Elements, Book I: First Principles, Propositions 1 and 47

Definitions定义#

2. A line is breadthless length.

2. 线是没有宽度的长度。

3. The extremities of a line are points.

3. 线的两端是点。

4. A straight line is a line which lies evenly with the points on itself.

4. 直线是同其上各点均匀地平放着的线。

5. A surface is that which has length and breadth only.

5. 面是只有长度和宽度的东西。

6. The extremities of a surface are lines.

6. 面的边缘是线。

7. A plane surface is a surface which lies evenly with the straight lines on itself.

7. 平面是同其上各直线均匀地平放着的面。

8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

8. 平面角是一个平面上两条相交且不在一条直线上的线彼此之间的倾斜。

9. And when the lines containing the angle are straight, the angle is called rectilineal.

9. 当夹着这个角的两条线都是直线时，这个角叫作直线角。

10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

10. 一条直线立在另一条直线上，若使相邻的两个角彼此相等，则这两个相等的角都是直角，而立着的那条直线叫作它所立于其上的那条直线的垂线。

11. An obtuse angle is an angle greater than a right angle.

11. 钝角是大于直角的角。

12. An acute angle is an angle less than a right angle.

12. 锐角是小于直角的角。

13. A boundary is that which is an extremity of anything.

13. 边界是任何东西的端际。

14. A figure is that which is contained by any boundary or boundaries.

14. 图形是被一条或几条边界所包围的东西。

15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;

15. 圆是由一条线所包围的平面图形，在这个图形内有一点，从这一点到这条线上所引的一切直线都彼此相等；

16. And the point is called the centre of the circle.

16. 而这一点叫作圆心。

17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

17. 圆的直径是经过圆心、两端都终止于圆周的任意一条直线，这样的直线也把圆二等分。

18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

18. 半圆是由直径和被它截下的圆周所包围的图形。半圆的中心与圆心相同。

19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

19. 直线形是由直线所包围的图形：三边形由三条直线包围，四边形由四条直线包围，多边形由四条以上的直线包围。

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

20. 在三边形中，三条边都相等的叫等边三角形，只有两条边相等的叫等腰三角形，三条边都不相等的叫不等边三角形。

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

21. 又，在三边形中，有一个角是直角的叫直角三角形，有一个角是钝角的叫钝角三角形，三个角都是锐角的叫锐角三角形。

22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

22. 在四边形中，各边相等且各角为直角的叫正方形；各角为直角而各边不全相等的叫长方形；各边相等而各角不是直角的叫菱形；对边、对角彼此相等，但各边不全相等、各角也不是直角的叫长斜方形。此外的四边形，都叫作不规则四边形。

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

23. 平行直线是在同一平面内、向两个方向无限延长，在任一方向上都不相交的直线。

Postulates公设#

Let the following be postulated:

设以下诸事为公设：

1. To draw a straight line from any point to any point.

1. 从任一点到任一点可以作一条直线。

2. To produce a finite straight line continuously in a straight line.

2. 一条有限直线可以沿直线连续地延长。

3. To describe a circle with any centre and distance.

3. 以任一点为圆心、任意距离为半径，可以作一个圆。

4. That all right angles are equal to one another.

4. 所有直角都彼此相等。

Common Notions公理#

1. Things which are equal to the same thing are also equal to one another.

1. 与同一事物相等的事物，彼此也相等。

2. If equals be added to equals, the wholes are equal.

2. 等量加等量，其和相等。

3. If equals be subtracted from equals, the remainders are equal.

3. 等量减等量，其差相等。

4. Things which coincide with one another are equal to one another.

4. 彼此重合的事物，彼此相等。

5. The whole is greater than the part.

5. 整体大于部分。

Proposition 1命题一#

On a given finite straight line to construct an equilateral triangle.

在一条已知的有限直线上作一个等边三角形。

Let AB be the given finite straight line.

设 AB 为已知的有限直线。

Thus it is required to construct an equilateral triangle on the straight line AB.

于是要求在直线 AB 上作一个等边三角形。

With centre A and distance AB let the circle BCD be described; [Post. 3]

以 A 为圆心、AB 为距离，作圆 BCD；〔公设三〕

again, with centre B and distance BA let the circle ACE be described; [Post. 3]

再以 B 为圆心、BA 为距离，作圆 ACE；〔公设三〕

and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1]

从两圆的交点 C 到点 A、B，连直线 CA、CB。〔公设一〕

Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15]

因为点 A 是圆 CDB 的圆心，所以 AC 等于 AB。〔定义十五〕

Again, since the point B is the centre of the circle CAE, BC is equal to BA. [Def. 15]

又因为点 B 是圆 CAE 的圆心，所以 BC 等于 BA。〔定义十五〕

But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB.

而已经证明 CA 等于 AB；所以直线 CA、CB 都等于 AB。

And things which are equal to the same thing are also equal to one another; [C. N. 1] therefore CA is also equal to CB.

而与同一事物相等的事物彼此也相等；〔公理一〕所以 CA 也等于 CB。

Therefore the three straight lines CA, AB, BC are equal to one another.

所以三条直线 CA、AB、BC 彼此相等。

Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB.

所以三角形 ABC 是等边三角形；并且它是在已知的有限直线 AB 上作出的。

(Being) what it was required to do.

这就是所要作的。

Proposition 47命题四十七#

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

在直角三角形中，对着直角的边上的正方形，等于夹着直角的两边上的正方形之和。

Let ABC be a right-angled triangle having the angle BAC right;

设 ABC 为直角三角形，角 BAC 为直角；

I say that the square on BC is equal to the squares on BA, AC.

我说，BC 上的正方形等于 BA、AC 上的两个正方形之和。

For let there be described on BC the square BDEC, and on BA, AC the squares GB, HC; [I. 46]

在 BC 上作正方形 BDEC，在 BA、AC 上作正方形 GB、HC；〔命题一·四十六〕

through A let AL be drawn parallel to either BD or CE, and let AD, FC be joined.

过 A 作 AL 平行于 BD 或 CE，连 AD、FC。

Then, since each of the angles BAC, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC, AG not lying on the same side make the adjacent angles equal to two right angles;

因为角 BAC、BAG 都是直角，所以在直线 BA 上的点 A 处，不在 BA 同侧的两条直线 AC、AG 所成的两个邻角之和等于两直角；

therefore CA is in a straight line with AG. [I. 14]

所以 CA 与 AG 在一条直线上。〔命题一·十四〕

For the same reason BA is also in a straight line with AH.

同理，BA 与 AH 也在一条直线上。

And, since the angle DBC is equal to the angle FBA: for each is right: let the angle ABC be added to each; therefore the whole angle DBA is equal to the whole angle FBC. [C. N. 2]

又因为角 DBC 等于角 FBA（两者都是直角），在两者上各加上角 ABC，于是整个角 DBA 等于整个角 FBC。〔公理二〕

And, since DB is equal to BC, and FB to BA, the two sides AB, BD are equal to the two sides FB, BC respectively; and the angle ABD is equal to the angle FBC; therefore the base AD is equal to the base FC, and the triangle ABD is equal to the triangle FBC. [I. 4]

又因为 DB 等于 BC，FB 等于 BA，两边 AB、BD 分别等于两边 FB、BC；而角 ABD 等于角 FBC；所以底 AD 等于底 FC，三角形 ABD 等于三角形 FBC。〔命题一·四〕

Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallels BD, AL. [I. 41]

而平行四边形 BL 是三角形 ABD 的二倍，因为它们有同一底 BD，并且在同一对平行线 BD、AL 之间。〔命题一·四十一〕

And the square GB is double of the triangle FBC, for they again have the same base FB and are in the same parallels FB, GC. [I. 41]

正方形 GB 是三角形 FBC 的二倍，因为它们也有同一底 FB，并且在同一对平行线 FB、GC 之间。〔命题一·四十一〕

Therefore the parallelogram BL is also equal to the square GB.

所以平行四边形 BL 也等于正方形 GB。

Similarly, if AE, BK be joined, the parallelogram CL can also be proved equal to the square HC; therefore the whole square BDEC is equal to the two squares GB, HC. [C. N. 2]

同样，若连 AE、BK，也可以证明平行四边形 CL 等于正方形 HC；所以整个正方形 BDEC 等于两个正方形 GB、HC 之和。〔公理二〕

And the square BDEC is described on BC, and the squares GB, HC on BA, AC.

而正方形 BDEC 是作在 BC 上的，正方形 GB、HC 是作在 BA、AC 上的。

Therefore the square on the side BC is equal to the squares on the sides BA, AC.

所以边 BC 上的正方形等于边 BA、AC 上的两个正方形之和。

Therefore etc. Q. E. D.

证完。