knowledge. The benefit to be derived from them is thus stated by Mr. Locke: "I have mentioned mathematics as a way to settle in the mind a habit of reasoning closely, and in train; not that I think it necessary that all men should be deep mathematicians; but that having got the way of reasoning, which that study necessarily brings the mind to, they might be able to transfer it to other parts of knowledge, as they shall have occasion."*

The greatest perspicuity is found to prevail in every part of these researches. By reasonings founded upon lines and figures represented to the eye, the clearest truths are conveyed to the understanding. In one

respect these studies claim the preeminence over all others; they reach the highest degree of evidence, by which a position is not only proved to be true, but the contrary position is reduced to an absurdity-This is demonstration.

"Such is the method of science, in which reason advances by a sublime intellectual motion from the simplest axioms to the most complicated speculation, and exhibits truth springing out of its first and purest elements, and rising from story to story in a most elegant progressive way into a luminous and extensive fabric. The certainty of self evidence attends it through every stage, and every link of the mathematical chain is of equal, that is, the utmost strength." Tatham's Chart and Scale of Truth, vol. i, p. 117.

* Conduct of the Understanding, vol. i, p. 339. "In geometria partem fatentur esse utilem teneris ætatibus : agitari namque animos, atque acui ingenia, et celeritatem percipiendi venire inde concedunt." Quint. lib. i, c. 10,

The name of mathematics was originally intended either to denote by way of eminence the high rank, which the sciences hold in the order of intellectual discipline, on account of their peculiar clearness and utility; or it was designed to convey an idea of their extent, as containing every kind of useful knowledge. According to their proper definition, they constitute the science of quantity, either as subject to measure or number. Their various branches are adapted to the common uses of life, and to the deepest and most abstract speculations. They are pure and mixed. The former consider quantity abstractedly, without any regard to matter, or particular bodies; the latter treat of quantity as subsisting in bodies, and consequently they are intermixed with the consideration of physics, or experimental philosophy.

Pure mathematics are Arithmetic, Algebra, Geometry, and fluxions: mixed consist chiefly of Mechanics, Pneumatics, Hydrostatics, Optics, and astronomy.

1. The experience of every day proves the utility. of the art, which teaches the properties of numbers, and the method of employing them in all calculations with ease and expedition. The nations, which want arithmetic, as is the case with some tribes of American savages, who can scarcely reckon to twenty, are sunk in the lowest ignorance and barbarism. It is not only the indispensable instrument of private accounts and commerce, but it lays the only just foundation for po litical knowledge, as to the population, revenues, balance of trade, coinage, and military power of nations.

2. Algebra is an Arabic word; and is that peculiar kind of calculation, in which the known as well as the

unknown quantities are expressed by the letters of the alphabet. It is the art of computing by symbols. Algebra is one of the most important and useful branches of pure mathematics, and may be justly considered the key to all the rest./ Geometry delights us by the sim-f plicity of its principles, and the elegance of its demon strations. Arithmetic is confined in its object, and partial in its application. But algebra, or the analytic art, is general and comprehensive, and may be applied with success in all cases were truth is to be obtained, and proper data can be established.

To trace this science to its origin, and to point out the various alterations and improvements which it has received, would exceed the limits of this work. It is of the highest antiquity, and has obtained the praise of all ages. The Greeks were acquainted with it, and applied it to the solution of certain curious and difficult problems; but it is to the moderns that we are principally indebted for the improvements of the art, and its great and extensive usefulness in every abstruse inquiry.

Algebra ought to be learned before geometry, because it facilitates the study of geometry; but geometry does not facilitate the study of algebra.

3. Geometry, whether derived from the Egyptians, or the Greeks, was originally, as its name denotes, the art of measuring the earth, or any distances or dimensions within it. In its present acceptation, it signifies the science of magnitude in general. Its application to the use and ornament of mankind is very important and extensive. Furnished with this assistance, geographers are enabled to ascertain the magnitude of the terraqueous globe, the extent of oceans, and the various divisions of the earth. Hence architects derive their

just measures and proportions for the construction of all kinds of buildings. By its assistance likewise surveyors measure land, and delineate the plans of towns. Hence fortification derives its strength, security, and systematic regularity, in the erection of forts, batteries, and all other military works; and hence the general is best enabled to draw the lines of regular encampments, or arrange his army in the most advantageous order of battle. From geometry is acquired an exact knowledge of perspective, and accuracy is given to maps and charts.

Trigonometry is a part of geometry, and is the art of finding the dimensions of the sides and angles of a trian- gle. It supplies fundamental rules for ascertaining every degree of distance and altitude. Without its aid, the magnitude of the earth, and the heavenly bodies, their distances, motions, and eclipses, would be utterly unknown. Its assistance is necessary to dialling, geography, navigation, and astronomy.

4. The most extensive, ingenious, and subtile of all the branches of pure mathematics are fluxions, which were entirely unknown to the ancients. They were invented by Sir Isaac Newton, one of the greatest mathematicians and philosophers that any age or nation has produced. Newton and Leibnitz contended for the honour of the invention, and it is probable that they both had made some progress in this new science before either knew what the other had done.

By means of fluxions we can resolve the most abtruse problems in pure and mixed mathematics. Since this noble invention some of the grand phenomena of the universe have been explained, and mechanical philosophy has attained a degree of perfection which algebra and geometry were not able to accomplish.

This doctrine has been applied by mathematicians to a variety of useful and important objects. The uses of fluxions are so many and so various that we cannot enumerate them in this work.

The doctrine of fluctions is founded upon this principle, that all magnitudes or quantities are supposed to be generated by motion. Thus, a line is supposed to be generated by the motion of a point, a surface by the motion of a line, and a solid by the motion of a surface. Algebra and geometry lend their aid to this sublime science.

Mixed mathematics, which constitute the pleasing and instructive branches of experimental philosophy; are next to be considered.

I. Mechanics is that science which treats of the motion and equilibrium of bodies. There are six simple instruments, which are called the mechanical powers; and by their combination, all machines, however complicated, are constructed. Their names are— the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. However small the strength of man, considered in itself, may appear, his ingenuity has supplied him with the means of remedying its defects: by the friendly aid of the mechanical powers he is enabled to conquer the obstacles, which are opposed to him; to subdue, or to arm himself with the elements; and to make air, water, and fire, subservient to the purposes of his necessity, or his ornament. Skill in mechanics constitutes the great distinction between savage and civilized life, whether we consider their application to minute or to great objects, as aiding the ingenious artist in the construction of a clock or a watch, or as assisting in driving down

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