tion, and a great variety by proportionally altering the given number and answer. A person incapable of doing this, or of proposing questions for himself, is equally incapable of receiving benefit from those proposed and not performed.

unmean

Perspicuity and conciseness, being the only qualities which render a work of this nature useful, have been the peculiar care of the author, and such a combination of both has been chosen as appeared most conducive to the instruction of the different capacities of youth by giv. ing explicit rules and difinitions, few and entertaining examples; as it is presumed that books designed for instruction, and to be useful, cannot be too plain if not too voluminous, nor too brief if not too obscure ; hence formality and logical precision has been sacrificed to convenience and utility: the whole being so designed as to leave that necsssary room for exercise of the scholar's genius, something inviting before him for conquest; lest from the bare performance of numerous ing and tedious operations, he might sink into absolute despondency, and acquire but a mere smattering of the science; since it is not the number of examples, with the performance of which he is fatigued, his alertness nor memory, that constitutes the mathematician, but a knowl edge drawn from the due observance of rules, and consideration of their use and propriety, combined with that judgment which this science is calculated to improve, rather than enlarge the memory. For sake of brevity long operations have been omitted, and the given and required numbers, with the essential parts of the example, signs, &c. arranged in due conformity to the rule. Also, to the arithmetical signs there have been some additions made, to render them more plain and definite, and to save the often repetition of long words. The number of different positions in which figures may be advantageously used in their application to common business, is endless; and to explain more of their varieties, than will lead the student into a useful knowledge of the science, or to give a multitude of examples while he is acquainted with but few rules, is exacting more time and expense than the acquirement deserves, in comparison with others.

It is not expected that the contents of this treatise will be fully comprehended by the same slight application with which Temple's Primer, or a trifling superficial knowledge of the simple rules, may be learned; for it is designed to contain nearly all the arithmetical knowl-' edge necessary in the common affairs of life. Though perhaps the same subjects here contained, may be learned in as short a time and with as much ease, as those contained in any other treatise of the kind. Yet obscurity on some particulars in a work of this nature is unavoida ble, and is, at the present state of improvement, inseparable from some of the subjects treated of; also error in the first impression of a mathe. matical work is equally unavoidable; and perhaps it may contain im proprieties, notwithstanding the careful attention which has been paid to remedy all the evils to which, from the nature of the science, it is subject; communications of which, or any improvement which may be suggested will be thankfully received.

With regard to the arrangement of rules, each instructor or pupil will follow his own taste; as it is of little importance what subject Nows he acquiren.ent of the four sin ple i uks and rule of three": "but

a systematic arrangement, such as would bring every subject into its proper channel, was designed; and the whole system of mathematics being but an application of the four simple rules of arithmetic, several subjects are introduced under those rules and the rule of three, which are by others treated of in a different manner; and such subjects as require an absolute combination of rules are introduced after those performed by a single rule, and each preceding or succeeding another, as it is supposed to be more or less useful. And with a view to facilitate the acquirement of arithmetic, and to bring more fully into use the most important subject it contains, (without violation of rule or system,) both integers and decimals, because they differ only in value, increase and decrease in the same ratio, &c. have been considered as belonging to the same rule. The contractions and application of rules were not designed for the strict attention of young scholars; but as they may be of utility to those who have acquired an adequate knowledge of the principle rules, it was thought expedient to insert them. Under numeration the tables of weight, measure, &c. with a variety of useful proportions, are introduced that the scholar might more easily become acquainted with them as they often occur throughout the work: under this head there are also several tables for finding the contents of divers kinds of figures, the specific gravity of. substances, &c. which would no where else so conveniently occur, with regard to utility. And under application of the simple rules is given a rule for easily extracting the roots of all powers, in lieu of that complicated extraction which has hitherto exacted so much time and labor. It is not expected that this rule will be very easily learned by a young scholar without an instructor. Proportion through the following treatise is supposed to be simple and direct, and a sufficient number of questions stated therein, which are usually termed inverse and compound, perhaps to render it satisfactory to the inquirer, that there can be no convenience experienced, but many disadvantages, from dividing proportion into so many different kinds as are by some thought necessary, and which only serve as so many needless embarrassments to the learner. Proportion is of such a nature that it cannot literally admit of a compound, (for, though the terms may be composed of several parts, it is an improportion until it be rendered simple ;) neither will it admit of being inverse, between a series of numbers, compared with another between which the proportion is direct; for, all the parts of any cause making the whole, must bear such proportion to its whole effect, as all the parts of another similar cause does to its whole effect, (or all its parts, conjunctly,) in every case. Duodecimals and all numbers whose division is not decimal, are in the following supposed to belong to the four compound rules, and under division a new method of dividing duodecimals, &c. is given. The knowledge of fractions, (so useful while our present system of weights, measures, &c. continue,) and that of the indispensable science of geometry being previously initiated in every convenient place, it was not deemed necessary to continue them to a great length as separate subjects; the design, of dispersing those subjects as above mentioned, is that persons perusing the following treatise may not easily as usual neglect those very important branches of mathematical knowledge, and to give to the rules that useful application so necessary to gain the attention of the fearn. r. Position, alliða

tion, progression, evolution, &c. are subjects of which a complete investigation, such as would sufficiently elucidate all their intricacies, to the perception of an ordinary capacity, would require the work of volumes; but from what is given, a sufficient knowledge, as it respects utility, may be acquired. Exchange, with different parts of the world, is very essential to the merchant and traveller; but under this head nothing more seems necessary, than the simple value of currencies, weights, &c. or the proportions between these and those to which they are to be reduced and a few examples; but since no two, of the best authors can be found to agree on this subject, it is very difficult to choose, and from the present instability of foreign monies, it is impossible to determine the true par of exchange, with all countries; there. fore a rule is given for ascertaining the value of all real coins, or of gold and silver, from their specific gravity. The touch on subjects forcign to arithmetic, may, in many instances, be found useful, in common business, as well as to inspire the industrious youth with a relish for further researches into the higher branches of mathematics. Where a rule in this treatise, on any material point, essentially deviates from that used by others, the common method is also inserted; that the scholar may be able to choose from the immediate contrast as it is designed not to obscure any light, heretofore thrown on the subject, nor leave any thing untouched, which has been previously handled, necessa ry to be noticed in arithmetic. Yet it must experience many inconveniences and local prejudices,on account of its wide, but supposed necessa ry, deviation from others; as all ranks of society have their aversions, prejudices and peculiarities; and as it is impossible to unbias the minds of those who have been accustomed to repose implicit confidence in the judgment of those from whom they received their elementary

tenets.

Arithmetic in its present state appears to be naturally divided into three parts. The first of which is numeration or the art or act of ascertaining and expressing any value or quantity in any form. The second, the art of performing operations with simple numbers. The third is arithmetic in compound numbers; which in process of time will be no more than nominal: for numbers must inevitably undergo great and valuable change; as the revolution, begun long before the introduction of the Arabic characters, which greatly advanced it, is not half completed; as may be readily seen from a review of the progress of arithmetic, and the present irregularity of our best numeral systems, on account of their many vulgar divisions of value and quantity, which are all we have to compute, with our Arabic decimals; and if the quantity partake of less than unity we must have recourse to parts irregular as the daily productions of chance, or be unable to ascertain the resulting value in known denominations. A perusal of the tables of weight, measure, &c. which we daily use, and a knowledge of decimals is all that is neces sary to convince every person of the inconvenience and impropriety of such divisions and of their more absurd different values bearing the same name, (as gallon, tun, second, &c.) The only divisions of any propriety now in use are such as increase in the same ratio; and those (if not decimal) falter and stumble at the integral point, where their disagreeable ratio is lost in decimal. The general reason why we ascertain the quantity is that its price may be determined; and until the 1

dimensions of all magnitudes or quantities be determined by a measure of length, whose division corresponds with that of the coin, in which the value of the quantities produced by a regular involution of the length, is given, and all ratios, between which there is any connexion, be correspondent, we shall be perpetually embarrassed with innumerable tedious reductions and intricate proportions. For this ratio, none can be compared with the decimal, to which are adapted, the characters by which we compute, all numbers from unity upward, and the American coin, which is (with others similarly divided) the best in the world; and which has, in our best American productions, been very much neglected, while the devoted labor of thousands has been be stowed on that, which is, only by pride and prepossession withheld from oblivion and which will, with all others, and, the hundreds of different and ill divided weights and measures, be lost in the aggrandizement of one complete standard of decimal coin, weight and measure; which will lessen the labor of acquirement in this science more than any other improvement it has ever received: the method of placing the decimal point being the only knowledge necessary, separate from that of performing operations in whole numbers.

A system as above mentioned of decimal division of quantity, which was calculated to explain the various operations of decimals, to show the propriety of corresponding divisions, &c. a sketch of arithmetical trigonometry, a demonstration of rules, and a few other subjects, have been omitted, as the size of the volume already exceeds what was de signed, and as the task of publishing has been unexpectedly lengthy and tedious, from the difficulty of correcting a work of this kind, and as the copy was better calculated for an octavo page, and therefore comprises more than usual in the same space; which may by some be thought an error, if so, it is hoped it will be a profitable one, and that the size, price or contents of the volume will not be dissatisfactory to its purchasers.

From a conviction of the inadequate knowledge contained in our present systems of arithmetic, and of the many disadvantages experienced therefrom, the author was prompted to the undertaking of the arduous task, which has been executed with a fidelity suited to its impor tance. And how far it may be deserving of that patronage and encour agement to which it is so liberally recommended by those gentlemen who have honored it with their names, for which the highest expressions of gratitude are tendered, and what part, it may contain, of that merit, which with unremitted diligence has been sought, is left to the candid decision of a liberal and impartial public.

Haverhill, Feb. 1811.