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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1,399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]]</div>
第1,405行 ⟶ 第1,408行:
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]]</div>
第1,419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]]</div>
|-|
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>then<br><div style="text-align: center;"><math>dx:du::1:3x^2</math>.</div>The expressions ''dx'', ''du'' are read differential of ''x'', differential of ''u'', and denote the rates of variation of ''x'' and ''u''.<br><br>If we multiply together the extremes and the means of the preceding proportion, we have<br><div style="text-align: center;"><math>du=3x^2dx</math>,</div>which signifies that the rate of increase of the function ''u'' is 3''x''<sup>2</sup> times that of the variable ''x''.<br><br>If we divide each member of the last equation by ''dx'', we have<br><div style="text-align: center;"><math>\frac{du}{dx}=3x^2</math>,</div>which expresses the ratio of the rate of variation of the function to that of the independent variable, and is called the ''differential coefficient'' of ''u'' regarded as a function of ''x''.|2=''Elements of Analytical Geometry and of The Differential and Integral Calculus'', Differential Calculus, Section I, Differentiation of Algebraic Functions, Proposition II - Theorem}}
<div style="text-align: center;">[[File:Elements of Analytical Geometry and of The Differential and Integral Calculus.pdf|page=131|300px]]</div>
第1,440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]]</div>
|-|
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>.</div>This sign ''∫'' is still retained ever by those whu reject the philosophy of Leibnitz. |2=''Elements of Analytical Geometry and of The Differential and Integral Calculus'', Integral Calculus, Section I, Integration of Monomial Differentials}}
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。
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