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).

Fibonacci Number Series

     After reading the post about Fibonacci's Number Series I was quite interested in the ramifications of his discovery and the possible applications of it. It is hard for me to comprehend how mathematical precision, certainty, and prediction could be found in something as wild and chaotic as the stock market or nature. I know it isn't necessarily Math History but I was interested in looking into all the different application of the Fibonacci Number Series for my independent study this week.
    Upon looking at the graph that was in your link in the Fibonacci lecture, I was amazed at the tendency of the stock to retrace itself with uncanny precision between those percentages which correlated to the percentages between some of the first few Fibonacci Numbers. Although the retracement wasn't picture perfect, it did seem to offer an educated guess at where the stock might go after reaching a certain percentage marker.
    After viewing the graph I decided to investigate other possibilities for the Fibonacci Number Series to be used. I was surprised at the application of the number series in nature. From the formation of galaxies to the formation of a Nautilus shell, which are both actually quite similar in formation and shape, just not size. As the Nautilus shell's spiral begins to fan out, its expansion, as the spiral begins to grow larger, can be measured and expressed in boxes of increasing size as seen below:
    
     http://ualr.edu/lasmoller/fibonacci.html?utm_source=http://ualr.edu/~lasmoller/fibonacci.html&utm_medium=700pxcustomerror404&utm_content=click&utm_campaign=custom404

    Another interesting application of the Number Series in nature is in flowers and the amount of petals that they contain. It was only after learning about the Fibonacci Series and its relevance in flowering plants that I realized that I haven't really seen alot of 4 petaled flowers, maybe a 4 leaf clover, but is that even considered a flower? Much of the Lilly family has 3 petals, Buttercups have 5 petals, some orchids don't exactly have distinguishable pedals, but mostly around 3 or so pedals on average. Even flowers with huge amounts of pedals such as daisies and other flowering plants from the Asteracae family have either 21, 55, or 89 petals, all Fibonacci numbers.
     Even in music, Fibonacci's Number Series seems to have relevance.  Many mathematicians agree that Mozart's Music shows strong indications of Fibonacci numbers even though there is no evidence to suggest that Mozart knew of Fibonacci or deliberately based music off of the series. In Mozart's Sonata No. 1 in C-major there are some striking Fibonacci numbers that stick out when counting "measures". The first section has 32 measures, and the last section has 68 measures, both Fibonacci numbers(http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm). This theme seems to be prevalent in modern day music and could just be due to the way the human ear recognizes and enjoys music. Even instruments bare a resemblance to the Fibonacci numbers, pianos especially. The scale consists of 13 keys, 8 of which are white and 5 of which are black, and split into groups of 3 and 2. It doesn't get more Fibonacci related than that.
    From spirals that make up our universe and Nautilus shells to stock reports and engineering marvels like the Parthenon. From flower's pedals to bunny rabbits subsequent generations, the Fibonacci Number Series is an interesting and somewhat mysterious mathematical tool that can make sense and order out of a seemingly chaotic world.

Greek Number System

Since Greek mathematicians, Philosophers, and thinkers seem to be one of the most prolific influences on western culture, and specifically the United States, I thought it would be fitting to begin by examining their number system and its roots.

      The Greek number system is largely similar to the Roman and Egyptian number systems. Identical to the Roman Number system because of the inheritance of many Greek ways by the Romans, such as religion, math and science. Similar to the Egyptian number system because of the large amount of trade done in that region during that time. In fact, although many believe that the Greek number system was developed by the Greeks between 475BC and 325BC, there has been some suggestion that in fact the System was adopted and amended by the Greeks after learning it in trade from the Egyptians (http://www.mlahanas.de/Greeks/Counting.htm).

      The Greek number system is a base 10 system, with actually two separate systems for naming and denoting of numbers. The first, using similar names as the second, but instead labeling numbers without symbols, but with the first letter of the name for that number. For example, the number 1 is called "Iota" and denoted by the letter "I". In the second system symbols are designated to represent numbers. For example, in the second system, the number 1 is called "Alpha" and is represented by an "A" or the symbol alpha seen in the first box on the left below.

     The latter system, called the "Ionian System", was used more often and was actually quite advanced and capable of some relatively complex calculations (at least for that time). Larger number were read by adding the value of the hundreds place with the value of the tens place and then with the value of the ones place, just as we do now days. For example, the number 523 is read as (500+20+3). Similarily, Φ+К+γ= ΦКγ. The first symbol in the previous additive equation is "Phi" and represents 5 in the hundreds spot. The second symbol in the sum is "Kappa" which represents 2 in the tens spot. Finally, the last symbol in the sum is "Gamma" which represents 3 in the ones spot. So, all together they represent "PhiKappaGamma" or 523.

     This Greek number system could also express fractions as well. Depending on the context, a sort of apostrophe type symbol following the number expressed a fraction. The apostrophe symbol signaled a 1 over whichever number or numbers was followed by the symbol. The "Ionian System" also allowed for more complex fractions by stating a number with a line over the top, this represented the numerator followed by the denominator which was again represented by an apostrophe symbol that followed the number

(http://www.math.tamu.edu/~dallen/history/gr_count/gr_count.html).