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. Robertson,

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 Woolsthorpe. For 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 the Scottish mathematician James Gregory, the Frenchman Gilles de Roberval (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. His 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 Blaise Pascal, during his stay in Paris from 1672 to 1676, where he made contact with men such as the French 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 Edition:

“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 with. Leibniz 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 half.

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.

Maclaurin acknowledged his debt to the English mathematician Brook Taylor (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 perspective 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 differences.”

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. A 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 Britannica.

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 Mathematica (Mathematical Principles of Natural Philosophy). 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:

“Newton’s 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 Jean Picard (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 lenses.

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 (Author) 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 after Halley. It took generations until the next periodic comet was identified.

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 German 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 Halley, 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 move.

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 motion in Tang Dynasty China. 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 recorded. 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 data.”

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 observations.

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 gravity.

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 Ptolemy:

“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 yang.”

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 China 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.

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.