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 immediately 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 enterprise 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 cultivation 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 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." Chart and Scale of Truth, vol, i, p. 117.

Tatham's

"In

* Conduct of the Understanding, vol. i, p. 339. 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 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. 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