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^ A good discussion of this so-called "Boethius question," which has been de-
bated for two centuries, is given by D. E. Smith and L. C. Karpinski in their Hitidu-
Arabic Numerals, 191 1, Chap. V.

2 Encyclopedic des sciences mathemaliqiies, Tome I, Vol. 2, 1907, p. 2. An il-
luminating article on ancient finger-symbolism is L. J. Richardson's " Digital
Reckoning Among the Ancients" in the Am. Math. Monthly, Vol 22,, 1816,
PP- 7-13-



THE MAYA

The Maya of Central America and Southern Mexico developed
hieroglyphic writing, as found in inscriptions and codices dating ap-
parently from about the beginning of the Christian era, that ranks
"probably as the foremost intellectual achievement of pre-Columbian
times in the New World." Maya number systems and chronology
are remarkable for the extent of their early development. Perhaps
five or six centuries before the Hindus gave a systematic exposition
of their decimal number system with its zero and principle of local
value, the Maya in the flatlands of Central America had evolved
systematically a vigesimal number system employing a zero and the
principle of local value. In the Maya number system found in the
codices the ratio of increase of successive units was not lo, as in the
Hindu system; it was 20 in all positions except the third. That is,.
20 units of the lowest order {kins, or days) make one unit of the next
higher order {uinals, or 20 days), 18 uinals make one unit of the third
order (tun, or 300 aays), 20 tuns make one unit of the fourth ordet
(katmt, or 7200 days), 20 katuns make one unit of the fifth order
(cycle, or 144,000 days) and finally, 20 cycles make i great cycle of
2,880,000 days. In Maya codices we find symbols for i to 19, ex-
pressed by bars and dots. Each bar stands for 5 units, each dot for
I unit. For instance,



I 2 4 5 7 II 19

The zero is represented by a symbol that looks roughly Hke a half-
closed eye. In writing 20 the principle of local value enters. It is
expressed by a dot placed over the symbol for zero. The numbers
are written vertically, the lowest order being assigned the lowest
position. Accordingly, 37 was expressed by the symbols for 17 (three
bars and two dots) in the kin place, and one dot representing 20,
placed above 17 in the uinal place. To write 360 the Maya drew
two zeros, one above the other, with one dot higher up, in third place
(1X18x20+0-1-0=360). The highest number found in the codices
is in our decimal notation 12,489,781.

A second numeral system is found on Maya inscriptions. It em-
ploys the zero, but not the principle of local value. Special symbols
are employed to designate the different units. It is as if we were to
write 203 as "2 hundreds, o tens, 3 ones," ^

' For an account of the Maya number-systems and chronology, see S. G. Morley
A II Introduction to the Study of the Maya Hierogliphs, Government Printing Office
^Vashinglon, 1915.

)0



70 A HISTORY OF MATHEMATICS

The Maya had a sacred year of 260 days, an ofl5cial year of 360
days and a solar year of 365+ days. The fact that 18x20=360
seems to account for the break in the vigesimal system, making 18
(instead of 20) uinals equal to i tun. The lowest common multiple
of 260 and 365, or 18980, was taken by the Maya as the "calendar
round," a period of 52 years, which is "the most important period in
Maya chronology."

We may add here that the number systems of Indian tribes in North
America, while disclosing no use of the zero nor of the principle of
local value, are of interest as exhibiting not only quinary, decimal, and
vigesimal systems, but also ternary, quarternary, and octonary sys-
tems.^

1 See W. C. Eells, "Number Systems of the North American Indians" in Amer-
ican Math. Monthly, Vol. 20, i9'i3, pp. 263-272, 293-299; also BibliotJieca malhe-
malica, 3 S., Vol. 13, 1913, pp. 218-222.



THE CHINESE i

The oldest extant Chinese work of mathematical interest is an
anonymous publication, called Chou-pei and written before the
second century, a. d., perhaps long before. In one of the dialogues the
Chou-pei is believed to reveal the state of mathematics and astronomy
in China as early as iioo b. c. The Pythagorean theorem of the right
triangle appears to have been known at that early date.

Next to the Chou-pei in age is the Chiu-chang Suan-shu ("Arith-
metic in Nine Sections"), commonly called the Chiu-chang, the most
celebrated Chinese Text on arithmetic. Neither its authorship nor
the time of its composition is known definitely. By an edict of the
despotic emperor Shih Hoang-ti of the Ch'in Dynasty "all books were
burned and all scholars were buried in the year 213 b. c." After the
death of this emperor, learning revived again. We are told that a
scholar named Chang T'sang found some old writings, upon which
he based this famous treatise, the Chiu-chang. About a century later
a revision of it was made by Ching Ch'ou-ch'ang; commentaries on
this classic text were made by Liu Hui in 263 a. d. and by Li Ch'un-
feng in the seventh century. How much of the "Arithmetic in Nine
Sections," as it exists to-day, is due to the old records ante-dating
213 B. c, how much to Chang T'sang and how much to Ching Ch'ou-
ch'ang, it has not yet been found possible to determine.

The "Arithmetic in Nine Sections" begins with mensuration; it
gives the area of a triangle as | 6 h, of a trapezoid as | {b +b')h, of a
circle variously as |c .^t/, led, |i- and -^^c-, where c is the circumference
and d is the diameter. Here tt is taken equal to 3. The area of a
segment of a circle is given as ^(ca+a-), where c is the chord and a
the altitude. Then follow fractions, commercial arithmetic including
percentage and proportion, partnership, and square and cube root of
numbers. Certain parts exhibit a partiality for unit-fractions. Divi-
sion by a fraction is effected by inverting the fraction and multiplying.
The rules of operation are usually stated in obscure language. There
are given rules for finding the volumes of the prism, cylinder, pyramid,
truncated pyramid and cone, tetrahedron and wedge. Then follow
problems in alligation. There are indications of the use of positive
and negative numbers. Of interest is the following problem because
centuries later it is found in a work of the Hindu Brahmagupta:

' All our information on Chinese mathematics is drawn from Yoshio Mikami's
The Development of Mathematics in China and Japan, Leipzig, 19 12, and from David
Eugene Smith and Yoshio Mikami's History of Japanese Mathematics, Chicago,
1914.

71



72 A HISTORY OF MATHEMATICS

There is a bamboo lo ft. high, the upper end of which is broken and
reaches to the ground 3 ft. from the stem. What is the height of the

break? In the solution the height of the break is taken =Y- t^ttt'

2X 10

Here is another: A square town has a gate at the mid-point of each
side. Twenty paces north of the north gate there is a tree which
is visible from a point reached by walking from the south gate 14
paces south and then 1775 paces west. Find the side of the square.
The problem leads to the quadratic equation a:2+(2o + i4)x— 2X10X
jy^^ ^Q_ The derivation and solution of this equation are not made
clear in the text. There is an obscure statement to the effect that
the answer is obtained by evolving the root of an expression which
is not monomial but has an additional term [the term of the first
degree (20 + i4)x]. It has been surmised that the process here re-
ferred to was evolved more fully later and led to the method closely
resembling Horner's process of approximating to the roots, and that
the process was carried out by the use of calculating boards. Another
problem leads to a quadratic equation, the rule for the solution of
which fits the solution of Uteral quadratic equations.

We come next to the Sun-Tsu Suan-ching ("Arithmetical Classic
of Sun-Tsu"), which belongs to the first century, a. d. The author,
SuN-Tsu, says: "In making calculations we must first know positions
of numbers. Unity is vertical and ten horizontal; the hundred stands
while the thousand lies; and the thousand and the ten look equally,
and so also the ten thousand and the hundred." This is evidently a
reference to abacal computation, practiced from time immemorial in
China, and carried on by the use of computing rods. These rods,
made of small bamboo or of wood, were in Sun-Tsu's time much longer.
The later rods were about i| inches long, red and black in color,
representing respectively positive and negative numbers. According
to Sun-Tsu, units are represented by vertical rods, tens by horizontal
rods, hundreds by vertical, and so on; for 5 a single rod suffice s. The
numbers 1-9 are represented by rods thus: I, II, 111, nil, mil, |,J|, 1||, ||||;
the numbers in the tens column, 10, 20, . . ., 90 are written thus:

_ =^ =^ =, ^, _1_, J_, =, =. The number 6728 is designated
by I IT ^^ TlT- ^^^ ^°^^ ^^^^ placed on a board ruled in columns,
and'were rearranged as the computation advanced. The successive
steps in the multiplication of 321 by 46 must have been about as
follows:

321 321 321

138 1472 14766

46 46 46

The product was placed between the multiplicand and multiplier.

The 46 is multiplied first by 3, then by 2, and last by i, the 46 being



THE CHINESE 73

moved to the right one place at each step. Sun-Tsu does not take up
divasion, except when the divisor consists of one digit. Square root
is explained more clearly than in the "Arithmetic in Nine Sections."
Algebra is involved in the problem suggested by the reply made by a
woman washing dishes at a river: "I don't know how many guests
there were; but every two used a dish for rice between them; every
three a dish for broth; every four a dish for meat; and there were 65
dishes in all. — Rule: Arrange the 65 dishes, and multiply by 12, when
we get 780. Divide by 13, and thus we obtain the answer."

An indeterminate equation is involved in the following: "There are
certain things whose number is unknown. Repeatedly divide by 3,
the remainder is 2 ; by 5 the remainder is 3 ; and by 7 the remainder is
2. What will be the number?" Only one solution is given, viz. 23.

The Hai-tao Suan-ching ("Sea-island Arithmetical Classic") was
written by Liu Hui, the commentator on the "Arithmetic in Nine
Sections," during the war-period in the third century, a. d. He gives
complicated problems indicating marked proficiency in algebraic
manipulation. The first problem calls for the determination of the
distance of an island and the height of a peak on the island, when two
rods 30' high and 1000' apart are in line with the peak, the top of the
peak being in line with the top of the nearer (more remote) rod, when
seen from a point on the level ground 123' (127') behind this nearer
(more remote) rod. The rules given for solving the problem are
equivalent to the expressions obtained from proportions arising from
the similar triangles.

Of the treatises brought forth during the next centuries only a few
are extant. We mention the "Arithmetical Classic of Chang Ch'iu-
chien" of the sbcth century which gives problems on proportion, arith-
metical progression and mensuration. He proposes the "problem of
100 hens" which is given again by later Chinese authors: "A cock
costs 5 pieces of money, a hen 3 pieces, and 3 chickens i piece. If
then we buy with 100 pieces 100 of them, what will be their respective
numbers?"

The early values of ir used in China were 3 and vio^ i-iu Hui
calculated the perimeters of regular inscribed polygons of 12, 24, 48,
96, 192 sides and arrived at 7r =3.14+. Tsu Ch'ung-chih in the fifth
century took the diameter 10^ and obtained as upper and lower limits
for TT 3.1415927 and 3.1415926, and from these the "accurate" and
"inaccurate" values 355/113, 22/7. The value 22/7 is the upper Umit
given by Archimedes and is found here for the first time in Chinese
history. The ratio 355/113 became known to the Japanese, but in
the West it was not known until Adriaen Anthonisz, the father of
Adriaen Melius, derived it anew, sometime between 1585 and 1625.
However, M. Curtze's researches would seem to show that it was
known to Valentin Otto as early as 1573.^

1 Bibliotheca malhemalica, 3 S., Vol. 13, 19 13, p. 264. A neat geometric construe-



74 A HISTORY OF MATHEMATICS

In the first half of the seventh century Wang Hs' iao-t'ung brought
forth a work, the Ch'i-ku Siian-ching, in which numerical cubic equa-
tions appear for the first time in Chinese mathematics. This took
place seven or eight centuries after the first Chinese treatment of
quadratics. Wang Hs'iao-t'ung gives several problems leading to
cubics: "There is a right triangle, the product of whose two sides is
706 -gQ, and whose hypotenuse is greater than the first side by 30 g"^.
It is required to know the lengths of the three sides." He gives the
answer 14 -^, 49 \, S'^-h ^^^ the rule: "The Product (P) being
squared and being divided by twice the Surplus (S), make the result
shih or the constant class. Halve the surplus and make it the lien-fa
or the second degree class. And carry out the operation of evolution
according to the extraction of cube root. The result gives the first
side. Adding the surplus to it, one gets the hypotenuse. Divide the
product with the first side and the quotient is the second side." This
rule leads to the cubic equation x^ +S/2.v^-|'^ =0, The mode of solu-
tion is similar to the process of extracting cube roots, but details of
the process are not revealed.

In 1247 Ch'in Chiu-shao wrote the Su-shu Chiu-chang ("Nine
Sections of Mathematics") which makes a decided advance on the
solution of numerical equations. At first Ch'in Chiu-shao led a mili-
tary life; he lived at the time of the Mongolian invasion. For ten
years stricken with disease, he recovered and then devoted himself to
study. The following problem led him to an equation of the tenth
degree: There is a circular castle of unknown diameter, having 4
gates. Three miles north of the north gate is a tree which is visible
from a point 9 miles east of the south gate. The unknown diameter
is found to be 9. He passes beyond Sun-Tsu in his ability to solve
indeterminate equations arising for a number which will give the
residues ri, r2, . ., r^ when divided by nii, m<i, ., Wn, respectively.

Ch'in Chiu-shao solves the equation -x*-|-7632oox- — 40642560000
= by a process almost identical with Horner's method. However,
the computations were very probably carried out on a computing
board, divided into columns, and by the use of computing rods.
Hence the arrangement of the work must have been different from
that of Horner. But the operations performed were the same. The
first digit in the root being 8, (8 hundreds), a transformation is ef-
fected which yields .r^ — 3 2oox^ — 30768oox'-^- 826880000.x -I-3820544-
0000=0, the same equation that is obtained by Horner's process.
Then, taking 4 as the second figure in the root, the absolute term
vanishes in the operation, giving the root 840. Thus the Chinese had

tion of the fraction ^ '^ | =3 +4^ -^ (7^+8-) is given anonymously in Grunert's Archiv
Vol. 12, 1849, p. 98. Using ^'Jl, T. M. P. Hughes gives in Nature, Vol. 93,
T914, p. no, a method of constructing a triangle that gives the area of a given
circle with great accuracy.



THE CHINESE



75



invented Homer's method of solving numerical equations more than
five centuries before Ruffini and Horner. This solution of higher
numerical equations is given later in the writings of Li Yeh and others.
Ch'in Chiu-shao marks an advance over Sun-Tsu in the use of o as a
symbol for zero. Most likely this symbol is an importation from
India. Positive and negative numbers were distinguished by the use
of red and black computing rods. This author gives for the first time
a problem which later became a favorite one among the Chinese; it
involved the trisection of a trapezoidal field under certain restrictions
in the mode of selection of boundaries.

We have already mentioned a contemporary of Ch'in Chiu-shao,
namely, Li Yeh; he lived far apart in a rival monarchy and worked
independently. He was the author of Tse-yuan Hai-ching ("Sea-
Mirror of the Circle-Measurements"), 1248, and of the I-ku Ven-liian,
1259. He used the symbol o for zero. On account of the inconven-
ience of writing and printing positive and negative numbers in dif-
ferent colors, he designated negative numbers by drawing a cancella-
tion mark across the symbol. Thus J_o stood for 60, JKo stood for
- 60. The unknown quantity was represented by unity which was
probably represented on the counting board by a rod easily distin-
guished from the other rods. The terms of an equation were written,
not in a horizontal, but in a vertical line. In Li Yeh's work of 1259,
as also in the work of Ch'in Chiu-shao, the absolute term is put in the
top line; in Li Yeh's work of 1248 the order of the terms is reversed,
so that the absolute term is in the bottom line and the highest power
of the unknown in the top line. In the thirteenth century Chinese
algebra reached a much higher development than formerly. This
science, with its remarkable method (our Horner's) of solving numer-
ical equations, was designated by the Chinese "the celestial element
method."

A third prominent thirteenth century mathematician was Yang
Hui, of whom several books are still extant. They deal with the
summation of arithmetical progressions, of the series 1+^+6 + . . +
(1-I-2-I-. . +n) =n{ji + i}{n+2)^6, 12+22-f . . +^2 =i?i(«+i)(»+i),
also with proportion, simultaneous linear equations, quadratic and
quartic equations.

Half a century later, Chinese algebra reached its height in the
treatise Suan-hsiao Chi-meng ("Introduction to Mathematical
Studies"), 1299, and the Szu-yiien Vu-chien ("The Precious Mirror
of the Four Elements"), 1303, which came from the pen of Chu
Shih-Chieh. The first work contains no new results, but exerted a
great stimulus on Japanese mathematics in the seventeenth century.
At one time the book was lost in China, but in 1839 it was restored
by the discovery of a copy of a Korean reprint, made in lOOo. The
'Vrecious Mirror" is a more original work. It treats fully of the
"celestial element method." He gives as an "ancient method" a



76



A HISTORY OF MATHEMATICS



triangle (known in the West as Pascal's arithmetical triangle), dis-
playing the binomial coefficients, which were known to the Arabs in
the eleventh century and were probably imported into China. Chu
shih-Chieh's algebraic notation was altogether different from our
modern notation. Thus, a +o +c +d was written



I
1*1

I



I
202

I O *2 O I

22 O 2



as shown on the left, except that, in the central position, we employ
an asterisk in place of the Chinese character t'ai (great extreme, ab-
solute term) and that we use the modern numerals in place of the
sangi forms. The square of a -\-b +c -f-J, namely, a- -\-b'^ +c^ +d^ + 2ab
+2ac+2ad+2bc+2bd+2cd, is represented as shown on the right.
In further illustration of the Chinese notation, at the time of Chu
Shih-Chieh, we give 1



'


• 1 -■■


*


hi-


*


-2


.-.=


,





'1


I




*




*







02


*







= 11





= .v








= xz








= 2






I







I













In the fourteenth century astronomy and the calendar were studied
They involved the rudiments of geometry and spherical trigonometry.
In this field importations from the Arabs are disclosed.

After the noteworthy achievements of the thirteenth century,
Chinese mathematics for several centuries was in a period of decline.
The famous "celestial element method" in the solution of higher
equations was abandoned and forgotten. Mention must be made,
however, of Ch'eng Tai-wei, who in 1593 issued his Suan-fa T'ung-
tsung ("A Systematised Treatise on Arithmetic"), which is the oldest
work now extant that contains a diagram of the form of the abacus,
called suan-pan, and the explanation of its use. The instrument was
known in China in the twelfth century. Resembling the old Roman
abacus, it contained balls, movable along rods held by a wooden
frame. The suan-pan replaced the old computing rods. The "Sys-
tematised Treatise on Arithmetic" is famous also for containing some
magic squares and magic circles. Little is known of the early history

^In the symbol for "xz" notice that the "i" is one space down (x) and one
space to the right (2) of *, and is made to stand for the product xz. In the symbol
for "2y3" the three o's indicate the absence of the terms y, x, xy; the small "2"
means twice the product of the two letters in the same row, respectively one space
to the right and to the left of *, i. e., 2 yz. The limitations of this notation are ob-
vious.



THE CHINESE



77



of magic squares. Myth tells us that, in early times, the sage Yii,
the enlightened emperor, saw on the calamitous Yellow River a divine
tortoise, whose back was decorated with the figure made up of the
numbers from i to 9, arranged in form of a magic square or lo-shu.



\^\ o 00000000 /



4


9

5

I


2

7
6


3
8



^ ' <>



The lo-shu.

The numerals are indicated by knots in strings: black knots repre-
sent even numbers (symbolizing imperfection), white knots repre-
sent odd numbers (perfection).

Christian missionaries entered China in the sixteenth century.
The Italian Jesuit Matteo Ricci (1552-1610) introduced European
astronomy and mathematics. With the aid of a Chinese scholar
named Hsu, he brought out in 1607 a translation of the first six books
of EucHd. Soon after followed a sequel to Euclid and a treatise on
surveying. The missionary Mu Ni-ko sometime before 1660 intro-
duced logarithms. In 17 13 Adrian Vlack's logarithmic tables to 11
places were reprinted. Ferdinand Verbiest ^ of West Flanders, a
noted Jesuit missionary and astronomer, was in 1669 made vice-
president of the Chinese astronomical board and in 1673 its president.
European algebra found its way into China. Mei Ku-ch'ing noticed
that the European algebra was essentially of the same principles as
the Chinese "celestial element method" of former days which had
been forgotten. Through him there came a revival of their own
algebraic method, without, however, displacing European science.
Later Chinese studies touched mainly three subjects: The determina-
tion of TT by geometry and by infinite series, the solution of numerical
equations, and the theory of logarithms.

We shall see later that Chinese mathematics stimulated the growth
of mathematics in Japan and India. We have seen that, in a small
way, there was a taking as well as a giving. Before the influx of
recent European science, China was influenced somewhat by Hindu
and Arabic mathematics. The Chinese achievements which stand
out most conspicuously are the solution of numerical equations and
the origination of magic squares and magic circles.

^ Consult H. Bosnians, Ferdinand Verbiest, Louvain, 191 2. Extract from Revue
des Questions scienlifiqiics, January-April, 1912.



THE JAPANESE *

According to tradition, there existed in Japan in remote times a
system of numeration which extended to high powers of ten and re-
sembled somewhat the sand counter of Archimedes. About 552 a. d.
Buddhism was introduced into Japan. This new movement was
fostered by Prince Shotoku Tatshi who was deeply interested in all
learning. Mathematics engaged his attention to such a degree that
he came to be called the father of Japanese mathematics. A little
later the Chinese system of weights and measures was adopted. In
701 a university system was established in which mathematics figured
prominently. Chinese science was imported, special mention being
made in the official Japanese records of nine Chinese texts on mathe-
matics, which include the Chou-pei, the Suan-ching written by Sun-
Tsu and the great arithmetical work, the Chiu-chang. But this eighth
century interest in mathematics was of short duration; the Chiu-chang
was forgotten and the dark ages returned. Calendar reckoning and
the rudiments of computation are the only signs of mathematical
activity until about the seventeenth century of our era. On account
of the crude numeral systems, devoid of the principal of local value
and of a symbol for zero, mechanical aids of computation became a
necessity. These consisted in Japan, as in China, of some forms of
the abacus. In China there came to be developed an instrument,
called the suan-pan, in Japan it was called the soroban. The importa-
tion of the suan-pan into Japan is usually supposed to have occurred
before the close of the sixteenth century. Bamboo computing rods
were used in Japan in the seventh century. These round pieces were



Online LibraryFlorian CajoriA history of mathematics → online text (page 9 of 62)