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most entirely of an uninterrupted process of reasoning; and as this process is identical in every subject, whether of necessary or contingent truth, no other study can be more conducive to the improvement of this faculty. A step of reasoning, or a syllogism, consists of a major and minor proposition, and a conclusion; and, by a law of our mental constitution, whether it be called judgment or the faculty of relative suggestion, the conclusion follows as a necessary consequence from these premises, in reasoning in any subject as well as in mathematics; so that reasoning is exactly of the same nature in the investigation both of necessary and contingent truth-with this difference, that in the former the chain of reasoning is of almost indefinite extent, and in the latter it is generally brief. There is, however, a difference in the fundamental principles. The premises in the former are incontrovertible, at least in pure mathematics, and generally in the other branches of this science; whereas, in subjects of contingent matter, the premises are generally only probable, and the probability of the conclusion must therefore be commensurate with that of the premises.
Synthetic Geometry, or the ordinary didactic method, affords, in the gradual exposition of geometrical truth, excellent specimens of the most clear and satisfactory reasoning; and that branch of it called Geometrical Analysis, affords, in addition, examples of the resolution of truth into its simple elementary principles. But analytical geometry, and the other analytical branches of the science, supply the best examples of the resolution of complex questions-a process which must be effected before the conditions can be comprised in symbolical expressions; they also accustom the mind to comprehensive views, and afford excellent specimens of subtle reasoning; and exercise the mind in the interpretation of the expressions of the final result. In these branches, a subordinate acquirement, made at the expense of much perseverance, is necessary; namely, the power of managing skilfully the concise and comprehensive algorithm employed in its researches, of which, however, that part of the operations that may be considered to be in some measure mechanical, will sometimes interrupt the chain of
reasoning, though in the theory the time thus spent by an expert analyst is comparatively small.
The application of the principles of the science to physical subjects, which constitutes the science of natural philosophy, in addition to the preceding kinds of intellectual exercise, affords examples of premises resting on probable evidence, and requires habits of close reflection and accurate observation, and also furnishes the finest specimens to be found in the whole range of human knowledge, of the methods of philosophical research, both inductive and deductive. In training the mind to such researches, it affords peculiar advantages; for although it is a subject of contingent matter, the rigorous nature of investigation operates as a salutary check against those fantastic speculations that result from the unrestrained excursions of the imaginative faculty, which, in original researches in other subjects, frequently produce extravagant theories, but which, from the unsettled state of the principles, may, with a little ingenuity, be made very plausible; whereas any such theory, in the former subject, would be certain to meet with speedy and complete refutation.
A knowledge of the methods of investigating necessary truth, is not inconsistent with a knowledge of the nature of moral evidence. An exclusive attention to any department of study, may to some extent disqualify the mind for appreciating truth in other departments. If the mere mathematician cannot appreciate minute degrees of moral evidence, neither can the mere student of probable truth appreciate the necessity of scientific rigour in mathematical science; and both might commit serious blunders in the department to which they are strangers; and the latter, if exclusively acquainted with those branches in which the premises are exceedingly doubtful, might, from the constant and bewildering uncertainty of his own conclusions, be liable to adopt a theory of universal scepticism. It is a truth readily assented to even by a mathematician, that of two contradictory propositions, that for which there is a preponderance of evidence, ought to be believed in preference to the other, although the amount of evidence fall far short of demon
A step of reasoning in mathematics is clear and satisfactory when once perceived, which is also the case in other subjects; for in them the vagueness or unsatisfactoriness accompanying any discussion properly conducted, originates, not in the reasoning, but in the uncertainty, and sometimes the multiplicity, of the principles involved. A distinction, however, must be made between difficulty and uncertainty; for they are not necessarily connected, at least if difficulty be estimated by the degree of exercise required of the higher faculties. The converse of this, however, that is, the union of difficulty with certainty of principles, is constantly experienced by the mathematician; for, such is the complexity arising from the multiplicity of the principles involved in some subjects, that, notwithstanding the certainty of its principles, and the perfection of its language, and the almost magical powers of the higher calculus, they have baffled the most resolute efforts of the most able and vigorous minds; and had its language been less perfect, there are many jects already thoroughly investigated, the difficulties of which would have been insurmountable.
It is an undoubted fact, that many men of reflecting minds have been devoted to mathematical study, which proves that there is an adaptation between it and minds of this complexion, or that it is fitted to afford their powers a sufficient exercise. Many celebrated mathematicians, too, have been very eminent for their attainments in general knowledge; in proof of which, it is merely necessary to mention the names of Eratosthenes, of almost universal acquirements -the learned Beda-the eloquent Pascal-Ramus, of uncommon acuteness and eloquence-Descartes-LeibnitzCondorcet-D'Alembert-Dr Clarke-Bishop HorsleyPlayfair and the all but universal Young; and the supe rior talents of many mathematicians, not so distinguished for varied attainments, is undeniable; as of Newton, Maclaurin, Lagrange, Laplace, and many others. In this science, too, there is great scope for the exercise of taste; for, since taste consists in the judicious selection of the fittest, and most agreeable, and most efficient means to accomplish an end, there must be an opportunity for its exercise in the discussion of scientific as well as of literary subjects; and
the qualities of unity, clearness, force, and elegance, thus belong to scientific, as well as to literary, composition.
Mathematics, like any other science, cannot afford information respecting the principles of other subjects; but it possesses this peculiar advantage, that every branch of science tends rapidly towards a state of perfection in proportion as it admits of mathematical investigation. There is still a difference of opinion, not regarding the truth, but regarding the self-evidence, of some of the fundamental principles even of geometry, and also to what extent those of theoretical mechanics are dependent upon experience; and to investigate these, or the axiomatic principles of any science, or to appreciate moral evidence, a habit of reflection and of reasoning is indispensable, and the judgment must be the final arbiter. Since the science of theoretical mathematics consists almost entirely of a continued chain of reasoning, it affords, in a given period of study, many more examples of this process than any other subject. A mind, therefore, disciplined, though not exclusively, by this invigorating study, and also improved by the study of other branches, will certainly be the best qualified for investigating either necessary or contingent truth.
In a pamphlet entitled "Thoughts on the Study of Mathematics, as a part of Liberal Education, by the Rev. William Whewell, Cambridge," we find the following passages on the value of mathematics as a means of mental discipline. "In reasoning, as in other arts, we are not masters of what we have to do, till we do it both well and unconsciously. Now, this advantage a judicious cultivation of mathematics supplies. It familiarises the student with the usual forms of inference, till they find a ready passage through his mind, while any thing which is fallacious and logically wrong, at once shocks his habits of thought, and is rejected. He is accustomed to a chain of deduction, where each link hangs upon the preceding; and thus he learns continuity of attention and coherency of thought. His notice is steadily fixed upon those circumstances only in the subject on which the demonstrativeness depends; and thus that mixture of various grounds of conviction, which is so common in other men's minds, is rigorously excluded from his. He knows
that all depends upon his first principles, and flows inevitably from them; that however far he may have travelled, he can at will go over any portion of his path, and satisfy himself that it is legitimate; and thus he acquires a just persuasion of the importance of principles, on the one hand, and, on the other, of the necessary and constant identity of the conclusions legitimately deduced from them."
HISTORY OF GEOMETRY.
The meaning of the term Geometry, in reference to its etymology, is the Measurement of the Earth. This was no doubt one of the original objects of the science of geometry; but its sphere soon became more widely extended, so that it is as applicable to the measurement of the heavens as to that of the earth.
Some of the elementary principles of geometry must have been known at a very remote period. Before a building of any considerable size could be erected, a plan of it must have been determined on, which could not be made without a knowledge of the simpler problems. This knowledge might have been attained, however, only by shrewd conjecture or tentative mechanical methods, instead of logical reasoning.
The opinions regarding the origin of geometry are various, but they concur in fixing it in Egypt. Some believe that it originated in the circumstance that the inundations of the Nile effaced the landmarks, and that it thus became necessary to assign annually to the proprietors their particular shares of land. The supposition of the obliteration of the landmarks, however, from this cause, is only an improbable conjecture. The historian Herodotus fixes its origin at the time when Sesostris intersected Egypt by numerous canals, and apportioned the country among the inhabitants; and Aristotle ascribes its origin to the Egyptian priests.
According to Plutarch, Thales of Miletus, who lived about six hundred years before the Christian era, measured the altitude of the pyramids of Egypt, and Amasis, the king, was astonished at his scientific skill; a fact which, if correct, proves the imperfect state of geometry at that time in Egypt. It is stated by Proclus that Thales could compute,