新东方初中几何教学特点 海韵教育欧几里得在几何原本给出的23个定义

　　欧几里得的《几何原本》这部关于几何学的著作在西方被用作教科书超过2000年，在逻辑和现代科学的发展中发挥了重要作用，其逻辑严谨性直到19世纪才被超越。

　　他一书中，开头便列出了23个定义，描述了点、线、平面、圆、钝角和锐角等数学术语的定义，随后才给出了5种基本概念、5大公设以及各种命题及证明，覆盖了平面、立体几何及初等数论。

1

• A point is that which has no part.
• 点是没有部分的。

2

• A line is breadthless length.
• 线段只有长度，没有宽度。

3

• The extremities of a line are points.
• 线段的两端为点。

4

• A straight line is a line which lies evenly with the points on itself.
• 直线是一条在自身上的点均匀分布的线。

5

• A surface is that which has length and breadth only.
• 面只有长度和宽度。

6

• The extremities of a surface are lines.
• 面由线围成。

7

• A plane surface is a surface which lies evenly with the straight lines on itself.
• 平面是其上直线均匀分布的面。

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.
• 平面角是指在一个平面内的两条线彼此相交而不在一条直线上的倾斜度。

9

• And when the lines containing the angle are straight, the angle is called rectilineal.
• 当包含角的两条线都是直线时，这个角叫直线角。

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.
• 当一条直线和另一条直线交成的邻角彼此相等时，这些角的每一个都是直角，而且称直线彼此垂直。

11

• An obtuse angle is an angle greater than a right angle.
• 大于直角的角叫钝角。

12

• An acute angle is an angle less than a right angle.
• 小于直角的角叫锐角。

13

• A boundary is that which is an extremity of anything.
• 边界是物体的边缘。

14

• A figure is that which is contained by any boundary or boundaries.
• 一个图形是由一个边界或几个边界所围成的。

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;
• 圆是由一条线包围成的平面图形，内有一点与这条线上的点连接成的所有线段都相等。

16

• And the point is called the centre of the circle.
• 该点称为圆心。

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.
• 圆的直径是任意一条经过圆心的直线且在两个方向被圆周所截线段，并且把圆二等分。

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.
• 半圆是直径和由它截得的圆弧所围成的图形，且半圆的心和圆心相同。

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.
• 直线形是由直线围成的，三边形是由一条直线围成的，四边形是由四条直线围成的，多边形是由四条以上直线围成的。

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.
• 在三边形中，三条边相等的，叫做等边三角形；只有两条边相等的，叫做等腰三角形；各边不等的，叫做不等边三角形。

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.
• 此外，在三边形中，有一个角是直角的，叫做直角三角形；有一个角是钝角的，叫做钝角三角形:有三个角是锐角的，叫做锐角三角形。

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.
• 在四边形中，四边相等且四个角是直角的，叫做正方形；角是直角，但四边不全相等的，叫做长方形；四边相等，但角不是直角的，做菱形；对角相等且对边也相等，但边不全相等且非直角的，叫做长菱形；其余的四边形叫做不规则四边形。

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.
• 平行直线是在同一平面内的直线，向两个方向无限延长，在不论哪个方向它们都不相交。