There is, besides, a more absolute precision in all the forms and in the language of geometrical disquisition. Pure Geotnetry is always precise and logical; it carries on its demonstrations by the exact comparison of ideas, adhering to the constant use of terms, the meanings of which are always verified by a reference to accurate definitions. Its reasonings proceed by means of syllogisms, in which, for the sake of brevity, the minor proposition is suppressed. But even in the proofs of those theorems of Algebra, in which little depends upon the employment of its peculiar symbols, the reasoning is seldom close and exact. The absurdities which have been published with a view of explaining the rule for Algebraic multiplication; the common method of shewing that the numerator and denominator of a fraction in its lowest terms are measures of the numerator and denominator, respectively, of every other equal fraction; the imperfect state in which the proof of the rule for finding the greatest common measure of two complex algebraic quantities, has been left by most elementary writers, as if they could only be accurate as far as Euclid is accurate, from whom they have copied, but who did not contemplate the nature of quantities expressed algebraically; the defects in the demonstration of the binomial theorem ; and many more examples might be adduced in support of the assertion made above. Nay, it is well known, that some propositions of the greatest importance in Algebra have never yet received a 1 satisfactory proof: and although mere metaphysical objections ought not to stop the progress of any science, it is time that these faults were remedied: the most eminent writers in this department, however, always appear to be in haste to quit the province of severe reasoning, and to exhibit their skill in the management of symbols. Thus it appears, that even when the same method is used in both, Geometry affords a better exercise, than Algebra, for the mental powers. That it exercises, without oppressing and fatiguing them, will scarcely be denied by any man, of even middling abilities, from his own experience. Different individuals may, indeed, find it more or less difficult to retain, and to recollect, the proofs of a long series of geometrical propositions; but fully to comprehend these proofs, at the time when they are considered, to * "Quoique les vérités mathématiques soient toutes d'une certitude parfaite, elles n'ont pas toutes le même degré d'évidence ce sont sur-tout les notions premières qu'il est difficile de porter au point de clarté desirable; mais on seroit arrêté dans la carrière, dès les premiers pas, si parce que certains principes fondamentaux restent enveloppés de quelque obscurité, on refusoit d'aller plus avant. Ainsi les anciens, parce qu'ils n'avoient pu parvenir à éclaircir entièrement la théorie des parallèles, n'ont point été pour cela retardés dans leurs recherches. Ainsi, quoique la notion de l'infini présentât aux modernes des difficultés, ils n'ont pas laissé de donner à l'analyse infinitésimale tout le développement qu'on pouvoit attendre de leur sagacité. Ainsi, enfin, l'obscurité dans laquelle est restée la notion des quantités négatives n'a nullement entravé la marche des algébristes." Carnot, Geom. de posit. p. 481. perceive the concatenation which binds the parts of the series together, and thus indirectly to gain all the essence of a system of logic, without the tediousness of its technical terms and rules, requires nothing more than common sense and sedulous application. What has been hitherto said, upon this subject of comparison, relates chiefly to the progress of the student in making himself master of the discoveries of other men. But it is not only in reading and digesting what has been written upon the mathematics, that his mind is disciplined; another most important employment of his faculties consists in the application of knowledge so acquired to cases which are new to him. Now the questions proposed to a learner to be answered algebraically, as a trial of his skill and talents, are usually of such a kind as not to demand any extraordinary exertion of his reason. He is not called upon to attempt the investigation of new and recondite theorems. The difficulty presented to him, is seldom more than the mere translation of the conditions of the question, into a language, the peculiarity of which is, that it is so concise as to exhibit several propositions in a small compass. This having once been effected, and it is seldom an arduous task to perform, the attention is then withdrawn from the things signified, and confined to the signs and from performing the mere operations of Algebra, it will scarcely be contended that any improvement of the reasoning faculties is to be derived. But the exercise of the understanding 1 is of a very different kind, when it is occupied in the solution of a geometrical problem. Whether it proceeds, strictly speaking, analytically, or whether it makes a particular construction, in the way of trial and conjecture, and then pursues the consequences of it, until they either end in the attainment of that which was proposed, or else indicate that some other method must be had recourse to, its faculties of judging, recollecting and inventing are continually exercised. Algebra is, doubtless, the more powerful and convenient instrument for use. "Idem omnino mihi," says EULER, "cum Newtoni Principia et Hermanni Phoronomia perlustare cœpissem, usu venit, ut quamvis plurium problematum solutiones satis percepisse mihi viderer, tamen parum tantum discrepantia problemata resolvere non potuerim." But the same causes, which give analytics their superiority in that respect, prevent them from being so valuable, considered as a mental discipline. The great praise, it may be further remarked, which has been bestowed upon the Mathematics as conducing to strengthen the mind, has proceeded from men, who lived when Geometry constituted the principal part of them: and those who have lately denied them this merit, seem to have been biassed in their estimate by a partiality for extended analytics. If the view which has here been taken of this subject be just, it should seem to be no disservice to our established system of education, to afford scope for the efforts of our junior students in an easy extension of those rudiments of knowledge which they learn from Euclid. It is impossible for them to enter upon a more fertile field than that of Geometry, which really seems to admit of the exercise of as much genius and invention as poetry itself: and after having thus strengthened their faculties, and accustomed themselves to the comparison of clear ideas, they will proceed with better success to the remaining part of their academical course. A permanent taste for the Mathematics will thus be formed, and a study, which is now too frequently thrown aside as soon as it has answered a temporary purpose, will become a valuable resource to amuse and adorn their future leisure. |