Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī

“ Let the square be represented by a quadrangle, the sides of which are unknown to us, though they are equal among themselves, as also the angles. This is the quadrate A D, which comprises the three roots and the four of numbers mentioned in this instance. In every quadrate one of its sides, multiplied by a unit, is its root. We now cut off the quadrangle H D from the quadrate A D, and take one of its sides H C for three, which is the number of the roots. ”

“ The verb جبر jabar of which the جبْرٌ jebr is derived, properly signifies to restore something broken, especially to cure a fractured bone. It is thus used in the following passage from Motanabbi (p. 143, 144, ed. Calcutt.)
“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. ”

“ You know that all mercantile transactions of people, such as buying and selling, exchange and hire, comprehend always two notions and four numbers, which are stated by the enquirer; namely, measure and price, and quantity and sum. The number which expresses the measure is inversely proportionate to the number which expresses the sum, and the number of the price inversely proportionate to that of the quantity. ”

“ Thus, if you wish to multiply the root of five by the root of ten, [6] multiply five by ten: the root of the product is what you have required.
If you wish to multiply the root of one-third by the root of a half; [7] you multiply one-third by a half: it is one-sixth: the root of one-sixth is equal to the root of one-third, multiplied by the root of a half.
If you require to multiply twice the root of nine by thrice the root of four, [8] then take twice the root of nine, according to the rule above given, so that you may know the root of what square it is. ”

“ One thing will be as much as thirty-one (115) parts of the sum of dirhems which you have; and just so much will be the legacy out of the two hundred; it is sixty-four dirhems and sixteen thirty-one parts.
“Suppose that some one emancipates a slave girl of the price of one hundred dirhems, and makes to some person a present of a slave girl, which is five hundred dirhems worth; the receiver cohabits with her, and her dowry is one hundred dirhems; the donor bequeaths to some other person as much as one-fourth of his capital.” ”

“ The science as taught by Mohammed ben Musa, in the treatise now before us, does not extend beyond quadratic equations, including problems with an affected square. These he solves by the same rules which are followed by Diophantus [5] , and which are taught, though less comprehensively, by the Hindu mathematicians [6] That he should have borrowed from Diophantus is not at all probable; for it does not appear that the Arabs had any knowledge of Diophantus’ work before the middle of the fourth century after the Hejira, when Abu’l-wafa Buzjani rendered it into Arabic [7] . ”

“ I observed that the numbers which are required in calculating by Completion and Reduction are of three kinds, namely, roots, squares, and simple numbers relative to neither root nor square.
A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending. ”

“ The verb جبر is not likewise used, when in an equation an integer is substituted for a fractional power of the unknown quantity: the proper expression for this is either the second or fourth conjugation of, or the second كمل, of تمّ.
The word مقابلة mokābalah is a noun of action of the verb قبل to be in front of a thing, which in the third conjugation is used in a reciprocal sense of two objects being opposite one another or standing face to face; and in the transitive sense of putting two things face to face, of confronting or comparing two things with one another. ”

“ Praised be God for his bounty towards those who deserve it by their virtuous acts: in performing which, as by him prescribed to his adoring creatures, we express our thanks, and render ourselves worthy of the continuance (of his mercy) , and preserve ourselves from change: acknowledging his might, bending before his power, and revering his greatness! He sent Mohammed (on whom may the blessing of God repose!) with the mission of a prophet, long after any messenger from above had appeared, when justice had fallen into neglect, and when the true way of life was sought for in vain. ”

“ 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; you may say, for instance, “squares are equal to roots,” or “squares are equal to numbers,” or “roots are equal to numbers.” [2]
(4) Of the case in which squares are equal to roots, this is an example, “A square is equal to five roots of the same;” ”

“ The product is a square, which is equal to a hundred and thirty-two dirhems and one root. Reduce this, according to what I have taught you, it will be right.
If the instance be: “A dirhem and a half to be divided among one person and certain persons, so that the share of the one person be twice as many dirhems as there are other persons;” [31] then the Computation is this: [32] You say, the one person and some persons are one and thing: it is the same as if the question had been one dirhem and a half to be divided by one and thing, and the share of one person to be equal to two things. ”

“ You say, therefore, thing multiplied by thing is a square positive; and ten by thing is ten things positive; and minus ten by thing is ten things negative. You now remove the positive by the negative, then there only remains a square. Minus ten multiplied by ten is a hundred, to be subtracted from the square. This, therefore, altogether, is a square less a hundred dirhems.
(19) Whenever a positive and a negative factor concur in a multiplication, such as thing positive and minus thing, the last multiplication gives always the negative product. ”

“ In some notes on the Arabic text which are appended to my translation, I have endeavoured, not so much to elucidate, as to point out for further enquiry, a few circumstances connected with the history of Algebra. The comparisons drawn between the Algebra of the Arabs and that of the early Italian writers might perhaps have been more numerous and more detailed; but my enquiry was here restricted by the want of some important works. Montucla, Cossali, Hutton, and the Basil edition of Cardanus’ Ars magna, were the only sources which I had the opportunity of consulting. ”

“ Suppose one of the parts to be thing, and the other ten minus thing. You multiply thing by itself, it is a square; then by two and seven-ninths, this makes it two squares and seven-ninths of a square. You afterwards multiply ten by ten; it is a hundred, which much be equal to two squares and seven-ninths of a square. Reduce it to one square, through division by nine twenty-fifths ”

“ When I considered what people generally want in calculating, I found that it always is a number.
I also observed that every number is composed of units, and that any number may be divided into units. ”