# Ancient Math

### From Math2033

I and a few fellow history enthusiasts decided that not enough research had been done into the field of ancient civilizations and the role of mathematics in them. Miles Bryant, Zach Gallestum, Armistead Freeman, and myself(Danny Gadeke) set off on a journey with the set goal to create a wiki detailing a few of our favorite ancient societies.

We met in the library several times deciding how far back in history we wanted to delve and which societies we wanted to historically chronicle the role of math in. We decided on Egypt, Greece, Romans, and Mesopotamia. I took the Egyptians, Zach took the Romans, Miles took the Mesopotamians, and Armistead took the Greeks. We created a wiki off site but it is divided into separate pages with each one containing information about the specific civilizations. I hope you enjoy our wiki site and hope it is as much fun to read as it was to create!

**Mathematics & Antiquity**
Welcome to a wiki devoted to the mathematics of yesteryear, very yesteryear. This wiki discusses and explores the ancients and their understanding and contributions to the world of math. In the 21st century, we often associate complex mathematics with modern day civilization and technology. However, we only arrived to our level of math with the help of those before us, thousands of years before us to be more precise. This wiki not only discusses the mathematics of the ancients, but also attempts to show how their understanding of math has had lasting impacts on modern day civilization.

In this far out world that we live in, the history of mathematics took a two paths. The first path was counting, and the second path shapes and space. Counting lead to numbers and shapes and space lead to Geometry.

In the numbers category whole numbers lead to fractions, and fractions lead to algebraic and more exotic numbers. Through this wing of mathematics we can learn about number theory and prime numbers.

Geometry, on the other hand studies regular polyhedra and other polyhedra, high dimensions, tilings, curves and spheres.

Mathematics gives rise to the quesiton of infinity. Infinity can be shown by the story of Achelese and the Tortus, where Achelese must do infinate things to past the Tortus, and it would take a finite amount of time.

As time ticked on things began to become more modern and the ancients were no longer. With modernization we see the rise of Algebra, which is an abstraction of an abstraction. For instance, we put an X instead of a number. The study of algebra gave rise to group theory. Our main man A-Kitabal-mukhtasar invinted algebra.

The next step in the history of mathematics was Descartes. Descartes's theory is best discribed by imagining a window pane with a spider on it. On a window pane, we can track the spiders movements. This was groundbreaking, because we can use Algebra and arithmatic to study geometry, and infinity was no longer the infinitly big, but also the infinitly small.

Descartes was important not only because he was groundbreaking, but also because of when his ideas lead to, that is the invention of Calculus. There is some debate as to who was the true invinter of Calculus. It was either Liebniz or Newton. Newton, however, is credited for it. Calculus can be thought of as the flowing of a river and an object becing carried by that river. In other words, Calculus is the study of moving objects, and it utilized infinate objects.

Questions arose such as does the Blancmange have a slope apart from a few corners?

The triumph of Calculus belongs to Fourier. Fourier studies the travel of heat, and examines questions such as, I have a line and a point, how many lines can I draw through the point that never cross the first line?

Hyperbolic Geometric dragons arose. They gave birth to a set theory. This leads to the quest for the foundations of mathematics, which leads to the fall of arithmatic and the birth of computers.

Mathematics is an abstraction from the world, and a further abstraction was finding hidden structure and rules.

NOTE: This web page is the translation of a lecture given by Professor Edmund Harris at the University of Arkansas on Ancient Mathematics in the fall of 2010. The notes of the lecture from which this page has been created belong to an irresponsible undergraduate who really has no idea what is going on.Long ago in ancient Iraq there exist a civilization known as Mesopotamia. It lay in between the mighty Tigres and Euphrates rivers, and it was rulled by Sumerians and Akkadians between 3000 B.C. and 2000 B.C. Archaeologists have worked and unearthed thousands of clay tablets marked with cuneiform that show us today that the people of Mesopotamia had a complex counting system. The system was a sexagesimal (base 60) counting system, had its own notation that allowed practice of arithematic, algebra, and geometry (Robyn Whilby and Paul Scott. 2003. "Babylonian Mathematics." Ebsco Publishing.). The counting system was essential to Messopotamian culture not for the sake of mathematics, but for the sake of culture.

**Mesopotamian Mathematics**
Mathematics in ancient Mesopotamia contributed to one's status. With the emergence of an irrigation system demographics exploded and social structure emerged. A society developed around the temple, which was the center of Mesopotamian society. As any complex society, Mesopotamian society needed an accounting system to keep track of grain, fish, dates ect. The accounting system that emerged began with small tokens of burnt clay spheres, cones, discs, cylinders ect. Tokens were originally placed into spherical clay containers, and the containers called "bullae" and the bulae's surface was marked according to what it contained. However, it was discovered that rather than placing tokens in the bullae and marking the surface, a clay tablet would do just as well. To increase variety, clay tablets were marked various ways. The people responsible for this accounting held a high social status (Jens Hoyrup. 2007. "the Roles of Mesopotamian bronze age mathematics tool for state formation and administration-carrier of teachers' professional intellectual autonomy." Springer Science + Business Media B.V.).
The Egyptians

**Overview**
Egypt is located in the northeast corner of Africa and is one of the oldest civilizations with origins beginning in 5000 B.C. Egypt's ability to survive in the desert is the due to the reliance on the great Nile River and the Mediterranean Sea. The Nile's flood pattern following almost a systematic schedule allowed the Egyptians to cultivate a thriving society in a time when many civilizations could not survive. The Egyptians are surrounded by deserts which made it a set of formidable obstacles for opposing invading forces to cross.

The leader of the Egyptian state was called the Pharaoh or "Great House". Pharaoh's were thought to be divine were bound by religious beliefs in regards to his enforcement of his policies. The Egyptians were a polytheistic religious society and the most famous of these gods was Amon-Re or the god of wind, fertility, and secrets. Egyptians believed that their gods has to be pleased through sacrificial offerings and they were worshiped in large cult temples. The Egyptians are most known for the burial rite of mummification or the removal and preservation of the internal organs for the after life. This is one of the most common rituals associated with ancient Egypt due to their ornate sarcophagus's. Mathematics in Egypt The Egyptians were some the earliest and best mathematicians the world had ever seen. As early as 4800 B.C. they had developed a yearly calendar, but in 4200 B.C. they developed a calendar that was 365 days long (12 months 30 days each and 5 feast days). The Egyptians out of need to maintain and organize their empire over time developed a counting system called counting glyphs that allowed for the addition,subtraction, multiplication, and division of numbers. All of this provided a systematic mechanism for collecting taxes, censuses, and maintaining a larger army.

1 =vertical stroke 10=heal bone 100=a snare 1,000=lotus flower 10,000=a bent finger 100,000=a burbot fish 1,000,000=a kneeling figure

Fractions first appear in Egypt in the year 3900 B.C. An Egyptian fraction is an expression of the sum of unit fractions and a Egyptian number is any number equal which can be expressed as the sum of an integer plus the sum of an Egyptian fraction. Fractions were first used to determine the amount of land conquered during war or land lost during famine. Mathematical Achievements The crowning mathematical achievement for ancient Egypt is the construction and correct alignment of the pyramids of Giza. They are aligned to several constellations and are off by just fractions of an inch. They are the burial chambers for several father and son pharaohs that was needed to assert themselves in the realm of the gods.

The Egyptians are accredited with the invention of Geometry and commonly used in the empire. Heroduts, the great Greek historian tells of a implication of Geometry by Rameses II ' divided the land into lots and gave a square piece of equal size, from the produce of which he exacted an annual tax. [If] any man's holding was damaged by the encroachment of the river ... [The Pharoah] ... would send inspectors [and surveyors] to measure the extent of the loss, in order that he pay in future a fair proportion of the tax at which his property had been assessed. Perhaps this was the way in which geometry was invented, and passed afterwards in to Greece."

**Important Thinkers**
Ahmes wrote the Rhind Mathematical Papyrus which is an ancient mathematical work that contains hundreds of algebraic and geometry problem written around 1650 B.C. He is the earliest mathematician to be known by name.
Plolemy was a Greek living Egypt in a time of Roman rule between 100-170 A.D. He is most works are the Mathematical Treatise, The Great Treatise,and Apostelmatika. He is known for hist astronomy in the Arab world and he estimated the Sun was at an average distance of 1210 Earth radii while the radius of the sphere of the fixed stars was 20,000 times the radius of the Earth.
Sources
http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt.html
http://www.math.buffalo.edu/mad/Ancient-Africa/egyptian-fractions.html
http://www.math.tamu.edu/~dallen/history/egypt/node2.html
[glyphs]
http://www.recoveredscience.com/const130egymathcontributions.htm
http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_mathematics.html
http://commons.wikimedia.org/wiki/File:Pyramids_of_Egypt1.jpg

**The Greeks**

**Overview**
Greece is a country located in southeastern Europe on the Balkan Peninsula. Math is old as time naturally some of the oldest civilizations known to man had huge contributions to the field of Mathematics, and the Greeks had their fair share of contributions. Some of the more famous Greek mathematicians include, Euclid (323-283BC), Archimedes (287-212BC), & Pythagoras (570-495BC); a few of their approaches to math are still used to this day.The Greeks were probably the most decisive in the founding of mathematics as we know it. Also there were two separate periods of Greek mathematics the first is the Classical (300-600BC) and the second is the Hellenistic (300BC-300AD). Math was very important to the Greeks because the they were seekers of knowledge. They used math in a lot of areas of their everyday lives including; building structures, constructing roadways and even in battle.

**Important Thinkers & Their Contributions**
Euclid is often considered to be the Father of Geometry. His work titled Elements has been one of the most important documents in the history of mathematics, especially regarding geometry. In his work Elements; Euclid came up with the principles that are now called Euclidean geometry, which is the earliest known systematic discussion of geometry. His earliest method was assuming a small amount of axioms and deducing many other theorems from these.Euclid was the first thinker to show how these propositions could fit into a logical proof. As stated earlier his Elements writings is one of the most influential documents om mathematics to date. Some of his teachings are still used to this day. Euclid was also the first known mathematician to use the Method of Exhaustion; which is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large.
Pythagoras was a Greek thinker that was involved in many forms of teaching and educating; one of his main focuses was on mathematics. Of coarse one his greatest achievements in mathematics is the Pythagorean Theorem. In this theorem he proves "that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a2 + b2 = c2," (Riedweg 2005). He also found a way to link music with mathematics with the Pythagorean Tuning. This is a system of musical tuning where the frequency of relationships of all intervals are based on the ratio 3:2. This method of tuning is based on a stack of intervals, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is considered to yield the same note. Starting from D for example (D-based tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down. The picture below illustrates the Pythagorean Theorem.

Archimedes is known for his works in Algebra and Geometry. He was a student at the Euclidian School after the death of Euclid. Archimedes is also known for a few of the inventions that he created and some of them are involved with mathematics, for example the Archimedes Screw which was used to raise water efficiently. Like Euclid before him; Archimedes use the Method of Exhaustion. He used this method as a way to figure out the area inside a circle by filling the circle with a polygon of greater number of sides. he quotient formed by the area of this polygon divided by the square of the circle radius can be made arbitrarily close to π as the number of polygon sides becomes large, proving that the area inside the circle of radius r is πr2, pi being defined as the ratio of the circumference to the diameter. He used this method for many other things as well.

**Influence On Today**
Features of all the great mathematicians mentioned above works are still used in one form of another.Euclidean geometry is still used in some aspects to this day. When a person surveys a road using a level to see how level a road is. Also Sphere Packing could be seen as a direct result of Euclidean geometry. Grocers use sphere packing when packing oranges into crates to avoid concerns about non-overlapping identical spheres filling a space. Pythagorean tuning is also still used to this day. Gothic Voices an English based vocal group used it when recording their album Music For The Lion-Hearted King in 1989. This method is also used in chamber music. Students are still taught the Pythagorean Theorem to this day; however it rarely has any real world applications these days. Some of Archimedes work is also still used to this day. Although he did not invent the lever in his work On the Equilibrium of Planes, he explains how the lever works and gives detailed application of how to use the lever. The lever is possibly one of the most influential inventions in history and without Archimedes explanation of the device it might not have been. Archimedes also developed the first pulley system that was used to move objects that were too heavy to move. These three great minds and many other ancient Greek thinkers had a huge influence on mathematics and without them there would still be great mysteries that about mathematics that would still need to be solved. Their teachings have stood the test of time and will continue to do so, possibly to influence the next great mathematical thinker.
http://math2033.uark.edu/wiki/index.php/File:Greek_1.jpg
http://math2033.uark.edu/wiki/index.php/File:Greek_2.png

**Roman Mathematics**

**Sons of the Wolf**
Traditionally founded in 753 BC, Rome would become one of the most enduring and long lasting civilizations in the Mediterranean world. From humble beginnings to the massive empire which spanned nearly a quarter of the world's population, the Romans had a good deal of help along the way with mathematics. Although it may seem unconventional or unorthodox, looking at the success of the Romans through mathematics may help to shed light on how and why they were so successful, along with understanding why and how mathematics plays a vital role in any civilization, even if it's inhabitants don't realize it! Often, the only credit the Romans get in the realm of mathematics is with killing Archimedes! Even Roman numerals are considered inferior to other ancient counting systems, and the lack of zero doesn't help their reputation either. However, consider the size and organization of the Roman republic and empire. The Roman world was vast, with hundreds of thousands of people carrying out daily transactions and building marvels of both the ancient and modern world. Considering the aforementioned, it is hard to argue that the Romans didn't have an understanding and appreciation of mathematics.

War, Math, and Weapons..The Perfect Combination
When one mentions the Romans, more often than not, you might dream up images of armored legionnaires arrayed in perfect order. Movies such as Gladiator, Ben Hur and Spartacus will more than likely find their way into your picturing or understanding of the Romans and their culture. However, you might not appreciate the mathematical concepts and applications behind the Roman way of war, which gave the Romans more than a few successes on the battlefield.

**A Deadly Checkerboard**
To the Romans, efficiency was everything in battle. Years of constant warfare had taught the Romans to be flexible in war and avoid rigid tactics. Years before a change in strategy, the Romans had often fought in closely arrayed lines of tightly packed soldiers, ready to clash with the enemy. However, the Romans had come to realize that a radical change was needed in how they fought. The answer to the Roman's problem came in the form of a new military unit called the Maniple. It is here where mathematics comes into the picture. A Maniple was a military unit consisting of 120 soldiers. Historically, three lines of infantry were used in a Roman legion. The Hastati, the first line of soldiers, were the younger and inexperienced men. Second, the Principiis, were older and more experienced troops. The third and final line, the Triarii, were the toughest and most experienced troops in the legion. So important were the Triarii that a popular phrase in Roman society was to say "It has come to the Triarii" to describe a particularly dire situation. Each of these lines consisted of several Maniples. The mathematical concept behind Maniples lies mostly in numbers and division. With a legion consisting of approximately 5,000 soldiers, it could be stressful and maddening to keep such a large amount of men together and to effectively deploy them. The Maniple was the perfect solution. Maniples allowed commanders to grasp the preparedness and tactics of their legion much more easily and effectively. Thanks to mathematics, a Roman commander could easily identify his strength and capability by means of Manipular organization. Below is a diagram describing how Maniples were so successful for the Romans, the use of Maniples and mathematics in Roman warfare was extremely organized. Trying to command a disorderly mob of 5,000 men could lead to chaos and disaster, but thanks to mathematical organization, the Romans were able to dominate their opponents in battle.

**The Ancient Roman Counting System**
But how did the Romans count? Most famously, and still in use today, Roman numerals were the method by which Romans denoted counting. Roman numerals have had a profound impact on modern society. Evidence of their enduring legacy can be seen on the faces of clocks, for instance, in which Roman numerals are marked from one through twelve, or I to XII. At first glance, the need for a counting system might not be seen as something that would be so important. The Romans built ancient wonders, buildings that could be heated and have running water, along with extensive roads and highways that are still in use today! Along with engineering, the Romans were known for taking a complete and thorough census of their population, which required more than just shouting "HERE!" in the general assembly. It is because of there amazing use of counting and the Roman numeral system that the Romans ever achieved such amazing results and thus deserves a closer look.

**The Abacus....Texas Instruments, Eat Your Heart Out**
Long before Texas Instruments developed the engineer students favorite tool, there was a little relatively obscure device called an abacus. The abacus was, for all intents and purposes, a hand held device f

The portable Roman abacus, like the one above, can fit in a modern day shirt pocket or Romans to count and keep records with; an ancient hand held calculator of sorts for the Roman engineer or businessmen.

Amazingly, the complex trade records, engineering calculations, and census records were done using only Roman numerals and abaci to record the results and keep track of thousands, if not hundreds of thousands, of transactions. The Romans made extensive use of accounting methods with these seemingly simplistic counting boards like the one pictured on the left. Considering the technical marvels and engineering wonders such as highways and boulevards, it is amazing that this small counting board was what helped many a Roman engineer get the job done!

**What about fractions?**
The Romans did indeed calculate fractions as not everything in their computations involved perfectly whole numbers. Although it may seem complicated to calculate fractions with Roman numerals, the Romans were innovators and found a way to remedy the problem. In order to calculate fractions, Romans simply wrote out, in their written language, the fraction and it's result. For example, two-sevenths was written as "duae septimae", or three-eights as "tres octavae". To aid in calculating fractions, Romans used a division called a "uncia". The uncia was 1/12, in the modern sense, and was especially helpful in that you can add twelfths to create halves, thirds or quarters, making the unit very versatile.

http://math2033.uark.edu/wiki/index.php/File:Roman_1.gif http://math2033.uark.edu/wiki/index.php/File:Roman_2.jpg http://math2033.uark.edu/wiki/index.php/File:Roman_3.jpg