Student lecturer Jennifer Hineline started today's discussion with an overview of the social climate of the Renaissance era, specifically the economic situation. Instead of traveling to buy and returning home to sell, merchants would hire others to do the traveling and buying. This complicated the accounting process, bringing about a need for mathematicians as accountants. Ms. Hineline also made a connection between the ease with which Arabic numerals can be altered and the practice of writing out numbers in other ways, which is still in practice today in the writing of checks.
After a brief discussion of some of the symbolic notation used by some Italian algebraists, Ms. Hineline explained Paolo Gerardi's algorithm for adding algebraic fractions. This was very similar to what we teach in beginning algebra, except that instead of necessarily finding the lowest common denominator, Gerardi would simply use the product of the denominators of all the addends. Ms. Hineline then discussed Maestro Dardi and his method of solving cubic equations. Unfortunately, Dardi's method only works when certain relationships exist between the coefficients.
Ms. Hineline proceeded to introduce a series of mathematicians from France, Germany, England and Portugal, paying special attention along the way to Nicolas Chuquet from France. Chuquet theorized that given two ratios, a/b and c/d, the ratio (a+c)/(b+d) would fall between the two original ratios. Ms. Hineline demonstrated Chuquet's method of estimating square roots using his ratio theory. Although he never used the word limit, Chuquet was using a variation of the squeeze theorem and a limiting process to find square roots. Chuquet realized that square roots that are not integers are irrational, and he acknowledged that his method would never allow him to find a square root exactly. But by taking this method far enough, we can come arbitrarily close to the actual root, and we can estimate a root to any desired degree of accuracy.
The discussion began with the "maestri d'abbaco" or abacists of Italy. The abacists were professional mathematicians that wrote texts to be used to teach sons of merchants in newly founded schools. It was the fourteenth century's new shipping technology that brought about international trading companies centered in major Italian cities. The merchants needed math tools for calculating and problem solving.
It was interesting to note that the abacists expanded on the Islamic methods by using abbreviations, symbolism, and expanded algebra into equations of degrees higher than the second. Luca Pacioli, an ordained Franciscan friar of the 1470's, became famous as a teacher. He gathered mathematical material for 20 years and wrote the most comprehensive mathematics text of the time. His 600 page text was one of the earliest mathematical texts to be printed.
Jennifer Hineline gave the presentation today after Dr. Barsky's comments on the upcoming Mathematics Association of America's sectional meeting. The topics covered consisted of double entry book keeping, the Abacists, history of the Rennaissance, different mathematical symbols used by the Abacists, Dardi of Pisa, Luca Pacioli, the "Summa de Arithmetica, Geometrica, Propotioni et Proportionalita," Nicolas Chuquet, the "Triparty," Christoff Rudolff, Michael Stifel, the "Wortrechnung," Robert Recorde, the "Whetstone of Witte," Pedro Nunez, and the "Libro de Algebra."
Among this wide array of topics, I found the most interest in Nicolas Chuquet and the "Triparty." According to the presentation, Nicolas Chuquet discusses the concept of convergence in part of his book, the "Triparty". This concept of convergence deals with finding roots. As Jennifer explained, to find the root of six, Chuquet used two fractions, one too large and one too small. You add the numerator and denominator of the one too small to the numerator and the denominator of the last one too large, and then check to see if this new fraction is too small or too large to be the desired square root. By continuing this method you will get closer and closer to the root of six. According to Katz, Chuquet had been aware of the irrationality of the root of six, and developed a "new recursive algorithm to calculate it". He also applied this method to polynomials, however he gives no proof of the method's correctness. I think that it would be a difficult method to prove. Although you stop at the desired root, you could continue the method indefinitely. Although this may not be a problem for us today, it may have posed a problem for providing proof of the method back then.
Today's lecture was algebra during the Renaissance. Jennifer broke it up in the various European countries; Italy, France, Germany, England and Portugal. She further elaborated on various mathematicians from each place. Master Dardi of Pisa came up with a form similar to the quadratic formula. Nicolas Chuquet found a system to find the square roots of integers.
I was most impressed with a quote from Robert Recorde who said that mathematics sharpens the mind. I believe this is 100% true. I was an ENFP on the Briggs personality profile. This means I am intuitive and feeling; I would rather guess on my gut instinct than logically process events or situations. I have retaken the test recently and the scores are less skewed to one side. I believe my pursuit of mathematics has made me a more balanced individual.
I also perked up when I heard that Luca Paciolo might have worked with Leonardo de Vinci. I read through Katz but I didn't find any references so I went to the Internet and looked. There I found a reference to an unpublished work that included recreational problems and proverbs that Pacioli worked on with de Vinci.
Jennifer started her presentation a with description of the commercial revolution that occurred during the thirteenth Century and its effects on mathematics. She showed to the class the how algebra was done by the Italian mathematicians Paolo Gerardi and Dardi of Pisa. She also spoke of Luca Pacioli, Nicolas Chuquet of France, Christoff Rudolff and Michael Stifel of Germany, Robert Recorde of England, and Pedro Nunez from Portugal.
The most fascinating mathematical part of the presentation for me was the approximation method for the square root of a number; the number we used was 6. This approximation was done by narrowing down the solution in between a "too large" value and "too small" value. This was done by adding the numerator and denominators of the mixed number values for the large and small boundaries. This new fraction, along with the whole number was squared and evaluated to be too large or small, then the process was repeated until a satisfactory value was found.
Today's lecture was on the first part of algebra in the Renaissance. Jennifer began by letting the class know that the changes in the European economy in the fourteenth century had an effect on mathematics. The cultural movement of the next two centuries was known as the Renaissance.
We learned Paolo Gerardi's rule for adding fractions. In his book Libro de ragioni of 1328, you will find the rule for adding the fractions 100/x and 100/(x + 5). He has it written out in words, and it was at this time that some of the abacists began to substitute abbreviations for unknowns.
Jennifer then spoke about Master Dardi of Pisa. He had expanded al-Khwarizmi's six quadratic equations to 198 equations up to the fourth degree, where zero was still not considered a solution. Many of these were reducible to one of the standard forms. He claimed to have a solution to cubic equations. Jennifer explained to the class that he was basically completing the cube, but that method only works for special cubic equations.
Next, Jennifer talked about Luca Pacioli, one of the last of the abacists. He taught mathematics all over Italy and wrote three books. Pacioli spent 20 years gathering materials to write a complete mathematical treatise, and in 1494 he completed a 600 page and the most comprehensive text of the time. This was also one of the earliest texts to be printed. It was entitled Summa de Arithmetica, Geometrica, Proportioni et Proportionalita.
Nicolas Chuquet was the next to be discussed. He was a French physician, not a mathematician, and he composed his Triparty in 1484. The first part of Triparty is concerned with arithmetic. In the second part, Chuquet applies his fraction rule to the calculation of square roots of numbers which are not perfect squares. The third part is more strictly algebraic; it deals with exponents. Jennifer then showed the class how to find the square root of six. To do this, you need to find a fraction between two fractions that you have chosen to start with, add the two and square the results. This tells you whether the new fraction is too large or too small. If it's too small, have it replace the old small fraction, and if it's too large, have it replace the old small fraction. Continue this process until your answer is very close.
Christoff Rudolff of Germany was discussed next. He was the first to use a symbol for "equals." It was a period. Rudolff also wrote Coss, which was the first German algebra, in Vienna. Rudolff also worked with solving equations but instead of using the six-fold classification of quadratic equations, he used an eight-fold classification. And again, he doesn't deal with negative roots or zero as a root.
Michael Stifel revised Rudolff's book in 1553. In 1544, he had written a text of his own, the Arithmetica Integra and this included the second version of the "Pascal Triangle."
The books written by Rudolff and Stifel were important in Germany, and they also influenced England. Robert Recorde wrote The Whetstone of Witte. His greatest contribution was = (the equal sign). He used this because a pair of parallels or lines of one equal length, "because no two things can be no more equal."