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Essay: Hindu and Arabian Period: AD 500 – 1199

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Cultural Events
Mohammed had several visions to lead him to become the Prophet and Founder of the Religion of Islam. He is not their God as he preached the all mighty strength of Allah as their God. He also preached the day of judgment and the necessity of a person to be obedient to the will of God ‘ Allah.
Catholicism in Ireland
Ireland embraced Catholicism in 697 as they were a pagan country for more than half a century. It is famously known that St. Patrick was the saint to first spread Catholicism in the country however it is also know that several monks arrived there first to baptize several Irish communities. St. Patrick however was probably the person to be able to spread it widely over the country. However, now the majority of the Irish have turned their backs to Christianity as several sexual abuse cases of Priests from the early 90’s surfaced and affected the beliefs of the peoples.
Charlemagne became the emperor of the empire of Rome for 14 years. Before that he ruled the Franks for 47 years. Even though he served for a relatively short time, he was able to strengthen the empire just before it was conquered. He is part of the long line of the Carolinian dynasty which is also full of kings and highly influential people. He is the son of Pepin the Short and grandson to Charles Martel. He was known as a generous and intelligent man at that time. He built a strong relationship with the Church which strengthened the government. Charlemagne helped the empire in the worlds of commerce, education and agriculture.
Utrecht Psatler
The Utrecht Psalter is a great master piece. It is a manuscript with several beautiful illustrations. This Psalter’s beauty is not known to many as it is hard to understand.
871 AD
Alfred the Great ‘ was born at Wantage Berkshire; married to Ealhswith of Mercia
He was a Anglo-Saxon king who defended England against Danish and found the first English navy
New legal code came into force during his reign
He assembled the laws of Offa and other predecessors
He was reigned as king at the age of 21
He was a strong-minded veteran at the head of remaining resistance to the Vikings in southern England
His concept of kingship extended beyond the administration of the tribal kingdom of Wessex into a broader context
To improve literacy he arranged and took part in the translation from Latin into Anglo-Saxon of a handful books for he thought it ‘most needful for men to know, and to bring it to pass’.if we have the peace, that all the youth now in England’.may be devoted to learning’
Because of his valiant defense of his kingdom against a stronger enemy, for securing peace with the Vikings and for his farsighted reforms in reconstruction of Wessex and beyond, that alone of all the English kings and queens he is known as ‘the Great’.
Schism of the Church
900 AD
Vikings discovered Greenland
Viking expansion. It was a quest for retaliation against continental Europeans for their previous invasions of Viking homelands, such as Charlemagne’s campaign in forcing Scandinavian pagans to convert to Christianity by killing any who refused to become baptized. Another is that the Viking population had exceeded the agricultural potential of their homeland. This is true only in western Norway.
Two areas along Greenland’s southwest coast were colonized by Norse settlers on 986
For Norse pastoral farming, Greenland was the best marginal
Settlers arrived during warm phase, when crops such as rye and barley could be grown
Greenland became a dependency of Norway’s king
912-973 AD
Emperor Otto I
He continued his father’s goal in unifying the Germans into single tribe and expand his power through aristocracy. To reduce the dukes who will be a threat to his position, he used strategic marriages and personal appointments to put members of his family in authority. He also used Church-State Alliance which gave him military and political services. Otto was said to be the savior of the Christendom after he defeated the Magyars which marked the end of Hungarian invasions of Western Europe in 955. Otto was crowned Emperor by Pope John XII in 962.
Founder of the Holy Roman Empire
He was the oldest son of Henry I the Fowler and Matilda of Ringelheim
‘the first of the German’s to be called the emperor of Italy’
Otto installed members of his family to the kingdom’s most important duchies through strategic marriages and personal appointments
In Germany, Otto transformed the Roman Catholic Church to strengthen the royal office and subjected its clergy to his personal control
In 955 Otto defeated Magyars, thus ending the Hungarian invasions of Western Europe. This earned Otto reputation as a savior of Christendom
Otto had conquered the Kingdom of Italy by 961 and extended his realm’s borders to the north, east, south
Otto sought to improve relations with the Byzantine Empire, which opposed his claim to emperorship. To resolve this, the Byzantine princess, Theophanu ,married Otto II in April 872
950 AD
Beginning of Dark Ages
The fall of the Holy Roman Empire marks the beginning of the Dark Ages. There are still debates on what is the real cause of the fall of the Empire but there are some theories and explanations that arise why the fall happened namely: decay owing to general malaise, monocasual decay, catastrophic collapse, and transformation.
According to Edward Gibbo (theorizing general malaise) the fall was inevitable because of the unsound foundation of the empire. He added that the Christianity was said to be the major cause of the monocasual decay. On the other hand, the catastrophic collapse explains that the fall was due to the migration of people that gives stress to the structure of the empire. Then, the transformation explains that the empire only underwent a series of transformations converting it to the medieval world.
The Period of steady demographic and economic growth in Western Europe
Demographic growth was supported by a food resources expansion
European agricultural production increased
Climate was warmer than in the early Middle Ages
Medieval Climate Optimum means longer growing seasons and the ability to cultivate lands further and expand the repertoire of crops
It was also in 950 when the revival of Christian trade in Mediterranean happened
Long-distance trade routes were dominated by Italian and Jewish merchants. The development of merchant guilds as sworn associations/confraternities of merchants to protect, avenge, bury members
973-983- Emperor Otto II
He is the son of Emperor Otto I. He became the successor at the age of 18 after his father died. Otto II continued his father’s ambitions. He defeated the other Ottonian dynasty which result to remove the Bavarian Line of Ottonians from the succession of the throne. He focused in invading the whole Italy but he did not succeed because of the defeat by the Muslims. He died at the age of 28 in 983.
990- Development of systematic musical notation
Guido d’Arezzo(995-1050) was the Italian monk who developed the systematic musical notation in 990. The system of musical notation was composed of four parallel lines and it uses notes instead of letters. He introduced solfegge which is a system of mnemonic (ut, re, mi, fa, sol and la) that represents the hexachord C, D, E, F, G and A. He also invented the Guidonian hand.
993- First canonizations of saints
The first saint who was canonized by Pope John XV in 993 was Saint Udalric [(890- july 4, 973) Bishop of Augsburg and Pope of Geramany]. He aimed to improve the moral and social condition of the ministry and to firmly implement the laws of the church. One of his goal was to build more church for the word of God to be reachable to the common people. He became the role model of his clergy and diocese because of his good example and practices
1003- Leif Erickson crosses Atlantic to Vinland
Leif Erickson was the son of the founder of Greenland, Erik the Red. According to some historians, on his way to Greenland, he stopped on the North American continent and discovered one portion of the land and called it Vinland. On the other hand, some said that he search for Vinland since an Islander trader told him about that region. Though there are many speculations, still he is said to be the first European to reach the continent before Christopher Columbus did in 1492.
The Tale of Genji
Tale of Genji was the World’s first novel, first modern novel and first Psyhchological novel written by Muraasaki Shikibu. It is also known as ‘The Pimp of Heian Court’. It is considered a Japanese literature classic.
The story is about Genji, the son of the Emperor to his favorite concubine Kiritsubo. However, Kiritsubo died early because the other concubines became jealous and made her life difficult. Genji grew up beautifully and would have made a fine crown prince but he lacked backing in the court so the emperor made him one of his retainers and called him Hikaru Genji or Shining Genji. When he was young the emperor married another woman named Fujitsubo who looked so much like Kiritsubo. Genji grew infatuated to Fujitsubo because of his longing for his mother and after a few years they had a son which would be the new Emperor named Reizei.
Although Genji is married to Aoi no Ue he pursued seven other women: Murasaki no Ue, the widow of the former Crown Prince, a married woman, the lover of his best friend, an almost 60 year old maid, a princess and the daughter of his political enemy. Because of his affair to the daughter of his political enemy, he was sent to Kyoto for exile. There he met Akshi no Kimi and they had a baby girl. His son Reizei became the new Emperor and Genji became Minister. Genji built the Rokujo Estate where he lived with all his ladies. He married another woman named Onna San no Miya. Onna was seduced by Kashiwaga and they had a son named Kaoru which would be the main character in the closing ‘The Ten Uji Chapters’. He felt that the heavens where punishing him so he went into self-exile and died.
School of Chartres
The school was famous and influencial. It was operated by the Chartres Cathedral. The School of Chartres is known to be the center of scholarships and an institution of higher learning. The school was established by Bishop Fulbert who is known as Father Socrates. The teaching in the school are based on the seven liberal arts, trivium and quadrivium. Pioneers of the Renaissance such as Bernard of Chartres, Thierry of Chartres, William of Conches, and Englishman John of Salisbury were attracted to the school.
Normans penetrate England
Norman, Breton and French armies began to invade England on September 28, 2066. The armies are led by Duke William II of Normandy also known as William the Conqueror. Normans are the people that gave Normandy its name. They are also descendants of conqueror Vikings and the natives Merovingians. In the Battle of Fulford, Edwin and Morcar were defeated by Harald Hardrada and Tostig Godwinson. In the Battle of Stamford Bridge, Harald and Tostig were defeated by Harold and in the Battle of Hastings William defeated Harold in October 14.
The invasion of Normans in England was an important part in English history as it resulted to a lot of consequences. One of which is Elite replacement where natives were removed from high governmental and ecclesiastical offices. Another consequence is English Emigration. Anglo-Saxons including nobles went to Scotland, Ireland and Scandinavia. The invasion also affected the language. Old English was replaced by Anglo-Norman which is a northern dialect of Old French. French words began to infiltrate into the English language. French names like William, Robert and Richard gradually became common. Immigration and Intermarriages are another consequence. Normans went and settled into England. Norman men and English women intermarry.
Macbeth defeated at Dunsinane
Macbeth is a Shakespeare play considered to be one of his darkest and most influential tragedies. It is about the effects or consequences of using evil methods in attaining power. Macbeth the protagonist is a Scottish general who received a prophecy from three witches that he would one day become the King of Scotland. His passion and ambition led him to kill the current King Duncan and took the throne for himself. He became a tyrannical ruler who is consumed by his guilt and paranoia. To protect his throne and to stop suspicions, he killed a lot of people. Lady Macbeth killed herself because of despair. In the battle of Dundee Macbeth was defeated by Malcolm Canmore. The battle happened in July 1054 in the plains of Gowrie. Dunsinane is the westmost part of the Sidlaw hills. Malcolm declares his good intentions for Scotland and invites everybody to his coronation.
Consecration of Westminster Abbey
Westminster Abbey was consecrated in December 28, 1065. It was made because of Religious and Sacred purposes. British monarch coronations are held in the Abbey. It is also the burial site of Royals or British Monarchs. Royal weddings also took place in the Abbey. The first coronation held in Westminster abbey is of William the Conqueror.
Chinese Use Movable Type to Print Books
The first movable type was created by an alchemist named Bi Sheng during the Song Dynasty. Each block of clay and glue has one Chinese character carved into it. The movable type was made because the block printing before it was too expensive and time consuming.
1088 – First Modern University
The University of Bologna is a university located in Bologna, Italy. The date of its founding is believed to be in 1088. As of 2013, the motto of the university is Alma mater studiorum. It has about 85,000 students in its 23 schools. It has branch centres in Imola, Forl??, Ravenna, Cesena and Rimini and one abroad in Buenos Aires. It has a school of excellence named Collegio Superiore di Bologna. It is known as the oldest university in continuous operation. It was the first to use the term ‘universitas’ for the groups of students and masters for the people in the institution.
The university received a charter coming from Frederick I Barbarossa in 1158. In the 19th century, Giosu?? Carducci lead a committee of historians who traced the founding of the University back to 1088, making it the oldest continuously-operating university in the world.
The University of Bologna arose surrounding mutual aid societies of foreign students called “nations” , grouped by nationality, for protection against city laws that imposed collective punishment on foreigners for their countrymen’s crimes and debts.
1096 – Start of first crusade
The First Crusade started as a pilgrimage in Germany and France. It ended as a military expedition by Roman Catholic Europe to regain the Holy Lands taken in the Muslim conquests of the Levant (632’661). It resulted in the recapture of Jerusalem in 1099. It has the primary goal of responding to a plea from Byzantine Emperor Alexios I Komnenos, who requested that Western volunteers to help repel the invading Seljuq Turks coming from Anatolia. It was launched on November 27 1095 by Pope Urban II. An additional goal soon became the principal objective’the Christian reconquest of the Holy Land and the sacred city of Jerusalem and the freeing of the Eastern Christians from Islamic rule.
During the crusade, knights and peasants from many nations of Western Europe travelled to Constantinople and then to Jerusalem. The Crusaders arrived at Jerusalem, launched an assault on the city, and captured it in July 1099. They killed many of the Muslim, Christian, and Jewish inhabitants.
1110 – Chinese Invent Playing Card
A Playing Card is a piece of specially prepared heavy paper, plastic-coated paper, thin cardboard, thin plastic or cotton-paper blend. It is invented in Imperial China. It is marked with distinguishing motifs. It is used as one of a set for playing card games. It is palm-sized for convenient handling.
It is found in China as early as the 9th century during the Tang Dynasty (618’907). Ouyang Xiu, a Song Dynasty scholar, asserted that playing cards and card games existed at least since the mid-Tang Dynasty. They associated their invention with the simultaneous development of using sheets or pages instead of paper rolls as a writing medium.
1125 – Commencement of Troubadour Music
The best-known troubadours lived in Southern France (1100 to 1300).Their verse and musical manuscripts provide an interesting insight into the rich culture of medieval communities, the crusades, and courtly life. Troubadour music is synonymous with themes of courtly love.
J.B. Beck, a musicologist, said that the troubadours came from a culture of nomadic singers known as histrions, mimes, and jongleurs (Whigham). Many of these people have noble backgrounds. The upper-class citizens sent their children to Catholic monastic schools where they were taught grammar, religious music and neumatic notation. Beck told that many of the students became good composers and musicians. After their formal education, these young men returned home to apply what they have learned to more secular themes.
Their beginning can be traced back to the 6th century. It is when Caesar of Arles wrote a decree banishing secular entertainers at the urging of church bishops. His text notes that they are responsible for infamous and diabolic songs of love.
1154 – Beginning of Plantagenet Reign
This dynasty is normally subdivided into three parts.
-The Angevins (1154-1216)
3 Kings, commencing with Henry 2nd (1154-1189)
-The Plantagenets (1216-1399)
5 Kings, commencing with Henry 3rd (1216-1272)
-The Houses of Lancaster and York (1399-1485)
6 Kings, commencing with 10 year old Henry 4th (1399-1413)
There are main events during the Plantagenet period. Punishments such as hanging, drawing, quartering and burning at the stake were invented in England during this period. The introduction of parliamentary democracy and the slow loss of dictatorial power of the English kings also happened in the Plantagenet era. The nonstop battle between English Kings and the Church in Rome for the lawful high ground and the unholy manners and corruption of the Church headquartered in Rome were also important events in this period. There were development in the universities, particularly Oxford and Cambridge as centres of free thought.The steady boost in power of Islamic forces and the devastating attempts of European Christian forces in their endeavours to save Jerusalem and their trading courses to the Far East also occurred in the Plantagenet period.
1165 – Maimonides: Mishneh torah
Rabbi Moshe ben Maimon also known in the Jewish world as Rambam and most of the world as Maimonides is one of the most principal figures in the chronicle of Torah scholarship. He is one of the towering symbols in medieval ,intellectual and religious life. He also did well in the fields of philosophy, science, medicine, exegesis and communal leadership apart from his law code.
The Mishneh Torah or The second law is a summary of the entire body of Jewish religious law or Torah. It is divided up into fourteen general sections further subdivided into books and then into numbered chapters and laws.
The Mishneh Torah has distinctive features. It covers the full range of Jewish law, as formulated for all ages and places. It completely reorganizes the laws in a clear and logical system.It presents the normative rulings without any discussion of how the decisions were reached and it opens with a section on organized philosophical theology .
1186 – Domesday Book; Tax Census Ordered by William the Conqueror
William I , William the Conqueror or William the Bastard ‘is the first Norman King of England, ruling from 1066 up to his death in 1087.
The Domesday Book is a manuscript that documents the great survey of much of England and parts of Wales completed in 1086. The survey was made for William the Conqueror. One purpose of the survey was know who held what and what taxes had been liable under Edward the Confessor. It was made containing information on who owned what throughout the country. It also tells him who owed him what in tax. Because the information was on record, nobody could dispute or argue against a tax demand. The book brought disaster to England.
The Domesday Book is correlated with William the Conqueror’s attempt to dominate Medieval England. together with castles throughout England, the Domesday Book was to give William huge authority in England.
Mathematical Events
‘A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero (shunya) minus zero is nothing. (kha).
A debt subtracted from zero is a fortune.
So a fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multiplied by itself is nothing.
The product or the quotient of two fortunes is one fortune.
The product or the quotient of two debts is one debt.
The product or the quotient of a debt multiplied by a fortune is a debt.
The product or the quotient of a fortune multiplied by a debt is a debt.’
Brahmagupta is astronomer and mathematician. He also became heads of certain observatory groups in India. The man also used his poetic side to express his scientific ideas such as the text above. He used certain terms to better explain certain algebra concepts such as the rule of signs. He is also known as one of the first Indians to use the term zero, which was new to them at that period.
Bede’s greatest scientific achievement was creating the western calendar. This calendar helped people and especially the Church to find when Easter should come every year. This was an important discovery as Christians wanted to celebrate the Resurrection of Christ in a uniformly manner. He was also one of the major contributors to the establishment of the term ‘AD’ as the normal system for dating or timing in Europe.
The term Zero was a concept mathematicians still had to argue on in the past. Not all civilizations used the term or the symbol of zero immediately as only a few civilizations were able to discover it on their own. The cities or civilizations to discover this on their own were Babylonians, Mayans and the Indians. Several other civilizations like China, Arabia and Europe followed upon the influence of the first discoverers. For the Hindu’s the Zero meant ‘nothing’ to them as soon they created symbol for it to supplement their mathematical analysis.
Muhammad Ibn Musa Al-Khwarizmi (810)
Muhammad Ibna Musa Al-Khwarizmi is a Persian mathematician who worked in The House of Wisdom in India. He developed and further explained several algorithms in the field of algebra and mathematics in general. He explained these by analysis and by geometrical examples. He also being called (by some) the ‘foundation’ and ‘cornerstone’ of the sciences. As he explained and developed concepts that we need to do the math we do today.
810 AD
House of Wisdom set up in Baghdad; Greek and Indian mathematical astronomy works are translated into Arabic
Hindu Numerals are the ten digits (0,1,2,3,4,5,6,7,8,9)
Digits such as 975 is read as a single number
850 AD
Mahavira- Indian Mathematician who contributed to the development of Algebra. He is also a Jain by religion
Was aware of the works of Jaina mathematicians and also the works of Aryabhata and Brahmagupta, and he improved their works
He was not an astronomer; was confined solely to mathematics & stands almost alone in history of Indian mathematics
Mahavira’s major contributions to mathematics include:
Arithmetic ‘ GSS was the first text in arithmetic. He made more simple classification of arithmetical operators. He gave all almost formulae in Geometric progression.
Permutations and Combinations- He was the first to give general formula; extended and systematized Jain works.
General formula for combinations as given by Mahavira
Algebra- He worked on quadratic, indeterminate and simultaneous equations. He demonstrated definite understanding of the concept of a quadratic equation having two roots.
Ellipse- the only Indian mathematician to refer to ellipse. Though he gave incorrect identity for are of ellipse, his formula for the perimeter is worth noting.
870 AD
Thabit Ibn Qurra Ibn Marwan al-Sabi al-Harrani ‘ born on 836 AD at Harran(Turkey). He was a member of Sabian sect, but Muhammad Ibn Musa Ibn Shakir, Muslim mathematician, was impressed by his knowledge of languages, and realizing his potential for a scientific career, he was selected to join scientific group that was patronized by the Abbasid Caliphs.
Contributed in mathematics and astronomy
He extended the concept of traditional geometry to geometrical algebra and proposed several theories that led to the development of spherical trigonometry, non-Euclidean geometry, integral calculus and real numbers.
He criticized some theorems of Euclid’s elements and improved it.
He applied arithmetical terminology to geometrical quantities and studied parabola and ellipse.
He was a great help in determining the surfaces and volumes of different types of bodies and constitute.
900 AD
Abu Kamil Shuja also known as al-Hasib al-Misri, which means the calculator of from Egypt
He was one of al-Khwarizmi’s immediate successor
Abu Kamil considered that he was building on the foundations of algebra as set up by al-Khwarizmi and he forms an important link in the development of algebra between al-Khwarizmi and al-Karaji
His work was the basis of Fibonacci’s books
He was not only important in development of Arabic algebra, but through Fibonacci, he is also fundamentally important in the introduction of Algebra in Europe
One important step forward in Abu Kamil’s algebra is his ability to work with higher powers of the unknown than x2. The powers are not given in symbols but are written in words, yet the powers tell us that Abu Kamil had begun to understand that we would write in symbols as xnxm=xn+m
For example: ‘square square root’ for x5, ‘cube cube’ for x6
He also worked on some books: (a) Book of fortune, (b) Book of the key to fortune, (c) Book on algebra, (d) Book on surveying and geometry, (e) Book of the adequate, (f) Book on omens, (g) Book of the kernel, (h) Book of the two errors, and (i) Book on augmentations and diminution. But only Book on algebra and Book on surveying and geometry survived
976: oldest examples of written numerals in Europe
The oldest examples of written numerals in Europe was introduced by Arabs in 900. This Arabic numerals were gathered in Codex Vigilanus compiled by Vigila (illustrator), Serraacino(friend) and Garcia(apprentice) from the Riojan Monastery. The said Codex contains Mozarabic, Visigothic and Carolingian elements in its illuminations.
980: Abu’wefa: constructions, trig tables
Abu’wefa (June 10, 940 – July 1, 998) major role in trigonometry was his development of the trigonometric table most specially sines and cosines. His fascination in mathematics and astronomy drew his attention to the works of Ptolemy most specifically the Almagest. The said book which shows the relationship between the angles and the sides of a right triangle triggers Abu’wefa to create the trigonometric ratios. Abu’wefa perfected Ptolemy’s chords through applying algebraic methods instead of geometric one. He also introduced trigonometric identity and the law of sines for spherical triangle . He made the Zif al-wadith (that which is clear), Kitab fi ma yahtaj ilayh al-kuttab wa l-‘ummal min ‘ilm al-hisab of 961-976(Book on What is Necessary from the Science – of Arithmetic for Scribes and Businessmen), the Kitab fi ma yahtaj ilayh al-sani ‘min al-a’mal al-handasiyha of 990(Book on What is Necessary from Geometric Construction for the Artisan), and Kitab al-kamil. The application of his development of the ‘half chord’ which gives more accurate measurements was in astronomy, surveying and navigation.
999: Pope Sylvester II (GERBERT)- arithmetic, pi approximated as ‘8= 2.83
Pope Sylvester II (946 ‘ May 12, 1003) was the French pope from April 2, 999 until his death in 1003. His real name is Gerbert of Aurillac who promoted the study of Arab or Greco-Roman arithmetic, mathematics, and astronomy. He also reintroduce the abacus and armillary sphere and present decimal numeral system using the Arabic numerals in Europe.
1000: Sridhara recognizes the importance of zero
He wrote the books P??t??ganita and the P??t??ganitas??ra or Tri??atik??. The latter contains solutions concerning mixtures, series, plane figures, volumes, shadows, and zero. The said book states that “If 0(zero) is added to any number,the sum is the same number; If 0(zero) is subtracted from any number,the number remains unchanged; If 0(zero) is multiplied by any number,the product is 0(zero)”. He also developed the formula: ax2+bx+c=0 and 4a2x2+4abx+4ac=0
Al-Karkhi or Al-Karaji
Al-Karkhi was born on May 6, 1826 and died on March 25, 1864. He’s a Persian mathematician and engineer. He uses arithmetical operations rather than geometrical operations in algebra and by doing so he freed algebra from geometrical operations. His contributions to math include the formula xnxm = xm+n. He also defined monomials such as x, x2, x3, … and 1/x, 1/x2, 1/x3 and gave the general structure of the Pascal’s Triangle. His other works are that of binomial theorem and binomial coefficients. The binomial theorem explains the expansion of powers in binomials. His three major works are Wonderful on calculation, Glorious on Algebra and Sufficient on calculation. The rules for arithmetic operations in addition, subtraction and multiplication of polynomials were derived from his work on algebra and polynomials.
Game of Rithmomachia
It is also known as The Battle of Numbers and The Philosopher’s Game. It is an early European mathematical board game similar to chess that is played by two. It is a strategy game that would enhance not only a person’s mathematical skills but also teach them moral values. The game is also an effective tool in teaching the number theory of Boethius. The number theory of Boethius stresses on the innate harmony and perfection of numbers and proportions.
There are four pieces to be used in the Game of Rithmomachia. Each piece have different moves and values.
Rounds: moves one square diagonally
Triangles: moves two squares vertically or horizontally
Squares: moves three squares vertically or horizontally
Pyramids: The White Pyramid needs a “36” Square, a “25” Square, a “16” Triangle, a “9” Triangle, a “4” Round, and a “1” Round, to get the Pyramid’s value of 91. The Black Pyramid needs a “64” Square, a “49” Square, a “36” Triangle, a “25” Triangle, and a “16” Round, to get the Pyramid’s value of 190.
There are different types of Capturing. When a piece is captured it changes sides and is removed in the game.
Meeting: A piece lands on another piece of the same value
Assault: A smaller value piece is equal to the larger value piece it wants to capture when it is multiplied by the number of vacant spaces between it and the larger number
Ambuscade: When the sum two pieces is equal to the enemy piece between them
Siege: All four sides of a piece is surrounded
There are also a variety of ways to determine the winner and when the game ends. There are Common and Proper victories.
Common Victories:
De Corpore (“by body”): The player captured a certain number of pieces that was agreed upon by the players
De Bonis (“by goods”): The player Captured pieces add up to or exceed the value agreed upon by the players
Proper Victories:
Victoria Magna (“great victory”): three pieces should be arranged in arithmetic progression.
Victoria Major (“greater victory”): Three pieces of a four piece set must be in a certain progression and another three piece must be in another progression.
Victoria Excellentissima (“most excellent victory”): A set of four pieces have all three types of mathematical progressions.
Omar Khayyam
Omar was born on May 18, 1048 and died on December 4, 1131. He was a Persian mathematician, philosopher, astronomer and poet. He wrote the Treatise on Demonstration of Problems of Algebra. The Pascal’s triangle could be seen in his writings. The Pascal’s triangle is binomial coefficients arranged in a triangular manner. Except for the edges the rest of the numbers in the triangle are the sum of the two numbers above them.
Omar had a different procedure in solving cubic equations. The first step is to draw two conic sections then draw a line perpendicular to the x-axis through the point of intersection. The solution of the problem would be the distance between the perpendicular and the second point which lies on the x-axis.
Bhaskara II
Bhaskara was the greatest mathematician of medieval India. He had various contributions to the world of mathematics. His main work is Siddhanta Shiromani or Crown of treatises. It has four parts that deals with different branches of math. Lilavati the first part is about arithmetic. Bijaganita the second part is about algebra. Goladhyana the third part is about spheres and Grahaganita the last part is about mathematics of the planets. He is best known for establishing the decimal number system we use today.
He contributed a lot to mathematics such as the concept of negative numbers and how to solve problems with such numbers. He also introduced the concept of infinity in dividing numbers with zero. Bhaskara also gave rules for zero such as A + 0 = A, A ‘ 0 = A and A x 0 = 0. The formulas in solving the area of sphere and volume of a sphere are also given by Bhaskara. The formula for the Area of sphere is 4 x area of a circle and the formula for the Volume of a sphere is area of a sphere x 1/6 of its diameter. His contribution to trigonometry are the formulas sin (?? + ??) = sin ?? cos ?? + cos ?? sin ?? and sin (?? – ??) = sin ?? cos ?? – cos ?? sin ??.
1125 – Earliest account of Mariner’s Compass
The Mariner’s Compass has an uncertain history. But there proofs that showed the people’s knowledge of magnetism in the early times. The evidences stated that house foundations were laid out according to magnetic north but it was not the first definite proof. The first definite reference to magnetism is from a Chinese book and it describes the lodestone as a stone that summons or attracts iron.
After all those proofs, there is a clear description of the magnetic compass. It was from Shen Kua who wrote a book containing a passage which describes geomancers, a kind of fortune teller long employed in China to determine the luck of proposed sites for structures. Such geomancers pursued their art by rubbing a lodestone against a steel needle, thus causing the needle to point south. The knowledge here shown of the principle of magnetic deviation proves that the Chinese already knew and studied the compass way before Shen Kua’s time.
1150 – Bhaskara: algebra
Bh??skara , also known as Bh??skar??ch??rya , Bh??skara the teacher , and Bh??skara II was an Indian mathematician and astronomer.
Bijaganita (Algebra) his work in twelve chapters, was the first text to recognize that a positive number has two square roots ,a positive and negative square root). Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y. Bhaskara’s method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called “Pell’s equation”) is of considerable importance.
His work Bijaganita is effectively a treatise on algebra and contains the following topics:
positive and negative numbers.
the unknown
determining unknown quantities
Surds (includes evaluating surds)
Kuttaka (for solving indeterminate equations and Diophantine equations)
simple equations (indeterminate of second, third and fourth degree)
simple equations with more than one unknown.
indeterminate quadratic equations (of the type ax2 + b = y2)
quadratic equations
quadratic equations with more than one unknown
operations with products of several unknowns.
extracted from http://www-history.mcs.st-and.ac.uk/Projects/Pearce/Chapters/Ch8_5.html
1175 – Averroes: Trigonometry, Astronomy
Ab?? l-Wal??d Mu’ammad bin ??A’mad bin Ru??d , or in his Latinized name, Averroes was a defender of Aristotelian philosophy against Ash’ari theologians led by Al-Ghazali.
In astronomy, Averroes claimed for a precisely concentric model of the universe, and explained sunspots and scientific reasoning talking about the occasional opaque colors of the moon. He also worked on the description of the spheres, and movement of the spheres.
On the other hand, Abu Arrayhan Muhammed Ibn Ahmad Al Biruni was the first scientist to give an accurate determination of latitude and longitude. His manuscripts on trigonometry where number became elements of function, were used by many Western scholars.
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