body by experience, and a subsequent process of abstraction. If we abstract from any body all the properties of the matter composing it, as its hardness, colour, weight, and so on, and retain merely its quality of extension in the three dimensions of length, breadth, and thickness, we have then a distinct conception of a geometrical solid, which possesses none of the properties of matter, except extension; and it therefore cannot possess a material existence. It is a different object from the space which it occupies; for in any limited portion of space, an indefinite number of such solids may exist, the one encompassing the other. Abstract now from any solid its thickness, and we then form the conception of a surface having only length and breadth. And if from a surface one of its dimensions be abstracted, as its breadth, we have then the idea of a line, which possesses only length, The intersection of two such lines is a point, which only marks position, and has neither length, breadth, nor thickness. It has been objected to this view of a mathematical point, that, as it has no magnitude, it can have no existence. It has certainly no material existence, but its existence is no less real on that account. Even a line or a surface occupies no portion of space. No number of points, however great, can fill any assignable portion of space, however small. And it has been remarked, that even a solid does not occupy exclusively any portion of space. They would exist independently of matter, but their existence is no less real, though immaterial. A system of geometry proceeds from simple, axiomatic, and incontrovertible principles, to the demonstration of new truths; and from the combination of truths previously known, new truths are continually evolved; and thus a system of geometrical science is established by a continued process of logical deduction. Some of the elementary truths in geometry are so obvious as to be almost self-evident; but many of them are of a different character, and are striking, and even beautiful, at least when the mind is habituated to contemplate abstract truth. Some propositions are, in some of their cases, axiomatic, but in other cases they require to be demonstrated. Without this distinction, the demonstrations would appear to be unnecessary; and, in such cases, they are perhaps more useful in completely obviating objections, than in producing conviction. UTILITY OF MATHEMATICS AS A MEANS OF EXTENDING OUR KNOWLEDGE OF THE MATERIAL WORLD. Our knowledge has been greatly extended by means of this science. Independently of the innumerable important and striking properties of magnitudes and relations of abstract quantities that it has made known, and which can be sufficiently appreciated only by the mathematician, it has unfolded a very extensive range of natural phenomena. It has investigated the principles of theoretical mechanics; the laws of the equilibrium and motion of fluids, fixed and elastic; the principles of vision, of electricity, and of magnetism; the theory of the propagation of sound and of light; and a variety of other subjects. But even the most abstruse branches, that appear to be incapable of any useful application, ought not to be neglected; for they may be applied at some future period, like the ancient doctrine of the conic sections, which for two thousand years was an object of mere curious speculation, till it became, in the hands of Newton, a most efficient means of unfolding the planetary motions. Without the aid of rules derived from this science, the navigator, relying only on his compass as a guide, could not with safety venture to any considerable distance on his element; intercourse with transmarine regions would be impossible; and, consequently, our knowledge of the globe which we inhabit would be very limited. We should probably still believe that its surface is an extended plane, and that it is supported on pillars; or, as was the opinion of some of the ancient philosophers, that its figure is cylindrical, like a drum. Without the aid of this science, our knowledge of celestial objects would be still more imperfect, and the consequences of our ignorance still more striking. We should still believe that these objects are equally distant from us, and, very probably, that they are distributed on the surface of an extensive crystalline sphere, performing a diurnal rotation about the earth, as the centre of the universe. We should also believe that some celestial phenomena, as eclipses and comets, are certain signs of a conflict of the elements of nature, or that 2 they are the portentous indications of the wrath of heaven, while contemplating to inflict, on superstitious mortals, some dire calamity, as war, famine, or pestilence. How different from these unsatisfactory and incoherent conjectures is that great achievement of this science-the clear and satisfactory exposition, on the most incontrovertible principles, of the complex though sublime and systematic mechanism of the heavens; by which the distances and magnitudes of the sun and planets have been measured, and also their weights, and even that of their satellites, ascertained; and by which the masses and distances of some of the stars or suns of other systems, though inconceivably remote, even in comparison with the great extent of our own system, will probably ere long be determined. PRACTICAL UTILITY OF MATHEMATICS. It will be unnecessary to make many observations on this subject, as the practical utility of mathematics is well known and indisputable. From its principles the rules of calculation and measurement are derived. It supplies the rules of the art of measuring distances, heights, surfaces, and solids, in artificers' work, guaging, land and marine surveying; it furnishes the principles of calculation in navigation, nautical and practical astronomy, of the arts of the optician and machinist, and also of the arts of carpentry and engineering, both civil and military. On its principles also depend the arts of planning, perspective, and of the construction of maps and charts. In short, whenever the construction of figures or computation is in requisition, the principles of mathematics are indispensable. EFFICIENCY OF MATHEMATICS AS AN INSTRUMENT OF MENTAL IMPROVEMENT. Considered as an instrument of intellectual improvement, it may be affirmed that mathematics cultivates chiefly the reasoning faculty. It also exercises the memory in a considerable degree; and it has a great tendency to form a habit of undivided and unremitting attention, which is indispensable for great success in any pursuit. Every branch in the theory of the science consists al 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 2 they are the portentous indications of the wrath of heaven, while contemplating to inflict, on superstitious mortals, some dire calamity, as war, famine, or pestilence. How different from these unsatisfactory and incoherent conjectures is that great achievement of this science-the clear and satisfactory exposition, on the most incontrovertible principles, of the complex though sublime and systematic mechanism of the heavens; by which the distances and magnitudes of the sun and planets have been measured, and also their weights, and even that of their satellites, ascertained; and by which the masses and distances of some of the stars or suns of other systems, though inconceivably remote, even in comparison with the great extent of our own system, will probably ere long be determined. PRACTICAL UTILITY OF MATHEMATICS. It will be unnecessary to make many observations on this subject, as the practical utility of mathematics is well known and indisputable. From its principles the rules of calculation and measurement are derived. It supplies the rules of the art of measuring distances, heights, surfaces, and solids, in artificers' work, guaging, land and marine surveying; it furnishes the principles of calculation in navigation, nautical and practical astronomy, of the arts of the optician and machinist, and also of the arts of carpentry and engineering, both civil and military. On its principles also depend the arts of planning, perspective, and of the construction of maps and charts. In short, whenever the construction of figures or computation is in requisition, the principles of mathematics are indispensable. EFFICIENCY OF MATHEMATICS AS AN INSTRUMENT OF MENTAL IMPROVEMENT. Considered as an instrument of intellectual improvement, it may be affirmed that mathematics cultivates chiefly the reasoning faculty. It also exercises the memory in a considerable degree; and it has a great tendency to form a habit of undivided and unremitting attention, which is indispensable for great success in any pursuit. Every branch in the theory of the science consists al |