代数表示法 | 从文字题到符号化的表示法演化简史

注：本文为 “代数表示法 ” 相关合辑。
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From Word Problems to Symbolism: A Brief History of Algebra

从文字题到符号化：代数学简史

Kangping Cheng, Richard A. Kilburn, and Steven D. Wallace, Middle Georgia State University
康平·程（Kangping Cheng）、理查德·A·基尔伯恩（Richard A. Kilburn）、史蒂文·D·华莱士（Steven D. Wallace），中乔治亚州立大学（Middle Georgia State University）

The authors describe some of the history of algebraic ideas and notation and discuss the relative newness of the transition in mathematics from rhetorical to syncopated to symbolic representations.
作者阐述了代数思想与符号表示的部分发展历史，并探讨了数学领域中从文字表述、简记表述到符号表述这一转变的相对新颖性。

It is common for math teachers to hear students complaining about symbolic manipulation; however, those complaints are minimal when compared to the uproar created by word problems. But there was once a time when all that existed was word problems. The concept or field of algebra was not always what we see now. The development of the symbolism used in mathematical expressions today did not arrive instantly. Algebra evolved through three different stages: Rhetorical, Syncopated, and Symbolic.
数学教师经常听到学生抱怨符号运算，但与文字题引发的强烈不满相比，这类抱怨微不足道。然而，曾有一段时期，数学中只存在文字题。代数这一概念或学科并非一直如我们现在所见的模样。如今数学表达式中使用的符号体系并非一蹴而就，代数学的发展经历了三个不同阶段：文字表述阶段（Rhetorical）、简记表述阶段（Syncopated）和符号表述阶段（Symbolic）。

Rhetorical Mathematics

文字表述阶段的数学

The debate around the definition of rhetoric is nuanced and has been ongoing for a very long time. A consensus dating from the time of Plato is that rhetoric involves persuasive arguments (Kronman, 1998). While many may not consider mathematical works as rhetoric, “It seems clear that… arithmeticians (more generally, mathematicians) practice an art of persuasion” (Kronman, 1998, p. 67). Regardless of whether mathematics can claim to belong to the realm of rhetoric, the term has been applied to the word problem style origins of algebra.
关于“修辞（rhetoric）”定义的争论错综复杂，且已持续很长时间。自柏拉图时代起，人们就普遍认为修辞涉及具有说服力的论证（Kronman，1998）。尽管许多人可能不认为数学著作属于修辞范畴，但“显然……算术家（更广泛地说，数学家）也在运用一种说服的艺术”（Kronman，1998，第67页）。无论数学是否能被归为修辞领域，“文字表述（rhetorical）”一词已被用于描述代数起源时的文字题形式。

Rhetorical algebra was first developed by the ancient Babylonians. In this stage equations were written in complete sentences like “The thing plus one equals three.” Thus, algebra originated in word problems. The Babylonians developed a positional number system that helped in solving their rhetorical algebraic equations. The ancient Egyptians likewise dealt with linear equations such as those found in the Rhind Papyrus written in 1650 BC (Boyer, 1991). The Rhind Papyrus contains rhetorical representations of linear equations that today would be represented with the following symbology: $x+ax=b$ and $x+ax+bx=c$ where $a$, $b$, and $c$ are known constants and $x$ is the unknown, which is referred to as “aha” or heap (Boyer, 1991).
文字表述阶段的代数最早由古巴比伦人创立。在这一阶段，方程以完整句子的形式呈现，例如“某数加一等于三”。因此，代数起源于文字题。巴比伦人发明了位值制记数法，这一记数法为求解文字表述的代数方程提供了帮助。古埃及人同样会处理线性方程，例如公元前1650年的《林德纸草书》（Rhind Papyrus）中就记载了这类方程（Boyer，1991）。《林德纸草书》中以文字形式表述的线性方程，在如今可通过以下符号表示：$x+ax=b$ 和 $x+ax+bx=c$，其中 $a$、$b$、$c$ 为已知常数，$x$ 为未知数，在当时被称为“aha”（意为“堆”）（Boyer，1991）。

Written around 250 BC, the book Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art (Boyer, 1991) is considered the most influential math book of all time in China. It has 246 math problems throughout its nine chapters. Chapter 8 deals with solving systems of linear equations. For example, the first question in this chapter, written in modern notation, is:
约公元前250年成书的《九章算术》（Chiu-chang suan-shu 或 The Nine Chapters on the Mathematical Art）（Boyer，1991），被认为是中国历史上最具影响力的数学著作。全书共九章，包含246道数学题。其中第八章涉及线性方程组的求解。例如，该章的第一道题用现代符号表示如下：

$$\{\begin{array}{ll}3x+2y+z=39& (1)\\ 2x+3y+z=34& (2)\\ x+2y+3z=26& (3)\end{array}$$

According to Shen et al. (1999), the original word problem states:
根据沈康身等人（Shen et al.，1999）的研究，该题的原始文字表述为：

Now given 3 bundles of top grade paddy, 2 bundles of medium grade paddy, [and] 1 bundle of low grade paddy. Yield: 39 dou of grain. 2 bundles of top grade paddy, 3 bundles of medium grade paddy, [and] 1 bundle of low grade paddy, yield 34 dou. 1 bundle of top grade paddy, 2 bundles of medium grade paddy, [and] 3 bundles of low grade paddy, yield 26 dou. Tell: how much [dou] does one bundle of each grade yield? (p. 399)
今有上禾三秉，中禾二秉，下禾一秉，实三十九斗；上禾二秉，中禾三秉，下禾一秉，实三十四斗；上禾一秉，中禾二秉，下禾三秉，实二十六斗。问：上、中、下禾实一秉各几何？（第399页）

Notice the complexity of this word problem. Imagine giving such a question to students today. Ultimately, the reader is asked to solve a system of three linear equations with three unknowns; however, without the modern use of symbolism the author introduced some symbolic mathematics as a way of finding the solution, arranging the coefficients of the unknowns and the constants from the right-hand side in a matrix like the following.
不难发现这道文字题的复杂性，试想将其交给如今的学生作答。本质上，该题要求求解一个三元一次线性方程组；但由于当时没有现代符号体系，作者采用了一种初步的符号化方法来求解——将未知数的系数与等号右侧的常数项整理成如下形式的矩阵（注：原文此处表述为“matrix like”，结合《九章算术》背景，实际为“方程术”中的“方阵”）。

$$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 2\\ 3& 1& 1\\ 26& 34& 39\end{array}\right]$$

The first column represents the coefficients and constant from equation (3); the second and third columns represent the coefficients and constants from equation (2) and (1), respectively. The method explained in the book to solve the system is essentially what today is taught as the elimination method for solving systems of linear equations.
第一列对应方程（3）的系数与常数项；第二列和第三列分别对应方程（2）和方程（1）的系数与常数项。书中阐述的求解方法，本质上与如今教授的“线性方程组消元法”一致。

Syncopated Mathematics

简记表述阶段的数学

Syncopated algebraic expression was first introduced in Diophantus’ book Arithmetica in the third century. This is a first step away from problems presented in rhetorical form, but it lacks all the characteristics of symbolic algebra. Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations (Boyer, 1991). The main difference between Diophantine syncopated algebra and modern algebraic notation is that Diophantine algebra did not have special symbols for operations, relations, or exponentials. For example, if an equation in modern algebra is written as $x^2+2x-1=10$, it would be written in Diophantus’ syncopated notation as ${\Delta }^v\underline{\alpha }\zeta \underline{\beta }\text{ m }M\underline{\alpha }\text{ io }M\underline{\underline{\iota }}$. The symbol representations are given in Table 1 (Derbyshire, 2006). Different from modern notation, the coefficients come after the variables and addition is represented by the group of terms, as in $(x^2\cdot 1+x\cdot 2)-x^0\cdot 1=x^0\cdot 10$.
简记表述形式的代数表达式最早出现在公元3世纪丢番图（Diophantus）的著作《算术》（Arithmetica）中。这是数学题脱离纯文字表述的第一步，但尚未具备符号代数的全部特征。丢番图首次使用符号表示未知数，并对数的幂、数量关系及运算采用缩写形式（Boyer，1991）。丢番图简记代数与现代代数符号的主要区别在于，前者没有专门表示运算、关系或指数的符号。例如，现代代数中的方程 $x^2+2x-1=10$，在丢番图的简记符号体系中会写作 ${\Delta }^v\underline{\alpha }\zeta \underline{\beta }\text{ m }M\underline{\alpha }\text{ io }M\underline{\underline{\iota }}$。各符号的含义如表1所示（Derbyshire，2006）。与现代符号不同的是，丢番图符号中系数位于变量之后，加法通过项的组合来表示，例如上述方程可理解为 $(x^2\cdot 1+x\cdot 2)-x^0\cdot 1=x^0\cdot 10$。

Table 1. Diophantine Syncopated Symbols

表1 丢番图简记符号

Symbol	What it means
$\underline{\alpha }$	1
$\underline{\beta }$	2
$\underline{\iota }$	10
𝚤̈𝜎	equals 等于
⋔	subtraction of everything following up to 𝚤̈𝜎 表示减去紧跟其后直至“𝚤̈𝜎”的所有项
M	the zeroth power of the unknown quantity 未知数的零次幂
$\zeta$	the unknown quantity 未知数
${\Delta }^{\nu }$	The second power of the unknown quantity 未知数的二次幂

Diophantus is often called “the father of algebra” for two reasons: first because he contributed greatly to number theory and mathematical notation; second, his book Arithmetica contains the earliest known use of syncopated notation. Ironically, Diophantus is most famous for a puzzle written about him around 500 AD by Metrodorus (Petkovic, 2009). The poem describes a few events in Diophantus’ life and illustrates the limitations of mathematics done by rhetoric. The poem can be translated as:
丢番图常被称为“代数之父”，原因有二：一是他对数论和数学符号体系的发展贡献卓著；二是他的著作《算术》中记载了目前已知最早的简记符号表述。颇具讽刺意味的是，丢番图最广为人知的却是公元500年左右梅特罗多鲁斯（Metrodorus）为他创作的一道谜题（Petkovic，2009）。这道谜题以诗歌形式描述了丢番图的部分生平，同时也体现了纯文字表述数学的局限性。诗歌译文如下：

Here lies Diophantus," the wonder behold…Through art algebraic, the stone tells how old:
God gave him his boyhood one-sixth of his life, One-twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun; In five years there came a bouncing new son.
Alas, this dear child of master and sage, Attained only half of his father’s full age.
When chill fate took him - an event full of tears -Heartbroken, his father lived just four more years.
How long did Diophantus live? (PurpleMath, 2023)
“丢番图长眠于此，见证奇迹……石碑以代数之术，诉说他的寿命：
上帝赐予他生命的六分之一为童年，十二分之一为青年，彼时胡须渐生；
又过了生命的七分之一，他才步入婚姻殿堂；五年后，可爱的儿子降临。
可惜这位贤哲之子，仅活到父亲寿命的一半，便被无情命运带走——令人悲痛不已；
儿子离世后，悲痛欲绝的父亲又活了四年。
请问丢番图活了多少岁？”（PurpleMath，2023）

To many beginning algebra students, this word problem would be overwhelming. It is a perfect demonstration of the need for something beyond rhetorical representations of mathematics to which Diophantus worked.
对许多初学代数的学生而言，这道文字题难度极大。它充分表明，数学需要突破纯文字表述的局限，而这正是丢番图所探索的方向。

Some historians of mathematics credit the title of “the father of algebra” to the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī. He provided an exhaustive explanation for the algebraic solution of quadratic equations with positive roots and was the first to teach algebra in an elementary form (Gandz, 1936). The word “algebra” comes from the Arabic word al-jabr which is written in the book The Compendious Book on Calculation by Completion and Balancing and translated from the Arabic title, Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala (Boyer, 1991, p.228). The book was written in the year 830 by the medieval Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī. The word “al-jabr” means restoration or completion, and it refers to the transposition of subtracted terms to the other side of an equation. The other word “muqabalah” means reduction or balancing which is the cancellation of like terms on opposite sides of the equation.
部分数学史学家将“代数之父”的称号归于波斯数学家穆罕默德·伊本·穆萨·花拉子米（Muhammad ibn Mūsā al-Khwārizmī）。他系统阐述了正根二次方程的代数解法，并首次以基础形式教授代数学（Gandz，1936）。“代数（algebra）”一词源自阿拉伯语“al-jabr”，该词出自花拉子米于公元830年撰写的著作《还原与对消计算概要》（The Compendious Book on Calculation by Completion and Balancing），其阿拉伯语原书名是 Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala（Boyer，1991，第228页）。“al-jabr”意为“还原”或“补全”，指将方程中被减去的项移到等号另一侧；“muqabalah”意为“对消”或“平衡”，指消去等号两侧的同类项。

Brahmagupta, a seventh century Indian mathematician, wrote the book Brahma Sphuta Siddhanta (Boyer, 1991). In this book, syncopated mathematics again appears as he solved the general quadratic equation for both positive and negative roots. Brahmagupta was the first to give a general solution to the linear Diophantine equation $ax+by=c$, where $a$, $b$, and $c$ are integers. Different from Diophantus, who only gave one solution, Brahmagupta gave all integer solutions (Boyer, 1991). In Brahmagupta’s work, addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend. Multiplication, exponentiation, and unknown quantities were represented by abbreviations of appropriate terms (Boyer, 1991).
公元7世纪的印度数学家婆罗摩笈多（Brahmagupta）撰写了《婆罗摩笈多悉檀多》（Brahma Sphuta Siddhanta）（Boyer，1991）。在该书中，简记表述形式的数学再次出现，婆罗摩笈多求解了一般二次方程，并得到了正根和负根。他还首次给出了线性丢番图方程 $ax+by=c$（其中 $a$、$b$、$c$ 为整数）的通解。与仅给出一个解的丢番图不同，婆罗摩笈多给出了该方程的所有整数解（Boyer，1991）。在婆罗摩笈多的著作中，加法通过数字并列表示，减法通过在减数上方加一个点表示，除法通过将除数置于被除数下方表示；乘法、乘方及未知数则通过相关术语的缩写来表示（Boyer，1991）。

Symbolic Mathematics

符号表述阶段的数学

Symbolic algebra is the stage in which full symbolism is used in a mathematical expression. Early steps toward this symbolism can be seen in the work of several Islamic mathematicians such as Ibn al-Banna around the thirteenth to fourteenth centuries and al-Qalasadi in the fifteenth century. But sixteenth century French mathematician François Viète, who has been called the “inventeur de l’algèbre modern,” is attributed with the invention of the symbolic algebraic notation that we use today (Ritter, 1895). Modern day high school students would recognize many of the equations he wrote in his 1591 book In Artem Analyticem Isagoge (translated as An Introduction to Analytical Arts). Viète published other works in the fields of trigonometry and number theory using modern notation, such as the trigonometric formula $\sin (a)\sin (b)=\frac{1}{2}(\cos (a-b)-\cos (a+b))$ (Ritter, 1895). The notation that we use today for addition, subtraction, and division are derived from Viète’s text (Ritter, 1895).
符号代数阶段的核心是在数学表达式中全面使用符号。早在13-14世纪的伊本·班纳（Ibn al-Banna）和15世纪的卡拉萨迪（al-Qalasadi）等伊斯兰数学家的著作中，就已出现符号化的初步尝试。但如今我们使用的符号代数体系，通常认为是由16世纪法国数学家弗朗索瓦·韦达（François Viète）创立的，他被誉为“现代代数发明者（inventeur de l’algèbre modern）”（Ritter，1895）。现代高中生能看懂他在1591年出版的《分析术入门》（In Artem Analyticem Isagoge，英译本名为 An Introduction to Analytical Arts）中写下的许多方程。韦达还在三角学和数论领域的其他著作中使用了现代风格的符号，例如三角恒等式 $\sin (a)\sin (b)=\frac{1}{2}(\cos (a-b)-\cos (a+b))$（Ritter，1895）。我们如今使用的加减除符号，也源自韦达的著作（Ritter，1895）。

It is evident that Viète represented the historical link between the syncopated and the symbolic forms of algebraic expression when we consider his representations for exponents and subscripts. His convention was to use capitalized letters with consonants for known values and vowels for the variables (Ritter, 1895). For example, he developed the rhetorical expressions of A quadratus, A cubus, and A quadrato-cubus, that he represented in the syncopated short-hand as Aq, Ac, and Aqc (Ritter, 1895). Viète’s conventions paved the way for the superscript notation for these, $A^2$, $A^3$, and $A^{2+3}=A^5$. On the other hand, B planum, B solidum, and B plano-solidum, also written as Bp, Bs, and Bps represent a series of values with subscripts $B_2$, $B_3$, and $B_5$. Indeed, he also devised syncopated shorthand for the radical notation we use today. For instance, \sqrt{\quad} ​, $\sqrt[3]{}$, and $\sqrt[5]{}$, were written as Rq, Rc, and Rqc, respectively (Ritter, 1895).
从韦达对指数和下标的表示方式中可明显看出，他是代数表达式从简记形式向符号形式过渡的关键人物。他的符号规则是：用大写辅音字母表示已知量，用大写元音字母表示变量（Ritter，1895）。例如，他将文字表述的“A 的平方（A quadratus）”“A 的立方（A cubus）”“A 的平方乘立方（A quadrato-cubus）”，用简记形式写作 Aq、Ac、Aqc（Ritter，1895）。韦达的这一规则为后来的上标符号奠定了基础，即演变为如今的 $A^2$、$A^3$ 和 $A^{2+3}=A^5$。此外，他用“B 的平面（B planum）”“B 的立体（B solidum）”“B 的平面乘立体（B plano-solidum）”，简记为 Bp、Bs、Bps，分别表示带有下标的一系列量 $B_2$、$B_3$、$B_5$。同时，他还为如今的根式符号设计了简记形式，例如将 \sqrt{\quad} ​（平方根）、$\sqrt[3]{}$（立方根）、$\sqrt[5]{}$（五次方根）分别写作 Rq、Rc、Rqc（Ritter，1895）。

It cannot be overlooked that the all-important equals sign, consisting of two parallel line segments, was introduced decades before Viète’s Analytical Arts by a Welsh mathematician and scientist named Robert Recorde. Recorde was a prolific writer in many areas including medicine, astronomy, and mathematics. At the time, most mathematics books were written in Latin, but Recorde marketed his texts for the British commoner who could only read in English. In his 1557 book, The Whetstone of Witte, Recorde argued that tedious repetition of the phrase “is equalle to” would be better conveyed using “a paire of paralleles” (Inglis-Arkell, 2017). He thought that “noe 2 thyngs, can be moare equalle” than a pair of parallel line segments (Inglis-Arkell, 2017, para. 5).
不容忽视的是，至关重要的“等号（=）”——由两条平行线构成——是在韦达《分析术入门》出版前数十年，由威尔士数学家、科学家罗伯特·雷科德（Robert Recorde）发明的。雷科德著作颇丰，涉及医学、天文学和数学等多个领域。当时大多数数学书籍以拉丁文撰写，而雷科德则面向仅能读懂英文的英国平民创作数学读物。在他1557年出版的《智力磨石》（The Whetstone of Witte）中，雷科德提出，反复使用“is equalle to（等于）”这一短语过于繁琐，用“一对平行线（a paire of paralleles）”来表示会更简洁（Inglis-Arkell，2017）。他认为，“没有任何两样东西比一对平行线更相等（noe 2 thyngs, can be moare equalle）”（Inglis-Arkell，2017，第5段）。

Even without the use of Recorde’s equal sign, Viète’s innovations for the formatting of equations allowed for a seamless integration of mathematical sentences into the prose of his writing while maintaining a separation between the two environments that a reader can easily follow. However, it was several decades later, in 1637, that René Decartes published La Géométrie, thereby perfecting and popularizing Viète’s work into the modern variable conventions for algebraic notation we recognize today (Watson, 2023). For instance, Descartes used capital letters for mathematical objects like points or line segments, reserving lowercase letters at the beginning of the alphabet for known numerical quantities, and, finally, letters at the end of the alphabet for unknown variables in equations (Watson, 2023). It was Descartes who represented powers of those variables as superscripts, often displaying polynomial equations on separate lines with like degree terms lined up vertically, making calculations easier to follow (Descartes, 1954/1637). The evolution from the syncopated to the symbolic is evident in Table 2, which juxtaposes the same equation using Viète’s and Descartes’ notation along with its modern translation. Notice that, in place of the independently developed equal sign, Viète and Decartes, instead use an abbreviation for the Latin word aequalis.
即便未使用雷科德发明的等号，韦达对方程格式的创新也实现了数学表达式与散文叙述的无缝融合，同时保持了两者的区分度，便于读者理解。然而，直到数十年后的1637年，勒内·笛卡尔（René Descartes）出版《几何学》（La Géométrie），才完善并推广了韦达的符号体系，形成了如今我们所熟知的现代代数变量规则（Watson，2023）。例如，笛卡尔用大写字母表示点、线段等数学对象；用字母表前半部分的小写字母表示已知数值；用字母表后半部分的小写字母表示方程中的未知数（Watson，2023）。正是笛卡尔首次用上标表示变量的幂次，并且常将多项式方程分行书写，让同次项垂直对齐，使计算过程更清晰易懂（Descartes，1954/1637）。表2对比了同一方程在韦达符号、笛卡尔符号与现代符号下的表示形式，从中可清晰看出代数从简记形式到符号形式的演变。需注意的是，韦达和笛卡尔均未使用独立发明的等号，而是采用拉丁语“aequalis（等于）”的缩写来表示相等关系。

Table 2. The Evolution of Symbolic Notation

表2 符号表示的演变

Viète Notation 韦达	$B\text{ in }{A}_c-D\text{ in }{A}_q+5AӕqC{A}_{qq}$
Descartes Notation 笛卡尔	$b{x}^3--dxx+5xᴔc{x}^4$
Modern Notation 现代	$b{x}^3-d{x}^2+5x=c{x}^4$

（注：表中“ӕq”为拉丁语“aequalis”的缩写，意为“等于”；韦达符号中“in”表示乘法，“$A_c$”“$A_q$”“$A_{qq}$”分别对应 “$A^3$” “$A^2$” “$A^4$”，笛卡尔符号在此基础上简化了幂次表示，现代符号进一步统一为上标形式并使用“=”表示相等。）

Another significant algebraic contribution to mathematics was Descartes’s development of the coordinate system that carries his name (Descartes, 1954/1637). The Cartesian coordinate system, with which every student who has ever graphed using $x$ and $y$ axes is familiar, was introduced in the early to mid-1600s. The common story is that while recovering from an illness, Descartes found himself watching a fly move about on the ceiling of his room. Wondering how to describe the movements to others, he realized that measuring the fly’s distance from two perpendicular walls would provide exact locations (Dobler & Klein, 2002). This development, to the chagrin of students ever since, allowed for the graphical representations of algebraic equations and paved the way for modern mathematics.
笛卡尔对数学的另一重大代数贡献，是创立了以他命名的坐标系（笛卡尔坐标系）（Descartes，1954/1637）。17世纪早中期出现的笛卡尔坐标系，是每个使用过 $x$ 轴和 $y$ 轴绘图的学生都熟悉的工具。广为流传的说法是，笛卡尔在一次生病休养期间，观察到一只苍蝇在天花板上移动，他思考如何向他人描述苍蝇的位置，进而意识到：测量苍蝇到房间内两面垂直墙壁的距离，即可确定其准确位置（Dobler & Klein，2002）。尽管这一发明让此后的学生们感到头疼，但它实现了代数方程的图形化表示，为现代数学的发展奠定了基础。

Historical Perspective

历史视角

To investigate why the mathematics and notations developed by people like Diophantus and Brahmagupta and were lost for much of time while those of Viète, Recorde, and Descartes have survived, it is useful to juxtapose the timeline of mathematics with the timeline of world history. What happened during the 1400s and 1500s that would permit the writings of these mathematicians to be remembered?
为何丢番图、婆罗摩笈多等人发展的数学理论与符号体系在很长一段时间内被遗忘，而韦达、雷科德、笛卡尔等人的成果却得以流传？将数学发展时间线与世界历史时间线并置分析，有助于解答这一问题。15至16世纪发生了什么，使得这些数学家的著作能被后人铭记？

The sharing of information between cities, languages, and cultures was extremely difficult for most of human history. However, a singular invention in the middle of the fifteenth century changed that. Some scholars suggest that the printing press in the mid 1400’s was one of the most significant inventions in human history (Ditmar, 2011). In fact, the proliferation of the printing press during the latter half of the fifteenth century, “transform[ed] the ways ideas were disseminated and the conditions of intellectual work” (Ditmar, 2011, p. 1133).
在人类历史的大部分时期，跨城市、跨语言、跨文化的信息传播极为困难。然而，15世纪中期的一项重大发明改变了这一局面。部分学者认为，15世纪中期出现的印刷术是人类历史上最重要的发明之一（Ditmar，2011）。事实上，15世纪后半叶印刷术的普及，“彻底改变了思想传播的方式和学术研究的条件”（Ditmar，2011，第1133页）。

Historically, this timeframe is very recent. The 1500s, as books were becoming cheaper and more available, saw the birth of Shakespeare and the unification of Scotland and England under a single monarch. In fact, Romeo and Juliet was published one year after Rene Descartes was born and only ten years before the founding of the Jamestown, Virginia in 1607. This means that no one watching the star-crossed lovers on stage had ever plotted a point on a plane or seen variable exponents as we know them. Harvard was founded only a few decades after the publication of Descartes’ work explaining these things. It is a great trivia question to ask why Harvard did not teach calculus when it was founded. This fact surprises many people but is easy to explain when one considers that calculus did not exist until 29 years after Harvard began.
从历史尺度来看，这一时期距今并不久远。16世纪，随着书籍价格降低、获取渠道增多，莎士比亚（Shakespeare）诞生，苏格兰与英格兰也实现了君主统一。事实上，《罗密欧与朱丽叶》（Romeo and Juliet）的出版时间，比勒内·笛卡尔出生晚一年，比1607年弗吉尼亚州詹姆斯敦（Jamestown）殖民地建立仅早十年。这意味着，当年观看这部爱情悲剧舞台演出的观众，从未在平面上绘制过点，也从未见过我们如今熟知的变量幂次符号。哈佛大学（Harvard）的创立时间，也仅比笛卡尔阐述这些符号体系的著作出版时间晚数十年。有一个有趣的 trivia 问题：为何哈佛大学创立初期不教授微积分？许多人对这一事实感到惊讶，但原因很简单——微积分在哈佛创立29年后才被发明出来。

Mathematics is both old and new. Many ideas existed in ancient times, but the sharing and building upon those ideas is relatively recent history. Concepts that are taught in every high school algebra classroom today were never known to figures of the past. Christopher Columbus never graphed a line. Leonardi DaVinci never factored a quadratic trinomial. Michelangelo never solved anything for $x$. Each of these individuals solved similar mathematical problems, but they did not have the assistance of modern mathematical symbolism in understanding or solving these problems. The concepts may be old, but the symbolic representations we use today did not exist until a short time ago. Without modern mathematical conventions for symbols and notation, we would be stuck in a world of nothing but word problems.
数学既有悠久的历史，又充满现代活力。许多数学思想在古代就已存在，但这些思想的传播与进一步发展，却是相对较近的历史。如今每个高中代数课堂上教授的概念，对过去的历史人物而言都是陌生的。克里斯托弗·哥伦布（Christopher Columbus）从未绘制过直线，莱昂纳多·达·芬奇（Leonardi DaVinci）从未分解过二次三项式，米开朗基罗（Michelangelo）从未求解过关于 $x$ 的方程。这些人物都曾解决过类似的数学问题，但他们在理解和求解过程中，并未借助现代数学符号体系。数学概念或许古老，但我们如今使用的符号表示形式，直到近代才出现。若没有现代数学符号与表示规则，我们仍会被困在只有文字题的数学世界中。

References

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The Origin of the Word Algebra: A Linguistic Exploration

“代数”一词的起源：语言学探索

2023-09-27 00:00:00

Have you ever wondered about the origin of the word algebra? Algebra is a branch of mathematics that deals with variables, equations, and operations. But how did it get its name?
你是否曾经好奇“algebra”（代数）一词的起源？代数是数学的一个分支，研究变量、方程和运算。但这个术语是如何得名的呢？

In this article, we’ll take a linguistic exploration into the origin of the word algebra. We’ll delve into its etymology and uncover the historical roots of this mathematical term. So let’s dive in and discover the fascinating story behind the word algebra!
在本文中，我们将从语言学角度探索“algebra”（代数）一词的起源。我们会深入研究它的词源，挖掘这个数学术语的历史根源。现在就让我们深入其中，揭开“algebra”（代数）一词背后引人入胜的故事！

What is the Meaning of the Word “Algebra”?

“Algebra”（代数）一词的含义是什么？

The word “algebra” has its roots in Arabic and dates back to the 9th century. It comes from the Arabic word “al-jabr,” which means “reunion of broken parts” or “restoration.” This term was used by the Persian mathematician Muhammad ibn Musa al-Khwarizmi in his influential treatise “Kitab al-jabr wa al-muqabala,” or “The Compendious Book on Calculation by Completion and Balancing.”
“Algebra”一词起源于阿拉伯语，可追溯至公元 9 世纪。它源自阿拉伯语词汇“al-jabr”，意为“破碎部分的重组”或“复原”。波斯数学家穆罕默德·伊本·穆萨·花拉子米（Muhammad ibn Musa al-Khwarizmi）在其颇具影响力的论著《Kitab al-jabr wa al-muqabala》（《代数学》，直译为“通过补全与平衡进行计算的简明著作”）中使用了该术语。

In its early usage, “algebra” referred to a specific branch of mathematics that dealt with equations and operations related to unknown quantities. It was primarily concerned with solving linear and quadratic equations, as well as performing calculations involving variables.
在早期用法中，“algebra”（代数）指数学中一个特定的分支，研究与未知量相关的方程和运算。它主要致力于求解一次方程（linear equations）和二次方程（quadratic equations），同时涉及含变量的计算。

Over time, the meaning of “algebra” expanded to encompass a broader range of mathematical topics and concepts. Modern algebra includes areas such as abstract algebra, linear algebra, and algebraic geometry, which go beyond basic equation-solving and delve into the structure and properties of mathematical systems.
随着时间的推移，“algebra”（代数）的含义不断扩展，涵盖了更广泛的数学主题与概念。现代代数包含抽象代数（abstract algebra）、线性代数（linear algebra）和代数几何（algebraic geometry）等领域，这些领域已超越基础的方程求解，深入研究数学体系的结构与性质。

The word “algebra” has also been borrowed by numerous languages, including English, French, and Spanish, to describe the field of mathematics related to equations and unknown quantities. It has become a fundamental term in mathematics education and is an essential tool for solving problems in various scientific, engineering, and financial disciplines.
“Algebra”一词还被多种语言借用，包括英语、法语和西班牙语，用于描述与方程和未知量相关的数学领域。它已成为数学教育中的基础术语，同时也是解决科学、工程和金融等多个学科领域问题的重要工具。

In the next section, we will explore the history and evolution of the word “algebra” and how it has influenced the development of mathematics as a whole.
在下一部分中，我们将探讨“algebra”（代数）一词的历史与演变，以及它如何影响了整个数学领域的发展。

The History and Evolution of the Word “Algebra”

“Algebra”（代数）一词的历史与演变

The word “algebra” has a long history, with its origins dating back to ancient civilizations. It encompasses a rich linguistic journey that spans centuries, reflecting the development and evolution of mathematical concepts over time.
“Algebra”一词拥有悠久的历史，其起源可追溯至古代文明。它经历了长达数世纪的丰富语言演变历程，反映了数学概念随时间的发展与演进。

Ancient Origins and Arabic Influence

古代起源与阿拉伯语影响

The word “algebra” traces its roots to the Arabic term “al-jabr,” which means “reunion of broken parts.” This term was first used by the Persian mathematician Muhammad ibn Musa al-Khwarizmi in the 9th century, during the Islamic Golden Age. Al-Khwarizmi’s groundbreaking book, “Kitab al-Jabr wal-Muqabala,” introduced systematic techniques for solving equations and laid the foundation for modern algebraic methods.
“Algebra”一词的根源可追溯至阿拉伯语术语“al-jabr”，意为“破碎部分的重组”。在伊斯兰黄金时代（Islamic Golden Age）的公元 9 世纪，波斯数学家穆罕默德·伊本·穆萨·花拉子米（Muhammad ibn Musa al-Khwarizmi）首次使用了该术语。花拉子米具有开创性的著作《Kitab al-Jabr wal-Muqabala》（《代数学》）介绍了求解方程的系统方法，为现代代数方法奠定了基础。

Latin Translation and European Development

拉丁语翻译与欧洲发展

During the Middle Ages, al-Khwarizmi’s work was translated into Latin by scholars in the Islamic world. The Latin translation of “al-jabr” became “algebra” and was introduced to Europe, where it gained popularity among mathematicians and scholars. This Latin term, “algebra,” became widely used in academic circles and marked the beginning of algebra as a distinct field of mathematics.
中世纪时期，伊斯兰世界的学者将花拉子米的著作翻译成拉丁语。“Al-jabr”的拉丁语译文为“algebra”，随后传入欧洲，并在数学家和学者群体中流行起来。“Algebra”这一拉丁语术语在学术界被广泛使用，标志着代数成为数学领域中一个独立的分支。

Further Refinement and Symbolic Notation

进一步完善与符号表示法

Throughout the Renaissance and Enlightenment periods, mathematicians such as François Viète, René Descartes, and Joseph-Louis Lagrange made significant contributions to the development of algebra. They introduced symbolic notation and algebraic symbolism, which revolutionized mathematical expressions and equations. This shift allowed for more complex and abstract mathematical ideas to be expressed and manipulated with ease.
在文艺复兴（Renaissance）和启蒙运动（Enlightenment）时期，弗朗索瓦·韦达（François Viète）、勒内·笛卡尔（René Descartes）和约瑟夫-路易·拉格朗日（Joseph-Louis Lagrange）等数学家为代数的发展做出了重要贡献。他们引入了符号表示法（symbolic notation）和代数符号系统（algebraic symbolism），这一变革彻底改变了数学表达式和方程的呈现方式。这种转变使得更复杂、更抽象的数学思想能够被轻松表达和运算。

Modern Interpretations and Applications

现代解读与应用

In the modern era, the word “algebra” has taken on broader interpretations and applications. It now encompasses various branches of mathematics, including abstract algebra, linear algebra, and algebraic geometry. Algebra is no longer limited to solving equations but has expanded to include the study of mathematical structures and their properties.
在现代，“algebra”（代数）一词有了更广泛的解读和应用。如今它涵盖了数学的多个分支，包括抽象代数（abstract algebra）、线性代数（linear algebra）和代数几何（algebraic geometry）。代数不再局限于方程求解，而是扩展到对数学结构及其性质的研究。

The evolution of the word “algebra” reflects the evolution of mathematical knowledge and the continuous exploration of mathematical concepts. It serves as a testament to the ingenuity and creativity of mathematicians throughout history, who have built upon the foundations laid by their predecessors and expanded the boundaries of algebraic thinking.
“Algebra”（代数）一词的演变，反映了数学知识的发展以及对数学概念的持续探索。它证明了历史上数学家们的智慧与创造力——他们在前辈奠定的基础上不断探索，拓展了代数思维的边界。

Connections and Influences of the Word “Algebra” in Other Languages

“Algebra”（代数）一词在其他语言中的关联与影响

The word “algebra” has its roots in Arabic, where it was initially referred to as “al-jabr,” meaning “reunion of broken parts.” However, the concept of algebra can be traced back to ancient Babylonian and Egyptian civilizations, where mathematical ideas similar to algebra were developed independently. Over time, the word and the concept of algebra spread to other cultures, influencing and being influenced by different languages and mathematical traditions.
“Algebra”一词起源于阿拉伯语，最初以“al-jabr”形式存在，意为“破碎部分的重组”。但代数的概念可追溯至古代巴比伦和埃及文明——在这些文明中，与代数相似的数学思想被独立发展出来。随着时间的推移，“代数”这一术语及其概念传播到其他文化中，与不同语言和数学传统相互影响。

Influence of Arabic on the Word “Algebra”

阿拉伯语对“Algebra”（代数）一词的影响

The word “algebra” itself is a testament to the influence of the Arabic language on mathematics. During the Islamic Golden Age, scholars in the Islamic world made significant contributions to mathematics, including the development of algebraic concepts and techniques. The word “al-jabr” became synonymous with this branch of mathematics and gradually made its way into European languages.
“Algebra”一词本身就证明了阿拉伯语对数学的影响。在伊斯兰黄金时代，伊斯兰世界的学者为数学做出了重大贡献，其中包括代数概念和方法的发展。“Al-jabr”成为了这一数学分支的代名词，并逐渐传入欧洲语言体系中。

Latin and European Influences

拉丁语与欧洲语言的影响

As algebra spread beyond the Islamic world, Latin became the primary language of scholarship in Europe. The Arabic term “al-jabr” was Latinized as “algebra,” retaining its meaning and giving rise to the modern word used in many languages today. Latin, being the language of academia in Europe for several centuries, played a crucial role in preserving and disseminating algebraic concepts across the continent. From Latin, the word “algebra” was borrowed into other European languages, including English, French, Spanish, and Italian.
当代数传播到伊斯兰世界以外的地区时，拉丁语成为欧洲学术领域的主要语言。阿拉伯语术语“al-jabr”被拉丁化为“algebra”，保留了原有的含义，进而形成了如今在多种语言中使用的现代术语。几个世纪以来，拉丁语一直是欧洲学术界的通用语言，在欧洲大陆保存和传播代数概念方面发挥了关键作用。“Algebra”一词从拉丁语进一步被借入其他欧洲语言，包括英语、法语、西班牙语和意大利语。

Influences in Eastern Languages

在东方语言中的影响

Alongside Latin and Arabic, algebra also had a significant impact on Eastern languages. In India, the word “bīja-gaṇita” meaning “seed calculation” or “analysis of seeds,” was used to refer to algebraic concepts. The word “algebra” influenced the development of this term, highlighting the interconnectedness of mathematical ideas across cultures. In China, the term “fangcheng” meaning “equation calculation” was used for algebraic concepts, demonstrating the unique linguistic and cultural influences on the understanding and expression of algebra.
除拉丁语和阿拉伯语外，代数对东方语言也产生了重要影响。在印度，“bīja-gaṇita”（意为“种子计算”或“种子分析”）一词被用来指代代数概念。“Algebra”（代数）一词对“bīja-gaṇita”的形成产生了影响，这体现了不同文化间数学思想的关联性。在中国，“方程”（意为“方程计算”）一词被用于描述代数概念，这一现象反映了语言和文化对代数理解与表达的独特影响。

Modern Expressions of Algebra

代数的现代表述

In contemporary mathematics, the word “algebra” has become a universal term used across different languages to denote the study of mathematical symbols and the manipulation of equations and formulas. While the linguistic expressions may vary, the core concepts and techniques of algebra remain consistent, allowing mathematicians worldwide to communicate and collaborate.
在当代数学中，“algebra”（代数）已成为一个通用术语，在不同语言中均用于表示对数学符号的研究以及对方程和公式的运算。尽管不同语言中的表述形式存在差异，但代数的核心概念和方法保持一致，这使得全球数学家能够顺畅地沟通与合作。

Understanding the origins and linguistic influences of the word “algebra” provides insight into the historical development of mathematical ideas and the interconnectedness of different cultures and languages in the field of mathematics.
了解“algebra”（代数）一词的起源及其语言影响，有助于深入认识数学思想的历史发展，以及不同文化和语言在数学领域中的相互关联。

Significance and Cultural Context of the Word “Algebra” in Mathematics

“Algebra”（代数）一词在数学领域的意义与文化背景

The word “algebra” has significant cultural and historical significance in the field of mathematics. It originated from the Arabic word “al-jabr,” which refers to the process of restoring or reuniting broken parts. Al-jabr was first introduced in the book “Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa al-muqābala” (The Compendious Book on Calculation by Completion and Balancing) by the renowned Persian mathematician Muhammad ibn Musa al-Khwarizmi in the 9th century.
“Algebra”（代数）一词在数学领域具有重要的文化和历史意义。它起源于阿拉伯语词汇“al-jabr”，指“复原”或“重组破碎部分”的过程。公元 9 世纪，著名波斯数学家穆罕默德·伊本·穆萨·花拉子米（Muhammad ibn Musa al-Khwarizmi）在其著作《Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa al-muqābala》（《代数学》，直译为“通过补全与平衡进行计算的简明著作”）中首次引入了“al-jabr”这一概念。

During the Islamic Golden Age, al-Khwarizmi’s work played a crucial role in the development and dissemination of algebraic concepts. His book introduced algebraic techniques for solving linear and quadratic equations, laying the foundation for the modern algebraic system. The word “al-jabr” eventually evolved into the Latin word “algebra” during the translation of Arabic mathematical texts into Latin in the 12th century.
在伊斯兰黄金时代，花拉子米的著作对代数概念的发展与传播起到了关键作用。他的著作介绍了求解一次方程（linear equations）和二次方程（quadratic equations）的代数方法，为现代代数体系奠定了基础。12 世纪，阿拉伯数学文献被翻译成拉丁语时，“al-jabr”最终演变为拉丁语词汇“algebra”。

The adoption of the word “algebra” in Western mathematics marked a significant turning point in the field. It signified the recognition and adoption of algebraic methods and notation as a distinct branch of mathematics. Over the centuries, algebra has become an essential tool in various fields, including physics, engineering, computer science, and economics. Its concepts and techniques have significantly shaped the development of mathematical thinking and problem-solving strategies.
在西方数学中采用“algebra”（代数）一词，标志着该领域迎来了一个重要转折点。它意味着代数方法和符号表示法作为数学的独立分支得到认可和采用。几个世纪以来，代数已成为物理、工程、计算机科学和经济学等多个领域的重要工具。其概念和方法极大地影响了数学思维和问题解决策略的发展。

Moreover, the word “algebra” and its associated concepts have been influenced by various cultures and languages. From its origins in Persia and its introduction to the Islamic world, algebra spread to Europe during the Renaissance and further evolved through interactions with scholars from different regions. This cross-cultural exchange contributed to the enrichment and refinement of algebraic ideas and approaches.
此外，“algebra”（代数）一词及其相关概念受到了多种文化和语言的影响。代数起源于波斯，随后传入伊斯兰世界，文艺复兴时期又传播到欧洲，并通过与不同地区学者的交流进一步发展。这种跨文化交流推动了代数思想和方法的丰富与完善。

In conclusion, the word “algebra” carries immense significance in the field of mathematics. Its historical and cultural context, stemming from al-Khwarizmi’s work and the subsequent translations and developments, highlights its importance in the development of mathematical knowledge. Today, algebra plays a fundamental role in various branches of mathematics and has wide-ranging applications in practical fields, making it an integral part of mathematical education and problem-solving techniques.
总之，“algebra”（代数）一词在数学领域具有极其重要的意义。其历史和文化背景源于花拉子米的研究，以及后续的翻译和发展，这凸显了它在数学知识发展中的重要性。如今，代数在数学的各个分支中发挥着基础性作用，并且在实际领域具有广泛应用，成为数学教育和问题解决方法中不可或缺的一部分。

Conclusion

结论

The word “algebra” has a rich history and fascinating linguistic roots. Its meaning, evolution, and connections in other languages highlight its significance in the field of mathematics. The word “algebra” carries cultural and historical context that adds depth to our understanding of this mathematical concept.
“Algebra”（代数）一词拥有丰富的历史和引人入胜的语言根源。它的含义、演变以及在其他语言中的关联，凸显了它在数学领域的重要性。“Algebra”（代数）一词所承载的文化和历史背景，为我们理解这一数学概念增添了深度。

As we continue to explore the world of algebra, let us appreciate the influence and evolution of this word and its profound impact on the field of mathematics. It serves as a reminder of the interconnectedness of language, culture, and knowledge.
在我们继续探索代数世界的过程中，让我们重视这一术语的影响与演变，以及它对数学领域的深远作用。它时刻提醒着我们，语言、文化与知识之间存在着紧密的联系。

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Where the Word “Algebra” Came From

“代数”（Algebra）一词的起源

2010-12-29 00:00:00

Today I found out the origins of the word “Algebra”.
今天，我弄清了“Algebra”（代数）一词的起源。

It all started back around 825 AD when a man named Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, the “father” of Algebra, wrote a book called “Kitab al-jabr wa al-muqabalah”. This roughly translates to “Rules of Reintegration and Reduction”. This work was specifically covering the branch of mathematics we now know as Algebra and was the most notable work on the subject during this period, covering such things as polynomial equations up to the second degree; introducing methods for reduction and balancing; and other such staple algebraic methods.
这一切要追溯到公元 825 年左右。被称为“代数之父”的阿布·阿卜杜拉·穆罕默德·伊本·穆萨·花拉子米（Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī）撰写了一本名为《Kitab al-jabr wa al-muqabalah》的著作，其大意可译为《重组与化简法则》。这部著作专门探讨了如今被我们称为“代数”的数学分支，是当时该领域最具影响力的作品。书中内容包括二次多项式方程（polynomial equations up to the second degree）、化简与平衡方法的介绍，以及其他核心代数方法。

It was so notable that it eventually found its way into Europe, becoming the first text book on the subject of Algebra in Europe. The Europeans eventually used the name “al-jabr” for the name of this subject (which in the translated Latin text version was “algebrae”, hence “algebra”).
这部著作的影响力如此之大，最终传播到了欧洲，成为欧洲第一本代数领域的教科书。欧洲人随后将“al-jabr”用作这一学科的名称——在拉丁语译本中，“al-jabr”变为“algebrae”，进而演变成如今的“algebra”（代数）。

“Al-jabr” more or less just means “reunion of broken parts”; basically describing the method for solving both sides of an equation.
“Al-jabr”的含义大致为“破碎部分的重组”，本质上描述的是求解方程两侧的方法。

Bonus Fact:

补充知识：
The word “algorithm” comes from none other than al-Khwarizmi’s name. If you distort the name slightly when you say it, you’ll get the connection.
“Algorithm”（算法）一词正是源自花拉子米（al-Khwarizmi）的名字。若你在发音时稍微调整一下这个名字的音节，就能发现二者的关联。

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī
阿布·阿卜杜拉·穆罕默德·伊本·穆萨·花拉子米

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代数表示法发展三阶段概览

一、表格

发展阶段	主要特征	时间跨度	关键突破
修辞代数（Rhetorical）	全文字表述方程，无任何符号	古代（公元前 1650 年-公元 3 世纪前）	发明位置记数法、解线性方程组、矩阵式系数排列
简字代数（Syncopated）	用符号/缩写表未知量/幂，无运算符号	3 世纪-15 世纪	首次引入未知量符号、解释二次方程解法、给出线性丢番图方程全解
符号代数（Symbolic）	完整符号体系，含运算/关系/幂符号	13 世纪-17 世纪	发明等号、确立变量/已知量符号规则、完善幂表示、创立笛卡尔坐标系

二、图示

三、重要贡献者及成果

贡献者	所属阶段	主要成果	对现代代数的影响
丢番图（希腊）	简字代数	首次用符号表未知量及幂的缩写	奠定“用符号简化表述”的思路，被称“代数之父”之一
花拉子米（波斯）	简字代数	著《代数学》，解释二次方程解法	“代数”一词的起源，普及代数问题求解思路
韦达（法国）	符号代数	创立变量/已知量符号规则，完善幂的缩写	搭建符号代数框架，为现代符号体系打基础
雷科德（英国）	符号代数	发明等号（=）	解决“等式表述繁琐”问题，成为数学基础符号之一
笛卡尔（法国）	符号代数	确立未知数符号（x/y/z）、幂上标表示；发明笛卡尔坐标系	完成符号代数优化，实现代数与几何融合，形成现代代数雏形

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via：

• From Word Problems to Symbolism:A Brief History of Algebra - Front Matter - Cheng-etal-Sp25.pdf
  https://www.ncctm.org/activities/the-centroid1/centroid-articles1/cheng-k-et-al-spring-2025/
• The Origin of the Word Algebra: A Linguistic Exploration - Symbol Genie
  https://symbolgenie.com/origin-of-word-algebra/
• Where the Word “Algebra” Came From
  https://www.todayifoundout.com/index.php/2010/12/the-origins-of-the-word-algebra/