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"Saper Vedere" - "Knowing How to See" (Leonardo Da Vinci’S Motto)

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Michael York Dr. Greenwald Math 1010 24 June 2011 Timeline of Leonardo Da Vinci's Accomplishments [relating to math] "Saper vedere" - "Knowing how to see" (Leonardo Da Vinci’s motto) -1452: Born in the town of Vinci, Italy. *-1469: Starts apprenticeship in the workshop of Andrea del Verrocchio, a famous artist who did landscape paintings using the perspective technique. Here, Da Vinci studies perspective himself. *-1478: Begins writing first notebooks; begins studying duplicates of tables of squares as well as basic arithmetic. *→1478: Draws a version of Archimedes' screw in an engineering plan for contracting a tower (right image) *←1480: Draws a parabolic section of a cone (left image) -1483: Hired by Duke of Milan as a military engineer *-1483: Studies algebra in Milan, meets the father of famous mathematician Girolamo Cardano. -1489: Begins studying anatomy *-1490: Studies the transformation of volumes *-1493: Becomes recognized as a Geometer *-1496: Befriends mathematician Luca Pacioli, and Da Vinci illustrates his book "Divina Proportione" - containing the infamous Vitruvian Man image. →→→ *~1496-1504: Da Vinci entered an intense mathematical study during which he studies Euclid, quadrature problems (squaring circles), geometry, kinematic figures, doubling cubes, and pyramids. -1502: Cesare Borgia hires Da Vinci as a military engineer *-1502: Begins fixed compass problems → → → *-1503: Paints the Mona Lisa, a painting famous for incorporating the golden mean to achieve aesthetic appeal. *-1503: Claims to have successfully squared a circle. It wasn't until later that Klaus Schroer recognized that Da Vinci encoded this knowledge in the picture of the Vitruvian Man. Squaring circles involves creating a square and a circle of equal area. -1503: Worked with Michelangelo to produce murals for the town hall of Florence. *-1509: Discovered the Lunes of Alhazen, a method for finding the areas of certain shapes. (Image to the left) ~1513-1516: Moves to Rome and begins working for the Pope *-1515: Determines the surface area of a cone (image to the right) *-1515: Contributes hundreds of quadrature problems for "De Luda Geometrico", which was the guide for Geometry at the time -1516: Hired by King Francis I of France, moves to Amboise, France -1519: Dies on May 2 in Cloux, France Annotated Bibliography: A Timeline of Leonardo da Vinci's Mathematical Work <http://www.hypatiamaze.org/leonardo/leo_timlin.html> This website contained a lot of the information I used for this timeline as it gave specific events relating to Da Vinci's mathematical feats. In addition to the math discoveries, this page also contains information about many mathematicians who Da Vinci met and many of his employers. I included this source because it conveys information simply and without a lot of "fluff", which makes it much easier to put into my own words. I also included this source because all of the information I used from it was verified by other sources; even though this website did not include an author or a publishing date, I still felt as though it was credible and useful. O'Connor, JJ. "Leonardo da Vinci". 1996. <http://www-history.mcs.st-andrews.ac.uk/Biographies/Leonardo.html> This article was an excellent supplement to the other sources as it filled in a lot of biographical information about Da Vinci for my project. This summary of his life contained information about his private life as well as his working life, which made understanding the mathematical part much easier. The article was written in 1996 by a teacher at the University of St. Andrews in Scotland, and I take it to be a credible source, as the author includes a bibliography and many other side notes about his research. Phillips, John. "Leonardo da Vinci: the Genius who Defined the Renaissance". Washington DC: National Geographic, 2006. Pg 36-38, 58-60 This book by John Phillips, is a source that details the life of Da Vinci, instead of just his works. This biography tells of his early life, apprenticeships, struggles, challenges, and much about his legacy and accomplishments. Much of this book was unusable as it was not focused on the math aspect of his works, however, I did find some things I was able to incorporate into this project. Published by National Geographic as part of its world history biography series, this book proved to be a good source for this project, especially the sections dealing with his notebooks and his legacy. Veltman, Kim. "Leonardo da Vinci: A Review". Leonardo; 2008, Vol. 41 Issue 4, p381-388 This article summarizes the knowledge gained about Da Vinci in the 20th century, demonstrating that most of what we know about this man wasn't known until very recently, making much of the theories relating to his work open to interpretation. Veltman also summarized the contributions to the fields of physics, mechanics, and perspective made by Da Vinci. Yet, as an undercurrent in the article, the author reminds the reader often that much remains to be understood about his works. The journal this article was featured in is called "Leonardo", a publication that serves as a channel of communication between artists who use mathematics and science in their work. I think this source is acceptable and very valuable to the project. PICTURES: Picture of Archimedes' screw <http://www.lhup.edu/~dsimanek/museum/people/leo1.jpg> This picture, sketched in a notebook of Da Vinci's, shows his model of Archimedes' screw, a device used to transfer water from a low point to a higher point. Picture of parabolic cone with compass <http://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Leonardo_parabolic_com pass.JPG/220px-Leonardo_parabolic_compass.JPG> This picture was sketched in one of Da Vinci's notebooks, showing his workings with investigating the parabolic area of a cone. Picture of the Vitruvian Man <http://leonardodavinci.stanford.edu/submissions/clabaugh/images/vm/leonardo.jpg> This picture was essential to my project as it is probably one of Da Vinci's best known drawings, found in Luca Pacioli's book "Divina Proportione". Picture of the fixed compasses <http://cache2.allpostersimages.com/p/MED/29/2947/VNVRD00Z/posters/leonardo-da-v inci-two-types-of-adjustable-opening-compass-paris-manuscript-h-1493-4.jpg> This picture shows the compass that Da Vinci was renowned for using in his workings with geometry. Example of a part of the Lunes of Alhazen method <http://www.hypatiamaze.org/leonardo/leo_lune2p2.gif> This picture shows the drawing method that Da Vinci used to determine the area of certain shapes. There are many more pictures that go along with this one as it is a series of drawings that makes up the method. Picture of parabolic cone on person's head <http://farm6.static.flickr.com/5216/5454202036_fe5066973c.jpg> I used this picture because it was a sketch from one of Da Vinci's notebooks showing a method he used to find the surface area of a cone. .
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