an essay on the history of mathematics

Intellectual Mathematics

How to write a history of mathematics essay

This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.

The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.

So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.

Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.

The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.

Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.

For many examples of how one might write a critique, see my book reviews categorised as “critical.”

Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.

Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?

Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.

When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?

Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.

The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.

Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.

Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn ? Or does mathematics work according to the model of Kitcher ? Or that of Lakatos or Crowe ? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska ? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?

Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.

Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.

In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book . It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper ) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis , 1621, or many other works listed in a bibliography by Schüling .

Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.

Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.

Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:

The table shows the state of the literature before and after your research project has been carried out.

A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.

You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.

An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.

Many areas of the history of mathematics have been thoroughly Described but never Explained.

What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.

Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.

See also History of mathematics literature guide .

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Victor Katz, editor

Table of Contents

Don't be turned off by the title. Victor Katz has gathered a diverse and fascinating selection of 26 essays on the history of mathematics and on ways to use it to teach mathematics, just like it says in the title. The title, though, does not capture the enthusiasm of the various authors, the depth and breadth of their topics, or their conviction that understanding and using history can enrich and improve the ways we teach mathematics.

Katz has divided the essays into five groups, proceeding from the more pedagogical in Part I to the more historical in Part 5. The first four parts consist of three to five essays each, and the fifth part consists of eleven.

The three essays in "Part I: General Ideas on the Use of History in Teaching" lay a foundation and motivation for the incorporation of history. Siu Man-Keung opens the work with some ways to include history without sacrificing mathematical content. Frank Swetz follows with an account of mathematical education from Mesopotamia through China to the Italian Renaissance.

Wann-Sheng Horng contributes "Euclid versus Liu Hui: A Pedagogical Reflection" to "Part II: Historical Ideas and their Relationship to Pedagogy." He gives a provocative comparison between the structural approach to mathematics that the Greeks used to the more operational approach of the Chinese, with special emphasis on Euclid's and Liu Hui's descriptions of the so-called Euclidean algorithm.

The third part of the book turns to "Teaching a Particular Subject Using History." Janet Heine Barnett shows how mathematical anomalies such as incommensurables, infinity and non-Euclidean geometries open mathematical minds and "prepare new intuitions." Evelyne Barbin gives a delightful account of how the meaning of "obvious" has evolved. For example, geometric proofs of proportionality may be beautiful or tedious, depending on your aesthetic, but those same theorems proved symbolically become obvious "in the sort of 'blind' way that algebraic calculations allow."

"The Use of History in Teacher Training" is the fourth part of the book. Ian Isaacs, V. Mohan Ram and Ann Richards remind us how important it is to the future of mathematics that elementary school teachers encourage, or at least not discourage, young mathematics students. They give specific examples of how they use history "to modify the belief systems and perceptions of these preservice teachers regarding the nature of mathematics and the purpose of school mathematics."

Maxim Bruckheimer and Abraham Arcavi remind us that we can't teach mathematics using history unless we have a repertoire of facts from the history of mathematics. They give us some anecdotes, including the thrilling and tragic story of Feuerbach wielding a sword and threatening to behead students who could not solve problems in class. They also challenge us to use original sources and share one of their worksheets based on the original works of Viète.

Katz put almost half of the essays in this collection into Part V, "The History of Mathematics." This section reflects the fact, of which Bruckheimer and Arcavi reminded us, that incorporating history into a mathematics course requires a knowledge of history as well as of mathematics.

In the first essay of the section, Eleanor Robson uses Mesopotamian mathematics to contend that writing arose first to record mathematics. This makes mathematics a function of civilization that predates even writing. She also emphasizes the importance of context when viewing mathematical artifacts, and emphasizes how mathematics is the product of the society from which it arises.

George Heine gives us a delightful example attributed to the Persian scholar ibn Sina (980-1037), but in the spirit of the Ancient Greek Nichomachus or the more modern Conway and Guy in their Book of Numbers . His example works for any square array of consecutive odd numbers, but Heine gives us the following case of a 5x5 array:

The highlighted entries sum, respectively, to 5 and 5 2 =25.

The highlighted entries sum to 5 3 =125 and 5 4 =625. In the third array, the sum on the opposite diagonal is also 125.

Torkil Heiede, in "The History of Non-Euclidean Geometry" traces how attempts to prove Euclid's Fifth Postulate grew into the geometrical revolution of Bolyai and Lobachevsky. He also gives a remarkable list of eight concise statements equivalent to Euclid's Fifth Postulate, all much simpler to state and some easier to believe than Euclid's version.

Some authors submit to an occasional temptation to change what is taught so that it can be more easily taught from an historical perspective. Others are sometimes a bit optimistic about when history adds understanding to the presentation of a topic. Early techniques in linear algebra, for example, are so burdened with now-obsolete notation that no presentation can be both understandable to the students and historically faithful.

Overall, this collection of essays goes well beyond the promise of its title. It presents a broad spectrum of ideas about how to use history in teaching, from things as basic as particular classroom activities to concepts as profound as different ways to consider the nature of mathematics. The perspective certainly is international. There are contributions from every continent, and only three of the 31 contributors are from the United States, matching the contributions from Israel, Italy and Portugal and one fewer than the number from France.

Beyond its title, though, this collection of essays captures, in a way that ordinary textbooks on the subject do not, some of the ways that the beauty and vitality of mathematics grows from its roots in history. In his essay, Heiede also asks "But does it matter if a teacher does not know about non-Euclidean geometry[?]" This wonderful book will convince you that it does matter. If we are to keep mathematics out of the museum, somewhere between the mastodon bones and the mummy, then we should keep it connected to its roots.

If only it had a better title...

See the table of contents in pdf format .

Dummy View - NOT TO BE DELETED

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The History of Mathematics: A Very Short Introduction

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(page xiv) p. xiv (page xv) p. xv Introduction

Published: February 2012
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Mathematics has a history that stretches back for at least 4,000 years and reaches into every civilization and culture. It might be possible, even in an introduction as very short as this book, to outline some key mathematical events and discoveries in roughly chronological order. Indeed, this is probably what most readers will expect. There can be several problems, however, with that kind of exposition.

The first is that such accounts tend to portray a whig version of mathematical history, in which mathematical understanding is generally perceived to progress onwards and upwards towards the splendid achievements of the present day. Unfortunately, those looking for evidence of progress tend to overlook the complexities, lapses, and dead ends that are an inevitable part of any human endeavour, including mathematics; sometimes failure can be as revealing as success. Besides, by defining present-day mathematics as the benchmark against which earlier efforts are to be measured, we can too easily come to regard the contributions of the past as valiant but ultimately outdated efforts. Instead, in looking to see how this or that fact or theorem originated, we need to see discoveries in the context of their own time and place.

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Essay on History of Mathematics

Students are often asked to write an essay on History of Mathematics in their schools and colleges. And if you’re also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic.

Let’s take a look…

100 Words Essay on History of Mathematics

Introduction to mathematics.

Mathematics, a universal language, has a rich history dating back thousands of years. It emerged from the practical needs of early civilizations, including counting, measuring, and understanding the natural world.

Mathematics in Ancient Civilizations

The Sumerians and Egyptians were among the first to use mathematics. They developed basic arithmetic and geometry around 3000 BC to support their complex societies.

Classical Greek Mathematics

Greek mathematicians, notably Euclid and Pythagoras, made significant contributions. They introduced logical reasoning and proofs, forming the basis of modern mathematics.

Mathematics in the Middle Ages

During the Middle Ages, Islamic scholars preserved Greek mathematical texts and expanded upon them, introducing algebra.

Modern Mathematics

In the modern era, mathematics has evolved rapidly, with the development of calculus, statistics, and numerous other branches. It continues to be a vital tool in various fields.

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Paragraph on History of Mathematics

250 Words Essay on History of Mathematics

Origins of mathematics.

Mathematics, as a discipline, had its genesis in ancient civilizations. The earliest evidence originates from the Sumerians, who developed a counting system around 4000 BCE. This rudimentary form of mathematics was primarily used for practical tasks, such as commerce, architecture, and astronomy.

Classical Era

The Golden Age of Greece brought a revolutionary shift, with mathematics becoming a subject of abstract thought. Pioneers like Pythagoras, Euclid, and Archimedes developed theories and principles that still form the basis of modern mathematics.

Medieval Mathematics

During the Dark Ages in Europe, the Islamic world became the torchbearer of mathematical knowledge. Scholars like Al-Khwarizmi introduced algebra, while others translated and preserved Greek mathematical texts.

Renaissance and Enlightenment

The Renaissance and Enlightenment periods witnessed a resurgence of mathematical exploration in Europe. This era saw the invention of calculus by Newton and Leibniz, and the establishment of mathematical proof as a cornerstone of the discipline.

The 19th and 20th centuries saw the development of more complex fields like set theory, abstract algebra, and topology. The advent of computers in the 20th century also introduced computational mathematics, which has become integral to various scientific disciplines.

The history of mathematics is a testament to humanity’s relentless quest for understanding the universe. It is a journey that continues to evolve, shaping our world in profound ways.

500 Words Essay on History of Mathematics

Introduction.

Mathematics, a universal language of logic and order, has been a part of human civilization for thousands of years. Its history is a rich tapestry of intellectual pursuit, spanning cultures, continents, and epochs.

The earliest evidence of mathematical knowledge dates back to the ancient civilizations of Egypt and Mesopotamia. The Egyptians developed a decimal system and used simple geometry for practical purposes, such as measuring fields and constructing pyramids. The Mesopotamians, particularly the Sumerians and Babylonians, are credited with introducing the sexagesimal system (base-60), which is still used in measuring time and angles.

Ancient Greek Mathematics

The ancient Greeks made significant contributions to mathematics. Pythagoras, Euclid, and Archimedes are among the most renowned Greek mathematicians. Pythagoras is best known for the Pythagorean theorem, while Euclid’s “Elements” remains a foundational text in geometry. Archimedes made significant contributions to geometry, calculus, and the understanding of the concept of infinity.

Mathematics in India and the Arab World

In India, the concept of zero and the decimal system were developed. Indian mathematicians also made significant contributions to algebra and trigonometry. The Arab world played a crucial role in preserving and expanding mathematical knowledge during the Middle Ages. Arab scholars translated Greek and Indian mathematical texts, and introduced algebra, algorithm, and trigonometry into Europe.

Mathematics in the Renaissance and Beyond

The Renaissance period witnessed a renewed interest in mathematics. This era saw the development of logarithms by John Napier and the introduction of analytical geometry by René Descartes. The 17th century marked the birth of calculus, independently developed by Isaac Newton and Gottfried Leibniz.

The 19th and 20th centuries brought about a rigorous formalization of mathematics. Georg Cantor’s work on set theory laid the foundation for modern mathematics. The introduction of abstract algebra and the development of non-Euclidean geometry further expanded the mathematical landscape. In the 20th century, the advent of computers led to the development of new fields like computational mathematics and cryptography.

The history of mathematics is a testament to the human quest for understanding the world. From ancient civilizations to the digital age, each epoch has left its mark on mathematical thought. As we continue to explore the universe and delve into the mysteries of quantum physics and artificial intelligence, mathematics will undoubtedly continue to evolve, offering new tools for our intellectual journey.

That’s it! I hope the essay helped you.

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Happy studying!

History Of Mathematics Essay

Mathematics is a certain way of thinking and doing that has been around since the dawn of humanity. Mathematics as a whole can be seen through history as a steady evolution, starting from simple hand calculations to modern-day computing machinery.

Mathematics allows people to understand not only what’s happening in the world but also why these things are happening. Mathematics has helped humanity to understand the laws of nature, predict where certain places are on Earth, travel into space and new worlds, estimate population size, track economies, etc. Mathematics is an integral part of human life – without it society would be all but lost.

Math was first used by humans as a system for counting things they wanted or needed. They started with simple things, like how many eggs are in a nest or how many fish are in a pond. Mathematics developed quickly to give us knowledge of geometry and measure the world around us. Mathematics was used to make advancements in astronomy; using math, we were able to calculate Earth’s circumference (Eratosthenes), create star catalogues (Hipparchus and Ptolemy), predict eclipses (Eudoxus and Calliphenes). Mathematics also helped us create calendars, which we use to this day.

The next step for Mathematics was during the Renaissance period with people like Fibonacci and Descartes. This is where the base of Mathematics as we know it today was laid. Mathematics became for the first time a discipline independent of astronomy and physics, having its own rules and proofs. Mathematics continued to be developed with mathematicians like Newton and Euler and then Mathematics finally reached what we know as ‘modern Mathematics’.

The history of Mathematics is an interesting topic that can be looked at from many different points of view. Mathematics has been used for thousands of years and, when Mathematics is so integral in all our lives, Mathematics’ history is important in understanding how we got to where we are today.

Mathematics, therefore, has become the basic tool of physical science and must be included in the social sciences. Mathematics is also indispensable in many forms of human endeavor (e.g., music, philosophy) that would not ordinarily be classed as scientific.

Mathematics has its roots in counting, calculation, measurement, and the formulation of quantitative laws. Mathematics is used in the study of all branches of the physical sciences, biological sciences, earth sciences, and social sciences; to solve problems in pure and applied arts; to formulate the rules governing games, sports, and gambling; to devise coding systems for transmitting messages or storing information; to carry out actuarial computations for insurance companies; and so on. Mathematics has made possible human progress by furnishing means for dealing with natural phenomena in ways that are precise rather than intuitively apparent.

The word mathematics comes from mathesis , a form of address derived from muses (Gr., “the patron goddesses of creative arts”), thus meaning literally “that which is learned.” The Greeks called mathematics, or sometimes “philosophy,” “the knowledge of things that are,” and the division of the quadrivium (arithmetic, geometry, astronomy, and music) recognized by ancient scholars may be interpreted as a classification of all branches of knowledge.

Mathematics is distinguished from other sciences in several ways: mathematicians seek to know pure truth without considering its application; mathematicians seek necessary truths whereas other scientists seek empirical laws; mathematics studies abstract patterns whereas science concerns itself with concrete objects.

The history of Mathematics can be seen as an ever-increasing series of abstractions. The earliest methods by which man obtained a measure for a quantity were based only on the properties of concrete objects such as a string or a stick. A length was determined by using the human body as the standard unit of measure. Only by degrees did man progress to the invention of simple tools such as the divided segment, marked stick, and marked pebble.

In Mathematics, history is important. Mathematics as a whole would not exist without history. Mathematics is the study of numbers and figures as far as we know it today.

Rigorous mathematics as it exists now was started in India by Aryabhata I, who lived from 476-550 CE . He introduced zero to mathematics and he attempted to solve quadratic equations. From India, mathematics went to China where it flourished until around 1200 A.D., when a general disinterest in Chinese Mathematics caused it to decline until its re-discovery during the Renaissance Period.

The first Mathematics book was written was by Euclid of Alexandria around 300 B.C.. In this book The Elements, Euclid set out to prove Pythagoras’ Theorem of right triangles by using a process known as deductive reasoning, or proof.

Only two other Mathematics books were written after this for about 1000 years – one by Al-Khowarizmi and one by Plato of Alexandria.

In 1400 A.D., Mathematics was brought to Europe from Africa by the Moors when they invaded Spain. It remained in Spain until 1492, then it spread throughout Western Europe. In 1545 A.D., Francois Viete wrote on imaginary numbers, which were a major focus of Mathematics at the time, Blaise Pascal had his first thoughts on what is now known as infinitesimal calculus in 1644 A.D.. Three years later, John Wallis published works on calculus, and this is the first known works on calculus. In 1665 Isaac Newton published his work on infinitesimal calculus, which was a major advancement from Pascal’s work.

In 1748 A.D., Mathematics took a big step forward when Leonard Euler’s Seven Bridges of Königsberg Problem was solved. This problem involving walking over bridges to cross rivers with different numbers of arches had been around since the mid-1700s. Leonhard Euler set out to solve it through real analysis by looking at what shapes were possible for traversing each bridge only once. He found that one shape worked for all seven bridges and proposed a solution in 1736 A.D.. His solution used something now called graph theory, which is the study of points that are joined by lines. Graph theory may sound familiar because it plays an important role in Mathematics today, but this was just the beginning.

Besides Mathematics becoming more general in its study, to include all possible Mathematics, Mathematics also became much more abstracted away from real-world problems and examples. This abstraction began in 1854 A.D., when George Boole published his work on symbolic logic. His work introduced numbers called 0 and 1 along with logical operators for not ( ), and ( & ) along with parentheses, making expressions like ((A & B) | ~C), which would be read as “A and B or C”. This system turned out to be very useful for Mathematics later.

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The History of Mathematics: From Ancient Times to Modern Era

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