Rene Descartes

  Rene Descartes seemed to offer some of the most profound and thought altering contributions to science and mathematics. Although his contemporaries did much for the advancement of thought in the 17th Century, Descartes's marriage of geometry and algebra seems to be one of the most monumental discoveries, as well as his dabbling with early calculus.

  Rene Descartes was born in what is actually modern day Descartes France. He was sent to college at a very young age, and because of his poor health, was allowed to sleep in until 11 a.m. or whenever he felt well enough to get up and begin his studies. He continued this practice for the majority of his life, until it was disrupted, at which point he literally died. After leaving school, he attended a military academy, as well as serving sometime as a soldier. He traveled around Europe for an extensive period of time, writing and staying relatively recluse. After pressure from many of his friends to publish and share his ideas, he moved to Holland where he finally maintained a more permanent residence and began to work.

  Descartes published a number of works, however holding one back, entitled La Monde, or The world, after hearing the retaliation of the Catholic church on Galileo for his blasphemous belief in the theory of Copernicus, which Descartes used in his book. Among the works he did publish was a book called, the Discourse on Method: Optics, Meteorology, and Geometry. In this he layed out many of his methods for testing and accumulation of evidence as well a the Cartesian Coordinate system. This system was the beginning of analytic geometry which allowed algebra to be visualized through geometry, and vice versa, geometry to be written as algebra.

  There are a number of accounts of how he made this discovery, one legend has it that he was laying in his bed (where he spent much of his time) and was watching a fly walk across the ceiling when he had the idea for his coordinates system

(http://ualr.edu/lasmoller/descartes.html?utm_source=http://ualr.edu/~lasmoller/descartes.html&utm_medium=700pxcustomerror404&utm_content=click&utm_campaign=custom404).  Another legend maintains that he was sleeping, when he had a fantastical dream that gave him the idea (http://oregonstate.edu/instruct/phl302/philosophers/descartes.html). I think that the fly story seems more plausible and likely. I can just imagine watching a fly walking in a sort of parabola across the wall and wishing to graph his journey and getting the idea for the use of coordinates.

  On top of analytic geometry and the beginnings of calculus, Descartes was a philosopher, and a scientist. In his work, La Methode, he lays out a very sound and logical approach to discovery, and in general, a way to live ones life. He arrived at four criteria for working on mathematics and science that I think could hold true in all walks of life. All four are quite long but I thought I would share the first of the four, "The first was never to accept anything as true if I had not evident knowledge of its being so; that is, carefully to avoid precipitancy and prejudice, and to embrace in my judgment only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it" Discours de la Méthode. 1637. Another quote from La Methode that I think sums up the general consensus of the other four criteria is,

"If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things"
Discours de la Méthode. 1637.

  Descartes was later summoned to teach a young Queen Christina of Sweden, who moved and lived at a very fast paced, scheduling their lessons for 5 a.m. Descartes was only there about a year before the shockingly early lessons, and the bitter wet cold of Stockholm got the better of him and he died of pneumonia in 1650 (http://oregonstate.edu/instruct/phl302/philosophers/descartes.html).

Nicolo Tartaglia

   After reading the story of the contest for solving the cubic equation, I was immediately interested in Nicolo Tartaglia, his life, his debates, and his other mathematical discoveries. After traveling to Italy I was also interested in his childhood, the towns he grew up in and lived, and where he taught and practiced mathematics.
    Tartaglia was born in Brescia Italy, he was born into a lower middle class family, and although his father was just a messenger, he was still well off enough to be educated for the first part of his life. However, after his father's death his family was catapulted into poverty and he spent the remainder of his early years being self taught. When he was a teenager he was injured by a french soldier' sabre to his jaw, the injury was nearly life threatening, and even though his mother could not afford medical treatment she was still able to nurse him back to health (http://www.gap-system.org/~history/Biographies/Tartaglia.html). The injury was concealed by Tartaglia's beard in later years, however, he had trouble talking his whole life.
    Tartaglia was obviously skilled at teaching himself, since he proved to be somewhat of a prodigy and was taken on as an apprentice to a patron who helped him further his education. He later became a mathematics teacher, holding positions at various schools until he finally arrived in Venice, where he became well known for participating in debates, leading us up to the point in his life when the famed contest between he and Fior over cubic equations occurred. Although Tartaglia prevailed, and discovered the method for solving them just before the debate, he did not however, publish, or reap any real benefit from having discovered the solution.
  Fior had only learned it from someone else, and had not discovered it like Tartaglia had, and was decisively less affective in the contest. Tartaglia did not publish his findings, instead sat on them, in hopes of saving them for a later date. This was his major flaw, since another Italian mathematician named Cardan was extremely interested in finding the solution and publishing it. Cardan managed to learn it form Tartaglia, but only under a condition of complete secrecy, however, after finding out that Fior's teacher had solved it first he felt that it wouldn't be breaking his oath to publish Fior's version.
  This obviously upset Tartaglia and bitter feud erupted between Tartaglia and Ferrari, Cardan's assistant. The two exchanged hate mail until their arguments finally culminated in a debate. Although Tartaglia was thought to be the more experienced mathematician, he was unpleasantly shocked by Ferrari's knowledge and skills. Ferrari and Cardan had done extensive work,Theorems and proofs for solving cubic equations and was well versed when it came time for the debate. Tartaglia left the debate after the first day, sensing an impending loss. This loss damaged his reputation and hurt his credentials as a teacher and mathematician, from which he never recovered.
   Tartaglia accomplished and published much in the way of mathematics on ballistic and artillery fire. His work would later be built on by Galileo and others. Although his intelligence did little in the way of pulling him out of the life he had started out in, he did contribute heavily to 16th century mathematics, and mathematics today(http://www2.stetson.edu/~efriedma/periodictable/html/ta.html).