numerical algebra. 4. The only known type of algebra which does not contain | The effect of these definitions is that the sum and the proarithmetical elements is substantially due to George Boole. duct of two quaternions are also quaternions; that addition Although originally suggested by formal logic, it is is associative and commutative; and that multiplication is Non- most simply interpreted as an algebra of regions in associative and distributive, but not commutative. Thus space. Let i denote a definite region of space; and e1e2=-e21, and if q, q′ are any two quaternions, qq' is generally let a, b, &c., stand for definite parts of i. Let a+b different from q'q. The symbole behaves exactly like 1 in denote the region made up of a and b together (the common ordinary algebra; Hamilton writes 1, i, j, k instead of eo, e, part, if any, being reckoned only once), and let aXb or ab mean e2, eз, and in this notation all the special rules of operation may the region common to a and b. Then a+a=aa=a; hence | be summed up by the equalities numerical coefficients and indices are not required. .The inverse i2=j2 = k2=ijk = −1. symbols, are ambiguous, and in fact are rarely used. Each symbol a is associated with its supplement à which satisfies the equivalences a+ā=i, aā=o, the latter of which means that a and a have no region in common. Finally, there is a law of absorption expressed by a+ab=a. From every proposition in this algebra a reciprocal one may be deduced by interchanging and X, and also the symbols o and i. For instance, x+y=x+xy and xy=x(x+y) are reciprocal. The operations + and X obey all the ordinary laws A, C, D (§ 3). 5. A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol aA. The sum of two weighted points Möbius's aA, BB is, by definition, the point (a+B)G, where G divides AB so that AG: GB=B:a. It can be proved by geometry that barycentric calculus. (aA+BB) + C = aA'+(BB+yC) = (a+B+y)P, where P is in fact the centroid of masses a, ẞ, y placed at A, B, ,C respectively. So, in general, if we put aA+BB+yČ+...+λL= (a+ß+y+...+λ)X. X is, in general, a determinate point, the barycentre of aA, BB, &c. (or of A, B, &c. for the weights a, B, &c.). If (a+ß+...+-λ) happens to be zero, X lies at infinity in a determinate direction; unless -aA is the barycentre of BB, YC,... XL, in which case aA+ßB+...+λL vanishes identically, and X is indeterminate. If ABCD is a tetrahedron of reference, any point' P in space is determined by an equation of the form (a+B+y+8) PaA+8B+C+D: a, B, y, d are, in fact, equivalent to a set of homogeneous coordinates of P. For constructions in a fixed plane three points of reference are sufficient. It is remarkable that Möbius employs the symbols AB, ABC, ABCD in their ordinary geometrical sense as lengths, areas and volumes, except that he distinguishes their sign; thus AB-BA, ABC=-ACB, and so on. If he had happened to think of them as " products," he might have anticipated Grassmann's discovery of the extensive calculus. From a merely formal point of view, we have in the barycentric calculus a set of " special symbols of quantity" or extraordinaries " A, B, C, &c., which combine with each other by means of operations + and which obey the ordinary rules, and with ordinary algebraic quantities by operations X and ÷, also according to the ordinary rules, except that division by an extraordinary is not used. Putting q=a+Bi+j+dk, Hamilton calls a the scalar part of q, and denotes it by Sq; he also writes Vq for Bi+vj+dk, which is called the vector part of q. Thus every quaternion may be written in the form q=Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions. The equations q'+x=q and y+q=q are satisfied by the same quaternion, which is denoted by q-q'. On the other hand, the equations q'x=q and yq=q have, in general, different solutions. It is the value of y which is generally denoted by q÷q'; a special symbol for x is desirable, but has not been established. If we put go-Sq'-Vq', then go is called the conjugate of q', and the scalar q'gogog' is called the norm of qʻ and written Nq'. With this notation the values of x and y may be expressed in the forms x = qoq/Ng', = y=qqo/Ng', In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion. Clifford's biquaternions are quantities +r, where q, r are quaternions, and , are symbols (commutative with quaternions) obeying the laws -, n2=n, En=no (cf. QUATERNIONS). 7. In the extensive calculus of the nth category, we have, From these first of all, n independent " units," ei, e2, ... en. are derived symbols of the type A1 = a1ei+age2+...+anen = Zae, Grassmann's extensive which we shall call extensive quantities of the first species calculus. = To multiply A1 by a scalar, we apply the rule and similarly for division by a scalar. All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units e, are taken to satisfy the equalities this reduces them to n(n-1) distinct values, exclusive of zero. == - it follows that A1B1-B1A1, and that A12=o. By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, S(VabVcd) = SadSbc-SacSbd dSabc=aSbcd-bScda+cSadb we may construct derived units of the third, fourth... nth | These may be compared and contrasted with such quaternion species. Every unit of the rth species which does not vanish formulae as is the product of different units of the first species; two such units are independent unless they are permutations of the same set of primary units e, in which case they are equal or opposite according to the usual rule employed in determinants. Thus, for instance €1.Є2e3 = €12.е3 = €1е2€ 3 = — €2€1€3 = €2€ 3Є1; and, in general, the number of distinct units of the rth species in the nth category (rn) is Cn,r. Finally, it is assumed that (in the nth category) eezes... en 1, the suffixes being in their natural order. = Let A, ZaE and B.=ZẞE" be two extensive quantities of species r and s; then if r+sn, they may be multiplied by the rule A,B,=2(as) EE where the products EE may be expressed as derived units of species (r+s). The product B,A, is equal or opposite to A,B, according as rs is even or odd. This process may be extended to the product of three or more factors such as A,B,Ct... provided that r+s+t+...does not exceed n. The law is associative; thus, for instance, (AB)C=A(BC). But the commutative law does not always hold; thus, indicating species, as before, by suffixes, A,B,C (-1)++C,B,A,, with analogous rules for other cases. H If r+s>n, a product such as E,E,, worked out by the previous rules, comes out to be zero. A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive. The new rule requires some preliminary explanation. If E is any extensive unit, there is one other unit E', and only one, such that the (progressive) product EE' = 1. This unit is called the supplement of E, and denoted by E. For example, when n = 4, |e1 = €2€34, |e1e2=e3e4, |e2eses = —€1, and so on. Now when r+s>n, the product E,E, is defined to be that unit of which the supplement is the progressive product EE,. For instance, if n=4, Er=е1€3, Es=е2€3€4, we have |E|E.=(-e2e4) ( −е1) = €1е2€4= |e3, consequently, by the rule of regressive multiplication, A,B,= 2αE,ΣẞE, = 2(aß)E,E,, where a, b, c, d denote arbitrary vectors. Linear 8. An n-tuple linear algebra (also called a complex number system) deals with quantities of the type A=2ae derived from n special units e1, e2... en. The sum algebras. and product of two quantities are defined in the first instance by the formulae Σαρ + Σβε = (α+β)e, Σα;; Χ Σβjej = Σαβ;)eie;, so that the laws A, C, D of § 3 are satisfied. The binary products ee,, however, are expressible as linear functions of the units et by means of a "multiplication table" which defines the special characteristics of the algebra in question. Multiplication may or may not be commutative, and in the same way it may or may not be associative. The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B. Peirce. Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra. If, in the extensive calculus of the nth category, all the units (including 1 and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2-1)-tuple linear algebra, which, however, is not wholly associative. It should be observed that while the use of special units, or extraordinaries, in a linear algebra is convenient, especially in applications, it is not indispensable. Any linear quantity may be denoted by a symbol (a1, a2, . . . an) in which only its scalar coefficients occur; in fact, the special units only serve, in the algebra proper, as umbrae or regulators of certain operations on scalars (see NUMBER). This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative. Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices. 9. In ordinary algebra we have the disjunctive law that if ab=0, then either a=o or b=o. This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general. One of the most important questions in where the regressive products E,E, are to be reduced to units of investigating a linear algebra is to decide the necessary relations species (r+s-n) by the foregoing rule. If A=ZaE, then, by definition, JA=Za|E, and hence A|(B+C)=AB+A|C. Now this is formally analogous to the distributive law of multiplication; and in fact we may look upon AB as a particular way of multiplying A and B (not A and B). The symbol AB, from this point of view, is called the inner product of A and B, as distinguished from the outer product AB. An inner product may be either progressive or regressive. In the course of reducing such expressions as (AB)C, (AB) {C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other. In the first two cases the product is said to be pure, in the third case mixed. A pure product is associative; a mixed product, speaking generally, is not. The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively. As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units. Only a few illustrations can be given here, Let a, b, c, d, e, f be quantities of the first species in the fourth category; A, B, C... quantities of the third species in the same category. Then (de)(abc) = (abde)c+(cade)b+(bcde) a = (ab) (AB) = (aA) (bB) — (aB)(bA) ablc (alc)b(blc)a, (abled) = (ac) (b|d) — (ald) (b|c). = between a and b in order that this product may be zero. Subsidiary algebras. 10. The algebras discussed up to this point may be considered as independent in the sense that each of them deals with a class of symbols of quantity more or less homogeneous, and a set of operations applying to them all. But when an algebra is used with a particular interpretation, or even in the course of its formal development, it frequently happens that new symbols of operation are, so to speak, superposed upon the algebra, and are found to obey certain formal laws of combination of their own. For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like. In most cases these subsidiary algebras, as they may be called, are inseparable from the applications in which they are used; but in any attempt at a natural classification of algebra (at present a hopeless task), they would have to be taken into account. Even in ordinary algebra the notation for powers and roots disturbs the symmetry of the rational theory; and when a schoolboy illegitimately extends the distributive law by this want of complete harmony. writing (a+b)=√a+vb, he is unconsciously emphasizing AUTHORITIES.-A. de Morgan, "On the Foundation of Algebra," Trans. Camb. P.S. (vii., viii., 1839-1844); G. Peacock, Symbolical Algebra (Cambridge, 1845); G. Boole, Laws of Thought (London, 1854); E. Schröder, Lehrbuch der Arithmetik u. Algebra (Leipzig, 1873). Vorlesungen über die Algebra der Logik (ibid., 1890-1895); A. F. Möbius, Der barycentrische Calcul (Leipzig, 1827) (reprinted in his collected works, vol. i., Leipzig, 1885); W. R. Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866); 64 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. H. Grassmann, Die lineale Ausdehnungslehre (Leipzig, 1844), | Magiore; ditta dal vulgo la Regula de la Cosa over Alghebra e Die Ausdehnungslehre (Berlin, 1862) (these are reprinted with valuable emendations and notes in his Gesammelte math. น. phys. Werke, vol. i., Leipzig (2 parts), 1894, 1896), and papers in Grunert's Arch. vi., Crelle, xlix. lxxxiv., Math. Ann. vii. xii.; B. and C. S. Peirce, Linear Associative Algebra," Amer. Journ. Math. iv. (privately circulated, 1871); A. Cayley, on Matrices, Phil. Trans. cxlviii., on Multiple Algebra, Quart. M. Journ. xxii.; J. J. Sylvester, on Universal Algebra (i.e. Matrices), Amer. Journ. Math. vi.; H. J. S. Smith, on Linear Indeterminate Equations, Phil. Trans. cli.; R. S. Ball, Theory of Screws (Dublin, 1876); and papers in Phil. Trans. clxiv., and Trans. R. Ir. Ac. xxv.; W. K. Clifford, on Biquaternions, Proc. L. M. S. iv.; A. Buchheim, on Extensive Calculus and its Applications, Proc. L. M. S. xv.-xvii.; H. Taber, on Matrices, Amer. J. M. xii.; K. Weierstrass," Zur Theorie der aus n Haupteinheiten gebildeten complexen Grössen,' Götting, Nachr. (1884); G. Frobenius, on Bilinear Forms, Crelle, lxxxiv., and Berl. Ber. (1896); L. Kronecker, on Complex Numbers and Modular Systems, Berl. Ber. (1888); G. Scheffers, "Complexe Zahlensysteme,' Math. Ann. xxxix. (this contains a bibliography up to 1890); S. Lie, Vorlesungen über continuirliche Gruppen (Leipzig, 1893), ch. xxi.; A. M'Aulay," Algebra after Hamilton, or Multenions," Proc. R. S. E., 1908, 28, p. 503. For a more complete account see H. Hankel Theorie der complexen Zahlensysteme (Leipzig, 1867); O. Stolz, Vorlesungen über allgemeine Arithmetik (ibid., 1883); A. N. Whitehead, A. N. Whitehead, A Treatise on Universal Algebra, with Applications (vol. i., Cambridge, 1898) (a very comprehensive work, to which the writer of this article is in many ways indebted); and the Encyclopädie d. math. Wissenschaften (vol. i., Leipzig, 1898), &c., §§ A 1 (H. Schubert), A 4 (E. Study), and B 1 c (G. Landsberg). For the history of the development of ordinary algebra M. Cantor's Vorlesungen über Geschichte der Mathematik is the standard authority. (G. B. M.) Etymology. C. HISTORY 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-muqābala, 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 geber, meaning “man." Since, how ever, 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 Franciscus Vieta (François Viète) named it Specious Arithmelic, 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. Greek algebra. 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 geometers, 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 (16211670). 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. Indian algebra. 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 an Astronomical 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 factors; 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 yāvattāvat, 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 yā and y by kā (from kālaka, 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 axby=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, posterity 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. Arabian 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-jebr 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, ranean. Europe. rendered conspicuous service by his translations of various Greek | lation of Arabic manuscripts. The first successful attempt to 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 Karhi, 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 x2"+ax"+b=0; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes. revive the study of algebra in Christendom was due to Leonardo of Pisa, an Italian merchant trading in the MediterHis travels and mercantile experience had led Algebra in him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical. But it contains little that is original, and although the work created a great sensation when it was first published, the effect soon passed away, and the book was practically forgotten. Mathematics was more or less ousted from the academic curricula by the philosophical inquiries of the schoolmen, and it was only after an interval of nearly three centuries that a worthy successor to Leonardo appeared. This was Lucas Paciolus (Lucas de Burgo), a Minorite friar, who, having previously written works on algebra, arithmetic and geometry, published, in 1494, his principal work, entitled Summa de Cubic equations were solved geometrically by determining Arithmetica, Geometria, Proportioni et Proportionalita. In it the intersections of conic sections. Archimedes' problem of he mentions many earlier writers from whom he had learnt the dividing a sphere by a plane into two segments having a pre-science, and although it contains very little that cannot be found scribed 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 Wefa (940-998), who succeeded in solving the forms xa and x1+ax3=b. 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 x3= =a and r3= 2a3), 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 Cordova, Dania and Granada. Gabir ben Aflah 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 íntellectual 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. In Europe the decline of Rome was succeeded by a period, lasting several centuries, during which the sciences and arts were all but neglected. Political and ecclesiastical dissensions occupied the greatest intellects, and the only progress to be recorded is in the art of computing or arithmetic, and the trans in Leonardo's work, yet it is especially noteworthy for the ratic Cubic equations having been solved, biquadratics soon followed suit. As early as 1539 Cardan had solved certain particular cases, but it remained for his pupil, Lewis (Ludovici) Ferrari, to devise a general method. His Biquadsolution, which is sometimes erroneously ascribed to equations. Rafael Bombelli, was published in the Ars Magna. In this work, which is one of the most valuable contributions to the literature of algebra, Cardan shows that he was familiar with both real positive and negative roots of equations whether rational or irrational, but of imaginary roots he was quite ignorant, and he admits his inability to resolve the so-called |