If a jurist, for example, wishes to explain the legal descent of property through different branches of a family, to one not familiar with the subject, he does not state the principle in an abstract form, using all the technical terms of his profession, which would be precisely analogous to the common method of teaching numbers; but he immediately presses the whole company into his service. He supposes this man, a son; another, a brother; a third, a grandson, and so on, till he has represented every branch of the family by some present person; and then in connexion with these persons, he illustrates his principle clearly, and the supposed family resume their former relations, with a perfect knowledge of the abstract principle to be communicated.

This method of communicating an abstract principle, is just as applicable to elementary arithmetick, as any other subject; and it is even more important in this application, than any other. No processes of reasoning for children are more complicated, than those of arithmetick. There are none, where the young mind requires more helps, to enable it to pursue its course, without distraction or interruption. Is it not because this facility in communicating knowledge has not been applied to arithmetick, that that study has been pronounced intrinsically difficult, and far beyond the capacity of young learners? And with this impression, the whole subject has been wrapt in mystery as unintelligible, as the hieroglyphicks of the Egyptian Magi.

This improvement will ultimately give to arithmetick its proper rank and dignity, among elementary studies. The third stage or abstraction, properly so called, loses sight altogether of any particular objects, and the mind reasons upon the numbers alone. This is where all have, heretofore, taken up the subject. And the evils of the method have been long and severely felt.

The third characteristick, which I proposed to examine, it will be recollected, is the following:

All those rules, which are merely artificial, and those formed for particular applications of the same general principle, have been discarded. Such rules make the largest, and by far the most difficult part of the common systems of arithmetick. Any arrangement, therefore, by which they disappear in form and name, will seem, to superficial observers, to change the identity of the subject. But such may be assured, that notwithstanding the great transformation in the looks of arithmetick, the whole subject remains.

The arrangement of the system by analysis and induction is according to principles, and not according to subjects, as in other books.*

A thorough knowledge of general principles, and the habit of analyzing, which this system is so em"In tracing

* Mr. Colburn shall state his own arrangement. the principles, several distinctions have been made, which have not generally been made. They are principally in division of whole numbers, and in division of whole numbers by fractions,

inently calculated to give, will better prepare the mind for the examples which occur in life, than the multiplication of complicated rules, so nearly alike, that much discrimination is required to distinguish them. The rules of Barter, Loss and Gain, Fellowship, Equation of Payments, and Alligation are not recognised by name, according to this arrangement. But the principles necessary to the solution

and fractions by fractions. There are some instances also of combinations being classed together, which others have kept sep

arate.

"As the purpose is to give the learner a knowledge of the principles, it is necessary to have the variety of examples under each principle as great as possible. The usual method of arrangement, according to subjects, has been on this acount entirely rejected, and the arrangement has been made according to principles. Many different subjects come under the same principle; and different parts of the same subject frequently come under different principles. When the principles are well understood, very few subjects will require a particular rule, and if the pupil is properly introduced to them, he will understand them better without a rule than with one. Besides, he will be better prepared for the cases which occur in business, as he will be obliged to meet them there without a name. The different subjects, as they are generally arranged, often embarrass the learner. When he meets with a name with which he is not acquainted, and a rule attached to it, he is frequently at a loss, when if he saw the example without the name, he would not hesitate at all.

"The manner of performing examples will appear new to many, but it will be found much more agreeable to the practice of men of business, and men of science generally, than those commonly found in books. This is the method of those that understand the subject. The others were invented as a substitute for understanding." [Sequel, preface, pp. vii. and viii.]

of questions, usually put under these rules, are fully illustrated. The learner finds himself solving all such questions, with the utmost facility, in the most philosophical manner, without even knowing, that such rules exist by name. I subjoin a few examples of questions, solyed by the complicated and artificial rules of the most popular books, and then add the same questions solved by analysis. Readers shall then judge, which method is most expeditious and philosophical.*

* I select examples from an "Arithmetick, by Daniel Adams, M. B." because they are fair specimens of the common method, and because this book has been more generally adopted in New-England, and more widely circulated over the whole continent, than any other. Between 1802 and 1815, "it had been through nine editions, and more than 40,000 copies of it had been circulated." At that time, it was stereotyped, and I fear my knowledge of numeration would not enable me to state the numbers, with which the publick have since been afflicted. This book owes its popularity precisely to that trait in its character, which ought, in the onset, to have condemned it to oblivion. It degrades the whole science of arithmetick to a mechanical art. As a discipline to the mind of the learner, therefore, it is useless, and worse than useless. It calls into exercise no power of the mind, but memory, and requires the practice of no virtues but faith and patience faith to believe all that is stated, for nothing is analyzed or proved; and patience to labour so long in the dark, without ever understanding, or coming to the light. The principles of the science, the development of which constitutes the chief excellence of any system, cannot be disentangled by the learner, from the useless forms, in which they are involved. The pupil never thinks of any thing but doing his "sums," and getting through the book. It is difficult to dwell with any complacency or patience on the fact, that so many of these books

A man bought 12 cords of wood at 3 dollars per cord, and paid for it with flour at 6 dollars per barrel. How many barrels of flour did he give?

It seems necessary to beg my readers not to spoil my illustration, by solving the above problem at once, before they have been taught to do it by rule. In the first place, this question belongs to a rule called "Barter," and it is proper they should commit to memory a definition of Barter. Here it is. "Barter is exchanging one commodity for another, and teaches merchants" (I suppose no one else has a right to know it) "so to proportion their quantities that neither shall sustain loss."

When this definition is well committed, my readers are permitted to learn by heart, the following

RULES.

"1. When the quantity of one commodity is given with its value, or the value of its integer, as also the value of the integer of some other commodity to be exchanged for it, to find the quantity of this commodity :-Find the value of the commodity of which the quantity is given, then find how

have been so long suffered to waste and pervert the precious time of those, who have but little time, at most, to bestow upon the subject. The public are intreated to look into the claims of this book to such overwhelming patronage; and to examine it thoroughly, both in regard to the knowledge it gives of the subject, and the discipline it affords the mind. For nothing but a misapprehension of the subject,-of the purposes of a school book, and the principles of the human mind, will excuse the mischief, it is allowed to do the community,