Thursday, December 8, 2011

James Clerk Maxwell 1831-1879



 Early Life

     James Clerk Maxwell was born in Scotland in 1831. Maxwell’s early education was given to him by his mother, she was a dedicated Christian and his studies included the Bible. He had an exceptional memory and he showed that by memorizing all of Psalm 119. By the time Maxwell was 8 years old he became bored with his simple toys and started to conduct small scientific investigations. He liked to reflect sunlight off of a tin plate and show his parents how he captured the light and could “control” where the sun was. His mother wrote a letter to her sister describing his curiosity. Here is a passage from the letter-

“He is a very happy man, and has improved much since the weather got moderate; he has great work with doors, locks, keys etc., and 'Show me how it doos' is never out of his mouth. He also investigates the hidden course of streams and bell-wires, the way the water gets from the pond through the wall and a pend or small bridge and down a drain...”

     Sadly his mother died when he was still 8 years old from abdominal cancer, as he would too later in his life. His father got him a tutor to continue on with his studies after she died but the boy tutor was just 16 and the arrangement wasn’t very successful. In 1841 it was decided that Maxwell would get a formal education at the Ednburgh Academy. He had poor health that kept him absent from school a lot, but he still managed to make excellent academic progress. When he was just 15 year old he published his first scientific paper called A Mathematical Analysis Involving the Ellipse. In 1847 he went to the University of Edinburgh and soon after published two more scientific papers. Maxwell continued to publish papers all throughout his life on a variety of subjects that included the mathematics of human perception of colors, the kinetic theory of gases, the dynamics of a spinning top, theories of soap bubbles, and many others. In 1850 Maxwell left Scotland and enrolled at Cambridge University. He had originally attended Peterhouse but transferred to Trinity College where he graduated four years later with a degree in mathematics.

Contributions

     James Maxwell showed that phenomena of magnetism, electricity, and light were just different manifestations of the same fundamental laws. He made improvements to the kinetic theory of gases. He also did research on radio waves, radar, and radiant heat, and explained them by a unique system of equations. These equations first appeared in their modern form of four partial differential equations; these were in Maxwell’s textbook he wrote in 1873 called A Treatise on Electricity and Magnetism. The equations were a result of when Maxwell first began his work in electromagnetism. He looked at Michael Faraday's theories of electricity and magnetic lines of force that led him to see the connections between the approaches of Reimann, Faraday, and Gauss. The equations he derived could describe and quantified the relationships between magnetism, electricity and the propagation of electromagnetic waves. Today these equations are known as Maxwell's Equations. An example of when he first used his equations was when he used them to calculate the speed of an electromagnetic wave; he found that the speed of that wave was almost identical to that of the speed of light. Because of this, he was the first person to purpose that light was actually an electromagnetic wave. This was a very big achievement because not only was light an electromagnetic wave but it could also be said that electricity, magnetism and light could be understood as aspects of the phenomenon of electromagnetic waves.

     In 1857 Maxwell decided to compete for the Adams Prize of 1857 when he found out the subject was on The Motion of Saturn’s Rings, which he had already theorized about ten years earlier. He spent two years on developing a theory to explain the physical composition of the rings. He showed that stability could be achieved only if the rings consisted of numerous small particles. He won the Adams Prize and almost a hundred years later his theory was confirmed by the Voyager 1 space probe.

    Maxwell also had a hand in the development of color photography. He invented the trichromatic process, which by using red, green, and blue filters he created the first color photograph. 


“Maxwell proposed that if three black-and-white photographs of a scene were taken through red, green and violet filters, and transparent prints of the images were projected onto a screen using three projectors equipped with similar filters, when superimposed on the screen the result would be perceived by the human eye as a complete reproduction of all the colours in the scene.” 


James Clerk Maxwell - Red, Green and Blue sliedes of a tartan ribbon, used to demonstrate colour photography




    The pictures are of a tartan ribbon and each of the pictures was made by using a black and white slide; the slides were exposed through red, green and blue filters giving them their own color. Maxwell also had one taken through a yellow filter but it was not used in the demonstration. 



“When brought into alignment, the three images (a black-and-  red image, a black-and-green image and a black-and-blue  image) formed a full color image. Thus demonstrating the principles of additive color." 


    Before he delved into color photography he was studying other types of optics and he wrote a paper called, On the Theory of Colour Vision, that was awarded the Rumford Medal. In the paper Maxwell talked about how color blindness was due to individuals not being able to recognize red light. His experiments for colored light were very well constructed and made use of a color box designed by Maxwell himself.

   Maxwell continued his work until he took ill with abdominal cancer in 1879 and was forced to resign his position at the Cavendish Laboratory, where he had been working since 1874. He died on November 5th 1879. James Clerk Maxwell had an amazing mind and made countless contributions to the many fields of science. He developed numerous theories that we still use over a hundred years later. 













Monday, November 28, 2011

Augustin-Jean Fresnel 1788-1827



     Fresnel was a 19th century French physicist. He was the son of an architect and was schooled by his parents but was considered to be a slow learner when at the age of 8 he still had difficulties reading. When he was sent to an actual school for a formal education at the age of 12 it was then that he was introduced to science and mathematics; he excelled greatly at both. 

     He decided he wanted to become an engineer and in 1806 he went to the School of Civil Engineering. After graduating, he worked for several years on engineering projects for the French government departments but due to Napoleon returned from Elba in 1815 he lost his post but only temporarily. During the time when he wasn’t working for the government was when he started to become interested in optics.

     Fresnel came up with formulas that could explain refraction, double refraction, reflection, and the polarization of light reflected from a transparent substance. The work he did supported the concept of transverse vibrations in light waves, which he evaluated mathematically and the wave theory. He also developed a method that ended up producing circularly polarized light. In 1819 he was nominated a commissioner of lighthouses and began to encourage that mirrors be replaced with compound lenses in lighthouses, which he invented a lens for the task. The type of special lens he constructed for lighthouses is called today the Fresnel lens. Here is a little information on those lenses and why he needed to construct them differently. 

     "Faced with the need to construct a large lens for a lighthouse of appropriate focal length, but unable to support the large weight of a double convex lens of that size, French physicist Augustin Fresnel (1788-1827) reasoned that it was the surface curvature which gave the focusing power. He reproduced the surface curvature of the thick lens in sections, maintaining the same focal length with a fraction of the weight. The lens strength in diopters is defined as the inverse of the focal length in meters.http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/fresnellens.html 



     The type of lens he invented is used in many lighthouses around the world and has a very distinctive shape, sort of resembling a beehive with a lamp in the center. 

http://en.wikipedia.org/wiki/Lighthouse 
     "The lens is composed of rings of glass prisms positioned above and below the lamp to bend and concentrate the light into a bright beam. The Fresnel lighthouse lens works so well that the light can be seen from a distance of 20 or more miles." http://micro.magnet.fsu.edu/optics/timeline/people/fresnel.html


     The lighthouses before he invented his lens, the ones that used mirrors, could only reflect light to small distances and reflect barely enough light in fog or misty nights.

     During his lifetime Fresnel was awarded for his work and was was a member of many societies. He read a memoir on diffraction of light in 1818, which he received the prize of the Academie des Sciences at Paris and in 1823 he was unanimously elected a member of the French Academy of Sciences. Two years after that in 1825 he also became a member of the Royal Society of London who later awarded him the Rumford medal. He died shortly after that in1827 of consumption on the 14th of July. 















Sunday, November 13, 2011

John Wallis (1616-1703


      John Wallis was an English mathematician born in Ashford, Kent, England.  He was first intended to be a doctor but his interested circled around mathematics. After receiving his Masters degree he entered the priesthood as a scribe for 6 years. He made many contributions to trigonometry, calculus, geometry and the analysis of infinite series along with other fields in English grammar, Theology, Logic, and Philosophy.. He was also interested in cryptography and he used his interest along with his knowledge in the area to decode Royalist messages for the Parliamentarians. This was during the time of the Civil War between the Royalists and Parliamentarians. Around that time he also joined a group of scientists that were later called the Royal Society and 6 years after that in 1649 he was appointed as chair of Geometry at Oxford University, where he stayed until his death in 1703.

     John Wallis among with all his brilliance was also an insomniac and used to do mental calculations in his head while lying in bed at night. He once calculated the square root of a 53-digit number and later recited it correctly from memory to 27-digit accuracy. He also designed a structure that was cleverly put together by using what we call today as structural analysis, solving a set of 25 simultaneous equations by hand.

“Wallis designed a structure that could span a square space while only being supported at the edges. It was made up of a pattern of short and identical interlocking pieces, each of which is attached to another piece at each end and supporting two other pieces in-between.” 




shelroof.jpg
http://www.soue.org.uk/souenews/issue4/shelroof.jpg




The structure didn’t need any screws, nails or glue since the pieces supported each other imposing only vertical forces.






      Wallis also introduced the symbol for infinity as being represented by \inftyThe Romans commonly used a similar symbol for a thousand. The reason why Wallis thought it was a good symbol to represent infinity was because the "ribbon" that was in the shape of an eight had no ending and just continued on in the same shape. This symbol was also sometimes depicted as another form of the ancient ouroboros snake symbol, which is a snake twisted into a horizontal eight that is eating its own tail that represented endlessness. 

      These examples are just a few of his many accomplishments that he made throughout his life and in the field of mathematics. To know more about his more complicated discoveries about mathematics this website does a good job at explaining them. 

http://scramble.hubpages.com/hub/John-Wallis-Mathematical-Biography



Websites used












Saturday, October 29, 2011

Luca Pacioli

        











Mathematical Works 

     Luca Pacioli was an Italian mathematician born somewhere in between the years 1445 and 1447, and died in 1517.  Some call him the father of accounting but that wasn’t all he contributed to. Throughout his life he wrote a number of books that dealt with a somewhat wide range of topics. One of the first books he wrote was Tractatus mathematicus ad discipulos perusinos (Mathematics for Perugian Pupils) and it contained 16 sections on merchant arithmetic; things like profit, exchange barter mixing metals and algebra. He dedicated the book to his students at the University that he worked at from 1477 to 1480. He also wrote a textbook Summa de arithmetica, geometria, proportioni et proportionalità  (Everything about Arithmetic, Geometry, and Proportions) that was used for schools in Northern Italy that consisted of a compilation of mathematical knowledge of his time and was the first printed work on algebra written in the vernacular, which was the spoken language of the day. It was also the book for which he was named the father of accounting for because it contained the first published description of bookkeeping called the double-entry accounting system. In one of his books De divina proportione the subject was mathematical and artistic proportion, along with the golden ratio and its application in architecture. This is the book that contains illustrations from Leonardo da Vinci.

Leonardo da Vinci drew the illustrations of the regular solids in De Divina Proportione while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back.”










Mathematics and Magic   









     Pacioli wrote a book called De viribus quantitatis (On The Powers Of Numbers) written between 1496 and 1508 that contains the first reference to card tricks, how to eat fire, how to juggle, solve puzzles etc. The book is divided into three sections: puzzles and tricks, mathematical problems, and a collection of proverbs and verses. In the book it also states that Leonardo de Vinci was left-handed. Unfortunately this work was never published and stayed in the archives of the University of Bologna until a mathematician came across a reference to it that was in a 19th century manuscript and it was finally published in 2007. 

Mathematics and Chess

    Pacioli also wrote an unpublished book about the game of chess De ludo scacchorum (On the Game of Chess). The manuscript was lost until 2006 when it was rediscovered in the library of Count Guglielmo Coronini, which contained 22,000-volumes of books. In the book are chess problems and chess pieces that scholars suspect that Leonardo da Vinci either drew or at the very least designed based on his long association with Pacioli.

Pictures from the manuscript De ludo scacchorum
http://digitaljournal.com/article/250880





































      If you would like to know more details about the manuscript and what it contained here is a link to an article that I thought was interesting. http://www.nytimes.com/2008/04/18/arts/18iht-leo.1.12127016.html

     Things like Pacioli's works that were lost for hundreds of years being recently found, kind of gives perspective on how many more documents and manuscripts there are yet to be unearthed.  


List of sites that were used to help create the written portion of this blog:

http://en.wikipedia.org/wiki/Luca_Pacioli#Mathematics

http://www.gap-system.org/~history/Biographies/Pacioli.html

http://www.codicesillustres.com/catalogue/de_divina_proportione/index.html

Sunday, October 23, 2011

Counters

     After reading the blog post I wanted to know more about the make shift abacus merchants used to tally up totals of sales that was called a counter/counting board. I was very curious on what that would look like and how exactly it would work. I’ve heard of it before but only concerning the origin of the name and why we call counters in a business a counter. Looking into it more and being able to understand how it works must require knowledge of Roman numerals.

I=1
V=5
X=10
L=50
C=100
D=500
M=1000

     These are the basic Roman numerals. Another part before being able to understand how a counter works is to understand how to add and subtract Roman numerals.  When they are in a row like, XXX, that means you add 10+10+10=30. Or CLX would mean 100+50+10=160. If a smaller numeral is placed before a bigger one then that is where you would subtract. Like if you have IV, which would be 5-1=4. You can use those two rules to write a larger number in Roman numerals. If you want to write 971 you would first break it down to convert one digit at a time to make it easier. By doing this its actually using the Hindu-Arabic place-value number system that eventually replaced counters and Roman numeral methods. 

971
=900+70+1
=C-M+L+X+X+I
=CMLXXI

   When looking at a counter a merchant would use, it would have been made out of stone or wood. You would use pebbles or beads to count with by placing them in the columns, usually grooved so the pebbles/beads wouldn’t roll away. The size would vary between the size of a table down to the size of a portable abacus. 

File:Rechentisch.png
                                            http://en.wikipedia.org/wiki/Counting_board

    
    The most simple counter would look something like this:
                                                                
                                                                       
                           All these examples of counting boards come from this site-    
                                   http://mathforum.org/dr.math/faq/faq.roman.html

     If a pebble was placed in the bottom part of the board it represented 1, 10, 100, or one 1000, all depending on its placement. If it was in the top part of the board it had its value multiplied by 5. So if a pebble was placed in column C, and C is 100, then its 100 x 5= 500.

                                   Example of what a number on a counter would look like.
     To decipher what this number is first break it down into steps. To write the number shown on the counter in Roman numerals you have to remember what values are represented by what numeral, if you cant remember the Roman numerals that equal 500, 50 and 5 (D, L, V) then just imagine the counter like this-

                                                                      D       L       V

     This number in Roman numerals would look like this- MMDLXIV. To decipher it into numbers it would be 2000+500+50+10+5-1=2564. This gets harder when the number gets bigger and more complicated, there can even be more columns added to hold bigger numbers. At that point in the middle ages most people actually turned the columns the other way and drew lines down the middle to accommodate more numbers being represented at once. Like this-





     For showing just the basic way they would calculate things, the vertical columns give a good depiction so it doesn't get too confusing. It took a long time to do these calculations the more complicated they got. So I'm sure it was a relief when people started switching from this method to the Hindu-Arabic numbers and calculation system, because it sure made things a lot simpler. 



These are the two sites I used to get information-

http://mathforum.org/dr.math/faq/faq.roman.html

http://en.wikipedia.org/wiki/Counting_board


Saturday, October 15, 2011

Ancient Egyptian Mathematics


     The ancient Egyptians had many uses for mathematics whether it was for building, cooking, working out taxes, counting money, measuring straight lines or time, or for knowing when the Nile would flood. They used mathematics all the time in their everyday lives. Some of the people who used basic math were priests/priestesses, tax collectors, shopkeepers and cooks. Builders, masons, surveyors, and priests who were in charge of builders used a higher form of mathematics. There were two scrolls that acted like mathematical textbooks for the Egyptians, the Rhind Papyrus and the Moscow Papyrus. 
Rhind Papyrus
http://www.gap-system.org/~history/HistTopics/Egyptian_mathematics.html


     The Rhind Papyrus has 87 problems within it, some problems involve geometry such as problem 50: a round field has diameter 9 khet. What is its area? Other problems like number 56 gives an equation to find the angle of the slope of a pyramids face.  Architects had also built into their structures right triangles that obeyed the theorem a²+ b²= c², a and b represent the two sides while c is the hypotenuse. Pythagoras himself studied in temples of the Nile Valley for 22 years; there could have been the source for the Pythagorean theorem.
     
     The Egyptians knew addition, subtraction but only some multiplication and division. They had an interesting way of getting around the dilemma of multiplication and division that their number system created. They had methods of multiplication and division that only involved addition and they only would multiply or divide by two. An example of this would be 3 x 2 + 3 = 9 instead of 3 x 3 = 9.
Here is what their numbers looked like-
"1 = I 2 = II 3 = III 4 = IIII
10 = Arch determinative 100 = Rope determinative 1,000 = Flower determinative 10,000 = Finger determinative 100,000 = Tadpole determinative 1,000,000 = Ma'at worshiping determinative
As for fractions, 'r' r was used for the word 'part'. This means that r-10 rArch determinative is equivalent to our 1/10.
The Egyptian sign 'gs' 1/2 determinative was used for the word 'side' or 'half' ½. The word 'hsb' 1/4 determinative meant 'fraction', but it came to mean 'part-4' or ¼. 'rwy' 2/3 determinative meant 'two parts out of three' 2/3, and 'khmt rw' 3/4 determinative, though rare, was 'three parts out of four' 3/4."


http://www.touregypt.net/featurestories/numbers.htm


 They had no symbol for zero because they had no use for it.
     One of their greatest achievements involving mathematics was the building of the Great Pyramid of Khufu from the Fourth Dynasty. It was very geometrically precise with an almost perfect square base and sides that only differ from each other by less than 20 centimeters. There are around 2,300,000 giant stone blocks in the pyramid that are so closely placed together that the blade of a knife can not be fit between them. Mathematically there are discoveries about this pyramid that can either be deemed as accidental, coincidental or that the Egyptians knew more about the world than history thought.  
"When using the Egyptian cubit the perimeter is 365.24 - the amount of days in the year. 
When doubling the perimeter, the answer is equal to one minute of one degree at the equator.
The apex to base slant is equal to 600th of a degree of latitude.
The height x 10 to the power of 9 gives approximately the distance from the earth to the sun.
The perimeter divided by 2 x the height of the pyramid is equal to pi - 3.1416.
The weight of the pyramid x 10 to the power of 15 is equal to the approximate weight of the earth.
When the cross diagonals of the base are added together, the answer is equal to the amount of time (in years) that it takes for the earth's polar axis to go back to its original starting point - 25,286.6 years.
The measurements of the King's Chamber gives 2,square root(5),3 and 3,4,5 which are basic Pythagorean triangles."


   Either way you look at it the Egyptians showed they had the means and the great minds to create a masterpiece that not even today we can replicate properly, even with our advanced mathematics and machinery.

http://www.touregypt.net/featurestories/numbers.htm
http://www.gap-system.org/~history/HistTopics/Egyptian_mathematics.html

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