History of Algebra Essay

Assorted derivations of the word “algebra. ” which is of Arabian beginning. have been given by different authors. The first reference of the word is to be found in the rubric of a work by Mahommed ben Musa al-Khwarizmi ( Hovarezmi ) . who flourished about the beginning of the ninth century. The full rubric is ilm al-jebr wa’l-muqabala. which contains the thoughts of damages and comparing. or resistance and comparing. or declaration and equation. jebr being derived from the verb jabara. to reunite. and muqabala. from gabala. to do equal. The root jabara is besides met with in the word algebrista. which means a “bone-setter. ” and is still in common usage in Spain. )

The same derivation is given by Lucas Paciolus ( Luca Pacioli ) . who reproduces the phrase in the transliterated signifier alghebra vitamin E almucabala. and ascribes the innovation of the art to the Arabians. Other authors have derived the word from the Arabic atom Al ( the definite article ) . and gerber. intending “man. Since. nevertheless. Geber happened to be the name of a famed Moorish philosopher who flourished in about the 11th or 12th century. it has been supposed that he was the laminitis of algebra. which has since perpetuated his name. The grounds of Peter Ramus ( 1515-1572 ) on this point is interesting. but he gives no authorization for his remarkable statements. In the foreword to his Arithmeticae libri couple et totidem Algebrae ( 1560 ) he says: “The name Algebra is Syriac. meaning the art or philosophy of an first-class adult male. For Geber. in Syriac. is a name applied to work forces. and is sometimes a term of honor. as maestro or physician among us.

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There was a certain erudite mathematician who sent his algebra. written in the Syriac linguistic communication. to Alexander the Great. and he named it almucabala. that is. the book of dark or cryptic things. which others would instead name the philosophy of algebra. To this twenty-four hours the same book is in great appraisal among the learned in the oriental states. and by the Indians. who cultivate this art. it is called aljabra and alboret ; though the name of the writer himself is non known. ” The unsure authorization of these statements. and the plausibleness of the predating account. have caused philologues 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 non algebra. is the right signifier. and entreaties to the authorization of the Arabian Avicenna. Although the term “algebra” is now in cosmopolitan usage. assorted other denominations were used by the Italian mathematicians during the Renaissance. Thus we find Paciolus naming it l’Arte Magiore ; ditta dal vulgo La Regula de la Cosa over Alghebra vitamin E Almucabala. The name l’arte magiore. the greater art. is designed to separate it from l’arte minore. the lesser art. a term which he applied to the modern arithmetic.

His 2nd discrepancy. la regula de la cosa. the regulation of the thing or unknown measure. appears to hold been in common usage in Italy. and the word cosa was preserved for several centuries in the signifiers coss or algebra. cossic or algebraic. cossist or algebraist. & A ; c. Other Italian authors termed it the Regula rei et nose count. the regulation of the thing and the merchandise. or the root and the square. The rule underlying this look is likely to be found in the fact that it measured the bounds of their attainments in algebra. for they were unable to work out equations of a higher grade than the quadratic or square.

Franciscus Vieta ( Francois Viete ) named it Specious Arithmetic. on history of the species of the measures involved. which he represented symbolically by the assorted letters of the alphabet. Sir Isaac Newton introduced the term Universal Arithmetic. since it is concerned with the philosophy of operations. non affected on Numberss. but on general symbols. Notwithstanding these and other idiosyncratic denominations. European mathematicians have adhered to the older name. by which the topic is now universally known.

It is hard to delegate the innovation of any art or scientific discipline decidedly to any peculiar age or race. The few fragmental records. which have come down to us from past civilisations. must non be regarded as stand foring the entirety of their cognition. and the skip of a scientific discipline or art does non needfully connote that the scientific discipline or art was unknown. It was once the usage to delegate the innovation of algebra to the Greeks. but since the decoding of the Rhind papyrus by Eisenlohr this position has changed. for in this work there are distinguishable marks of an algebraic analysis.

The peculiar problem—a pile ( hau ) and its 7th makes 19—is solved as we should now work out a simple equation ; but Ahmes varies his methods in other similar jobs. This find carries the innovation of algebra back to about 1700 B. C. . if non earlier. It is likely that the algebra of the Egyptians was of a most fundamental nature. for otherwise we should anticipate to happen hints of it in the plants of the Grecian aeometers. of whom Thales of Miletus ( 640-546 B. C. ) was the first.

Notwithstanding the prolixness of authors and the figure of the Hagiographas. all efforts at pull outing an algebraic analysis rom their geometrical theorems and jobs have been bootless. and it is by and large conceded that their analysis was geometrical and had small 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 foreword and 13 books. is now lost. but we have a Latin interlingual rendition of the first six books and a fragment of another on polygonal Numberss by Xylander of Augsburg ( 1575 ) . and Latin and Greek interlingual renditions by Gaspar Bachet de Merizac ( 1621-1670 ) .

Other editions have been published. of which we may advert Pierre Fermat’s ( 1670 ) . T. L. Heath’s ( 1885 ) and P. Tannery’s ( 1893-1895 ) . In the foreword to this work. which is dedicated to one Dionysius. Diophantus explains his notation. calling the square. regular hexahedron and 4th powers. dynamis. cubus. dynamodinimus. and so on. harmonizing to the amount in the indices. The unknown he footings arithmos. the figure. and in solutions he marks it by the concluding s ; he explains the coevals of powers. the regulations for generation and division of simple measures. but he does non handle of the add-on. minus. generation and division of compound measures.

He so proceeds to discourse assorted ruses for the simplification of equations. giving methods which are still in common usage. In the organic structure of the work he displays considerable inventiveness in cut downing his jobs to simple equations. which admit either of direct solution. or autumn into the category known as undetermined equations. This latter category he discussed so assiduously that they are frequently known as Diophantine jobs. and the methods of deciding them as the Diophantine analysis ( see EQUATION. Indeterminate. ) It is hard to believe that this work of Diophantus arose spontaneously in a period of general stagnancy.

It is more than probably that he was indebted to earlier authors. whom he omits to advert. and whose plants are now lost ; however. but for this work. we should be led to presume that algebra was about. if non wholly. unknown to the Greeks. The Romans. who succeeded the Greeks as the head civilised power in Europe. failed to put shop on their literary and scientific hoarded wealths ; mathematics was all but neglected ; and beyond a few betterments in arithmetical calculations. there are no material progresss to be recorded. In the chronological development of our topic we have now to turn to the Orient.

Probe of the Hagiographas of Indian mathematicians has exhibited a cardinal differentiation between the Greek and Indian head. the former being pre-eminently geometrical and bad. the latter arithmetical and chiefly 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. The earliest Indian mathematician of whom we have certain cognition is Aryabhatta. who flourished about the beginning of the sixth century of our epoch.

The celebrity of this uranologist and mathematician remainders on his work. the Aryabhattiyam. the 3rd chapter of which is devoted to mathematics. Ganessa. an high uranologist. mathematician and scholiast of Bhaskara. quotes this work and makes separate reference of the cuttaca ( “pulveriser” ) . a device for set uping the solution of undetermined equations. Henry Thomas Colebrooke. one of the earliest modern research workers of Hindu scientific discipline. presumes that the treatise of Aryabhatta extended to determinate quadratic equations. undetermined equations of the first grade. and likely of the 2nd.

An astronomical work. called the Surya-siddhanta ( “knowledge of the Sun” ) . of unsure writing and likely belonging to the 4th or fifth century. was considered of great virtue by the Hindus. who ranked it merely second to the work of Brahmagupta. who flourished about a century subsequently. It is of great involvement to the historical pupil. for it exhibits the influence of Greek scientific discipline upon Indian mathematics at a period prior to Aryabhatta. After an interval of about a century. during which mathematics attained its highest degree. at that place 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 authors reference may be made of Cridhara. the writer of a Ganita-sara ( “Quintessence of Calculation” ) . and Padmanabha. the writer of an algebra. A period of mathematical stagnancy so appears to hold possessed the Indian head for an interval of several centuries. for the plants of the following writer of any minute base but small in progress of Brahmagupta.

We refer to Bhaskara Acarya. whose work the Siddhanta-ciromani ( “Diadem of anastronomical System” ) . written in 1150. contains two of import chapters. the Lilavati ( “the beautiful [ scientific discipline or art ] ” ) and Viga-ganita ( “root-extraction” ) . which are given up to arithmetic and algebra. English interlingual renditions 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 notes by W. D. Whitney ( 1860 ) . may be consulted for inside informations.

The inquiry as to whether the Greeks borrowed their algebra from the Hindus or frailty versa has been the topic of much treatment. There is no uncertainty that there was a changeless traffic between Greece and India. and it is more than likely that an exchange of green goods would be accompanied by a transference of thoughts. Moritz Cantor suspects the influence of Diophantine methods. more peculiarly in the Hindu solutions of undetermined equations. where certain proficient footings are. in all chance. of Grecian beginning. However this may be. it is certain that the Hindu algebraists were far in progress of Diophantus.

The lacks of the Grecian symbolism were partly remedied ; minus was denoted by puting a point over the subtrahend ; generation. by puting bha ( an abbreviation of bhavita. the “product” ) after the factom ; division. by puting the factor under the dividend ; and square root. by infixing Ka ( an abbreviation of karana. irrational ) before the measure. The terra incognita was called yavattavat. and if there were several. the first took this denomination. and the others were designated by the names of colorss ; for case. ten was denoted by ya and Y by Ka ( from kalaka. black ) .

A noteworthy betterment on the thoughts of Diophantus is to be found in the fact that the Hindus recognized the being of two roots of a quadratic equation. but the negative roots were considered to be unequal. since no reading could be found for them. It is besides supposed that they anticipated finds of the solutions of higher equations. Great progresss were made in the survey of undetermined equations. a subdivision of analysis in which Diophantus excelled. But whereas Diophantus aimed at obtaining a individual solution. the Hindus strove for a general method by which any indeterminate job could be resolved.

In this they were wholly 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 peculiar instance of the last equation. viz. . 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 boring solution which was published in 1658. and afterwards in 1668 by John Pell in his Algebra. A solution was besides given by Fermat in his Relation.

Although Pell had nil to make with the solution. osterity has termed the equation Pell’s Equation. or Problem. when more justly it should be the Hindu Problem. in acknowledgment of the mathematical attainments of the Brahmans. Hermann Hankel has pointed out the preparedness with which the Hindus passed from figure to magnitude and frailty versa. Although this passage from the discontinuous to continuous is non genuinely 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 Numberss or magnitudes. so the Brahmans are the existent discoverers of algebra.

The integrating of the scattered folks of Arabia in the seventh century by the stirring spiritual propaganda of Mahomet was accompanied by a meteorologic rise in the rational powers of a hitherto obscure race. The Arabs became the keepers of Indian and Grecian scientific discipline. whilst Europe was rent by internal discords.

Under the regulation of the Abbasids. Bagdad became the Centre of scientific idea ; doctors and uranologists from India and Syria flocked to their tribunal ; Greek and Indian manuscripts were translated ( a work commenced by the Caliph Mamun ( 813-833 ) and competently continued by his replacements ) ; and in about a century the Arabs were placed in ownership of the huge shops of Greek and Indian acquisition. Euclid’s Elementss were foremost translated in the reign of Harun-al-Rashid ( 786-809 ) . and revised by the order of Mamun. But these interlingual renditions were regarded as progressive. and it remained for Tobit ben Korra ( 836-901 ) to bring forth a satisfactory edition.

Ptolemy’s Almagest. the plants of Apollonius. Archimedes. Diophantus and parts of the Brahmasiddhanta. were besides translated. The first noteworthy Arabian mathematician was Mahommed ben Musa al-Khwarizmi. who flourished in the reign of Mamun. His treatise on algebra and arithmetic ( the latter portion of which is merely extant in the signifier of a Latin interlingual rendition. discovered in 1857 ) contains nil that was unknown to the Greeks and Hindus ; it exhibits methods allied to those of both races. with the Grecian component predominating.

The portion devoted to algebra has the rubric al-jeur wa’lmuqabala. and the arithmetic Begins with “Spoken has Algoritmi. ” the name Khwarizmi or Hovarezmi holding passed into the word Algoritmi. which has been farther transformed into the more modern words algorism and algorithm. meaning a method of calculating Tobit ben Korra ( 836-901 ) . born at Harran in Mesopotamia. an complete linguist. mathematician and uranologist. rendered conspicuous service by his interlingual renditions of assorted Grecian writers.