and authority in a tender and judicious manner, acts under a sense of the inspection, and with a view to the approbation of his Maker; is daily aspiring after immortality, and ripening fast for its joys; and having sustained his part with consistency to the closing scene of life, quits the stage with a modest and graceful dignity; this is the best, the wisest, and the happiest old man. Dodsley's Preceptor, vol. ii, p. 379, &c.

Therefore the whole of youth, manhood, and old age, which is spent in this manner, is the best and happiest life, -the genuine result of RIGHT reason.

They who thus conduct themselves are sensible that virtue is the best exercise and greatest improvement of their understandings, and constitutes the health, stength, and beauty of the mind. They are convinced that every deviation from this standard has a tendency to vice, misery, and folly; and that every advance towards it is an approach to wisdom, perfection, and happiness. The advantages, which such persons derive from logic in the improvement of their minds, and the regulation of their conduct, shows its most important use and entitles it to the highest praise.

CHAPTER II.

THE MATHEMATICS.

"NATURE, says Mr. Bonnycastle, bountiful and wise in all things, has provided us with an infinite variety of scenes, both for our instruction and entertainment; and, like a kind and indulgent parent, admits all her children to an equal participation of her blessings. But, as the modes, situations, and circumstances of life are various, so accident, habit, and education, have each their predominating influence, and give to every mind its particular bias. Where examples of excellence are wanting, the attempts to attain it are few; but eminence excites attention, and produces imitation. To raise the curiosity, and to awaken the listless and dormant powers of younger minds, we have only to point out to them a valuable acquisition, and the means of obtaining it. The active principles are immedi ately put into motion, and the certainty of the conquest is ensured from a determination to conquer. Of all

the sciences which serve to call forth this spirit of enter prise and inquiry, there is none more eminently useful than the mathematics. By an early attachment to these elegant and sublime studies we acquire a habit of reasoning, and an elevation of thought, which fixes the mind, and prepares it for every other pursuit. From a few simple axioms, and evident principles, we proceed gradually to the most general propositions, and remote analogies: deducing one truth from another, in a chain of argument well connected and logically pursued; which brings us at last, in the most satisfactory manner, to the conclusion, and serves as a general direction in all our inquiries after truth."

"And it is not only in this respect that mathematical learning is so highly valuable; it is likewise equally estimable for its practical utility. Almost all the works of art, and devices of man, have a dependence upon its principles, and are indebted to it for their origin and perfection. The culti vation of these admirable sciences is therefore a thing of the utmost importance, and ought to be considered as a principal part of every liberal and well regulated plan of education. They are the guide of our youth, the perfection of our reason, and the foundation of every great and noble undertaking."

Mathematics are calculated to produce effects highly beneficial to the mind. They make us fix our attention steadily upon the objects placed before us, and are therefore very properly recommended as the best remedy to cure an unsteady and volatile disposition. They teach us a method of clear and methodical reasoning, and coincide both in principles and rules with sound logic. They give a manly vigour to our understanding, and free us from doubt and uncertainty on the one hand, and credulity and rash presumption on the other. They incline us to a due assent conformable to the nature of things, and subject us to the government of strict reason. These studies are calculated to teach exactness and perspicuity in definition, connexion and conclusiveness in argument, carefulness in observation, patience in meditation; and from no exercises can the scholar go better prepared and disciplined to the pursuit of the higher branches of knowledge. The benefit to be derivad 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 pre-eminence 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.

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 arithemetic, as is the case with some tribes of American savages, who can scarcely reckon to twenty, are sunk in the lowest ignorance and

* 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.

barbarism. It is not only the indispensable instrument of private accounts and commerce, but it lays the only just foundation for political 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 simplicity of its principles and the elegance of its demonstrations. Arithemetic 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 where 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 triangle. 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 fluxions 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.

1. Mechanics is that science which treats of the motion and equilibrium of bodies. There are six simple instru ments, 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