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{{tex}}
{{Book
| author = {{w|伟烈亚力}}口译<br>{{w|李善兰}}笔述<br>{{w|伊莱亚斯·罗密士}}原著
第9行 ⟶ 第11行:
'''《代微积{{ruby|拾|shè}}级》''',由英国汉学家、来华传教士{{w|伟烈亚力}}(Alexander Wylie,1815年4月6日-1887年2月10日)口译,清代数学家{{w|李善兰}}(1811年1月22日-1882年12月9日)笔述;1859年由上海{{w|墨海书馆}}(The London Missionary Society Press)出版。原著为美国数学家{{w|伊莱亚斯·罗密士}}(Elias Loomis,1811年8月7日-1889年8月15日)于1851年出版的'''《解析几何和微积分初步》(''Elements of Analytical Geometry and of The Differential and Integral Calculus'')'''。
据考证,李善兰翻译所用的底本,应为1852年再版的版本,而非1851年的初版(内容差别较大)。原著在1874年改版,在此之前有不下20个版次(包括1856年的第6版、1859年的第10版、1864年的第17版、1868年的第19版等),畅销程度可见一斑。
1872年(明治5年),这本书被福田半译至日文,但书中并未使用李善兰创立的对应符号系统(参见[[#符号]]),而是使用原著的符号。
需要注意的是,这本书中的定义以今天的眼光来看并不严谨。一方面,当时的微积分学基础并不牢固,(ε, δ)-极限定义还未出现,书中仍然只使用未严格定义的无穷小量来定义概念;另一方面,这本书本身就是入门教材,不应过于晦涩难懂。
<gallery>
Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 1).
Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 2).
Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 3).
</gallery>
第27行 ⟶ 第33行:
* [[:File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 3).pdf|卷十七-卷十八(积分)]]([[Media:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 3).pdf|源文件]])
* [[:File:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|英文原著]]([[Media:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|源文件]])
== 原文 ==
* [[/卷一|卷一]]([[/卷一/英文|英文]])
== 符号 ==
第143行 ⟶ 第152行:
| 癸
! {{lang|el|Κ}}
| {{ruby|𠯆|
! {{lang|el|κ}}
| 女
第1,393行 ⟶ 第1,402行:
在卷十的开头,可以看到那句经典名句:
{{from|微分之數有二,一曰常數,一曰變數;變數以天地人物等字代之,常數以甲乙子丑等字代之。<br><br>凡式中常數之同數俱不變。如直線之式爲<math>地\xlongequal{\
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 2).pdf|page=4|300px]][[File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 2).pdf|page=3|300px]]</
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English=
原文如下:
{{from|1=A<span style{{=}}"font-size:
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|page=124|300px]][[File:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|page=125|300px]]</
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第1,413行 ⟶ 第1,421行:
同样是卷十,可以看到微分的定义:
{{from|'''函數與變數之變比例俱謂之微分''',用彳號記之。如 <math>戌\xlongequal{\
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 2).pdf|page=19|300px]][[File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 2).pdf|page=18|300px]]</
|-|
English=
原文如下:
{{from|1=(170.) '''The rate of variation of a function or of any variable quantity is called its ''differential''''', and is denoted by the letter ''d'' placed before it. Thus, if<br><div style="text-align: center;"><math>u=x^3</math>,</
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|page=131|300px]]</
注意,时至今日,微分号为了与量区别而记作<math>\mathrm{d}</math>,书中的微分号仍然是斜体<math>d</math>。
</tabber>
书中所称“Differential Coefficient”,译作“微系数”,今天称作'''导数(Derivative)'''。无论是中英文,这种古老的说法都已很罕见。
== 积分 ==
第1,431行 ⟶ 第1,442行:
在卷十七,定义了积分:
{{from|積分爲微分之還原,其法之要,在識別微分所由生之函數。如已得 天<sup>二</sup> 之微分爲 二天{{RareChar|𢓍|⿰彳天}},則有 二天{{RareChar|𢓍|⿰彳天}},即知所由生之函數爲 天<sup>二</sup>,而 天<sup>二</sup> 即爲積分。<br><br>'''已得微分所由生之函數爲積分。'''而積分或有常數附之,或無常數附之,既不能定,故式中恒附以常數,命爲{{RareChar|𠰳|⿰口丙}}。{{RareChar|𠰳|⿰口丙}}或有同數,或爲〇,須攷題乃知。<br><br>來本之視微分,若函數諸小較之一,諸小較并之,即成函數。故微分之左係一禾字,指欲取諸微分之積分也。如下式 禾二天{{RareChar|𢓍|⿰彳天}}<math>\xlongequal{\
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 3).pdf|page=4|300px]][[File:Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 3).pdf|page=3|300px]]</
|-|
English=
原文如下:
{{from|1=A<span style{{=}}"font-size: 70%;>RTICLE</span> (291.) T<span style{{=}}"font-size: 70%;>HE</span> Integral Calculus is the reverse of the Differential Calculus, its object being to determine the expression or function from which a given differential has been derived.<br><br>Thus we have found that the differential of ''x''<sup>2</sup> is 2''xdx'', therefore, if we have given 2''xdx'', we know that it must have been derived from ''x'', or plus a constant term.<br><br>(292.) '''The function from which the given differential has been derived, is called its ''integral''.''' Hence, as we are not certain whether the integral has a constant quantity or not added to it, we annex a constant quantity represented by C, the value of which is to be determined from the nature of the problem. <br><br>(293.) Leibnitz considered the differentials of functions as indefinitely small differences, and the sum of these indefinitely small differences he regarded as making up the function; hence the letter S was placed before the differential to show that the sum was to be taken. As it was frequently required to place S before a compound expression, it was elongated into the sign ''∫'', which, being placed before a differential, denotes that its integral is to be taken. Thus,<br><div style="text-align: center;"><math>\int 2xdx = x^2+C</math>.</
实际上,书中的积分号仅仅写作<span style="font-family:serif;">''∫''</span>,没有写得像今天的<math>\int</math>那么长。
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|page=228|300px]]</
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