Download The Development of Algebraic Methods of Problem

The Development of Algebraic Methods
of Problem-Solving in Japan in the Late Seventeenth
and the Early Eighteenth Centuries
Annick M. Horiuchi
REHSEIS (CNRS), 49 rue Mirabeau, F-75016 Paris, France
I. Introduction
The rapid growth of mathematical knowledge during the Edo period (1600-1868)
is one of the most remarkable features of the history of science in Japan before the
modernization of the Meiji era. The chief outcomes of this long-standing tradition
were:
- the accumulation of a significant body of high-standard mathematical works
jealously kept within private academies where they were communicated to small
numbers of selected disciples,
- the wide diffusion of mathematical practice through the publication of popular
textbooks and the activity of schoolmasters and itinerent teachers.
To appreciate the extent of this development, one must recall that, at the turn
of the 17th century, mathematics meant little more than elementary computations
performed with the abacus. The situation was modified to a considerable extent
after the Japanese turned their attention to ancient Chinese scientific works. In less
than half a century, Japanese mathematics developed from the primary art of
computation which served the practical needs of merchants, craftsmen and lowgrade warriors into a discipline appealing to a scholarly audience.
Before dealing with the original contribution of Japanese mathematicians, it is
to be noted that the assimilation of Chinese mathematics was quite effectively
prompted by the novel practice of leaving several problems unsolved at the end of
the books as a challenge to other mathematicians. Almost all the difficult problems
discussed in Chinese works were integrated in this way into the Japanese corpus.
A further impetus was added in 1658 by the discovery and the subsequent reprint
of a 13th century Chinese treatise [Zhu Shijie 1299], the level of which far surpassed
those of previously available works. The 13th century in the history of Chinese
mathematics was a very productive period (often described as its golden age) when
significant achievements were made, most particularly in the area of algebraic
devices [Li Yan and Du Shiran 1987, chap. 5]. The Japanese scholars' attention
soon focused on the tengen (or tianyuan in Chinese) method which Zhu Shijie used to
solve a range of tricky problems.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
© The Mathematical Society of Japan, 1991
1640
Annick M. Horiuchi
The tengen/tianyuan method of solving problems was basically similar to the one
which is called algebra in the West. The Japanese scholars spent many years before
getting at the meaning of the word tengen (literally "celestial origin"), the name given
by the Chinese to the "unknown". The level of the difficulties involved was such
that the contemporary Chinese mathematicians who were unaware of the past
achievements had to wait for the introduction of the western algebra in the 18th
century to rediscover the meaning of the method. [Martzloff 1987, pp. 105-106]
Japanese mathematicians worked on problems which were modelled on the
older Chinese problems. These problems dealt with concrete situations and were
expressed in numerical terms; their solution included both the numerical result and
the procedure (jutsu in Japanese), the sequence of the operations to perform on the
abacus or with counting-rods to get to the result.
The proliferation of small problems of this kind throughout the Edo period and
the increasingly artificial character of most of them led some historians to stress the
artistic and recreational character of Japanese mathematics and the mathematicians'
indifference to the 'utility' of their art [Mikami, 1921].
This point of view relied chiefly on the examination of one side of Japanese
mathematics: the problems. But one cannot ignore the fact that the "utility" of
mathematics in the past very often depended on the general and efficient methods
and tools which were obtained through solving particular problems. Therefore,
the issue of the utility of Japanese mathematics cannot be settled before having
examined the other side of the mathematicians' work, that of elaborating methods
of solving problems.
The aim of this paper is to discuss some prominent features of the development
of these methods in the late 17th and the early 18th centuries. I will focus on the
achievements of Seki Takakazu (7-1708) and Takebe Katahiro (1664-1739), two
major mathematicians of this period. I will discuss the way Seki improved the
Chinese algebraic methods and stress the importance of the Chinese root-extraction
procedure in the course of his research. I will then turn to one of Takebe's main
contributions, the introduction of the infinite power series in the scope of algebraic
calculation.
II. SEKI's Study of Algebraic Devices
Let us begin with Seki's synthesis of earlier methods of problem-solving. The
synthesis, achieved by 1683, took the form of a trilogy. Each treatise was devoted
to a particular method of problem-solving [Hirayama et al 1974, sects. 6, 7 and 8].
I will examine only the last treatise where Seki expounds an original method of
problem-solving which can be understood as an extension of the Chinese tengen/
tianyuan method.
Let us consider first the main features of this Chinese method which had a
considerable influence on the development of algebraic devices in Japan. Here is
an example of a problem requiring the tengen method for its solution:
"Given a rectangularfieldof 8 mu 5 fen and 5 li [1 mu = 240 square bu; 1 fen =
0.1 mu; 1 li = 0.1 fen]. We only say that the sum of the length and the width is
92 bu. Find the length and the width". [Zhu Shijie 1299, chapter kaifang shisuo]
The Development of Algebraic Methods of Problem-Solving
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The solution followed a regular pattern: first, a quantity was set up as the
unknown. Then two different algebraic expressions for a certain quantity were built
up (in the problem above, the area was expressed as x(92 — x) and 2052). The
equation was derived by subtracting one of the expressions from the other. Finally,
the numerical value of the unknown was determined by extracting the root of the
equation, digit by digit.
One basic feature of the tengen/tianyuan method was the use of counting rods
to carry out the polynomial calculation as well as the extraction of the root. The
solution itself, as I have said in the introduction, consisted of the sequence of
operations to be performed with this instrument. The instructions given by the
author were like: "Put one rod for the unknown", "Multiply by itself four times",
"Add it to the area", etc.
The way the counting-rods were to be handled could be reconstructed from
specific symbols representing the polynomials obtained on the counting-board (a
large sheet of paper with horizontal and vertical lines drawn on it). These symbols
were inserted in the solution after each instruction. Polynomial expressions as well
as equations of one unknown were represented by the column of their coefficients
arranged in the order of increasing powers of x (see Fig. 1):
|o||||
}
HI
Ex: 3x - 16x + 209
209 (constant term)
—16 (coefficient of x)
3 (coefficient of x2)
2
Fig. 1. The Chinese notation for polynomials
Let us now turn to Seki's improvement of the Chinese method. The need for an
improvement originated in the publication in 1671 of fifteen unsolved problems by
Sawaguchi Kazuyuki, a Kyoto mathematician. Sawaguchi's problems were so intricate that none of them could be handled with the Chinese method. This is an
example of Sawaguchi's problems:
"We have now A, B and C, such that each is a cube.
We say first that the volumes of A and B altogether make 137,340 tsubo and
also that the volumes of B and C altogether make 121,750 tsubo.
We say in addition that the square root of the edge of A, the cube root of
the edge of B and the fourth root of the edge of C altogether make 1 shaku
2 sun. Find the sides of A, B and C." [Sawaguchi 1671, Problem 4].
The third treatise of Seki's trilogy gave a global solution to two questions
implicitly raised by Sawaguchi's problems which can be formulated as follows:
How can one proceed to the calculation when the quantities involved cannot be
1642
Annick M. Horiuchi
HI—UHI /
-M
Il h
h
(315/-4fc) (constant term)
(-12/I 5 )
(coefficient of x)
3bhl
(coefficient of x 2 ]
4
m»
Ex: 3lbhx2 - 12h5x + (315/ - Ah)
Fig. 2. Seki's notation for polynomials with literal coefficients.
The literal and the numerical parts were dealt with separately. The latter part was
transcribed by using the rod-numerals; the former was written beside these numerals using
Chinese characters.
In the example above, the Chinese characters meaning length, breadth, height and number
four have been replaced by the letters /, b and h and the arabic numeral 4. h5 is represented
h
by because Japanese mathematicians used to consider the number of times a quantity was
multiplied by itself, that is the power minus one unit.
expressed in terms of one unknown? How can one eliminate the unknown within
two equations?1
Seki answered by adding supplementary steps to the tengen pattern. In the
course of these steps, additional unknowns were introduced and eliminated. Seki's
improvement can be characterized by two main features. First, the general pattern
of the tengen was maintained by considering one unknown at each stage and by
integrating the other unknown quantities into the data. Second, the calculation
was reduced to a single process of eliminating one unknown within two given
algebraic equations.
Seki's ability to cope with such general contexts and questions was closely
related to his use of adequate notations to represent polynomials and equations
with literal coefficients (see Fig. 2). Seki's notations were obtained by extending the
traditional representation of polynomials with numerical coefficients. The rules of
calculation with the new notations remained unchanged. [Hirayama et al. 1974,
Sect. 29].
This step has been described by many historians as a shift from an "instrumental" algebra into a "written" algebra. But this description is misleading in at
least one respect. In fact, the solution maintained its algorithmic character and
consisted of a rhetorical description of the operational instructions to be performed
on an imaginary counting-board. In the solution, the notation was only used to
represent the polynomial expressions obtained at each main step of the calculation.
The calculation itself was performed somewhere else and was not made explicit. We
do not even know if Seki performed the calculation on the paper or went on using
the counting-board in some manner. In this respect, his calculation still preserved
an instrumental character.
1
Regarding this point, an extension to the case of four unknowns was achieved in China as
early as the beginning of the 14th century [Zhu Shijie, 1303]. Japanese mathematicians of
Edo period did not know about this extension.
The Development of Algebraic Methods of Problem-Solving
1643
Additional features should be noted about Seki's use of this notation. Seki
described the procedure of elimination in very general terms, by considering sets of
two equations with arbitrary coefficients.
The coefficients were represented by means of Chinese characters taken from a
series of twelve characters (the ten "stems" and the twelve "branches"; kanshi) in the
same way as alphabet letters symbolise numbers in western algebra.
Many studies have been devoted to Seki's method of elimination by which he
solved problems through cancelling a certain determinant [Mikami 1913]. Leaving
aside detailed analysis of this method, we note that Seki's interest in the general
rules of calculating the determinant was closely connected to his systematic use of
Chinese characters as symbols in place of particular numbers.
To conclude the first part of this study, I will add that the whole trilogy of Seki's
reveals a clear shift of his concern from particular problems to general methods of
solution. In all the treatises composing the trilogy, Seki displays both his intention
and his ability to reduce any particular problem to general processes of calculation.
This tendency towards generalization, however, was not entirely new in the
Sino-Japanese tradition. The search for general procedures of problem-solving had
always been part of the Chinese tradition, as is well exemplified by the classification
adopted in one of the oldest mathematical treatises in China, the Nine Chapters of
Mathematics [Chemla 1988; Wu 1986]. Seki's originality lay rather in departing
from the general tendency of his time to favour particular problems, and in his
introduction of a range of efficient tools to describe the methods in general terms.
As a consequence, Seki's attention gradually concentrated on general objects like
equations and polynomials.
III. 'Defective' Problems and Seki's Reflection
on Root-Extraction Procedures
Let us now examine more closely Seki's study of equations. This part of Seki's
achievement, which took place shortly after the trilogy, is particularly interesting
in that it brings forward a quite different way of discussing properties of equations
and roots from the one so far better-known to modern mathematicians. A particular
computational device plays a central role in Seki's research: the root-extraction
procedure.
Seki's interest in equations stemmed from a particular, quite pragmatic preoccupation: the search for methods to correct what he called 'defective' problems
(byôdai). According to Seki's definition [Hirayama et al. 1976, Sect. 9], problems
were "defective" or wrongly stated if they led either to more than one acceptable
solution or to none. Defective problems had to be corrected by changing the terms
of the problem.
In the course of his research, Seki's interest shifted to the equations themselves
and he thought out a general method of transforming an equation with more than
one root into one equation with a single root [Hirayama et al. 1974, Sect. 8]. This
was done by changing the value of one of the coefficients in the initial equation.
Annick M. Horiuçhi
1644
Before discussing Seki's method, we must note that the notion of "equation' did
not have a strict equivalent in traditional Chinese methematics. Instead, we find the
concept of "configuration for extraction" (kaifangshi), referring to the numbers set
up on the counting-board to perform the extraction. The configuration itself was
similar to the polynomial expression (Fig. 1). Likewise, instead of the concept of
"root", we find that of "quotient", which referred to the location on the countingboard of the result of the extraction. The coefficients of the equation were similarly
called by the names of their respective locations on the counting-board,
The extraction procedure which Seki used in this context was a general procedure which allowed him to compute successively all the real roots of a given
equation.
This procedure, in fact, was an outcome of a long tradition of research in China
[Li Yan and Du Shiran 1987, Sect. 5.2] but let us concentrate here only on the
procedure as was set forth in Seki's treatise [Hirayama et al. 1976, Sect. 8]. The
whole extraction was built on one basic pattern of computation involving the
coefficients of the equation (see Table 1), in which one can easily recognize the
so-called Horner-Ruffini process. The successive configurations can be interpreted
as a gradual alteration of the coefficients of (1) into the coefficients of (2) where (2)
is satisfied by y with y = x — a.
f(x) = m + nx + lx2 + px 3 = 0
0)
<p(y) = f(y + a) = o
(2)
The above interpretation of the root-extraction pattern as a substitution of the
unknown was not explicitly stated by Seki in this context. But all the improvements
he introduced suggest that Seki did have a similar explanation of it.
Let us examine more closely how Seki used this procedure to find all the roots
of a given equation. To begin, the first root of the equation, let us call it a, was
sought by carrying out successively this basic pattern with several quotients
Table 1. The basic pattern of Seki's root extraction procedure
(2)
(1)
Quotient
a
a
Constant term
Coefficient of x
Coefficient of x2
Coefficient of x3
m
n
I
P
m + (n + la + pa2). a = m -+- na + la2 + pa3
n + (l + pa).a = n + la + pa2
l + p.a
P
(4)
(3)
a
a
2
3
m + na + la + pa
n + la + pa2 + (/ + 2pa). a = n + 2/a + 3pa2
I + pa + p.a = l + 2pa
P
m + na + la2 + pa3
n + 2la + 3pa2
l + 2pa + p. a = l + 3pa
P
The calculation is carried out for each configuration from the bottom up.
The Development of Algebraic Methods of Problem-Solving
1645
(positive or negative), suitably chosen so that the number in the top line gets closer
and finally becomes equal to zero. The first root was then derived by adding the list
of quotients.
The equation cp(y) = 0 corresponding to this last configuration would therefore
be inferior by one unit compared to the initial equation.
Seki's originality lay in his idea of iterating the previous process to the new
equation. Assuming for example thatfcis a root of the new equation <p(y) = 0, which
means that at the end of the iterated process with b as quotient, the top line of the
last configuration gets to zero, then the number a + b would be the second root of
the initial equation.
The central role played by this root-extraction procedure in Seki's reflection on
equations and roots is particularly obvious in the way he tried to solve the aforementioned question of removing the "excess" roots.
Seki's method was based on a general criterion which was to be fulfilled by the
coefficients of any equation having a single root. This criterion which is actually a
condition of existence of a double root was the outcome of a computation in which
the extraction procedure played a central role.
The fact that a is the single root of (1) meant for Seki that at the end of the
extraction procedure, the top two lines of the last configuration (the configuration
(4) in Table 1) were equal to zero.
\m H- na + la2 + pa3 = 0
[n + 21a + 3pa2 = 0.
The criterion was then derived by eliminating a within these two equations:
27m2p2 -I- 4m/3 -I- 4w3p - limnlp - n2!2 = 0.
As shown by this example, the root-extraction procedure in Seki's mathematics
was clearly something more than a device for computing the roots of an equation.
This procedure was automatically involved in any discussion pertaining to equations, coefficients or roots. No hypothesis on the roots or any property of the
equation could be stated without referring to this procedure. "Theory of extraction"
would thus be the most suitable name for this part of Seki's research.
IV. Takebe's Method of Computing the Length of An Arc
Let us turn now to another significant response given by the Japanese mathematicians to a very ancient problem of trigonometry, the problem of finding a
general procedure for calculating the length of an arc a in terms of the sagitta s and
the diameter of the circle d (see Fig. 3).
We shall focus here on the work of Seki's disciple, Takebe Katahiro, whose
contribution provides the crowning step of a long tradition of research in China
and Japan [Li Yan and Du Shuran, §3.2]. In 1720, Takebe expressed for the
first time the square of the length of the arc of the circle as an infinite power
series of the sagitta. Takebe's work displays an unprecedented intensity in the
application of algebraic methods and especially of the root-extraction procedure.
1646
Annick M. Horiuchi
Fig. 3
To begin with, let us look at the infinite procedure of calculating the quantity
(a/2)2 as formulated in Takebe's treatise [Takebe 1722, sect. 12]. a, s and d are
respectively the length of the arc, the sagitta and the diameter of the circle.
"The fundamental procedure runs as follows:
Sagitta and diameter are multiplied. This gives the approximate value of the
square of half the arc. [sd~]
Divide by three the square of the sagitta. This gives the first discrepancy.
[*i = *73]
Put down the first discrepancy and multiply by the sagitta. Divide by the
diameter. Then, multiply by 8 and divide by 15. This gives the second discrepancy. [X2 = X±. (s/d). (8/15)]
Put down the second discrepancy. Multiply by the sagitta. Divide by the
diameter. Multiply by 9. Divide by 14. This gives the third discrepancy. fX3 =
X2 (s/d). (9/14)]
Put down the third discrepancy. Multiply by the sagitta. Divide by the
diameter. Multiply by 32. Divide by 45. This gives the fourth discrepancy.
IX4 = X3 (s/d). (32/45)]
[...]
The successive discrepancies are added to the approximate value of the
square of half the arc. This gives the fixed value of the square of half the arc."
[Takebe 1722, Chap. 12]
Leaving aside the highly empirical method through which Takebe found his
way to this procedure, we shall focus on his extension of the scope of algebraic
calculation.
The first thing to examine is the original commentary he added to the explanation of his method. In this commentary, Takebe took as his starting point a common
distinction between 'exhaustible' numbers (tsukuru kazu) and 'inexhaustible' ones
(tsukizaru kazu). Numbers were exhaustible (respectively inexhaustible) if they had
a limited (respectively unlimited) decimal expression. He then extended this distinction to procedures and formed the following programme:
The Development of Algebraic Methods of Problem-Solving
1647
"Numbers like 1/4 and 1/5 are exhaustible numbers. 1/3 and 1/7 are examples
of inexhaustible numbers. Addition, subtraction, and multiplication are exhaustible procedures. Division and extraction are inexhaustible procedures. The
perimeter of the square or the areas of rectangles have an exhaustible form. The
circumference and the segment of the circle have an inexhaustible form. Thus,
just as the forms of the arc and the circle are inexhaustible, the procedure
involved is also inexhaustible. Since the procedure is inexhaustible, so are the
resulting numbers."
[Takebe 1722, Chap. 12]
Takebe's commentary aimed at conferring a legitimate status to the infinite
procedure, a thing which must have been quite unfamiliar and uncomfortable to
most of his contemporaries. The link he established between the infinite procedures
and the numbers with an infinite decimal expression may be considered as his
primary argument for integrating the new procedure into the existing mathematical
corpus.
The role played by this link was not only pedagogical. As can be seen in Takebe's
later works, this analogy between the decimal numbers and the infinite series
allowed him to venture into still more new fields of research.
This last point is exemplified by the second method of calculating the length of
arc which Takebe devised shortly after the first. The method rested on the idea of
deriving an infinite series by performing the root-extraction on an equation with
literal coefficients.
We shall limit our discussion to the first step of the process which contains the
main idea. Takebe resorted to a classical device consisting in inscribing polygonal
lines inside the arc and getting approximate values of the arc by calculating the
length of the polygonal lines (see Fig. 4).
Now, let us call a0 the length of the arc to be calculated and s 0 , the sagitta in
terms of which the arc is to be expressed.
As can easily be seen, the sagitta s1 of the arc a1 = a0/2 satisfies the following
equation (1).
- s 0 d + 4dx-4x2
=0
(1)
Fig. 4
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Annick M. Horiuchi
The extraction procedure is performed on (1) exactly as if the coefficients were
numerical. As a result of this extraction s1 is expressed as an infinite power series
of s0/d.
Sl
=
So /4
+ s02/16d + s 0 3 /32d 2 + 5s 0 4 /256d 3 + 7s 0 5 /512d 4 + 21s 0 6 /2048d 5 + •••
The same process is then performed on (2) in which the constant term — s±d is
given as an infinite series of s0/d.
-s1d + 4dx-4x2 = 0
(2)
s2 could then be expressed in terms of s0/d by performing the root-extraction
procedure on (2). We note that in the course of the calculation successive terms of
the infinite series are handled as if they were successive digits of a decimal number.
This example shows clearly the central role played by the aforedescribed analogy
in Takebe's second method of calculating the arc. His extension of the extraction
procedure to cases where coefficients are literal and even infinite series, as well as
his way of handling the infinite series, were direct outcomes of this analogy.
From a historical point of view, this later aspect of Takebe's research had a
fundamental significance in offering a very effective method of constructing infinite
series. Moreover, in his explanation of the second method, Takebe clearly displayed
his belief that the previously described extraction procedure could be applied to an
equation of any degree. He also suggested that the general law which governs the
successive terms of the series was resulting from the extraction procedure [Takebe
s.d, p. 24b]. Both points bring Takebe's method very close to the idea of the binomial
theorem.
Conclusion
Even though the analyses above do not cover the whole scope of mathematical
research in the Edo period, they show well enough the growing interest in algebraic
devices during the 17th and the 18th centuries in Japan. We have seen how the
Chinese algebraic devices, including the procedure of extraction, have been extended and enriched by Seki and Takebe. By studying these devices, they not only
sought to solve the largest amount of problems, but also strove to clarify and make
explicit the general patterns of methods of problem-solving. In my view, this
outstanding feature of the Japanese tradition has more claim to the utilitarian cause
than has usually been recognized.
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