than in other sciences, merely because its language is so remarkably concise and comprehensive; for if this circumstance causes any difference, it must increase, not diminish, the difficulty.

COMPARISON BETWEEN THE ALGEBRAICAL AND GEOMETRICAL

METHODS.

It appears frequently that the investigation of a subject by the geometrical method is exceedingly difficult, and remarkably simple by analytical methods. This, however, generally happens when the subject may be considered as merely of an elementary character in the analytical department; for this branch properly begins at that point, at which common Geometry-from its being a comparatively feeble, and, generally, in the higher problems, an inadequate instrument of investigation-ought to terminate. In applying these two methods to mathematical subjects, the most useful mode of proceeding would certainly be to confine the geometrical method to the more elementary portions of a subject, so far as it is applicable without introducing prolixity, both on account of its greater simplicity, and the peculiar elegance of many of its results; and also because there are many students who cannot devote sufficient time to enable them to acquire that degree of proficiency in the analytical department, which is indispensable to the successful cultivation, to any considerable extent, of the interesting and extensive subjects to which it is applied. The geometrical method, however, even in regard to many subjects, to the investigation of which it is adequate in other respects, is inferior to the analytical, in the generality of its results. A property, for instance, may be proved by the former method to belong to some curves; but it is by the latter method that the more general truth is established, that this property is possessed by no other curves: the former method affords one means of reaching a certain point, the latter shows by what and by how many ways it may be attained; the one is only particular, the other is general, in its conclusions.

As to the comparative difficulty of the two methods,

it is one of the most certain truths that that difficulty is as unlimited in degree, as the field for investigation is unbounded in extent; with, however, this important distinction, that by means of the superior power of the analytical method, it is possible, with a certain expenditure of intellectual labour, to penetrate to an incomparably greater extent into the unexplored regions of science. A subject must undergo a mathematical analysis before its conditions can be combined and a solution obtained; and if it is a physical subject, it must undergo a physical as well as a mathematical analysis. Since, therefore, the analysis of a subject generally constitutes the chief difficulty in its investigation, and the chief exercise of the mind, and since those subjects to which the analytical method is applied are usually most complex, it is an inevitable consequence that the analytical departments afford in these, the most important points, a superior exercise to the mind. It no doubt sometimes happens that in the application of this method in original researches, a considerable part of the labour consists in the transformation of expressions; but in such cases these operations are not always performed according to rule, for, sometimes, to effect them, it is necessary to investigate new methods of resolution, which alone not unfrequently require much address and inventive power of no ordinary kind to accomplish; and thus a series of difficulties of different kinds must be surmounted before the ultimate object of research can be attained.

How mistaken, then, is the opinion that the analytical methods are mere mechanical processes; that difficulties of the most abstruse and recondite character can be solved by means of it by the smallest intellectual effort! Were this opinion true, how great would be our admiration of those minds that invented and fabricated such a miraculous instrument of science, compared with the power of which the supernatural agency of the rod of the magician is impotent! The Higher Analysis is certainly a very efficient instrument, but even its power is frequently inadequate to surmount difficulties that lie on every hand as immoveable obstacles to the progress of the mathematician. What constitutes one of the peculiar difficulties of Mathematical

Science, is, that, from the certainty of its principles, it is capable of being indefinitely extended; and the system already formed is of such formidable magnitude, that the student must acquire a great proficiency in a very extended course of study, before he can engage, with the prospect of great success, in original researches.

OF THE PRACTICAL UTILITY OF ALGEBRA.

By the principles of Algebra, rules are investigated for the solution of questions in Compound Interest, and in Certain and Contingent Annuities, and in various other important branches. The importance of this science, in a practical point of view, may be comprehended in the statement, that the rules in many of the practical branches of Mathematics and Physical Science are expressed by Algebraical Formulas, which are well adapted for this purpose by their conciseness and precision; as these rules are in many instances incapable of intelligible enunciation in common language.

HISTORY OF ALGEBRA.

The term Algebra is of Arabic origin, being composed of the two words al and gebr; but the title of the earliest European treatises on this science was Alghebra e Almukabala, which means resolution and composition. The origin of the science, like that of some others, is involved in much uncertainty; whether we are indebted to the Greeks or Hindoos for its first principles, is still an undetermined question.

About the middle of the fourth century, the arithmetic of the ancients contained only the four fundamental rules, and the extraction of the square and cube roots, when Diophantus (350 A. D.), one of the most distinguished Grecian mathematicians of the Alexandrian school, invented a new class of arithmetical problems, and the very ingenious methods of solution employed by him have a great resemblance to those of Algebra. But in the first six books of his works, which are all that remain of the thirteen com

posed by him, he has not given any proof of having been acquainted with the simplest notation of the science; for, instead of adopting letters to denote quantities, he merely uses words, or their contractions.

A more improved notation, and a more advanced knowledge of the science, is found in the Algebra of Arya-Batta, an Indian (fifth century), which it is probable he derived from some work superior to that of Diophantus; but whether of an earlier or later date, is uncertain. It is probable, however, that the first rudiments of the method are due to Diophantus, and that the Hindoos soon after advanced the science a step further, by extending the method and improving the notation.

Diophantus had many ingenious commentators, and of that number was the learned and unfortunate Hypatia, the daughter of the eminent philosopher Theon. Among the modern commentators may be mentioned Bachet de Meziriac, Fermat, Maclaurin, and Euler.

The progress of this science among others soon met with a serious obstruction, which almost threatened its total extinction, in consequence of the invasion of Egypt, and the capture of Alexandria, by a fanatical and ignorant army of Arabs, commanded by one of the successors of Mahomet, Caliph Omar, who barbarously expelled from the country the artists and philosophers assembled there from every quarter, destroyed the astronomical instruments, and an immense quantity of observations, and consumed by fire the invaluable library of Alexandria.

Although it was from Arabia that Algebra was first introduced into Europe, yet the Arabians have no claim as original inventors of the science, and neither have the Persians; for treatises of the science existed before it was known to either of these people.

The first treatise of Algebra that appeared in Europe was that of Lucas de Burgo, a Minorite friar, in Tuscany (1480); and from it may be obtained an idea of the very imperfect state of the science about the end of the fifteenth century. It appears from it that neither the Arabian nor European algebraists used either symbols or signs to denote the quantities or the operations, excepting some contractions

for the words representing their names. At this time the highest subject in the science was Quadratic Equations; and algebraists made use of the double values of the unknown quantity only when they were positive, but rejected the negative and imaginary roots.

The solution of one of the cases of a complete cubic equation was discovered (1505) by Scipio Ferrei, and the solution of the other cases was afterwards effected by Tartaglia. In a masterly work on Algebra, published by Cardan (1545), the whole theory and solution of Cubic Equations is given, which had been confidentially communicated to him, even under an oath of secrecy, by Tartaglia, who severely resented this breach of faith. The ungenerous spirit that prevailed in former times among learned men of monopolising knowledge, was not merely a serious obstacle to its diffusion, but also to its advancement.

In this work, Cardan, a person as remarkable for singularity as for ingenuity, demonstrates the theorems relating to Cubic Equations geometrically. He discovered the method. of transforming equations; the nature and number of their roots, which he distinguished into true and fictitious; and the rules for the signs of the roots of quantities. He also gave some examples of the application of Algebra to the solution of geometrical problems-a method in which, however, he had been anticipated by Regiomontanus and others; and he improved the notation of the science, by sometimes introducing letters of the alphabet. But it was still a common practice with the Italian algebraists to use the initials and m for plus and minus; the letter R to denote root; and the contractions of words for quantities.

Raphael Bombelli first proved in his Algebra (1595) that the imaginary results in the irreducible case of Cubic Equations produce real roots; for it appeared by his demonstration, that, after extracting the cube root of the binomial surds composing the preceding expressions, the imaginary parts destroyed each other; these roots, however, could be extracted only in particular cases. The paradox of the roots in the irreducible case, which are real roots, being represented by imaginary results, was long the torture of algebraists; and it did not finally vanish from the science