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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 8 – MONDAY 27TH SEPTEMBER Cubic equations It could rightly be argued that (once we accept imaginary numbers) the quadratic formula tells us all we need to know about how to solve quadratic equations. Since this was known a long time ago, it did not take long for ancient mathematicians to turn their attention to solving cubic equations. However this proved far more difficult, with a general solution not being produced until the 16th century. An excavated Babylonian tablet contains not only a list of perfect squares and cubes, but also the values of n3 + n2 for values of n up to 50. It is thought that they were used to solve cubic equations of the form x3 + x2 = a (see exercise 2.5.12 in the text). Significant progress on finding a general solution was made by Omar Khayyam (who is more famous for his poetry) during the 11th century, who extended some ideas of Archimedes to classify all cubic equations into fourteen different types, and provided geometric solutions that involved the intersection of conic sections. Ex. It was during the 16th century however that steady progress was made by a succession of Italian mathematicians, which ultimately led to a general solution being determined (see pages 319‐328). In the early 1500’s, Scipioni del Ferro (1465‐1526) discovered a method for solving all equations of the form x3 + px = q for positive values of p and q. However he kept the method secret until just before he dies, at which point he confided in his student Antonio Fiore. Armed with this knowledge, Fiore challenged Niccolo Tartaglia (1500‐1557) to a problem solving contest, as Tartaglia also claimed to be able to solve cubics. The winner was to receive 30 banquets from the loser. In the resulting contest, which lasted 30 days, Fiore was not able to solve any of Tartaglia’s problems, since they were all of the form x3 + px2 = q. Hence Tartaglia won the contest, though he refused the prize. Tartaglia’s triumph was of great interest to the eminent physician and mathematician Girolamo Cardano (1501‐1576), who decided to try and coax the method of solution from Tartaglia in the hope of publishing it in a book he was writing. Eventually Tartaglia told Cardano of his method, but under the strict instruction that he was notto publish it. Six years later though, Cardano did just that, claiming that it was the work of del Ferro. This started a three year feud between the two, which led Tartaglia to challenge Cardano to a problem solving contest. The challenge was accepted on Cardano’s behalf by his student Lodovico Ferrari (1522‐1565), who prevailed in the contest, causing Tartaglia to lose much of his reputation. Solving the generalized cubic equation Ex. Quartic equations Once the problem of solving cubic equations was resolved, it did not take long before quartics were conquered as well, this time with no dispute as to who to deserved the credit, as it was solely due to the work of Ferrari. Look ahead to quintics It would seem that this process of finding general solutions to higher order equations has no end, and so mathematicians tried to find one for quintic equations. However for 300 years no progress was made, until work by Paolo Ruffini (1765‐1822) and Niels Abel (1802‐1829) led to the theorem that there is no general solution to polynomials of degree higher than four. This result was then expanded upon by Evariste Galois (1811‐1832), who showed exactly which quintics could be solved using radicals. ...
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 Spring '08
 EVINSON
 Calculus, Equations

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