PART III. ALG EBRA. ALGEBRA is an universal method of reasoning on quantity by means of general characters. It is applied to the resolution of all kinds of problems wherein quantity is concerned ; for which purpose it does not require that rules should be previously laid down, but teaches how to discover or invent them, and that by the force of reasoning, from a bare contemplation of the conditions and relations of the quantities, as expressed in the problem under consideration. Algebra likewise shews how to demonstrate or prove the rules, theorems, and conclusions, thus investigated. It is impossible to do justice to this elegant and useful branch of learning by any description; all that is attempted in this place is to give the learner a general, although necessarily an inadequate, conception of the subject, which it is hoped will nevertheless be of service to him in his progress. The quantities which occur in Algebra are represented by the small alphabet, or other convenient symbols, which, standing for no particular value themselves, are made the arbitrary representatives of the quantities in question: this general mode of representation is attended with very great advantages; one of which is that the solution of a problem, by the general method of Algebra, furnishes an answer to every particular problem of the kind that can possibly be proposed, by merely substituting the numbers concerned in the particular problem, in the place of their corresponding letters in the answer to the general one. Letters and symbols being made the representatives of quantities, are managed like numbers, and consequently like them are subject to the fundamental rules of common arithmetic; and their relations and operations are denoted by the same marks or signs. When a problem is to be resolved, the first thing necessary to be done is to translate it out of common into algebraic language, by substituting letters for all the quantities concerned, both known and unknown; namely, by putting one of the initial letters for each known quantity, and final letters for the unknown ones, and expressing the conditions and relations of the quantities by their proper signs; this is the composition: when this is effected, we shall have an expression, wherein one or more quantities are declared equal to some other quantities; this expression is called an equation. Having made the composition, the next thing to be done is to find the value of the unknown quantities contained in the equations, which process is called the resolution: thus each unknown quantity must be disentangled from all known ones connected with it on the same side of the sign of equality, which is effected in each case by a process the contrary to that by which they are connected. A known quantity connected by addition, is taken away by subtraction; namely, by subtracting it from both sides of the equation; if it be connected by subtraction, it is taken away by addition; if by multiplication, it is taken away by division, &c. &c. always employing a contrary process: and thus the unknown quantity is at last found by itself on one side of the equation, and known ones only on the other; thus the value of the unknown quantity is found, for (as the equation thus reduced implies) it is equal to the known ones, connected together according to the import of their signs. Of the origin and early history of Algebra nothing is known. The Algebra of Diophantus, a Greek of Alexandria, who is supposed to have flourished about the second century after Christ, is the earliest work on the subject that has descended to us: in this work Diophantus has not given the principles and elements, but confined himself to questions depending on the most curious properties of numbers, which require considerable skill and address to resolve them; whence it is inferred, that Algebra must have been cultivated by the ancient Greeks, and had arrived at a considerable degree of perfection before the time of Diophantus. The destruction of the Alexandrian Library, A. D. 642, by which the sciences suffered an irrecoverable shock, most probably deprived us of many valuable writings, which would undoubtedly have thrown considerable light on the subject. But although Diophantus was the earliest author that is known to have written on Algebra, whence it can hardly be supposed that the ancient Greeks were unacquainted with the science, it was not from them, but from the Arabians, that the knowledge of Algebra was acquired by the western nations. The Italians were the first Europeans who cultivated this branch of science. Leonard de Pisa, and several others of that country, are said to have possessed great knowledge in the various methods of resolving problems, as known among the Arabians. M. Bossut affirms, that Leonard de Pisa flourished as early as the beginning of the thirteenth century, and that from a manuscript of his, discovered and quoted by Cossali, it appears he understood the method of solving not only cubic equations, but also those of higher powers, capable of being reduced to the second and third orders". The Germans were acquainted with Algebra at an early period, as appears from a treatise on Plane and Spherical Trigonometry, by the famous Regiomontanus, written about the year 1464; wherein some of the problems concerning rectilineal triangles are accompanied with an algebraic solution". The analytical works of Leonard de Pisa remaining in manuscript were scarcely known, even in Italy, so that Lucas De Burgo' is universally considered as the first who wrote professedly on Algebra in Europe, at least whose works were printed. His great work, entitled Summa de Arithmetica et Geometria, &c. was published in 1494 at Venice, and is considered as a very complete and masterly work on the sciences treated of, as they then stood. After naming several authors from whom he acquired the knowledge of the sciences, he proceeds to treat of Numbers figurate, odd, even, perfect, prime, composite, and many others; then of Numeration or Notation, Addition, Subtraction, Multiplication, Division, Progression, Evolution, &c. performing and proving his operations by various methods; by casting out the nines, sevens, &c. he extracts the square and cube roots, after the manner now in use, denoting a root by the initial R. He * See, Origine, Transporto in Italia e primi Progressi in Essa del Algebra, &c. 1797. quoted in Bossut's General History of the Mathematics. London, 1803. * See the introduction to Dr. Hutton's Mathematical Tables, p. 3. * Lucas Paciolus, commonly called Lucas De Burgo, because he was born at Borgo San Sapocha in Tuscany, was a Cordelier, or Minorite Friar, and the first who occupied the mathematical professorship founded at Milan, by Lewis Sforsa, called the Black. He translated Euclid into Latin, or rather revised the translation of Campanus, which he enriched by many learned annotations. He wrote several treatises on Arithmetic, Algebra, Geometry, Perspective, Music, Architecture, &c. which were published between the years 1470 and 1500. treats of Vulgar Fractions, the Rule of Three, Loss and Gain, with other rules used by merchants, in the same way we do : he then proceeds to Algebra, ascribing the invention to the Arabians; he shews the method then in use of denominating the powers and roots, and the necessary abbreviations required in practice. He treats of Proportions and Proportionalities, Arithmetical and Geometrical, accompanied with a copious collection of questions relating to numbers in continued proportion. Single and double Position are next unfolded by nearly the same method as at present; then follow the common operations of Algebra, proving that like signs give plus, and unlike minus, both in multiplication and division. He treats of the Extraction of Roots, Surds, simple and quadratic Equations, completing the Square and extracting the Root, all by the methods at present in use; he resolves Equations of the simple fourth power, and of the fourth combined with the second, treating the latter the same as quadratics. In the third case of quadratics, which has two positive roots, he uses both, but takes no notice of the negative roots which occur in both the other cases. He offers no solution of any other forms of affected Equations besides those above-mentioned, and says that no general rule for that purpose was then known. The remaining part of this work is on Geometry. The state of Algebra, at the time of its reception into Europe, is supposed to be fully exhibited in this book: the solution of quadratics, restricted to the use of the affirmative roots, is the highest pitch to which the science is here carried; so that if the method of solving higher equations was known to the Arabs, as has been asserted, the Europeans did not learn it from them". It * The instance we have quoted from Bossut cannot be considered as an ex |