math history essay

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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Who Invented Math? The History of Mathematics

Did math start with the Greeks, or has it always existed?

Who invented math? It’s a deceptively complex question—a lot harder than 2+2. Math has been around forever, but we are always learning more about it.

Short answer: Many different people invented math, including ancient societies and many famous mathematicians who came along later.

The long answer: It depends on what kind of math you’re asking about. Below is a look at the history of mathematics and the people who contributed to developing math as we know it today.

What is math?

• Was math invented or discovered?

So, who invented math?

• Videos about the invention of math

According to Britannica Kids , math is the study of numbers. It’s a kind of language that we use every day to calculate distances, tell time, build things, and so on.

Mathematicians think about math in two areas: pure and applied. Pure math is studying math for its own sake. Figuring out how to solve a particular algorithm or tackling a theory, for example. Applied math is using math to solve real-life problems, like building a house or predicting an earthquake.

There are lots of different types of math: arithmetic, algebra, geometry, trigonometry, statistics, and more.

So, since math is already a part of the world, the first question is, can math be invented at all?

Was math discovered or invented?

Some mathematicians think that math is invented, as people name aspects of math or create new ways of solving problems . Other people think that math is always there— the concepts and ideas exist in nature, just waiting for us to discover them .

Here’s a look at the history of math and many of the societies and people who contributed to its development.

Early Societies

Math has evolved over thousands of years , with input from thousands of mathematicians. We don’t know exactly how prehistoric humans dealt with math problems (like counting how many berries they picked, or figuring out the distance between two places), but researchers believe that people were using addition, multiplication, and other math concepts in early China, India, and Mesopotamia.

In fact, the oldest clay tablets we have with math inscribed on them are more than 4,000 years old. They’re from Mesopotamia. We also have Egyptian papyrus sheets with math written on them. So, there’s evidence of math from the two oldest societies in the world.

Around 1800 B.C.E., the ancient Babylonians developed a number system based on the number 60 (it’s still used today to think about angle measurement). They were the first people we know of to use actual numbers to represent amounts.

It’s clear that, considering the pyramids and their society, the Egyptians used math . They definitely understood geometry and even had a formula for calculating the volume of a truncated pyramid.

The Ancient Greeks

There’s more information about who invented (or discovered) math concepts as human society evolved. The Greeks , more than 2,500 years ago, started doing more advanced math. Plato, Euclid, and Archimedes are still remembered for their mathematical achievements. For example, Pythagoras studied triangles and he invented what we learn about triangles, called the Pythagorean theorem.

We also know that in ancient Greece, math became something to study, and mathematicians started thinking about specific theories and building on one another’s work.

After Ancient Greece

After ancient Greece, mathematicians continued making new discoveries and new theories and solving new problems. In 17th-century England, Sir Isaac Newton developed the field of calculus on his own. At the same time, in Germany, Gottfried Leibniz was also involved in developing calculus. Some mathematicians have created problems and hypotheses that have never been solved, like Bernhard Riemann , who created the Riemann hypothesis , which has been attempted but never proven.  

And throughout history, women have also studied math and invented math concepts. For example, Emmy Noether gained recognition for her innovations in advanced algebra, and Katherine Johnson calculated and analyzed flight paths for spacecraft that sent astronauts to the moon. Mathematicians of color who have made significant contributions to mathematics include Fern Hunt , who created math models to describe different kinds of movement, and Mark Dean , a mathematician and computer scientist who holds patents on the computer that all PCs are based upon.

As math has evolved, people are building on what we know to create new types of math and new ways to use math, like applying math to build computers and create game theory , a branch of applied mathematics. So, maybe the question isn’t who invented math, but what will math invent next?

Videos About the Invention of Math

Use these videos to explore how different math concepts came about.

The Origin of Numbers

How Old Is Zero?

Where Do Math Symbols Come From?

Who Invented Algebra?

Who Invented Geometry?

Who Invented Trigonometry?

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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.

Also check:

• 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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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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Using history to teach mathematics: an international perspective.

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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 .

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History of mathematics in mathematics education: Recent developments in the field

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• Published: 10 November 2022
• Volume 54 , pages 1407–1420, ( 2022 )

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• Renaud Chorlay 1 ,
• Kathleen Michelle Clark   ORCID: orcid.org/0000-0003-1399-0165 2 &
• Constantinos Tzanakis 1 , 3  

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Over the last 40 years, exploring the possible interrelations between the history of mathematics and mathematics education has gradually emerged as an interdisciplinary domain of educational research and practice, and entered a maturity stage during the last two decades. This development calls for an account of the general issues concerning the rationale and main themes underlying research and applications in its context, and the foundational issues addressed as a result of its interdisciplinary character. Therefore, after explaining the perspective characterizing work in this domain (what has been called the HPM perspective), we present the main themes along which current research is conducted and we analyze the main issues and concerns raised by current research in this domain, with due reference to recent publications. Papers in this issue further develop the key research strands shaped by these main issues and concerns from a variety of perspectives; in particular, papers address the theoretical points related to the interdisciplinary character of this domain (e.g., the role of history in promoting and developing STEM education more profoundly), and the design of innovative teaching approaches based on original sources (e.g., how non-didacticized resource material can motivate mathematically rich tasks without requiring too specialized knowledge in the history of mathematics). These papers also suggest fresh avenues for research, some bearing on potential methodological connections between mathematics, education, and history (e.g., the possibility of different readings of original sources that reveal the complex interrelations among historical knowledge, teaching objectives, and pedagogical practices), and some bearing on the implementation, evaluation, and dissemination of designs connected to the history of mathematics, in teaching at all levels of education (e.g., development of curricular material based on original texts, offering new learning opportunities in relation to core topics in university mathematics). We conclude with a brief description of each contributed paper.

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History of Science, Epistemology and Mathematics Education Research

Introduction: Integrating History and Epistemology of Mathematics in Mathematics Education

Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-Art and Looking Ahead

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1 Introduction

In the last few decades, exploring the possible relevance and significance of the history of mathematics in the context of mathematics education has evolved into an interdisciplinary domain of educational research and practice. Benefitting from or/and contributing to research both in mathematics education and the history of mathematics, work in this domain has provided innovative teaching approaches, has led to the design and production of helpful resource material and teaching aids, and has raised important theoretical issues concerning the nature of mathematics, and its teaching and learning in regard to its historical development Footnote 1 . This domain constitutes a meeting point of mathematics, history, and education, as three a priori distinct disciplines having their own epistemological and methodological characteristics and particularities. Therefore, it is expected that working and engaging in this domain requires facing many challenges that are not easy to resolve and which call for a constructive dialogue among the corresponding scientific communities based on mutual understanding and respect (Chorlay & de Hosson, 2016 , Sect. 8.2, 8.4; Wang et al., 2018 ).

On this basis, one can distinguish between two closely interrelated lines of investigation, depending on whether history of mathematics—regarded as body of knowledge resulting from the professional endeavor of historians—is considered as a form of knowledge of mathematics which, as any form of knowledge of mathematics, supports research in mathematics education; or as a resource for didactic intervention in all aspects of mathematics education (curriculum design, classroom implementation, resource material, teaching aids, teacher education, etc.).

Along the first line, similarities between the professional endeavors of historians of mathematics and researchers in mathematics education have been highlighted:

Contrary to what research mathematicians do, the object of their investigation is not mathematics, and this object is not studied primarily mathematically. Rather, they study how agents engage with mathematics, in a context which can be described; mathematics is necessary to make sense of this engagement and this context, but cannot possibly be the only background tool (Chorlay & de Hosson, 2016 , p. 159).

Artigue ( 1990 ) even argued for the “need” of epistemology and history of mathematics for mathematics education researchers, not only for them to become more knowledgeable about mathematics (as both a body of knowledge and an activity carried out along characteristic rules), but to enable them to distance themselves from tacit and contingent “epistemological representations” reflecting personal trajectories and professional communities. In this context, history of mathematics is regarded not only as a source of knowledge of mathematics, but as an opportunity to experience “ dépaysement ” (Barbin et al., 2020 ), “otherness” (Radford & Santi, 2022 ): “There are times in life when the question of knowing if one can think differently than one thinks, and perceive differently than one sees, is absolutely necessary if one is to go on looking and reflecting at all” (Foucault, 1990, as cited in Chorlay & de Hosson 2016 , p. 187).

The second, more action-oriented line of investigation has long been called the HPM domain ( HPM is an abbreviation for H istory and  P edagogy of  M athematics ) because it has formed the core and main concern of the approaches adopted towards integrating elements from the history and epistemology of mathematics in mathematics education, namely, designing, implementing, and evaluating teaching approaches at all levels (including teacher education), producing educational aids and resource material, promoting and implementing interdisciplinary teaching, and connecting empirical educational research along these lines with research in other domains of mathematics education. This line of investigation emerged to a considerable extent in the context of the HPM group Footnote 2 . It has been an active field of research especially in the last two or three decades. A comprehensive account of the work done up to 2000 is presented in a highly collaborative and influential ICMI Study edited by Fauvel and van Maanen ( 2000 ), whereas a sufficiently comprehensive survey of the literature after 2000 is given by Clark et al. ( 2016 , Sect.3), and a detailed discussion of selected examples is given in the collective volume edited by Clark et al. ( 2018b ).

Research in this area has entered a maturity stage during the last two decades. This allows us to give an account of some general issues concerning the rationale and main themes underlying fundamental research and applications in its context, as well as the foundational issues that are or have to be addressed as a result of its interdisciplinary character. In particular, this special issue consists of contributions to this field that report on recent research along the above-mentioned two lines of investigation, at the levels of both theory and application.

Therefore, this paper is structured as follows: In Sect.  2 we address the question of why research in this area is important. In Sect.  3 we present an overview of research in this area. Specifically, we first present the rationale and the main themes along which research is conducted (and hence, the way this special issue is also structured), and then proceed to formulate, analyze, and comment on the main issues and concerns raised by recent research. Due to space limitations, we often refer the reader to the literature for further details. Finally, since the contributed papers in this special issue further develop the key lines of research shaped by these main issues and concerns from a variety of perspectives, Sect.  4 provides a brief description of the contributed papers, grouped according to the main themes of research presented in the preceding section on which they chiefly focus.

2 Connecting mathematics education and the history of mathematics

Perhaps it is not an exaggeration to say that above all sciences it is mathematics that places the greatest emphasis on and relies on the need for logical, rational, and intellectual rigor and consistency in our attempt to explore and understand better some aspects of the world we live in. And this is one of the main intrinsic reasons that mathematics has often been considered as a collection of definitions, axioms, theorems, and proofs, which—especially under the influence of the axiomatic approach and formalism as a philosophical thesis—became a common way of its presentation. Conceiving mathematics in this way, at least implicitly, suggests that mathematical knowledge coincides with the ‘polished’ products of humans engaged in mathematical activities. This emerging image is by no means specific to today’s mathematics and, beyond differences in style, this impersonal ‘coldness’ rings true, for instance in Babylonian numerical tables and lists of worked-out problems, as well as in Euclidean deductions.

On the other hand, however, mathematics (like any other scientific discipline) is a living human intellectual enterprise with a long history and a vivid present, where “…knowledge of any value is never possession of information, but ‘know-how’. To know mathematics means to be able to do mathematics” (Memorandum, 1962 , p. 189). This perspective implies that mathematical knowledge is delimited not only by the circumstances in which it becomes a deductively structured corpus of knowledge, but also by the procedures that originally led or may lead to it. Therefore, what is acceptable as mathematical knowledge—like any other form of human knowledge—is not absolute, but ‘time-dependent’. It is a continuously evolving, complex system of ways of thinking, reflecting, and acting (Radford & Santi, 2022 , Sect. 2.3). Therefore, it is potentially subject to change. In other words, historicity is one of its epistemological characteristics (Barbin et al., 2020 , p. 335; Radford et al., 2007 , p. 107; see also Grabiner, 1974 , Sects. 1, 5; Radford et al., 2014 , pp. 105–106). Historicity can be experienced when learners engage in some activity in response to a sample of mathematics from the past. It is at this point that mathematics and history meet in a way beneficial for mathematics education: The integration of historical and epistemological issues in mathematics education is a way for exposing mathematics in the making, which may help learners to understand the following characteristics of mathematics:

it has undergone changes over time, underscored by shifting views of what mathematics is and how it should be taught and learnt;

it has been in fruitful dialogue with other scientific disciplines, technology, philosophy, and the arts;

it has constantly stimulated and supported scientific, technical, artistic, and social developments; and

it is the result of contributions from many different cultures (Clark et al., 2019 , p. 3).

And it is at this point that the conception of mathematics as a human intellectual activity for the acquisition of knowledge either by the individual, or/and collectively together with the historical character of this knowledge becomes important for supporting the doing, learning, and teaching of specific pieces of mathematics and for appreciating the relation of mathematics with other intellectual and cultural pursuits all along its historical development (Clark et al., 2019 , p. 4; Fried, 2014a , Sect. 21.3.3; Radford, 1997 , Sects. 4, 5). In this perspective (that permeates both lines of investigation mentioned in Sect.  1 and for which the name HPM perspective has been coined; see Clark et al., 2018a , Sect. 1.1 for more details) mathematics is conceived both as a logically structured collection of intellectual products and as processes of knowledge production that lead to these products. As a consequence, learning mathematics becomes not only the process by which the learner gets acquainted with and competent in handling the symbols and the logical syntax of theories and in accumulating new results presented as finished products. It also encompasses acquaintance with the implicit motivations, the sense-making actions and the reflective processes of mathematicians that aim at the construction of meaning by linking old and new knowledge, and by extending and enhancing existing conceptual frameworks (Fauvel & van Maanen, 2000 , p. 202).

Of course, though the above-mentioned connections between past and present mean that possible similarities between past and present knowledge exist, still there are also crucial differences and dissimilarities, due to the very different scientific, social, and cultural conditions in which this knowledge has emerged. However, ignoring either of the two will lead to a very limited and incomplete picture of mathematics (Nooney, 2002 , p. 4; see also Thomaidis & Tzanakis, 2022 , Sect. 2): Neglecting similarities will restrict mathematics simply to the (currently considered) acceptable results of mathematicians’ activity deprived of the motivation behind them and without a deeper understanding of the processes that led to them. On the other hand, neglecting dissimilarities will lead to a distorted and biased view of the processes that led to current mathematical knowledge. The past will be forced “…through a sieve keeping out ideas foreign to a modern way of looking at things and letting through those that can be related to modern interests” (Fried, 2011 , p. 16; see also Grattan-Guinness, 2004a , 2004b ). Essentially this is what has been called a Whig or anachronistic approach to the past (Butterfield, 1965 Footnote 3 ). In this view, the past is studied in the light of our present knowledge (Kragh, 1989 , ch. 9), so that the present becomes the measure of the past and therefore, “…what one considers significant in history is precisely what leads to something deemed significant today” (Fried, 2001 , p. 395).

Therefore, although any similarities between past and present knowledge may serve as a motivation, guide, or resourceful aid for understanding specific pieces of mathematics and issues about mathematics, it is absolutely necessary to do so in a context that takes into account the quite different milieu in which teaching and learning takes place today. In other words, an effective development of a historical perspective in mathematics education necessarily has to take into account carefully that teaching and learning of mathematics today takes place under different social conditions, in the context of different cultural tradition(s), addressed to individuals and groups with varying characteristics and needs, and delimited by a variety of educational constraints (imposed by the curriculum, the instruction level, the learners’ orientation, etc.). As a consequence, any educational approach aimed at the teaching and learning of a specific piece of knowledge along these lines can be efficient only by achieving a subtle balance between staying faithful to the historical developments, while paying due attention to the current conditions under which this teaching and learning takes place (Thomaidis & Tzanakis, 2022 , Sect.  2). This is a difficult endeavor facing many challenges that do not admit easy nor general valid-for-all answers. Perhaps, this is the leitmotiv of research in and implementations of the HPM perspective, which, implicit as it was during the early development of this field, gradually became explicit, especially in the last 20 years or so.

3 Research in the HPM domain

3.1 motivation, main themes, and rationale.

Introducing elements from the history of mathematics in mathematics education has been advocated since the second half of the 19th century, by important mathematicians and historians such as De Morgan, Zeuthen, Poincaré, Klein, Tannery, and later Loria and Toeplitz, who showed an active interest in promoting the role the history of mathematics can play in mathematics education, though not all to the same degree and with the same motivation and rationale (Clark et al., 2019 , Sect. 3; Furinghetti, 2020 ; Jahnke et al., 2022 , and the bibliographies in these papers). Moreover, there were also some works referring to the history of mathematics in actual practice at school (Furinghetti, 2000 , pp. 49–50; 2019 , p. 110; 2000 , pp. 969–970).

At the beginning of the 20th century this interest was revived as a consequence of the debates on the foundations of mathematics and became stronger after the New Math reform in the period 1960–1980, considering history a natural possible way to conceive mathematics as an evolving human activity, and in this way to help improve its teaching and learning (Lakatos, 1976 ; Memorandum, 1962 , pp. 190–191; NCTM, 1969 ; see also Barbin et al., 2020 , pp. 333–334 and references therein).

This led to the formation of the HPM Group in the 1970s. In the following decades the establishment of this group greatly stimulated and supported the interest and educational research in this area at an international level (Fasanelli & Fauvel, 2006 ), leading to the 4-year ICMI Study providing a survey of the work done in this domain and reporting on the main issues for further research, captured in the comprehensive collective volume cited in Sect.  1 above (Fauvel & van Maanen, 2000 ). This volume became a landmark in establishing and making widely visible the potential significance of the history of mathematics in mathematics education, stimulating and enhancing the international interest of the educational community, and inspiring and motivating further research and actual implementations in education Footnote 4 . In particular, several collective volumes and special issues of research and practitioner journals appeared after this ICMI study Footnote 5 .

Therefore, aspects of the intensive research activity in this area in the last several years, as also recorded in important regularly organized international meetings Footnote 6 , their proceedings, and collective volumes that resulted from them, deserve to be presented in a special issue of a journal like ZDM – Mathematics Education . This gives us the opportunity to communicate to a wide international readership recent research on the multifaceted role the history of mathematics can play in mathematics education at all levels of instruction, including teacher education. In this connection, a clear indication of the main areas of research interest and activity in the HPM domain are the recurrent and closely interrelated themes along which the main international meetings in the HPM domain mentioned above have been organized in the last two decades, which are as follows:

Theoretical and/or conceptual frameworks for integrating history in mathematics education; exploring how reflecting on the history of mathematics could enrich didactical research.

History and epistemology of mathematics in students’ and teachers’ mathematics education at all levels of instruction: Design and/or assessment of classroom experiments and teaching/learning materials (preferably based on empirical data), considered from various perspectives, e.g., cognitive, didactical, pedagogical, affective, etc.

Original historical sources and their educational effects: classroom implementations; enhancing and deepening reflections on the teaching and learning of mathematics.

Surveys on the existing uses of history or epistemology in curricula, textbooks, and/or classrooms in primary, secondary, or tertiary levels, and in teacher education.

History and epistemology of mathematics as a tool for an interdisciplinary approach in the teaching and learning of mathematics by unfolding its productive interrelations with science, technology, and the arts.

Although recent publications (including the papers in this issue) report on empirical investigations, surveys, and reflective and critical studies on the above themes, this does not mean that introducing historical elements into mathematics education should be regarded as a panacea to all problems in mathematics education. Instead, it should be seen as a possible way to improve the teaching and learning of (specific parts of) mathematics and to enable deeper reflection on the nature of mathematics and mathematical activity, which, in view of the discussion in Sect.  2 , is worth exploring critically and carefully. Moreover, from this perspective and as research and implementations in this area are interdisciplinary in character, this fact (also reflected in the contributed papers either implicitly, or in some cases explicitly) stresses that a renewed discourse among the corresponding communities is desirable or even necessary; that is, among researchers in mathematics education, historians of mathematics, mathematics teachers, and mathematicians, and possibly historians, philosophers, and teachers of science and other disciplines (Siu, 2015 , p. 43; cf. Thomsen et al., 2022 , Sect. 7). Actually, being at the interface of mathematics, history, and education, the HPM domain points to, raises, and emphasizes several nontrivial questions and issues, as follows: Which history is suitable, pertinent, and relevant to mathematics education, why, and with which role(s)? In particular, can scholarly historical knowledge be useful for mathematics education, or is it necessary that it be didactically transposed (as is done for scholarly mathematical knowledge), and how this can be done? (Clark et al., 2018a ; Furinghetti et al., 2006 ; Jahnke, 2014 ; Jankvist, 2014 ; Kjeldsen, 2011a , 2012a , Thomaidis & Tzanakis, 2022 ). Furthermore, do similarities between past mathematicians’ creative work (including struggles or qualms among them) and students’ ways of learning mathematics exist? What are the limitations imposed by the different scientific, sociocultural, and historical conditions between modern learners and past mathematicians? And to what extent could any such eventual similarities be beneficial for mathematics education and for understanding and exploring further the historical development (Bråting & Pejlare, 2015 ; Furinghetti & Radford, 2008 ; Jankvist, 2014 ; Schubring, 2011 ; Pejlare & Bråting, 2019 ; Thomaidis & Tzanakis, 2007 )? At a more practical level, what can be the goal(s) and the objective(s) of a small-scale didactical intervention where historical elements have been integrated? And to what extent do or should such interventions commit to history (Barbin, 2022 )? Can such interventions be designed so that they can be implemented at a larger scale? How can relevant large-scale didactical research be encouraged, enabled, and enlightened (Clark, 2019 , p. 49)? Should historical elements be used in the same way and with similar goals in classroom teaching and in teacher education? If not, what are the differences, and the decisive factors shaping them that should be considered in designing teaching activities and providing teaching aids and resources? How can such interventions be evaluated and assessed and to what extent and in what sense do they contribute to the teaching and learning of mathematics (Clark et al., 2022 )? In particular, whether and in which ways does a historical perspective contribute to or become necessary for the mathematical and pedagogical development of pre- and in-service mathematics teachers at all levels (Clark, 2019 , pp. 49–50; Jankvist et al., 2020 ; Siu, 2015 , p. 44)?

These questions and further theoretical discussions on the right scale, the right goals, and the appropriate assessment procedures of didactical innovations in this domain can be helped and promoted by tools, methods, and theoretical constructs developed in the context of research in mathematics education (see, e.g., Agterberg et al., 2022 ; Barnett, 2022 ; Bernardes & Roque, 2018 ; Chorlay, 2022 ; Gosztonyi, 2022 ; Jankvist, 2011 ; Jankvist & Kjeldsen, 2011 Kjeldsen, 2012b ; Kjeldsen & Blomhøj, 2012 ; Moustapha-Corrêa et al., 2022 ; Spies & Witzke, 2018 ). More generally, the need for empirical studies on the actual impact of such didactical innovations calls for a fruitful dialogue with researchers in mathematics education working outside this domain (Chorlay & de Hosson, 2016 ). Similarly, new perspectives in historical research, as well as trends in mathematical research, call for an up-to-date scholarly discussion between historians of mathematics, researchers in mathematics education, and mathematicians on the relationship between the history of mathematics (and the history of sciences) and mathematics education (Barbin et al., 2020 , p. 340; Fried, 2014b , p. 10ff; Radford et al., 2014 ); for instance, by putting emphasis on work collectives (their usually tacit norms, their shared practices and boundaries, the competition among collectives, the various sociological structures—the school, a theory, a discipline, etc.), exploring the interactions between ‘high-mathematics’ and ‘low-mathematics’ users (professional mathematicians, physicists, computer scientists, engineers, economists, calculators (i.e., persons who calculate), instrument makers etc.), reappraising non-Western mathematics (with new perspectives on proof, algorithms, the notion of ‘problem’), etc.

Along these lines, further questions at a more fundamental level can be and have been raised: Can the communities of mathematics educators and historians of mathematics cooperate harmoniously by benefitting from the epistemological characteristics, aims, commitments, and methodologies specific to each discipline, or is this prevented by strong constraints due to crucial differences between these epistemological characteristics, aims, commitments, and methodologies, thus becoming a task that is almost unfeasible (Chorlay & de Hosson, 2016 ; Fried, 2001 , 2007 , 2011 ; Kjeldsen, 2011b ; Radford et al., 2014 , pp. 94–98)? In particular, can historical research and practice inspire, support, or supply explanatory frameworks and working tools for research in mathematics education? And conversely, can research and practice in mathematics education inspire, support, and broaden research in the history of mathematics? And if so, how this can be achieved (Furinghetti et al., 2006 ; Jankvist, 2014 , Sect. 27.2, 27.7; Schubring 2011 , p. 90; Thomaidis & Tzanakis, 2022 ; see also Barnett, 2022 ; Demattè & Furinghetti, 2022 , Sect. 6; Moustapha-Corrêa et al., 2022 )?

All these questions and issues that in one form or another have been addressed in the literature, reveal that in realizing the HPM perspective in practice, several intertwined factors related to research in history and mathematics education come into play, which cannot be ignored. Moreover, although there is no general consensus about their answers and much work is still to be done, we hope that the papers in this issue will contribute to their better understanding and will motivate and stimulate further work to this end.

3.2 On recent contributions and main current concerns

In the past few decades many mathematics education researchers and teachers all over the world have seriously considered and followed the general ideas outlined in the previous sections about the evolutionary nature of mathematical knowledge and the significance of adopting a historical perspective in unveiling these ideas, by exploring in one way or another the related questions and issues mentioned in the previous subsection. A number of recent surveys and overviews of the work done so far have appeared, addressing key issues from various perspectives and guiding the reader to the relevant literature. Below, we refer briefly to an indicative sample.

Jankvist ( 2009 ) focused on why and how history of mathematics may or should be used or integrated Footnote 7 in mathematics education according to the emphasis on history being a goal , or a tool , classifying the corresponding arguments accordingly, and categorizing the corresponding teaching and learning approaches into three categories from a methodological point of view, namely, the illumination , module , and history-based approaches. Furinghetti ( 2004 , 2020 ) made the distinction between what she called the two main streams for exploring the role of the history of mathematics in mathematics education; these are “history for constructing mathematical knowledge” through an approach to the roots around which mathematical knowledge developed, and “history for promoting mathematics” as an activity of a community, thus appreciating mathematics as an integral part of human intellectual history and cultural development. Furinghetti and Radford ( 2008 ) and Schubring ( 2011 ) reviewed the work in this domain from the perspective of the connections and contrasts between the historical development and students’ learning in a modern classroom. Fried (2014) argued that recent attempts for bringing the history of mathematics into mathematics education fall under three central themes, which he called “motivational”, “curricular”, and “cultural” themes. He also further argued that such attempts—especially under the cultural theme—could help enrich, deepen, and widen the main aims of mathematics education itself (this idea of transforming the main aims of mathematics education by connecting it with the history of mathematics is further elaborated by Fried ( 2018a ). Clark et al. ( 2018a , pp. 1–2) argued that the interdisciplinary nature of work in the HPM domain is a result of the multifarious interrelations among history, mathematics, and education. They surveyed the work in this domain in relation to what they consider as the main issues of research in its context (ibid., p. 2) as expressed via the following questions: Which history is suitable, pertinent, and relevant to mathematics education? Which role can the history of mathematics play in mathematics education? To what extent has the history of mathematics been integrated in mathematics education (curricula, textbooks, educational aids and resource material, teacher education)? How can this role be evaluated and assessed and to what extent does it contribute to the teaching and learning of mathematics? From another perspective, Barbin et al. ( 2020 ) classified the contributions associated with experimental and theoretical work in the HPM domain into three interrelated types, namely, epistemological, cultural, and didactical. They further pointed to the need for a productive dialogue among the corresponding communities, with emphasis on the interdisciplinary character of history in an educational context, together with the need for developing appropriate and effective theoretical and conceptual frameworks, and conducting more in-depth empirical studies.

A more detailed study of all these surveys (and other publications, of course) indicates that despite the variety of perspectives from which research in the HPM domain is conducted, in the implementation of the results of international research both at the theoretical level and in the applications in actual didactical practice, the following strongly interconnected concerns play a central role (Clark et al., 2019 , Sect. 5).

It is of significant importance to realize carefully designed and systematically applied empirical investigations in order to examine critically and evaluate the eventual effectiveness of a historical perspective in mathematics education on improving the teaching and learning of mathematics, as well as students’ and teachers’ awareness of mathematics as a discipline and their disposition towards it. Several such investigations have appeared, but much more work is still necessary that will shed more light on the issues briefly presented in the previous sections (see for instance, the surveys by Jankvist, 2007 , 2012 ; also, Barbin, 2018 ; Clark et al., 2018b , parts III–V; Jankvist, 2011 ; Katz & Tzanakis, 2011 , chs. 9–11, 13–16, 19; more references in Clark et al., 2019 , p. 15).

For any integration of the history and epistemology in mathematics education to be possible at all, an adequate pre- and in-service teacher education in this area is necessarily presupposed, hence research on how to achieve this task is central. This point was recognized quite early and emphasized repeatedly (Fauvel & van Maanen, 2000 , ch. 4 and Sect. 3.2; see also Barbin et al., 2011 ; Clark, 2019 , pp. 48–50; Gazit, 2013 , Sect. 4; Huntley & Flores, 2010 , Sect. 1; Siu, 2015 , p. 44). In this connection, many efforts have been made to educate teachers (Arcavi & Isoda, 2007 ; Bruckheimer & Arcavi, 2000 ; Clark, 2011 ; Clark et al., 2018b , chs. 4, 11, 14, 18; Povey, 2014 ; Smestad, 2011 ; Waldegg, 2004 ) and explore their attitudes, beliefs, and/or teaching approaches and possible ways they could be modified (Alpaslan et al., 2014 ; Buchholtz & Schorcht, 2019 ; Charalambous et al., 2009 ; Chevalarias, 2019 ; Furinghetti, 1997 , 2007 ; Moyon, 2022 , Sect. 2; Philippou & Christou, 1998 ; Smestad, 2011 ; Spies & Witzke, 2018 ; Vicentini et al., 2019 ). So far, however, there are no generally accepted results in relation to these issues. Further research is still needed to explore ways to educate teachers in this respect that, among other things, will be convincing enough concerning the effectiveness of introducing historical elements into teaching.

Closely related to (b) and of equal significance is the design, production, availability, and dissemination of diverse didactical source material in a variety of forms, e.g., anthologies of original sources, annotated bibliographies, description of teaching sequences or modules that could serve as a source of inspiration and/or as generic examples for classroom implementation, educational aids of various types, appropriate websites, etc. This has also been stressed in the literature (e.g., Fauvel & van Maanen, 2000 , pp. 212–213; Panasuk & Horton, 2012 , p. 16; Pengelley, 2011 , pp. 3–4), and although the need for such material has been satisfied to a considerable extent in the last 20 years or so (for an indicative useful sample see Clark et al., 2018a , p. 12; Clark et al., 2019 , pp. 14–15), further empirical investigation of the usefulness and appropriateness of this material is needed (Barnett, 2022 ; Clark et al., 2022 ; Danielsen et al., 2018 ; Moyon, 2022 ; Schorcht, 2018 ). For instance, an in-depth systematic exploration is needed of how teachers design, select (or reject), and/or use, modify, or misuse various resources in their teaching (Chorlay, 2022 ), why they do it and what this may reflect (their beliefs, a global state of the educational system in which they work, their knowledge of history of mathematics, their professional competencies, etc.).

Finally, in close relation to (a), (b), and (c), it is very important to identify and unfold theoretical ideas and concepts that underlie or enrich empirical investigations, classroom implementations, and the production of didactical material, and to develop them carefully into coherent theoretical frameworks and methodological schemes that will serve as a foundation and/or orientation for further research and applications in this area. Several works, including the following, have been published in this connection, some of which have already been cited in the previous sections: Bråting & Pejlare, 2015 ; Clark et al., 2018b , chs. 2–4, 8; Chorlay & de Hosson, 2016 ; Fauvel & van Maanen, 2000 , ch. 2; Fried, 2001 , 2007 , 2011 , 2018a , 2018b , 2022 ; Furinghetti & Radford, 2008 ; Grattan-Guinness, 2004a , 2004b ; Hanna et al., 2010 ; Jahnke, 1994 , 1996 , 2014 ; Jankvist, 2009 , 2011 Jankvist & Kjeldsen, 2011 ; Kjeldsen, 2011c , 2012b 2018 , 2019 ; Kjeldsen & Blomhøj, 2012 ; Kjeldsen & Petersen, 2014 ; Radford & Santi, 2022 ; Radford et al., 2014 ; Schubring, 2011 ; Thomaidis & Tzanakis, 2007 , 2022 ; Tzanakis & Thomaidis, 2012 .

Elaborating on the many issues raised by the above concerns will improve the field’s understanding of the benefits that possibly underlie the integration of the history of mathematics in mathematics education.

4 Structure and content of this issue

The papers in this issue are directly related to one or more of the four central concerns outlined in the previous subsection. Taken as a whole they refer in one way or another to issues in the HPM domain relevant to all levels of education, including pre- and in-service teacher education. Seen individually, each paper’s main focus is situated within one of the five themes listed in Subsection 3.1 . Therefore, for clarity we ground each set of papers according to the five themes. However, as these themes are strongly interrelated, almost all papers contribute to more than one of them.

In the first part on theme 1, six papers address important theoretical issues pertaining to historical research, and exploring how reflection on the history of mathematics could enrich didactical research.

Jahnke et al. ( 2022 ) present overviews of the writings of H. Poincaré, F. Klein, and H. Freudenthal on the role of the history of mathematics in mathematics education, since these important mathematicians’ views have inspired or influenced subsequent research in mathematics education in general and the HPM domain in particular. The comparative study of these writings reveals these scholars’ understanding of the role of the history of mathematics in mathematics education as a clear recommendation for history becoming part of mathematics teacher education, as well as history’s placement within human culture.

Fried’s ( 2022 ) paper links historical scholarship with the role of primary sources in mathematics education by considering an important historical text via which one sees a past mathematician looking at his own past and advocating the importance of studying such texts as an aid to refine one’s own sense of possible postures towards historical material. This activity encourages self-reflection, and enables Fried to describe in detail different types of approach to the past (Fried, 2018b ) that give a better understanding of a modern teacher’s position in trying to integrate historical material into teaching. This self-reflection constitutes an important justification for the history of mathematics in mathematics education as a catalyst that could widen, deepen and enrich the main goals of mathematics education (Fried, 2007 , 2014a , 2018a ).

Thomaidis and Tzanakis ( 2022 ) argue that for understanding the interrelations between the history of mathematics and mathematics education, many relevant concepts are complementary rather than antagonistic. By analyzing a known students’ misconception in elementary arithmetic and algebra in relation to an alleged mistake in Euler’s Elements of Algebra , they illustrate how didactical and historical research can be mutually supportive, call for a constructive discourse between the two communities, and point to the need for a didactical transposition of historical knowledge that complements rather than distorts it.

Theoretical reflections on inquiry-based mathematics form the core of Gosztonyi’s ( 2022 ) paper. She proposes an analysis of the structure of Clairaut’s “ Éléments de Géométrie ” on the basis of an interdisciplinary project in which problem-centered texts from various cultures are analyzed, and ongoing didactical research on the “Hungarian guided discovery approach” where “series of problems” are central in designing and implementing teaching sequences. With the detailed presentation of two examples from this book, Gosztonyi shows how this approach aids reflection on constructing and analyzing inquiry-based teaching sequences in the context of mathematics education.

Radford and Santi ( 2022 ) aim to contribute to diminishing the gap between theoretical and empirical research in the HPM domain, by reconsidering and reconceptualizing mathematics learning and mathematical knowledge that students encounter at school. The former is theorized as a critical encounter with mathematical knowledge occurring via intertwined objectification and subjectification processes (Radford, 2021 ). Thus, the crucial educational point at stake is to offer students occasions for this encounter. In this perspective history becomes a necessary part and parcel of mathematics education. These ideas are exemplified by the analysis of two arithmetical problems from two original texts, and its actual implementation as a teaching-learning activity.

Approaching original texts is a real challenge for both teachers and learners because of a strong tendency to ‘translate’ these texts into modern mathematics, thus making difficult to understand and appreciate the historical development and one’s own conceptualizations (Barbin, 2022 ; Fried, 2022 ). In this vein, Guillemette & Radford ( 2022 ) aim to go beyond the classical opposition between historians’ History and Whig History (Grattan-Guinness, 2004b , Sect. 3.8). Drawing on inputs from linguistic theories, they highlight the theoretical and ethical potential of a “dialogical” perspective in which understanding an original text is seen as one of the ways to enter responsively in a dialogue with another human Being. They illustrate their position with an example from an empirical study and argue that proceeding along these lines reveals another possibility for approaching an original text, that of an educator.

The four papers in the second part (on theme 2), report on recent classroom experiments of introducing a historical perspective, and detailed discussion on the selection and design of the corresponding teaching material.

Clark et al. ( 2022 ) report on parts of a large-scale research project in progress, with focus on the use of curricular materials based on primary historical sources (Primary Source Projects; PSPs). Leaning upon a variety of theoretical tools, the authors provide a qualitative analysis of the data relevant for students’ understanding of the mathematical metadiscursive rules, students’ worldviews about mathematics and how these may change under the influence of PSPs, and for examining students’ reports on the challenges and benefits of learning from primary sources resulting from their engagement with PSPs.

Demattè and Furinghetti ( 2022 ) report on an empirical study with upper secondary school students, who were given problems to solve from three Renaissance Italian abbacus treatises, aiming to explore their response to these texts and the concepts and solution strategies they used. The choice and implementation of the original excerpts focused both on their mathematical content and the characteristics of the societies these treatises served. It stimulated several students’ interest in reflecting on cultural and scientific issues beyond the task of solving a problem and allowed the authors to explore their knowledge and beliefs concerning important pieces of mathematics.

Moustapha-Corrêa et al. ( 2022 ) present their “ historical zoom-in and zoom-out ” (ZIZO) approach for mathematics teacher education. That is, based on questions and problems capable of engendering and/or inducing concepts and results, the goal of bringing teachers in contact with the complex network of historical, social, and subjective processes leading to mathematical knowledge. ZIZO is structured according to teachers’ alternating immersion in specific details of original sources and in historical overviews of the ideas in focus. The paper reports on implementing this approach with a group of in-service teachers with a focus on its influence on the teachers’ awareness of what mathematics is and what teaching and learning mathematics means.

De Varent and Décamp ( 2022 ) report on an empirical study with grade 10 students, concerning the design and implementation of a teaching sequence based on a paleo-Babylonian cuneiform tablet showing the calculation of a square’s area. In the paper they analyze from a history-as-a-goal perspective (Jankvist, 2009 ) the specifics of an unusual teaching sequence designed by a historian of mathematics. Both qualitative and quantitative data allow for a study of the tensions and discontinuities between the original design choices, the actual implementation by the historian, and the impact on students.

Original historical sources are an important means for introducing a historical perspective in actual didactical practice, and for enhancing and deepening meta-level reflections on the teaching and learning of mathematics. On the other hand, several nontrivial methodological, epistemological, and didactical questions and problems are related with their use, which are far from being settled (Barnett et al., 2014 ; Chorlay, 2016 ; Fauvel & van Maanen, 2000 , ch. 9; Furinghetti et al., 2006 ; Jankvist, 2014 ; Pengelley, 2011 ). The two papers in the third part (on theme 3) are specifically focused on particular aspects of these effects, and their dependence on design choices.

Having as a leitmotiv the unavoidable question, “why is it worthy to use original sources in teaching and learning mathematics when there are so many good contemporary textbooks prepared for dealing with the same topics using today’s mathematical language, notation, and terminology?”, Barnett ( 2022 ) focuses on the PSPs mentioned above. By using Sfard’s commognitive approach, she analyzes three PSPs and discusses the new learning opportunities they offered, which are not shared by standard textbooks, at the same time pointing to the constraints imposed in this way.

Chorlay ( 2022 ) reports on a case study in which high school mathematics teachers autonomously designed and implemented classroom sessions based on an excerpt from Euler’s Elements of Algebra . The paper focuses on teaching practices and not on the effect on students, while the historicity of the source is not regarded as a key factor. An interesting suggestion is that the dissemination of non-didacticized resource materials can motivate the design of mathematically rich tasks in ordinary teaching contexts without requiring the teachers to be specifically knowledgeable in the history of mathematics. This paper also suggests a complementary line of research, on the didactic potential of using historical sources in the classroom for purposes other than introducing a historical perspective in education (Chorlay, 2016 ).

Two of the three papers in the fourth part on theme 4 provide information on the existing uses of history in curricula, textbooks, and didactical sources.

In the new high school mathematics curriculum in France (2019) the history of mathematics is one of its three main guiding threads. However, an important long-lasting issue is the conflict between the requirement to integrate the history of mathematics into teaching, and many teachers’ concerns or reluctance to do so because of their lack of knowledge of this history (Clark et al., 2018a , pp. 6–7; Fauvel & van Maanen, 2000 , p. 203; Siu, 2006 , pp. 268–269). Motivated by this conflict, Barbin ( 2022 ) explores the complex interrelations among historical knowledge, teaching objectives, and pedagogical practices. She illustrates these complex interrelations by analyzing different possible readings of two particular texts (of Diophantus and Euler), and two case studies in the context of teacher education based on original texts.

Also motivated by this reform, Moyon ( 2022 ) presents his survey conducted with French mathematics teachers, and a comparative study of textbooks used before and after the reform as to what they include concerning a typical important case, namely, Fibonacci. Though teachers appreciate the significance of the history of mathematics for its teaching and learning, they are hindered by lack of teaching time and lack of confidence on their historical expertise and textbooks provide insufficient, or inappropriate support for this. Moyon presents two cases of how the material in current textbook examples could be transformed so that this integration becomes historically accurate and didactically appropriate, and suggests how teachers could be helped to (re)design the tasks for the available textbooks.

The third paper by Thomsen et al. ( 2022 ) explores the main features of the “classical” subject of illuminating the mathematics learner through historical knowledge, with the “modern” subject of using digital technologies to support mathematics teaching and learning. To this end they survey the relevant literature in the last 30 years and examine to what extent it draws on theoretical constructs from mathematics education research and for what purposes. Based on a multifaceted theoretical framework, the authors analyze the surveyed publications, presenting in detail three examples of how this has been realized.

In the last part, two papers touch upon aspects of theme 5. Siu ( 2022 ) focuses on the promising and actively explored STEM approach in education, in which, however, mathematics seems to play an understated role (Maass et al., 2019 , p. 869). Siu expresses this concern and argues that throughout history and across cultural traditions “one witnesses STEM at work”. In his perspective “STEM embodies a spirit of exploration and a way of thought, combined with mathematical thinking and scientific spirit, to organize theories through experiments and observations in the search for knowledge and further innovation” (ibid, Sect. 2). Siu argues that historical knowledge can support this view and provides several important examples to this end, also highlighting their appearance under similar forms in different cultures.

Understanding the interrelations between mathematics and physics, especially by paying attention to their historical development, is a non-trivial subject, with far reaching educational implications and with ongoing discourse from several interconnected perspectives (Karam, 2015 ; Tzanakis, 2016 ). Liu ( 2022 ) takes the stance that these interrelations can be understood by adopting a structuralist view . To support this stance, he comments on some important historical examples, and reports on a historically motivated empirical study with college students about one of Galileo’s inclined plane experiments, classifying students’ beliefs and attitudes towards these interrelations as they are revealed via this didactical activity.

For developments up to 2000, see Fasanelli & Fauvel ( 2006 ). For an account of the various activities in this domain, see Clark et al. ( 2019 , Sect. 3 and the appendix).

The well-known abbreviated title for the International Study Group on the relations between the  H istory and  P edagogy of  M athematics , the HPM  Group , which began as a Working Group at the second ICME (International Congress on Mathematics Education) in 1972, and at ICME-3 in 1976 became the first study group affiliated to ICMI (International Commission on Mathematical Instruction), together with the International Group for the Psychology of Mathematics Education (PME).

“It is part and parcel of the Whig interpretation of history that it studies the past with reference to the present” (ibid., p. 11).

For an indicative list of activities and publications after this ICMI study, see Clark et al., ( 2018a , Sect. 1.2) and Clark et al. ( 2019 , Sect.  3 and appendix); for a list of (both old and recent) ‘core publications’ in the HPM domain that offers newcomers in the field the possibility to begin a survey of the literature see Katz et al. ( 2014 , Sect. 3), whereas, for a sufficiently comprehensive and up to date bibliographical survey see Clark et al. ( 2016 ).

These include Barbin ( 2018 ); Barbin & Bénard ( 2007 ); Barnett et al. ( 2014 ); Bekken & Mosvold ( 2003 ); Clark & Thoo ( 2014 ); Clark et al., ( 2018b ); Furinghetti et al. ( 2007 ); Katz ( 2000 ); Katz & Michalowicz ( 2005 ); Katz & Tzanakis ( 2011 ); Katz et al. ( 2014 ); Knoebel et al. ( 2007 ); Matthews ( 2014 ); Siu & Tzanakis ( 2004 ); Sriraman ( 2012 ); Stedall ( 2010 ).

Such as the International Congress on Mathematical Education (ICME) and the accompanying HPM Satellite Meeting , the European Summer University on History and Epistemology in Mathematics Education (ESU), and the Congress of the European Research in Mathematics Education (CERME).

There has been a long discussion on different expressions conveying different connotations and meanings concerning the connections of history to mathematics education (hence, different understandings of these connections), including history used in, integrated in, introduced in, and permeating the teaching and learning of mathematics (e.g., Barbin, 2022 , Sect. 1.2; Siu & Tzanakis, 2004 , p. viii).

Agterberg, D. A., Oostdam, R. J., & Janssen, F. J. J. M. (2022). From speck to story: relating history of mathematics to the cognitive demand level of tasks. Educational Studies in Mathematics , 110 , 49–64.

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Undergraduate essay prize.

The British Society for the History of Mathematics is pleased to launch its famous undergraduate essay prize in the history of mathematics for 2023-2024. 

The essay should be written on any aspect from the history of mathematics, in any historical or geographical region. The competition is open to undergraduate or taught masters students at any university in the UK or the Republic of Ireland during the 2023-24 academic year. Research students are not eligible. If you have any queries about your eligibility, please contact Dr Ciarán Mac an Bhaird, BSHM Education Officer ( [email protected] ).

The absolute maximum length for submissions is 3000 words (excluding references), and previous winners or runners up of the undergraduate prize cannot submit another entry.

The award for the best essay is £150 and the winning entry will be considered for publication in our journal, The British Journal for the History of Mathematics . There will be as many run-up prizes awarded as we find appropriate (£50 each).  

The winner and runners-up will also benefit from free BSHM subscription for a year.  

Submit your entry here.

Deadline is 28th June 2024, midnight.  We look forward to reading your submissions!

Previous winners

2022/23:  Eleanor Brittain , University of Cambridge, ‘Wh.1754: The Crossing of Mathematics, Religion and Art’.  

2022/23 Runner-up:  James Hayes , University of Galway, ‘Hilbert's Space: How David Hilbert's Mentorship Shaped Mathematics’.

2022/23 Runner-up:  Felix Lindsay-Smith , University of Lincoln, ‘A preliminary investigation regarding the obscurity of Omar Khayyam's method of computing cubic roots’.

2021/22:  Zakkai Goriely , Oxford University, ' British Mathematical Reformers in the Nineteenth Century: Motivations and Methods '. 

2021/22 Runner-up:  Ffinlo Wright , University of St Andrews, ' A Turning Point for Game Theory: Situating John F. Nash in the History of Economic Thought '. 

2020/21:  Ellen Flower , Oxford University,  The ‘analysis’ of a century: Influences on the etymological development of the word 'analysis' in a mathematical context to 1750 .

2020/21:  George Waters , London School of Economics,  Exploring the use of mathematics to obtain consensus

2020/2021 runner up: Aoife Kearins , Trinity College Dublin,  Proofs, Partnerships and the Penny Post: How the Development of the U.K. Postal Service Made Mathematics Collaborative.

2019/20:  Quantifying the unquantifiable: the role of mathematicisation of philosophy during seventeenth century Scotland,  by  Natasha Bailie , Queen's University, Belfast 

2019/2020 runners up:  Emma Lepinay , University of Oxford:  Treatise of Algebra: John Wallis’ journey towards understanding complex numbers .

Molly Chung , University of St Andrews:  The effects of rivalry on mathematical development in Europe.

Arman Jena , London School of Economics:  The 20th century development of Linear Programming in USA and USSR . 

Sasha Ramani , University of York:  Deep rooted evidence of the use of mathematical principles in music .

We did not award the prize in 2018-19.

The winner of the 2017-18 Essay Prize was  Kamilla Rekvenyi of the University of St Andrews, for her essay, Paul Erdös’ Mathematics as a Social Activity. Kamilla was awarded her prize at the BSHM Meeting at Gresham College on 24th October, at 4pm. Her paper was subsequently published in the British Journal for the History of Mathematics, and can be seen here  https://www.tandfonline.com/doi/abs/10.1080/26375451.2019.159303 .

2016/17: Eli Hymson of the University of Exeter for an essay entitled The Unreasonable Effectiveness of Foundationless Mathematics . The judging panel also highly commended  entries by Siddhi Doshi (essay title: The evolution of the game theory as a foundation in economic analysis ) and Stefan Kitic (essay title: What is “money” and does its use require “mathematics”? )   of the London School of Economics.

2015/16: Michael Seal (London School of Economics), 'Was there a Revolution in Analysis in the Early 19th Century?' and Brigitte Stenhouse (University of Oxford), 'How Financial Instability Influenced the Mathematical Publications of Mary Somerville' .

2014/15: Edwin Reynolds (University of Oxford), ‘To What Extent Were the Contributions of Cauchy to the Development of Rigour in Analysis Influenced by Those of Lagrange?'

2013/14: Remus Stana (University of Glasgow), 'Mathematics in Nazi Germany' .

2012/13: Ryan Stanley (University of Exeter), 'Dedekind, Cantor and the rigour of calculus' .

2011/12: Stephanie Crampin (University of Oxford), 'The contribution of Évariste Galois to the founding of group theory' and Nicole Johannesen (University of St Andrews), 'The application of mathematical understanding in the ancient Olympic Games'.

MA 410: History of Mathematics: Sample Paper Topics

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Here are some topics suggested by your teacher:

• History of perfect numbers from Euclid to the day before yesterday • The influence of social needs on the uses of mathematics • History of the binomial theorem • The quadratic equation • An insight into Islamic mathematics • Goldbach and his famous conjecture • The history of logarithms and logarithm tables • Geometrical mathematical recreations • Prime numbers • The history of trigonometry • Charles Babbage and his engines • Some revolutionary curves • Angle trisection • Relationship of mathematics to music • Comparison of algebra texts from 1900 to today • American mathematics prior to 1875 • How our view of the early history of mathematics has changed in the last 100 years • The dark ages were dark for mathematics, too • Is mathematics discovered or invented? • How new was the new math? • Female mathematicians in history • The books that most changed our present geometry texts • The remarkable Bernoulli family • The rise of periodical literature and how it affected mathematics • The changing nature of function • The calculus priority dispute • The history of linear programming • Quality control, with emphasis on control charts • The role of cryptology in World War II • Navigation and mathematics • The Platonic solids • Mathematics and the early Rabbis • History of Hindu mathematics • The history of pi

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Study Like a Boss

History of Math

Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions.

This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems. This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory.

Indeed, mathematics is nearly as old as humanity itself; evidence of a sense of geometry and interest in geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. The earliest records of advanced, organized mathematics date back to the ancient Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC.

There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs. The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number.

For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers to be added. Multiplication was based on successive doublings, and division was based on the inverse of this process.

The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express all other fractions. For example, the fraction E was the sum of the fractions 3 and *. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. In geometry, the Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of figures such as bricks, cylinders, and pyramids.

To find the area of a circle, the Egyptians used the square on U of the diameter of the circle, a value of about 3. 16-close to the value of the ratio known as pi, which is about 3. 14. The Babylonian system of numeration was quite different from the Egyptian system. In the Babylonian system-which, when using clay tablets, consisted of various wedge-shaped marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics.

The number 60, however, was represented by the same symbol as 1, and from this point on a positional symbol was used. That is, the value of one of the first 59 numerals depended henceforth on its position in the total numeral. For example, a numeral consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 602 + 27 60 + 10. This principle was extended to the representation of fractions as well, so that the above sequence of numbers could equally well represent 2 60 + 27 + 10 (†), or 2 + 27 (†) + 10 (†-2).

With this sexagesimal system (base 60), as it is called, the Babylonians had as convenient a numerical system as the 10-based system. The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation (Equation). They could even find the roots of certain cubic equations. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest.

They could solve complicated problems using the Pythagorean theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, arranged so that c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic and some geometric progressions, as well as sequences of squares. They also arrived at a good approximation for . In geometry, they calculated the areas of rectangles, triangles, and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders.

However, the Babylonians did not arrive at the correct formula for the volume of a pyramid. The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. According to later Greek accounts, this development began in the 6th century BC with Thales of Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance of studying numbers in order to understand the world.

Some of his disciples made important discoveries about the theory of numbers and geometry, all of which were attributed to Pythagoras. In the 5th century BC, some of the great geometers were the atomist philosopher Democritus of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of circles are equal to areas of certain triangles. This discovery is related to the famous problem of squaring the circle-that is, constructing a square equal in area to a given circle.

Two other famous mathematical problems that originated during the century were those of trisecting an angle and doubling a cube-that is, constructing a cube the volume of which is double that of a given cube. All of these problems were solved, and in a variety of ways, all involving the use of instruments more complicated than a straightedge and a geometrical compass. Not until the 19th century, however, was it shown that the three problems mentioned above could never have been solved using those instruments alone.

In the latter part of the 5th century BC, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. That is, the two lengths are incommensurable. This means that no counting numbers n and m exist whose ratio expresses the relationship of the side to the diagonal. Since the Greeks considered only the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express this ratio of diagonal to side. (This ratio, , would today be called irrational.

As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also discovered a method for rigorously proving statements about areas and volumes by successive approximations. Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and who also wrote on optics, astronomy, and music.

The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century BC on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes. The century that followed Euclid was marked by mathematical brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes used a method of discovery, based on theoretically weighing infinitely thin slices of figures, to find the areas and volumes of figures arising from the conic sections.

These conic sections had been discovered by a pupil of Eudoxus named Menaechmus, and they were the subject of a treatise by Euclid, but Archimedes’ writings on them are the earliest to survive. Archimedes also investigated centers of gravity and the stability of various solids floating in water. Much of his work is part of the tradition that led, in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman soldier during the sack of Syracuse.

His younger contemporary, Apollonius, produced an eight-book treatise on the conic sections that established the names of the sections: ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry until the time of the French philosopher and scientist Ren Descartes in the 17th century. After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable stature. The writings of Hero of Alexandria in the 1st century AD show how elements of both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside the logical edifices of the great geometers.

Very much in the same tradition, but concerned with much more difficult problems, are the books of Diophantus of Alexandria in the 3rd century AD. They deal with finding rational solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis). Paralleling the studies described in pure mathematics were studies made in optics, mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and Archimedes, also wrote on astronomical topics.

Shortly after the time of Apollonius, Greek astronomers adopted the Babylonian system for recording fractions and, at about the same time, composed tables of chords in a circle. For a circle of some fixed radius, such tables give the length of the chords subtending a sequence of arcs increasing by some fixed amount. They are equivalent to a modern sine table, and their composition marks the beginnings of trigonometry. In the earliest such tables-those of Hipparchus in about 150 BC-the arcs increased by steps of 71, from 0 to 180.

By the time of the astronomer Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his Almagest a table of chords in a circle for steps of 3, which, although expressed sexagesimally, is accurate to about five decimal places. In the meantime, methods were developed for solving problems involving plane triangles, and a theorem-named after the astronomer Menelaus of Alexandria-was established for finding the lengths of certain arcs on a sphere when other arcs are known.

These advances gave Greek astronomers what they needed to solve the problems of spherical astronomy and to develop an astronomical system that held sway until the time of the German astronomer Johannes Kepler. Medieval and Renaissance Mathematics Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of the preceding centuries was established in various centers of Greek learning. The preservation of such works as have survived to modern times began with this tradition. It was continued in the Islamic world, where original developments based on these masterpieces first appeared.

After a century of expansion in which the religion of Islam spread from its beginnings in the Arabian Peninsula to dominate an area extending from Spain to the borders of China, Muslims began to acquire the results of the foreign sciences. At centers such as the House of Wisdom in Baghdad, supported by the ruling caliphs and wealthy individuals, translators produced Arabic versions of Greek and Indian mathematical works. By the year 900 AD the acquisition was complete, and Muslim scholars began to build on what they had acquired.

Thus mathematicians extended the Hindu decimal positional system of arithmetic from whole numbers to include decimal fractions, and the 12th-century Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the algebra of polynomials of Muhammad al-Khwarizmi. Al-Karaji included polynomials with an infinite number of terms. (Al-Khwarizmi’s name, incidentally, is the source of the word algorithm, and the title of one of his books is the source of the word algebra.

Geometers such as Ibrahim ibn Sinan continued Archimedes’ investigations of areas and volumes, and Kamal al-Din and others applied the theory of conic sections to solve optical problems. Using the Hindu sine function and Menelaus’s theorem, mathematicians from Habas al-Hasib to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical trigonometry. These did not become mathematical disciplines in the West, however, until the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.

Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations. The Latin West acquired much of this learning during the 12th century, the great century of translation. Together with translations of the Greek classics, these Muslim works were responsible for the growth of mathematics in the West during the late Middle Ages. Italian mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for their knowledge.

Although the late medieval period saw some fruitful mathematical considerations of problems of infinity by writers such as Nicole Oresme, it was not until the early 16th century that a truly important mathematical discovery was made in the West. The discovery, an algebraic formula for the solution of both the cubic and quartic equations, was published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The discovery drew the attention of mathematicians to complex numbers and stimulated a search for solutions to equations of degree higher than 4.

It was this search, in turn, that led to the first work on group theory (Group) at the end of the 18th century, and to the theory of equations developed by the French mathematician variste Galois in the early 19th century. The 16th century also saw the beginnings of modern algebraic symbolism (Mathematical Symbols), as well as the remarkable work on the solution of equations by the French mathematician Franois Vite. His writings influenced many mathematicians of the following century, including Pierre de Fermat in France and Isaac Newton in England. Europeans dominated in the development of mathematics after the Renaissance.

During the 17th century, the greatest advances were made in mathematics since the time of Archimedes and Apollonius. The century opened with the discovery of logarithms by the Scottish mathematician John Napier, whose continued utility prompted the French astronomer Pierre Simon Laplace to remark, almost two centuries later, that Napier, by halving the labors of astronomers, had doubled their lifetimes. (Although the logarithmic function is still important in mathematics and the sciences, logarithmic tables and their instrumental form-slide rules-are of much less practical use today because of electronic calculators.

The science of number theory, which had lain dormant since the medieval period, illustrates the 17th-century advances built on ancient learning. It was Arithmetica by Diophantus that stimulated Fermat to advance the theory of numbers greatly. His most important conjecture in the field, written in the margin of his copy of the Arithmetica, was that no solutions exist to an + bn = cn for positive integers a, b, and c when n is greater than 2. This conjecture stimulated much important work in algebra and number theory but is still unproved. Two important developments in pure geometry occurred during the century.

The first was the publication, in Discourse on Method (1637) by Descartes, of his discovery of analytic geometry, which showed how to use the algebra that had developed since the Renaissance to investigate the geometry of curves. (Fermat made the same discovery but did not publish it. ) This book, together with short treatises that had been published with it, stimulated and provided the basis for Isaac Newton’s mathematical work in the 1660s. The second development in geometry was the publication by the French engineer Grard Desargues in 1639 of his discovery of projective geometry.

Although the work was much appreciated by Descartes and the French philosopher and scientist Blaise Pascal, its eccentric terminology and the excitement of the earlier publication of analytic geometry delayed the development of its ideas until the early 19th century and the works of the French mathematician Jean Victor Poncelet. Another major step in mathematics in the 17th century was the beginning of probability theory in the correspondence of Pascal and Fermat on a problem in gambling, called the problem of points.

This unpublished work stimulated the Dutch scientist Christiaan Huygens to publish a small tract on probabilities in dice games, which was reprinted by the Swiss mathematician Jakob Bernoulli in his Art of Conjecturing. Both Bernoulli and the French mathematician Abraham De Moivre, in his Doctrine of Chances in 1718, applied the newly discovered calculus to make rapid advances in the theory, which by then had important applications in the rapidly developing insurance industry.

Without question, however, the crowning mathematical event of the 17th century was the discovery by Sir Isaac Newton, between 1664 and 1666, of differential and integral calculus (Calculus). In making this discovery, Newton built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as Descartes, Francesco Bonaventura Cavalieri, Johann van Waveren Hudde, and Gilles Personne de Roberval.

About eight years later than Newton, who had not yet published his discovery, the German Gottfried Wilhelm Leibniz rediscovered calculus and published first, in 1684 and 1686. Leibniz’s notation systems, such as dx, are used today in calculus. The remainder of the 17th century and a good part of the 18th were taken up by the work of disciples of Newton and Leibniz, who applied their ideas to solving a variety of problems in physics, astronomy, and engineering. In the course of doing so they also created new areas of mathematics.

For example, Johann and Jakob Bernoulli invented the calculus of variations, and French mathematician Gaspard Monge invented differential geometry. Also in France, Joseph Louis Lagrange gave a purely analytic treatment of mechanics in his great Analytical Mechanics (1788), in which he stated the famous Lagrange equations for a dynamical system. He contributed to differential equations and number theory as well, and he originated the theory of groups. His contemporary, Laplace, wrote the classic Celestial Mechanics (1799-1825), which earned him the title the French Newton, and The Analytic Theory of Probabilities (1812).

The greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in these areas. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas.

That is, Newton’s own accounts were based on kinematics and velocities, Leibniz’s explanation was based on infinitesimals, and Lagrange’s treatment was purely algebraic and founded on the idea of infinite series. All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not resolved until the following century. In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus.

He based his approach only on finite quantities and the idea of a limit. This solution posed another problem, however; that of a logical definition of real number. Although Cauchy’s explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W. R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. This definition is still taught, but other definitions were given at the same time by the German mathematicians Georg Cantor and Karl T.

W. Weierstrass. A further important problem, which arose out of the problem-first stated in the 18th century-of describing the motion of a vibrating string, was that of defining what is meant by function. Euler, Lagrange, and the French mathematician Jean Baptiste Fourier all contributed to the solution, but it was the German mathematician Peter G. L. Dirichlet who proposed the definition in terms of a correspondence between elements of the domain and the range. This is the definition that is found in texts today.

In addition to firming the foundations of analysis, as the techniques of the calculus were by then called, mathematicians of the 19th century made great advances in the subject. Early in the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a whole new field for analysis, one that was developed in the work of Cauchy, Weierstrass, and the German mathematician Georg F. B. Riemann. Another important advance in analysis was Fourier’s study of infinite sums in which the terms are trigonometric functions.

Known today as Fourier series, they are still powerful tools in pure and applied mathematics. In addition, the investigation of which functions could be equal to Fourier series led Cantor to the study of infinite sets and to an arithmetic of infinite numbers. Cantor’s theory, which was considered quite abstract and even attacked as a disease from which mathematics will soon recover, now forms part of the foundations of mathematics and has more recently found applications in the study of turbulent flow in fluids. A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry.

In non-Eculidean geometry, more than one parallel can be drawn to a given line through a given point not on the line. Evidently this was discovered first by Gauss, but Gauss was fearful of the controversy that might result from publication. The same results were rediscovered independently and published by the Russian mathematician Nikolay Ivanovich Lobachevsky and the Hungarian Jnos Bolyai. Non-Euclidean geometries were studied in a very general setting by Riemann with his invention of manifolds and, since the work of Einstein in the 20th century, they have also found applications in physics.

Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era. While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1. In his doctoral dissertation he gave the first satisfactory proof of the fundamental theorem of algebra.

Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying. Of more importance for algebra itself than Gauss’s proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems.

A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many, but not all, of the properties of the real numbers. Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann Gnther Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange.

Galois applied this work deeply to provide a theory of when polynomials may be solved by an algebraic formula. Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to a geometric theory of differential equations by means of continuous groups of transformations known as Lie groups.

In the 20th century, algebra has also been applied to a general form of geometry known as topology. Another subject that was transformed in the 19th century, notably by Laws of Thought (1854), by the English mathematician George Boole and by Cantor’s theory of sets, was the foundations of mathematics (Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantor’s theory. One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set ( Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising.

They left open the question, however, of whether other paradoxes might arise in these restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent. ) Particularly disturbing is the result, proved in 1931 by the American logician Kurt Gdel, that in any axiom system complicated enough to be interesting to most mathematicians, it is possible to frame propositions whose truth cannot be decided within the system.

At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at Gttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilbert’s address at Gttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century.

These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the Hilbert problems has been solved, mathematicians all over the world await the details of the story with impatience. Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (Computer).

Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. Babbage’s imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible. This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics.

It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors.

The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois. Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications.

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The Architecture of Modern Mathematics: Essays in History and Philosophy

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This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research, and how a number of historical accounts can be deepened by embracing philosophical questions.

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Advertisement

What Solar Eclipse-Gazing Has Looked Like for the Past 2 Centuries

Millions of people on Monday will continue the tradition of experiencing and capturing solar eclipses, a pursuit that has spawned a lot of unusual gear.

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By Sarah Eckinger

• April 8, 2024 Updated 12:37 p.m. ET

For centuries, people have been clamoring to glimpse solar eclipses. From astronomers with custom-built photographic equipment to groups huddled together with special glasses, this spectacle has captivated the human imagination.

Creating a Permanent Record

In 1860, Warren de la Rue captured what many sources describe as the first photograph of a total solar eclipse . He took it in Rivabellosa, Spain, with an instrument known as the Kew Photoheliograph . This combination of a telescope and camera was specifically built to photograph the sun.

Forty years later, Nevil Maskelyne, a magician and an astronomy enthusiast, filmed a total solar eclipse in North Carolina. The footage was lost, however, and only released in 2019 after it was rediscovered in the Royal Astronomical Society’s archives.

Telescopic Vision

For scientists and astronomers, eclipses provide an opportunity not only to view the moon’s umbra and gaze at the sun’s corona, but also to make observations that further their studies. Many observatories, or friendly neighbors with a telescope, also make their instruments available to the public during eclipses.

Fredrik Hjalmar Johansen, Fridtjof Nansen and Sigurd Scott Hansen observing a solar eclipse while on a polar expedition in 1894 .

Women from Wellesley College in Massachusetts and their professor tested out equipment ahead of their eclipse trip (to “catch old Sol in the act,” as the original New York Times article phrased it) to New London, Conn., in 1922.

A group from Swarthmore College in Pennsylvania traveled to Yerbaniz, Mexico, in 1923, with telescopes and a 65-foot camera to observe the sun’s corona .

Dr. J.J. Nassau, director of the Warner and Swasey Observatory at Case School of Applied Science in Cleveland, prepared to head to Douglas Hill, Maine, to study an eclipse in 1932. An entire freight car was required to transport the institution’s equipment.

Visitors viewed a solar eclipse at an observatory in Berlin in the mid-1930s.

A family set up two telescopes in Bar Harbor, Maine, in 1963. The two children placed stones on the base to help steady them.

An astronomer examined equipment for an eclipse in a desert in Mauritania in June 1973. We credit the hot climate for his choice in outfit.

Indirect Light

If you see people on Monday sprinting to your local park clutching pieces of paper, or with a cardboard box of their head, they are probably planning to reflect or project images of the solar eclipse onto a surface.

Cynthia Goulakos demonstrated a safe way to view a solar eclipse , with two pieces of cardboard to create a reflection of the shadowed sun, in Lowell, Mass., in 1970.

Another popular option is to create a pinhole camera. This woman did so in Central Park in 1963 by using a paper cup with a small hole in the bottom and a twin-lens reflex camera.

Amateur astronomers viewed a partial eclipse, projected from a telescope onto a screen, from atop the Empire State Building in 1967 .

Back in Central Park, in 1970, Irving Schwartz and his wife reflected an eclipse onto a piece of paper by holding binoculars on the edge of a garbage basket.

Children in Denver in 1979 used cardboard viewing boxes and pieces of paper with small pinholes to view projections of a partial eclipse.

A crowd gathered around a basin of water dyed with dark ink, waiting for the reflection of a solar eclipse to appear, in Hanoi, Vietnam, in 1995.

Staring at the Sun (or, How Not to Burn Your Retinas)

Eclipse-gazers have used different methods to protect their eyes throughout the years, some safer than others .

In 1927, women gathered at a window in a building in London to watch a total eclipse through smoked glass. This was popularized in France in the 1700s , but fell out of favor when physicians began writing papers on children whose vision was damaged.

Another trend was to use a strip of exposed photographic film, as seen below in Sydney, Australia, in 1948 and in Turkana, Kenya, in 1963. This method, which was even suggested by The Times in 1979 , has since been declared unsafe.

Solar eclipse glasses are a popular and safe way to view the event ( if you use models compliant with international safety standards ). Over the years there have been various styles, including these large hand-held options found in West Palm Beach, Fla., in 1979.

Parents and children watched a partial eclipse through their eclipse glasses in Tokyo in 1981.

Slimmer, more colorful options were used in Nabusimake, Colombia, in 1998.

In France in 1999.

And in Iran and England in 1999.

And the best way to see the eclipse? With family and friends at a watch party, like this one in Isalo National Park in Madagascar in 2001.

Our Coverage of the Total Solar Eclipse

Anticipation and Anxiety Build:  Across parts of the United States, Mexico and Canada, would-be eclipse-gazers are on the move for what could be a once-in-a-lifetime event .

Awaiting a Moment of Awe:  Millions of people making plans to be in the path of the solar eclipse know it will be awe-inspiring. What is that feeling ?

The Eclipse Chaser:  A retired astrophysicist known as “Mr. Eclipse” joined “The Daily” to explain why these celestial phenomena are such a wonder to experience .

Historic Photos:  From astronomers with custom-built photographic equipment to groups huddled together with special glasses, here’s what solar eclipse-gazing has looked like for the past two centuries .

Hearing the Eclipse:  A device called LightSound is being distributed to help the blind and visually impaired experience what they can’t see .

Animal Reactions: Researchers will watch if animals at zoos, homes and farms act strangely  when day quickly turns to night.