ELEMENTARY ALGEBRA, WITH BRIEF NOTICES OF ITS HISTORY. SECTION II. OF ALGEBRA (continued). BY ROBERT POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE, LONDON: OF ALGEBRA. (Continued.) In the former part of the sketch, an attempt has been made to exhibit a brief account of the origin and early progress of the science of Algebra, as far as it was known and cultivated by the Greeks and the Hindus. The pre-eminence of having invented the most perfect science of Numerical and Universal Arithmetic is justly assigned to the latter; while the former stand unrivalled in their earliest achievements in the science of Pure Geometry. In this second part of the sketch, the account is continued of the cultivation of this science by the Arabians and Persians; and very briefly, with some notices of the great improvements of the science among the nations of Western Europe. In less than two centuries after the rapid and desolating wars of ihe Mohammedan power, their successors began to turn their attention to the remains of the writings on literature and science which had escaped the general destruction. It appears that the scientific writings both of the East and the West had become objects of attention in the reign of the Caliph Almansur, who in A.1. 141, A.D. 762, had transferred the seat of government of the Mohammedan empire to Bagdad. Both the Caliph Almansur and his immediate successors, Haroun Alraschid and Almamun, were munificent patrons of learning and learned men, and zealously encouraged the cultivation of the arts and sciences. The scientific writings of the Hindus and the Greeks were diligently sought, and translated into Arabic, and at present there are extant Arabic translations of works on science, the originals of which, in Greek and Sanscrit, are not known to be in existence. The oldest treatise on Algebra in the Arabic language was composed by Abu Abdallah Mohammed Ben Musa,' at the command of Almamun, who succeeded to the Caliphat A.D. 813. Haji Khalfa, in 1 in the year 1831, the original Arabic of this treatise was edited, and translated into English by Mr. F. Rosen, and published by the Oriental Translation Fund. The manuscript which Mr. Rosen used is preserved in the Bodleian Library at Oxford, bearing the date of its transcription in the year 473 of the Hejira, which corresponds to 1342 A.D. Three other treatises on Arithmetic and Algebra are bound up with it in the same volume, which is marked “CMXVIII. Hunt, 214, fol." This copy of Ben Musa's Algebra is the only one Mr. Rosen knew to be extant. On the margin of the manuscript are written some notes in a very small character. Some of them are marked as being taken from a commentary by Al Mozaihafi, probably the same person whose Introduction to Arithmetic is one of the three treatises above mentioned. his Bibliographical work, has recorded in two places that Mohammed Ben Musa was the first Mussulman who had ever written on the solution of problems by “the rules of completion and reduction." Both the title of the manuscript and the author's preface agree in stating it to be the work of Mohammed Ben Musa, and the mention of the Caliph Almamun in the preface, establishes the identity of the author. It appears also from a marginal note (written in a different hand) on the first and second pages of the manuscript, that this was the first work on Algebra composed by a Mohammedan. Arabian writers agree that Mohammed Ben Musa was the first who made known to the Arabians the Hindu method of computation by numbers, and it is highly probable that his knowledge of the science of Algebra was derived from the same source. The concurrent testimony of Arabian writers seems to be still further confirmed by the fact, that the Arabians wrote their figures from left to right, after the manner of the Hindus, but contrary to the order of writing their language, which is from right to left. It may further be added, there appears no evidence of facts that the Arabians had any knowledge of the Arithmetics of Diophantus anterior to the middle of the fourth century of the Hejira. It is related by Ebn Aladami in the preface to his Astronomical Tables, that Mohammed Ben Musa abridged, at the request of Almamun (but before his accession to the Caliphat), the Sindhind, or Astronomical Tables, translated by Mohammed Ben Ibrahim al Fazari, from the work of an Indian astronomer who visited the Court of Almansur at Bagdad, in the year of the Hejira 156, A.D. 777. The Algebra of Mohammed Ben Musa is elementary, exhibiting the fundamental operations, including mensuration of triangles, 1 The following note explains their notion of the relation of the diameter and circumference of a circle :-“In any circle, the product of its diameter multiplied by 3. will be equal to the periphery. This is the rule generally followed in practical life, though it is not quite exact. The geometricians have two other methods. One of them is, that you multiply the diameter by itself; then by 10, and hereafter take the root of the product ; the root will be the periphery. The other method is used by the astronomers among them : it is this, that you multiply the diameter by 62832, and then divide the product by 20000; the quotient is the periphery. Both methods come nearly to the same effect.” The following note appears on the margin of this passage in the Arabic text :-“This is an approximation, not the exact truth itself ; nobody can ascertain the exact truth of this, and find the real circumference ; for the line is not straight so that its exact length might be found. This is called an approximation, in the same manner as it is said of the square roots of irrational numbers that are an approximation, and not the exact truth. The best method here given is, that you multiply the diameter by 31 ; for it is the easiest and quickest." The first and third rules of Ben Musa are found in the Lilavati, chap. vi, sec. 201, and the second in chap. xii., sec. 40 of the Arithmetic of Brahmegupta. The quotient of the numbers 62832 divided by 20000 in the Algebra of Ben Musa. is the same as that of 3927 divided by 1250 in the Lilavati. 1 circles, &c., but is not extended beyond the solution of simple and quadratic equations. With respect to this treatise, Mr. Rosen, in his preface, has made the following remarks : “But under whatever obligation our author may be to the Hindus, as to the subject matter of his performance, he seems to have been independent of them in the manner of digesting and treating it; at least, the method which he follows in expounding his rules, as well as in showing their application, differs considerably from that of the Hindu mathematical writers. Bhascara and Brahmegupta give dogmatical precepts, unsupported by argument, which, even by the metrical form in which they are expressed, seem to address themselves rather to the memory than to the reasoning faculty of the learner. Mohammed gives his rules in simple prose, and establishes their accuracy by geometrical illustrations. The Hindus give comparatively few examples, and are fond of investing the statement of their problems in rhetorical pomp. The Arab, on the contrary, is remarkably rich in examples, but he introduces them with the same perspicuous simplicity of style which distinguishes his rules. In solving their problems, the Hindus are satisfied with pointing at the result, and at the principal intermediate steps which lead to it. The Arab shows the working of each example at full length, keeping his view constantly fixed upon the two sides of the equation, as upon the two scales of a balance, and showing how any alteration in one side is counterpoised by a corresponding change in the other." The author's preface to his work contains the following passage: " The learned, in times which have passed away, and among nations which have ceased to exist, were constantly employed in writing books on the several departments of science, and on the various branches of knowledge, bearing in mind those who were to come after them, and hoping for a reward proportionate to their ability, and trusting that their endeavours would meet with acknowledgment, attention, and remembrance ; content as they were, even with a small degree of praise; small, if compared with the pains which they had undergone, and the difficulties which they had encountered in revealing the secrets and obscurities of science. “Some applied themselves to obtain information which was not known before them, and left it to posterity; others commented upon the difficulties in the works left by their predecessors, and defined the best method of study), or rendered the access to science) easier, or placed it more within reach ; others again discovered mistakes in preceding works, and arranged that which was confused, or adjusted what was irregular, and corrected the faults of their fellow-labourers, without arrogance towards them, or taking pride in what they did themselves." Mohammed Ben Musa, at the beginning of his treatise, gives no |