Sacred Mathematics
Japanese Temple GeometryBy Fukagawa Hidetoshi Tony RothmanPrinceton University Press
Copyright © 2008 Princeton University
All right reserved.
ISBN: 978-0-691-12745-3
Chapter One
Japan and Temple Geometry Temple bells die out. The fragrant blossoms remain. A perfect eve ning! -Basho
Temples
No visitor to a foreign country has failed to experience the fascination and unease that accompanies an encounter with unknown traditions and customs. Some visitors attempt to overcome their fears, while the majority quickly retreats to familiar shores, and in this lies a distinction: Those who embrace culture shock are travelers; those who do not are tourists.
The most profound culture shock comes about when one is confronted by a different way of thinking. Most of us can hardly imagine walking into a Western church or cathedral to encounter stained glass windows covered by equations and geometrical figures. Even if we can conceive of it, the thought strikes us as alien, out of place, perhaps sacrilegious. Yet for well over two centuries, Japanese mathematicians-professionals, amateurs, women, children-created what was essentially mathematical stained glass, wooden tablets adorned with beautiful geometric problems that were simultaneously works of art, religious offerings, and a record of what we might call "folk mathematics." The creators of these sangaku-a word that literally means "mathematical tablet"-hung them by the thousands in Buddhist temples and Shinto shrines throughout Japan, and for that reason the entire collection of sangaku problems has come to be known as "temple geometry," sacred mathematics.
In this book you will be invited not only to encounter temple geometry but to appreciate it. There is a bit of culture shock to be overcome. A single glance at a sangaku is enough for one to realize that they were created by a profoundly different esthetic than the Greek-inspired designs found in Western geometry books. On a deeper level, one learns that the methods Japanese geometers employed to solve such problems differed, sometimes significantly, from those of their Western counterparts. Ask any professional mathematician whether the laws of mathematics would be the same in another universe and he or she will reply, of course. Real mathematicians are Pythagoreans-they cannot doubt that mathematics exists independently of the human mind. At the same time, during their off hours, mathematicians frequently speculate about how different mathematics could look from the way it is taught in Western schools.
Temple geometry provides a partial answer to both questions. Yes, the rules of mathematics are the same in East and West, but yes again, the traditional Japanese geometers who created sangaku saw their mathematical world through different eyes and sometimes solved problems in distinctly non-Western ways. To learn traditional Japanese mathematics is to learn another way of thinking.
Traditional Japanese mathematics, and with it temple geometry, arose in the seventeenth century under a nearly unique set of circumstances. In 1603, three years after defeating his rival daimyo- warlords-at the battle of Sekigahara, Tokugawa Ieyasu became shogun of Japan, establishing the Tokugawa shogunate. (A contemporary depiction of the battle of Sekigahara can be seen in the color plate 1.) The Tokugawa family ruled Japan for the better part of three hundred years, until 1868, when a decade after Commodore Matthew C. Perry forcefully opened Japan to the West, the shogunate collapsed.
One of Ieyasu's first moves after Sekigahara was to establish his headquarters at a small fortress town in central Japan, a town that became known as Edo (pronounced "Yedo")-today's Tokyo. For that reason the rule of the Tokugawa is also known as the Edo period. During the first years of the Tokugawa shogunate, Ieyasu (who, although living until 1616, officially remained shogun only until 1605) consolidated power by confiscating the lands of other warlords, but nevertheless continued many of the foreign policies of his pre decessor, the great daimyo Toyotomi Hideyoshi (1537-1598). At the turn of the seventeenth century, Japan carried on substantial trade with foreign countries, both Eastern and Western. Nagasaki on the island of Kyushu had become the base for the "southern barbarians" to import their goods, as well as to print translations of Western literature, much of it religious.
Foreign missionaries had by then been in Japan for over fifty years. In autumn of 1543, three Portuguese were shipwrecked off Kyushu. The misfortune proved decisive in terms of Japan's relations with outsiders, for the men were carrying arquebuses, which were rapidly adopted by the Japanese warlords. Of equal or greater importance was that, within a few years of the fateful shipwreck, Portuguese merchants and Jesuit missionaries began to arrive, seeking both trade and converts. The Jesuits were particularly successful, converting as many as two hundred thousand Japanese over the next forty years and becoming de facto rulers of the Nagasaki region.
All of this alarmed the proponents of Buddhism and raised the distrust of Hideyoshi himself; he in 1587 took direct control of Nagasaki and issued two edicts designed to curb the spread of Christianity. But the Spanish soon arrived, with Spanish merchants vying with Portuguese for trade and Franciscans vying with Jesuits for converts. In 1596, after a Spaniard supposedly boasted that the missionaries were merely the vanguard of an Iberian conquest, Hideyoshi ordered the execution of twenty- six priests and converts. The warlord, though, had other affairs on his mind, in particular the conquest of China, and he failed to pursue a resolution of the growing tensions between the Japanese and Westerners.
The tensions were resolved, in a particularly decisive and brutal fashion, at the very end of Tokugawa Ieyasu's life and in the two decades that followed. In 1614 Ieyasu reissued an earlier edict with which he summarily ordered that all Christian missionaries leave the country, that places of worship be torn down, and that the practice of Christianity be outlawed. But other internal affairs intervened and Ieyasu died in 1616 without having taken much action. After his death, though, persecution of Christian converts began in earnest and by 1637, according to some estimates, three hundred thousand converts had apostasized or been killed. Throughout the 1630s Ieyasu's grandson, Togukawa Iemitsu, issued a series of decrees that offered rewards for the identification of kirishitan, forbade the sending of Japanese ships overseas, and forbade any Japanese from traveling abroad, on pain of death.
By 1641 the last of the Portuguese merchants had been expelled, leaving only the Dutch. The Dutch had arrived comparatively late to Japan, in 1609, and had shown more interest in trade than mission. For that reason they were allowed to remain after the expulsion of the Iberians. The Japanese, however, by now utterly suspicious of Westerners, put severe strictures on the Dutch presence: The representatives of the Dutch East India Company were forced to move onto a small, man-made island called Deshima in Nagasaki harbor (see color plate 2 and plate 1.2). The fan-shaped island, originally created for the Portuguese, measured only 200 by 70 meters. A wall surrounded Deshima, posted with signs warning the Japanese to keep away, and it was entirely cut off from the mainland except for a bamboo water pipe and a single, guarded bridge. On this oasis, twenty or so members of the East India Company lived among the few warehouses, sheep, pigs, and chickens, and awaited the summer arrival of the Dutch ships. Upon making port, captains locked all Bibles and Christian literature into barrels, while Japanese laborers unloaded cargo.
That, for the next two hundred years, constituted Japan's trade with the West, and so began the policy of what would eventually become known as sakoku, "closed country." It is impossible to claim that sakoku was one hundred percent effective; certainly trade with Korea and China continued. Two Japanese did escape to Holland around 1650 in order to study mathematics. We know the scholars only by their adopted names, Petrius Hartsingius and Franciscus Carron, the former at least having achieved some distinction. Whether they ever returned to Japan we do not know. One doctor, Nakashima Chozabruo, traveled abroad with a Dutch trader and risked beheading to come home. According to tradition, the local daimyo spared Nakashima's life because he healed one of the warlord's injured pigeons.
Such scraps of information do lead one to believe that by any ordinary standards the isolation from the West was nearly complete. In terms of mathematics, it is extremely unlikely that anyone in Japan learned about the creation of modern calculus by Newton and Leibnitz later in the seventeenth century, and there is certainly no evidence from sangaku problems and traditional Japanese mathematics texts that its practitioners understood the fundamental theorem of calculus.
One should not conclude from this state of affairs that sakoku had entirely negative consequences. To the contrary, the policy was so successful at eliminating foreign conflicts that the 250 years of the Edo period became known as the "Great Peace." Moreover, with the stability provided by the Tokugawa shogunate, Japanese culture experienced a brilliant flowering, so much so that the years of the late seventeenth century are referred to as Genroku, Renaissance. At the time a gentleman was expected to cultivate skills in "medicine, poetry, the tea ceremony, music, the hand drum, the noh dance, etiquette, the appreciation of craft work, arithmetic and calculation ... not to mention literary composition, reading and writing. There are other things as well ..."
We do not have space here to delve into the riches of Genroku culture, but one should recognize that during this era many of the arts for which Japan is renowned attained their highest achievements: Noh dance flourished; the great dramatist Chikamatsu Monzaemon (1653-1725) produced plays for both the Kabuki and puppet theatres; tea ceremonies, flower arranging, and garden architecture were on the ascendant, as well as painting in several schools, including the ubiquitous ukiyo-e, or "floating world" prints that illuminated the demimonde of courtesans and erotic love and fairly defined the entire epoch. Ukiyo-e prints were made using wood blocks, not because the Japanese lacked movable type, which had been imported from Korea during Hideyoshi's day, but because printers preferred the calligraphic and artistic possibilities afforded by block printing. Poetry was not to be eclipsed, especially haiku, which achieved some of its greatest expression in the works of Matsuo Basho (1644-1694), who long ago achieved worldwide renown.
What is strikingly absent in the standard reviews of Japanese cultural achievements of the period is any mention of science or mathematics. And yet the isolation that produced such a distinctive esthetic in the arts certainly had no less an impact on these fields. The stylistic form of the impact on geometry will gradually become apparent to readers who delve into the mathematical aspects of this book, but it isn't coincidental that many sangaku problems resemble origami designs, nor that the practice of hanging the tablets began precisely during the Genroku, for, as we will see shortly, it was in the mid- to-late seventeenth century that traditional Japanese mathematics began to flourish.
Regardless of the formal developments in mathematics at the time, Western readers invariably want to know how the strange custom of hanging tablets in shrines and temples arose. In the context of Japan, it was fairly natural. Shintoism, Japan's native religion, is populated by "eight hundred myriads of gods," the kami, whose spirits infuse everything from the sun and moon to rivers, mountains, and trees. For centuries before sangaku came into existence, worshippers would bring gifts to local shrines. The kami, it was said, love horses, but horses were expensive, and a worshipper who couldn't afford to offer a living one might present a likeness drawn on a piece wood instead. In fact, many tablets from the fifteenth century and earlier depict horses.
And so it would not have seemed extremely strange to the Japanese to hang a mathematical tablet in a temple. We cannot say exactly in what year, or even decade, the tradition began, but the oldest surviving sangaku has been found in Tochigi prefecture and dates from 1683, while the nineteenth-century mathematician Yamaguchi Kanzan, whose travel diary we excerpt in chapter 7, mentions an even older tablet dating from 1668; that one is now lost. Over the next two centuries the tablets appeared all over Japan, about two-thirds in Shinto shrines, one-third in Buddhist temples. We do not know how many were originally produced. From sangaku mentioned in contemporary mathematics texts, we are certain that at least 1,738 have been lost; moreover, only two percent of the tablets recorded in Yamaguchi's diary survive. So it is reasonable to guess that there were originally thousands more than the 900 tablets extant today. The practice of hanging sangaku gradually died out after the fall of the Tokugawa shogunate, but some devotees continued to post them as late as 1980, and sangaku continue to be discovered even now. In 2005, five tablets were found in the Toyama prefecture alone. The "newest" one was discovered by Mr. Hori Yoji at the Ubara shrine and dates from 1870. Two problems in chapter 4 are taken from the tablet and we present a photo of it in the color section, color plate 13.
Most sangaku contain only the final answer to a problem, rarely a detailed solution. (In Sacred Mathematics we usually give both answers and solutions, many drawn from traditional Japanese texts.) Apart from considerations of space, there seems to have been a certain bravado involved: Try this one if you dare! Nevertheless, as you will discover yourself from reading the inscriptions, the presenters of sangaku also took the spiritual, and even religious, aspect of the practice seriously, seeing nothing odd in offering a tablet to God in return for progress in mathematics. But just who were the creators of sacred mathematics? Sangaku are inscribed in a language called Kanbun, which used Chinese characters and essentially Chinese grammar, but included diacritical marks to indicate Japanese meaning. Kanbun played a role similar to Latin in the West and its use on sangaku would indicate that whoever set down the problems was highly educated. The majority of the presenters, in fact, seem to have been members of the samurai class. During the Edo period most samurai were not charging around the countryside, sword in hand, but worked as government functionaries; many became mathematicians, some famous ones. Nevertheless, the inscriptions on the tablets make clear that whole classes of students, children, and occasionally women dedicated sangaku. So the best answer to the question "Who created them?" seems to be "everybody."
While contemplating this lesson, let us paint a fuller picture of the context in which sangaku were created by backing up as far as possible and briefly exploring the development of Japanese mathematics.
The Age of Arithmetic
The early history of Japan is inextricably bound up with that of China, from which it imported much of its culture, the Buddhist religion, as well as its system of government. This is true of Japanese mathematics as well; however, our knowledge about the state of mathematics in Japan prior to the eighth century is almost nonexistent. Perhaps the only definite piece of information from the earliest times is that the Japanese had some system of exponential notation that could be used for writing high powers of ten, similar to what Archimedes employed in the Sand Reckoner. Traditionally, the system was in place before the legendary Jimmu founded Japan in the seventh century b.c., but the date and the exact nature of the system are open to dispute.
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Excerpted from Sacred Mathematicsby Fukagawa Hidetoshi Tony Rothman Copyright © 2008 by Princeton University. Excerpted by permission.
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