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 com pound, (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 be tween 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 learn. r. Position, allia

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 arithme. tic. 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 Sir, respectfully yours, JOSEPH DANA. Newburyport, Feb. 2, 1811.

THE Subscribers having examined in manuscript a new system of Arithmetic, entitled THE COLUMBIAN ARITHMETICIAN, are satisfied that it is better calculated than any other system extant to facili tate the pogress of youth in the science of numbers, The ease, perspicuity and correctness in elucidating the fundamental rules-the il.. lustrations 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.
LEONARD WHITE.

JOSHUA DODGE.

WILLIAM BATCHELDER.
JONATHAN C. HILL.

Haverhill, Feb. 15, 1811.

STEPHEN MINOT.
JOHN VARNUM.
JOSEPH HOVEY.
SAMUEL MERRILL.
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. ASAHEL HUNTINGTON, A. M. Topsfield.
Rev. MOSES DOW, A. м. Beverly.

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 wherein 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 fally 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 man. The 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; botli of whom equally feel the many disadvantages of a great variety of examples proposed, to the exclusion of more necessary rules; since the higher branches of mathematics, 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