# The Scientific Revolution

# CONTEXT

In a mischievous challenge to current ideas of historical continuity historian David Wootton claims that the Scientific Revolution was ‘*the rejection of moribund Scholastic Aristotelianism*‘ and that ‘*Modern science was invented between 1572, when Tycho Brahe saw a nova, or new star, and 1704, when Newton published his Opticks*‘^{[2]} but this conclusion relies on a narrow interpretation of both science and its predecessors. The customary convenient historical bookends for the period are Copernicus’s published claims for a heliocentric planetary system published in 1543 and the masterly *Principia* of Newton in 1687. Certainly today’s science involves a community of experts with research programs that have established sophisticated evidence-based theories that can make reliable predictions. But, it is argued elsewhere (see **Reason & science**) that the underlying use of reason divested of all extraneous accretions, which is at the core of science, came to us from Classical times – in other words, that the science of the modern era *grew out of* ancient modes of thought rather than *replacing* them.

Our current use of the word ‘science’ and our idea of scientists dates back only as far as the 19th century, **the word ‘scientist’** first appearing in 1834 created by the English naturalist-theologian William Whewell to distinguish those who sought knowledge of nature from those who sought knowledge in other disciplines. Only in the 19^{th} century did publicly funded scientists emerge who did not depend on patronage or their own resources: before this time ‘science’ (Latin *scientia* pl. *scientiae*) simply meant ‘knowledge’. What we now know as ‘science’ fell, in the 16^{th} and 17^{th} centuries, under the general name of *natural philosophy*.

The sphere of intellectual study occupied by the natural philosopher was much wider than that of the scientist today being an attempt to integrate all aspects of reasoning about the the world and therefore including theology and metaphysics. Much more than today these thinkers tried to establish the connectedness of everything rather than viewing objects and systems in isolation. It is only by embracing their worldview that we can appreciate the significance of their achievements. This article attempts to paint something of the experience of our precursor scientists.

## From antiquity to the Scientific Revolution

Theophrastus was followed by Strato at the Lyceum in Athens before the seat of learning moved to Alexandria which became the centre of learning for western Europe.

In the eastern (Byzantine) Roman empire learning was fostered in major cities like Antioch and Constantinople. A university-like school was founded as the Pandidakterion at Constantinople in 425 by emperor Theodosius II. It had 31 chairs in what would become the traditional Medieval liberal arts of law, philosophy, medicine, arithmetic, geometry, astronomy, music, rhetoric and languages (mainly Latin and Greek) and persisted into the 15th century. Though the Pandidakterion is sometimes regarded as the first ‘university’ it lacked the corporate structure of students and masters that defined the medieval universities of Western Europe to which the Latin term *universitas* was first applied.

In the West the Roman world had converted to Christianity under Emperor Constantine in the 4^{th} century CE. But Christians and Romans alike looked back with deep respect to the intellectual tradition of thought that had flowed from ancient Greece as distilled into the works of the great thinkers Plato and Aristotle. For example, Christian theology was united with the thought of Plato in St Augustine’s (354-430 CE) monumental *City of God*). Plato’s Theory of Forms, his owrld of timeless, perfect, and eternal truths was equated with the Christian heaven. The much later recovery of Aristotle’s works from the Muslim world, and their translation from the Arabic into Latin, permitted a similar unification with Christian theology by St Thomas Aquinas (1225-1274) in his great treatise on Catholic theology *Summa Theologica*).

Today we associate scientific thinking with the attempt to eliminate from our reasoning anything that might interfere with or prejudice our conclusions. The Pre-Socratic philosophers minimized the dependence on supernatural explanations that were prevalent at that time. Aristotle, in his *Physics*, demonstrated the kind of detached critical thinking needed to penetrate those subjects that would later become domains of study on the curricula of educational institutions. He attempted to explore rational thought unfettered by tradition, ideology, religion, or authoritarian doctrine and his philosophy was devoid of the transcendental mysticism evident in Plato. Aristotle was perhaps the world’s greatest polymath, originating or making major contributions to the studies of aesthetics, ethics, logic (especially syllogistic deductive logic), metaphysics, physics, politics, rhetoric, biology (especially zoology), his was a comprehensive system of Western philosophy built on the genius of his teacher Plato.

Aristotle had no inkling of modern science so how is it that, until the mid 17^{th} century his teaching remained vitually unchallenged? English philosopher Bertrand Russell (1872-1970) offers us two valuable insights into Aristotle’s method and legacy ‘… after his death it was two thousand years before the world produced any philosopher who could be regarded as approximately his equal … he is the first to write like a professor: his treatises are systematic … he is a professional teacher not an inspired prophet … his work is critical, careful, pedestrian, without any trace of Bacchic enthusiasm … the errors of his predecessors were the glorious errors of youth attempting the impossible; his errors are those of age which cannot free itself of habitual prejudices’.^{[3]} Plato’s philosophy was expressed in elegant Greek prose. Much of Aristotle’s work is lost and what remains is in the form of lecture notes although we are told that he too, in works now lost, demonstrated a mastery of the Greek language. Russell adds … ‘Aristotle is the last Greek philosopher who faces the world cheerfully; after him, all have, in one form or another, a philosophy of retreat’.^{[4]}

Gradually and inevitably the discoveries of modern science would surpass those of the ancients. This happened in the early 17th century when the sense of progress resulting from new scientific and technological discoveries created a forward-looking mentality. But for nearly 1000 years thinkers had looked backwards for intellectual and moral guidance. Wisdom could only be obtained through the interpretation and understanding of the great works of the past. The common man looked to the Christian Bible for not only moral truth, but a literal account of Creation, cosmology, and the history of mankind. However, men of learning needed a greater breadth of knowlege, albeit founded in religion, and this was inherited from Roman learning with its foundation in Greek philosophy and traditions. To overcome the deep-seated traditional influences of the past required active rebellion.

The progressive overthrow of several key ideas can be traced through the years of the Scientific Revolution. Medicine had acquired from the time of Hippocrates (c. 460-c.370 BCE), the ‘Father of Medicine’ ancient texts known as the *Hippocratic Corpus*. These texts, which outlined the state of medical knowledge in his time, were later acquired by the influential Greco-Roman physician Galen (129-200 CE) who promoted the theory of the four humours (blood, phlegm, yellow bile, black bile) whose proportions and condition were not only responsible for the health but also for the four major personality types, the sangine, choleric, phlegmatic, and melancholic. From Aristotle, and the Greek tradition associated with Empedocles (c. 490 – c. 430 BCE) the struggle to determine the ultimate composition of matter had resolved into the four elements of Earth, Air, Fire, and Water and the atomistic theory of Leucippus (5th BCE) and Democritus (c. 460 – c. 370 BCE). To Plato’s student Eudoxus we can attribute the idea of the firmament as consisting of concentric spheres. Greek mathematics had flourished with the genius of people like Pythagoras, Archimedes, and Euclid, the latter living in Hellenistic Alexandria at the time of King Ptolemy I (323–283 BCE) who was himself an authority on astronomy and world geography up to the Early Modern period. The Ptolemaic map of the world remained unchanged until the 15^{th} century era of European maritime exploration while it was the Ptolemaic system of planetary cycles and epicycles that so challenged the astronomers of the Scientific Revolution. Aristotle was himself triumphantly proved wrong in various ways by the new empiricists. Among his errors were claims for spontaneous generation and that bodies fell at speeds relating to their weight.

Gradually ancient ideas like these dissolved or were reformulated into a different, more sophisticated, and more evidence-based picture of the Earth and the heavens.

#### Carolingian Renaissance

There were two revivals of learning prior to the Renaissance, firstly that of Charlemagne (called the the Carolingian Renaissance) in the 9^{th} century when the court of Emperor Charlemagne in Aachen became a centre of learning and culture. There was a period of formal governance through monastic life across Europe based on Roman traditions along with the birth of Cathedral schools that were to evolve into our modern universities. By the end of the Roman empire only a few popular fragments of the former Greek academic output were available.

#### Muslim Renaissance

In the 12^{th} century there was another revival of learning and culture, including the growth of towns, cities, and more stable government, as the population doubled, and there was a higher standard of living and steady food supply, perhaps a consequence of the Medieval Warm Period. It was at this time that Aristotle’s works, which had been lost to the West, but translated from Greek into Arabic by the Muslim world (and supplemented by Arabic learning) to be translated into Latin. These works, along with many other scientific, philosophical and other masterpieces, were rediscovered by Western scholars, mostly in Spain, where they were translated from the Arabic into Latin. Cathedral schools developed formal university curricula based on religious studies, medicine, the law, and Aristotelian philosophy in an educational tradition that became known as Scholasticism. Cultural achievements were eroded during the Great Plague of 1348 that swept across Europe killing towards half the population.

Humanism was in part about freeing the human intellect from dogma and authoritarian tradition to breathe the oxygen of liberated knowledge and experience opened up to the creative imagination. But this took place within religious tradition. Humanists were mostly in religious orders, mainly Franciscan and Dominican, and are not to be confused with the 20th century movement of secular humanism. Several key factors can be seen to contribute to this release of the human intellect. Firstly there was the invention of the printing press with its capacity to disseminate information. ‘*By 1500, there were about 1000 in operation, and between thirty and forty thousand titles had been printed, representing roughly ten million books*‘.^{[5]} Secondly there were the world-expanding voyages of discovery as Europeans spread into western Asia, sub-Saharan Africa, then India, the Americas and finally the far East in a circumnavigation of the Earth, thus unshrouding the mystery of the world beyond and revealing myriad new plants, animals and, above all, the need to reconcile accepted doctrines with those of other peoples and cultures.

All of life was religious life – birth, marriage, death, the afterlife, and the meaning of everything – was steeped in religious tradition and thinking. The transition from polytheism to monotheism had resolved many intellectual questions but monotheism itself was racked by internal dissent. The opening of the sixteenth century was marked by protestation against Catholic practices, a reformation movement that became Protestantism, itself fracturing into Lutheran, Calvinist and more while the Catholic Church responded with its own program of ‘Counter-Reformation’ creating Jesuit orders that would spur the Scientific Revolution and the spread of scholarship to China, India, and the Americas. The order of the world could be seen in everything from mathematics to the design of nature and was clear evidence of the work of God. But for several hundred years Europe became a conglomeration of rival states warring over religious ideology.

## The Early Modern World

Life for most people in the Early Modern world was one of toil in the fields made meaningful through religious observance. For those with the time to contemplate existence by study, the universe was the marvel of God’s orderly creation filled with his purpose and rich in meaning. The enquiring mind could investigate God’s design that was so evident in all living creatures, the orderly behaviour of the Sun, Moon, stars and planets of the firmament. One reason for study was the discovery of the many secret signs in the world that indicated God’s presence.

### Humanism

The intellectual ethos of the Scientific Revolution began with a Renaissance in the Italian city states of the 13th and 14th centuries, subsequently spreading to north-west Europe. Universities at this time provided essentially vocational training for men destined to enter the professions of Medicine, Theology, and Law but rediscovery of the Classical learning of Greece and Rome resulted in an intellectual movement now known as Renaissance humanism which placed an emphasis on the ‘humanities’ (Latin *humanitas*) which included subjects like rhetoric, history, philosophy, and literature. Works of the Roman scholar Cicero (106-43 BCE) were particularly influential. Petrarch (1304-1374) was a Florentine poet and scholar who studied at Montpellier (1316-1320) and Bologna (1320-1323) and a leading inspirational figure in the movement. He travelled widely, his letters as literary work achieving wide acclaim. In his *Secretum Meum* (My Book) that the full flowering of human creativity and intellectual imagination was the manifestation of God-given qualities and therefore to be encouraged. He is credited with the rediscovery of Cicero’s letters and the coining of the concept of the Dark Ages.

Among the humanists were writers whose work remains vital and interesting today. Dante Alighieri (1265-1321) was an Italian poet from Ravenna and author of the Divine Comedy, a journey through Hell, Purgatory, an Paradise, acclaimed as the greatest poem of the Middle Ages like the work of his fellow poet Bocaccio (1313-1375) (known for *The Decameron*) his most famous work was written in the vernacular rather than the usual Latin. Niccolo Machiavelli (1469-1527) was a senior official in the Florentine Republic sometimes heralded as the ‘Father of Political Science’. His cynical vision of political life as represented in ‘*The Prince*‘ (1513) was struck a chord down the generations. As the movement passed northwards other figures would emerge, among them the Dutchman Erasmus (1466-1536) from Rotterdam whose Latin and Greek editions of the New Testament and other works, wwould be infuential in the coming Protestant Reformation and Catholic Counter-Reformation.

The teaching of the Church encouraged a spiritual and contemplative life as preparation for the eternal life hereafter. The early Church regarded the rationalism of the Classical era with suspicion since it had given rise to confusion and conflict. Secure foundations for life came from faith, not reason. The new emphasis, while retaining the old religious faith, was on the individual and secularism.

St Thomas Aquinas (1225-1274) was an Italian Dominican friar and priest who restored respect for rational enquiry through his promotion of natural theology, the idea that the order we see in the world around us is a manifestation of God’s order, of intelligent design, and that we must use reason to discover this order. His greatest work, *Summa Theologiae* was the pinnacle of Scholasticism^{[7]}, was a reconciliation of the thought of Aristotle (‘The Philosopher’) with Christian theology. Supposedly at the request of Thomas Aquinas, William of Moerbeke (1215-1235 – c. 1286), a Dominican Flemish monk working in the Greek Pelopponese, translated the complete works of Aristotle directly from the Greek into Latin when the Byzantine Empire was under Roman Latin rule. Many of the copies of Aristotle in Latin at that time had originated from Spain as translations of Arabic texts in the library of Toledo, a provincial capital in the Caliphate of Cordoba. Many of these had, in turn, passed through Syrian versions rather than being translated from the originals.

Marsilio Ficino (1433-1499) was an Italian scholar, astrologer and priest who encouraged the revival of Neoplatonism, the uniting of Platonic ideas with those of Christianity, notably the merging of heaven and the Platonic eternal world of Forms.

Through the 15^{th} century Greek became more widely written and studied but by the 16^{th} century the humanist movement had began to fade as more writers resorted to the use of the vernacular and the Scientific Revolution gathered momentum.

The intellectual cosmological synthesis of these times was, as we shall see, in various ways more integrated, coherent, and meaningful than the fragmented and detached world we know today with its limited sense of purpose and absence of cosmic meaning. The humanists based this social movement of renewal on the recovery of Roman learning. An outstanding genius and product of these times was the polymath Leonardo da Vinci (1452-1519) who soaked up the knowledge of his day while making detailed observations of the natural world and contributions to academic disciplines like anatomy, engineering, and architecture while at the same time being widely acknowledged as one of the greatest painters of all time (Mona Lisa c. 1503-1506). He worked in Florence, Rome, Bologna and Venice before ending his days in France. Leonardo was the embodiment of the expression ‘Renaissance Man’, celebrating knowledge, both art and science, engaging in critical experimentation and observation, and the exercise of the intellect while avoiding dogma and superstition.

Humanists both respected and challenged the scientific and cultural world of the ancients. The contradictions and complexity of polytheism, so much a part of the Greco-Roman world, had now resolved into the more manageable monotheism. The gradual acceptance, among intellectuals, of the scientific mode of thought was a reaction to the chaos and cruelty of superstition but it was awakened by the rediscovery of ancient culture with its liberation of the human spirit.

## Earth, Air, Fire, Water

A central task for the natural philosopher is to provide a naturalistic account of the physical matter or substance of the universe. Is it one kind of thing manifest in different ways, or is it simply many different things? Is substance infinitely divisible or it composed of indivisible minute particles? If made up of particles are these particles all the same or are they different, and if so in what way? How does matter combine to form the many abjects we see around us? How is the variety we see around us to be reduced or classified into more basic or fundamental elements? How do we account for transformations of substance – solid to liquid, water to gas etc.? Indeed, why do things change at all, and why do they move as they do – for example, why do unsupported objects fall to the ground rather than floating up into the sky?

Today we so easily forget, or ignore, the enormity of these questions to which we still struggle to find satisfactory answers today.

The ancient learning of Pre-Socratic and Classical philosophers bequeathed to the Middle Ages the idea, via Empedocles and Aristotle, that the physical world could be divided into four fundamental elements – Earth, Air, Fire, and Water. Over the period of the Scientific Revolution the Classical cosmology and this characterisation of matter would undergo extensive revision.

Following Aristotle it was assumed that the heavy elements, earth and water, were drawn down towards the centre of the Earth, which was a point of rest at the centre of the universe. Light elements, air and fire, would therefore move upwards. This explained why a candle flame and its smoke smoke were directed upwards and rocks and rain downwards as they ‘strived’ to find their natural place. This natural order was to be refined or replaced although Aristotelian ‘powers’ would never go away completely. With the Bible as accepted and literal authority the age of the Earth was taken as beeing about 6000 years. Though rock strata containing seashells were observed away from the sea and there was evidence of cataclysmic events these were mostly related to biblical events, most notably Noah’s flood. However, observations on ocean currents, volcanoes and other geological phenomena were recorded in Athanasius Kircher’s *Subterranean World* (1665) an Thomas Burnet’s *Sacred Theory of the Earth* (1680s) and mysterious magnetic attraction investigated in *On the Magnet *(1600) by William Gilbert (1544-1603) in which he named north and south poles as he noted that compass needles were attracted towards terrestrial not celestial poles and he noted for the first time that the Earth was itself a vast magnet together with some further, idiosyncratic by today’s standards, observations on attractive forces.

The selection of fire as one of the four fundamental elements of the universe might seem strange to us but it seemed to be at the heart of physical transformation, of volcanic eruption and, with the emission of great heat, the conversion of objects into smoke and different forms of matter. This was the domain of alchemy and the chymists. Another signature characteristic of the Scientific Revolution was the attempts to convert base metals like lead into gold (chrysopoeia). Along with the pursuit of the mystical and magical, the potions and medicaments, came careful experiment and observation into dyes, oils, perfumes, pharmaceuticals, pigments, salts in a tradition harking back to 4th century Hellenistic Egypt. One major goal was to find the Philosopher’s Stone, a physical agent that would serve as a transforming agent. Profitable discoveries could be witheld as trade secrets.

## The cosmological order, the connectedness of all things

For most people life in the Middle Ages was one of relentless toil in the fields but what were the intellectual changes that were taking place among the more privileged, what was their mind-set? The Early Modern educated people lived in a world of cosmological order and interconnectedness. There were several key ideas contributing to this perception:

#### The sidereal or supra-lunal world, the sublunar world, and the microcosm

The popular image of the universe in 1600 consisted of three major parts: everything beyond the Moon (the sidereal world), eveything under the Moon (the sublunar world), and human beings (microcosm). The study of these integrated worlds – God, nature, man – was the task of the natural philosopher who developed a cosmic perspective. It was only in the 19th century that academic study resulted in subject specialization that excluded a synoptic overview, possibly a consequence of the decreased relevance of an integrating deity.

The categories of knowledge were those largely inherited from the ancients, subjects like philosophy, physics,^{[6]} astronomy, medicine, music, geography, poetry, and mathematics, all valuable sources of knowledge and not divided into science and humanities although high on the agenda in the universities was law, medicine, and theology.

#### Great Chain of Being

First there was the **Great Chain of Being**, the overall structure of the universe including the meaning and purpose of all its contents, especially the role of humanity within this structure and the relationship between man and God. This image of existence pervaded all thought and is described under the heading ‘Vallue’ **here**.

#### Magia naturalis

For men of learning the task was to decipher the many connections that existed in the cosmic order and, if possible, to harness that order for human benefit. This mode of study was called *magia naturalis* or mastery of nature, pejoratively referred to as ‘magic’. Aristotelians divided the world into two kinds of properties, the *manifest or primary qualities* available to the sense organs (hot, cold, wet, dry, bitter, salty, smooth) as anything that stimulated the senses. Some objects, like magnets (lodestones) and the pull of the Moon on the tides, had hidden qualities that could not be detected by the senses. The connections of the cosmos, the hidden properties of things, needed to be discovered, and to assist this process God had provided clues. So, for example, there was a silent forceful connection between the sunflower and the Sun. The sunflower demonstrated this by the shape and colour of its flower that followed the path of the Sun across the sky through the day. The Sun, most prominent of the bodies in the firmament, was yellow, the colour of the glistening noble metal gold. The walnut was associated with the brain, and so on in what was known as the ‘*doctrine of signatures*‘ which demonstrated the power of similitude. The book *De Signaturis* (1609) by Oswald Cross of Prague declares that ‘*God has stamped upon each plant legible characters to disclose the uses*‘. The world was full of such meanings and hidden purposes waiting to be revealed. One observation in music was the sympathetic vibrations of objects at a distance that occurred in harmonic intervals of sound. Aristotle had said that this required a medium for the transmission of vibration which became the *spiritus mundi*, an invisible object connecting all things.

#### The mechanical philosophy

The perception of **cause** as efficient cause (consider colliding billiard balls) and the mechanical predictability of the clocks of the period created a popular metaphor of the universe as a machine, mechanical processes also being associated with the operation of bodies and biological systems. Roman Lucretius had spoken of the *machina mundi* to express the order and regularity of the universe. The ancient metaphor of the world as an organism whose parts were integrated to achieve purposeful objectives was replaced by the metaphor of a lifeless and purposeless mechanism. God was like a watchmaker. But there were still the mysteries of ‘action at a distance’ – the way magnets attract and fire heats without any visible connection between the source object and the objects it affects. Newtonian mechanics would describe with mathematical precision how these mysterious forces behaved but without explaining what they were.

#### Intelligent design

Regardless of individual interests almost all thinkers of the period embraced the idea of a close connection between God, man, and the natural world. Religion was not a matter of opinion and personal choice; these natural philosophers were Christians whose theology expressed facts as certain as those of science. God’s Creation, the Book of Nature, was a supplement to the Book of Scripture; it was an object for sustained study and interpretation since it revealed the wonder of God’s design. ‘Early modern thinkers, like their Medieval forebears, looked out on a world of connections and a world full of purpose and meaning as well as of mystery, wonder, and promise.’^{[1]}

## Superstition & science

Through the Middle Ages a deep and pervasive religious belief was practiced in a world alongside what learned men called ‘superstition’. Uneducated people attended church services whose liturgy promoting the one true god was given in Latin. But their daily lives were still populated with nature gods, evil spirits, sacred groves and trees, demons, goblins, magic, mysterious chemical potions, ghosts, and the occult. Catholicism worked hard to first eliminate and then absorb the old animistic and polytheistic beliefs of the past. Old women were slaughtered by the thousand for being wiches. To reduce the impact of pagan beliefs the Catholic Church combined days of Christian celebrations with those of pagan festivals. Chemistry was in its earliest stages. In accounting for substance we are familiar with the elegance of matter as elements of the Periodic Table all forged into neat mathematical compounds. Though the atomism of Democritus was carried over from the ancients the elemental explanation of the material world was in terms of Earth, Air, Fire, and Water.

The new thinking of the Scientific Revolution was a reaction to the terrors and excesses of superstition in all aspects of academic thought, but it was largely a critical attack from within the framework of accepted state religion. Expressed crudely, there was a supernatural exorcism as academic disciplines were subjected to the scrutiny of disinterested critical reason. The outcome amounted to a transition from: numerology to mathematics; physic to medicine; alchemy to physics and chemistry; and astrology to astronomy.

The spirit of direct enquiry into nature needed a fresh start to escape the deference to the ancient intellect, mostly the influence of Aristotle ‘The Philosopher’, but also the grip of religious interpretation. ‘… *men did not study flowers or birds for their own sakes or in order to learn from them new insights into reality; they went to them solely for illustrations of moral or metaphysical dogmas, accepted on authority and believed to be dicinely ordained; they sought in nature not knowledge but edification, not Enlightenment but the exemplification of preconceived ideas*‘.(R. p. 3)

### Astrology to Astronomy

Today the night sky is, for most of us, of little consequence, obscured by the lighting of our cities. But before electricity each night would reveal to all the beauty and wonder of points of distant twinkling light rotating in the eternal black silence. From the earliest times the regular movement of the heavens has triggered human curiosity.

Babylonian star charts were inherited by ancient Greek astronomers. Plato observed mathematical harmony in the firmament its perfect regularity also being admired by the Pythagoreans who thought it reflected the mathematical ideas of its Creator and served as a model for orderly human government. The movement of the firmament was a circle, the most perfect shape. Since motion around a circle is without beginning or end it symbolises timelessness and the eternal. Aristotle divided the universe into a superlunar domain which included everything beyond the moon and a sublunar world that included everything below. Looking up to the unchanging perfection and mathematical precision of movement of the superlunary heavens Aristotle thought it must consist of a pure and elemental substance which he called the *aither* being a different element than the sublunary elements earth, air, fire and water out of which the ever-changing sublunary realm was composed.

The stars remained fixed in relation to one-another, passing across the sky each night but rising about four minutes earlier each night such that over the weeks the constellations (patterns of stars named after objects and people like Taurus (bull), Orion (the hunter), Libra (scales), and Pisces (fish)) seemed to move across the sky taking one year to returning to their original position. The Sun moved more slowly changing its position relative to the starry background and taking a year to return to its original position. The Moon would rise about 50 minutes later each night, taking a month (29 days) to return to its original position. Then there were seven planets (meaning ‘wanderers’), the Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn, that moved irregularly against the starry backdrop, their movement passing through a narrow band of 12 constellations known as the zodiac.

The ancient viewed the heavens as made up of concentric spheres with the Earth it the centre. Objects naturally move towards the centre of the universe, falling downwards. However, this model failed to explain some heavenly movements as did the system of circles within circles devised by Ptolemy (c. 100-170 CE) to overcome these difficulties. As learning passed to the Arabic world these problems occupied Islamic mathematicians and astronomers until inherited by Medieval European astronomers.

#### Nicholas Copernicus (1473-1543)

Nicholas Copernicus was a canon at a cathedral in today’s Poland. He had studied medicine in Padua, and developed an interest in astronomy while studying canon law at Bologna. Here he calculated that the Earth rotated around the Sun along with the other planets, except for the moon which orbits the Earth. In this heliocentric system the Earth rotated on its axis once a day creating the illusion of a rotating firmament and the Sun passing through the zodiac and that the irregular movement of the planets was a consequence of the relative motion between them and the Earth. Copernicus worked steadily on his thesis but it was only in 1543, shortly before he died, that his work *On the Revolutions of the Heavenly Orbs* was published. His conclusions were questioned because they were counterintuitive. How could the Earth be spinning when we have no sense of motion?

Astronomy calculated the positions of the objects on the celestial sphere and astrology tried to explain their effects on th Earth although the two were interrelated. Celestial studies attempting to make predictions, especially of the weather that was so important to farmers. This resulted in a proliferation of almanacs that included calendars, tables of lunar cycles, dates of eclipses, dates of passage of planets through the constellations etc. Medical astrology related the planetary configurations at time of birth to medical treatments and peoples’ temperaments as related to the humours they acquired at birth: this being called the ‘complexion’. Suitable dates for important life choices were calculated this way and strange occurrences like the sudden appearance of comets could be interpreted as portents. All things were signs, symbols, and messages hidden in nature and there to be deciphered.

#### Johannes Kepler (1571-1630)

Astronomers continued making discoveries. Johannes Kepler, who was originally an assistant to Tycho de Brahe, calculated that the positions of the planets are best accounted for if they move in an ellipse rather than a circle. He thus broke with the 2000-year conviction expressed in Plato’s idea of celestial pefection. But the factors controlling the movement and positioning of planets remained a mystery, Kepler postulating an *anima motrix*, a power in the Sun that propelled the planerts on their way.

#### Galileo Galilei (1564-1642)

Italian Galileo Galilei heard about the Dutch invention of the telescope and, building his own, began a phase of observations and new discoveries in 1609 that included: the craters, mountains, and valleys of the Moon, the four moons orbiting Jupiter, the rings of Saturn, and the way Venus had phases like the Moon. He published these findings in the *Starry Messenger* (1610). His findings proved unpopular with the Church discouraging open communication but the discoveries continued. Pierre Gassendi (1592-1655) obserbved the transit of Mercury across the face of the Sun in 1631, the detail of Saturn’s rings and largest Moon by Christiaan Huygens (1629-1695) in 1656 and four more discovered by Gian Cassini (1625-1712). Star maps and lunar maps followed.

#### Isaac Newton (1643-1727)

From the most ancient times we had looked *up* to the heavens because here was the eternal and unchanging perfection of the firmament. Though Aristotle had regarded the supralunary world as perfect, eternal and unchanging he had advocated more research into living organisms (see **Socrates, Plato, Aristotle** and **Theophrastus**) and the sublunary world of coming-to-be and passing-away. The Middle Ages at first retained the four elements with attention directed to both the celestial to the terrestrial.

Planetary movement remained a mystery philosopher-scientist Rene Descartes who made original contributions in algebra and geometry, and promoting the mechanical model of the universe. Descartes postulated celestial vortices of particles driving the planets along and this was, at first, sufficient to convince the young English natural philosopher Isaac Newton but by the 1680s he was thinking in terms of an attractive force existing between Sun and planets based on the kind of attraction observed in magnets. In his most famous work, *Principia Mathematica*, he gave a mathematical derivation of planetary motion as the laws of universal gravitation, the inverse-square law that challenged any remnant Aristotelian distinction between the celestial and terrestrial. But Newton’s hidden force, like Aristotle’s ‘hidden qualities’ continued to tantalize the mind. Newton was content to describe gravity’s effects without wishing to penetrate further into what gravity itself *was *– and what was its *cause*. For Newton God was both the Creator and universal Agent.

Newton was the first scientist to be buried in Westminster Abbey.

### Physic to medicine

For a more in-depth historical account of plants and medicine see **plant science 1**

Education at universities of the Middle Ages was directed towards three major disciplines – Theology, Medicine, and Law – which led, in turn, to socity’s most highly esteemed professions. Physics and astronomy lay on the path to these major occupations, chemistry as we know it today had no classical precedent and therefore did not emerge as an academic discipline until the 18th century. Medicine as taught in 1500 was a synthesis of the teachings of Greek Hippocrates, Greco-Roman Galen and medicinal learning that had also accumulated in the Arabic world as synthesised by Ibn Sina (Avicenna). Botany as a discipline diverged from the medical faculties with the appointment of botany professors to 16^{th} century university medicinal gardens at places at the medical faculties of universities in Padua and Pisa. A corner had been turned though as herbals began to accumulate herbal pharmacopia-like information and organisations like London‘s *Royal College of Physicians* (est. 1518) gradually became more commonplace.

Ancient Romans had, perhaps surprisingly given the barbarity of some of their practices, resisted the dissection of human bodies, regarding it as a violation. But by 1300 human anatomical dissection was routine training for medical students at universities like Padua and Bologna although only the bodies of criminals and other undesirables.

#### Andreas Versalius’s (1514-1564)

Coinciding with the 1543 publication of of Copernicus’s *On the Revolutions* was Paduan surgeon Andreas Versalius’s *On the Structure of the Human Body* leading to the construction of steep anatomy theatres adopted by other countries as the medical torch was passed on from Italy to, for example, the Netherlands Leiden University (1596) which would later become the medical centre for Europe under Herman Boerhaave (1668-1738). He was preceded by Flemish nobleman Joan Baptista van Helmont (1579-1644), a rebellious but stimulating and influential thinker of his day.

#### William Harvey (1578-1657)

Of special note too is William Harvey’s (1578-1657) work on the circulation of the blood published in 1628 with careful empirical description of the arteries and veins.

#### Robert Hooke (1635-1703)

The world was suddenly viewed at a different scale when Robert Hooke (1635-1703) built an improved model of the microscope in the 1660s generating research into the reproductive cycles of both animals and plants and illustrations showing the cellular structure of both animals and plants. How did new individuals arise from sperm and eggs – were passed from generation to generation as tiny individuals (preformationism) or were completely new individual produced with each generation (epigenesis)? Plants anatomy was launched by London’s Hehemiah Grew (1641-1712) and Italian Marcello Malpighi (1628-1694).

Aspects of chemistry also merged from medicine as chemical medicine. In Germany of the following century Johannes Hartmann (1568-1631) became the first professor of chemiatria (chemical medicine) at the University of Marburg in 1609 and classes in chymistry were soon given at the medicinal *Jardin du Roi* in Paris.

#### Paracelsus (1493-1541)

Early modern chymists were renowned for their preparation of medicinal extracts. Most famous of these men was the Swiss Paracelsus who resisted the medicine that had been passed down the ages from Egyptian, Greco-Roman, and Arabic sources preferring to follow instead a system based on Germanic folk beliefs. He began to view the world as fundamentally chemical – the formation of rain, body metabolism, the minerals of the Earth, healing medicines – all was chemistry. But although physics and astronomy were an established part of the university curriculum chemistry would not be included until the 18^{thth} century since it could account for the way that substances could be broken down and reconstituted.

### Medicine to botany

The academic interest in plants during the Early Modern period began as a preoccupation with their medicinal properties. This was a continuation of a tradition dating back to at least ancient Egypt. Only Theophrastus (c. 371–c.287 BCE), who followed Aristotle as Head of the Lyceum in ancient Athens, showed an interest in plants for their own sake rather than their utility. His study of plant structure, function, and relationships (independent from the herbalists or rhizotomi) initiated the academic discipline of plant science. Following Theophrastus interest in plants once again focused on utilitarian and medicinal concerns. Plant science as a subject independent from medicine did not subsequently reappear until the Italian Renaissance and the appointment of professors of botany as administors of physic gardens associated with the medical faculties of universities in Pisa (1544), Padua and Florence (1545). These medicinal gardens are generally interpreted as the world’s first botanic gardens although as medicinal gardens they are probably more accurately referred to as the first stage in the development of botanic gardens of the modern era.

Though professors of botany were still essentially physicians and apothecaries their interest in accurate description led to the first collections of dried specimens pressed and attached to card to be labelled and shelved in buildings called herbaria. Specimens were used for scientific comparison, study and, above all, description. Following the development of the printing press among the first printed books were the herbals. These were descriptions of plants, including their medicinal and other properties and their publication would span the period c. 1470–1670 before evolving, on the one hand, into a medicinal pharmacopeia and, on the other hand, the formal compendia of plant descriptions known as Floras.

Associated with herbals were woodcut pictures that would, over time, evolve into accurate botanical illustrations. The Age of Discovery and exploration was introducing new plants that needed description, especially those from the New World. By the mid 17^{th} century the first simple attempts at global inventories of plants were being made with major advances in botanical nomenclature and increasing standardization of terminology for the now more closely observed features of plant morphology and anatomy which assisted identification. However, the formalization of an internationally-agreed fully standardized method of plant description and classification would not arrive until the work of the great naturalist Linnaeus, published in the middle of the 18^{th} century. It was only towards the end of the period that studies in plant sexuality such as the observation of fertilization by Camerarius (1665–1721) and the structure of pollen by Englishman Nehemiah Grew (1641-1712) opened up this field. The serious business of plant collection and study was just beginning in earnest at Leiden and Amsterdam during the Dutch Golden age at the end of the 17th and beginning of the 18th centuries at a time when the collection of plants for garden decoration was also gathering pace.

### Numerology to mathematics & engineering

Nowhere could the underlying order of the universe be encountered more obviously than in the world of numbers and mathematics. A vegetarian fraternity headed by Pythagoras of Samos (c. 570 – 495 BC) would greatly influence the future of mathematics. Pythagoras benefitted from travel to Egypt, Greece, and possibly India, but spent most of his working life in Magna Graeca (the ancient name for southern Italy and Sicily). He had a major influence on fellow Greek philosopher Plato (c. 428-348 BCE) who founded a university-like Academy in 387 BCE with the motto ‘*Let none unversed in geometry enter here*‘.

The Pythagoreans were a mystical and spiritual sect believing in the transmigration of souls. Number infused the world and was redolent with symbolism and meaning in a tradition that was just as evident among early modern intellectuals. Pythagoreans considered four a perfect number and we can get an impression of the way numerology was integrated into Medieval life by tracing some of its associations.

#### Four

According to Indian, Persian, Greek, and Roman traditions of medicine, especially those espoused by Hippocrates and the Islamic physicians, the human body reflected the four elements of the world. Number 4 was the number of stability, order, and justice. Within mathematics four is the only cardinal numeral in the English language that has the same number of letters as its number value, it is the smallest composite number, its proper divisors being 1 and 2.

To the ancients there were four elements. These were the simplest substances out of which all else was composed (earth, air, fire, and water). Galen (c. 129- 200 CE) claimed that these elements were used by Hippocrates to describe the association of the human body with four bodily fluids or humours: yellow bile (fire), black bile (earth), blood (air), and phlegm (water) from which are derived the four temperaments (melancholic, choleric, phlegmatic, sanguine) these, in turn, were related to the four qualities hot and cold, wet and dry. The rational behaviour of the heavens was assumed to influence our bodily functions through the medium of medical astrology. The planets, for example, could have an influence on particular human organs. Astrology was associated more with medical diagnosis than with the prognostication we associate with horoscopes today. Astrological diagnoses began to wane in the 17^{th} century but th theory of ‘humours’ we associate with Hippopcrates and Galen continued well into the 18^{th} century. Attention from professional physicians was, of course, a luxury som most households and communities maintained catalogues of home remedies that were a mixture of common sense, superstition, and occult practice.

But the association went much deeper. There were four seasons (spring, summer, autumn, winter), four phases of the moon, four causes (material, efficient, formal, final), the four corners of the world (north, south, east, west) and therefore four winds. Existence itself is comprised of four dimensions (time in one dimension, and space in the three dimensions of length, width, and height).

Similar catalogues of numerical significance were assembled for other numbers and we see remanants of these in today’s astrology.

Following the example of the Oxford Calculators (c. 1300) who had explored the application of mathematics to motion Galileo applied mathematical abstraction to the idealized movement of bodies regarding air resistance and friction as ‘imperfections’. He famously wrote that the universe ‘… *is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a word of it*‘. The application of mathematics would be a feature of the Scientific Revolution along with a satisfaction in establishing effects, *what things do*, rather than *why *they do so in the old Aristotleian search for causes.

Following in the great civilizations of the past Renaissance Italian natural philosophers were deeply engrossed with the problems of supplying cities with water, that is, with hydraulics and fluid dynamics – the design, construction, and operation of aqueducts, canals, fountains, and sewers. Advances were made in the study of the atmosphere and barometric pressure notably the air pump of Robert Boyle and Robert Hooke.

## Commentary

The Scientific Revolution emphasized naturalistic explanations over the traditionally accepted dogmas of the church and ancients: while most leading figures were god-fearing, they questioned the literal biblical account of the cosmos, Aristotelian philosophy, and Scholasticism. It would be several decades into the 17^{th} century before scientific knowledge would surpass that of the ancients and thinking would look to the developments of the future rather than the authority of the past.

Natural philosophers wanted to give knowledge a secure grounding. The new emphasis was on scientific knowledge which was acquired through the application of rigorous logic to measurable (mathematical) and repeatable experiments and observations in what became known as the ‘scientific method’. The knowledge acquired using this methodology was cumulative since it built on and enriched the scientific knowledge of the past. The evidence for both the effectiveness and improving capacity of the scientific method came from the benefits of science-based new technologies that raised living standards and the general quality of life. This rapid technological change contrasted with the former uniformity of agrarian life by creating a new sense of constant progress and improvement that has persisted to the present day.

As a movement of the 16th and 17th centuries the success of the Scientific Revolution, its logic and science, were later used as tools to challenge the social order through a period of revolutions known as the 18th century Enlightenment.

The Scientific Revolution is often presented largely as a new system of physics, astronomy, and mathematics culminating in Newton’s *Principia* of 1687 and, by using its own methodology, replacing the old Aristotelian cosmic order of two spheres (and much besides). But this was also an era whose intellectual curiosity was stimulated by a maritime search for spices that, in an Age of Discovery, led across the Atlantic to the New World of the Americas and beyond into the Indian and Pacific Oceans. In another challenge to inward-looking and traditional ideas of the Old World, foreign cultures and global exploration directed the European mind outwards, opening it to new ideas and possibilities.

It is hard to avoid a Eurocentric interpretation despite Chinese, Hindu, and Muslim elements and ancient Egypt and Mesopotamia, many echoes of Greek thought beyond Aristotle, such as Pythagoras, Archimedes, Aristarchus, Euclid and many other Greek intellectuals. If not a period of novelty then the Scientific Revolution was certainly a period of renaissance and accelerated change whose effects, through technology, were dispersed through society. After a brief period of popularity postmodern ideas claiming science as uncertain knowledge – essentially a mythic narrative created to maintain power structures – is now discredited. Science, as a body of knowledge, can only survive when administered by a community of scholars who record its conclusions and the findings of new research. The earliest known scientific society was founded in Naples as The *Academy of Secrets* (*Academia Secretorum Naturae*) established by Giambattista della Porta (1535-1615) who wrote a boook called *Magiae Naturalis*. This was a society of the well-to-do with intellectual curiosity about the world and many societies were to follow. Della Porta was also member of the early seventeenth century *Academy of the Lynx-Eyed *(*Accademia Dei Lincei*, est. 1603) which had Galileo as a member. The Royal Society of London was chartered in 1662 publishing the first edition of its *Philosophical Transactions* in 1665 while the Parisian *Academie Royale des Sciences* had its first meeting in 1666 and royal patronage supported a chemical laboratory, astronomical observatory, and botanical garden. The *Royal Observatory* at Greenich in London was founded in 1675.

The natural philosophy of the Scientific Revolution was a manifestation of insatiable human curiosity and the desire to understand the composition and functioning of the world. Their discoveries were not platitudes about the general nature of the world and existence, or intuitions about human nature. Their developed ideas and subsequent benefits were a consequence of deep thought and prolonged work that still today requires effort to understand. Among the major achievements were Copernican heliocentrism, Newton’s *Principia* and its universal laws of physics, and Harvey’s circulation of the blood. The technology of quadrants, telescopes, microscopes, navigation, shipbuilding and mother wethods to master the sea. A taxonomy of animals and plants that flooded into Europe. The impression of the era of the Scientific Revolution can be one of ignorance, disorder and religious persecution but there was another side. The thinking person of these times found themselves in a coherent world system of orderly religious interconnection rich in meaning and purpose that we might contrast with a kind of bland and fragmented diversity today. We can contrast the extremities of this world, its God and the Devil, good and evil, plenty and famine, health and pandemic illness).

Among what has changed we can note the wonder in things that has been transferred from God to nature itself. Scientific traditions and schools of thought have become much more centred on ideas than people and traditions. Questions of meaning and purpose have been largely abandoned. The natural philosopher came from all walks of life and belief but was generally a broadly educated man of private means has been replaced by men and women carrying out highly specialized work as students or professional scientists. The technological and therefore economic benefits have changed the course of history but left science overall as a fragmented academic domain. Desire to avoid intellectual systems, ideologies, and religious dogma has made us cautious of integrated knowledge and outlooks. Today, through modern technology, we can control nature more than ever before and yet we are also more detached from it than ever before. Paradoxically we have an intimate knowledge of nature but are physically separated from it.

This was a world driven by purpose and rich with meaning. The world bequeathed by Aristotle to the Neoplatonists of Hellenized Egypt, although a Christian world, was structured in an orderly, graded hierarchical way as a Great Chain of Being or *scala naturae* (see Grand narratives). For Aristotle this order arose from within nature itself (see **purpose**) for Christians it was imposed by God from without, earthly beings destined to strive for an escape from the material world as the soul yearned for the spiritual like the Platonic timeless and eternal world of forms, both here on Earth and in the afterlife. The orderly universe was a manifestation of the orderly and rational form of governance that should be followed by humans. Aristotle’s world placed objects in a meaningful relationship to other objects, giving them a universal context. This was the tenor of natural philosophy.

The solar system and planets had a visceral effect on humanity so the behaviour of objects in the heavens was directly related to human well-being.

The relationship between complexity of social organization and political and economic power is reflected in the regions and countries where mathematical innovation took place. Complex societies employed an intelligentsia of bureaucrats and academics and the geography of mathematical achievement has followed the geography of political and economic dominance.

The presence of a strong mathematical community is a measure of intellectual aspiration within that community since mathematics is generally recognized as making high intellectual demands.

Applied mathematics that depends on computational capacity has made major progress as once demanding calculations moved from abacus to slide rule to personal calculators. Analogue computers address continuously changing physical phenomena such as electrical, mechanical, or hydraulic quantities while digital computers represent varying quantities symbolically, as their numerical values change.

## Timeline

The Scientific Revolution (roughly 1543-1687) was a historical period that merged with the Renaissance humanist revival of Greco-Roman culture and learning and the global European maritime exploration that we know as the Age of Discovery. This was also a time when the Church was rent by the Protestant Reformation and Catholic Counter-Reformation. These changes that took place between the 15^{th} and 17^{th} centuries, especially the emphasis on science, logic, combined with a sense of progress and improvement, found later expression in the 18^{th} century Age of Enlightenment when Revolutions gave social expression to the influential intellectual ideals and ideas developed through this period with its desire for science and logic. Life-changing technologies had emerged together with a scientific mind-set that would flow into the Industrial Revolution.

#### Ancient world

Naturalistic explanations developed by the Pre-Socratic philosophers; Aristotle derives a detached empiricism based on syllogistic (deductive) logic and free of Plato’s transcendental mysticism (mostly in his *Physica*). Also to be challenged were: Ptolemaic geography and astronomy; Empedocles’s division of matter into Earth, Air, Fire, and Water; Eudoxus’s system of the heavens consisting of concentric spheres; the medicine and anatomy of Hippocrates and Galen

**1348** – Great Plague

**c. 1425** – Emergence of perspective painting

**c. 1428** – Increasing use of engraved prints

**c. 1440** – Johannes Gutenberg – invention of the printing press which, when combined with engraved illustrations in perspective, transforms scientific communication

**1462** – Publication of of compendium *Epitome of Ptolemy’s Almagest*

**1469** – Publication of the influential *Corpus Hermeticum*

**1473-1543** – Nicolaus Copernicus

**1472** – Peuerbach’s *New Theory of the Planets*

**1492** – Columbus lands in the New World

**1493-1541** – Paracelsus

**1494-1555** – Georg Agricola

**1514** – Copernicus’s private circulation of *Commentariolus*, a precursor to his later heliocentric theory

**1517**– Luther launches the Reformation

**1518** – *London College of Physicians* given a royal charter as a guild and learned society

**1522** – Magellan expedition circumnavigates the world

**c. 1530** – Paracelsus develops iatrochemistry a form of alchemy devoted to extending life, thus a kind of pharmacy. He probably coined the word ‘chemistry’

————–

**1543** – Nicolaus Copernicus publication *De Revolutionibus Orbium Coelestium* (*On the Revolutions of the Heavenly Spheres*) often taken, with Versalius’s work, as the commencing date for the Scientific Revolution

**1543** – Versalius’s *De Humani Corporis Fabrica* a treatise on human anatomy – supercedes Galen’s work and serves as a forerunner to the definitive *Gray’s Anatomy* first published in 1858

**1545** – Girolamo Cordano – *The Great Art* – advances in algebraic method

**1545** – *Council of Trent* sets off the Counter-reformation

**1545** – Gerolamo Cardano – conceives the idea of complex numbers

**1546-1601** – Tycho Brahe

**1550** – Jyeshtadeva – Indian Kerala school mathematician, writes the Yuktibhāṣā, the world’s first calculus text with detailed derivations of many calculus theorems and formulae

**1551** – Founding of Jesuit University *Collegio Romano*

**1554** – Giovanni Benedetti – opposes Aristotle’s theory of falling bodies

**1556** – Georgius Agricola –* De Re Metallica*, a comprehensive account of mining and metallurgy

**1561-1626** – Francis Bacon

**1564-1642** – Galileo Galilei

**1566** – Pedro Niunez – compendium on navigation and its instrumentation

**1569** – Mercator’s cartographic projection published

**1571-1630** – Johannes Kepler – laws of elliptic planetary motion

**1572**– Tycho Brahe observes new star as a supernova blazing for 18 months thus demonstrating that the superlunary sphere was not unchanging as Aristotle had claimed

**1572** – Rafael Bombelli – writes Algebra treatise and uses imaginary numbers to solve cubic equations

**1576** – Tycho Brahe massive observatory begins construction at Hven in Denmark

**1582** – Conversion from Julian to Gregorian calendar

**1591** – Francois Viete invention of Analytical Trigonometry,essential to the study of physics and astronomy

**1591** – Galileo demonstrates that a 1 pound weight and a 100 pound weight, dropped at the same moment from the leaning tower of Pisa, hit the ground at the same time thus refuting Aristotele’s claim that the rate of fall of an object is dependent upon its weight

**1596** – Gresham College founded as a precursor to the Royal Society of London

**1596-1650** – Rene Descartes – Philosophy and laws of reflection

**1597** – Andreas Libavius – publishes Alchemia, possibly the first chemistry textbook

**1600** – William Gilbert’s – *De Magnete* is a nexemplary empirical study of electrical phenomena

**1600** – Italian Dominican friar Giordano Bruno is tried for heresy by the Roman Inquisition for denial of core Catholic doctrines including eternal damnation, the Trinity, the divinity of Christ, the virginity of Mary, and transubstantiation. Found guilty he was burned at the stake but was remembered as a martyr for science – although the extent to which it was his scientific and astronomical views were the main heresy is debated

**1604** – Kepler – made Imperial Mathematician in 1603 now publishes *Astronomiae Pars Optica* – optics

**1605** – Sir Francis Bacon- in The Proficience and Advancement of Learning outlines his scientific method

**1605** – Michal Sedziwój – publishes A New Light of Alchemy proposing a ‘food of life’ within air, now known as oxygen

**1607** – Galileo – demonstrates that a fired projectile follows a parabolic path

**1608** – Hans Lippershey – produced first refracting telescope in the Netherlands. Built by lens-makers in the spectacle industry. Claims first patent followed by Jacob Metius. Already in 1609 Galileo built himself one for astronomical observations

**1609** – Kepler – *Astronomia Nova* – Laws of Planetary Motion and Mars following an elliptic orbit

**1610** – Galileo – Sidereal Messenger – various significant astromical observations

**1614** – John Napier – development of logarithms in *Mirifici Logarithmorum Canonis Descriptio*

**1614** – John Napier discusses Napierian logarithms in Mirifici Logarithmorum Canonis Descriptio

**1615** – Jean Beguin – publishes Tyrocinium Chymicum, a chemistry textbook containing the first ever chemical equation

**1618**– Thirty Years War begins

**1619** – Kepler – *Harmonia Mundi *(Harmonies of the World) presents his Third Law concerning planetary orbits, the final step in rejection of the Aristotelian system

**1619** – René Descartes – discovers analytic geometry though Fermat claims independent discovery

**1620** – Francis Bacon – *Novum Organum* in which he challenges Aristotelian Scholasticism by developing a scientific method based on inductive reasoning, experiment, and observation

**1623** – Galileo in *The Assayer* attacks Aristotle and the Scholastics in favor of mathematics and experimentation by considering the cases especially of of statics, dynamics, and his theory of matter

**1627-1691** – Robert Boyle – the first modern chemist

**1627** – Kepler – Rudolphine Tables, world’s most accurate astronomical tables including Tycho Brahe’s observations and Kepler’s laws of planetary motion

**1628** – William Harvey – *Anatomical Exercises on the Movement of the Heart and Blood* or *De Motu Cordis* describing experiments demonstrating the circulation of the blood in a reinterpretation of the role of the heart as a pump that superceded the old doctrines of Greco-Roman physician Galen

**1629-1695** – Christiaan Huygens

**1629**– Fermat – develops a rudimentary differential calculus

**1631** – Pierre Gassendi – observes the transit of the planet Mercury across the disc of the Sun

**1632** – Galileo – *Dialogue Concerning the Two Chief World Systems* – a comparison of the Ptolemaic and Copernican systems often interpreted as the final break with the Classical view of the natural order

**1632-1723** – Antonie van Leeuwenhoek – develops a powerful single lens microscope

**1632** – Galileo – in his *Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican* summons all the evidence for the Copernican system

**1633** – Galileo – summoned by the Inquisition in Rome and suspected of heresy by supporting the Copernican system. Forced to recant and placed under house arrest for life, the *Dialogue …* and *Two Chief World Systems …* becoming prohibited works though he saw it translated into Latin for other Europeans

**1637** – Descartes – *Discourse on Method* published together with his *Geometry* (which discusses the motion can be represented by the curve of a graph and defined by its relation to planes of reference) also a description of scientific method. This and his *Meditations* had a profound and lasting influence on European thought. His skepticism derives ‘I think therefore I am’ from systematic doubt

**1637**– Fermat – claims proof of Fermat’s Last Theorem

**1638** – Galileo’s – *Discours on Two New Sciences* published in Protestant Leiden and addressing the problem of motion the second science dealing with the strength of materials

**1639** – Englishman Jeremiah Horrocks makes the first observation of the transit of Venus across the Sun

**1641** – Descartes – his *Meditations* presents the case for the separation of mind and body, the dualism of a *res cogitans* (thinking thing or mind) and *res extensa* (extended thing or matter)

**1642** – Blaise Pascal – invention of mechanical calculator

**1644** – Evangelista Torricelli (1607–1647) – produces an operational mercury barometer

**1644** – Descartes – *Principles of Philosophy* presents arguments for the Mechanical Philosophy, a universe of matter united across space and time by principles of motion and mechanical connections of contact, impact, pressure, causation

**1646** – Thomas Browne coins the word ‘electricity’

**1647** – Johannes Hevelius – *Selenographia* a detailed and illustrated description of the surface of the Moon

**1647** – Otto von Guericke constructs the first air pump

**1648** – Jan Baptista van Helmont proposes that chemistry is foundational to physiology, his posthumous publication *Ortus medicinae* is a transitional work between alchemy and chemistry influencing Robert Boyle. Describes many experiments and an early version of the law of conservation of mass

**1648** – first meeting of the *Oxford Philosophical Society* which advocated the formation of the Royal Society of London

**1651** – Thomas Hobbes, philosopher and student of geometry, atomism, and optics publishes his political theory *The Leviathan* supporting mechanistic concepts and the benefits of the state in maintaining a rule of law

**1653** – *Montmor Academy* of Paris meets for the first time as an important semi-private scientific society formed under patron Habert de Montmor. This would be one of many subsequent intellectual salons. Members included Huygens and Roberval of the *Académie des Sciences*

**1654** – Otto von Guericke – invents vacuum pump, greatly improved by Robert Hooke in 1658

**1654** – James Ussher – biblical scholar argues that having fully analyzed evidence in the Bible Holy Writ the date of Creation was 23 October 4004 BCE at 9.00 am

**1654** – Pascal & Fermat – devise the theory of probability

**1646-1716** – Gottfried Liebniz – calculus (with Newton); the pinwheel calculator

**Otto von Guericke** – demonstrates vacuum pressure by using teams of horses to try and pull apart two metal hemispheres from which air had been removed

**1660** – Antonie van Leeuwenhoek publishes illustrations of micro-organisms drawn using his new microscope

**1660** – Otto von Guericke – electrostatic generator

**1661** – Robert Boyle – *The Sceptical Chymist* examines the distinction between chemistry and alchemy and includes early concepts of atoms, molecules, and chemical reactions. Effectively launches modern chemistry arguing against Aristotelian and Paracelsian methods and distinguishes between acids and bases

**1662** – Robert Boyle – Boyle’s law experimentally based description of the behavior of gases, specifically the relationship between pressure and volume in a closed system

**1662** – formation of the *Royal Society of London* by Royal Charter

**1665** – Robert Hooke – *Micrographia* includes exquisite etchings of microscopic sttructures including the famous ‘Hooke’s flea’

**1665** – Newton – develops the fundamental theorem of calculus and his version of infinitesimal calculus

**1665** – Publication of the first edition of the *Philosophical Transactions of the Royal Society of London*

**1666** – The Great Fire of London

**1666** – Foundation of the Parisian *Académie des Sciences* directed by Colbert with the patronage of Louis XIV. Positions are funded appointments not paying members who propose and elect new members as in London

**1666** – Robert Boyle’s *Origin of Form and Qualities* is a precursor to studies of matter on the atomic level

**1668** – Italian Francesco Redi demonstrates in Generation of Insects that, contra Aristotle, insects and other forms of life are not generated spontaneously

**1668-1738** – Herman Boerhaave – ‘Father of Physiology’ Europe’s leading physician from Leiden University

**1669** – Newton – builds his first reflecting telescope with an eyepiece and concave mirror

**1671** – Jean Picard – *Mesure de la terre* (Measure of the Earth)provides measurements for the length of a meridian. His work on the pendulum clock would aid Newtonian speculation on the shape of the earth

**1672** – Newton – notes in the *Philosophical Transactions* that white light is composed of a spectrum of colors (the rainbow), each color with a measurable angle of refraction

**1673** – Christiaan Huygens – publishes *Horologium oscillatorium* (The oscillation of pendula), a study of the pendulum clock

**1673** – Leibniz – develops his own version of infinitesimal calculus

**1674** – John Mayow – proposes that particles in the air are needed for combustion and that particles are also transmitted by the lungs to the blood

**1675** – Newton – theory of light, the prismatic decomposition of white light into a spectrum of colours

**1675** – Ole Roemer – uses astronomical observations to show that the speed of light is finite

**1677** – Antoni van Leeuwenhoek observes spermatozoa through a microscope arguing they are the source of reproductive matter

**1678–1761** – Pierre Fauchard – the founder of dentistry

**1678** – Edmond Halley publishes a catalogue of the stars in the Southern Hemisphere

**1679** – Denis Papin (1647–1712) – invention of the steam digester, forerunner of the steam engine

**1680s** – Leibniz – works on symbolic logic

**1683** – Elias Ashmole – donates his collections and library for the foundation of the *Ashmolean Museum* in Oxford, the first public museum in England

**1684** – Leibniz’s – paper indicating notation for the calculus of infinitesimals

**1687** – Isaac Newton – *Principia Mathematica* – the most influential and revered work of the Scientific Revolution establishing the foundational principles and the categories of force, mass, and acceleration as evidenced in three ‘laws of motion’ and principle of universal gravitation to give rise to what became known as classical mechanics

—————-

**1690** – Christiaan Huygens – *Treatise on Light* a wave theory contra Newton’s particulate theory

**1690** – John Locke – *Essay Concerning Human Understanding* knowledge of the nature must be contingent not necessry (probable not certain) and is based in sense experience not innate ideas – claiming that the mind is a *tabula rasa* or blank slate

**1697** – Samuel Clarke’s translation of Rohault’s *Treatise on Physics as a System of Natural Philosophy* (1671) becomes a university textbook

**1698** – Thomas Savery – patents an operational steam engine. In *The Miner’s Friend; or, An Engine to Raise Water by Fire* (1702) he claims it can pump water out of mines

**1700** – Liebniz founds the *Berlin Academy of Science*

**1704** – Newton publishes *Opticks*

**1708** – Abraham Darby I 1678–1717 – famous father of three generations Darbys produces high-grade iron in a blast furnace fuelled by coke rather than charcoal

**1712** – Thomas Newcomen (1664–1729) – produces the first practical steam engine water pump and becomes a central figure of the Industrial Revolution

**1712** – Flamsteed’s *Historia Coelestis Britannica* logs the positions of 3000 stars, many more than listed in Brahe’s catalogue

**1716** – Establishment of the *Societa Botanica Florentina*

**1730** – John Hadley invents the reflecting octant, precursor to the sextant

## Mathematics timeline

This timeline is an adaptation of that given in Wikipedia.The first records of mathematical activity date back to early Egypt and Mesopotamia also India and China. But maths underwent a major transition in Ancient Greece where proofs were first demonstrated and geometry took precedence over arithmetic. During the European Dark Age further mathematical progress was made by Arab mathematicians during an Islamic Golden Age when translations of Greeks mathematics became available. Only with the Renaissance was there a revival of mathematics in Europe that culminated in the 17^{th} century as a Heroic Age of mathematics. Being a symbolic ‘language’ mathematics has always involved the development of the most efficient systems of notation.

### Mesopotamia, Egypt, India, and China

Evidence for mathematical principles and procedures appear in the early civilizations of Mesopotamia, Egypt, the Indus Valley and China. All had arithmetic and geometry, the Egyptians and Babylonians using fractions by about 2000 BCE. China had decimal fractions about 1500 years before Europe. Babylonians were aware of Pythagoras’s theorem before Pythagoras while Babylonians used a sexagesimal (base 60) for numbers and fractions which persisted into the 17th century CE. Indo-Arabic numerals began in India around 3rd century BCE. But none used formal proofs or demonstrations of logical underpinnings. Related to commerce, surveying and building but also recreational puzzles.

### Ancient Greece

Deductive proofs began with the ancient Greeks, the first attributed to Thales of Miletus beginning with agreed axioms and definitions connected by logic. Greeks worked mostly with geometry with little number theory and a little trigonometry as an extension of geometry used in astronomy – even how the Earth moved round the Sun. This was done by Euclid (4th BCE) studied by all subsequent major mathematicians and the Pythagoreans (6th BCE) who discovered irrational numbers. Archimedes (3rd BCE) excelled at geometry and number theory and the preliminary ideas relating to future calculus as one of the greatest mathematicians of all time. Appolonius (2nd BCE) carried out foundational studies in conic sections (ellipse, parabola etc.) so important later in astronomy.

### Islamic Golden Age

400-1200 CE was a European Dark Age with mathematical work being done by Islamic scholars after the establishment of Islam by Mohammed in the 7th century CE building on ancient texts and used Hindu/Arabic (Indian) numerals and decimal fractions c. 800-1200 CE. The word ‘Algebra’ is derived from the Arabic ‘Al-Jabr’ in geometric form c. 825 cubic equations by Omar Khayam who was not surpassed in Europe until 16th century.

### European Renaissance & Scientific Revolution

Around 1000 CE Hindu/Arabic numerals are first used in Europe, the Fibonnacci series produced the Liber Abaci in 1200s ittook300years to get these numerals accepted. In 1450 universities could cope with addition and subtraction but multiplication and division was only available in Italy (Europeans were still using cumbersome Roman numerals).

Only in 1585 did Belgian Simon Stevin advocate base 10 (decimal) fractions although his notation was clumsy and ignored until the 1620s. Scotsman John Napier invented logarithms in 1614 noting that you could convert multiplication into addition (products to sums), quotients into differences etc. but his base was clumsy so Briggs in 1624 developed base 10 logs. Today’s symbolic algebra (that of today) was launched in Europe in 16th century, moving from rhetorical to syncopated algebra as mathematics in general was gathering momentum. Geometry and algebra now become united with the modern-day coordinate system established by Frenchman Fermat and Descartes (method) in the 1630s. Could find the solutions to algebraic equations by graphing them.

### The 17th century European Heroic Century for mathematics

The 17th century in Europe is known as the Heroic Century for mathematics as in 1600 it passes from an emphasis on geometry to calculus, as formal analysis in 1700 marking the transition from ancient to modern mathematics – a major transition. In 1650 there was a foundation of Greek geometry, symbolic algebra, Hindu/Arabic numerals, logarithms and the combination of algebra and geometry to create coordinate systems and graphs. This set the scene for the birth of calculus via people like Kepler, Fermat, Borrow at Cambridge and others. The two towering figures of Newton and Liebniz (the notation we use today) developed their own versions, Liebniz publishing in 1684. From now on advances were given momentum by the application of calculus to physics, astronomy, and engineering using the calculus tools of functions, curves, series, derivatives, and integrals. The work of Bernoulli was passed on to people like Leonard Euler (1707-1783) the greatest mathematician of the 18th century.

### The 19th century challenge to foundations

By the 19th century the foundations of mathematics were under challenge. Euclidian geometry’s parallel postulate led to three angles adding up to 180. Gauss when 15 was convinced it couldn’t be done and in hyperbolic space it was less than 180 used by Einstein and Riemann’s elliptic geometry ‘fat’ triangles. By 1870 the hyperbolic geometry was clearly accepted and Euclid equalled. Gauss (differential geometry and number theory) was one of the greatest mathematicians of the 19th century with contemporaries Lobachevsky (1829) and Bolyai (1832) adding to the work of the day. Fourier worked with trigonometric series and now the foundations of calculus began to look shaky. They depended on limits which depended on real numbers (as yet undefined), reals were based on rationals, rationals on integers, integers on natural numbers. By 1900 the arithmetization of analysis was in place (Gauss, Cauchy, Riemann, Weierstrass, Dedekind, Peano) establishing the foundations of analysis. Cantor developed the idea of ‘the set of all the natural numbers’ that some infinities are bigger than others.

### 20th century global mathematics

In the last 100 years much more mathematics has been created than in the last 5000, much of it extremely difficult. Of note are the incompleteness theorems of Kurt Gödel (1906-1978) challenged the foundations of mathematics in 1931. That the formal axiomatic system of natural numbers is impossible to prove within that system, that the system is complete. It is also impossible to prove (within the system) the statement that ‘this system is consistent’. American Claude Shannon (1916-2001) in 1948 founded information theory, transmission of information (words, images, sounds) using 1s and 0s.

## Timeline

### BCE

#### Mesopotamia, Egypt, India, and China

**c. 3400** – first numeral system, and a system of weights and measures. Sumeria

**c. 3100** – earliest decimal system allows indefinite counting by way of introducing new symbols Egypt

**c. 2800** – earliest use of decimal ratios in a uniform system of ancient weights and measures, the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 gramsIndus Valley Civilization

**2700 **– precision surveying Egypt

**2400** – precise astronomical calendar used into the Middle Ages Egypt

**c. 2000** – Sexagesimal 60-base numeral system. First known computation of π to 3.125.Babylon

**c. 2000** – carved stone balls with all of the symmetries of Platonic solids Scotland

**1800** – findings volume of a frustum Egypt, Moscow Mathematical Papyrus

**c. 1800** – Berlin Papyrus 6619 (Egypt, 19th dynasty) contains a quadratic equation and its solution

**1650** – Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of π at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order linear equations

**1046 to 256** – arithmetic and geometric algorithms and proofs China, attributed to Zhoubi Suanjing

**c. 1000** – simple fractions but only unit fractions are used (i.e. those with 1 as the numerator) and interpolation tables are used to approximate the values of the other fractions

**c.500** – Yajnavalkya, in *Shatapatha Brahmana* describes the motions of the sun and the moon, and advances a 95-year cycle to synchronize the motions of the sun and the moon Vedic India

**c. 8th century** – Yajur Veda (one of four Hindu Vedas) has mathematical reference to infinity ‘if you remove a part from infinity or add a part to infinity, still what remains is infinity’

**800** – Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of two correctly to five decimal places

**624 – 546** – Thales of Miletus has various theorems attributed to him

**c. 600** – the other Vedic “Sulba Sutras” (“rule of chords” in Sanskrit) use Pythagorean triples, contain of a number of geometrical proofs, and approximate π at 3.16.

**second half of 1st millennium**– The Lo Shu Square, the unique normal magic square of order three, was discovered in China

**530**– Pythagoras studies propositional geometry and vibrating lyre strings; his group also discovers the irrationality of the square root of two.

**c. 500** – Indian grammarian Pānini writes the Astadhyayi, which contains the use of metarules, transformations and recursions, originally for the purpose of systematizing the grammar of Sanskrit.

**470 BC – 410** – Hippocrates of Chios utilizes lunes in an attempt to square the circle.

**5th century** – Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places.

**490 BC – 430** Zeno of Elea Zeno’s paradoxes

**5th c. ** Theodorus of Cyrene, Democritus, Hippias, Archytas, Plato, Theaetetus (mathematician)

**c. 400** – Jaina mathematicians in India write the Surya Prajinapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually

**408 BC – 355** Eudoxus of Cnidus, Antiphon, Bryson of Heraclea, Xenocrates

**4th century** –Dinostratus, Autolycus of Pitane, Aristaeus the Elder, Callippus

**370**– Eudoxus states the method of exhaustion for area determination

**350** – Aristotle discusses logical reasoning in Organon

**330**– the earliest known work on Chinese geometry, the Mo Jing, is compiled

**3rd century Aristarchus of Samos, Heraclides of Pontus, Menaechmus**

**300 BC – Jain mathematicians in India write the Bhagabati Sutra, which contains the earliest information on combinations. **

**300 BC – Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in Catoptrics, and he proves the fundamental theorem of arithmetic. **

**c. 300 BC – Brahmi numerals (ancestor of the common modern base 10 numeral system) are conceived in India. **

**370 BC – 300 BC – Eudemus of Rhodes works on histories of arithmetic, geometry and astronomy now lost **

**300 BC – Mesopotamia, the Babylonians invent the earliest calculator, the abacus. **

**c. 300 BC – Indian mathematician Pingala writes the Chhandah-shastra, which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of a binary numeral system, along with the first use of Fibonacci numbers and Pascal’s triangle. **

**c. 3rd century Nicomedes (mathematician), Philon of Byzantium, Chrysippus, Conon of Samos, Dionysodorus, Apollonius of Perga, Diocles (mathematician) **

**202 BC to 186 BC – Book on Numbers and Computation, a mathematical treatise, is written in Han Dynasty China. **

**260 BC – Archimedes proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3. **

**c. 250 BC – late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number). **

**240 BC – Eratosthenes uses his sieve algorithm to quickly isolate prime numbers. **

**225 BC – Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola. **

**206 BC to 8 AD – Counting rods are invented in China. **

**c. 2nd century Zenodorus (mathematician), Perseus (geometer), Zeno of Sidon, Hypsicles, Theodosius of Bithynia, Posidonius**

**150 BC – Jain mathematicians in India write the Sthananga Sutra, which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. **

**150 BC – A method of Gaussian elimination appears in the Chinese text The Nine Chapters on the Mathematical Art. **

**150 BC – Horner’s method appears in the Chinese text The Nine Chapters on the Mathematical Art. **

**150 BC – Negative numbers appear in the Chinese text The Nine Chapters on the Mathematical Art. **

**190 BC – 120 BC – Hipparchus develops the bases of trigonometry. **

**50 BC – Indian numerals, a descendant of the Brahmi numerals (the first positional notation base-10 numeral system), begins development in India. **

**final centuries BC – Indian astronomer Lagadha writes the Vedanga Jyotisha, a Vedic text on astronomy that describes rules for tracking the motions of the sun and the moon, and uses geometry and trigonometry for astronomy. **

**1st Century Geminus, Cleomedes, Heron of Alexandria, (Hero) the earliest fleeting reference to square roots of negative numbers, Theon of Smyrna, Nicomachus, Menelaus of Alexandria Spherical trigonometry**

**50 BC – 23 AD Liu Xin**

**CE**

**1st to 2nc century Zhang Heng, Cai Yong**

**c. 2nd century – Ptolemy of Alexandria wrote the Almagest. **

**3rd century – Sporus of Nicaea, Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes Arithmetica, one of the earliest treatises on algebra. **

**263 – Liu Hui computes π using Liu Hui’s π algorithm. **

**300 – the earliest known use of zero as a decimal digit is introduced by Indian mathematicians. **

**3rd to 4th century – Porphyry (philosopher), Serenus of Antinouplis. **

**300 to 500 – the Chinese remainder theorem is developed by Sun Tzu. **

**300 to 500 – a description of rod calculus is written by Sun Tzu. **

**4th to 5th century – Theon of Alexandria, Pappus of Alexandria states his hexagon theorem and his centroid theorem, Hypatia, Proclus, Domninus of Larissa, Marinus of Neapolis, Anthemius of Tralles**

**c. 400 – the Bakhshali manuscript is written by Jaina mathematicians, which describes a theory of the infinite containing different levels of infinity, shows an understanding of indices, as well as logarithms to base 2, and computes square roots of numbers as large as a million correct to at least 11 decimal places. **

**450 – Zu Chongzhi computes π to seven decimal places. This calculation remains the most accurate calculation for π for close to a thousand years. **

**500 – Aryabhata writes the Aryabhata-Siddhanta, which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of sine and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees). **

**5th to 6th century – Eutocius of Ascalon, 490 – 560 Simplicius of Cilicia.**

**6th century – Aryabhata gives accurate calculations for astronomical constants, such as the solar eclipse and lunar eclipse, computes π to four decimal places, and obtains whole number solutions to linear equations by a method equivalent to the modern method. **

**6th century – Yativṛṣabha, Varāhamihira.**

**535 – 566 Zhen Luan.**

**550 – Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system. **

**7th century – Bhaskara I gives a rational approximation of the sine function.**

**7th century – Brahmagupta invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems. He also develops methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon.**

**628 – Brahmagupta writes the Brahma-sphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed. It also gives rules for manipulating both negative and positive numbers, methods for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta’s identity, and the Brahmagupta theorem.**

**602 – 670 Li Chunfeng**

**Islamic Golden Age**

**8th century – Virasena gives explicit rules for the Fibonacci sequence, gives the derivation of the volume of a frustum using an infinite procedure, and also deals with the logarithm to base 2 and knows its laws. **

**8th century – Shridhara gives the rule for finding the volume of a sphere and the formula for solving quadratic equations. **

**773 – Kanka brings Brahmagupta’s Brahma-sphuta-siddhanta to Baghdad to explain the Indian system of arithmetic astronomy and the Indian numeral system. **

**773 – Al-Fazari translates the Brahma-sphuta-siddhanta into Arabic upon the request of King Khalif Abbasid Al Mansoor. **

**9th century – Govindsvamin discovers the Newton-Gauss interpolation formula, and gives the fractional parts of Aryabhata’s tabular sines. **

**810 – The House of Wisdom is built in Baghdad for the translation of Greek and Sanskrit mathematical works into Arabic. **

**820 – Al-Khwarizmi – Persian mathematician, father of algebra, writes the Al-Jabr, later transliterated as Algebra, which introduces systematic algebraic techniques for solving linear and quadratic equations. Translations of his book on arithmetic will introduce the Hindu-Arabic decimal number system to the Western world in the 12th century. The term algorithm is also named after him. **

**820 – Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra. **

**c. 850 – Al-Kindi pioneers cryptanalysis and frequency analysis in his book on cryptography. **

**c. 850 – Mahāvīra writes the Gaṇitasārasan̄graha otherwise known as the Ganita Sara Samgraha which gives systematic rules for expressing a fraction as the sum of unit fractions. **

**895 – Thabit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations. He also generalized the Pythagorean theorem, and discovered the theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other). **

**c. 900 – Abu Kamil of Egypt devises new notation**

**940 – Abu’l-Wafa al-Buzjani extracts roots using the Indian numeral system. **

**953 – The arithmetic of the Hindu-Arabic numeral system at first required the use of a dust board (a sort of handheld blackboard) because “the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded.” Al-Uqlidisi modified these methods for pen and paper use. Eventually the advances enabled by the decimal system led to its standard use throughout the region and the world. **

**953 – Al-Karaji is the “first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He started a school of algebra which flourished for several hundreds of years”. He also discovered the binomial theorem for integer exponents, which “was a major factor in the development of numerical analysis based on the decimal system”.**

**975 – Al-Batani extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions.**

**c. 1000 – Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree. **

**c. 1000 – Abu-Mahmud al-Khujandi first states a special case of Fermat’s Last Theorem. **

**c. 1000 – Law of sines is discovered by Muslim mathematicians, but it is uncertain who discovers it first between Abu-Mahmud al-Khujandi, Abu Nasr Mansur, and Abu al-Wafa. **

**c. 1000 – Pope Sylvester II introduces the abacus using the Hindu-Arabic numeral system to Europe. **

**1000 – . He used it to prove the binomial theorem, Pascal’s triangle, and the sum of integral cubes. He was “the first who introduced the theory of algebraic calculus”.**

**c. 1000 – Ibn Tahir al-Baghdadi studied a slight variant of Thabit ibn Qurra’s theorem on amicable numbers, and he also made improvements on the decimal system. **

**1020 – Abul Wáfa gave the formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the parabola and the volume of the paraboloid. **

**1021 – Ibn al-Haytham formulated and solved Alhazen’s problem geometrically. **

**1030 – Ali Ahmad Nasawi writes a treatise on the decimal and sexagesimal number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner. **

**1070 – Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations. **

**c. 1100 – Omar Khayyám “gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections”. He became the first to find general geometric solutions of cubic equations and laid the foundations for the development of analytic geometry and non-Euclidean geometry. He also extracted roots using the decimal system (Hindu-Arabic numeral system). **

**12th century – Indian numerals have been modified by Arab mathematicians to form the modern Hindu-Arabic numeral system (used universally in the modern world). **

**12th century – the Hindu-Arabic numeral system reaches Europe through the Arabs. **

**12th century – Bhaskara Acharya writes the Lilavati, which covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations. **

**12th century – Bhāskara II (Bhaskara Acharya) writes the Bijaganita (Algebra), which is the first text to recognize that a positive number has two square roots.**

**12th century – Bhaskara Acharya conceives differential calculus, and also develops Rolle’s theorem, Pell’s equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places. **

**1130 – Al-Samawal gave a definition of algebra: “[it is concerned] with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.” **

**1135 – Sharafeddin Tusi followed al-Khayyam’s application of algebra to geometry, and wrote a treatise on cubic equations that “represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry”.**

**1202 – Leonardo Fibonacci demonstrates the utility of Hindu-Arabic numerals in his Liber Abaci (Book of the Abacus). **

**1247 – Qin Jiushao publishes Shùshū Jiǔzhāng (Mathematical Treatise in Nine Sections). **

**1248 – Li Ye writes Ceyuan haijing, a 12 volume mathematical treatise containing 170 formulas and 696 problems mostly solved by polynomial equations using the method tian yuan shu. **

**1260 – Al-Farisi gave a new proof of Thabit ibn Qurra’s theorem, introducing important new ideas concerning factorization and combinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 that have also been joint attributed to Fermat as well as Thabit ibn Qurra. **

**c. 1250 – Nasir Al-Din Al-Tusi attempts to develop a form of non-Euclidean geometry. **

**1303 – Zhu Shijie publishes Precious Mirror of the Four Elements, which contains an ancient method of arranging binomial coefficients in a triangle. **

**14th century – Madhava is considered the father of mathematical analysis, who also worked on the power series for π and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of calculus. **

**14th century – Parameshvara, a Kerala school mathematician, presents a series form of the sine function that is equivalent to its Taylor series expansion, states the mean value theorem of differential calculus, and is also the first mathematician to give the radius of circle with inscribed cyclic quadrilateral. **

**1400 – Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places.**

**c. 1400 – Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots, which is a special case of the methods given many centuries later by [Paolo] Ruffini and [William George] Horner.” He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include The Key of arithmetics, Discoveries in mathematics, The Decimal point, and The benefits of the zero. The contents of the Benefits of the Zero are an introduction followed by five essays: “On whole number arithmetic”, “On fractional arithmetic”, “On astrology”, “On areas”, and “On finding the unknowns [unknown variables]”. He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine. **

**15th century – Ibn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general**

**15th century – Nilakantha Somayaji, a Kerala school mathematician, writes the Aryabhatiya Bhasya, which contains work on infinite-series expansions, problems of algebra, and spherical geometry. **

**1424 – Ghiyath al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons. **

**1427 – Al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones. **

**1464 – Regiomontanus writes De Triangulis omnimodus which is one of the earliest texts to treat trigonometry as a separate branch of mathematics. **

**1478 – An anonymous author writes the Treviso Arithmetic. **

**1494 – Luca Pacioli writes Summa de arithmetica, geometria, proportioni et proportionalità; introduces primitive symbolic algebra using “co” (cosa) for the unknown. **

**1501 – Nilakantha Somayaji writes the Tantrasamgraha. **

**1520 – Scipione dal Ferro develops a method for solving “depressed” cubic equations (cubic equations without an x2 term), but does not publish. **

**1522 – Adam Ries explained the use of Arabic digits and their advantages over Roman numerals. **

**1535 – Niccolò Tartaglia independently develops a method for solving depressed cubic equations but also does not publish. **

**1539 – Gerolamo Cardano learns Tartaglia’s method for solving depressed cubics and discovers a method for depressing cubics, thereby creating a method for solving all cubics. **

**1540 – Lodovico Ferrari solves the quartic equation. **

**1544 – Michael Stifel publishes Arithmetica integra. **

**1545 – Gerolamo Cardano conceives the idea of complex numbers. **

**1550 – Jyeshtadeva, a Kerala school mathematician, writes the Yuktibhāṣā, the world’s first calculus text, which gives detailed derivations of many calculus theorems and formulae. **

**1572 – Rafael Bombelli writes Algebra treatise and uses imaginary numbers to solve cubic equations. **

**1584 – Zhu Zaiyu calculates equal temperament. **

**1596 – Ludolf van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons. **

**17th century**

**1614 – John Napier discusses Napierian logarithms in Mirifici Logarithmorum Canonis Descriptio. **

**1617 – Henry Briggs discusses decimal logarithms in Logarithmorum Chilias Prima. **

**1618 – John Napier publishes the first references to e in a work on logarithms. **

**1619 – René Descartes discovers analytic geometry (Pierre de Fermat claimed that he also discovered it independently). **

**1619 – Johannes Kepler discovers two of the Kepler-Poinsot polyhedra. **

**1629 – Pierre de Fermat develops a rudimentary differential calculus. **

**1634 – Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle. **

**1636 – Muhammad Baqir Yazdi jointly discovered the pair of amicable numbers 9,363,584 and 9,437,056 along with Descartes (1636). **

**1637 – Pierre de Fermat claims to have proven Fermat’s Last Theorem in his copy of Diophantus’ Arithmetica. **

**1637 – First use of the term imaginary number by René Descartes; it was meant to be derogatory. **

**1643 – René Descartes develops Descartes’ theorem. **

**1654 – Blaise Pascal and Pierre de Fermat create the theory of probability. **

**1655 – John Wallis writes Arithmetica Infinitorum. **

**1658 – Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle. **

**1665 – Isaac Newton works on the fundamental theorem of calculus and develops his version of infinitesimal calculus. **

**1668 – Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbolic segment. **

**1671 – James Gregory develops a series expansion for the inverse-tangent function (originally discovered by Madhava). **

**1671 – James Gregory discovers Taylor’s Theorem. **

**1673 – Gottfried Leibniz also develops his version of infinitesimal calculus. **

**1675 – Isaac Newton invents an algorithm for the computation of functional roots. **

**1680s – Gottfried Leibniz works on symbolic logic. **

**1683 – Seki Takakazu discovers the resultant and determinant. **

**1683 – Seki Takakazu develops elimination theory. **

**1691 – Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations. **

**1693 – Edmund Halley prepares the first mortality tables statistically relating death rate to age. **

**1696 – Guillaume de L’Hôpital states his rule for the computation of certain limits. **

**1696 – Jakob Bernoulli and Johann Bernoulli solve brachistochrone problem, the first result in the calculus of variations. **

**1699 – Abraham Sharp calculates π to 72 digits but only 71 are correct. **

**18th century**

**1706 – John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places. 1708 – Seki Takakazu discovers Bernoulli numbers. Jacob Bernoulli whom the numbers are named after is believed to have independently discovered the numbers shortly after Takakazu. 1712 – Brook Taylor develops Taylor series. 1722 – Abraham de Moivre states de Moivre’s formula connecting trigonometric functions and complex numbers. 1722 – Takebe Kenko introduces Richardson extrapolation. 1724 – Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in Annuities on Lives. 1730 – James Stirling publishes The Differential Method. 1733 – Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid’s fifth postulate were false. 1733 – Abraham de Moivre introduces the normal distribution to approximate the binomial distribution in probability. 1734 – Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations. 1735 – Leonhard Euler solves the Basel problem, relating an infinite series to π. 1736 – Leonhard Euler solves the problem of the Seven bridges of Königsberg, in effect creating graph theory. 1739 – Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients. 1742 – Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach’s conjecture. 1748 – Maria Gaetana Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana. 1761 – Thomas Bayes proves Bayes’ theorem. 1761 – Johann Heinrich Lambert proves that π is irrational. 1762 – Joseph Louis Lagrange discovers the divergence theorem. 1789 – Jurij Vega improves Machin’s formula and computes π to 140 decimal places, 136 of which were correct. 1794 – Jurij Vega publishes Thesaurus Logarithmorum Completus. 1796 – Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge. 1796 – Adrien-Marie Legendre conjectures the prime number theorem. 1797 – Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms. 1799 – Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers). 1799 – Paolo Ruffini partially proves the Abel–Ruffini theorem that quintic or higher equations cannot be solved by a general formula. **

**19th century**

**1801 – Disquisitiones Arithmeticae, Carl Friedrich Gauss’s number theory treatise, is published in Latin. 1805 – Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set of observations. 1806 – Louis Poinsot discovers the two remaining Kepler-Poinsot polyhedra. 1806 – Jean-Robert Argand publishes proof of the Fundamental theorem of algebra and the Argand diagram. 1807 – Joseph Fourier announces his discoveries about the trigonometric decomposition of functions. 1811 – Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration. 1815 – Siméon Denis Poisson carries out integrations along paths in the complex plane. 1817 – Bernard Bolzano presents the intermediate value theorem—a continuous function that is negative at one point and positive at another point must be zero for at least one point in between. 1822 – Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane. 1822 – Irisawa Shintarō Hiroatsu analyzes Soddy’s hexlet in a Sangaku. 1823 – Sophie Germain’s Theorem is published in the second edition of Adrien-Marie_Legendre’s Essai sur la théorie des nombres 1824 – Niels Henrik Abel partially proves the Abel–Ruffini theorem that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. 1825 – Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis. 1825 – Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat’s Last Theorem for n = 5. 1825 – André-Marie Ampère discovers Stokes’ theorem. 1828 – George Green proves Green’s theorem. 1829 – János Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry. 1831 – Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green. 1832 – Évariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory. 1832 – Lejeune Dirichlet proves Fermat’s Last Theorem for n = 14. 1835 – Lejeune Dirichlet proves Dirichlet’s theorem about prime numbers in arithmetical progressions. 1837 – Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructability of regular polygons. 1837 – Peter Gustav Lejeune Dirichlet develops Analytic number theory. 1841 – Karl Weierstrass discovers but does not publish the Laurent expansion theorem. 1843 – Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem. 1843 – William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative. 1847 – George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what is now called Boolean algebra. 1849 – George Gabriel Stokes shows that solitary waves can arise from a combination of periodic waves. 1850 – Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points. 1850 – George Gabriel Stokes rediscovers and proves Stokes’ theorem. 1854 – Bernhard Riemann introduces Riemannian geometry. 1854 – Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space. 1858 – August Ferdinand Möbius invents the Möbius strip. 1858 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions. 1859 – Bernhard Riemann formulates the Riemann hypothesis, which has strong implications about the distribution of prime numbers. 1870 – Felix Klein constructs an analytic geometry for Lobachevski’s geometry thereby establishing its self-consistency and the logical independence of Euclid’s fifth postulate. 1872 – Richard Dedekind invents what is now called the Dedekind Cut for defining irrational numbers, and now used for defining surreal numbers. 1873 – Charles Hermite proves that e is transcendental. 1873 – Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points. 1874 – Georg Cantor proves that the set of all real numbers is uncountably infinite but the set of all real algebraic numbers is countably infinite. His proof does not use his diagonal argument, which he published in 1891. 1882 – Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge. 1882 – Felix Klein invents the Klein bottle. 1895 – Diederik Korteweg and Gustav de Vries derive the Korteweg–de Vries equation to describe the development of long solitary water waves in a canal of rectangular cross section. 1895 – Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis. 1895 – Henri Poincare publishes paper “Analysis Situs” which started modern topology. 1896 – Jacques Hadamard and Charles Jean de la Vallée-Poussin independently prove the prime number theorem. 1896 – Hermann Minkowski presents Geometry of numbers. 1899 – Georg Cantor discovers a contradiction in his set theory. 1899 – David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry. 1900 – David Hilbert states his list of 23 problems, which show where some further mathematical work is needed. 20th century 1901 – Élie Cartan develops the exterior derivative. 1901 – Henri Lebesgue publishes on Lebesgue integration. 1903 – Carle David Tolmé Runge presents a fast Fourier transform algorithm 1903 – Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem. 1908 – Ernst Zermelo axiomizes set theory, thus avoiding Cantor’s contradictions. ng>1908 – Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky – Plemelj formulae. 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem. 1912 – Josip Plemelj publishes simplified proof for the Fermat’s Last Theorem for exponent n = 5. 1915 – Emmy Noether proves her symmetry theorem, which shows that every symmetry in physics has a corresponding conservation law. 1916 – Srinivasa Ramanujan introduces Ramanujan conjecture. This conjecture is later generalized by Hans Petersson. 1919 – Viggo Brun defines Brun’s constant B2 for twin primes. 1921 – Emmy Noether introduces the first general definition of a commutative ring. 1928 – John von Neumann begins devising the principles of game theory and proves the minimax theorem. 1929 – Emmy Noether introduces the first general representation theory of groups and algebras. 1930 – Casimir Kuratowski shows that the three-cottage problem has no solution. 1930 – Alonzo Church introduces Lambda calculus. 1931 – Kurt Gödel proves his incompleteness theorem, which shows that every axiomatic system for mathematics is either incomplete or inconsistent. 1931 – Georges de Rham develops theorems in cohomology and characteristic classes. 1933 – Karol Borsuk and Stanislaw Ulam present the Borsuk–Ulam antipodal-point theorem. 1933 – Andrey Nikolaevich Kolmogorov publishes his book Basic notions of the calculus of probability (Grundbegriffe der Wahrscheinlichkeitsrechnung), which contains an axiomatization of probability based on measure theory. 1940 – Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory. 1942 – G.C. Danielson and Cornelius Lanczos develop a fast Fourier transform algorithm. 1943 – Kenneth Levenberg proposes a method for nonlinear least squares fitting. 1945 – Stephen Cole Kleene introduces realizability. 1945 – Saunders Mac Lane and Samuel Eilenberg start category theory. 1945 – Norman Steenrod and Samuel Eilenberg give the Eilenberg–Steenrod axioms for (co-)homology. 1946 – Jean Leray introduces the Spectral sequence. 1948 – John von Neumann mathematically studies self-reproducing machines. 1948 – Alan Turing introduces LU decomposition. 1949 – John Wrench and L.R. Smith compute π to 2,037 decimal places using ENIAC. 1949 – Claude Shannon develops notion of Information Theory. 1950 – Stanisław Ulam and John von Neumann present cellular automata dynamical systems. 1953 – Nicholas Metropolis introduces the idea of thermodynamic simulated annealing algorithms. 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron. 1955 – Enrico Fermi, John Pasta, Stanisław Ulam, and Mary Tsingou numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior. 1956 – Noam Chomsky describes an hierarchy of formal languages. 1957 – Kiyosi Itô develops Itô calculus. 1957 – Stephen Smale provides the existence proof for crease-free sphere eversion. 1958 – Alexander Grothendieck’s proof of the Grothendieck–Riemann–Roch theorem is published. 1959 – Kenkichi Iwasawa creates Iwasawa theory. 1960 – C. A. R. Hoare invents the quicksort algorithm. 1960 – Irving S. Reed and Gustave Solomon present the Reed–Solomon error-correcting code. 1961 – Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer. 1961 – John G. F. Francis and Vera Kublanovskaya independently develop the QR algorithm to calculate the eigenvalues and eigenvectors of a matrix. 1961 – Stephen Smale proves the Poincaré conjecture for all dimensions greater than or equal to 5. 1962 – Donald Marquardt proposes the Levenberg–Marquardt nonlinear least squares fitting algorithm. 1962 – Gloria Conyers Hewitt becomes the third African American woman to receive a PhD in mathematics. 1963 – Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory. 1963 – Martin Kruskal and Norman Zabusky analytically study the Fermi–Pasta–Ulam–Tsingou heat conduction problem in the continuum limit and find that the KdV equation governs this system. 1963 – meteorologist and mathematician Edward Norton Lorenz published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour and strange attractors or Lorenz Attractor – also the Butterfly Effect. 1965 – Iranian mathematician Lotfi Asker Zadeh founded fuzzy set theory as an extension of the classical notion of set and he founded the field of Fuzzy Mathematics. 1965 – Martin Kruskal and Norman Zabusky numerically study colliding solitary waves in plasmas and find that they do not disperse after collisions. 1965 – James Cooley and John Tukey present an influential fast Fourier transform algorithm. 1966 – E. J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix. 1966 – Abraham Robinson presents non-standard analysis. 1967 – Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory. 1968 – Michael Atiyah and Isadore Singer prove the Atiyah–Singer index theorem about the index of elliptic operators. 1973 – Lotfi Zadeh founded the field of fuzzy logic. 1975 – Benoît Mandelbrot publishes Les objets fractals, forme, hasard et dimension. 1976 – Kenneth Appel and Wolfgang Haken use a computer to prove the Four color theorem. 1978 – Olga Taussky-Todd is awarded the Austrian Cross of Honour for Science and Art, 1st Class, the highest scientific award of the government of Austria. 1981 – Richard Feynman gives an influential talk “Simulating Physics with Computers” (in 1980 Yuri Manin proposed the same idea about quantum computations in “Computable and Uncomputable” (in Russian)). 1983 – Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat’s Last Theorem. 1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed. 1985 – Louis de Branges de Bourcia proves the Bieberbach conjecture. 1986 – Ken Ribet proves Ribet’s theorem. 1987 – Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places. 1991 – Alain Connes and John W. Lott develop non-commutative geometry. 1992 – David Deutsch and Richard Jozsa develop the Deutsch–Jozsa algorithm, one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. 1994 – Andrew Wiles proves part of the Taniyama–Shimura conjecture and thereby proves Fermat’s Last Theorem. 1994 – Peter Shor formulates Shor’s algorithm, a quantum algorithm for integer factorization. 1995 – Simon Plouffe discovers Bailey–Borwein–Plouffe formula capable of finding the nth binary digit of π. 1998 – Thomas Callister Hales (almost certainly) proves the Kepler conjecture. 1999 – the full Taniyama–Shimura conjecture is proven. 2000 – the Clay Mathematics Institute proposes the seven Millennium Prize Problems of unsolved important classic mathematical questions. 21st century 2002 – Manindra Agrawal, Nitin Saxena, and Neeraj Kayal of IIT Kanpur present an unconditional deterministic polynomial time algorithm to determine whether a given number is prime (the AKS primality test). 2002 – Yasumasa Kanada, Y. Ushiro, Hisayasu Kuroda, Makoto Kudoh and a team of nine more compute π to 1241.1 billion digits using a Hitachi 64-node supercomputer. 2002 – Preda Mihăilescu proves Catalan’s conjecture. 2003 – Grigori Perelman proves the Poincaré conjecture. 2004 – Ben Green and Terence Tao prove the Green-Tao theorem. 2007 – a team of researchers throughout North America and Europe used networks of computers to map E8. 2009 – Fundamental lemma (Langlands program) had been proved by Ngô Bảo Châu. 2010 – Larry Guth and Nets Hawk Katz solve the Erdős distinct distances problem. 2013 – Yitang Zhang proves the first finite bound on gaps between prime numbers. 2014 – Project Flyspeck announces that it completed proof of Kepler’s conjecture. 2014 – Using Alexander Yee’s y-cruncher “houkouonchi” successfully calculated π to 13.3 trillion digits. 2015 – Terence Tao solves The Erdös Discrepancy Problem 2015 – László Babai found that a quasipolynomial complexity algorithm would solve the Graph Isomorphism Problem 2016 – Using Alexander Yee’s y-cruncher Peter Trueb successfully calculated π to 22.4 trillion digits**

#### Glossary

**Analogue** – analogue signals are continuous while digital signals are discrete – soanalog technologies record waveforms while digital technologies convert analog signals into numbers

**Frustum** – in geometry, a frustum is the portion of a solid that lies between one or two parallel planes cutting it. A right frustum is a parallel truncation of a right pyramid

**Integer** – a number that can be written without a fractional component e.g. 4,0, -2010 while 0.75 and 3/2 are not

**Irrational number** – a real number that cannot be written as a simple fraction, like pie. numbers that are not the ratio of two whole numbers

**Natural numbers** – are those used for counting (cardinal numbers) and ordering (ordinal numbers)

**Prime number** – a natural number>1 that has no positive divisors other than 1 and itself. A natural number that is >1 that is not a prime number is called a composite number

**Rational number** – any number that can be expressed as the fraction (quotient) of two integers, a numerator p and a non-zero denominator q. Since q maybe equal to 1 then every integer is a rational number

**Real number ** – a value that represents a quantity along a line

**Whole number** – a number without fractions; an integer

## Commentary & sustainability analysis

Though archaeological indications of numeracy date back into prehistory, mathematics developed in urban environments as a means of facilitating commerce, surveying, engineering, architecture, and building construction, and where academic interests could be shared by a community of people with like interests. This included a special interest in the motions of other-worldly astronomical objects of the night sky, a possible world of the gods. Though, at times, it was clearly treated as a form of intellectual recreation. These are also the kind of communities for which there is a written historical record.

The relationship between complexity of social organization and political and economic power is reflected in the regions and countries where mathematical innovation took place. Complex societies employed an intelligentsia of bureaucrats and academics and the geography of mathematical achievement has followed the geography of political and economic dominance.

The presence of a strong mathematical community is a measure of intellectual aspiration within that community since mathematics is generally recognized as making high intellectual demands.

Applied mathematics that depends on computational capacity has made major progress as once demanding calculations moved from abacus to slide rule to personal calculators. Analogue computers address continuously changing physical phenomena such as electrical, mechanical, or hydraulic quantities while digital computers represent varying quantities symbolically, as their numerical values change.

## Key points

- Copernicus replaced the former geocentric (Earth-centred) planetary system with a heliocentric (Sun-centred) one

- Kepler’s confirmation of planetary elliptic, not circular, orbits

- Newtonian development of the laws of motion and gravity

- Newtonian development of the laws of motion and gravity

- What we now know as science was, until the 19th century, known as natural philosophy

- A scientific ethos was inherited from the Pre-Socratic philosophers who sought to explain phenomena based on naturalistic (non-supernatural) explanations within the physical world

- The ancient metaphor of the world operating like an organism is replaced by the metaphor of the world as a machine

- By about 1550 the boom in metals, mostly mined in C Europe, succumbed to the metal trade from the New World

- Gerhardus Mercator (1512-1594) developed the Mercator map projection – a way of projecting the round Earth onto a flat map

- In 1656 Christiaan Huygens built the first pendulum clock

- Englishman Francis Bacon (1561-1626) called for science to become an applied discipline that used experiment, observation, and inductive reasoning with conclusions shared by a scientific community rather than outmoded Aristotelian deductive logic applied by isolated researchers. His thoughts were published in his influential Novum Organum (New Instrument of Science) of 1620. This was a reaction to excessive speculation and the construction of dubious grand explanatory and theoretical systems. Science was for ‘The glory of the Creator and the relief of man’s estate

- News, gossip, and research were exchanged between scientists of the 17th century through a ‘republic of letters’

- Perspective painting and engraved prints combined with the power of the printing press in 1450 transformed scientific communication

- : perceived as a move away from abstract speculation, hypotheses, and ideas to an empirical grounding in the world of physical objects by means of experiment and observation

- Newton’s Principia united the former two Aristotelian spheres through the theory of universal gravity while making a major contribution to science in many other areas including optics but especially mathematics and, with Liebniz, derivation of the calculus

- In the Classical Era Aristotle, in his Physics emphasised the application of rigorous logic to careful observation. This manner of thinking was especially pronounced in the West

- The early modern period was held together by a cosmology of connection through the Great Chain of Being (scala naturae), mastery of nature (magia naturalis), an all-enveloping medium (spiritus mundi), the macrocosm and microcosm, and the power of similitude

- The urge for exploration encouraged by European landing in the New World, led to developments in shipbuilding, navigation instrumentation, and cartography

- Compass and quadrant determined direction and latitude but rich rewards were offered for a way of determining longitude (only solved by Englishman John Harrison in the 18th century

- The barometer and thermometer were invented and improved

- From the 17th to 18th centuries scientific societies emerged (many of the first in Italy where there was, for example, a Florentine Academy sponsored by the Medici) in increasing numbers with communication through letters replaced by printed journals. Active research groups later developed across to northwest Europe

- From the 17th to 18th centuries scientific societies emerged (many of the first in Italy where there was, for example, a Florentine Academy sponsored by the Medici) in increasing numbers with communication through letters replaced by printed journals. Active research groups later developed across to northwest Europe

- By the 19th century science had become professional with the scientists formally trained in universities creating more standardized principles and practices with scientific study a social vocation

**UNIVERSITY FOUNDATIONS**

Bologna – 1088

Oxford – c. 1096

Salamanca - 1134

Paris – 1160

Cambridge – 1209

Padua – 1222

Naples – 1224

Siena – 1240

Montpelier - 1289

Lisbon – 1290

Coimbra – 1290

Madrid - 1293

Rome – 1303

Perugia – 1308

Florence – 1321

Pisa – 1343

Prague – 1348

Vienna - 1365

St Andrews - 1410

Glasgow – 1451

Aberdeen - 1495

**EARLY BOTANIC GARDEN FOUNDATION**

Pisa - 1544

Padua - 1545

Florence - 1545

Valencia - 1567

Bologna - 1568

Leiden - 1587

Montpellier - 1593

Leipzig - 1597

Oxford - 1621

Paris - 1635

Berlin - 1646

Uppsala - 1655

Edinburgh - 1670

Chelsea Physic G. - 1673

Amsterdam - 1682

St Petersburg - 1714

#### The Scientific Revolution

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First published on the internet – 1 March 2019