卷一·第 1–19 问　几何之始：体、面、线、点、角、圆与三角形

BOOK I. GEOMETRY

卷一　几何

§1. What is geometry?

第一问　何谓几何？

It is the science of magnitudes and of their quantities in relation to one another, and the knowledge of the properties of their forms and of the figures that exist in bodies. Through it the science of number is carried from the particular to the universal, and the science of astronomy from reckoning and conjecture to truth.

几何是关于诸量、以及诸量彼此之间多少关系的学问，也是认识诸量形状的性质、认识物体之中所存诸形的学问。借着它，数的学问由个别提升到普遍，天文的学问由推算与猜测提升到真实。

§2. What is a body?

第二问　何谓物体？

It is that which exists to the sense of touch and subsists by itself. It has taken its share of space and filled as much of the dimensions of place as equals its own size, and it prevents any other possible thing from sharing its space and its place with it.

物体是触觉所能感到、凭自身而存在的东西。它占取了自己那一份空间，充满了与自身大小相等的那一份处所的维度，并且不让其他任何可能之物与它共享它的空间与处所。

§3. What are the dimensions of place?

第三问　处所的维度是什么？

They are three, called length, breadth and depth. These names do not belong to the dimensions in themselves, so as never to change; they are relative. If you call one of them length, the one crossing it is breadth, and the one crossing both is depth. Custom has it that the longer of the first two is called length and the shorter breadth, and the one extending downward depth; if it extends upward it is called height.

维度有三个，称为长、宽、深。这些名称并不属于维度本身、永不改变，而是相对的：你若把其中一个叫作长，与它相交的那一个就是宽，与这两者都相交的那一个就是深。习惯上，前两者中较长的叫作长，较短的叫作宽，向下延伸的叫作深；若是向上延伸，就叫作高。

§4. What are the six directions?

第四问　六个方向是什么？

They are the senses of these dimensions on both their sides together. One sense of length is the front and the other the back; one sense of breadth is the right and the other the left; one sense of depth is above and the other below.

六个方向就是这些维度两头的指向合在一起。长的一头是前，另一头是后；宽的一头是右，另一头是左；深的一头是上，另一头是下。

§5. What is a surface?

第五问　何谓面？

A body is necessarily finite in all its directions, and its limit is the surface. It is likened to the roof of a house because it lies above it, and it is also called basīṭ ('spread out') because it is as if spread over the body. It is length and breadth only, falling short of the body by one dimension, depth; for if it had depth it would also be a body, whereas we have taken it as the body's limit. When a body is coloured, dull and not transparent, the colour seen is in its surface, since what lies beneath it is not perceived by sight. This makes the surface easy to conceive, and we can readily find one when water and oil are put together in a vessel: they do not mix, and they touch each other at a surface between them. Surfaces are of two kinds, the one plane and the other not plane.

物体必定在它的一切方向上都有尽头，它的尽头就是面。面被比作房屋的屋顶，因为它在物体的上面；它又叫作「巴西特」（意为「铺开的」），因为它好像铺在物体之上。面只有长和宽，比物体少一个维度，就是深；因为它若有深，它也就成了物体，而我们是把它当作物体的尽头来看的。一个物体若有颜色、暗而不透明，看得见的颜色就在它的面上，因为面以下的东西眼睛看不到。这使面容易设想；我们也很容易找到一个面：把水和油一同放进一个器皿里，两者并不相混，而是在它们之间的一个面上彼此相接。面有两种，一种是平的，另一种不平。

§6. What is a line?

第六问　何谓线？

If a surface is finite, its limit is necessarily a line. A line is length without breadth, falling short of the surface by one dimension, breadth; for if it had breadth it would be a surface, whereas we have assumed it to be the surface's limit. It is easily conceived from the line that can be seen in a flask holding water and oil, and from the boundary between shadow and sunlight lying side by side on the ground. All of this can also be conceived with thin paper, so that, once trained in conceiving through the senses, one may proceed step by step to what is imagined and grasped by the intellect.

面若有尽头，它的尽头必定是线。线是有长而无宽的，比面少一个维度，就是宽；因为它若有宽，它就成了面，而我们是把它设定为面的尽头的。线很容易设想：可以从盛着水和油的玻璃瓶里看得见的那条线来想，也可以从地上并排的阴影与日光之间的分界来想。这一切也都能借薄纸来设想，好让人在借感官设想上受过训练之后，一步一步进到想象之中、由理智把握的东西。

§7. What is a point?

第七问　何谓点？

If a line is finite, its limit is a point, which falls short of the line by one dimension, length. So a point has neither length, nor breadth, nor depth; it is the limit of limits, and therefore has no parts. It is conceived through the senses by the tip of a sharp needle. The line, the surface and the point all exist only in a body, which is their bearer; taken alone, abstracted from it, they can be conceived only in the intellect.

线若有尽头，它的尽头就是点；点比线少一个维度，就是长。所以点既无长，也无宽，也无深；它是尽头的尽头，因此没有部分。借感官，可以用一根尖针的针尖来设想它。线、面、点都只存在于物体之中，物体是承载它们的；若单独拿出来、从物体中抽离，它们只能在理智中设想。

§8. What are a plane surface and a straight line?

第八问　何谓平面与直线？

Of each of them, the straight one is the shortest between its limits. If lines lying in a surface all fall in line with one another, the surface is plane; and likewise, if the points on a line fall in line with one another, the line is straight.

两者之中，直的那一种都是它两端之间最短的。若位于一个面上的诸线彼此全都对成一直，这个面就是平面；同样，若一条线上的诸点彼此对成一直，这条线就是直线。

§9. What is an angle?

第九问　何谓角？

It is the termination of a surface at a point, where two straight lines enclose it, meeting one another but not in a straight line. For this reason it is called rectilinear, since if one or both of the lines were not straight it would not be called rectilinear.

角是面在一点上的收束：两条直线围住它，彼此相遇，却不成一条直线。因此它叫作直线角；因为两条线中若有一条或两条不是直的，它就不叫直线角了。

§10. How many kinds of angles are there?

第十问　角有几种？

If a straight line falls on another straight line and the two angles it makes on either side of it are equal, each of them is called a right angle, and the line standing on the other is perpendicular to it. If the two angles are not equal, the line is not a perpendicular, and the greater of the two angles is called obtuse and the smaller acute.

一条直线落在另一条直线上，若它在自己两侧所成的两个角相等，这两个角各叫作直角，立在另一条线上的这条线就叫作那条线的垂线。若两个角不相等，这条线就不是垂线，两个角中较大的叫作钝角，较小的叫作锐角。

§11. What is a figure?

第十一问　何谓形？

It is a form enclosed by one line or more.

形是由一条或多条线围成的形状。

§12. What is a circle?

第十二问　何谓圆？

It is a figure in a plane surface, enclosed by one line called the circumference, also called the circuit. Inside it is a point called the centre, and all the straight lines drawn from it to the circumference are equal to one another.

圆是平面上的一种形，由一条线围成，这条线叫作圆周，也叫作周线。圆内有一点叫作圆心，从它引到圆周的一切直线彼此相等。

§13. What are the diameter and the chord?

第十三问　何谓直径与弦？

Every straight line lying in a circle and ending at both its extremities on the circumference is called the diameter of the circle if it passes through the centre, and it divides the circle into two halves. If it does not pass through the centre it is called a chord, and it divides the circle into two unequal parts. A chord is shared by two arcs, one of them greater than the other.

凡位于圆内、两端都止于圆周的直线，若经过圆心，就叫作圆的直径，它把圆分成两半；若不经过圆心，就叫作弦，它把圆分成不相等的两部分。一条弦为两段弧所共有，其中一段比另一段大。

§14. What is the sagitta?

第十四问　何谓矢？

It is the part of the diameter lying between the midpoint of an arc and the midpoint of its chord. If the arc is greater than a semicircle, the sagitta is more than half the diameter; if the arc is less than a semicircle, the sagitta is less than half the diameter.

矢是直径上介于一段弧的中点与它的弦的中点之间的那一段。弧若大于半圆，矢就大于直径的一半；弧若小于半圆，矢就小于直径的一半。

§15. What is the greatest sine?

第十五问　何谓最大正弦？

It is the half-diameter of the circle, and it is also called the whole sine.

最大正弦就是圆的半径，它也叫作全正弦。

§16. What is the right sine?

第十六问　何谓直正弦？

It is half the chord of twice the arc. Or, if you like, it is the perpendicular let fall from one end of the arc onto the diameter that passes through its other end. Whenever you see 'sine' without qualification, know that the right sine is meant.

它是两倍之弧的弦的一半。或者，你若愿意，可以说它是从弧的一端向经过弧另一端的直径所作的垂线。凡见到不加限定的「正弦」，要知道指的就是直正弦。

§17. What is the versed sine?

第十七问　何谓正矢？

It is the sagitta of twice the arc. Or, if you like, call it the line joining the beginning of the arc to the foot of its sine opposite it. The greatest of the versed sines is twice the whole sine, just as the greatest of the right sines is the whole sine.

正矢是两倍之弧的矢。或者，你若愿意，可以把它叫作连接弧的起点与对面正弦垂足的那条线。正矢中最大的是全正弦的两倍，正如直正弦中最大的是全正弦。

§18. What are the complement of an arc and the complement of a sine?

第十八问　何谓弧之余与正弦之余？

The complement of an arc is what, added to it, completes a quarter of the circle with it; so if the arc is subtracted from 90, its complement remains. The complement of a sine is what, added to it, makes the sum equal to the whole sine.

弧之余，是加到弧上便与它合成四分之一圆的那一段；所以从九十中减去弧，剩下的就是弧之余。正弦之余，是加到正弦上使总和等于全正弦的那一段。

§19. What are the kinds of triangles?

第十九问　三角形有哪几种？

A triangle has three sides and three angles. With respect to its sides it receives three names. The first is equilateral, when its three sides are equal; the second is isosceles, when two of its sides are equal and the third is greater or smaller; the third is scalene, when no two of its sides are equal at all. With respect to its angles it also receives three names. The first: when it has a right angle, it is named after it right-angled; the second: when it has an obtuse angle, it is named after it obtuse-angled; the third: when it has neither a right nor an obtuse angle, it is called acute-angled.

三角形有三条边、三个角。就边而论，它有三个名称：第一是等边三角形，三边相等；第二是等腰三角形，两边相等，第三边较大或较小；第三是不等边三角形，任何两边都不相等。就角而论，它也有三个名称：第一，有一个直角时，就依它称为直角三角形；第二，有一个钝角时，就依它称为钝角三角形；第三，既无直角也无钝角时，称为锐角三角形。