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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).pdf|page=1|
Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 2).pdf|page=1|
Elements of Analytical Geometry and of The Differential and Integral Calculus (Chinese, 1859, 3).pdf|page=1|
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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>。
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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]]</
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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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