East Kingdom Gazette

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Covering the Eastern Realm of the SCA
Updated: 42 min 14 sec ago

King Kenric’s Pennsic Challenge / Défi de Pennsic du Roi Kenric

Wed, 2016-07-13 18:33

En français

In an effort to encourage crossing over to different fields at the war, I would like to announce the following:

If you compete in armored combat, rapier, archery, or thrown weapons, and you compete in all war point activities which you are eligible for in at least two of these areas, I would like to recognize your efforts with a token.

(For archery and thrown weapons, that means throwing or shooting each shoot/throw as many times are you are allowed.)

If you compete in all war point activities for three of these areas, there will be a slightly different token.

If you compete in all four, well, one, you’re insane. But if you do, I’ll have to think of something because I don’t really expect it…

Kenric rex

 

En français
Traduction par Baron Godfroy de Falaise

Dans le but d’encourager la participation aux différentes activités martiales à la Grande guerre, j’annonce ce qui suit :

Si vous participez au combat en armure, à l’escrime, au tir à l’arc ou aux armes de jet, et si vous participez à toutes les activités auxquelles vous êtes éligible et qui contribuent aux points de guerre dans au moins 2 catégories, j’aimerais vous récompenser en vous donnant un gage particulier.

À titre d’exemple, pour le tir à l’arc et les armes de jet, cela implique de tirer ou de lancer aussi souvent que vous y êtes autorisé.

Si vous participez à 3 catégories, un gage un peu différent vous sera octroyé. Et si par la plus grande des chances, vous en veniez à participer à 4 activités, eh bien….vous êtes complètement fou!

Mais si vous le faites réellement, je devrai trouver quelques chose, car je ne m’attend pas à cette possibilité!

Kenric Rex


Filed under: En français, Pennsic Tagged: Kenric and Avelina, Kenric's challenge, Pennsic, Pennsic 45, pennsic war, pennsic war points, war point challenge, War Points

Call for Teachers/Presenters – Theleme at Penn

Tue, 2016-07-12 19:00

For one day, the Rare Books Library of the University of Pennsylvania will welcome the free learners of the SCA. In the same spirit as the voyages of discovery previous event please come and talk about your research, your failed attempts, your successes, and get feedback on your work and ideas for future projects.

Have questions? Want to present? E-mail Lissa.
All topics are welcome, though classes dedicated to the making of books (parchment, paper, ink, pigments, calligraphy, illumination, printing, bookbinding, etc.) are especially desired.

A list of presentations and posters currently being offered is available online at the event website.

We hope to see you there!!


Filed under: Events

Curia to be Held at Great Northeastern War

Wed, 2016-07-06 21:37

East Kingdom Curia  will be held Saturday, 9 July 2016, 8:00 a.m. at Great Northeastern War in the Province of Malagentia (Hebron ME).

Following is the agenda, prepared 5 July 2016, 6:10 p.m.

  1. Curia Opening
  2.  Old Business
  3. New Business
    1. 3.1Laws of the East Kingdom Clarification that, in cases of conflict, the SCA, Inc. Governing Documents take precedence over EK Law.
    2. 3.2Glossary Revisions to several terms to match the definitions in SCA, Inc. Governing Documents.
    3. 3.3Polling Procedures Revision to VII.C.2.f. to clarify the polling where there are more than two candidates or choices.
    4. 3.4King’s and Queen’s Equestrian Champions Tourney(s) Revision to VIII.A.8. to clarify preferred scheduling within the Summer Reign.
    5. 3.5Award of the Burdened Tyger Relocation of IX.D.3. to IX.E.2.
    6. 3.6The Company of Fellowship New section IX.E.13., an award to recognize local branches.
    7. 3.7Discussion of a possible new armigerous order to include and provide recognition for martial activities other than rattan, rapier, and target archery.
    8. 3.8Discussion of potential East Kingdom 50-Year celebratory event.
  4. Officer Reports
  5. Curia Closure

EK Law Section III.I. The Agenda for the Curia Regis 1. Any items that The Crown chooses to add to the agenda after the Curia has been called will be added to the agenda under “New Business”. 2. If a Curia notice has been sent according to East Kingdom Law, but another Curia needs to be held before the previously announced one, any items of business held over from the earliest Curia will be automatically added to the agenda of the subsequent Curia under “Old Business”.


Filed under: Announcements, Law and Policy Tagged: curia

Arts & Sciences Research Paper #10: Fibonacci – A Master By Any Name

Sun, 2016-07-03 21:10

Our tenth A&S Research Paper comes to us from Lady Rosina von Schaffhausen, of the Shire of Quintavia. She introduces us to a fascinating figure from the 13th century – the mathematician Leonardo of Pisa, known most familiarly to us as Fibonacci.  (Prospective future contributors, please check out our original Call for Papers.)

Fibonacci – A Master By Any Name

The Latin phrase filius bonacci, in the first line of the Liber Abbaci manuscript (above) gave rise to Leonardo da Pisa’s modern nickname, Fibonacci. (National Library of Florence)

Imagine being an Italian merchant in the early 13th century, traveling around the Mediterranean. You visit fascinating places, eat new and unusual foods, see many exotic sights, and trade many of the goods passing through the region.

However, you have a problem. The basic addition and subtraction you need to do to keep your account books you can handle, using the tools you have available, Roman numerals and an abacus. But doing any sort of multiplication or division is difficult. And you need to multiply, or divide, or sometimes both, to do all sorts of important things. You need them to determine how much cinnamon your pepper is worth, how much of your profits each of your investors should receive, how much your cut is, to calculate currency exchange, and to determine how much interest you have earned on the loan your city forced you to give them to build their navy. The methods you know seem much more difficult to deal with than the Arab merchants’ system.

On your next stop at home, a friend is raving about the new system of Hindu reckoning in a book written by one of your compatriots, and you resolve to find a copy and learn this new system…

Contents
Leonardo’s Pisa
Leonardo’s Life
Leonardo’s Work
Teaching Europe Math
Blazing New Trails
Conclusion
Bibliography

As commerce began expanding in the Middle Ages in the Mediterranean, the Italian city-states vied for control of lucrative trade routes. Throwing off the control of the Holy Roman Empire, the northern cities began running their own governments and conducting trade. They built navies and were the first governments to go into debt to finance their expenditures, sometimes by forcing their citizens to loan them money. They established customs houses at home and in a number of ports abroad, as well as forming the first corporations. Many of the cities coined their own money, and the first banks started offering interest on deposits.

While the first universities were being founded at this time, they did not add algebra to their curricula until the middle of the 16th century [3, p. 107]. It was the needs of the merchants that drove the eventual revolution in European mathematics. The need for better and easier bookkeeping, computation, and problem-solving methods brought the Hindu-Arabic number system and Arabic algebra into use throughout Europe. Leonardo of Pisa, today known as Fibonacci, was a pivotal figure in the process of changing over from Roman numerals to the system we use today. The practical example problems he included in his best known work, Liber Abbaci, displayed the ability of the Hindu-Arabic number system and algebra to solve many of the pressing problems of the medieval merchants.

Fibonacci is best known today for his famous sequence, 1, 2, 3, 5, 8, 13, 21, … where successive numbers are found by adding the previous two together.  However, this sequence is not original to him. Since the concept of plagiarism did not exist in the form it does today, many of Fibonacci’s example problems, including this sequence, can be found in identical or similar versions in earlier texts from the Islamic world, India, and even China. The main purpose of Fibonacci’s writing was to educate merchants and surveyors on techniques that were largely unknown or forgotten in Europe. Thus he was influential in bringing in our modern number system, more so than other scholars of his day. In addition, he wrote some impressive original mathematics, which was unheard of in Western Europe at the time. Most of all, he left legacies of education and research in mathematics that lasted long after his death. While today we remember one sequence, in the middle ages and Renaissance, many math texts acknowledged a large debt to one Leonardo of Pisa.

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Leonardo’s Pisa

Leonardo of Pisa, known today as Fibonacci, was born in Pisa around 1170. Around this time Pisa was a republic, one of many Italian city-states vying for control of trade in the Mediterranean. At sea, Pisa contended with the other maritime republics of Genoa, Amalfi, and Venice. On land, their main rivals were Florence and Lucca. Pisa is situated on the Arno River on the western coast of the Italian Peninsula, just south of the top of the “boot” of Italy. From Roman times until the 15th century, when the river silted up and Pisa lost its port, Pisa was a prime commercial center in Tuscany.

The 13th century on the Italian peninsula. Pisa is just northeast of Corsica, on the western coast of what is now Italy. (Map by srinaldipds at SlideShare.)

While other cities mostly concentrated on going east toward Constantinople and the Holy Land for their trade, at first Pisa mostly went west. The city established trading ports, conquered towns, and took over islands in places such as Corsica, Sardinia, and Carthage. During the First Crusade in 1099 Pisa was instrumental in the campaign, and used the opportunity to establish trading centers in the eastern Mediterranean. While other states did the same, using this advantage Pisa became a serious international power. For a time Pisa surpassed Venice as the foremost merchant and military ally of the Byzantine Empire.

At home, Pisa used her spoils to begin building a beautiful city center, including a cathedral and baptistery. As a very young child, Leonardo may have watched the first three floors of the famous bell tower being built and start to lean. Construction on the Leaning Tower, begun in 1173, was stopped in 1178 as the structure, built on unstable ground, began its famous tilt.  The tower stayed at that height throughout Leonardo’s lifetime, as building would not resume for nearly 100 years and took nearly 200 to complete.  In 1284, the Pisan navy was nearly completely destroyed by Genoa at the Battle Meloria, ending Pisa as a naval power. The city managed to keep up some independent trade until she came under the rule of Florence in 1406. Then the river silted up, ending Pisa’s ability to trade easily.

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Leonardo’s Life

Fibonacci signs his name in Liber Abbaci as Leonardo of Pisa of the Family Bonaci, which might also be translated as Son of Bonaci. However, since his father’s name was Guglielmo, this translation is incorrect.  Some scholars think that Bonaci or Bonacio was the name of an ancestor, as a reference to a famous ancestor was a common practice in Italy at this time. The name Fibonacci was first used in the 19th century and has become the most common one used for him today. He signed Liber Quadratorum as Leonardo Pisano, or Leonardo of Pisa. However, in his later work Flos (1225) and in a legal document (1241) [3, p. 149] his name is listed as Leonardo Pisano Bigollo. Some scholars think Bigollo means traveler, good-for-nothing, or absent-minded, while others think these translations are incorrect. Considering that the legal document is honoring Leonardo, Bigollo could not possibly have been meant as an insult.

Most of what we know about Leonardo’s life comes from the introduction to his book Liber Abbaci. When Leonardo was a boy, his father Guglielmo was working in the customs house of the Pisan trading port of Bugia (now Bejaïa) on the Barbary coast of Africa, east of Algiers. Guglielmo had young Leonardo sent over to Africa, most likely just after finishing grammar school, so the boy could learn mathematics. This was a trading center in a Muslim region, so Leonardo learned as much as he could from scholars visiting from many places. Modern scholars believe he learned to read and possibly write Arabic [5, pp. xviii-xx]. He studied a variety of mathematical systems and methods, and decided that the “Indian method”, as he called it, was superior to the rest. Leonardo continued his studies on business trips around the Mediterranean, including Egypt, Syria, Greece, Sicily, and Provence. He later wrote his first book, Liber Abbaci, or Book of Calculation, in 1202 to spread this method, which is our modern number system [6, p. 15-16].

Leonardo’s first book got him noticed not just in Pisa, but also by the Holy Roman Emperor Frederick II. At this time Pisa was nominally part of the Holy Roman Empire, but had begun governing itself even before a previous emperor permitted the city to function under its own governance. Frederick II was known as the Wonder of the World. At this time, the first universities were still very new, and many scholars worked for wealthy patrons. Frederick encouraged scholarship in his court, and even wrote a treatise himself. Leonardo was introduced to Frederick’s court likely around 1225. Some scholars in the court posed Leonardo some challenge problems, another common practice of the day, used to determine the abilities of scholars. Leonardo was able to solve the challenges posed to him, and his writing on the solutions and related mathematics encompass a large portion of his original mathematics. In the 1220s Leonardo wrote most of his other works, including those on the challenge questions. He wrote a revision of his first book Liber Abacci, dedicating it to the scholar who first wrote about him to the court, Michael Scott. In 1241, at the end of his life, Leonardo was presented with an annual stipend from the city of Pisa [3, pp. 148-149] for his contributions to the city.

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Leonardo’s Work

Leonardo wrote a number of impressive texts in the course of his career.  The first and foremost is Liber Abbaci (1202; 1228), mentioned above [6], a compilation of arithmetic and the algebra that was known in his day. Next Leonardo wrote De Practica Geometriae (1220, meaning Practical Geometry), a book on a variety of geometry problems including practical ones on land area and surveying [5]. Liber Quadratorum (1225, meaning Book of Squares), on advanced algebra and number theory, contains some impressive original mathematics [4]. These three are available in English translation. Flos (1225, meaning flower) and Epistola ad Magistrum Theodorum (date unknown, meaning letter to Master Theodore), both on indeterminate equations like those in Liber Quadratorum, have not yet been translated into English. Two of Fibonacci’s works have been lost. One is a tract on Book X of Euclid’s Elements.  The other is Libro di minor guisa (date unknown, meaning probably The Book of the Lesser Method) on commercial arithmetic.  We will go into more details later on some of these books.

Folio 124r of the Codex magliabechiano, a manuscript of Liber Abbaci preserved in the Biblioteca Nazionale di Firenze. The “Fibonacci Sequence” is shown in the column at the right hand side. (Image courtesy Wikimedia Commons.)

Fibonacci’s impressive texts come in two varieties.  The first are texts designed to explain mathematics to the common person, as well as to show its usefulness.  De Practica Geometrie, Liber Abbaci and Libro di minor guisa fall into this category.  Liber Abaci is an encyclopedic work which, together with Euclid’s Elements, contains most of the mathematics known in the world at that time.  The other two texts are shorter and focused more on application. The second group of texts contain Leonardo’s original mathematics. Flos, Liber Quadratorum, and Epistola ad Magistrum Theodorum contain his solutions to problems posed to him by masters in the court of the Holy Roman Emperor Frederick II. In addition, in these books Leonardo expanded on the challenge questions and wrote original proofs of some ancient Greek knowledge as well as general solution methods for the types of problems posed. The tract on Book X of Euclid’s Elements was likely expanded from a chapter in Liber Abbaci. Leonardo’s popular works are more extensive than other European texts at this time, and no other European mathematician between the fall of the Roman Empire and the Renaissance has a body of original research.

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Teaching Europe Math

Leonardo’s longest and most influential work was Liber Abaci, first written in 1202 and revised in 1228. The revised version is the one that was recently translated to English, and Leonardo not only made corrections but added some of his original mathematics at the end. In the introduction he stated that his goal was to bring the “Hindu numerals” to the Italian people. He succeeded, at least in planting the seed, since it took was not until after the invention of the printing press for that our modern number system to fully take took hold in Europe. At this time, Leonardo was not the only European familiar with Arab mathematics. Hindu Arabic numerals were known in Europe at least as far back as the 10th century. Gerbert d’Aurilac, later Pope Sylvester II, used them as number symbols but not in calculations. In the 12th century other scholars began translating Arab works into Latin, as well as writing their own texts. However, these translations and texts were aimed at other scholars. Leonardo was the first to deliberately focus on mathematics useful for everyday purposes. Soon after Liber Abbaci appeared, other popular arithmetics did also, but few approached the sheer magnitude of Leonardo’s compilation of arithmetic, algebra, geometric proofs, some of his original research, and a wide variety of practical and impractical examples.

While the goal of Liber Abbaci is to spread the Hindu Arabic number system, the book contains a wide variety of mathematics. The first chapter explains the basics of the number system. Chapters 2-5 deal with the operations of addition, subtraction, multiplication and division while Chapters 6 and 7 concern operations with fractions. Chapters 8-12 deal with various “word” problems, many of which would naturally arise from business situations of the time. Other problems are abstract, intended to display different solution methods of a problem or to provide further examples on a solution method. Occasionally Leonardo throws in a whimsical problem. One of these is the famous “rabbit problem” from which we get the Fibonacci Sequence. Chapter 13 is on algebra, namely on methods for solving linear equations. Chapter 14 is on extracting roots and arithmetic operations on roots. The 15th chapter deals with geometry and some applications to algebra, namely quadratic equations. The last two chapters are material from Leonardo’s work Flos added in the revision.

Folio 139v-140r of the Liber Abbaci. (Biblioteca Nazionale Centrale, Florence, Italy)

Most of the mathematics in Liber Abbaci can be solved in terms of modern mathematics with elementary school arithmetic, basic first year high school algebra, and geometry. The main exception to this is that most schools no longer teach algorithms for square root extraction, and I have never heard of any that taught cube root extraction. Most of the application problems in the book can be modeled with linear equations, which in modern terms are equations that can be manipulated into the form Ax+B=C, where x represents the unknown in the problem and A, B, and C are numbers. However, variables as we know them today, as well as almost all modern math symbols, were not used until the late 16th/early 17th centuries. Because of this Leonardo’s solution methods vary from somewhat different to quite different from modern methods. The main methods he used for these problem types are formalized guess-and-check systems called single false position and double false position. In addition, Leonardo uses what he calls “the direct method” which is basically the same as our modern algebraic manipulation, only using words instead of symbols. Most of Leonardo’s paragraphs-long solutions could be condensed to a few lines using modern symbols.

While Leonardo displays a wide range of mathematics and shows a facility for original mathematics in other works, Liber Abbaci is a summary of the existing useful arithmetic and algebra of the time. Leonardo had a wide range of sources at his fingertips that he used liberally for this book. Plagiarism was not seen in the same light as today, so many of the example problems can be found elsewhere before Liber Abbaci was written. This includes the Fibonacci sequence, which appears in several Indian texts going back to at least 200 BC. In some cases Leonardo acknowledges that he obtained problems elsewhere, but does not always mention names. One example is “A Problem on the Same Thing Proposed to Us by A Master near Constantinople” [6, p. 290]. Other problems come from a variety of texts. Books that Leonardo used included al Khwārizmī’s Algebra text (the oldest known algebra text for which the subject was named) and Euclid’s Elements.

Given the massive scope of Liber Abaci, there are a number of interesting items within its pages. First, Fibonacci used a fraction system from North Africa where he went to school that seems bizarrely complicated. That is, until one notices that it is incredibly useful for old fashioned monetary and measuring systems that do not work as well with modern decimals. For example, using this fraction system with the current U.S. system of measurement, 1 yard, 2 feet and 3 inches could be expressed as

yards. In some problems with solutions that cannot be expressed as fractions, Leonardo found approximations using this fraction notation that are very much like our modern decimals.

Second, Leonardo partially ignored the existence of negative numbers and considers equations with negative solutions to have no solutions, unless they are part of a merchant account, in which case he considered them debits in the account. He did, however, give directions on how to operate with negative numbers [6, pp. 417-419]. Leonardo never admitted the possibility of square roots of negative numbers for any reason.

Third, Liber Abbaci contributed to the financial developments in Italy in multiple ways. Leonardo used a form of present value analysis in some problems, which is the foundational concept of modern financial calculations and decisions [8]. This is the first known use of this method to compare two investments. In addition, the mathematics of minting precious metals into coins in Liber Abbaci may have contributed to the more consistent coinage available in subsequent centuries.

Fourth, Leonardo included an early version of the classic “two trains” problems that students love to hate. However, instead of having two trains leave different cities at the same time, he calculated when two ships would meet [6, p. 280]. One scholar examining a 15th century French arithmetic text claims that that text contains possibly the earliest known example of a “two trains” problem [13], but Leonardo’s is 200 years earlier.

While Liber Abbaci is an impressive book and has been admired by scholars throughout the centuries, its immense size would be overwhelming to a merchant who just wants to know what his goods are worth in the coin of the city he’s in that week. Most likely for this reason Leonardo wrote another book on the basics of our number system specifically for merchants. This is referred to elsewhere as di minor guisa (the minor work), which probably refers to the fact that the book was shorter or contained fewer topics. However, no known copies of this text survive. For a long time scholars wondered why many arithmetic textbooks in the Renaissance had an acknowledgement in them to Leonardo of Pisa, but the books were not similar to any of Leonardo’s surviving texts. Recent scholarship shows a plausible link between the missing Libro di minor guisa and the “abbacus” texts used for instruction in Italy in the 14th and 15th centuries [3, Ch. 8]. Recently found manuscripts from the late 13th century and early 14th centuries have some elements in common with later arithmetic texts and some with Liber Abbaci, providing the missing link between Leonardo’s work and the tradition of mathematics instruction in later medieval and Renaissance Italy.

The third of Leonardo’s explanatory texts is De Practica Geometrie, or Practical Geometry. Practical geometry was the term used at the time for surveying or land measurement. This profession had been very important and well-regarded in Roman times. The art declined during the so called Dark Ages, but began reviving on discovery and transcription of Roman surveyors’ tracts in the ninth and tenth centuries. These works then became common through the 11th century, but were more for teaching than for practical use. In addition to the duties of the Roman surveyors, medieval surveyors also verified weights and measures. [12]

Most other practical geometries of this period contained three sections on measuring heights, areas and volumes, while Hugh of St. Victor [5, p. xxv; 9] replaced the section on volumes with measurement of the heavens. Leonardo included all four of these topics [5, p. xxv], but one seems to have been largely lost between Leonardo’s time and when the existing manuscripts were copied. [5, p. xxv] The other thing that sets Leonardo’s text apart from prior practical geometry texts is his inclusion of the theory underlying the practice. Theoretical validation of surveying procedure became a goal in later texts, sometimes displacing the practice. De Practica Geometrie was translated into Italian a couple of times in period, as well as included in various compilations with other geometry texts in Italian.

For his surveying manual, Leonardo used a wide variety of geometry texts from the ancient Greeks and the Arab world. One text he used was Euclid’s On Division of Figures, which has since been lost. However, a century ago a scholar took an outline of Euclid’s text that still remained and found that Leonardo’s use of it matched very closely. This scholar then used Leonardo’s text to fill in the missing pieces and come up with a plausible reconstruction of Euclid’s missing text [1]. The scholar who recently translated De Practica Geometrie into English opines that it is not only a useful compilation of Greek and Arabic geometry, but is a practical analog of Euclid’s Elements: a stand-alone text containing everything a surveyor would need to solve the mathematics problems inherent in their work.[Back to Top]

Blazing New Trails

In addition to being an excellent explainer of mathematics to the masses, Leonardo of Pisa was also a research mathematician. Between the fall of Rome and the Renaissance, almost all of the writings left by European mathematicians were translations of Greek and Arabic works, expository texts like Leonardo’s Liber Abbaci, or textbooks or lists of problems to use in teaching students. Leonardo, however, solved some original mathematics problems and produced new solutions to some previously solved problems. While some of these texts are on other lines of inquiry, some of these were based on the challenge problems he was given in the court of Frederick II. One of these, Liber Quadratorum, or The Book of Squares, is currently in English translation [4].

Leonardo solved two of the challenge problems and later wrote Liber Quadratorum on the solutions and related problems, but never finished or published this book. It contains 24 propositions written in a style similar to Euclid’s Elements, which Leonardo references in the text. The text focuses on whole number and fraction solutions to indeterminate equations, or Diophantine equations, which are equations with multiple variables, and thus may have multiple solutions. The Greek mathematician Diophantus was the first to study these equations. For example, one of Leonardo’s challenge problems was equivalent to finding x,y,”and” z such that x2+5=y2 and y2+5=z2. In the process of finding a solution, Leonardo stated and proved a special case of Lagrange’s identity, although scholars argue over whether his proof is original or based on Arab material. Also, Leonardo wondered what other numbers besides 5 could be used in these equations and the equations would still have solutions. He proved some interesting facts about these numbers, called congruent numbers [17], based on the Latin word that Leonardo used for them. The mathematics scholars of the Tuscan school that arose from studying Leonardo’s works were very interested in congruent numbers. Modern mathematicians have not yet determined a general rule for which numbers are congruent numbers.

Another impressive result in Liber Quadratorum is that

for any whole numbers n and m where n > m. A conclusion Leonardo reaches in the proof is that no square can be a congruent number. This result is equivalent to the fact that the area of a Pythagorean triangle, a right triangle with whole number side lengths, cannot be a square. The proof about the Pythagorean triangle was one of Fermat’s greatest achievements.

Leonardo’s other higher level works, some of which include some original work, include Flos and Epistola ad Magistrum Theodorum, and a now lost text on Book X of Euclid’s Elements. Flos was included in the revision of Liber Abbaci as the last two chapters. The first of the two chapters is also a discussion on Book X of Euclid’s elements, and is basically a discussion on how to deal with numbers that can be expressed using square roots, among other things. Unfortunately we do not know how Leonardo’s lost text would have differed from Flos. The second chapter is on solving quadratic equations, which contain a square of the variable in them. Leonardo solved these using completing the square, since methods commonly taught in high school today such as the quadratic formula were not possible without the symbols we now use. He also proved some facts about squares and includes some of his challenge problems. Leonardo wrote Epistola ad Magistrum Theodorum, or Letter to Master Theodore, to one of the scholars offering challenge problems, with more material on those challenge problems. Leonardo’s works began traditions of scholarship in algebra and in investigating congruent numbers both in Tuscany and in Germany.

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Conclusion

Leonardo of Pisa, today called Fibonacci, wrote a number of impressive texts containing the bulk of the practical mathematics known in his day, plus some original mathematics research. His goal in writing his most impressive text was to bring our modern number system to Europe. While other mathematicians were also writing and translating books about this system, the legacy of mathematics scholarship and education that sprang from Leonardo’s works testify to the influence he had in making this happen. His legacy of education continued in the arithmetic and algebra textbooks in Italy and nearby areas until the Renaissance. His legacy of scholarship continued through schools of mathematicians in his native Tuscany and in Germany. Even today mathematicians are studying questions that he pursued in his research. It is no wonder that Leonardo of Pisa is considered the greatest mathematician of the Middle Ages.

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Bibliography

1. Archibald, Raymond Clare, Euclid’s book On divisions of figures…with a restoration based on Woepcke’s text and on the Practica geometriae of Leonardo Pisano, Cambridge University Press, 1915.

2. Berlinghoff, William P. and Fernando Q. Gouvea, Math through the Ages: A Gentle History for Teachers and Others, Oxton House Publishers and the Mathematical Association of America, 2004.

3. Devlin, Keith, The Man of Numbers: Fibonacci’s Arithmetic Revolution, Walker & Company, New York, 2011.

4. Fibonacci, Leonardo, The Book of Squares / Leonardo Pisano Fibonacci; an annotated translation into modern English by L.E. Sigler, Academic Press, Boston, 1987.

5. Fibonacci, Leonardo, Fibonacci’s De Practica Geometrie, ed. and tr. by Barnabas Hughes, Springer Science + Business Media, LLC, 2008.

6. Fibonacci, Leonardo, Fibonacci’s Liber abaci : a translation into modern English of Leonardo Pisano’s Book of calculation, tr. by L.E. Sigler, Springer-Verlag, New York 2002.

7. Gies, Frances and Joseph, Leonard of Pisa and the New Mathematics of the Middle Ages, Crowell, New York, 1969.

8. Goetzmann, William N., “Fibonacci and the Financial Revolution”, The Origins of Value: The Financial Innovations That Created Modern Capital Markets, Goetzmann and Rouwenhorst, eds., Oxford University Press, New York, 2005

9. Hugh of St. Victor, Practica Geometriae, tr. by Frederick A. Homann, Marquette University Press, Milwaukee, WI, 1991.

10. Menninger, Karl A., Number Words and Number Symbols, tr. by Paul Broneer, MIT Press, Cambridge, MA.

11. Mucillo, Maria, “Fibonacci, Leonardo”, Medieval Science, Technology, and Medicine: An Encyclopedia, 2005.

12. Pikulska, Anna, “Agrimensores”, Medieval Science, Technology, and Medicine: An Encyclopedia, 2005.

13. Schwartz, Randy K., “‘He Advanced Him 200 Lambs of Gold’: The Pamiers Manuscript,” Convergence (July 2012), DOI:10.4169/loci003888, http://www.maa.org/press/periodicals/convergence.

14. Suzuki, Jeff, A History of Mathematics, Prentice Hall, 2002 (ska Master William the Alchemist).

15. Swetz, Frank, Capitalism and Arithmetic, Open Court, La Salle, IL, 1987.

16. Swetz, Frank, Ed., The European Mathematical Awakening: A Journey Through the History of Mathematics from 1000 to 1800, Dover Publications, Mineola, NY, 2013.

17. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2010, Sequence A003273

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Filed under: A&S Research Papers, Arts and Sciences Tagged: a&s, Arts and Sciences, mathematics

Preliminary Announcement: King& Queens Arts& Sciences Competition Format

Mon, 2016-06-20 17:50

This announcement is meant to give artisans advanced knowledge of the format so they can plan their work accordingly. However, please note that this announcement is incomplete, and a more complete announcement, along with details about preregistration, will be constructed once the event date and site are chosen. If you have questions, please e-mail Lissa.

 -Mistress Elysabeth Underhill (Lissa), Queens Champion

-Master Magnus Hvalmagi, Kings Champion

Entrants may enter 1- 3 items into the competition, but the championship will be judged as a body of work. Individual entries will not be scored. No item should have won a previous King’s or Queen’s Championship, and each item to be judged should have been made within three years of the competition. The items can be from a single discipline or from multiple disciplines, however, entries which tell a coherent story about a people, time and place are encouraged. The winner of the competition is the Queen’s Champion of Arts and Science. The King determines the King’s Champion of Arts and Science.
The judging will be conducted face-to-face using a judging rubric which will be made publicly available before the competition (and one that is similar to the rubric that has been used for the past two years and is available on the kingdom arts and sciences website). Entrants will have the opportunity to stand by their table and judges will be asked to talk to entrants before reading their documentation.  Thus entrants should prepare a brief overview presentation about their project to be delivered to both judges and interested event attendees.  Absentee entries will not be accepted.  Judging will be done as a single team, drawing from various areas of expertise. The top entries will then be announced and then a second round of face-to-face judging will occur with the King and Queen present.

Documentation is required to compete in the King and Queen’s Arts and Sciences Championship. Entrants are encouraged to use the judging rubric (which will be updated shortly) to help them determine what information to include in their documentation. However, due to the large number of entrants this competition typically receives, the length of the documentation provided must be limited to allow time for judging. Entrants are asked to compose a 1/2 to 1 page abstract or summary which provides an overview of their entire entry (their body of work). The primary documentation for the entire entry should be no longer than 6 pages, not including references. Entrants are encouraged to include a table of contents and section headings to make reading the documentation easier for the judges, and judges will be asked to read both the summary and primary documentation in full. Appendixes may be used to convey supplemental information, things such as images, tables, charts, excerpts from historic texts, detailed descriptions of processes undertaken, etc. Judges may look at these appendixes if they desire, and entrants can refer judges to information in their appendixes during their presentation. However, entrants should plan to include all information that is critical to an understanding of their project in their primary documentation.

Documentation is not easy to write! If an entrant desires help with writing their documentation, or wants feedback on documentation already written, please e-mail the current queen’s champion, Mistress Elysabeth Underhill (Lissa) at Lissa, no later than 3 weeks before the competition date, and she will try to find a volunteer to assist you. If you are interested in assisting with providing documentation feedback to entrants prior to the competition, please e-mail Lissa as well.


Filed under: Announcements, Arts and Sciences, Uncategorized

History on Display at the SCA 50 Year Celebration

Mon, 2016-06-20 16:50

The Æthelmearc Gazette has posted an article with many photos from the history deisplay at the SCA 50 Year Celebration.  The article can be found at their website here.


Filed under: Tidings Tagged: SCA 50 Year

Premiering the East Kingdom Video for SCA 50th Year Celebration

Sun, 2016-06-19 14:30

For the SCA’s 50th Year Celebration currently occurring in Indiana, the East Kingdom produced a video overview of its history, people and activities.  The East’s booth at 50th Year was a major undertaking overseen by Countess Marguerite and supported by the work of many Easterners.  The people who created this video are listed in the credits at the end.  Many familiar faces and a few familiar voices are featured.

 


Filed under: Tidings

A Reminder About Pennsic Herald’s Point

Fri, 2016-06-17 17:37
Many people wait until Pennsic to visit Herald’s Point to try to register their name and armory.  Mistress Alys Mackyntioch reminds us that for Easterners, this is not the best strategy.  The East gets between 150 and 200 submissions from Pennsic, substantially more than any other Kingdom, and substantially more than is possible to review in a single month.  We can do a meaningful review of only about 50 submissions a month (again, substantially more than any other Kingdom), so submissions done near the end of Pennsic may not get to the first level of review until November. If you want to get a name or armory done, do it now before Pennsic. Requests for assistance can be sent by e-mail to pantheon@eastkingdom.org or blue.tyger@eastkingdom.org or diademe@eastkingdom.org.   Also, Herald’s Point is always looking for volunteers.  Even if you know little or nothing about heraldry, we need people to draw and color armory submissions, help greet people and manage the line, and do administrative work to free up heralds for consults.
Filed under: Announcements, Heraldry, Pennsic Tagged: heraldry, Pennsic

Pennsic Volunteers Needed

Fri, 2016-06-17 15:50

Lady Sabina Lutrell, the East Kingdom Minister of Lists is looking for your help. The Pennsic Inspection Point and Marshals Tent on the battlefield is probably the second busiest place at Pennsic besides the gate and they need our help. Inside the tent, we need lots of hands to check authorization cards, ID, and Pennsic medallions. Do you have an hour or two to spare so that everyone’s vacation is a little bit more enjoyable?

We could really use your help. A sign up sheet has been created. Please click on the link below to sign up. No experience is necessary. Share with your friends. Sign up for one shift, or multiple shifts.

I look forward to working with you!

https://www.volunteersignup.org/MHFF9

 


Filed under: Announcements, Pennsic Tagged: Pennsic, volunteers

Updated East Kingdom Awards Overview and Other Award Resources

Wed, 2016-06-15 21:25

The East Kingdom Awards Overview (created by Tola knitýr) has been updated to reflect the addition of the new AoA awards (Apollo’s Arrow, Silver Wheel, Silver Tyger, Silver Brooch) as well as the Golden Lance.

East Kingdom Awards

 

East Kingdom Awards (printable version)

Other Award Resources

How to Write an Awards Recommendation by Queen Avelina II
How Web Pollings and Recommendations Work by Lady Raffaella Mascolo
How the Awards Process Works by Queen Avelina II
On the Scheduling of Awards by Brennan II, Princeps

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Filed under: Court Tagged: awards, chart, resources

Polling Order Recommendations Due July 9th / Recommandations des Ordres nécessitant devoir faire par le 9 juillet

Wed, 2016-06-15 07:01

En français
Recommendations for TRH Brion and Anna’s first order polling are due on July 9, 2016.

The orders that require polling are: 

Peerages (society level awards): Order of Chivalry (heavy weapons), Order of the Laurel (arts & sciences), Order of the Pelican (service), Order of Defense (rapier combat)

Orders of High Merit (East Kingdom): Order of the Silver Crescent (service), Order of the Maunche (arts & sciences), Order of the Tygers Combatant (heavy weapons), Order of the Sagittarius (archery), Order of the Golden Rapier (rapier combat), Order of the Golden Lance (equestrian).

Recommendations for awards that do not require polling (including Awards of Arms and the new Armigerous (“Silver”) Orders) may be submitted to the Crown at any time.

Full descriptions of all East Kingdom awards can be found in East Kingdom Law available on-line here (.pdf document), in Section IX Awards, starting on page 23. Additional information about the new Armigerous Orders can be found online here.

Anyone may recommend any person for any award. You do not need to be a member of an order to recommend someone for that order or award.

An excellent summary of how to write a good recommendation letter is available on the East Kingdom Wiki by clicking here.

Her Majesty Avelina also wrote this excellent article on how to recommend someone for an award.

You may submit recommendations for any award by using the EK Awards Web Form. Click here to access the form.

En français
Traduction par Ekaterina Solov’eva Pevtsova

Les recommandations pour les Ordres nécessitant un vote à leurs Altesses Royales Brion et Anna doivent être reçues au plus tard le 10 Juillet 2016.

Les Ordres nécessitant un vote sont :

Pairs (Ordres de la SCA) : Ordre de la Chevalerie/Chivalry (Combat en armure) Ordre du Laurier / Laurel (Arts et Sciences), Ordre du Pélican/ Pelican (Service) et l’Ordre de la Défense / Defense (Escrime).

Ordres de Haut Mérite (Ordres du Royaume de l’Est) : L’ordre du Croissant d’Argent / Silver Crescent (service), de la Rapière dorée / Golden Rapier (escrime), de la Manche / Maunche (Arts et Sciences), du Tigre Combattant / Tygers Combatant (combat en armure), du Sagittaire / Sagittarius (tir à l’arc) et de la Lance dorée / Golden Lance (équestre )

Les recommandations ne nécessitant pas de vote (ce qui inclut les décernement d’armes (Award of Arms) ainsi que les nouveaux Ordres non-votant) peuvent être envoyés à la Couronne en tout temps.

La description de toustes les reconnaissances du Royaume de l’Est se retrouve dans la Loi du Royaume de L’Est ici (en document .pdf) dans la section IX, Awards, débutant à la page 23. Les informations additionnelles pour les nouveaux ordres non-votant sont disponible ici.

Tous peuvent recommander une personne pour une reconnaissance. Il n’est pas nécessaire de faire partie d’un Ordre pour pouvoir recommander une personne pour cet Ordre ou cette reconnaissance.

Une très bonne description expliquant comment écrire une bonne recommandation est disponible sur le wiki du Royaume de l’Est en cliquant ici.

Sa Majesté Avelina écrit également cet excellent article sur la façon de recommander quelqu’un pour une reconnaissance.

Vous pouvez soumettre vos recommandations pour toutes les reconnaissances en utilisant le formulaire EK Awards Web Form en cliquant ici.


Filed under: Announcements, Court, En français, Official Notices, Tidings Tagged: award recommendations, awards, polling deadlines, polling orders, pollings

King’s and Queen’s Archery Championship

Mon, 2016-06-13 17:07

On a sunny field in Quintavia, 84 archers shot for the honor of serving Their Majesties as Royal Archery Champions. Queen Avelina proclaimed that Her Champion would be the winner of the tournament, and that only an archer who could out-shoot Her could hold that title. King Kenric declared that he would select His Champion based on who impressed Him during the competition. The Captain General of Archers and outgoing King’s Champion, Mistress Jehannine de Flandres, reminded all participants of the duties of the Champions and said that anyone who wished to withdraw from the final round could do so with honor.

The day’s tests of skill were designed with a Chinese theme. Outgoing Queen’s Champion, Master Li Kung Lo, explained that Emperor Chin has called forth all mercenaries to defeat his aunt, the evil Empress Wu.

She has used her magic to cause dragons and other creatures to do her bidding. As dragons are avatars of the gods, it would anger the gods if any dragons were shot, thus any hits on dragons throughout the day would score negative 1 point. All participants were able to shoot the full course of 8 targets, at the end of which the finalists would be selected.

The tournament began with a long-distance shoot at Empress Wu with an ensorceled dragon at her feet.

The second shoot was at increasing distances; archers could only proceed to the next distance if they did not miss. There followed a sumptuous lunch, after which competitors formed into small groups and proceeded to each of 6 stations. Three stations had timed shoots: Rats and Rat King, Egg-Stealing Monkeys, and Bats. The other three stations were 6 shots, untimed: Empress Wu being overpowered by a Dragon (a friend/foe shoot), an Assassin, and a Hunting Shoot (at 3D animals).

The following top scorers declined to participate in the finals:
Rupert the Unbalanced (71)
Colin Ursell (69)
Kusunoki Yoshimoto (67)
Phillip the Facetious (59)
Meruit Kieransdottir (57)
Maxton Gunn (55)

There was a 5-way tie of archers with a total score of 38. Two of these archers — Kira Asahi and Cosimo di Venezia — were no longer on-site. The other three — Nest verch Tangwistel, Alec Craig and Julienne Ridley — shot-off for the 16th spot, and Julienne prevailed.

The final round was a version of the now-traditional head-to-head pairing. In this story, Emperor Chin has reneged on his promise to pay his mercenaries. So, each archer had to knock down 5 “coins”, and the first to shoot Chin would win and advance to the next opponent.

Top 16 Quarter-Finals Semi-Finals Finals 1. Godric of Hamtun (73) Godric Godric Peter (winner) 16. Julienne Ridley (38) Treya Devillin Godric (2nd) 2. Peter the Red (66) Peter Peter 15. Rolland Ian MacPherson (39) Stefan Squirrelsbane 3. Nathaniel Wyatt (65) Wyatt 14. Elizabeth Hawkwood (39) Squirrelsbane 4. Miles Boweman (63) Osmond 13. Osmond de Berwic (41) Devillin 5. Devillin MacPherson (55) 12. Hawkmoon (42) 6. Mark Squirrelsbane (55) 11. Otto Gottlieb (42) 7. Stefan O’Raghaillagh (54) 10. Ygraine of Kellswood (43) 8. Ryan MacWhyte (48) 9. Treya min Teanga (46)

YouTube: King’s and Queen’s Archery Championship Finals

Photos and text by Mistress Ygraine of Kellswood, additional photos by Eleanora Stewart.  Video by Baroness Arlyana van Wyck.

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Filed under: Archery, Events

Pennsic Pre-Registration Deadline: June 18th

Sat, 2016-06-11 11:34

Paid registration for Pennsic ends on Saturday, June 18th at 11:59pm EDT.  The size of the land allotted to a group is determined by the number of people who preregister by this deadline. Unpaid online registration is available until July 8th.

Reminder: The last day of pre-registration almost always has technical difficulties, we suggest not waiting until then.

 Preregistration is available at this website.


Filed under: Pennsic