mer confider quantity abftractedly, without any regard to matter, or particular bodies; the latter treat of quantity as fubfifting in bodies, and confequently they are intermixed with the confideration of phyfics, or experimental philofophy.

Pure Mathematics are Arithmetic, Algebra, Geometry, and Trigonometry: mixed confift chiefly. of Mechanics, Pneumatics, Hydroftatics, Optics, Aftronomy, and Navigation.

ARITHMETIC.

This art teaches the powers and properties of numbers, and the method of calculation with exactness and expedition. It is likewife defined the fcience of difcrete quantity, to diftinguish it from Geometry, which is by the logicians termed the science of continuous quantity. The four fundamental rules of Arithmetic are Addition, Subtraction, Multiplication, and Divifion. Confidering their univerfal application, it is a pleafing circumfrance, that thefe rules, which are the moft ufeful, are the eafieft of all. For purpofes of a commercial and feientific kind, many other rules have been contrived, which are modifications of the former, fuch as Practice, Vulgar and Decimal Fractions, the Extraction of the Square and Cube Roots, the Rules of Proportion, &c. &c. This art from its obvious utility commenced in the earlieft ftate of fociety, and received gradual improvements as commerce was advanced, and the fciences were cultivated.

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cultivated.

The prefent mode of computation, and the characters we ufe to denote numbers, were introduced into Europe, in the tenth century, by the Saracens, when they had poffeffion of Spain. The method of reckoning by tens was common to them with the Egyptians, Greeks, and Romans, and the practice moft probably took its rife from the obvious mode of counting by the fingers. The ftate of calculation feems. to afford fome criterion of the progrefs of civilization, for it has been remarked by travellers, that certain tribes of American favages, who can fcarcely count twenty, are funk in the loweft ignorance and barbarifm.

This art is cultivated with fufficient diligence by thofe, whofe occupations render it their intereft to be expert in the application of its rules. To the merchant and the banker it is indifpenfably neceffary, and to all ranks in fociety from the peafant to the peer it is highly ufeful. From the firft entrance of a young man into fociety, he will find the advantage of keeping fome account of what he receives and pays; as he may reap the double advantage of fecuring himfelf from the impofitions of the fraudulent, and from the folly of fpending more than he can afford. This practice may give a check, to his expenditure, fuggeft to him the propriety of not exceeding his income, and put it in his power to be generous as well as juft. Thus economy, produced by regularity of accounts, may be the parent of beneficence. Such a habit of

regularity

regularity is far from being inconfiftent with the liberal notions of a gentleman, although not reconcileable to the heedlefs extravagance of the fpendthrift. The obfervation of every day proves its utility in every fituation of life, and without the conftant comparison of receipts with payments, communities as well as individuals must eventually be involved in embarraffment. It can never be neglected except by thofe, who are determined to proceed blindly and inconfiderately to certain diftrefs and ruin.

Arithmetic is the measure of almost every thing valuable. It is not only the indispensable guardian of private accounts, but the guide to political knowledge, as to the population, revenues, balance of trade, coinage, and military power of nations. Dr. Johnfon well obferved, "that nothing amufes more harmlessly than computation, and nothing is more often applicable to real bufinefs or fpeculative inquiries. A thoufand ftories, which the ignorant hear and believe, die away when the computift takes them within his grafp. Numerical inquiries give entertainment in folitude by the prac tice, and reputation in public by the effect."

ALGEBRA

Is the method of calculating all kinds of quantities by means of general figns, that is, generally

Seward's Biog. vol. ii. p. 348.

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fpeaking,

fpeaking, by letters of the alphabet, which, for the fimplicity and diftinétnefs of their founds and characters, are much more commodious for this purpofe, than any other marks. In the infancy of fcience, numbers and things were expreffed by their names at full length; but afterwards thefe were abridged, and the initials of words were used inftead of them. Afterwards the letters of the alphabet were employed as the figns of all kinds of quantity, and ather marks were gradually introduced to exprefs all kinds of combinations, fo as to entitle this art to the diftinguishing name of univerfal Arithmetic, and the Arithmetic of figns.

Algebra, properly fpeaking, confifts of two parts: firft, the method of calculating inagnitudes or quantities, as reprefented by letters or other characters; and fecondly, the manner of applying thefe calculations in the folution of problems.

The first bufinefs of Algebra, as applied to the refolution of problems, is to tranflate the problem out of the common into the algebraic language, by expreffing all the conditions and quantities, both known and unknown, by, their proper characters arranged in an equation, or feveral equations if neceffary; and treating the unknown quantity, whether it be a number or a line, or any thing. elfe, in the fame way as if it were a known

Hutton's Dictionary, Saunderfon's Algebra, 2 vols, 4to. and Maclaurin,

one:

one: this forms the compofition. Then the refolution, or analytic part, is the difentangling the unknown quantity from the feveral others with which it is connected, fo as to retain it alone on one fide of the equation, while all the other, or known quantities, are collected on the other fide, and fo giving the value of the unknown one. And as this difentangling of the quantity fought is performed by the converfe of the operations, by which it is connected with the others, taking them always backwards in the contrary order, it hence becomes a fpecies of the analytic art, and is called the modern analysis, to diftinguish it from the antient analyfis, which chiefly related to Geometry.

The characters ufed in Algebra are,

more, as 3+4, is 3 added to 4.

lefs, as 4-3, is 3 taken from 4.

multiplied by, as 3x4, is 3 multiplied by 4.
divided by, as 3÷4, is 3 divided by 4.
equal to, as a=4, is a equal to 4.

An Equation is a fentence expreffed in algebraic characters, afferting the equality of two quantities. The propofition that S and 4 is equal to 12, is an equation, and is written thus 8+4=12.

An equation in any of whofe terms the first power of one unknown quantity only is contained, is called a fimple equation, or an equation of one dimenfion, as a+3=10.