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Abstract. Sangaku is a uniquely Japanese form of doing mathematics in which Euclidian geometry is presented artistically and with great elegance.
Mathematics in Japan was traditionally confined to the study of Chinese classics. The government saw mathematics merely as a tool helpful to certain areas of administration and hence the mathematical education was offered to a handful of officials. The curriculum focused on the problem of taxation and consisted primarily in methods for calculating surfaces and basic arithmetic. The government office that oversaw this purely practical mathematical education was the aptly named “The Ministry of Arithmetical Intelligence.”
In absence of any proper teachers, the cornerstone of mathematical education was the Chinese classic - “Nine Chapter on Mathematical Art”(1). The nine chapter is question are:
“Nine Chapter on Mathematical Art” (Image 1)
“Nine Chapter on Mathematical Art” (Image 2)
“Nine Chapter on Mathematical Art” (Image 3)
“Nine Chapter on Mathematical Art” (Image 4)
At the start of the XVII century Tokugawa Ieyasu(3) becomes shōgun and established a dynasty that will rule until the end of the XIX century. One of the policies that marked the rule of the Tokugawa shogunate(4) was Sakoku (鎖国) - “Closed country” - a strict isolationist policy that lasted from 1633 to 1853 when it was abolished under the American threat of war. During this period a distinct Japanese kind of mathematics - Wasan(5) - emerges and, due to almost complete isolation of the country, is developed independently of Western mathematics” - Josan.
Below are some books that illustrate the general mathematical activity of that period.
The first Japanese treaties on mathematics - Jinkōki[1] is written in 1643.
Jinkōki - Calculating on soroban (Japanese abacus).
Kigenkai
A lot of text were practical in nature such as Kiku yoho kuden shiroku[2] - a text on surveying which was worked on the series of successive scholar up to the late XVIII century.
Shoho kongen[3] - a treatise on geomtery.
Riches of Jinko-ki[4]
Riches of Jinko-ki - Calculating the weight of an elephant using the Archimedean method.
Sanpo dojimon - “Questions Children Ask About Mathematics”[5] - a classic of Japanese pedagogy of mathematics.
Sanpo dojimon - How far is the sun?
Sanpo dojimon - How high is the kite?
Sanpo dojimon - Calculating distance.
Sanpo dojimon - Use of theodolite.
Sanpo dojimon - Use of theodolite.
The strict isolationist period of Sankoku was a period of cultural flourishing of Japan. Arts like tea ceremony, ikebana, haiku, igo, etc. were either established or were reaching new heights. The warrior class - the samurai (侍) - became the new de facto aristocracy. No longer needed as soldiers, the government reassigned many to civil administration. However, many were unsatisfied with new bureaucratic assignments and decided (if permitted) to become itinerant teachers. The society was ready for the explosion of learning and Japan would soon become a country with one of the highest literacy rate. Like monasteries and churches in Europe, Buddhist and Shino monasteries and temples have become centers of learning and academic gatherings of scholars.
The great expansion of learning during the Edo period was additionally helped by the notion of Bunburyodo (文武両道) - The Way of Sword and Pen” - a samurai ideal of concurrent development of martial skill and scholarly and artistic pursuits.
gemini://tilde.club/~filip/images/sangaku/bunburyodo.jpg
The government helped foster this ideal both by promoting cultural activities and by forbidding mercantile and other ignoble work. One of the important pieces of legislation in which this can clearly be seen is Buke shohatto (武家諸法度) of 1615. Among it points are:
Special aspect of Wasan is Sangaku(6) - geometrical problems (often with solutions) and theorems calligraphically written on wooden tables which were often painted and engraved. They were offered to Shinto shrines and Buddhist temples both to announce the new discovery and to thank the gods for the newfound knowledge.
The subject matter of Sangaku is closest to what we label as Euclidian geometry. Wasan didn't know of analitical geometry, differential and integral calculus. The text of Sangaku was written in Kanbun(7) - a Japanese from of Classical Chinese.
Sangaku was practiced by men of all ages and social classes.
Sangaku by 16 year old Tanabe Sogetosi.
After the fall of the Tokugawa shogunate, the new government forbade Wasan, and with it Sangaku, as it thought it hindered the rapid modernization it set its sights on. Teacher who thought Wasan had their licenses taken and in some cases were imprisoned. About 900 Sangaku from the Edo period have survived.
Some examples of Sangaku:
Euclidian geometry beautifully and artistically presented should be reason enough. Areas in which Truth and Beauty are simultaneously manifested are always worthy of attention, specially as they are less frequent in modern times than they ought to be given the possibilities available.
If more reasons are needed, the following two quotes present views worth considering.
"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." - Sacred Mathematics: Japanese Temple Geometry[6]
...I always think how different everything would be if we in the Orient had developed our own science. Suppose for instance that we had developed our own physics and chemistry: would not the techniques and industries based on them have taken a different form, would not our myriads of everyday gadgets, our medicine, the products of our industrial art - would they not have suited our national temper better than they do? In fact our conception of physics itself, and even the principles of chemistry, would probably differ from that of Westerners; and the facts we are now taught concerning the nature and function of light, electricity, and atoms might well have presented themselfes in a different form. - Junichiro Tanizaki[7]
Problem 1: Let r_1, r_2, r_3, and r_4 be the radii of circles O_1, O_2, O_3, and O_4. Show that 1/r_1+3/r_3=3/r_2+1/r_4.
Problem 2: Find r_n.
Problem 3: Peacocks tail. Let the light blue circles be of radii R, red of radii t, and blue of radi t'. Show that t=t'=R/6.
Sangaku problem 3 - Peacocks tail.
Problem 4: Express the radius of the smaller circle in terms of the radius of the largest.
Sangaku probmel 4 - Solution: R/5.
One of the most beautiful theorems of plain Euclidian geometry is the “Japanese theorem” so called as it was clearly a staple of Wasan.
Japanese theorem: For every triangulation of a cyclic polygon, the sum of inradii of triangles is constant.
If Wasan continued to develop one can only speculate what interesting mathematics it would discover. While behind the global mathematical knowledge it did arrive at some result much earlier. A good example is Soddy's hexlet.
Soddy's hexlet (1937): Given three spheres that are tangent to one another in three distinct points, there exist six spheres that are tangent to them and form a chain.
Here is a Sangaku demonstrating the Soddy's hexlet well over a century before Soddy.
Samukawa Shrine Sangaku from 1822. demonstrating the Soddy's hexlet.
A modern form that might be likened to Sangaku are so-called “Proofs without words.”
1/3 is a sum of powers of 1/4.
Enthusiast for history of mathematics in Japan are working to preserve the remaining Sangaku as well as to make replicas. They also create new ones that in the same spirit of the original Sangaku.
gemini://tilde.club/~filip/images/sangaku/today.jpg
gemini://tilde.club/~filip/images/sangaku/new-3.jpg
gemini://tilde.club/~filip/images/sangaku/new-2.jpg
The practice is still lingering on among pupils:
gemini://tilde.club/~filip/images/sangaku/new-1.jpg
gemini://tilde.club/~filip/images/sangaku/new-4.jpg
(1) “Nine Chapter on Mathematical Art” (九章算術) - One of the earliest surviving mathematical texts from China whose last major version dates from the II century. ). Unlike ancient Greek mathematics that focuses on deduction from the initial set of axions, it lays out an approach to mathematics that focuses on finding the most general problem solving method. Its influence on mathematical education in East Asia is comparable to that of Euclid's Elements in Europe.
(2) Note that the Chinese text precedes Gauss for about 1500 years.
(3) Tokugawa Ieyasu (徳川家康) (1543 - 1616), first shōgun of the Tokugawa shogunate.
(4) Tokugawa shogunate, Edo shogunate or Edo period, lasted from 1603 until the Meiji Restoration in 1868. Coming after the long period of intense strife (Sengoku Jidai - “Warring States period”) it sought to maintain peace and stability by adhering to strict Confucian principles of social order.
(5) Wasan (和算) - Japanese mathematics or literally “Japanese calculation.”
(6) Sangaku (算額) - Literally “calculation tablet.”
(7) Kanbun (漢文) - Form of Classical Chinese used in Japan from the Nara period to the middle of the XX century. It was the general literary and administrative writing style.
[1] Yoshida Mitsuyoshi, Jinkōki (1643).
[2] Shimizu Sadanori, Kiku yoho kuden shiroku.
[3] Matsunaga Yoshisuke, Shoho kongen.
[4] Anon., Riches of Jinko-ki (1778).
[5] Murai Chuzen, Sanpo dojimon - “Questions Children Ask About Mathematics” (1784).
[6] Fukagawa Hidetoshi and Tony Rothman, “Sacred Mathematics: Japanese Temple Geometry”.
[7] Junichiro Tanizaki, “In Praise of Shadows” (1977).
<1> Japanese Mathematics in the Edo Period - National Diet Library of Japan
