Rodolfo :: essays research papers

The Three Wise Men. I remember growing up for part of my child hood living in Mexico City. I was about 4 years old when I moved to Mexico to live close to my grand parents and other relatives. In Mexico City there is a fairy tale tradition that is still practice these days. The fairy tale tradition is that of (Los Tres Reyes Magos) The Three Wise Men or also known as The Three Wise Kings. Their names are Melchor, Gaspar, and Baltazar and they ride on camels, Gaspar is the black wise man and they speak all languages because they are universal. The date is January 6th, when the three wise men come to your house and bring gifts for you. The only requirements are that you have to be a child, have to be sleeping the night of January 6th when they come to drop of your gifts and the last requirement is that you have to believe in them in order for them come. My parents would tell me about them, the three wise men would come to my house to bring me gifts, the night of January 6th if I was sleeping. I had to be under good behavior in order for them to come and bring me gifts or else they wouldn’t bring me nothing because the three wise men knew if you were good or bad. The gifts that the wise men bring are toys, clothing, school supply, shoes, bikes, and other stuff for the children. My parents told my brother and I about the wise men and what they do. My brother and I believed our parents because we were kids and why would your parents lie to us. My brother and I weren’t the only ones that believed in the three wise men, there are still kids that are young enough to believe in them these days, in Mexico City. The three wise men are magical because they can shrink small to the size that they can enter your residence through a small crack or small opening for example like the small space under a door. They are magical because they can shrink their size or they can turn themselves into light balls and enter your house through a window like regular light would go though a clear glass window. You would have to be sleeping and would have had to leave them one of your shoes or a shoe that belongs to you with the list of what you wanted inside the shoe.

Non Financial Factors

TABLE OF CONTENT INTRODUCTION1 TESCO’S RATIO ANALYSIS2 SUMMARY TESCO’S RATIO13 COMPARATIVE ANALYSE – Tesco’s Vs Marks and Spencer’s________________ _______14 CRITICAL ANALYSIS OF TESCO PLC__________________________________________ 21 CONCLUSION? BIBLIOGRAPHY? APPENDIX 1 –TESCO’S PLC APPENDIX 2- MARKS AND SPENCER’S- CONSOLIDATED STATEMENTS I-Introduction This report will evaluate the financial performance of Tesco’s and comparing it to Marks and Spencer’s has the purpose of evaluating the company's worthiness as investment.As a well knowing company around the world and having an important background in the retail environment Tesco’s is one of the largest supermarkets in the world. Present in 14 countries around Europe, Asia and North America. Tesco’s is always dealing in the financial world, providing also bank and insurance services. ‘Tesco was founded in 1919 by Jack Cohen from a market stall i n London’s East End. Over the years our business has grown and we now operate in 14 countries around the world, employ over 500,000 people and serve tens of millions of customers every week.We have always been committed to providing the best shopping experience. Today we continue to focus on doing the right thing for our customers, colleagues and the communities we serve. ’ (Tesco 2012). The first section of this report, which is the main body, will use financial statements from 2010, 2011 and 2012, along with standard financial ratio analysis to develop a clear picture of Tesco’s financial performance comparing to the competitor. The second section includes a comparative analysis of the competitor strategy and also a conclusion on the performance and health of Tesco PLC based on the years 2010, 2011 and 2012.The third section, presents a critical analysis containing the non-financial factors and risks impacting on the future of Tesco PLC. II-Tesco’s rati o analysis: Ratio Analysis simplifies the financial statement and helps in future planning. It also helps us to inform the entire story of changes and current performance of the company. Ratios highlight all the different factors linked with successful and unsuccessful business. It is a powerful tool of financial analysis in the company. By using Ratio analysis it is easy to evaluate and understand financial health and trend of the business and possible future forecast of the company.Currency = ? (000) The return on capital employed is an important measure of a company's profitability. If ROCE is higher than the company is sound healthy. In 2010 Tesco had 11. 52% ROCE which increase steadily in 2011 and 2012 respectively 12. 93 and 12. 64. So there is a possible reason for this change is that profit increase. It determines management's ability to generate earnings from a company's total pool of capital. Company’s gross profit margin ratio shows that there is slightly differen ce between 2010 and 2012 which shows there was no any major change in their prices.In 2011 the company recorded a gross profit margin ratio of 8. 30%. The positive trend in this margin shows that the company is on profitability trend and therefore is a good investment option. So there is a possible reason for this change the higher cost of production. Operating Margin often refer to simply as a company's profit margin, there is no major change during the period from 2010 to 2012. Activity ratio: 1. Assets Turnover: Asset Turnover= Sales revenue/Capital employed During the last three years Tesco has improved gradually returning continuously in 2011 and 2012 turnover was respectively 2. 4 and 2. 06 . For most companies, their investment in net assets represents the largest component of their total assets. There are no significant changes in asset turnover. Liquidity ratio: Liquidity is a very important ratio for money lenders, suppliers and potential investors to access. According to the Tesco annual statement the result from 2010 to 2012 shows that the current ratio was less than 1 which has a problem to meet their liability in short term. Tesco’s assets are less and its liabilities are quite high which indicates company’s weak current ratio and liquidity problem.Quick ratio is a more conservative (safer) measure of liquidity. A higher quick ratio implies greater safety. According to the acid test ratio Tesco’s acid test ratio was not good because it is below the standard. The liabilities have increased because of increased loan 2010, 2011 and 2012 respectively. In the year 2010 Receivable days was 12. 10 days but after that in 2011 and 2012 fiscal year respectively it was increased to 13. 86 days and 15. 02 days, which is showing their position is not good to collect receivable earlier.It could affect business as well because customers always prefer a long time to pay back whatever they have taken on credit. 2012 = 3598/59278*365 = 22. 15 days It takes Tesco’s approximately 19 to 22 days taken to sell its product from the time it acquire it. Inventory days increased continuously since 2010 to 2012. The possible reasons could be the company’s sales are not good. Capital Gearing: The term â€Å"capital gearing† or â€Å"leverage† normally refers to the proportion of relationship between equity share capital including reserves and surpluses to preference share capital and other fixed interest bearing funds or loans.As the higher a company's degree of leverage as the more the company is considered risky. In, Tesco’s gearing scenario gearing was decreased in 2010 and 2011 separately from 0. 51 to 0. 43, and it was standstill 0. 43 in 2012, which indicates the company improving financially. So there are possible reasons for this change, long term is decreasing in comparison with capital employed. Return on assets: . The profitability ratio here measures the relationship between net profit and assets. Return on assets= Net profit before interest and tax / Total asset*100Return on asset (ROA) indicator of how profitable a company is relation to its total asset . ROA gives us an idea of Tesco how efficient management is sat using its asset to generate earning. In 2010 return on asset was 7. 51% after that there was a decrease till 2012 to 5. 54 %. Tesco PLC has recorded in decreasing sharply value of P/E with values of 14. 12, 12. 12, and 8. 74, being recorded for 2010, 2011, and 2012 respectively (Yahoo Finance 1st Nov 2010,2011,2012). A number of factors could be possible vary due to decreasing in P/E including increased competitiveness for capital in market. Yahoo Finance 2012) 2. Earnings per share: The Earning per Share (EPS) considers the profits that could be paid to each ordinary shareholder. The increase in profit resulted in the increase in EPS. Earnings per share: Earnings o holders / No of o shares in issue 2010 = 29. 33p 2011 = 34. 43p 2012 = 36. 75p The c ompany recorded EPS increased in 2010, 2011 and2012 respectively. There could be number of reason for increasing earnings per share. Possible reason could be the increase in profit, increasing in loan. But it would not be the long term sustainability. 3.Dividend: Dividend per share (DPS) is the sum of declared dividends for every ordinary share issued. DPS is the total dividends paid out over an entire year divided by the number of outstanding ordinary shares issued. Tesco financial statements indicate that dividend yield for the company has been rising in the last five years. The company recorded dividend yields of 3. 15%, 3. 56 %p and 4. 59% for 2010, 2011 and 2012 respectively (Yahoo Finance 1st Nov 2010, 2011, 2012). This is an indication that investor willing to invest in the company have a chance of receiving better dividend in the future. Yahoo Finance 2012) In 2011 company’s debt/equity ratio was higher to1. 04, which is not very good indication for the company. Becau se it heavily depends on loan is not a good policy for any business. But it was reduced the following years in 2011 and 2012 respectively 0. 77 and 0. 77. Debt to Equity: Debt to equity = Non-current interest bearing debt: Equity It is used to determine how easily a company can pay interest expenses on outstanding debt. In 2010 company’s interest coverage was 5. 99 times which increased in 2011 to 10. 47 times but in 2012 decreased slightly to 9. 5. The company’s profit has increased to pay their interest easily. Interpretation and ratio analysis conclusion: In the year 2012 Tesco‘s activity, profitability, liquidity ratio, financial gearing, and investment ratio was comparing with the previous year ratio. In the activity ratio net assets turn increased. Liquidity ratio was quite reasonable due to the economic condition and creditor days decreased which was not good for the company. Financial gearing was not satisfactory and finally, investment ratio increased ma rgin which indicates revenue.The organization managed to increase its return on capital and assets turn over remarkably. Tesco has slightly increased its receivable and payable credit payment period currently showing its financial position. On the other side, it can also be an opportunity for the customers to attract more customers as they always prefer to hold back as much as possible. There is no major difference in the net profit and gross profit margin that means Tesco did not bring any change in its prices and there was not any external pressure from government or competitors.Liquidity of Tesco shows not a major decline over the past 3 years even though it is below 1 which is quite risky condition because current ratio below 1 means liabilities are more and assets are very less. If there will be major decline in the business, the company will not be able to pay their short term liabilities. The Interim report shows that they are reducing the gearing but we Tesco improved its sh ares value by having an increase in the dividends per share and share price. Investors will be attracted by this but this will not stay for long. Yahoo Finance, 2012) III-Comparative Analyse – Tesco’s Vs Marks and Spencer’s We can use Ratio Analysis to do a comparative analysis and seeing our performance with respect to our competitors. For this I have taken Marks and Spencer Group PLC and compared it with Tesco PLC to see the Standing of my company with another company. This helps us to know our strengths and weaknesses in all the areas of the business. Summary of Comparative Results between M & S and Tesco (2010-2012) Revenue and Operating Profits:The revenues earned by the company and the level of operating profit does tell us the size , capacity and type of player the company is in market. The Tesco’s Operating profit s increase over the years but if we see the table below M & S, they reduced the operating cost, but the revenue increased constantly as well. Chart : Tesco & M and S Revenue Comparison The Comparison of Tesco and Marks & Spencer tells us that Tesco is a much bigger company and has a much higher turnover. But through its policies we see that the level of Operating profit of Tesco is higher because of its strong Optimization policies and procedures.Ratio’s comparison between M& S and Tesco: Tesco’s and M & S ratio analysis: Ratio Analysis helps us to inform the entire story of changes and current performance of the company. = 12. 93% 2012 = 3985/ (13731+17801) *100 = 12. 64 % The return on capital employed is an important measure of a company's profitability. If ROCE is higher than it the company is sound healthy. If we see the chart we can M is in stronger condition. 2011= 9740. 30/ (2677. 40+2456. 50) =2. 46 2012= 9934. 30/ (2778. 80+2489. 10) = 2. 49 = 0. 43 2012 = (1460. 10-681. 90) /2005. 40 0. 38 Earnings per share: Earnings o holders / No of o shares in issue 2010 = 29. 33p 2011 = 34. 43p 2012 = 36. 75p M & S 2010 = 33. 50 p 2011 = 38. 80 p 2012 = 32. 50 p The increase in earnings per which is attractive point for investors. Tesco Earning per share increased on 2012 while M Earning per share decreased. 2012 = 2489. 10/2778. 80 = 0. 89% Tesco debt/equity ratio was higher to 1. 04 %, which reduced the following years in 2011 and 2012 respectively 0. 77% and 0. 77%. While M & S was 1. 40 % on 2010 & it’s got bit better on following years.IV-Critical Analysis of the non- financial factors and risks for Tesco PLC In today's worldwide competitive environment organisations have to compete with others regarding a wide range of fields like product quality, delivery, reliability, after-sales services, brand, customer care and feedbacks†¦ (Chairman, FTSE 100 Company, 2003) The financial ratio analysis done above, is very useful as it summarises all the necessary information in order to understand the health of a company, covering profit, liquidity, growth and risk of a com pany.But it is also essential to look at the non financial factors that can have a huge impact on a company’s future potential. V-Conclusion Taking into consideration the ratio analysis applied to Tesco’s between 2010 and 2012 what can be noticed is that the company had some variation. According to level of risk, Tesco’s is less risky than M&S in terms of investment considering that in 2010, 2011 and 2012 had as gearing ratios: 1. 04 %, 0. 77% and 0. 77% respectively and M & S for the same period 1. 40 %, 0. 92% and 0. 89%. As much higher is the gearing ratio more vulnerable is the company to downturns.With an improvement of its shares value by having an increase in the dividends per share and share price, Investors will be attracted by this but this will not stay for long. Moreover considering how much cash flow is available for each pound invested, which is demonstrated by the dividend yield, Tesco’s in 2010 had a variation from 3. 15% to 4. 59% in 2012 which is positive for the business. On the other hand, Tesco’s reacted negatively into the full analysis of profitability, efficiency and effectiveness, liquidity and investor ratios.As an example, the investment per share had a decrease of 5. 38 from 2010 to 2012 and also receivable days had a considerable increase which is a negative impact. Despite of having lower prices than M&S with strong position in UK and also in other continents, Tesco’s might be a good investment in the future, depends on its performance and long-term investment for the follow years. However currently it is not an investment to be considered. Bibliography London Stock Exchange (2012). Tesco PLC ORD SP. London Stock Exchange (2012). Marks and Spencer Group PLC ORD 25P.Available at:http://www. londonstockexchange. com/exchange/prices/stocks/summary/fundamentals. html? fourWayKey=GB0031274896GBGBXSET1 Mark and Spencer (2010)-Annual Report and Financial Statements. Available at http://corporate. marksandspencer. com/documents/publications/2010/annual_report_2010 http://corporate. marksandspencer. com/documents/publications/2011/annual_report_2011 http://corporate. marksandspencer. com/documents/publications/2012/annual_report_2012 (Yahoo Finance, 2012) http://www. bizmove. com/finance/m3b3. htm APPENDIX 1 APPENDIX 2 APPENDIX 1 APPENDIX 2

The History of Algebra

Various derivations of the word algebra, which is of Arabian origin, have been given by different writers. The first mention of the word is to be found in the title of a work by Mahommed ben Musa al-Khwarizmi (Hovarezmi), who flourished about the beginning of the 9th century. The full title is ilm al-jebr wal-muqabala, which contains the ideas of restitution and comparison, or opposition and comparison, or resolution and equation, jebr being derived from the verb jabara, to reunite, and muqabala, from gabala, to make equal. (The root jabara is also met with in the word algebrista, which means a bone-setter, and is still in common use in Spain.) The same derivation is given by Lucas Paciolus (Luca Pacioli), who reproduces the phrase in the transliterated form alghebra e almucabala, and ascribes the invention of the art to the Arabians. Other writers have derived the word from the Arabic particle al (the definite article), and gerber, meaning man. Since, however, Geber happened to be the name of a celebrated Moorish philosopher who flourished in about the 11th or 12th century, it has been supposed that he was the founder of algebra, which has since perpetuated his name. The evidence of Peter Ramus (1515-1572) on this point is interesting, but he gives no authority for his singular statements. In the preface to his Arithmeticae libri duo et totidem Algebrae (1560) he says: The name Algebra is Syriac, signifying the art or doctrine of an excellent man. For Geber, in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. There was a certain learned mathematician who sent his algebra, written in the Syriac language, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of algebra. To this day the same book is in great estimation among the learned in the oriental nations, and by the Indians, who cultivate this art, it is called aljabra and alboret; though the name of the author himself is not known. The uncertain authority of these statements, and the plausibility of the preceding explanation, have caused philologists to accept the derivation from al and jabara. Robert Recorde in his Whetstone of Witte (1557) uses the variant algeber, while John Dee (1527-1608) affirms that algiebar, and not algebra, is the correct form, and appeals to the authority of the Arabian Avicenna. Although the term algebra is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance. Thus we find Paciolus calling it lArte Magiore; ditta dal vulgo la Regula de la Cosa over Alghebra e Almucabala. The name larte magiore, the greater art, is designed to distinguish it from larte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms coss or algebra, cossic or algebraic, cossist or algebraist, c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square. The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in algebra, for they were unable to solve equations of a higher degree than the quadratic or square. Franciscus Vieta (Francois Viete) named it Specious Arithmetic, on account of the species of the quantities involved, which he represented symbolically by the various letters of the alphabet. Sir Isaac Newton introduced the term Universal Arithmetic, since it is concerned with the doctrine of operations, not affected on numbers, but on general symbols. Notwithstanding these and other idiosyncratic appellations, European mathematicians have adhered to the older name, by which the subject is now universally known. Continued on page two.   This document is part of an article on Algebra from the 1911 edition of an encyclopedia, which is out of copyright here in the U.S. The article is in the public domain, and you may copy, download, print and distribute this work as you see fit. Every effort has been made to present this text accurately and cleanly, but no guarantees are made against errors. Neither Melissa Snell nor About may be held liable for any problems you experience with the text version or with any electronic form of this document. It is difficult to assign the invention of any art or science definitely to any particular age or race. The few fragmentary records, which have come down to us from past civilizations, must not be regarded as representing the totality of their knowledge, and the omission of a science or art does not necessarily imply that the science or art was unknown. It was formerly the custom to assign the invention of algebra to the Greeks, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this work there are distinct signs of an algebraic analysis. The particular problem---a heap (hau) and its seventh makes 19---is solved as we should now solve a simple equation; but Ahmes varies his methods in other similar problems. This discovery carries the invention of algebra back to about 1700 B.C., if not earlier. It is probable that the algebra of the Egyptians was of a most rudimentary nature, for otherwise we should expect to find traces of it in the works of the Greek aeometers. of whom Thales of Miletus (640-546 B.C.) was the first. Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra. The first extant work which approaches to a treatise on algebra is by Diophantus (q.v.), an Alexandrian mathematician, who flourished about A.D. 350. The original, which consisted of a preface and thirteen books, is now lost, but we have a Latin translation of the first six books and a fragment of another on polygonal numbers by Xylander of Augsburg (1575), and Latin and Greek translations by Gaspar Bachet de Merizac (1621-1670). Other editions have been published, of wh ich we may mention Pierre Fermats (1670), T. L. Heaths (1885) and P. Tannerys (1893-1895). In the preface to this work, which is dedicated to one Dionysius, Diophantus explains his notation, naming the square, cube and fourth powers, dynamis, cubus, dynamodinimus, and so on, according to the sum in the indices. The unknown he terms arithmos, the number, and in solutions he marks it by the final s; he explains the generation of powers, the rules for multiplication and division of simple quantities, but he does not treat of the addition, subtraction, multiplication and division of compound quantities. He then proceeds to discuss various artifices for the simplification of equations, giving methods which are still in common use. In the body of the work he displays considerable ingenuity in reducing his problems to simple equations, which admit either of direct solution, or fall into the class known as indeterminate equations. This latter class he discussed so assiduously that they are often known as Diophantine problems, and the methods of resolving them as the Diophantine analysis (see EQUATION, Indeterminate.) It is difficult to believe that this work of Diophantus arose spontaneously in a period of general stagnation. It is more than likely that he was indebted to earlier writers, whom he omits to mention, and whose works are now lost; nevertheless, but for this work, we should be led to assume that algebra was almost, if not entirely, unknown to the Greeks. The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in arithmetical computations, there are no material advances to be recorded. In the chronological development of our subject we have now to turn to the Orient. Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical. We find that geometry was neglected except in so far as it was of service to astronomy; trigonometry was advanced, and algebra improved far beyond the attainments of Diophantus. Continued on page three.   This document is part of an article on Algebra from the 1911 edition of an encyclopedia, which is out of copyright here in the U.S. The article is in the public domain, and you may copy, download, print and distribute this work as you see fit. Every effort has been made to present this text accurately and cleanly, but no guarantees are made against errors. Neither Melissa Snell nor About may be held liable for any problems you experience with the text version or with any electronic form of this document. The earliest Indian mathematician of whom we have certain knowledge is Aryabhatta, who flourished about the beginning of the 6th century of our era. The fame of this astronomer and mathematician rests on his work, the Aryabhattiyam, the third chapter of which is devoted to mathematics. Ganessa, an eminent astronomer, mathematician and scholiast of Bhaskara, quotes this work and makes separate mention of the cuttaca (pulveriser), a device for effecting the solution of indeterminate equations. Henry Thomas Colebrooke, one of the earliest modern investigators of Hindu science, presumes that the treatise of Aryabhatta extended to determinate quadratic equations, indeterminate equations of the first degree, and probably of the second. An astronomical work, called the Surya-siddhanta (knowledge of the Sun), of uncertain authorship and probably belonging to the 4th or 5th century, was considered of great merit by the Hindus, who ranked it only second to the work of Brahmagupta, who flourish ed about a century later. It is of great interest to the historical student, for it exhibits the influence of Greek science upon Indian mathematics at a period prior to Aryabhatta. After an interval of about a century, during which mathematics attained its highest level, there flourished Brahmagupta (b. A.D. 598), whose work entitled Brahma-sphuta-siddhanta (The revised system of Brahma) contains several chapters devoted to mathematics. Of other Indian writers mention may be made of Cridhara, the author of a Ganita-sara (Quintessence of Calculation), and Padmanabha, the author of an algebra. A period of mathematical stagnation then appears to have possessed the Indian mind for an interval of several centuries, for the works of the next author of any moment stand but little in advance of Brahmagupta. We refer to Bhaskara Acarya, whose work the Siddhanta-ciromani (Diadem of anastronomical System), written in 1150, contains two important chapters, the Lilavati (the beautiful [science or art]) and Viga-ganita (root-extraction), which are given up to arithmetic and algebra. English translations of the mathematical chapters of the Brahma-siddhanta and Siddhanta-ciromani by H. T. Colebrooke (1817), and of the Surya-siddhanta by E. Burgess, with annotations by W. D. Whitney (1860), may be consulted for details. The question as to whether the Greeks borrowed their algebra from the Hindus or vice versa has been the subject of much discussion. There is no doubt that there was a constant traffic between Greece and India, and it is more than probable that an exchange of produce would be accompanied by a transference of ideas. Moritz Cantor suspects the influence of Diophantine methods, more particularly in the Hindu solutions of indeterminate equations, where certain technical terms are, in all probability, of Greek origin. However this may be, it is certain that the Hindu algebraists were far in advance of Diophantus. The deficiencies of the Greek symbolism were partially remedied; subtraction was denoted by placing a dot over the subtrahend; multiplication, by placing bha (an abbreviation of bhavita, the product) after the factom; division, by placing the divisor under the dividend; and square root, by inserting ka (an abbreviation of karana, irrational) before the quantity. The unknown was ca lled yavattavat, and if there were several, the first took this appellation, and the others were designated by the names of colours; for instance, x was denoted by ya and y by ka (from kalaka, black). Continued on page four. This document is part of an article on Algebra from the 1911 edition of an encyclopedia, which is out of copyright here in the U.S. The article is in the public domain, and you may copy, download, print and distribute this work as you see fit. Every effort has been made to present this text accurately and cleanly, but no guarantees are made against errors. Neither Melissa Snell nor About may be held liable for any problems you experience with the text version or with any electronic form of this document. A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them. It is also supposed that they anticipated discoveries of the solutions of higher equations. Great advances were made in the study of indeterminate equations, a branch of analysis in which Diophantus excelled. But whereas Diophantus aimed at obtaining a single solution, the Hindus strove for a general method by which any indeterminate problem could be resolved. In this they were completely successful, for they obtained general solutions for the equations ax( or -)byc, xyaxbyc (since rediscovered by Leonhard Euler) and cy2ax2b. A particular case of the last equation, namely, y2ax21, sorely taxed the resources of modern algebraists. It was proposed by Pierre de Fermat to Bernhard Frenicle de Bessy, and in 1657 to all mathematician s. John Wallis and Lord Brounker jointly obtained a tedious solution which was published in 1658, and afterwards in 1668 by John Pell in his Algebra. A solution was also given by Fermat in his Relation. Although Pell had nothing to do with the solution, posterity has termed the equation Pells Equation, or Problem, when more rightly it should be the Hindu Problem, in recognition of the mathematical attainments of the Brahmans. Hermann Hankel has pointed out the readiness with which the Hindus passed from number to magnitude and vice versa. Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of algebra, and Hankel affirms that if we define algebra as the application of arithmetical operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of algebra. The integration of the scattered tribes of Arabia in the 7th century by the stirring religious propaganda of Mahomet was accompanied by a meteoric rise in the intellectual powers of a hitherto obscure race. The Arabs became the custodians of Indian and Greek science, whilst Europe was rent by internal dissensions. Under the rule of the Abbasids, Bagdad became the centre of scientific thought; physicians and astronomers from India and Syria flocked to their court; Greek and Indian manuscripts were translated (a work commenced by the Caliph Mamun (813-833) and ably continued by his successors); and in about a century the Arabs were placed in possession of the vast stores of Greek and Indian learning. Euclids Elements were first translated in the reign of Harun-al-Rashid (786-809), and revised by the order of Mamun. But these translations were regarded as imperfect, and it remained for Tobit ben Korra (836-901) to produce a satisfactory edition. Ptolemys Almagest, the works of Apolloniu s, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated. The first notable Arabian mathematician was Mahommed ben Musa al-Khwarizmi, who flourished in the reign of Mamun. His treatise on algebra and arithmetic (the latter part of which is only extant in the form of a Latin translation, discovered in 1857) contains nothing that was unknown to the Greeks and Hindus; it exhibits methods allied to those of both races, with the Greek element predominating. The part devoted to algebra has the title al-jeur walmuqabala, and the arithmetic begins with Spoken has Algoritmi, the name Khwarizmi or Hovarezmi having passed into the word Algoritmi, which has been further transformed into the more modern words algorism and algorithm, signifying a method of computing. Continued on page five. This document is part of an article on Algebra from the 1911 edition of an encyclopedia, which is out of copyright here in the U.S. The article is in the public domain, and you may copy, download, print and distribute this work as you see fit. Every effort has been made to present this text accurately and cleanly, but no guarantees are made against errors. Neither Melissa Snell nor About may be held liable for any problems you experience with the text version or with any electronic form of this document. Tobit ben Korra (836-901), born at Harran in Mesopotamia, an accomplished linguist, mathematician and astronomer, rendered conspicuous service by his translations of various Greek authors. His investigation of the properties of amicable numbers (q.v.) and of the problem of trisecting an angle, are of importance. The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the conic sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals (see NUMERAL), arithmetic and astronomy (q.v..) It thus came about that while some progress was made in algebra, the talents of the race were bestowed on astronomy and trigonometry (q.v..) Fahri des al Karbi, who flourished about the beginning of the 11th century, is the author of the most important Arabian work on algebra. He follows the methods of Diophantus; his work on indeterminate equations has no resemblance to the Indian methods, and contains nothing that cannot be gathered from Diophantus. He solved quadratic equations both geometrically and algebraically, and also equations of the form x2naxnb0; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes. Cubic equations were solved geometrically by determining the intersections of conic sections. Archimedes problem of dividing a sphere by a plane into two segments having a prescribed ratio, was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin. The determination of the side of a regular heptagon which can be inscribed or circumscribed to a given circle was reduced to a more complicated equation which was first successfully resolved by Abul Gud. The method of solving equations geometrically was considerably developed by Omar Khayyam of Khorassan, who flourished in the 11th century. This author questioned the possibility of solving cubics by pure algebra, and biquadratics by geometry. His first contention was not disproved until the 15th century, but his second was disposed of by Abul Weta (940-908), who succeeded in solving the forms x4a and x4ax3b. Although the foundations of the geometrical resolution of cubic equations are to be ascribed to the Greeks (for Eutocius assigns to Menaechmus two methods of solving the equation x3a and x32a3), yet the subsequent development by the Arabs must be regarded as one of their most important achievements. The Greeks had succeeded in solving an isolated example; the Arabs accomplished the general solution of numerical equations. Considerable attention has been directed to the different styles in which the Arabian authors have treated their subject. Moritz Cantor has suggested that at one time there existed two schools, one in sympathy With the Greeks, the other with the Hindus; and that, although the writings of the latter were first studied, they were rapidly discarded for the more perspicuous Grecian methods, so that, among the later Arabian writers, the Indian methods were practically forgotten and their mathematics became essentially Greek in character. Turning to the Arabs in the West we find the same enlightened spirit; Cordova, the capital of the Moorish empire in Spain, was as much a centre of learning as Bagdad. The earliest known Spanish mathematician is Al Madshritti (d. 1007), whose fame rests on a dissertation on amicable numbers, and on the schools which were founded by his pupils at Cordoya, Dama and Granada. Gabir ben Allah of Sevilla, commonly called Geber, was a celebrated astronomer and apparently skilled in algebra, for it has been supposed that the word algebra is compounded from his name. When the Moorish empire began to wane the brilliant intellectual gifts which they had so abundantly nourished during three or four centuries became enfeebled, and after that period they failed to produce an author comparable with those of the 7th to the 11th centuries. Continued on page six. This document is part of an article on Algebra from the 1911 edition of an encyclopedia, which is out of copyright here in the U.S. The article is in the public domain, and you may copy, download, print and distribute this work as you see fit. Every effort has been made to present this text accurately and cleanly, but no guarantees are made against errors. Neither Melissa Snell nor About may be held liable for any problems you experience with the text version or with any electronic form of this document.