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1、第一段翻译(2)：what is the exact value of the number pai?a mathematician made an experiment in order to find his own estimation of the number pai.in his experiment,he used an old bicycle wheel of diameter 63.7cm.he marked the point on the tire where the wheel was touching the ground and he rolled the whee
2、l straight ahead by turning it 20 times.next,he measured the distance traveled by the wheel,which was 39.69 meters.he divided the number 3969 by 20*63.7 and obtained 3.115384615 as an approximation of the number pai.of course,this was just his estimate of the number pai and he was aware that it was
3、not very accurate.数的精确值是什么？一位数学家做了实验以便找到他自己对数的估计。在试验中，他用了一直径63.1厘米的旧自行车轮。他在车轮接触地面的轮胎上做了标记，而且将车轮向前转动20次。接下来，他测量了车轮经过的距离，是39.69米。他用3969除20*63.7得到了数的近似值3.115384615。当然，这只是对数的估计值，并且他也意识到不是很准确。第二段翻译(5)：one of the first articles which we included in the History Topics section archive was on the history of
4、pai.it is a very popular article and has prompted many to ask for a similar article about the number e.there is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one of pai.the number e is,compared to
5、pai,a relative newcomer on the mathematical scene.我们包括在“历史专题”部分档案中的第一篇文章就是历史上的，这是一篇很流行的文章，也促使许多人想了解下一些有关数e的类似文章。这两个数字的历史发展中有着很大的反差并且在许多方面写数e的历史是比写的历史更为艰巨的任务。与相比，数e在数学界相对较晚。第三段翻译(24)：the path to the development of the integral is a branching one,where similar discoveries were made simultaneously by d
6、ifferent people.the history of the technique that is currently known as integration began with attempts to find the area underneath curves.the foundations for the discovery of the integral were first laid by Cavalieri,an Italian Mathematician,in around 1635.Cavalieris work centered around the observ
7、ation that a curve can be considered to be sketched by a moving point and an area to be sketched by a moving line.积分发展的道路是一个分支，不同的人在同一时间作了类似的发现。目前众所周知的积分这一历史方法最初是为了求出曲线下方的面积。积分的的第一奠基人是Cavalieri（卡瓦列里），一位意大利数学家，时间大约为1635年。Cavalieri（卡瓦列里）的工作集中在观察，即一个曲线可以被视为是移动的点所勾勒且和面积由移动的线勾勒出。第四段翻译(35)：Pierre De Ferma
8、ts method for finding a tangent was developed during the 1630s,and though never rigorously formulated,is almost exactly the method used bu Newton and Leibniz.lacking a formal concept of a limit,Fermat was unable to properly justify his work.however,by examining his techniques,it is obvious that he u
9、nderstood precisely the method used in differentiation today. in order to understand Fermats mathod,it is first necessary to consider his technique for finding maxima.Fermats first documented problem in differentiation involved finding the maxima of an equation,and it is clearly this work that led t
10、o his technique for finding tangents.找到一个切线的Pierre De Ferma（皮埃尔德费马）方法发展于1630，尽管从来没有严格的规定，却几乎是被Newton（牛顿）和Leibniz（莱布尼茨）完全采用的方法。缺乏一个正式的概念限制，Fermat（费马）无法严格地证明他的工作是正确的。然而，通过查看他的技术，很显然，他准确地明白今天在微分中使用的方法。为了理解Fermat（费马）方法，首先要考虑的是他的方法是寻找最大值。Fermat（费马）第一个记录在微分的问题中涉及找到一个极大等式，很显然这项工作导致了他寻找切线的方法。第五段翻译(39)：the n
11、otation of Leibniz most closely resembles that which is used in modern calculus and his approach to discovering the inverse relationship between the integral and differential will be examined.though Newton independently arrived at the same conclusion,his path to discovery is slightly less accessible
12、 to the modern reader.Leibniz（莱布尼茨）符号最接近用于现代的微积分，他发现积分和微分之间的逆关系的方法也会被审查。虽然Newton（牛顿）独立地得出同样的结论，但他的发现途径略少接触到现代读者。第六段翻译(46)：both Torricelli and Barrow considered the problem of motion with variable speed.the derivative of the distance is velocity and the inverse operation takes one from the velocity t
13、o the distance.hence an awareness of the inverse of differentiation began to evolve naturally and the idea that integral and derivative were inverses to each other were familiar to Barrow.in fact,although Barrow never explicitly stated the fundamental theorem of the calculus,he was working towards t
14、he result and Newton was to continue with this direction and state the Fundamental Theorem of the Calculus explicitly.Torricelli（托里拆利） 和 Barrow（巴罗）都在考虑变速运动的问题。距离衍生出速度，逆运算就可以使得速度到距离的一个成为可能。因此微分的逆的意思开始自然演变，积分和微分是相互的逆的构想对于Barrow（巴罗）很熟悉了。事实上，虽然Barrow（巴罗）从未明确表示微积分基本定理，他一直在向着这个结果努力，Newton（牛顿）也在继续着这个方向并明确地
15、说明了微积分基本定理。第七段翻译(48)：for Newton integration consisted of finding fluents for a given fluxion so the fact that integration and differentiation were inverses was implied. Leibniz used integration as a sum,in a rather similar way to Cavalieri.he was also happy to useinfinitesimalsdx and dy where Newton
16、 used x and y which were finite velocities.of course neither Leibniz nor Newton thought in terms of functions,however,but both always thought in terms of graphs.for Newton the calculus was geometrical while Leibniz took it towards analysis.由于Newton（牛顿）积分包括对一个给定的流数找变数，因此这样一个事实即积分和微分是相互的逆就被暗示了。Leibniz（莱布尼茨）用一种与Cavalieri（卡瓦列里）相当类似的方式将积分视作为一个和。他还高兴地在Newton（牛顿）使用有限的速度X和Y 的地方使用“无穷的”dx和dy。当然， Leibniz（莱布尼茨）和Newton（牛顿）都不是在函数方面思考，然而，都常在图形方面思考。对于Newton（牛顿）微积分
