Category Archives: Greek mathematics

Differential & Integral Calculus – The Math of Change

Most will remember their first experience with calculus. From limits to derivatives, rates of changes, and integrals, it was as if the heavens had opened up and the beauty of mathematics was finally made clear. There was, in fact, more to the world than routine numerical manipulation. Numbers and symbols became the foundational building blocks with which theories could be written down, examined, and shared with others. The language of mathematics was emerging and with it a new realm of thinking. For me, calculus marked the beginning of an intellectual awakening and with it a new way of thinking. It is therefore perhaps worthy to examine the early development of our modern calculus and to provide a more concrete historical context.

The method of exhaustion. Image: Margaret Nelson, illustration for New York Times article “Take it to the Limit” by Steven Strogatz.

The distinguishing feature of our modern calculus is, undoubtedly, its unique ability to utilize the power of infinitesimals. However, this power was only realized after more than a millennium of intense mathematical debate and reformation. To the early Greek mathematicians, the notion of infinity was but a paradoxical concept lacking the geometric backing necessary to put it on a rigorous footing. It was this initial struggle to provide both a convincing and proper proof for the existence and usage of infinitesimals that led to some of the greatest mathematical development this world has ever seen. The necessity for this development is believed to be the result of early attempts to calculate difficult volumes and areas of various objects. Among the first advancements was the use of the method of exhaustion. First used by the Greek mathematician Eudoxus (c. 408-355 BC) and later refined by the Chinese mathematician Liu Hui in the 3rd century AD[1], the method of exhaustion was initially used as a means of “sandwiching” a desired value between two known values through repeated application of a given procedure. A notable application of this method was its use in estimating the true value of pi through inscribing/circumscribing a circle with higher degree n-gons.[1] With the age of Archimedes (c. 287-212) came the development of heuristics – a practical mathematical methodology not guaranteed to be optimal or perfect, but sufficient for the immediate goals.[2] Followed by advancements made by Indian mathematicians on trigonometric functions and summations (specifically work on integration), the groundwork for modern limiting analysis began to unfold and thus the relevance for infinitesimals in the mathematical world.

Isaac Newton. Image: Portrait of Isaac Newton by Sir Godfrey Kneller. Public domain.

By the turn of the 17th century, many influential mathematicians including Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis, and others had already been applying the results on infinitesimals to the study of tangent lines and differentiation.[2] However, when we think of modern calculus today the first names to come to mind are almost certainly Isaac Newton and Gottfried Leibniz. Before 1650, much of Europe was still in what historians refer to as the Hellenistic age of mathematics. Prior to the contributions of Newton and Leibniz, European mathematics was largely an “informal mass of various techniques, methods, notations, and theories.”[2] Through the creation of a more structured and algorithmic approach to mathematics, Newton and Leibniz succeeded in transforming the heart of the mathematical system itself giving rise to what we now call “the calculus.”

Both Newton and Leibniz shared the belief that the tangent could be defined as a ratio but Newton insisted that it was simply the ratio between ordinates and abscissas (the x and y coordinates respectively in the plane in regular Euclidean geometry).[2] Newton further added that the integral was merely the “sum of the ordinates for infinitesimals intervals in the abscissa” (i.e., the sum of an infinite number of rectangles).[4] From Leibniz we gain the well-known “Leibniz notation” still in use today. Leibniz denoted infinitesimal increments of abscissas and ordinates as dx and dy and the sum of infinitely many infinitesimally thin rectangles as a “long s” which today constitutes our modern integral symbol ò.[2] To Leibniz, the world was a collection of infinitesimal points and that infinitesimals were ideal quantities “less than any given quantity.”[3] Here we might draw the connection between this description and our modern use of the greek letter e (epsilon) – a fundamental tool in modern analysis in which assertions can be made by proving that a desired property is true provided we can always produce a value less than any given (usually small) epsilon.

From Newton, on the other hand, we get the groundwork for differential calculus which he developed through his theory on Fluxionary Calculus first published in his work Methodus Fluxionum.[2] Initially bothered by the use of infinitesimals in his calculations, Newton saught to avoid using them by instead forming calculations based on ratios of changes. He defined the rate of generated change as a fluxion (represented by a dotted letter) and the quantity generated as a fluent. He went on to define the derivative as the “ultimate ratio of change,” which he considered to be the ratio between evanescent increments (the ratio of fluxions) exactly at the moment in question – does this sound familiar to the instanteous rate of change? Newton is credited with saying that “the ultimate ratio is the ratio as the increments vanish into nothingness.”[2/3] The word “vanish” best reflects the idea of a value approaching zero in a limit.

The derivative of a function.

The derivative of a function.

Contrary to popular belief, Newton and Leibniz did not develop the same calculus nor did they conceive of our modern Calculus. Both aimed to create a system in which one could easily manage variable quantities but their intial approaches varied. Newton believed change was a variable quantity over time while for Leibniz change was the difference ranging over a sequence of infinitely close values.[3] The historical debate has therefore been, who invented calculus first? The current understanding is that Newton began work on what he called “the science of fluents and fluxions” no later than 1666. Leibniz on the other hand did not begin work until 1673. Between 1673 and 1677, there exists documented correspondence between Leibniz and several English scientists (as well as Newton himself) where it is believed that he may have come into contact with some of Newton’s   unpublished manuscripts.[2] However, there is no clear consensus on how heavily this may have actually influenced Leibniz’s work. Eventually both Newton and Leibniz became personally involved in the matter and in 1711 began to formally accuse each other of plagiarism.[2/3] Then in the 1820’s, following the efforts of the Analytical society, Leibnizian analytical calculus was formally accepted in England.[2] Today, both Newton and Leibniz are credited for independently developing the foundations of calculus but it is Leibniz who is credited with giving the discipline the name it has today: “calculus.”

The applications of differential and integral calculus are far reaching and cannot be overstated. From modern physics to neoclassical economics, there is hardly a discipline that does not rely on the tools of calculus. Over the course of thousands of years of mathematical development and countless instrumental players (e.g. Newton and Leibniz), we now have at our disposal some of the most advanced and beautifully simple problem solving tools the world has ever seen. What will be the next breakthrough? The next calculus? Only time will tell. What is certain is that the future of mathematics is, indeed, very bright.

Works Cited

[1]Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). “A comparison of Archimdes’ and Liu Hui’s studies of circles”. Chinese studies in the history and philosophy of science and technology 130. Springer. p. 279. ISBN 0-7923-3463-9., Chapter , p. 279

[2]“History of Calculus.” Wikipedia. Wikimedia Foundation, n.d. Web. 14 Mar. 2015.

[3]“A History of the Calculus.” Calculus History. N.p., n.d. Web. 14 Mar. 2015.

[4] Valentine, Vincent. “Editor’s Corner: Voltaire and The Man Who Knew Too Much, Que Sera, Sera, by Vincent Valentine.” Editor’s Corner: Voltaire and The Man Who Knew Too Much, Que Sera, Sera, by Vincent Valentine. ISHLT, Sept. 2014. Web. 15 Apr. 2015.

Reflections on Zeno’s Paradox —— A Problem about Geometric Series, or Not?

Our knowledge of mathematics develops along with the long history of human civilization. Ancient Greece is usually considered as the cradle of western civilization and the birthplace of mathematics. Here I will discuss the famous Zeno’s Paradox, an intellectual legacy we inherited form those great thinkers in ancient Greece, whose philosophical thinking has been energetic and attractive since ancient times; Then I will have a brief introduction about the “solving” of the paradox using geometric series; In the end I will show that, in some sense, the paradox has not been fully unraveled, by reference to another problem proposed by contemporary scholars. I believe the charm of mathematics will be presented after these efforts.

The ancient Greek philosopher Zeno once created quite a few paradoxes to show his skepticism about some common phenomena. He thought plurality and change were not a universal truth, and in particular, motion was only our illusion. Among his paradoxes that survived today, most of them have equivalent math models. So I will pick up one of them, “Achilles and the Tortoise”, to represent his logic.

The problem is like this: Achilles, the most famous Achaean warrior in Homer’s Iliad, the “swift-footed” hero, is chasing a tortoise. Suppose the initial distance between them is 100-meters, and Achilles’ speed is 10 m/s while the tortoise’s speed is 1m/s. After the chasing begins, Achilles will spend 10 seconds to finish a first the 100-meters. Then he will be at the spot where the tortoise was, at 10 seconds ago; In this period (10 seconds), the tortoise also proceeds 10 meters. Then, to finish the second distance, 10 meters, Achilles spends 1 second, while in the same period, the tortoise proceeds 1 meter; Then it goes on, every time Achilles reaches the tortoise’s previous spot, he still needs to chase more because in that period the tortoise proceeds to another further spot. Hence, Zeno concludes, in this case, Achilles will never overrun the lucky tortoise, which is a very bizarre conclusion against our common sense.

This paradox raised in history of great interest. Many scholars tried to give an answer or explanation, including Aristotle, Archimedes, Thomas Aquinas, etc. The joint efforts of philosophers and mathematicians did not succeed immediately. Without the help of rigorous mathematical tools, their solutions cannot resist questioning from skepticism. To make it more clear, philosophical thinking alone could hardly solve this problem; even if it accomplished so, to convince others to believe this will be no less difficult. Immanuel Kant in his Critique of Pure Reason mentioned that rationality is not omnipotent. It has its own structure of a priori knowledge, and after itself combined with a posterior experience, it becomes useful knowledge, which guides our cognition. However, due to the nature of human’s longing for perfection, eternal, and universality (I would like to add “infinite” here), we are inclined to abuse our rationality and expands it to areas that it in fact does not apply. This is to say, human rationality arises from very specific experience, and is applicable there; but due to our preference, we create some concepts (like “perfection”, “eternal” and “universality” I mentioned above), which is non-existent in real life and also beyond rationality’s realm. But we are so confident and accustomed to our rationality that we apply it to those concepts generated by ourselves, without noticing it is not applicable there. After the abuse, confusion subsequently follows.

I really admire Kant’s genius in his noticing that a critique of human reasoning itself is very much needed. And I would use his theory to help form my personal understanding about this problem. But I will leave it here and deal with it later, after the introduction of the rigorous mathematical proving with respect to this problem.

Thanks to the invention of calculus and the epsilon-delta language, we now have the rigorous mathematical tool to deal with problems about infinity. A brief solution is to use geometric series. With respect to the “Achilles and Tortoise” problem we mentioned above, the time that Achilles needed to catch up with the tortoise can be represented as:


This means Achilles could overrun the tortoise after approximately 11.11 seconds. Thus, the sum of a series with infinite terms, are quite possibly finite, which may be beyond our predecessors’ understanding. But, does this problem stops here? Some modern scholars believes not. Why, because we are not sure what is Zeno’s true meaning. This is to say, the result of the formula may not answer Zeno’s question. Let me here give an example, which is called Thomson’s Lamp: suppose there is such a lamp with a toggle switch. After you start the game, it’s switched one after 1 minute, then switched off after half minute, then on after fourth minute, then off after eighth minute, and so goes on. The sum all the time we spend in the game is 2 minutes, according to the same method about sum of geometric series above. Then, one question follows: After exactly two minutes, is the lamp on or off?

This time we find it’s also very difficult to answer this variation of Zeno’s paradox, even if we know geometric series. And because of this, I believe to use geometric series could give a result, but could not solve the problem about the process, which may be Zeno’s real point. And Kant’s argument gives me guidance in understanding this paradox. Also, there is a scholar making this point more explicitly: according to Pat Corvini, this paradox arises from “a subtle but fatal switch between the physical and abstract”. When we expand our mathematical abstractions to the physical world, even it’s applicable almost everywhere, with respect to some concepts, it’s quite unimaginable and confusing. This time, we may still need mathematics as well as philosophy, to finally solve this paradox.


Binmore, K. G. & Voorhoeve, A. (2003). Defending Transitivity against Zenois Paradox, Philosophy and Public Affairs, Vol.31(3), pp.272-279

John, L. (2003). Key Contemporary Concepts from Abjection to Zeno’s Paradox, Ebrary, Inc.

Wikipedia, Zeno’s paradoxes,

Parallel Lines

I learned the parallel postulate in middle school. The best known equivalent of the postulate is attributed to Scottish mathematician John Playfair, and it says that “in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.”

The reason that I have a special impression on this postulate may be probably due to a popular metaphor in my middle school period. That metaphor related the parallel lines with the mutual feelings between girls and boys: when a girl and a boy cannot stay together, or they do not develop a mutual affection, we say that they are like two parallel lines. No matter what the two parallel lines “do”, they cannot have an interaction. Similarly, for the two unlucky people, no matter what they do, they can never fall in love with each other. I have to say this metaphor describes a tragic situation and sometimes I do not feel satisfied with the “tragic” destinies of the two parallel lines. Fortunately, as my mathematical knowledge grows, I do find that in some other branches of geometry, the seemingly unbreakable law in Euclidean geometry no longer holds. Among the new branches are hyperbolic geometry and elliptic geometry, which will be the main topic of this blog.

Before we talk about non-Euclidean geometry, let me have a brief introduction to the differences between non-Euclidean geometry and Euclidean geometry. The fundamental difference between them lies in the parallel postulate. We already stated a widely adopted equivalent of parallel postulate in the beginning of this article. For two thousand years after Euclid’s work was published, many mathematicians either tried to prove this “fifth postulate” (in Euclid’s Element) or tried to show that it’s not necessarily true. Actually, even in Euclid’s own book, this parallel postulate was left unproved; Also, unlike the first four postulates, the fifth postulate — the “tragic” parallel postulate, was not being used to prove his following theorems in the book. A breakthrough in this topic came out in the 18th century. A Russian mathematician,  Nikolai Lobachevsky, developed the hyperbolic geometry. His most famous contributions are in two aspects: he convincingly showed that Euclid’s fifth postulate cannot be proved, and he presented hyperbolic geometry to the world.

Multiple parallel lines in hyperbolic geometry. Image: Vladimir0987, via Wikimedia Commons.

In the original parallel postulate, we said for any given line R and point P, there is exactly one line through P that does not intersect R; i.e., parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, rendering the parallel postulate invalid. Hyperbolic geometry may be against common sense at first glance, because usually, our recognition about the shape of a space is limited to Euclidean space. However, hyperbolic geometric space does exist, one example is the saddle space with a constant negative Gaussian Curvature. Hyperbolic space is possible in dimensions that are larger than or equal to two. It is curved — the reason why it differs from Euclidean planes — and is characterized by a constant negative curvature. Euclidean spaces are always with zero curvature. To make it more vivid in my own words (which very likely will not be so rigorous), if we observe a small region in the hyperbolic plane, it looks like just a concave plane. And when you draw a triangle in this concave plane, the sum of its inner angles is always less than 180 degrees. This is also a proved theorem in hyperbolic geometry.

In elliptic geometry, we have the following conclusion: “Given a line L and a point p outside L, there exists no line parallel to L passing through p, and all lines in elliptic geometry intersect.” This means we can never find any parallel lines in elliptic geometry. This kind of geometry together with hyperbolic geometry, perfectly form a counter example of the parallel postulate’s assumption “there is one and only one parallel line…”: in elliptic geometry, there is more than one parallel line, and in hyperbolic geometry, there are none. Examples of elliptic geometry are more common in our real life than hyperbolic geometry. One example is the surface of Earth. A line in such a space becomes a great circle (a circle centered at earth’s core). When you draw a line through point P and if P is away from line (great circle) L, the new line you get will be a new great circle, and it will always have two intersections with great circle L, because any two great circles on the surface of sphere will have two intersections.

Here we have three pictures visualizing the relationship between Euclid’s geometry, hyperbolic geometry and elliptic geometry.

Image: Joshuabowman and Pbroks, via Wikimedia Commons.

The establishment of non-Euclidean geometry is the outcome of many generations’ collective endeavors. For example, classical era’s scholar Proclus commented some attempts to prove the postulate, esp. Those attempts tried to deduce it from the previous four postulates; Arab mathematician Ibn al-Haytham in the 10th century, tried to prove the theorem by contradiction; in the Age of Enlightenment Italian mathematician Giordano Vitale and Girolamo Saccheri both contributed new approaches to this problem although they finally failed; Gauss and Nikolai Lobachevsky (we already mentioned him above) also joined the sequence — the latter finally finished this task by establishing a new geometric branch. This mansion was built over such a long time and I am fortunate to feel part of its grandeur and beauty.

So for those suitors who complain their misfortune that their dream lovers and they are like two parallel lines, I think you are too pessimistic. You can imagine yourself being in a elliptic geometric space. Then as long as you try your best, you will always have an intersection with the other line. I am not sure whether this will convince those guys and give them confidence. For me, I am now feeling happy and believe that everything is possible in our real world, just like that everything is possible in mathematics. The story about seemingly very simple parallel lines do make me feel the power and beauty of mathematics.


  5. H. S. M. Coxeter(1942) Non-Euclidean Geometry, University of Toronto Press, reissued 1998 by Mathematical Association of AmericaISBN 0-88385-522-4.
  6. Hazewinkel, Michiel, ed. (2001), “Elliptic geometry”Encyclopedia of MathematicsSpringerISBN978-1-55608-010-4
  7. Weisstein, Eric W.“Hyperbolic Geometry”MathWorld.

Archimedes’ principle

Background of Archimedes

Archimedes was born in 287 BC into a wealthy family of nobility and his hometown was a small village near to Greece. He also had a great father who was a great astronomer and mathematician. His father was very friendly to his children, thus Archimedes was greatly influenced by his father. This made him take a keen interest in mathematics, astronomy and ancient Greek geometry when he was a child. In 267 BC, Archimedes was 11 years old. At that time, his father sent him to Alexandria, Egypt and let him learn mathematics with Euclid’s student. Alexandria, located in the mouth of the Nile, was the knowledge and cultural center of the world at that time. There were also a lot of scholars and professionals in various fields. During his stay in this city, Archimedes met many mathematicians, and he learned a lot of knowledge and skills from them. This knowledge made a major impact for his scientific career and is also the basis of his science research in the future.

Achievements of Archimedes

Fig.1 Archimedes' principle

Fig.1 Archimedes’ principle. Image: Yupi666, via Wikimedia Commons.

Archimedes is considered by most great mathematicians as one of the greatest mathematicians of all time. And he had a lot of important achievements because of his early life of learning in Alexandria, Egypt. He was very good at learning, and this skill made him to find a way to solve areas, surface areas, volumes and other many geometrical objects. In addition to geometrical objects, he also had a important achievement in buoyant force. I think that Archimedes’ principle is his most important achievement. This principle told us the basic rule of buoyant force. According to Wikipedia, “the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.”[1] We also can express this principle by using a formula such that F = G (F is the buoyancy and G is the weight of the liquid that the object displaces ). Another different expression is  (ρ is the density of liquid, g is the acceleration of gravity, V is the volume of liquid).

Story of Archimedes’ principle

Fig.2 Archimedes run to palace

Fig.2 Archimedes runs to the palace. Image: Public domain, via Wikimedia Commons.

According to legend, one day, the king of Greece asked his craftsman to make a gold crown for him. However, the king suspected his crown was not made of real gold after his craftsman completed the crown. The king was afraid that his craftsman pocketed his gold. Although the weight of the crown is equal to the weight of gold that he give to the craftsman, but he could not destroy the crown and check. This question stumped the king and his chancellor. At that time, Archimedes had a very good relationship with the king of Greece and he was already very famous in Greece. After he listened to the suggestion of his minister, the king was going to invite Archimedes to test the crown. In the beginning, Archimedes also had no idea how to solve this problem. One day, he was about to bathe in his bath tub. When he got into the bath tub and saw the water spill, he suddenly had a good idea to solve the king’s problem. He thought that he could measure the displacement of a solid in the water and use this method to determine if the crown was made of real gold. And then, he excitedly jumped out from bath tub and ran to the king’s palace. He even forgot to wear clothes and he said “Eureka! Eureka!” (Eureka means “Found it!”) When he arrived at the palace, he immediately began to test the crown. He put the same weight of pure gold and crown into the two bowls that filled with full water and to compare the water of overflow. Then he found that the bowl with real crown overflowed more water than another bowl. It means that crown was made of other metals. This proves that the craftsmen deceived the king. The significance of the test is not that whether the goldsmith deceive the king, but Archimedes discovered the Archimedes’ principle.


Archimedes’ principle is a very important theory for the world. We can see that many modern inventions were made by using this theory, like big ships, submarine and so on. Thus Archimedes was really a great scientist, and he has made an indelible impact for social progress and human development.



Method of Exhaustion

When asked to find area of any given triangle, anyone will be able to solve it with the known formula for area of a triangle which is (1/2)(base)(height). Anyone can do the same with a square because the formula for the area of a square is (base)(height). This is basic geometry/algebra that students learn and is part of the curriculum. If you were asked to solve the area of a curved object, would you be able to solve for the area without using Calculus? Would you be able to find the area without using any type of formula? How can the area be solved for any object with curves, like a circle? Greek mathematicians used a technique called the method of exhaustion, which is precursor to Calculus. This method is no longer commonly used to solve problems but this method if very similar to Riemann Integration. The idea of this method is to take an approximation of an object repeatedly to end up with an area, which would otherwise be difficult to find.

The method of exhaustion was developed by Greek mathematicians in order to find the area of a shape. At the time this method was used, there was no known formula for the area of a circle. This method helped find a formula for the area of a circle that we use today. What was needed was to find an approximation for pi. They did not know the constant they needed to multiply by. The way the Greeks would use the method of approximation was to inscribe a polygon with a known area inside the circle. A square inside of a circle is how this might look like if you are trying to imagine it. The next step would be to add a side to the square, to create a pentagon inside the circle.  With this process we could solve for the area of the pentagon and then we will know some of the area of the circle. But there is still the area of outside of the pentagon and inside of the circle that is still unknown so we would then repeat the process. The next step would be to add another length to the pentagon to create a hexagon inside of the circle. The Greeks would use this process over and over again until they got the n-polygon close enough to the circle. If this process is correctly constructed, the difference in area between the n-polygon inscribed and that of the circle will become really small. This would result in a more accurate estimation of the area of the circle. The possible values for the area of the circle are systematically exhausted by the polygon.

The method of exhaustion using polygons was used to solve the area of a circle. Image: Leszek Krupinski, via Wikimedia Commons.

It is important to remember that Greeks did not have the same tools we have today. How were they able to construct such accurate polygons that were inscribed within the circles? Well the answer is that they used a straight edge and a compass. This would only work for specific polygons. “In 1837, mathematician Carl Gauss was able to prove that a n-polygon could be constructed with a compass and straight edge if and only if n is the product of a power of 2 and any number of the Fermat primes m = 22k,” wrote Chelsea E. DeSouza in The Greek Method of Exhaustion: Leading the Way to Modern Integration. The first polygons that would able to be constructed would be 3=2+1, 4=22, 5=22+1, 6= 2(2+1), 8 =23, 10=2(22+1), 12=22(3), 16=24, 24+1. Not all polygons were able to be constructed with a compass and straight edge.

Though it is a long process, it is possible to find the area of a circle with polygons inscribed within. Can the same be done with other curved shapes, like parabolas? Parabolas can be inscribed with polygons like triangles to evaluate the area of the parabola. In the third century, Archimedes came up with 24 propositions that were about parabolas. At the end of these propositions, there is a proof that says, the region inside of the parabola and a line is 4/3 the area of a triangle inscribed inside the parabola. Archimedes split up the region inside the parabolas into triangles to solve for the area.

Solving the area of a parabola without integration would look like this. Image: Public domain, via Wikimedia Commons.


Using integration to find the area under a curve could be difficult at times but when you realize what the Greeks developed to find the area, integration seems like a quicker method. The method of exhaustion is a process that took some time to solve. This method is easy to learn because it is just using the area formulas for polygons.


The Mysterious Pythagoras

               The first time I heard of the Pythagorean theorem was in my high school geometry class in ninth grade. As a young child, I never thought of it as anything else but a strange name. It was simply another formula that I needed to memorize in order to get a good grade on the upcoming test. Many of you can relate to this, as I’m sure everybody was required to learn about the theorem. Everyone has been through the “drill and kill” method where “A squared plus B squared equals C squared” can be said without effort. But there is actually a heavy history behind the name Pythagorean, and it derives from a historical influential Greek named Pythagoras. Surprisingly, the life of Pythagoras can be seen as a legend or myth with ambiguity. As I studied and read about him, I was amazed at how influential he was and the people that he moved with his teachings. Every country he visited, he obtained followers who were inspired and motivated to better themselves and contribute to mathematics, philosophy, and music. Truly it is unquestionable why his name has been discreetly implanted into mathematics and our modern day culture.

Sculpture of Pythagoras in the Capitoline Museums, in Rome. Image: Galilea, via Wikipedia

Sculpture of Pythagoras in the Capitoline Museums, in Rome. Image: Galilea, via Wikipedia

             There is great controversy among historians about the origins and early life of Pythagoras. All scholars can agree that he was born in the island of Samos. Aristoxenus, a Greek philosopher and pupil of Aristotle, stated that Pythagoras lived in Samos for about 40 years and then left during the reign of Polycrates. As we trace it back, it could be estimated that he was born around the year 570 BC what is today Italy in what was then a part of Greece At around 530 BC, he emigrated to Croton and spent the rest of his life there until his death at around 510 BC. Throughout his early life, he had the opportunity to travel and learn from many different cultures and obtained their knowledge. He traveled to Egypt where he learned about their religious practices and geometry. He also visited the Phoenicians, the Chaldeans, and the Magians to learn about arithmetic, astronomy, and religious practices. Along with knowledge about religion, he had the privilege to be taught ethics by a Delphic priestess named Themistoclea. Unquestionably, Pythagoras was a revered individual who knew how to communicate with others and connect with them. The talent and knowledge he obtained throughout his life had influenced many and created an exclusive secret society called the Pythagoreans. This society made Pythagoras a God-like figure and they strictly practiced his teachings. They consisted of a group of individuals who studied philosophy and were serious mathematicians and scientists.

Pythagoras in the center teaching music. Image from “The School of Athens” by Raphael, via Wikipedia.

Great Teacher or Renowned Mathematician?
             Surprisingly, there is no clear record of Pythagoras being a mathematician and his famous theorem being proved by him. I mentioned previously that he obtained knowledge in mathematics, but there is actually no clear record of his works contributing to geometry. Although this is the case, Aristotle’s pupil Eudemus wrote a history of geometry in the fourth century. He did not mention anything about the Pythagorean theorem, but he did note that Pythagoras played a role in the use of geometry in education for people who were considered freemen and not slaves. Proclus, who at the time preserved and commentated the works of Plato and Euclid, later rebuked Eudemus’ proposal that Pythagoras contributed to mathematics. So when it came down to it, Pythagoras was not known as a geometer during the time of Plato or Aristotle. But because of his influence and teaching towards the Pythagoreans, it is likely that they attributed their works to him. It may even be possible that his followers gave homage to his name when the Pythagorean theorem was founded. There are unfortunately many more claims and theories about Pythagoras that can either support him or not support him as a mathematician. Because of all of these different opinions and perspectives, it is difficult to say whether or not he actually contributed anything to geometry. But it is safe to say that he was a highly religious person and taught a particular way of life. He believed that the soul was immortal and went through a series of reincarnations. He taught a peculiar strict way of life that emphasized on dietary restrictions, religious ritual, and rigorous self-discipline. His expertise in religious ritual gave notice to many and he was recognized as someone who could be at two different places at the same time. Unquestionably, because of his acquired education and communication skills, he was popular among his peers and followers.

            There is so much more that can be discussed about Pythagoras and the more I read about him, the more fascinated I become. It’s amazing to me how one individual can be so popular that no one really knows his true origins or who he was exactly. Clearly he has made himself exalted in the form of his name among mathematics and other fields of study. I never thought that something that I used in almost all of my math classes would be connected with so much history and mystery. I feel even more impressed by the Pythagorean theorem as I researched deeper about Pythagoras and his background. There is definitely more information that can be gathered about him and I’ve only shared just a small portion of what he actually is. For further reading, I recommend looking into his secret organization and their discoveries. In closing, my take home message is to encourage everyone to look deeper into the math formulas and theorems taught in our education. It will definitely give deeper insight and provide meaning to them to the point where we can all appreciate frameworks of our mathematical world.


“Euclid – Master of us all”

Statue of Euclid in the Oxford Museum of Natural History, Courtesy of Lawrence OP on Flickr.

Statue of Euclid in the Oxford Museum of Natural
History, Courtesy of Lawrence OP on Flickr.

After talking about Euclidean and non-Euclidean geometry in class, I wanted to know more about Euclid and his life. I went to a couple different sources, and found an awesome biography from the MacTutor History of Mathmatics archive that gave me a lot of interesting information.

Euclid of Alexandria was born around 325 BC, but we don’t know exactly where he was born. We do know, however, that he died about 265 BC in Alexandria, Egypt. We refer to him as Euclid of Alexandria to avoid confusion with Euclid of Megara, who lived nearly 100 years earlier and was a student of Socrates. (Wikipedia, 2014) We don’t know a whole lot about Euclid’s life. One source of information that exists (although not believed to be very credible) comes from some Arabian authors who claim that Euclid was the son of Naucrates and that he was born in a city called Tyre (present day Lebanon). But, as mentioned, most mathematical historians believe that the claim was invented by the authors. (O’Connor & Robertson, 1999)

Math historians have varying opinions as to the existence of Euclid the mathematician. The most widely held is that Euclid actually was a real person, and that he really did write “The Elements” and the other works published in his name. The second idea is that he was a leader of a team of mathematicians in Alexandria that all contributed in writing the works attributed to Euclid. Some think that this team even continued to publish “The Complete Works of Euclid” in Euclid’s name after his death. The third hypothesis is that Euclid of Alexandria was a creation of this team of mathematicians in Alexandria who used the name Euclid, having derived it from Euclid of Megara. (O’Connor & Robertson, 1999) There exists a great deal of evidence suggesting that Euclid, whether that be an individual person or a team of mathematicians, founded a prestigious mathematics school in Alexandria.

The authors of the article, JJ O’Connor and EF Robertson, point out that while the third hypothesis is unlikely, we see the example of Bourbaki in the 20th century. However, the members were renowned mathematicians in their own right. If “Euclid” was a secret team of competent mathematicians, we don’t know who they were.

I find the argument that Euclid led a team of mathematicians most convincing. I believe Euclid was a person because of the stories and history associated with the man, but I believe that the work he performed was too much for any one man to produce without a team behind him. The Elements became a textbook that was used for centuries after his death, and I just think it’s unlikely that a work that comprehensive is something that came from just one person.

Other pieces of evidence that lead me to believe in the idea that Euclid was an actual person was the different accounts of Euclid’s relationship with others, especially mathematicians. Pappus, known as “the last of the great Greek geometers”, (O’Connor & Roberton, Pappus of Alexandria, 1999) said that Euclid was “… most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself.” The fact that Pappus gave such a specific description makes it seem unlikely that Euclid was merely a creation of other mathematicians of that time period. One other story that the authors recounted was originally told by Stobaeus, who was a compiler of works from many ancient Greek authors (Wikipedia, 2014). He said, “…someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid “What shall I get by learning these things?” Euclid called his slave and said “Give him three pence since he must make gain out of what he learns.” My favorite story, and one that I had actually heard before in several other circles, is that of the interaction between Euclid and Ptolemy. Proclus says, “they say that Ptolemy once asked him if there were a shorter way to study geometry than the Elements, to which he replied that there was no royal road to geometry.” I loved that. I’ve often heard that same phrase applied to other subjects, athletics, and religious pursuits. Indeed, there is no “royal road” to anything that is of worth.

The Elements, like we’ve discussed in class, is Euclid’s greatest claim to fame. In the article, Robertson and O’Connor quote Sir Thomas Heath, who was responsible for translating the works of many ancient Greek authors, including Euclid to English. Heath said, “This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time. … Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency…” (O’Connor & Robertson, Euclid of Alexandria, 1999) Indeed, Euclid’s name will always be reverenced as one of the formative thinkers in all mathematics for his work on one of the greatest textbooks ever produced by men. However, I maintain my belief that The Elements was not the work of one man, but the work of many. I also believe that it should in no way detract from the respect and praise given to Euclid, the individual, as one of the greatest mathematicians of all time.

The Beauty of The Elements


A stone statue by Joseph Durham depicting the famous mathematician Euclid. Image: Garrett Coakley via flickr.

When I was in high school, I eventually learned about the mathematical subject known as geometry. Unlike most schools though, instead of our teacher having us sit down and listen to them talk about the subject, our teacher had each and every one of us go to the library and rent a copy of Euclid’s Elements (Book 1). From that point on till the end of the first semester, each day we would separately read from Euclid’s Elements and then try to prove to our teacher each and every postulate using Euclid’s methods. It wasn’t until recently that I discovered that most children do not learn about geometry in this fashion and how unique of an experience I had. While I can see some of the possible advantages behind the new ways people learn about geometry, I still believe that Euclid’s The Elements has its own advantages that some of these other sources don’t.

One of the most noticeable things about The Elements is that each and every one of Euclid’s postulates build exceptionally well off of each other. While I see proofs building off of each other in most other texts books, there is just something about the way it is done in The Elements that feels much smoother. Perhaps the big advantage with a book like The Elements is that it was never meant to be a “text” book but rather a book for people who are interested in learning about geometry. Because of this, it doesn’t have to continually throw out real world examples or ask the reader to try to use this proof in specific scenarios. Instead, The Elements will just make a statements, go about proving that statement, and then go straight into making another statement and most likely prove it using the previously proven statement.

Another difference between The Elements and other geometry books which I believe makes it far superior is the general way in which it goes about solving proofs. Nowadays, most geometry books will use a popular form of algebra and a number system to solve equations. However, Euclid’s Elements is fully self-contained and takes nothing for granted. Because this book was created in a time where people didn’t necessarily have access to other sources, everything that is necessary to understand what is being stated in this book is there; including its own algebraic system. This self-contained version of algebra within The Elements uses simple comparisons between lines and shapes to each other which replaces constants and variables found in other forms of algebra (which is also explained in the book) to prove that the different statements that are being made are true. These comparisons in combination with previously proven statements allows The Elements to create proofs of all different kinds. While the algebraic like system Euclid’s Elements uses to solve equations may be a little difficult to get one’s mind around sometimes it makes the proofs within its pages much more difficult to refute than other geometry books.

So, why do we not use this book to teach students about geometry today? Perhaps the biggest reason and most obvious is that The Elements is a difficult book to read. Unlike most textbooks today, it doesn’t use numbers and doesn’t give examples. However, just because current day geometry books are easier to teach with and easier for students to understand does not mean that they are better books. Perhaps the final reason that I believe The Elements is such a great geometry book compared to others is that the reader must want to learn about geometry if they wish to get anywhere in Euclid’s Elements.  But, if they are able to get through Euclid’s Elements, they will have a much stronger fundamental idea of geometry than from other textbooks. While it is easy to state the fact that someone who survives being stranded in the wilderness will have a better idea of how to survive in the wild than someone who hasn’t, it doesn’t change the fact that it is true.

Going back to my classroom experience, I thoroughly enjoyed going through the proofs in The Elements and I would spend most of my lunch time going to my teacher and proofing more of Euclid’s Statements. After about 2 weeks of starting the book, I had finished it. After that point, I spent the rest of time in class helping other students understand The Elements. Unfortunately, most of the other students had a hard time getting through that semester and only a few other students were able to understand it in a similar fashion as myself. However, those among us who did understand Euclid’s The Elements had no troubles passing the second semester of class which was going back to the more common form of geometry. In conclusion, I believe that Euclid’s Elements is a fantastic book that does more for geometry than any other book out there and, if someone is really interested in geometry, they should do their best to read through and understand The Elements if they want the best foundation in geometry they can have.




Transcendental Numbers: Beyond Algebra

The history of mathematics has been fraught with disappointments for mathematicians. This is particularly true in regard to the expected and continued failure of numbers, and math in general, to be pure and graceful. The Pythagoreans, in 6th century BCE Greece, venerated the whole numbers with an almost religious devotion because of their purity, and believed that the universe could be described by using only whole numbers. Unfortunately, math is not as pure as the Pythagoreans thought, which was revealed first by Hippasus of Metapontum when he discovered an undeniable proof for the existence of irrational numbers. Incidentally, Pythagoras had poor Hippasus drowned because of this. (The tale of the drowning of Hippasus may be merely a legend, like much of what is “known” about the Pythagoreans, due to a lack of reliable sources from the period.) Another impurity of numbers was wrestled with for millennia in the form of the square roots of negative numbers, a problem that was only put to rest with the advent of the imaginary number i. However, numbers turned out to be even weirder than previously imagined, because transcendental numbers were discovered by Gottfried Wilhelm Leibniz, in 1682.

In order to understand transcendental numbers, we need to understand algebraic numbers, or numbers that are not transcendental. An algebraic number is any number that is the solution to a polynomial with rational coefficients. Rational numbers are numbers that can be written as the ratio of two integers. All rational numbers are algebraic numbers, for instance the number 2 is a rational number because it can be written as 2/1. It is also an algebraic number because it is the root of the polynomial X – 2 = 0, which is a polynomial with only rational coefficients. While all rational numbers are algebraic, not all are algebraic numbers are rational, for example, √ (2) is an irrational number, but it is also algebraic because it is the solution to X² – 2 = 0. Strangely, the imaginary number i, although it is not real, is an algebraic number since it is the root of the polynomial X² + 1 = 0.

Transcendental numbers are numbers that cannot be written as the root to a polynomial with rational coefficients. All transcendental numbers are irrational. Leibniz coined the term “transcendental” in his 1682 paper in which he proved that the sin function is not an algebraic function.  Leonhard Euler (1707- 1783) was the first to generally define transcendental numbers in the modern sense, although it was Joseph Liouville, in 1844, who definitively proved the existence of the first transcendental number. That number is now called the Liouville Constant, and it is .110001000… with a 1 in every n! place after the decimal. The Liouville Constant was specifically constructed by Liouville to be a transcendental number. However, Charles Hermite first identified a transcendental number that was not created for that purpose in 1873. That number was the constant e, or Euler’s number, and is the base of the natural logarithm.

A famous transcendental number, called “Champernowne’s Number,” was discovered in 1933 and named after David G. Champernowne. It is formed by concatenating all the natural numbers behind the decimal point 0.12345678910…. Although, easily the most famous transcendental number is pi, which was proved to be transcendental by Ferdinand von Lindemann in 1882.

Pi. Image: Travis Morgan, via Flickr.

Pi. Image: Travis Morgan, via Flickr.

Georg Cantor, in the1870’s, proved that there are as many transcendental numbers as real numbers, a concept that is mind-boggling since the real numbers are uncountable. However, only a few numbers have ever been definitively proven to be transcendental, because it is extremely difficult to prove that any given number is transcendental.

Along with irrational and imaginary numbers, transcendental numbers have challenged and frustrated mathematicians throughout the ages. Undoubtedly, Pythagoras would be horrified by transcendental numbers, or maybe he would just drown anyone who tried to tell him about them. Today, however, transcendental numbers are embraced by mathematicians as a deep and important part of math.


The conchoid of Nicomedes and other engineering geometries

Before I became a Math major, I was an Engineering major, and the things that interested me the most in engineering were ‘non standard’ ways of generating power. Solar power is a classical example of ‘non standard’. Another thing that is really interesting is that ancient geometric principles are being used to generate that power. Though not all of those ways of generating power are considered power generation to most people; some are like the use of the conchoid Nicomedes to develop a heliostat to help generate solar power, and others are like the downtown library in Salt Lake City which uses an understanding of geometry and physics to heat and cool a massive structure.

The conchoid of Nicomedes can be thought of as a curve r=b*secant θ in polar coordinates.  This family of curves was discovered by Nicomedes, who was an ancient Greek mathematician. Wolfram/Alpha describes the conchoid as “…the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family for 0<a/b<1, a/b=1, and a/b>1. (For a=0, it obviously degenerates to a circle.)


ConchoidofNicomedesCurves The Conchoid of Nicomedes ¹

The conchoid of Nicomedes was popular during the 17th century, at least with mathematicians. Currently it is going through a slight revival in popularity, at least with heliostat designers.  For those of you wondering what a heliostat is, heliostat means stationary sun. It is used to project the suns rays wherever you might want them to be pointing, such as a solar energy collector.

An example of the conchoid of Nicomedes being used as a solar concentrator.²

How this works is that there are a series of gimbaled mounts (6) for the mirrors (3) on the poles holding them up and the motion of a single bar (5) is enough to get the whole array to face a new direction. The mirrors will focus the rays of the sun onto the solar collector (14)  and off you go with all that nice fresh solar energy. The solar collector is at the apex of the focus point of the conchoid and all the mirrors are following the path of the line of the conchoid, or put another way each mirror is tangential to a corresponding point on the curve of the conchoid. It really is an interesting way to use this type of math to harness power.

File:Salt Lake City Public Library -IMG 1754.JPG The downtown Library in SLC ³

The Salt Lake City library (downtown) also harnesses the power of the sun but most people don’t seem to realize it. The entire southwest side of the library is shaped like an enormous lens and if you can go in and look at what it is, it really is amazing. That lens captures the warmth of the sun and is used for both heating the entire structure, but also is used for cooling in the summer time. During the winter its operation is pretty basic: capture was much heat as possible. This heating can be calculated by summing the total energy captured by the area of the glass and dividing that total by the total area covered in glass to determine the energy captured by the windows. This equation is true for any window but the shape of the lens really helps out, because of the magnification equation.5 This equation uses the ratio of image and object distance to determine magnification. In this case an image is not being attempted to be made, but the magnification is still able to be used to magnify the available heat.

During the summer the true genius of this piece of geometrical engineering happens.  There are vents that run parallel to the surface of the water (they are dark brown). Those vents open and a corresponding set of vents on the roof line also open. The heat generated by the “lens” then escapes through the roof vents and creates a lower pressure inside the lens that the air from the lower vents rushes to fill. The air from the lower vents has been cooled by passing over the reflecting pool, and more air is pulled from the library creating a natural gentle breeze. This whole set up uses the heat of the sun to cool a huge space, all using ancient geometric principles.