ATLAS EXCLUSIVE Fjordman: A History of Mathematical Astronomy, Part 3

Art! Music! Love! History! Fjordman!

Yes. The inestimable Fjordman is back in the Atlassphere, bringing us the third in his series on Mathematical Astronomy. Atlas – keeping you well rounded and properly educated. Food for the heart, soul, mind and spirit.

The first two parts
of his history of mathematical astronomy were published at Jihad
. The fourth part will be published at The Brussels Journal.

Fjordman: A History of Mathematical Astronomy, Part 3

Isaac Newton, perhaps the greatest scientist the world has ever seen,
was born in Woolsthorpe, a village in Lincolnshire, England, into a family of farmers. His father owned property and animals and was
not poor, but he was illiterate. Newton lost his father before birth. His mother
soon remarried, and Isaac was effectively separated from her during most of his
childhood, left in the care of his maternal grandmother. Some biographers trace the emotional instability he sometimes
demonstrated as an adult back to insecurities he experienced in his childhood.
Unlike his father he got an education. At the grammar school in Grantham he
gained a firm command of Latin. In the words of biographer James Gleick:

“He was born into a world of darkness, obscurity and magic; led a strangely pure and obsessive life, lacking parents, lovers and friends; quarreled bitterly with great men who crossed his path; veered at least once to the brink of madness; cloaked his work in secrecy; and yet discovered more of the essential core of human knowledge than anyone before or after. He was the chief architect of the modern world. He answered the ancient philosophical riddles of light and motion, and he effectively discovered gravity. He showed how to predict the courses of heavenly bodies and so established our place in the cosmos. He made knowledge a thing of substance: quantitative and exact. He established principles, and they are called his laws. Solitude was the essential part of his genius. As a youth he assimilated or rediscovered most of the mathematics known to humankind and then invented calculus – the machinery by which the modern world understands change and flow – but kept this treasure to himself. He embraced his isolation through his productive years, devoting himself to the most secret of sciences, alchemy. He feared the light of exposure, shrank from criticism and controversy, and seldom published his work at all.”

Since the young Isaac showed promise at school he was in June 1661 sent to matriculate at Trinity College at the University of Cambridge, which in 1664 for the first time had a
professor of mathematics, the talented English scholar Isaac Barrow (1630-1677). Barrow had studied Greek and Latin, theology, medicine,
history and astronomy. Between 1655 and 1659 he traveled across Europe as far as
Constantinople, then under Turkish Muslim rule. His ship came under attack from
pirates along the way. He was one of the individuals who made great progress
toward developing the methods of calculus. Newton attended his first lectures at
Cambridge, and Barrow encouraged him and later examined him in the Elements of
Euclid. Barrow had issued a complete edition of the Elements in
Latin in 1655 and in English in 1660.

Newton studied extensively on his own as well, absorbing the recent
work of men such as Galileo in addition to the traditional Aristotelian
philosophy. He was largely self-taught in mathematics and essentially mastered the entire achievement of seventeenth-century
mathematics, from François Viète to René Descartes, by the 1660s. He read Descartes’s difficult masterpiece La Géométrie from
1637 and Wallis’s Arithmetica Infinitorum.

John Wallis (1616-1703), the gifted English mathematician who introduced the
symbol ∞ for infinity, was the author of numerous books and contributed to the development of calculus. He was proficient in Latin, Greek and Hebrew
and studied logic. According to J. J. O’Connor and E. F.
Wallis contributed substantially to the origins of calculus and was the
most influential English mathematician before Newton. He studied the works of
Kepler, Cavalieri, Roberval, Torricelli and Descartes, and then introduced ideas
of the calculus going beyond that of these authors. Wallis's most famous work
was Arithmetica infinitorum which he published in 1656….Wallis developed
methods in the style of Descartes analytical treatment and he was the first
English mathematician to use these new techniques.”

The Great
Plague of 1665, the last major outbreak of plague in England, killed more than
one in every six Londoners. In Cambridge, the university closed down for two
years, which proved to be a fruitful period for Newton scientifically at his
home in WoolsthorpeFor most of the following period when he was forced to stay at his
home he laid the foundations of the calculus, some of his ideas in mathematical
astronomy and most of the material later elaborated in his Opticks. These
innovations were not published for many years to come.
The analytical
geometry invented by the French mathematicians René Descartes and
Pierre de Fermat earlier
in the seventeenth century was probably a necessary precondition for the
invention of integral calculus by Newton and Leibniz a few decades later. Fermat
himself took steps in that direction. Other notable pioneers in the history of
calculus include Scottish mathematician James Gregory, the Frenchman
Gilles de
(1602-1675) and above all the Italian Bonaventura Cavalieri (1598-1647). In 1629,
with the help of Galileo, Cavalieri secured the chair of mathematics at the
University of Bologna. He is chiefly remembered for his work on “indivisibles.”
Building on work by Archimedes he investigated the method of construction by
which areas and volumes of curved figures could be found.

As historian of mathematics Victor J. Katz states, “Newton
and Leibniz are considered the inventors of the calculus, rather than Fermat or
Barrow or someone else, because they accomplished four tasks. They each
developed general concepts – for Newton the fluxion and fluent, for Leibniz the
differential and integral – which were related to the two basic problems of
calculus, extrema and area. They developed notations and algorithms, which
allowed the easy use of these concepts. They understood and applied the inverse
relationship of their two concepts. Finally, they used these two concepts in the
solution of many difficult and previously unsolvable problems. What neither did,
however, was establish their methods with the rigor of classical Greek geometry,
because both in fact used infinitesimal quantities.”

The German
polymath Gottfried Wilhelm Leibniz (1646-1716) was a prominent
philosopher in addition to being one of the greatest mathematicians of all time.
rational philosophy embraced history, theology, linguistics, biology and
geology. Born in Leipzig, he entered the University of Leipzig as a law
student in the early 1660s. He taught himself Latin and
read the classics in that language, but also widely employed the French
language. The Dutch polymath Christiaan Huygens
brought him
to the frontiers of mathematical research, for instance the work of Fermat and
during his stay in Paris from 1672 to 1676, where he made contact with men such
as the
philosopher Nicolas Malebranche (1638-1715).
<Leibniz conducted an extensive correspondence with the leading intellectual
figures in Europe and met fellow rational philosopher Baruch Spinoza, although
they did not always see eye to eye in matters of religion. He believed that the
principles of reasoning could be reduced to a symbolic system, an algebra of
thought, in which controversy could be settled by calculations. Dirk J. Struik
elaborates in A Concise History of Mathematics, Fourth Revised

“He was one of
the first after Pascal to invent a computing machine; he imagined steam engines,
studied Chinese philosophy, and tried to promote the unity of Germany. The
search for a universal method by which he could obtain knowledge, make
inventions, and understand the essential unity of the universe was the
mainspring of his life. The scientia generalis he tried to build had many
aspects, and several of them led Leibniz to discoveries in mathematics. His
search for a characteristica generalis led to permutations, combinations,
and symbolic logic; his search for a lingua universalis, in which all
errors of thought would appear as computational errors, led not only to symbolic
logic but also to many innovations in mathematical notation. Leibniz was one of
the greatest inventors of mathematical symbols. Few men have understood so well
the unity of form and content. His invention of the calculus must be understood
against this philosophical background; it was the result of his search for a
lingua universalis of change and of motion in particular. Leibniz found
his new calculus between 1673 and 1676 in Paris under the personal influence of
Huygens and by the study of Descartes and Pascal.”
The invention
of calculus resulted in a protracted and heated priority dispute between the
followers of Newton and Leibniz. The consensus among most mathematical historians
today is that both of them should be considered independent co-founders of
calculus, but their methods were not identical. Leibniz’s notation and his
calculus of differentials prevailed because it was easier to work
nevertheless had great respect for Newton’s intellect. In Berlin he is alleged
to have told the Queen of Prussia that in mathematics there was all previous
history and then there was Newton; and that Newton’s was the better

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The leading mathematician in Britain in the eighteenth century was
Colin Maclaurin (1698-1746) from Scotland, a disciple of Newton, with whom he had been personally
acquainted from visits to London. Maclaurin studied at the University of Glasgow and became the world’s
youngest professor at nineteen at the University of Aberdeen. On the
recommendation of Newton, he was made a professor of mathematics at the
University of Edinburgh in 1725. Maclaurin published the first systematic exposition of Newton’s methods and put his
calculus on a rigorous footing. In 1740 he shared, with the Swiss mathematicians
Leonhard Euler and Daniel Bernoulli, the prize offered by the French Academy of
Sciences for an essay on tides.

acknowledged his debt to the English
mathematician Brook (1685-1731). Taylor was
born into a wealthy family at Edmonton north of London and studied law at the
University of Cambridge. He inherited a love of music and painting from his
strict father and investigated the mathematics of vibrating
strings and the mathematical principles of
in painting. He is especially remembered for Taylor’s theorem and Taylor
series and added to mathematics a new branch now called the “calculus of finite
In 1669 Isaac
Barrow resigned his professorship at Cambridge in favor of Newton. Newton’s life from 1669
to 1687 when he was Lucasian professor was a highly productive period.
comet appeared in 1680 which was observed by Halley, Robert Hooke and Newton.
Comets were known to appear every now and then and often considered bad omens,
but each one was believed to be unique. Yet in 1680, European astronomers
observed two comets with intervals of a few weeks. In England, John Flamsteed
thought that comets might behave like planets, orbiting the Sun, and that the
latter comet was the same as the first, now on its way back.

John Flamsteed (1646-1719) was born in Denby, Derbyshire in
England. His father was a prosperous maltster, a lucrative business as malted
grain could be used for malt beer or whisky. John studied astronomical
science by himself in the 1660s and was ordained a clergyman
in 1675
. “Instruments were of immense importance to Flamsteed. They bulk very large in his
autobiographical accounts of his life, and they form the central theme of his
Preface to the Historia. Early in his life he learned to grind lenses. He was
constantly concerned with making and improving instruments–a sextant, a
quadrant, a mural arc of 140 degrees, telescopes, the graduation and calibration
of the scales and micrometer-screws.” He was appointed as the first Astronomer
Royal when the Greenwich Observatory was constructed outside of London. John
Flamsteed had a stormy working relationship with Edmund Halley and Isaac Newton.
The complete version of his meticulous observations of nearly 3000 stars was
published posthumously in 1725 as the Historia Coelestis

The Englishman Robert Hooke, a brilliant instrument maker and
technician but not an equally great mathematician, “became Newton’s goad,
nemesis, tormentor, and victim.” In 1679, Newton learned of his idea that orbital motion could be
explained by a combination of a linear inertial component along the orbit’s
tangent and a continual falling inward toward the center.

Robert Hooke had not been the first to propose the inverse-square law
of attraction and for him it was only a guess. For Newton it appeared logical
and mathematically inevitable: Every material object in the universe attracts every other object
with a force proportional to their masses and inversely proportional to the
square of the distance between their centers.

Galileo had said that bodies fall with constant acceleration no
matter how far they are from the Earth. Newton sensed that this must be wrong,
and estimated that the Earth attracts a falling apple 4,000 times as powerfully
as it attracts the Moon. If the ratio, like brightness, depended upon the
square of distance, that might be roughly correct. Since the distance to
the Moon is about 60 times that of the Earth’s radius then the Earth’s gravity
might be 3,600 times (60 times 60) weaker there than at the Earth’s surface. He
also arrived at the inverse-square law by an inspired argument based on Kepler’s
laws of planetary movements.

Encouraged by
his friend Edmond Halley, who had seen some of his promising initial ideas,
Isaac Newton began to develop his work in greater detail. In 1687 he finally
published his resulting masterpiece, the Philosophiae Naturalis Principia
(Mathematical Principles of Natural
). The first of Principia’s three books set forth the science
of motion, the second the conditions of fluid resistance and their consequences,
and the third the system of the world, explanations of tides, the motions of the
Moon and comets, the shape of the Earth etc. Authors James E. McClellan and
Harold Dorn explain:
celestial mechanics hinges on the case of the earth’s moon. This case and the
case of the great comet of 1680 were the only ones that Newton used to back up
his celestial mechanics, for they were the only instances where he had adequate
data. With regard to the moon, Newton knew the rough distance between it and the
earth (60 Earth radii). He knew the time of its orbit (one month). From that he
could calculate the force holding the moon in orbit. In an elegant bit of
calculation, using Galileo’s law of falling bodies, Newton demonstrated
conclusively that the force responsible for the fall of bodies at the surface of
the earth – the earth’s gravity – is the very same force holding the moon in its
orbit and that gravity varies inversely as the square of the distance from the
center of the earth. In proving this one exquisite case Newton united the
heavens and the earth and closed the door on now-stale cosmological debates
going back to Copernicus and Aristotle.” The French astronomer,
priest and engineer

(1620-1682) was born in La Flèche, studied at the Jesuit College there and
became involved in astronomy with Pierre Gassendi in Paris. Picard became a
major figure in the development of scientific cartography. He corresponded with
leading men of science such as Christiaan Huygens, Ole Rømer, the Danish natural
philosopher Erasmus Bartholin (1625-1698) and the Dutch mathematician Jan
Hudde (1628-1704), who also
served as a city council member in Amsterdam and worked with philosopher and
lens grinder Baruch Spinoza on the construction of telescopic

According to biographers J. J. O’Connor and E. F. Robertson,
All the instruments he used to carry out this work were fitted with
telescopic sights which gave him values correct to 10 seconds of arc (Tycho
Brahe had only attained an accuracy of 4 minutes of arc) and he produced a value
for the radius of the Earth which was only 0.44% below the correct result. The
use of these techniques meant that Picard was one of the first to apply
scientific methods to the making of maps. He produced a map of the Paris region,
and then went on to join a project to map France. His data on the Earth was used
by Newton in his gravitational theory.”

Newton suffered from periods of depression and had a serious
nervous breakdown in 1693. He became Warden of the Royal Mint in 1696
in London and as such a highly paid government official with less interest in
research, but he was a capable administrator and a president of the Royal
Society. He lived in an island nation and explained how the Moon and
the Sun tug at the seas to create tides, but it is possible that he never set
eyes on the ocean.

When he died in London in 1727 he was given a state funeral, the
first for a subject whose attainment lay in the realm of the mind. The visiting
French writer calling himself Voltaire was amazed by his kingly
funeral. He was buried in Westminster Abbey. Even Newton had to build on the work of his predecessors,
which is why he made his famous statement that “
If I have seen further it
is by standing on the shoulders of Giants.” Yet arguably no single human being
has ever changed the way we view the world more than him. As James Gleick
James Gleick
puts it:

“Newton’s laws are our laws. We are Newtonians, fervent and devout,
when we speak of forces and masses, of action and reaction; when we say that a
sports team or political candidate has momentum; when we note the inertia of a
tradition or bureaucracy; and when we stretch out an arm and feel the force of
gravity all around, pulling earthward. Pre-Newtonians did not feel such a force.
Before Newton the English word gravity denoted a mood – seriousness,
solemnity – or an intrinsic quality. Objects could have heaviness or lightness,
and the heavy ones tended downward, where they belonged. We have assimilated
Newtonianism as knowledge and as faith. We believe our scientists when they
compute the past and future tracks of comets and spaceships. What is more, we
know they do this not by magic but by mere technique. ‘The landscape has been so
totally changed, the ways of thinking have been so deeply affected, that it is
very hard to get hold of what it was like before,’ said the cosmologist and
relativist Hermann Bondi. ‘It is very hard to realise how total a change in
outlook he produced.’”

The English astronomer and mathematician Edmond Halley (1656-1742)
became the first major scholar to work squarely within the Newtonian school of
thought. He was born into a prosperous London family, made astronomical
observations at Oxford and was inspired by John Flamsteed at the newly
established Royal Observatory at Greenwich. In 1676 he sailed for the island of
St. Helena, then the southernmost territory under British rule, and spent a year
to produce a chart of stars of the Southern Hemisphere. Halley encouraged,
personally oversaw and paid for the publication of Newton’s groundbreaking
Principia in 1687.

For the second edition of the Principia in 1695 he agreed to
calculate comet orbits. He realized that the comets of 1531, 1607 and 1682 had
similar orbits, and deduced that they were the same comet turning around the Sun
in an elliptical orbit. This was the first calculation of a cometary orbit ever
made. Halley found the time to participate in many non-astronomical activities
as well, to create an improved diving bell, study magnetic variation and serve
as a sea captain. He enhanced our understanding of trade winds, tides,
cartography, naval navigation and mortality tables. He succeeded John Flamsteed
as Astronomer Royal.

The comet which is now called Halley’s Comet had been seen by others
before him. There are Chinese records of it going back to 240 BC and the
Bayeux Tapestry, which commemorates the Norman Conquest of England in 1066,
depicts an apparition of it. Yet nobody had recognized these comets as the same
one returning and calculated its orbit. This is why it is properly named
Halley. It took generations until the next periodic comet was

Johann Franz Encke (1791-1865),
born in Hamburg, Germany, studied mathematics and astronomy at the University of
Göttingen under the great genius Carl Friedrich Gauss.
Encke followed a suggestion by the French astronomer and prolific
comet discoverer Jean-Louis
Pons (1761-1831), who suspected that a
comet he had spotted was the same as one discovered by him in 1805. Encke sent calculations to Gauss, Olbers and Bessel and
predicted its return for 1822. This comet is known today as Encke’s
Comet, but Encke himself always referred to it as “Pons' Comet.” Its orbital
period of just 3.3 years caused a sensation and made Encke famous as the discoverer of short
periodic comets. The
nineteenth century astronomer
Wilhelm Olbers devised the first satisfactory
method of calculating cometary orbits.

Oskar Backlund (1846-1916) was educated at the
University of Uppsala in his native Sweden, but spent his career in the Russian
Empire at the Dorpat Observatory (now Tartu, Estonia) and the Pulkovo
Observatory. He computed the orbit of Encke’s Comet and used it to estimate the
masses of Mercury and Venus. He concluded that its motion was affected by
nongravitational forces and an unknown effect that coincided with the sunspot
cycle. The study of comet tails in the twentieth century led to the prediction
of the existence of the solar wind.

In 1718
based on his own observations as well as those made by Flamsteed, compared star
positions with the more limited star catalog created by Hipparchus and Ptolemy
in Antiquity. Most of the positions matched reasonably well, but some stars such
as Arcturus were so far away from their recorded ancient positions that the
discrepancy could not be because of slight inaccuracies; it had to be because
the stars really had moved relative to us. Tycho Brahe was convinced that stars
are fixed on their spheres and smoothed these anomalies away, but Halley lived
in the Newtonian universe where mutual gravity affects the movement of objects
and was willing to consider the possibility that stars can actually
There are
those who suggest that the Chinese astronomer and Buddhist
monk Yi
Xing (AD 683-727), born Zhang Sui, was the first to describe proper stellar
motionin Tang Dynasty
hina. There are many claims that the Chinese did this or that centuries
before Western scholars, some of them credible, others less so, but Yi Xing’s
alleged discovery is plausible; he was a gifted man who made one of the first
known clockwork escapement mechanisms.
Su Song
(1020-1101), a Chinese bureaucrat, astronomer, engineer and statesman in the
Song Dynasty, around 1090 made a large water-driven astronomical clock in the capital city of
Kaifeng, an
impressive mechanical device by eleventh century standards. His work included a
star map based on a new survey of the heavens, the oldest printed star map ever
Books printed with wooden blocks were fairly widespread in China already at this
time. Here
is a quote from the book Science and
Technology in World History
Second Edition

“Although weak in astronomical theory, given the charge to search for
heavenly omens, Chinese astronomers became acute observers….who produced
systematic star charts and catalogues. Chinese astronomers recorded 1,600
observations of solar and lunar eclipses from 720 BCE, and developed a limited
ability to predict eclipses. They registered seventy-five novas and supernovas
(or 'guest' stars) between 352 BCE and 1604 CE….With comets a portent of
disaster, Chinese astronomers carefully logged twenty-two centuries of cometary
observations from 613 BCE to 1621 CE, including the viewing of Halley's comet
every 76 years from 240 BCE. Observations of sunspots (observed through dust
storms) date from 28 BCE. Chinese astronomers knew the 26,000-year cycle of the
precession of the equinoxes. Like the astronomers of the other Eastern
civilizations, but unlike the Greeks, they did not develop explanatory models
for planetary motion. They mastered planetary periods without speculating about
orbits. Government officials also systematically collected weather

The Chinese apparently never calculated the orbits of any of the many
comets they had observed. They had a large mass of observational data yet never
used it to deduct mathematical theories about the movement of planets and comets
similar to what Kepler and others did in Europe. Newton’s Principia was
written a few generations after the introduction of the telescope, which makes
it seductively simple to believe that the theory of universal gravity was
somehow the logical conclusion of telescopic astronomy. Yet this is not at all
the case. As we have seen, Kepler’s initial work was based on pre-telescopic

What would have happened if the telescope had been invented in China?
Would we then have had a Chinese Newton? This is far from certain. Chinese
culture never placed much emphasis on law, either in the shape of man-made law
or natural law. If the Chinese had invented the telescope it is likely that they
would have used it to study comets, craters on the Moon etc. This would clearly
have been valuable; any culture that used telescopes would undoubtedly have
generated new knowledge with the device, but not necessarily a law of universal

In his excellent book Cosmos, scholar John North points out
that in China, where astronomy was intimately connected with government and
civil administration, interest in cosmological matters was not markedly
scientific in the Western sense of the word and did not develop any great
deductive system of a character such as we see in Newton, or even Aristotle or

“The great scholar we know as Confucius (551 BC-478 BC) did nothing to
help this situation – if in fact it needed help. Primarily a political reformer
who wished to ensure that the human world mirrored the harmony of the natural
world, he wrote a chapter on their relation, but it was soon lost, and a number
of stories told of him give him a reputation for having no great interest in the
heavens as such….The all-pervading Chinese view of nature as animistic, as
inhabited by spirits or souls, gave to their astronomy a character not unknown
in the West, but at a scholarly level made it markedly less well structured. At
a concrete level, we come across such Chinese doctrines as that there is a cock
in the Sun and a hare in the Moon – the hare sitting under a tree, pounding
medicines in a mortar, and so forth. At a more abstract level there is the
notorious all-encompassing doctrine of the yin and the yang, a
form of cosmology that is to Aristotelian thinking as yin is to

Naturally occurring regularities and phenomena could be observed, of
course, but the Chinese did not generally deduct universal natural laws from
them, possibly because their view of nature was that reality is too subtle to be
encoded in general, mathematical principles. In European astronomy phenomena
such as comets, novae and sunspots that did not readily lend themselves to
treatment in terms of laws were taken far less seriously than those that were.
The history-conscious Chinese, on the other hand, kept detailed and plentiful
records of all such phenomena, records which still remain a valuable source of
astronomical information.

The Chinese could clearly produce talented individuals, but their
work was often not followed up. The Imperial bureaucracy was hampered by many
obstacles to the free and unfettered pursuit of scientific knowledge, especially
due to excessive secrecy and regulation in the study of mathematics and
astronomy. By making this study a state secret, Chinese authorities drastically
reduced the number of scholars who could, legitimately or otherwise, study
astronomy. This restriction greatly reduced the availability of the best and
latest astronomical instruments and observational data. The Rise of Early
Modern Science
by Toby E. Huff:

“The fact remains that virtually every move made by the astronomical
staff had to be approved by the emperor before anything could be done, before
modifications in instrumentation or traditional recoding procedures could be put
into effect. It is not surprising, therefore, that despite the existence of a
bureau of astronomers staffed by superior Muslim astronomers (since 1368), Arab
astronomy (based as it was on Euclid and Ptolemy) had no major impact on Chinese
astronomy, so that three hundred years later when the Jesuits arrived in China,
it appeared that Chinese astronomy had never had any contact with Euclid's
geometry and Ptolemy’s Almagest. Moreover, contrary to Needham’s
arguments, more recent students of Chinese astronomy suggest that Chinese
astronomy was perhaps not as advanced as Needham suggested and that ‘Chinese
astronomers, many of them brilliant men by any standards, continued to think in
flat-earth terms until the seventeenth century.’ If we consider the study of
mathematics, in which the metaphysical implications of abstract thought may be
less obvious to outsiders and which may therefore give scholars more freedom of
thought, we encounter an institutional structure equally detrimental to the
advancement of science.”

Astronomy in the Islamic world stagnated and never managed to leave
behind its Earth-centered Ptolemaic structure, as Europeans eventually did, but
Muslims were familiar with Greek knowledge and geometry. The sphericity of the
Earth had been known to the ancient Greeks since the time of Aristotle and was
never seriously questioned among those who were influenced by Greek knowledge in
the Middle East, in Europe and to some extent in India. The myth that medieval
European scholars believed in a flat Earth is of modern origin.

I have consulted several balanced, scholarly works on the matter.
Even a pro-Chinese book such as A Cultural History of Modern Science in
by Benjamin A. Elman admits that Chinese scholars still believed in a
flat Earth in the seventeenth century AD, when European Jesuit missionaries
introduced new mathematical and geographical knowledge to China:

“For instance, the first translated edition of Matteo Ricci’s map of
the world (mappa mundi), which was produced with the help of Chinese
converts, was printed in 1584. A flattened sphere projection with parallel
latitudes and curving longitudes, Ricci’s world map went through eight editions
between 1584 and 1608. The third edition was entitled the Complete Map of the
Myriad Countries on the Earth
and printed in 1602 with the help of Li
Zhizao. The map showed the Chinese for the first time the exact location of
Europe. In addition, Ricci’s maps contained technical lessons for Chinese
geographers: (1) how cartographers could localize places by means of circles of
latitude and longitude; (2) many geographical terms and names, including Chinese
terms for Europe, Asia, America, and Africa (which were Ricci's invention); (3)
the most recent discoveries by European explorers; (4) the existence of five
terrestrial continents surrounded by large oceans; (5) the sphericity of the
earth; and (6) five geographical zones and their location from north to south on
the earth, that is, the Arctic and Antarctic circles, and the temperate,
tropical, and subtropical zones.”

The ancient
Greeks developed spherical trigonometry as an important tool. One of the most
prominent pioneers was Hipparchus in the mid-second century BC, who made very
good estimates of the Earth-Moon distance. Trigonometry in the Western fashion
was virtually unknown in East Asia until the seventeenth century AD, when it was
introduced to China via Jesuit missionaries from Western Europe. This was
further introduced to Japan in the eighteenth
century, later supplemented by translations via Dutch traders in

Japan received much scientific and technological information from the
mainland via Korean immigrants during the sixth, seventh and eighth centuries
AD. Confucianism, Buddhism and iron technology all came to Japan from China.
They also took over some of China’s flaws, for instance with ranking astrology
and divination higher in the scale of human wisdom than calendar-making.
Yet Japan evolved not in the direction of a centralized monarchy but of
what might be termed feudal anarchy. The clan was an enlarged patriarchal family
and the nation the most enlarged family of all. Shinto religious practices, with
no fixed doctrines or canonical strictures, coexisted easily with Buddhism. The
emperor was formally at focus, but powerful families such as the Fujiwara clan often held the real power for long periods of time.

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