Arithmetic, is a science which teaches the method of computing numbers, and explains their nature and peculiarities. At what time it was invented, is altogether unknown ; though the four first fundamental principles, viz. addition, subtraction, multiplication, and division, have always, in a certain degree, been practised by different nations.

The Greeks were among the first who brought arithmetic to perfection ; and they are supposed to have originally made use of pebbles in their calculations. The most complete method of numbering now used in this country, was introduced into Europe by the Arabians, when they were in possession' of Spain. These people, however, acknowledged that they derived their information from the Indians. How the latter became acquainted with it, we are entirely ignorant. The earliest treatises extant upon the theory of arithmetic, are, the 7th, 8th, and 9th books of Euclids Elements, in which he treats of proportion ; of prime and composite numbers. NIcoma-chus, the Pythagorean, also wrote concerning the distinctions and divisions of numbers into classes, as plain, solid, triangular, etc.; in which he explained some of the leading peculiar! ties of the several kinds. learning advanced in Europe, the knowledge of numbers also increased, and the writers on this subject soon became numerous. Ramus was the first who, in his Treatise on Arithmetic, published in 1586, used decimal periods, for reducing the square and cube roots to fractions ; but the greatest improvement which the art of computation ever received, was from the invention of logarithms, the honour of which is due to John Napier, Baron of Merchiston, in Scotland, who published his discovery about the beginning of the 17th century.

Arithmetic may now be consi-dered as having advanced to a degree of perfection which, in former times, could scarcely have been conceived, and to be one of those few sciences which have left little room for farther improvement. It is, however, a serious and almost general complaint, that few children, while at school, make any tolerable progress in arithmetic ; and that the generality, after having spent several years under the tuition of a master, are incapable of applying the few rules which they may have learned, to the useful purposes of life. A little reflection will suffice to convince us, that not much benefit is to be derived from the usual mode of instruction. A few elementary principles are acquired by role, and therefore quickly forgotten ; because the most essential particulars, viz. the reasons on which these rules are founded, and their extensive use in the various concerns of society, are generally omitted. Teachers, as well as writers, cannot be wholly exempted from the charge of having, in some degree, contributed to this evil; for, by stating the rules without their corresponding reasons, they act upon mechanical principles, and thus encourage the idea, that demonstrations in every instance are useless, and in some, impossible.

Every young arithmetician should remember, that before he forms any particular question or numerical proposition, it is absolutely ne--ary to consider whether the terms be directly proportionate to each other ; for otherwise he will be liable to commit gross errors. Although in buying and selling price increases or decreases in the same relative proportion as the quantity of goods, yet in geometry, natural philosophy, etc. those things which at first bight appear to be in simple proportion to each other, may, on a mature investigation, prove the contrary. Pre-viously, therefore, to the solution of questions respecting these sciences, he should be made acquainted with those elementary principles-on which they are founded.

Another material error committed in the interior schools, is the admission of boys under the age of ten or twelve, often for the sake of early fees, though they are incapable of being instructed by reasoning with them. Hence we are decidedly of opinion, that this is one of the negative, modern im-provements ; and that the earliest period of fixing the attention of youth on scientific objects, is, according to their individual capacities, from the twelfth to the fifteenth year of their age.

Among the latest, and most instructive works on this subject, we enumerate the following: - " An Introduction to Arithmetic and Algebra ;" by T. MaNNINg, two volumes, 8vo. 10s. boards. Riving-tons, 1793.—" Arithmetical Questions, on a New Plan;" by W. Butler, 8vo. edit. 2d. 4s. boards. Dilly, 1797. - "The Arithmeticiants Guide;" by W. Taylor, 12mo. 2s. fid. bound. Baldwin, 1788.

Arithmetic beside the elementary works formerly stated, the following will be found peculiarly serviceable, both to masters and pupils, namely, Mr.Charles Vyse's " Tutor's Guide, " 12mo. ; - his " Key to the Tutor's Guide, ' 12mo, - Mr. Ewing's "Institutes of Arithmetic;" - Mr. MoLINEUx's " Question Book, or Practical In-troduction to Arithmetic, " 12mo. - his " Scholars Question Book, " also in 12mo.; - Mr. Bonnycas-tle's " Introduction to Arithme-tic" 8vo.; - and, lastly, Mr. Keith's " Complete Practical Arithmetician, " etc. 12mo. 2d edit.