Authors: Michela Fontana
One day in 1590, he received a visit from a
shidafu
named Qu Rukui but better known as Qu Taisu, who had already called at the Jesuits’ residence in Zhaoqing the previous year and had been greatly impressed by Ricci’s wisdom and charisma. Obsessed with alchemy and convinced that the missionary was a great adept capable of teaching him its secrets, he decided to travel to Shaozhou on learning that the Jesuit had moved there. Three years younger than Ricci, Qu Taisu was the son of the eminent mandarin and illustrious scholar Qu Jingchun, who had held the very important post of minister of rites and had died at an early age. Qu Taisu had studied for a long time and passed the imperial examinations but held no position in the bureaucracy.
Ricci was now familiar with the system of examinations that opened the way to a career in the bureaucracy. Selection took place in three stages. Held annually in the most important cities of every province, the first examination earned successful candidates the title of
xiucai
, or “budding talent,” which Ricci regarded as equivalent to “bachelor,” the first academic degree in medieval universities. This qualification brought a number of privileges, including exemption from certain duties obligatory for the rest of the population and permission to wear a special gown with a black hat and boots. The “budding talents” were a sort of reservoir of excellent students, about one hundred thousand in the country as a whole at the time, the best of whom were subsequently selected for participation in the second-level examinations held every three years in the provincial capitals. Successful candidates, about 10 percent of the total, became
juren
—provincial graduates or “literati recommended [to the court]” (translated by Ricci as
licenziato
or “licentiate”)—who were qualified to hold positions on the lower rungs of the bureaucratic ladder. Held again every three years but this time in Beijing, the third and highest examination conferred the qualification of
jinshi
—metropolitan graduate or “literatus presented [to the court],” which Ricci equated with “doctor”—and permitted access to the upper echelons of the bureaucracy.
Even though admission to the Confucian schools and the examinations was open to all in principle, it was in practice only wealthy families such as those of great landowners that could afford to support their sons for all the years required to obtain an official post. Young men often sat the examinations repeatedly, willing to invest many years of their lives in the enterprise. The stakes were very high and the competition correspondingly fierce, as some of the privileges enjoyed by those obtaining good positions in the bureaucracy were extended to members of their immediate family, whose well-being thus also depended on the candidate’s success. When a son passed an examination, the family celebrated on a lavish scale and erected a wooden arch before their front door. The young man’s name and those of all the other successful candidates were then carved on massive stone tablets in the courtyards of the Confucian temples.
Qu Taisu passed the first two levels of examinations with flying colors and became a
juren
. But then, finding himself alone with no authoritative guidance after his father’s sudden death, he abandoned any ambition for a career in the bureaucracy and ceased his studies. He devoted his energies to pursuing the impossible dream of alchemy and spent years traveling in search of experts capable of teaching him how to transmute base elements, accommodated everywhere with all the respect due to the son of an important
guan
. Having squandered practically all of his inheritance over the years and being heavily in debt, he was now living in Nanxiong with his concubine, who had stayed with him after the death of his wife.
In accordance with custom, Qu Taisu visited Ricci bearing gifts. After prostrating himself three times as required, he invited the Jesuit to a solemn banquet and asked him to become his master. The young scholar’s social position made this a sign of great respect, especially as the master-pupil relationship was understood to last for the rest of one’s life. Ricci presented Qu Taisu with valuable gifts in turn to show him that Jesuits did not teach for gain but solely for the love of wisdom. He then agreed to become his master but surprised him by declaring that he would not teach him the secrets of transmutation but rather a discipline that would help him to cultivate his mind, namely mathematics. Qu Taisu accepted with great curiosity.
Doing Sums with Brush and Paper
Ricci had noted that the Chinese used the abacus,
5
a rectangular frame of wood with beads that could be slid along wires, in order to make arithmetical calculations. Merchants, shopkeepers, literati, and officials used this with great skill to add, subtract, and multiply but found it much harder to divide and extract square and cube roots. Commonly used in China from the eleventh century on, the abacus was a descendant of the tablets employed for calculation in antiquity by the Chinese and even earlier by the Babylonians, followed by the Greeks, Indians, and Romans, which had evolved with different characteristics in the various countries over the centuries.
In its form as a counting board,
6
the abacus had long been supplanted in Europe by the more flexible and effective method of written calculations using Hindu-Arabic numerals. Commonly used today, this base ten positional numeral system
7
was invented in India and was introduced into Europe by the Arabs sometime around the tenth century. It made it possible to perform arithmetical operations, including multiplication and division, more easily than with the abacus and the outmoded Roman numerals. The disputes between “abacists” and “algorists”—advocates respectively of the old systems of calculation and the new methods,
8
soon established as an invaluable tool for merchants, bankers, architects, and everyone required to keep systematic accounts—were now a thing of the past.
Written calculation was not used in China despite the existence of a decimal positional numeral system conceptually similar to ours and originating long before the analogous system was adopted in India.
9
The symbols representing the numbers from one to nine were made up of vertical or horizontal dashes, and zero was an empty circle, as in the West. This acquired the new name of
ling
—meaning “dew drop” in ancient Chinese—during the Ming era.
10
When Ricci saw Qu Taisu using the antiquated abacus—“that device of threaded beads”
11
—to do sums, he said he would teach him a new system of calculation and set about doing so with brush and paper. The scholar discovered that the new written method was effectively an improvement, especially as a record remained of the various operations required to arrive at the final result, thus making it much easier to check the calculation performed. Moreover, written calculation made it possible to carry out more complex arithmetical operations than the abacus.
The value of the Western method in the eyes of Qu Taisu and the other literati wishing to learn it from Ricci was, however, not only practical. The ability to perform “calculations with the brush,” as the new method soon came to be called by the Chinese, enabled mandarins to acquire another cultural advantage with respect to the other social classes in the field of mathematics, where they had never enjoyed supremacy. All merchants were skilled in reckoning, more so indeed than the mandarins, but none of them knew the Western art of written calculation.
Teaching Qu Taisu confirmed Ricci’s suspicion that the scholars’ cultural background was deep but almost exclusively humanistic. Even though arithmetic was part of the basic school syllabus in China and the literati studied some aspects of it in greater depth during their later education, mathematics was regarded in the Ming era as an inferior branch of knowledge with respect to literary studies, and it was not one of the set subjects in the imperial examinations. While technical questions on astronomical calculations and the calendar were occasionally included,
12
this was not standard practice.
As a result of this situation, the numerous mathematicians of the Ming era mostly belonged to the lower classes. For example, the author of one of the most important mathematical works of the period—namely the
Suanfa Tongzong
, or
Systematic Treatise on Arithmetic
, published in 1592 and including a detailed description of how the Chinese abacus worked—was Cheng Dawei, born into a family of merchants and employed as a clerk in local administration. The only government official to become a skilled mathematician was Gu Yingxiang,
13
appointed governor of Yunnan halfway through the sixteenth century and subsequently minister of justice.
Ricci believed that mathematics was not only little studied in China but was also very backward with respect to developments in Europe. The reality was very different from the image formed by the Jesuit, who had only limited knowledge of his hosts’ culture, was influenced by his preconceptions, and knew nothing of the history of Chinese science. There were in fact many important mathematicians in China during the period when Ricci lived there, and the scientific discoveries achieved can be considered significant, as contemporary historians of mathematics have pointed out. Peter Engelfriet, who has studied the influence of the work of Ricci and the other Jesuits on the development of mathematics in China, observes for example that Cheng Dawei described a method of solving systems of equations with many unknowns that would have been something new to Ricci if the missionaries had shown any interest in Chinese mathematics.
14
It is, however, true that mathematics, and science in general, was going through a phase of decline with respect to the past during the Ming era,
15
and there was no scientific community comparable to the one existing in the West at the time.
16
Scholars taking an interest in mathematics in China were comparatively isolated and had great difficulty obtaining specialized texts because the classical works of Chinese mathematics had disappeared from the libraries.
17
Even the most famous work, namely the
Jiuzhang suanshu
, or “Nine Chapters on the Mathematical Art”—which dated back to the early centuries of the Christian era, boasted countless commentators, and had been printed for the first time in 1084, four centuries before the first printed version of Euclid’s
Elements
was published in Latin in Venice in 1482—was no longer available in its complete version. Scholars sometimes even embarked on great journeys in the hope of finding copies of long-out-of-print treatises on algebra and arithmetic in remote libraries. Cheng Dawei himself had collected the books for his studies by traveling through China for more than twenty years, clear proof that those who studied mathematics in China were spurred by passion rather than by any hope of institutional recognition. The only exceptions were the imperial mathematicians and astronomers responsible for performing the calculations necessary for drawing up the calendar, who enjoyed positions of great prestige but were in many cases devoid of intellectual curiosity or any real skill, as Ricci was to see for himself.
18
Euclidean Geometry and the
Achievements of Chinese Mathematics
Ricci was unaware of China’s mathematical achievements in previous ages, when Chinese scholars had made many breakthroughs long in advance of their Western colleagues. One of the many instances of this regarded pi, the ratio of the circumference of a circle to its diameter, for which the sixth-century Chinese mathematician Zu Chongzhi succeeded, with the help of his son, in calculating an approximate value down to the tenth decimal place with a precision that no one in the West was to surpass for eleven centuries. This result was so significant that a region of the moon was named after this Chinese mathematician.
Other important achievements were made in algebra in the Song era between the tenth and thirteenth centuries, regarded as the “Chinese Renaissance,” and in the immediately subsequent Yuan era. Rather than the abacus, the Chinese of the time used counting rods of bamboo, ivory, or iron, with black for negative numbers and red for positive, in more advanced calculations. Placed on a tabletop divided into squares, they were used to perform complex operations such as extracting square and cube roots and solving algebraic equations and systems of equations.
19
All officials of the period carried a bag of rods with them ready for use and were so quick in manipulating them that, according to a writer of the ninth century, it was impossible to follow the movements of their hands.
20
The works of algebra printed in those years also contained highly advanced results. The
Shushu jiuzhang
(“Mathematical Treatise in Nine Chapters”) published in 1245 by the mathematician and imperial official Qin Jiushao—whose passion for numbers did not save him from having repeated charges of corruption lodged with the censors—included numerical equations higher than the third order and a method for calculating the square roots of large numbers that was not discovered in the West until six centuries later.
The
Siyuan yujian
(“Precious Mirror of Four Elements”) of 1303, considered the highest expression of Chinese algebra of the period, included equations up to the fourteenth order and presented an approximate way of calculating solutions that appeared in the West as Horner’s method only five centuries later. The “four elements” mentioned in the title were the heavens, the earth, man, and matter, the names used to designate the four unknowns in an equation. Another feature of this volume was the representation on its cover in Chinese characters of a set of numbers arranged in triangular form and showing the coefficients of the successive binomial powers up to eight, which appears to have been discovered two hundred years earlier. Also known as Pascal’s triangle, this was introduced in Europe by Niccolò Tartaglia in the sixteenth century, and the typical image presenting the binomial coefficients in triangular form appeared for the first time on the cover of a book by the German Peter Bienewitz, in Latin Petrus Apianus, in 1527.