History of Algebra
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 wa’l-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 l’Arte Magiore; ditta dal vulgo la Regula de la Cosa over Alghebra e Almucabala. The name l’arte magiore, the greater art, is designed to distinguish it from l’arte 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.
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 rom 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 which we may mention Pierre Fermat’s (1670), T. L. Heath’s (1885) and P. Tannery’s (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. 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 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 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 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 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 -)by=c, xy=ax+by+c (since rediscovered by Leonhard Euler) and cy2=ax2+b. A particular case of the last 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 nothing to do with the solution, osterity has termed the equation Pell’s 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. Euclid’s 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.
Ptolemy’s 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 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 wa’lmuqabala, 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.
University/College: University of Arkansas System
Type of paper: Thesis/Dissertation Chapter
Date: 30 September 2016
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