A History of Japanese Mathematics/Chapter 14

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We have already spoken at some length in Chapter IX of the possible connection, slight at the most, between the mathematics of Japan and Europe in the seventeenth century. The possibility of such a connection increased as time went on, and in the nineteenth century the mathematics of the West finally usurped the place of the wasan. During this period of about two centuries, from 1650 to the opening of Japan to the world, knowledge of the European mathematics was slowly finding its way across the barriers, not alone through the agency of the Dutch traders at Nagasaki, but also by means of the later Chinese works which were written under the influence of the Jesuit missionaries. These missionaries were men of great learning, and they began their career by impressing this learning upon the Chinese people of high rank, Matteo Ricci (1552—1610), for example, with the help of one Hsü Küang-ching (1562—1634), translated Euclid into the Chinese language in 1607, and he and his colleagues made known the Western astronomy to the savants of Peking. It must be admitted, however, that only small bits of this learning could have found a way into Japan. Euclid, for example, seems to have been unknown there until about the beginning of the eighteenth century, and not to have been well known for two and a half centuries after it appeared in Peking.

Some mention should, however, be made of the work done for a brief period by the Jesuits in Japan itself, a possible influence on mathematics that has not received its due share of ​attention.^[1] It is well known that the wreck of a Portuguese vessel upon the shores of Japan in 1542 led soon after to the efforts of traders and Jesuit missionaries to effect an entry into the country. In 1549 Xavier, Torres, and Fernandez landed at Kagoshima in Satsuma. Since in 1582 the Japanese Christians sent an embassy carrying gifts to Rome, and since it was claimed about that time that twelve thousand^[2] converts to Christianity had been received into the Church, the influence of these missionaries, and particularly that of the "Apostle of the Indies," St. Francis Xavier, must have been great. In 1587 the missionaries were ordered to be banished from Japan, and during the next forty years a process of extermination of Christianity was pursued throughout the country.

In none of this work, not even in the schools that the Jesuits are known to have established in Japan, have we a definite trace of any instruction in mathematics. Nevertheless the influence of the most learned order of priests that Europe then produced, a priesthood that included in its membership men of marked ability in astronomy and pure mathematics, must have been felt. If it merely suggested the nature of the mathematical researches of the West this would have been sufficient to account for some of the renewed activity of the seventeenth century in the scientific circles of Japan. That the influence of the missionaries on mathematics was manifested in any other way than this there is not the slightest evidence.

It should also be mentioned that an Englishman named William Adams lived in Yedo for some time early in the seventeenth century and was at the court of Iyeyasu. Since he gave instruction in the art of shipbuilding and received honors at court, his opportunity for influencing some of the practical mathematics of the country must be acknowledged. There is also extant in a manuscript, the Kikujutsu Denrai nо Maki, a story that one Higuchi Gonyemon of Nagasaki, a ​scholar of merit in the field of astronomy and astrology, learned the art of surveying from a Dutchman named Caspar, and not only transmitted this knowledge to his people but also constructed instruments after the style of those used in Europe. Of his life we know nothing further, but a note is added to the effect that he died during the reign of the third Shogun (1623—1650). A further note in the same manuscript relates that from 1792 to 1796 a certain Dutchman, one Peter Walius(?) gave instruction in the art of surveying, but of him we know nothing further.

In the eighteenth century the possibility that showed itself in the seventeenth century became an actuality. European sciences now began to penetrate into Japanese schools, either directly or through China. In the year 1713, for example, the elaborate Chinese treatises, the Li-hsiang K'ao-chêng and the Su-li Ching-Yün, which had been compiled by Imperial edict, were published in Peking. Of these the former was an astronomy and the latter a work on pure mathematics, and each showed a good deal of Jesuit influence. These books were carly taken to Japan, and thus some of the trend of European science came to be known to the scholars of that country. There was also sent across the China Sea the Li-suan Ch'üan-shu in which Mei Wen-ting's works were collected, so that Japanese mathematicians not only came into some contact with Europe, but also came to see the progress of their science among their powerful neighbors of Asia. Takebe, for example, is said to have studied Mei's works and to have written some monographs upon them in 1726.^[3]

Nakane Gonkei (1662—1733) also wrote, about the same time, a trigonometry and an astronomy (see Fig. 64) based on the European treatment,^[4] the result certainly of a study of Mei Wen-ting's works and possibly of the Su-li Ching-Yün. ​
Fig. 64. From Nakane Genkei's astronomy of 1696. His pupil Kōda Shin-yei, who died in 1758, also wrote upon the same subject. The illustrations given from the works on surveying by Ogino Nobutomo in his Kiku Genpō Chōken of 1718 (Fig. 65), and Murai Masahiro in his Riochi Shinan of ​
Fig. 65. From Ogino Nobutomo's Kiku Genpō Chōken (1718).

​about the same time (Fig. 66) show distinctly the European influence.

Later writers carried the subject of trigonometry still further. For example, in Lord Arima's Shūki Sampō of 1769 there appear some problems in spherical trigonometry, and in Sakabe's Sampō Tenzan Shinan-roku of 1810—1815 the work is even more advanced. Manuscripts of Ajima and Takahashi upon the same subject are also extant. Yegawa Keishi's treatise

Fig. 66. From Murai Masahiro's Riochi Shinan.

on spherical trigonometry appeared in 1842. Some of the illustrations of the manuscripts on surveying are of interest, as is seen in the reproductions from Igarashi Atsuyoshi's Shinki Sokurio hō of about 1775 (Fig. 67) and from a later anonymous work (Fig. 68).

The European arithmetic began to find its way into Japan in the eighteenth century, but it never replaced the soroban by the paper and pencil, and there is no particular reason why it should do so. Probably the West is more likely to ​return to some form of mechanical calculation, as evidenced in the recent remarkable advance in calculating machinery, than is the Eastern and Russian and much of the Arabian mercantile life to give up entirely the abacus. Napier's rods, however, appealed to the Japanese and Chinese computers, and books upon their use were written in Japan. Arithmetics on the foreign plan were, however, published, Arizawa Chitei's Chūsan Shiki of 1725 being an example. In this work Arizawa speaks of the "Red-bearded men's arithmetic," the Japanese of

Fig. 67. From Igarashi Atsuyoshi's Shinki Sokurio hō. the period sometimes calling Europeans by this name,—the title Barbarossa of the medieval West. Senno's works of 1767 and 1768 were upon the same subject, not to speak of several others, including Hanai Kenkichi's Seisan Sokuchi as late as the Ansei (1854—1860) period. (See Fig. 69.) It is a matter of tradition that Mayeno Ryōtaku (1723—1803) received an arithmetic in 1773 from some Dutch trader, but nothing is known of the work. Mayeno was a physician, and in 1769, at the age of forty-six, he began those linguistic studies that made him well known in his country. He translated several Dutch works, including a few on astronomy, but we have no ​
Fig. 68. From an anonymous manuscript on surveying. ​evidence of his having studied European mathematics. Nevertheless one cannot be in touch with the scientific literature of a language without coming in contact with the general trend
Fig. 69. From Hanai Kenkichi's Seisan Soknchi, showing the Napier rods.

of thought in various lines, and it is hardly possible that Mayeno failed to communicate to mathematicians the nature of the work of their unknown confrères abroad.

​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/275 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/276 ​customs and country up to this time had been practically unknown to the European world. As a result he published in 1824 his De Historia Fauna Japonica, and in 1826 his Epitome Lingua Faponica. He later published his Catalogus Librorum Japonicorum, Isagoge in Bibliothecam Japonicam, and

Fig. 70. Native Japanese surveying instrument. Early nineteenth century.

Bibliotheca Japonica, besides other works on Japan and its people. It is thus apparent that by the close of the first quarter of the nineteenth century Japan was fairly well known to the outer world, and that foreign science was influencing the work of Japanese scholars.

​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/278 ​In the Tempō Period (1830—1844) Koide Shūki translated some portions of Lalande's work on astronomy, and showed to the Astronomical Board the superiority of the European calendar, but without noticeable effect.^[5]

In 1843 Iwata Seiyō published his Kubō Shinkei Shinō (a work relating to the telescope) in which he made use of European methods in astronomy.^[6]

Fig. 71. Native Japanese surveying instrument.
Early nineteenth century.

In 1851 Watanabe Ishin published a work on Illustrating the Use of the Octant, in which he even adopted the Latin term as appears by the title,—Okutanto Yōhō Ryaku-zusetsu. He was followed by Murata Tsunemitsu in 1853 on the use of the sextant. An octant had been brought from Europe in 1780, ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/280 ​never been printed in our country. If anyone who cares to copy them will apply to me I shall be glad to lend them to him and to give him detailed information as to their use." He gave the logarithms of the numbers 1—130 to seven decimal places, by way of illustration. He may possibly have

Fig. 72. From an anonymous logarithmic table in manuscript. ​XIV. The Introduction of Occidental Mathematics.

2/O

had some Dutch work on the "logarithm,"

or

possibly

he

subject, since

he knew the word

had the Peking

tables of 1713

and 1721. Sakabe further says: "The

ratios involved in spherical triangles, as given in the Li-suan Ctiuan-shu, are so numerous that it Since addition and subtraction are is tedious to handle them.

easier than multiplication and division, Europeans require their calculations involving the eight trigonometric lines J to be made

subtracting logarithms. They do not know, however, how to obtain the angles when the three sides 2 are given, or how to get the sides from the three angles, by

by means of adding and

the use of logarithms alone." The first extensive logarithmic table was printed by Koide

Another one was published by 1865) in 1844. in Keishi in which the logarithms were given 1857, Yegawa and in to the same 10,000,3 year an extensive table of up Shuki (1797

natural trigonometric functions

Mori Masakado,

in their

was published by Okumura and

Katsu-yen Hio.

Although the tables were used more or less in the first half of the nineteenth century, the theory of logarithms remained unknown for a long time after it was understood in China. Ajima, Aida, Ishiguro, and Uchida Gokan seem to have been first to pay any attention to the nature of these numbers, but few explanations were put in print until Takemura Kothe

his work in 1854. Since Uchida used only logarithms to the base 10, his theory as to developing them

kaku published is

very complicated. 4

It is quite probable that some suggestion leading to the study of center of gravity found its way in from the West. Seki seems the first to have had the idea in Japan, and it appears in

his investigation of the

volume of the

revolution of circular arcs. 1

I.

e.,

the

six

common

coversed sine. 2

Of

3

ENDO, Book ENDS, Book

a spherical triangle. III, p. III,

135.

p. 143.

solids

generated by the

Arima touches upon the

subject

functions together with the versed sine and the ​in the Shaki Sampō of 1769, and Takahashi Shiji also mentions it. But it was not until after the publication of Hashimoto's work in 1830, and after there was abundant opportunity for European influence to show itself, that the problem became at all popular. From that time on it was the object of a great deal of attention, the solids becoming at times quite complicated. For example, the center of gravity was studied for such a solid as a segment of an ellipsoid pierced by a cylindrical hole, and for a group of several circular cones, each piercing the others.

Similarly we may be rather sure that the study of various roulettes, including the cycloid and epicycloid, came from some hint that these problems had occupied the attention of mathematicians in the West. This does not detract from the skill shown by Wada Nei, for example, but it merely asserts that the objects, not the methods of study, were European in source, For the method, the ingenuity, and the patience, all credit is due to the Japanese scholars.

The same remark may be made with respect to the catenary and various other curves and surfaces. The catenary first appears in Hagiwara's work above mentioned, and the problem was subsequently solved by Ōmura Isshū and Kagami Mitsuteru, being attacked by approximating, step by step, the root of a transcendental equation, a treatment very complicated but full of interest. The treatment is purely Japanese, even though the idea of the problem itself may have found its way in through Dutch avenues.

In the nineteenth century there were a number of scholars in Japan who possessed more or less reading knowledge of the Dutch language. One of these was Uchida Gokan whose name has just been mentioned in connection with logarithms. He even called his school by the name "Matemateka."^[7] Of him Tani Shomo wrote, in the preface of a work published in 1840,^[8] these appreciative words: "Uchida is a profoundly ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/284 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/285 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/286 ​seems to be no basis, but it shows that even in the nineteenth century the Western methods of computation were not at all well known.

About the middle of the century the European methods began to find definite place in Japanese works, if not in the

Fig. 73. From Hanai Kenkichi's Seisan Sokuchi (1856).

schools. The first of these works was Hanai Kenkichi's Seisan Sokuchi (Short Course in Western Arithmetic), published in 1856 (Fig. 73), and Yanagawa Shunzō's Yōsan Yōhō (Methods ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/288 ​Grōndbeginzels der Hoogere Meetkunde which was published in Rotterdam in 1794. This translation seems not to be known.

Of the conic sections some intimation of the subject may have reached Japan in the seventeenth century, but it evidently was taken, if at all, only as a hint. The Japanese studied the ellipse very zealously, always by their own peculiar

Fig. 74. From a manuscript by Wake Yukismasa.

method, but the parabola and hyperbola seem never to have attracted the attention of the old school. To be sure, the parabola enters into a problem about the path of a projectile in Yamada's Kaisanki of 1656, but it seems never to have been noticed by subsequent writers. The graphs of these curves are found in certain astronomical works, as in Yoshio's Yensei Kanshō Zusetsu of 1823 where they are used in illustrat​ing the orbits of comets, but they do not enter into the works on pure mathematics. This very fact is evidence against any influence from without affecting the native theories.

We have already spoken of the change of the Board of Translation to the Institute for the Investigation of European Books. Six years after this change was made the Kaiseijo School was founded (1863), in which every art and science was to be taught. A department of mathematics was included, and in this Kanda Kõhei was made professor. He it was who made the first decisive step towards the teaching of European mathematics in Japan, and from his time on the subject received earnest attention in spite of the small number of students in the department.

The year 1868 is well known in the West and in Japan as a year of great import to the world. This was the year of the political revolution that overthrew the Tokugawa Shogunate, that put an end to the feudal order, and that restored the Imperial administration. Yedo, the Shogun's capital, became Tōkyō, the seat of the Empire. The year is known to the West because it marked the coming of a new World Power. What this has meant the past forty years have shown; what it is to mean as the centuries go on, no one has the slightest conception. To Japan the year marks the entrance of Western ideas, many of which are good, and many of which have been harmful. The art of Japan has suffered, in painting, in sculpture, and especially in architecture. The exquisite taste of a century ago, in textiles for example, has given place to a catering to the bad taste of moneyed tourists. And all of this has its parallel in the domain of mathematics, in which domain we may now take a retrospective view.

What of the native mathematics of Japan, and what of the effect of the new mathematics? What did Japan originate and what did she borrow? What was the status of the subject before the year 1868, and what is its status at the present and its promise for the future?

Looked at from the standpoint of the West, and weighing the evidence as carefully and as impartially as human imper​fections will allow, this seems to be a fair estimate of the ancient wasan:—

The Japanese, beginning in the seventeenth century, produced a succession of worthy mathematicians. Since these men studied the general lines that interested European scholars of a generation earlier, and since there was some opportunity for knowing of these lines of Western interest, it seems reasonable to suppose that they had some hint of what was occupying the attention of investigators abroad. Since their methods of treatment of every subject were peculiar to Japan, either her scholars did not value or, what is quite certain, did not know the detailed methods of the West. Since they decried the European learning in mathematics, it is probable that they made no effort to know in detail what was being done by the scholars of Holland and France, of England and Germany, of Italy and Switzerland.

With such intimation as they may have had respecting the lines of research in the West, Japan developed a system of her own for the use of infinite series in the work of mensuration. She later developed an integral calculus that was sufficient for the purposes of measuring the circle, sphere, and ellipse. In the solution of higher numerical equations she improved upon the work of those Chinese scholars who had long anticipated Horner's method in England. In the study of conics her scholars paid much attention to the ellipse but none to the parabola and hyperbola.

But the mathematics of Japan was like her art, exquisite rather than grand. She never developed a great theory that in any way compares with the calculus as it existed when Cauchy, for example, had finished with it. When we think of Descartes's La Géométrie; of Desargues's Brouillon proiect, of the work of Newton and Leibnitz on the calculus; of that of Euler on the imaginary, for example; of Lagrange and Gauss in relation to the theory of numbers; of Galois in the discovery of groups,—and so on through a long array of names, we do not find work of this kind being done in Japan, nor have we the slightest reason for thinking that we ought to find it. ​Europe had several thousand years of mathematics back of her when Newton and Leibnitz worked on the calculus,—years in which every nation knew or might know what its neighbors were doing; years in which the scholars of one country inspired those of another. Japan had had hardly a century of real opportunity in mathematics when Seki entered the field. From the standard of opportunity Japan did remarkable work; from the standpoint of mathematical discovery this work was in every way inferior to that of the West.

When, however, we come to execution it is like picking up a box of the real old red lacquer, not the kind made for sale in our day. In execution the work was exquisite in a way wholly unknown in the West. For patience, for the everlasting taking of pains, for ingenuity in untangling minute knots and thousands of them, the problem-solving of the Japanese and the working out of some of the series in the yenri have never been equaled.

And what will be the result of the introduction of the new mathematics into Japan? It is altogether too early to foresee, just as we cannot foresee the effect of the introduction of new art, of new standards of living, of machinery, and of all that goes to make the New Japan. If it shall lead to the application of the peculiar genius of the old school, the genius for taking infinite pains, to large questions in mathematics, then the world may see results that will be epoch making. If on the other hand it shall lead to a contempt for the past, and to a desire to abandon the very thing that makes the wasan worthy of study, then we cannot see what the future may have in store. It is in the hope that the West may appreciate the peculiar genius that shows itself in the works of men like Seki, Takebe, Ajima, and Wada, and may be sympathetic with the application of that genius to the new mathematics of Japan, that this work is written.