Pathway to Japanese Literature

We now raise this title and wish to discuss some observations we have made.
Some may argue:
"Mathematics and other sciences stand entirely opposed to art; no mutual relationship has ever existed between them.
They might even be completely unrelated."
That may indeed be true.
Generally speaking, such a tendency likely exists.

In our nation's art world today, there exist many influential masters beginning with Takeuchi Seihō - yet none of these great artists show any evidence of proficiency in mathematics or other sciences.
Even female poets like Kujō Takeko and Yanagihara Byakuren do not simultaneously practice science.
If we seek individuals who are both conventional artists and mathematicians/scientists simultaneously - while not entirely nonexistent - they would likely prove nearly impossible to find.

When arguing from this type of relationship, one may say there is no direct connection between mathematics, science, and art.
We have no objection to this; indeed, anyone would wholeheartedly agree.

Despite circumstances being as they are, we have dared to posit their relationship as our subject of inquiry.
Reckless it may be called, and reckless it is.
However reckless it may be, we must clarify this.
We believe there must be an extremely close relationship in terms of historical development.
It is only when one comprehends this that the vibrant flow of meaning becomes visible.
If one cannot even [“shimo” or “sura”] comprehend this, the true significance of history’s flow cannot be grasped.
We dare to advance our argumentative steps toward this point.

〈I. Waka and Haiku〉

In our country, many mathematicians emerged during the Edo period.
Since there were numerous individuals, when examining this matter in detail, various diverse elements likely existed.
Nevertheless, regarding Edo-period mathematicians, it cannot be denied that artistic elements were abundantly evident.
We had previously addressed this matter in the chapter titled "Mathematics in Japanese Cultural History."

Among these, what stood out most strikingly was that among Edo-period mathematicians, the cultivation of waka poetry and haiku had been remarkably widespread. Attempts to record arithmetic through song, such as Imamura Chishō’s *Inkika Sanka* (1640), could be seen from an early period.

Although works that made such attempts throughout entire texts were difficult to find in great numbers elsewhere, those that included some poetry within their pages could be found in abundance. These examples could often be found in published mathematical books since their early days.

As for whether great masters such as Seki Takakazu, the Tatebe brothers, and Matsunaga Yoshisuke were involved in such matters, evidence can scarcely be obtained today.
This was likely because their era was somewhat earlier, resulting in few surviving historical materials.
When we consider mathematicians of the Bakumatsu period, there was scarcely any who remained uninvolved with either waka or haiku—such became the prevailing state.
Yamaguchi Kazu was a native of Echigo Suibara who traveled widely throughout the provinces. Examining his travel records reveals that he documented mathematical problems encountered during his journeys while also enthusiastically including poems and haiku.
Through this, one can discern that person’s tastes.
When Sakuma Tsuzumi, a mathematician from Oshu Miharu, journeyed through various provinces, he compiled a booklet of calligraphy, paintings, and poetry composed for him by people in the places he visited.
Among this person’s works exists a poetry collection titled *Chiyomigusa*.

Shiraishi Nagatada, author of works such as Sha Mei Sanpu, left behind numerous waka poems.
Baba Masamichi and his son Masamune were known as Sonnichian in haikai circles and served as masters of the art.
In Nikkō Kiyotaki stands a monument commemorating their haikai contributions.

Kawakita Chōrin studied under Uchida Gokan as a mathematician of the Seki school tradition, yet in haikai he followed the teachings of the Baba school; retiring at Mount Fuji's base in his later years, he composed many poems about its scenery.
Hagiwara Teisuke of Jōshū appeared the unlikeliest practitioner of arts as a meticulous mathematics researcher, yet nevertheless actively engaged in haikai poetry.
While poets remained rare among wasan mathematicians, those practicing waka or haikai proved remarkably numerous.
Naturally, compiling them all lies beyond possibility.

〈II. The Current State of Sangaku〉

It is common knowledge that shrines and temples across the provinces were adorned with numerous votive tablets.
Many of these votive tablets had been inscribed with mathematical problems and techniques, decorated with colors such as circles and triangles to beautify them, and prepared in the same format as other votive tablets before being dedicated at worship halls.
It was said their numbers had been quite substantial in the early Meiji years, but now—gradually falling into ruin with no new dedications—their count had drastically diminished.
Yet even then, examples could still be found scattered throughout various locations.
Let us here attempt to describe those examples.

In the Tokyo vicinity, examples include Fuchu’s Rokusho Myojin Shrine and Ōmiya’s Hikawa Shrine; slightly further afield, one can count Chiba’s Chibadera Temple, Narita’s Fudō, Shibayama’s Enma, Kazusa’s Kanōzan, Jōshū Maebashi’s Hachimangū Shrine, and Usui Pass’s Kumano Shrine at the Jōshū-Shinano border. Myōgi Shrine in Jōshū also contains these tablets. A number of tablets remain extant at Ichinomiya. These are based on my own firsthand observation. There is also a large tablet at Shiogama Shrine.

In the Keihan region, remnants can be found at Kyoto’s Kitano Tenmangū Shrine, Yasui Shrine, and Fushimi Mikamiya Shrine, while Kyoto’s Gion—that is, Yasaka Shrine—houses the oldest surviving mathematical votive tablet, dating from the Genroku era (1688–1704). These are in Kyoto, but in Osaka there is a large tablet at Sumiyoshi.

When heading slightly west into Banshū, they could be found at Onoe Shrine, Tatsuno Hachimangū, and other locations.
Himeji Hachimangū Shrine also housed them.
In Miyajima in Aki there was a mathematical votive tablet, while at Tsurubane Shrine in Hiroshima surviving examples remained.
At Dōgo Hachimangū in Iyo, mathematical votive tablets existed in great numbers—likely unmatched in quantity anywhere else today.

Proceeding further west into Kyushu, they could be found at Hakozaki and Sumiyoshi Shrines in Fukuoka, at Akizuki in Chikugo, at Yanagawa, and also at Suwa Shrine in Nagasaki.

All of these were extant, but among those said to have existed until recent years, there were some that could no longer be found.
For instance, Uzen Haguroyama was one such case; while some claimed to have seen tablets at Banshu Ishi no Hoden in recent years, upon actually visiting, none could be found, and shrine priests professed no knowledge of them.
Given this situation, it cannot be definitively stated that none of the mathematical votive tablets I personally observed had been destroyed or removed within a few years.

In Tokyo as well, it is said that [mathematical votive tablets] existed at Yushima Tenjin Shrine before the earthquake disaster, but they were of course burned down. Even before the earthquake disaster, though those in Tokyo City—such as at Shiba Atago—had originally existed in great numbers, not a single tablet could be seen, and shrine priests remained unable to provide answers when inquired with. Within Tokyo City, mathematical votive tablets appear to have vanished earliest.

During the Bakumatsu period, there were quite a number of people coming to view these votive tablets, and as they attracted attention, they were carefully preserved; however, after the Restoration, though rare new dedications occurred—examples being those at Kyoto’s Kitano Tenmangū, Narita, and Ōmiya—their numbers had drastically diminished compared to before. Newly dedicated tablets gradually decreased in number, while others were lost to fires, destruction, weathering, or removal; though some relics remain today, they will become increasingly scarce and rare in the future—a natural progression.

〈III. The Custom of Mathematical Votive Tablets〉

The practice of Japanese mathematicians fashioning mathematical problems and techniques into votive tablets and dedicating them at shrines and temples was a very ancient one.

As previously mentioned, the tablet currently existing at Kyoto’s Gion was dedicated in Genroku 4 (1691) by one Hasegawa Rinkan and contained solution methods for two problems that fellow disciple Yamamoto Munenobu had presented when dedicating a tablet at Fushimi Mikamiya Shrine in Tenna 3 (1683).

The mathematical votive tablet by Yamamoto Munenobu at Fushimi Mikamiya Shrine naturally no longer remains today.

Regarding this mathematical votive tablet at Gion, the *Supplemented and Revised History of Japanese Mathematics* stated that it had been dedicated by a certain Tōkai-bō,

Genroku 4 (1691)... Tōkai-bō solved the votive tablet problem posed by Yamamoto Munenobu and dedicated it at Kyoto’s Gion Shrine... Tōkai-bō—his true name remained unknown; presumably an art name...

However, this was incorrect—it was clearly the dedication by Hasegawa Rinkan as signed.
Thus, the notation "Tōkai-bō" referred to none other than the name of the monk who had acted as intermediary for the dedication, and moreover, what was written as "Tōkai-bō" was a misreading of "Tōbai-bō."
This mathematical votive tablet was the oldest extant example; however, through its existence, it became clearly evident that the custom of dedicating such tablets had been practiced from the Tenna to Genroku eras.

However, this custom had been practiced from even earlier times. That matter appears in *Fearless Correction of Mathematical Methods*. *Fearless Correction* was created by Murase Yoshimasu in the first year of Enpō (1673). It states that within the text there was someone who dedicated a mathematical votive tablet at Fudō-dō in Musashi Province’s Meguro Village—Katooka Toyotada’s *Algorithmic Direct Explanation* also contained solution techniques for this problem, and *Fearless Correction* separately created its own solution techniques.

*Algorithmic Direct Explanation* was created in the tenth year of Kanbun (1670), and Fudō-dō’s mathematical votive tablet is known to predate this work.

According to *Fearless Correction*, it appeared that mathematical votive tablets had been dedicated even earlier than this. This fact was recorded in *Supplemented and Revised History of Japanese Mathematics*. It therefore became evident that the custom of dedicating mathematical votive tablets had existed since around the Kanbun era (1661–1673), demonstrating its origins in an early phase of Japanese mathematics' development.

Iwamoto Goyū’s *Kōko Sogen*, created in the eighth year of An’ei (1779), also contained discussions regarding mathematical votive tablets within its text.

By this era, the custom of dedicating mathematical votive tablets appears to have flourished.

Fujita Sadatoshi compiled *Shinpei Sanpō* (Divine Wall Mathematics) and *Zoku Shinpei Sanpō* (Sequel to Divine Wall Mathematics), gathering mathematical votive tablets that had been dedicated at various shrines and publishing them.
These two works were produced in 1789 (Kansei 1) and 1806 (Bunka 3). Though nominally credited to his son Kagen under Sadatoshi’s editorial oversight, since Sadatoshi’s epitaph states that he personally compiled them, I believe this was indeed the case.
Fujita Sadatoshi initially used the characters 定資 for his given name before changing them to 貞資 late in life.

The very creation of these original and sequel volumes of *Shinpei Sanpō* ("Divine Wall Mathematics") makes clear how vigorously the custom of dedicating mathematical votive tablets flourished during Fujita Sadatoshi's era.

The well-documented clash between Aida Yasuaki and Fujita Sadatoshi—wherein Aida established his Saijō-ryū faction and persistently challenged the Seki school through heated disputes—reportedly began when Kamiya Sadanori, a Fujita disciple acquainted with both men, arranged their meeting. During this encounter, Fujita allegedly demanded revisions to flawed portions of a sangaku tablet Aida had previously dedicated at Asakusa Kannon Temple. Aida's refusal to comply transformed this scholarly critique into open academic warfare.

This was likely a superficial matter. In truth, I wish to view this as a serious conflict in which Aida Yasuaki—a genius operating outside established academic circles—rebelled against the Seki school's faction. Be that as it may, the fact that this major incident in mathematical history—where two titans, Aida Yasuaki and Fujita Sadatoshi, competed fiercely against one another for nearly two decades—is said to have been triggered by wording on a mathematical votive tablet demonstrates just how highly such tablets were valued by mathematicians of that era. That books such as *Shinpei Sanpō* were created and published was only natural.

Needless to say, numerous mathematical votive tablets were also dedicated outside the Fujita school.

The title *Shinpei Sanpō* essentially signified "a collection of mathematical problems dedicated in the wall spaces of shrines."
Later, Shiraishi Nagatada created *Shamon Sanpu*, which similarly meant "a collection of mathematical problems sworn before shrines"—carrying the same significance as *Shinpei Sanpō*.
Uchida Gokan’s *Kokon Sankan* (古今算鑑) was likewise a published compilation gathering dedicated mathematical problems.
Publications of this kind were far from scarce.

Among manuscripts there existed numerous compilations that gathered mathematical votive tablets from various locations. Such works included *Gakudai Shūroku* (Compilation of Votive Tablet Problems), and to enumerate each one here would naturally prove too tedious. Their number was exceedingly large.

The quantity of mathematical votive tablets recorded in these publications and manuscripts was already considerable; however, since even among extant tablets there existed those not found in these documents, the actual number of dedicated tablets had likely been far greater than anticipated. Thus they likely fell into disuse unrecorded.

It seems abundantly evident that mathematical votive tablets enjoyed tremendous popularity.

The practice of fashioning mathematical problems into votive tablets and displaying them in the wall spaces of shrines and temples was likely a sight found nowhere else but in our Japan.
Even when examining works by Western historians of mathematics that document Japanese mathematical votive tablets, one finds no mention of examples from other countries.

In a country abundant in poetic sentiment where mathematicians of all ranks collectively studied waka and haikai poetry, I contend that mathematical problems being frequently fashioned into votive tablets with added coloration and crafted as artworks was by no means a mere coincidence.

This is how I see it, but might this be merely my own biased perspective? I entrust this matter to the reader's discernment.

〈IV. The Beauty of Japan〉

Japanese mathematical books contained numerous illustrated volumes. Even *Jinkōki* (塵劫記), one of the earliest works of its kind, incorporated large illustrations. The *Jinkōki*’s images even attempted color printing, serving as precursors to Japanese color woodblock prints—a fact demonstrating that these mathematical texts hold significant value in art history.

Works such as *Kaisanki* (改算記) and *Sanpō Ketsugishō* (算法闕疑抄) likewise recorded abundant illustrations. Even in later eras, textbook-style mathematical books continued to feature copious pictorial elements.

There also exists a mathematical book titled *Ehon Kufu no Nishiki* (絵本工夫之錦), but one could infer its contents from the title alone. In this aspect too, one perceives a richness of artistic sensibility.

Furthermore, beyond these, numerous discussions concerning mathematical games and similar subjects were presented, with countless examples demonstrating how mathematics was treated in a profoundly artistic manner.

I maintain that Japanese mathematics abounds with such artistic elements. These elements did not first emerge when mathematics developed during the Edo period. We must also recognize that this tendency had been markedly apparent since earlier times.

In medieval times, there was scarcely any mathematics worthy of the name; yet in that very era, the remarkable development of family crests must have been an exceptionally striking phenomenon.
Regarding the development of family crests, Mr. Numata Raisuke’s valuable research has made matters exceedingly clear—the use of geometric shapes as familial identifiers constituted an eminently artistic approach.
China may have possessed something resembling family crests, but none rivaled Japan’s in vividness, of course.
That crested formal wear became ceremonial attire must be considered unique to Japan.
The fact that games such as stepchild standing were practiced from the late Heian period onward also demonstrates that mathematical elements were incorporated into playthings—here too we observe mathematics’ artisticization.

In a nation where mathematical elements were artisticized before mathematics itself had yet developed, when the time came for mathematics to develop and artistic mathematics was practiced, it should by no means seem strange that such great progress was made.

When we travel through the countryside and see thatched-roof farmhouses scattered among the foothills and forests, we are captivated by their serene elegance.
They are beyond comparison with the clumsy, tasteless things known as 'culture houses' that are in vogue these days.

Even when visiting a moss-covered shrine surrounded by ancient forests, one was struck by its profoundly solemn sacred aura.
Even if it was but a tiny shrine of one *tsubo* or half—no larger than three square meters—passing before it would naturally bow one’s head unbidden.
In the structure of its shrine buildings and the arrangement of trees, one sensed an ineffably divine quality.

I believe Japan’s architectural beauty is by no means inferior to that of other nations. Of course, Japan has no stone-built grand palaces or gaudy skyscrapers. Even Buddhist temples—unlike those of that country—are entirely wooden structures and may lack in grandeur. Yet they are elegant and rich in aesthetic value.

The ancient architectures of Egypt and India, as well as the renowned artistic architectures of Greece and the West, are by no means inferior. I do not claim that Japanese architecture surpasses them. I perceived whether there existed something that could stand among those world-class architectures without being inferior to such an extent.

Recently, Bachelor of Letters Fujita Motoharu published a work titled *Nihon Minka-shi* (A History of Japanese Folk Houses). He enumerated what kinds of folk houses existed in our country and clarified the origins of their development. His perspective was acute and his observations appeared precisely targeted; yet even regarding roof shapes, he noted that it seemed as though every conceivable form found across the world had been gathered within this single nation, with remarkable elegance as their defining characteristic. Maintaining this elegant architectural style of our folk houses henceforth would prove difficult indeed. It was sad, if one had to put it that way.

Yet even in folk house architecture, the fact that such elegant and excellent works were created naturally in the past—without any attributable creator—stems from a national character rich in artistic sensibility. Though adopting foreign styles may lead to temporary lapses into decadent tastes, it can be readily expected that before long, they will be recast into something uniquely graceful, manifesting a different aesthetic distinctiveness.

Even in painting and sculpture—though their beginnings undoubtedly involved much learning from China and India—countless unique and splendid works, clearly unseen in China, India, or elsewhere, were created.
It is widely renowned that Sesshū, during his sojourn in China, far surpassed those under whom he studied; yet when one who has encountered many Chinese paintings later views Sesshū’s works, even an amateur cannot help but feel their undeniable superiority.

Yamato-e painting, which had developed since medieval times, did not follow the Chinese style but instead formed a unique composition.

The flourishing production of ukiyo-e woodblock prints during the Edo period was an entirely indigenous development, with their advancement in color printing techniques having no parallel in other nations.
It was therefore only natural that once these became known in the West, they came to be highly treasured.

When viewing masterpieces by Edo-period painting masters alongside renowned Western works, one notices only differences in style and technique—never does the feeling arise that theirs alone possess value while ours are vastly inferior.
One cannot help but feel deeply moved by their distinctive refinement.

Japanese paintings indeed possess certain elements of excellence.

〈V. Western Transmission〉

There must certainly be a profound foundation in the aesthetic sensibility of the Japanese people that produced such paintings.

Even when examining sculptures like those by Unkei, one finds truly exceptional works.
I once encountered at Ueno Museum—though it was a replica—a piece of such remarkable quality that I could not help but marvel endlessly.
At that time, I still possessed no cultivated eye for sculpture.
Yet its vivid realism and overflowing spiritual essence compelled me to feel deeply moved despite myself.
Japan holds such masterpieces within its heritage.
They may have received some influence from Greek sculpture along their path.
Yet the impression they convey differs markedly from what one gleans from photographs of ancient Greek works.
Though their learning passed through China and India, I ultimately concluded that Japan's ancient sculptures stand as supremely accomplished achievements.

After that, I had opportunities to see many old Buddhist statues in various places—some attributed to Gyōki Bosatsu, others to Kōbō Daishi.

I do not know whether they were actually works by Gyōki or Kūkai, but it remains an undeniable fact that there existed remarkably splendid pieces of sculpture.
The many works displayed at Nara Museum included numerous truly excellent pieces; having actually seen how Japanese sculpture demonstrated considerable skill from early times, we were profoundly moved.

According to remarks by up-and-coming art historian Mr. Dan Inō, our country’s ancient sculptures had naturally received Greek influence.
Although they received this influence, when speaking of its origin, it came from the Gandharan art of India; transmitted across Central Asia with the spread of Buddhism, then entered China, and finally reached our country.
The path of its transmission had been made exceedingly clear, leaving no room for doubt.

It went without saying that Gandharan Buddhist art had taken lessons from Greece, but by that time Greek art was already in decline.
It retained scarcely any vitality.
One might think that Bactria—a Greek colony in Central Asia where Hellenic influence endured longest, known in China as Da Xia—would have propagated Greek culture eastward given its history. Yet in reality, the Greeks in this country appeared to have been few in number, and to what extent they actually developed or sustained Greek art and culture remained highly questionable. Recent excavations of its capital’s ruins had reportedly yielded nothing of significance.

For these reasons, when Gandharan Buddhist art emerged, the extent of its influence from Greek sculpture and other elements was likely quite limited.

Then, one could generally infer the nature of these connections transmitted through Central Asia, China, and Korea.

Numerous artifacts remain in the ruins of Gandhara as well as Central Asia, China, and other regions; one ought to examine them.
In truth, there exist no works of particular excellence.
They proved markedly inferior when compared to sculptures from Greece's golden age, and paled even beside those created after reaching Japan.
While Japanese sculpture had indeed been influenced and instructed by Greek works through Gandharan Buddhist art as intermediary, once it passed into Japanese hands, they produced far superior works to those of their very teachers.
Though they learned from others, they achieved still more outstanding results through transcending mere imitation.

This could only be attributed to the fact that the artistic capacity of the Japanese people stood second in potency solely to that of Greece.
Without such capacity, these realities would have been fundamentally inconceivable.

Mr. Dan states thus as the outcome of his new research.
The reason I had witnessed many ancient Japanese sculptures and been struck by their subtle beauty was here completely resolved.

〈VI. Hōryū-ji Temple〉

Hōryū-ji Temple in Yamato was an ancient structure from the time of Empress Suiko, though one theory held that it had been destroyed and rebuilt during Emperor Tenji’s reign.
Moreover, the two theories of reconstruction versus non-reconstruction were exceedingly entangled and not easily resolved.

Setting that aside for now, while Korean craftsmen likely oversaw Hōryū-ji Temple’s construction, it went without saying that the vision of its founder Prince Shōtoku had also been incorporated; moreover, the layout of its halls and pagodas followed a form unseen in Buddhist temple architecture across China and Korea.

This was the opinion of Dr. Itō Chūta, the foremost authority in Eastern architectural history, and thus naturally deserved trust without reservation.
Our ancestors during Hōryū-ji Temple’s ancient construction period had already ceased being mere imitators of foreign cultures.

The columns of Hōryū-ji Temple narrow at both ends, some swelling in a flask shape at the middle.
This truly remarkable feature also appears in ancient Greek architecture under the name entasis (entasis). Columns bearing this entasis were considered visually harmonious—said to embody one of those divine inspirations gracing Greek art—and were regarded as innovations achievable only through Hellenic artistic sensibilities. Therefore, when columns displaying identical entasis were discovered in Hōryū-ji’s construction outside Greece, it had become nearly established doctrine that this must indicate transmission of Greek architectural styles.

However, Mr. Dan Inō also raised questions regarding these.
In Greece, entasis had been similarly applied during the flourishing period of Greek art.
As it entered its period of decline, entasis gradually disappeared.
Gandharan art in India was influenced by Greece, but not during its flourishing period.
They transmitted the art of its period of decline.
When examining Gandharan architecture, not a single instance of similar entasis could be discovered.
The same held true for Buddhist architecture in Central Asia, China, and Korea.

Despite circumstances being thus, when that Buddhist architecture came to our country and took form in Hōryū-ji Temple's construction, columns bearing entasis once again appeared here.
This was truly suspicious.

When considering this fact, acknowledging that Greek techniques had been transmitted proved far too strained an interpretation. The swelling of Hōryū-ji's columns could not have originated from external influences; we must recognize there existed other formative circumstances at work.

The reason Greek columns bear entasis lies in their architectural progression: originally practicing wooden construction before advancing to stone. In wooden columns, the dimensions at both ends had to be predetermined, but in an era without planes—when carving was done with axes—it would have been natural for the middle section to develop such a swelling.
This could also be seen as having naturally tended to occur.
In this manner, entasis emerged in wooden columns, and it is thought that even when transitioning to stone construction, this form was preserved.

In Japan as well, precisely because it was entirely wooden architecture, one can see that columns with entasis were likely created under similar circumstances.
Therefore, they did not necessarily create the entasis of the columns through learning Greek styles.

This interpretation constitutes Mr. Dan’s new perspective.
This new perspective demolishes what had been nearly an established theory; yet when compared to the old view—which posited Greek transmission without considering the actual conditions of its pathway, instead carelessly constructing its argument from mere superficial resemblances between Greece and Japan at opposite ends of the world—we find this new interpretation eminently worthy of approval.
Regarding this matter, there may yet be arguments both for and against from various authorities; nevertheless, we shall tentatively adhere to this interpretation.

If we consider that the entasis of columns was created separately in Greece and Japan—being unseen in other nations—and that this entasis proves exceptionally pleasing to the eye, then its employment by our ancestors alongside Greece may stand as sufficient evidence of their acute and refined aesthetic sensibility.

〈VII. Art and Science (1)〉

Japan, as noted above, was a country excelling in art from various perspectives and was by no means mediocre in its artistic endeavors.
In architecture, painting, or sculpture, it remained an undeniable fact that Japan had learned everything from China and India, and through their intermediary indeed came into contact with Greek traditions.
This fact could never be denied, nor was there any need to deny it.
To China, India, Greece, and Korea as well, we commemorated with gratitude how we had been richly supplied with various arts and sciences alongside broader cultural matters, receiving profound instruction.
These countries were all our teachers.
We must never forget the beneficence of these teachers.
Nor would we ever forget them.

However, our ancestors did not accept everything as mere imitation according to their teachers' instructions in all matters.
There was nothing they did not transform, assimilate, and imbue with new elements.
And through this process, they produced results superior to what they had learned.

In this manner, from the Dynastic period through the medieval samurai era, exceptional art was created in our country, and truly splendid works were produced.

Though I had not mentioned it previously, our nation's swordsmithing techniques achieved an unparalleled development during this era as well.

Yet during this period, neither mathematics nor other sciences developed into proper academic disciplines.
They remained mere technical skills.

However, with the advent of the Edo period, mathematics did develop.
Various sciences likewise developed.
These developments may safely be deemed entirely products of the Edo period.

In our Japan, art first developed, and subsequently mathematics along with various sciences flourished.
It was only after art had undergone a long period of development that science began to advance.
Because this period was excessively long, it attracted exceptional attention.

At this point, we contemplate:
Could one not assert that a people capable of creating excellent art are endowed with the aptitude to produce excellent science as well, given the opportunity?
I find myself incessantly haunted by such thoughts.

Conversely, could it not be that a country without art cannot establish distinguished science?

Our Japan, which possessed outstanding technical prowess, was also exceptional in mathematics and science.

In summary, I have argued that in our Japan, first the various arts—architecture, painting, sculpture, and others—flourished; that in art we surpassed even the teachers from whom we learned to achieve exceptional results, demonstrating that our artistic capability was by no means ordinary; that following this artistic development, mathematics and other sciences subsequently advanced; and that scientific progress cannot be expected in the absence of art.

However, when considered from another angle, if one delves deeply into things like mathematics and science, people tend to become overly argumentative.
Human qualities diminish.
If human qualities diminish, consequently artistic qualities will also be lost.
Even when observing art, as one's ability to discern whether forms and colors resemble nature may develop increasingly, they merely stop at distinguishing formal resemblances and cease to inquire where the true value of art lies.
The more scientific knowledge advances, the more one loses the capacity to appreciate art.

Such a tendency likely exists.
One might even say science and art cannot coexist—nay, that coexistence proves fundamentally impossible.

I once visited Nihonmatsu.
In Nihonmatsu, there was a person who loved composing haiku and pursued it with fervor.
The former wasan mathematicians of the Nihonmatsu area were no exception—they too composed haiku zealously, and their tanzaku poem strips still survive today. Yet haiku masters remain wholly unimpressed by these surviving works.
Haiku should not be composed in such a logically rigid manner.
They insisted emotions must flow more plainly and directly.
Mathematicians remain mathematicians through and through—even when composing haiku, they could only produce verses befitting mathematicians—or so it was said.

This criticism was naturally based on evaluating the compositions of two or three individuals and does not constitute a comprehensive argument.
However, such censure may perhaps prove generally unavoidable.
Even should rare exceptions exist, falling prey to these shortcomings likely follows common patterns.

I do not of course deny that such phenomena occur.
I deem this circumstance permissible.
Yet I do not believe my prior assertions would thereby undergo material alteration.
They remain wholly distinct matters.

〈VIII. Art and Science (2)〉

How did this compare with other nations? I wished to attempt to expound on this briefly.

Among Persia's renowned poets stands Omar Khayyam.
No Persian poet surpasses him in fame.
While Sassanid-era Persia produced figures like Firdawsi who achieved renown through epic poetry, among Islamic-era Persian poets none could equal Omar Khayyam.
In our country too, this man's poetry has occasionally been highly praised.

Now, what kind of person was this Omar Khayyam? He was, of course, no mere poet.
He is also renowned as an algebraist.

In Cajori’s history of mathematics, regarding this person,

One of those who contributed most to establishing the method of solving algebraic equations through the intersection of conic sections was Omar Khayyam (c. 1045–1123), a poet from Chorassan.
He classified cubic equations into two types—trinomial and quadrinomial—divided each type into families, and further categorized each family into genera.
He discussed each genus separately but relied on a unified approach.
He believed that cubic equations could not be solved through calculation and that quartic equations could not be solved even using geometry.
He did not consider negative roots.
Moreover, he often did not succeed in finding all positive roots.

In saying this, since we have also listed what Omar Khayyam failed to achieve, he might not seem particularly capable. However, achieving such success in that era was undoubtedly remarkable.

It must be noted that the situation in which an algebraist from the Islamic world such as Omar Khayyam employed geometric methods for solving equations was entirely different in nature from that in China, where they established approximate solution methods comparable to Horner’s method.

Omar Khayyam’s birthplace of Khorasan was a region of Persia.
This region produced many astronomer-mathematicians under Islamic rule.

In Cajori’s work, too, it is stated that the writings of Omar Khayyam and others belong to the period when Islamic mathematics reached its zenith, and that from this point onward, it began to decline.

The science of the Islamic world reached its zenith of development during Omar Khayyam’s time and thereafter entered a period of decline—a circumstance Dr. Sarton eloquently elucidates in his *Introduction to the History of Science*.

The fact that a mathematician of Omar Khayyam's stature was simultaneously Persia's foremost poet—indeed, one of the greatest poets across all ages, not merely in Persia—must utterly demolish the notion that mathematicians cannot become great poets.

Though a mathematician, Omar Khayyam faced no criticism that his poetry was overly logical or suffered from such tendencies.

Leonardo da Vinci and Omar Khayyam stood together as the most compelling exemplars of this synthesis.

That da Vinci was a renowned painter, sculptor, and architect is well known.

He also secured an indispensable position in the history of science.
Moreover, he was versed in mathematics.
In mathematics—it goes without saying—he was not a first-rate figure.
Yet he remains a figure who cannot be omitted from mathematics' historical records.

Cajori’s *History of Elementary Mathematics* notes that da Vinci focused on constructing inscribed regular polygons—some of his proposed methods being mere approximations possessing practical value but lacking theoretical interest.
He believed constructions like inscribing a regular heptagon within a circle were precise, though they were of course approximate.
Da Vinci also attempted constructions using a single compass opening—a method previously proposed by Pappos of Greece and attempted during Islam’s golden age by Abul Wefa. When da Vinci and others later pursued it further, this technique finally gained renown.

Moreover, performing constructions using means other than ruler and compass was something that Leonardo da Vinci also engaged in. An example of this would be his approach to the circle squaring problem using a cylinder.

That Leonardo da Vinci—a great artist and architect—left behind this kind of achievement in mathematics may perhaps seem disappointingly few in number. However, as an instance where mathematical achievements could coexist with being a great artist, it is entirely instructive.

Leonardo da Vinci was a master of art and simultaneously a master of science; consequently, there are a great many biographies and similar works that discuss this. Perhaps there is no need to enumerate these here. But unless I provide at least a general account, the circumstances will remain unclear.

I now resolved to quote a footnote from Cajori’s *History of Elementary Mathematics* and commit this to record.
We adopted this approach provisionally to express our regard for fellow translators.

Leonardo da Vinci was an Italian born in the village of Vinci near Florence.
His father was a notary public, and he was born an illegitimate child.
Though his father had a legal wife, Leonardo was taken into that household and spent his childhood in solitude.
At around age ten, he moved to Florence where his father's office was located and became an apprentice to the renowned painter Verrocchio.
It was from this point that the painter's genius began to fully manifest.
That he later developed an interest in scientific matters is said to have stemmed from this teacher's influence.
In 1482 at age thirty, he entered service under the Duke of Milan where he engaged in designing bronze statues and painting murals; following Milan's fall, he went to France and received royal patronage.

Leonardo da Vinci notably applied geometry in practice and believed that "only through the application of mathematics does knowledge become precise." He studied perspective as "the science of divine light." In mechanics, he achieved such feats as the parallelogram of forces and dynamic studies of bird flight; he diligently pursued the invention of flying machines; made numerous discoveries in physics; and became a pioneering figure in research later pursued by Galileo and others. However, many of these studies were left unfinished.

Moreover, Leonardo da Vinci’s theories on geological matters such as mountain formation were highly worthy of attention. Because Leonardo da Vinci created anatomical diagrams for anatomists, it is said that he also possessed profound knowledge of anatomy. Leonardo da Vinci, whether in art or science, could not help but leave deep imprints of his genius everywhere.

When we consider the figure of Leonardo da Vinci, we see the finest example of art and science being completely intertwined.
If there are those who believe that science and art cannot coexist, let them study this great man’s biography in meticulous detail.
That doubt will immediately vanish through this.

The German painter Albrecht Dürer (1471–1528), who emerged around the same time as Italy’s Leonardo da Vinci and was renowned as an artist, also attempted constructions such as inscribing regular polygons within circles, just as da Vinci had done.
While there were instances where Leonardo da Vinci considered approximate constructions to be accurate, Albrecht Dürer always clearly stated that such constructions were approximate and that his own were precise.
He stands as the pioneer of such practices.

Dürer, like da Vinci, performed constructions using a single opening of the compass.

Albrecht Dürer was born in Nuremberg, Germany, the son of a Hungarian blacksmith.
He studied woodcarving and painting, then traveled to Venice where he gained significant artistic insights.
In both painting and woodcarving, he achieved extraordinary success and created numerous masterpieces.
He became a deeply devoted friend to Martin Luther, the renowned Reformation leader.

The most significant of Dürer's mathematical treatises, completed in 1525, systematically expounded perspective techniques and graphical solution methods.

Dürer conceived of drawing connected polygons that form the sides of regular and semi-regular polyhedrons on paper, then folding along their edges to construct these polyhedrons—a method he pioneered.

Albrecht Dürer also expounded on epicycloids and performed their construction; while there are some traces of such curves in Greek mathematics, what Dürer discussed truly begins from this time.
Moreover, even after that period, there were none who discussed it again until France’s Desargues and De La Hire expounded upon it.
Both belonged to the seventeenth century.

In the West, it was also Dürer who first expounded on magic squares, which can be seen in his renowned painting *Melencolia*.
This painting was created in 1514.

Dürer, in connection with art, also expounded on higher-order plane curves.

The fact that Albrecht Dürer—a German painter and sculptor—emerged around the same time as Italy’s great artist Leonardo da Vinci and left behind such achievements in the history of mathematics as we now speak of them becomes an essential point that must never be overlooked when considering the interplay between art and science.

Next we cite Britain’s Christopher Wren (1632–1723).
Wren was an architect renowned for constructing London’s St. Paul’s Cathedral, yet he was also deeply versed in mathematics.

The Great Fire of London occurred in 1666. Though Wren was then a professor of astronomy at Oxford University, he actively participated as an architectural engineer in London’s reconstruction efforts, devoting himself to rebuilding over fifty structures including St. Paul’s Cathedral alongside temples, government offices, and major buildings.

The reason Christopher Wren’s name became renowned as a great architect is ultimately due to this.
Had Wren not encountered the Great Fire of London, it is said he likely would have remained solely a mathematician and astronomer—or perhaps that would indeed have been the case.
But without knowledge and insight in architecture, that great undertaking could not have been accomplished.

Wren first received his education at Westminster School, entered Oxford University around age seventeen or eighteen, then became a graduate. At age twenty-one in 1653, he obtained a Bachelor of Arts (M.A.), spent the next four years as a research fellow, was appointed Professor of Astronomy at the University of London, and transferred to the position of Professor of Astronomy at Oxford in 1661, remaining in that post until 1673.
From 1680 to 1682, he also served as President of the British Academy of Scholars.

In mathematics, he conducted research on laws concerning the collision of rigid bodies, methods for polishing parabolic mirrors, perspective techniques, and determining the arc length of curves, and also discovered two sets of generatrices for a hyperboloid.
He determined the length of the cycloid and discovered its center of gravity.
It is said that he had also conducted some research on the law of gravity prior to Newton.

The few examples we have cited here illustrate individuals who were artists and architects as well as masters of mathematics and science; however, it must be said that the development of perspective methods in the West was something born from the union of painting and mathematics.

Regarding this matter, the achievements of Leonardo da Vinci and Dürer had been discussed previously.

Moreover, among painters, sculptors, or engravers, those who engaged in this kind of research were frequently observed.

Pietro Franceschi was also one such individual.
Since he was said to have died in 1492, he would have preceded Leonardo da Vinci.
This man also left manuscript writings concerning perspective method.
As for da Vinci himself, he conducted research on perspective method and discussed matters such as shadow play; indeed, he demonstrated clear progress beyond his predecessors.

From this period onward, research in optics (or the science of light) concerning painting methods also advanced; Descartes came to discover the principles of rainbows, and other developments such as projection methods were devised.

In short, in early modern Europe, there were great artists who were simultaneously mathematicians and scientists; this led to remarkable advancements in the principles of painting techniques from a mathematical perspective. On the other hand, there were also those who approached painting and sculpture with mathematics and science's precise analytical eye—thereby enabling them to preserve art's original divine essence while simultaneously achieving formal accuracy.
The reason Western paintings came to emphasize perspective methods and proportional accuracy so vehemently—developing in a natural manner—was ultimately due to this.
When viewed purely from an artistic perspective—setting aside for now whether this relationship was advantageous or disadvantageous—the realization of such circumstances precisely formed the robust foundation upon which scientific culture in the West would develop exceptionally.

That Western discourse focused so exclusively on direct interactions between art and science stemmed precisely from the exceptional closeness of their relationship. How did this appear when considered against broader historical currents?

In the Western Middle Ages, there existed no science worthy of the designation. Even mathematics offered scarcely anything deserving serious attention.

Nor did art show any marked development during this period. Western art first achieved true flourishing during the Renaissance era. It was precisely then that Leonardo da Vinci—previously mentioned—emerged, while Albrecht Dürer of Germany likewise belonged to this epoch.

In Renaissance Italy, great painters such as Raphael, Michelangelo, and Tintoretto emerged.
It goes without saying that countless master artists later appeared and produced many masterpieces, but there had never been a time when such splendid great masters flourished as during that revival period.
It was as if history had concentrated its great geniuses into a single moment.

In the Western Middle Ages prior to this period, Gothic-style architecture—such as majestic stone castles constructed atop hills and church buildings in various cities—was vigorously built, with many structures deserving of praise for their beauty.

In literature as well, works such as Dante’s *The Divine Comedy* that conveyed divine essence were created.
According to recent studies, Dante was profoundly influenced by Arabic literature, and even the romantic style involving the beautiful Beatrice has precedents in Islamic literary tradition; yet despite this, Dante’s poetry remains an unparalleled masterpiece.

While we cannot delve into exhaustive detail here, it goes without saying that even during the medieval period—an era still culturally underdeveloped—significant artistic achievements emerged prior to scientific advancements; indeed, in the West too, art was first cultivated before scientific development made great strides.

This Western trajectory does not form a clear temporal parallel with Japan’s medieval age—where art alone flourished for centuries—only for scientific progress to emerge belatedly in the Edo period.
The circumstances are fundamentally distinct.
Yet after art first achieved its monumental leap forward, developments in mathematics and science followed, progressing ever onward until they seemed unstoppable.

In the West, great art was born, and with it emerged great science.

That mathematics in Greece developed into an exceptionally logical system was truly remarkable.
In astronomy and calendrical science too, remarkably significant developments were achieved.
There was no field—whether medicine or natural history—that did not attain substantial development in Greece.
That Greece's scientific advancement manifested great genius stands evident as ten eyes witness and ten fingers point.
Nor was it only science and mathematics that flourished in Greece.
Both Greek architecture and sculpture were truly magnificent.

The history of Greek culture began with the poetic works of the poet-saint Homer.
The elegance of those poetic works was known to all.
Behold how skillfully Greek mythology was constructed.
Later, exceptionally outstanding tragic works were also produced.

The stone architecture of Greek temples erected rows of columns and roofed them with gently sloping linear forms; the overlapping sections of these roofs were simple and not at all displeasing.
While they likely learned from Egypt’s grand stone structures—though lacking Egyptian architecture’s bulkiness—they also possessed an elegance absent in Egypt.

Greek marble sculptures represent the culmination of ancient art and are utterly without parallel.
They excel in formal accuracy and possess strength within their elegance; we find ourselves deeply struck with reverence even when viewing photographs of these stone carvings.
In art, the genius of Greece far surpassed that of the advanced nations.
Greece, which had surpassed other advanced nations in art, similarly far surpassed them in mathematics and science as well.

In Greece, literature and art had flourished most during the period when Athens extended its influence following the Persian Wars.
The prosperity of philosophy too belonged to this era.
Though mathematics and various sciences during this time contained elements worthy of attention, they had not yet attained their zenith.

At this moment, the peerless hero Alexander the Great suddenly emerged.
Greece expanded its reach toward Persia and the East in pursuit of great development.
However, even this newly forged great empire crumbled with the emperor’s unforeseen demise, and the influence of Athens—once the cultural center—had been lost in an instant, never to be wielded again.
While this marked the end of philosophy’s golden age, both literature and art—in tandem with Athens’ period of national expansion—plummeted completely into decline from this point onward.
Even works created after this period, while certainly possessing excellence where it existed, could not begin to rival those from the golden age.

However, whether in mathematics or science, they instead achieved their most outstanding development thereafter.
That Euclid created *Elements*, that Archimedes and Apollonius emerged, that astronomers like Eudoxus and Aristarchus appeared, and that the great arithmetician Diophantus arose—all occurred after this.
Although in Greece, where great art was created, mathematics and science also achieved extraordinary development, it was art that first took form, followed thereafter by mathematics and science.

We think it was entirely natural that this sequence occurred.

Unlike Greece—which remained politically fragmented as a nation—Rome established a vast empire whose political and military power stood unrivaled in momentum, thereby intimidating and subjugating even Greece itself; yet unlike Greece, it may be said that Rome failed to create a culture distinctly its own.
In Rome—whether in literature, architecture, sculpture, philosophy, or science—all were appropriated from Greece.

However, it is widely accepted that there was nothing worthy of note in Roman philosophy.
They learned sculpture and other arts from Greece, yet ultimately failed to attain its divine essence.

In Rome, even in mathematics, they could not fully adopt Greek mathematical systems.
To say that not a trace remained would suffice.

Of course, it goes without saying that Rome advanced in military studies and excelled in constructing roads and aqueducts; however, in mathematics and the sciences, they merely possessed a modicum of practical knowledge and achieved no original developments.

That mathematics and various sciences in Rome—which had learned Greece’s superior art yet failed to surpass it—existed in such a state was indeed only natural.

There too, we sense something that suggests an intrinsic relationship between art and science.

On this issue, there are still countless things we wish to discuss.
However, we have already consumed too much space.
At this juncture, we must reluctantly curtail our discussion.

〈9. Conclusion〉

Let us now turn our attention to reflect upon our own country.
In art, our country consistently created works surpassing what it learned from others, and in both science and mathematics alike did it ever follow this course.
We had already learned there must exist an intrinsic relationship between art and science, and came to understand that science cannot prosper in a country devoid of art.
If this holds true, then in our nation—renowned as a land of art and abundantly rich in artistic sensibility—whether in mathematics or the sciences, they advance boundlessly like the rising sun surging through heaven's expanse; this momentum shall never cease. I firmly believe discerning readers have already fully grasped how this reality stands far removed from our groundless fabrications.

As for the future of mathematics in our country, even from this perspective, it could be said to hold great promise.

We are convinced that we are not mistaken in viewing matters thus.