|
|
The Elements are mainly a
systematization of earlier knowledge of geometry. Its superiority
over earlier treatments was rapidly recognized, with the result
that there was little interest in preserving the earlier ones, and
they are now nearly all lost.
其组成部分主要是一个系统化的早期知识的几何。其优势迅速早期治疗是公认的，因此，很少有兴趣维护以前的，现在几乎所有的损失。
Books I-IV discuss plane geometry.
Many results about plane figures are proved, e.g., If a triangle
has two equal angles, then the sides subtended by the angles are
equal.
图书一至四讨论平面几何。许多结果飞机数字证明，例如，如果一个三角形有两个平等的角度，然后双方subtended的角度都是平等的。毕达哥拉斯定理的证明。
[ 5 ]
图书的VX处理数论，几何与数字处理的通过他们的代表与各线段的长度。概念，如素数，合理的和不合理的数字介绍。在无穷的素数的证明。
The Pythagorean theorem is proved.[5]
Books V-X deal with number theory,
with numbers treated geometrically via their representation as line
segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime
numbers is proved.
Books XI-XIII concern solid
geometry. A typical result is the 1:3 ratio between the volume of a
cone and a cylinder with the same height and base.
The parallel postulate: If two lines
intersect a third in such a way that the sum of the inner angles on
one side is less than two right angles, then the two lines
inevitably must intersect each other on that side if extended far
enough.
[edit] Axioms
Euclidean geometry is an
axiomatic system, in which all theorems ("true statements") are derived from a small
number of axioms. Near the beginning of the first book of the
Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of
constructions:[6]
欧几里德几何是一个不言自明的制度，所有定理（ “真正的声明”
）是来自一个小数目公理。即将开始的第一本书的元素，欧几里得给五个假设（公理）的平面几何，在建筑方面： [ 6
]
Let the following be postulated:
to draw a straight line from any point to
any point.
To produce [extend] a
finite straight line continuously in a straight line.
To describe a circle with any center and distance
[radius].
That all right angles are equal
to one another.
The parallel postulate: That, if a straight line falling on two
straight lines make the interior angles on the same side less than
two right angles, 平行的假设：如果一条直线下降两个直线使内部角度的同一侧不到两成直角，
两直线，如果无限期地生产，满足于这方的角度是小于两个直角。
虽然欧几里得的声明的假设只有明确声称存在的建设，他们还采取是独一无二的。
the two straight lines, if produced indefinitely, meet on that side
on which are the angles less than the two right angles.
Although Euclid's statement of the
postulates only explicitly asserts the existence of the
constructions, they are also taken to be unique.
The Elements also include
the following five "common notions":
Things that equal the same thing
also equal one another.
If equals are added to equals,
then the wholes are equal.
If equals are subtracted from
equals, then the remainders are equal.
Things that coincide with one
another equal one another.
The whole is greater than the
part.
[edit] The parallel
postulate
Main article: Parallel postulate
To the ancients, the parallel
postulate seemed less obvious than the others. Euclid himself seems
to have considered it as being qualitatively different from the
others, as evidenced by the organization of the Elements:
the first 28 propositions he presents are those that can be proved
without it.
Many alternative axioms can be
formulated that have the same logical consequences as the parallel
postulate. For example Playfair's axiom states:
Through a point not on a given
straight line, at most one line can be drawn that never meets the
given line.
|
|
