卷一·第 20–46 问　三角形诸名、四边形、平行线、相切与比例

§20. What are the perpendicular and the base?

第二十问　何谓垂线与底？

The perpendicular is the line drawn from one of the angles of a triangle and standing at right angles on one of the sides of the triangle; the base is the side on which the perpendicular falls.

垂线是从三角形的一个角引出、与三角形的一条边成直角而立的线；底是垂线所落的那条边。

§21. What is the 'fall of the stone' (masqaṭ al-ḥajar)?

第二十一问　何谓「落石点」？

It is the point of the base on which the perpendicular falls. Some people give the name 'fall of the stone' to the smaller of the two parts into which the perpendicular divides the base, but that usage is far from what the word and its meaning require.

它是垂线落在底上的那一点。有些人把垂线分底所成两段中较短的那一段叫作「落石点」，但这种用法与这个词和它的意思所要求的相去甚远。

§22. Do the sides of a triangle have special names other than these?

第二十二问　三角形的边，除了这些之外还有专门的名称吗？

The side opposite the obtuse or the right angle is called the greatest side, and in the right-angled triangle in particular it is also called the diagonal and the chord. As for the two remaining sides in both, if they are unequal, the smaller is called the shorter of the two lesser sides and the greater the longer of the two lesser sides.

钝角或直角所对的边叫作最大边；特别是在直角三角形里，它又叫作「对角线」和「弦」。至于两种三角形中其余的两条边，若不相等，较小的叫作两小边中的短边，较大的叫作两小边中的长边。

§23. What are the kinds of quadrilaterals?

第二十三问　四边形有哪几种？

The first is the square, which is right-angled, equilateral and has equal diagonals, the diagonals being the two lines joining opposite angles. The second is the oblong, which is right-angled but has unequal sides, only the opposite sides being equal in pairs. The third is the rhombus, which is equilateral but has unequal diagonals and is not right-angled. The fourth is the rhomboid, whose diagonals are unequal and of which only the opposite sides are equal in pairs.

第一种是正方形，它是直角的、等边的、对角线相等的，对角线就是连接相对两角的两条线。第二种是长方形，它是直角的，但各边不相等，只有相对的边两两相等。第三种是菱形，它是等边的，但对角线不相等，也不是直角的。第四种是似菱形，它的对角线不相等，也只有相对的边两两相等。

All figures other than these are called trapezia, except the polygons whose sides and angles are equal; these are given names derived from the number of their sides, such as the pentagon, the hexagon, the heptagon and so on.

除此以外的一切形都叫作不规则形；只有边与角都相等的多边形例外，它们按边的数目得名，如五边形、六边形、七边形，依此类推。

§24. What are parallel lines?

第二十四问　何谓平行线？

They are lines lying in one plane, the distance between which does not vary, and which, if produced straight on indefinitely in both directions, never meet.

平行线是位于同一平面上、彼此间的距离不变的线；若向两个方向笔直地无限延长，它们永不相遇。

§25. What are vertically opposite angles?

第二十五问　何谓对顶角？

If two straight lines cut each other, four angles arise between them, and each two of them that meet only at the point are called opposite, and they are equal.

两条直线相交时，它们之间生出四个角；其中只在交点上相遇的每两个角，叫作相对的角，它们相等。

§26. What are alternate angles?

第二十六问　何谓错角？

If a straight line falls on two parallel straight lines, the angle on one side of it at one of the lines is alternate to the angle on its other side at the other line, and the two are equal.

一条直线落在两条平行直线上时，它一侧在其中一条线上的角，与它另一侧在另一条线上的角互为错角，两者相等。

§27. What are the angles that arise from a triangle?

第二十七问　由三角形生出的角是什么？

If one of the sides of a triangle is produced straight on, there arises from it, outside the triangle, an angle called exterior, and each of the two angles not adjacent to it is called opposite to it.

把三角形的一条边笔直延长，就在三角形外生出一个角，叫作外角；不与它相邻的那两个角，各叫作与它相对的角。

§28. What is the exterior angle of two parallel lines?

第二十八问　两平行线的外角是什么？

If the line falling on the two parallel lines is produced straight on, it makes outside them an angle equal to the angle facing it on its side at the other line, and this is called exterior.

把落在两平行线上的那条线笔直延长，它在两线之外所成的角，等于在另一条线上、同一侧与它相对的那个角，这个角叫作外角。

§29. What is a parallelogram?

第二十九问　何谓平行四边形？

It is a four-sided plane figure in which each two opposite sides are parallel. The line joining two opposite angles in it is called its diagonal.

它是一个四边的平面形，其中每两条相对的边都平行。其中连接两个相对之角的线叫作它的对角线。

§30. What is a complement?

第三十问　何谓余形？

If on the diagonal of a parallelogram there lie two parallelograms whose diagonals are parts of its diagonal, and the two touch each other, then the two pieces of it that remain on either side are called complements, each of them a complement.

若在一个平行四边形的对角线上有两个平行四边形，它们的对角线是它的对角线的一部分，而且两者彼此相接，那么在两侧剩下的两块就叫作余形，每一块是一个余形。

§31. What is a gnomon?

第三十一问　何谓磬折形？

It is the two complements together with one of the two parallelograms, and it is marked by an arc of a circle drawn across the three.

它是两个余形连同那两个平行四边形中的一个，用一段横跨这三块的圆弧来标示。

§32. How is a line multiplied by a line?

第三十二问　线怎样乘线？

It is moving one of them along the other until a right-angled surface results, enclosed by those two lines. If they are equal, that surface is a square; if they are unequal, it is an oblong.

就是让其中一条沿着另一条移动，直到生成一个由这两条线围成的直角的面。两条线若相等，这个面就是正方形；若不相等，就是长方形。

§33. What is contact?

第三十三问　何谓相切？

Contact occurs between two unequal circles on the inside, between two circles of any size on the outside, and between a straight line and a circle, when they meet without cutting each other.

相切发生在两个不等的圆之间（在内侧），发生在任何大小的两个圆之间（在外侧），也发生在一条直线与一个圆之间，都是相遇而不相割的时候。

§34. What is a figure circumscribing a circle?

第三十四问　何谓圆的外切形？

It is a figure constructed outside the circle such that each of its sides touches it.

它是作在圆外的形，它的每一条边都与圆相切。

§35. What is a figure inscribed in a circle?

第三十五问　何谓圆的内接形？

It is a figure lying inside the circle, the circle passing through all its angles.

它是位于圆内的形，圆经过它所有的角。

§36. What is the angle contained by an arc?

第三十六问　何谓弧所含之角？

It is the angle equal to the angle formed by two lines drawn from the two ends of the arc and meeting.

它是一个角，等于从弧的两端引出两条线、相交所成的那个角。

§37. How much is the circumference of a circle reckoned to be when its diameter is 1?

第三十七问　直径为一时，圆周算作多少？

It is 3 1/7, according to Archimedes' efforts to approximate the truth: he confined it between two magnitudes, so that it is greater than the smaller of them and smaller than the greater. If half the diameter is multiplied by half the circumference, the area comes to 1/2 and 2/7 of 1.

是三又七分之一；这是依阿基米德逼近真值的功夫：他把它夹在两个量之间，使它大于其中较小的、小于其中较大的。若以半径乘半圆周，面积得一的二分之一与七分之二。

§38. What are a part and multiples?

第三十八问　何谓分与倍？

If one magnitude measures another a number of times and exhausts it, the smaller is called a part of it; the greater is called a multiple of it by that number. Multiples are also called 'doubles'.

一个量若量另一个量若干次而恰好量尽，较小的就叫作较大的一个分；较大的就叫作较小的按那个次数的倍。倍又叫作「双倍」。

§39. What is a ratio?

第三十九问　何谓比？

It is a relation between two things of the same kind, by which the size of the one is known from the other when it is related to it; just as a man is called a father when related to his son, and a son when related to his father. Thus one of the two things is, for example, half of the other, and the other double the first.

比是同类两物之间的一种关系，凭着它，把一物与另一物相比时，就可由另一物知道它的大小；正如一个人对他的儿子而言叫作父亲，对他的父亲而言叫作儿子。这样，两物之一比如是另一物的一半，另一物则是前者的两倍。

§40. What is proportion?

第四十问　何谓比例？

It is the equality of two ratios or more, and at the least it holds among three magnitudes. For example, if the ratio is a fifth, the first magnitude is a fifth of the second and the second a fifth of the third.

比例是两个或更多的比相等，它至少存在于三个量之间。例如，比若是五分之一，第一个量就是第二个的五分之一，第二个又是第三个的五分之一。

§41. [What is discontinuous proportion?]

第四十一问　［何谓间断比例？］

They are four magnitudes, the ratio of the first to the second being as the ratio of the third to the fourth, whether the second is equal to the third or not. One of their properties is that the product of the first by the fourth equals the product of the second by the third, crosswise along the diagonals, so that the two diagonals face each other. Division, however, comes out equal only along the sides: that is, the quotient of the second divided by the first equals the quotient of the fourth divided by the third, and the quotient of the third divided by the first equals the quotient of the fourth divided by the second.

它是四个量：第一比第二，如同第三比第四，不论第二是否等于第三。它们的一个性质是：第一乘第四等于第二乘第三，沿着对角线交叉相乘，使两条对角线彼此相对。而除法只有沿着边才得出相等：就是说，第二除以第一所得，等于第四除以第三所得；第三除以第一所得，等于第四除以第二所得。

§42. What are the antecedent and the consequent?

第四十二问　何谓前项与后项？

The antecedent is that one of the two magnitudes of a ratio which is mentioned first and is related to the other; the consequent is the one mentioned last, to which the antecedent is related.

前项是一个比的两个量中先说出、并被拿去与另一个相比的那一个；后项是后说出、前项与之相比的那一个。

§43. What is the inversion of a ratio?

第四十三问　何谓反比？

It is the ratio of the second to the first, which in our example is five times, as the ratio of the fourth to the third. The inversion of a ratio is also called its converse.

它是第二比第一，在我们的例子里是五倍，如同第四比第三。比的反比也叫作它的逆比。

§44. What is the alternation of a ratio?

第四十四问　何谓更比？

It is the ratio of the first to the third as the ratio of the second to the fourth; in our example both are the ratio of a third.

它是第一比第三，如同第二比第四；在我们的例子里，两者都是三分之一的比。

§45. What is the composition of a ratio?

第四十五问　何谓合比？

It is the ratio of the sum of the first and the second to the second as the ratio of the sum of the third and the fourth to the fourth; in our example both are the ratio of 1 and a fifth.

它是第一与第二之和比第二，如同第三与第四之和比第四；在我们的例子里，两者都是一又五分之一的比。

§46. What is the separation of a ratio?

第四十六问　何谓分比？

It is the ratio of the excess of the first over the second to the second, as the ratio of the excess of the third over the fourth to the fourth. Since in our example the first is smaller than the second, this separation between them can only be made after inversion, that is, taking the ratio of the second to the first, so that the second becomes first in the ratio, as antecedent. This ratio is then the ratio of four times.

它是第一超出第二之数比第二，如同第三超出第四之数比第四。在我们的例子里，第一小于第二，所以这种分比只能在反比之后才作，就是取第二比第一，使第二成为比中的第一项，作为前项。这样，这个比就是四倍的比。