greatest clearness of thought and force of reasoning. The mathematical sciences, and particularly Arithmetick, Geometry, and Mechanicks, abound with those advantages; and if there were nothing valuable in them, for the uses of human life, yet the very speculative parts of this sort of learning, are well worth our study; for, by perpetual examples, they teach us to conceive with clearness, to connect our ideas in a train of dependence, to reason with. strength and demonstration, and to distinguish between truth and falsehood. Something of these sciences should be studied by every man, who pretends to learning."

When, therefore, we consider the influence of arithmetical studies, in disciplining the mind; when we estimate the utility of the knowledge to be gained, in the transaction of the various business of life; and, especially, when we view the subject as lying at the foundation of the whole science of mathematicks; or rather as the instrument, or key, without which we cannot proceed to the higher branches of the science, it rises to no small dignity among elementary studies. To all, it is important, to the man of business and the scholar, it is essential. There is little danger, therefore, of examining too closely into the character of our books upon the subject. And there is, perhaps, as little danger of exposing too plainly the weakness and deformity of the bad, or of overestimating the value of the good.

The system of Arithmetick, to which I have before alluded, and which it is proposed to examine, as I proceed, as a specimen of inductive instruction, was published a year or two since, "by Warren Colburn."* It is contained in two small volumes, entitled "First Lessons in Arithmetick upon the plan of Pestalozzi," and "Arithmetick, being a Sequel

* It gives me great pleasure thus publickly to acknowledge my obligations to Mr. Colburn, not only for the light, he has afforded me upon the subject of Arithmetick, but for what has been reflected from that subject to others, which have been before noticed.

The "First Lessons" profess to be "upon the plan of Pestalozzi." Some account therefore, of this remarkable man, will enable readers to judge, how far Mr. Colburn is indebted to him, for his system of Arithmetick. Pestalozzi was born at Zurich in 1746. His parents were too obscure for him to inherit much consequence or notice on their account. He early became interested in the subject of education, and viewing the miserable condition of the lower classes of the people in his neighbourhood, he resolved to devote himself to elementary instruction, as the most direct and effectual means of improving their situation and prospects. From the time he commenced instructer, he was so exclusively devoted to his employment, that he seemed to live only for that object.

He made bold innovations upon the established principles of instruction, and probably on that account, did not at first receive such notice, as his exertions merited. But the ardour of his interest was not cooled by neglect. The aid of a few friends, who were attracted by the reasonableness of his principles of instruction, and an inefficient patronage from the government of his Canton, enabled him to establish a school, which gave some celebrity to his name, and at length gained the assistance of some very warm and able friends. Pestalozzi was at length united

to First Lessons." Waving here the question of independent authorship, which Mr. Colburn might with some propriety claim, I shall enter, at once, in

with Mr. de Fellenberg, who from similar motives had established a school at Hoffwyl. This school has attracted considerable notice in Europe, and has been approved, and encouraged by some of the most distinguished men of the age.

The object of Mr. de Fellenberg was, to find a plan for the education of the poorer classes of society, at the least expensc. Agriculture, therefore, constituted an essential part of the education. But the principles of government and instruction, adopted at his school, succeeded so well, that pupils were sent from many of the principal families in every part of Europe. In conjunction with, and under the patronage of Mr. de Fellenberg, who was a gentleman of some fortune, Pestalozzi was enabled to carry his improvements in the principles of instruction, into more complete operation. It would be foreign to my present purpose, however interesting the subject, to go into the detail of that establishment. We are interested, at present, only in the method of instruction.

It was a fundamental principle in their system, never to suffer a pupil to pass over, what he did not thoroughly comprehend. The course of instruction was so conducted, as to give accurate and well defined ideas upon the subject to be taught. For this object, the instructer gave lessons in the field; and upon subjects, which there presented themselves. This manner excited and kept up a lively interest in the learner, because he saw at once the use and application of what he was learning. The instructer was thus spared the perplexing question, "cui bono ?" which so constantly arises in the pupil's mind, and which can so seldom be satisfactorily answered. "Questions continually occurred respecting the measures of capacity, length, weight, and their fractional parts; the cubic contents of a piece of timber, or a stack of hay, the time necessary to perform any particular task, under such or such circumstances, &c. &c." The boys en

to an examination of the general principles of the system; it being a much more interesting question, what the system is, than whose it is. The system

deavoured to find the solution of arithmetical and mathematical problems without writing, and at the same time to proceed with the mechanical process, in which they might happen to be engaged. This method of instruction, among improvements in other branches, gave rise to the plan of Arithmetick, invented by Pestalozzi. He began with the most simple combinations upon small numbers, and proceeded to the more difficult, as the learner acquired strength to encounter them. The language of figures, and their use in the solution of questions involving large numbers, were reserved for a later and more difficult stage in their progress. These hints constitute the principal assistance, which Mr. Colburn derived from Pestalozzi, in forming his system of Arithmetick. He has adopted the arrangement of Pestalozzi in some of the combinations, but he has rejected it in others, and developed all, by the selection and composition of examples, in which he derived no assistance from him. Pestalozzi undoubtedly discovered the applicability of the inductive method to communicating knowledge, whether he knew it by that name or not, and applied that method in teaching the science of numbers. Mr. Colburn, with hints from him, has applied the same method to teaching the same subject, but in a manner somewhat peculiar to himself. Both, in common with all the philosophers since Bacon, are indebted to him for telling them how to learn, and how to teach. And it would, perhaps, be better if Mr. Colburn would say at once, "Arithmetick upon the plan of Bacon," rather than adopt any name, which can only reflect, what it has received from him. The identity of the principles of this method of instruction, with the inductive method of acquiring knowledge, taught by Bacon, has never been established and inculcated by those, who have adopted the method as a basis for their books. [For a more full account of the establishment at Hoffwyl, see Ed. Rev. Oct. 1819, and Simond's Switzerland. Vol. ii. pp. 198, 194 and 330-340.]

is new, and widely different, from any thing before published in this country. These circumstances, together with the importance of the subject, and the happy illustration, it affords, of inductive instruction ; seem to require a pretty detailed account of it; I shall confine myself, however, in my remarks, merely to general principles, except so far as detail is essential to their illustration.

The distinctive traits in the character of the system will be at once seen, by examining it under the following principal divisions

I. It teaches all the combinations in Arithmetick, with numbers so small, that the mind of the pupil can perfectly comprehend them.

II. Every new combination is introduced by practical examples upon concrete numbers.

III. All those rules, which are merely artificial, and those formed for particular applications of the same general principle, have been discarded.

The first principle above stated gives rise to the division of the subject into the "First Lessons" and "Sequel." The solution of every arithmetical problem requires two processes; first, to analyze the question and determine the relation of the several numbers; and then to reason upon those numbers, in a manner peculiar to the science, till the result required is attained. These two processes must be performed in the solution of every problem; but when the numbers are so small, as not to require the aid of a written numeration, they are both performed