History Of Mathematics PDF - Tarlac State University

Bachelor of Secondary Education Department Reaccredited Level IV by the Accrediting Agency of Chartered College of the Philippines BACHELOR OF SECONDARY EDUCATION DEPARTMENT 1|Page 2|Page History of Mathemati...

Bachelor of Secondary Education Department Reaccredited Level IV by the Accrediting Agency of Chartered College of the Philippines BACHELOR OF SECONDARY EDUCATION DEPARTMENT 1|Page 2|Page History of Mathematics M 100 Ezra Gil S. Lagman, LPT BSECE Graduate with 18 units Professional Education Tarlac State University Civil Service Examination Professional Level Passer LET September 2018 Topnotcher (91.2%) 9th Placer Secondary – Math Contact No. 09399215341 3|Page Course Description: The course presents the humanistic aspects of mathematics which provides the historical context and timeline that led to the present understanding and applications of the different branches of mathematics. Course Outline: I. : Course Introduction / Prehistoric Mathematics II : Sumerian and Babylonian Mathematics III: Egyptian Mathematics IV: Ancient Greek Mathematics V : Hellenistic Mathematics VI: Roman Mathematics VII: Mayan Mathematics VIII: Chinese Mathematics IX : Indian Mathematics X : Arab / Islamic Mathematics XI : Notable Mathematicians from 16th Century to 21st Century Purpose and Rationale In line with the introduction of an alternative learning system, the College of Teacher Education, as part of its commitment in supporting equity of access to Higher Education for all students, has developed this module for use by both teacher and students to support in building their skills needed to access quality education, in addition to learning the historical context and timeline that led to the mathematics that we know today. The purpose of this module is to introduce to you student how mathematics begun and how it changed through the years and the different nations who contributed to its development. This module will also highlight the contributions of 4|Page various mathematicians throughout history that we still use today. No copyright infringement was intended in the development of this module. This learning module will: Introduce the students to the history of mathematics Identify the contributions of Sumerians, Babylonians, Egyptians, Greek, Romans, Mayan, Chinese, Indians and Arabs in the development of mathematics Make the students appreciate the life of notable mathematicians Identify the contributions in mathematics of various mathematicians Instruction to the User This module consists of several chapters and subtopics starting from Mathematics during the prehistoric times until the last chapter where we discuss some notable mathematicians who contributed to the development of mathematics. The lectures, images and examples were taken from the following sources: Victor J. Katz: A History of Mathematics: An Introduction 3rd Edition Luke Hodgkin: A History of Mathematics: From Mesopotamia to Modernity Uta C. Merzbach and Carl B. Boyer: A History of Mathematics Third Edition David M. Burton: The History of Mathematics: An Introduction, 6th Edition Clifford A. Pickover: The Math Book From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics Mastin, L. (2020). The Story of Mathematics. Retrieved from https://www.storyofmathematics.com/ https://www.mathopenref.com/ https://mathshistory.st-andrews.ac.uk/Biographies The exercises and post-test were made by the author. Enjoy reading the history of mathematics. 5|Page CHAPTER 1 Prehistoric Mathematics Learning Objectives: By the end of this chapter you should be able to: Identify the methods used by people before the development of mathematics Explain the development of mathematics before history was written Our prehistoric ancestors would have had a general sensibility about amounts, and would have instinctively known the difference between, say, one and two antelopes. But the intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of “two” took many ages to come about. There are isolated hunter-gatherer tribes in Amazonia which only have words for “one”, “two” and “many”, and others which only have words for numbers up to five. In the absence of settled agriculture and trade, there is little need for a formal system of numbers Early man kept track of regular occurrences such as the phases of the moon and the seasons. Some of the very earliest evidence of mankind thinking about numbers is from notched bones in Africa dating back to 35,000 to 20,000 years ago. But this is really mere counting and tallying rather than mathematics as such. Pre-dynastic Egyptians and Sumerians represented geometric designs on their artifacts as early as the 5th millennium BCE, as did some megalithic societies in northern Europe in the 3rd millennium BCE or before. But this is more art and 6|Page decoration than the systematic treatment of figures, patterns, forms and quantities that has come to be considered as mathematics. Mathematics proper initially developed largely as a response to bureaucratic needs when civilizations settled and developed agriculture – for the measurement of plots of land, the taxation of individuals, etc – and this first occurred in the Sumerian and Babylonian civilizations of Mesopotamia (roughly, modern Iraq) and in ancient Egypt. According to some authorities, there is evidence of basic arithmetic and geometric notations on the petroglyphs at Knowth and Newgrange burial mounds in Ireland (dating from about 3500 BCE and 3200 BCE respectively). These utilize a repeated zig-zag glyph for counting, a system that continued to be used in Britain and Ireland into the 1st millennium BCE. Stonehenge, a Neolithic ceremonial and astronomical monument in England, which dates from around 2300 BCE, also arguably exhibits examples of the use of 60 and 360 in the circle measurements, a practice which presumably developed quite independently of the sexagesimal counting system of the ancient Sumerian and Babylonians. Here are some examples that mathematics was already practiced in various ways in history 1. Ant Odometer - Ants are social insects that evolved from vespoid wasps in the mid-Cretaceous period, about 150 million years ago. After the rise of flowering plants, about 100 million years ago, ants diversified into numerous species. The Saharan desert ant, Cataglyphis fortis, travels immense distances over sandy terrain, often completely devoid of landmarks, as it searches for food. These creatures are able to return to their nest using a direct route rather than by retracing their outbound path. Not only do they judge directions, using light from the sky for orientation, but they also appear to have a built-in ‘"computer” that functions like a pedometer that counts their steps and allows them to measure exact distances. An ant may travel as for as 160 feet (about 50 meters) until it 7|Page encounters a dead insect, whereupon it tears a piece to carry directly back to its nest, accessed via a hole often less than a millimeter in diameter. By manipulating the leg lengths of ants to give them longer and shorter strides, a research team of German and Swiss scientists discovered that the ants “count” steps to judge distance. For example, after ants had reached their destination, the legs were lengthened by adding stilts or shortened by partial amputation. The researchers then returned the ants so that the ants could start on their journey back to the nest. Ants with the stilts traveled too for and passed the nest entrance, while those with the amputated legs did not reach it However, if the ants started their journey from their nest with the modified legs, they were able to compute the appropriate distances. This suggests that stride length is the crucial factor. Moreover, the highly sophisticated computer in the ant's brain enables the ant to compute a quantity related to the horizontal projection of its path so that it does not become lost even if the sandy landscape develops hills and valleys during its journey. 2. Primates Count - Around 60 million years ago, small, lemur-like primates had evolved in many areas of the world, and 30 million years ago, primates with monkeylike characteristics existed. Could such creatures count? The meaning of counting by animals is a highly contentious issue among animal behavior experts. However, many scholars suggest that animals have some sense of number. H. Kalmus writes in his Nature article “Animals as Mathematicians”. 8|Page There is now little doubt that some animals such as squirrels or parrots can be trained to count— Counting faculties have been reported in squirrels, rats, and for pollinating insects. Some of these animals and others can distinguish numbers in otherwise similar visual patterns, while others can be trained to recognize and even to reproduce sequences of acoustic signals. A few can even be trained to tap out the numbers of elements (dots) in a visual pattern. The lack of the spoken numeral and the written symbol makes many people reluctant to accept animals as mathematicians. 3. Cicada – Generated Prime Numbers - Cicadas are winged insects that evolved around 1.8 million years ago during the Pleistocene epoch, when glaciers advanced and retreated across North America- Cicadas of the genus Magicicada spend most of their lives below die ground, feeding on the juices of plant roots, and then emerge, mate, and die quickly. These creatures display a startling behavior: Their emergence is synchronized with periods of years that are usually the prime numbers 13 and 17. (A prime number is an integer such as 11, 13, and 17 that has only two integer divisors: 1 and itself) During the spring of their 13th or 17th year, these periodical cicadas construct an exit tunnel. Sometimes more than 1.5 million individuals emerge in a single acre; ibis abundance of bodies may have survival value as they overwhelm predators such as birds that cannot possibly eat them all at once. Some researchers have speculated that the evolution of prime-number life cycles occurred so that the creatures increased their chances of evading shorter-lived predators and parasites. For example, if these cicadas had 12-year life cycles, all predators with life cycles of 2, 3,4, or 6 years might more easily find fire insects. Mario Markus of the Max Planck Institute for Molecular Physiology in Dortmund, Germany, and his coworkers discovered that these kinds of prime-number cycles arise naturally from evolutionary mathematical models of interactions between predator and prey. In order to experiment, they first assigned random life-cycle durations to their computer-simulated populations. After some time, a sequence of mutations always locked tbe synthetic cicadas into a stable prime-number cycle. Of course, this research is still in its infancy and many questions remain. What is special about 13 and 17? What predators or parasites have actually existed to drive the cicadas to these periods? Also, a 9|Page mystery remains as to why, of the 1,500 cicada species worldwide, only a small number of the genus Magicicada are known to be periodical 4. Ishango Bone - In 1960, Belgian geologist and explorer Jean de Heinzelin de Braucourt (1920-1998) discovered a baboon bone with markings in what is today the Democratic Republic of the Congo. The Ishango bone, witii its sequence of notches, was first thought to be a simple tally stick used by a Stone Age African. However, according to some scientists, the marks suggest a mathematical prowess that goes beyond counting of objects. The bone was found in Ishango, near the headwaters of the Nile River, the home of a large population of upper Paleolithic people prior to a volcanic eruption that buried the area. One column of marks on the bone begins with three notches that double to six notches. Four notches double to eight. Ten notches halve to five. This may suggest a simple understanding of doubling or halving. Even more striking is the fact that numbers in other columns are all odd (9, 11, 13,17, 19, and 21). One column contains the prime numbers between 10 and 20, and the numbers in each column sum to 60 or 48, both multiples of 12. A number of tally sticks have been discovered that predate the Ishango bone. For example, the Swaziland Lebombo bone is a 37,000-year-old baboon fibula with 29 notches. A 32,000^ear-old wolf tibia with 57 notches, grouped in fives, was found in Czechoslovakia. Although quite speculative, some have hypothesized that the markings on the Ishango bone form a kind of lunar calendar for a Stone Age woman who kept 10 | P a g e track of her menstrual cycles, giving rise to the slogan “menstruation created mathematics.” Even if the Ishango was a simple bookkeeping device, these tallies seem to set us apart from the animals and represent the first steps to symbolic mathematics. The full mystery of the Ishango bone can't be solved until other similar bones are discovered. 5. Quipu - The ancient Incas used quipus (pronounced “key^poos”), memory banks made of strings and knots, for storing numbers. Until recently, the oldest- known quipus dated from about A-D. 650. However, in 2005, a quipu from the Peruvian coastal city of Caral was dated to about 5,000 years ago. The Incas of Soufri America had a complex civilization with a common state religion and a common language. Although they did not have writing, they kept extensive records encoded by a logical-numerical system on the quipus, which 11 | P a g e varied in complexity from three to around a thousand cords. Unfortunately, when the Spanish came to South America, they saw the strange quipus and thought they were the works of the Devil. The Spanish destroyed thousands of them in the name of God, and today only about 600 quipus remain. Knot shapes and positions, cord directions, cord levels, and color and spacing represent numbers mapped to real-world objects. Different knot groups were used for different powers of 10. The knots were probably used to record human and material resources and calendar information. The quipus may have contained more information such as construction plans, dance patterns, and even aspects of Inca history. The quipu is significant because it dispels the notion that mathematics flourishes only after a civilization has developed writing; however, societies can reach advanced states without ever having developed written records. Interestingly, today there are computer systems whose file managers are called quipus, in honor of this very useful ancient device One sinister application of the quipu by the Incas was as a death calculator. Yearly quotas of adults and children were ritually slaughtered, and ffiis enterprise was planned using a quipu. Some quipus represented the empire, and the cords referred to roads and the knots to sacrificial victims. 12 | P a g e 6. Dice - Imagine a world without random numbers. In the 1940s, the generation of statistically random numbers was important to physicists simulating thermonuclear explosions, and today, many computer networks employ random numbers to help route Internet traffic to avoid congestion. Political poll-takers use random numbers to select unbiased samples of potential voters. Dice, originally made from the anklebones of hoofed animals, were one of the earliest means for producing random numbers. In ancient civilizations, the gods were believed to control the outcome of dice tosses; thus, dice were relied upon to make crucial decisions, ranging from the selection of rulers to the division of property in an inheritance. Even today, the metaphor of God controlling dice is common, as evidenced by astrophysicist Stephen Hawking’s quote, “Not only does God play dice, but He sometimes confuses us by throwing them where they can’t be seen.” The oldest-known dice were excavated together with a 5,000-year-old backgammon set from the legendary Burnt Gity in southeastern Iran. The city represents four stages of civilization that were destroyed by fires before being abandoned in 2100 B.G. At this same site, archeologists also discovered the earliest-known artificial eye, which once stared out hypnotically from the face of an ancient female priestess or soothsayer Assignment Research other findings about mathematics during prehistoric times 13 | P a g e CHAPTER 2 Sumerian and Babylonian Mathematics Learning Objectives: By the end of this chapter you should be able to: Identify the contributions of Sumerians and Babylonians in the development of mathematics Write numbers using the Sumerian and Babylonian Number System Differentiate the Sumerian and Babylonian Number System from our current number system Introduction The fourth millennium before our era was a period of remarkable cultural development, bringing with it the use of writing, the wheel, and metals. As in Egypt during the first dynasty, which began toward the end of this extraordinary millennium, so also in the Mesopotamian Valley there was at the time a high order of civilization. There the Sumerians had built homes and temples decorated with artistic pottery and mosaics in geometric patterns. Powerful rulers united the local principalities into an empire that completed vast public works, such as a system of canals to irrigate the land and control flooding between the Tigris and Euphrates rivers, where the overflow of the rivers was not predictable, as was the inundation of the Nile Valley. The cuneiform pattern of writing that the Sumerians had developed during the fourth millennium probably antedates the Egyptian hieroglyphic system. The Mesopotamian civilizations of antiquity are often referred to as Babylonian, although such a designation is not strictly correct. The city of Babylon was not at first, nor was it always at later periods, the center of the culture associated with the two rivers, but convention has sanctioned the informal use of the name “Babylonian” for the region during the interval from about 2000 to roughly 600 14 | P a g e BCE. When in 538 BCE Babylon fell to Cyrus of Persia, the city was spared, but the Babylonian Empire had come to an end. “Babylonian” mathematics, however, continued through the Seleucid period in Syria almost to the dawn of Christianity. Then, as today, the Land of the Two Rivers was open to invasions from many directions, making the Fertile Crescent a battlefield with frequently changing hegemony. One of the most significant of the invasions was that by the Semitic Akkadians under Sargon I (ca. 2276 2221 BCE), or Sargon the Great. He established an empire that extended from the Persian Gulf in the south to the Black Sea in the north, and from the steppes of Persia in the east to the Mediterranean Sea in the west. Under Sargon, the invaders began a gradual absorption of the indigenous Sumerian culture, including the cuneiform script. Later invasions and revolts brought various racial strains—Ammorites, Kassites, Elamites, Hittites, Assyrians, Medes, Persians, and others—to political power at one time or another in the valley, but there remained in the area a sufficiently high degree of cultural unity to justify referring to the civilization simply as Mesopotamian. In particular, the use of cuneiform script formed a strong bond. Laws, tax accounts, stories, school lessons, personal letters—these and many other records were impressed on soft clay tablets with styluses, and the tablets were then baked in the hot sun or in ovens. Such written documents were far less vulnerable to the ravages of time than were Egyptian papyri; hence, a much larger body of evidence about Mesopotamian mathematics is available today than exists about the Nilotic system. From one locality alone, the site of ancient Nippur, we have some 50,000 tablets. The university libraries at Columbia, Pennsylvania, and Yale, among others, have large collections of ancient tablets from Mesopotamia, some of them mathematical. Despite the availability of documents, however, it was the Egyptian hieroglyphic, rather than the Babylonian cuneiform, that was first deciphered in modern times. The German philologist F.W. Grotefend had made some progress in the reading of Babylonian script early in the nineteenth century, but only during the second quarter of the twentieth century did substantial accounts of Mesopotamian mathematics begin to appear in histories of antiquity. 15 | P a g e Cuneiform Writing The early use of writing in Mesopotamia is attested to by hundreds of clay tablets found in Uruk and dating from about 5,000 years ago. By this time, picture writing had reached the point where conventionalized stylized forms were used for many things: for water, for eye, and combinations of these to indicate weeping. Gradually, the number of signs became smaller, so that of some 2,000 Sumerian signs originally used, only a third remained by the time of the Akkadian conquest. Primitive drawings gave way to combinations of wedges: water became and eye. At first, the scribe wrote from top to bottom in columns from right to left; later, for convenience, the table was rotated counter clockwise through 900, and the scribe wrote from left to right in horizontal rows from top to bottom. The stylus, which formerly had been a triangular prism, was replaced by a right circular cylinder—or, rather, two cylinders of unequal radius. During the earlier days of the Sumerian civilization, the end of the stylus was pressed into the clay vertically to represent 10 units and obliquely to represent a unit, using the smaller stylus; similarly, an oblique impression with the larger stylus indicated 60 units and a vertical impression indicated 3,600 units. Combinations of these were used to represent intermediate numbers. Sexegesimal System Starting as early as the 4th millennium BCE, they began using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. Over the course of the third millennium, these objects were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text. A rudimentary model of the abacus was probably in use in Sumeria from as early as 2700 – 2300 BCE. 16 | P a g e Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand the five fingers on the other hand. Unlike those of the Egyptians, Greeks and Romans, Babylonian numbers used a true place- value system, where digits written in the left column represented larger values, much as in the modern decimal system, although of course using base 60 not base 10. Thus, in the Babylonian system represented 3,600 plus 60 plus 1, or 3,661. Also, to represent the numbers 1 – 59 within each place value, two distinct symbols were used, a unit symbol ( ) and a ten symbol ( ) which were combined in a similar way to the familiar system of Roman numerals (e.g. 23 would be shown as ). Thus, represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context. It has been conjectured that Babylonian advances in mathematics were probably facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 – in fact, 60 is the smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12 pence, 2 x 12 hours, etc). 17 | P a g e Babylonian Clay Tablets We have evidence of the development of a complex system of metrology in Sumer from about 3000 BCE, and multiplication and reciprocal (division) tables, tables of squares, square roots and cube roots, geometrical exercises and division problems from around 2600 BCE onwards. Later Babylonian tablets dating from about 1800 to 1600 BCE cover topics as varied as fractions, algebra, methods for solving linear, quadratic and even some cubic equations, and the calculation of regular reciprocal pairs (pairs of number which multiply together to give 60). One Babylonian tablet gives an approximation to √2 accurate to an astonishing five decimal places. Others list the squares of numbers up to 59, the cubes of numbers up to 32 as well as tables of compound interest. Yet another gives an estimate for π of 3 1⁄8 (3.125, a reasonable approximation of the real value of 3.1416). The idea of square numbers and quadratic equations (where the unknown quantity is multiplied by itself, e.g. x2) naturally arose in the context of the measurement of land, and Babylonian mathematical tablets give us the first ever evidence of the solution of quadratic equations. The Babylonian approach to solving them usually revolved around a kind of geometric game of slicing up and rearranging shapes, although the use of algebra and quadratic equations also appears. Babylonian Clay Tablets 18 | P a g e Plimpton 322 Clay Tablet Plimpton 322 refers to a mysterious Babylonian clay tablet featuring numbers in cuneiform script in a table of 4 columns and 15 rows. Eleanor Robson, a historian of science, refers to it as "one of the world's most famous mathematical artifacts." Written around 1800 B.C., the table lists Pythagorean triples-that is, whole numbers that specify the side lengths of right triangles that are solutions to the Pythagorean theorem a2 + b2 = c2. For example, the numbers 3, 4, and 5 are a Pythagorean triple. The fourth column in the table simply contains the row number. Interpretations vary as to the precise meaning of the numbers in the table, with some scholars suggesting that the numbers were solutions for students studying algebra or trigonometry- like problems. Plimpton 322 is named after New York publisher George Plimpton who, in 1922, bought the tablet for $10 from a dealer and then donated the tablet to Columbia University. The tablet can be traced to the Old Babylonian civilization that flourished in Mesopotamia, the fertile valley of the Tigris and Euphrates rivers, which is now located in Iraq. To put the era into perspective, the unknown scribe who generated Plimpton 322 lived within about a century of King Hammurabi, famous for his set of laws that Included "an eye for an eye, a tooth for a tooth." According to biblical history, Abraham, who is said to have led his people west from the city of Ur on the bank of the Euphrates into Canaan, would have been another near contemporary of the scribe. The Babylonians wrote on wet clay by pressing a stylus or wedge into the clay. In the Babylonian number system, the number 1 was written with a single stroke and the numbers 2 through 9 were written by combining multiples of a single stroke. 19 | P a g e  Plimpton 322 Clay Tablet Exercise: Write the following numbers using the Babylonian Number System a. 25 f. 45 b. 134 g. 234 c. 62 h. 96 d. 87 i. 100 e. 54 j. 89 20 | P a g e CHAPTER 3 Egyptian Mathematics Learning Objectives: By the end of this chapter you should be able to: Identify the contributions of Egyptians in the development of mathematics Write numbers using ancient Egyptian Number System Determine the importance of the Rhind and Moscow papyrus during ancient times Introduction About 450 BCE, Herodotus, the inveterate Greek traveller and narrative historian, visited Egypt. He viewed ancient monuments, interviewed priests, and observed the majesty of the Nile and the achievements of those working along its banks. His resulting account would become a cornerstone for the narrative of Egypt’s ancient history. When it came to mathematics, he held that geometry had originated in Egypt, for he believed that the subject had arisen there from the practical need for resurveying after the annual flooding of the river valley. A century later, the philosopher Aristotle speculated on the same subject and attributed the Egyptians’ pursuit of geometry to the existence of a priestly leisure class. The debate, extending well beyond the confines of Egypt, about whether to credit progress in mathematics to the practical men (the surveyors, or “rope- stretchers”) or to the contemplative elements of society (the priests and the philosophers) has continued to our times. As we shall see, the history of mathematics displays a constant interplay between these two types of contributors. In attempting to piece together the history of mathematics in ancient Egypt, scholars until the nineteenth century encountered two major obstacles. The first was the inability to read the source materials that existed. The second was the 21 | P a g e scarcity of such materials. For more than thirty-five centuries, inscriptions used hieroglyphic writing, with variations from purely ideographic to the smoother hieratic and eventually the still more flowing demotic forms. After the third century CE, when they were replaced by Coptic and eventually supplanted by Arabic, knowledge of hieroglyphs faded. The breakthrough that enabled modern scholars to decipher the ancient texts came early in the nineteenth century when the French scholar Jean-Francois Champollion, working with multilingual tablets, was able to slowly translate a number of hieroglyphs. These studies were supplemented by those of other scholars, including the British physicist Thomas Young, who were intrigued by the Rosetta Stone, a trilingual basalt slab with inscriptions in hieroglyphic, demotic, and Greek writings that had been found by members of Napoleon’s Egyptian expedition in 1799. By 1822, Champollion was able to announce a substantive portion of his translations in a famous letter sent to the Academy of Sciences in Paris, and by the time of his death in 1832, he had published a grammar textbook and the beginning of a dictionary. Ancient Egyptian Number System The early Egyptians settled along the fertile Nile valley as early as about 6000 BCE, and they began to record the patterns of lunar phases and the seasons, both for agricultural and religious reasons. The Pharaoh’s surveyors used measurements based on body parts (a palm was the width of the hand, a cubit the measurement from elbow to fingertips) to measure land and buildings very early in Egyptian history, and a decimal numeric system was developed based on our ten fingers. The oldest mathematical text from ancient Egypt discovered so far, though, is the Moscow Papyrus, which dates from the Egyptian Middle Kingdom around 2000 – 1800 BCE. 22 | P a g e It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BCE (and probably much early). Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there was no concept of place value, so larger numbers were rather unwieldy. Rhind Papyrus The Rhind Papyrus is considered to be the most important known source of information concerning ancient Egyptian mathematics. This scroll, about a foot (30 centimeters) high and 18 feet (5.5 meters) long, was found in a tomb in Thebes on the east bank of the river Nile. Ahmes, the scribe, wrote it in hieratic, a script related to the hieroglyphic system. Given that the writing occurred in around 1650 B.C., this makes Ahmes the earliest-named indiVIdual in the history of mathematics! The scroll also contains the earliest-known symbols for mathematical operations-plus is denoted by a pair of legs walking toward the number to be added. In 1858, Scottish lawyer and Egyptologist Alexander Henry Rhind had been visiting Egypt for health reasons when he bought the scroll in a market in Luxor. The British Museum in London acquired the scroll in 1864. Ahmes wrote that the scroll gives an "accurate reckoning for inqUIring into things, and the knowledge of all things, mysteries... all secrets." The content of the scroll concerns mathematical problems involving fractions, arithmetic progressions, algebra, and pyramid geometry, as well as practical mathematics useful for surveying, building, and accounting. The problem that intrigues me the most is Problem 79, the interpretation of which was initially baffling. 23 | P a g e Today, many interpret Problem 79 as a puzzle, which may be translated as "Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats (measures) of wheat. What is the total of all of these?" Interestingly, this indestructible puzzle meme, involving the number 7 and animals, seems to have persisted through thousands of years! We observe something quite similar in Fibonacci's Liber Abaci (Book of Calculation), published in 1202, and later in the St Ives puzzle, an Old English children's rhyme involving 7 cats. Multiplication, for example, was achieved by a process of repeated doubling of the number to be multiplied on one side and of one on the other, essentially a kind of multiplication of binary factors similar to that used by modern computers (see the example at right). These corresponding blocks of counters could then be used as a kind of multiplication reference table: first, the combination of powers of two which add up to the number to be multiplied by was isolated, and then the corresponding blocks of counters on the other side yielded the answer. This effectively made use of the concept of binary numbers, over 3,000 years before Leibniz introduced it into the west, and many more years before the development of the computer was to fully explore its potential. 24 | P a g e Practical problems of trade and the market led to the development of a notation for fractions. The papyri which have come down to us demonstrate the use of unit fractions based on the symbol of the Eye of Horus, where each part of the eye represented a different fraction, each half of the previous one (i.e. half, quarter, eighth, sixteenth, thirty-second, sixty-fourth), so that the total was one- sixty-fourth short of a whole, the first known example of a geometric series. Unit fractions could also be used for simple division sums. For example, if they needed to divide 3 loaves among 5 people, they would first divide two of the loaves into thirds and the third loaf into fifths, then they would divide the left over third from the second loaf into five pieces. Thus, each person would receive one- third plus one-fifth plus one-fifteenth (which totals three-fifths) 25 | P a g e The Egyptians approximated the area of a circle by using shapes whose area they did know. They observed that the area of a circle of diameter 9 units, for example, was very close to the area of a square with sides of 8 units, so that the area of circles of other diameters could be obtained by multiplying the diameter by 8⁄9 and then squaring it. This gives an effective approximation of π accurate to within less than one percent Moscow Papyrus Another important papyrus, known as the Golenishchev or Moscow Papyrus, was purchased in 1893 and is now in the Pushkin Museum of Fine Arts in Moscow. It, too, is about eighteen feet long but is only one-fourth as wide as the Ahmes Papyrus. It was written less carefully than the work of Ahmes was, by an unknown scribe of circa. 1890 BCE. It contains twenty-five examples, mostly from practical life and not differing greatly from those of Ahmes. Some of the problems are unreadable or too damaged to translate. The mathematical problems the scribes could solve, as illustrated in the Rhind and Moscow Papyri, deal with what we today call linear equations, proportions, and geometry. For example, the Egyptian papyri present two different procedures for dealing with linear equations.First, problem 19 of the Moscow Papyrus used our normal technique to find the number such that if it is taken 1 1/2 times and then 4 is added, the sum is 10. In modern notation, the equation is simply (11/2)x + 4 = 10. The scribe proceeded as follows: “Calculate the excess of this 10 over 4. The result is 6. You operate on 1 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4. Behold, 4 says it. You will find that this is correct. Namely, after subtracting 4, the scribe noted that the reciprocal of 1 1/2 is 2/3 and then multiplies 6 by this quantity. 26 | P a g e Exercise Write the following numbers using ancient Egyptian Number System a. 3111 b. 653 c. 48 d. 885 e. 93 27 | P a g e CHAPTER 4 Ancient Greek Mathematics Learning Objectives: By the end of this chapter you should be able to: Identify the contributions of Greeks in the development of mathematics Write numbers using Herodianic Numerals Write numbers using Ionic Numerals Identify the Three Geometrical Problems presented during the Greek Era Explain Zeno’s Paradox of Achilles and the Tortoise Identify the contributions of Thales, Pythagoras, Democritus and Plato to Mathematics A report from a visit to Egypt with Plato by Simmias of Thebes in 379 bce (from a dramatization by Plutarch of Chaeronea (first–second century ): “On our return from Egypt a party of Delians met us and requested Plato, as a geometer, to solve a problem set them by the god in a strange oracle. The oracle was to this effect: the present troubles of the Delians and the rest of the Greeks would be at an end when they had doubled the altar at Delos. As they not only were unable to penetrate its meaning, but failed absurdly in constructing the altar so they called on Plato for help in their difficulty. Plato replied that the god was ridiculing the Greeks for their neglect of education, deriding, as it were, our ignorance and bidding us engage in no perfunctory study of geometry; for no ordinary or near- sighted intelligence, but one well versed in the subject, was required to find two mean proportionals, that being the only way in which a body cubical in shape can be doubled with a similar increment in all dimensions. This would be done for them by Eudoxus of Cnidus; they were not, however, to suppose that it was this the god desired, but rather that he was ordering the entire Greek nation to give up war and its miseries and cultivate the Muses, and by calming their passions through the practice of discussion and study of mathematics, so to live with one another that their relationships should be not injurious, but profitable.” 28 | P a g e As the quotation and the (probably) fictional account indicate, a new attitude toward mathematics appeared in Greece sometime before the fourth century BCE. It was no longer sufficient merely to calculate numerical answers to problems. One now had to prove that the results were correct. To double a cube, that is, to find a new cube whose volume was twice that of the original one, is equivalent to determining the cube root of 2, and that was not a difficult problem numerically. The oracle, however, was not concerned with numerical calculation, but with geometric construction. That in turn depended on geometric proof by some logical argument, the earliest manifestation of such in Greece being attributed to Thales. This change in the nature of mathematics, beginning around 600 BCE, was related to the great differences between the emerging Greek civilization and those of Egypt and Babylonia, from whom the Greeks learned. The physical nature of Greece with its many mountains and islands is such that large-scale agriculture was not possible. Perhaps because of this, Greece did not develop a central government. The basic political organization was the polis, or city-state. The governments of the city-state were of every possible variety but in general controlled populations of only a few thousand. Whether the governments were democratic or monarchical, they were not arbitrary. Each government was ruled by law and therefore encouraged its citizens to be able to argue and debate. It was perhaps out of this characteristic that there developed the necessity for proof in mathematics, that is, for argument aimed at convincing others of a particular truth. Because virtually every city-state had access to the sea, there was constant trade, both in Greece itself and with other civilizations. As a result, the Greeks were exposed to many different peoples and, in fact, themselves settled in areas all around the eastern Mediterranean. In addition, a rising standard of living helped to attract able people from other parts of the world. Hence, the Greeks were able to study differing answers to fundamental questions about the world. They began to create their own answers. In many areas of thought, they learned not to accept what had been handed down from ancient times. Instead, they began to ask, and to try to answer, “Why?” Greek thinkers eventually came to the 29 | P a g e realization that the world around them was knowable, that they could discover its characteristics by rational inquiry. Hence, they were anxious to discover and expound theories in such fields as mathematics, physics, biology, medicine, and politics. And although Western civilization owes a great debt to Greek society in literature, art, and architecture, it is to Greek mathematics that we owe the idea of mathematical proof, an idea at the basis of modern mathematics and, by extension, at the foundation of our modern technological civilization. Earliest Greek Mathematics Unlike the situation with Egyptian and Babylonian mathematics, there are virtually no existing texts of Greek mathematics that were actually written in the first millennium BCE. What we have today are copies of copies of copies, where the actual written documents date from not much earlier than 1000 CE. And even then, the earliest complete texts (of which these are copies) are not from earlier than about 300 BCE. Attic or Herodianic Numerals The ancient Greek numeral system, known as Attic or Herodianic numerals, was fully developed by about 450 BCE, and in regular use possibly as early as the 7th Century BCE. It was a base 10 system similar to the earlier Egyptian one (and even more similar to the later Roman system), with symbols for 1, 5, 10, 50, 100, 500 and 1,000 repeated as many times needed to represent the desired number. Addition was done by totalling separately the symbols (1s, 10s, 100s, etc) in the numbers to be added. 30 | P a g e Greek Ionic Numerals From what fragments exist from ancient times, and even from some of the copies, we do know that the Greeks represented numbers in a ciphered system using their alphabet, from as far back as the sixth century BCE. To represent thousands, a mark ( ‘ ) was made to the left of the letter alpha through theta (e.g. ‘α for 1000). Larger numbers still were written using the letter M to represent myriads (10,000) with the number of myriads written above: (e.g.Mβ = 20,000). Thales’ Intercept Theorem most of Greek mathematics was based on geometry. Thales, one of the Seven Sages of Ancient Greece, who lived on the Ionian coast of Asian Minor in the first half of the 6th Century BCE, is usually considered to have been the first to lay down guidelines for the abstract development of geometry, although what we know of his work (such as on similar and right triangles) now seems quite elementary. Thales established what has become known as Thales’ Theorem, whereby if a triangle is drawn within a circle with the long side as a diameter of the circle, then the opposite angle will always be a right angle (as well as some other related properties derived from this). He is also credited with another theorem, also known as Thales’ Theorem or the Intercept Theorem, about the ratios of the line 31 | P a g e segments that are created if two intersecting lines are intercepted by a pair of parallels (and, by extension, the ratios of the sides of similar triangles). Three Geometrical Problems Three geometrical problems in particular, often referred to as the Three Classical Problems, and all to be solved by purely geometric means using only a straight edge and a compass, date back to the early days of Greek geometry: “the squaring (or quadrature) of the circle”, “the doubling (or duplicating) of the cube” and “the trisection of an angle”. These intransigent problems were profoundly influential on future geometry and led to many fruitful discoveries, although their actual solutions (or, as it turned out, the proofs of their impossibility) had to wait until the 19th Century. 32 | P a g e Zeno’s Paradox of Achilles and the Tortoise For more than a thousand years, philosophers and mathematicians have tried to understand Zeno's paradoxes, a set of riddles that suggest that motion should be impossible or that it is an illusion. Zeno was a pre - Socratic Greek philosopher from southern Italy. His most famous paradox involves the Greek hero Achilles and a slow tortoise that Achilles can never overtake during a race once the tortoise is given a head start. In fact, the paradox seems to imply that you can never leave the room you are in. In order to reach the door, you must first travel half the distance there. You'll also need to continue to half the remaining distance, and half again, and so on. You won't reach the door in a finite number of jumps! Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (112 + 1/4 + 1/8 +... ). One modem tendency is to attempt to resolve Zeno's paradox by insisting that the sum of this infinite series 1/2 + 1/4 + 1/8 is equal to 1. If each step is done in half as much time, the actual time to complete the infinite series is no different than the real time required to leave the room. However, this approach may not provide a satisfying resolution because it does not explain how one is able to finish going through an infinite number of points, one after the other. Today, mathematicians make use of infinitesimals (unimaginably tiny quantities that are almost but not quite zero) to provide a microscopic analysis of the paradox. Coupled with a branch of mathematics called nonstandard analysis and, in particular, internal set theory, we may have resolved the paradox, but debate continues. Some have also argued that if space and time are discrete, the total number of jumps in going from one point to another must be finite. 33 | P a g e Notable Greek Mathematicians 1. Thales of Miletus The first individuals with whom specific mathematical discoveries are traditionally associated are Thales of Miletus (circa 625– 547 B.C.) and Pythagoras of Samos (circa 580–500 B.C.). Thales was of Phoenician descent, born in Miletus, a city of Ionia, at a time when a Greek colony flourished on the coast of Asia Minor. He seems to have spent his early years engaged in commercial ventures, and it is said that in his travels he learned geometry from the Egyptians and astronomy from the Babylonians. To his admiring countrymen of later generations, Thales was known as the first of the Seven Sages of Greece, the only mathematician so honored. In general, these men earned the title not so much as scholars as through statesmanship and philosophical and ethical wisdom. Thales is supposed to have coined the maxim “Know thyself,” and when asked what was the strangest thing he had ever seen, he answered “An aged tyrant.” Ancient opinion is unanimous in regarding Thales as unusually shrewd in politics and commerce no less than in science, and many interesting anecdotes, some serious and some fanciful, are told about his cleverness. On one occasion, according to Aristotle, after several years in which the olive trees failed to produce, Thales used his skill in astronomy to calculate that favourable weather conditions were due the next season. Anticipating an unexpectedly abundant crop he bought up all of the olive presses around Miletus. When the season came, having secured control of the presses, he was able to make his own terms for renting them out and thus realized a large sum. Others say that Thales, having proved the point that it was easy for philosophers to become rich if they wished, sold his olive oil at a reasonable price. As we have seen, the mathematics of the Egyptians was fundamentally a tool, crudely shaped to meet practical needs. The Greek intellect seized on this rich body of raw material and refined from it the common principles, thereby making 34 | P a g e the knowledge more general and more comprehensible and simultaneously discovering much that was new. Thales is generally hailed as the first to introduce using logical proof based on deductive reasoning rather than on experiment and intuition to support an argument. Modern reservations notwithstanding, if the mathematical attainments attributed to Thales by such Greek historians as Herodotus and Proclus are accepted, he must be credited with the following geometric propositions. An angle inscribed in a semicircle is a right angle. A circle is bisected by its diameter. The base angles of an isosceles triangle are equal. If two straight lines intersect, the opposite angles are equal. The sides of similar triangles are proportional. Two triangles are congruent if they have one side and two adjacent angles, respectively, equal. Because there is a continuous line from Egyptian to Greek mathematics, all of the listed facts may well have been known to the Egyptians. For them, the statements would remain unrelated, but for the Greeks they were the beginning of an extraordinary development in geometry. Conventional history inclines in such instances to look for some individual to whom the “miracle” can be ascribed. Thus, Thales is traditionally designated the father of geometry, or the first mathematician. Although we are not certain which propositions are directly attributable to him, it seems clear that Thales contributed something to the rational organization of geometry—perhaps the deductive method. For the orderly development of theorems by rigorous proof was entirely new and was thereafter a characteristic feature of Greek mathematics. 2. Pythagoras Pythagoras is often referred to as the first pure mathematician. He was born on the island of Samos, Greece in 569 BC. Various writings place his death between 500 BC and 475 BC in Metapontum, 35 | P a g e Lucania, Italy. His father, Mnesarchus, was a gem merchant. His mother's name was Pythais. Pythagoras had two or three brothers. Pythagoras was well educated, and he played the lyre throughout his lifetime, knew poetry and recited Homer. He was interested in mathematics, philosophy, astronomy and music, and was greatly influenced by Pherekydes (philosophy), Thales (mathematics and astronomy) and Anaximander (philosophy, geometry). Pythagoras left Samos for Egypt in about 535 B.C. to study with the priests in the temples. Many of the practices of the society he created later in Italy can be traced to the beliefs of Egyptian priests, such as the codes of secrecy, striving for purity, and refusal to eat beans or to wear animal skins as clothing. Ten years later, when Persia invaded Egypt, Pythagoras was taken prisoner and sent to Babylon (in what is now Iraq), where he met the Magoi, priests who taught him sacred rites. Iamblichus (250-330 AD), a Syrian philosopher, wrote about Pythagoras, "He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians..." In 520 BC, Pythagoras, now a free man, left Babylon and returned to Samos, and sometime later began a school called The Semicircle. His methods of teaching were not popular with the leaders of Samos, and their desire for him to become involved in politics did not appeal to him, so he left. Pythagoras settled in Crotona, a Greek colony in southern Italy, about 518 BC, and founded a philosophical and religious school where his many followers lived and worked. The Pythagoreans lived by rules of behavior, including when they spoke, what they wore and what they ate. Pythagoras was the Master of the society, and the followers, both men and women, who also lived there, were known as mathematikoi. They had no personal possessions and were vegetarians. Another group of followers who lived apart from the school were allowed to have personal possessions and were not expected to be vegetarians. They all worked communally on discoveries and theories. Pythagoras believed: All things are numbers. Mathematics is the basis for everything, and geometry is the highest form of mathematical studies. The physical world can understood through mathematics. The soul resides in the brain, and is immortal. It moves from one being to another, sometimes from a human into an animal, through a series of 36 | P a g e  reincarnations called transmigration until it becomes pure. Pythagoras believed that both mathematics and music could purify. Numbers have personalities, characteristics, strengths and weaknesses. The world depends upon the interaction of opposites, such as male and female, lightness and darkness, warm and cold, dry and moist, light and heavy, fast and slow. Certain symbols have a mystical significance. All members of the society should observe strict loyalty and secrecy. Because of the strict secrecy among the members of Pythagoras' society, and the fact that they shared ideas and intellectual discoveries within the group and did not give individuals credit, it is difficult to be certain whether all the theorems attributed to Pythagoras were originally his, or whether they came from the communal society of the Pythagoreans but the Pythagoreans always gave credit to Pythagoras as the Master for: 1. The sum of the angles of a triangle is equal to two right angles. 2. The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it. 3. Constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means. 4. The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy the all things are numbers, since number to him meant the ratio of two whole numbers. 5. The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two. 6. Pythagoras taught that Earth was a sphere in the center of the Kosmos (Universe), that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus. The over-riding dictum of Pythagoras’s school was “All is number” or “God is number”, and the Pythagoreans effectively practised a kind of numerology or 37 | P a g e number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or “wandering stars”; etc. Odd numbers were thought of as female and even numbers as male. The holiest number of all was “tetractys” or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans’ intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands. The reports of Pythagoras' death are varied. He is said to have been killed by an angry mob, to have been caught up in a war between the Agrigentum and the Syracusans and killed by the Syracusans, or been burned out of his school in Crotona and then went to Metapontum where he starved himself to death. At least two of the stories include a scene where Pythagoras refuses to trample a crop of bean plants in order to escape, and because of this, he is caught. The Pythagorean Theorem He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal 38 | P a g e to the sum of the square of the other two sides (or “legs”). Written as an equation: a2 + b2 = c2. The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (32 + 42 = 52), as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer “Pythagorean triples”. The Pythagorean Theorem is a cornerstone of mathematics, and continues to be so interesting to mathematicians that there are more than 400 different proofs of the theorem. 3. Democritus The Heroic Age in mathematics produced half a dozen great figures, and among them must be included a man who is better known as a chemical philosopher. Democritus of Abdera (ca. 460 370 BCE) is today celebrated as a proponent of a materialistic atomic doctrine, but in his time he had also acquired a reputation as a geometer. He is reported to have travelled more widely than anyone of his day—to Athens, Egypt, Mesopotamia, and possibly India—acquiring what learning he could, but his own achievements in mathematics were such that he boasted that not even the “rope-stretchers” in Egypt excelled him. He wrote a number of mathematical works, not one of which is extant today. The key to the mathematics of Democritus is to be found in his physical doctrine of atomism. All phenomena were to be explained, he argued, in terms of indefinitely small and infinitely varied (in size and shape), impenetrably hard atoms moving about ceaselessly in empty space. The physical atomism of Leucippus and Democritus may have been suggested by the geometric atomism of the Pythagoreans, and it is not surprising that the mathematical problems with which Democritus was chiefly concerned were those that demand some sort of infinitesimal approach. 39 | P a g e The Egyptians, for example, were aware that the volume of a pyramid is one- third the product of the base and the altitude, but a proof of this fact almost certainly was beyond their capabilities, for it requires a point of view equivalent to the calculus. Archimedes later wrote that this result was due to Democritus but that the latter did not prove it rigorously. This creates a puzzle, for if Democritus added anything to the Egyptian knowledge here, it must have been some sort of demonstration, albeit inadequate. Perhaps Democritus showed that a triangular prism can be divided into three triangular pyramids that are equal in height and area of the base and then deduced, from the assumption that pyramids of the same height and equal bases are equal, the familiar Egyptian theorem. This assumption can be justified only by the application of infinitesimal techniques. If, for example, one thinks of two pyramids of equal bases and the same height as composed of indefinitely many infinitely thin equal cross-sections in one-to-one correspondence (a device usually known as Cavalieri’s principle, in deference to the seventeenth century geometer), the assumption appears to be justified. Such a fuzzy geometric atomism might have been at the base of Democritus’s thought, although this has not been established. In any case, following the paradoxes of Zeno and the awareness of incommensurables, such arguments based on an infinity of infinitesimals were not acceptable. Archimedes consequently could well hold that Democritus had not given a rigorous proof. The same judgment would be true with respect to the theorem, also attributed by Archimedes to Democritus, that the volume of a cone is one- third the volume of the circumscribing cylinder. This result was probably looked on by Democritus as a corollary to the theorem on the pyramid, for the cone is essentially a pyramid whose base is a regular polygon of infinitely many sides. Democritean geometric atomism was immediately confronted with certain problems. If the pyramid or the cone, for example, is made up of indefinitely many infinitely thin triangular or circular sections parallel to the base, a consideration of any two adjacent laminae creates a paradox. If the adjacent sections are equal in area, then, because all sections are equal, the totality will be a prism or a cylinder and not a pyramid or a cone. If, on the other hand, adjacent sections are unequal, the totality will be a step pyramid or a step cone and not the smooth-surfaced figure one has in mind. This problem is not unlike the difficulties with the incommensurable and with the paradoxes of motion. 40 | P a g e Perhaps, in his On the Irrational, Democritus analyzed the difficulties here encountered, but there is no way of knowing what direction his attempts may have taken. His extreme unpopularity in the two dominant philosophical schools of the next century, those of Plato and Aristotle, may have encouraged the disregard of Democritean ideas. 4. Plato The time of Plato (429–347 BCE) saw significant efforts made toward solving the problems of doubling the cube and squaring the circle and toward dealing with incommensurability and its impact on the theory of proportion. These advances were achieved partly because Plato’s Academy, founded in Athens around 385 BCE, drew together scholars from all over the Greek world. These scholars conducted seminars in mathematics and philosophy with small groups of advanced students and also conducted research in mathematics, among other fields. There is an unverifiable story, dating from some 700 years after the school’s founding, that over the entrance to the Academy was inscribed the Greek phrase , meaning roughly, “Let no one ignorant of geometry enter here.” A student “ignorant of geometry” would also be ignorant of logic and hence unable to understand philosophy. The mathematical syllabus inaugurated by Plato for students at the Academy is described by him in his most famous work, The Republic, in which he discussed the education that should be received by the philosopher-kings, the ideal rulers of a state. The mathematical part of this education was to consist of five subjects: arithmetic (that is, the theory of numbers), plane geometry, solid geometry, astronomy, and harmonics (music). The leaders of the state are “to practice calculation, not like merchants or shopkeepers for purposes of buying and selling, but with a view to war and to help in the conversion of the soul itself from the world of becoming to truth and reality. It will further our intentions if it is pursued for the sake of knowledge and not for commercial ends. It has a great 41 | P a g e power of leading the mind upwards and forcing it to reason about pure numbers, refusing to discuss collections of material things which can be seen and touched.” In other words, arithmetic is to be studied for the training of the mind (and incidentally for its military usefulness). Again, a limited amount of plane geometry is necessary for practical purposes, particularly in war, when a general must be able to lay out a camp or extend army lines. But even though mathematicians talk of operations in plane geometry such as squaring or adding, the object of geometry, according to Plato, is not to do something but to gain knowledge, “knowledge, moreover, of what eternally exists, not of anything that comes to be this or that at some time and ceases to be.” So, as in arithmetic, the study of geometry—and for Plato this means theoretical, not practical, geometry—is for “drawing the soul towards truth.” The next subject of mathematical study should be solid geometry. Plato complained in the Republic that this subject has not been sufficiently investigated. This is because “no state thinks [it] worth encouraging” and because “students are not likely to make discoveries without a director, who is hard to find.” In any case, a decent knowledge of solid geometry was necessary for the next study, that of astronomy, or, as Plato puts it, “solid bodies in circular motion.” Again, in this field Plato distinguished between the stars as material objects with motions showing accidental irregularities and variations and the ideal abstract relations of their paths and velocities expressed in numbers and perfect figures such as the circle. It is this mathematical study of ideal bodies that is the true aim of astronomical study. Thus, this study should take place by means of problems and without attempting to actually follow every movement in the heavens. Similarly, a distinction is made in the final subject, of harmonics, between material sounds and their abstraction. The Pythagoreans had discovered the harmonies that occur when strings are plucked together with lengths in the ratios of certain small positive integers. But in encouraging his philosopher-kings in the study of harmonics, Plato meant for them to go beyond the actual musical study, using real strings and real sounds, to the abstract level of “inquiring which numbers are inherently consonant and which are not, and for what reasons.” That is, they should study the mathematics of harmony, just as they should 42 | P a g e study the mathematics of astronomy, and should not be overly concerned with real stringed instruments or real stars. Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes, which he maintained were the basis for the whole universe, and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles, and which for Plato represented fire), the octahedron (composed of 8 triangles, representing air), the icosahedron (composed of 20 triangles, and representing water), the cube (composed of 6 squares, and representing earth), and the dodecahedron (made up of 12 pentagons, which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”). Exercise Write the following numbers using Greek Ionic Numerals and Attic Numerals 1. 75 2. 156 3. 432 4. 1432 5. 8060 43 | P a g e