Saturday, November 01, 2003

A History of Mathematics and Mathematical Astronomy

The Fjordman Report

The noted blogger Fjordman is filing this report via Gates of Vienna.
For a complete Fjordman blogography, see The Fjordman Files. There is also a multi-index listing here.

This essay contains new material. A slightly shorter version was originally published in four parts at various sites. See: Part 1, Part 2, Part 3, and Part 4.

Astronomy and Natural Philosophy in Ancient Times

While writing this history, the MacTutor History of Mathematics website maintained by John J. O’Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland proved invaluable to me. They have created extensive online biographies of hundreds of mathematicians and astronomers from around the world from Antiquity until the turn of the twenty-first century. I have found the biographical information they provide to be generally reliable and have therefore widely consulted their Internet entries when searching for background information, in addition to the entries at the Encyclopædia Britannica Online.

It is difficult to speak of “science” in prehistoric times. Perhaps the closest we can get is the systematic study of the heavens. Archaeoastronomy is the intersection between astronomy and archaeology. The patterns of stars in the night sky were far more familiar to people in ancient times than they are to us, who often suffer from light pollution from electric lights.

Paintings on cave walls and ceilings from prehistoric times, often depicting large wild animals, have been found in South Africa, Southeast Asia, Australia and South America, but some of the oldest and most spectacular ones have been discovered in Europe. Hundreds of cave paintings were created in the Chauvet Cave in south France from around 30,000 BC onward. Lascaux is the setting of a complex of caves in southwestern France with beautiful prehistoric cave paintings and spectacular drawings of bulls, horses and other animals. They were painted during the Upper Paleolithic, the final phase of the Old Stone Age, and are estimated to be more than 16,000 years old. Other paintings exist in the Cave of Altamira in Spain, dating back to at least 13,000 BC. German researcher Michael Rappenglueck believes that he has found a prehistoric map of the night sky among the Lascaux paintings. This is plausible, but we simply don’t know for sure what the function of these artistic drawings was.

According to Paul Mellars in The Oxford Illustrated History of Prehistoric Europe, “One possibility is that some of the major centres of art production (such as Lascaux in south-west France, or Altamira in northern Spain) served as major ritualistic or ceremonial centres — perhaps the scene of important ceremonies during regular annual gatherings by the human groups. Alternatively (or in addition) the production of the art could have been in the hands of particular chiefs or religious leaders who used the creation of the art, and associated ceremonial, to reinforce and legitimate their particular roles of power or authority in the societies. Clearly, all this lies in the realm of speculation. What is clear is that cave art is not uniformly distributed throughout Europe, and is concentrated in areas which (on other, independent archaeological grounds) are known to have contained some of the highest and densest concentrations of human populations.”

During the last Ice Age, much of the Northern Hemisphere was covered by thick ice sheets. Central Europe resembled the tundra of present-day Siberia. At the height of the Last Glacial Maximum around 20,000 BC, land temperatures were about 20 °C lower than they are today. After the end of the Ice Age (ca. 13,000 BC) came the gradual establishment of a milder climate similar to today’s from 9500 to 8000 BC. Because of this, the flora and fauna of the European continent changed rapidly, with species such as the woolly mammoth disappearing. The ice smelting following the retreat of the great glaciers changed the face of Europe dramatically. Much of what was then dry land is now underwater and vice versa. The same goes for other regions in Asia and the Americas as the sea level rose considerably worldwide.

During the Neolithic or New Stone Age, settled communities adopted agriculture, starting in the Balkans close to the Near East. One well-preserved natural mummy from the Copper Age, the intermediate stage between the Stone and Bronze Ages when early metal tools were developed, was found in 1991 in the Alps between Italy and Austria. The mummy from about 3300 BC, named Ötzi the Iceman, apparently died a violent death. He had many small tattoos, a cloak made of woven grass, a pair of leggings, a loincloth and excellent shoes. His coat was made of the hide of the domestic goat and he had a bearskin cap and a belt of calf’s leather.

Ötzi’s equipment consisted of 18 different types of wood and demonstrates that he and his contemporaries possessed excellent knowledge of natural materials and herbs. He carried a dagger with a flint blade, a bow and arrow set and above all a fine copper axe. While the Alpine region had rich copper deposits, only the wealthy could at this time afford copper tools, which indicates that the Iceman’s family were reasonably well-off. Ötzi himself may have been a shepherd who also had to be able to hunt and repair his clothing and equipment.

By the fourth millennium BC, people had been living in fixed dwellings for some time in much of Central Europe, and food was procured from farming and animal rearing. Among the plants that were cultivated were naked wheat, einkorn, emmer wheat, barley, poppy, flax and peas. In addition to the traditional hunting, gathering and fishing, domesticated cattle, pigs, sheep and goats were used as sources of meat and provided leather, milk and possibly wool.

The Goseck Circle in north-central Germany dates back to ca. 5000 BC, one of a number of similar structures in Central Europe. It is proof that Neolithic Europeans observed the heavens with greater accuracy than was previously supposed and is one of a rising number of archaeological finds aided by aerial photography. John North in his book Cosmos, 2008 Edition, writes about early European astronomy. Many attempts have been made to reconstruct the belief systems of the peoples responsible for these astronomical monuments:

“There are numerous indications of cults of the Sun and Moon, not all of them stemming from the orientation and planning of large monuments. One of the most interesting finds was that made in 1902 at Trundholm on Zealand (Sjælland, Denmark), of a Bronze Age horse-drawn disk, dating perhaps from roughly 1400 BC. There can be little doubt that this had solar significance. The Sun is shown being pulled by a horse in several crude Swedish rock carvings of much the same date. An equally rich discovery, this time from Germany, was of a disk of bronze 32 centimeters in diameter, studded with gold shapes that related to the heavens in some way. Found near Nebra at the end of the twentieth century and now known as the Nebra disk, it came more specifically from Mittelberg — a modest hill in the Ziegelroda Forest, between Halle and Erfurt. It seems to have been discovered within a pit inside what had once been a Bronze Age palisade and complex of defensive ditches.”

The Nebra sky disk from ca. 1600 BC was at first assumed to be a forgery (of which there are unfortunately quite a few in museums around the world), but closer studies eventually revealed it to be most likely authentic. The Trundholm disk or Sun chariot dates from around 1400 BC and shows a horse-drawn vehicle with spoked wheels. The whole group measures 60 centimeters in length, and the disk has a bright side covered with gold leaf. Horse-drawn chariots with spoked wheels are associated with the second phase of the Indo-European expansion and spread across Eurasia all the way to China in the second millennium BC.

According to the book Indo-European Poetry and Myth by Martin L. West, the words for “Sun” are related in nearly all the Indo-European branches. Ancient Greek writers observed what they took to be Sun-worship among other Indo-European speaking peoples such as the Persians and the Thracians, and the Germanic tribes were attributed a form of solar worship by Roman writers. The Slavs, too, were regularly credited with Sun-and Moon-worship by chroniclers and clerics. The Sun appears in Russian folklore in female persona as “Mother red Sun.” The chorus in the Greek play Oedipus Tyrannus (Oedipus the King) by Sophocles (ca. 430 BC) swears to Oedipus by the Sun god Helios that they wish him no harm. The Franks in the seventh century AD, although converted to Christianity, still had the habit of swearing by the Sun. Oaths by the Sun, Moon etc. are mentioned in Old Irish and Norse literature as well.

There is much evidence for the circular Sun being associated with a wheel, or that the Sun-god has a chariot drawn by a horse. The Trundholm disk is not unique; fragments of similar Sun-disks have been found elsewhere in northern Europe. There are also solar disks from the second millennium BC further south, in Greece and the Aegean region. There seems to be a mythology related to a many-legged animal, perhaps as an expression of speed and stamina. Slovak and Russian folklore tells of an eight-legged horse that draws the Sun. Although he has no apparent solar association it is conceivable that there is a connection from this creature to Odin’s treasured eight-legged horse Sleipnir in Norse mythology.

From roughly 4500-2500 BC, a belt of megalithic monuments (large stone structures) stretched along the Atlantic coastlands of Western Europe, Britain, the Iberian Peninsula and certain western Mediterranean islands. In Sardinia, numerous nuraghes or towers of large stones were built in the second millennium BC or earlier, many of which still exist today. Some of their entrances may have had lunar or solar orientations, but their usage is uncertain.
T-shaped megaliths are known from prehistoric Menorca, but some of the most impressive megalithic monuments can be found on the western Mediterranean island of Malta. Seven megalithic temples are known on Malta and the neighboring island of Gozo. The oldest of these massive Late Neolithic Maltese stone structures date from 3600 BC, centuries before the ancient Egyptian state existed, and their construction continued into the third millennium BC.

Nine megaliths in a remote part of Dartmoor, Devon in south England have been carbon-dated to around 3500 BC. They may predate Stonehenge, but both sites feature large standing stones that are aligned to mark the rising of the midsummer Sun and the setting of the midwinter Sun. There is a persistent myth that the people who built Stonehenge, the famous prehistoric monument on Salisbury Plain in south England, were Celtic-speaking Druids, but this is wrong. According to archaeological data provided from twentieth century excavations, Stonehenge was built in three main phases, the first one beginning with the digging of ditches around 3100 BC. The final phase of construction took place around 1600-1500 BC, although it is possible that it was used as a cultic site and place of worship well into the later Iron Age.

Celtic is an Indo-European tongue. As we have seen, the IE expansion began, most likely from the northern Black Sea region of Eastern Europe, in the centuries before and especially after 3000 BC. The various Indo-European branches gradually started emerging after this date. The IE expansion had not yet reached far western Europe at this point, which makes it unlikely that those who built Stonehenge, at least the beginning stages of it, spoke Celtic.

The Iron Age began in the centuries prior to and mainly after 1000 BC, during which time the Celtic expansion across much of the European Continent reached its greatest extent. There are indications that the Celts enjoyed a military advantage from their early adoption of iron weapons. Nicholas Ostler explains in Empires of the Word: A Language History of the World:

“Gaulish owed its success, or rather the success of the lineages that spoke it, to their distinctive equipment, notably wheeled vehicles drawn by horses, and to the magnificent products of their smiths, especially ironwork for warriors’ swords, helmets and ring-mail armour. A linguistic note confirms this. The words for ‘iron’ in Greek (sideron), Latin (ferrum) and Celtic (isarno) have separate origins, but the Germanic word (e.g. Gothic eisarn, Old English isern, iren) appears to have been borrowed from Celtic. This is unsurprising, since the Celts were evidently the middlemen for the transmission of ironworking to the north of Europe. (Tacitus even mentions (Germania, xliii) that the Cotini, a Gaulish tribe, paid tribute to the German Quadi in iron ore. He adds typically, ‘quo magis pudeat — the more shame to them’: they should have been able to use the iron to turn the tables.)”

Many Neolithic peoples around the world systematically observed the heavens, particularly the motions of the Moon and the Sun, and sometimes created astronomically aligned monuments that served as seasonal calendars and places of religious worship. In the eyes of the historians of science James E. McClellan and Harold Dorn: “In the case of Neolithic astronomy, we are dealing not with the prehistory of science, but with science in prehistory.”

Be that as it may, it was difficult to create a continuity of scientific studies in non-literate cultures. The true history of science therefore begins after the introduction of writing, and this crucial innovation was introduced to Europe from the Middle East. In the Fertile Crescent agriculture was gradually established after 10,000 BC, with settlements at the Neolithic town of Jericho near the Dead Sea dating back to perhaps 9000 BC. The early success of Chatal Huyuk or Çatalhöyük, a settlement in central Anatolia that existed from ca. 7200 to after 6000 BC, is thought to have resulted from its trade in the dark volcanic glass known as obsidian.

The greatest change in the history of the Near East came with a people called the Sumerians in southern Mesopotamia, “the land between the rivers” Euphrates and Tigris. During the Uruk period (ca. 4000 to 3100 BC), they are credited with many “firsts” in human history, from creating the first writing system to the first monumental statues in an urban setting. Their origin is unknown and their language has no proven connection to any other language, living or dead, yet they produced lasting literature such as the Epic of Gilgamesh.

The settlement of Eridu in southern Mesopotamia (present-day southern Iraq) was founded before 5000 BC. The trading center at Tell Brak in northern Mesopotamia had large buildings around 3800 BC and housed many thousands of people. Hamoukar in northeastern Syria was thriving after 4000 BC. American excavations in collaboration with Syrian authorities have found many clay balls at Hamoukar that were meant to be fired from slings, with evidence from about 3500 BC of the oldest known case of large-scale organized warfare. We don’t know who this war involved, but by 3400 BC pottery of the Uruk type predominated there.

There were undoubtedly other cities or proto-cities in the Fertile Crescent, stretching from western Iran and Mesopotamia into Syria and Anatolia. Uruk wasn’t alone, but by 3300 BC it contained a population of tens of thousands of people, larger than any other known settlement in the world at that time. The city was certainly unique in its historical impact. The “Uruk Expansion” during the fourth millennium BC spread its cultural influence to other regions.

With the growing complexity of society and ensuing expansion of bureaucracy came the development of a system for recordkeeping which evolved into cuneiform script, the world’s first undisputed writing system. Writing was used by the Elamites in Iran, but it probably evolved under Sumerian influence. “Most intriguing is the possibility that Uruk influenced early Egypt, where in the late fourth millennium a number of cultural characteristics similar to those of southern Mesopotamia appeared: niched mudbrick architecture, decorative clay cones, some pottery styles, cylinder seals, and certain artistic motifs.”

In the book Egypt: The World of the Pharaohs, scholar Stefan Wimmer comments on the fact that in ancient Egypt in contrast to Mesopotamia, hieroglyphs emerged almost fully formed in the generations before the unification of the Egyptian state around 3100 BC. The Egyptians had direct or indirect contact with Mesopotamia just as this region was developing writing.

In Europe, the Minoans in Crete employed a form of writing, maybe inspired by the Egyptians, by 2000 BC. Some historians support idea diffusion as an explanation for why different societies adopted writing shortly after the Sumerians. Those who believe in an independent evolution of writing in Egypt or the Indus Valley in India will point to the fact that these writing systems do not outwardly resemble Sumerian cuneiforms, yet it remains possible that these peoples imported the very concept of writing from Mesopotamia. While ancient China was not as isolated as Chinese scholars like to claim, an independent development there should be considered a possibility. If we assume that the Maya and other Mesoamericans had no extensive early contacts with Eurasia that are currently unknown to us then writing was probably independently invented at least a couple of times in human history.

The Austrian historian of science Otto Eduard Neugebauer (1899-1990) was born in Innsbruck and studied engineering and physics at the University of Graz. He devoted much of his life to the history of mathematical astronomy in ancient civilizations and his influence was great. He deeply shaped our understanding of the astronomical and mathematical knowledge of Mesopotamia and Egypt, Greco-Roman Antiquity, India, Islam and the European Middle Ages. In 1939 with the rise of the Nazis he moved to the United States, where he joined the mathematics department at Brown University and the Institute for Advanced Study in Princeton. Neugebauer’s last scientific paper was published when he was 90 years old.

In his fine and well-researched book A History of the Ancient Near East ca. 3000-323 BC, Second Edition, scholar Marc Van De Mieroop states that in Uruk, “a sexagesimal system, relying on units with increments of ten and six, was used to account for animals, humans, and dried fish, among other things. A bisexagesimal system, which diverges from the previous one as its units also show increments of two, was used for processed grain products, cheese, and fresh fish. Volumes of grain or surfaces of fields were measured differently.”

This sexagesimal (base 60) system was adopted and passed on by the successive cultures that dominated Mesopotamia down to the Persians and the Greeks and from them on down to us. We retain sexagesimal numbers today in our system for measuring time (60 minutes to an hour) and angles (60 minutes in a degree and 360 degrees in a circle), but it dates back in a straight line to the civilization of the ancient Sumerians more than five thousand years ago.

An ancient Egyptian astronomical interest can be detected in the alignment of their temples, but rarely on the level of sophistication eventually achieved in Mesopotamia. The ceilings of the tombs of rulers from the Middle Kingdom onward, for instance in the impressive Valley of the Kings in Luxor, contain drawings that could be described as simple celestial maps, yet as author John North states in Cosmos, “except in the case of the calendar it does not seem to have occurred to them to seek for any deeply systematic explanation of what they observed. For all that they were in possession of a script, they seem to have produced no systematic records of planetary movements, eclipses, or other phenomena of a plainly irregular sort.”

At some point after 2000 BC, Sumerian ceased to be an actively spoken language, yet continued to be studied because it was associated with learning, a bit like Latin in medieval Europe. Sumerian, the world’s first written language, also became the first “classical” language: “The tools of bureaucracy, script, seals, measures, and weights, all continued to develop in later Near-Eastern history based on the foundation laid in the Uruk period. To a certain extent, these elements are what defines the ancient Near East: cuneiform writing on clay tablets, the cylinder seal, and the mixture of decimal and sexagesimal units in numerals.”

If you believe that the Biblical Abraham was an historical person (not everybody does), he is supposed to have been born in the city of Ur in the 1900s or 1800s BC, when Mesopotamia came to be dominated by peoples speaking Semitic languages such as Akkadian. We do know that one of the most famous kings in Mesopotamian history, Hammurabi, ushered in what we call the “Old Babylonian period,” the beginning of Babylon’s political dominance over southern Mesopotamia for the next 1500 years. He died around 1750 BC. The function of his influential law code, the Code of Hammurabi, has been much debated; some scholars claim that it was primarily intended as a monument presenting him as an exemplary and just king.

As author Marc Van De Mieroop writes, “From the beginning of writing the administrators of Babylonia showed their mathematical abilities when measuring fields, harvests, numbers of bricks, volumes of earth, and many other things that were of importance to bureaucrats. The tools to calculate these had to be taught but, as with literature, the skills displayed in the school texts show a much higher level than needed in daily practice.”

The gods were believed to speak through objects and events in the natural world, including animal entrails, dreams and celestial phenomena. Omens were important for every level of Mesopotamian society, yet astronomical observations did not become the major focus of divination until after 1500 BC. Mesopotamian bureaucrats and astronomers/astrologers gradually amassed detailed information about the movement of the planets. The quantity and quality of observations improved greatly in the Neo-Assyrian period after the eighth century.

Assurbanipal, the last major king of the brutal Neo-Assyrian Empire, in the seventh century BC collected an extensive Mesopotamian library in his capital, Nineveh. A now fragmentary record of texts that were acquired in the year 648 BC listed some 2000 clay tablets and 300 writing boards, that is, wooden or ivory boards covered with wax and inscribed with a cuneiform text. These were bought or confiscated mainly from the private libraries of Babylonian priests and exorcists. Manuscripts were copied as well. Nineveh was sacked by the Medes in 612 BC. The first modern excavations of this region began with the British pioneering archaeologist Austen Henry Layard (1817-1894) in the nineteenth century AD.

By the fifth century BC, Babylonian celestial divination had expanded to embrace horoscopic astrology, which used planetary positions at the moment of the date of birth to predict individual fortunes. According to the science historian James Evans, “While horoscopic astrology was certainly of Babylonian origin (as, indeed, the Greek and Roman writers always claimed), it was elaborated into a complex system by the Greeks. Thus, the familiar and fantastically complicated system of horoscopic astrology with dozens of conflicting rules does not descend from remote antiquity. Rather it is a product of Hellenistic and Roman times.”

The Maya in Mesoamerica devoted much attention to divination and amassed detailed studies of the movements of the Sun, Moon and planets over long periods of time. The Inca elites in pre-Columbian South America, too, elaborated special forms of divination. The Chinese had their own ideas about the stars and divination from an early date, but may have absorbed additional ideas influenced by Babylonian astrology during the Han Dynasty by way of India.

Babylonian astronomy and astrology reached India from Mesopotamia at least with the Persian conquests of northwest India by the fifth century BC, along with alphabetic writing systems. Contact with Greek mathematical astronomy came after Alexander the Great’s conquests of this region and through trade with the Roman Empire in the first and second centuries AD. Early medieval Indians were influenced by Greek spherical trigonometry.

The Chinese lunisolar calendar with its twelve Zodiac signs (the rat, ox, tiger etc.) is still used for marking East Asian holidays such as the Chinese New Year. The earliest divinations are found on the inscribed oracle bones and turtle shells from the city of Anyang in northeastern China, widely regarded as the cradle of Chinese civilization. They concern matters of significance to the king and the state. G.E.R Lloyd elaborates in The Ambitions of Curiosity:

“Since the knowledge claimed for divination concerned the future, it held out the promise of influencing it — a prospect that state authorities could hardly ignore. The legitimacy or otherwise of the practices were not just matters of the rationality of aims and methods, for issues of state control, or its subversion, could be at stake. The unauthorised casting of the horoscope of a Roman Emperor was high treason (cf. Barton 1994) — just as in late imperial China private studies of astronomy and astrology could be criminal offences.”

In ancient Greece, cosmic regularities were seen as unchanging. In China, order in the Heavens could not be taken for granted. The Emperor acted as a mediator between Heaven and Earth. Because of this, the regulation of the calendar and the interpretation of celestial signs were matters of vital importance to the Emperor himself as the bearer of the Mandate of Heaven. The Astronomical Bureau existed for more than 2,000 years, but since everything was regulated by the Imperial court, astronomical instruments and findings were frequently treated as state secrets, which sometimes hampered scientific progress in Chinese astronomy.

While not as lasting as the stone pyramids built by the Egyptians, the mud brick ziggurats of the Sumerians must nevertheless have been impressive structures. They made a profound impression on the ancient Hebrews, who memorialized the Babylon ziggurat as the Tower of Babel, a monument to the insolent pride of humans. The White Temple at Uruk dates from around 3200 BC. The Sumerians seem to have been the first to set up monumental statues in their cities and sanctuaries. One of the earliest is the white marble female head found at Uruk in the sacred precinct of the goddess Inanna, who is mentioned in the Epic of Gilgamesh.

In the earliest Greek literature we find traces of a prehistoric Indo-European astral religion. In Homer’s the Iliad, Achilles’ shield is likened to the Earth, which is surrounded by an ocean-river, the source of all water and of the gods. In the Odyssey, the starry heaven is said to be of bronze or iron and supported on pillars. The early Greeks may not have understood that Venus was a planet rather than two different bodies, the Morning Star and the Evening Star.

To the ancient Greeks, the planets were “wandering stars.” Our word planet comes from a Greek verb meaning to wander. The modern names for the five naked-eye planets are the names of Roman divinities which were more or less equivalent to a number of Greek gods. Most people today probably know this. What many of them don’t know is that some of the Greek names themselves may have been derived from ancient Babylonian divinities.

Mars was often associated with war because of its reddish color, which can be spotted through naked-eye observations; the ancient Egyptians called it the Red One. However, there are other parallels that are unlikely to be accidental. In ancient Mesopotamia, Ishtar was the Babylonian and Assyrian counterpart of Inanna, the moody Sumerian goddess of love and fertility, identified with the planet Venus. To the Romans, Venus was the goddess of love and fertility, their equivalent of the Greek goddess Aphrodite, who was also a symbol of love and fertility.

In Roman mythology, Jupiter was king of the gods, the equivalent of Zeus in the Olympic pantheon of ancient Greece. The name “Zeus” is Indo-European. James Evans elaborates in The History and Practice of Ancient Astronomy, which is excellent on pre-telescopic astronomy in Europe and the Middle East but contains nothing on East Asia or the Americas:

“Marduk was the most important god of Babylon. His star is the planet Jupiter. The fact that the Babylonians associated the planet Jupiter with the chief god of their pantheon is an interesting parallel to Greek practice. Moreover, Venus was associated with Ishtar, the goddess of love and fertility, and Mars with Nergal, the god of war and pestilence. These parallels are too striking to be due to chance. The Greek associations are probably the result of Hellenization of earlier Mesopotamian associations. The divine associations came into use by the time of Plato. For the early Greeks, the Sun, Moon, and fixed stars were far more important than were the planets. The motion of the Sun was intimately connected with the annual cycle of agricultural labors. The phases of the Moon governed the reckoning of months. And the heliacal risings and settings of the stars told the time of year. So, it is not surprising that Hesiod’s Works and Days (ca. 650 B.C.), which contains a good deal of practical lore about the Sun, Moon, and stars, makes no mention of the planets.”

In the eyes of Walter Burkert, a few similarities between the Epic of Gilgamesh and Homeric poetry can no longer be ignored. He is nevertheless careful to point out that philosophy in the modern sense was a Greek invention as much as was deductive proof in mathematics. As Ibn Warraq puts it in Defending the West, “what emerges is something entirely distinctive: what we call Greek civilization. The very strength of this civilization lay in its ability to learn from and improve upon the ideas, art, and literature of the Near East, Persia, India, and Egypt.”

According to the website of the American Institute of Physics, “Sky-watchers in the ancient Middle East, Central America, and China made many observations. From their tables of numbers, they devised schemes to predict future movements in the heavens. But the explanations that the Babylonians, Mayans, and early Chinese sky-watchers devised for these movements were no more than colorful myths. Scientific cosmology — the search for a picture of the universe that would make sense with no mention of divine beings — began with the Greeks. They sought to look beyond the patterns of numbers to something fundamental….Greek philosopher-scientists set themselves the task of envisioning the universe as a set of physical objects….All generation and corruption occurred in the ‘sublunar’ region, below the Moon and above the Earth.”

A turning point in history was the ancient Greek invention of scientific theory or “natural philosophy.” This process began on the then-fertile western coast of Anatolia or Asia Minor (present-day Turkey), in the region known as Ionia. It is traditionally said to have started with Thales of Miletus, who flourished in the decades after 600 BC. Authors James E. McClellan and Harold Dorn elaborate in Science and Technology in World History, Second Edition:

“We do know that he came from Miletus, a vibrant trading city on the Ionian coast of Asia Minor, and that he was later crowned as one of the seven ‘wise men’ of archaic Greece, along with his contemporary, the lawgiver Solon….Thales’s claims about nature were just that, his claims, made on his own authority as an individual (with or without other support). Put another way, in the tradition stemming from Greek science, ideas are the intellectual property of individuals (or, less often, close-knit groups) who take responsibility and are assigned credit (sometimes by naming laws after them) for their contributions. This circumstance is in sharp contrast with the anonymity of scientists in the ancient bureaucratic kingdoms and, in fact, in all pre-Greek civilizations.”

Anaximander of Miletus was a Greek philosopher who flourished in the first half of the sixth century BC and apparently was a pupil of Thales. Anaximander is often mentioned as being the first person to develop a cosmology, that is, a systematic philosophical view of the universe. He wrote treatises on geography and astronomy and believed eclipses to be a result of blockage of the apertures in rings of celestial fire. Anaximenes of Miletus was another prominent Pre-Socratic philosopher and a younger contemporary of Anaximander. Together they contributed substantially to the transition from magical explanations of nature to non-magical ones in ancient Greece. Anaximenes thought that the Earth was flat, a view that was soon challenged by the mathematician Pythagoras and his followers, the Pythagoreans.

The Milesian thinkers used logic and reason to criticize the ideas of other individuals and saw the need to defend their theories, thus beginning a tradition of rational and critical assessment which remains alive to this day. It appears as if these pioneering Ionian philosophers identified the basic structure of the universe as material. Thales seems to have suggested that there must be something underlying matter in the universe, out of which everything else is composed. His ideas were developed further by his successors. Thales suggested that water was the primary substance whereas Anaximenes believed air to be the primeval element.

The philosopher Heraclitus worked in the years before and after 500 BC. In his view the heavenly bodies are bowls filled with fire; an eclipse occurs when the open side of a bowl turns away from us. He argued for a world without beginning or end, of constant change as well as stability. According to Plato, Heraclitus was the first person to compare our world to a river and the inventor of the famous maxim that we can never step into the same river twice.

Heraclitus held that change is perpetual, that everything flows. Parmenides, a Greek philosopher from Elea in southern Italy, in the decades after 500 BC countered with the radical notion that change is an illusion. Parmenides held that the multiplicity of existing things, their changing forms and motion, are simply different appearances of a single eternal reality. He adopted the radical position that change is impossible. His doctrine was highly influential; others felt compelled to argue against it. The Heraclitean-Parmenidean debate raised fundamental questions about the senses and how we can know things with certainty.

Zeno of Elea (ca. 490-425 BC) was a Greek mathematician and a pupil of Parmenides. Zeno’s Paradoxes, such as the famous race between Achilles and the tortoise where the tortoise wins, were important in the development of the notion of infinitesimals. Anaxagoras and the Pythagoreans, with their development of incommensurables, may have been the targets of his arguments. If you believe Plato, Zeno and Parmenides visited Athens around 450 BC where they met the young Socrates. Whether this alleged meeting took place is not universally accepted by historians, but it is chronologically conceivable that it may have happened.

In ancient India, Jains and others did philosophical work on the concept of infinity. In Europe, Zeno’s work on the subject had repercussions right down to the invention of set theory by the German mathematician Georg Cantor (1845-1918) in the late nineteenth century AD. As historian David C. Lindberg states in The Beginnings of Western Science, Second Edition:

“These theories of Anaximander and Heraclitus do not seem particularly sophisticated (fifty years after Heraclitus the philosophers Empedocles and Anaxagoras understood that eclipses were simply a case of cosmic shadows), but what is of critical importance is that they exclude the gods. The explanations are entirely naturalistic; eclipses do not reflect personal whim or the arbitrary fancies of the gods, but simply the nature of fiery rings or of celestial bowls and their fiery contents. The world of the philosophers, in short, was an orderly, predictable world in which things behave according to their natures. The Greek term used to denote this ordered world was kosmos, from which we draw our word ‘cosmology.’ The capricious world of divine intervention was being pushed aside, making room for order and regularity; chaos was yielding to kosmos. A clear distinction between the natural and the supernatural was emerging; and there was wide agreement that causes (if they are to be dealt with philosophically) must be sought only in the natures of things. The philosophers who introduced these new ways of thinking were called by Aristotle physikoi or physiologoi, from their concern with physis or nature.”

The eminent historian of archaeology Bruce Trigger in Understanding Early Civilizations: A Comparative Study offers a comparison between seven early civilizations: ancient Mesopotamia and Egypt, pre-Imperial China, the Maya and their Mesoamerican neighbors, the Aztecs, the Incas in South America and the Yoruba and Benin peoples of West Africa. Surprisingly, the cosmologies of these very different peoples exhibited a few similarities:

“The sky and underworld planes were the exclusive realms of the gods and the dead, while the earth was shared by living people and the supernatural. These levels were interconnected, most often at the centre and around the edges of the terrestrial realm, through hills, trees, caves, and temples. The gods and supernatural energy were able to move through these gateways, conveying life-giving powers from the purely supernatural realms to the human one and back again. The earth was generally believed to be a flat plane, round or square in outline, at most a few thousand kilometers across, and surrounded by a salt-water ocean. Each early civilization and usually each city-state believed itself to be located at the centre of the terrestrial plane, which had been created especially for its benefit.”

The idea of a spherical cosmos can be attributed to sixth- and fifth-century BC Greek philosophers such as Pythagoras and Parmenides. Pythagoras and his disciples had suggested that the Earth was spherical, not flat, by 500 BC. The sphere was considered the most perfect solid, although the shadow of the Earth cast on the Moon during an eclipse later added observational credibility to this theory. The concept of a spherical Earth was apparently rather rare and was not generally developed independently by non-European cultures. It had become widely adopted by the time of Aristotle and was never forgotten during the Middle Ages by those in western (but not eastern) Eurasia who were familiar with Aristotle’s writings.

Pythagoras, who lived in the late sixth century BC, was a spiritual teacher who believed that the pursuit of philosophical, musical and mathematical studies provided a moral basis for the conduct of life. He left no written record, so the mathematical doctrines of his school can only be surmised from the works of other Pythagorean writers. Pythagoras and his followers were fascinated with music and studied the properties of vibrating strings and musical harmonies. They believed that similar mathematical harmonies could be found in the universe as a whole.

The Pythagoreans pioneered the mathematical approach to nature. Their approach was in stark contrast to that of the materialists, among whom the atomists were most prominent. The materialism of the sixth century was extended in the fifth century BC by the atomist Leucippus, who was probably from Miletus although very little is known about his life, and by his pupil Democritus (ca. 460-370 BC). Democritus was born in the city of Abdera in mainland Greece, at the northern end of the Aegean Sea. He inherited considerable wealth from his father and spent years travelling in Eastern lands, which means Egypt and Persia and possibly India, as well as within the Greek world. Democritus argued that all matter is made up of imperishable, indivisible elements of different sizes and shapes called atoma or “indivisible units.” During collisions they rebound or stick together because of hooks and barbs on their surfaces. Underlying the changes in the perceptible world there was thus both constancy and change; change was caused by different combinations of permanent atoms.

Atomism was also supported by Epicurus (341-270 BC), who grew up in the Athenian colony of the island of Samos and studied philosophy under followers of Democritus and Plato. Epicureanism advocated a materialistic — its critics say hedonistic — philosophy where good was identified with friendship, pleasure and the absence of pain. Epicurus banished fear of the gods, death and eternal punishment. His ethical system proved popular and was influential over the next centuries and into the Roman era. The long Latin poem De rerum natura (“On the nature of things”) by the Roman poet and philosopher Lucretius (died ca. 50 BC), which has survived to us virtually intact, defends the Epicurean thought system, including atomism.

These speculations about the physical nature of matter culminated in the influential ideas of the philosopher Empedocles (ca. 490-430 BC). He was born in the city of Agrigentum in Sicily, which had been founded by Greek colonists in the sixth century and was a wealthy center of culture before being sacked by the Carthaginians in 406 BC. He is the first person recorded as having said that the speed of light, while very great, is finite, a claim that was not empirically confirmed until more than two thousand years later. He has also been credited with introducing an early, although not fully developed, theory of evolution. Legend has Empedocles ending his life while trying to prove his immortality by leaping into the crater of the active volcano Mount Etna in Sicily. Above all, he is remembered for his belief that all substances are composed of four elements in different ratios: air, earth, fire and water.

In the fifth century BC, the physician Hippocrates and his followers correlated the four elements of Empedocles with four bodily humors: blood, phlegm, yellow bile and black bile. This humoral doctrine was supported by the Greek physician Galen in the Roman Empire and remained highly influential in Europe well into modern times. Traditional medicine worldwide stressed health maintenance through regulation of diet, exercise and lifestyle.

According to Roy Porter in The Greatest Benefit to Mankind: A Medical History of Humanity, “From Hippocrates in the fifth century BC through to Galen in the second century AD, ‘humoral medicine’ stressed the analogies between the four elements of external nature (fire, water, air and earth) and the four humours or bodily fluids (blood, phlegm, choler or yellow bile and black bile), whose balance determined health. The humours found expression in the temperaments and complexions that marked an individual. The task of hygiene was to maintain a balanced constitution, and the role of medicine was to restore the balance when disturbed. Parallels to these views appear in the classical Chinese and Indian medical traditions.”

The atomists had responded to the challenge from the Milesian philosophers by stating that the material world is composed of tiny particles, but they faced the challenge of explaining how these random atoms could assume any lasting, coherent pattern or structure in nature. Their theories were criticized by Aristotle for some logical inconsistencies and for their seeming inability to explain qualities such as color, taste, odor etc. The belief in atomism was not shared by Aristotelians and was always a minority view among the ancient Greeks.

Atomism experienced a renaissance of sorts in seventeenth century Europe. The breakthrough for “modern” atomism, now with more experimental evidence in its favor, took place in the nineteenth century AD, staring with the English chemist and meteorologist John Dalton (1766-1844) in the early 1800s. The first subatomic particle, which proved that atoms were not truly “indivisible” after all, was the electron, finally identified in 1897 by the Englishman Joseph John “J. J.” Thomson (1856-1940). Thomson’s student, the New Zealand-born physicist Ernest Rutherford (1871-1937), discovered the atomic nucleus and the proton a few years later, and the English physicist James Chadwick (1891-1974) discovered the neutron in 1932. Later in the twentieth century, many other subatomic particles were identified.

Meton of Athens, a Greek geometer, worked with another astronomer, Euctemon, to make a series of observations of the solstices. In 432 BC he introduced a calendar based on a 19-year cycle into the Athenian luni-solar calendar. He had observed that a period of 19 solar years (6,940 days) is almost exactly 235 lunar months. This Metonic cycle forms the basis of the Jewish calendar and is used to determine the date for the Christian Easter. We do not know whether this was inspired by similar advances made earlier in Mesopotamia. Meton’s calendar never seems to have been adopted, but his observations proved useful to later astronomers.

Eudoxus of Cnidus, who had studied with followers of Pythagoras in Italy and in Athens under Plato at his Academy, created the first serious geometrical model of planetary motion. He based it entirely on spherical motions, but there is reason to believe that he personally viewed this as a purely mathematical model, unlike many of those who followed him. Aristotle regarded the sphere of the fixed stars as a real, material sphere. Lindberg elaborates:

“Greek astronomy took a decisive turn in the fourth century with Plato (427-348/47) and his younger contemporary Eudoxus of Cnidus (ca. 390-ca. 337 B.C.). In their work we find (1) a shift from stellar to planetary concerns, (2) the creation of a geometrical model, the ‘two-sphere model,’ for the representation of stellar and planetary phenomena, and (3) the establishment of criteria governing geometrical theories designed to account for planetary observations. Let us consider these achievements in some detail. The two-sphere model devised by Plato and Eudoxus conceives of the heavens and the earth as a pair of concentric spheres. To the celestial sphere are affixed the stars, and along its surface move the sun, the moon, and the remaining five planets. The daily rotation of the celestial sphere accounts for the observed daily rising and setting of all the celestial bodies.”

Eudoxus was a gifted mathematician who was largely responsible for some of the finest sections of the Elements, a treatise of 13 books written by the Greek mathematician Euclid after 300 BC. Euclid was most likely born a few years before Archimedes of Syracuse (ca. 287-212 BC). It is usually assumed that Euclid taught and wrote at the Library of Alexandria.

Aristotle (384-322 BC) came from a privileged family in northern Greece, his father being royal physician to the king of Macedonia. From 343 BC Aristotle became tutor for the son of the Macedonian king, a young man who became known as Alexander the Great (356-323 BC) when he began his expansive conquests. The Museum and Library in Alexandria at the Mediterranean coast of Egypt was founded around 300 BC by Ptolemy I Soter (ca. 367-283 BC), the Macedonian general who became ruler of Egypt after the death of Alexander. “Museum” here means a “Temple of the Muses,” a location where scholars could meet and discuss philosophical and literary ideas. The period from 600 to 300 BC is called the Hellenic period whereas the period after Alexander’s conquests is known as the Hellenistic era.

Heraclides of Pontus, a younger contemporary of Plato in the fourth century BC who studied with Plato and Aristotle in Athens, suggested that the Earth rotates on its axis once in twenty-four hours. He is the first person we know to have held this view. This hypothesis would explain the daily rising and setting of the Sun and the celestial bodies, but it was rejected by most of his contemporaries. They considered it implausible because it violated sensory evidence indicating that the Earth is stationary. Heraclides also wrote on many of the usual topics that a Greek philosopher would have written on, including literature, history and music.

Philolaus (ca. 470-385 BC), a Greek Pythagorean philosopher from southern Italy, wrote the book On Nature in which “ The cosmos comes to be when the unlimited fire is fitted together with the center of the cosmic sphere (a limiter) to become the central fire. Philolaus was the precursor of Copernicus in moving the earth from the center of the cosmos and making it a planet, but in Philolaus’ system it does not orbit the sun but rather the central fire.”

The Greek astronomer and mathematician Aristarchus of Samos (ca. 310-230 BC) compared the Earth-to-Sun distance with the Earth-to-Moon distance and figured the former to be twenty times the latter. The correct ratio is about 400:1, but he apparently understood that since the Sun was far away it had to be much larger than the Earth. It may have been this realization that led him to suggest that the Sun was the center of the universe. As Lindberg states, “It is usually assumed that Aristarchus also gave the other planets sun-centered orbits, although the historical evidence does not address this point. In all likelihood, Aristarchus’s idea was a development of Pythagorean cosmology, which had already removed the earth from the center of the universe and put it in motion around the ‘central fire.’“

His heliocentric theory was overwhelmingly rejected in Antiquity because it seemingly violated common sense, everyday observations and Aristotelian physics. If the Earth orbits the Sun, why doesn’t everything that is not nailed down go flying off on its own? Heliocentrism was successfully revived two thousand years later by the Polish astronomer Nicholas Copernicus (1473-1543), who may have been aware of Aristarchus’s ideas.

Archimedes was the first mathematician to derive quantitative results from the creation of mathematical models of physical problems on Earth, for example the basic principle of hydrostatics. He discovered what is still known as the Archimedean principle: the weight of a body wholly or partially immersed in a fluid is reduced by an amount equal to the weight of the fluid that the body displaces. The principle of the lever was known before this, but no-one had created a mathematical model for it before Archimedes. He is also credited with the invention of the Archimedes screw, a screwpump which is still employed in modern factories to move powdery substances. It was used for many centuries in irrigation as an apparatus for raising water. His genius as an engineer of practical military devices kept the invasion forces at bay for months, but he was allegedly killed by a Roman soldier after the capture of Syracuse, Sicily in 212 BC, even though the commander wanted to spare his life.

Johan Ludvig Heiberg (1854-1928), a philologist and historian of mathematics at the University of Copenhagen, Denmark, inspected a manuscript in Constantinople in 1906 which contained previously unknown mathematical works by Archimedes. A long-lost text shows that he had begun to discover some of the principles of calculus. Eudoxus had paved the way for Archimedes’ later study of volumes and surfaces in his work On the sphere and cylinder.

The Greek scholar Eratosthenes (276-194 BC) knew that the Sun was never vertically overhead in Alexandria; by June 21 it was off by more than 7 degrees, or 1/50 of a full circle. Yet at Syene (Aswan) in southern Egypt close to the Tropic of Cancer it appears directly overhead at the summer solstice. The Earth currently has an axial tilt of about 23.5 degrees, which was estimated by Eratosthenes. This is the cause of the seasons as our planet moves around the Sun; when the Northern Hemisphere receives the maximum amount of sunlight, in areas south of the Tropic of Cancer a vertical stick will cast no shadow at noon whereas in regions north of the Arctic Circle there will be 24 hours of sunlight. In December, when the Earth is tilted the other way vis-à-vis the Sun, the Southern Hemisphere enjoys summer.

Knowing the rough distance from Syene to Alexandria, Eratosthenes could find the Earth’s circumference by assuming that this constitutes 1/50 of the full circle and that the Sun is very far away. There were a few sources of error, but his methods were theoretically sound and his result was certainly in the right range. Exactly how accurate his value of 250,000 stades was is a matter of debate, as there were several “stades” of different lengths in use. Eratosthenes’ estimate of the size of the Earth was remembered in the Middle East and in Europe, where learned people knew from the writings of the ancient Greeks that the Earth is round. East Asians, on the other hand, did not know this prior to modern contact with Europeans.

It is believed that the first definition of a conic section was due to the Greek mathematician Menaechmus (380-320 BC), a pupil of Eudoxus and a friend of Plato. Menaechmus “is famed for his discovery of the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base.” Major progress in the study of conics was made by Apollonius of Perga, the last of the great mathematicians of the Hellenistic period, active just before 200 BC. Victor J. Katz explains:

“Apollonius was born in Perge, a town in southern Asia Minor, but few details are known about his life. Most of the reliable information comes from the prefaces to the various books of his magnum opus, the Conics. These indicate that he went to Alexandria as a youth to study with successors of Euclid and probably remained there for most of his life, studying, teaching, and writing. He became famous in ancient times first for his work on astronomy, but later for his mathematical work, most of which is known today only by titles and summaries in works of later authors. Fortunately, seven of the eight books of the Conics do survive, and these represent in some sense the culmination of Greek mathematics. It is difficult for us today to comprehend how Apollonius could discover and prove the hundreds of beautiful and difficult theorems without modern algebraic symbolism. Nevertheless, he did so, and there is no record of any later Greek mathematical work that approaches the complexity or intricacy of the Conics.”

We know very little about the bureaucrats who were the originators of Babylonian mathematical astronomy. This is in sharp contrast to the individualism of Greek society, where the different thinkers criticized their rivals by name. Moreover, while the Babylonians developed a sophisticated system of arithmetical computations for predicting astronomical phenomena such as eclipses, their purposes were strictly calendrical. As far as we know, they never visualized the motions of the planets in terms of geometrical or mechanical models.

According to James Evans, “one of the critical developments of this period was the rise of Greek geometry, which led rapidly to the mathematization of Greek astronomy. Notable geometers of this period were Euclid, Archimedes, and Apollonius of Perga. Apollonius (ca. 225 B.C.) seems to have been the first to experiment with combinations of deferent circles and epicycles in an attempt to provide an explanation for the motions of the planets, Sun, and Moon. The work on the solar and lunar theories was carried to a high level by Hipparchus (ca. 140 B.C.). For the first time in Greek astronomy, it became possible to make quantitative predictions of the future positions of the Sun and Moon, as in the prediction of eclipses.”

Hipparchus, one of the greatest mathematical astronomers in history, was born in Nicaea in Bithynia but spent much of his life in Rhodes. His recorded observations span the years 147 to 127 BC. He ranged over all aspects of contemporary astronomy — mathematical, observational and instrumental. He was the probable inventor of stereographic projection, the crucial element of the astrolabe, and played a major role in the development of trigonometry. He made his own star catalog and was the first to introduce a system for measuring the brightness of stars, with six levels of magnitude. Today’s system essentially follows the same logic.

He is often credited with the discovery of the precession of the equinoxes. Twice a year, at equinox, the Sun rises exactly in the east and sets exactly in the west. Hipparchus compared older astronomical observations to his own ones and concluded that those intersections had moved slightly after a few centuries. This is due to the equatorial bulge of the Earth, which is caused by the centrifugal force of its rotation. The attraction of the Moon and Sun on the bulge is the “nudge” which makes the Earth precess. Through each such cycle, lasting nearly 26,000 years, the direction in the sky to which the axis points goes around a big circle.

Because of this, the “pole star” used by ancient Greek sailors was a different one from the North Star we currently have with respect to the backdrop of the stars. Your Zodiac sign now corresponds to the position of the Sun relative to constellations as they appeared over 2200 years ago. Since then, the signs have slipped nearly one-tenth of the way around the sky to the west relative to the stars. For instance, those born between March 21 and April 19 consider themselves to be Aries. Yet today, the Sun is no longer within the constellation of Aries during much of that period; from March 11 to April 18 it is in the constellation of Pisces.

Hipparchus calculated that the Moon’s mean distance from the Earth is 63 times the Earth’s radius. The true value is about 60 times. He was also a crucial figure in the Greek borrowing of astronomical results and mathematical procedures from the Babylonian tradition. This development was facilitated by the fact that following the conquests of Alexander the Great, in the second century BC Mesopotamia was, like Egypt, ruled by a Greek-speaking dynasty.

The most influential Greek astronomer was without question Claudius Ptolemaeus, or Ptolemy, who made astronomical observations from Alexandria in Roman-ruled Egypt during the years AD 127-141. His name indicates that he was from a Greek family as well as a citizen of Rome. His work represented the culmination of Greek scholarship in several disciplines. His great astronomical treatise, later known as the Almagest, dominated astronomical thought in Europe and the Mediterranean region up to and including Copernicus in the sixteenth century, and even longer than that in the Middle East. It included and superseded earlier astronomical works, above all those by Hipparchus from the second century BC. His Tetrabiblos (“Four books”) was a standard astrological text for centuries.

He benefited from the work of Menelaus of Alexandria (ca. AD 70-140), one of the last notable Greek geometers who applied spherical geometry to astronomy. Ptolemy reports on Menelaus’s observations of lunar occultations of stars. The first definition of a spherical triangle is contained in his Sphaerica, a three-book treatise from around AD 100 in which Menelaus developed the spherical equivalents of Euclid’s propositions for planar triangles.

While geocentric (Earth-centered) Ptolemaic astronomy is widely familiar, many people don’t know that he was an excellent geographer for his time as well. The recovery of Ptolemy ‘s Geography around AD 1295 revolutionized Byzantine cartography, just as it revolutionized Western European cartography when it was translated into Latin about a century later. It was very popular among Renaissance humanists during the fifteenth century. In addition to this, his Optics was arguably the most important work on that subject in Antiquity. Euclid’s Optics was almost wholly geometrical with little concern for theories of vision. Ptolemy used Euclid’s law of reflection, but went far beyond him with a theory of refraction as well.

Diophantus of Alexandria worked in Roman Egypt in the third century of our era. He is usually assumed to have been a Greek, but very little is known about his life. His collection of books known as the Arithmetica, a landmark work in the history of algebra and number theory with the so-called Diophantine equations, is believed to have been completed ca. AD 250. Dirk J. Struik explains in A Concise History of Mathematics, Fourth Revised Edition:

“Their skillful treatment of indeterminate equations shows that the ancient algebra of Babylon or perhaps India not only survived under the veneer of Greek civilization but also was improved by a few active men. How and when it was done is not known, just as we do not know who Diophantus was — he may have been a Hellenized Babylonian….In Diophantus we find the first systematic use of algebraic symbols. He has a special sign for the unknown, for the minus, for reciprocals. The signs are still of the nature of abbreviations rather than algebraic symbols in our sense (they form the so-called ‘syncopated’ algebra); for each power of the unknown there exists a special symbol. There is no doubt that we have here not only, as in Babylon, arithmetical questions of a definite algebraic nature, but also a well-developed algebraic notation which was greatly conducive to the solution of problems of greater complexity than were ever taken up before.”

Astronomy and Mathematics during the Middle Ages

The Byzantine Empire did an invaluable job in preserving Classical knowledge and pioneered the creation of hospitals in western Eurasia, yet relatively few original scientific works of lasting importance were produced there during the Middle Ages. Authors James E. McClellan III and Harold Dorn sum up the established wisdom when they state that “Byzantium never became a center of significant original science.” It remained a somewhat autocratic state. The development of parliaments, autonomous cities and universities that took place in the Christian West did not happen in the Christian East, which was under near-constant siege by Muslims from the seventh century until it was finally destroyed by them in the mid-1400s.

There are a few exceptions, especially from the Eastern Roman Empire in what can still be described as Late Antiquity. John Philoponus (ca. AD 490-570) was a Christian neo-Platonist who taught in Alexandria. He may have been a member of the Monophysite sect, which held that Jesus Christ had only one nature, not two at the same time (human and divine). Philoponus was the most original of the ancient commentators on Aristotle and attacked what he perceived as logical inconsistencies in the pagan concept of an eternal, uncreated world.

Edward Grant writes that “Philoponus rejected many of Aristotle’s theories and replaced them with well-thought-out new theories, which exerted a significant influence on medieval natural philosophers and even influenced Galileo in the seventeenth century. Philoponus rejected Aristotle’s explanation of projectile motion and replaced it with an impressed force theory that marked a significant step toward the principle of inertia. Philoponus also believed — contrary to Aristotle — that finite motion could occur in a vacuum. Finally, it is noteworthy that Philoponus, once again in opposition to Aristotle, argued that if you drop two unequal weights from the same height, they will reach the ground at approximately the same time, an experiment that Galileo is alleged to have performed from the Leaning Tower of Pisa.”

After the Roman period, the legacy of Greek Antiquity was passed on to medieval times, to Byzantium, the Middle East and to Europe. Scholar F. R. Rosenthal states that “Islamic rational scholarship, which we have mainly in mind when we speak of the greatness of Muslim civilisation, depends in its entirety on classical antiquity…in Islam as in every civilisation, what is really important is not the individual elements but the synthesis that combines them into a living organism of its own…Islamic civilisation as we know it would simply not have existed without the Greek heritage.”

According to Islamic jurists, Muslims should not stay for too long in the lands of non-Muslims if they cannot live a proper Muslim life there. Muslims had little knowledge of or interest in any Western or non-Muslim languages, the knowledge of which was considered unnecessary or even suspect. Consequently, the translators of Greek and other non-Muslim scientific works to Arabic were never Muslims. They were Christians of the dominant Eastern denominations plus a few Jews and Sabians. The language of culture for these Christians was Syriac (Syro-Aramaic or Eastern Aramaic) and their liturgical language was Greek.

The Baghdad-centered Abbasid Dynasty, which replaced the Damascus-centered Umayyad Dynasty after AD 750, was closer to pre-Islamic Persian culture and influenced by the Sassanid Zoroastrian practice of translating works and creating libraries. Even Dimitri Gutas admits this in his pro-Islamic book Greek Thought, Arab Culture. There was still, for a while, many Zoroastrians, Christians and Jews around and they held a disproportionate amount of expertise in the medical field. According to author Thomas T. Allsen, Middle Eastern medicine in Mongol ruled China was “almost always” in the hands of Nestorian Christians.

One prominent translator was Hunayn ibn Ishaq (AD 808-873), called Johannitius in Latin. He was a Nestorian (Assyrian) Christian who had studied Greek in Greek lands, presumably in the Byzantine Empire, and eventually settled in Baghdad. He, his son and his nephew translated into Arabic, sometimes via Syriac, Galen’s medical treatises, Hippocratic works and texts by Aristotle, Plato and others. He also wrote several treatises of his own making.

Thabit ibn Qurra (ca. 836-901) was a member of the peculiar Sabian sect of star worshippers whose elites had adopted much of ancient Greek culture. His native language was Syriac but he knew Greek and Arabic well. He worked for years in Baghdad where he produced influential Arabic translations or revised earlier ones of Ptolemy’s Almagest and works by Archimedes and Apollonius. Later Arabic versions developed from his version of Euclid’s Elements. He was an original mathematician who contributed to geometry and number theory.

Al-Kindi (died ca. AD 873), commonly known as “the Philosopher of the Arabs,” lived in Baghdad and was close to several Abbasid Caliphs. Al-Kindi was a natural philosopher and mathematician who did significant work on optics and made notable contributions to cryptography. Al-Farabi (ca. 875-950), “perhaps the greatest” Muslim philosopher in the eyes of scholar Rémi Brague, emphasized human reason and was more original than many of his successors. The attempt to reconcile Islam with Greek philosophy was to last for several centuries but ultimately prove unsuccessful due to persistent religious resistance.

Ibn Rushd, or Averroes (1126-1198), was born in Cordoba, Spain (Andalusia). He faced trouble for his freethinking ways and is today often hailed as a beacon of “tolerance,” yet he was also an orthodox jurist of sharia law and served as an Islamic judge in Seville. He approved, without reservation, the killing of heretics in a work that was wholly philosophical in nature. His attempts to combine Aristotelian philosophy and Islam had a major influence on Latin scientists but he was practically forgotten in the Islamic world. Most freethinkers were at odds with Islamic orthodoxy and frequently harassed for this.

Moses Maimonides (1135-1204), a Jewish rabbi, philosopher and physician, was born in Islamic-ruled Spain, but had to flee the country when the devout Berber Almohades invaded from Morocco and attacked Christians and Jews in a classical Jihad fashion. He eagerly read Greek philosophy, some of which was available in Arabic with commentaries. His The Guide for the Perplexed was still read in the seventeenth century, and his attempts at reconciling Aristotelian philosophy with Biblical Scripture influenced leading Christian philosophers.

Basic algebra was known to the ancient Egyptians, the Babylonians in Mesopotamia in the second millennium BC, to the Chinese, the Indians and other cultures. Yet with the exception of the work Diophantus and some contributions by scholars in medieval East Asia, India and the Middle East, the history of algebra seemingly made surprisingly little progress for several thousand years until Renaissance Europe, after which modern algebra was born. The solutions to linear and quadratic equations were known to the ancient Babylonians, but the solution to the general cubic equation did not come until Renaissance Italy.

According to the book Mathematics Across Cultures: The History of Non-Western Mathematics, “Throughout these 3000 years, the Greeks, Indians, Chinese, Muslims, Hebrews and Christians seem to have done no more than present their own versions of solutions to linear and quadratic equations, which were well known to the Babylonians. Diophantus, Bhaskara, Jia Xian, al-Khwarizmi, Levi ben Gerson and Leonardo of Pisa, all provide examples of this. However, the history of algebra is subtler than that, with small changes accumulating slowly. There were many concepts which needed to mature before the breakthrough in algebra was ready to occur. Independence from geometry, a comfortable symbolic notation, a library of polynomial identities, and the tool of proof by induction, were all stepping stones in the history of algebra.”

The talented Bhaskara (1114-1185), also known as Bhaskara II or Bhaskaracharya, was the leading mathematician and astronomer in late medieval India. He was born into a Brahmin family in the south and served as head of the astronomical observatory at Ujjain, a major mathematical center in northern India where many Indian mathematical astronomers had lived before him. In addition to astronomy, Bhaskara II worked on number systems and algebra.

Jia Xian (ca. 1010-1070) in China invented what has become known as Pascal’s triangle, which was discovered independently by Blaise Pascal in France centuries later. It was also utilized by other Chinese mathematicians such as Zhu Shijie (ca. 1260-1320). The knowledge of Pascal’s triangle is one example of how the Chinese originated, but did not follow up, discoveries or inventions that later became key elements of Western science and technology.

Levi ben Gerson or Gersonides (1288-1344) was a Jewish rabbi, philosopher and astronomer who lived all his life in the south of France and was highly regarded in the Christian majority community. His surveying device called Jacob’s Staff, which was popular with sailors who used it for navigational purposes, was similar to a device which had been employed for several centuries in China. It is not currently known whether the idea was carried along trade routes from East Asia or whether it was an independent invention in Europe.

Arguably the most important mathematician in the Islamic world was Muhammad ibn Musa al-Khwarizmi (ca. 780-850). He or his ancestors probably came from Khwarizm, the region south of the Aral Sea in Central Asia, yet he spent much of his life in Baghdad, synthesizing Babylonian with Greek methods. According to David C. Lindberg, his Algebra “contains no equations or algebraic symbols, but only geometrical figures and Arabic prose, and it would not be recognized as algebra by a mathematics student of the twenty-first century. Its achievement was to deploy Euclidean geometry for the purpose of solving problems that we would now state in algebraic terms (including quadratic equations).” This book circulated in Europe and contributed in the long run to the development of a true symbolic algebra there.

Author John Derbyshire states in his Unknown Quantity: A Real and Imaginary History of Algebra that “It is a shame we do not know who first used a symbol for the unknown, but since Diophantus used it so well, so early, we ought to honor him for that. Probably someone of whom we have no knowledge, nor ever will have any knowledge, was the true father of algebra. Since the title is vacant, though, we may as well attach it to the most worthy name that has survived from antiquity, and that name is surely Diophantus.” Derbyshire attaches only medium-level importance to al-Khwarizmi work: “For one thing, al-Khwarizmi has no literal symbolism — no way to lay out equations in letters and numbers, no sign for the unknown quantity and its powers.” Muslim algebraists spelled out their problems in words.

Another gifted mathematician from the medieval Middle East was the Persian scholar Omar Khayyam (1048-1131), also renowned for the Rubaiyat poems attributed to him. He was definitely not an orthodox Muslim and he loved wine. He compiled astronomical tables and contributed to reform of the Persian calendar by introducing ideas from the Hindu one. The result was superior to the Julian calendar and at least comparable in accuracy to the Gregorian one. Khayyam was the first to solve some cubic equations and to see the equivalence between algebra and geometry, yet further progress in this field did not take place in the Islamic world.

Our numeral system dates back to India during the early post-Roman era. It came to Europe via the medieval Middle East, which is why these numbers are called “Arabic” numbers in European languages, yet even Muslims admit that they imported them from India. Labeling them “Arabic” numerals is this therefore deeply misleading. Calling them the “Hindu-Arabic” number system could be acceptable, but the preferred term should be “Indian numerals.”

The Maya in Mesoamerica developed a place-value number system with the zero at least as early as Indians did in Eurasia, but this great innovation sadly did not influence people elsewhere. According to Michael P. Closs in Mathematics Across Cultures, “There is reason to credit the Maya with the first invention of a zero symbol. It is absent in the surviving epi-Olmec texts but is very common in the Maya inscriptions. Zeros are found in many chronological counts in the Dresden Codex where they occur in positional contexts just as other numerals. Most Maya glyphs come in several variants and the same is true of the zero sign. The zeros in the codices are identifiable as shells and are always painted red. In most cases, the zero shells are stylized and simplified. In the inscriptions, the most common form of the zero is shaped somewhat like a three quarter portion of a Maltese cross.”

The ruthless Aztecs who dominated Mexico from the 1300s onward used hand, heart and arrow symbols to represent fractional distances when calculating areas of land. The Maya visualized the Milky Way as a road, a river or a serpent. Mesoamerican and especially Mayan mathematics is one pre-Columbian scientific achievement that can be compared favorably to developments in the Old World, but the mainstream development of mathematics happened in the Eurasian civilizations. The Maya seem to have concentrated their computational efforts largely in the field of planetary astronomy. They did good work in this regard, but the almost super-human achievements attributed to them by certain modern writers are exaggerated.

The zero can be used as an empty place indicator, to show that 2106 is different from 216. The ancient Babylonians had a place-value number system with this feature, but base 60. The second use of zero is as a number in the proper sense, the way we use it now. A few historians believe that the Indian use of zero evolved from innovations by Greek astronomers. Symbols for the first nine numbers of our number system have their origins in the Brahmi system of writing in India, which dates back at least to the third century BC when Indians had been introduced to the Semitic alphabetic script employed by the Persian Empire. More important than the form of the symbols is the notion of place value, and here the evidence is weaker.

The Chinese had a multiplicative system with the base 10, probably derived from the Chinese counting board, a checker board with rows and columns. Numbers were represented by little rods made from bamboo or ivory. The abacus was introduced in China around the fourteenth century. Somewhere around or before AD 600 (the place and date remains uncertain) Indians dropped symbols for numbers higher than 9 and began to use symbols for 1 through 9 in our familiar place-value arrangement. The question remains why Indians dropped their own multiplicative system and introduced the place-value system, including a symbol for zero. We currently don’t know for sure. Victor J. Katz elaborates in his A History of Mathematics:

“It has been suggested, however, that the true origins of the system in India may be found in the Chinese counting board. Counting boards were portable. Certainly, Chinese traders who visited India brought them along. In fact, since southeast Asia is the border between Hindu culture and Chinese influence, it may well have been the area in which the interchange took place. Perhaps what happened was that the Indians were impressed with the idea of using only nine symbols, but they took for their symbols the ones they had already been using. They then improved the Chinese system of counting rods by using exactly the same symbols for each place value rather than alternating two types of symbols in the various places. And because they needed to be able to write numbers in some form, rather than just have them on the counting board, they were forced to use a symbol, the dot and later the circle, to represent the blank column of the counting board. If this theory is correct, it is somewhat ironic that Indian scientists then returned the favor and brought this new system back to China early in the eighth century.”

A decimal place-value system for integers definitely existed in India by the eighth century AD, possibly earlier. Although decimal fractions were used in China, in India there is no early medieval evidence of their use. It was the Muslims who “completed the Indian written decimal place-value system by introducing these decimal fractions.”

There is evidence of the transmission of pre-Ptolemaic Greek astronomical knowledge to India, possibly along the Roman trade routes. The earliest known Indian work containing trigonometry dates from the fifth century AD. The Gupta period from the fourth to seventh centuries was a golden age for Indian civilization, with a flourishing of art and literature.

Indian astronomy did not delve into the physics of celestial movements; it remained backward-looking, focused on astrology and computation. Nevertheless, scholars produced a series of high-level textbooks (siddhanta or “solutions”) covering the basics of astronomy, using Greek planetary theory. The Aryabhatiya from 499 by Aryabhata (AD 476-550) was an important text which summarized Hindu mathematics up to that point, covering arithmetic and algebra plus plane and spherical trigonometry. Aryabhata apparently held the unorthodox view that the Earth rotates daily on its axis. The first artificial satellite from the Republic of India, launched into orbit with the aid of the Soviet Union in 1975, was named after him.

Next to Aryabhata, Brahmagupta (AD 598-ca. 665) was the most accomplished Indian mathematical astronomer. He came from the region of Rajasthan in the northwest and was associated with the observatory at Ujjain in north-central India, an important reference point for geographers. He made advances in algorithms for square roots and the solution of quadratic equations. His main work was Brahmasphuta-siddhanta, published around AD 630.

As Katz writes, “in 773 an Indian scholar visited the court of al-Mansur in Baghdad, bringing with him a copy of an Indian astronomical text, quite possibly Brahmagupta’s Brahmasphutasiddhanta. The caliph ordered this work translated into Arabic….The earliest available arithmetic text that deals with Hindu numbers is the Kitab al-jam’wal tafriq bi hisab al-Hind (Book on Addition and Subtraction after the Method of the Indians) by Muhammad ibn-Musa al-Khwarizmi (ca. 780-850)…In his text al-Khwarizmi introduced nine characters to designate the first nine numbers and, as the Latin version tells us, a circle to designate zero. He demonstrated how to write any number using these characters in our familiar place-value notation. He then described the algorithms of addition, subtraction, multiplication, division, halving, doubling, and determining square roots, and gave examples of their use.”

One Latin manuscript begins with the words “Dixit Algorismi,” or “al-Khwarizmi says.” The word “algorismi” through some misunderstandings became a term referring to arithmetic operations and the source of the word algorithm. A number of Sanskrit works and terms were introduced to Europe via Arabic translations. “Zero” derives from sifr, Latinized into “zephirum.” The word sifr itself was an Arabic translation of Sanskrit sunya, or “empty.”

Rabbi Abraham ben Meir ibn Ezra, or Abenezra (ca. 1090-1167), a Jewish philosopher and Biblical commentator, left Spain before 1140 to escape persecution of Jews and Christians by the regime of the Muslim Almohads. He wrote treatises which helped to bring the Indian symbols to the attention of some European scholars, but it took more time for Indian numerals to become fully adopted by Europeans. Another Spanish Jewish mathematician was Abraham bar Hiyya (ca. 1065-1136), who lived in Barcelona. His writings were among the first scientific works to be written in Hebrew and he helped to introduce Islamic algebra to Europe.

Leonardo of Pisa (ca. 1170-1240), often known as Fibonacci (son of Bonaccio), was the first great Western mathematician after the decline of ancient Greek science. The son of a merchant from the northern Italian city of Pisa with personal contacts in North Africa, Leonardo is most famous for his masterpiece the Liber abbaci or Book of Calculation. The word abbaci (from abacus) does not refer to a computing device but to calculation in general. The first edition appeared in 1202, and a revised one was published in 1228. This work enjoyed a wide European readership and contained rules for computing with the new Indian numerals. The examples were often inspired by examples from Arabic-language treatises, but filtered through Leonardo’s own considerable creative genius. Indian numerals faced powerful opposition for generations but were gradually adopted during the Renaissance period, especially by merchants. Their practical advantages compared to the more cumbersome Roman numerals were simply too great to ignore in the long run.

Roman numeral s are nevertheless still used for specialized purposes in the modern Western world, for instances to indicate the order of rulers such as Queen Elizabeth II, the second, or King Louis XIV, the fourteenth; in the publishing industry for copyright dates or on coins and clock faces. The symbol for one is I, for five V and for ten X. Placing any smaller number in front of any larger number indicates subtraction; IV means 4, VIII 8 and XXIV is 24. C stands for centum, the Latin word for 100. We use this term in words such as “century” for one hundred years and “centimeter” for one hundredth of a meter. A centurion was a professional military officer in the Roman army who commanded the smallest unit of a Roman legion, which nominally consisted of one hundred men but sometimes less in practice. “M” equals one thousand and comes from the Latin mille, meaning one thousand, which we retain in millennium (one thousand years) and millimeter (one thousandth of a meter). The year 2013 written by Roman numerals would be MMXIII, while 1939 would be MCMXXXIX.

Victor J. Katz sums up the state of global mathematics around the year 1300, with a special emphasis on the major Eurasian civilizations, Europe, India, China and the Middle East:

“European algebra of this time period, like its Islamic counterpart, did not consider negative numbers at all. India and China, however, were very fluent in the use of negative quantities in calculation, even if they were still hesitant about using them as answers to mathematical problems. The one mathematical subject present in Europe in this time period which was apparently not considered in the other areas was the complex of ideas surrounding motion. It was apparently only in Europe that mathematicians considered the mathematical question of the meaning of instantaneous velocity and therefore were able to develop the mean speed rule. Thus the seed was planted which ultimately grew into one branch of the subject of calculus nearly three centuries later. It appears that the level of mathematics in these four areas of the world was comparable at the turn of the fourteenth century. Although there were specific techniques available in each culture that were not available in others, there were many mathematical ideas and methods common to two or more.”

If the level of knowledge was comparable across the major regions of Eurasia by the early fourteenth century, why was modern mathematics developed in Europe? In the Islamic world, mathematical sciences and natural philosophy tended to be classified as “foreign sciences” and treated with some suspicion, not integrated into the core curriculum at places of learning. In Europe there was a growing body of universities where natural sciences were viewed more favorably and where students enjoyed much more free inquiry and legal protection. The Islamic world did not develop calculus, analytic geometry or heliocentric astronomy.

In China, the education system was a part of the Imperial bureaucracy, which did not encourage studies in science but memorization of ancient literary classics and Confucian philosophy. Those who did mathematical work usually did so in isolation, independent of and often unknown to each other, and their work was in many cases not followed up. This does not mean that Chinese mathematicians did not make valuable contributions, but just like in the Islamic world this often happened more in spite of than because of the education system.

The practical handbook Jiuzhang Suanshu (Nine Chapters on the Mathematical Art) is the longest surviving Chinese mathematical work, and prominent Chinese mathematicians, among them Liu Hui in AD 263, published commentaries on it. Zu Chongzhi (ca. AD 429-500) calculated p to seven decimals, the most accurate known estimate in the world until the Persian scholar Jamshid al-Kashi (ca. 1380-1429) surpassed this. The Chinese were proficient in solving many kinds of algebraic problems. One of the most dynamic periods of mathematics in China was the thirteenth century, with men such as Qin Jiushao (ca. 1202-1261), but further progress stagnated after this, just before advances in Europe accelerated.

Katz states that “Chinese scholars were primarily interested in solving problems of importance to the Chinese bureaucracy. Although some development of better techniques evidently occurred over the centuries, to a large extent ‘progress’ was stifled by the general Chinese reverence for the past. Hence even incorrect methods from such works as the Jiuzhang suanshu were repeated through the centuries. Although the thirteenth-century mathematicians exploited the counting board to the fullest, its very use imposed limits. Equations remained numerical, so the Chinese were unable to develop a theory of equations comparable to the one developed several centuries later in the West….Finally, in the late sixteenth century, with the arrival of the Jesuit priest Matteo Ricci (1552-1610), Western mathematics entered China and the indigenous tradition began to disappear.”

Madhava of Sangamagramma (ca.1350-1425) was the founder of the Kerala School of astronomy and mathematics in South India, which did some interesting work during the fourteenth to sixteenth centuries, but we currently possess little proof of transfer of ideas to other regions. Moreover, in the words of author John North, “Indian religious tradition was a powerful controlling force, not only of content, but also of form and of the ways of learning by rote. As a result, a typical work of eighteenth-century astronomy can be easily mistaken for one of the previous millennium. We are reminded of the situation in China, markedly different from that in the West.” There was never any strong native drive in India to link astronomy with other systems of knowledge, for instance physics, as happened in Europe.

The most important Asian mathematician of the early modern era before massive Western influence was arguably Japan’s Seki Kowa or Seki Takakazu (ca. 1642-1708). He was a leading figure in the wasan (“Japanese calculation”) movement. During the Edo period, ardent enthusiasts turned beautiful geometrical solutions into finely illustrated wooden tablets called sangaku that adorned the walls of local temples and shrines. Authors Fukagawa Hidetoshi and Tony Rothman write in Sacred Mathematics: Japanese Temple Geometry:

“Most of Seki’s works were published posthumously by his disciples and, because Japanese mathematicians traditionally deferred to their masters, this has always made it difficult to know precisely what he did and did not do….Of samurai descent, he was adopted in infancy by the noble family of Seki Gorozayemon and went by that surname. Later, he worked in the treasury of the Koufu clan, whose head was Lord Tokugawa Tsunashige.…there is no question that he was the first to develop the theory of determinants, a decade before Leibnitz. He also discovered the so-called Bernoulli numbers before Jacob Bernoulli, and Horner’s method 150 years before Horner, although in this he was anticipated by the Chinese.”

While there are a few exceptions, Katz concludes that “Nevertheless, the locus of the history of mathematics after the fourteenth century was primarily in Europe.” Almost all advances in mathematics between the fourteenth and the mid-twentieth century happened there.

The term “medieval” has, somewhat unfairly, come to carry negative connotations for many Westerners. Renaissance humanists in Western Europe viewed everything in between the fall of Rome in the fifth century AD and the revival of the Classical heritage in the fourteenth century as an unenlightened age which they labeled the Middle Ages. Much later, historians such as Jacob Burckhardt (1818-1897) from Switzerland and George Voigt (1827-1891) from Germany devoted considerable time to the epoch which was dubbed the “Renaissance,” or “rebirth,” and they reinforced the impression of the previous era as a “Dark Age.”

There is no doubt that there was prolonged unrest and urban disintegration following the collapse of Roman authority, accompanied by major population movements across the European continent, yet even during these troubled times, Charles Martel and the Carolingians managed to halt the Islamic invasion in France in the eighth century and for some time rebuilt a stronger state. Meanwhile, Roman Christianity spread among the pagans.

Boethius (ca. 480-525) was born in Rome and has been called “the last of the Romans, the first of the scholastics.” Like Augustine before him, he believed that the application of reason to theology was essential. According to Edward Grant, “Boethius began a trend that would eventually revolutionize Christian theology and transform it into a rationalistic and analytical discipline.” He knew Greek and wrote on philosophy, logic, music and mathematics, and helped to preserve a little bit of the knowledge from Antiquity during the Early Middle Ages.

Saint Isidore of Seville (ca. 560-636) and the Venerable Bede (ca. 672-735) also contributed to the modest storehouse of philosophical knowledge that was available in Western Europe before the twelfth century. The theologian Isidore was born into a prominent family in Roman Spain and served as Archbishop of Seville, then under Visigothic rule, for years. His encyclopedia Etymologies was one of the most popular books before the printing press, covering the seven liberal arts, medicine, law, timekeeping, theology, anthropology, geography, cosmology, mineralogy and agriculture. He was not an original thinker, but his work contained some useful bits of information in an age when this was in short supply.

The Venerable Bede was an accomplished English (Anglo-Saxon) monk and historian. At the age of seven he entered the monastery of Monkwearmouth in northeast England, near the modern city of Newcastle. He is especially remembered for his Ecclesiastical History of the English People, which constitutes the chief source of information for modern scholars about early Britain. He also helped popularize the system of dating events from the birth of Christ. Bede’s work is a fine example of decent medieval scholarship, but he was not typical, as most monks spent more time in the fields and farms or in administration than on being scholars.

Monks from Ireland, which was very early converted to Christianity following the disintegration of the Western Roman Empire, played a major role in keeping alive what remained of learning in the West during the Early Middle Ages. John Scotus Eriugena (ca. 810-877), the Irish philosopher and theologian who served King Charles the Bald of France, wrote a significant treatise titled On the Division of Nature. According to Edward Grant, “Eriugena’s emphasis on reason was given institutional roots in eleventh-century Europe with the development of the cathedral schools that emerged in various European cities.” Grant believes that “…medieval theology was a systematic, rationalistic discipline.”

Emperor Charlemagne brought in Alcuin, a distinguished scholar and headmaster of the cathedral school at York in present-day England, to serve as his educational adviser. Alcuin had studied with an Irish teacher and was assisted by several Irish clerics. Authors John McKay, Bennett Hill and John Buckler elaborate:

“At his court at Aachen, Charlemagne assembled learned men from all over Europe. The most important scholar and the leader of the palace school was the Northumbrian Alcuin (ca 735-804). From 781 until his death, Alcuin was the emperor’s chief adviser on religious and educational matters. An unusually prolific scholar, Alcuin prepared some of the emperor’s official documents and wrote many moral exempla, or ‘models,’ which set high standards for royal behavior and constitute a treatise on kingship. Alcuin’s letters to Charlemagne set forth political theories on the authority, power, and responsibilities of a Christian ruler. Aside from Alcuin’s literary efforts, what did the scholars at Charlemagne’s court do? They copied books and manuscripts and built up libraries. They used the beautifully clear handwriting known as ‘caroline minuscule,’ from which modern Roman type is derived. (This script is called minuscule because unlike the Merovingian majuscule, which had letters of equal size, minuscule had both upper- and lowercase letters.) Caroline minuscule improved the legibility of texts and meant that a sheet of vellum could contain more words and thus be used more efficiently. With the materials at hand, many more manuscripts could be copied.”

Although this Carolingian revival was initially motivated primarily by concerns about the low level of clerical literacy, it welcomed the natural sciences as well. Astronomy, for instance, was relevant for timekeeping and the calendar and for determining the correct date of Easter.

As David C. Lindberg says, “The importance of the copying of classical texts is demonstrated by the fact that our earliest known copies of most Roman scientific and literary texts (also Latin translations of Greek texts) date from the Carolingian period. The recovery and copying of books, combined with Charlemagne’s imperial edict mandating the establishment of cathedral and monastery schools, contributed to a wider dissemination of education than the Latin West had seen for several centuries and laid a foundation for future scholarship.”

There was some revival of interest in mathematics after Gerbert d’Aurillac (ca. 945-1003), who became Pope Sylvester II of the Catholic Church in the year 999. Grant states that his students “disseminated his love of learning and his teaching methods throughout northern Europe. As a consequence, logic became a basic subject of study in the cathedral schools of Europe. And, in the twelfth and thirteenth centuries, would become ever more deeply entrenched in the curricula of the cathedral schools and then the universities of Europe.”

Saint Anselm of Canterbury (1033-1109) used logical techniques in his approach to Christian theology. He was born in a Burgundian town on the frontier with Lombardy, Italy. Once he arrived in Normandy his interest was captured by the Benedictine abbey at Bec with its famous school. Anselm managed to write a good deal of philosophy and theology in addition to his teaching, administrative duties, and extensive correspondence as an adviser to rulers and nobles. In 1093 Anselm was enthroned as Archbishop of Canterbury in England.

The French scholar Peter Lombard (ca. 1095-1160) wrote the treatise Four Books of Sentences, which became the basic textbook in all schools of theology in the Latin West until the seventeenth century. Between 1150 and 1500, only the Bible was read and discussed more than the Sentences. After education at Bologna he taught theology at the school of Notre Dame, Paris. Here he came into contact with Peter Abelard and the mystic Hugh of Saint-Victor (1096-1141), who were among the most influential theologians of the time.

European Christians re-conquered Toledo in Spain and Sicily from the Muslims in 1085 and 1091, respectively. Gerard of Cremona (ca. 1114-1187), the prolific Italian translator from Arabic to Latin of works on science and natural philosophy, lived for years at Toledo and was aided by a team of local Jewish interpreters and Latin scribes. David C. Lindberg argues that Alhazen’s Book of Optics probably was translated during the late twelfth century by Gerard or somebody from his school; it was known in thirteenth century Europe. Many ancient works initially translated from Arabic by Gerard and his associates, among them Ptolemy’s Almagest, were later translated directly from Greek into Latin from Byzantine manuscripts.

The basic principle of the astrolabe, a model of the heavens, was a discovery of the ancient Greeks, although it is possible that related devices were used by Babylonian astronomers. Stereographic projection, one way among several of mapping a sphere onto a flat surface, was probably known to the great mathematical astronomer Hipparchus in the second century BC and was certainly in use by Roman times. The first treatise on the astrolabe in the modern sense was probably written by Theon of Alexandria (ca. AD 335-405), a teacher of mathematics who made an influential edition with added comments of Euclid’s Elements.

It is true that the use of astrolabes was partly reintroduced to medieval Europe via Islamic-ruled Spain. This instrument became widely popular during the Renaissance. James E. Morrison is the author of the book The Astrolabe. As he says, “Astrolabe manufacturing was centered in Augsburg and Nuremberg in Germany in the fifteenth century with some production in France. In the sixteenth century, the best instruments came from Louvain in Belgium. By the middle of the seventeenth century astrolabes were made all over Europe.”

The oldest surviving, moderately sophisticated scientific work in the English language is the Treatise on the Astrolabe, written by the English poet, philosopher, courtier and diplomat Geoffrey Chaucer (ca. 1343-1400) for his son. Chaucer’s most famous work, The Canterbury Tales, is studded with astronomical references. The English used in Chaucer’s time is often called Middle English to distinguish it from the Germanic language known as Old English spoken during the Anglo-Saxon era of the Early Middle Ages. After the Norman Conquest in 1066, many words used by the French-speaking ruling elites affected the English language.

It should be noted that while it was a very popular device, the astrolabe was not a precision instrument even by medieval standards. Its popularity stemmed from the fact that approximate solutions to astronomical problems could be found by a mere glance at the instrument. The invention of the pendulum clock and more specialized and useful scientific devices such as the telescope from the seventeenth century on soon replaced the astrolabe in importance.

Nevertheless, its medieval reintroduction via the Islamic world did leave some traces. Quite a few star names in use in modern European languages, for instance Aldebaran or Algol, can be traced back to Arabic or Arabized versions of earlier Greek names. Today astronomers frequently identify stars by means of Bayer letters, introduced by the German astronomer Johann Bayer (1572-16259) in his celestial atlas Uranometria from 1603. In this system, each star is labeled by a Greek letter and the Latin name of the constellation in which it is found.

It is undoubtedly true that there were translations from Arabic and that these did have some impact in Europe, leaving traces in star names and some mathematical and alchemical terms. Yet far too much emphasis is currently placed on the translations themselves and too little on how the knowledge contained within these texts was actually used. After the translation movement it is striking to notice how fast Europeans surpassed whatever scholarly achievements had been made in the medieval Middle East based on largely the same material.

Moreover, it is simply not the case that these translations “rescued” the Classical heritage, which had survived largely intact among Byzantine, Orthodox Christians. When Western, Latin Christians wanted to recover the Greco-Roman heritage they translated Greek historical works and literature in addition to philosophy, medicine and astronomy, and copied works by Roman authors and poets in Latin which had been totally ignored by Muslims. The permanent recovery of the full body of Greco-Roman learning and literature was undertaken as a direct transmission from Greek-speaking Orthodox Christians to Western, Latin Christians.

The greatest translator from Greek to Latin was the Flemish scholar William of Moerbeke (ca. 1215-1286), a contemporary of the prominent German scholar Albertus Magnus. He was fluent in Greek and made very accurate translations, still held in high regard today, from Byzantine originals and improved earlier translations of the works of Aristotle and many by Archimedes, Hero of Alexandria and others. William of Moerbeke was a Roman Catholic friar of the Dominican order and had personal contacts at the top levels of the Vatican, including several popes. Thanks in part to his efforts, by the 1270s Western Europeans had access to Greek works that were never translated into Arabic, for instance Aristotle’s Politics. This benefited his Italian friend Thomas Aquinas for his major work the Summa Theologica.

The translation movement began in the eleventh century, continued during the Renaissance and culminated in its final and arguably most important phase stretching into the sixteenth century with the introduction of the printing press. This invention vastly increased the circulation of books as well as the accuracy of their reproduction. The much cheaper printed copies were of particular importance in the sciences, since university students could get mathematical texts where the diagrams and tables in a given book were the same in all copies.

As the historian of medieval science Edward Grant says, “The advantages of printed books over hand-copied books cannot be overestimated. The printed book transformed learning in Europe not only because it introduced uniform standards, but also because it greatly increased the speed by which learning was disseminated. The invention of printing from movable type may have been the most important contribution to the advance of civilization made in the second millennium. The transition from hand-copied documents to printed documents was far more revolutionary than the transition from the typewriter to the computer.”

It was a major stroke of historical luck that printing was introduced in Europe at exactly the same time as the last vestige of the Roman Empire fell to Muslim Turks. Texts that had been preserved in Constantinople for a thousand years could now be permanently rescued. Elizabeth L. Eisenstein writes in her monumental The Printing Press as an Agent of Change:

“The classical editions, dictionaries, grammar and reference guides issued from print shops made it possible to achieve an unprecedented mastery of Alexandrian learning even while laying the basis for a new kind of permanent Greek revival in the West.…We now tend to take for granted that the study of Greek would continue to flourish after the main Greek manuscript centers had fallen into alien hands and hence fail to appreciate how remarkable it was to find that Homer and Plato had not been buried anew but had, on the contrary, been disinterred forever more. Surely Ottoman advances would have been catastrophic before the advent of printing. Texts and scholars scattered in nearby regions might have prolonged the study of Greek but only in a temporary way.”

Rémi Brague is a French professor and specialist of medieval religious philosophy. According to his book The Legend of the Middle Ages: Philosophical Explorations of Medieval Christianity, Judaism, and Islam, philosophy was always marginal in the Islamic world and was never institutionalized there as it was in the European universities. In his view, theology as such is a Christian specialty. He even claims that “‘theology’ as a rational exploration of the divine (according to Anselm’s program) exists only in Christianity” and states that “You can be a perfectly competent rabbi or imam without ever having studied philosophy. In contrast, a philosophical background is a necessary part of the basic equipment of the Christian theologian. It has even been obligatory since the Lateran Council of 1215.”

Demand usually precedes the presence of a product on the market and it is the demand that needs to be explained. As Brague notes, translations are made because someone feels that a certain text contains information that people need. The real intellectual revolution in Europe began well before the wave of translations in Toledo and elsewhere. This was demonstrated by the American jurist Harold J. Berman in his important 1983 book Law and Revolution. The efforts of the Catholic Church to make a new system of law required refined tools, which meant that the West sought out Aristotle’s and other Greek works on logic and philosophy.

The “Papal Revolution” starting in the eleventh century was an effort to apply ancient Greek methods of logic to the remnants of Roman law dating back to Late Antiquity and the reforms of the very active Eastern Roman Emperor Justinian the Great. Justinian’s revision of existing Roman law, the Corpus Juris Civilis (Body of Civil Law) was compiled in Latin in the 530s AD and later influenced medieval Canon Law.

While they did utilize Roman law and Greek logic, medieval Western scholars through their efforts created a new synthesis which had not existed in Antiquity. Prominent among them was the twelfth century Italian scholar Gratian, a monk who taught in Bologna. His great work, commonly known as the Decretum, appeared around 1140 as a synthesis of church law. Harold J. Berman writes in Law and Revolution: The Formation of the Western Legal Tradition:

“Every person in Western Christendom lived under both canon law and one or more secular legal systems. The pluralism of legal systems within a common legal order was an essential element of the structure of each system. Because none of the coexisting legal systems claimed to be all inclusive or omnicompetent, each had to develop constitutional standards for locating and limiting sovereignty, for allocating governmental powers within such sovereignty, and for determining the basic rights and duties of members….Like the developing English royal law of the same period, the canon law tended to be systematized more on the basis of procedure than of substantive rules. Yet after Gratian, canon law, unlike English royal law, was also a university discipline; professors took the rules and principles and theories of the cases into the classrooms and collected, analyzed, and harmonized them in their treatises.”

With the papacy of the dynamic and assertive Gregory VII (1073-1085), the Roman Catholic Church entered the Investiture Struggle, a protracted and largely successful conflict with European monarchs over control of appointments, investitures, of Church officials. Author Edward Grant explains in God and Reason in the Middle Ages:

“Gregory VII began the process that culminated in 1122 in the Concordat of Worms (during the reign of the French pope, Calixtus II [1119-1124]), whereby the Holy Roman Emperor agreed to give up spiritual investiture and allow free ecclesiastical elections. The process manifested by the Investiture Struggle has been appropriately called the Papal Revolution. Its most immediate consequence was that it freed the clergy from domination by secular authorities: emperors, kings, and feudal nobility. With control over its own clergy, the papacy became an awesome, centralized, bureaucratic powerhouse, an institution in which literacy, a formidable tool in the Middle Ages, was concentrated. The Papal Revolution had major political, economic, social, and cultural consequences. With regard to the cultural and intellectual consequences, it ‘may be viewed as a motive force in the creation of the first European universities, in the emergence of theology and jurisprudence and philosophy as systematic disciplines, in the creation of new literary and artistic styles, and in the development of a new consciousness.’…the papacy grew stronger and more formidable. It reached the pinnacle of its power more than a century later in the pontificate of Innocent III (1198-1216), perhaps the most powerful of all medieval popes.”

The power of the secular states grew, too, but the separation between Church and state endured because the Papal Revolution had established a virtual parity between them. It was the internal dynamism of medieval Europe that drove the European intellectual renaissance and the recovery of Classical learning. As Brague writes in The Legend of the Middle Ages:

“Like all historical events, it had economic aspects (lands newly under cultivation, new agricultural techniques) and social aspects (the rise of free cities). On the level of intellectual life, it can be understood as arising from a movement that began in the eleventh century, probably launched by the Gregorian reform of the Church.…That conflict bears witness to a reorientation of Christianity toward a transformation of the temporal world, up to that point more or less left to its own devices, with the Church taking refuge in an apocalyptical attitude that said since the world was about to end, there was little need to transform it. The Church’s effort to become an autonomous entity by drawing up a law that would be exclusive to it — Canon Law — prompted an intense need for intellectual tools. More refined concepts were called for than those available at the time. Hence the appeal to the logical works of Aristotle, who was translated from Greek to Latin, either through Arabic or directly from the Greek, and the Aristotelian heritage was recovered.”

From the eleventh century onward, more political stability and an extension of the money economy to include the countryside combined with technological improvements such as the spread of water wheels and windmills generated a rapid growth of the European population.

As David Lindberg explains in The Beginnings of Western Science, “exact figures are not available, but between 1000 and 1200 the population of Europe may have doubled, tripled, or even quadrupled, while the city-dwelling portion of this population increased even more rapidly. Urbanization, in turn, provided economic opportunity, allowed for the concentration of wealth, and encouraged the growth of schools and intellectual culture. It is widely agreed that a close relationship exists between education and urbanization. The disappearance of the ancient schools was associated with the decline of the ancient city; and educational invigoration followed quickly upon the reurbanization of Europe in the eleventh and twelfth centuries.” Because of this, “An educational revolution was in progress, driven by European affluence, ample career opportunities for the educated, and the intellectual excitement generated by teachers such as Peter Abelard. Out of the revolution emerged a new institution, the European university, which would play a vital role in promoting the natural sciences.”

Peter Abelard, or Pierre Abélard (1079-1142), was a French scholastic philosopher and theologian who was very influential in his time as a thinker and teacher. He was a poet and a musician as well, and became famous for his luckless love affair with Héloïse. First and foremost, however, he was the most significant logician of his age and arguably the greatest rationalist of the twelfth century. He believed that it was necessary to use logic and reason to defend the Faith. Author Rémi Brague considers him to be “ one of the greatest French philosophers, to be placed beside Descartes or Bergson.”

While university-educated people were a miniscule fraction of the total European population, their cumulative influence should not be underestimated. A striking number of the leading scholars in early modern Europe, from Copernicus to Galileo and Newton, had studied at these institutions. They emerged gradually out of preexisting schools, but it is customary to say that Bologna had achieved university status by 1150, Paris by about 1200 and Oxford before 1220. Later universities were generally modeled on one or another of these three.

This network constituted a crucial innovation compared to other civilizations at the time. Although the Scientific Revolution began in the seventeenth century with the systematic use of the experimental method and a more critical view of the knowledge of the ancients, exemplified by individuals such as Galileo, the initial institutional basis for these developments was laid with the natural philosophers of the medieval universities.

The English Franciscan friar William of Ockham (ca. 1287-1347), sometimes spelled Occam, was among the most prominent philosopher-theologians of the High Middle Ages, next toFind all the books, read about the author, and more.See search results for this author Are you an author? Learn about Author Central next tonext to John Duns Scotus (ca. 1266-1308) and a handful of others. He studied theology at the University of Oxford before 1320 and was called to the Papal court at Avignon, France in 1324 to answer charges of heresy. His logical textbook, the Summa Logicae (“Summary of Logic”), was written around 1323. In 1328 he moved to Munich where he remained until his death. Around 1320 he developed what has become known as Ockham’s Razor. He was not the first person to mention the principle of parsimony but he promoted it so systematically that it has been named after him. Briefly formulated it says “Don’t multiply entities beyond necessity,” in other words make as few assumptions as possible and prefer the simplest explanation that fits the available data. This remains a crucial principle of scientific logic.

The French scholastic philosopher and economist Nicole Oresme (ca. 1320-1382) was one of the most original mathematicians of the European Middle Ages. He was born in Normandy near the city of Caen, studied and taught at the University of Paris, was appointed dean in 1364 of the Cathedral of Rouen and was elected Roman Catholic Bishop of Lisieux in 1377. Oresme was “ one of the most eminent scholastic philosophers, famous for his original ideas, his independent thinking and his critique of several Aristotelian tenets. His work provided some basis for the development of modern mathematics and science. Furthermore he is generally considered the greatest medieval economist. By translating, at the behest of King Charles V of France, Aristotle’s Ethics, Politics, and On the Heavens, as well as the pseudo-Aristotelian Economics, from Latin into French, he exerted a considerable influence on the development of French prose, particularly its scientific and philosophical vocabulary.”

By the fourteenth century a commercial revolution had begun in Western Europe where the “new capitalists” could remain at home and hire others to travel to various ports as their agents. This led to the creation of international trading companies centered in major cities, and these companies needed more sophisticated mathematics than their predecessors did because they had to deal with letters of credit, bills of exchange, promissory notes and interest calculations. Double-entry bookkeeping began as a way of keeping track of these various transactions. A new class of “professional” mathematicians grew up in response to these growing needs, the maestri d’abbaco or abacists. Italian abacists and merchants were instrumental in teaching Europeans the Hindu-Arabic decimal place-value system.

Luca Pacioli (ca. 1446-1517) was one of the last of the abacists and a Franciscan friar. He gathered materials for some 20 years and in 1494 completed the most comprehensive mathematics text in Europe of the time, Summa de arithmetica, geometria, proportioni et proportionalita. This summary lacked originality, but its comprehensiveness and the fact that it was printed caused it to be widely circulated. Pacioli was well-connected. His friends included the architect Leon Battista Alberti and the polymath Leonardo da Vinci, whose interest for mathematics was reflected in his art and his studies of proportions and perspective.

Niccolò Tartaglia (1499-1557) was largely self-taught in mathematics and got the nickname Tartaglia (“Stammerer”) from a serious injury he had suffered as a boy at the hands of a French soldier. He settled in Venice in 1534 as a teacher of mathematics, wrote on the application of mathematics to artillery fire and was thus a pioneer in the science of ballistics. By 1535 he had discovered a general method for solving cubic equations. Scipione del Ferro (1465-1526) had made some progress on the cubic previously. Gerolamo Cardano or Cardan (1501-1576), who was trained as a physician and was a lecturer of mathematics in Milan, in 1539 contacted Tartaglia to publish his solution. Tartaglia refused at first but eventually confided in him. He extracted an oath from Cardano that he would never publish Tartaglia’s mathematical discoveries as he planned to publish them himself at a later date.

Cardano began working with the problem of cubic equations, assisted by his gifted student Lodovico Ferrari (1522-1565). Over the next years he worked out the solutions and the justifications to all the various cases of the cubic. Ferrari solved the fourth degree (quartic) equation as well. Tartaglia still hadn’t published anything and Cardano was eager that the solutions should be made available. In 1545 he published his great work Ars Magna, Sive de Regulis Algebraicis (The Great Art, or On the Rules of Algebra), devoted to the solution of cubic and quartic equations. Tartaglia was furious, even though Cardano did mention him as one of the discoverers of the method. Cardano’s influential masterpiece Ars Magna contained much else of interest, including a solid understanding of the use of negative numbers.

Rafael Bombelli (1526-1572), an engineer from Bologna, was the first person to write down the rules for addition, subtraction and multiplication of complex numbers. Algebraic notation was gradually replacing the strictly verbal accounts of the Muslims, and Bombelli contributed to this change. According to Katz, “although in the questions concerning multiple roots of cubic equations Bombelli did not achieve as much as Cardano, nevertheless Bombelli’s Algebra marks the high point of the Italian algebra of the Renaissance.”

The Italian humanist Federico Commandino (1509-1575) had studied Latin and Greek. He spent years of his life publishing improved Latin translations with commentaries of Greek texts by Archimedes, Ptolemy, Euclid, Aristarchus, Pappus, Apollonius and Hero of Alexandria. He contributed greatly to the survival of these works, which were eagerly studied by leading figures in Europe. Some medieval translations made before the printing press were linguistically flawed and had not been done by skilled mathematicians, as Commandino was.

Pappus of Alexandria (ca. AD 290-350) worked in Roman Egypt, maybe as a teacher. His compendium Synagoge (“Collection”) was not original, but it preserved some Greek mathematical texts that would otherwise have been lost. According to the Encyclopædia Britannica online, “Pappus’s Synagoge first became widely known among European mathematicians after 1588, when a posthumous Latin translation by Federico Commandino was printed in Italy. For more than a century afterward, Pappus’s accounts of geometric principles and methods stimulated new mathematical research.” His influence is conspicuous in the work of René Descartes, Pierre de Fermat and Isaac Newton, among many others.

As these examples demonstrate, Italy during the Renaissance period was the leading nation in Europe in science and mathematics, but the northern peoples were making advances, too. Robert Recorde (1510-1558), a Welsh physician and mathematician and graduate from the University of Oxford in England, was the first author of mathematical works in Britain during the Renaissance. His book The Whetstone of Witte from 1557 introduced the equals sign =.

The Flemish mathematician and engineer Simon Stevin (1548-1620) made substantial contributions to a change in mathematical thinking with his well-though-out notation for decimal fractions and his role in erasing the Aristotelian distinction between number and magnitude. Decimals had been used by the Chinese and Muslims before, but not by Europeans. Stevin introduced this in 1585 with De Thiende (The Art of Tenths). The Scottish scholar John Napier and others took up his notation and developed it into that used today. Stevin also published L’arithmétique in French, a work containing arithmetic and algebra.

Born in the city of Bruges in Flanders, Stevin became a bookkeeper with a firm in Antwerp and traveled in Poland, Prussia, Norway and other parts of northern Europe in the 1570s. He left the southern Netherlands, then under Spanish rule, and in 1583 entered the University of Leiden. He became an advisor to Prince Maurice of Nassau (1567-1625) and was responsible for meeting the growing need of the Dutch nation for trained engineers, merchants and navigators. He wrote textbooks in Dutch rather than the traditional Latin. Inspired by Archimedes, Stevin wrote works on mechanics and arguably founded the science of hydrostatics. In 1586, before Galileo did the same, he reported that different weights fall a given distance in the same time after experiments conducted from a church tower in Delft.

Ancient Greek planetary theory was brought into its final, successful form with Ptolemy’s masterpiece in Alexandria in the second century AD. Author James Evans explains:

“The original title was something like The 13 Books of the Mathematical Composition of Claudius Ptolemy. Later the work may simply have been known as Megale Syntaxis, the Great Composition. The superlative form of the Greek megale (great) is megiste. Arabic astronomers of the early Middle Ages joined to this the Arabic article al-, giving al-megiste, which was later corrupted by medieval Latin writers to Almagest….The Almagest is one of the greatest books in the whole history of the sciences — comparable in its significance and influence to Euclid’s Elements, Newton’s Principia, or Darwin’s Origin of Species.”

The mathematician al- Battani (ca. 850-929) made measurements of the stars and planets. The Persian astronomer Abul Wafa (940-998) was a capable mathematician who made good trigonometric tables. Ibn Yunus (950-1009), an Egyptian mathematician, made reliable observations of the Moon and described many lunar eclipses. A lunar crater has been named in his honor and another one in honor of Ibn al-Zarqali, Latinized as Arzachel, an eleventh-century Andalusian astronomer who was partly responsible for the so-called Toledan Tables. These were accurate for their time and were later translated into Latin and used in Europe.

The Mongols under the leadership of Hulegu Khan (ca. 1217-1265), a grandson of the feared and powerful Mongol conqueror Genghis Khan (ca. 1162-1227), sacked Baghdad in 1258. Hulegu believed that many of his military successes were due to the advice of astronomers who were also astrologers (astrology was important in Mongol culture) and was persuaded to found the Maragha observatory in Iran by the Persian mathematician Nasir al-Din al-Tusi (1201-1274). Muslim achievements in astronomy thus peaked after the Mongol conquests.

Astronomy in the Islamic world remained fundamentally Ptolemaic and Earth-centered, but Ptolemy did have critics regarding certain technical details. The planetary models developed by Maragha astronomers such as Ibn al-Shatir of Damascus (ca. 1304-1375) showed some originality. Some mathematical constructions similar to those of Ibn al-Shatir later turned up in the work of Nicolaus Copernicus, who may have learned of them while studying in Italy.

The Maragha observatory from 1259 was destroyed already in the early 1300s. According to author Toby E. Huff, “The fact that the Maragha observatory not only stopped functioning within fifty years but soon thereafter was completely obliterated suggests that there were very strong antipathies against it and its activities” because of their alleged association with astrology, which was considered a challenge to the omnipotence of Allah. The observatory as a scientific institution failed to take root in the Islamic world due to religious resistance. As historian Bernard Lewis states, in the Ottoman Empire the observatory in Constantinople/Istanbul created by Taqi al-Din (1526-1585) “was razed to the ground by a squad of Janissaries, by order of the sultan, on the recommendation of the Chief Mufti.”

Among major regions or civilizations, the two with the most similar medieval starting points were the Middle East and Europe. Greek geometry was virtually unknown in East and Southeast Asia. This constituted a major disadvantage for Chinese, Japanese and Korean scholars in optics and astronomy. The only regions where clear glass was extensively made were the Middle East and Europe. Clear glass was used by Europeans to create eyeglasses for the correction of eyesight and later for the creation of microscopes and telescopes, facilitating the birth of modern medicine and astronomy. The Maya in pre-Columbian Mesoamerica did not know how to make glass and could consequently not have made glass lenses for microscopes or telescopes. Muslims could have done so, but they didn’t. Likewise, medieval Europeans invented mechanical clocks while Muslims did not, despite a similar starting point.

The best Muslim scholars could be capable observational astronomers and a few made minor adjustments to Ptolemaic theory, but none of them ever made a huge conceptual breakthrough comparable to that provided by Copernicus when he put the Sun, not the Earth, at the center of our Solar System. Combined with the pre-telescopic work of Tycho Brahe and Johannes Kepler, Ptolemaic astronomy was in reality outdated in Europe even before Galileo and others had introduced telescopic astronomy by 1610. In contrast, Muslims resisted Copernican heliocentrism in some cases into the twentieth century. One of those who rejected it was the influential Islamic activist and alleged reformist Jamal-al-Din al-Afghani (ca. 1838-1897).

The Scientific Revolution

The greatest scientific advances in human history began in Europe following the Renaissance period. The translation of Greek and Latin Classical works, dictionaries and texts culminated in printed editions at the turn of the sixteenth century. Many of these were written, translated or edited by Byzantine émigré scholars such as Cardinal Johannes Bessarion (1403-72), a Byzantine Greek with excellent personal connections who promoted the study of the classics of Greek literature and philosophy in Western Europe after the Ottoman conquests.

The number of competent, practicing astronomers in late medieval European universities far exceeded the number who had been active at any stage during Greek Antiquity. One of them was the Austrian Georg Peurbach, who was born near the city of Linz on the river Danube. He travelled through Europe around 1450 and after that accepted a chair at the University of Vienna. Peurbach published observations as well as a textbook on trigonometric calculation.

The German astronomer Johannes Müller was born near Königsberg in Bavaria and Latinized his name to Regiomontanus. He was a talented mathematician who worked squarely in the Ptolemaic tradition. He was educated at the Universities of Leipzig and Vienna. In Italy he perfected his Greek and read widely in Bessarion’s library. The Epitome of the Almagest was completed around 1463 but printed in 1496. In 1464 Regiomontus published the first systematic European work on trigonometry as a subject divorced from astronomy. Between 1467 and 1471 he worked in Hungary for King Matthias Corvinus (1443-1490), a great patron of the arts and sciences and son of the Hungarian general John Hunyadi (ca. 1400-1456) who led the Christian European defenses against the aggressive Turks. In 1471 Regiomontanus relocated to Nuremberg where he started a business of publishing mathematical and astronomical books, the first printing establishment specifically dedicated to scientific works.

According to David Lindberg in The Beginnings of Western Science, “Reunion of the Roman Catholic and Greek Orthodox churches after centuries of schism brought Bessarion to Vienna in the 1460s, where he became the friend and patron of two young professors of astronomy: Georg Peurbach (1423-61) and Peurbach’s student Johannes Regiomontanus (1436-76). Motivated by the fall of Constantinople to the Turks in 1453, Bessarion campaigned to save as much of the Greek intellectual legacy as possible. One result was a Latin Epitome of the Almagest, produced by Peurbach and Regiomontanus — essentially a commentary that improved on all previous commentaries on Ptolemy’s Almagest and ‘provided Copernicus with a crucial conceptual stepping stone to the emergence of the heliocentric theory.’“

Nicolaus Copernicus (1473-1543) was born into a merchant family in the town of Torun on the Vistula River in north-central Poland, south of the Baltic seaport of Gdansk (Danzig) then the country’s largest city. He was educated at the university in Cracow or Kraków, which was for a long time the capital of the Kingdom of Poland, where he studied Latin, mathematics, astronomy, geography and philosophy. He spoke both Polish and German in addition to Greek and Italian, but the language of learning in Europe was Latin. Copernicus continued his education in Renaissance Italy between 1496 and 1503, studying Canon Law at the University of Bologna as well as medicine and astronomy at Padua, and finally took a degree at the University of Ferrara before returning to his native land.

Before 1514 he had written a small manuscript called the Commentariolus (“Little Commentary”), outlining some of his revolutionary ideas, but this text was hand written and exclusively distributed to a few personal friends. Had it not been for encouragement from his Austrian pupil, the astronomer Georg Joachim von Lauchen, or Rheticus (1514-1574) from the University of Wittenberg, Copernicus’s masterpiece might never have been published.

Copernicus did carry out occasional planetary observations, but his system initially had little quantitative advantage over the Ptolemaic one. He preferred it because of its beauty and harmony. His model showed that the swiftest planet, Mercury, took up the orbit closest to the Sun whereas the slowest naked-eye planet, Saturn, fell farthest away. According to legend, Copernicus received a copy of his printed work De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) on his own deathbed in 1543. His book was considered an important astronomical text even by many of those who did not believe in heliocentrism.

To Copernicus, his model was better than that of Ptolemy because it was more elegant. As James Evans notes, his theory contained a mixture of radical innovation and traditional astronomy: “To launch the Earth into orbit was a bold move. The Sun-centered theory does have great explanatory advantages. And it does turn the whole solar system into a unified whole, as Copernicus himself stressed. But in the technical details of his planetary theory, Copernicus remained a part of the Ptolemaic tradition. Nearly every detail of his model has a corresponding element in Ptolemy’s model. It was for this reason that Kepler was later to say that Copernicus would have done better if he had interpreted nature, rather than Ptolemy.”

The ancient Greek astronomer Aristarchus of Samos was the first known person to maintain that the Earth revolves around the Sun. His ideas are mentioned in several preserved works such as Archimedes’s Sand Reckoner, which Copernicus was probably familiar with, and in the writings of the influential Greek historian Plutarch (ca. AD 46-120).

The heliocentric theory of Aristarchus from the third century BC was overwhelmingly rejected in Antiquity because it seemingly violated common sense, everyday observations and Aristotelian physics. If the Earth orbits the Sun, why can birds fly equally well in all directions? Another problem was that of stellar parallax, a relative change in the position of stars over a six month period as the Earth orbits the Sun, which could not be observed. Aristarchus stated that the stars are extremely far away and that the parallax is consequently too small to be observed. This is exactly the same answer that Copernicus gave much later.

Seleucus of Seleucia, a Hellenistic astronomer from Mesopotamia in the second century BC, was virtually the only ancient scholar who is recorded as having supported Aristarchus’s heliocentric theory. Another possible, but much more controversial claim for pre-modern support of heliocentrism is the mathematical astronomer Aryabhata in early medieval India.

In the opinion of author Michael Hart, “The enormous scientific advances made by Europeans between 1540 and 1700 — we might call them collectively ‘the Scientific Revolution’ — dwarf even the noteworthy advances made by the ancient Greeks. Indeed, they are greater than all the scientific advances made throughout the entire world in all prior ages combined. One might truly say that science, as an organized human activity, came into existence in this period. What had previously been a sporadic activity, one occasionally indulged in by isolated persons, became an important, continuing project of a small but important community.”

This begs the question: What triggered the Scientific Revolution? Europe of the Late Middle Ages was increasingly prosperous. The rediscovery of the writings of the ancient Greeks and the introduction of the printing press were necessary conditions for the developments that took place, but not sufficient ones. Why were there more people engaged in mathematics and science during this age than before? Michael H. Hart speculates whether the introduction of the heliocentric hypothesis was responsible for this seemingly abrupt change in mentality.

While Copernicus did remove the Earth from the central place it had previously enjoyed in the cosmos it is important to realize that this did not automatically imply a lesser status since in the Greek worldview, the heavens were eternal and perfect whereas the Earth was the region of corruption and imperfection. As writer Rémi Brague notes, “The Copernican hypothesis, far from being considered a wound, was felt as a flattering promotion: instead of crouching in a dungeon, man was henceforth the inhabitant of a neighborhood as chic as the sun’s.”

Nevertheless, there is no doubt that the heliocentric theory triggered a mental revolution, first in Europe and eventually throughout the world. It wasn’t just another mathematical construct; it radically altered man’s place in the universe and — most importantly — his relations to God. This is why it triggered so much resistance of a non-scientific nature. In ancient times, many peoples had commonly assumed that the Earth was the center of creation, and often that their particular city, country or nation was the center of the Earth itself. The heliocentric theory permanently removed the cradle of mankind from the center of the universe.

In the centuries that followed, Western astronomers proved that the Sun was just one of countless stars and that the Milky Way is just one among countless billions of galaxies in the known universe. The most radical cosmologists in the twenty-first century claim that our entire observable universe is just one among many different universes. Our tiny planet is more insignificant in the cosmos than a grain of sand is to us. Copernicus himself would no doubt have been shocked by this realization, but he started the process which eventually led to it.

The so-called Copernican Revolution was indeed a revolution, but it was a shift that nevertheless took generations to mature. Long after Copernicus had died there were still leading European astronomers who questioned his ideas, sometimes also on a scientific basis.

The nobleman Tyco Brahe was born Tyge Brahe (1546-1601) in Scania (Skåne) in the far south of Sweden, which was then a part of the Kingdom of Denmark. He attended the universities of Copenhagen and Leipzig and traveled through the German-speaking lands, studying in the cities of Wittenberg, Rostock and Basel. He lost his nose in a duel and wore an artificial nose made from silver and gold. With financial aid from the King of Denmark he set up Europe’s finest observatory called Uraniborg on the little island of Hven near Copenhagen, equipped with exceptionally large instruments plus an alchemical laboratory. In 1572 Brahe recorded the first modern European observation of a supernova, which undermined the Greek concept of a fixed sphere of unchanging stars and the Aristotelian division between the corrupt and ever changing sublunary world and the perfect and immutable heavens. He was also able to show that a comet he spotted in 1577 was further away than the planet Venus.

Tycho Brahe was influenced by some technical elements of the Copernican theory but developed an alternative geo-heliocentric system in which the planets all went around the Sun while the Sun moved around a stationary Earth. He opposed Copernicus for several reasons, one of them being his attachment to Aristotelian physics, another being the lack of observable stellar parallaxes. He changed observational practice profoundly. Whereas earlier astronomers had been content to observe the positions of the planets and the Moon only at important points of their orbits, Tycho and his assistants observed these bodies throughout their orbits. As a result, he detected a number of orbital anomalies that had never been noticed before.

With his mural quadrant and other naked-eye instruments, Tycho recorded the positions of hundreds of stars and followed the motions of planets over decades. His mass of data was invaluable for later astronomers. Tycho’s measurements were the most accurate ever made until telescopes came on the scene.” As science historian John North states in his book Cosmos, “Much of his early work is reliable to three or four minutes of arc, and his later accuracy is often better than a minute of arc for star positions, and hardly much less for those of the planets. This was better, by a factor of five or even ten, than the level of accuracy of the best Eastern astronomers, even than that of Ulugh Beg’s observatory in Samarqand.”

Ulugh Beg (ca. 1394-1449) was the grandson of the brutal and influential Islamic conqueror Timur, often known as Tamerlane (1336-1405). His father had captured the city of Samarkand in Central Asia where Ulugh Beg proceeded to build an observatory. With improved instruments and careful observations, he made a new star catalog with unprecedented accuracy for its time and even corrected some errors in Ptolemy’s calculations.

The Holy Roman Emperor Rudolf II arguably contributed to the religious tensions that culminated in the destructive Thirty Years’ War (1618-48), but during his reign (1576-1612), Prague was a cultural center of East Central Europe, with a flowering of art and architecture. According to authors Robert Bideleux and Ian Jeffries, “Prague became one of the musical capitals of Europe, thanks to Rudolf’s patronage of orchestras, choirs and polyphonic music. Last but not least, he built up a major art collection, including works by Leonardo da Vinci, Raphael, Tintoretto, Titian, Brueghel, Dürer, Holbein and Cranach. Intellectually, Rudolf’s Prague was home to the Czech founder of meridian astronomy, Tadeas Hajek of Hajek, the German mathematician and astronomer Johannes Kepler and the Danish astronomer Tycho Brahe, while in medicine Jan Jessenius pioneered dissection at Prague University.”

Tycho Brahe’s patron died and the new king was less positive. In 1597 he left Denmark and in 1600 went to Prague to the post as Imperial Astronomer at the court of Rudolf II. The great German astronomer and mathematician Johannes Kepler (1571-1630) joined him there from Graz in Catholic-ruled Austria, where anti-Protestant policies were making his life difficult.

Kepler was a deeply religious Christian man who made many references to God in his works. At the University of Tübingen he had studied mathematics, Greek and Hebrew. Teaching was in Latin. He appears to have accepted quite early that the Copernican system was true, but through his work he was to discredit some ideas that Copernicus himself had maintained from the ancient Greeks, above all the notion that the planets move in perfect circles.

Although Tycho had a large mass of observational material available he had not developed completed theories and was not forthcoming with his data at first. When Kepler arrived to work with him, Brahe’s principal assistant was luckily observing Mars. Of the naked-eye planets, only Mercury and Mars have eccentricities large enough to make the departures of their orbits from perfect circles apparent from naked-eye observations, but Mercury is close to the Sun and difficult to observe. In essence, this means that Kepler’s discovery of the elliptical nature of planetary orbits could only have been made through a study of Mars.

After Tycho died in 1601, Kepler succeeded him as Imperial Mathematician. The process of calculating the orbit of Mars was immensely laborious but was completed around 1605. Meanwhile, Kepler made very valuable contributions to optical theory.

His discoveries were published in 1609 in Astronomia nova (New astronomy). It was a difficult book to read as he had not done a good job of highlighting his most important results. Kepler’s first planetary law stated that the planets move in elliptical paths with the Sun at one focus, not in perfect circles as had been assumed by the Greeks since Eudoxus of Cnidus. The second law stated that a line joining a planet to the Sun sweeps out equal areas in equal intervals of time as the planet describes its orbit. Kepler’s third law, published in 1619, stated that the square of the time of an orbit equals the cube of the mean distance from the Sun.

The Earth is in a stable orbit because its forward motion exactly counterbalances the gravitational pull of the Sun at this distance. As gravitational pull decreases with distance, so does orbital speed. Kepler’s laws imply that the closer a planet is to the Sun the faster it will move. Mercury, the innermost planet in our Solar System, has an average speed of 47.87 km/s and an orbital period of about 88 Earth days around the Sun. In Roman mythology, Mercury was the god of commerce and travel and the Roman counterpart of Hermes, the swift messenger of the Greek gods. Jupiter’s average distance from the Sun is over 778 million kilometers, more than five times that of the Earth (5.2 Astronomical Units). Its mean orbital speed is 13.07 km/s. Neptune, which is about 30 times as far from the Sun as is the Earth (30.1 AU), has an orbital period of almost 165 Earth years and a speed of merely 5.43 km/s.

According to James Evans, “The area law, which is equivalent to the principle of conservation of angular momentum, provided the crucial clue that the force exerted on a planet by the Sun is directed radially inward toward the body of the Sun itself, and not tangentially around the orbit, as Kepler and his contemporaries had supposed. The elliptical shape of the orbit and, even more directly, the harmonic law provided the clues that the attractive force exerted by the Sun on a planet varies as the inverse square of the distance between them. For the ancient Greeks, planetary astronomy had been a branch of mathematics. Kepler’s constant goal was to provide a physical basis for astronomy. In his Mathematical Principles of Natural Philosophy of 1687, Isaac Newton showed how to deduce Kepler’s laws of planetary motion from his own laws of motion and the law of universal gravitation: it was Newton who realized Kepler’s dream of making planetary astronomy into a branch of physics.”

Galileo Galilei introduced the telescope to astronomy and discovered the largest moons of Jupiter. Kepler wrote an enthusiastic reply to Galileo’s 1610 work the Sidereal Messenger. The discovery of a new astronomy resulted in the complete destruction of the old physics. The leaders of this development were men such as Galileo, René Descartes and Isaac Newton.

We should be careful about projecting a too modern understanding of the concept of “science” onto the activities of ancient scholars. As Edward Grant reminds us, “Science in the ancient world was a tenuous and ephemeral matter. Most people were indifferent to it, and its impact was meager. It was a very small number of Greek thinkers who laid the foundations for what would eventually become modern science. Of that small number, a few were especially brilliant and contributed monumentally to the advancement of science.”

This is not said to dismiss them or downplay the significance of what they did. On the contrary, we should be all the more impressed by their achievements given how few they were and the limited resources they had at their disposal. Nevertheless, while the ancient Greek contribution was substantial and important it was only one of the major components of what would become modern science when this was finally created in early modern Europe.

Hipparchus, with his star catalog from the second century BC, was the first to introduce a system for measuring the brightness of stars according to how bright they appear as seen by observers here on Earth, with six levels of magnitude and the brightest ones in class 1. Today’s system essentially follows the same logic. This system is called apparent magnitudes, as opposed to absolute magnitude which measures an object’s intrinsic brightness and is directly related to the star’s energy output, or luminosity, regardless of how far away from us it is. If Vega is assigned magnitude zero then Sirius, the brightest star in the night sky, has an apparent magnitude of —1.47; Venus at its brightest is about —4.5, the Full Moon —12.6 and the Sun —26.7. Uranus at most reaches around 5.5, barely within the limit for what can be spotted through naked-eye observations. The binocular limit is about 10, which means that you can see Neptune at 8, yet Pluto at 14 is too faint to be seen without more powerful telescopes.

Aristotle’s ideas were not always treated uncritically in ancient times. Strato of Lampsacus (ca. 335-269 BC), the third head of the Lyceum after Aristotle himself and his successor Theophrastus, criticized Aristotle’s lack of attention to the speeding up and slowing down of bodies as they begin or end their motion. The sixth-century AD Eastern Roman natural philosopher John Philoponus added to this debate regarding Aristotle’s theories of motion.

Galileo’s dynamics and kinematics of motion drew substantially from fourteenth-century developments at the Universities of Oxford and Paris. Nicole Oresme did notable work on projectile motion and was critical of some of Aristotle’s views on the subject. He was taught by the highly influential French priest and philosopher Jean Buridan (ca. 1300-1358) at the University of Paris in the 1340s. Buridan’s views on logic were close to those of William of Ockham. The impetus theory of Philoponus was reaffirmed by Buridan and Oresme.

In the eyes of David C. Lindberg, a few breakthroughs often attributed to the seventeenth century had ancient or medieval roots. In some cases we can find that “revolutionary achievements in many disciplines were built on medieval foundations and out of resources provided by the classical tradition. Revolution does not demand total rupture with the past.”

The mechanical philosophy, a centerpiece of the Scientific Revolution, was aided by a revival of Epicurean atomism from ancient Greece which had been preserved through the writings of Roman authors, passed down with the Classical tradition and Christianized. It was employed in the seventeenth century by men such as Pierre Gassendi, Robert Boyle and Isaac Newton.

Another important branch of science where there was some continuity was optics. Robert Grosseteste (ca. 1170-1253), born in Suffolk, England, was a talented scholastic philosopher. Before 1230 he was probably teaching at the very young University of Oxford. He composed short works regarding optics and experimented with mirrors. In 1235, he was appointed Bishop of Lincoln. Prior to that, he had served as a theology lecturer to the Franciscans. “ In his investigations of rainbows, comets, and other optical phenomena, he notably made use of both observational data and mathematical formulations. Moreover, Grosseteste was an early proponent of the need for experimental support of scientific theories (a view he clearly passed on to Bacon during his tutelage), and carried out numerous experiments with mirrors and lenses.” This deviation from traditional Aristotelian philosophy has earned him, with some justification, the reputation of being a pre-modern supporter of the scientific method.

Grosseteste’s student Roger Bacon (ca. 1220-1292) doesn’t deserve the label as the “founder of experimental science” that is sometimes bestowed upon him, but he was an influential advocate of gathering empirical evidence in the sciences and he practiced what he preached.

Another medieval pioneer from the thirteenth century was the talented French natural philosopher Peter Peregrinus of Maricourt, who used a simple version of the experimental method in his work Epistola de magnete from 1269. This was the first extant written account of the polarity of magnets and proved exceedingly popular in the generations that followed.

One of those who quoted it was the English scholar Thomas Bradwardine (ca. 1290-1349), who was educated at Merton College, Oxford. Bradwardine was a noted mathematician as well as theologian and died in London of the plague during the Black Death after having served briefly as Archbishop of Canterbury. Later in the fourteenth century the author Geoffrey Chaucer would rank him next to Saint Augustine and Boethius in importance.

Mr. Lindberg has successfully demonstrated that while Kepler in the early 1600s formulated the first modern optical theory, he built it on the foundations of optical studies dating back to Ptolemy and developed further by medieval scholars such as Alhazen, Kamal al-Din, Theodoric of Freiberg, Robert Grosseteste and Roger Bacon. All of these men had at least some rudimentary understanding of the experimental method, even Ptolemy in the Roman era, yet this did not then evolve into a methodical use of it or became widely adopted by others.

It is almost certainly possible to find isolated cases where Chinese, Indian or other thinkers employed unsophisticated versions of the scientific method; what was entirely novel in seventeenth century Europe was a sustained program of experimentation, successfully promoted by influential men such as the English philosopher and writer Francis Bacon. That was new and revolutionary. David Lindberg writes in The Beginnings of Western Science:

“Peter Peregrinus of Maricourt manipulated magnets in order to gain an understanding of their properties and behavior — discoveries that anticipated many of those that would subsequently be made in the seventeenth century by William Gilbert, often identified as one of the founders of experimental science. And who could deny the status of experimental scientist to the thirteenth-century Franciscan friar Paul of Taranto, who initiated an alchemical tradition characterized methodologically by laboratory manipulation of substances in the attempt to discover the pathway to transmutation?…If all of this is true, what credit is left for Francis Bacon (1561-1626), popularly celebrated as the founder (or a founder) of experimental science? This Bacon (no descendent of Roger) argued, in books filled with references to empiricism and experiment, for the experimental interrogation of nature. However, what he and the Baconian tradition of the seventeenth century gave us was not a new method of experiment, but a new rhetoric of experiment, coupled with full exploitation of the possibilities of experiment in programs of scientific investigation.”

Edward Grant, on the other hand, chooses to emphasize that “Science in the late ancient and medieval periods, was, however, radically different from modern science. Although some interesting experiments were carried out, they were relatively rare occurrences and certainly did not constitute a recognized aspect of scientific activity. Few claims were tested objectively. The experimental method did not yet exist. The mathematical sciences, however, were presented with the same kind of vigor as a modern treatise in mathematical physics.”

In mathematics it is not at all difficult to find certain elements of continuity. Euclid’s Elements from around 300 BC incorporated even older Greek, Egyptian and Babylonian geometry and has been studied by Western students of mathematics until the present day.

Before 200 BC, the Conics by the great Hellenistic Greek geometer Apollonius of Perga introduced ideas and terminology that influenced later scholars from Ptolemy to Newton. As John North states, “He did for the geometry of conic sections (parabola, hyperbola, line-pair, circle, and ellipse) what Euclid had done for elementary geometry. He set out his own work, and much of that done by his predecessors, in a strikingly logical way. He also showed how to generate the curves using methods strongly reminiscent of those used in modern algebraic geometry. Those methods were to prove enormously important to astronomy, in the century of Kepler, Newton, and Halley — who studied Apollonius’s text closely.”

According to author Stephen Gaukroger in his book The Emergence of a Scientific Culture it is wrong to compare the totally unprecedented efforts of the Scientific Revolution in early modern Europe, which is the Scientific Revolution, to the more limited creative periods in other civilizations and time periods. The difference is not merely one in degree, but in kind:

“Scientific developments in the classical and Hellenistic worlds, China, the medieval Islamic world, and medieval Paris and Oxford, share a distinctive feature. They each exhibit a pattern of slow, irregular, intermittent growth, alternating with substantial periods of stagnation, in which interest shifts to political, economic, technological, moral, or other questions. Science is just one of a number of activities in the culture, and attention devoted to it changes in the same way attention devoted to the other features may change, with the result that there is competition for intellectual resources within an overall balance of interests in the culture. The ‘Scientific Revolution’ of the early-modern West breaks with the boom/bust pattern of all other scientific cultures, and what emerges is the uninterrupted and cumulative growth that constitutes the general rule for scientific development in the West since that time….This form of scientific development is exceptional and anomalous. The question is, then, not why the Scientific Revolution didn’t occur in any of the other cases of rich, innovative scientific cultures, but why it occurred in the West.”

De Magnete (“On the Magnet”) from the year 1600, written by the English physician and natural philosopher William Gilbert (1544-1603), became the standard work on electrical and magnetic phenomena at a time when Western European nations were engaged in long sea voyages and needed knowledge about the workings of the magnetic compass. Gilbert likened the polarity of the magnet to the polarity of the Earth itself. His work included descriptions of some of his own experiments as well as data that had been obtained by others. Galileo was very interested in Gilbert’s magnetic researches as well as his experimental methodology.

The Italian natural philosopher, astronomer and instrument maker Galileo Galilei (1564-1642) was educated at home and with monks before he enrolled at the university in his native city of Pisa in 1581. He left without a degree, but continued to do independent research. He had to make a living by teaching and tried to generate additional income by inventing various devices, among them an early thermometer which was only a limited success. He was appointed professor of mathematics at Pisa in 1589. From 1592 to 1610 he held a similar position at Padua. During these years he studied falling bodies but did not publish his findings at that time. Had Galileo died in the early 1600s he would have been virtually unknown today.

Whether he literally dropped weights from the Leaning Tower in Pisa, as legend has it, is debatable, but the Flemish scholar Simon Stevin had conducted similar experiments in 1586. His results were published and may have been known to Galileo. Galileo demonstrated that heavy and light objects, aside from the effects of air resistance, fall at the same rate. Aristotle claimed that heavy objects fall more rapidly than lighter ones. Galileo showed that a falling object increases its speed in a steady fashion, i.e. undergoes uniform acceleration. At sea level on our planet this gravitational acceleration roughly equals 9.8 m/s² for all objects. He established the principle of inertia whereby a moving object in the absence of external forces will continue to move with undiminished speed. Aristotle claimed that it would slow down.

What set Galileo apart from most scholars of ancient and medieval times and clearly separated him from Copernicus a few decades earlier was that he embraced the philosophy of the “experiment” as a controlled situation in which specific phenomena could be produced and studied at will. This was, as we have seen, not necessarily an entirely new idea, but it was taken up with a new enthusiasm in the late 1500s and early 1600s. A greater emphasis on empiricism and experiment could be detected from Italy via the Low Countries to England.

William Harvey (1578-1657), a physician from Elizabethan England who studied at the University of Padua after 1598 under Galileo’s anatomist colleague Hieronymus Fabricius (1537-1619), became the first person to correctly describe the full circulation of the blood in the human body. Jan Baptist van Helmont (1579-1644), a Brussels-born Flemish physician, in the early 1600s carried out a famous experiment by growing a tree in a pot. Through careful measurements he deduced that its increased mass came from the water, which was partly correct; it also came from atmospheric carbon dioxide and sunlight through photosynthesis.

After 1609 Galileo became heavily involved in astronomy with the newly invented telescope. The sensational observations he published in 1610, including his discovery of the four largest moons of Jupiter, brought him instant fame. His Dialogue Concerning the Two Chief World Systems from 1632 with its favorable view of the Copernican system got him into the famous controversy with the Roman Inquisition. While under house arrest near Florence he completed his final masterwork, Two New Sciences (1638), which was published in Leiden in the Netherlands. Here he summarized investigations he had conducted decades earlier regarding the strength of materials and the motion of objects and anticipated Newton’s laws of motion.

Galileo was a firm believer in the application of mathematics to physics, not only for heavenly objects such as planets but for the study of motion of everyday objects here on Earth. It was his instinct for mathematical modeling of physical phenomena combined with his groundbreaking experiments and systematic use of the scientific method that earned him the well-deserved reputation he now enjoys for being a, if not the, founder of modern physics.

There is an immense gulf between societies in which science was the domain of a handful of wise men and our society with thousands of specialists seeking to contribute to a coherent understanding of all natural phenomena. Modern Europe contributed a unique institutional invention in world history: a large, organized body of scholars seeking explanations of all natural phenomena by a common method based on observation, experiment and reason. Europeans were aided in this by better scientific instruments, physical as well as mathematical ones. Authors Nathan Rosenberg and L.E. Birdzell Jr. explain in How The West Grew Rich:

“What made the difference to the creation of organized science was that the experimental method was adopted by a number of researchers, and their common method united them in a community of working scientists. Post-Galilean natural science could specialize and departmentalize into physics, astronomy, chemistry, geology, biology, and a host of narrower specialties because all of them shared a common method of determining scientific truth. A geologist or biologist could use the teachings of physics or chemistry in geological or biological research without feeling the need (or even the possibility) of checking their validity. The general acceptance of the experimental method made it possible for hundreds and even thousands of specialists to build the results of their individual research into a single store of information, usable across all sciences. The introduction of the printing press greatly speeded the cumulation of this body of knowledge….Thus the West, alone among the societies of which we have knowledge, succeeded in getting a large number of scientists, specialized by different disciplines, to cooperate in creating an immense body of tested and organized knowledge whose reliability could be accepted by all scientists.”

In his 1996 book The Scientific Revolution, professor of sociology Steven Shapin questions whether there was such a phenomenon at all. Most historians hold that a new pattern did emerge during the seventeenth century. Scholars questioned and occasionally ridiculed some of the ancient wisdom that had been inherited from previous ages. Many “new men” were critical of the established universities, which were seen as having invested too much faith in past authority, particularly in Aristotle. This was in sharp contrast to the Renaissance, which was preoccupied with Antiquity and had relished ancient knowledge; the older, the better.

A new pattern of organized science based on experiment emerged, less tied to Aristotelian natural philosophy and open for reports from non-academics. This development found its institutional basis in the scientific societies. During the 1660s the Royal Society of London for the Improvement of Natural Knowledge, the longest continuously enduring scientific society in the world often known simply as the Royal Society, was created. Similar set-ups followed from Berlin to St. Petersburg. European scholars developed networks and created increasingly sophisticated descriptions of their experiments for others to trust in or duplicate and verify for themselves. Here we find the seeds of what would gradually evolve into scientific journals.

The universities eventually caught up with this, too. The influential Dutch professors Willem Gravesande (1688-1742) and Pieter van Musschenbroek (1692-1761), the inventor of the Leiden jar, at Leiden University gave the first sustained university courses in natural philosophy illustrated with experiments. By 1750 many professors were introducing experiments into their courses. Charles-Augustin de Coulomb used long, thin magnetic needles with well-defines poles to establish the law of magnetic force. Antoine Lavoisier used his self-made apparatus for experiments on gases, respiration and heat in the 1770s and 80s.

The word “scientist” was not seriously used much before 1840, and not widely used until the twentieth century. “Man of science,” “thinker” or “scholar” should be considered preferred labels for those who worked with natural philosophy and what we call science prior to this. By the mid-eighteenth century, probably no more than 300 people in the world could be classified as scientists. By the year 1800 there were perhaps a thousand, by the mid-nineteenth century 10,000 and by 1900 maybe 100,000. The overwhelming majority of these were still Europeans or people of European origins. The European population itself grew rapidly at the same time, but the percentage of scientists grew even faster. Science during this period finally made the transition from being a gentleman’s hobby to being a well-populated profession.

Algebra in the Islamic world had been entirely rhetorical, with no symbols for the unknown. Everything was written out in words. In fifteenth century Italy some of the abacists began to use abbreviations for unknowns. Innovations spread faster after the introduction of printing.

The brilliant French algebraist François Viète (1540-1603), Latinized as Franciscus Vieta, was a lawyer by profession. He developed the first systematic use of algebraic symbols. In their online biography, J. J. O’Connor and E. F. Robertson state that “François Viète was a French amateur mathematician and astronomer who introduced the first systematic algebraic notation in his book In artem analyticam isagoge. He was also involved in deciphering codes.” He demonstrated the value of symbols to represent quantities. “While Viète had come only part way toward modern symbolism, the crucial step of allowing letters to stand for numerical constants enabled him to break away from the style of examples and verbal algorithms of his predecessors. He could now treat general examples rather than specific ones and give formulas rather than rules.” He was not the first person ever to employ letters of the alphabet to denote numerals, but after 1590 he popularized this concept.

Omar Khayyam had seen a relationship between geometry and algebra, but the decisive breakthrough came with Descartes and Fermat in France. Just like Viète, Pierre de Fermat in Toulouse was a busy lawyer. His mathematical work was communicated to friends by means of letters, often with little in the way of formal proof. He was fluent in several languages, among them Italian, Greek and Latin, and made major contributions to geometric optics, modern number theory, probability theory and analytic geometry. According to Victor J. Katz, “Descartes, along with Thomas Harriot and Albert Girard, reworked some of Viète’s algebraic ideas into a theory of equations….Fermat also was responsible for the first new work in number theory since Leonardo of Pisa, while Pascal, along with Girard Desargues, made some of the earliest contributions to the subject of projective geometry.”

In the margin of a Latin translation of the Arithmetica by Diophantus, Fermat claimed to have found a beautiful theorem which became famous and known as Fermat’s Last Theorem. Mathematicians struggled with it for centuries until finally in 1995 the English mathematician Andrew Wiles (born 1953) appeared to have solved it. There are some historians of mathematics who question whether Fermat ever had the proof that he claimed to have.

The French philosopher and mathematician René Descartes was a key figure of the Scientific Revolution. He had studied at a Jesuit college in Anjou: Classical subjects, Aristotle and mathematics as well as poetry, riding and fencing. He enlisted in a military school and in the 1620s travelled through Europe, ending up in the Netherlands to enjoy the liberty of a society where he could be an original thinker without fear of the Catholic Counter-Reformation. He later went for brief visits back to France. He was not alone in doing this. The French-born mathematician Albert Girard (1595-1632), originally a lute player, was a Calvinist who fled to the Netherlands as a religious refugee. Descartes contributed substantially to optics and meteorology and devised a universal method of deductive reasoning based on mathematics, his most famous statement being Cogito ergo sum (I think, therefore I am).

Viète had used vowels for the unknowns and consonants for known quantities. Descartes introduced the convention we use today of employing letters at the beginning of the alphabet (a, b, c etc.) to represent known quantities and letters at the end of the alphabet to represent unknown ones (x, y, z). He created the familiar Cartesian coordinate system with axes labeled x, y and z, which made it possible to express positions in two or three dimensions. This was later extended to include negative numbers. Algebra could now be linked with geometry, a development which had repercussions right down to Einstein’s general theory of relativity.

A few ideas of coordinate geometry had been anticipated by Nicole Oresme in fourteenth century France, but his work did not explicitly link algebra and geometry. Coordinates had been used in Antiquity by Hipparchus and Ptolemy in astronomy and geography, but historians give René Descartes and Pierre de Fermat shared credit for the invention of analytic geometry. Traditionally, algebra had been treated as completely separate from geometry. This tradition began breaking down with the work of François Viète in the late sixteenth century. Both Fermat and Descartes built on Viète’s ideas. Author Marvin Jay Greenberg writes:

“Descartes was the first to publish in 1637, as an appendix (La Géométrie, in three parts) to his very influential Discourse on Method, his philosophical method for finding and recognizing correct knowledge. Fermat never did publish his work; instead, he communicated his results in private letters to a few colleagues, and his work was made public only in 1679, fourteen years after he died. Curiously, although both these men were outstanding mathematicians, mathematics was not their profession. Fermat was a jurist who did mathematics as a hobby. He is best known for his work in number theory….Fermat also discovered the basic idea of the differential calculus before Newton and Leibniz. Descartes contributed to other sciences besides mathematics, but he was primarily a philosopher whose writings had a great impact on the way educated people viewed the world. Both men initially introduced their algebraic methods in order to solve problems from classical Greek geometry, recognizing that the new methods had great potential to solve other problems. Their successors over many decades realized that potential. Descartes’ stated goal was to provide general methods, using algebra, to ‘solve any problem in geometry.’“

Gambling is older than civilization itself. Ancient peoples would use fruit stones, sea shells or pebbles to get random results for games or fortune telling. East Asian dice evolved into Chinese dominoes. The heel bone of a running animal such as deer and sheep has been found, sometimes polished and engraved, in sites from Mesopotamia and Egypt. The ancient Greeks and Romans used sheep anklebones as well as the more developed cubical spotted dice. Yet no senior mathematician before Renaissance Europe apparently took an interest in dice. Italian scholars like Galileo Galilei and Gerolamo Cardano were the first people we know of who showed a serious interest in attempting to apply mathematics to games of chance.

The French natural and religious philosopher Blaise Pascal (1623-1662) was a highly influential mathematician. Descartes visited Pascal and the two argued about the existence of a vacuum, which Descartes did not believe in. Using the newly invented barometer, Pascal in 1648 observed that the pressure of the atmosphere decreases with height and deduced that a vacuum exists above the Earth’s atmosphere. In mathematics he is especially famous as the co-founder of probability theory, a branch of mathematics which is of vital importance to modern physics and economics. This started in 1654 when he corresponded with Fermat about problems related to a game of dice. Blaise Pascal famously underwent a profound religious experience and pledged the rest of his life to the service of Christianity. He argued his case for the rational belief in God by stating that if God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.

The Dutch astronomer Christiaan Huygens (1629-1695) learnt of the work on probability carried out by Pascal and Fermat and in 1657 wrote a small work De Ratiociniis in Ludo Aleae on the calculus of probabilities, which was the first printed text on the subject.

Abraham de Moivre (1667-1754) was born in the Champagne region, but as a Protestant he left Catholic France for England after the revocation of the Edict of Nantes in 1685. He became a member of the Royal Society in London and friendly with Edmond Halley and Isaac Newton. His major work was The Doctrine of Chances, published in 1718 with expanded editions in 1738 and 1756. This text was much more detailed than the work on probability by Huygens, partly because of the rapid general advances in European mathematics since then. De Moivre gives not only general rules but also detailed applications of these rules.

The Russian Andrey Kolmogorov (1903-1987) is above all remembered for his brilliant papers published on probability, starting with About the Analytical Methods in Probability Theory (in German) in 1931, the year he became a professor at Moscow University. In 1933 he published his magnum opus Foundations of the Theory of Probability. In over 300 research papers and books he covered almost every area of mathematics except number theory. A man of wide-ranging interests he also did serious research into the physics of turbulence, founded Kolmogorov complexity theory and was fascinated with the poetry of Alexander Pushkin. He devised an imaginative secondary school curriculum featuring mathematics, European Classical music, hiking, poetry and activities intended to promote Greek and Renaissance values to Soviet students rather than Marxist indoctrination. This made him a few enemies.

Jeremiah Horrocks (ca. 1618-1641), an astronomer born near Liverpool, England, was one of very few people to predict a transit of Venus which he successfully observed through his small telescope in 1639. Based on these observations and aided by Kepler’s laws of planetary motions, Horrocks estimated the distance between the Earth and the Sun to be 95 million kilometers. This was too short, but it represented a major step in the right direction and constituted the best scientific estimate anybody in the world had made until that time.

Kepler’s laws aided other European astronomers, too. As author John Gribbin writes in his book The Scientists, “In 1671 the French astronomer Jean Richer (1630-1696) travelled to Cayenne, in French Guiana, where he made observations of the position of Mars against the background of ‘fixed’ stars at the same time that his colleague in Paris, the Italian-born Giovanni Cassini (1625-1712), made similar observations. This made it possible to work out the distance to Mars, and, by combining this with Kepler’s laws of planetary motion, to calculate the distance from the Earth (or any other planet in the Solar System) to the Sun. The figure Cassini came up with for the Sun-Earth distance, 140 million km, was only 7 percent less than the accepted modern value (149.6 million km), and gave the first accurate indication of the scale of the Solar System. Similar studies of Venus during the transits of 1761 and 1769 (predicted by Halley) led to an improved estimate of the Sun-Earth distance (known as the Astronomical Unit, or AU) of 153 million km, close enough to the modern value for us to leave the later improvements in the measurements as fine tuning, and accept that by the end of the eighteenth century astronomers had a very good idea of the scale of the Solar System.”

Giovanni Cassini, a professor of astronomy at the University of Bologna, Italy, was invited by King Louis XIV of France to join the recently formed Académie Royale des Sciences and assumed the directorship of the Paris Observatory after it was completed in 1671. In the 1670s and 80s he discovered four of Saturn’s moons: Iapetus, Rhea, Tethys and Dione. He was able to measure Jupiter’s and Mars’s rotational periods and determined the parallax of Mars, which allowed the first realistic calculation of the distance to Mars and the Earth-Sun distance. He went on to establish a Cassini dynasty which dominated the Paris Observatory until the French Revolution. Giovanni Cassini is generally credited, perhaps at the same time as the Englishman Robert Hooke, with the discovery of the Great Red Spot on Jupiter around 1665.

Jupiter rotates about once every ten hours and its clouds are in perpetual motion. Its belts provide a framework for turbulent and colorful swirling cloud patterns and rotating storms similar in structure to the hurricanes or cyclones we are familiar with. Its Great Red Spot is a hurricane-like storm of swirling gases larger than the Earth. In 1690, Cassini noticed that the speeds of Jupiter’s clouds vary with latitude, an effect called differential rotation. Near the poles the rotation period of its atmosphere is longer than at the equator. Furthermore, clouds at different latitudes circulate in opposite directions, creating visually fascinating patterns.

Isaac Newton, perhaps the greatest scientist the world has 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. 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. His mother wanted him to be a farmer, but he was terrible at it. His maternal uncle, a clergyman who had studied at Cambridge, persuaded his reluctant sister that the talented boy Isaac should attend the university. 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 gifted 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 talented English mathematician who introduced the symbol 8 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 of Descartes and Fermat was a necessary precondition for the invention of integral calculus. Fermat himself took certain steps in that direction. Other notable pioneers in the field include the Scottish mathematician James Gregory, the French scholar Gilles de Roberval (1602-1675) and Bonaventura Cavalieri (1598-1647). Cavalieri was born in Milan, Italy. Through a lecturer at Pisa he was initiated in the study of geometry. He developed a general rule for the focal length of lenses and constructed a hydraulic pump for his monastery. He wrote dozens of letters to Galileo and in 1629, with Galileo’s help, secured the chair of mathematics at the University of Bologna. Cavalieri 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 notable 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 view among most mathematical historians is currently 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.

Are you an author? Learn about Author Central 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. See search results for this author James Gleick (Author) Find all the books, read about the author, and more.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. 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 English polymath Robert Hooke (1635-1703), a brilliant instrument maker and technician but not an equally gifted mathematician, “became Newton’s goad, nemesis, tormentor, and victim.” Both of them were great scholars who could also be quarrelsome men. 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. 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 logically 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 been allowed to see some of his promising ideas, Newton began to develop his work in greater detail. In 1687 he 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. Science historians James E. McClellan and Harold Dorn elaborate:

“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, where France was to play a leading role. He corresponded with prominent men of science such as Christiaan Huygens, Ole Rømer and the Dutch mathematician Jan Hudde (1628-1704), who also served as a city council member in Amsterdam and worked with the Dutch Jewish philosopher and lens grinder Baruch Spinoza on the manufacture 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.”

Isaac Newton 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. He 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 president of the Royal Society.

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. He was buried in Westminster Abbey. The visiting French writer calling himself Voltaire was amazed by his kingly funeral. 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 universe 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.’“

Mathematics and Astronomy after Newton

The English astronomer and mathematician Edmond Halley (1656-1742) became the first major scholar to work squarely within the Newtonian school of thought. Born into a prosperous London family, he made astronomical observations at Oxford and was inspired by Flamsteed at the newly established Royal Observatory at Greenwich. In 1676 he sailed for the Atlantic 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 in world history. Halley found the time to participate in non-astronomical activities, too, to create an improved diving bell, study magnetic variation and serve as a sea captain. He enhanced our understanding of trade winds, tides, 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) was born in Hamburg, Germany, and studied mathematics and astronomy at the University of Göttingen under the genius Carl Friedrich Gauss. During Encke ‘s directorship the work at the Berlin Observatory concentrated on the calculation of the orbits of asteroids. Encke followed a suggestion by the French prolific comet discoverer Jean-Louis Pons (1761-1831), who suspected that a comet he had spotted was the same as one seen by him in 1805. Encke sent calculations to Gauss, Olbers and Bessel and predicted its return for 1822. This comet is known as Encke’s Comet, but Encke himself always referred to it as “Pons’ Comet.” Its orbital period of just over 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. Studies of comet tails eventually aided the prediction in the twentieth century 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 possessed a large mass of observational data, yet never used this 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 assume that his theory of universal gravity was somehow the logical and inevitable conclusion of telescopic astronomy. Yet this is not at all the case. Kepler’s initial work was based on Brahe’s 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 placed less emphasis than the Western one on law, be that 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 people that used telescopes would have generated much new knowledge with the device, but not necessarily a law of universal gravitation.

The Chinese made promising beginnings in the secular observation of nature. Like their Korean and Japanese neighbors they were great practical engineers and entrepreneurs, but they never completed an ideological framework for the scientific project comparable to Greek natural philosophy, or developed an organized program for promoting the scientific method.› Visit Amazon’s Edward Grant Page

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 philosophy 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 knowledge was further brought to Japan in the eighteenth century and eventually supplemented by translations via Dutch traders there.

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.

The Dutch astronomer Christiaan Huygens argued that if Sirius is as bright as the Sun it must be 27,664 times further away. By 1685, Newton used a new technique devised by the prominent Scottish astronomer and mathematician James Gregory (1638-1675) to show that the stars must lie at much greater distances from the Sun than had previously been supposed.

Gregory was born near the city of Aberdeen in Scotland and studied there and well as at the University of Padua in Italy. He returned to London in 1668 and was soon appointed to the University of St. Andrews in Scotland and from 1674 to the University of Edinburgh as a professor of mathematics. As a mathematician he contributed to the development of calculus. In Optica Promota (“The Advance of Optics”) from 1663, Gregory introduced photometric methods to estimate (but not directly measure) distances to the stars and a description of a practical reflecting telescope known as the Gregorian telescope. He also pointed out the possible use of transits of Venus and Mercury to determine the distance to the Sun, something which was successfully done after his death.

Isaac Newton had to show that the stars were so far away that their gravitational attractions on one another were minimal. This was important to him as he wondered why the world does not collapse on itself, under gravity. We currently believe that the universe is expanding, but this was demonstrated by Western astronomers in the twentieth century, long after Newton died.

Some cosmologists have suggested the possibility that the universe may in fact collapse at some point in the future due to gravity and end in a singularity. Although this Big Crunch, the opposite of the Big Bang, is one possible, hypothetical fate of the universe it is not the only conceivable future and according to observational evidence not necessarily the most likely one. Indeed, supernova observations from the late 1990s and afterward could indicate that the expansion of the universe is not being slowed down by gravity but is rather accelerating.

The English astronomer James Bradley (1693-1762) discovered stellar aberration while looking for stellar parallax. Since the diameter of the Earth’s orbit is roughly 300 million kilometers, nearby stars should appear to move compared to more distant ones over a six-month period as the Earth orbits the Sun. They do, but because the stars are extremely far away from us this effect is difficult to measure. While studying the star Gamma Draconis, Bradley in 1728 succeeded in detecting a slight annual variation in the apparent positions of stars, but in the opposite direction from what was expected. This is caused by the aberration of light, a result of the finite speed of light and the forward movement of the Earth in its orbit around the Sun. This finding provided the first direct evidence for the Copernican theory.

The fact that the speed of light is finite had been known in Europe since the work of the astronomer Ole Rømer from Denmark in 1676. Combined with an increasingly accurate knowledge of the Earth-Sun distance, James Bradley could correctly calculate that it takes sunlight more than eight minutes to travel from the Sun to observers here on the Earth.

The German scholar Friedrich Wilhelm Bessel (1784-1846) was born in Minden and despite his limited education became one of the leading astronomers of his generation, a rigorous observer as well as a capable mathematician. He was a contemporary of the mathematician Carl Friedrich Gauss and was brought into the field of astronomy in 1804 when he contacted the great German astronomer Wilhelm Olbers concerning a paper he had written on Halley’s Comet. In 1836 he published a theory which stated that comets consist of volatile matter.

Bessel’s contributions covered most of contemporary astronomy, especially precision measurements. Aided by better instruments plus his own personal skills he won the race to become the first person to measure stellar parallax and by implication the distance to a star other than the Sun. He spent years observing and accurately pinpointing the positions of thousands of stars, systematizing the observations of James Bradley. His efforts culminated in 1838 when he calculated the distance to the star 61 Cygni as more than 10 light-years. This has later been modified to about 11.4 light-years. Bessel chose a star with unusually large proper motion, correctly deducing that it was probably close to us by astronomical standards.

The Baltic German astronomer Friedrich Georg Wilhelm von Struve (1793-1864) and the Scottish astronomer Thomas Henderson (1798-1844) independently measured stellar parallax at almost the same time, but Bessel’s measurements were most convincing. Henderson, from Dundee, Scotland, measured the distance to Alpha Centauri, the nearest stellar system to our own at 4.4 light-years away from the Sun. He was also appointed the first Astronomer Royal for Scotland. Struve was the first in a line of four generations of distinguished astronomers from his family. In addition to his work on stellar parallax he is especially remembered for his observations of double stars and for contributions to surveying; the Struve Geodetic Arc, stretching from Hammerfest in northern Norway to the Black Sea, is named after him.

During his many years at the Königsberg Observatory in Prussia, Bessel’s students included the German-born astronomer Friedrich Wilhelm August Argelander (1799-1875), who had a German mother and a Finnish father. Argelander established the study of variable stars as an independent branch of astronomy and was appointed director of the Turku Observatory in southwest Finland in 1823 and of the Helsinki Observatory in the same country in 1832. He then moved to Bonn, Germany and published an extensive star catalog there in the 1850s.

The Scottish astronomer David Gill (1843-1914) was born in Aberdeen, Scotland and educated at the University of Aberdeen, but he spent many years of his career in South Africa where he also made geodetic surveys. He was trained and worked as a watchmaker, and his interest in timekeeping led to an interest in astronomy. Gill used the parallax of Mars to re-determine the distance to the Sun with such precision that his value was used for almanacs until 1968. He carried out observations of stellar parallax to measure distances to other stars.

The Hipparcos satellite, an acronym for High Precision Parallax Collecting Satellite and named for Hipparchus, of the European Space Agency (ESA) in the early 1990s carried out measurements of star positions and parallaxes. Astronomers used it to measure the distances to over 2.5 million of the nearest stars up to 500 light-years (150 parsecs) away. ESA’s Gaia Mission will create even more accurate three-dimensional maps over many stars after 2012.

The Scottish theologian and mathematician John Napier (1550-1617) studied at the University of St Andrews, the oldest university in Scotland, and spent several years in Continental Europe. He invented logarithms, a mathematical device which simplified and speeded up manual calculations and aided the work of scholars for centuries. This inspired the invention of the slide rule during the 1600s, excellent for multiplication and division and the calculation of powers and roots. The Apollo lunar program in the USA as late as the 1970s kept slide rules as backups for their electronic computers. Napier also improved the decimal notation introduced by the Flemish mathematician Simon Stevin. Michael Stifel (1486-1567) invented logarithms independently, using a different approach. He was a monk at the Augustinian monastery at Esslingen, Germany, who became an early Protestant follower of Martin Luther.

The Swiss mathematician, clockmaker and astronomer Joost Bürgi (1552-1632) also independently invented a system of logarithms which he published in 1620, but Napier has the priority due to his publication in 1614 of his Mirifici Logarithmorum Canonis Descriptio. Napier’s system was extended and improved by his admirer Henry Briggs (1561-1630), an English professor of geometry at Gresham College in London who visited Napier at Edinburgh in 1615. Briggs is especially remembered for his publication of tables of logarithms to the base 10, first one in 1617 and later the Arithmetica Logarithmica in 1624.

Adriaan Vlacq (1600-1667), a book publisher born in Gouda in the Netherlands, extended the earlier work of Briggs and in 1628 published the first full table of logs from 1 to 100,000, calculated to ten places. The Slovenian mathematician Jurij Vega (1754-1802) attended school in Ljubljana and became professor of mathematics at the Austrian Imperial Artillery School in Vienna. As an artillery officer he fought against the Muslim Turks near Belgrade in 1788, but he is best known for his accurate tables of logarithms based on those of Vlacq.

Basel in Switzerland had been a free city and a center of learning for centuries. The sciences flourished there as it did in the Dutch and Flemish cities. The Bernoulli merchant family were originally religious refugees of the Protestant faith fleeing from persecution by the Spanish rulers of the southern Netherlands who had come to Basel from Antwerp via Amsterdam.

The Bernoullis were to become the world’s most successful mathematical family. The founding member and arguably also the most gifted representative of this famous scientific family was the brilliant mathematician Jacob Bernoulli (1654-1705), whose father was an important citizen of Basel. Jacob Bernoulli was one of the early students of probability theory, on which subject he wrote the Ars Conjectandi (The Art of Conjecturing), published posthumously by his nephew in 1713. This work contains the theorem of Bernoulli and the Bernoulli numbers, which are among the most interesting number sequences in mathematics, particularly important in number theory. As author Dirk J. Struik states:

“Indeed it is difficult to find in the whole history of science a family with a more distinguished record. This record begins with two mathematicians, Jakob (James, Jacques) and Johann (John, Jean) Bernoulli. Jakob studied theology, Johann studied medicine; but when Leibniz’s papers in the Acta eruditorum appeared both men decided to become mathematicians. They became the first important pupils of Leibniz. In 1687 Jakob accepted the chair of mathematics at Basel University, where he taught until his death in 1705. In 1695 Johann became professor at Groningen; upon his older brother’s death he succeeded him in the chair at Basel, where he remained for forty-three more years. Jakob began his correspondence with Leibniz in 1687. Then, in a constant exchange of ideas with Leibniz and with each other — often with bitter rivalry between them — the two brothers began to discover the treasures contained in Leibniz’s pioneering venture. The list of their results is long and includes not only much of the material now contained in our elementary texts on differential and integral calculus, but also material on the integration of many ordinary differential equations.”

Jacob’s younger brother Johann Bernoulli (1667-1748) was born and died in Basel, but spent some time teaching at the University of Groningen in the Netherlands. Johann was tutored in mathematics by his brother and developed a mastery of the new Leibnizian calculus. Johann’s life was always full of controversy. Both brothers are considered the inventors of the calculus of variations, but their relationship changed from one of collaborators to one of often bitter rivals. Johann Bernoulli became the mentor of his even more brilliant pupil, Leonhard Euler.

The Bernoulli family produced good mathematicians for generations. Johann’s son Daniel Bernoulli (1700-1782) was born in Groningen but spent decades teaching as a professor at the University of Basel in Switzerland. He stayed with Euler in St. Petersburg, and his Hydrodynamica from 1736 contains Bernoulli’s law on hydraulic pressure and a kinetic theory of gases. This work inspired Euler in his studies of the dynamics of fluids.

The extensive use of hydraulic construction in the Roman Empire and in medieval times led to innovations such as aqueducts and the waterwheel, but this added little to Archimedean mathematical theory. Leonardo da Vinci brought some updates, and Blaise Pascal formulated the law of isotropic pressure. Daniel Bernoulli’s work Hydrodynamica expressed a new synthesis between the conceptions of hydrostatics and hydraulics. Hydrodynamics progressed during the nineteenth century, but despite the practical orientation of some theorists it still didn’t always meet hydraulic and other engineering needs. The full development of hydrodynamics into a mathematical discipline took place during the twentieth century.

The Swiss scholar Leonhard Euler (1707-1783) is widely recognized as one of the greatest mathematicians of all time. He was certainly the most productive measured in published pages, although the eccentric and prolific Hungarian Jewish mathematician Paul Erdos (1913-1996) beat him in the number of published papers. Erdos, too, was interested in combinatorics, graph theory and number theory and the “contributions which Erdös made to mathematics were numerous and broad. However, basically Erdös was a solver of problems, not a builder of theories.” The mathematical community was obviously much smaller in the eighteenth century than it is today; it would be difficult for even the most gifted scholar to have such a wide-ranging influence and dominate the entire, now very diverse field of mathematics as Euler did in his time, but his achievements are nevertheless impressive.

Euler graduated with honors from the University of Basel and got a post at the newly formed St. Petersburg Academy of Sciences, created by Tsar Peter the Great (1672-1725) as part of his modernization efforts of the Russian state. The forceful and ruthless Peter traveled unofficially with a group of 250 Russian officials on a tour of Western European capitals and was particularly impressed with the power and dynamism of the Dutch and the English. Sweden was at this time a regional power in the Baltic Sea which had to be defeated. The Swedes fought well during the Great Northern War (1700-21), but Peter the Great implemented sweeping military reforms and eventually gained the upper hand thanks to his country’s much larger population base. At the Battle of Poltava in the Ukraine in 1709, Tsar Peter won a decisive victory over the forces of King Charles XII (1682-1718) of Sweden.

In 1703, Peter the Great decided to build a new and magnificent capital city and founded Saint Petersburg next to the Baltic Sea, which remained the capital of the Russian Empire until Soviet times, after the Russian Revolution in 1917. Russia had now definitely become a European Great Power to be reckoned with and moved a little closer to the European mainstream with the influx of Western Enlightenment ideas, although these never fully penetrated Russian society. However, serfdom “became more oppressive” under his rule, and modernization efforts were undertaken with forced labor and achieved at great human cost.

Euler left St. Petersburg in 1741 to take up a post which he had been offered by Frederick the Great (1712-1786) of Prussia at the Berlin Academy of Sciences, founded on the advice of Leibniz. He returned to Russia in 1766 at the invitation of Empress Catherine the Great (1729-1796), whose succession to the throne marked a return to the Westernizing policies.

Leonhard Euler was extremely prolific and made contributions to almost every field of mathematics that existed in his day, from hydraulics to artillery, but a large part of his activity was devoted to astronomy. The tremendous prestige of his textbooks settled forever many moot questions of notation on calculus and algebra. Lagrange, Laplace and Gauss followed Euler in their works. Euler was able to dictate articles and letters to his sons until the day of his death in 1783. As The Oxford Guide to the History of Physics and Astronomy states:

“Yet productivity was perhaps the least important of Euler’s claims to mathematical distinction. One of his great contributions was his clarity….He contributed to every branch of mathematics of his day except probability. He achieved much in the realm of number theory. He arguably founded graph theory and combinatorics when he solved the Königsberg Bridge problem in 1736….in addition Euler contributed to ordinary and partial differential equations, the calculus of variations, and differential geometry….Euler made major contributions to every branch of mechanics. The motion of mass points, celestial mechanics, the mechanics of continuous media (mechanics of solids and nonviscous fluids, theories of materials, hydrodynamics, hydraulics, elasticity theory, the motion of a vibrating string, and rigid-body kinematics and dynamics), ballistics, acoustics, vibration theory, optics, and ship theory all received something important from him. If Beethoven did not need to hear to compose music, Euler did not need to see to create mathematics. He began to go blind in one eye in 1738 and became totally blind thirty years later. This only increased his productivity, since total blindness relieved him of academic chores like proofreading and eliminated unwanted visual distractions. Euler did not miss eyes for another reason; he had a prodigious memory.”

An important insight in the history of science was formulated by the French mathematician, biologist and astronomer Pierre-Louis de Maupertuis (1698-1759). Maupertuis had studied in Paris and in Basel with Johann Bernoulli. He became a leading member of the Berlin Academy of Sciences in 1741 and in 1744 enunciated the principle of least action. He hoped that the principle might unify the laws of the universe and prove the existence of God.

Basically this concept says that nature is lazy, for instance that light always travels in straight lines. It turned out to be hugely important in quantum mechanics. According to scholar Alan Gabbey this principle “enjoyed an improved mathematical treatment by William Rowan Hamilton (1834, 1835), whose transformation of Lagrange’s equations was modified and generalized by Carl Gustav Jacobi in the form now known as the Hamilton-Jacobi Equation (1837). In turn, the Hamilton-Jacobi Equation found fruitful application in the establishment of the quantum mechanics of Louis de Broglie (1923) and Erwin Schrödinger (1926).”

Euler further developed the principle of least action and “pointed the way for the work of Joseph Lagrange (1736-1813), which in turn provided the basis for a mathematical description of the quantum world in the twentieth century.” Lagrange was born Giuseppe Lodovico Lagrangia in Turin, Italy, where he lived during the early years of his life. He replaced Euler when the latter left Berlin for Saint Petersburg in 1766 and spent twenty productive years in that city. By 1786 he moved to France where he eventually became known as Joseph-Louis Lagrange. Lagrange was a much better mathematician than de Maupertuis and provided the concept of least action with a thorough mathematical foundation. He also worked with analysis and with analytical and celestial mechanics. He is particularly famous for defining the Lagrange points, which are of great practical significance in space exploration today.

The French mathematician and naturalist Alexis Clairaut (1713-1765) was a prodigy, educated at home by his father who taught mathematics. He became the youngest person ever elected to the Paris Academy in 1731. There he joined a group who supported the natural philosophy of Newton. Clairaut helped Marquise du Châtelet (1706-1749), a woman mathematician and mistress of Voltaire, translate Newton’s Principia into French. Together with Maupertuis he took part in an expedition to Lapland to measure a degree of longitude. In 1743, Clairaut published a book confirming the Newton-Huygens belief that the Earth is flattened at the poles. He further built on Colin Maclaurin’s work on tides and hydrostatics.

The French mathematician and philosopher Jean le Rond d’Alembert (1717-1783) was born and died in Paris. He is above all remembered for his work on fluid mechanics and for being co-editor with Denis Diderot (1713-1784) of the highly influential Encyclopédie, but he did astronomical work, too, and in 1749 carried out the first valid derivation of the precession of the equinoxes. His contemporary and rival Clairaut also did work in mathematical astronomy.

Throughout the eighteenth century, Newtonian theory advanced. After having mastered the main orbits of the planets around the Sun, European scholars began to focus on the smaller, but not negligible, effects called “ perturbations “ caused by the gravity of other bodies. For example, the planets Jupiter and Saturn modify the motions of each other about the Sun. Euler helped to develop the mathematical techniques needed to compute perturbation effects.

The French mathematical astronomer Pierre-Simon Laplace (1749-1827) extended the work of his predecessors in his multi-volume Mécanique Céleste (Celestial Mechanics) (1799-1825). He was born in the Calvados area in Normandy next to the English Channel, a region famous for its cider, an alcoholic beverage made from fermented apple juice. His father was in the cider business and fairly well-off. Laplace was a survivor who managed to keep his life through the Reign of Terror under Maximilien Robespierre (1758-1794) during the French Revolution, unlike his chemist friend Antoine Lavoisier who was beheaded by the guillotine.

Laplace was preoccupied with issues related to probability throughout his life and used his considerable mathematical skills to analyze the orbits of the planets and moons in our Solar System and their influence on each other. In his 1796 book Exposition du Système du Monde he summarized for lay people the general knowledge about astronomy in his day and advanced his version of the “nebular hypothesis,” the idea that the Solar System formed from a cloud of gas and dust. All the planets move around the Sun in the same direction and plane, which is a strong indication that they were formed at the same time through the same physical processes. Comets had different orbits, which indicated that their origin might be different.

Newton himself had suggested that divine intervention might be necessary after some centuries to keep the system stable. Pierre-Simon Laplace through his calculations showed that the Solar System is reasonably stable, although a few of his conclusions had to be modified in the late twentieth century. In the words of science historian John North:

“Leibniz had taunted Samuel Clarke with the imperfection of a Newtonian universe, which he said would need winding up by God from time to time, so implying that God was an inferior artisan. The stability question was a true test of mathematical prowess. Laplace made much use of Lagrange’s method of introducing variations into the six elements of a planet’s orbit — the eccentricity, direction of aphelion, and other parameters that define it — and in 1773 he was able to prove that, even if one planet’s elements are perturbed by another planet, its mean distance from the Sun will not change appreciably, even over millennia. Over the next few years Laplace followed this with more complex theorems relating the distances, eccentricities, and angles of the orbital planes, and again these seemed to point in the same direction: the solar system is highly stable….In more recent studies, the frictional effects of the tides have been introduced into the account, and again it has been found necessary to qualify Laplace’s claims, but the skeleton of his analysis remains, a remarkable testimony to the achievements of Newton’s great successors in the century following his death.”

Johann Carl Friedrich Gauss (1777-1855), maybe the greatest mathematician who has ever lived, was born in 1777 to a poor family in Brunswick, Germany. He was a child prodigy and his skills eventually gained him the patronage of the Duke of Brunswick, which granted him the opportunity to study at the University of Göttingen. In 1800 Ceres, the first asteroid, was discovered by the Italian astronomer Giuseppe Piazzi (1746-1826) but was soon lost from sight. Reflecting his interest in astronomy, Gauss developed a new method for calculating orbits which succeeded in tracking it. This feat earned him a rising European-wide fame.

In 1807 he accepted a post at the observatory in Göttingen. He began a thorough geodesic survey of the city of Hanover in 1818 which was not completed until 1832. At Göttingen he became acquainted with physics professor Wilhelm Weber. The two shared an interest in electricity and magnetism which they explored together. Gauss’ laws describing magnetic and electric fluxes served as part of the foundation on which the brilliant Scottish mathematical physicist James Clerk Maxwell developed his electromagnetic theory. Gauss outlived two wives and two of his six children and suffered from bouts of depression. Victor J. Katz writes:

“The patronage of the duke lasted until he was killed in battle against France in 1806 and the duchy was occupied by the French army. Fortunately for science, the French general had been given explicit orders to look out for Gauss’s welfare. Thus Gauss was able to stay in Brunswick until he accepted a position at Göttingen in the following year as Professor of Astronomy and Director of the Observatory. Gauss remained at Göttingen for the remainder of his life, doing research in pure and applied mathematics as well as astronomy and geodesy. Gauss was never particularly happy with teaching classes, because most of the students were uninterested in, and ill-prepared for, mathematics, but he was willing to work privately with any actively interested student who approached him. Compared to his predecessor Euler and his French contemporary Cauchy, Gauss ultimately published little, his collected works occupying only (!) 12 volumes. Nevertheless, his mathematical papers in various fields are of such profundity that they have influenced the progress of the subject to the present day.”

Augustin-Louis Cauchy (1789-1857) brought a new insistence on precision and rigor to mathematics. His life began in Paris a month after the French Revolution had started with the Storming of the Bastille prison on 14 July 1789. The learned Cauchy family once had Laplace and the chemist Claude Louis Berthollet as neighbors. A staunch Roman Catholic, he made an appeal to the pope on behalf of the stricken people during the famine in Ireland in 1846. He was a prolific writer in mathematical physics and astronomy and established the calculus on the basis of the limit concept familiar today. Cauchy wrote 789 papers, a quantity exceeded only by Euler and Arthur Cayley. Many mathematical theorems have been named after him.

Cauchy’s notion of convergence had been developed in essence before by the Bohemian mathematician and theologian Bernard Bolzano (1781(1781-10-05)-1848), associated with the University of Prague, and by the Portuguese mathematician José Anastácio da Cunha (1744-1787). However, they worked in secondary languages, not in the mathematical centers of France and Germany, and it was out of Cauchy’s work that today’s notions developed. Cunha worked with algebraic analysis and differential calculus and witnessed how the Great Lisbon Earthquake in 1755 reduced the city to rubble before it was rebuilt in a beautiful way. In 1778 he was sentenced by the Inquisition to three years in prison for heretical views and for being a follower of Voltaire. Prison ruined his health and probably contributed to his premature death.

The great astronomer William Herschel (1738-1822) was born in Hanover, Germany, where his father was an oboist and brought up his sons to be musicians. William eventually ended up as an organist in England, and his sister Caroline soon joined him there. His interest in music led him to mathematics and from there on to astronomy. While this was an unusual career path it was not a unique one; there has been a perceived connection between music, mathematics and astronomy in Europe at least since the Pythagoreans in ancient Greece.

William Herschel is credited with the discovery of Uranus in 1781, the first planet to be identified with the telescope, and naturally became famous after this. He discovered it with a fine telescope of his own making. He was a keen observer of nebulae and believed that the Milky Way, which he described in greater detail than anybody had done before him, was composed of millions of individual stars. Through his unprecedented observational work in the late eighteenth century he “reached farther into space than anyone had before and began to outline the structure of our galaxy, but his speculative cosmology did not attract disciples.”

The German astronomer Johann Elert Bode (1747-1826), director of the Berlin Observatory, determined the new planet’s orbit and gave it the name Uranus. He collected the known observations of it and found that it had been observed before Herschel, among others by the English astronomer John Flamsteed, yet nobody had realized that it was a planet. Along with Johann Daniel Titius (1729-1796), a German astronomer and professor at Wittenberg, Bode is known for the Titius-Bode Law which relates the mean distances of the planets from the Sun to a simple mathematic progression of numbers. It was discovered in 1766 by Titius and published by Bode. While it was taken seriously and contributed to the search for even more planets it was discredited with the discovery of Neptune, which did not fit into its pattern.

Caroline Herschel (1750-1848) became William’s valued assistant in England and the first notable woman astronomer. She was granted a salary from the king, just like her brother, and could with some justification be viewed as a professional astronomer. She personally discovered eight comets and together with the Scottish science writer Mary Somerville (1780-1872) became the first honorary woman member of the Royal Society of London in 1835.

The next notable woman in the history of astronomy came from North America. Maria Mitchell (1818-1889) was born to Quaker parents in Nantucket, Massachusetts in the USA. Her enlightened father was a great influence on her life; she developed her love of astronomy from him. In 1847 she discovered a telescopic comet, too faint to be seen by the naked eye. At that point, the only previous woman to discover a comet had been Caroline Herschel.

Uranus can be spotted by a person with good eyesight under ideal conditions, yet no ancient culture had identified it as a planet as far as we know. Neptune, on the other hand, is so distant and faint that it cannot be seen through naked-eye observations. Galileo had apparently noticed it through his telescope as early as 1612 but had mistaken it for a star. Neptune was recorded several times without being recognized for what it was, among others by Jérôme Lalande (1732-1807), a French astronomer who made accurate tables of planetary positions and taught men such as the astronomer Jean Baptiste Joseph Delambre (1749-1822). The same goes for the Englishman John Herschel (1792-1871), son of Wilhelm Herschel, in 1830.

Neptune was eventually discovered following detailed mathematical calculations in the 1840s undertaken to explain certain anomalies that had been detected in the orbit of Uranus. In 1845 the mathematical physicist François Arago (1786-1853), director of the Paris Observatory, persuaded the French mathematical astronomer Urbain Le Verrier (1811-1877) to start working on the problem. He was appointed a teacher of astronomy at the École Polytechnique in Paris in 1837. Le Verrier had carefully studied the orbit of the innermost planet Mercury, which exhibits anomalies that could not be explained before the general theory of relativity (spacetime curves noticeably around massive objects such as the Sun) and were thought to be caused by a hypothetical planet between Mercury and the Sun called Vulcan. Le Verrier explained the irregular orbit of Uranus by assuming the presence of a more distant planet.

On Sept. 23, 1846, after only an hour of searching, the German astronomer Johann Galle (1812-1910) at the Berlin Observatory found the new planet very close to where it had been predicted to exist. Galle was born in Saxony, Prussia, and educated at Berlin. He served as assistant director under Encke from 1835 until 1851, studied the rings of Saturn and suggested a method of measuring the scale of the Solar System by observing the parallax of asteroids.

Unknown to Le Verrier, similar calculations based on studies of the orbit of Uranus were made at the same time by the English mathematician and astronomer John Couch Adams (1819-1892). Adams was initially self-taught in mathematics but later gained admission to the University of Cambridge. He had sent his calculations to the Cambridge Observatory, but these had not been properly followed up by George Airy (1801-1892), an otherwise competent, but not brilliant, Astronomer Royal who modernized the Greenwich Observatory.

This situation triggered a long and fruitless debate over who should receive credit for the discovery of Neptune. Many astronomers credit both Urbain Le Verrier and John Couch Adams as co-discoverers because they had worked simultaneously and independently on the same problem and Adams had in fact begun his work first. Others credit Le Verrier alone as the actual discovery by Galle was based on his calculations, not those made by Adams.

The discovery of Neptune was rightly hailed as a great triumph for science. Two centuries after Newton, European mathematicians could calculate orbits with such precision that they could scientifically predict the existence of a planet. However, while the predictions of Le Verrier and Adams were reasonably correct in 1846 they would have been significantly less so a few years later since they didn’t get everything right regarding Neptune’s obit. As with many other discoveries, it was based on a combination of good work and a little bit of luck.

Mathematical Proof and Scientific Logic

Charles Murray’s book Human Accomplishment includes rankings of influential personalities in Western, Chinese and Indian philosophy. The men at the top — Aristotle, Confucius and Sankara, respectively — are there because in some sense they defined what it meant to be Western, Chinese or Indian. The same is not true of artists, no matter how great they are.

Who does Murray personally consider to be the most accomplished individual who ever lived? “ Aristotle. He more or less invented logic, which was of pivotal importance in human history (and no other civilization ever came up with it independently). He wrote the essay on ethics (‘Nicomachean Ethics’) that to my mind contains the bedrock truths about the nature of living a satisfying human life. He made huge contributions to aesthetics, political theory, methods of classification and scientific observation. Who else even comes close?”

Georges Lemaître (1894-1966), a Belgian priest, introduced his primeval “cosmic egg” in the 1920s. In 1948 the American Ralph Alpher (1921-2007) together with the Ukrainian-born American physicist George Gamow (1904-1968) outlined a theory of how the first elements formed in the early universe (Big Bang nucleosynthesis). Later that year, collaborating with scientist Robert Herman (1914-1997), an American-born son of a Russian Jewish immigrant just like Alpher himself, Alpher predicted the existence of a cosmic background radiation resulting from the Big Bang. Two individuals who didn’t realize what they had found in the 1960s stumbled across evidence of this radiation and received the 1978 Nobel Prize in Physics for the discovery. The contributions of Alpher and Herman were overlooked. A Nobel Prize cannot be shared by more than three individuals, nor can it be awarded posthumously.

This Big Bang model was modified in the 1980s with the introduction of the concept of an early period of cosmic inflation by the American cosmologist Alan Guth (born 1947) and the Russian-born physicist Andrei Linde (born 1948). Alpher’s contributions have unquestionably helped shape the way we currently look at the universe, yet he is not mentioned in Murray’s Human Accomplishment and is entirely overlooked in a number of other works on scientific history. By contrast, Aristotle’s physical ideas have all been discredited centuries ago, yet most educated people have heard of Aristotle. Does that mean that Aristotle is overrated and is primarily famous for being wrong? Not quite so. His biological works have stood the test of time, and his personal contributions to the development of scientific logic are profound.

Aristotle’s six works on logic are known collectively as the Organon, which means “tool” or “instrument.” This reflects the awareness that logic was not a science in itself but a tool for rationally analyzing the world. The vocabulary of logic, syllogism, types of logical fallacy, the elements of deductive reasoning and a long list of terms for analyzing propositions date back to Aristotle. The power of his logic was so great that the importance of logic overrode empiricism for centuries. The balance was restored when it was combined with experiment. Francis Bacon published his Novum Organum (“The New Organon”) in 1620 with the Baconian method. During the Scientific Revolution, supporters of the experimental method frequently criticized what they considered blind adherence to Aristotelian philosophy.

Perhaps the most impressive aspect of Aristotle’s work is the sheer scale of it, and how he extended his investigations to include all natural phenomena. There was no known equivalent to him in ancient Egypt, Mesopotamia, India, China or Mesoamerica. He enjoyed an exceptional prestige throughout the Middle Ages in the Middle East and Europe, so much so that even his errors prevailed into the modern era, especially in physics and astronomy. Many theologians and natural philosophers simply referred to him as “the Philosopher.” Yet his tremendous influence resulted not merely from intellectual subservience on the part of medieval scholars, but also “from the overwhelming explanatory power of Aristotle’s philosophical and scientific system. Aristotle prevailed through persuasion, not coercion.”

The nature of mathematical proof is related to Aristotelian logic, but mathematical logic predates it in time. The development of mathematical proof started among the ancient Greeks in the sixth century BC with Thales and Pythagoras. Other ancient civilizations practiced mathematics in an intuitive and experimental manner to solve practical problems. It was the Greeks who came to insist that geometric statements should be established by careful deductive reasoning rather than by trial and error. The logico-deductive method can provide conclusions/new knowledge by means of reasoning step by step from established knowledge. In his excellent book A History of Mathematics, Second Edition, Victor J. Katz states:

“Aristotle believed that logical arguments should be built out of syllogisms, where ‘a syllogism is discourse in which, certain things stated, something other than what is stated follows of necessity from their being so.’ In other words, a syllogism consists of certain statements that are taken as true and certain other statements that are then necessarily true. For example, the argument ‘if all monkeys are primates, and all primates are mammals, then it follows that all monkeys are mammals’ exemplifies one type of syllogism, while the argument ‘if all Catholics are Christians and no Christians are Moslem, then it follows that no Catholic is Moslem’ exemplifies a second type. After clarifying the principles of dealing with syllogisms, Aristotle notes that syllogistic reasoning enables one to use ‘old knowledge’ to impart new.”

Aristotle distinguished between the basic truths that are peculiar to each particular science and the ones that are common to all. The former are often called postulates, the latter as axioms. Yet “Although Aristotle emphasized the use of syllogisms as the building blocks of logical arguments, Greek mathematicians apparently never used them. They used other forms, as have most mathematicians down to the present. Why Aristotle insisted on syllogisms is not clear. The basic forms of argument actually used in mathematical proof were analyzed in some detail in the third century B.C.E. by the Stoics, of whom the most prominent was Chrysippus (280-206 B.C.E.). This form of logic is based on propositions, statements that can be either true or false, rather than on the Aristotelian syllogisms.”

Aristotle was a member of Plato’s Academy for twenty years. When he returned to Athens in 335 BC he founded a rival school in another Athenian gymnasium, the Lyceum. Athens had by then acquired educational leadership within the Greek world. One of the visitors there was Zeno (ca. 335-263 BC), who was born in Citium in Cyprus and founded the influential Stoic school of philosophy. Men should face the world with “stoic calm.” The Stoics “looked upon the passions as essentially irrational, and demanded their complete extirpation. They envisaged life as a battle against the passions, in which the latter had to be completely annihilated. Hence their ethical views end in a rigorous and unbalanced asceticism.”

According to Lindberg, “Zeno of Citium arrived in Athens about 312 and subsequently began to teach in the stoa poikile (painted colonnade) in a corner of the Athenian agora, thus founding a school of what came to be called ‘Stoic’ philosophy. Epicurus, an Athenian citizen born on the island of Samos, returned to Athens about 307, purchased a house and garden, and there founded a school of ‘Epicurean’ philosophy that survived into the Christian era. The Academy, the Lyceum, the Stoa, and the Garden of Epicurus — the four most prominent schools in Athens — all developed institutional identities that enabled them to survive their founders. The Academy and the Lyceum seem to have had continuous existence until the beginning of the first century B.C. (perhaps until the sack of Athens by the Roman general Sulla in 86 B.C.). It is often claimed that the Academy survived until it was closed by the Emperor Justinian in A.D. 529. The truth seems to be that Neoplatonists (so-called because of their departure from or reinterpretation of various Platonic doctrines) refounded the Academy in the fifth century A.D. and managed to keep it alive until about 560 or later.”

The Megarians were followers of Euclid of Megara (ca. 430-360 BC), a pupil of Socrates. They were interested in logical puzzles and influenced Stoic logic. Many Greek mathematicians followed the forms of argument delineated by Stoic philosophers such as Chrysippus of Soli, who was a systematizer of Stoic philosophy and in Antiquity was considered a logician comparable in stature to Aristotle himself. His works have mostly been lost, but he aided the popularity that Stoicism enjoyed during Hellenistic and Roman times, with prominent adherents such as the Roman Emperor Marcus Aurelius (AD 121-180).

Parmenides was one of the first philosophers to use an extended argument for his views rather than merely proposing a vision of reality, yet he never formulated his principles in a systematic manner. Zeno of Elea’s arguments, or “Zeno’s Paradoxes,” could establish a claim by showing that its opposite leads to absurd consequences. This line of argumentation is known as reductio ad absurdum. It is likely that Zeno consciously used it in a systematic way.

As author Marvin J. Greenberg says, “The orderly development of theorems with proofs about abstract entities became characteristic of Greek mathematics and was entirely new. This was the first major revolution in the history of mathematics. How this revolution came about is not well understood by historians. Among Greek philosophers, dialectics, the art of arguing well, which originated in Parmenides’ Eleatic school of philosophy, played an important role. And undoubtedly proofs were an outgrowth of the need to convince others in a debate.”

The Greeks’ basic political organization was the polis or city-state. Its government could be democratic or monarchical but was usually ruled by law, and thus its citizens were motivated to learn the skills of argument and debate. The Sophists were teachers of rhetoric, prepared to teach anything anyone was ready to pay for. Plato criticized them for this as his teacher Socrates did not accept payment, but the Sophists did contribute to a culture of argumentation. This political atmosphere favored the development of proof in mathematics. To contrast between (mere) persuasion and demonstration, Aristotle defined a logic proceeding from self-evident primary premises via valid deductions to incontrovertible conclusions.

As G.E.R Lloyd puts it in The Ambitions of Curiosity, the development of scientific logic and mathematical proof may seem natural and inevitable to us today, “But when we reflect that neither the Chinese nor any other ancient mathematical tradition did so, there would appear to be more to it than mere intellectual attractiveness. What more may be answered in part, I suggest, by the negative models provided by the styles of argument cultivated in those other peculiarly Greek institutions of the law-courts (dikasteria) and political assemblies. It was dissatisfaction with the merely persuasive arguments used there that led some philosophers and mathematicians to develop their alternative.”

The concept of mathematical proof reached its highest state of development in Greek geometry. Among Euclid’s most important predecessors was the astronomer Hippocrates of Chios (ca. 470-410 BC), who compiled the first significant Greek work on the elements of geometry. Archytas of Tarentum (ca. 400-350 BC) was a Greek mathematician, political leader, Pythagorean philosopher and musical theorist from southern Italy and a friend of Plato. Euclid may have borrowed from his work in his own treatment of number theory.

Although Euclid gave the work an overarching structure and added some material of his own, he is first and foremost famous for creating a brilliant synthesis of the work of others. Almost nothing is known about him personally, but he lived in the early Hellenistic period. It is assumed that he taught at the Museum and Library at Alexandria, founded around 300 BC.

As Katz states, “The most important mathematical text of Greek times, and probably of all time, the Elements of Euclid, written about 2300 years ago, has appeared in more editions than any work other than the Bible….Biographies of many famous mathematicians indicate that Euclid’s work provided their initial introduction into mathematics, that it in fact exited them and motivated them to become mathematicians. It provided them with a model of how ‘pure mathematics’ should be written, with well-thought-out axioms, precise definitions, carefully stated theorems, and logically coherent proofs. Although there were earlier version of Elements before that of Euclid, his is the only one to survive, perhaps because it was the first one written after both the foundations of proportion theory and the theory of irrationals had been developed in Plato’s school and the careful distinctions always to be made between number and magnitude had been propounded by Aristotle. It was therefore both ‘complete’ and well organized.”

There were a few notable logicians during the Middle Ages in the Middle East, more in Western Europe but very few in Byzantium. The Spanish (Majorcan) priest and mystic Ramón Lull, or Raymond Lully (ca. 1232-1315), helped to develop the Catalan language. His Ars magna, generalis et ultima (1501; “Great, General and Ultimate Art”) represents an attempt to symbolize concepts and derive propositions that form various combinations of possibilities. These notions influenced Pascal and Leibniz. Gottfried Wilhelm Leibniz in his 1666 publication De Arte Combinatoria proposed the idea of an algebra of logic, but later developments happened along somewhat different lines. Author Marvin Jay Greenberg writes:

“George Boole and Augustus de Morgan began to carry out his idea. Boolean algebra is now the foundation for computer arithmetic and is very important in pure mathematics. In 1879 Gottlob Frege brought quantifiers into logic, introducing what is now known as the predicate calculus, but with terrible notation. Most of the currently used notation and methods of mathematical logic stem from the society of logicians founded in the 1880s by Giuseppe Peano along with Mario Pieri. They emphasized the importance of a formal symbolic language for mathematics to remove the ambiguities of natural languages, to make mathematics utterly precise, and to permit the mathematical study of entire mathematical theories. Many years later, this formalization also enabled the programming of computers to do mathematics. The discovery and validation of non-Euclidean geometries, together with Georg Cantor’s invention of set theory and Karl Weierstrass’ rigorous presentation of analysis, caused mathematicians to study axiomatics seriously for the first time. It was not until 1889 that axioms for the arithmetic of natural numbers were satisfactorily formulated — by Peano, based on Richard Dedekind’s set-theoretic development using the successor function (and influenced by earlier algebraic work of Hermann Grassmann).”

Hermann Grassmann (1809-1877) was a German polymath educated at the University of Berlin who was most famous in his day as a scholar of Sanskrit. His basic idea of a general calculus of vectors was first published in 1844, followed by a reworked version in 1862. According to author John Derbyshire, “He defined such concepts as linear dependence and independence, dimension, basis, subspace, and projection. He in fact went much further, working out ways to multiply vectors and express changes of basis, thus inventing the modern concept of ‘an algebra’ in a much more general way than Hamilton with his quaternions. All this was done in a strongly algebraic style, emphasizing the entirely abstract nature of these new mathematical objects and introducing geometrical ideas as merely applications of them.”

Grassmann’s work was studied by Giuseppe Peano and by the Frenchman Élie Cartan (1869-1951), the son of a poor blacksmith. Cartan showed unusual ability and was able to obtain state funds for his education. He lectured at the Universities of Montpellier, Lyon and Nancy before moving to Paris and did valuable work on Lie algebras and group theory. His son Henri Cartan (1904-2008) became a distinguished scholar in his own right and a member of the group of mathematicians writing under the collective pseudonym Nicolas Bourbaki.

The French mathematician Jean-Pierre Serre (born 1926) was a student of Henri Cartan. He had attended the École Normale Supérieure and the Sorbonne in Paris. In 2003, Serre was awarded the first Abel Prize by the Norwegian Academy of Science and Letters “ for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.” He was a source of inspiration for other great names such as the German-French scholar Alexander Grothendieck (born 1928), who made profound contributions to algebraic geometry. Serre became the first person to win the Fields Medal, the Wolf Prize as well as the Abel Prize, three of the most prestigious awards in mathematics. As of 2010, the only other person to win all three is John Griggs Thompson, born in Kansas in the USA in 1932 and educated at the Universities of Yale and Chicago. Thompson shared the Abel Prize in 2008 with the Belgian-French mathematician Jacques Tits (born 1930) for their “profound achievements in algebra and in particular for shaping modern group theory.”

William Rowan Hamilton (1805-1865) was born in Dublin, Ireland of Scottish parents and lived his whole life in that city. He could read Hebrew, Latin and Greek at the age of five. By 1822 his mathematical abilities had advanced to such an extent that he discovered a significant error in Laplace’s treatise Celestial Mechanics. He formed a lasting friendship with the English Romantic poet William Wordsworth (1770-1850) and made major contributions to optics. Hamiltonian mechanics became even more appreciated with quantum mechanics in the twentieth century. In 1843 he introduced quaternions, algebra with hyper-complex numbers. His many years of work on a theory of quaternions “would greatly impact the development of the modern system of vector analysis and is sometimes utilized today for computer graphics, attitude control systems, and the control theory used in engineering.”

Mathematics has historically been overwhelmingly created by men, often by very young men. The French mathematician Évariste Galois (1811-1832) died from wounds suffered in a duel before his twenty-first birthday, but had nevertheless contributed notable works whose full importance was grasped years after his premature death. As one online biography states, “ Galois ‘ complete works fill only 60 pages, but he will be remembered.” In 1830 he developed group theory, which was to prove of critical importance for mathematical physics, especially in the development of quantum mechanics during the twentieth century.

His countryman Joseph Liouville (1809-1882) discovered transcendental numbers — numbers that are not the roots of algebraic equations having rational coefficients. Liouville was taught by the physicist André-Marie Ampère and his work was extremely wide ranging, from mathematical physics to astronomy. He published Galois’s notes in 1846. The engineer Camille Jordan (1838-1922), born in Lyon and educated at the École Polytechnique, was highly regarded for his work in algebra, group theory and Galois theory. He was the first person to develop a full understanding of the importance of the theories of Évariste Galois.

The Norwegian mathematician Niels Henrik Abel (1802-1829) died in poverty while still in his twenties. He was able to attend the prestigious Cathedral School in Christiania (Oslo), where his teacher encouraged him. After a brief university education there he received modest financial support from the government and managed to travel abroad in 1825. In Berlin he met the engineer August Leopold Crelle (1780-1855), who found the courage after his encounter with Abel to publish a mathematical journal — Crelle’s Journal — that could compete with the best French ones. The mathematician and logician Joseph Diaz Gergonne (1771-1859) from Nancy, France, had established his own mathematics journal in 1810, with a special focus on geometry. While Paolo Ruffini (1765-1822), who taught at the University of Modena in Italy, had worked on the quintic equation, Abel made the final breakthrough:

“Abel proved the impossibility of solving the general quintic equation by means of radicals — a problem which had puzzled mathematicians from the time of Bombelli and Viète (a proof of 1799 by the Italian Paolo Ruffini was considered by Poisson and other mathematicians as too vague). Abel now obtained a stipend which enabled him to travel to Berlin, Italy, and France. But, tortured by poverty most of his life and unable to get a position worthy of his talents, Abel established few personal mathematical contacts and died (1829) soon after his return to his native land….Abel’s investigations on elliptic functions were conducted in a short but exciting competition with Jacobi….Legendre, who had spent so much effort on elliptic integrals, had missed this point entirely and was deeply impressed when, as an old man, he read Abel’s discoveries.”

The Frenchman Adrien-Marie Legendre (1752-1833) was one of the leading mathematicians in Europe at the turn of the nineteenth century. While he made many valuable personal contributions, some of his work was later perfected by others, among them Abel, Jacobi and Galois. Carl Gustav Jacobi (1804-1851) was a German (Prussian) mathematician, born of Jewish parents, who studied in Potsdam and at the University of Berlin to be able to teach mathematics, Greek and Latin. In 1829 Jacobi met Legendre and other French mathematicians such as Fourier when he made a visit to Paris, and he visited Gauss in Göttingen. He became an influential and inspiring teacher and made contributions to the theory of elliptic functions.

The Norwegian mathematician Marius Sophus Lie (1842-1899) studied at the University of Christiania, but afterward spent much of his time in Germany where he collaborated with leading figures such as Felix Klein. He is remembered for developing Lie algebra and Lie groups, which are important in quantum physics and quantum mechanics. Peter Ludwig Mejdell Sylow (1832-1918), another mathematician from Christiania (now Oslo) in Norway, in the 1870s collaborated with Lie to prepare an edition of Abel’s complete works.

The German mathematician Karl Weierstrass (1815-1897) was a pioneer of modern analysis and theory of functions and added rigor to many problems in mathematics. His father wanted him to study law to secure a position in the Prussian civil service, but Weierstrass spent his time at the University of Bonn on fencing and beer-drinking. He then studied mathematics at the University of Münster and began teaching at small schools. In 1853 he wrote up an original result and sent it to a well-read professional journal, Crelle’s Journal. His fame spread quickly after that. When the Russian-born Sonya Kovalevskaya (1850-1891), one of the earliest woman mathematicians of some note, came to Germany from Russia to study mathematics, Weierstrass privately tutored her. Many students benefited from his teaching.

The leading English mathematician Arthur Cayley (1821-1895) while training to be a lawyer went to Dublin to hear William Rowan Hamilton lecture on quaternions. He practiced law in London until 1863, but eventually focused on his mathematical talents. In 1863 he was appointed professor of mathematics at Cambridge. He published over 900 papers covering nearly every aspect of modern mathematics, including non-Euclidean and n-dimensional geometry. According to the Encyclopædia Britannica online, “ Cayley made important contributions to the algebraic theory of curves and surfaces, group theory, linear algebra, graph theory, combinatorics, and elliptic functions. He formalized the theory of matrices.”

The English Jewish mathematician James Joseph Sylvester (1814-1897), a friend of Cayley, coined the term “matrix” used in an algebraic context in 1850. He taught many students privately, one of whom was the famous English nurse Florence Nightingale, who used her knowledge of statistics in her medical work. In 1878, while being employed at the new Johns Hopkins University in the USA, Sylvester founded the American Journal of Mathematics.

Richard Dedekind (1831-1916) is especially remembered for his redefinition of irrational numbers in terms of Dedekind cuts. Dedekind was, like Gauss before him, born in Brunswick (Braunschweig) in north-central Germany. He became the last pupil of Carl Friedrich Gauss in 1852 before the latter retired from teaching. He thereafter studied at the University of Berlin, where he was a contemporary of Bernhard Riemann. In 1858 he began teaching at the Polytechnic in Zürich. He was a friend of the German mathematician Peter Gustav Lejeune Dirichlet (1805-1859) and was fond of holidays in Switzerland, the Austrian Tyrol or the Black Forest region in southern Germany. He met the great German scholar Georg Cantor (1845-1918) in 1874 while staying in the scenic town of Interlaken in the Swiss Alps. Dirichlet made contributions to number theory, analysis and mechanics and was a lifelong friend in Carl Jacobi. He taught at the universities of Breslau, Berlin and Göttingen.

Cantor ‘s Danish father was a successful merchant in St Petersburg. His Russian mother was very musical, and Georg inherited musical talents from his parents. Cantor spent much time in mathematical discussions with his friend Dedekind, and spent his final years with little food because of the war conditions in Germany. He suffered from periods of mental illness and depression. In 1917 he entered a sanatorium for the last time, before he died of a heart attack.

Discussions of the infinite date back to Antiquity, and were continued by individuals such as the German scholastic philosopher Albert of Saxony (ca. 1316-1390), a pupil of Jean Buridan. Bernhard Bolzano, a (1781-10-05)Bohemian Catholic priest, mathematician and logician at the University of Prague, published the work Paradoxes of the Infinite, which was admired by Dedekind, Cantor and the American logician Charles Sanders Peirce (1839-1914). In 1874, Cantor published an article in Crelle’s Journal which marks the birth of set theory. Weierstrass and Dedekind supported him, but he faced opposition from the German mathematician Leopold Kronecker (1823-1891). Dirk J. Struik explains in A Concise History of Mathematics:

“Cantor, who taught at Halle from 1869 until 1905, is known not only because of his theory of the irrational number, but also because of his theory of aggregates (Mengenlehre). With this theory Cantor created an entirely new field of mathematical research, which was able to satisfy the most subtle demands of rigor once its premises were accepted….Cantor developed a theory of transfinite cardinal numbers based on a systematical mathematical treatment of the actually infinite….Cantor also defined transfinite ordinal numbers, expressing the way in which infinite sets are ordered. These discoveries of Cantor were a continuation of the ancient scholastic speculations on the nature of the infinite, and Cantor was well aware of it. He defended St. Augustine’s full acceptance of the actually infinite, but had to defend himself against the opposition of many mathematicians who refused to accept the infinite except as a process expressed by 8….Cantor finally won broad acceptance when the enormous importance of his theory for the foundation of real function theory and of topology became more and more obvious — this especially after Lebesgue in 1901 had enriched the theory of aggregates with his theory of measure.”

The Frenchman Henri Léon Lebesgue (1875-1941) made groundbreaking contributions to integration theory. The integral calculus of Newton and Leibniz had been put on a rigorous mathematical foundation by Riemann, but the Lebesgue integral, published in the year 1900, was applicable under much more general conditions. Building on the work of his countryman Émile Borel (1871-1956), Lebesgue formulated the theory of measure in 1901. Borel was a decorated member of the French Resistance during the Second World War. The German Jewish mathematician Felix Hausdorff (1868-1942) did not leave Nazi Germany after Kristallnacht in 1938, when the persecution of Jews escalated. In 1942, when he could no longer avoid being sent to a concentration camp, he committed suicide together with his wife. Though Hausdorff’s definition of measure never played as important a role in probability theory as those of Borel and Lebesgue it turned out to be very useful in chaos theory.

Maurice Fréchet (1878-1973) from France is credited as the founder of the theory of abstract spaces and made contributions to statistics and probability. He taught at the Universities of Poitiers, Strasbourg and Paris. “ Fréchet was also a pioneer topologist (topology is the branch of mathematics dealing with the properties of figures that remain unchanged upon elastic deformation) and contributed notably to statistics and to differential and integral calculus.” Jacques Hadamard (1865-1963) was professor of astronomy at the University of Bordeaux.

The German scholar Gottlob Frege (1848-1925) was one of the founders of modern mathematical logic and analytic philosophy. He had a major influence on the prominent Austrian and British philosopher Ludwig Wittgenstein (1889-1951), born in Vienna in the Habsburg Empire to a family with a Jewish background who practiced Christianity. His father was an industrialist and patron of the arts. Composers Johannes Brahms, Richard Strauss, Gustav Mahler and the Catalan conductor Pablo Casals (1876-1973) frequented the family, and Wittgenstein senior collected works of artists like the French sculptor Auguste Rodin (1840-1917) and the Austrian painter Gustav Klimt (1862-1918). In 1911 Ludwig Wittgenstein went to Cambridge to study with Bertrand Russell. His philosophical influences include the Danish theologian Søren Kierkegaard and Arthur Schopenhauer. His major works are Tractatus Logico-Philosophicus from 1922 and Philosophical Investigations from 1953. Wittgenstein spent many years at Cambridge University in England where he eventually died.

The gifted Italian mathematician Giuseppe Peano (1858-1932) was born and raised in the Piedmont region of northern Italy, an area justly renowned for its fine wines. He spent most of his career teaching at the University of Turin where he became a lecturer of infinitesimal calculus in 1884 and a professor in 1890. He was above all a pioneer in the development of a symbolic logic and the use of the axiomatic method, stressing the necessity of rigor.

Alfred North Whitehead (1861-1947) was an English mathematician and philosopher who wrote a Universal Algebra (1898) based on Grassmann, Boole and Hamilton. Bertrand Russell (1872-1970) was a prominent and controversial British philosopher, author and social reformist who made valuable contributions to the development of mathematical logic. In 1950 he was awarded the Nobel Prize in Literature. Part of Peano’s logic notation was adopted by Russell and Whitehead in their Principia Mathematica (1910-13), which had a complicated but precise symbolism. Like Hilbert’s approach, however, it failed in its ultimate purpose.

According to Marvin Jay Greenberg, “The most influential foundational works in logic in the early twentieth century were the Principia Mathematica of Bertrand Russell and A. N. Whitehead; the work of David Hilbert with his associates Wilhelm Ackermann, Paul Bernays, and John von Neumann; and the contributions of Thoralf Skolem. By formalizing all rules of reasoning and axioms in a purely symbolic language, mathematicians were able to study entire branches of their subject, such as Peano arithmetic and elementary geometry and Zermelo-Fraenkel set theory. They were then able to prove theorems about those branches — theorems that are called metamathematical because they are about mathematical theories, not about numbers or geometric figures or sets. The most important metamathematical theorems are the completeness and incompleteness theorems of Kurt Gödel from the early 1930s, which revolutionized our thinking about the nature of mathematics. Also vitally important in the 1930s were the equivalent determinations of the class of effectively computable number-theoretic functions by Alan Turing, Alonzo Church, Emil Post, and Gödel.”

The German-born Jewish mathematician Abraham Fraenkel (1891-1965) was a Zionist who, after leaving Kiel, taught at the Hebrew University of Jerusalem from 1929. Building on the work of German scholar Ernst Zermelo (1871-1953) from 1908 he helped to create axiomatic set theory in 1922. Thoralf Skolem (1887-1963), a Norwegian mathematician and professor at the University of Oslo, made further contributions in the field of mathematical logic and set theory, as did the German mathematician Wilhelm Friedrich Ackermann (1896-1962). Paul Bernays (1888-1977) from Switzerland collaborated with David Hilbert and made significant contributions to the development of mathematical logic and the philosophy of mathematics.

The English mathematician and logician Augustus De Morgan (1806-1871) published his Formal Logic in 1847. Fellow Englishman George Boole (1815-1864) became the inventor of Boolean logic, the basis of modern digital computer logic. Emil Post (1897-1954) was an American logician associated with Columbia University in New York City. Another prominent mathematician and logician in the United States, Alonzo Church (1903-1995) of Princeton University, was one of the founders of theoretical computer science along with the great English mathematician, cryptanalyst and computer scientist Alan Turing (1912-1954).

The German mathematician Carl Louis Ferdinand von Lindemann (1852-1939) was the first to prove that p is transcendental, i.e. p is not the root of any algebraic equation with rational coefficients. He served as a supervisor for doctoral thesis of David Hilbert, Hermann Minkowski and Arnold Sommerfeld. Erik Ivar Fredholm (1866-1927) from Sweden did work on spectral theory and was professor of theoretical physics at the University of Stockholm. He founded modern integral equation theory, and his mathematical efforts inspired Hilbert.

Felix Klein (1849-1925), born in Düsseldorf, Germany to a Prussian family, was active in many branches of mathematics and a highly influential teacher, yet he is best known for his work on the connections between geometry and group theory. After 1870, during the period of German unification under the leadership of the “Iron Chancellor” Otto von Bismarck, he collaborated with the Norwegian mathematician Marius Sophus Lie, who introduced him to the group concept which had been pioneered by mathematicians such as the Norwegian Niels Henrik Abel and the Frenchman Évariste Galois. “Klein’s synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development” and gave a unified approach to geometry which is now the standard accepted view.

Emmy Noether (1882-1935) was the daughter of a Jewish mathematician. She came from the town of Erlangen in Bavaria and took classes at the university there. In 1915 she was invited by Hilbert and Klein to the University of Göttingen. Often regarded as the most important woman in the history of mathematics, she made many contributions to abstract algebra. Noether’s Theorem, proved by her in 1915, “establishes a quite remarkable connection between the symmetries of a physical system and its conserved quantities, a connection that has acquired particular importance for the entire body of modern physics because of the emphasis all contemporary theories put on both symmetries and conservation laws.”

The Hungarian János Bolyai (1802-1860), the Russian Nikolai Lobachevsky (1792-1856) and above all the brilliant German mathematician Bernhard Riemann (1826-1866) in the mid-nineteenth century founded non-Euclidean geometry, which we will deal with later when looking at the general theory of relativity. Following the development non-Euclidean geometry in late nineteenth century Europe came a renewed interest in Euclid’s geometry.

The most successful attempt to set up a complete set of axioms from which Euclidean geometry could be derived was David Hilbert (1862-1943). He rose to prominence after Felix Klein called him to Göttingen, the preeminent university for mathematics in Germany and perhaps the world prior to the rise of the Nazis. He was rivaled only by the French mathematical physicist Henri Poincaré as the leading mathematician during his lifetime. He made outstanding contributions to many mathematical fields; the infinite-dimensional spaces used in quantum mechanics, Hilbert spaces, are named after him. His treatise Grundlagen der Geometrie (Foundations of Geometry) from 1899 “provided brand new important insights into the foundations of geometry.” There were many axiom schemes developed in this period to clarify various areas of mathematics. Hilbert’s work was the culmination of this process where the ideas of Aristotle and Euclid were reconfirmed as the model for pure mathematics.

At the International Congress of Mathematicians in Paris in 1900, Hilbert challenged his fellow scholars with a list of twenty-three unsolved problems in mathematics that turned out to be some of the most important of the twentieth century. Many of his problems have now been solved. In the year 2000, the Clay Mathematics Institute in the United States offered a million-dollar prize to anyone who can solve seven problems considered to be among the most challenging for the new century. One of them, the Poincaré conjecture, has been solved.

The Russian Jewish mathematician Grigori Perelman was born in 1966 in Saint Petersburg. In 2002-2003 the former child prodigy published on the Internet the solution to the Poincaré conjecture. He was aided by earlier progress made by fine mathematicians such as Richard Hamilton (born 1943) and William Thurston (born 1946) from the USA but made the final breakthrough in solitude. In 2006, Perelman was awarded the prestigious Fields Medal but declined to accept it. In 2010 it was announced that he had met the criteria to receive the first Clay Millennium Prize Problems award of one million US $. He is reported to have said that “I’m not interested in money or fame. I don’t want to be on display like an animal in a zoo.”

Hilbert’s axiomatic approach was criticized by Luitzen Egbertus Jan Brouwer (1881-1966), a Dutch mathematician and philosopher from the University of Amsterdam who contributed substantially to the emerging field of topology and founded the philosophy of intuitionism in opposition to Hilbert’s formalistic ideas. As writer Roger G. Newton states in From Clockwork to Crapshoot: A History of Physics, “In Hilbert’s publicly stated view, the success of intuitionism would destroy mathematics. The intuitionist movement has now lost momentum, but Hilbert’s axiomatic program was dealt a mortal blow from another direction, by the work of the Austrian mathematician Kurt Gödel (1906-1978). Gödel’s famous theorem states that in every consistent and sufficiently strong formal axiomatic system there would necessarily arise statements that could be neither proved nor disproved within that system.”

Kurt Gödel was born in Austria-Hungary. Although a non-Jew he found the situation unacceptable after the Anschluss in 1938, when Austria became a part of Nazi Germany, and relocated to the USA. While sometimes mentally unstable he was a brilliant logician. He made his deepest impact on mathematics in 1931 at the University of Vienna. According to the Stanford Encyclopedia of Philosophy online he “ founded the modern, metamathematical era in mathematical logic. His Incompleteness Theorems, among the most significant achievements in logic since, perhaps, those of Aristotle, are among the handful of landmark theorems in twentieth century mathematics. His work touched every field of mathematical logic, if it was not in most cases their original stimulus. In his philosophical work Gödel formulated and defended mathematical Platonism, involving the view that mathematics is a descriptive science, and that the concept of mathematical truth is an objective one.”