HAVING perused in manuscript the "Columbian Arithmetician," we esteem it a work of much ingenuity and originality, well calculated for the improvement of youth in the science of arithmetic. Its consistent perspicuity, brevity and variety of matter, will no doubt render it useful. We would therefore cheerfully recommend it to public patronage.. JOSHUA JEWETT. Rowley, Aug. 28, 1810. SAMUEL ADAMS. SAMUEL JACKSON. Extract of a letter from Joseph Dana, Esq. Preceptor of the Newburyport Academy. DEAR SIR, I HAVE examined the COLUMBIAN ARITHMETICIAN, with as much attention as I have been able, and am well satisfied that it contains a greater variety of important matter than any work of the kind with which I have been acquainted: and have no doubt but that it will be found highly interesting and useful to instructors and to proficients in Arithmetic, and that it will reflect honor on the author, and our country. I am Şir, respectfully yours, JOSEPH DANA. Newburyport, Feb. 2, 1811. THE Subscribers having examined in manuscript a new system of Arithmetic, entitled THE COLUMBIAN ARITHMETICIAN, are sat isfied that it is better calculated than any other system extant to facili. tate, the progress of youth in the science of numbers, The ease, per spicuity and correctness in elucidating the fundamental rules-the illustrations in proportion, and the new method of extracting roots---The utility of the Tables of Exchange and the exemplifications of the science of Geometry, discover, not only great improvements upon former treatises, but contain a variety of ingenious and original matter; and we can recommend it to be peculiarly useful in Schools and Academies, and highly valuable to the community, and sincerely wish the author that approbation and encouragement so justly due to his talents and industry. BAILEY BARTLETT. JOSHUA DODGE. WILLIAM BATCHELDER. Haverhill, Feb. 15, 1811. STEPHEN MINOT. JAMES C. MERKILL. THE following gentlemen, among others, whose letters being lengthy we have not sufficient room to insert, have perused the Columbian Arithmetician, and given recommendations similar to the above. Rev. ISAAC BRAMAN, A. M. Rowley. Rev. JONATHAN ALLEN, A, M. Bradford. Rev. JOHN SMITH, A. M. Salem. N. H. PREFACE. AS perfection in any art or science can never be attained, there is always room for improvement. And notwithstanding the valuable improvements, which have been recently made in the science of Arithmetic, the author of this treatise, after careful examination of the most celebrated publications of the kind, was thoroughly convinced that the progression might still advance with increasing light and utility; which is the only apology he has to offer for this humble attempt to promote the more general knowledge of so useful a science. Some may suppose, taking into consideration the diverse characters employed and the numerous treatises written, that the subject is now exhausted, and inaccessible to further improvement; but numbers are the only perfect instruments we can handle, and the only means of ascertaining the true effect of any given cause, and until this can be done in the best possible manner, and no difficulty or obscurity attend any arithmetical operation, improvement awaits the science. Resting assured that but few will doubt the possibility or necessity of improvement in every art and science, and that no amendment, however small it may appear, can, in a land which has hitherto been so highly favorable to growing science, be ungratefully viewed, shall with diffidence proceed to point out some particulars, to which it has been the general aim of the author to conform, and where in it appears improvements have been, and may continue to be made of which, if there be none found in this, it is confessed to be, not for want of pains taken, or condition of the science to receive. ner. There are two capacities which arithmetic has particularly to address, in point of utility; the first, is the youth, just beginning to count, whose abilities are insufficient fully to comprehend a rule, (from which in a short time he might receive singular advantages,) and whose progress can be promoted only by his instructor and by examples wrought out, or the different parts of the operation placed in a conspicuous manThe second, is a person who has made some proficiency in the study, and would inquire into the various parts of the operation, apply it in practice, or review any part of his acquirement without an instructor; all which he will best perform by an apt rule, and still receive benefit, or at least no inconvenience from examples wrought out in the most simple manner. The former needs a variety of subjects, from which he may choose, and the latter, the same, to rouse his attention, and stimulate him to application; both of whom equally feel the many disadvantages of a great variety of examples proposed, to the exclu sion of more necessary rules; since the higher branches of mathe. matics, are but an application of arithmetic, and each succeeding rule but practice on the foregoing. But if more examples than were thought sufficient, be deemed necessary, the scholar may very easily double the number, having the answer to each given, by once inverting each ques 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. : 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 unmeaning and tedious operations, he might sink into absolute despondency, and acquire but a me.e 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 knowledge 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 applica tion 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 knowledge 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 mathematical 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 follows the acquirement of the four simple rules 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 mathemat. ics being but an application of the four simple rules of arithmetic, sev. eral 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 propertion 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 trea tise 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 learner. Position, alliga tion, progression, evolution, &c. are subjects of which a complete in vestigation, 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; therefore 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 foreign to arithmetic, may, in many instances, be found useful, in common business, as well as to inspire the industrious youth with a esh 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 de signed 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 necessary, 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. The third is 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. arithmetic in compound numbers; which in process of time will be no more than nominal: for numbers must inevitably undergo great and val uable 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 clance, 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 |