name or definition of the science, nor does he give a formal explanation of the terms al jebr and al mokabalah, by which he designates certain operations peculiar to the solution of equations. However, he repeatedly employs these two terms combined for the name of this entire branch of mathematical science. As the former of these terms has been introduced into the several languages of Europe, and is now used as the name of the science of universal arithmetic, some explanation appears necessary. The following particulars are chiefly taken from the notes on Mr. Rosen's translation.

The Arabic verb jabar, of which the substantive al jebr (algebra) is a derivative, properly and correctly means, to restore something broken, especially to set right a fractured bone, or a dislocated joint. As the Arabian Moors for ages maintained their conquests in Spain until the capture of Granada, in 1492, it might be expected that many words of Arabic origin would be found in the language of the Spanish people. Hence, in the Spanish and Portuguese languages, is the word Algebrista, unquestionably derived from the Arabic words al jebr, and employed as synonymous with the word surgeon. In its primary sense, however, it was probably restricted in its use to the person, who in England was formerly called a bone-setter.'

In mathematical language, the verb jabar means to make perfect or complete any quantity that is incomplete, or subject to diminution. As applied to equations, it denotes to add to one side of an equation as much as is removed from the other, so as to make the latter complete. As, for instance, if ten less one thing be equal to four things. Then jebr is the removal of one thing from one side of the equation, and then adding one thing to four things on the other side; so that one side is made complete (or wholly positive) by the removal of one thing; and the equality is restored by adding one thing to the four things on the other side.

The word Mokabalah is a noun of action, from a verb which means to be in front of a thing, or to compare two things with one another. In mathematical language it is employed to express a comparison between positive and negative quantities. As applied to equations, it denotes the removal of equal quantities from both sides of an equation; also the withdrawal of equal quantities affected with contrary signs from the same side of an equation; as, for example, if thing added to square be equal to square added to four, then thing is equal to four;

Mr. Rosen illustrates the meaning of the word by a passage in Arabic from Motanabbi, of which he gives the following translation in English :-"O Thou on whom I rely in whatever I hope, with whom I seek refuge from all that I dread; whose bounteous hand seems to me like the sea, as thy gifts are like its pearls: pity the youthfulness of one whose prime has been wasted by the hand of Adversity, and whose bloom has been stifled in the prison. Men will not heal a bone which thou hast broken, nor will they break one which thou hast healed."

also, if ten added to thing less ten, be equal to 4, then thing is equal to 4.

This science has therefore been called by the name of these two rules al jebr and al mokabalah; namely, the rule of jebr or restoration, and of mokabalah or reduction.1

Both Mohammed Ben Musa and other Arabian writers on Algebra did not adopt the Hindu notations. They have no symbols, arbitrary or abbreviated, to denote known or unknown quantities, either positive or negative. Neither have they any symbols to denote the elementary operations of the science, but express everything by words and phrases at length.

The Arabian algebraists designated the unknown quantity by the word shai, thing. A positive quantity they characterised by the word zaid, additional; and a negative quantity by nakis, deficient, without any discriminating mark for either of them. The first power of an unknown quantity was also named root, in reference to other powers. The square was termed mal, possession or wealth; and the cube, câb; and these two terms, mal and cab, were combined by the sums of the indices of the powers, as compound names of higher powers of the unknown quantity. Their method of solving equations was different

1 Haji Khalfa, in his biographical work, gives the following explanation. Jebr is the adding to one side what is negative on the other side of an equation, owing to a subtraction, so as to equalise them. Mokabalah is the removal of what is positive from either sum, so as to make them equal.

He gives further illustration of this by an example; "For instance, if we say, "Ten less one thing equal to four things.' Then Jebr is the removal of the subtraction, which is performed by adding to the minuend an amount equal to the subtrahend; hereby the ten is made complete, that which was defective in them being restored. An amount equal to the subtrahend is then added to the other side of the equation; as in the above instance, after the ten have been made complete, one thing must be added to the four things, which thus become five things. Mokabalah consists in withdrawing the same amount from quantities of the same kind on both sides of the equation; or, as others say, it is the balancing of certain things against others, so as to equalise them. Thus, in the above example, the ten are balanced against the five with a view to equalise them. This science has therefore been called by the name of these two rules-namely, the rule of jebr, or restoration, and of mokabalah, or reduction, on account of the frequent use that is made of them."

* The following is an example of their method of solving simple equations :"Two numbers, the difference of which is two dirhems; you divide the small one by the great one, and the quotient is equal to half a dirhem.

Then the computation is this: Multiply thing and two dirhems by the quotient, that is a half. The product is half a thing and one dirhem, equal to thing. Remove now half a dirhem on account of the half dirhem on the other side, the remainder is one dirhem equal to half a thing. Double it; then you have thing, equal to two dirhems. This is one of the two numbers, and the other is four."

In modern notation,

from that of the Hindus. The equation was prepared by two preliminary operations, by means of which all the terms on both sides of the equation were made positive, before their rules were applied to effect the solution.

The numbers which are required in calculating by "completion and reduction" are described to be of three kinds; namely, roots, squares, and simple numbers relative neither to root nor square. A root is any quantity which is to be multiplied by itself. A square is the whole amount of the root multiplied by itself. A simple number is any number which may be pronounced without reference to root or square.

A number belonging to one of these three classes may be equal to a number of another class; for instance, squares are equal to roots, or squares are equal to numbers, or roots are equal to numbers. By combining these three kinds-namely, roots, squares, and numbersthere arise three forms of quadratic equations: (1) squares and roots equal to numbers; (2) squares and numbers equal to roots; (3) roots and numbers equal to squares. These, in the modern algebraic notation, are expressed by x2+px=q, x2+q=px, and x=px+q. The form x2+px= -g was excluded from their classification of quadratic equations. The Arabians held a less general notion of equations than the Hindus, as they only knew equals to be the equality of two or more absolute positive quantities. They had no conception of the equation x2+px+q=0, the sum of three positive quantities being equal to zero, otherwise than an absurdity, or an impossibility.

In the solution of quadratic equations; in the first place, they reduced the coefficient of the square to unity, where it was necessary. The next step was to relieve the equation of negative quantities by the operation of al jebr. This name was given to the operation because it was only after negative terms had been removed that the equation itself became capable of solution by the application of their rules. After al jebr followed al mokabalah, which consisted in comparing the equal numbers on the same side, and on both sides of the equation; and as far as it could be done, to cancel them. After this operation, an equation of the second degree was reduced to one of the three specified forms, and adapted for solution.1

The Hindus, on account of their more complete algebraic notation, and their wider view, did not find it necessary to carry back an equation to the special form of two positive equals. On that account they

1 As an example of the first form of quadratic equations, "roots and squares are equal to numbers; one square and ten roots of the same, amount to thirty-nine dirhems;" that is to say, "what must be the square, which when increased by ten of its own roots, amounts to thirty-nine?" The solution is this: you halve the number of the roots, which in the present instance yields five. You multiply by itself; the product is twenty-five. Add this to thirty-nine, the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots,

had no particular names for the two preliminary operations which the Arabians had described by the names al jebr and al mokabalah.

These two operations (one of which has given the name to the science) are precisely what are enjoined, without distinctive appellations of them, in the introduction of the Arithmetics of Diophantus. He directs that if the quantities be positive on both sides, like are to be taken from like until one species be equal to one species; but if, on either side or on both, any species be negative, the negative species as positive must be added to both sides, so that all the quantities become positive on both sides of the equation; after which, like are again to be taken from like, until one species remain on each side (Def. 11).

Among the writers subsequent to Mohammed Ben Musa, may be mentioned Abufaraj, the author of a treatise on computation. He lived in the twelfth century of the Christian era, and noticed a book of numerical computation which Mohammed Ben Musa amplified, and which he declared "is a most expeditious and concise method, and testifies the ingenuity and acuteness of the Hindus."

The celebrated Abu Yusef Alkindi, contemporary with the astrologer Abu Masher, in the third century of the Hejira, was versed in the sciences of Greece, India, and Persia, and was also the author of several treatises on numbers. His pupil, Ahmed Ben Mohammed of Sarkiri in Persia, who flourished about the middle of the third century of the Hejira, was the author of a complete treatise of computation, embracing Arithmetic and Algebra. About the same time a treatise on Algebra was composed by Abu Hanifah of

which is five; the remainder is three. This is the root of the square you sought for; the square itself is nine.

In modern notation, the equation is x2+10x=39.

The following is another example from M. Rosen's translation :

"I have divided ten into two parts; I have then multiplied each of them by itself, and when I had added the products together, the sum was fifty-eight dirhems.

Computation. Suppose one of the two parts to be thing, and the other, ten minus thing. Multiply ten minus thing by itself; it is a hundred and a square minus twenty things. Then multiply thing by thing; it is a square. Add both together. The sum is a hundred, plus two squares minus twenty things, which are equal to fifty-eight dirhems. Take now the twenty negative things from the hundred and the two squares, and add them to fifty-eight; then a hundred, plus two squares, are equal to fifty-eight dirhems and twenty things. Reduce this to one square by taking the moiety of all you have. It is then fifty dirhems and a square, which are equal to twenty-nine dirhems and ten things. Then reduce this, by taking twenty-nine from fifty; there remains twenty-one and a square, equal to ten things. Halve the number of roots, it is five; multiply this by itself, it is twenty-five; take from this the twenty-one which are connected with the square, the remainder is four. Extract the root, it is two. Subtract this from the moiety of the roots, namely, from five, there remains three. This is one of the portions; the other is seven."

:

Dainawar, who lived till about the 290th year of the Hejira. At a later period flourished Abulwafa of Buzjani, a distinguished mathematician, whose death happened in the 388th year of the Hejira. He wrote several treatises on computation, and a commentary on the Algebra of Mohammed Ben Musa, also on another of less repute and of later date by Abu Yahya. He was the first Arabian who translated the Arithmetics of Diophantus from Greek into Arabic. This work was executed nearly two centuries after the Arabians had become acquainted with the sciences of the arithmetic and astronomy of the Hindus.

It was in the reign of Haroun Alraschid that the astronomy of the Greeks became known to the Arabians by a translation of Ptolemy's Almagest.

In the year 1851, Dr. Woepcke published the original Arabic of a treatise on Algebra, composed by Omar Alkhayyami, with a French translation.1 The years of the birth and death of the author are unknown. It is known, however, that he was brought up with two youths who afterwards became distinguished. One of them was the Vizier of Malak Shah, the last of the Seljukian Sultans, who died about A.D. 1072. His work is divided into five parts. The introduction contains a preface, and the definitions of the science, &c. Next, the solution of equations of the first and second degree. Thirdly, the construction of cubic equations. Fourthly, the discussion of equations involving fractions, and having powers of the unknown quantity in the denominators. And lastly, some additions. In treating the subject of equations, he always connects the solution with some geometrical construction, or the converse. For cubic equations, however, he was obliged to confine himself to the latter course. He gives the same threefold division of quadratic equations as Mohammed Ben Musa, and exhibits a classification of cubic equations.2 The constructions of these equations are effected in different ways, some of them requiring the properties of the conic sections, as the example (p. 62). "The equation, a cube and

1 The original Arabic of this work, with a translation in the French language, was printed at Paris in 1851, entitled, "L'Algèbre d'Omar Alkhayyami, publiée, traduite et accompagnée d'extraits de manuscrits inédits, par F. Woepcke, Docteur agrégé a l'Université de Bonn, Membre de la Société Asiatique de Paris."

2 The following are examples of the classes of cubic equations, each of which consists at least of two terms on one side and one term on the other, expressed in modern notation.

(1) a3+cx2=bx: (2) x3+bx=a: (3) x3+cx2=bx+a:

(4) x3+cx2+bx=a: cx2+bx+a=x3: x3+cx2+a=bx: x3+bx+a=cx2. These different classes arise from the method adopted by the Arabians of con. sidering an equation when reduced to consist only of positive quantities. In the text, these classes are expressed in words, as, for instance, the first example above, x3+cx2=bx, is described, “A cube and squares equal to roots."