Explore the reasons why Mercutio is killed and the consequences of his death. Why is this such an important event in the play Essay Example

Explore the reasons why Mercutio is killed and the consequences of his death. Why is this such an important event in the play? Essay In this essay I will be closely examining how act III scene i is a pivotal scene in the play Romeo and Juliet and I will be comparing how the two different directors have chosen to portray this scene. The opening of the scene in both versions of the play warns the audience of a key part of the play and seizes the audiences attention. They all show how Benvolio is anxious and worried about Mercutio and how it is a hot day and tempers may rise when people become restless. Benvolio says I pray thee, good Mercutio lets retire This signals to the audience that Benvolio fears that his friend may be killed or face death under the princes decree earlier in the play. This decree states that if anyone else is killed then so will the heads of each house. However the mood of Mercutio contrasts with Benvolios mood. He is up tight and ready for a fight. Thou art like one of these fellows that, when he enters the confines of a tavern, claps me his sword upon the table, and says God send me no need of thee! and by the operation of the second cup draws him on the drawer, when indeed there is no need.. This is Mercutio being argumentative and showing the audience that he is ready for a fight. Shake speare and the directors are hyperbolising or over exaggerating Mercutios views of Benvolio. Mercutio is accusing Benvolio of being scared of the Capulets. This is a significant part of the play. We will write a custom essay sample on Explore the reasons why Mercutio is killed and the consequences of his death. Why is this such an important event in the play? specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Explore the reasons why Mercutio is killed and the consequences of his death. Why is this such an important event in the play? specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Explore the reasons why Mercutio is killed and the consequences of his death. Why is this such an important event in the play? specifically for you FOR ONLY $16.38 $13.9/page Hire Writer The two different directors portray the play in several different ways contrasting much of the time, but sometimes there are similarities between the two. The first version of the play that we watched was Zeffirellis version which uses unknown actors to make sure that no-one could relate them to previous films. He also used colour connotations so that it is easy for people to differentiate between the two houses; he used grey for the Montagues and orange for the Capulets. The costumes were typical of the 11th century portraying the original setting for when Shakespeare set the play. Juxtaposition However, Baz Luhrman gave similar connotations but not with colour. The Capulets were portrayed as very religious Catholics and had Cuban or Spanish accents. They had bright colourful clothing though, showing very distinctive characteristics. The Montagues were business, city folk and centred on money, creating a contrast between the two families already. Music also plays a big part in both films, Zeffirelli uses classical music with violins and flutes creating an airy effect and gives the impression that we are in the 11th century. However, this differs with the newer version of Romeo and Juliet directed by Baz Luhrman as he uses upbeat modern music with some computer generated sounds which people of today can relate to and form a relationship with their own lives. They each portray the key scene differently. Zeffirelli shows the fight as an accident and comedic. He chose a hot mid-day and set the fight in the market where the previous fight we saw took place. We know that it is hot because Mercutio is in a cold trough cooling down. Mercutio and Tybalt begin the fight and it becomes clear that there are many obstacles to contend with as they use weapons, such as the chairs, hay stacks and others objects. The key thing with this scene is that the fight is an accident, we are shown an example of this when Tybalt has the chance to kill Mercutio. This moment is made light hearted when Mercutio whistles and Tybalt covers up and pretends to not realise what Mercutio is doing. It is shown as a joke or a mock fight. The audience are aware that characters in the crowd are joking and laughing and not looking intensely or trying to hinder the fight. Luhrman shows the fight as more violent, contrasting with the previous fight scene. The fight scene is extremely aggressive to demonstrate the emotion and resentment of both of the men. The scene also shows the audience that this scene was a fundamental part of the play, and this grabs the attention of the audience. The audience would notice that Mercutio is trying to shoot fish in the modern film, showing his frustration. This is showing a new side to him, a more dangerous side, the audience have seen this before when they saw him give a speech previously in the play in which he had a premonition about Queen Mab. He was restless and impatient however, he did not hurt anyone whilst he was like this. This proves his name correct as he is a Mercurial and dangerous character. This is also reflected in the weather when it changes to cloudy and dark, this is the director using pathetic fallacy. Another example of this is the hot weather reflecting in the characters behaviour. The previous scene set at the petrol station is shown as comedic as there are several key points that make the audience find this early scene humorous, this makes us contrast it with the more violent Act III Scene i. The way in which the director plays with time speeding up and slowing the shots makes the audience find the shots funny. Also Luhrman uses the angles of the camera to make the audience laugh. He creates funny camera shots with the angles by emphasising things that the audience would not expect. One of the best ways in the petrol station of seeing this is where all the characters interact with each other for example the lady with the handbag makes the audience laugh as she has an amusing face when she realises what is happening. Another example is when he emphasises Tybalts shoes. In all of the plays Tybalt makes an extravagant entrance but these fluctuate slightly. The audience are reminiscing about the party when Tybalt is infuriated about Romeo being at the party. In Zeffirellis version Tybalt enters with an army of servants and his friends. He opens with words to aggravate Romeo more than he already was by saying Romeo, the love I bear for thee can afford so better term that this: thou art a villain. This makes the audience sit on the edge of their seat as the tension builds up. The audience sense that is a major part. Tybalt is very hesitant and wants to have his friends to support him and cheer him on. This is signalled when he says in Shakespeares version, Follow me close, for I will speak to them. This tells the audience that Tybalt wants his friends to protect him if anything serious does start. Tybalt is one of the minorities keeping the fight between the two houses alive. Zeffirelli uses this scene to make people laugh with the funny movements of the way of fighting. Luhrman uses the arrival of Tybalt as the key part as well. People can feel that there will be something happening between Mercutio and Tybalt as the stance of the characters is showing that they feel threatened and intimidated. This is also signalled by the way in which Tybalt arrives; slowly in a car with his friends and guards by his side, he exits the car and slams the door even before any speaking takes place; the audience sees that Tybalt is infuriated. The director uses this as the basis of the entire scene; he makes sure that some of the audience remember this part of the scene when they have finished viewing the entire film making it a memorable scene. Dramatic irony makes up a substantial amount of this play. This is when then the audience know something that most of the main characters do not. Zeffirelli uses dramatic irony throughout the entire play even up until the end. An example of this is at the end of the play when Juliet has taken the sleeping potion and the audience know that she is asleep and Romeo enters. He enters the tomb and sees her lying there and kills himself, this example shows the power of the audience and their status in the play which is called Dramatic Irony in which Shakespeare uses to affect the play. He uses this in many films. Luhrman also uses a lot of dramatic irony to make the audience become involved in the play; this encourages them to feel for the characters. An example of this is during the key scene when Romeo says, Tybalt, the reason that I have to love thee doth much excuse the appertaining rage to such a greeting. This is Romeo saying that he loves Tybalt but he cannot say why he loves him. The audience think about why Romeo loves Tybalt and remember the marriage scene that we saw previously. Both directors use the way that Tybalts speaking from the start of the play to show that this influences the future. This would have been similar to the audiences lives and make the play more credible to them. This was another common occurrence in Shakespeares plays where he uses the current ideas, for example the views on fate to influence and change his plays. Romeo is the main character in this play. His emotions change in all the versions of the play. He changes from a calm and collected person to an aggressive violent person willing to do anything for love. Zeffirelli shows Romeo as calm when the audience first see him, he is a gentle person who keeps his life private. Then at the pivotal scene he changes to an antagonistic and violent character, shown when he kills Tybalt. This shows the audience how love and emotions can change the future and fate, connecting the audience of the 11th century. This also relates to the fact how Romeo had premonitions about his death also relating to the 11th century beliefs on fate and how peoples lives were mapped out before birth. Luhrman uses Romeo to show the changes love can create for people and how fate plays a big part in the play. The premonitions are not emphasised enough as he does not feel them relevant to the time period it is shown to be in. Romeo is still show in the classical way of true love and concerned about his love with Juliet but he has other worries in this portrayal such as his gang that he hangs around with. This shows the similarities between the two plays and how they decided on the way to portray Romeo as an accomplished and excellent idea. Both directors show Romeo as a careful and calm character, who changes into an aggressive person. This never changes in the two films we analysed. (However, the difference is that Luhrman decided to show less thought on showing other characters with less importance and more emphasis on the changes in Act III scene i to significant characters.) Both directors change the way in which characters react. The list below shows you how Zeffirelli and Luhrman direct the main characters in much of the same way. Tybalt changes to a guilty person after the unintentional murder of Mercutio. Mercutio does not really change, however he becomes more violent and more aggressive male. He also becomes threatened more easily if he or his friends are insulted. Romeo turns to a murderous person who was willing to kill anyone for love. Lord Capulet is a hesitant character that changed his mind numerous times about his daughter marrying. He changes from a protector to a tyrant. Lord Montague is not seen much during the play but toward the end he becomes involved in the uniting of the two families. Benvolio changes and becomes more involved with the events that are occurring with other main characters. Friar Lawrence changes very late in the play. He becomes deceitful and betrays the two families which were key to his existence as a priest. He lies about the wedding between Romeo and Juliet and agrees to marry Paris and Juliet. The Nurse betrays Juliets trust and agrees with her master about the wedding between Juliet and Paris which goes against her former character when she formed a strong relationship with Juliet. Juliet she becomes deeply engrossed in her lust and love for Romeo. She changes and phoney death of herself and becomes private only confessing to Friar Lawrence about this. She changes her trust of the nurse. Finally Shakespeares plays can be pivotal at Act III Scene i. One of the main reasons for this is that Shakespeare usually wrote his plays with vii acts in them and vi scenes per act, therefore to change the play will be easier half way through as the audience will be understanding and will have seen some of the play. Another reason for this to be the changing point is that the characters all change their behaviour in many ways even if they changed slightly. It changes the whole plays future relating again to the 11th and 17th century beliefs on fate. This highlights the importance of this key scene and the way in which both directors show it and also the way in which the play was written. In conclusion both directors portray the pivotal scene with similar characteristics, for example clothing and characteristics of certain key characters. However, they are set in very different time periods, one in 17th century and one in modern day-. The fight and the way it is portrayed are fundamental to the events that follow. One of the reasons is that the play is peaceful up until this point with no main characters killed. Romeo is anti-violent. Another reason that this point is pivotal is because the play is looking good up to this point. Romeo is in love and Juliet is in love creating a romantic feel to the play. Finally this scene is pivotal because Shakespeare has written the play to show many different possibilities and routes which can occur.

Definition and Examples of Deep Reading

Definition and Examples of Deep Reading Deep reading is the active process of thoughtful and deliberate reading carried out to enhance ones comprehension and enjoyment of a text. Contrast with skimming or superficial reading. Also called slow reading. The term deep reading was coined by Sven Birkerts in The Gutenberg Elegies (1994): Reading, because we control it, is adaptable to our needs and rhythms. We are free to indulge our subjective associative impulse; the term I coin for this is deep reading: the slow and meditative possession of a book. We dont just read the words, we dream our lives in their vicinity. Deep Reading Skills By deep reading, we mean the array of sophisticated processes that propel comprehension and that include inferential and deductive reasoning, analogical skills, critical analysis, reflection, and insight. The expert reader needs milliseconds to execute these processes; the young brain needs years to develop them. Both of these pivotal dimensions of time are potentially endangered by the digital cultures pervasive emphases on immediacy, information loading, and a media-driven cognitive set that embraces speed and can discourage deliberation in both our reading and our thinking.(Maryanne Wolf and Mirit Barzillai, The Importance of Deep Reading. Challenging the Whole Child: Reflections on Best Practices in Learning, Teaching, and Leadership, ed. by Marge Scherer. ASCD, 2009) [D]eep reading requires human beings to call upon and develop attentional skills, to be thoughtful and fully aware. . . .Unlike watching television or engaging in the other illusions of entertainment and pseudo-events, deep reading is not an escape, but a discovery. Deep reading provides a way of discovering how we are all connected to the world and to our own evolving stories. Reading deeply, we find our own plots and stories unfolding through the language and voice of others.(Robert P. Waxler and Maureen P. Hall, Transforming Literacy: Changing Lives Through Reading and Writing. Emerald Group, 2011) Writing and Deep Reading Why is marking up a book indispensable to reading? First, it keeps you awake. (And I dont mean merely conscious; I mean  awake.) In the second place, reading, if it is active, is thinking, and thinking tends to express itself in words, spoken or written. The marked book is usually the thought-through book. Finally, writing helps you remember the thoughts you had, or the thoughts the author expressed.(Mortimer J. Adler and  Charles Van Doren, How to Read a Book. Rpt. by Touchstone, 2014) Deep Reading Strategies [Judith] Roberts and [Keith] Roberts [2008] rightly identify students desire to avoid the deep reading process, which involves substantial time-on-task. When experts read difficult texts, they read slowly and reread often. They struggle with the text to make it comprehensible. They hold confusing passages in mental suspension, having faith that later parts of the text may clarify earlier parts. They nutshell passages as they proceed, often writing gist statements in the margins. They read a difficult text a second and a third time, considering first readings as approximations or rough drafts. They interact with the text by asking questions, expressing disagreements, linking the text with other readings or with personal experience.But resistance to deep reading may involve more than an unwillingness to spend the time. Students may actually misunderstand the reading process. They may believe that experts are speed readers who dont need to struggle. Therefore students assume that their own reading difficulties must stem from their lack of expertise, which makes the text too hard for them. Consequently, they dont allot the study time needed to read a text deeply.(John C. Bean, Engaging Ideas: The Professors Guide to Integrating Writing, Critical Thinking, and Active Learning in the Classroom, 2nd ed. Jossey-Bass, 2011 Deep Reading and the Brain In one fascinating study, conducted at Washington Universitys Dynamic Cognition Laboratory and published in the journal Psychological Science in 2009, researchers used brain scans to examine what happens inside peoples heads as they read fiction. They found that readers mentally simulate each new situation encountered in a narrative. Details about actions and sensation are captured from the text and integrated with personal knowledge from past experiences. The brain regions that are activated often mirror those involved when people perform, imagine, or observe similar real-world activities. Deep reading, says the studys lead researcher, Nicole Speer, is by no means a passive exercise. The reader becomes the book.(Nicholas Carr, The Shallows: What the Internet Is Doing to Our Brains. W.W. Norton, 2010 [Nicholas] Carrs charge [in the article Is Google Making Us Stupid? The Atlantic, July 2008] that superficiality bleeds over into other activities such as deep reading and analysis is a serious one for scholarship, which is almost entirely constituted of such activity. In this view engagement with technology is not just a distraction, or another pressure on an overloaded academic, but is positively dangerous. It becomes something akin to a virus, infecting the key critical engagement skills required for scholarship to function. . . .What is . . . not clear is if people are engaging in new types of activity that replace the function of deep reading.(Martin Weller, The Digital Scholar: How Technology is Transforming Scholarly Practice. Bloomsbury Academic, 2011)

Billy Bristol Assignment Example | Topics and Well Written Essays - 1000 words

Billy Bristol - Assignment Example There are two examples to support my answer. The profit is generated by deducting the total operating expenses from the gross profit. The gross profit is arrived at by deducting the cost of sales from the net sales. Since collections include revenues from prior or future accounting periods, cash collection is not the best basis for determining current period net income. Current period $10,000 collection for sales generated during the prior accounting period does not affect the current period net income (Berry, 2011). Likewise, expense payments may include expenses for future accounting periods. Consequently, an adjusting entry is made to include only the current (accrued) portion of the total expense payments in the computing the current accounting period’s net income. A payment of $1,200 insurance expense 2 years should adjust to only include $600 for the current period insurance expense. Based on the above financial statement analysis ratios, Brisbane fared financially bette r than Perth. Brisbane’s 38 % Gross profit margin is higher than Perth’s 25 gross profit margin. A higher gross profit ratio indicates a better financial or operations performance. Brisbane’s 7 % profit margin is higher than Perth’s profit margin. A higher profit ratio indicates a better financial or operations output. Brisbane’s 3.65 times current ratio is higher than Perth’s 2.92 times current ratio. This clearly shows that Brisbane’s has more current assets allocated to pay for the currently maturing liabilities.

Choose a topic from your personal knowledge and experience Essay

Choose a topic from your personal knowledge and experience - Essay Example The most important thing when learning English as a second language is to learn to be a good listener. Effective listening is very important since it helps you to understand some of the words and concepts that may appear challenging. I have realised that when you carefully listen to others, you may be in a position to quickly understand what they are trying to say. When you listen carefully, you are also able to ask questions to the speaker so that he or she may explain the points that you may require clarification. I have also observed that if you are a good listener, mutual understanding is likely to be created with the person you are communicating with. If there is mutual understanding between the two parties involved, it is quite easy to exchange information. From my own experience, I have observed my level of understanding of different terms has significantly improved following the adoption of this strategy. I can now easily understand the main subject of discussion without aski ng the speaker to repeat what he or she has said. The other important point I must emphasize in this particular case is that you should not shun the way of behaviour of the people in the host country if you are an international student. This can only create hostility and it is counterproductive. Therefore, I have noted that you must be as cooperative as much as possible in order to avoid conflicts of interest. If you treat other people as important, there are likely chances of learning quickly their culture as well as language. From my own perspective, I have rapidly developed cordial relationships with other students than I ever anticipated. When you are learning English as a second language, I have realised that learning by observing other people is very effective in as far as acquiring new knowledge is concerned. One important thing I

Effective Tools and Tips for Getting the Most from Your Work Relations Assignment

Effective Tools and Tips for Getting the Most from Your Work Relationships by Mathew Gilbert - Assignment Example Mathew Gilbert has written extensively on business, spirituality, and psychology topics. He has also served as an editorial director at the renowned Institute of Noetic Sciences. Apart from authoring Communication Miracles at Work, Gilbert is also the author of The Workplace Revolution. This book provides its readers with the ability to identify some of the obstacles to effective communication in the workplace. This book also outlines some of the ways of breaking bad communication habits and how employees can communicate effectively at their workplace to achieve harmony in the workplace. The major topics in this book include; the role that stress plays in ineffective communication, influencing corporate culture on the ability of employees to get along with one another, gender issues, and effective communication skills for navigating a variability of real-world situations (Gilbert 34). The author, Gilbert also offers practical advice to its readers that can be applied to any job situation. In essence, this book practically demonstrates that improving communication within an organization is the key to having and enjoying a better day-to-day work experience. The author starts off by first of all talking about the work itself and how it has gone through changes over the years. In his opinion, this author claims that most companies were initially straitlaced, with lots of earnest men dressed in starched white shirts and conservative ties undertaking narrow but important roles with steadfast commitment (Gilbert 9). There existed specific rules, clear chains of commands and a general drone of commerce. This defined the times when companies like IBM, Ford and General Electric ruled the Western World. Unfortunately, this is no more since more women are in the workforce, there is more autonomy for employees, there are more teamwork and partnering, there is flextime and job shares, and the ever-growing multicultural diversity. As a matter of fact, the modern day workplace bears little resemblance to the one in which our fathers made a life commitment to work in (Gilbert 10).  Ã‚  

Contribution Of Indian Mathematics History Essay

Contribution Of Indian Mathematics History Essay Mathematics is the study of numbers, and counting, and measuring, but that is only the beginning. Mathematics involves the study of number patterns and relationships, too. It is also a way to communicate ideas, and perhaps more than anything, it is a way of reasoning that is unique to human beings. Mathematics plays a vital role in the modernization of this civilization. It is everywhere and affects the everyday lives of people. Although it is abstract and theoretical knowledge, it emerges from the real world. It is also a way to communicate and analyze ideas, a tool for organizing and interpreting data and above all, perhaps a method of logical reasoning unique to man. Mathematics is a necessary part of other sciences. In the words of Physicist Richard Feynan (2002) Nature talks to us in the language of mathematics that is numbers, mathematical rules and equations help us to make sense of the world around us (The Book of Popular Science). Mathematics in some form or other has been s ince the early age of human civilization. But its use in todays world has assumed great importance, since without its application higher technology cannot be mastered and harnessed for increasing production of goods and services and promoting human welfare. Over the centuries there has been spectacular progress in the development of mathematics as a branch of knowledge. And without the application of mathematics on a wide scale no country can march forward in line with the general progress of human knowledge and thought. Therefore learning of mathematics and promoting the horizons of knowledge by advanced researches in mathematics should be over emphasized. Thus, mathematics is an important and inseparable part of human life. It has been existed and developed since the ancient era and the aim of this article is to give a brief review of a few of the outstanding innovations introduced by Indian mathematics from ancient times to modern as Indias contribution in the field of mathematic s is immense and it should always be studied from a thoughtful perspective. Key Words: INTRODUCTION: India was the motherland of our race and Sanskrit the mother of Europes languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity of self-government and democracy. In many ways, Mother India is the mother of us all. Will Durant, American Historian 1885-1981 Mathematics is an important field of study. Mathematics is vital as it helps in developing lots of practical skills, in fact study of mathematics itself include the concepts related to the routine lives of human. It not only develops mathematical skills and concepts, it also helps in developing the attitudes, interest, and appreciation and provides opportunities to develop ones own thinking. So, mathematics is undoubtedly a discipline which is imperative to know and study. Fig. 1, Importance of MathematicsC:UsersnaveenDesktopUntitled.png Mathematics has played a very significant role in the development of Indian culture for millennia. Mathematical ideas that originated in the Indian subcontinent have had a thoughtful impact on the world. In ancient time, mathematics was mainly used in an auxiliary or applied role. Thus mathematical methods were used to solve problems in architecture and construction (as in the public works of the Harrappan civilization) in astronomy and astrology (as in the Jain mathematicians) and in the construction of Vedic altars (as in the case of the Shulba Sutras of Baudhayana and his successors). By the sixth or fifth century BCE, mathematics was studied for its own sake, as well as for its application in other fields of knowledge. In fact there does not seem to have been a time in Indian history when mathematics was not being developed. Recent work has unearthed many manuscripts, and what were previously regarded as inactive periods in Indian mathematics are now known to have been very activ e. The picture is yet not complete, and it seems that there is much more to do in the field of the history of Indian mathematics. The challenges are twofold. First, there is the task of locating and identifying manuscripts and of translating them into a language that is more familiar to modern scholars. Second there is the task of interpreting the significance of the work that was done. The time is ripe to make a major effort to develop as complete a picture as possible of Indian mathematics. The importance of mathematics in India can be seen by a well-known verse in Sanskrit of VedangJyotish (written 1000 BC) as: This verse means that As the crown on the head of a peacock and as the gem on the hood of a snake, so stands Mathematics crowned above all disciplines of knowledge. This fact was well known to intellectuals of India that is why they gave special importance to the development of mathematics, right from the beginning. Indian mathematicians made great strides in developing arithmetic, algebra, geometry, infinite series expansions and calculus. Indian works, through a variety of translations, have had significant influence throughout the world. Mathematics in ancient times (3000 to 600 BCE) The oldest evidence of mathematical knowledge to Indians is being found in Indus Valley Civilization. The metallic seals found in the excavations of Mohan-Jo-Daro and Harrapan indicates that the people of this civilization had the knowledge of numbers. It is also clear from the pottery and other archaeological remains that they had the knowledge of measurement and geometry even in crude form. The Indus valley civilization is considered to have existed around 3000 BCE. Two of its most famous cities, Harappa and Mohenjo-Daro, provide evidence that construction of buildings followed a standardized measurement which was decimal in nature. Here, we see mathematical ideas developed for the purpose of construction. This civilization had an advanced brick-making technology (having invented the kiln). Bricks were used in the construction of buildings and embankments for flood control. The study of astronomy is considered to be even older, and there must have been mathematical theories on which it was based. Even in later times, we find that astronomy motivated considerable mathematical development, especially in the field of trigonometry. Much has been written about the mathematical constructions that are to be found in Vedic literature. In particular, the Shatapatha Brahmana, which is a part of the Shukla Yajur Veda, contains detailed descriptions of the geometric construction of altars for yajnas. Here, the brick-making technology of the Indus valley civilization was put to a new use. Supplementary to the Vedas are the Shulba Sutras. These texts are considered to date from 800 to 200 BCE. Four in number, they are named after their authors: Baudhayana (600 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE). The sutras contain the famous theorem commonly attributed to Pythagoras. The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers. It is interesting that the mathematics of this period seems to have been developed for solving practical geometric problems, especially the construction of religious altars. However, the study of the series expansion for certain functions already hints at the development of an algebraic perspective. In later times, we find a shift towards algebra, with simplification of algebraic formulate and summation of series acting as catalysts for mathematical discovery. Jain Mathematics (600 BCE to 500 CE) Just as Vedic philosophy and theology inspired the development of certain aspects of mathematics, so too did the rise of Jainism. Jain cosmology led to ideas of the infinite. This in turn, led to the development of the notion of orders of infinity as a mathematical concept. By orders of infinity, we mean a theory by which one set could be deemed to be more infinite than another. In modern language, this corresponds to the notion of cardinality. For a finite set, its cardinality is the number of elements it contains. However, we need a more sophisticated notion to measure the size of an infinite set. In Europe, it was not until Cantors work in the nineteenth century that a proper concept of cardinality was established. Besides the investigations into infinity, this period saw developments in several other fields such as number theory, geometry, computing, with fractions. In particular, the recursion formula for binomial coefficients and the Pascals triangle were already known in this period. The period 600 CE coincides with the rise and dominance of Buddhism. In the Lalitavistara, a biography of the Buddha which may have been written around the first century CE, there is an incident about Gautama being asked to state the name of large powers of 10 starting with 10. He is able to give names to numbers up to 10 (tallaksana). The very fact that such large numbers had names suggests that the mathematicians of the day were comfortable thinking about very large numbers. It is hard to imagine calculating with such numbers without some form of place value system. Brahmi Numerals, The place-value system and Zero No account of Indian mathematics would be complete without a discussion of Indian numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahmi numerals that seem to have made their appearance in 300 BCE. But Brahmi numerals were not part of a place value system. They evolved into the Gupta numerals around 400 CE and subsequently into the Devnagari numerals, which developed slowly between 600 and 1000 CE. By 600 CE, a place-value decimal system was well in use in India. This means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its position. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. The importance of a place-value system need hardly be emphasized. It would suffice to cite an often-quoted remark by La-place: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the magnificence of the achievement the more when we remember that it escaped the genius of Archimedes and Apolloniu s, two of the greatest men produced by ancient times. A place-value system of numerals was apparently known in other cultures; for example, the Babylonians used a sexagesimal place-value system as early as 1700 BCE, but the Indian system was the first decimal system. Moreover, until 400 BCE, The Babylonian system had an inherent ambiguity as there was no symbol for zero. Thus it was not a complete place-value system in the way we think of it today. The elevation of zero to the same status as other numbers involved difficulties that many brilliant mathematicians struggled with. The main problem was that the rules of arithmetic had to be formulated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defined and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined efforts of many minds. It is interesting to note that it was not until the seventeenth century that zero was being used in Europe. The Classical Era of Indian Mathematics (500 to 1200 CE) The most famous names of Indian mathematics belong to what is known as the classical era. This includes Aryabhata I (500 CE) Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhatta II (1000 CE) and Bhaskarachrya or Bhaskara II (1200 CE). During this period, two centers of mathematical research emerged, one at Kusumapura near Pataliputra and the other at Ujjain. Aryabhata I was the dominant figure at Kusumapura. One of Aryabhatas discoveries was a method for solving linear equations of the form ax + by = c. Aryabhata devised a general method for solving such equations, and he called it the kuttaka (or pulverizer) method. It should be noted that Aryabhatas studied linear equations because of his interest in astronomy. Amongst other important contributions of Aryabhata is his approximation of Pie to four decimal places (3.14146) and work on trigonometry. The other major centre of mathematical learning during this period was Ujjain, which was home to Varahamihira, Brahmagupta and Bhaskaracharya. The text Brahma-sphuta-siddhanta by Brahmagupta, published in 628 CE, dealt with arithmetic involving zero and negative numbers. As with Aryabhata, Brahmagupta was an astronomer, and much of his work was motivated by problems that arose in astronomy. He gave the famous formula for a solution to the quadratic equation. Brahmagupta also studied quadratic equation in two variables and sought solutions in whole numbers. This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refined the kuttaka method of Aryabhata and Brahmagupta. The Lilavati is impressive for its originality and diversity of topics. Brahmagupta discovered a method, which he called samasa, by which; given two solutions of the equation a third solution could be found. Brahmaguptas lemma was known one thousand years before it was rediscovered in Europe by Fermat, Legendre, and others. This method appears now in most standard text books and courses in number theory. The name of the equation is a historical accident. Mathematics in South India Mahavira is a mathematician belonging to the ninth century who was most likely from modern day Karnataka. He studied the problem of cubic and quartic equations and solved them for some families of equations. His work had a significant impact on the development of mathematics in South India. His book Ganita- sara- sangraha amplifies the work of Brahmagulpta and provides a very useful reference for the state of mathematics in his day. Another notable mathematician of South India was Madhava from Kerala. Madhava belongs to the fourteenth century. He discovered series expansions for some trigonometric functions such as the sine, cosine and arctangent that were not known in Europe until after Newton. In modern terminology, these expansions are the Taylor series of the functions in question. Madhava gave an approximation to Pie of 3.14159265359, which goes far beyond the four decimal places computed by Aryabhata. Madhavas work with series expansions suggests that he either discovered elements of the differential calculus or nearly did so. Mathematics in the Modern Age In more recent times there have been many important discoveries made by mathematicians of Indian origin. We shall mention the work of three of them: Srinivasa Ramanujan, Harish-Chandra, and Manjul Bhargava. Ramanujan (1887- 1920) is perhaps the most famous of modern Indian mathematicians. Though he produced significant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modular forms. In an important paper published in 1916, he initiated the study of the Pie function. Ramanujan proved some properties of the function and conjectured many more. As a result of his work, the modern arithmetic theory of modular forms, which occupies a central place in number theory and algebraic geometry, was developed by Hecke. Harish-Chandra (1923- 83) is perhaps the least known Indian mathematician outside of mathematical circles. He began his career as a physicist, working under Dirac. In his thesis, he worked on the representation theory of the group SL2 (C). This work convinced him that he was really a mathematician, and he spent the remainder of his academic life working on the representation theory of semi-simple groups. For most of that period, he was a professor at the Institute for Advanced Study in Princeton, New Jersey. His Collected Papers published in four volumes contain more than 2,000 pages. His style is known as meticulous and thorough and his published work tends to treat the most general case at the very outset. This is in contrast to many other mathematicians, whose published work tends to evolve through special cases. Interestingly, the work of Harish-Chandra formed the basis of Langlandss theory of automorphic forms, which are a vast generalization of the modular forms considered by R amanujan. CONCLUSION: The present mathematical knowledge has not dropped as a bolt from the blue, nor a product of some magical tricks. The apparently ready-made knowledge and results have been achieved after centuries of efforts, often painful, by hundreds of mathematicians and historians through the ages. Lots of discoveries and inventers contributed to the fruits, facilities and luxuries which we enjoy today were the contribution of Indian mathematicians. From the notion of zero to the modern concept of computational number theory, their contribution is significant. It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognize this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realize what was so clear in front of them. Kim Plofker from Netherland says that Indian mathematical science is extremely important and has a significant effect on the worlds knowledge as it is today. The lack of available resources has kept us under informed about the developments that have taken place in India. It is the need of the hour to carry forward the legacy of great mathematicians so as to encourage and nurture the glorious tradition of the country in mathematics. The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of ancient times, Archimedes and Apollonius.

Words and Meaning :: Philosophy of Language

Words and Meaning How do words bear meaning? The notion that a word means what it stands for – its denotation - will be examined and found wanting because logical analysis is only able to illuminate limited areas of language. It will be then suggested that metaphysical speculations about the sort of entities named by words are at best unhelpful. The idea that words get their meaning from the way they are used in public discourse will then be introduced as potentially more useful, although some problems with this approach will also be noted. Finally it will be suggested, very briefly, that an answer to this question may best be found in the common human condition – how we operate in the world using language. It is attractive to assume that the meaning of a word is the entity it denotes. There are many cases where this definition will do. For example in the sentence, John sat at the table. ‘John’ denotes a person and ‘table’ denotes an object. This seems straightforward. There are sentences, however, where the meaning is apparently clear but where the entities are not so clear cut. The sentence below has a clear meaning: The singing was divine but the acting was wooden. The proposition carried by this sentence is easily understood. However, the entities ‘singing’ and ‘acting’ are not so clear. They are ongoing actions not so clearly defined as tables and chairs. Furthermore, the metaphorical qualifiers ‘divine’ and ‘wooden’ do not help do not sharpen the meaning. Is ‘divine’, for example, merely a fanciful replacement for ‘enjoyable’? A whole conversation about the nature of singing and acting might follow such an utterance. Denotation and questions of logical form do not seem to be helpful in explaining the meanings of words in ordinary talk although human beings do seem to be impelled towards rational discussion. We habitually give reasons for things. For example, a discussion about ‘the greatest footballer ’ often finishes with extensive debate about what the defining criteria might be (a verbal dispute about connotation). Subsequently the argument often then turns to which player best meets these criteria (arguments to establish denotation). Whatever the case the denotation for ‘the greatest footballer’ is problematical. Much of this kind of discourse is based upon opinions. These opinions may or may not be true. This in no way affects the meanings intended by the speakers.