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.”

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





 

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

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 19^th 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

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. 

                                            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

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 =  2 =  3 =  4 = 

10 =  100 =  1,000 =  10,000 =  100,000 =  1,000,000 = 

As for fractions, 'r'  was used for the word 'part'. This means that r-10  is equivalent to our 1/10.

The Egyptian sign 'gs'  was used for the word 'side' or 'half' ½. The word 'hsb'  meant 'fraction', but it came to mean 'part-4' or ¼. 'rwy'  meant 'two parts out of three' 2/3, and 'khmt rw' , 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

"When us

The Number Zero

     When reading the blog post about representing numbers, specifically the base-ten place value system, it got me thinking about the number zero and how if you put it in the middle of a number like 321 (three hundred and twenty one), that number becomes 3201 (three thousand two hundred and one). One little number that is supposed to mean nothing completely changes another number. It sparked my curiosity on what was there before zero came to be a number and how the actual zero we use today was created. 


     In the ancient civilizations when a place-value number system was finally set up one would think that there would have to be a zero involved, but there wasn't. The Babylonians didn't use the number "0" for over 1000 years. Their number system was based on 60, not 10 like ours, and from 1700 BC to around 400 BC they didn't distinguish between numbers like 246 and 2406, instead it would have to be interpreted by what context it was used in. During 400 BC is when they decided to use two angled wedge symbols in the place for zero, or where there was supposed to be an empty space as they saw it. This was all written in cuneiform, which was symbols pressed into soft clay tablets. So the number 246 to change it to 2406 they would put 24 " 6. Everyone did not use this technique though. At an ancient Mesopotamian city called Kish, which today is south-central Iraq, around 700 BC they used three hook like symbols ''' to indicate an empty space. Other cities around that time also used a different method by just placing one hook ' for an empty place. 


     Around that time the Greeks would actually start using the symbol "0" but only a few astronomers would use it and eventually it faded out. The next appearance would be in India around 650 AD. Before then Indian mathematicians used a filled in dot to indicate zero or an empty space. Even in 650 AD there is debate on if the actual symbol "0" was really being used, but by 876 AD there was enough historical evidence to mark that as a genuine    date. Around that time the Mayans also developed a place-number value system that contained a zero, and unlike the Babylonians they used a system to base 20. Although historians have also found that the Mayans were using zero long before they even set up a place-valued number system; they very well could have been the first ones to use it. The Europeans adopted the Indian number system though and by 1200 they were using the sign zero but that’s all it was to them, a sign not a number like 1,2, or 3 and not tell much later did zero earn its spot as an actual number.


     So there really is no solid evidence on who created the actual “0”. You could say that zero got created by a number of civilizations and cultures all put together and you would be right. Some had more influence than others but it took a long time and a lot of ideas and techniques put together to get the zero we have today. 




Here are two sources where I got my information from. 

http://www.scientificamerican.com/article.cfm?id=what-is-the-origin-of-zer

http://www.gap-system.org/~history/HistTopics/Zero.html