History of Algebra Essay

Various derivations of the vocalize algebra, which is of Arabian origin, convey been inclined by different writers. The initial mention of the cry is to be found in the title of a relieve oneself by Mahommed ben Musa al-Khwarizmi (Hovargonzmi), who flourished more or less the beginning of the 9th light speed. The full title is ilm al-jebr wal-muqabala, which contains the ideas of income tax return and comparison, or opposition and comparison, or re firmness of purpose and compare, jebr being derived from the verb jabara, to reunite, and muqabala, from gabala, to make equal. The topic 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 innovation of the craft to the Arabians. opposite writers sport derived the word from the Arabic particle al (the definit e article), and gerber, inwardness man. Since, however, Geber happened to be the name of a celebrated Moorish philosopher who flourished in to the highest degree the 11th or 12th century, it has been speculate that he was the founder of algebra, which has since perpetuated his name. The turn out of Peter Ramus (1515-1572) on this point is interesting, but he gives no authorization for his singular statements. In the predate 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 utilise to men, and is sometimes a term of honour, as master or limit among us.There was a au thuslytic learned mathematician who sent his algebra, scripted in the Syriac langu suppurate, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things, which others would alternatively call the doctrine of algebra. To this day the same book is in great(p) 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 k without delayn. 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, period ass Dee (1527-1608) affirms that algiebar, and not algebra, is the correct form, and appeals to the authority of the Arabian A guiltnna. 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 greet it from larte minore, the lesser art, a term which he applied to the youthful arithmeti c.His second variant, la regula de la cosa, the rule of the thing or unmapped quantity, appears to have been in common use in Italy, and the word cosa was preserve for several centuries in the forms coss or algebra, cossic or algebraic, cossist or algebraist, &c. early(a) Italian writers termed it the Regula rei et census, the rule of the thing and the harvest-time, or the root and the square. The principle primal this expression is probably to be found in the fact that it careful the limits of their attainments in algebra, for they were unable to reckon equations of a higher degree than the quadratic equation 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 bear on with the doctrine of trading operations, not affected on numbers, but on universal sy mbols. Notwithstanding these and other idiosyncratic appellations, European mathematicians have adhered to the older name, by which the written report is now universally known.It is difficult to assign the invention of both 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 skip of a science or art does not necessarily advert that the science or art was unknown. It was formerly the custom to assign the invention of algebra to the classics, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this get there are distinct signs of an algebraic analysis.The particular problema heap (hau) and its seventh makes 19is solved as we should now solve a simple equation but Ahmes varies his methods in other comparable 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 plant of the classic 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 rom their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had undersized or no affinity to algebra. The first existent get going 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 doomed, but we have a Latin shift 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 (1621-1 670).Other editions have been published, of which 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 stern powers, dynamis, cubus, dynamodinimus, and so on, according to the sum in the indices. The unknown he price 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, tax write-off, 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 tact in reducing his problems to simple equations, which admit either of direct solution, or fall into the mob known as enigmatical 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 impromptu 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 ameliorations 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 unsounded distinction between the G reek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical. We find that geometry was neglected besides in so far as it was of suffice to astronomy trig was advanced, and algebra improved far beyond the attainments of Diophantus. The early 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 prone to mathematics. Ganessa, an eminent astronomer, mathematician and scholiast of Bhaskara, quotes this work and makes separate mention of the cuttaca (pulve get holdr), a device for effecting the solution of indefinite equations. Henry Thomas Colebrooke, one of the earliest modern investigators of Hindi science, presumes that the treatise of Aryabhatta extended to determinate quadratic equations, indeterminate equations of th e 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 flourished 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 breakup of about a century, during which mathematics attained its highest level, there flourished Brahmagupta (b.A. D. 598), whose work empower 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 ( vinyl ether 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 meth ods, more particularly in the Hindu solutions of indeterminate equations, where certain technical monetary value 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 contraction 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 called 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).A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the introduction of two grow 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 -)by=c, xy=ax+by+c (since rediscovered by Leonhard Euler) and cy2=ax2+b. A particular case of the demise equation, namely, y2=ax2+1, sorely taxed the resources of modern algebraists. It was proposed by Pierre de Fermat to Bernhard Frenicle de Bessy, and in 1657 to all mathematicians. 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 postcode to do with the solution, osterity 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 sc ience, whilst Europe was rent by internal dissensions.Under the rule of the Abbasids, capital of Iraq 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 Apollonius, 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 o f 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 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.