Italian Mathematics of the 16th Century

To connect the last post to this one, I will quote a passage from Victor Katz’s A History of Mathematics textbook.

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.)  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 eleiminated 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.

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.

In November of 2008 I wrote about some of the history of algebraic symbols, most of which was taken directly from Jeff Miller’s website Earliest Uses of Various Mathematical Symbols.

The story of the Italian mathematicians of the 16th century and their efforts to solve cubic and quartic equations is fascinating and filled with academic intrigue…

Solving the Cubic

November 21, 2008 by richbeveridge

Italy was a center of mathematical activity after the publication of Fibonacci’s Liber Abaci (1202) and the Treviso Arthmetic of 1478.  These books formed the foundation for European mathematics.

At some point in the early 1500′s, an Italian mathematician named Scipione del Ferro determined a general solution for what is known as the depressed cubic equation.  This is a cubic equation without any x² terms.  The general form is : x³+px=q.  As it turns out, any cubic equation of the form x³+bx²+cx+d=0 can be written as a depressed cubic, but that came later.

At the time, mathematicians didn’t publish their results, but, instead, kept them secret so that they could win the problem contests that were common at the time in Italy.  As a result, 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.  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.  Tartaglia was a superior mathematician to Fior, but didn’t know how to solve the cubic equation yet.  So, in the time before the contest, he worked feverishly to find a solution.  Finally, he found the same solution that del Ferro had found thirty years before and was able to win the contest.

Word of Tartaglia’s victory spread among mathematicians and Giralamo Cardano decided to see if he could get Tartaglia to reveal his secret to the solution of the cubic.  At first Tartaglia 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.  Eventually, Cardano learned that del Ferro had found the solution first.  Cardano also determined the derivation of the formula that Tartaglia had shown him for himself.  So, in 1545, Cardano published the solution of the general cubic in his book Ars Magna.

It is interesting that Cardano encountered the square roots of negative numbers in working with his “cubic formula.”  These numbers, which today are called imaginary, or complex numbers were almost completely unknown at the time.  Cardano was initially flummoxed by these numbers that seemingly had no physical meaning.  However, in true trailblazing spirit, Cardano wrote in his book that although these numbers were unfamiliar, “nevertheless, we will operate,” with them using rules similar to those used for the standard number system.

Before the publication of Ars Magna one of Cardano’s students named Ludovico Ferrari found a solution for the quartic equation.  That is, an equation of the form x⁴+bx³+cx²+dx+e=0.  This solution was also published in Ars Magna which became known throughout Europe as the foundational text of classical European algebra.

The next step in the story took place over the ensuing 250-300 years.  Finally, in the early 1800′s it was shown that the quintic or fifth degree equation was not solvable by formula.

The stories of these mathematicians is told in An Imaginary Tale, by Paul Nahin and The Equation that Couldn’t be Solved by Mario Livio.

Although it’s somewhat out of historical sequence, you can read about the work of Abel and Galois in the early 1800s here.

The History of Algebra

I wrote a few posts back in October of 2008 about the history of algebra and the transmission of Hindu and Arab knowledge to Europe.

Al-Khwarizmi and Algebra

October 24, 2008 by richbeveridge

Abu Ja’far Muhammad ibn Musa Al-Khwarizmi was an Arab mathematician who lived and worked in Baghdad during the 8^th and 9^th centuries.  Not much is known of his life.  He is primarily recognized as the author of two books, one on arithmetical computation and one on algebra.  The word algebra comes from the title of his most famous book Hisab al-jabr w’al-muqabala or The Compendious Book on Calculation by Completion and Balancing.  In this book Al-Khwarizmi discussed methods of solving linear and quadratic equations, although he used prose writing to do this.  The notation that we use today in algebra class was developed by European mathematicians between 1400-1800.

Al-Khwarizmi’s other notable book was about the Hindu-Arabic place-value number system that we learn about in elementary school.  The Hindu used 10 symbols to represent the numbers 0-9 with positional or place values to represent larger numbers.  The other important subject of this book were the methods of calculation that we learn in elementary school.  These “algorithms” for calculation in the number system allow us to add, subtract, multiply and divide any collection of numbers.  The Arab title of the book is not known.  It was translated into Latin as Algoritmi de numero Indorum or, in English, Al-Khwarizmi on the Hindu Art of Reckoning.  The word “algorithm” comes from the Latin representation of Al-Khwarizmi’s name.

The usefulness of Al-Khwarizmi’s work is explained by the author himself

… what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

Here is a link to a biography of Al-Khwarizmi

Here is a link to a Wikipedia article on his book

Next week I’ll write about the importance of Al-Khwarizmi’s work to European mathematicians.

Fibonacci

October 28, 2008 by richbeveridge

Last week, I wrote about the Arab mathematician Al-Khwarizmi and the origin of the word algebra.  The European and American cultures received much more than just a word from Arab mathematicians.  In the 12th century, a man named Guilielmo Bonacci lived in Bejaia, Algeria, where he represented the interests of the traders of Pisa, Italy.  His son, Leonardo, was tutored in the Hindu-Arabic mathematics that Al-Khwarizmi had written about several hundred years before.  Leonardo, who became known as Fibonacci, wrote in his book Liber Abaci:

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.

It was this book, Liber Abaci, written in 1202, that led to the adoption of the Hindu-Arabic numeral system by the Europeans.  At the time, Europeans were still using Roman numerals, which are cumbersome for calculation.  The Hindu-Arabic numerals and the system of calculation that goes along with them are very useful in both business and science.

The second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.

In 1478, the portions of Liber Abaci that dealt with trade and currency conversion were translated into Italian, so that people who didn’t read Latin could understand the ideas as well.  This book, known as the Treviso Arithmetic, spread these ideas to a much wider audience than was possible through a book written in Latin.  This allowed the Italian merchants to keep much better accounts, enabling trade to flourish.

Previously, calculations had been made using an abacus, but with the Hindu-Arabic numbers and calculation system, people had a much easier time keeping track of their money.  Before the advent of the calculation system, merchants had kept a table in their business that was used as a makeshift abacus in order tally up sales totals.  This table was known as a counter.  Although the counters in today’s businesses are not directly used to tally things up, the machines that do this job are kept on the “counter.”

Fibonacci is probably best known for a series of numbers that bears his name – the “Fibonacci Series.”  This series is derived from a problem in Liber Abaci and is created by adding the two numbers in the series to obtain the next.  It starts with ones, which are added to get 2.  Then the 1+2=3, then 2+3=5, and 3+5=8 and so on.  The series begins 1,1,2,3,5,8,13, 21,…  There are many applications where this seemingly trivial series of numbers is useful.

One of the more surprising uses of the Fibonacci Series is in stock trading, where investors use the ratio of Fibonacci Numbers to analyze the behavior of stock, bond or commodity markets.

Here is a link from the Forbes Investopedia

Applications from the Ancient World

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

The Greek knowledge of mathematics and its applications is vast.  I will touch on a few topics here before discussing the other cultures I mentioned last week.  A lot of effort in the ancient world was focused on moving heavy objects, often in an effort to build structures of various sorts.  The systems of compound pulleys mentioned in the passage above that were used in Greece and Rome are often referred to “engines” and individuals who built and worked these devices were known as “engineers.”

Eratosthenes is famous for using mathematics to measure the size of the earth in 240BC.  Using the angle of the sun at the summer solstice, he was able to calculate the earth’s circumference.  The picture in the previous link helps a lot to understand the reasoning used in this calculation.  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.

Mayan Mathematics

The Mayan mathematical system was very closely tied to their calendar, which is, of course based on astronomical calculations and events.  The Mayans actually used two calendars – one of 365 days and the other of 260 days.

The 365 day calendar was divided into 18 twenty day months with a brief ceremonial month of 5 days at the end of each year.  This 5 day month is often considered to be an “intercalary” period which must be used in all calendar systems to keep them accurate.

Our leap year is an example of intercalation, but even just adding one day every four years makes a difference over 1,000 or 1,500 years, pushing the calendar ahead of the season.  Our Gregorian calendar was developed after the Catholic church found themselves celebrating Easter in late winter instead of early spring.

The Mayans used their calendars to keep track of the motion of the celestial bodies.  One of the more common astronomical observations they made was the transits of Venus.  Venus crosses between the earth and the sun (here is a great picture from 2004) twice in eight years every 120 years or so.  Knowledge of and the ability to predict this event indicates a high level of sophistication in terms of understanding time and celestial patterns.

Most of the knowledge of what the Mayans actually did was lost when Bishop Diego de Landa ordered all Mayan texts burned in 1562.  Currently, there are only three known remaining Mayan texts (or codices) still in existence – The Dresden Codex, The Paris Codex and The Madrid Codex – each one named after the European city in which it is stored.

The Mayans combined their advanced astronomical knowledge with skills in construction to build elaborate ceremonial structures which would catch the rays of sunlight at particular times of year.  Mayan observatories were built so that particular windows established a fixed point of observation so that as celestial bodies moved in and out of the line of sight, the astronomers could more easily keep track of the movement.

Many Mayan structures are also aligned with the position of the sunrise on the horizon for the equinoxes and winter and summer solstice.

Egyptian Mathematics

Our knowledge of Egyptian mathematics comes mainly from the problems contained in the Rhind Papyrus and the Moscow Papyrus, two Egyptian documents that were purchased by Europeans in Egypt in the 19th century.  Each one contains a variety of problems designed to help teach mathematical methods for calculation, determining ratios, simple algebra, mathematical conversions and geometry.

Surveying was also an important part of Egyptian mathematics.  Every year the Nile would flood its banks and many properties would be inundated with water.  In order to know whose fields were where, the Egyptians needed to be able redetermine the outlines of their fields.

Babylonian Mathematics

The Babylonian culture existed in what is present day Iraq about 4,000 years ago.  Their number system is known to us because of the clay tablets on which they recorded information, including bookkeeping and mathematics.

The Babylonian texts are essentially collected problems for students to work through:

The Babylonians were strong believers in word problems. Apart from a few procedure texts for finding things like square roots, most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. Students had to find lengths of canals dug, weights of stones, lengths of broken reeds, areas of fields, numbers of bricks used in a construction, and so on.

Chinese Mathematics

The mathematicians of ancient China were concerned with surveying and calculating distances such as the height of a tree, the depth of a valley, the width of a river, or the distance to an island.  There was also fairly advanced solving of systems of linear equations as well as higher degree polynomial equations.  Medieval Chinese mathematicians are also credited with The Chinese Remainder Theorem which is a fairly involved piece of advanced mathematics that was devised to solve a problem of remainders – for example

There are things of an unknown number which when
divided by 3 leaves 2, by 5 leave 3, and by 7 leave 2. What is the (smallest) number?

There are a lot of topics here to explore, focus on something that interests you and research it a little more.  Please try to have something posted by Monday February 7th.  Let me know if you have any questions.

The Pythagoreans

Back in October of 2008 I wrote about the Pythagoreans – which is what the group led by Pythagoras was called.

Here’s a link to that post.

I’ll write next week about some of the applications of mathematics in the ancient world.

Representing Numbers

The way different cultures have chosen to represent numbers can be a very interesting topic.  Many people forget that, like our English language, the base-ten place value system is a somewhat arbitrary cultural choice that we have made as a system to represent quantity.

Other cultures use a variety of strategies to represent quantity.  Our system is a Base 10 place value system which was also used by the ancient Hindu and Chinese cultures.

The Mayans used a Base 20 place value system, the Babylonian/Mesopotamians used a Base 60 place value system.

The Egyptian and Roman systems were additive systems in which the symbols represented a constant value, whereas in a place value system, the value of the symbol depends on its place.

For instance the “4″ in 945,721 represents 40,000 because it’s in the 10,000s place, but the “4″ in 278,451 represents 400 because it’s in the 100s place.

The Mayan and Mesopotamian systems were similar except that they used bases other than ten.  The Math history site at St. Andrews University in Scotland has great information on a lot of these topics.

The Mayan Number System

Here is a picture of the Mayan number system

You can see that the dots represent ones and the lines represent fives.  The Mayans used a vertical positional system to represent numbers bigger than 19.  For instance 20 would be represented as a dot above the shell symbol for zero, representing one in the “20s” place and zero in the ones place.

You can check out the Wikipedia page on Mayan numbers here, and the St. Andrews page here.

The Babylonian Number System

The Mesopotamian number system used symbols for one and for ten to build numbers up to sixty and then switched to a positional place system for numbers over sixty.  Here is a picture of the Babylonian/Mesopotamian numerals

Here is a link to a Google search that shows different forms of the Babylonian Numerals.  Again the Babylonian system was a place value system, so that for numbers larger than 59, they would move to the left and put the “one” symbol in the “sixties” place and whatever other symbols were necessary in the ones place.

Here are links for the Wikipedia article, and the St. Andrews site.

The Roman and Egyptian Systems

The Roman and Egyptian number systems were mainly additive systems with symbols representing the powers of ten grouped together to represent whatever quantity was necessary.

Wikipedia has info about the Egyptian symbols here and on Roman numerals here.

St. Andrews has a site on the Egyptian numerals here.

The Roman numerals were used throughout much of Europe until the Middle Ages.  Trading between Italy and North Africa introduced Italians to the Arabic number system, which then spread throughout Europe.  We’ll spend more time on this topic later because it’s an important time in European mathematics.

The Hindu-Arabic Numerals

The Arabic 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 622CE which says

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.

He wrote this from the monastery at Qenneshre/Keneshra which is today a part of Syria in the upper Euphrates valley.

The Vedic Number System

The origin of the Vedic or Indian number system lies thousands of years in the past.  The Vedic texts are called Sulbasutras and contain religious geometric constructions that are quite advanced mathematically.

The Indian number system evolved over several thousand years until the base ten place value system was developed during the 6th century CE.

The approximation of √2 from the Sulbasutras is very accurate and quite extraordinary.

Here is the St. Andrews article on the development of numbers in ancient India and Pakistan.

The Chinese Number System

The Chinese number system is also very ancient.  The oldest known record of Chinese numerals comes from the 15th century BCE.  It was an additive system similar to the Roman or Egyptian systems.

A thousand years later in the 5th century BCE the Chinese began to use counting boards and counters made of bamboo or ivory which were placed in the cut out portions of the counting board.  Shallow depressions were made in the counting boards to gather the counters in and each position represented a different power of ten.

This process later led to the development of abacus.  Counters and abacuses were also used in Europe until the introduction of the Hindu-Arabic system.  Merchants had counting boards set up in their shops and would tally up customers purchases using their counting board or “counter.”

Today, we often still use the counter in a shop to tally up our purchases, although the calculation is generally done by a machine that sits on the counter!

Also, in ancient China, the development of mathematics is best recorded in the book known as The Nine Chapters on the Mathematical Art.  There is a variety of important mathematical ideas contained in this text, which is about 2,000 years old.

The St. Andrews article on the development of the Chinese number system is here.

History of Math

I will use this blog to post information and links related to the History of Mathematics.

Probably the first thing to do is post a link to the Math History site at St. Andrews.