一半，是第三的一半的一半，一半重复了两次；第一是第四的一半的一半的一半，一半重复了三次；到第五，就重复了四次。若你取它们之间的比不是一半，而是三分之一、四分之一，或其他任何分与倍，也是一样。

§51. What is a compound ratio?

第五十一问　何谓复比？

It is like the ratio duplicated by repetition, except that the latter is composed of two equal ratios, such as half of a half, and this of two different ratios, such as a quarter of a fifth in the first example. For if there is a ratio between two magnitudes and another magnitude is placed between them, the first ratio is composed of the ratio of one of them to the middle and of the ratio of the middle to the other, just as the distance between two towns is composed of the distances of the stages along the way. Composition is sometimes expressed as duplication, so that one says that the ratio of the first to the third is as the ratio of the first to the second duplicated by the ratio of the second to the third; but speaking of composition is better. An example: the ratio of 2 to 12 is that of a sixth. If we place 4 between them as a middle term, that ratio is composed of the ratio of 2 to 4, which is that of a half, and of the ratio of 4 to 12, which is that of a third; and half of a third is a sixth, whether we say half of a third or a third of a half. Again, if we invert, the ratio of 12 to 2 is that of six times, composed of the ratio of 12 to 4, which is that of three times, and of the ratio of 4 to 2, which is that of twice; for three times twice, or twice three times, is six times.

它像重复而成的二重比，只是二重比由两个相等的比组成，如一半的一半；复比则由两个不同的比组成，如第一个例子里的五分之一的四分之一。因为两个量之间若有一个比，又在它们之间放进另一个量，那么原来的比就由其中一个量比中间量的比、与中间量比另一个量的比复合而成，正如两城之间的距离由沿途各站的距离合成。复合有时也说成二重，于是有人说第一比第三，如同第一比第二的比以第二比第三的比加以二重；但说「复合」较好。举个例子：二比十二是六分之一的比。我们若在两者之间放进四作为中项，这个比就由二比四（一半的比）与四比十二（三分之一的比）复合而成；一半的三分之一是六分之一，说一半的三分之一也好，说三分之一的一半也好。再者，若反过来，十二比二是六倍的比，由十二比四（三倍的比）与四比二（两倍的比）复合而成；因为三倍的两倍，或两倍的三倍，都是六倍。

§52. What is the height of a figure?

第五十二问　何谓形的高？

The height of a figure is the greatest of the perpendiculars let fall from the angles of the figure onto its base or onto the base produced.

形的高，是从形的各角向它的底、或向底的延长线所作诸垂线中最大的那一条。

§53. What are similar triangles?

第五十三问　何谓相似三角形？

They are triangles in which every angle of the one is equal to its counterpart in the other, so that their corresponding sides are also proportional.

它们是这样的三角形：一个的每一个角都等于另一个中与之相应的角，因此它们相应的边也成比例。

§54. What is the ratio having a mean and two extremes?

第五十四问　何谓有中项与两端的比？

It occurs in a straight line divided into two parts such that the ratio of the smaller to the greater is as the ratio of that greater to the sum of both, that is, to the whole line.

它出现在一条分成两段的直线上：较小一段比较大一段，如同这较大一段比两段之和，也就是比全线。

§55. What is reciprocity in ratio?

第五十五问　何谓比的互反？

It is when the second and the third lie on one side. It is evident in the weights of the qarasṭūn, that is, the steelyard: the ratio of the distance of the hook from the point of suspension to the distance of the counterpoise from it is as the ratio of the weight of the counterpoise to the weight that balances it in the pan.

就是第二与第三位于同一侧。这在「卡拉斯通」也就是杆秤的重量上很明显：钩到提纽的距离比秤锤到提纽的距离，如同秤锤的重量比在秤盘里与它平衡的重量。

§56. What are 'power' and 'length'?

第五十六问　何谓「幂」与「长」？

Power is the square, and length is its side. So when it is said that such-and-such a line is equal in power to two lines such-and-such, it means that its square is equal to their two squares.

幂就是正方形，长就是它的边。所以说某条线在幂上等于某某两条线，意思是它的正方形等于那两条线的两个正方形之和。

§57. What is a cube?

第五十七问　何谓立方体？

It is a solid figure, like a die, enclosed by six squares on its six sides.

它是一个立体形，像骰子一样，六面由六个正方形围成。

§58. What is a prism?

第五十八问　何谓棱柱？

It is a figure enclosed on its sides by three square or oblong surfaces, and above and below by two triangles. Sometimes the square is a rhombus, or the oblong a rhomboid.

它是这样的形：侧面由三个正方形或长方形的面围成，上下由两个三角形围成。有时正方形换作菱形，或长方形换作似菱形。

§59. What is a right cylinder?

第五十九问　何谓直圆柱？

It is an elongated round body whose two bases, the lower and the upper, are two equal and parallel circles. Its axis is the shortest line between the centres of the two bases, and its side is the shortest line between their circumferences. It is generated by revolving a line, standing perpendicular on the two bases, between their circumferences.

它是一个长圆的物体，上下两底是两个相等且平行的圆。它的轴是两底圆心之间最短的线；它的侧线是两底圆周之间最短的线。它由一条垂立于两底之上的线，沿两底圆周旋转而生成。

§60. What is an oblique cylinder?

第六十问　何谓斜圆柱？

It is one whose axis does not stand perpendicular on the plane of the base [corrected: MS has 'stands perpendicular']. Sometimes the bases of a cylinder are not circles but two similar and equal figures, such as two squares or two triangles or similar polygons.

它的轴不垂立于底面之上〔抄本作「垂立」，据文义改正〕。有时圆柱的两底不是圆，而是两个相似而相等的形，如两个正方形、两个三角形或类似的多边形。

§61. What is a cone?

第六十一问　何谓圆锥？

It is a body that begins from a base, which is a circle or any figure whatever, and ends at a point. It has a cylinder whose base is the base of the cone and the centre of whose other base is the vertex of the cone. If that cylinder is right, the cone is called a right cone; if it is oblique, an oblique cone. A cone is always a third of its cylinder. The axis of a cone is the straight line between its vertex and the centre of its base, and its side is the line between its vertex and the circumference of the base.

它是一个从一个底开始、止于一点的物体，底是圆或任何一种形。它有一个圆柱，圆柱的底就是圆锥的底，圆柱另一底的中心就是圆锥的顶点。那圆柱若是直的，圆锥就叫作直圆锥；若是斜的，就叫作斜圆锥。圆锥总是它的圆柱的三分之一。圆锥的轴是顶点与底面中心之间的直线，它的侧线是顶点与底圆周之间的线。

§62. What are the sections of a cone?

第六十二问　圆锥的截线有哪些？

If a cone is cut by a plane passing through its vertex, this produces in the cone a rectilinear triangle. If the plane cuts it parallel to its base, the section is a circle. If it cuts it parallel to one of its sides, the section is called a parabola. If the cutting plane is not parallel to the side but meets it inside the cone towards the base, either as it is or when produced straight on, the section is called an ellipse; and if it meets it towards the vertex when produced in that direction, the section is called a hyperbola. There are no sections of the cone other than these, except in the oblique cone: if a plane not parallel to the base cuts it, and its angles are equal to the angles of the base, the section is also a circle.

圆锥若被一个经过顶点的平面所截，就在圆锥中生出一个直线三角形。平面若平行于底而截它，截口就是圆。若平行于它的一条侧线而截它，截口叫作抛物线。截面若不平行于侧线，而是在圆锥内部朝底的方向与侧线相遇——不论照原样还是笔直延长之后——截口叫作椭圆；若朝顶点的方向延长之后才与侧线相遇，截口叫作双曲线。除此以外，圆锥没有别的截线，只有斜圆锥例外：若一个不平行于底的平面截它，而所成的角等于底的角，截口也是圆。