ful instrument, after the growth of true benevolence, in the progressive improvement of the human race. MATHEMATICAL REASONING. Mathematical reasoning, as in fact all other, is divided into two great and opposite methods of demonstration—the analytical and inductive. The one would prove the principles of a machine by taking it to pieces and examining its parts; the other by puting those parts together. At the head of one stands algebra, and of the other geometry. Algebra assumes the conditions of a proposition as it is, and, analyzing it, arrives at its elements. Geometry takes those elements, and, putting them together, step by step deduces a conclusion which cannot be resisted. And these two methods comprehend in general all the varieties of demonstration-moral or physical-which human wisdom has devised, from the philosophers of the academy to those of the institute. Any other than these appeals not to reason, but to the fallible testimony of the senses. In fact, all treatises. upon logic teach nothing, except terms, which may not be found in the elementary propositions of geometry; and when the youth, who in his collegiate course has mastered the mathematics, comes at the close of it to peruse some book of logic, he smiles with contempt at what appears to him an inferior method of reasoning. What form of syllogismthe sophism excepted-has he not found in Euclid? : The mode of reasoning, and the things reasoned about, which give no result but the exactness of truth, claim a superiority for this system over every other. Observation deceives; consciousness itself errs; but demonstration never. This method of investigation, however, though in general applied only to physical objects, may be transferred to any upon which the mind can be employed. It was the opinion of Mr. Locke that moral as well as mathematical science may be reduced to a demonstration. The improvements in moral investigation seem fast leading to this result; and Mr. Locke, like some other great minds, has, I believe, published a truth which posterity may see accomplished though we may not.* If it be true then that mathematics include a perfect system of reasoning, whose premises are self * What is a demonstration but a series of connected truths with a conclusion drawn from them? Now these truths may be derived from any source, and may be exhibited in any form, provided the connection is kept up and the conclusion clearly drawn. Mathematics assumes truths drawn from the relations of figure and extension; natural philosophy those drawn from experiment. Moral science deduces its conclusions from testimony and consciousness. The demonstration in either case is the same. But as the facts of mathematics are at once obvious to the senses and incapable of denial, the demonstrations are the most perfect. For this very reason, they furnish the best means of studying logic. If a man wished to learn the art of engraving, would he go to the worst engraver in the land? No to the best. Then shall he not learn reasoning in its best form? evident, and whose conclusions are irresistible, an there be any branch of science or knowledge better adapted to the discipline and improvement of the understanding? It is in this capacity, as a strong and natural adjunct and instrument of reason, that this science becomes the fit subject of education with all conditions of society-with all conditions of society, whatever may be their ultimate pursuits. Most sciences, as indeed most branches of knowledge, address themselves to some particular tastes or subsequent avocations; but this, while it is before all as a useful attainment, especially adapts itself to the cultivation and improvement of the thinking faculty, alike necessary to all who would be governed by reason or live for usefulness. HE ORDER OF REASONING. But, by teaching geometry first* and algebra subsequently, an inversion of the usual order, these sciences prevent the very method by which the human mind in its progress from childhood to age develops its faculties. What first meets the observation of a child? Upon what are his earliest investigations. employed? Next to color, which exists only to the sight, figure, extension, dimension, are the first objects which he meets and the first which he examines. * This idea is derived from a discourse delivered by Mr. Grund, a teacher of mathematics in Boston. He ascertains and acknowledges their e.istence; then he perceives plurality, and begins to enumerate ; finally, he begins to draw conclusions from the facts to the whole, and makes a law from the individuals to the species. Thus he has obtained figure, extension, dimension, enumeration, and generalization. This is the teaching of nature; and hence, when this process becomes embodied in a perfect system, as it is in geometry, that system becomes the easiest and most natural means of strengthening the mind in its early progress through the fields of knowledge. Long after the child has thus begun to generalize and deduce laws, he notices objects and events, whose exterior relations afford him no conclusion upon the subject of his contemplation. Machinery is in motion-effects are produced. He is surprised -examines and inquires. Analysis is begun, and he reasons backward from effect to cause. This is algebra, the metaphysics of mathematics, and the second step in the order of nature; and through all its varieties, from arithmetic to the integral calculus, it furnishes a grand armory of weapons for acute philosophical investigation. But algebra advances one step further; by its peculiar notation it exercises, in the highest degree, the faculty of abstraction, which, whether morally or intellectually considered, is always connected with the loftiest efforts of the mind. Thus this science when taught subsequently t geometry, comes in to assist the faculties in their progress to the ultimate stages of reasoning; and the more these analytical processes are cultivated, the more the mind looks in upon itself, estimates justly and directs rightly those vast powers which are to buoy it up in an eternity of future being. The minds of nations, as well as of individuals, have pursued the same order; generations have their infancy and age, and the great public mind of the world has cultivated its understanding and aggregated its knowledge by the same processes which are natural and necessary to individuals. Thus, the philosophers of ancient Greece perfected plain geometry, and Euclid is still a text-book in modern schools.* But not so with analysis; the Greeks knew not the numerals,† and their whole arithmetic was exceedingly imperfect, while in algebra they were but beginners, having scarcely advanced beyond equations * Geometry, like most other sciences, is supposed to have had its origin among the Chaldeans, or Egyptians. But however that may have been, their knowledge upon the subject must have been slight; for it was Pythagoras, in the year five hundred and ninety before Christ, who discovered the fundamental proposition that the square of the hypothenuse is equal to the sum of the squares of the other two sides. Euclid appeared in the year three hundred B. C. His object was to systematize the scattered discoveries in science, and clothe them in the strictest form of reasoning; and he did it with such success, that no book of science ever attained the duration and celebrity of Euclid's elements. They were for many centuries taught exclusively in many schools, and translated and commented upon in all languages.- Vide Bossuet's History of Mathematics. The Greeks, and all other ancient nations, used letters and other characters for arithmetical operations, but of arithmetic itself they knew very little. |