of ideas, is being constantly extended in all departments of human knowledge.

§ 274. Although there are few protracted controversies in regard to any truths which have been established by demonstrations, many of these truths are understood by few; and of those who understand them, some know them as truths which have been demonstrated, and generally admitted by mathematicians; others from having demonstrated them by original processes of their own; and others, from a study of the demonstrations of previous discoverers.

Among those not acquainted with the great systems of truth developed by mathematical reasoning, erroneous opinions in regard to the objects which they explain, are scarcely less frequent than on moral subjects. The knowledge of quantity in its manifold amounts, forms, positions. combinations, and divisions, constitutes a large and important portion of human ideas. Some have more of this knowledge, and others less, according to the skill and diligence with which they prosecute the attainment of it; the length of time during which their investigations are prosecuted; and the aids and facilities which they possess for prosecuting them successfully and rapidly.

§ 275. The capacity for pursuing mathematical reasonings belongs to all men who possess common sense; and is a department of the general faculty of reason, All who possess ordinary intellectual powers, may be mathematicians; and most may excel in the prosecution of mathematical studies, if they early engage in them, and earnestly pursue them for considerable periods of time. Ideas of quantity are common to all men, and all are accustomed to some degree of reasoning in respect to the objects of those ideas. Just in proportion to the extent and accuracy of their reasonings, is both the amount of their knowledge, and the development of their powers of acquiring it. Some become masters of Arithmetic, and acquire ability to reason correctly and successfully on all propositions which belong to this art; and to solve rapidly and accurately all its problems. Others extend their knowledge to Geometry and Algebra, and become masters of those sciences; and others still, go into the higher departments of mathematical reasoning, and explore the whole field of its ideas. This, however, is done by few, and is the work of a life.

§ 276. Those who have devoted a large portion of their lives to mathematical studies, have in some cases obtained a development of their powers of reasoning which has excited universal admiration. Euclid and Archimedes among the ancients, and Newton and La Place among the moderns, are examples of this. The accuracy and extent of their reasonings strike us with amazement, and give us new views of the majesty and might of the human intellect. The propositions which they demonstrated, and the problems which they resolved, are among the most glorious triumphs of reason, and have unlocked the most profound mysteries of creation and Providence. They are not only points of illumination, but fountains and sources from which flow out vast rivers of knowledge. Each of their great discoveries is the revelation of new fields of thought and inquiry, and of new departments of interesting and useful knowledge,

But these great men do not stand alone on the high places of the universe. They soared with an eagle's wing, and searched for the sublime and true with the steadfast gaze of the eagle's eye. They seized on points of observation, and developed sublime objects of reason which the human mind had never reached before; and became the high priests of their profession and benefactors of their race. But others, without number, have gone after them, and stand with them on those same exalted heights; fully equal to them in every thing excepting the honors of discovery. And the tides of discovery which commenced with them, are rolling on, and on; receiving fresh impulses from other minds, and leading to endless progressions of thought, and developments of truth without end.

§ 277. Mathematical knowledge is in the highest degree useful, by means of the numerous practical purposes which it answers; and the attainment of it is generally prosecuted with reference to those purposes. Men study Arithmetic that they may avail themselves of its principles and methods, in the various arts of civilized life, and in the various branches of industry. The same is true, to a great extent, of the higher branches of mathematics. They have their practical uses, which make them necessary to particular classes of men, and in the prosecution of particular branches cf business. Every branch of mathematics has been created to meet he exigencies of the human race, and to minister in some way to the promotion of human happiness. This

science is not therefore in any of its departments a merely speculative art, in which the mind operates without any reference to practical results; but is a means for the accomplishment of certain ends which are subservient to human happiness. In subserviency to these ends, and with a degree of devotion and intensity of interest proportioned to their importance, mathematical studies ought to be pursued. It is not possible for all to be profound mathematicians, neither is it necessary. The same principles apply to the exclusive prosecution of this science, and its affiliated arts, which apply to other sciences and other arts. It is not necessary for every man to be a skillful physician, or a profound divine, or an able lawyer. The great purposes of society are best answered by having a distribution of arts and sciences and professions to different persons, enabling each to be in a degree the servant of all, and all to be the servant of each; and thus binding the whole together by mutual kind offices, all of which are necessary both to the individual and to the whole community.

§ 278. It is often an occasion of regret that one individual cannot know every thing, and do every thing, and have all the springs of enjoyment in himself. But this is entirely foreign to the plan of the Creator; and under the existing system of things can never be attained. It is the plan of God to make men instruments of good to each other; and to make each, necessary to all; and all, necessary to each. In the carrying out of this plan, each individual has his limited faculties, and the whole are furnished with objects for the exercise of their powers, so numerous, so extensive, and so useful both to the individual and to others, that they cannot attain their greatest conceivable prosperity and hapJiness in any department of rational enjoyment, without the most extensive possible divisions of labors and offices, and the most extensive combinations of laborers. Just in proportion as labors are divided, and the entire powers of individuals concentrated on particular objects, will be their success in the prosecution of their objects; and in proportion to the. rumber of such laborers, and the variety of useful arts which they prosecute, will be the aggregate of happiness attained by the entire community.

A similar arrangement no doubt prevails in heaven. It is an essential condition of the greatest happiness of finite beings; and must therefore prevail in all worlds, and in all stages of the development both of virtue and happiness.

Instead of repining, therefore, at the limitation of our faculties, or indulging one feeling of regret that we cannot monopolize every thing, and be independent of other beings, we ought rather to rejoice that we are harnessed into the society of millions, and capable of participating jointly with other holy beings, in the production of one common stock of infinite and eternal happiness, the whole of which will be at once the happiness of each and of all.

§ 279. Besides the direct purposes of mathematical knowledge in assisting us to make proper estimates of things, the study of this branch of learning is highly useful as a means of mental discipline, and of the improvement of the mental faculties for general purposes. A course of study and instruction in Arithmetic, Algebra, Geometry, and other branches of Mathematics, produces increased powers of attention, discrimination, comparison, and judgment generally. This effect is so clear and decided, and so general, that hardly an instance can occur in which it is not apparent both to the student, and to others who are acquainted with him, and have opportunities of observing his intellectual powers and habits, before and after such courses of study. An observation of this fact has led to the introduction of mathematical studies into all systems of liberal education; and this department of learning is deemed to a considerable extent necessary, as a means of mental culture and discipline, independently of any practical purposes beyond this, to which it can be applied. The prosecution of this class of studies is useful as a means of intellectual improvement generally, but especially so, as a means of improving our powers of attention, discrimination, and judgment.

In order, however, to be of the greatest use as a means of intellectual improvement, mathematics ought to be studied in connection with other departments of learning. A mere mathematician is not likely to be a good general reasoner, or a powerful thinker, on general subjects. His habits of attention will be too exclusive, and his ranges of thought too much restricted, to answer in the best manner the purposes of general reasoning. But if mathematics are studied in connection with other and different branches of learning and philosophy, and a proportionable attention is devoted to each, the greatest and most symmetrical development of the mental faculties will be secured, and all the legitimate purposes of mental discipline be most effectually accomplished.

CHAPTER XX.

NATURE AND OBJECTS OF PHILOSOPHICAL REASONING.

§280. The second variety of synthetical reasoning is that which belongs to the category of reality, and which relates to things considered as causes and effects. It may appropriately be denominated Philosophical Reasoning, because it is the sole instrument of philosophical discoveries, and the foundation and source of every department of philosophical knowledge. Like mathematical reasoning, it is principally synthetical, and has for its primary object the discovery of truth; and subordinately to this the communication of it, by the method of its discovery.

Theory of Causes and Effects.

That

Ideas of causality and dependence are obtained at an early period of life, and are common to all men. We infer them directly from the sensations and consciousness, which constitute the first conditions of all our knowledge, and from all our subsequent ideas and affections. From sensations, we form ideas of something which experiences sensations; from thoughts, of some thing which thinks; and from resistence, of some thing which resists; and so on. which thinks is the cause of thought; that which resists is the cause of resistance; and so of all other known states and operations. Our primitive ideas of causes, therefore, relate to them as things which feel, think, resist, and sustain, the other states, and perform the other operations with which we become acquainted. Our perception of things as causes is derived from our ideas of things, as sensations, ideas, resistences, and other states and operations. We do not conceive of the being that thinks, as differing from thought in the manner that one entity or substance differs from another; but we conceive of the thinking being, as thinking; the feeling being, as feeling; and the resisting being, as resisting; and of the cause in all these cases, as having a permanent existence, and the phenomena as being states or effects of that existence. In the case of sensations of resistance; we infer from the sensation two causes, one