Sangaku

Japanese Mathematical Development

during the Edo Period with Applications to Modern JuJitsu

A Japanese Cultural Study

By Ed Birrane

Table of Contents

1 TIMELINE 3

2 INTRODUCTION 4

3 MATHEMATICS 5

4 MATHEMATICAL PARADISE LOST 6

5 THE ERA OF GENROKU 10

6 WASAN AND YOSAN 11

7 SANGAKU 15

8 MODERN, WESTERN JU-JITSU 19

9 CONCLUSION 22

10 SAMPLE SANGAKU 23

1 Timeline

Dates (AD) Description

1338 Start of the Ashikaga Shogunate. This shogunate begins the dark age of Japanese science.

1573 The end of the Ashikaga Shogunate.

1600 Tokugawa Ieyasu wins the battle of Sekigahara, defeating the Hideyori loyalists.

1603 The start of the "Edo Period" in Japan, under the Tokugawa shogunate.

1615 Ieyasu captures Osaka castle, effectively eliminating all political opposition, and ushering in centuries of peacetime in Japan.

1627 Koyo Yoshido wrote the Jinko-ki (translated as small and large numbers), a work that quickly became synonymous with "arithmetic" throughout Japan.

1633 Shogun Lemitru officially forbids travel outside of Japan. Trade is only allowed with China and the Netherlands through the sole port of Nagasaki.

1639 The start of sakoku ("national seclusion") in Japan.

1642 The birth of Kowa Seki, the greatest mathematician of the Japanese 17th century.

1683 Earliest reported Sangaku tablet, in Tochigi Prefecture. Their rise is not accidentally related to Kowa Seki's influence on Japan.

1854 The period Sakoku comes to a forceful end when Commodore Perry docks US warships in Japan.

1867 The official end of the "Edo" period in Japan.

2 Introduction

"What is the sound of one hand clapping?" asks the Buddhist Koan invented by Hakuin Ekaku Zenji (1686-1768) . To some, the answer to this riddle is an outstretched palm thrust foreword followed by the sacred "Mu". To others, the riddle is unanswerable, setting the questioner on a journey far more rewarding than the immediate question. While the answer to the riddle may never be known, the Zen pupil may, eventually, come to understand the Koan. An overriding principle of "ancient" Japanese theology, science, and religion stresses the journey to enlightenment as much as the actual attainment of enlightenment. Within that culture learning was a process - not an instantaneous event.

There is a tendency in the West to seek immediate gratification. This tendency may explain why internal Koans such as "one hand clapping" are so often misunderstood. The idea of a learning process for seemingly atomic concepts strikes a Westerner as repetitive and inefficient. However, these methods of education in ancient Japan were neither repetitive nor inefficient. Often, they were integral to keeping certain arts and knowledge alive.

This cultural study seeks to explain the rediscovery of knowledge in ancient Japan and how that rediscovery was aided by the view of learning as a process and not an instantaneous event. Specifically, this cultural study will show the evolution of mathematical prowess in Japan from 6 BC to the 19th century.

Zenji was a critically influential Zen teacher --

3 Mathematics

One obvious method of understanding the educational environment of an ancient culture is by looking at how knowledge advances within that culture. Generally, a common measure of a culture's intellectual potential is the collective mathematical ability of the population. Just as some types of knowledge lend themselves to change, others fiercely resist it. Mathematics is called the "universal language" precisely because it deals with provable theorems that are not subject to interpretation. Mathematics is only considered "creative" at the cutting edge of knowledge and even that creativity is fleeting as hypotheses are proven and disproved. It is precisely that durability of mathematical knowledge that makes it a good measure of a culture's advancement.

Knowledge within Japanese culture, from medicine to war craft, often and easily incorporated mystery and magic in ways that seem simple-minded to the average Westerner. This flows from a preference for internalizing discovery and enlightenment through self-meditation -- from the belief that answers come from within. This is, of course, quite different from the "scientific method" that has guided scientific thought in the West. There exists a question, then, of how an ancient culture so steeped in mystery, philosophy, and ethnicity could develop strong ability in a discipline as firm and scientific as mathematics.

Japanese culture was able to blend creativity, national pride, and the universal language of mathematics. Mathematical theorems were no longer means to an end and "calculate the area of this circle" became as much of a spiritually enriching exercise as answering the Koan "what is the sound of one hand clapping", or trimming a bonsai tree. To understand, then, mathematical development in feudal Japan we must understand the cultural mystery, philosophy, and ethnicity of feudal Japan.

4 Mathematical Paradise Lost

Some of the most advanced mathematical concepts of the ancient world were developed in China decades (and sometimes centuries) before they were discovered in the West. Reliable accounts indicate that Chinese mathematical prowess migrated into Japan along with Buddhism in the mid-6th century BC. "Judging from the works that were taught at official schools at the start of the eighth century, historians infer that Japan had imported the great Chinese classics on arithmetic, algebra and geometry."

The earliest of these classics is the Chou-pei Suan-ching , which contains an example of the "Pythagorean theorem" centuries before Pythagoras. This famous diagram, from the Chou-pei is shown in Figure 4-1.

Figure 4?1 - The hsuan-thu

A much more influential book on mathematics, the Chiu-chang Suanshu contains more complicated instruction, such as finding areas of different shapes (circles, triangles). The Chiu-chang Suanshu may have been written as early as the 3rd century BC. If this is the case, this document contains the first documented mention of negative numbers. "Other important Chinese math texts include the Mathematical Classic of Sun Tzu (Sun Tzu Suan Ching) written in the 3rd century A.D., and The Ten Mathematical Manuals (Suanjing Shi Shu). The 13 century text, Detailed Analysis of the Mathematical Rules in the Nine Chapters (Hsiang Chieh Chiu Chang Suan Fa), proved the theory know as "Pascal's Triangle" 300 years before Pascal was born."

China exported a wealth of mathematical information into Japan before the turn of the millennium. This knowledge was, itself, lost in China over the centuries. For example, "the despotic emperor Shih Huang-ti of the Ch'in dynasty (221-207 B.C.) ordered the burning of books in 213 B.C."

Despite a wealth of learning imported from China mathematical prowess did not take root in Japan. For almost two thousand years the country languished in a mathematical dark age. During the Ashikaga shogunate (1338-1573) it was said that there could hardly be found in all of Japan a person who could divide .

It is incredible, from a modern perspective, to fathom the loss of such knowledge, and to envision the kind of future that could have unfurled for Japan had the Japanese better cultivated their gifts of mathematical knowledge from China. Of course, a modern view does not take into account the fight for survival that marked each day in feudal Japan. Even China eventually lost its mathematical knowledge at the hands of war and tyranny.

Japan is, of course, famous for the various open-handed and weapons-based martial arts that formed through centuries of bloody civil feuds. In order for "luxury" knowledge to evolve people needed to first stop worrying about sudden death. This end to life-and-death struggle began to abate in the 17th century.

In 1600, Tokugawa Ieyasu won the battle of Sekigahara, defeating supporters of Hideyori, his political opponent. This defeat seriously crippled his political opposition and by 1603 he was appointed shogun by the emperor. In victory, Ieyasu was generous to all of his daimyo (loyal vassals who had supported him before the battle of Sekigahara) and gave them strategic pieces of land. A tight controller of the country, however, Ieyasu declared that all of his daimyo spend every second year in the capital of Edo (modern day Tokyo). Such a decree imposed huge expenses on the daimyo, preventing the formation of any significant opposition . Thus began, in 1603, the "Edo Period" in Japan.

Ieyasu captured Osaka castle in 1615, destroying the rival Toyotomi clan and effectively destroying the remainder of his enemies. Japan was unified under his shogunate. For the first time in cultural memory Japan knew peace. The Edo period would establish a Tokugawa line that would last for almost three hundred years.

In 1615 this new, peaceful Japan had a long road ahead of it. There was no knowledge foundation to build upon, as the country was just emerging from a scientific dark age that had lasted almost two thousand years. Throughout the Edo period there existed no college or universities in Japan. Teaching, as it evolved, occurred in private schools or, more often, Shinto shrines and Buddhist temples. Over time, the general population learned to read, write, and use the abacus and the Edo period encompassed Japan's greatest renaissance (the Genroku era). In order to reach that renaissance, however, Japan needed to go through some transitions.

Scientific American Article :

Which translates to "Arithmetic Classic of the Gnomon and the Circular Paths of Heaven"

Which translates to "Nine Chapters on the Mathematical Art"

The raise of the daimyo was what contributed to the fall of the Ashikaga Shogunate (http://www.wikipedia.org/wiki/Ashikaga_Shogunate)

5 The Era of Genroku

Japan achieved its renaissance in pieces, transitioning slowly from the poverty-stricken fiefdoms of the past two thousand years. All cultural and academic pursuits not required for everyday survival were in sore need of development. All of the hard sciences enjoyed by Europe at the time were largely unknown. Artistic expression, certainly not valued as a farming aid, was not developed or valued in the "previous" society.

Without doubt, the cultural and scientific vacuum in Japan was in danger of being filled from external sources. In order to combat the possible dilution of Japanese tradition and culture, the Tokugawa shogun Lemitsu forbade travel outside of Japan in 1633. This edict effectively stopped all trade with Japan, with the small exception of trade with China and the Netherlands and then only through the port at Nagasaki. Additionally, all foreign books were banned from the country and massive book burnings occurred in that time period.

The decree of the shogun was completely implemented by 1639 when Japan entered into its period of sekoku ("national seclusion"). This period would last almost as long as the Edo period, ending only when Commodore Matthew C. Perry forced open Japan's borders in 1854.

This situation could have been similar to the devastating loss of knowledge in the 3rd century BC in China. Far from crippling Japan's growth, however, the period of sekoku allowed Japan the national introspection and meditation required to grow from within. This allowed a cultural renaissance that not only respected Japanese traditions, religion, and culture, but also to develop arts that were unique in the world. This period of renaissance was known as the era of Genroku. This era established many of the cultures and traditions that are casually associated with Japan today.

The Edo period saw the creation of the four-caste system most often associated with Japan, composed of the samurai, peasant, artisan and merchant castes. The samurai caste, no longer required to fight for a living, began to expand their pursuits to include literature, philosophy, and the arts. During this era haiku developed into a fine art form. No and Kabuki theatre styles reached the pinnacle of their development. The tea ceremony and flower arranging also reached new heights in cultural status and importance .

6 Wasan and Yosan

The concept of sankoku was important in maintaining the integrity of Japanese culture and tradition in areas of development affected by culture and tradition. Mathematics, however, is often called the 'universal" language because it is not affected by culture and tradition. A mathematical proof is a proof regardless of the background of the mathematician. Yet, Japan included hard sciences, like mathematics, in its era of national seclusion and did, indeed, attempt to affect its mathematical development.

The first significant development in the area of Japanese mathematics occurred in 1627 when the mathematician Koyo Yoshida wrote a booklet called the Jinko-Ki (literally translated as "small and large numbers"). This booklet became the most widely read (indeed, one of the only available) manuscripts on mathematics and quickly the term Jinko-ki became synonymous with the term "arithmetic"

. Figure 6?1 - The Jinko-Ki

Koyo Yoshida was the pupil of Kambei Mori, who prospered around 1600. Mori's work was centered primarily on mathematics surrounding the abacus. Together, the works of Mori and Yoshida changed the focus of "mathematics" from logic to computation. This new focus of mathematics as a discipline rooted in computation, and not logical musing, marked the resurgence of Japanese mathematics apart from Japanese mathematical philosophy. This "new math" was termed Wasan ("native Japenese mathematics") and was most probably directly fathered by the works of Yoshida and Mori.

Wasan was in direct contrast to Yosan ("Western" mathematics). While Mori and Yoshida performed their initial work in the 1620's, shogun Lemitsu instituted sokoku in the 1630's. It is certainly possible that external knowledge influenced Wasan in the early years, there is no doubt that after 1639 Western influences in mathematics were abolished. The fact that there exist two different terms for the two different "types" of mathematics (an otherwise "universal" language) enforces this separation.

The establishment of Wasan provided the computational foundation for real mathematical development within Japan. The period of sekoku ensured that this development would occur without foreign influence.

The late 17th and early 18th century saw the serious growth of math in Japan under the direction of Kowa Seki (1642-1718). Seki is often described as the "Newton" or "Leibnitz" of Japan. In truth, he may have been more influential to math than either Newton or Leibnitz. Scholars agree that his "theory of determinants" is more powerful than Leibnitz's, and Seki's theory pre-dates Leibnitz's by more than a decade. Seki was one of the most powerful advancers of Wasan in the ancient Japanese world.

Figure 6?2 - Kowa Seki

"Seki was the first person to study determinants in 1683. Ten years later Leibniz, independently, used determinants to solve simultaneous equations although Seki's version was the more general. Seki also discovered Bernoulli numbers before Jacob Bernoulli. He wrote on magic squares, again in his work of 1683, having studied a Chinese work by Yank Hui on the topic in 1661. This was the first treatment of the topic in Japan.

In 1685, he solved the cubic equation 30 + 14x - 5x2 - x3 = 0 using the same method as Horner a hundred years later. … Secrecy surrounded the schools in Japan so it is hard to determine the contributions made by Seki, but he is also credited with major discoveries in the calculus which he passed on to his pupils."

One of Seki's most important contributions to Wasan was the enri. The enri ("circle principal") was a method used to calculate the area of a circle. Whereas the "West" had the "Method of Exhaustion" which used n-sided polygons to approximate a circle, enri divided a circle into n rectangles. The method used by Seki is a crude form of integral calculus that was later extended to work with spheres and ellipses. Until recently, there was scholarly argument that Seki has invented Calculus itself:

"Some Japanese historians advocated that the infinitesimal calculus was invented by Seki, Takebe and a few others in the form of Enri, but this claim was an exaggeration and has now been abandoned. Despite its originality, the achievements of Wasan form a group of fragmentary results, as outlined in this article. Even some fundamental concepts in analysis, including the variable, the function and differentiation, never appeared, to say nothing of the fundamental theorem of calculus, that is the inverse relationship between integration and differentiation, and the mathematical natural philosophy developed by Descartes, Newton, Leibniz and others."

Wasan traditionally deals more with circular objects while Yosan focused more on rectangular objects. It is unclear if the circular shape preference existed in Wasan because of more advanced circular theory, or if the more advanced circular theory existed because of the interest. Perhaps it was the Japanese culture's preference for parallels with nature (which seems to prefer smooth not sharp shapes), or the martial arts, which prefer circular, over rectangular, motion. Regardless, Japanese mathematical geometrical problems are skewed towards problems dealing with circles far more than Western geometrical problems.

Scientific American Article :

7 Sangaku

By the end of the 17th century Wasan was firmly entrenched in Japanese culture, and it was growing just as all other artistic and scientific pursuits were growing during the era of Genroku. It is little surprise, then, that this time period saw the introduction of cultural artifacts of Wasan, the output becoming more popularized than simply equations in manuscripts.

In 1683, in the Tochigi prefecture, the earliest known Sangaku tablet was created. A Sangaku tablet is a wooden tablet , usually hung from the ceilings of Shinto or Buddhist temples, upon which colorful mathematical theorems were painted. These theorems dealt predominantly with Euclidian geometry and, true to Wasan preferences, mostly dealt with circles and ellipses. Most Sangaku contain only theorems and not their proof. The tablets, did, however, contain the theorem's presenter and the date of the carving.

Some of the Sangaku describe simple theorems, understood by any modern high school student. Other theorems require proof using math (such as integral calculus) that, according to understood history, was not present when the tablets were created. Still other tablets contain theorems that are unable to be proven using the most advanced modern mathematics.

Given that Sangaku refer mostly to the circular geometry problems surrounding Seki's enri principle it should be no surprise that they appeared in Japan during Seki's lifetime. Scholars wonder whether Seki and Sangaku were indicative of the classic "chicken and egg" problem, although many agree that it is probable that Seki influenced Sangaku far more than Seki was influenced by Sangaku.

The method of displaying theorems on wood carvings hung in temples has its history in Shintoism. In the "religion" of Shintoism there exist over 800 myriads of gods collectively known as the Kami. Tradition relates that the Kami had a love of horses and often required (or were gifted) horses as temple offerings. Shintoists who were poor, however, could not afford to give up their horses (if they had them at all) to temple offerings. As such, impoverished worshipers took to carving likenesses of horses onto wooden tablets and using those tablets as offerings to the Kami. Sangaku are found in both Shinto shrines and Buddhist temples, but are found in twice as many Shinto shrines as Buddhist temples. It is probable that Sangaku mimicked the practice of impoverished Shintoists to display hand-made horse carvings. It is known that some Sangaku were created by the poorer castes in the Edo period.

Some Sangaku theorems are elementary and solved in only a few lines -- advanced mathematicians would hardly memorialize such simple problems. A Sangaku from the Mie Prefecture was inscribed with the name of a merchant, one of the lowest castes in the Edo period. Still other Sangaku were inscribed with the names of women and children (aged 12 - 14) as their authors. Clearly, Sangaku was something available to all people in Edo Japan.

The samurai remained the dominant creators of Sangaku, consistent with their status of the educated and artistic caste in Japan. A majority of Sangaku are inscribed in Kambun, an archaic Japanese dialect related to Chinese. Kambun was the equivalent of Latin in Europe, used during the Edo period for scientific works and known predominantly by only the most educated castes. Seeing a tablet inscribed in Kambun gives a strong indication that the author was a samurai (or otherwise highly educated).

While Sangaku may have derived from Shinto horse carvings, they were not enjoyed only by the poor or lower castes. Also, while Shinto carvings had a clear religious significance (offerings to the Kami), Sangaku had no such immediate religious affiliation. Without a clear religious affiliation, there exists a question of why Sangaku are only found in Shinto and Buddhist temples.

The Edo period saw an increase in education and artistic development, but there existed no colleges and universities in Edo Japan. Learning occurred only in private schools and temples, and it is conceivable that temples received financial compensation for their tutelage. Sangaku could have been advertisements for the types of education, and quality of mathematicians, residing at the temple. The colorful and intricate tablets surpassed what was necessary to simply convey formulaic knowledge. Their artistry, absent any provable religious significance, leads scholars to believe that these works were advertisements. It is probable that these tablets were used in the education of the temple congregation.

However, the desire to popularize the education of Euclidian geometry is not the only impetus to create Sangaku. Popularizing the education of mathematics ensured an interest in the generalized science of mathematics. As more students were created more teachers were created and the cutting edge of the science was developed and kept alive. Japan never wanted to fall back into the scientific dark ages before the Edo period.

A 19th century diary penned by Kazu Yamaguchi lists his count of Sangaku encountered on his travels throughout Japan. He counted thousands of tablets uniformly distributed throughout Japan, in both rural and urban districts. Clearly, the Sangaku had done their job. For hundreds of years their popularity grew through the temples and shrines of Japan, informing the countryside and popularizing mathematics. The fact that the tablets rarely included the proof to the theorem carved on them may have increased their appeal.

Today over 880 Sangaku are known to exist.

Figure 7?1 - Location of Sangaku Discoveries

8 Modern, Western Ju-Jitsu

Being a student of JuJitsu in the modern, Western world is a bit like being a mathematical student during the Edo period of Japan. The modern Western world does not present the daily struggle to survive as was the case in feudal Japan. Modern opponents have guns not staffs or open hands and there is a much smaller probability of using martial art knowledge for self-defense. Certainly learning the full art of JuJitsu, once the feudal Japanese self-defense style, is "overkill" for the modern self-defense experience. The study and creation of Sangaku towards the end of the Edo period may have seemed similar to the art of JuJitsu practiced in the modern West. Sangaku theorems did not provide immediate societal benefit (they did not help crops grow), did not necessarily translate into large sums of money, and were being done by everyone in Japan from women and children to merchants and samurais. No individual Sangaku author was the sole supporter of the mathematical knowledge of the day. Collectively, however, these authors kept alive the interest and mystery of the discipline. Maintaining societal interest in the art was a critical component of ensuring the long-term viability of the art.

Martial arts are popular in Japan, with many students learning various arts from the earliest ages. Such martial disciplines are ingrained in the culture. However, there were times when martial arts were banned from practice. The multilayered history of martial arts styles in Japan is a cultural study in its own right. Many modern styles of martial art in Japan exist as modifications to older styles required to keep the style alive and popular. Dr. Jigoro Kano created Judo as a way of keeping JuJitsu popular as a sport. Regardless, Japan will likely never be devoid of martial art training.

The West, however, has a much different history regarding the martial arts. Interest in various styles will increase or decrease according to the popular culture. It is more important in the West to strive for the popularity of the martial arts as a cultural movement, a sporting movement, and a self-defense movement. While, certainly, if one student leaves the discipline of JuJitsu, the art form will continue within the West. However, if one student is especially avid and extroverted in the art, they have a true ability to enhance the popularity and exposure of the art.

In this sense, JuJitsu in the modern western world is like the ancient Sangaku: both were disciplines meant to enrich their studiers. Both strived to keep knowledge alive while also popularizing themselves in an alien culture. Both disciplines could be positively affected by even small numbers of students and teachers.

Even the method of solving a Sangaku problem is similar to learning a specific JuJitsu technique. The tablet often did not contain the proof of the theorem, just the theorem itself. Often, a JuJitsu student is shown a technique (roughly where to stand, where the Uke should land, what genre of submission may be entered) without specific instructions. Beginning mathematical students might need help understanding where to start a mathematical proof just as beginning JuJitsu students may need help fitting in to certain positions, achieving the appropriate kuzushi, shifting weight correctly, etc… Eventually, an accomplished mathematician may be given only a theorem and expected to prove it without aid from the author, as a sign of prowess. This is similar to our Nidan black belt tests where Shodans are required to fit into various techniques from various "set-ups". The Shodan is given a result, and must reach that result on their own, as a sign of his or her prowess.

The ability to creatively express oneself in the execution of a JuJitsu technique is a powerful draw to the martial art, and introduces personalized mutations required to evolve and perfect the discipline. While Sangaku problems may not have shown the same ability to evolve and mutate (mathematics does not evolve in that sense), proofs to Sangaku theorems certainly may show certain styles and personalization.

American Article :

9 Conclusion

Japan was able to remove itself from a scientific dark age while maintaining its national pride and culture. This process was achieved in part through the patient skills of several generations of scholars throughout the Genroku period. This goal was also achieved through the introspective labor of countless hobbyists who maintained the popularity of the sciences through the artistry of things like the Sangaku. The development, then, of the science of mathematics in Japan, and probably all sciences in Japan, was a process, not an event. Once theorems were made, they came alive, needing to be fed by public interest less they perish as China's mathematical works had perished two thousand years before.

In our modern culture of scientific popularity it is the more subtle arts, such as JuJitsu, that require that same sustenance. A study of how mathematics was brought to flourish in a culture not conducive to its existence is also a study of how martial arts may be brought to flourish in a modern, Western world. Just as an ancient farmer struggling to survive could not understand how the "Pythagorean Theorem" could help him eat, so many modern Westerns cannot understand how a martial art can help them evolve. As practitioners of the arts, we must create our own Sangaku in the hopes of ever expanding our skills and pathways to inner peace. As practitioners of JuJitsu we must find way to ever increase the public interest in our art in ways that maintain the integrity of the art.

10 Sample Sangaku

All of these examples can be found at:

Sangaku problems typically involve multitudes of circles within circles or of spheres within other figures. This problem is from a sangaku, or mathematical wooden tablet, dated 1788 in Tokyo Prefecture. It asks for the radius of the nth largest blue circle in terms of r, the radius of the green circle. Note that the red circles are identical, each with radius r/2. (Hint: The radius of the fifth blue circle is r/95.) The original solution to this problem deploys the Japanese equivalent of the Descartes circle theorem.

Here is a simple problem that has survived on an 1824 tablet in Gumma Prefecture. The orange and blue circles touch each other at one point and are tangent to the same line. The small red circle touches both of the larger circles and is also tangent to the same line. How are the radii of the three circles related?

This striking problem was written in 1912 on a tablet extant in Miyagi Prefecture; the date of the problem itself is unknown. At a point P on an ellipse, draw the normal PQ such that it intersects the other side. Find the least value of PQ. At first glance, the problem appears to be trivial: the minimum PQ is the minor axis of the ellipse.

This beautiful problem, which requires no more than high school geometry to solve, is written on a tablet dated 1913 in Miyagi Prefecture. Three orange squares are drawn as shown in the large, green right triangle. How are the radii of the three blue circles related?

In this problem, from an 1803 sangaku found in Gumma Prefecture, the base of an isosceles triangle sits on a diameter of the large green circle. This diameter also bisects the red circle, which is inscribed so that it just touches the inside of the green circle and one vertex of the triangle, as shown. The blue circle is inscribed so that it touches the outsides of both the red circle and the triangle, as well as the inside of the green circle. A line segment connects the center of the blue circle and the intersection point between the red circle and the triangle. Show that this line segment is perpendicular to the drawn diameter of the green circle.

This problem comes from an 1874 tablet in Gumma Prefecture. A large blue circle lies within a square. Four smaller orange circles, each with a different radius, touch the blue circle as well as the adjacent sides of the square. What is the relation between the radii of the four small circles and the length of the side of the square? (Hint: The problem can be solved by applying the Casey theorem, which describes the relation between four circles that are tangent to a fifth circle or to a straight line.)

From a sangaku dated 1825, this problem was probably solved by using the enri, or the Japanese circle principle. A cylinder intersects a sphere so that the outside of the cylinder is tangent to the inside of the sphere. What is the surface area of the part of the cylinder contained inside the sphere?

This problem is from an 1822 tablet in Kanagawa Prefecture. It predates by more than a century a theorem of Frederick Soddy, the famous British chemist who, along with Ernest Rutherford, discovered transmutation of the elements. Two red spheres touch each other and also touch the inside of the large green sphere. A loop of smaller, different-size blue spheres circle the "neck" between the red spheres. Each blue sphere in the "necklace" touches its nearest neighbors, and they all touch both the red spheres and the green sphere. How many blue spheres must there be? Also, how are the radii of the blue spheres related?

Hidetoshi Fukagawa was so fascinated with this problem, which dates from 1798, that he built a wooden model of it. Let a large sphere be surrounded by 30 small, identical spheres, each of which touches its four small-sphere neighbors as well as the large sphere. How is the radius of the large sphere related to that of the small spheres?