The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics----analytic geometry and calculus, which belong to the higher mathematics.
2.1 数学、方程与比例 Mathematics, Equation and Ratio
New Words & Expressions:
algebra 代数学
geometrical 几何的
algebraic 代数的
identity 恒等式
arithmetic 算术, 算术的 measure 测量，测度
17世纪工业的快速发展推动了经济技术的进步， 从而遇到需 要处理变量的问题。从常量到变量的跳跃产生了两个新的数 学分支-----解析几何和微积分，他们都属于高等数学。
现在高等数学里面有很多分支，其中有数学分析，高等代数， 微分方程，函数论等。
Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.
1－A What is mathematics
回顾： 1. 如果没有运用数学， 任何一个科学技术分支都不可能
正常的发展 。 2. 符号在数学中起着非常重要的作用，它常用于表示概
念和命题。
1－B Equation
An equation is a statement of the equality between two equal numbers or number symbols.
Equations are of two kinds---- identities and equations of condition. An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation.
数学来源于人类的社会实践，比如工农业生产，商业活动， 军事行动和科学技术研究。
反过来，数学服务于实践，并在各个领域中起着非常重要的作 用。 没有应用数学，任何一个现在的科技的分支都不能正常 发展。
From the early need of man came the concepts of numbers and forms. Then, geometry developed out of problems of measuring land , and trigonometry came from problems of surveying. To deal with some more complex practical problems, man established and then solved equation with unknown
Now , since electronic computers are developed promptly and used widely, the role of computation becomes more and more important. In our times, computation is not only used to deal with a lot of information and data, but also to carry out some work that merely could be done earlier by logical deductions, for example, the proof of most of geometrical theorems. 数学结论主要由逻辑推理和计算得到。在数学发展历史的 很长时间内，逻辑推理一直占据着数学方法的中心地位。 现在，由于电子计算机的迅速发展和广泛使用，计算机的地 位越来越重要。现在计算机不仅用于处理大量的信息和数据， 还可以完成一些之前只能由逻辑推理来做的工作，例如，证 明大多数的几何定理。
Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition
term 项，术语
trigonometry 三角学
variable 变化的，变量
1－A What is mathematics
Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.
含有字母的恒等式对其中字母的任一组数值都成立。
一个等式若仅仅对其中一个字母的某些值成立，或对其中两个 或者多个字母的若干组相关的值成立，则它是一个条件等式， 简称方程。因此3x-5=7仅当x=4 时成立，而2x-y=0，当x=6,y=2 时成立，且对x, y的其他许多对值也成立。
A root of an equation is any number or number symbol which satisfies the equation. To obtain the root or roots of an equation is called solving an equation.
There are various kinds of equations. They are linear equation, quadratic equation, etc.
方程的根是满足方程的任意数或者数的符号。求方程根的过程 被称为解方程。
方程有很多种，例如： 线性方程，二次方程等。
To solve an equation means to find the value of the unknown term. To do this , we must, of course, change the terms about until the unknown term stands alone on one side of the equation, thus making it equal to something on the other side. We then obtain the value of the unknown and the answer to the question.
The conclusions in mathematics are obtained mainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathematics methods was occupied by the logical deductions.
等式是关于两个数或者数的符号相等的一种描述。
等式有两种－恒等式和条件等式。算术或者代数恒等式都是等 式。这种等式的两端要么一样，要么经过执行指定的运算后变 成一样。
An identity involving letters is true for any set of numerical values of the letters in it. An equation which is true only for certain values of a letter in it, or for certain sets of related values of two or more of its letters, is an equation of condition, or simply an equation. Thus 3x-5=7 is true for x=4 only; and 2x-y=10 is true for x=6 and y=2 and for many other pairs of values for x and y.
om 公理
numerical 数值的, 数字的
conception 概念，观点 operation 运算
constant 常数
postulate 公设
logical deduction 逻辑推理 proposition 命题
division 除，除法
subtraction 减，减法
formula 公式
numbers , thus algebra occurred.