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.”

A History of Astrophysics and Cosmology

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 was originally published in five parts at various sites: Part 1, Part 2, Part 3, Part 4, and Part 5

The introduction of the telescope in Western Europe in the 1600s revolutionized astronomy, but it did not found it as a discipline. Astronomy had existed in some form for thousands of years prior to this. It is consequently impossible to assign a specific date to its beginning. This is not the case with astrophysics. People in ancient and medieval times might speculate on the material makeup of stars and celestial bodies, but they had no way of verifying their ideas.

Anaxagoras of Clazomenae in the fifth century BC was the first Pre-Socratic philosopher to live in Athens. He championed many controversial theories, including his claim that the stars are fiery stones. He allegedly got this idea when a meteorite fell near Aegospotami. He assumed that it came from the Sun, and since it consisted largely of iron he concluded that the Sun was made of red-hot iron. Not a bad guess for his time, yet he had no way of proving his claims. Neither did Asian or Mesoamerican observers. Some sources indicate that Anaxagoras was charged with impiety, as most ancient Greeks still shared the divine associations with the heavenly bodies, but political considerations may have played a part in this process as well.

As late as in 1835 Auguste Comte (1798-1857), the French philosopher often regarded as the founder of sociology, stated that humans would never be able to understand the composition of stars. He was soon proved wrong by two new techniques — spectroscopy and photography.

The English chemist William Hyde Wollaston (1766-1828) in 1800 formed a partnership with his countryman Smithson Tennant (1761-1815), whom he had befriended at Cambridge. Tennant discovered the elements iridium and osmium, extracted from platinum ores, in 1803. The platinum group metals — platinum, ruthenium, rhodium, palladium, osmium and iridium — have similar chemical properties. Osmium (Os, atomic number 76) is the heaviest natural element with a density of more than 22.6 kg/dm3, twice as much as lead at 11.3 kg/dm3.

Platinum (Pt, atomic number 78) and its dense sister metals are very rare in the Earth’s crust. It had been introduced to Europe from South American mines in the 1740s by men such as the Spanish explorer Antonio de Ulloa (1716-1795). Wollaston was the first person to produce pure, malleable platinum and became wealthy from supplying Britain with the precious metal. The Wollaston Medal, granted by the Geological Society of London, is named after him.

The German chemist Martin Klaproth (1743-1817) was born in Wernigerode in Prussian Saxony and worked as an apothecary for years before continuing his career as a professor of chemistry at the newly established University of Berlin. He discovered uranium as well as zirconium (Zr, a.n. 40) in 1789. Uranium (symbol U, atomic number 92) was named for the planet Uranus, which had been discovered just prior to this. Wollaston detected the elements palladium in 1803 and rhodium in 1804. He named palladium (Pd, a.n. 46) after the asteroid Pallas, which had been discovered a year earlier by the German astronomer Olbers and was initially believed to be a planet, until the full extent of the asteroid belt had been grasped.

The birth of spectroscopy, the systematic study of the interaction of light with matter, followed shortly after the creation of scientific chemistry in Europe. William Hyde Wollaston in 1802 noted some dark features in the solar spectrum, but he didn’t follow this insight up. In 1814, the German physicist Joseph von Fraunhofer (1787-1826) independently discovered these dark features (absorption lines) in the optical spectrum of the Sun, which are now known as Fraunhofer lines. He carefully studied them and noted that they exist in the spectra of Venus and the stars, too, which meant that they had to be a property of the light itself.

In the 1780s a Swiss artisan, Pierre-Louis Guinand (1748-1824), began experimenting with the manufacture of flint glass, and in 1805 managed to produce a nearly flawless material. He passed on this secret to Fraunhofer, who worked in the secularized Benedictine monastery of Benediktbeuern. Fraunhofer improved upon Guinand’s techniques and began a more systematic study of the mysterious spectral lines. To the stronger ones he assigned the letters A to Z, a system which is also used today. Yet it was left to two other German scholars to prove the full significance of these unique lines, corresponding to specific chemical elements.

Robert Bunsen (1811-1899) is often associated with the Bunsen burner, a device found in many chemistry laboratories around the word, but the truth is that he made a few alterations to it rather than inventing it. He was born in Göttingen, where his father was a professor of languages. He obtained his doctorate in chemistry at the University of Göttingen and spent years traveling through Western Europe. He eventually settled at the scenic university town of Heidelberg in south-west Germany, where he taught from 1852 until his retirement. In the late 1850s, Bunsen began a new and very fruitful collaboration there with the physicist Kirchhoff.

Gustav Kirchhoff (1824-1887), the son of a lawyer, was born and educated in Königsberg, Prussia, on the Baltic Sea, now the Russian city of Kaliningrad. He graduated from Albertus University there in 1847 and relocated to the rapidly growing city of Berlin. After 1850 he became acquainted with Bunsen, who urged him to follow him to Heidelberg. Kirchhoff in 1859 coined the term blackbody to describe a hypothetical perfect radiator that absorbs all incident light and emits all of that light when maintained at a constant temperature. His findings proved instrumental to Max Planck’s quantum theory of electromagnetic radiation from 1900. He is above all remembered for his collaboration with Bunsen around 1860.

They demonstrated in 1859 that all pure substances display a characteristic spectrum. Together, Bunsen and Kirchhoff assembled the flame, prism, lenses and viewing tubes necessary to produce the world’s first spectrometer. They identified the alkali metals cesium (chemical symbol Cs, atomic number 55) and rubidium (Rb, a.n. 37) in 1860-61, showing in each case that these new elements produced line spectra that were unique for them, a chemical “fingerprint.” The dark lines in the solar spectrum show the selective absorption of light, caused by the transition of an electron between specific energy levels in an atom, in the gases of various elements that exist above the Sun’s surface. In the first qualitative chemical analysis of a celestial body, Kirchoff in the 1860s identified 16 different elements from the Sun’s spectrum and compared these to laboratory spectra from known elements here on Earth.

The great physicist George Gabriel Stokes (1819-1903) attended school in Dublin, Ireland, but later moved to England and Cambridge University. He theorized a reasonably correct explanation of the Fraunhofer lines in the solar spectrum, but he did not publish it or develop it further. According to the Molecular Expressions website, “ Throughout his career, George Stokes emphasized the importance of experimentation and problem solving, rather than focusing solely on pure mathematics. His practical approach served him well and he made important advances in several fields, most notably hydrodynamics and optics. Stokes coined the term fluorescence, discovered that fluorescence can be induced in certain substances by stimulation with ultraviolet light, and formulated Stokes Law in 1852. Sometimes referred to as Stokes shift, the law holds that the wavelength of fluorescent light is always greater than the wavelength of the exciting light. An advocate of the wave theory of light, Stokes was one of the prominent nineteenth century scientists that believed in the concept of an ether permeating space, which he supposed was necessary for light waves to travel.”

Fluorescence microscopy has become an important tool in cellular biology. The Polish physicist Alexander Jablonski (1898-1980) at the University of Warsaw was a pioneer in fluorescence spectroscopy. Stokes was a formative influence on subsequent generations of Cambridge men and was one of the great names among nineteenth century mathematical physics, which included Michael Faraday, James Joule, Siméon Poisson, Augustin Cauchy and Joseph Fourier. The English mathematician George Green (1793-1841), known for Green’s Theorem, inspired Lord Kelvin and devised an early theory of electricity and magnetism that formed some of the basis for the work of scientists like James Clerk Maxwell.

Astrophysics as a scientific discipline was born in mid-nineteenth century Europe, and only there; it could not have happened earlier as the crucial combination of chemical and optical knowledge, telescopes and photography did not exist before. In case we forget what a huge step this was, let us recall that as late as the sixteenth century AD in Mesoamerica, the region with the most sophisticated American astronomical traditions, thousands of people had their hearts ripped out every year to please the gods and ensure that the Sun would keep on shining.

Merely three centuries later, European scholars could empirically study the composition of the Sun and verify that it was essentially made of the same stuff as the Earth, only much hotter. Within the next few generations, European and Western scholars would in less than a century proceed to explain how the Sun and the stars generate their energy and why they shine. By any yardstick, this represents one of the greatest triumphs of the human mind in history.
Photography was born in France in the 1820s with Joseph-Nicéphore Niépce, who teamed up with the painter Louis Daguerre. As Eva Weber writes in her book Pioneers of Photography, “In March 1839 Daguerre personally demonstrated his process to inventor and painter Samuel Morse (1791-1872) who enthusiastically returned to New York to open a studio with John Draper (1811-1882), a British-born professor and doctor. Draper took the first photograph of the moon in March 1840 (a feat to be repeated by Boston’s John Adams Whipple in 1852), as well as the earliest surviving portrait, of his sister Dorothy Catherine Draper.”

The American physician Henry Draper (1837-1882), son of John Draper, was a pioneer in astrophotography. In 1857 he visited Lord Rosse, or William Parsons (1800-1867), famous for his construction in Ireland in the 1840s of the most powerful reflecting telescope in the Victorian period, frequently called the Leviathan. It remained the world’s largest telescope until the early twentieth century, but was often shut down due to the wet Irish weather. Most major ground-based observatories after it have been built in the clear air of a remote mountaintop, from the peaks of Hawaii to the dry mountains of Chile in South America.

Henry Draper eventually became a passionate amateur astronomer. After reading about the work on star spectra carried out by William Huggins and Joseph Lockyer he built his own spectrograph. He died at a young age, but his widow established the Henry Draper Memorial. This funded the Henry Draper Catalog, a massive photographic stellar spectrum survey.

The first successful daguerrotype photography of the Sun was made in 1845 by the French physicists Louis Fizeau and Léon Foucault, who are mainly remembered for their accurate measurements of the speed of light. Warren de la Rue (1815-1889), a British-born astronomer, astrophotographer and chemist educated in Paris, designed a special telescope dubbed the photoheliograph. On an expedition to Spain in 1860 during a total solar eclipse, his images demonstrated clearly that the corona is a phenomenon associated with the Sun.

The technique of spectral analysis caught on after the work of Robert Bunsen and Gustav Kirchhoff. One of those who quickly took it up was the great English chemist William Crookes (1832-1919), who discovered the metal thallium (Tl, atomic number 81) in 1862. The Englishman William Huggins (1824-1910) built a private observatory in South London and tried to apply this method to other stars. Through spectroscopic methods he showed that they are composed of the same elements as the Sun and the Earth. He collaborated with his friend William Allen Miller (1817-1870), a professor of chemistry at King’s College, London.

According to his Bruce Medal biography, “Huggins was one of the wealthy British ‘amateurs’ who contributed so much to 19th century science. At age 30 he sold the family business and built a private observatory at Tulse Hill, five miles outside London. After G.R. Kirchhoff and R. Bunsen’s 1859 discovery that spectral emission and absorption lines could reveal the composition of the source, Huggins took chemicals and batteries into the observatory to compare laboratory spectra with those of stars. First visually and then photographically he explored the spectra of stars, nebulae, and comets. He was the first to show that some nebulae, including the great nebula in Orion, have pure emission spectra and thus must be truly gaseous, while others, such as that in Andromeda, yield spectra characteristic of stars. He was also the first to attempt to measure the radial velocity of a star. After 1875 his observations were made jointly with his talented wife, the former Margaret Lindsay Murray.”

The Austrian mathematical physicist Johann Christian Doppler (1803-1853) was born in Salzburg, the son of a stonemason, and studied in Vienna. In 1842 he proposed that observed frequency of light and sound waves is dependent upon how fast the source and observer are moving relative to each other, a phenomenon called the Doppler Effect. For instance, most of us have heard how the sound of a car or a train changes in frequency as it moves toward us and then away from us. A more correct explanation of the principle involved was published by the French physicist Armand-Hippolyte-Louis Fizeau in 1848. The Doppler Effect has proved to be an invaluable tool for astronomical research. Most notably, the motions of galaxies detected through this manner led to the conclusion that the universe is expanding.

In 1864, probably as a result of discussions with his countryman William Huggins, the astronomer Joseph Norman Lockyer (1836-1920), originally from the town of Rugby in the West Midlands of England, obtained a spectroscope. In 1868 he was able to confirm that bright emission lines from prominences of the Sun could be seen at times other than during total solar eclipses. The same technique had been demonstrated by the French astronomer Pierre Janssen (1824-1907). Janssen was born in Paris, where he studied mathematics and physics, and took part in a long series of solar eclipse-expeditions around the world. Lockyer and Janssen are credited with independently discovering helium (chemical symbol He, atomic number 2) in 1868 through studies of the solar spectrum. Helium, from the Greek Helios for the Sun, remains the only element so far discovered in space before being identified on the Earth. Lockyer was also the founder of the leading British scientific journal Nature in 1869.

While the center of astronomy was still in Western Europe, Europeans overseas were starting to leave their mark, above all in North America. The US physicist Henry Rowland (1848-1901) did notable work in spectroscopy, and the American astronomer Vesto Slipher (1875-1969) was the first person to measure the enormous radial velocities of spiral nebulae. As the excellent reference book The Oxford Guide to the History of Physics and Astronomy states:

“In 1868, however, Huggins found what appeared to be a slight shift for a hydrogen line in the spectrum of the bright star Sirius, and by 1872 he had more conclusive evidence of the motion of Sirius and several other stars. Early in the twentieth century Vesto M. Slipher at the Lowell Observatory in Arizona measured Doppler shifts in spectra of faint spiral nebulae, whose receding motions revealed the expansion of the universe. Instrumental limitations prevented Huggins from extending his spectroscopic investigations to other galaxies. Astronomical entrepreneurship in America’s gilded age saw the construction of new and larger instruments and a shift of the center of astronomical spectroscopic research from England to the United States. Also, a scientific education became necessary for astronomers, as astrophysics predominated and the concerns of professional researchers and amateurs like Huggins diverged. George Ellery Hale, a leader in founding the Astrophysical Journal in 1895, the American Astronomical and Astrophysical Society in 1899, the Mount Wilson Observatory in 1904, and the International Astronomical Union in 1919, was a prototype of the high-pressure, heavy-hardware, big-spending, team-organized scientific entrepreneur.”

George Ellery Hale (1868-1938), a university-educated solar astronomer born in Chicago, represented the dawn of a new age, not only because he was American and the United States would emerge as a leading center of astronomical research (although scientifically and technologically speaking it was a direct extension of the European tradition), but at least as much because he personified the increasing professionalization of science and astronomy.

The telescope Galileo used in the early 1600s, although revolutionary at the time, was a simple refractor. The sheer weight of the glass lens makes a refracting telescope larger than one meter in diameter impractical. The introduction of the reflecting mirror telescope by Newton in 1669 paved the way for virtually all modern ones. Hale built the largest telescope in the world no less than four times: Once at Yerkes Observatory, then the 60-and 100-inch reflectors at Mt. Wilson and the 200-inch reflector at Mt. Palomar. As an undergraduate student at the Massachusetts Institute of Technology, Hale co-invented the spectroheliograph, an instrument to photograph outbursts of gas at the edge of the Sun, and discovered that sunspots were regions of lower temperatures but strong magnetic fields. He hired Harlow Shapley and Edwin Hubble and encouraged research in astrophysics and galactic astronomy.

There is still room for non-professional astronomers; good amateurs can occasionally spot new comets before the professionals do. Yet it is a safe bet to say that never again will we have a situation like in the late eighteenth century when William Herschel, a musician by profession, was one of the leading astronomers of his age. From a world of a few enlightened and often wealthy gentlemen in the eighteenth century would emerge a world of trained scientists in the twentieth; the nineteenth century was a transitional period. As the example of William Huggins demonstrates, amateur astronomers were to enjoy a final golden age.

The English entrepreneur William Lassell (1799-1880) had made good money from brewing beer and used some of it to indulge his interest in astronomy, employing very good self-made instruments at his observatory near the city of Liverpool. Liverpool was the fastest-growing port in Europe, and the world’s first steam-hauled passenger railway ran from Liverpool to Manchester in 1830. The Industrial Revolution, where Britain played the leading role, was a golden age for the beer-brewing industry. The combination of beer and science is not unique; the seventeenth-century Polish astronomer Hevelius came from a brewing family, and the English scientific brewer James Joule seriously studied heat and the conservation of energy.

In 1846 William Lassell discovered Triton, the largest moon of Neptune, shortly after the planet had itself been mathematically predicted by the French mathematician Urbain Le Verrier and spotted by the German astronomer Johann Gottfried Galle. Lassel later discovered two moons around Uranus, Ariel and Umbriel; a satellite of Saturn, Hyperion, was spotted by him as well as the American father-and-son team William Bond (1789-1859) and George Bond (1825-1865). William Bond was a clockmaker in Boston who became a passionate amateur astronomer. In 1848, with his son George, he discovered Hyperion. They were among the first in the USA to use Daguerre’s photographic process for astrophotography.

The US astronomer Asaph Hall (1829-1907) discovered the two tiny moons of Mars, Deimos and Phobos, in 1877 and calculated their orbits. While only a few kilometers in diameter, the moons could be seen by viewers using smaller telescopes, which means their discovery owed as much to Hall’s observational skills as to his equipment. Asaph Hall was the son of a clockmaker and worked for a while with George Bond at the Harvard College Observatory.

Photos taken by the European Space Agency’s Mars Express spacecraft of Phobos, the larger of the two tiny, potato-shaped Martian moons, have showed potential landing sites for Russia’s unmanned Phobos-Grunt mission, which is designed to bring samples of the Martian moon back to the Earth after 2012. The Russian Space Agency intends to include a Chinese Mars orbiter, Yinghuo-1, together with the mission. It will be China’s first interplanetary probe. China in 2003 became only the third nation to achieve human spaceflight, after the Soviet Union/Russia and the United States, and has plans for manned missions to the Moon.

The largely self-taught American astronomer Edward Barnard (1857-1923), originally a poverty-stricken photographer, made his own telescope and after some notable observations joined the initial staff of the Lick Observatory in 1887. He introduced wide-field photographic methods to study the structure of the Milky Way. The faint Barnard’s Star, which he discovered in 1916, had the largest proper motion of any known star. At a distance of about six light-years it is the closest neighboring star to the Sun next to the members of the Alpha Centauri system, around 4.4 light-years away. In 1892 he observed Amalthea, the first Jovian moon to be discovered since the four largest ones described by Galileo Galilei in 1610. The astronomer Charles Perrine (1867-1951), born in the USA but based in Argentina for many years, discovered two additional moons around Jupiter: Himalia in 1904 and Elara in 1905.

The Swiss natural philosopher Pierre Prévost (1751-1839), the son of a clergyman from Geneva, Switzerland who served as a professor of physics at Berlin, showed in 1791 that all bodies radiate heat, regardless of their temperature.

Early estimates of stellar surface temperatures gave results that were far too high. More accurate values were obtained by using the radiation laws of the Slovenian physicist Joseph Stefan from 1879 and the German physicist Wilhelm Wien from 1896. Stefan calculated the temperature of the Sun’s surface to 5400 °C, the most sensible value until then. The Stefan-Boltzmann Law, named after Stefan and his Austrian student Ludwig Boltzmann, suggests that the amount of radiation given off by a body is proportional to the fourth power of its temperature as measured in Kelvin units.

The surface temperature is not necessarily dependent upon the size of the star (the core temperature is a different matter). You can easily find red supergiants with many times the mass of the Sun, but with a surface temperature of less than 4000 K, compared to the Sun’s 5800 or so K. The surface temperature of a bright red star is approximately 3500 K, whereas blue stars can have ones of tens of thousands of degrees. Dark red stars have surface temperatures of about 2500 K. Blue stars are extremely hot and bright and live short lives by astronomical standards. The bright star Rigel in the constellation of Orion is a blue supergiant of an estimated 20 solar masses, shining with tens of thousands of times the Sun’s luminosity.

If you heat an iron rod with an intense flame it will first appear “red hot.” Heated a little more it will seem orange and feel hotter and then yellow after that. After more heating, the rod will appear white-hot and brighter still. If it doesn’t melt, further heating will make the rod appear blue and even brighter and progressively hotter. The same basic principle applies to stars, too.

Stars, molten rock and iron bars are approximations of an important class of objects that physicists call blackbodies. An ideal blackbody absorbs all of the electromagnetic radiation that strikes it. Incoming radiation heats up the body, which then reemits the energy it has absorbed, but with different intensities at each wavelength than it received. This pattern of radiation emitted by blackbodies is independent of their chemical compositions. Authors Neil F. Comins and William J. Kaufmann III explain in Discovering the Universe, Eighth Edition:

“Ideal blackbodies have smooth blackbody curves, whereas objects that approximate blackbodies, such as the Sun, have more jagged curves whose variations from the ideal blackbody are caused by other physics. The total amount of radiation emitted by a blackbody at each wavelength depends only on the object’s temperature and how much surface area it has. The bigger it is, the brighter it is at all wavelengths. However, the relative amounts of different wavelengths (for example, the intensity of light at 750 nm compared to the intensity at 425 nm) depend on just the body’s temperature. So, by examining the relative intensities of an object’s blackbody curve, we are able to determine its temperature, regardless of how big or how far away it is. This is analogous to how a thermometer tells your temperature no matter how big you are.”

Photography made it possible to preserve images of the spectra of stars. The Catholic priest and astrophysicist Pietro Angelo Secchi (1818-1878), born in the city of Reggio Emilia in northern Italy, is considered the discoverer of the principle of stellar classification. He visited England and the USA and became professor of astronomy in Rome in 1849. After the introduction of spectrum analysis by Kirchhoff and Bunsen, Secchi was among the first to investigate the spectra of Uranus and Neptune. On an expedition to Spain to observe the total solar eclipse of 1860 he “definitively established by photographic records that the corona and the prominences rising from the chromosphere (i.e. the red protuberances around the edge of the eclipsed disc of the sun) were real features of the sun itself,” not optical illusions or illuminated mountains on the Moon. In the 1860s he began collecting the spectra of stars and classified them according to spectral characteristics, although his particular system didn’t last.

The Harvard system based on the star’s surface temperature was developed from the 1880s onward. Several of its creators were women. The US astronomer Edward Pickering (1846-1919) at the Harvard College Observatory hired female assistants, among them the Scottish-born Williamina Fleming (1857-1911) and especially Annie Jump Cannon (1863-1941) and Antonia Maury (1866-1952) from the USA, to classify the prism spectra of hundreds of thousands of stars. Cannon developed a classification system based on temperature where stars, from hot to cool, were of ten spectral types — O, B, A, F, G, K, M, R, N, S — that astronomers accepted for world-wide use in 1922. Maury developed a different system.

Edward Pickering and the German astronomer Hermann Karl Vogel (1841-1907) independently discovered spectroscopic binaries — double-stars that are too close to be detected through direct observation, but which through the analysis of their light have been found to be two stars revolving around one another. Vogel was born in Leipzig in what was then the Kingdom of Saxony, and died in Potsdam in the unified German Empire. He studied astronomy at the Universities of Leipzig and Jena, joined the staff of the Potsdam Astrophysical Observatory and served as its director from 1882 to 1907. Vogel made detailed tables of the solar spectrum, attempted spectral classification of stars and also made photographic measurement of Doppler shifts to determine the radial velocities of stars.

Another system was worked out in the 1940s by the American astronomers William Wilson Morgan (1906-1994) and Philip Keenan (1908-2000), aided by Edith Kellman. They introduced stellar luminosity classes. For the first time, astronomers could determine the luminosity of stars directly by analyzing their spectra, their “stellar fingerprints.” This is known as the MK (after Morgan and Keenan) or Yerkes spectral classification system after Yerkes Observatory, the astronomical research center of the University of Chicago. Morgan’s observational work helped to demonstrate the existence of spiral arms in the Milky Way.

Maury’s classifications were not preferred by Pickering, but the Danish astronomer Ejnar Hertzsprung (1873-1967) realized their value. As stated in his Bruce Medal profile, “Hertzsprung studied chemical engineering in Copenhagen, worked as a chemist in St. Petersburg, and studied photochemistry in Leipzig before returning to Denmark in 1901 to become an independent astronomer. In 1909 he was invited to Göttingen to work with Karl Schwarzschild, whom he accompanied to the Potsdam Astrophysical Observatory later that year. From 1919-44 he worked at the Leiden Observatory in the Netherlands, the last nine years as director. He then retired to Denmark but continued measuring plates into his nineties. He is best known for his discovery that the variations in the widths of stellar lines discovered by Antonia Maury reveal that some stars (giants) are of much lower density than others (main sequence or ‘dwarfs’) and for publishing the first color-magnitude diagrams.”

The American astronomer Henry Norris Russell (1877-1957) spent six decades at Princeton University as a student, professor and observatory director. From 1921 on he made annual visits to the Mt. Wilson Observatory. “He measured parallaxes in Cambridge, England, with A.R. Hinks and found a correlation between spectral types and absolute magnitudes of stars — the Hertzsprung-Russell diagram. He popularized the distinction between giant stars and ‘dwarfs’ while developing an early theory of stellar evolution. With his student, Harlow Shapley, he analyzed light from eclipsing binary stars to determine stellar masses. Later he and his assistant, Charlotte E. Moore Sitterly, determined masses of thousands of binary stars using statistical methods. With Walter S. Adams Russell applied Meghnad Saha’s theory of ionization to stellar atmospheres and determined elemental abundances, confirming Cecilia Payne-Gaposchkin’s discovery that the stars are composed mostly of hydrogen. Russell applied the Bohr theory of the atom to atomic spectra and with Harvard physicist F.A. Saunders made an important contribution to atomic physics, Russell-Saunders coupling (also known as LS coupling).”

Herztsprung had discovered the relationship between the brightness of a star and its color, but published his findings in a photographic journal which went largely unnoticed. Russell made essentially the same discovery, but published it in 1913 in a journal read by astronomers and presented the findings in a graph, which made them easier to understand. The Hertzsprung-Russell diagram helped give astronomers their first insight into the lifecycle of stars. It can be regarded as the Periodic Table of stars. The Indian astrophysicist Meghnad Saha (1893-1956) provided a theoretical basis for relating the spectral classes to stellar surface temperatures.

Changes in the structure of stars are reflected in changes in temperatures, sizes and luminosities. The smallest ones, red dwarfs, may contain less than 10% the mass of the Sun and emit 0.01% as much energy. They constitute by far the most numerous types of stars and have lifespans of tens of billions of years. By contrast, the rare hypergiants may exceed 100 solar masses and emit hundreds of thousands of times more energy than the Sun, but they also have lifetimes of just a few million years. Those that are actively fusing hydrogen into helium in their cores, which means most of them, are called main sequence stars. These are in hydrostatic equilibrium, which means that the outward radiation pressure from the fusion process is balanced by the inward gravitational force. When the hydrogen fuel runs out, the core contracts and heats up. The star then brightens and expands, becoming a red giant.

The Eddington Limit, named after the English astrophysicist Arthur Eddington, is the point at which the luminosity emitted by a star is so extreme that it starts blowing off its outer layers. It was thought to be reached in stars around 120-140 solar masses. In the early stages of the universe, extremely massive stars containing hundreds of solar masses may have been able to form because they contained practically no heavy elements, just hydrogen and helium. Wolf-Rayet stars are very hot, luminous and massive objects that eject significant proportions of their mass through solar wind per year. They are named after the French astronomers Charles Wolf (1827-1918) and Georges Rayet (1839-1906), who discovered their existence in 1867.

It is highly likely that there is an upper limit to how large stars can become, but we do not yet know precisely how big this limit is. It was once believed to be around 150 solar masses for stars existing today, but astronomers have come across one at more than 300 solar masses. The star R136a1 is the most massive one ever observed and also has the highest luminosity of any star found to date — almost 10 million times greater than the Sun. It was discovered in 2010 inside two young clusters of stars by a European research team led by Paul Crowther, professor of astrophysics at the University of Sheffield in England. The theoretical models we currently operate with cannot fully explain the evolution of such extremely massive objects.

A common, medium-sized star like the Sun will remain on the main sequence for roughly 10 billion years. The Sun is currently in the middle of its lifespan, as it formed 4.57 billion years ago and in about 5 billion years will turn into a red giant. Even today, the Sun daily emits an estimated 30% more energy than it did when it was born. The so-called faint young Sun paradox, proposed by Carl Sagan and his colleague George Mullen in the USA in 1972, refers to the fact that the Earth apparently had liquid oceans, not frozen ones, for much of the first half of its existence, despite the fact that the Sun probably was only 70 percent as bright in its youth as it is now. Scientists have not yet reached an agreement on why this was the case.

The magnetic field of the Sun can be probed because in the presence of a magnetic field the energy levels of atoms and ions are split into more than one level, which causes spectral transition lines to be split as well. This is called the Zeeman Effect after the Dutch physicist Pieter Zeeman (1865-1943). Spectroscopic studies of the Sun by the American astronomer Walter Adams (1876-1956) with Hale and others at the Mt. Wilson Solar Observatory led to the insight that sunspots are regions of lower temperatures and stronger magnetic fields than their surroundings. The spectroheliograph for studying the Sun was developed independently by George Ellery Hale and by the talented French astrophysicist Henri-Alexandre Deslandres (1853-1948), working at the Paris Observatory around 1890. Sunspots appear dark to us because they are cooler than other solar regions, but in reality they are red or orange in color.

Richard Carrington and Edward Sabine in Britain in the 1800s had suggested a possible link between the occurrences of solar flares and observations of auroras and geomagnetic storms on the Earth. It takes a day or two for the charged particles of the solar wind to travel from the Sun to the Earth. This is obviously very fast, yet significantly slower than the roughly 8 minutes and 20 seconds that it takes for light to travel the same distance. This indicated that something other than light travels from the Sun to us. Following work by Kristian Birkeland from Norway, the English geophysicist Sydney Chapman and the German astronomer Ludwig Biermann, Eugene Parker in the USA created a coherent model for the solar wind in 1958.

Progress in mapping the Sun’s magnetic field was made in the mid-twentieth century by an American father-and-son team. The prominent solar astronomer Harold D. Babcock (1882-1968) studied spectroscopy and the magnetic fields of stars. Horace W. Babcock (1912-2003) was his son. The two Babcocks were the first to measure the distribution of magnetic fields over the solar surface. These fields change polarity every 11-year cycle, indicating that solar activity varies with a period of around 22 years. They developed important models of sunspots and their magnetism. In the early 1950s, Horace Babcock was the first person to propose adaptive optics, a methodology that provides real-time corrections with deformable mirrors to remove the blurring of ground-based astronomical images caused by turbulence in the Earth’s atmosphere. Adaptive optics works best at longer wavelength such as infrared.

To learn more about the Sun’s interior, astronomers record its vibrations — a study called helioseismology. In principle this is related to how geophysicists use seismic waves to study the interior of the Earth. While there are no true sunquakes, the Sun does vibrate at a variety of frequencies, which can be detected. Its magnetic field is believed to be created as a result of its rotation and the resulting motion of the ionized particles found throughout its body.

As we have seen, it was possible for European astrophysicists in the late 1800s to detect the presence of elements such as hydrogen in the Sun, but they did not yet know how big a percentage of its mass consisted of hydrogen. In the 1920s, many scientists still assumed that it was rich in heavy elements. This changed with the work of the English-born astronomer Cecilia Payne, later named Payne- Gaposchkin (1900-1979) when she married a Russian astronomer. Her interest in astronomy was triggered after she heard Arthur Eddington lecture on relativity. She joined the Harvard College Observatory in the USA. By using spectroscopy, Payne worked out that hydrogen and helium are the most abundant elements in stars. Otto Struve (1897-1963), a Russian astronomer of ethnic German origins, called her thesis Stellar Atmospheres from 1925 “the most brilliant Ph.D. thesis ever written in astronomy.”

The Irish astronomer William McCrea (1904-1999) and the German astrophysicist Albrecht Unsöld (1905-1995) independently established that the prominence of hydrogen in stellar spectra indicates that the presence of hydrogen in stars is greater than that of all other elements put together. Unsöld studied under the German theoretical physicist Arnold Sommerfeld at the University of Munich and began working on stellar atmospheres in 1927.

The English mathematical physicist James Jeans (1877-1946) worked on thermodynamics, heat and aspects of radiation, publishing major works on these topics and their applications to astronomy. The English astrophysicist and mathematician Arthur Milne (1896-1950) did research in the 1920s on stellar atmospheres, much of it with his English colleague Ralph H. Fowler (1889-1944). This led to the determination of the temperatures and pressures associated with spectral classes, explaining the origin of stellar winds. The astronomer Marcel Minnaert (1893-1970) was forced to flee from his native Belgium to the Netherlands for taking part in Flemish activism. From 1937 to 1963 he was director of the Utrecht Observatory, where he and his students did quantitative analysis of the solar spectrum.

Newton had speculated on the energy source of the Sun. He assumed that it loses mass by emitting light particles and suggested that incoming comets could provide it with more mass to compensate for this. The French physicist Claude Pouillet (1791-1868) in 1837 calculated a decent estimate of the energy emitted by the Sun. However, this would require a mass almost the equivalent of the Earth’s Moon to hit the Sun every year, which was clearly not the case.

In 1854, Hermann von Helmholtz suggested that the Sun was contracting and converting potential energy into radiated energy. This Kelvin-Helmholtz mechanism of gravitational contraction, named after Lord Kelvin and Helmholtz, is relevant for planets like Jupiter, which emits approximately twice as much energy as it receives from the Sun. Its energy comes from radioactive elements in its core and from an overall contraction amounting to a few centimeters per century. Although the required rate of solar contraction was 91 meters per year, this mechanism would have implied an impossible reduction of the Sun’s diameter with 50% over 5 million years. Nineteenth-century physicists were partially right, though; the initial release of gravitational energy ignites nuclear fusion in stars by heating up their cores.

In everyday language we say that stars “burn,” but this should not be taken literally. Burning a fuel — solid, liquid or gas — is called combustion, a chemical process normally involving oxygen in which a substance reacts rapidly with oxygen and gives off heat. The source of oxygen is called the oxidizer. Rocket engines, internal combustion engines and jet engines all depend on the burning of fuel to produce power. The combustion of liquid hydrogen and liquid oxygen is a commonly used reaction in rocket engines. The result is water vapor, H2O. The reason why water cannot burn is because water is already “burnt,” chemically speaking.

In a common fireplace or campfire, the carbon in the wood will combine with oxygen gas in the air to produce heat and carbon dioxide, CO2. Chemical combustion was considered a possible source of solar energy by European scientists in the 1800s, but was eventually rejected as it would have burnt away the entire Sun in a few thousand years. Solar energy is produced in a radically different way, not by combing various elements through normal chemical processes, but by producing new chemical elements through nuclear processes.

The discovery of radioactivity in 1896 suddenly provided a new source of heat. In 1905 Albert Einstein generalized the law of conservation of energy with his famous mass-energy equivalence formula E = mc2, where E stands for energy, m for mass and c for the speed of light in a vacuum. Since the speed of light is very great, the formula implies that very little mass is required to generate huge amounts of energy. The English physical chemist Francis William Aston with his mass spectrograph in 1920 made precise measurements of different atoms. He found that four individual hydrogen nuclei were more massive than a helium nucleus consisting of four nuclear particles. Arthur Eddington argued that these measurements indicated that by converting hydrogen nuclei to helium and releasing about 0.7% of the hydrogen’s mass as energy in the process, the Sun could shine for billions of years.

The great English astrophysicist Arthur Stanley Eddington (1882-1944) was born to Quaker parents and earned a scholarship to Owens College, Manchester, in 1898. He turned to physics and went to Trinity College at the University of Cambridge. He spent seven years (1906 to 1913) as chief assistant at the Royal Observatory at Greenwich. He took inspiration from the Hertzsprung-Russell diagram, made important investigations of stellar dynamics and became an influential supporter of the view that the spiral nebulae are external galaxies. Eddington’s greatest contributions concerned astrophysics. He dealt with the importance of radiation pressure, the mass-luminosity relation and investigated the internal structure and evolution of stars. He wrote several books, some of them for the general reader. His The Internal Constitution of the Stars from 1926 was extremely influential to a generation of astrophysicists. He was one of the first to provide observational support for Einstein’s general theory of relativity from 1916 and explain it to a mass audience. Eddington was also among the first to suggest that processes at the subatomic level involving hydrogen and helium could explain why stars generate energy, but it was left for other scientists to work out the details.

The Sun has a mass of about 1.989×1030 kg, roughly 333 thousand times more than the mass of the Earth, and a mean density of 1408 kg/m³ or 1.408 kg/dm³, a little bit more than water. Its equatorial radius (distance from its center to its surface) is 695,500 kilometers, approximately 109 times Earth’s radius. The energy per time put out by the Sun, its luminosity, is more than 3.8 x 1026 Joules per second (or Watts). The amount of mass that the Sun converts into energy equals more than 4 million metric tons, or 4 billion kg, per second.

The theoretical physicist George Gamow (1904-1968) was born in the seaport city of Odessa in the Russian Empire (now the Ukraine) on the northern shore of the Black Sea. His father came from a military family and was a teacher of Russian literature in high school; his mother’s father was Archbishop of Odessa from the Orthodox Church. At the University of Leningrad he studied briefly under the Russian mathematician Alexander Friedmann, who was interested in the mathematics of relativity. After completing his Ph.D. in 1928, Gamow worked on quantum mechanics at Göttingen, Copenhagen and Cambridge. He couldn’t endure the brutal oppression under the Communist dictator Joseph Stalin, but fled the Soviet Union and moved to the United States in 1934. Gamow introduced nuclear theory into cosmology.

According to classical physics, two particles with the same electrical charge will repel each other. In 1928, Gamow derived a quantum-mechanical formula that gave a non-zero probability of two charged particles overcoming their mutual electrostatic repulsion and coming very close together. It is now known as the “Gamow factor.” The Dutch-Austrian nuclear physicist Fritz Houtermans (1903-1966) together with his British colleague Robert Atkinson (1898-1982) in 1929 predicted that the nuclei of light atoms such as hydrogen could fuse through quantum tunneling, and that the resultant atoms would have slightly less mass than the original constituents. This loss in mass would be released as vast amounts of energy.

The final major piece of the puzzle was the structure of the atom itself. When the neutron had been detected by the Englishman James Chadwick in 1932, physicists finally had sufficient information about the atomic nucleus to calculate the details of how hydrogen can fuse to become helium. This nuclear fusion process was worked out independently by two German-born physicists in the late 1930s: Hans Bethe in the USA and Carl von Weizsäcker in Berlin.

Carl Friedrich Freiherr von Weizsäcker (1912-2007) was born in Kiel, Germany to a prominent family; his father was a German diplomat, and his elder brother would later become German President. He studied physics and astronomy in Berlin, Göttingen and Leipzig (1929-1933) and was supervised by prominent nuclear physicists such as Werner Heisenberg and Niels Bohr. After the Second World War he was appointed head of a department at the Max Planck Institute for Physics in Göttingen, and from 1957 to 1969 he was Professor for Philosophy at the University of Hamburg in northern Germany.

Hans Bethe (1906-2005) studied at the Universities of Frankfurt and Munich, where he earned his Ph.D. under the great German theoretical physicist Arnold Sommerfeld in 1928. He was forced to leave Germany after Adolf Hitler and the Nazi Party came to power there in 1933 since his mother was Jewish, although he had been raised as a Christian by his father. Bethe was at Cornell University in the USA from 1935 to 2005 and became an American citizen in 1941. Weizsäcker was a member of the team that performed nuclear research in Germany during WW2, while Bethe became the head of the theoretical division at Los Alamos during the development of nuclear weapons in the United States. In stellar physics, both men described the proton-proton chain, which is the dominant energy source in stars such as our Sun or smaller, and the carbon-nitrogen-oxygen (CNO) cycle. Author John North writes:

“It was not until 1938, when attending a Washington conference organized by Gamow, that he was first persuaded to turn his attention to the astrophysical problem of stellar energy creation. Helped by Chandrasekhar and Strömgren, his progress was astonishingly rapid. Moving up through the periodic table, he considered how atomic nuclei would interact with protons. Like Weizsäcker, he decided that there was a break in the chain needed to explain the abundances of the elements through a theory of element-building. Both were stymied by the fact that nuclei with mass numbers 5 and 8 were not known to exist, so that the building of elements beyond helium could not take place….Like Weizsäcker, Bethe favored the proton-proton reaction chain and the CNO reaction cycle as the most promising candidates for energy production in main sequence stars, the former being dominant in less massive, cooler, stars, the latter in more massive, hotter, stars. His highly polished work was greeted with instant acclaim by almost all of the leading authorities in the field.”

Bengt Strömgren (1908-1987) was an astrophysicist from Denmark, the son of a Swedish astronomer. He studied in Copenhagen and stayed in touch with the latest developments in physics via Niels Bohr’s Institute there. Strömgren did important research in stellar structure in the 1930s and calculated the relative abundances of the elements in the Sun and other stars.

The nineteenth century German astronomer Friedrich Bessel was the first to notice minor deviations in the motions of the bright stars Sirius and Procyon, which he correctly assumed must be caused by the gravitational attraction of unseen companions. The existence of these bodies was later confirmed. Bessel was also first person to clearly measure stellar parallax in 1838, an achievement which was independently made by the Baltic German astronomer Friedrich Georg Wilhelm von Struve and the Scottish astronomer Thomas Henderson.

In 1862 the American telescope maker Alvan Graham Clark (1832-1897) discovered the very faint companion Sirius B. Because the companion was about twice as far as Sirius from their common center of mass, it had to weigh about half as much (like a child twice as far from the center of a see-saw balancing an adult). The American astronomer Walter Sydney Adams (1876-1956) in 1915 identified Sirius B as a white dwarf star, a very dense object about the size of the Earth but with roughly the same mass as the Sun. Our Sun will eventually end up as a white dwarf billions of years from now, after first having gone through a red giant phase where it will expand greatly in volume and vaporize Mercury, Venus and possibly the Earth.

The astrophysicist Subrahmanyan Chandrasekhar (1910-1995) was born in Lahore into a Tamil Hindu family and got a degree at the University of Madras in what was then British-ruled India. After receiving a scholarship he studied at the University of Cambridge in England and came to the University of Chicago in the United States in 1937, where he remained for the rest of his life. NASA’s Chandra X-ray Observatory from 1999 was named after him. He is remembered above all for his contributions to the subject of stellar evolution.

He was the nephew of the physicist Chandrasekhara Venkata Raman (1888-1970) from Madras, whose discovery of the Raman Effect in 1928, the change in wavelength of light when it is deflected by molecules, “greatly impacted future research regarding molecular structure and radiation.” Raman was knighted by the British in 1929 and won the Nobel Prize in Physics in 1930, the first non-European to win a science Nobel. Rabindranath Tagore (1861-1941), the great Bengali writer and artist from India, had earlier been awarded the Nobel Prize for Literature in 1913. Chandrasekhar shared the Nobel Prize in Physics in 1983.

In 1930, Chandrasekhar applied the new quantum ideas to the physics of stellar structure. He realized that when a star like the Sun exhausts its nuclear fuel it will collapse due to its own gravity until stopped by the Pauli Exclusion Principle, which prevents electrons from getting too close to one another. Stars more massive than the Chandrasekhar Limit of 1.44 solar masses do not stabilize at the white dwarf stage but become neutron stars. The upper limit for a neutron star before it collapses further is called the Oppenheimer-Volkoff Limit after J. Robert Oppenheimer (1904-1967) from the USA and the Russian-born Canadian physicist George Volkoff (1914-2000). The O-V Limit is less certain, but is estimated at approximately three solar masses. Stars of greater mass than this are believed to end up as black holes.

The process of combining light elements into heavier ones — nuclear fusion — happens in the central region of stars. In their extremely hot cores, instead of individual atoms you have a mix of nuclei and free electrons, or what we call plasma. The term “plasma” was first applied to ionized gas by Irving Langmuir (1881-1957), a physical chemist from the USA, in 1923. It is the fourth and by far the most common state of matter in the universe in addition to those we are familiar with from everyday life on Earth: solid, liquid and gas. Extreme temperatures and pressure is needed to overcome the mutual electrostatic repulsion of positively charged atomic nuclei (ions), often called the Coulomb barrier after the French natural philosopher Charles de Coulomb, who formulated the laws of electrostatic attraction and repulsion.

While their work represented a huge conceptual breakthrough, the initial theories of Weizsäcker and Bethe did not explain the creation of elements heavier than helium. Edwin Ernest Salpeter (1924-2008) was an astrophysicist who emigrated from Austria to Australia, studied at the University of Sydney and finally ended up at Cornell University in the USA, where he worked in the fields of quantum electrodynamics and nuclear physics with Hans Bethe. In 1951 he explained how with the “triple-alpha” reaction, carbon nuclei could be produced from helium nuclei in the nuclear reactions within certain large and hot stars.

The fusion of hydrogen to helium by the proton-proton chain or CNO cycle requires temperatures in the order of 10 million degrees Celsius or Kelvin. Only at those temperatures will there be enough hydrogen ions in the plasma with high enough velocities to tunnel through the Coulomb barrier at sufficient rates. There are no stable isotopes of any element with atomic masses 5 or 8; beryllium-8 (4 protons and 4 neutrons) is highly unstable and short-lived. Only at extremely high temperatures of around 100 million K can the sequence called the triple-alpha process take place. It is so called because its net effect is to combine 3 alpha particles, which means standard helium-4 nuclei of two protons and two neutrons, to form a carbon-12 nucleus (6 protons and 6 neutrons). In main sequence stars, the central temperatures are too low for this process to take place, but not in stars in the red giant phase.

Further advances were made by the English astrophysicist Fred Hoyle (1915-2001). He was born in Yorkshire in northern England and educated in mathematics and theoretical physics at the University of Cambridge by some of the leading scientists of his day, among them Arthur Eddington and Paul Dirac. During World War II he contributed to the development of radar. With the German American astronomer Martin Schwarzschild (1912-1997), son of the astrophysicist Karl Schwarzschild and a pioneer in the use of electronic computers and high-altitude balloons to carry scientific instruments, he developed a theory of the evolution of red giant stars. Hoyle stayed at Cambridge from 1945 to 1973. In addition to his career in physics he was known for his popular science works and wrote novels, plays and short stories. He attributed life on Earth to an infall of organic matter from space. He was controversial throughout his life for supporting many highly unorthodox ideas, yet he made indisputable contributions to our understanding of stellar nucleosynthesis and together with a few others convincingly demonstrated how heavy elements can be created during supernova explosions.

The English astrophysicist Margaret Burbidge (born 1919) was educated at the University of London. She worked in the USA for a long time, but also served as director of the Royal Greenwich Observatory in her native Britain. She studied the spectra of galaxies, determining their masses and chemical composition, and married fellow Englishman Geoff Burbidge (1925-2010). He was educated at the University of Bristol and at University College, London, where he earned a Ph.D. in theoretical physics. The American astrophysicist William Alfred Fowler (1911-1995) earned his B.S. in engineering physics at Ohio State University and his Ph.D. in nuclear physics at the California Institute of Technology. He and his colleagues at Caltech measured the rates of nuclear reactions of astrophysical interest. After 1964, Fowler worked on problems involving supernovae and the formation of lighter chemical elements.

Building on the work of Hans Bethe, Hoyle in 1957 co-authored with Fowler and the husband-and-wife team of Geoffrey and Margaret Burbidge the paper Synthesis of the Elements in Stars. They demonstrated how the cosmic abundances of all heavier elements from carbon to uranium could be explained as the result of nuclear reactions in stars. Yet out of these four individuals, William Fowler alone shared the Nobel Prize in Physics in 1983 for work on the evolution of stars. By then Fred Hoyle was known for, among many other controversial ideas, attributing influenza epidemics to viruses carried in meteor streams.

The Canadian scientist Alastair G. W. Cameron (1925-2005) further aided our understanding of these stellar processes. Astrophysicists spent the 1960s and 70s establishing detailed descriptions of the internal workings of stars. Chushiro Hayashi (,1920-2010), educated at the University of Tokyo, together with his students made valuable contributions to stellar models. He found that pre-main-sequence stars follow what are now called “Hayashi tracks” downward on the Hertzprung-Russell diagram until they reach the main sequence. He was a leader in building astrophysics as a discipline in Japan. The Armenian scientist Victor Ambartsumian (1908-1996) was a pioneer in astrophysics in the Soviet Union, studied stellar evolution and hosted international conferences to search for extraterrestrial civilizations.

The important subatomic particles that are called neutrinos entered physics as a way to understand beta decay, the process by which a radioactive atomic nucleus emits an electron. Experiments showed that the total energy of the nucleus plus the electron was less than that of the initial nucleus. The Austrian physicist Wolfgang Pauli in 1930, trusting the principle of energy conservation, proposed that an unknown particle carried off missing energy. If it existed it had to be electrically neutral, possess nearly zero mass and move at close to the speed of light. Enrico Fermi named it the neutrino, meaning “little neutral one” in Italian.

Because of their extremely weak interactions with matter, neutrinos are difficult to detect; billions of them are thought to be going through your body every second. Their existence was confirmed in 1956 through experiments with tanks containing hundreds of liters of water by the scientists Frederick Reines (1918-1998) and Clyde Cowan (1919-1974) in the USA. This great achievement was decades later rewarded with a well-deserved Nobel Prize in Physics.

Scientists realized that nuclear reactions in stars should produce vast amounts of neutrinos, which might provide us with valuable information about places and physical processes that are otherwise hard to observe. Whereas light is easily absorbed as it moves through space, neutrinos rarely interact with anything and unlike many other particles have no charge, so they travel in a straight line from their source without being deflected by magnetic fields. In 1967, the physicist Raymond Davis, Jr. (1914-2006) installed a large tank of cleaning fluid in a deep gold mine in South Dakota in the United States. It was a prototype of sensitive detectors that were normally placed in abandoned mineshafts or other places deep below the Earth’s surface, in sharp contrast to optical telescopes placed on dry mountaintops.

Neutrino observatories have since then been built in many remote places, from the bottom of the Mediterranean Sea to IceCube at the Amundsen-Scott South Pole Station in Antarctica. The Russian researcher Moisey Markov (1908-1994) in the Soviet Union around 1960 suggested using natural bodies of water as neutrino detectors. By the 1980s, Russians realized that they had a massive tank of pure water in their own backyard: Lake Baikal. It contains 20 percent of the world’s unfrozen freshwater and has been isolated from other lakes and oceans for a very long time, leading to the evolution of a unique local flora and fauna. Russians regard it as their Galápagos. The neutrino telescope there operates underwater all year round.

Several tons of lead from an ancient Roman shipwreck has been transferred from a museum on the island of Sardinia to the Italian national particle physics laboratory at Gran Sasso. Once destined to become water pipes, coins or ammunition for the slingshots of Roman soldiers the lead in the ingots, which has lost almost all traces of its radioactivity, will instead form part of experiments to nail down the mass of neutrinos. The Kamiokande detector in the Japanese Alps has been important for similar studies. Raymond Davis and the great Japanese physicist Masatoshi Koshiba (born 1926) shared the 2002 Nobel Prize in Physics for work on neutrinos.

In the 1990s, Japanese and American scientists obtained experimental evidence indicating that neutrinos have non-zero mass, yet it is extremely small even compared to electrons. Neutrinos are light elementary particles, but because there are so many of them their tiny masses can add up to influence the overall distribution of galaxies. A neutrino’s mass is currently believed to be no more than 0.28 electron volts, less than a billionth of the mass of a hydrogen atom, but the value is not yet established with certainty and may turn out to be slightly higher than this.

From the 1960s to about 2002, scientists struggled to explain why the number of observed neutrinos from the Sun appeared to be less than had been predicted. The mystery of the “missing solar neutrinos” was solved when it was understood that neutrinos can change type, and that certain types are more challenging to detect than others. After these adjustments had been made, the number of observed solar neutrinos closely matched theoretical predictions, which indicated that our understanding of nuclear processes within stars like the Sun is reasonably accurate. As the American neutrino physicist John N. Bahcall (1934-2005) put it:

Link Text 1% error in the [Sun’s central] temperature corresponds to about a 30% error in the predicted number of neutrinos; a 3% error in the temperature results in a factor of two error in the neutrinos. The physical reason for this great sensitivity is that the energy of the charged particles that must collide to produce the high-energy neutrinos is small compared to their mutual electrical repulsion. Only a small fraction of the nuclear collisions in the Sun succeed in overcoming this repulsion and causing fusion; this fraction is very sensitive to the temperature. Despite this great sensitivity to temperature, the theoretical model of the Sun is sufficiently accurate to predict correctly the number of neutrinos.”

Neutrinos have emerged as an important tool for astrophysicists. 1987 was a landmark year in neutrino astronomy, with the first naked-eye supernova seen since 1604. That event, called SN1987A, took place in our galactic neighbor the Large Magellanic Cloud. The two most sensitive neutrino observatories in the world, one in Japan and another in the USA, detected a 12-second burst of neutrinos roughly three hours before the supernova became optically visible, which, again, seemed to match theoretical predictions for such events pretty well.

In 1911 the American astronomer Edward Pickering differentiated between low-energy novae, often seen in the Milky Way, and novae seen in other nebulae (galaxies) like Andromeda. By 1919, the Swedish astronomer Knut Lundmark (1889-1958) had realized that low-energy novae occur commonly whereas the brighter novae, which are vastly more luminous, occur rarely. The challenge was to explain the difference between them. In 1981, Gustav A. Tammann from Switzerland estimated that three supernovae occur every century in the Milky Way, yet most of them go undetected owing to obscuring interstellar material.

A nova (pl. novae) is a nuclear explosion caused by the accretion of hydrogen from a nearby companion onto the surface of a white dwarf star, which briefly reignites its nuclear fusion process until the hydrogen is gone. From the Earth we will see what appears to be a nova (“new” in Latin), but in reality it is an old star undergoing an eruption. It is possible for a star to become a nova repeatedly as this process does not destroy it, unlike a supernova event which obliterates a massive star in a cataclysmic explosion. A supernova explosion can release extraordinary amounts of energy and for a limited period outshine an entire galaxy.

If a white dwarf gains so much more additional mass that it exceeds the Chandrasekhar Limit of about 1.44 solar masses, electron degeneracy pressure can no longer sustain it. The star will then collapse and explode in a so-called Type Ia supernova. Since this limit is held to be constant, these supernovas have been used as standard candles to measure cosmic distances. Observations of such supernovas were used in 1998 to demonstrate that the expansion of our universe appears to be accelerating. However, some observations indicate that such events can be triggered by two white dwarves colliding, which might make them slightly less reliable as uniform standard candles since the weight limit could be less constant than was once believed.

The neutron was discovered in 1932. Shortly after this, the German-born Walter Baade (1893-1960) and the Swiss astronomer Fritz Zwicky (1898-1974), both eventually based in the USA, proposed the existence of neutron stars. Zwicky had a number of brilliant teachers at the ETH in Zürich, including Herman Weyl, Auguste Piccard and Peter Debye, but left Switzerland for the USA and the California Institute of Technology in 1925 to work with Robert Millikan.

Zwicky was not as systematic a thinker as Baade, but he could have excellent intuitive ideas. He was a bold and visionary scientist, but also eccentric and not always easy to work with. He stated that “Astronomers are spherical bastards. No matter how you look at them they are just bastards.” His colleagues did not appreciate his often aggressive attitude, but he was friendly toward students and administrative staff. In the words of the English-born physicist Freeman Dyson, “Zwicky’s radical ideas and pugnacious personality brought him into frequent conflict with his colleagues at Caltech. They considered him crazy and he considered them stupid.”

Educated at Göttingen, Walter Baade worked at the Hamburg Observatory in Germany from 1919 to 1931 and at the Mount Wilson Observatory outside of Los Angeles, California, from 1931 to 1958. During the World War II blackouts, Baade used the large Hooker telescope to resolve stars in the central region of the Andromeda Galaxy for the first time. This led to the realization that there were two kinds of Cepheid variable stars and from there to a doubling of the assumed scale of the universe. The German American astronomer Rudolph Minkowski (1895-1976) joined with him in studying supernovae. He was a nephew of the German Jewish mathematician Hermann Minkowski, who did important work on four-dimensional spacetime.

The optician Bernhard Schmidt (1879-1935) was born off the coast of Tallinn, Estonia, in the Baltic Sea, then a part of the Russian Empire. He spoke Swedish and German and spent most of his adult life in Germany. During a journey to Hamburg in 1929 he discussed the possibility of making a special camera for wide angle sky photography with Walter Baade. He then developed the Schmidt camera and telescope in 1930, which permitted wide-angle views with little distortion and opened up new possibilities for astronomical research. Yrjö Väisälä (1891-1971), a meteorologist, astronomer and instrument maker from Finland, had been working on a related design before Schmidt but left the invention unpublished at the time.

Zwicky and Baade introduced the term “supernova” and suggested that these events are completely different from ordinary novae. They proposed that after the turbulent collapse of a massive star, the residue of which would be an extremely compact neutron star, there would still be a large amount of energy left over. According to the book Cosmic Horizons:

Baade knew of several historical accounts of ‘new stars’ that had appeared as bright naked eye objects for several months before fading from view. The Danish astronomer Tycho Brahe, for example, had made careful observations of one in 1572. Zwicky and Baade thought that such events must be supernova explosions in our own Galaxy. At a scientific conference in 1933, they advanced three bold new ideas: (1) massive stars end their lives in stupendous explosions which blow them apart, (2) such explosions produce cosmic rays, and (3) they leave behind a collapsed star made of densely-packed neutrons. Zwicky reasoned that the violent collapse and explosion of a massive star would leave a dense ball of neutrons, formed by the crushing together of protons and electrons. Such an object, which he called a ‘neutron star,’ would be only several kilometers across but as dense as an atomic nucleus. This bizarre idea was met with great skepticism. Neutrons had only been discovered the year before. The notion that an entire star could be made of such an exotic form of matter was startling, to say the least.”

Astronomers readily accepted supernovas but remained doubtful about neutron stars for many years, believing that such strange objects were unlikely to exist in real life. To transform protons and electrons into neutrons, the density would have to approach the incredible density of an atomic nucleus, about 1017 kg/m3. A neutron star of twice the mass of our Sun would have a diameter of only 20 kilometers and would therefore fit inside any major city on Earth. Despite the name, these objects are probably not composed solely of neutrons. As authors Neil F. Comins and William J. Kaufmann III state in their book Discovering the Universe:

“Its interior has a radius of about 10 km, with a core of superconducting protons and superfluid neutrons. A superconductor is a material in which electricity and heat flow without the system losing energy, whereas a superfluid has the strange property that it flows without any friction. Both superconductors and superfluids have been created in the laboratory. Surrounding a neutron star’s core is a layer of superfluid neutrons. The surface of the neutron star is a solid, brittle crust of dense nuclei and electrons about ?-km thick. The gravitational force of the neutron star is so great at its surface that climbing a bump there just 1-mm high would take more energy than it takes to climb Mount Everest. Neutron stars may also have atmospheres, as indicated by absorption lines in the spectrum of at least one of them.”

Neutron stars were first observed in the 1960s with the rapid development of non-optical astronomy. In 1967 the astrophysicist Jocelyn Bell (born 1943) and the radio astronomer Antony Hewish (born 1924) at Cambridge University in England discovered the first pulsar. They were looking for variations in the radio brightness of quasars and discovered a rapidly pulsating radio source. The radiation had to come from a source not larger than a planet. The Austrian-born, USA-based astrophysicist Thomas Gold (1920-2004) soon identified these objects as rotating neutron stars, pulsars, with extremely powerful magnetic fields that sweep around many times per second as the stars rotate, making them appear as cosmic lighthouses.

Antony Hewish won the Nobel Prize in Physics in 1974, the first one awarded for astronomical research, although his graduate student Bell made the initial discovery. He shared the Prize with the prominent English radio astronomer Martin Ryle (1918-1984), who helped develop radar countermeasures for British defense during World War II and after the war became the first professor of radio astronomy in Britain. Ryle became a leading opponent of the Steady State cosmological model proposed by the English astrophysicist Fred Hoyle.

The process of converting lower-mass chemical elements into higher-mass ones is called nucleosynthesis. One or more stars can be formed from a large cloud of gas and dust. As it slowly contracts due to gravity, the condensation releases energy which in turn heats up the central region of the cloud. The protostar continues to contract until the core temperature reaches about 10 million K, which constitutes the minimum temperature required for normal hydrogen-to-helium fusion to begin. A main sequence star is then born. When a star exhausts its hydrogen supply the pressure in its core falls and it begins to shrink, releasing energy and heating up further. The next step is core helium-to-carbon fusion, the triple-alpha process, which requires a central temperature of about 100 million K. Helium fusion also produces nuclei of oxygen-16 (8 protons and 8 neutrons) and neon-20 (10 protons and 10 neutrons).

At core temperatures of 600 million K, carbon-12 can fuse to form sodium-23 (11 protons, 12 neutrons) and magnesium-24 (12 protons, 12 neutrons), but not all stars can reach such temperatures. Stars with higher masses fuse more elements than stars with lower ones. High-mass stars have more than 8-9 solar masses; intermediate ones 0.5 to 8 solar masses and low-mass stars contain merely 0.08 to 0.5 solar mass. After exhausting its central supply of hydrogen and helium, the core of a high-mass star undergoes a sequence of other thermonuclear reactions at increasingly faster pace, reaching higher and higher temperatures.

When helium fusion ends in the core of a star with more than 8 solar masses, gravitational compression collapses the carbon-oxygen core and drives up the temperature to above 600 million K. Helium fusion continues in a shell outside of the core, and this shell is itself surrounded by a hydrogen-fusing shell. At 1 billion K oxygen nuclei can fuse, producing silicon-28 (14 protons, 14 neutrons), phosphorus-31 (15 protons, 16 neutrons) and sulfur-32 (16 protons, 16 neutrons). Each stage goes faster and faster. At 2.7 billion K, silicon fusion begins. Every stage of fusion adds a new shell of matter outside the core, creating something resembling the layers of a massive onion. The outer layers are pushed further and further out.

Energy production in big stars can continue until the various fusion processes have reached nuclei of iron-56 (26 protons, 30 neutrons), which has one of the lowest existing masses per nucleon (nuclear particle, proton or neutron). The mass of an atomic nucleus is less than the sum of the individual masses of the protons and neutrons which constitute it. The difference is a measure of the nuclear binding energy which holds the nucleus together. Iron has the most tightly bound nuclei next to 62 Ni, an isotope of nickel with 28 protons and 34 neutrons, and consequently has no excess binding energy available to release through fusion processes.

No star, regardless of how hot it is, can generate energy by fusing elements heavier than iron; iron nuclei represent a very stable form of matter. Fusion of elements lighter than this or splitting of heavier ones leads to a slight loss in mass and a net release of nuclear binding energy. The latter principle, nuclear fission, is employed in nuclear fission weapons (“atom bombs”) by splitting large, massive atomic nuclei such as those of uranium or plutonium, while nuclear fusion of lighter nuclei takes place in hydrogen bombs and in the stars.

When a star much more massive than our Sun has exhausted its fuel supplies it collapses and releases enormous amounts of gravitational energy converted into heat. It then becomes a (Type II) supernova. When the outer layers are thrown into interstellar space, the material can be incorporated into clouds of gas and dust (nebulae) that may form new stars and planets. The remaining core of the exploded star will become a neutron star or a black hole, depending upon how massive it is. It is believed that the heavy elements we find on Earth, for instance gold (Au, atomic number 79) are the result of ancient supernova explosions and were once a part of the Solar Nebula that formed our Solar System almost 4.6 billion years ago.

Without any nuclear fusion reactions to create the temperatures and pressures needed to support the star, gravity takes over and the star collapses in a matter of seconds. Fowler and colleagues calculated that the energy generated within the collapsing star is so great that it provides the conditions needed to create all the elements heavier than iron. As the outer layers of such a star collapse and fall inwards they are met by a blast wave rebounding from the collapsing core. The meeting of these two intense pulses of energy creates a shock wave that is so extreme that iron nuclei absorb progressive numbers of neutrons, building all the heavier elements from iron to uranium. The blast wave continues to spread outwards, and in its final and perhaps finest flourish it creates a supernova explosion that blows the star apart.”

The Ukrainian-born astrophysicist Iosif Shklovsky (1916-1985), who became a professor at Moscow University and a senior Soviet Union authority on radio astronomy and astrophysics, proposed that cosmic rays from supernovae might have caused mass extinctions on Earth. The hypothesis is difficult to verify even if true, but such explosions are among the most violent events in the universe, and a nearby (in astronomical terms) supernova could theoretically cause such a disaster. Shklovsky made theoretical and radio studies of supernovae.

Since a star that dies passes along its heavier elements, this means that each successive generation contains a higher percentage of heavy elements than the former one. The Sun is a member of a generation of stars known as Population I. An older generation is called Population II. A hypothetical Population III of extremely massive, short-lived stars is thought to have existed in the early universe, but as of 2010 no such objects have been observed in distant galaxies. This constitutes an area of active astronomical research. If such stars are not found then we have to adjust our theoretical models. Astrophysicists currently believe that the young universe consisted entirely of hydrogen and helium with trace amounts of lithium and beryllium, all created through Big Bang or primordial nucleosynthesis. All other chemical elements have been created later through stellar nucleosynthesis and supernova explosions.

Although it took only about a decade for nuclear fission to go from weapons to be used for peaceful purposes in civilian power plants, this transition has been much slower for nuclear fusion. The American physicist Lyman Spitzer Jr., a graduate of the Princeton and Yale Universities, in 1951 founded the Princeton Plasma Physics Laboratory, a pioneering program in thermonuclear research to harness nuclear fusion as a clean source of energy. In Britain, the English Nobel laureate George Paget Thomson and his team began researching fusion. In the Soviet Union, similar efforts were led by the Russian physicists Andrei Sakharov and Igor Tamm. In 1968, a team there led by the Russian physicist Lev Artsimovich (1909-1973) achieved temperatures of ten million degrees in a tokamak magnetic confinement device, which after this became the preferred device for experiments with controlled nuclear fusion.

Progress has been made at sites in the USA, Europe and Japan, but no fusion reactor has so far managed to generate more energy than has been put into it. ITER (International Thermonuclear Experimental Reactor), an expensive international tokamak fusion research project with European, North American, Russian, Indian, Chinese, Japanese and Korean participation, is scheduled to be completed in France around 2018. There is substantial disagreement over how close we are to achieving commercially viable energy production based on nuclear fusion. Pessimists say we are still a century away, while optimists point out that promising advances have been made in recent years using high-energy laser systems.

The American astronomer Gerry Neugebauer (born 1932), son of the great Austrian historian of science Otto Neugebauer, did valuable pioneering work in infrared astronomy. He spent his entire career at the California Institute of Technology. Together with the US experimental physicist Robert B. Leighton (1919-1997), also at Caltech, he completed the first infrared survey of the sky. Leighton is also known for discovering five-minute oscillations in local surface velocities of the Sun, which opened up research into solar seismology. The American physicist Frank James Low (1933-2009) became a leader in the emerging field of infrared astronomy after inventing the gallium-doped germanium bolometer in 1961, which allowed the extension of observations to longer wavelengths than previously possible. He and his colleagues showed that Jupiter and Saturn emit more energy than they receive from the Sun.

Jupiter’s diameter is 142,984 kilometers, more than 11 times that of the Earth. It would take over one thousand Earths to fill up its volume. Jupiter alone contains almost two and a half times as much mass as the rest of the planets in our Solar System combined, but it would nonetheless have needed 75-80 times more mass to become a star. The lowest mass that an object can have and still be hot enough to sustain the fusion of regular hydrogen into helium in its core is about 8% or 0.08 of the Sun’s mass. Jupiter contains merely 0.001 solar masses.

Bodies with 13-75 times Jupiter’s mass fuse deuterium, a rare isotope of hydrogen, into helium, and those with between 60 and 75 times Jupiter’s mass also fuse lithium-7 nuclei (three protons and four neutrons) into helium. Yet this will only occur briefly in astronomical terms due to the limited supply of these materials. Such objects are called brown dwarf s or “failed stars” and are intermediate between planets and stars. Brown dwarfs are not literally brown. They were first hypothesized in 1963 by astronomer Shiv Kumar. The American astronomer Jill Tarter (born 1944) proposed the name in 1975. She later became the director of the Center for SETI Research, which looks for evidence of intelligent life beyond the Earth.

The astronomer Frank Drake, born 1930 in Chicago in the United States, in 1961 devised the Drake Equation, an attempt to calculate the potential number of extraterrestrial civilizations in our galaxy. He has participated in an on-going search for signals of intelligent origin. While this line of work was initially associated with searching for radio waves from other civilizations, more recently those engaged in these matters have started looking for other types of signals, above all optical SETI. Very brief, but powerful pulses of laser light from other planetary systems can potentially carry immense amounts of concentrated information across vast distances of many light-years. Obviously, if extraterrestrial civilizations do exist, it is quite conceivable that they may be scientifically more sophisticated than we are today and may possess some forms of communication technology that are totally unknown to humans.

The search for intelligent extraterrestrial life is not uncontroversial, especially when it comes to so-called “Active SETI” signals, where we beam signals into space in addition to passively recording signals we receive. The famous English mathematical physicist Stephen Hawking believes that intelligent aliens are likely to exist, but fears that a visit by them might have unfortunate consequences for us. “ We only have to look at ourselves to see how intelligent life might develop into something we wouldn’t want to meet,” he argues. It is possible to imagine an alien civilization of nomads, looking to conquer and colonize whatever planets they can reach, instead of peaceful interstellar “philosopher kings.” Others find it implausible that intelligent aliens would travel across vast astronomical distances merely to colonize us.

The possibility of life beyond the Earth has been discussed for centuries. The English bishop and naturalist John Wilkins (1614-1672), who proposed a decimal system of weights and measures that foreshadowed the metric system, in The Discovery of a World in the Moone (1638) suggested that the Moon is a habitable world. Wilkins worked in the turbulent age of Oliver Cromwell and the English Civil War and was associated with men who went on to found the Royal Society. He was not the first person to entertain such views, which had been suggested by some ancient Greek authors. No lesser figure than Johannes Kepler had written a story The Dream (Somnium) where a human observer is transported to the Moon.

The author Bernard de Fontenelle (1657-1757), born in Rouen, Normandy, in northern France, in 1686 published Conversations on the Plurality of Worlds (Entretiens sur la pluralité des mondes), which supported the heliocentric model of Copernicus and spoke of the possibility of life on other planets. The colorful German (Hanoverian) storyteller Baron von Münchhausen (1720-1797), who had fought against the Turks for the Russian army, in his incredible and unlikely tales allegedly claimed to have personally visited the Moon.

In From the Earth to the Moon (1865) by the great French science fiction author Jules Verne, three men travel to the Moon in a projectile launched from a giant cannon. William Henry Pickering (1858-1938) from the USA, brother of Edward Pickering and otherwise a fine astronomer, in the 1920s believed he could observe swarms of insects on the Moon’s surface.

In the 1870s the Italian scholar Giovanni Schiaparelli had observed geological features on Mars which he called canali, “channels.” This was mistranslated into English as artificial “canals,” which fueled speculations about the possibility of intelligent life on that planet.

The English author H. G. Wells in 1898 published the influential science fiction novel The War of the Worlds. In it, the Earth is invaded by technologically superior Martians who eventually succumb not to our guns, but to our bacteria and microscopic germs, which we had evolved immunity against but they had not. In 1938, when commercial radio was in its first generation, a drama adoption of Wells’ novel caused panic in the USA as thousands of radio listeners believed that it depicted a real, ongoing invasion. The man behind the broadcast, the American director Orson Welles (1915-1985), also wrote, directed, produced and acted in Citizen Kane from 1941, hailed as one of the best films from Hollywood’s Golden Age.

One of the earliest science fiction films, inspired by the writings of Jules Verne and H. G. Wells, was the black and white silent movie A Trip to the Moon from 1902 by the French filmmaker Georges Méliès (1861-1938). The Austrian-born motion-picture director Fritz Lang (1890-1976) created the costly silent film Metropolis in Germany in 1927. In the commercially successful Hollywood production E.T. the Extra-Terrestrial from 1982, directed by the influential American Jewish film director and producer Steven Spielberg (born 1946), a boy befriends a stranded, but friendly extraterrestrial and helps him to return home.

The American planetary scientist and science writer Carl Sagan (1934-1996), born in New York City, won enormous popularity as well as some criticism as a popularizer of astronomy and was a contributor to NASA’s Mariner, Viking, Voyager and Galileo expeditions to the planets. “ He helped solve the mysteries of the high temperatures of Venus (answer: massive greenhouse effect), the seasonal changes on Mars (answer: windblown dust), and the reddish haze of Titan (answer: complex organic molecules).” The Ukrainian astrophysicist Iosif Shklovsky in the Soviet Union was one of the first major scientists to propose serious examination of the possibility of extraterrestrial life. His book Intelligent Life in the Universe was translated and expanded by Carl Sagan, whose father was a Russian Jewish immigrant.

From infrared radiation was discovered in 1800 and ultraviolet radiation the year after, European scientists mapped the electromagnetic spectrum — radio waves, microwaves, terahertz radiation, infrared radiation, visible light, UV, X-rays and gamma rays — until the French physicist Paul Villard had found gamma radiation in the year 1900. Astronomers began observing in all wavelengths in the twentieth century, primarily in the second half of it.

Typically, only 2% of the light striking film triggers a chemical reaction in the photosensitive material. Photographic film and plates have been replaced in favor of the highly efficient electronic light detectors called charge-coupled devices (CCDs). CCDs respond to 70% or more of the light falling on them, and their resolution is currently better than that of film. A CCD is divided into an array of small, light-sensitive squares called picture elements or pixels. A megapixel is one million pixels. This is the same basic technology that is used in digital cameras for the mass consumer market. The Canadian physicist Willard Boyle (born 1924) and the physicist George E. Smith (born 1930) from the USA invented the first charge-coupled device in 1969. They shared the 2009 Nobel Prize in Physics in recognition of the tremendous importance their invention has had in the sciences, from astronomy to medicine.

Like radio and gamma ray astronomy, X-ray astronomy took off after WW2. Herbert Friedman (1916-2000), who spent most of his career at the US Naval Research Laboratory, using rocket-borne instruments found that the Sun weakly emits X-rays. The Nobel Prize- winning astrophysicist Riccardo Giacconi was born in Genoa in northern Italy in 1931 and earned his Ph.D. in cosmic ray physics at the University of Milan before moving to the USA. He there became one of the major pioneers in the discovery of cosmic X-ray sources, among them a number of suspected black holes. Giacconi and his American colleagues built instruments for X-ray observations that were launched into space. The first widely accepted black holes, such as the object called Cygnus X-1, were detected in the 1960s and 70s.

A black hole is an object with such a concentrated mass that no nearby object can escape its gravitational pull since the escape velocity, the speed required for matter to escape from its gravitational field, exceeds that of light. In the seventeenth century it had been established by Ole Rømer that light has a great, but finite speed, and Isaac Newton had introduced the concept of universal gravity. The idea that an object could have such a great mass that even light could not escape its gravitational pull was proposed independently in the late 1700s by the English natural philosopher John Michell (1724-1793) and the French mathematical astronomer Pierre-Simon Laplace, yet their ideas had little impact on later developments.

John Michell had studied at Cambridge University in England, where he taught Hebrew, Greek, mathematics and geology. He devised the famous experiment, successfully undertaken by Henry Cavendish in 1797-98 after his death, which measured the mass of the Earth. In addition to this, Michell is considered one of the founders of seismology. His interest for this subject was triggered by the powerful earthquake that destroyed the city of Lisbon, Portugal, in 1755. He showed that the focus of that earthquake was underneath the Atlantic Ocean.

Modern theories of black holes emerged after Einstein’s general theory of relativity from late 1915. The astrophysicist Karl Schwarzschild (1873-1916) was born in Frankfurt am Main in Germany to a Jewish business family and studied at the Universities of Strasbourg and Munich. From 1901 until 1909 he was professor at Göttingen and director of the Observatory there, and in 1909 he became director of the Astrophysical Observatory in Potsdam. On the outbreak of World War 1 in August 1914 he volunteered for German military service in Belgium and then Russia, where he contracted an illness that caused his death at the age of 42.

While on the Russian front, he completed the first two exact solutions of the Einstein field equations, which had been presented in November 1915. For a nonrotating black hole, the “ Schwarzschild radius “ (Rg) of an object of mass M is given by a formula where G is the universal gravitational constant and c is the speed of light: Rg = 2GM/c2. The Schwarzschild radius defines the spherical outer boundaries of a black hole, its event horizon. Ironically, Schwarzschild himself apparently did not believe in the physical reality of such objects.

According to scholar Ted Bunn, “Almost immediately after Einstein developed general relativity, Karl Schwarzschild discovered a mathematical solution to the equations of the theory that described such an object. It was only much later, with the work of such people as Oppenheimer, Volkoff, and Snyder in the 1930s, that people thought seriously about the possibility that such objects might actually exist in the Universe. (Yes, this is the same Oppenheimer who ran the Manhattan Project.) These researchers showed that when a sufficiently massive star runs out of fuel, it is unable to support itself against its own gravitational pull, and it should collapse into a black hole. In general relativity, gravity is a manifestation of the curvature of spacetime. Massive objects distort space and time, so that the usual rules of geometry don’t apply anymore. Near a black hole, this distortion of space is extremely severe and causes black holes to have some very strange properties.”

If the mass creating a black hole was not rotating then the black hole does not rotate, either. Nonrotating ones are called Schwarzschild black holes. When the matter that creates a black hole possesses angular momentum, that matter collapses to a ring-shaped singularity located inside the black hole between its center and the event horizon. Such rotating black holes are called Kerr black holes, after the mathematician Roy Kerr (born 1934) from New Zealand, who calculated their structure in 1963. A black hole is empty except for the singularity.

The American physicist John Archibald Wheeler (1911-2008) is credited with having popularized the terms black hole and wormhole, a tunnel between two black holes which could hypothetically provide a shortcut between their end points. While a popular concept in science fiction literature, it has so far not been proven that wormholes actually exist. Wheeler received many prestigious awards, among them the Wolf Prize in Physics. He was also an influential teacher of many other fine American physicists, among them Richard Feynman as well as Charles W. Misner (born 1932) and Kip Thorne (born 1940), who are both considered among the world’s leading experts on the astrophysical implications of general relativity.

The physicist Yakov B. Zel’dovich (1914-1987), born in Minsk into a Jewish family, played a major role in the development of thermonuclear weapons in the Soviet Union and was a pioneer in attempts to relate particle physics to cosmology. Together with Rashid Sunyaev (born 1943) he proposed the Sunyaev-Zel’dovich effect, an important method for determining absolute distances in space. Sunyaev has developed a model of disk accretion onto black holes and of X-radiation from matter spiralling into such a hole. Working in Moscow, Sunyaev led the team which built the X-ray observatory attached to the pioneering Soviet (and later Russian) MIR space station, which was constructed during the late 1980s and early 1990s.

The English theoretical physicist Stephen Hawking was born in Oxford in 1942 and studied at University College, Oxford. Hawking then went on to Cambridge University to do research in cosmology. In the 1970s he predicted that black holes, contrary to previous assumptions, can emit radiation and thus mass. This has become known as Hawking or Bekenstein-Hawking radiation after Jacob Bekenstein (born 1947), an Israeli Jewish physicist at the Hebrew University of Jerusalem. His work combined relativity, thermodynamics and quantum mechanics. Unlike Einstein’s theories, which have been empirically verified repeatedly, it is possible that Hawking’s ideas about black hole evaporation may never be directly observed.

Stephen Hawking, in collaboration with the English mathematical physicist Roger Penrose (born 1931), developed a new mathematical technique for analyzing the relation of points in spacetime. Hawking became a scientific celebrity, and his serious physical handicap contributed to the general public’s fascination with his person. His popular science book A Brief History of Time from 1988 has sold millions of copies in dozens of languages. In the twenty-first century, following the development of sophisticated electronic computers, complex computer simulations can be used to study the conditions in and around black holes. They are now believed to be dynamic, evolving, energy-storing and energy-releasing objects.

The Dutch-born American astronomer Maarten Schmidt (born 1929) earned his bachelor’s degree at the University of Groningen in the Netherlands and his Ph.D. under the great astronomer Jan Hendrik Oort at the University of Leiden in 1956. Schmidt was one of many prominent Dutch-born astronomers of the twentieth century, a respectable number for such a small nation. Some of them, like Oort, Willem de Sitter, Jacobus Kapteyn and Hendrik C. van de Hulst, remained in the Netherlands, whereas others like Sidney van den Bergh and Gerard Kuiper moved to North America. Sidney van den Bergh (born 1929) has worked on everything from star clusters to cosmology, but the research for which he is best known is in the classification of galaxies, the study of supernovae and the extragalactic distance scale.

Maarten Schmidt joined the California Institute of Technology. In 1963 he studied the spectrum of an object known as 3C 273 and found that it had a very high redshift, indicating that it was extremely far away from us. He investigated the distribution of such quasars (quasi-stellar objects) and discovered that they were more abundant when the universe was younger. The American radio astronomer Jesse Greenstein (1909-2002) collaborated with him in this work. Other quasars were soon found, but it was difficult to explain how they could generate enough energy to shine so brightly at distances of billions of light-years. Quasars were among the most distant, and by extension oldest, objects ever observed in the universe.

The English astrophysicist Donald Lynden-Bell (born 1935) was educated at the University of Cambridge in Britain, where he eventually became professor of astrophysics. He made significant contributions to the theories of star motions, spiral structure in galaxies, chemical evolution of galaxies and the distributions and motions of galaxies and quasars. In 1969 he proposed that black holes are at the centers of many galaxies and provide the energy sources for quasars, powered by the collapse of great amounts of material into massive black holes.

A black hole can attract gas from its neighbors, which then swirls into it in the form of an extremely hot accretion disk. Matter that spirals into it emits copious quantities of X-rays and gamma-rays, which can be detected by us although the black hole itself cannot be directly observed since light cannot escape it. There are several classes of black holes; some are just a few solar masses, formed from the collapse of a large star. There may be an intermediate class of such bodies, too, but most if not all galaxies, including our own Milky Way, are believed to harbor supermassive black holes of millions or even billions of solar masses in their centers. These objects are believed to constitute a key force in shaping the lifecycle of galaxies.

Most astronomers today believe that quasars are created by supermassive black holes that are growing, perhaps forming when two large galaxies collide. Previously, black holes were generally seen as the endpoints of evolution, but a detailed survey has found that giant black holes were already common 13 billion years ago. The universe’s first, probably extremely massive, stars collapsed after a few million years. In a remarkably short period of time and in a process that is not yet fully understood, smaller black holes apparently merged into supermassive centerpieces of star-breeding galaxies and evolved into galactic sculptors.

General relativity allows for the existence of gravitational waves, small distortions of spacetime geometry which propagate through space. However, just like black holes this was initially believed by most scientists to describe purely mathematical constructs, not actually existing physical phenomena. Significant gravitational waves are thought to be generated through the collision and merger of dense objects such as stellar black holes or neutron stars.

In 1974 the American astrophysicists Russell Hulse (born 1950) and Joseph Taylor (born 1941) discovered a pair of pulsars (neutron stars) in close orbit around each other. They shared the 1993 Nobel Prize in Physics for their studies of this pair, whose behavior deviates from that predicted by Newton’s theory of gravity. Einstein’s general theory of relativity predicts that they should lose energy by emitting gravitational waves, in roughly the same manner as a system of moving electrical charges emits electromagnetic waves.

These two very dense bodies are rotating faster and faster about each other in an increasingly tight orbit. The change is tiny, but noticeable, and is in agreement with what it should be according to the general theory of relativity. This is seen as an indirect proof of the existence of gravitational waves. We have to wait until later in the twenty-first century for a direct demonstration of their existence, or to revise our theories if it turns out that they do not exist.

The Polish astronomer Bohdan Paczynski (1940-2007) was born in Vilnius, Lithuania, educated at Warsaw University in Poland and in 1982 moved to Princeton University in the USA. He was a leading expert on the lives of stars. Because gravity bends light rays or, rather, appears to do so because it bends the fabric of space itself, an astronomical object passing in front of another can under certain conditions focus its light in a manner akin to a telescope lens. Paczynski showed that this effect could be applied to survey the stars in our galaxy. This is called gravitational microlensing. The possibility of gravitational lensing had been predicted by Einstein himself, but Paczynski worked out its technical underpinnings. He also championed the idea that gamma ray bursts originate billions of light-years away.

Gravitational lensing has today emerged as a highly useful tool in astronomical research. The phenomenon at the root of gravitational lensing is the deflection of light by gravitational fields predicted by Albert Einstein’s general relativity. The deflection has well-known observable effects, such as multiple images and the magnification of images.

Gamma ray bursts are short-lived but extremely powerful bursts that can briefly shine hundreds of times brighter than a regular supernova. They were discovered in the late 1960s, at the height of the Cold War, by military satellites designed to detect gamma radiation pulses from nuclear weapons tests. In 2008, NASA’s Swift satellite detected such an explosion 7.5 billion light-years away that was so powerful that its afterglow was briefly visible to the naked eye, making it the most distant object ever seen by human eyes without optical aid.

Gamma ray bursts are among the most powerful explosions in the universe. The light from the most distant such event yet recorded reached our world from more than 13 billion light-years away in 2009. That explosion, which lasted just a little more than a second, released roughly 100 times more energy than our Sun will release during its entire lifetime. Most likely, it originated from a dying star with far greater mass than the Sun in the younger universe.

Even after the introduction of the telescope it took centuries for Western astronomers to work out the true scale of the universe. The English astronomer and architect Thomas Wright (1711-1786) suggested around 1750 that the Milky Way was a disk-like system of stars and that there were other star systems similar to it, only very far away from us. Soon after, Immanuel Kant in 1755 hypothesized that the Solar System is part of a huge, lens-shaped collection of stars and that similar such “island universes” exist elsewhere, too. Kant’s thoughts about the universe, however, were philosophical and had little observational content.

Johann Heinrich Lambert (1728-1777), a Swiss-born mathematician, astronomer and philosopher, “ provided the first rigorous proof that p (the ratio of a circle’s circumference to its diameter) is irrational, meaning that it cannot be expressed as the quotient of two integers.” Lambert, the son of a tailor, was largely self-educated and early in his life began astronomical investigations with instruments he built for himself. He made a number of innovations in the study of heat and light and corresponded with Kant, with whom he shares the honor of being among the first to believe that certain nebulae are disk-shaped galaxies like the Milky Way.

William Herschel’s On the Construction of the Heavens from 1785 was the first quantitative analysis of the Milky Way’s shape based on careful telescopic observations. William Parsons in Ireland with the largest telescope of the nineteenth century, the Leviathan of Parsonstown, was after 1845 able to see the spiral structure of some nebulae, what we call spiral galaxies. Already in 1612 the German astronomer Simon Marius had published the first systematic description of the Andromeda Nebula (Galaxy) from the telescopic era, but he could not resolve it into individual stars. The decisive breakthrough came in the early twentieth century.

The Mount Wilson Observatory in California was founded by George Ellery Hale. He offered the young astronomer Harlow Shapley (1885-1972) a research post. Shapley had earned a Ph.D. at Princeton University in 1913 and in 1921 became director of the Harvard College Observatory. Before 1920 he had made his greatest single scientific discovery: That our galaxy was much bigger than earlier estimates by William Herschel made it out to be, and that the Sun was not close to its center. He didn’t get everything right, though. In the Great Debate with fellow American astronomer Heber Curtis (1872-1942) he argued that the mysterious spiral nebulae were merely gas clouds that were a part of the Milky Way and that everything consisted of one large galaxy: our own. Curtis, on the other hand, claimed that the universe consisted of many galaxies comparable to our own. Shapley was reasonably correct regarding the size of our galaxy, but Curtis was right that our universe is composed of multiple galaxies.

Jacobus Kapteyn (1851-1922) started the highly productive twentieth century Dutch school of astronomers. He studied physics at the University of Utrecht in the Netherlands, spent three years at the Leiden Observatory and thereafter founded and led the study of astronomy at the dynamic University of Groningen from 1878 to 1921. Kapteyn observed that many of the stars in the night sky could be roughly divided into two streams, moving in nearly opposite directions. This insight led to the finding of the galactic rotation of the Milky Way.

The Swedish astronomer Bertil Lindblad (1895-1965) was a graduate of the University of Uppsala and directed the Stockholm Observatory in Sweden from 1927-65. He studied the structure of star clusters, but his most important work was regarding galactic rotation. His efforts led directly to Jan Oort’s theory of differential galactic rotation. He confirmed Shapley’s approximate distance to the center of our galaxy and estimated its total mass. Oort at the University of Leiden in 1927 confirmed Lindblad’s theory that the Milky Way rotates, and their model of galactic rotation was verified by the Canadian astronomer John Plaskett (1865-1941), originally a mechanic employed by the University of Toronto physics department. Following the lead of Kapteyn, Lindblad and Oort, the Dutch astronomer Hendrik C. van de Hulst (1918-2000) and others in the 1950s mapped the clouds of the Milky Way and delineated its spiral structure. “Van de Hulst made extensive studies of interstellar grains and their interaction with electromagnetic radiation. He wrote important books on light scattering and radio astronomy. He investigated the solar corona and the earth’s atmosphere.”

The job of cataloging individual stars and recording their position and brightness from photographic plates at the Harvard College Observatory was done by a group of women, “human computers” working with Edward Pickering, among them the American astronomer Henrietta Swan Leavitt (1868-1921). The concept of “standard candles,” stars whose brightness can be reliably calculated and used as benchmarks to measure vast astronomical distances, was introduced by Leavitt for Cepheid variable stars. She became head of the photographic photometry department, and during her career she discovered more than 2,400 variable stars. This work aided Edwin Hubble in making his groundbreaking discoveries.

Scientists are flawed like other people. Newton could be a difficult man to deal with, yet he was undoubtedly one of the greatest geniuses in history. Henry Cavendish was a brilliant experimental scientist as well as painfully shy, and Nikola Tesla was notoriously eccentric. Judging from the many stories about him, Edwin Hubble had an ego the size of a small country, but that doesn’t change the fact that he was a great astronomer whose work permanently altered our view of the universe. He was a sociable man who partied with movie stars like Charlie Chaplin and Greta Garbo and with famous writers such as Aldous Huxley.

His contemporary Milton L. Humason (1891-1972), despite a very limited formal education, was a meticulous observer. In 1919, Hubble joined the Mount Wilson Observatory. The 2.5 meter Hooker telescope there was completed before 1920, at which point it was the largest telescope in the world. Using this, Hubble identified Cepheid variable stars in Andromeda. This allowed him to show that the distance to Andromeda was greater than Shapley’s proposed extent of the Milky Way. Hubble demonstrated that there are countless galaxies of different shapes and sizes out there, and that the universe is far larger than anybody had imagined. He then formulated Hubble’s Law and introduced the concept of an expanding universe. “ His investigation of these and similar objects, which he called extragalactic nebulae and which astronomers today call galaxies, led to his now-standard classification system of elliptical, spiral, and irregular galaxies, and to proof that they are distributed uniformly out to great distances. (He had earlier classified galactic nebulae.) Hubble measured distances to galaxies and with Milton L. Humason extended Vesto M. Slipher’s measurements of their redshifts, and in 1929 Hubble published the velocity-distance relation which, taken as evidence of an expanding Universe, is the basis of modern cosmology.”

The Austrian physicist Christian Doppler described what is known as the Doppler Effect for sound waves in the 1840s and predicted that it would be valid for other kinds of waves, too. An observed redshift in astronomy is believed to occur due to the Doppler Effect whenever a light source is moving away from the observer, displacing the spectrum of that object toward the red wavelengths. Hubble discovered that the degree of redshift observed in the light coming from other galaxies increased in proportion to the distance of those galaxies from us.

The work of Walter Baade in the 1940s and the American astronomer Allan Sandage (born 1926) in the 1950s resulted in revisions of the value of Hubble’s Constant and by extension the age of the universe. Sandage earned his doctorate under Baade and went on to determine the first reasonably accurate value for age of the universe. “ He has calibrated all of the ‘standard candles’ to determine distances of remote galaxies and has several times presented (often with Gustav Tammann) revised estimates of the value of the Hubble constant.”

Hubble’s observational work led the great majority of scientists to believe in the expansion of the universe. This had a huge impact on cosmology at the time, among others on the Dutch mathematician and astronomer Willem de Sitter (1872-1934). De Sitter had studied mathematics at the University of Groningen. A chance meeting with the Scottish astronomer David Gill led to an invitation to work at the Observatory at the Cape of Good Hope. After four years there, de Sitter returned to the Netherlands and became a mathematical astronomer, earning his doctorate under Jacobus Kapteyn. He spent most of his career at the University of Leiden, where he expanded its fine astronomy program. He performed statistical studies of the distribution and motions of stars but is best known for his contributions to cosmology.

According to writers J. J. O’Connor and E. F. Robertson, “Einstein had introduced the cosmological constant in 1917 to solve the problem of the universe which had troubled Newton before him, namely why does the universe not collapse under gravitational attraction. This rather arbitrary constant of integration which Einstein introduced admitting it was not justified by our actual knowledge of gravitation was later said by him to be the greatest blunder of my life. However de Sitter wrote in 1919 that the term ‘… detracts from the symmetry and elegance of Einstein’s original theory, one of whose chief attractions was that it explained so much without introducing any new hypothesis or empirical constant.’“ In 1932, Einstein and de Sitter published a joint paper in which they proposed the Einstein-de Sitter model of the universe. This is a particularly simple solution of the field equations of general relativity for an expanding universe. They also prophetically argued in this paper that there might be large amounts of matter which does not emit light and has not been detected.

The cosmologist Georges Lemaître (1894-1966) from Belgium was a Catholic priest as well as a trained scientist. The combination is not unique. The Italian astronomer Angelo Secchi was a priest and the creator of the first modern system of stellar classification; the Bohemian scholar Gregor Mendel, too, was a priest and the founder of modern genetics. World War I interrupted Lemaître’s studies. Serving as an artillery officer he witnessed one of the first poison gas attacks in history. After the war he studied physics and was ordained as an abbé.

In 1925 he accepted a professorship at the Catholic University of Louvain near Brussels. He reviewed the general theory of relativity and his calculations showed that the universe had to be either shrinking or expanding. Lemaître argued that the entire universe was initially a single particle — the “primeval atom” — which disintegrated in a massive explosion, giving rise to space and time. He published a model of an expanding universe in 1927 which had little impact then, but in 1930, following Hubble’s work, Lemaître’s former teacher at Cambridge University, Arthur Eddington, shared his paper with de Sitter. Albert Einstein confirmed that Lemaître’s work “fits well into the general theory of relativity.”

Unknown to Lemaître, another person had independently come up with overlapping ideas. This was the Russian mathematician Alexander Friedmann (1888-1925), who in 1922 had published a set of possible mathematical solutions that gave a non-static universe. Already in 1905 he wrote a mathematical paper and submitted it to the German mathematician David Hilbert for publication. In 1914 he went to Leipzig to study with the Norwegian physicist Vilhelm Bjerknes, the leading theoretical meteorologist of the time. He then got caught up in the turbulent times of the Russian Revolution in 1917 and the birth of the Soviet Union.

Friedmann’s work was hampered by a very abstract approach and aroused little interest at the time of publishing. Lemaître attacked the issue from a much more physical point of view. Friedmann died from typhoid fever in 1925, but he lived to see the city Saint Petersburg renamed Leningrad after the revolutionary leader and Communist dictator Vladimir Lenin (1870-1924). The astrophysicist George Gamow studied briefly under Alexander Friedmann, but he fled the country in 1933 due to the increasingly brutal repression of the Communist regime, which directly or indirectly killed millions of its own citizens during this time period.

Although Lemaître’s “primeval atom “ was the first version of this theory of the origin of the universe, a more comprehensive model was published in 1948 by Gamow and the cosmologist Ralph Alpher (1921-2007) in the USA. The term “Big Bang” was coined somewhat mockingly by Fred Hoyle, who did not believe in it. Gamow decided as a joke to include his friend Hans Bethe as co-author of the paper, thus making it known as the Alpher, Bethe, Gamow or alpha-beta-gamma paper, after the first three letters of the Greek alphabet. It can be seen as the beginning of Big Bang cosmology as a coherent scientific model.

Yet this joke had the practical effect of downplaying Alpher’s contributions. He was then a young doctoral student, and when his name appeared next to those of two of the most famous astrophysicists in the world it was easy to assume that he was a junior partner. As a matter of fact, he made very substantial contributions to the Big Bang model whereas Hans Bethe, brilliant though he was as a scientist, in this case had contributed very little. Ralph Alpher in many ways ended up being the “forgotten father” of the Big Bang theory.

Alpher published two papers in 1948. In another with the scientist Robert Herman (1914-1997) he predicted the existence of a cosmic background radiation as an “echo” of the Big Bang. Sadly, astronomers did not bother to search for this proposed echo at the time; radio astronomy was then still in its infancy. Alpher and Herman went on to calculate the present temperature corresponding to this energy. The remnant glow from the Big Bang must still exist in the universe today, although greatly reduced in intensity by the expansion of space.

The cosmic microwave background radiation, which is considered one of the strongest proofs in favor of the Big Bang theory, was accidentally discovered by Robert Wilson (born 1936) and Arno Penzias (born 1933) in the USA in 1964. Yet they did not initially grasp the full significance of what they had found, whereas Alpher and Herman were totally ignored when Wilson and Penzias received the Nobel Prize in Physics in 1978. In the early 1960s the Canadian James Peebles (born 1935) together with Robert H. Dicke (1916-1997) and David Todd Wilkinson (1935-2002) from the USA had also predicted the existence of the cosmic background radiation and planned to seek it just before it was found by Penzias and Wilson.

An alternative Steady State model was developed in 1948 by the Englishman Fred Hoyle together with Thomas Gold (1920-2004), an Austrian American astrophysicist born in Vienna to a wealthy Jewish industrialist, and Hermann Bondi, (1919-2005), an Anglo-Austrian who was also brought up in Vienna and arrived at Cambridge, Britain in 1937 and worked with Hoyle on radar during WW2. The Steady State model declined in popularity after the discovery of the cosmic microwave background radiation, the clearest evidence discovered so far indicating that something like the Big Bang really happened back in a very distant past.

Although it may sound counterintuitive at first, quantum physicists operate with the concept of vacuum energy, energy that is intrinsic to space itself and can create “virtual” subatomic particles. According to the quality magazine Scientific American, “Far from being empty, modern physics assumes that a vacuum is full of fluctuating electromagnetic waves that can never be completely eliminated, like an ocean with waves that are always present and can never be stopped. These waves come in all possible wavelengths, and their presence implies that empty space contains a certain amount of energy—an energy that we can’t tap, but that is always there. Now, if mirrors are placed facing each other in a vacuum, some of the waves will fit between them, bouncing back and forth, while others will not. As the two mirrors move closer to each other, the longer waves will no longer fit—the result being that the total amount of energy in the vacuum between the plates will be a bit less than the amount elsewhere in the vacuum. Thus, the mirrors will attract each other, just as two objects held together by a stretched spring will move together as the energy stored in the spring decreases. This effect, that two mirrors in a vacuum will be attracted to each other, is the Casimir Effect.” It was first predicted in 1948 by the Dutch physicist Hendrick Casimir (1909-2000).

According to our best current models, in its first 30,000 years the universe was radiation-dominated, during which time photons prevented matter from forming clumps. In the early stages there was an ongoing process of particles and antiparticles annihilating each other and spontaneously coming into being from radiation. Lucky for us there was a tiny surplus of ordinary matter, otherwise matter as we know it could not have existed. After this period the universe became matter-dominated, when clumps of matter could form. Most astrophysicists today believe that during the first 379,000 or so years, matter and energy formed an opaque plasma called the primordial fireball. The cosmic microwave background radiation (CMB) is believed to be the greatly redshifted remnant of the universe as it existed about 379,000 years after the Big Bang. It therefore contains the oldest photons in the observable universe.

Starting from this point, spacetime expansion had caused the temperature of the universe to fall below 3000 K, enabling protons and electrons to combine to form hydrogen atoms, at which point the universe became transparent. Following billions of years of expansion the CMB radiation is now very cold, less than three degrees above absolute zero. This nearly perfect blackbody radiation shines primarily in the microwave portion of the electromagnetic spectrum and is consequently invisible to the human eye, but it is isotropic and fills the universe in every direction we can observe. Only with very sensitive instruments can cosmologists detect minute fluctuations in the cosmic microwave background temperature, yet these tiny fluctuations were of critical importance for the formation of stars and galaxies.

In 1989, NASA launched its Cosmic Background Explorer ( COBE ) satellite into space under the leadership of American astrophysicist John C. Mather (born 1946). Detectors on board the COBE satellite were designed by a team led by American astrophysicist George Smoot (born 1945) and were sensitive enough to measure minute fluctuations, corresponding to the presence of tiny seeds of matter clumping together under the influence of gravity. A follow-up mission was the Wilkinson Microwave Anisotropy Probe satellite — WMAP — from the USA. Mather and Smoot shared the 2006 Nobel Prize in Physics for providing a view of the CMB in unprecedented detail. The European Planck space observatory was launched 2009 and will study the cosmic microwave background radiation in even greater detail over the entire sky.

Alan Guth (born 1947) is a leading American theoretical physicist and cosmologist, born to a middle-class Jewish couple in New Jersey. He graduated from the Massachusetts Institute of Technology (MIT) in 1968 and held postdoctoral positions at Princeton University, Columbia University, Cornell University and the Stanford Linear Accelerator Center. He was initially interested in elementary particle physics but later shifted to cosmology, bridging the gap between the very big and the very small. In the 1980s he proposed that the expansion of the universe was propelled by a repulsive anti-gravitational force generated by an exotic form of matter. “ Although Guth’s initial proposal was flawed (as he pointed out in his original paper), the flaw was soon overcome by the invention of ‘new inflation,’ by Andrei Linde in the Soviet Union and independently by Andreas Albrecht and Paul Steinhardt in the US.”

Andrei Linde (born 1948) is a prominent Russian theoretical physicist, originally educated at Moscow State University and the Lebedev Physical Institute in what was then the Soviet Union. He eventually moved to the West at the end of the Cold War, first as a staff member of CERN in Western Europe, then as a professor of physics at Stanford University in the USA. The idea of an inflationary multiverse (reality consisting of many universes with different physical properties) was proposed in 1982. According to the concept of cosmic inflation championed by Guth and Linde, during fractions of a second the young universe underwent exponential expansion, doubling in size at least 90 times. As the magazine Discover states:

Much of today’s interest in multiple universes stems from concepts developed in the early 1980s by the pioneering cosmologists Alan Guth at MIT and Andrei Linde, then at the Lebedev Physical Institute in Moscow. Guth proposed that our universe went through an incredibly rapid growth spurt, known as inflation, in the first 10-30 second or so after the Big Bang. Such extreme expansion, driven by a powerful repulsive energy that quickly dissipated as the universe cooled, would solve many mysteries. Most notably, inflation could explain why the cosmos as we see it today is amazingly uniform in all directions. If space was stretched mightily during those first instants of existence, any extreme lumpiness or hot and cold spots would have immediately been smoothed out. This theory was modified by Linde, who had hit on a similar idea independently. Inflation made so much sense that it quickly became a part of the mainstream model of cosmology.”

According to the models we operate with there are four basic forces at work in the universe: The strong and weak nuclear forces, electromagnetism and gravity. Of these, gravity is by far the weakest one and is therefore more or less ignored when dealing with particles at the subatomic level, yet it is the most important force when dealing with the universe at large.

The strong nuclear force is the most powerful force in nature, but it was the last to be understood. It binds quarks together to make subatomic particles such as protons and neutrons and holds together the atomic nucleus. The electromagnetic force holds atoms and molecules together. As those who have played with magnets will know, like charges (+ +, or — -) repel one another whereas opposites attract. Protons have a positive electrical charge and must therefore feel a very strong repulsive electromagnetic force from neighboring protons. So why does the nucleus hold together? Because the strong nuclear force is so powerful that it cancels out these forces. Yet just like the weak nuclear force it only works over very short distances, which is why large atomic nuclei, bigger than uranium, tend to be unstable and radioactive. As Neil F. Comins and William J. Kaufmann III state in their book Discovering the Universe:

“Their influences extend only over atomic nuclei, distances less than about 10-15 m. The strong nuclear force holds protons and neutrons together. Without this force, nuclei would disintegrate because of the electromagnetic repulsion of their positively charged protons. Thus, the strong nuclear force overpowers the electromagnetic force inside nuclei. The weak nuclear force is at work in certain kinds of radioactive decay, such as the transformation of a neutron into a proton. Protons and neutrons are composed of more basic particles, called quarks. A proton is composed of two ‘up’ quarks and one ‘down’ quark, whereas a neutron is made of two ‘down’ quarks and one ‘up’ quark. The weak nuclear force is at play whenever a quark changes from one variety to another….at extremely high temperatures the electromagnetic force, which works over all distances under ‘normal’ circumstances, and the weak force, which only works over very short distances under the same ‘normal’ circumstances, become identical. They are no longer separate forces, but become a single force called the electroweak force. The experiments verifying this were done at the CERN particle accelerator in Europe in the 1980s.”

Physicists have found that at sufficiently high temperatures, these various forces begin to behave in the same way. If you believe the current theoretical models, from the Big Bang until 10—43 seconds afterward, a moment which has been called Planck time, all the four forces are believed to have been united as one. Before the Planck time, the universe was so hot and dense that the known laws of physics do not describe the behavior of spacetime, matter and energy. At this point, matter as we think of it did not yet exist, but the temperature of the tiny and rapidly expanding universe may have been in the order of 1032 K. After this, gravity separated from the three other forces, which then separated from each other a little later.

The physicist Abdus Salam (1926-1996) from present-day Pakistan together with Sheldon Lee Glashow (born 1932) and Steven Weinberg (born 1933) from the United States shared the 1979 Nobel Prize in Physics for their work in this field. Following up their theoretical models from the 1970s, the Dutch accelerator physicist Simon van der Meer (born 1925) was awarded the Nobel Prize in Physics in 1984 together with the Italian particle physicist Carlo Rubbia (born 1934) for work at CERN and for contributions to the discovery of the W and Z particles, short-lived particles with masses around 100 times the mass of a proton. Rubbia was born in Gorizia, a small town in northeastern Italy next to Slovenia, and was educated at the University of Pisa. Simon van der Meer was born and grew up in The Hague in the Netherlands and worked a few years for the Philips Company before joining CERN.

The twentieth century brought two great new theories of physics: general relativity and quantum mechanics. Albert Einstein introduced the general theory of relativity and was also, along with the great German physicist Max Planck, a co-founder of quantum physics, although he famously had serious reservations about this field in later years. It is often said that Einstein “disproved” or “overturned” Newton’s theories of gravitation, but this is misleading. Newton’s theories work reasonably well for objects that are not extremely massive or move with velocities that approach the speed of light. It would be more accurate to say that Newton’s work on gravity should be considered a special case of general relativity.

The general theory of relativity is one of the best-tested theories of modern physics. It is not “wrong,” although it could be incomplete. There may well be phenomena awaiting discovery that it fails to explain, for example related to the concepts of dark matter and dark energy in cosmology. Yet any modified theory of gravity must contain all that is good about general relativity within itself, just like Einstein’s theory carried Newton’s theory within itself.

The problem is that while the theory of relativity normally works quite well for describing large objects in the universe it is next to useless when dealing with what happens on the subatomic level. This is where quantum mechanics takes over. It, too, has so far been quite successful at predicting empirically observed behavior. Physicists therefore use two sets of rules, one for the very large and one for the very small. The challenge is to bridge these two.

The difficulty in reconciling quantum mechanics (describing the weak, strong and electromagnetic forces) and general relativity (describing gravitation) is that the three former forces are quantized whereas general relativity is not, as far as we know today. The weak, strong and electromagnetic forces are transmitted by particles; photons are the quanta of electromagnetism; gluons are the exchange particles between quarks involved in the strong nuclear force and the W and Z bosons are the particles involved in the weak nuclear force.

A hypothetical particle, the graviton, has been suggested as the force carrier for gravity analogous to the photon, but it has not yet been detected, and no theory of quantum gravity has succeeded. Gravity as described in the general theory of relativity is based on a continuous rather than quantized force; the distortion of spacetime by matter and energy creates the gravitational force, which is to say that gravity is a property of space itself.

To combine gravity and the other three fundamental forces into one comprehensive “Theory of Everything,” some scientists try to imagine that the universe consists of more than the traditional four dimensions we are familiar with when we think of spacetime (three of space plus time). Theories that attempt to mathematically describe this new formulation of the universe are called superstring theories. There are several versions which assume that spacetime has 10 dimensions, or 11 according to M-theory. In addition to the four traditional ones are six that are rolled up into such tiny volumes that we cannot detect them directly.

The notion that the universe could be described with more than four dimensions in order to unify the fundamental forces of gravity and electromagnetism was suggested in the 1920s. The German mathematical physicist Theodor Kaluza (1885-1954), who came from a Catholic Christian family and studied at the University of Königsberg, tried to combine the equations for general relativity with Maxwell’s equations for electromagnetism using five dimensions. Einstein encouraged him and himself spent the last three decades of his career on a fruitless attempt to create a unified theory of gravity and electromagnetism. The Swedish Jewish physicist Oskar Klein (1894-1977), son of the chief rabbi of Stockholm, came up with the idea that extra dimensions may be physically real, but curled up and extremely small. Kaluza-Klein theory lost favor with quantum mechanics, but was later extended with string theory.

Superstring theories assert that what we perceive as a particles are actually tiny vibrating strings, with different particles vibrating at different rates. The interactions between these strings create all of the properties of matter and energy that we can observe. Calabi-Yau manifolds are six-dimensional spaces that, according to string theory, lurk in the tiniest regions of spacetime, down at the Planck length where the quantized nature of gravity should become evident, at 1.6 x 10-35 m, which is almost unimaginably tiny even compared to an atomic nucleus. The Planck time, 10-43 seconds, is the time it would take a photon traveling at the speed of light to cross a distance equal to the Planck length.

The German mathematician Erich Kähler (1906-2000) defined a family of manifolds with certain interesting properties. The Italian American Eugenio Calabi (born 1923) in the following generation identified a subclass of Kähler manifolds and conjectured that their curvature should have a special kind of simplicity. Shing-Tung Yau (born 1949), a mathematician from China based in the USA, proved the Calabi conjecture in 1977. These types of spaces are called Calabi-Yau manifolds. For believers in string theory they form a critical element of the explanation of what appear to us as a variety of natural forces and subatomic particles. They are six-dimensional, but the extra dimensions are “folded up” out of sight from our vantage point in the macroscopic world. At least, that is how the theory goes.

The Italian physicist Gabriele Veneziano (born 1942), working at CERN in Western Europe in the 1960s, made early contributions to the field, but interest in string theory took off in a major way in the 1980s and 1990s. The Englishman Michael Green (born 1946), currently professor of theoretical physics at Cambridge University, together with his American colleague John Schwarz (born 1941) in 1984 extended string theory, which treats elementary particles as vibrations of minute strings, into “superstring” theory. It incorporated a novel relationship called supersymmetry that placed particles and force carriers on an equal footing.

Many leading scholars have since joined this debate. They include Leonard Susskind (born 1940), a Jewish professor of theoretical physics at Stanford University in the United States, Juan Maldacena (born 1968) from Buenos Aires, Argentina who is now a professor at the Institute for Advanced Study at Princeton in the USA, the Iranian American Cumrun Vafa (born 1960) at Harvard University in the USA as well as David Olive (born 1937) and Peter Goddard (born 1945), both mathematical physicists from Britain. The American physicist Joseph Polchinski (born 1954) in the 1990s introduced a novel concept called D-branes.

By the mid-1990s there were as many as five competing string theories, which nevertheless had many things in common. The great American mathematical physicist Edward Witten (born 1951), a professor at the Institute for Advanced Study at Princeton, during a conference in 1995 provided a completely new perspective which was named “M-theory.” According to Witten himself, M stands for magic, mystery or matrix. Before M-theory, strings seemed to operate in a world with 10 dimensions, but M-theory would demand yet another spatial dimension, bringing the total to 11. The extra dimension Witten added allows a string to stretch into something like a membrane or “brane” that could grow to an enormous size.

Edward Witten, widely hailed as one of the greatest scientists of his generation, comes from a Jewish family. His father was a physicist specializing in gravitation and general relativity. Edward Witten was educated at the Brandeis, Princeton and Harvard Universities in his native USA and became a professor at the Institute for Advanced Study at Princeton. His early research focused on electromagnetism, but he developed an interest in what is now known as superstring theory and made very valuable contributions to Morse theory, supersymmetry, knot theory and the differential topology of manifolds. Although primarily a physicist he was nevertheless awarded the prestigious Fields Medal in 1990 for his superb mathematical skills.

Neil Turok (born 1958), a white South African, together with Paul Steinhardt (born 1952), director of the Princeton Center for Theoretical Science in the USA, devised a controversial cosmological model in 2002. They proposed the “ cyclic model “ in which the universe was born multiple times in cycles of fiery death and rebirth. Their idea is based on a mathematical model in which our universe is a three-dimensional membrane or “brane” embedded in four-dimensional space. The Big Bang was caused when our brane crashed against a neighboring one; our universe is just one of many universes in a vast “multiverse.” Enormous “branes” representing different parts of the universe(s) collide every few hundreds of billions of years.

The American physicist Brian Greene (born 1963), a professor at Columbia University in the USA, has done much to popularize these new string theories. According to Greene, “Just as the strings on a cello can vibrate at different frequencies, making all the individual musical notes, in the same way, the tiny strings of string theory vibrate and dance in different patterns, creating all the fundamental particles of nature. If this view is right, then put them all together and we get the grand and beautiful symphony that is our universe. What’s really exciting about this is that it offers an amazing possibility. If we could only master the rhythms of strings, then we’d stand a good chance of explaining all the matter and all the forces of nature, from the tiniest subatomic particles to the galaxies of outer space.”

Superstring theories are consistent with what we know, but critics, of which there are still quite a few, claim that they are too mathematically abstract to predict anything which can be experimentally tested and verified, as should be possible with a proper scientific theory. Its supporters claim that the theories suggest that there should be a class of particles called supersymmetric particles, where every particle should have a partner particle.

CERN, the European Organization for Nuclear Research, has opened their Large Hadron Collider (LHC), the world’s largest and highest-energy particle accelerator, near Geneva on the border between Switzerland and France. There are those who hope that the LHC will be able to detect signs of supersymmetric particles. If so, this finding will not by itself prove superstring theory, but it would constitute a piece of circumstantial evidence in its favor.

Critics who complain that string theory is unnecessarily complex with very little experimental evidence in its favor have a point. It does seem rather drastic to go from four to eleven dimensions, thereby nearly tripling the amount of dimensions in the universe. Yet just because a theory is complex and seemingly counter-intuitive does not necessarily mean that it is wrong, as quantum mechanics and the theory of relativity showed us in the twentieth century.

One humorous illustration of how hard it is to imagine extra dimensions was provided by the English writer Edwin A. Abbott (1838-1926). His satirical novel Flatland: A Romance of Many Dimensions from 1884 is narrated by a being who calls himself “Square” and lives in Flatland, a world populated by two-dimensional creatures with a system of social ranks, where creatures with more sides rank higher and circles highest of all. Women are merely line segments and are subject to various social disabilities. In a dream, Square visits the one-dimensional Lineland, and is later visited by a three-dimensional Sphere from Spaceland. The Sphere tries to convince Square of the existence of a third dimension and mentions Pointland, a world of zero dimensions, populated by a single creature who is completely full of himself.

Perhaps we are all a bit like Square, who finds it very hard to imagine extra dimensions. And most of us have encountered individuals who live in Pointland, occupied only by themselves.

Despite all this progress, countless questions remain unanswered. As Alan Guth notes, even if the present form of the Big Bang theory with inflation should turn out to be correct, it says next to nothing about exactly what “banged,” what caused it to bang or what happened before this event. “ Link Text actually find it rather unattractive to think about a universe without a beginning. It seems to me that a universe without a beginning is also a universe without an explanation.”

Another major question is whether the expansion that our universe appears to be experiencing at the moment will continue indefinitely, or whether there is enough mass to slow it down and eventually reverse it, causing the universe to collapse onto itself in a “Big Crunch.” The Swiss astronomer Fritz Zwicky already in 1933 stumbled upon observations indicating that there is more than visible matter out there and that this “dark matter” affects the behavior of galaxies.

In the 1970s the American astrophysicist Jerry Ostriker (born 1937) along with James Peebles discovered that the visible mass of a galaxy is not sufficient to keep it together. The astronomer Vera Rubin (born 1928) studied under Richard Feynman, Hans Bethe, George Gamow and other prominent scholars in the United States. She became a leading authority on the rotation of galaxies. She teamed up with astronomer Kent Ford (born 1931) and began making Doppler observations of the orbital speeds of spiral galaxies. Her calculations based on this empirical evidence showed that galaxies must contain ten times as much mass as can be accounted for by visible stars. She realized that she had discovered evidence for Zwicky’s proposed “ dark matter,” and her work brought the subject to the forefront of astrophysical research. Rubin is an observant Jew and sees no conflict between science and religion.

In the 1990s, two competing groups began observing a certain type of supernovas as a way to study the expansion of the universe. In 1998 a team led by Saul Perlmutter (born 1959) at Lawrence Berkeley National Laboratory in California completed a search for type Ia supernovas, supplemented by a second team led by Brian P. Schmidt (born 1967) and Adam Riess (born 1969). To everyone’s surprise, their observations indicated that the expansion was not slowing down due to gravitational attraction, as many had suspected, but was speeding up. Further results have confirmed that the expansion of the universe appears to be accelerating.

Astronomers now estimate that out of the total mass-energy budget in our universe, a meager 4% consists of ordinary matter that makes up everything we can see, such as stars and planets, whereas 21% is dark matter. A full 75% consists of “dark energy,” an even more puzzling entity than dark matter. The US cosmologist Michael S. Turner coined the term to describe the mysterious force which seems to work like anti-gravity. In the view of Turner dark energy, the causative agent for accelerated expansion, is diffuse and a low-energy phenomenon. It probably cannot be produced at particle accelerators; it isn’t found in galaxies or even in clusters of galaxies. “ Dark energy is just possibly the most important problem in all of physics. The only laboratory up to the task of studying dark energy is the Universe itself.”

Petr Horava is a Czech string theorist who is currently a professor of physics at the University of California, Berkeley, and co-author with Edward Witten on articles about string and M-theory. He has proposed a modified theory of gravity, with applications in quantum gravity and cosmology. “I’m going back to Newton’s idea that time and space are not equivalent,” Horava says. At low energies, general relativity emerges from this underlying framework and the fabric of spacetime restitches. He likens this emergence to the way some exotic substances change phase. For example, at low temperatures liquid helium’s properties change dramatically into a “superfluid.” Cosmologist Mu-In Park of Chonbuk National University in Korea believes that this gravity could be behind the accelerated expansion of the universe.

A few scientists have controversially proposed resurrecting the discredited light-bearing ether of nineteenth century physics. Niayesh Afshordi, an Iranian -born USA-based physicist, suggests a model where space is filled with an invisible fluid — ether — as predicted by some proposed quantum theories of gravity such as Horava’s. Black holes may give off feeble radiation, as suggested by many quantum theories of gravity. Afshordi calculates that this radiation could heat the ether and, like bringing a pot of water to a boil, generate a negative pressure of “anti-gravity” throughout the cosmos. This would have the consequence of speeding up cosmic expansion, but it took billions of years for black holes to heat up the ether sufficiently. Another, less exotic alternative theory called Modified Newtonian Dynamics has been introduced by the Israeli astrophysicist Professor Mordehai Milgrom. This proposal has received the backing of some notable scientists, but so far only a minority of them.

Perhaps “dark matter” will turn out to be a new class of particles that behave very differently from the kind of matter we are most familiar with. Perhaps “dark energy” will in hindsight turn out to be a fancy name for something that does not exist, a twenty-first century equivalent of phlogiston. Or perhaps we will discover new insights that will fundamentally alter our understanding of gravity, the age of the universe and the fabric of spacetime. Whatever the truth turns out to be, the terms “dark matter” and “dark energy” remind us that scientists cannot yet explain all properties of the visible universe according to known physical laws.

In the late 1800s, many European scholars sincerely believed that they understood almost all of the basic laws of physics. They had reason for this optimism as the previous century had indeed produced enormous progress, culminating in the new science of thermodynamics and the electromagnetic theories of Maxwell. Max Planck was once told by one of his teachers not to study physics since all of the major discoveries in that field had allegedly been made. Lucky for us he didn’t heed this advice but went on to initiate the quantum revolution. We have far greater knowledge today than people had back then, but maybe also greater humility: We know how little we truly understand of the universe, and that is probably a good thing.