sign in sign up
<<
Home , Binomial theorem, Mental image, Probability, Seki Takakazu

Gottfried Willhelm Leibniz Naomi Ecanow

Leibniz’s life

To many, intellect acknowledged across the continent is an attractive pursuit. Who would not want to under- stand the most complex and as well as become learned in business and the law while also being naturally sociable? Such an individual would have the pick of his profession and would make his mark in history. Gottfried Wilhelm Leibniz is, by many accounts, such an individual. In his time and still today, prodigy has been a word used to describe Leibniz. He entered the University at age fifteen, and followed by a PhD at age twenty. When he wrote anti-war pamphlets which attempted to demoralize Louis XIV, even the French people listened. He was recruited by dukes and emperors on important historical projects. He could think, hypothesize, and write for days. Despite all this, he is not a household like Galileo, , or even his biggest rival, Newton. Nonetheless, Leibniz’s work behind the curtains has prominently influenced the history, law, mathematics, and physics taught today.

Gottfried Wilhelm Leibniz was born in 1646 to a family of politically high stature. His maternal grandfather was a lawyer while his father was a professor of at the University in Leipzig. In once sense, Leibniz was destined for greatness before he could speak, though he certainly went above and beyond to make a name for himself by his own actions. Although his father died when Leibniz was just six years old, Leibniz took after his father’s niche in academia and philosophy throughout his life. Legend has it that Leibniz could be identified as a genius as a toddler. No one, however, denies that Leibniz’s potential was evident when he began at Nicolai School. He was formally taught Latin during elementary school, but this did not satisfy the young Leibniz. Most of his education can be traced back to a self taught education when he read the books in his father’s library on his own time. Leibniz decided to teach himself Greek, and he soon became proficient within a few year’s time. With a knack for schooling like his, it is no surprise that Leibniz started at the University of Leipzig at age fifteen. Here he was finally challenged and his base grew significantly as he studied Hobbes, Descartes, Galileo and Bacon. Though he would eventually learn math and physics, he began with a path learning philosophy, law, and politics, much resembling his father’s legacy. Leibniz’s undergraduate studies led him to his first philosophical task: to improve upon Arisotle’s . Leibniz, as a young philosopher, out his path: he wrote his thesis, De P rincipio Individui; spent time on his master’s degree in philosophy; and set out for his doctorate. Unfortunately, the university felt Leibniz was not ready. Getting a PhD at age twenty was too young, according to the University of Leipzig. However, this did not phase Leibniz. He simply transferred to the University of Altdorf, who gave him his doctorate in law as well as offered him a full professorship.

Of course, the self-assured new PhD turned down the University of Altdorf’s offer of professorship to pursue other interests in law and government. Instead, he became the secretary for the Nuremberg alchemical society, and soon thereafter worked for Baron von Boineburg, whom he befriended in graduate school. In his first jobs he built a name for himself as he invented for perfecting civil law and settling religious disputes between the Holy Roman Empire and the rest of . He also acted as a political lawyer, fighting for historic Italian homes on behalf of the government, and eventually a Councilor of the Supreme Court. He then wrote his first two books, T he Study of Law and T he Reform of the Corpus Juris. Soon Leibniz was discovered by archbishops, including Johann Christian and Phillip von Schonborn, as well as duke John Fredrick, which landed him the job of Director of the Library at Hanover, though he originally wanted a job at Paris’s Academy of instead. During this time as a librarian, Leibniz was recruited on the project to Harz mountain. Observing the mountain’s shape, he accurately predicted that the earth was originally all molten. He also proposed a major feat – using water and wind as power to fuel their work. Though this revolutionary did not work in practice at the time, Leiniz’s inspired the modern windmill. Around this time, Leibniz’s inquiries into the area of modern mathematics began, and he left the library to begin traveling around Europe. Upon traveling to London and Paris, Leibniz corresponded with the most progressive mathematical legends of his time and became the traveling philosopher and dedicated thinker he is known for today. In his last fifteen years, Leibniz dedicated himself to philosophy and wrote his most famous pieces including T he New Essays, the T heodicy, and Monadology. His greatest presents to the German people was the Brandenburg Society at the Berlin Academy of Science. After helping to found the institution, he served as their first president. Additionally, to help the society start a journal, he published most of the articles for the first edition. The Brandenburg Society later fostered Euler, Lambert, and Lagrange in their work. Without the society, Germany certainly would not have played the large role it did in advancing modern mathematics. Throughout his life, Leibniz had hundreds of writings and more than 600 correspondents. In his last few years, Leibniz was diagnosed with gout, during which time he set out to

2 write a Universal Encyclopedia. Yet, in a turn of events, Leibniz died at the age of 70, without ever being married and with his secretary being the only attendant at his funeral. However, this unfortunate ending to his life had no bearing on Leibniz’s lasting legacy.

Though the collection of works which can be attributed to Leibniz is enormous, most of his were never formally published. Some have suspected that Leibniz left many of his papers unpublished because he was waiting to write the perfect paper, which he never felt he achieved. Furthermore, most of his works which were known either could not be understood or could not be proved by even the world’s greatest thinkers for centuries. Because of this, many of Leibniz’s theories remained undiscovered for centuries. This is perhaps one explanation for Leibniz’s lack of fame among the average population. Luckily, this changed with the work of R.E. Raspe in the early 1800s. Raspe was the first to publish Leibniz’s unpublished and lost works after discovering them, mostly in the library at Hanover. However, it was not until Johann Eduard Erdmann’s publication of Leibniz’s theories in 1839 when Leibniz once again gained fame. However, his gap in Leibniz’s influence raised an important question: did modern logic, which resembled much of Leiniz’s philosophy and physics, form independently of Leibniz’s influence? This question remains unanswered today. As fits the recurring theme of Leibniz’s legacy, no one knows for sure how much influential credit Leibniz deserves.

Leibniz’s mathematical works

Though Leibniz’s name may be unrecognizable to many, his mathematical work is known to all with a high school education in the form of . His fascination with mathematics began in his undergraduate studies. He wished to discover what he described as ”logical ,” those these were proofs in nature. Leibniz began by mixing with engineering, to create the first calculator. Leibniz’s final machine could also multiply, divide, and take roots, whereas previous calculators could only add and subtract. Eventually, this earned him respect among great mathematicians of his time as well as a fellowship to the Royal Society which originally disregarded him. This opened up avenues for Leibniz to work with the elite mathematical society of the seventeenth century. Colleagues began writing Leibniz with problems to solve and famous manuscripts to read. This propelled Leibniz’s into his greatest years of mathematical progress.

Leibniz first delved into mathematics by comparing the proportions of circles as compared to squares. He studied the elements of a curve, whereupon his interests broadened to all naturally curved shapes, and soon he happened upon the unanswered question of his time: what is area bound by a natural shape? Ages before, Archimedes discovered that he could calculate the area under a parabola between the origin and some value x with using the formula (1/3)t3. However, even Archimedes himself did not know why. Thus, the search for a rule pertaining to all curved functions, rather than just the standard parabola, was inevitable, and Leibniz was determined to have a role. His first step towards finding the answer was to work with, what he called, infinitesimals. Infinitesimals referred to the idea that a number could approach zero without ever truly becoming zero. Today, infinitesimals are called limits. It was a revolutionary idea – and the key to unlocking the calculus – but the idea was not only Leibniz’s. Several mathematicians of Leibniz’s time had been working with infinitesimals already, such as Newton, which would be a source of conflict for Leibniz. Nonetheless, Leibniz’s personal quest for the formula to find the area bounded by a curve continued.

Though this question was geometric in nature, Leibniz approached his work with a more philosophically logical and analytical mindset. He thought of a curve as a histogram or ”picket-fence” made of rectangles with equal widths lined up in height order. This would make finding the area underneath easy, since one could simply add the areas of the rectangles. This picture seems inaccurate at first glance, since some portion of the corner of each rectangle will extend beyond the curve’s true area. However, Leibniz noticed that the portion of the rectangles which extended beyond the curve became smaller. If the width could somehow become zero, Leibniz thought, no portion of the rectangle would go beyond the curve. This would also imply that the rectangles had no area – unless the widths approached zero without actually becoming zero. Leibniz expanded on this

3 with mathematics until he was left with Archimedes’ formula. Only now, Leibniz had derived the formula using his own formula d(xn) = nxn−1. Thus, by combining his mental image with the of infinitesimals, Leibniz’s calculus was born.

The next chapter in mathematics history had begun, and mathematical darkness was over. Yet, Leibniz was not satisfied because he knew mathematicians could not build upon his discovery yet. Many of Leibniz’s mathematical colleagues were working for their own pleasure, changing the symbols they used for their mathe- matical operations with their changing mood each day. Leibniz recognized the importance of having a standard mathematical language. Thus, he began to create modern mathematical notation, one of his few discoveries attributed only to him. He began by perfecting the binary system, which uses unique of zero and one to express letters. Modern electronics would not be possible without this perfected binary system. Next, he attempted to describe the rational curve. Up until the seventeenth century, mathematicians studied curves by beginning with some curve and attempting to describe it with variables. Leibniz flipped this logic and instead began with a formula and plotted its points. With this he invented the word to describe the relationship between the collection of points. Finally, he devised new symbols to represent the new operations invented with calculus. His new symbols were first published in 1675. Leibniz established dx to indicate taking the of a function and R to indicate taking the integral of a function. He also developed standardized methods for finding the of curves made up of the or quotient of functions, such as the , as well as the derivative of a function raised to a power. Though Leibniz did not make the official connection between limits and derivatives, discoveries which has advanced mathematics to its modern could not have been possible without Leibniz’s notation.

4 Collaboration with other scholars

Leibniz’s work was completed in isolation. Through his many travels and 600 letter correspondents, Leibniz collaborated with some of mathematical history’s greatest . His outside influences began with his un- dergraduate studies at Leipzig. He was influenced by lectured given by Erhard Weigel and other established philosophers, mathematicians, and lecturers. However, one teacher, Jacob Thomasius, recognized Leibniz’s po- tential and urged him to spend his life thinking and innovating. Furthermore, though he could not collaborate with him, Descartes’s work on physical motion really fascinated Leibniz in his undergraduate years. Descartes’s was mechanicalism, which attributed movement to an aggregation of organized movement of smaller parts. Leibniz knew this theory was not correct, yet he refused to believe Newton’s contemporary alternative of gravity. Leibniz instead made his own theories loosely based off of Descartes’s. He thought the basic parts of an object moved mathematically according to some outside force. Furthermore, he believed objects contained monad, an intangible substance which allowed them to combust. He called his theory , which laid the ground work for the of potential energy, kinetic energy, and momentum. Together with Newton’s gravity, these theories later developed into modern physics. He expanded upon his physics theories when asked to solve Arnauld’s about whether a body changes upon being broken. Through solving this contradiction, Leibniz corresponded with Arnauld through letters and was influenced by his ideas. After joining the Royal Society and traveling to London, Leibniz’s list of academic colleagues grew quickly. While in London, he was tutored by , who had worked with Newton, and he corresponded with Henry Oldenburg and Tschirnhaus. Along with James Gregory, Leibniz developed an infinite to describe . Later Leibniz wrote letters with Grandi and Varignon about the harmonic series and theorized with Bernoulli about the possibility of taking the logs of negative .

Within a seemingly infinite list of Leibniz collaborators, Newton stands out. The race between Newton and Leibniz to discover the calculus might be the most well know aspect of Leibniz’s life. The Great Sulk, as the bitter rivalry is sometimes called, is a prime example of the recurrent theme of a fight for ownership between mathematicians. The two began corresponding in 1676, when Newton sent Leibniz a letter to state his theories about series without any intermediate work or proofs to supplement them. This was typical of the time, as mathematicians often sent each other answers to new mathematical theorems without showing their work, which they kept secret. When Leibniz received the letter, he had already begun to work on the calculus and was relieved to find that Newton was working on the separate subject of series, or so he thought. The two mathematicians continued to exchange letters regarding series and ”methods of fluxion,” and their relationship was peaceful. However, Newton was unaware that some of his earlier letters took a long time to reach Leibniz through mail. This led Newton to grow suspicious of his younger correspondent, who he thought was taking the extra time in between to discover the work behind Newton’s theories and claim them as his own. In Leibniz’s final letter, he described his calculus to Newton, who bitterly stated that Leibniz’s ideas were unoriginal. Nonetheless, Leibniz published his calculus in Acta Eruditorium before Newton ever published, though Newton had technically invented the calculus first. The ambiguity in who was the first formal inventor of calculus was disastrous. Of the seventeenth century mathematics, the second inventor was always regarded with suspicion and often had his career destroyed. Without knowing it, Leibniz had entered a battle he could not win.

Newton himself did not begin the formal accusations that Leibniz plagiarized him. Instead, Newton’s loyal student, Fatio de Duiller, exclaimed that Leibniz had claimed Newton’s ideas as his own. Ashamed and embarrassed, Leibniz wrote an article in Acta Eruditorium claiming his innocence. When Newtonian followers continued to ridicule Leibniz, he requested a formal trail with the Royal Society. This was Leibniz’s biggest mistake. While learning math in London years ago, he had studied with several of Newton’s students and collaborators, which raised questions of whether Leibniz had learned Newton’s methods without realizing it. Furthermore, Leibniz had previous accusations of plagiarism. For example, Leibniz presented his thoughts on series to the Royal Society, and Pell pointed out this was already published by Francois Regnaud in his ”Series

5 of Differences” paper, though Leibniz had come up with the ideas separately. Being German also did not help Leibniz’s case, as he was easily accused of stealing calculus ideas on behalf of the German government as a political advantage. Finally, Newton himself was the president of the Royal Society which tried Leibniz. It was inevitable that Leibniz was found guilty, which ruined Leibniz’s reputation. Leibniz was demoralized, and his optimistic views of the mathematical society were buried by .

Historical events that marked Leibniz’s life.

While the Great Sulk was Leibniz’s own personal war, wars waged between countries vying for power during Leibniz’s time. The Thirty Years War ended at the time of Leibniz’s birth. This war was a reflection of a disjointed Germany. Before the war, Germany was a collection of a few hundred individual states with their own rulers, who answered to the Seven German Princes, who were below the Holy Roman Emperor who ruled them all. Though all the states were loyal to the Emperor, this one Emperor was too distant from any one community to control anyone. As a result, the northern states became Protestant while the southern states stayed Roman Catholic. When the Holy Roman Empire began to persecute the northern Protestants, the Protestants rebelled and war broke out in the whole region. Before long, half of Europe was involved: Austria and Spain fought with the Empire, and Sweden and France fought with the Protestants. Eventually, the Protestants gained the upper hand and the Holy Roman Empire was forced to sign the Treaty of Westphalia. The treaty granted tolerance to all forms of Christianity and made the government more secular by granting more power to each state’s separate prince. Though the war ended on a peaceful note, Germany was far from organized. The previous few hundred German states were further divided into more than 360 states, each with their own laws, currency, and trade. The war had devastated Germany, killing about a third of its population in battle. Villages and farms had been burned down by soldiers, causing famine. And, with the Treaty of Westphalia, France and Sweden gained control of important German ports and providences, foreshadowing their attempts to conquer more of Europe in the near future. Leibniz was born at the end of a war, but

6 the struggles of his people would last beyond his lifetime. Perhaps this explains why Leibniz proffered his philosophy that universal harmony occurs with freedom but not necessarily with order.

Peace remained throughout Leibniz’s childhood. As Germany rebuilt its interior, it joined the global trade . This led to a new social ladder, as merchants became a new middle class. Interestingly, this implied a decline in noble wealth and power. Peace, however, was short-lived as consecutive wars began as soon as he became an adult. An emerging world power, the Ottoman Turks, had been marching through the Mediterranean and European countries conquering land and spreading Islam. In 1683, they reached Vienna. Vienna held major importance the primary port to the straits of Europe. As soon as the Ottomans attacked, Austria, Germany, and Poland sent their armies to defend their fellow Europeans. Though they defeated the Ottoman giant, fighting a battle within German walls destroyed some of the efforts to rebuild Germany from the prior Thirty Years War prior. Soon after, Germany became involved in wars fought outside of their land. France was a growing threat to the rest of Europe, so Germany joined the League of Augsburg, comprised of nations who were willing to donate their armies in defense of a country fighting France. Germany further helped the cause when it welcomed French immigrants fleeing from religious persecution. However, Germany soon became dangerous for its own Protestants, causing mass exodus. One large group traveled to North America, forming the German colony in Pennsylvania. The opposite was happening in Hapsburg Austria and Prussia. These German states declared independence after the Thirty Years War, and their governments were becoming progressive. The Hapsburgs were welcoming Slavs and Magyars who were assimilating with the Germans, forming a new population and culture. Austria would also become the first German state to be ruled by a women emperor. Over in Prussia, a new kingdom was established when the royal Hohenzollern family took the Thirty Years War as an opportunity to rule Prussia as their own. Unlike most power hungry rulers of the seventeenth century, the Hohenzollerns made Prussia’s government a bureaucracy. Germany’s many wars and unstable social environment shed light on the strength of Leibniz’s patriotism; in fact, the atmosphere of war throughout Germany inspired one of Leibniz’s greatest works, the T heodicee, which claimed that the world was imperfect. He forgave his chaotic world by seeing it as a reflection of the laws of physics whose existence protect the universe from even greater chaos. It is ironic that a society whose peace was crumbling could build such progressive science.

7 Significant historical events around the world during Leibniz’s life

Germany’s struggles reflected the struggles of Europe in general. Leibniz was born to a Europe which had been dominated by Spain for decades. Spain’s dominance would last much longer though. There was inflation throughout Europe, and Spain was expelling its Jews and Muslims, the populations whose work the government later realized drove its economy. Finally, in Spain’s last attempt to take over England, the king sent an expensive fleet of ships to England’s coast. When the armada was easily defeated by Queen Elizabeth’s army, Spain was left in financial ruin and its days of European dominance were over. Thus dominance of the western world was up for grabs, but which country would take control? This question would affect the history of all the European powers throughout Leibniz’s lifetime.

In England, a revolution was on its way. When the last Tudor of England died without an heir to the throne, the Stuart family took over. They believed in divine power and sought rule by absolute monarchy. Unfortunately for them, the English parliament had already been established. Over the next several years, the Stuart rulers disregarded parliament’s laws and tried to disband its members. This clash drove parliament to a point of rebellion, causing full-blown civil war from 1642 - 1651. Parliament formed the offensive army, which included the puritan clergy. Historically, they became known as the Roundheads. Against them were the Cavaliers, who supported absolutism. Against all , the Roundheads won the civil war. They brought the king to trial and and executed him, making England the first country where its citizens successfully rejected its king. With the parliament as the ruling body, the English government became a . With this reformed model of government came other freedoms radical for their time. The government began promoting privileges for the masses that were previously only available to the elite: a voice in government even for peasants; religious freedom; and educational opportunities for everyone. The government eventually welcomed back a king but checked his power regularly. Eventually the English adopted a Bill of Rights and political parties formed, which allowed for government representation of more people. Unlike most of Europe, England’s seventeenth century war improved the country and made it a prime example of a government ruled by its people, which would inspire the formation of the United States government.

A similar dislike for royalty did not have the same positive in France. Right before Leibniz’s birth, Louis XIV inherited the throne as a toddler, at the age of five. Since he spent most of his time as powerful royalty, it is no surprise that the king began taking advantage of his country when gaining full power over the government at age twenty-eight. A lasting symbol of his power can be seen today in the ornate Versailles castle he built for himself with the government’s money. Nobles and commoners alike grew angry with Louis XIV and formed the Fronde, a huge protest by all the social classes against the royalty. Despite internal disapproval, Louis XIV used the government to formed the largest army in Europe and eventually became Europe’s next great power.

Russia had an equally ambitious king, though he was met with much more acceptance from his people. Russia’s Peter the Great set out to expand Russia’s power, hoping to reach the same influence over Europe as France had. The king’s vision for his country began when he traveled Europe, learning about new European technologies, scientific discoveries, and historic armies. At the time, Russia was not taken seriously by the rest of the world. Peter the Great had a different vision for his country, and he decided the first step was to westernize Russian culture. To build his army, he forced nobles and peasants alike to serve as soldiers. Seeing the global market which was now forming between Europe and Asia, he realized Russia would also need to innovate to gain the world’s respect. Peter the Great the great took control of the church and emphasized modern European education for all citizens rich and poor. Finally, empowered by a new army and a thriving economy, Peter the Great set his eye on gaining ports which would connect Russia to the rest of the European market. He first attempted to fight the Ottoman Empire for a port in the Black Sea. When this failed, he turned to Sweden’s ports. After two battles, Russia won their ports and build St. Petersburg around it as a symbol of Russia’s new place in Western civilization. Leibniz was fascinated by Peter the Great’s ambition. In the later years of his life, Leibniz collaborated with the Tsar, which helped to bring modern science to Russia.

8 Italy, on the other hand, experienced the opposite end of the political spectrum while its academic connection to Europe flourished. With a stable government, the seventeenth century marked a golden age for Italian academics. The Renaissance, which began 150 years earlier, was at its peak. Society was transitioning from Medieval to Modern, and with it came a new interest in Greek and Roman classics. Because Italy was beginning to trade with the Muslim world, were able to learn Greek and Roman teachings, which had been preserved by the Muslims but lost by the Western world. Academics learned in Latin and were encouraged to be experts in multiple fields. A new emphasis on human experience rather than religion helped to form the new field of the Humanities. A zest for exploration lead voyagers to discover foreign lands and invent new areas of academic study or reexamine old ideas about philosophy, science, and mathematics. This new Italian culture produced Galileo, Michelangelo, and de Vinci. These were the erudite giants whose shoulders Newton would claim to stand on, and who provided much of the inspiration for Leibniz’s hunger for knowledge.

The Western world was expanding beyond Europe. The new obsession was a world market with large ships and new tradable goods, as well as the race to control the newly discovered lands on the non-Western end of the trade routes. Additionally, the printing press could now be widely used for secular books and even smaller academic papers. For better or worse, the rest of the world was infiltrated by European influence. This allowed Europe to thrive fiscally (that is, until inflation occurred), while the global market doomed the non-Western countries. Beginning with Africa, slave trade became a thriving business. With shiploads of African natives being sent to the Americas and Europe, the populations in African countries began to dwindle. Several African kings saw this as an opportunity to create kingdoms with the leftover villages. These kings traded their gold and silver as well as their own people for European weapons used to maintain power over their people. This lead to civil wars, making Africa an unsafe continent. The African people also had the Europeans to fear, as France, England, Denmark, and Portugal raced to take control of African coasts. One famous conquest was the Portuguese capture of Cape Town, which would explode into what centuries later became known as apartheid in South Africa.

The Europeans were not able to gain this same political momentum in the Asian countries. The Korean and Japanese governments banned trade with Europe altogether when merchants began spreading Christianity, which the Korean and Japanese governments felt was a threat. China regulated trade with Europe for similar , though its limited connections with Europe had a measurable impact. In China, the Machu Dynasty was declining, which lead to a revolution. Eventually, peace was reestablished when the Qing dynasty took rule. Parallel to France, the Qing dynasty established its dominance throughout central Asia as well as the world through its trade of silk and porcelain, which became popular commodities in Europe and the Americas. Conversely, India was open to trade with Europe and its economy thrived as a result. The Mughal Empire, which ruled India at the time, established power throughout the world by introducing cotton to the market. Much of its power also resided in the Spice Islands, though this region became a target for European colonialism. First the Dutch took control of the East India Company, and later the French took control of the company. When civil war between the Muslims and Hindus erupted in India, India’s economy and influence declined.

One of the most notable developments of Leibniz’s time was the colonization of the Americas. After several expeditions, the European powers set their sight on what they considered uninhabited, opportunity-laden land. First, North America was taken by the English. Their conquest began with the pilgrims’ settlements on the East coast. The North American colonies introduced potatoes, corn, tobacco, and tomatoes to Europe. Their market, which later became known as the Columbian Exchange, became instantly profitable. In Canada, the French took most of the land, beginning with Quebec. Canada’s population grew much quicker than North America’s, but their ties to French taxes and rule lasted much longer. Later, the Spanish took most of Central and South America. Europe’s new continent created a Price Revolution, which lead to ideas of capitalism and entrepreneurship. It was the beginning of the market realized today.

9 Significant mathematical progress during the Leibniz’s lifetime

It is not quite coincidence that multiple individuals discovered calculus simultaneously in the seventeenth century given the major findings right before. Kepler had just calculated formulas for the areas of 92 shapes; Descarte began to apply these shape theories to for lines, curves, and circles; and Fermat discovered the . The next step was to find a mathematics to tie all these ideas together. All the mathematicians of Leibniz’s time were set up for discoveries in calculus. Though they lived with countries between them, mathematical collaboration was at its height. Society was motivated to build institutions for thinking, such as the Royal Society, the Science Academy, and growing number of universities modeled off the Arabic House of Wisdom. The years between 1646 and 1716 were a time of academic culture, and numerous legendary mathematical discoveries arose from it.

Even in countries disconnected from Europe, calculus methods were being discovered. Though most are unknown, the prominent Japanese mathematician, , has well documented work. Shockingly, his life’s work parallels Leibniz’s biography. Takakazu discovered the properties of accurately and in isolation. He also created his own notation, called tengenjutsu notation, though his notation was created to make solving systems of equations more organized. However, while Leibniz concentrated his work around calculus, Takakazu mostly worked with polynomials. Takakazu’s greatest discovery, called the general theory of elimination, was the first successful method for solving complex systems of equations by eliminating one at a time. His work was not introduced to Europe for another hundred years when another mathematician invented the method separately.

Moreover, new fields of science were arising due to new advances in technology. The telescope was becoming highly complex, making the field of astronomy more accurate. On the other end of the scale, Leuwenhoek’s microscope made biological studies on the microscopic scale possible. Observing science through these size extremes meant recording data with endless lists of digits, which made calculations tedious. This obstacle was addressed by the of the logarithm by John Napier and Henry Briggs. Logarithms made expressing these huge numbers much simpler. Furthermore, multiplying these numbers became possible, as simply needed to add the bases of the logarithms. Due to this discovery, many new patterns in science became visible, such as radioactive decay or demography. This was apparent to Edmund Halley who constructed the first mortality table. By recording the birth and death rates of the population in Breslau, Poland, Halley was able to graph the relationship between the two phenomena to create a logarithmic curve. His work made it possible to predict a population’s growth and decay, a method used regularly in biology and anthropology. A French mathematician named Marin Mersenne also worked with a notation which attempted to simplify numbers, specifically prime numbers. Unlike logarithms, however, his work was based on a century-old dis- cussion. It had been falsely established that all prime numbers could be expressed by the formula 2x − 1 for some number x. Though this had long been disproved, mathematicians were still intrigued by the concept and wanted to find out which prime numbers could be expressed with this formula. Mersenne joined this search and claimed to have proven the longest list of numbers which fit the expression to date. They were 2,3,5,7,13,17,19,31,67,127, and 257, while all other numbers yielded by the expression 2x −1 which fell between the numbers in this list were not prime. These were called the Mersenne primes. Mathematicians later found that Mersenne had been wrong about the numbers 67 and 257, which were composite, while he missed the prime numbers 61, 89, and 107. The final list of correct Mersenne numbers were not proven and finalized until the early twentieth century. Despite this, Mersenne’s work was among the best of his time, and he inspired the work of many other mathematicians to challenge and prove old mathematical paradigms.

In predecessor to the calculus, a natural concept called the cycloid fascinated the mathematical world. A cycloid is formed when the path of one point on a wheel is traced as the wheel moves. The point which originates from a circle forms a parabola as it moves. Because the seventeenth century mathematicians were searching for properties of a curve when the area and for a circle was known, some thought the cycloid was the key to finding the area under a parabola, though their assumption did not quite hit on the solution,

10 which was the calculus. However, during Leibniz’s lifetime, another mathematician named Christopher Wren noticed that the length between the endpoints of a cycloid was always four times the diameter of the circle which traced the cycloid. The study of the cycloid got closer to the calculus in France, where mathematician Gilles Personne de Roberval analyzed the area underneath cycloids, of the cycloids, and arc lengths, as a cycloid itself is an arc. However, Leibniz himself was finally able to describe the cycloid with an equation after discovering the calculus, thus ending the mystery behind the properties of cycloids. Nonetheless, the cycloid led to the modern architecture and engineering of Leibniz’s time.

Though the aforementioned Newton-Leibniz controversy was well known, each mathematician did have his own work entirely recognizable from the other. Newton added plenty of new observations which allowed mathematics to progress. One might argue his most original contribution was the theorem. When expanding two terms being added and together raised to some power, the coefficients of their unique terms have a distinct pattern which reflect Pascal’s triangle and can be expressed in terms of . Newton noticed this pattern and produced a notation which expressed these coefficients in an organized way. The impact of his notation certainly made it easier to expand expressions composed of added variables and raised to some number. Perhaps more importantly, Newton’s would play a huge role in and probability. Therefore, it is no surprise that the came about at around the same time. The probability theory was a product of the correspondence between the great mathematicians Fermat and Pascal. Their theory describes how many times a specific chance event will occur out of a list of possible events, which can be expressed by dividing the total number of outcomes by the number of favorable outcomes within those total outcomes. Their theory, of course, would become the basis for the subject of statistics as well as countless fields within science and economics.

The fashionable task of Leibniz’s time was to describe geometry using algebra, as did calculus. Girard Desargues introduced this inquiry to engineering through his phenomenon called projective geometry. He imagined a shape as two dimensional. The two dimensional shape could be seen from a different three dimensional angle by transferring it by its individual points to a different three dimensional graph. Furthermore, when the shapes’ sides were casted into space while becoming thinner at the same rate, they would eventually meet at one point, forming a three dimensional figure. The point at which the sides meet is at a distance which can be described by a . Furthermore, when two similar shapes of different sizes stood one in front of the other, the point at which their sides met could be predicted. However, what happened when the two shapes were congruent? For one shape to project onto the other, neither shapes’ sides could thin. Thus the sides of the shapes did not meet at one point because the lines projected from the corners of the shapes were parallel. Desargues did not see this as a dead end. Instead, he theorized some ”point at infinite” where parallel lines intersect if a fourth dimension was considered. With the development of calculus, similar discoveries were made in quick succession. Though his analysis of limits, L’Hopital proved that the composed of two functions divided was equal to the of their derivatives if both functions in the numerator and denominator both approached zero or both approached infinite at the same limit. Around the same time, the Bernoulli brothers were able to solve the Brachistochrone problem. The problem, which had existed unsolved

11 since the time of the Greeks, proposed that a particle moving due to gravity can reach its final destination in the least amount of time by taking a single definite path. Using calculus, Jakob and Johann Bernoulli (the final answer being Johann’s work) showed how to calculate this quickest path. It turned out, the answer was the parabola formed by the cycloid. Newton’s and Leibniz’s work on series gave mathematicians a new of pi. During Leibniz’s lifetime, Abraham Sharp made the first great stride by calculating 71 correct decimals of pi. Then, in the last decade of Leibniz’s life, declared his converging series for pi along with the first 100 decimals of pi, as depicted below.

Connections between history and the development of mathematics

One pattern emerges while comparing Leibniz and the development of mathematics with what was happening in the world around his time: when the seventeenth century emerged, mathematics throughout all of Europe snowballed into its modern, complex form. Yet prior to this sudden phenomenon, mathematical progress had been nearly dormant throughout Europe since the time of the Romans. So, why was there a sudden fascination with mathematics? Juxtaposing this boom in mathematics with European history, one might see two themes emerging beside it: a global economy and political wars. A global economy began the exchange of mathematics. The fundamental ideas of geometry and algebra had begun and developed in the hands of the Greeks and Romans. Yet, when these civilizations disappeared, their ideas, including mathematics, was sure to have been buried with them. Luckily, the Indians and Muslims preserved mathematics while adding their own progress. Fast forward to the Italian Renaissance, when Italy begins to trade with the Muslim countries. At the same time, the Italians were in an age of discovery. Thus, it was only natural that Italian scholars would be fascinated by the mathematics, which were discussed with Muslim merchants and officials. Additionally, the Italians had been learning Greek and Latin as part of their academic renaissance, enabling them to read the ancient mathematics Italy was learning from the Muslims in their original language and symbols. Whether by design or pure coincidence, the Renaissance had prepared the Italians to receive mathematics. For more than one hundred years, Italian scholars coveted Muslim mathematics and absorbed it into their society, while Europe developed political ties and new technology to make trading with countries throughout the world possible. Finally, at about the time of Leibniz’s birth, Italy began trading with the major European powers. Of course, one intangible good traded was the new Italian mathematics. This completed the cycle, returning mathematics to Europe and beginning the mathematical revolution leading toward the calculus.

The most remarkable aspect of mathematical progress during Leibniz’s time was the rapid pace at which mathematics advanced. This progress can be largely attributed to the widespread collaboration between great mathematicians of different countries. The widespread use of the printing press was one invaluable catalysis. Originally, the printing press was invented about two hundred years earlier to reproduce the bible. Writers in

12 the sixteenth century changed this dynamic by using the printing press for secular writings. The use of the printing press slowly evolved until it could be used for even small journal articles in the seventeenth century. Thus, mathematicians were able to print and distribute their works throughout Europe in the form of articles. With mathematical discoveries more easily circulating, mathematicians from opposite sides of Europe were able to build upon each others’ discoveries. With many great minds from several countries working on the same mathematical concept simultaneously, it is no wonder there was duplication, one one hand, and an explosion of ideas, on the other hand, that mirrored the surrounding political . It also cannot be discounted that the wars facilitated collaboration in some cases between foreign mathematicians; because the renaissance valued educated individuals who had specialties in many subjects, mathematicians such as Leibniz were chosen to serve as diplomats. As diplomats, these seventeenth and eighteenth century mathematicians travelled to other countries often on behalf of their government. Surreptitiously, this exposed mathematicians to the mathematical progress in their neighboring and even enemy countries. For example, Leibniz himself met the Royal Society of Paris as well as philosophers Malebranche, who would influence his philosophical ideas, when he traveled to France on behalf of the German government in an attempt to convince Louis XIV to attack Egypt rather than Germany.

It is noteworthy that most major mathematical discoveries of Leibniz’s time were made by Europeans. This makes sense in light of China, , and Korea’s bans on trade with Europe. Though this trade disconnect between Europe and Asia certainly kept Asian mathematicians from becoming a part of the seventeenth and eighteenth century mathematical timeline, there are a few other possibilities to explain why. Perhaps the mathematics preserved by the Arabs never reached Asia as it did Europe, creating a standstill in mathematical progress in those countries. Conversely, another possibility was that Europe never learned about or recorded the mathematical progress which was occurring in Asia. This scenario is plausible seeing that Takakazu’s work in Japan was not discovered by the rest of the world until more than one hundred years later. Hence, there is a remote possibility that the solutions to today’s unsolved mathematical theorems may have been discovered centuries ago, yet they are lost with the undocumented mathematics of those countries.

Remarks

The history that surrounded Leibniz’s life and contributions marked a great transition across the globe. With an appreciation for discoveries in academia came several modern-style universities, societies, and journals. New technologies were transforming social structures and overall quality of life. The world was becoming connected as it had never been before. Though calculus might have been invented without Leibniz, it certainly could not have been taught universally and applied to other fields without Leibniz’s notation to organize and discuss the new branch of mathematics. What is also unique about Leibniz is how many subjects he aimed to reform, including philosophy, physics, civil law, religion, and the modern politics of his time. The impressive fact that Leibniz attempted to write an encyclopedia of the universe should reflect upon his zealous intent to comprehend the world. If he was not going to change mathematics, his work would have likely contributed to the growth of another field of study. Leibniz exhibited a unique mindset that ensured his name would be written in history.

References

1. Leibniz, Gottfried Wilhelm, and Leroy E. Loemker. Philosophical Papers and Letters. A Selection Trans- lated and Edited. 2nd ed. Norwell: Kluwer Academy, 1989. Print.

2. Leibniz, Gottfried Wilhelm, and George R. Montgomery. Discourse on Metaphysics, and the Monadology. Mineola: Dover Publications, 2005. Print.

13 3. http://stanford.library.usyd.edu.au/archives/win2011/entries/leibniz-logic-influence/

4. http://www.britannica.com/biography/Gottfried-Wilhelm-Leibniz

5. www.math.uh.edu/ tomforde/calchistory.html

6. pages.cs.wisc.edu/ sastry/hs323/calculus.pdf

7. http://www.larsoncalculus.com/calc10/content/biographies/leibniz-gottfried-wilhelm/

8. http://poncelet.math.nthu.edu.tw/disk5/js/linkage/m7.pdf

9. The Mechanical Universe: Integration. Dir. Mark Rothschild. Prod. Peter F. Buffa. Perf. Professor David Goldstein. Annenberg/CPB Project, 1985. DVD.

10. http://www.egs.edu/library/gottfried-wilhelm-leibniz/biography/

11. http://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.html

12. http://www-history.mcs.st-and.ac.uk/Societies/Berlin.html

13. http : //www.gwleibniz.com/britannicapages/westphaliatreaty/westphaliatreaty.html

14. http://espace.library.uq.edu.au/view/UQ:266787/AfterWestphalia.pdf

15. http://www.qdg.org.uk/pages/1701-to-1713-77.php

16. https://sites.google.com/site/group11louishist/the-war-of-the-spanish-succession

17. Sanders, Dr. Jeff. ””The Battle of Vienna – 1683”” Web log post. A Moment in History with Jeff Sanders. American Policy Roundtable, 30 Sept. 2010. Web. 27 Sept. 2015.

18. http://www.storyofmathematics.com/17th.html

17. Wikipedia contributors. ”Timeline of mathematics.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 24 Sep. 2015. Web. 2 Oct. 2015.

18. https://primes.utm.edu/mersenne/

19. Guichard, David. ”Calculus: Early Transcendentals.” Calculus: Early Transcendentals. Whitman College, 11 Sept. 2015. Web.

20. http://mathworld.wolfram.com/CartesianCoordinates.html

21. http : //www.math.uiuc.edu/ r − ash/BP T/BP T Ch1.pdf

22. http://www.umiacs.umd.edu/ ramani/cmsc828d/ProjectiveGeometry.pdf

23. http://www.purplemath.com/modules/binomial.htm

24. http://quadrivium.info/MathInt/Notes/Cycloid.pdf

25. http://fac.comtech.depaul.edu/jciecka/Halley.pdf

26. http : //www.hep.caltech.edu/ fcp/math/variationalCalculus/variationalCalculus.pdf

14 27. http : //personalpages.to.infn.it/ zaninett/pdf/machin.pdf

28. http://www.ndl.go.jp/math/e/s1/2.html

29. http://www.maa.org/news/math-news/seki-takakazu-an-unknown-giant-among-world-mathematicians

30. Ellis, Elisabeth Gaynor., Anthony Esler, and Burton F. Beers. Prentice Hall World History. Boston, MA: Pearson Prentice Hall, 2009. N. pag. Print.

31. Wikipedia contributors. ”Riemann sum.” Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 26 Sep. 2015. Web. 2 Oct. 2015.

32. www.earlymoderntexts.com

33. http://remus.shidler.hawaii.edu/genes/Saxony/Leipzig/home.htm

34. ”The Calculus Controversy.” Dynamic Calculus. State of New South Wales, Department of Education and Training, 2008. Web. 01 Oct. 2015.

35. www.lib.utexas.edu

36. www.gwleibniz.com

15