The History of Math - Clatsop Community College | Chart ... History of Math Richard W. Beveridge ... coins from the period 300 BC – 400 AD. ... important text summarizing much of

The History of Math

Richard W. BeveridgeClatsop Community College

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The Oldest Math Textbook

• Clay tablets from Mesopotamia have been dated as early as 3000 B.C.

• Geometry, surveying, architecture, commercial record-keeping.

• The Sumerian cultures used a base 60 number system.

Sumerian Mathematics

Sumerian Mathematics

This represents

414212963.121600010

360051

60241 ≈+++

This is an excellent approximation

for the square root of 2

414213562.1≈

Egyptian Mathematics

• The Egyptian Mathematics that is known to us is a result of the Papyrus or text.

• Egyptian scribes used the papyrus to record problems and solutions of various kinds.

• Surveying, record-keeping, simple algebra

Egyptian Numerals


• The Egyptian method for multiplication shows an interesting link to today’s binary number representation.


• To multiply: 41*5941 591 592 1184 2368 47216 94432 1888


41 = 32+8+1

41*59 = 32*59+8*59+1*59= 1888+472+59= 2419

Mayan Mathematics

• The Mayan culture used a base 20 counting system with dots representing one and bars representing five.

• Since the Mayans used a place value system, they also had a symbol for zero.

• The Mayans had a very complex calendricalsystem and advanced knowledge of astronomy.

Mayan Mathematics

Mayan Mathematics

• The Mayan used mathematics for astronomy to keep track of the passage of time.

• This is trickier than it seems. Our Gregorian Calendar was developed around 1500AD to correct the inefficiencies of the Roman calendar.

Mayan Mathematics

• The Mayans had 18 months of 20 days each and one “intercalary” month of 5 (or so) days.

• Intercalation is necessary to keep a calendar in proper relation to the celestial bodies.

Mayan Mathematics

• The Mayans were aware of and kept track of the “transits of Venus.” These are astronomical events that follow a somewhat intricate pattern and present evidence of a fairly sophisticated understanding of time and astronomy.

Mayan Mathematics

• A transit of Venus is when Venus passes directly between the Earth and the Sun.

• These occur eight years apart every 121.5 or 105.5 years.

• The last was in 2004 and the next will be in June 2012.

Mayan Mathematics

Greek Mathematics

• The written Greek number system represented each number by using the first letter of the word for the number. It was an additive system much like the later Roman system.

• Greeks began to use papyrus for writing in the 5th century BC. Before this time, there is no written record.

Greek Mathematics

• Thales of Miletus (7th century BC) is generally recognized as the first Greek philosopher and the first Greek mathematician.

• Pythagoras (6th century BC) is also a founding figure of Greek mathematics.

Greek Mathematics

• Pythagoras was born on the Greek island of Samos in the Aegean Sea off the coast of present day Turkey.

• He traveled to Egypt and Mesopotamia in his youth, eventually settling in Crotona in southern Italy.

Greek Mathematics

• The Pythagoreans were a very secretive group that rose up around Pythagoras.

• Their learning was centered around the topics of Music, Geometry, Arithmetic, and Trigonometry.

Greek Mathematics

• Part of the belief system of the Pythagoreans was a brand of numerology proposing that each number had certain mystical qualities.

• They also believed that all numbers were rational numbers (fractions) that could be represented as the ratio of two whole numbers.

Greek Mathematics

• In exploring geometrical figures and using their knowledge of the “Pythagorean Theorem,” the Pythagoreans were familiar with the idea of the square root of 2.

• Similar to the Babylonian conception of the square root of 2 as the diagonal of a square.

Greek Mathematics

• The Pythagoreans went one step beyond the Babylonians in that while the Babylonians approximated the square root of 2, Hippasus of Metapontum actually proved that the square root of 2 cannot be a rational number.

Greek Mathematics• “They say that the man (Hipassus) who first divulged the

nature of commensurability and incommensurability to men who were not worthy of being made part of this knowledge, became so much hated by the other Pythagoreans, that not only they cast him out of the community; they built a shrine for him as if he were dead, he who had once been their friend. Others add that even the gods became angry with him who had divulged Pythagoras’ doctrine; that he who showed how the icosagon…can be inscribed within a sphere, died at sea like an evil man. Others still say that the same misfortune happened on him who spoke to others of irrational numbers and incommensurability. Hyamblicus (or Iamblichus of Chalkis), De vita pythagorica 246-247”

Greek Mathematics

• Around 250 BC Eratosthenes used trigonometry to find a very good approximation for the circumference of the earth.

Greek Mathematics

Greek Mathematics• Eratosthenes saw that the angle he measured would be

the same as the angle from the center of the earth that measures out an arc between the two cities where they sit on the surface of the earth. He knew the distance between the two cities along the surface of the earth and calculated that the angle that swept out the arc between them was 7.2°. He then saw that this angle was 1/50 of the whole circumference (50*7.2°=360°). So, he multiplied the distance between the cities by 50 and came out with a remarkably accurate value for the circumference of the earth.

Greek Mathematics

• Eratosthenes is also known for the Sieve of Eratosthenes – a number sieve used to “catch” prime numbers while the composite numbers fall through.

• This is still one of the best ways to find prime numbers.

Greek Mathematics

• Archimedes was a master of the engine, catapult and compound pulleys.

• These were used against the Roman general Marcellus in the siege of Syracuse in 212 BC.

Greek Mathematics• And yet even Archimedes, who was a kinsman and friend of

King Hiero, wrote to him that with any given force it was possible to move any given weight; and emboldened, as we are told, by the strength of his demonstration, he declared that, if there were another world, and he could go to it, he could move this [world]. Hiero was astonished, and begged him to put his proposition into execution, and show him some great weight moved by a slight force. Archimedes therefore fixed upon a three masted merchantman of the royal fleet, which had been dragged ashore by the great labors of many men, and after putting on board many passengers and the customary freight, he seated himself at a distance from her, and without any great effort, but quietly setting in motion with his hand a system of compound pulleys, drew [the ship] towards him smoothly and evenly…

• - from Plutarch

Greek Mathematics

• Euclid of Alexandria ~ (325 BC – 265 BC)

• Very little is known of Euclid’s life apart from his monumental work The Elements.

• The Elements contains the essential ideas of plane and solid geometry and some early work on number theory.

Indian Mathematics

• The sulbasutras from the period of 1500 –800 BC contain detailed ritual geometry for the construction of altars. An approximation for the square root of two appears in these texts.

Indian Mathematics

• Brahmi numerals are the oldest known Indian number system.

• This was not a place value system, but had similarities to the Roman or Greek system.

• These numerals are found in caves and on coins from the period 300 BC – 400 AD.

Indian Mathematics

• The Jains of the 2nd century BC were known to explore ideas related to number theory, arithmetic, geometry, fractions, simple equations, cubic equations, quarticequations, and permutations and combinations.

Indian Mathematics

• By the 5th century AD, the Jains had developed the ideas of trigonometry to handle questions of astronomy (astrology).

• Aryabhata was known as the author of an important text summarizing much of the existing knowledge.

Indian Mathematics

• The Nagari (or Devanagari) Numerals were developed during the period around the 6th century AD, with the zero appearing in the 7th century.

Arab Mathematics• The Indian numerals were transmitted to the

Middle East and North Africa by trade with the civilizations of India and Pakistan (the Indus and Ganges river systems).

• One of the first mentions of the Hindu/Indian/Vedic numerals is a letter from Severus Sebokht in the year 622AD. He lived in a monastery at Qenneshre/Keneshra which is today a part of Syria in the upper Euphrates valley.

Arab Mathematics• “I will omit all discussion of the science of the

Indians, … , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value.”

Al-Khwarizmi

• Al-Khwarizmi was a 9th century Arab mathematician. He is best known for two books he is credited with writing.

• The word “algebra” comes from the title of one of his books Hisab al-jabr w’al-muqabala or The Compendious Book on Calculation by Completion and Balancing.

Al-Khwarizmi

• The title of Al-Khwarizmi’s other book is not known, but it was this book that explained the use of the Hindu symbols and the algorithms for calculation that we learn in elementary school.

• The word algorithm comes from Al-Khwarizmi’s name.

Fibonacci

• Leonardo of Pisa was born around 1170 AD, most likely in Pisa, Italy.

• His father was a trader and represented the interests of the merchants of Pisa in the port of Bugia in what is today Algeria.

Fibonacci• When my father, who had been appointed by his

country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.

• From Liber Abaci

Fibonacci

• Fibonacci wrote his book Liber Abaci in 1202 AD. This was to transform European mathematics and commerce over the next 300 years.

• At the time, Europeans still used Roman numerals which were fairly cumbersome to calculate with.

Fibonacci

• For calculation, Europeans used counter boards with places for chips of differing value.

• This “counter” survives today as our merchants put the machine that makes the calculation on the counter in their place of business.

Italian Mathematics

• Liber Abaci (1202) was written in Latin.

• After the development of the printing press, it was translated into Italian.

• This translation is known as the Treviso Arithmetic (1470) and led to an explosion of mathematical knowledge in Italy in the 16th century.

Italian Mathematics• From Victor Katz – “A History of Mathematics”

• “The Italian abacists of the fourteenth century were instrumental in teaching the merchants the Hindu-Arabic decimal place value system and the algorithms for using it. As is usual when a new system replaces an old traditional one, there was great resistance to the change. For many years account books were still kept in Roman numerals. It was believed that the Hindu-Arabic numerals could be altered too easily, and thus it was risky to depend on them alone in recording large commercial transactions. (The current system of writing out the amounts on checks in words dates from this time.) ”

Italian Mathematics• “The advantages of the new system, however,

eventually overcame the merchants initial hesitation. The old counting board system required accountants to carry around not only a board but a bag of counters, while the new system required only pen and paper and could be used anywhere. In addition, using a counting board required that preliminary steps in the calculation be eliminated as one worked toward the final answer. With the new system, all the steps were available for checking when the calculation was finished. (Of course, these advantages would have meant nothing had not a steady supply of cheap paper been recently introduced.) The abacists instructed entire generations of middle-class Italian children in the new methods of calculation, and these methods soon spread throughout the continent.”

Italian Mathematics• “In addition to the algorithms of the Hindu-Arabic

number system, the abacists taught their students methods of problem solving using the tools of both arithmetic and Islamic algebra. The texts written by the abacists, of which several hundred different ones still exist, are generally large compilations of problems along with their solutions…Recall that Islamic algebra was entirely rhetorical. There were no symbols for the unknown or its powers or for the operations performed on these quantities. Everything was written out in words. The same was generally true in the works of the early abacists and the earlier Italian work of Leonardo of Pisa [Fibonacci]. Early in the fifteenth century, however, some of the abacists began to substitute abbreviations for the unknowns.”

Italian Mathematics

• Calculating Table by Gregor Reisch: from the Margarita Philosophica, 1508

The Quadratic Formula

• Methods for solving quadratic equations which arose in situations involving areas as well as problems involving both multiplication and addition of unknowns had been developed by the Babylonians, Hindu, Greek and Arab mathematicians.

Italian Mathematics

In the early 1500’s Scipione del Ferro found a general solution for the depressed cubic equation.

qpxx =+3

In 16th century Italy, mathematicians didn’t publish their results, keeping them secret so that they could win the contests that were common at the time in Italy. del Ferro didn’t tell anyone about his discovery until shortly before his death in 1526. He then revealed the secret to a student of his named Antonio Maria Fior.

Italian Mathematics

• In 1535, Fior used this knowledge to challenge a better mathematicain named Niccolo Fontana to a problem contest.

• Fontana was known as “Tartaglia,” (the stutterer) because of a speech impediment caused by an old sword wound to his jaw.

Italian Mathematics• Tartaglia was a superior mathematician to Fior,

but didn’t know how to solve the cubic equation yet.

• In the time before the contest, he worked to find a solution.

• He finally found the same solution that del Ferro had found thirty years before and was able to win the contest.

Italian Mathematics• Word of Tartaglia’s victory spread among

mathematicians.

• Giralamo Cardano decided to see if he could get Tartaglia to reveal his secret solution of the cubic.

• Tartaglia first refused, but then told Cardano the formula, but not how to derive it.

• He also asked Cardano to promise not to reveal the result to anyone else.

Italian Mathematics

• Cardano eventually learned that del Ferro had found the solution first.

• Cardano also derived the formula that Tartaglia had shown him for himself.

• In 1545, Cardano published the solution of the general cubic in his book Ars Magna.

Italian Mathematics

Before the publication of Ars Magna one of Cardano’s students named Ludovico Ferrari found a solution for the quartic equation.

An quartic equation is an equation of the form:

0234 =++++ edxcxbxax

This solution was also published in Ars Magna which became known throughout Europe as the foundational text of classical European algebra.

Italian Mathematics• For 250 years European mathematicians would

work to find a formula to solve the fifth degree equation.

• In the early 1800s Niels Abel and Evariste Galois independently showed that it was impossible to find such a formula.

• Their work led to the development of Modern Algebra (also known as Abstract Algebra).

French Mathematics

• Francois Viete (1540-1603) is one of the first internationally recognized French mathematicians.

• He is credited with devising a scheme in which unknown quantities are represented by vowels and constant quantities are represented by consonants.

French Mathematics

• Rene Descartes later adopted this notation but used letters near the end of the alphabet for unknowns (x,y,z) and letters from the beginning of the alphabet for constants (a,b,c).

French Mathematics

• Marin Mersenne (1588-1648) was a French monk best known for his work in prime numbers.

• Mersenne also worked to determine that the frequency of a vibrating string is related to the length, tension, cross section and density of the material.

French Mathematics

• Rene Descartes (1596-1650)is known for developing the Cartesian coordinate system.

• Up to this time Geometry and Algebra had been separate fields of endeavor.

French Mathematics

French Mathematics

• The idea that the vertical and horizontal components of a geometrical figure could be described by an algebraic equation was a transformational breakthrough that led to the development of the Calculus.

French Mathematics

Pierre Fermat (1601-1665) is

known primarily for “Fermat’s

Last Theorem.” This theorem

says that given an equation in the

form ncnbna =+

there are no whole number

solutions for a,b and c unless 2=n .

French Mathematics

If , then this is the

Pythagorean Theorem .

2=n222 cba =+

Whole number solutions to this

equation are known as

Pythagorean Triples. Some

common examples are {3,4,5}

{6,8,10} {5,12,13} {7,24,25}

French Mathematics

Fermat famously wrote that he

had found a neat proof of this

statement, but was unable to fit it

in the margin of the book.

Mathematicians tried for 350

years to prove Fermat’s Last

Theorem until finally Andrew

Wiles succeeded in 1995.

French Mathematics

• Blaise Pascal (1623-1662)

• Pascal is best known for “Pascal’s Triangle.”

• This triangle illustrates counting relationships important in Discrete Mathematics, Probability and Statistics.

Pascal’s Triangle

Pascal’s Triangle

Development of the Calculus

• With the development of Analytic Geometry, mathematicians now had many questions to answer in analyzing a curve.


• Finding the equation of a line tangent to a curve

• Finding the maximum and minimum values for a curve

• Finding the area of a region bounded by a curve

• Finding the volume of a solid of revolution






• A number of mathematicians including Descartes and Torricelli had considered these questions separately over the years, but it would fall to Isaac Newton and Gottfried Leibniz to produce systematic solutions to these questions.

Isaac Newton

• Isaac Newton (1643-1727) entered Trinity College, Cambridge in 1661.

• There he learned his mathematics from textbooks by Viete, Descartes and John Wallis.

• Acccording to DeMoivre, Newton’s interest in mathematics came from his inability to understand an astrology text.

Isaac Newton

• In 1665 Cambridge University closed down due to an outbreak of plague.

• During this period (1665-1667) Newton returned to his family home in Lincolnshire and worked out the foundations of differential and integral calculus.

Isaac Newton

• One of the primary results of Newton’s work was the idea that finding the tangent line to a curve and finding the area under a curve were inverse procedures.

• While other mathematicians had successfully worked on these ideas, Newton’s genius was in seeing the relationship between these two ideas.

Gottfried Leibniz

• Gottfried Leibniz (1646-1716) studied in Paris under Christiaan Huygens after his university studies in Germany (Leipzig)

• Between 1672-1676, Leibniz developed a version of the procedures of calculus that had also been developed by Newton.


• There was controversy over the priority of the development of the calculus because Newton had developed his methods first, but not published them.

• Leibniz developed his equivalent methods independently shortly thereafter.


• It is generally acknowledged that Leibniz’s “differential” notation was superior and that the insistence of the English mathematicians on using Newton’s “fluxion” notation inhibited English mathematics in the ensuing years.


• It is also true that Newton’s physics was superior to that studied in continental Europe.

Mathematics of the 18th century

• The Swiss

• Jacob Bernoulli (1654-1705)

• Johann Bernoulli (1667-1748)

• Leonhard Euler (1707-1783)


• Jacob and Johann Bernoulli were brothers who were 12 years apart.

• They each produced a great deal of research on various problems related to calculus and analytic geometry.


• Leonhard Euler studied with the Bernoulli brothers in Basel.

• Johann Bernoulli’s son Nicolaus was the court mathematician in St. Petersburg, Russia, but died suddently in 1726.

• Euler agreed to take the position in St. Petersburg.

Leonhard Euler

• Euler was a masterful mathematician and worked applying mathematics and calculus to a wide variety of physical phenomena.

• Other mathematicians were often amazed by Euler’s abilities.

Leonhard Euler

• French mathematician Laplace is quoted as saying: “Read Euler, he is the master of us all.”

• Another French mathematician (Arago) is quoted as saying: “He calculated without any apparent effort, as other men breathe, or as eagles sustain themselves in the air.”

Leonhard Euler

• Euler contributed to the development of cartography, science education, magnetism, fire engines, ship building and navigation, artillery and ballistics, calculation of planetary orbits, the motion of the moon and optics, acoustics, the wave theory of light, hydraulics and music.

Leonhard Euler

One of Euler’s best known works is the relationship between the

values π,,,1,0 ie .

01=+πie

Euler can be seen as the fulcrum point that catapulted mathematics

from the days of Descartes, Huygens and Newton to the work

of Gauss, Fourier and Galois.

Birth of Modern Number Theory

• Adrien-Marie Legendre (1752-1833)

• Throughout the late 1700’s Legendre, Lagrange and Euler laid the groundwork for Gauss’ groundbreaking DisquisitionesArithmeticae in 1801

• Many of these ideas are based the concept of “residue” or remainder.

Birth of Modern Number Theory

• Though seemingly a part of “pure” or abstract mathematics, residues (remainders) are the basis for digital cryptography and computer security.

Carl Friedrich Gauss

• Carl Friedrich Gauss (1777-1855)

• Gauss was a mathematical prodigy whose talent was recognized at a young age.

• When Gauss was 7, his teacher asked the class to add the numbers from 1-100 so he could get some work done without being disturbed.


• A few minutes later, Gauss brought the answer to his teacher

• 1+2+3+4+….+99+100=5050• 1+100=101• 2+99=101• 3+98=101….


• Continue this pattern until we arrive at 50+51=101.

• This makes fifty pairs of numbers each adding up to 101.

• 50*101=5050


• Although Gauss’ DisquisitionesArithmeticae is concerned with abstract ideas, in later life Gauss concerned himself almost exclusively with applications.

• He was director of the observatory in Gottingen, Germany and also developed one of the earliest working telegraphs.

Evariste Galois (1811-1832)

• Galois is famous for his work proving that there is no formula to solve equations of the fifth degree or higher.

• His work led to the development of Abstract Algebra (also known as Modern Algebra).

• High school algebra is generally known as “Classical Algebra.”

19th Century Mathematics

• An explosion of new ideas and applications

• Non-Euclidean geometries lead to spherical geometry and hyperbolic geometry.


• Dirichlet and Dedekind continue the work of Gauss in analysis and number theory.

• Cauchy and Weierstrass gave the techniques of calculus a theoretical basis –150 years after Newton and Leibniz.


• Maxwell and Hertz develop equations to explain the behavior of electromagnetic waves.

Georg Cantor

• Georg Cantor (1845-1918) developed the foundations for modern mathematical logic.

• Cantor showed that though there are an infinity of whole numbers – there are more real numbers.

Georg Cantor

• The key to Cantor’s ideas is the ancient concept of a one-to-one mapping between objects.

• This goes back to the ancient efforts to create symbols and ideas for numbers.

• These early number systems were one-to-one maps to represent groups of objects.

Georg Cantor

• Cantor saw that there were one-to-one mappings for the whole numbers to the even numbers and whole numbers to fractions, but a one-to-one mapping for whole numbers to ALL numbers was not possible.

• This is the idea behind Cantor’s Diagonal Proof.

Kurt Gödel

• Kurt Gödel (1906-1978) was from a German-Austrian family and studied mathematics in Vienna, receiving his doctorate in 1930.

• Gödel is best known for his incompleteness theorems.

Kurt Gödel

• Throughout the early 20th century mathematicians had worked to systematize mathematics in order to firmly establish its logical foundation.

• Gödel’s Incompleteness Theorem showed that this was not possible.

Kurt Gödel

• Gödel’s work on incompleteness was focused on the idea of symbolic systems and rules – like our number system, or systems of algebra, languages and so forth.

Kurt Gödel

• The incompleteness theorem says that no symbolic system that is totally comprehensive can also be logically consistent – “there are always exceptions.”

• Also, any system that is logically consistent is therefore, by definition -incomplete.

Kurt Gödel

• The simplest example of this is the Liar’s Paradox.

• The statement “I always lie.” is irresolvable as being true or not true.

• Captain Kirk and Spock use this to defeat an android in an episode of the early Star Trek series.

Benoit Mandelbrot

• Benoit Mandelbrot (1924-2010) was born in Warsaw into a family of Jewish-Lithuanian roots.

• The family emigrated to Paris in 1936 because of the rise of Nazism in Poland.

Benoit Mandelbrot

• Mandelbrot received his PhD in Mathematical Science from the University of Paris in 1952.

• In 1958, Mandelbrot moved to New York to work at the IBM Thomas J. Watson Research Center in Yorktown Heights.

Benoit Mandelbrot

The starting place for studying Mandelbrot’s work is often the

“Mandelbrot Set” – a set of numbers in the Complex Plane

created by a formula.

CzZ += 2

Benoit Mandelbrot

Benoit Mandelbrot

Benoit Mandelbrot

• Mandelbrot researched a wide range of phenomena in applied fields including cotton prices and financial market behavior, thermodynamics, turbulence phenomena, acoustics and white noise, and cosmology.

Benoit Mandelbrot

• In each discipline, Mandelbrot saw the importance of self-similarity and behavior that contradicted the classical Gaussian statistical distribution.

• i.e. - “fat tails.”

Benoit Mandelbrot

Benoit Mandelbrot

• Just prior to Mandelbrot’s death, he addressed some of the issues of using classical statistical methods for computer trading in financial markets.

• These methods “usually” work.

Benoit Mandelbrot

• The idea of a “fat tail” is that events that classical statistics says should happen only every 10,000 years can actually happen three or four days in a row.

Benoit Mandelbrot

• This concept is closely related to the ideas of Nicholas Nassim Taleb, author of “The Black Swan” and “Fooled by Randomness”

Benoit Mandelbrot

• Mandelbrot was primarily a computer based mathematician. Earlier mathematicians had worked on the same ideas he did, but, because they didn’t have computers, the ideas they studied were called “useless monstrosities.”

The Future of Mathematics

• Computer based mathematics is a very important field, however, without the “hand-tools” of mathematics, it is less effective.

• Computers must be programmed using the ideas of classical algebra – variables and relationships.

The Future of Mathematics

• The ideas behind Mandelbrot’s work are based on simple rules that create complex processes.

• Without computers, it wasn’t possible to study processes this complex, but now it is and we can see that many complex phenomena are the result of simple rules.

Thanks for the pictures to :

Encyclopedia Britannica

University of Connecticut

Wikimedia

squarecirclez.com

Stephen Hartshorne

testfrenzy.com

James Schumaker Univ. of Hawaii

Jorge Rodriguez Cal State LA

Associated Press

New York Times

Gerard Pacillo

Math Forum at Drexel University

The History of Math - Clatsop Community College | Chart ... History of Math Richard W. Beveridge ... coins from the period 300 BC – 400 AD. ... important text summarizing much of

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