The Path of Numbers

What if higher mathematics—filled with intricate polynomials of high degree and complex geometric problems—were a popular art form and entertainment for samurai, merchants, women, and teenagers? Or even a religious practice? An art where the goal was not just to solve calculations correctly, but to do so beautifully, harmoniously, in a way that conveyed an idea or a philosophy—like haiku or ukiyo-e? Surely, such mathematics would not have been as practical in the physical sciences as European mathematics. But could it have functioned in this way at all? As it turns out, yes—it did. Today, we delve into wasan (和算), the Japanese art of refined mathematical calculations from the Edo period (1603–1868). It was a passion, a ritual, and an intellectual pursuit that matched the level of European achievements of the 17th and 18th centuries—and in some cases, even surpassed them by decades.

On the bustling streets of Edo, among merchant stalls and incense-scented temples, wasan was woven into the everyday life of its people. In teahouses, masters of calligraphy and mathematics engaged in heated discussions over puzzles inscribed on thin wooden tablets—sangaku (算額). Imagine a young samurai, between sword-training sessions, tackling a problem involving inscribed circles. He slides wooden counting rods (sangi) across the board, searching for harmony in numbers, while his rivals—merchants and monks—watch in silence. Perhaps he will find a solution so elegant that it deserves to be left at the temple, so future generations may admire his genius. Elsewhere, a wealthy merchant hangs his sangaku at a shrine, proving that his keen eye extends beyond trade and into mathematics. Perhaps, in the shadow of the temple pillars, a young girl gazes at a problem meticulously inscribed in calligraphy, silently working through it in her mind, determined to solve it better than her father. For, in an unusual exception for Japan at the time, women were not entirely excluded from mathematics—if they were literate, they could explore its mysteries.

European mathematics developed in close connection with physics—Newton needed calculus to describe gravity, while Leibniz sought tools to express the dynamics of the world. In Japan, wasan followed a different path. It focused on the harmony of forms, on visual and conceptual elegance. This is why it was in Japan—long before Europe—that the value of π was calculated with unprecedented precision, why Seki Takakazu independently discovered the determinant in 1683—a mathematical tool that would not be formally described in Europe until a decade later. This is also why Japan developed methods for solving equations of staggering degrees—some reaching as high as… 1458! If wasan had a secret, it was not just mathematical precision but also the fusion of science with beauty. And although Japan eventually embraced Western mathematics during the Meiji period, wasan remains a fascinating testament to an era when numbers were not just tools—they were an art.

Edo, the City of Numbers – Mathematics in Everyday Japan

The sun slowly rose over Edo, casting a warm golden light on the wooden rooftops. The city awakened—crowds flowed through the streets, merchants praised their wares, the sounds of wooden cart wheels and traders' shouts blended into the morning clamor. The air was filled with the scent of steaming rice, smoked fish, and incense drifting from the temples. It was a world of movement, rituals, and daily struggles—but also a world of numbers, for Edo was a city where mathematics permeated life in surprising ways.

At one of the bustling marketplaces, under a wooden awning, a fabric vendor sat intently working on his soroban—a Japanese abacus—calculating the total cost of silk bolts purchased by a customer. His fingers moved the wooden beads with practiced ease, almost instinctively, and after a brief moment, he raised his head and announced the sum without a hint of doubt. Next to him, a tea merchant did the same—giving correct change was an everyday skill, but in Edo, numbers were more than just business. People were fascinated by them. Even a small child watching his father playfully moved beads on a miniature soroban, pretending to make calculations like the adults.

Further down the street, in a teahouse filled with the soft rustling of kimonos and the fragrance of fresh tea, two students engaged in quiet discussion. On the table lay a scroll covered in neatly calligraphed characters and geometric diagrams—a mathematics problem from a recent wasan treatise.

"If you have two intersecting circles and inscribe a third one within them, how do you calculate its radius?" one asked, tapping the drawing with his finger. His companion smiled and reached for a brush. With a practiced hand, he drew a few additional lines on a sheet of paper and began explaining the proof. Several curious patrons turned their heads toward them—discussions of mathematics were as captivating as tales of warriors and politics (but far safer).

Another scene unfolded in the courtyard of a temple. Among the prayer plaques called ema, one stood out—a sangaku, a mathematical puzzle written on a wooden board. A group of young students, fascinated by the problem, whispered among themselves, trying to find the solution. An elderly monk, well-versed in wasan, smiled and approached them.

"A beautiful puzzle," he said, pointing to the elegantly written equations. "But look here—doesn’t this seem familiar? Isn’t it a variation of a theorem recorded in the books of Master Seki Kōwa?" The students leaned in closer, their minds ablaze with a passion for learning.

Wasan was not just a field for scholars. In Edo’s courtyards, children solved simple puzzles by counting bamboo sticks or measuring the length of wooden planks. In merchant houses, young apprentices learned how to calculate percentages to prepare for managing family businesses. Even samurai, who rarely unsheathed their swords in times of peace, turned to mathematics as a form of intellectual self-discipline. And remarkably, in a society where women had limited access to formal education, mathematics was one of the few domains where some of them could participate—solving complex problems that modern mathematicians classify as "advanced university-level mathematics of the 21st century."

Edo was a city where numbers were not an abstraction—they were a part of daily life, an art, a challenge, and an intellectual game. And even if not everyone saw it as extraordinary, somewhere in a quiet temple corner, in a teahouse, or at a marketplace, someone was discovering a mathematical truth that Europe would not encounter for decades.

While it was ultimately European (and later American) mathematics that shaped modern science, in the 17th century, the rivalry between wasan and Western mathematical thought would not have had a clear winner. In the isolated Japan of Edo, ideas were emerging that not only rivaled but sometimes even preceded the achievements of European scholars.

What is Wasan?

While in Europe mathematics was becoming a tool for scientists, engineers, and natural philosophers, in Edo-period Japan, a unique, almost artistic branch of mathematics developed—wasan (和算). This word can be translated as “harmonious calculations” or simply “Japanese mathematics”. The concept of harmony (和) was not incidental—it represented both order and serenity, but it was also an ancient name for Japan itself (before Nihon, Japan was referred to as Wa—a topic we explore in detail here: Why Do We Say "Japan" While the Japanese Say "Nihon"? From Oyashima to Zipangu – A Millennia-Long Game of Telephone). Wasan was not merely a system of calculation; it was an expression of national identity, a way of thinking, and a distinct approach to mathematics that contrasted with the rationalist tendencies of the West.

Wasan and Yōsan – Two Mathematical Paths of the World

When Japan opened up to the West during the Meiji era (1868–1912), European mathematics, known as yōsan (洋算, "Western calculations"), quickly supplanted the native wasan. Why? The key difference lay in their approach to learning.

Wasan developed independently, cut off from Western influences until the 19th century. While its roots traced back to Chinese computational methods, Japanese mathematicians—especially Seki Takakazu (Seki Kōwa)—infused it with a unique character. Wasan became an autonomous mathematical system, where geometry played a central role.

Yōsan (Western mathematics), on the other hand, evolved in close connection with natural sciences—it was a tool for describing reality and predicting phenomena, from the movement of planets to fluid dynamics. Wasan, in contrast, had little direct connection to physics or astronomy; its development was driven primarily by an internal fascination with numbers and forms.

The Mathematical Art – The Aesthetic of Wasan

While in Europe, mathematics was the domain of scholars, in Japan, it was a beauty accessible to everyone. One of the most extraordinary aspects of wasan was that it was not confined to academic elites—solving mathematical puzzles was a hobby enjoyed by samurai, merchants, monks, and even, though less frequently, wealthier farmers. The best evidence of this is the sangaku (算額)—wooden tablets inscribed with geometric problems, which were displayed in temples and shrines as mathematical offerings to the deities.

Wasan problems often emphasized the elegance of the solution. Japanese mathematicians avoided overly complex algebraic equations in favor of harmonious geometric constructions, where one theorem naturally followed from another. For this reason, wasan is sometimes compared to calligraphy or poetry—what mattered was not just finding the answer, but the way it was reached.

Mathematics as a Part of Everyday Life

Wasan permeated Japanese society at a level unmatched in Europe. It was not only the domain of scholars, nor merely a practical tool for commerce, agriculture, and administration—it was also a popular pastime and a form of art. Mathematical literacy was essential for merchants, who had to quickly calculate commodity prices, predict profits and losses, and plan transportation routes. Samurai, too, managing the finances of their fiefs, used wasan daily. However, wasan was more than just a practical skill—it became a part of religious ceremonies, an artistic expression of admiration for harmony, and even a form of entertainment for teenagers and adults who were not professionally involved in mathematics.

Over time, levels of mastery in wasan were developed—from a basic level, which included the four fundamental arithmetic operations, to an advanced, master level, where problems required knowledge of differential and integral calculus (yes—there was a Japanese equivalent of these methods as well!).

Interestingly, wasan was not limited to men. Women and children also engaged in it with enthusiasm. In historical Japan, learning arithmetic was one of the few fields where they could develop themselves independently of social norms (meaning it was accessible both to the literate lower class and to women).

Although wasan ultimately gave way to Western mathematics, its influence on Japanese ways of thinking about numbers and space remains profound. It was not merely an archaic system but rather an alternative branch of mathematical evolution, one that followed its own path, developing in parallel with European science.

Even today, one can find sangaku preserved in Japanese temples—testaments to an era when mathematics was not merely a tool but an art—the art of numbers, form, and harmony.

The History of Wasan

Origins: Chinese Roots and Celestial Calendars

The history of wasan began long before the Edo period, in an era when Japan drew inspiration from Chinese scholarship. As early as the Nara (710–794) and Heian (794–1185) periods, the imperial court collected Chinese mathematical texts, and scholars attempted to adapt these methods to local needs. At this time, mathematics was closely tied to astronomical calendars—precise calculations were crucial for determining festival dates, eclipses, and lunar cycles, all of which influenced religious rituals and agriculture.

The first independently written Japanese mathematical texts appeared later, during the Kamakura (1185–1333) and Muromachi (1336–1573) periods. While they still relied on Chinese models, they gradually incorporated original solutions and interpretations. It is believed that even at this early stage, the art of calculation extended beyond mere administrative or religious needs and began to influence daily life, particularly among merchants and samurai officials.

The Golden Age of Wasan – Mathematics in the Edo Period (1603–1868)

When the Tokugawa shogunate took power in 1603, Japan entered an era of peace and stability. Edo became not only the political capital but also a center of scientific and artistic development. During this time, wasan flourished, solidifying its identity and reaching a level comparable to European mathematical achievements.

This period also saw the establishment of schools and various mathematical traditions, as well as the creation of advanced wasan treatises—some so sophisticated that they were not understood even in Europe for many years.

Seki Takakazu – The Genius of Independent Mathematics

If wasan had its Newton or Leibniz, it would undoubtedly be Seki Takakazu (Seki Kōwa, 1642–1708). This extraordinary scholar, born into a samurai family, revolutionized Japanese mathematics and gave it a new direction.

Seki was self-taught, which in itself was remarkable—he gained knowledge by studying available texts and then expanded upon them, introducing groundbreaking methods. He was the one who independently discovered the determinant—a key concept in linear algebra, which in Europe would not be described until years later by Gottfried Leibniz. He also developed techniques that resembled differential calculus, and his students continued the research—who knows what breakthroughs they might have achieved had Japan not remained isolated?

Seki Takakazu earned the title of "sansei" (算聖) – "the saint of mathematics", much like Sen no Rikyū was the "saint of tea" and Matsuo Bashō the "saint of haiku."

Mathematical Schools in Wasan

After Seki Takakazu, mathematical schools began to emerge across Japan, each developing different aspects of wasan. His students and later masters wrote mathematical treatises, formalizing and expanding knowledge. Some of the most significant texts of this period include:

• Hatsubi Sanpō ("The Foundations of Mathematics") – the only book written by Seki Takakazu.
• Katsuyō Sanpō – a collection of mathematical essays by Seki’s students.
• Jinkōki ("The Art of Calculation") – the most popular mathematics book of the Edo period, written by Yoshida Mitsuyoshi in 1627.

Mathematics was no longer just an elite discipline—merchants, craftsmen, monks, and even women and children began engaging in wasan as a form of entertainment and intellectual exercise.

Sangaku – Mathematical Challenges in Temples

One of the most fascinating expressions of wasan’s cultural impact was the tradition of sangaku (算額)—wooden tablets inscribed with complex geometric problems, which were hung in temples and shrines as offerings.

Mathematics in Japan had a spiritual dimension—many scholars viewed numbers and geometric figures as keys to uncovering the order of the world. Sangaku were a form of mathematical offering to the deities—if someone discovered a beautiful solution to a geometric problem, instead of keeping it to themselves, they offered it to the temple as a gift.

Interestingly, many sangaku were created by ordinary people—merchants, craftsmen, and even children. This demonstrated that wasan was not just a high-level intellectual pursuit but had become a widespread hobby, a form of intellectual challenge, and a way to spend leisure time.

The Decline of Wasan: The Meiji Era and Western Mathematics

Everything changed in the second half of the 19th century, when Japan opened up to the West and began a rapid modernization process. The Meiji government (1868–1912) sought to catch up with European powers as quickly as possible, including in the fields of science and education.

Although wasan was fascinating and highly developed, it was not suited to modern applications. It was not used in physics, engineering, or astronomy, and so it was replaced by Western mathematical methods, which were better suited for technological advancement.

In 1872, Western mathematics was made mandatory in Japanese schools, and traditional wasan gradually faded into obscurity. However, this did not mean its complete disappearance—to this day, some schools still preserve the sangaku tradition, and researchers and mathematics enthusiasts in Japan continue to rediscover this remarkable era of numerical experimentation.

The Principles of Wasan and Its Differences from Western Mathematics

The Philosophy and Applications of Wasan

One of the most striking aspects of wasan was its fundamentally different philosophy compared to Western mathematics.

In 17th-century Europe, mathematics was increasingly being used as a tool to describe physical phenomena—the best example of this is Isaac Newton, who developed calculus to explain gravity. In contrast, Japanese mathematicians were not particularly interested in applying their work to the natural sciences.

Wasan evolved as an intellectual art, a form of abstract mental exercise. It was not intended for engineering or scientific applications—its purpose was to refine the mind and to create beautiful, elegant geometric solutions.

Rather than asking, "What can we use this for?", Japanese mathematicians asked, "Is this elegant? Is the structure of this proof beautiful?" This is why so much emphasis was placed on the harmony of shapes and symmetry, as seen in sangaku—temple tablets featuring geometric puzzles.

Wasan also had no role in economics or accounting—unlike European mathematics, which was developed by figures like Descartes and Pascal, who were interested in probability theory, financial analysis, and mechanics.

A Visual Approach to Mathematics – Geometry as Art

Wasan was deeply rooted in geometry, and mathematical figures were treated almost like calligraphy—harmony and aesthetic perfection were essential.

In 17th–18th century Europe, mathematics increasingly shifted toward abstract algebraic language—the development of calculus, matrices, and complex numbers led to a world where equations became increasingly detached from visual interpretations.

Wasan, on the other hand, remained faithful to geometric thinking—Japanese mathematicians sought to solve as many problems as possible using proportions, constructions, and visual methods of analysis. The following examples illustrate the distinctive approach of wasan compared to European mathematics:

• Problems frequently involved circles inscribed in triangles, tangents, and symmetry—figures were drawn with great precision, and solutions were presented in an aesthetically pleasing way.
• Proofs were expected to be not only correct but also "visually satisfying"—the layout of calculations was adjusted to appear as harmonious as possible.
• Sangaku were not merely collections of equations—their arrangement and composition were often designed to complement the aesthetics of the temple where they were displayed.

Today, one might say that wasan was akin to what in the 21st century is called "visual mathematics"—the use of diagrams and intuitive methods rather than pure algebraic formalism.

The Use of Wooden Counting Rods (Sangi) in Wasan Calculations

While Europeans in the 17th century were developing modern algebraic notation—such as Descartes’ system for equations and Leibniz’s notation for calculus—the Japanese relied on sangi (算木), a system of wooden counting rods arranged on special boards to perform mathematical operations.

How Did Sangi Work?

• Each rod represented a digit, and their arrangement allowed for mathematical operations.
• The system resembled Roman abacuses or more advanced versions of counting boards.
• They were used for addition, subtraction, and solving systems of equations—especially useful for geometry and algebraic calculations.
•  

Sangi was one of the key reasons why wasan algebraic notation differed from Western notation—instead of writing equations in a linear format (e.g., y = 3x + 5), Japanese mathematicians preferred geometric representations of numbers.

Advanced Achievements of Wasan

Despite Japan’s isolation during the Edo period, wasan achieved mathematical results that in many cases rivaled or even preceded European discoveries.

The Independent Discovery of Bernoulli Numbers

Bernoulli numbers—a special sequence of numbers used in summing powers of integers (e.g., 1k + 2k + 3k + …)—were formally described by Swiss mathematician Jakob Bernoulli in 1713.

However, one year earlier, in 1712, the same formula had already been recorded in Japan by Seki Takakazu and his students in the treatise Katsuyō Sanpō.

Accurate Approximations of π

Wasan mathematicians, using tedious calculations with geometric methods and sangi rods, were able to achieve remarkably precise approximations of π. However, in Japanese mathematical terminology, it was not called "pi", but rather "en no hido" (円の比例), meaning "the proportion of a circle".

Takebe Katahiro, a disciple of Seki Takakazu, calculated en no hido to 41 decimal places, one of the most precise approximations in the world at that time (18th century)!
(Of course, "41 decimal places" is a simplified way of expressing this—Japanese mathematicians recorded such numbers using different types of fractions, rather than the Western-style notation of "3.1415927").

Polynomial Equations of Extremely High Degrees

Japanese mathematicians were remarkably ambitious—they worked on solving algebraic equations of degrees so high that European mathematicians had no known methods for tackling them.

One wasan mathematician attempted to find approximate solutions for an equation of the 1458th degree—an extraordinary level of complexity even by modern standards.

Unlike in Europe, polynomial equations in wasan were often solved using graphical and visual methods rather than Western analytical formulas.

Soddy’s Hexlet Theorem—Discovered 100 Years Before the West

In 1936, British chemist Frederick Soddy published a theorem in the journal Nature, describing the properties of six spheres arranged in space between two other spheres—a result now known as Soddy’s Hexlet Theorem.

However, Japanese documents from 1822 contain a proof of the same theorem, originally recorded at Samukawa Shrine by Irisawa Shintarō Hiroatsu—a merchant who studied wasan in his free time.

Sangaku – Mathematical Puzzles in Temples

What Were Sangaku?

Beneath the wooden torii gates, amid the aromatic scent of incense wafting through temple courtyards, hung extraordinary tablets—sangaku (算額, "calculation tablets").

Carved from wood, adorned with precise geometric diagrams, and inscribed with elegant kanji characters, they were more than just offerings to the gods.

They were challenges left for future minds, mathematical mysteries waiting to be solved.

Sangaku were geometric puzzles that Japanese mathematicians of the Edo period (1603–1868) dedicated to temples and Shintō shrines.

They were not merely forms of prayer or expressions of gratitude—they were also records of intellectual achievement. Each tablet contained a problem, and often, its solution, meticulously calligraphed as if mathematics itself were a form of artistic expression.

The message was clear: here is a problem worthy of attention, a proof that our knowledge and skill in wasan can reach the heights of human intellect. Today, around 1,000 sangaku tablets survive in Japan. Some of them, having hung in temples for hundreds of years, still contain unsolved puzzles that continue to fascinate modern researchers.

The Social Role of Sangaku

Sangaku were more than just mathematical puzzles—they had educational, social, and even religious significance. Their solving was not limited to scholars—samurai, merchants, craftsmen, and even women and children participated in these intellectual competitions.

It was a form of entertainment open to anyone with enough patience and sharpness of mind to tackle the intricate challenges. In Edo-period Japan, learning mathematics was not restricted to the elite—wasan was widespread across all social classes. Temple tablets were both a way to popularize mathematical knowledge and a tool for demonstrating intellectual superiority.

• For samurai, who sought new paths of self-improvement during peacetime, sangaku offered the perfect challenge.
• For merchants, they were a way to refine skills necessary for trade and financial calculations.
• For scholars, they were a chance to immortalize their names in history.

Sangaku also served a spiritual function—offering a new mathematical puzzle to the temple deities was an act of gratitude for acquired knowledge and a prayer for further enlightenment.

Mathematics and religion intertwined in a unique way, and solving difficult mathematical problems became almost a mystical act.

Examples of Famous Sangaku Puzzles

Sangaku tablets contained geometric problems of varying difficulty—from basic exercises for beginners to challenges that still require advanced mathematical methods today.

The Pythagorean Theorem in the Wasan Style

One of the most popular sangaku problems presents the classic case of a right triangle, but in a form unique to wasan. On a wooden tablet, a complex diagram is displayed where right triangles are arranged in an origami-like pattern, and their relationships are derived using Japanese mathematical techniques rather than Western algebraic methods.

Puzzles on Inscribed and Circumscribed Circles

Another common theme in sangaku involved circles inscribed or circumscribed within triangles and other polygons. One example features a series of concentric circles, each tangentially touching others in an intricate pattern—the challenge being to calculate their radii or the ratio of their areas.

Problems Involving Volumes and Surface Areas

Sangaku was not limited to two-dimensional geometry—some problems dealt with volumes of solids, surface area ratios, and ways to divide three-dimensional space. Surprisingly, some of these challenges closely resemble the work of Newton on calculus and infinitesimal analysis.

Some sangaku puzzles remain unsolved to this day. Perhaps their authors intentionally designed them to demand greater intellectual effort—or they were waiting for someone worthy enough to solve them.

The Legacy of Sangaku and Wasan

Today, sangaku remains a valuable cultural and mathematical heritage of Japan. They stand as evidence that mathematics in the Edo period was not merely a practical tool—it was an art form, a passion, and even a means of connecting with the divine.

And while Japan adopted Western mathematics in the Meiji era, the temple tablets still bear witness to a time when numbers and geometric figures were as beautiful as poetry or painting.

Conclusion

Although the era of wasan ended with Japan’s modernization, its spirit still endures—in centuries-old mathematical puzzles and in modern research into this remarkable tradition.
Wasan was more than just mathematics—it was an intellectual challenge, a pastime, and even a part of spiritual life in Edo-period society.

It brought temples to life with sangaku, united samurai, merchants, and scholars in the pursuit of numerical harmony, and its great masters—Seki Takakazu, Takebe Katahiro, and others—discovered mathematical truths parallel to those of their European counterparts. Today, wasan is no longer taught, and its notation system and problem-solving methods remain largely unknown.

However, modern technology is opening new frontiers in its study and preservation.
Digital archives allow us to store hundreds of ancient mathematical texts, while AI-powered deep learning tools are revolutionizing the analysis of historical documents. One of the biggest challenges is recognizing Edo-period kanji, which have changed in appearance over centuries, making automatic classification and transcription difficult. Yet the greatest obstacle is not just technology. Research on wasan faces a larger threat—the aging generation of scholars and a lack of successors. Fewer and fewer academics study this field, and if action is not taken, we risk losing a vital part of Japan’s intellectual heritage.

The key task for the future is to create open-access databases, educational resources, and popularize wasan—ensuring it is preserved for future generations. Because while the history of wasan ended in Meiji-era classrooms, Japanese mathematics still has a chance to survive—in a digital world where numbers transcend the boundaries of time.

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"Strong Japanese Women"

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    未開    ソビエライ

 An enthusiast of Asian culture with a deep appreciation for the diverse philosophies of the world. By education, a psychologist and philologist specializing in Korean studies. At heart, a programmer (primarily for Android) and a passionate technology enthusiast, as well as a practitioner of Zen and mono no aware. In moments of tranquility, adheres to a disciplined lifestyle, firmly believing that perseverance, continuous personal growth, and dedication to one's passions are the wisest paths in life. Author of the book "Strong Women of Japan" (>>see more)

Personal motto:

"The most powerful force in the universe is compound interest." - Albert Einstein (probably)

  Mike Soray

   (aka Michał Sobieraj)

未開    ソビエライ

 An enthusiast of Asian culture with a deep appreciation for the diverse philosophies of the world. By education, a psychologist and philologist specializing in Korean studies. At heart, a programmer (primarily for Android) and a passionate technology enthusiast, as well as a practitioner of Zen and mono no aware. In moments of tranquility, adheres to a disciplined lifestyle, firmly believing that perseverance, continuous personal growth, and dedication to one's passions are the wisest paths in life. Author of the book "Strong Women of Japan" (>>see more)

Personal motto:

"The most powerful force in the universe is compound interest." - Albert Einstein (probably)

Mike Soray

(aka Michał Sobieraj)

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