The discussion was started by talking about art in the Renaissance. The idea of perspective in a painting began to be used in the Renaissance. To achieve realism, objects further away must be made to appear smaller. The painter Leon Battista Alberti (1404-1472) wrote a text on the subject of geometry as it relates to perspective in painting.

The main topic centered around solving the "cubic" problem. Several mathematicians of the fifteenth and sixteenth century built upon the work of the Islamic mathematicians. We discussed Scipione del Ferro (1465-1526) who discovered an algebraic method for solving the cubic equation x ^3 + cx = d. Del Ferro taught Antonio Fiore. Niccolo Tartaglia (1499-1557) claimed that he discovered the solution to the cubic equations of the form x^3 + bx^2 = d. Tartaglia told Gerolamo Cardano his secret, however Cardano published the work when he discovered that it had earlier been discovered by del Ferro. It is interesting to follow the long history of one problem.

After Dr. Barsky's commentary on the lack of a Nobel prize for mathematics and the mathematician of the day (Vickery), David Trigg began to talk about how the third dimension was represented in the art of this time period. The topics covered consisted of Copernicus and Kepler in Astronomy, the addition of perspective to make two dimensional art appear as three dimensional, Scipione Del Ferro, Antonio Fiore, Niccolo Tartaglia, Gerolamo Cardano and the "Artis Magnae", Libre de Ludo Aleae, Raphael Bombelli, and Simon Stevin.

Among the topics covered, Gerolamo Cardano interested me the most. According to the presentation, Scipione del Ferro first solved the cubic equation x^3 + Cx = d, and he passed the solution on to his student Antonio Fiore. Because neither of them had published this, word traveled around Italian mathematicians that this equation had been or would be solved, and Tartaglia said that he had solved the equation. In rebuttal, Fiore challenged Tartaglia publicly, but Tartaglia defeated him in argument and emerged as the winner. Cardano later published del Ferro's version of solving the cubic equation, and today mathematicians refer to it as Cardano's formula. David Trigg then demonstrated a version of this: x^3 + 45 = 98, which he solved by letting x = (p-q) and pq = 45/3. He then plugged these into the equation to get (p-q)^3 + 45(p-q) =98. David kept expanding the equation until he reached p^3 = {98 + sqrt [98^2 - (4 * -3375)]}/2. Dr. Barsky then added that Cardano had been the first to figure out that the solution to this cubic equation would also give you a solution to other cubic equations, and this set him apart from the others. According to David's presentation, Cardano predicted his own death and made it come true by committing suicide. Although Katz does not directly speak of this, Katz mentions that many unfortunate events occurred toward the end of his life that may have had some effect on him. Katz mentions that Cardano had been unsuccessful in his attempt to predict King Edward VI's horoscope, his wife died in 1546, his son was executed for the murder of his own wife in 1560, and the Inquisition judged him on accusations of heresy. At the time of the Inquisition, heretics and those judged as enemy of the church would be subject to torture or other means of punishment by the Inquisition. Brother Torquemada had been given the power of the Inquisition by Royalty, therefore Cardano must have been at their mercy. The Inquisition could have done anything to him. According to Katz, Cardano received a lenient sentence, however what did lenient mean at the time, and what did it mean to Cardano?

The Renaissance was a time of expansion, not just in economy, but also in cartography and art. It was during this time that perspective in art became important. Using one point perspective and two point perspective, three dimensional drawings were done. In mathematics, algebra was expanding. It was during this time that the first solution to the general cubic equation (x^3+cx=d) was developed. The first guy to come up with the solution was Scipione del Ferro. Later on, Niccolo Tartaglia found the solution. Each man only shared his solution with one person. Some of the other mathematicians from that time were Bombelli and Stevin. We did a problem of Bombelli's, but I got a little lost on it. Stevin was the first to introduce decimal fractions.

Cardano was the first to print the solution to the cubic equation. He had convinced Tartaglia to tell him the solution. No one had printed it before, because during that time the only way to get a teaching position at a university was by winning a contest. They didn't have tenure. Thus, having knowledge that others didn't could help you out. It seems to me that this method kind of crippled the flow of mathematics. If mathematicians were not willing to share new information or solutions, then there is the possibility that some great achievements went unnoticed. Also, it made it hard for mathematicians to collaborate, which can be very beneficial. Cardano had promised not to print Tartaglia's solution, but he was able to see del Ferro's solution which had been done before Tartaglia. Thus, Cardano printed del Ferro's solution. David demonstrated the method using the equation x^3+45x=98, instead of using a general c and d. He let x=P-Q and PQ=45/3. Then he substituted P-Q in for all x's in the equation and expanded it. The he substituted Q for 15/P. The P terms and fractions with P in the denominator cancel out. This leaves you with P^3 terms and fractions with P^3 in the denominator and numbers. By multiplying by P^3 and moving all terms to one side of the equation, you get a P^6 term, a P^3 term and a number. If you substitute z for P^3, you will have a quadratic equation to the second power. Then you just have to solve that, put P^3 back in for z and take the cube root of the answer. From this solution, one was to assume that you could use any c and d, from the equation x^3+cx=d, and get an answer. Cardano did have solutions with imaginary numbers, but chose to call them fictitious.

David started up where Jennifer left off. He covered Scipione del Ferro, Antonio Fiore, Niccolo Tartaglia, who wrote Libre de Ludo Aleae, and Gerolamo Cardano - the guy who committed suicide in order to be correct in his prediction of his own death.

Simon Stevin was also talked about along with his unique way of showing decimal place values. It was hard for me to believe that someone would actually use the method described because of the increased possibility of error it allows. I'm glad we use the notation that we do today and not this. ( I wouldn't stand a chance in mathematics.)

It also seems to me that the "solve for P&Q" method of solving cubic equations uses almost the same expansion that we used to find cube roots. I am finding that the higher up in mathematics I get, the more it (math) looks like something I've had before.

David's lecture on algebra in the renaissance began with a brief discussion of some of the many areas that improvements in mathematics had helped break new ground, such as the use of curved meridians in cartography and perspective in painting. Next he discussed the development of the solution for the general cubic equation, including some of the intrigue and politics surrounding it. I found this additional information very interesting, especially the part about the competitiveness among the mathematicians and the public challenges to obtain or retain academic positions.

The solution for the cubic equation, which came to known as Cardano's formula after it was first published by Gerolamo Cardano, even though it seems it was first discovered by both del Ferro and Tartaglia, was a solution to equations of the form x^3 + cx = d. But since any cubic equation can be rewritten in this reduced form, the solution can be extended to general cubic equations. The solution involves substituting an expression of the form a-b for x and setting the product ab = c/3. The equation can then be reduced to a quadratic of the form z^2 + qz + r = 0, with z = a^3. Cardano also accounted for the existence of negative roots and complex roots, which he called fictitious numbers. I found this part of the lecture particularly interesting and thought he did a good job presenting it.

The next mathematician discussed was Rafael Bombelli. Bombelli, while not a professional mathematician, wrote an important text on algebra that was aimed at the teaching of students. In this text Bombelli formalized the rules governing imaginary numbers and was thus able to help complete the picture concerning quadratic, cubic, and quartic equations with non-real roots.

Today David lectured on algebra in the Renaissance. I thought that the solution to the cubic equation had already been proved but today I learned more about x^3 + cx = d. Scipione del Ferro had discovered an algebraic method of solving this cubic equation, yet he did not publish his findings. It was to his advantage to keep it a secret. Before the death of del Ferro, he disclosed the solution to Antonio Fiore but Fiore did not do much with it. Another mathematician, Niccolo Tartaglia had said that he had in fact discovered the solution. Fiore challenged Tartaglia to a public contest and Tartaglia won. It had been declared that he had discovered the solution to the cubic equation. Gerolamo Cardano (who was our mathematician of the day on September 24, 1996), found that del Ferro and Tartaglia had the same solution, and to this day, the formula to the cubic equation is known as Cardano's formula. David showed solutions for x^3 + 45x = 98, and the proof of the general case (which works for any c and d). These were straight forward.

David talked about Raphael Bombelli. He was an engineer who had never had formal college training. He wrote a book called Algebra. This book began with elementary material and worked its way up to solving third and fourth degree equations. Bombelli also worked with imaginary numbers. In his notation he would write 2p di m3 for what we call 2 + 3i and 2m di m3 for our 2 - 3i. He is credited with multiplying by the complex conjugate.

This lecture was wrapped up with a bit of information on Simon Stevin. His contribution to the mathematical world was the creation of notation for decimal fractions. When writing a decimal expression with a denominator, in a circle above or after each digit, he wrote the power of ten assumed as the divisor.

On the handout that was given, we were to find the dimensions of the square inscribed in a triangle. It took a while for me to catch on to this, but I finally got it.