# 代微积拾级

1859，清咸丰九年（己未）
1851（原著）

——《代微積拾級》

《代微积shè级》，由英国汉学家、来华传教士伟烈亚力（Alexander Wylie，1815年4月6日－1887年2月10日）口译，清代数学家李善兰（1811年1月22日－1882年12月9日）笔述；1859年由上海墨海书馆（The London Missionary Society Press）出版。原著为美国数学家伊莱亚斯·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

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。

## 作者

[取自《清史稿·列傳二百九十四·疇人二》]

## 符号

### 字母

A a Α α F B 呷（jiǎ） 甲 唃（jiǎo） 角（jiǎo） 㖤（hán） 𠮙（yǐ） 乙 吭（kàng） 亢（kàng） 函 𠰳（bǐng） 丙 呧（dǐ） 氐（dǐ） 𭡝（hán） 叮 丁 房 涵 𱒐（wù） 戊 吣（xīn） 根 𠯇（jǐ） 己 𠳿（wěi） 尾 周[表注 1] 𱓒（gēng） 庚 𱕍（jī） 箕（jī） 訥（nè）（讷）[表注 2] 㖕（xīn） 辛 呌（dǒu） 斗（dǒu） 彳（wēi）[表注 3] 𠰃（rén） 壬 吽（níu） 牛 禾（jī）[表注 4] 𱓩（guǐ） 癸 𠯆（nǔ） 女 吇（zǐ） 子 嘘（xū） 虛 吜（choǔ） 丑 𠱓（wēi）[表注 5] 危 𠻤（yín） 寅 㗌（shì） 室 𠰭（mǎo） 卯 壁 㖘（chén） 辰 喹（kuí） 奎 𱒄（sì） 巳 嘍（lóu）（喽） 吘（wǔ） 午 喟（wèi）[表注 6] 胃 味 未 𭈾（mǎo） 昴（mǎo） 呻 申 嗶（bì）（哔） 畢（毕） 唒（yǒu） 酉 嘴（zī） 觜（zī） 㖅（xù） 戌 嘇（shēn）（𰇼） 咳（hài） 亥 𠯤（jǐn） 井 𭈘（wù） 物 𠺌（guǐ） 𱒆（tiān） 天 柳 哋（dì） 地 㕥（rén） 人

### 运算符及其他引进符号

LaTeX HTML 说明
$\displaystyle{ \bot }$[表注 7] [表注 8] 正也，加也[表注 9]
$\displaystyle{ \top }$[表注 10] [表注 11] 負也，減也[表注 12]
$\displaystyle{ \times }$[表注 13] ×[表注 14] 相乘也
$\displaystyle{ \div }$[表注 15] ÷[表注 16] 約也，或作$\displaystyle{ - }$
$\displaystyle{ :::: }$ :::: 四率比例也
$\displaystyle{ () }$ () 括諸數為一數也，名曰括弧
$\displaystyle{ \sqrt{} }$[表注 17] 開方根也
$\displaystyle{ = }$ = 等於[表注 18]
$\displaystyle{ \lt }$ < 右大於左
$\displaystyle{ \gt }$ > 左大於右
$\displaystyle{ 0 }$ 0 無也
$\displaystyle{ \infty }$[表注 19] 無窮也

### 注释

1. 圆周率
2. 自然常数，疑似应为e。
3. 音同“微”，指微分号。
4. 音同“积”，指积分号。
5. 书中表格与Ρ颠倒。
6. 书中表格与Μ颠倒。
7. \bot
8. U+22A5，&bottom;&bot;&perp;&UpTee;&#8869;
9. 此处并非垂直符号$\displaystyle{ \perp }$\perp）或⟂（U+27C2，&#10178;）。不使用加号+（U+002B，&#plus;&#43;）是因为易与汉字数字“十”混淆。此为“上”的古字（丄，U+4E04，&#19972;）。
10. \top
11. U+22A4，&top;&DownTee;&#8868;
12. 不使用减号−（U+2212，&minus;&#8722;）或连字暨减号-（U+002D，&#45;）是因为易与汉字数字“一”混淆。此为“下”的古字（丅，U+4E05，&#19973; ）。
13. \times
14. U+00D7，&times;&#215;
15. \div
16. U+00F7，&divide;&#247;
17. \sqrt{}
18. 与原符号不同，书中为防止与汉字数字“二”混淆而被拉长。
19. \infty

## 函数

[取自《代微積拾級·卷十 微分一·例》]

ARTICLE (157.) IN the Differential Calculus, as in Analytical Geometry, there are two classes of quantities to be considered, viz., variables and constants.

Variable quantities are generally represented by the last letters of the alphabet, x, y, z, etc., and any values may be assigned to them which will satisfy the equations into which they enter.

Constant quantities are generally represented by the firstletters of the alphabet, a, b, c, etc., and these always retain the same values throughout the same investigation.

Thus, in the equation of a straight line,
$\displaystyle{ y=ax+b }$,
the quantities a and b have but one value for the same line, while x and y vary in value for every point of the line.

(158.) One variable is said to be a function of another variable, when the first is equal to a certain algebraic expression containing the second. Thus, in the equation of a straight line
$\displaystyle{ y=ax+b }$,
y is a function of x.

So, also, in the equation of a circle,
$\displaystyle{ y=\sqrt{R^2-x^2} }$;
and in the equation of the ellipse,
$\displaystyle{ y=\frac{B}{A}\sqrt{2Ax-x^2} }$.
(159.) When we wish merely to denote that y is dependent upon x for its value, without giving the particular expression which shows the value of x, we employ the notation
$\displaystyle{ y=F(x) }$, or $\displaystyle{ y=f(x) }$,
or$\displaystyle{ x=F(y) }$, or $\displaystyle{ x=f(y) }$,
which expressions are read, y is a function of x, or x is a function of y.

[取自Elements of Analytical Geometry and of The Differential and Integral Calculus, Differential Calculus, Section I, Definitions and First Principles]

“凡此变数中函彼变数，则此为彼之函数”（One variable is said to be a function of another variable, when the first is equal to a certain algebraic expression containing the second.）这句话，往往被讹传为出现在李善兰另一译作《代数学》中，其实并非如此，出处就是《代微积拾级》。

## 微分

[取自《代微積拾級·卷十 微分一·論函數微分·第二款》]

(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
$\displaystyle{ u=x^3 }$,
then
$\displaystyle{ dx:du::1:3x^2 }$.
The expressions dx, du are read differential of x, differential of u, and denote the rates of variation of x and u.

If we multiply together the extremes and the means of the preceding proportion, we have
$\displaystyle{ du=3x^2dx }$,
which signifies that the rate of increase of the function u is 3x2 times that of the variable x.

If we divide each member of the last equation by dx, we have
$\displaystyle{ \frac{du}{dx}=3x^2 }$,
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.

[取自Elements of Analytical Geometry and of The Differential and Integral Calculus, Differential Calculus, Section I, Differentiation of Algebraic Functions, Proposition II － Theorem]

## 积分

[取自《代微積拾級·卷十七 積分一·總論》]

ARTICLE (291.) THE 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.

Thus we have found that the differential of x2 is 2xdx, therefore, if we have given 2xdx, we know that it must have been derived from x, or plus a constant term.

(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.

(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,
$\displaystyle{ \int 2xdx = x^2+C }$.
This sign is still retained ever by those whu reject the philosophy of Leibnitz.

[取自Elements of Analytical Geometry and of The Differential and Integral Calculus, Integral Calculus, Section I, Integration of Monomial Differentials]