Configurations versus Equations:

A Notational Difference 陣式和方程式 : 符號的異同

程貞一

聖迭戈加州大學物理系

There are historians of mathematics and sinologists who view the presentation of the early Chinese works in equations with mathematical signs such the equal ‘’, plus ‘’, minus ‘’ signs and letter symbols such as x, y, z is equivalent to read modern concept of equations into old work.

有些漢學家和科技史學者認為用現今數學符號系統 : 等號 ‘’ , 加號‘’ ,減號‘’ , 和字母符號 : x, y, z,書寫中國古代數學 , 就等于把近代“ equation” (即“方程式” ) 的觀念帶入到古代中國數學中。

In this talk, I would like to suggest misunderstandings such as these can be easily clarified, if one generalizes one’s concept of notation from written mode to include mechanical mode. 在此我建議如果把數學符號觀念由書寫符號系統闊大到包含機械符號系統 , 這類誤解自然就消失了。

Ever since antiquity, there were in mathematics, the written mode and the mechanical mode of representing technical and quantitative mathematical concepts and relations, and in executing algorithm and derivations. These two different modes lead to the creation of written and mechanical notational systems, respectively.

Notation in the written mode began probably with the use of a‘hard’pen to create impressions on tablets such as the cuneiform writings, or on turtle shells and ox bones such as the shell-bone inscriptions [1] (jiăgŭwén 甲骨文 ), and with the use of a ‘colored’ pen to write on papyrus, bamboo strips (zhújiăn 竹簡 ), parchment, silk fabrics or papers for expressing mathematical concepts and for performing computations.

[[1]. The translation “shell-bone inscriptions” is advised based on the technical terms “甲骨文” coined by Chinese Linguistic communities. The translation “oracle-bone inscriptions” in the West initiated by sinologists is misleading, depicting only one aspect of the inscriptions. The turtle shells and ox bones were the favored writing media during the Shāng 商 dynasty and their use probably begun in antiquity.

Notation in mechanical mode, on the other hand, began with tools, other than a pen for performing computations. In this mode, one uses the devised objects such as rods in chóusuàn 籌算 or axial-sliding balls in zhūsuàn 珠算 to compose configurational layouts for expressing mathematical thoughts and for performing computations.

Positional Numerals in Written Notation (left column) and in Mechanical Notation Displayed on Abacus in Zhūsuàn 珠算 and in Chóusuàn 籌算 on a Computational Board.

In many early civilizations, the written mode was favored in their early stage of mathematic developments both for performing calculations and for keeping records of calculations. Consequently, written notations were used exclusively

However, in the Chinese civilization early computations were carried out only with mechanical tools. The written mode was used only for describing and recording of the computations. Hence, notation both in written and in mechanical modes were used. Not until the 17th century did computations in written mode became common.

In order to study early mathematics in Chinese civilization, especial its algorithm and derivations, it is imperative to have certain familiarities with chóusuàn 籌算 and the rod notational system used by the Chinese mathematicians before the 17th century. This is because mathematical concepts and computations are expressed all in mechanical notation not in written notation.

Solving Mathematical Problems In the ‘written mode’

One uses written symbols to compose equations for expressing mathematical thoughts and to execute algorithms in performing mathematical computations

One uses devised objects such as rods, balls, rotations, or signals to compose configurations, patterns or sequences for expressing mathematical thoughts and to execute algorithms in performing mathematical computations.

Solving Mathematical Problems In the ‘Mechanical mode’

A practical way to examine the difference in notation between written and mechanical modes, Let us consider an actual problem from Problem 1 of Chapter 8 ‘Fāng Chéng’ 〈方程〉‘ Square Array’ from the Jiŭ-Zhāng Suàn-Shù 《九章筭術》 (Nine Chapters in Mathematics) which is a problem with three linear algebraic relations for three unknowns

Rectangular-Array Configuration (jŭzhènshì 矩陣式 ) Linear Equations Set

3x 2y z 39

2x 3y z 34

x 2y 3z 26

It is seen that the basic layouts are similar; the layout in current written notation is horizontal but the array-configuration layout is vertical. By rotating the written linear equations 90 degrees clockwise, one would then have the coefficients and constants of the linear equations arranged exactly as those given in the array configuration of the linear equations.

Instead of the letter symbols x, y, and z, the three variables are identified by their corresponding rows in the rectangular-array configuration. The first row is identified with the unknown x, the second row with the unknown y and so on until the row before the last row reserved for the constant terms. By construction, each constant in the last row equals the sum of all the entries above within the same column of the constant. The plus ‘’ and minus ‘’ signs are specified by the coefficients and the constant occupying the configuration.

For equations of higher degrees, let us consider the problem of square root extraction a problem from Problem 20, Chapter 9 of the Nine Chapters in Mathematics.

Columnar Configuration (hánglièshì 行列式 ) 2nd-Degree Equation

In the written notation, the degrees of the variable are denoted by the different powers of the letter x for the variable. In the columnar configurational representation, the degrees of the variable are denoted by the positional orders in the column with respect to the constant term of the problem at the bottom of the column.

These comparisons reveal that configurations and equations have the same function in their respective representations. There is actually a one-to-one correspondence between the two representations. This implies that one should be able to transform between the two representations without changing the content of the mathematics.

This demonstrated that once one’s concept of notation is generalized from written mode to include mechanical mode. One would then realized that the concept of equation is not a new mathematical concept, it appeared earlier as configurations in mechanical notations in Chinese mathematical work.

The presentation of the early Chinese works in equations with mathematical signs such the equal ‘’, plus ‘’, minus ‘’ signs and letter symbols such as x, y, z is merely a one-to-one transfer of notation from mechanical mode to the written mode.

It should also be noted mechanical notation is an important part of mathematics because history has shown that the development of mathematical tools was an important part of mathematical progresses. Attempts to improve calculating tools began actually very early in Chinese civilization.

Among the driving forces for improving calculating tools are speed, repetitive calculations, versatilities, etc. Facing the growing needs in commerce for faster calculations, the early Chinese invented suànpán 算盤 , a mechanical tool with axial-sliding balls (or beats) for faster calculations. During the Táng 唐 and Sòng 宋 dynasties, suànpán developed into an important calculating tool primarily in commerce.

To aid zhūsuàn, a number of algorithmic rhymes and verses, originated from the chóusuàn, were modified and adopted for zhūsuàn. These rhymes and verses were usually recited subconsciously during calculations to facilitate the recalling of algorithms and to speed up repetitive calculations. Such algorithmic rhymes and verses can be viewed as instructions for operating mechanical devises in performing repetitive computations.

For further improvement in handling mechanical calculations, it would be desirable, if such instructions as well as the mechanical notation can be built in the mechanical devises instead of carrying out manually and tracking mentally as in both zhūsuàn and chóusuàn.

The next generation of calculating tools involved wheels and gears instead of rods and balls. With such wheel-gear calculating machines, the mechanical notation were expressed in terms of patterns of rotational sequences of the interlocking geared wheels. The first machine capable of addition was a water-powered armillary clock invented in 1092 by astronomer Sū Sòng 蘇頌 (1020-1101).

In his documents Jìn-Yíxiàng Făzhuàng《進儀象法狀》 (Report on the Armillary Clock), Sū Sòng 蘇頌 stated, in discussing the interlocking wheel-and-gear operating system of the clock, that the machine “makes measurements and perform calculations” (qìdù suànshù “ 器度算數” ); and “keeps time and reporting time by drumming drums and ringing bells” (sīchén jígŭ yáolíng 司辰擊鼓搖鈴 )”.

The first mechanical device capable of adding numbers in Europe was also a calculating clock invented in 1623 by Wilhelm Shickard (1592-1635), a professor of mathematics at the University of Tübingen. Twenty years before the invention of the ‘Pascaline’ adding device in 1643 by the French mathematician Blaise Pascal (1623-1662).

The adding function was based on the movement of six dented wheels, geared through a wheel which with every full turn allowed the wheel located at the right to rotate 1/10th of a full turn. By design, it was able to handle up to six-digit numbers. The adding feature was devised to help performing multiplication with a set of Napier's cylinders included in the upper half of the machine. An overflow mechanism rang a bell.

In modern electronic calculations, one uses the binary numeral system for implementing calculations. The “zero” and “one” digits are designated by the “off” and “on” signals respectively. Notation adopted in such electronic calculations is expressed in signal-patterns which are built in integrated-circuits in chips and in microprocessor for calculators and computers.

By the 17th century, Europe began to develop innovative devises for assisting calculations. The first such a devises was the Napier’s bones consisting of strips of wood or bones written with multiplication tables which was built in 1617 by the Scottish mathematician John Napier (1550-1617) for performing arithmetic operations.

Through this device, he also developed logarithms to convert multiplication into addition. Later in 1633, the slide rule was invented based on logarithms by the English mathematician William Oughtred (1574-1660).

分析中西古代數學運算方式和推導思路上的異同 , 顯然可見由於運算方式上的差異 , 中西建立了不同符號體系和發明了不同運算工具 ; 由於推導思路上的差異 , 中西創建了不同的推理系統和精煉出不同的運算程序。但是歷史証實 , 這些異同對世界數學的發展曾多次出現相互推動和相互補助的功能。

“方程式”是用書寫表達代數法總結數學問題的方式 , 以便在筆算中進行開根演算求解。在觀念上“方程式”與“陣式”完全一樣。因為“陣式”是用算籌表達代數法總結數學問題方式 , 以便在籌算中進行開根演算求解。由此可見 ,陣式和方程式之間的差異並不在數學觀念上 , 而在符號系統上。陣式應用于機械符號系統中然而方程式應用于書寫符號系統中。把陣式改變為方程式 , 僅僅是一個一對一的符號轉換 , 不牽涉到數學的內容。