ANALYSIS, from avaitt;co, resolvo, is that procedure in Mat henzatics, by which a proposition is traced up, through a chain of necessary dependence, to some known operation, or some admitted principle. It is alike appli cable to the investigation of truth in a theorem, or the discovery of the construction of a problem. Analysis, as its name imports, is thus a sort of inverted form of solution. Assuming the hypothesis advanced, it re mounts, step by step, till it has reached a source already explored. The reverse of this process, constitutes Syn thcsis, or Composition ; which is the mode usually em ployed for explaining the elements of science. Analysis, therefore presents the medium of invention ; while syn= thesis naturally directs the course of instruction.
fhe successive advances in geometry were assuredly not the result of chance ; they clearly evince the appli cation at all times of sonic kind, however imperfect, of anaLytical research. But the science had made very considerable progress, before the mode of proceeding was examined, and reduced to a form of simplicity and elegance. This capital improvement is ascribed to Plato, who thereby rendered analysis a most relined instrument of discovery. Socrates, by introducing the study of logic, had indeed led the way. The celebrated species of argument, which that sage so happily turned against the sophists, was only a sort of indirect demon stration, or rcductio ad absurdum, and quite analytical in its structure; but, setting out from false premises, it inevitably terminated in absurd or contradictory princi ples ; and thus most effectually exposed the dogmatism and captious subtleties of his antagonists.
Plato first employed the method of analysis, in search ing for a solution of the celebrated problem of the du plication of the cube. Another problem of the same order of difficulty—the trisection of an angle—soon fol lowed; and these questions, being found to surpass the powers of ordinary geometry, gave occasion to the extension of the science, to the discovery of the conic sections, and of the rudiments of the theory of curve lines. Mathematics were always respected, and eagerly
cultivated, in the groves of the academy ; the disciples of the Platonic school, who migrated to Sicily and Alex andria, continued to prosecute those studies with still greater ardour ; and the age of Archimedes and Apol lonius forms the most brilliant aria in the history of science. The various analytical researches of the Greek geometers, are justly esteemed models of simplicity, clearness, and unrivalled elegance ; and, though mise rably defaced and mutilated by the riot of time and barbarism, they will yet be regarded by every person capable of appreciating their merits, as some of the finest and happiest monuments of human ingenuity. It is a matter of deep regret, that algebra, or the modern analysis, from the facility of its operations and the cer tainty of its results, has contributed, especially on the continent, to vitiate the taste, and destroy the proper relish for the strictness and purity, so conspicuous in the ancient mode of demonstration. The study of geo metrical analysis appears admirably fitted to improve the intellect, by training it to habits of precision, ar rangement, and close application. If the taste so acquired be not allowed to gain undue possession of the mind, it may be transferred with eminent advantage to algebra, which needs reform in almost every part, and which has shot up hastily to maturity, without acquiring compact ness or symmetry of form.
To give some idea of the nature of geometrical ana lysis, we shall now select a few specimens of the most elementary kind. We mean afterwards, in a separate article, to treat the subject at considerable extent.