And when by the exercise of no common share of acuteness, these elementary and essential principles are arranged in their natural order; the mass of examples must be carefully examined and assorted, for the development of each principle, and for each combination of principles. The work is still but progressing. The examples thus assorted according to the principles involved in them, must be again examined and arranged with reference to the young mind, which is to encounter them. A neglect or failure in this point would be as fatal as in any other. From this view of the subject, it would seem no humble labour, to produce a consistent book upon such a plan. And the author may congratulate himself, and the publick may well congratulate him, if he gets through such a work, without making some, nay, many mistakes.

Upon this plan, the pupil learns the reasoning, and not the technical name for it. And, I am much mistaken, if the child or youth, who has carefully analyzed every example in this little book, (which claims to be only first lessons,) and found the answer in his own way, has not a better knowledge of fractions in all their combinations, and in fact, of every principle of arithmetick, than it would be possible for him to gain, by reading the most elaborate treatise on the synthetick plan. The little reasoner will not dare to say he has learned Addition, Subtraction, Multiplication, and Division; Interest, Single Rule of Three, and Double Rule of Three; for

he has probably never heard of half these terms. But propose him a question under either of those rules, involving only numbers within his comprehension, and he will analyze the problem, and perform every operation in the solution distinctly, and give you the correct result. And if he is called to it, he will explain the why and wherefore of every step in the process.

Now, when parents leave off insisting, that their children's memories shall be burthened with a confusion of rules, which they do not in the least understand, and which it requires all the energy of the young mind to retain; and when they become contented, that their powers of mind are developing in their natural order, and as fast as the God of nature intended they should be developed, we may expect this subject to become more interesting to young learners, and to be more scientifically and successfully taught.

After the power of attention is strengthened, and habits of discrimination and analysis are in some degree formed, by examples on small numbers; the next thing to be learned is, a knowledge of the arbitrary signs or figures, and their use in facilitating our reasoning upon large numbers. This is taught in the "Sequel," which adds what necessary to comThe "Sequel" is

is

plete the science of arithmetick. divided into two parts. The first contains examples only; and those arranged, as in the "First Lessons," in the order of their dependence upon each other.

And if the principles, by which the examples are to be solved, have no dependence, they are arranged in the reversed order of the difficulty a learner will "be likely to encounter, in their solution. This arrangement enables the learner to bring the increased strength of his mind, at each advance, to bear upon the more difficult parts of his subject. The second part contains an analytical development of the principles, and is to be studied in connexion with the first. When the learner has performed the examples in the first part, which involve a principle, he is turned to the second part, and there sees the same principle developed in an abstract form, till at length he arrives at a rule, which he can now comprehend, because he has learned all the variety of particular examples, to which the rule is applicable. The rule is now, no more than a verbal generalisation of what he has already learned and it is the last thing he arrives at in order, instead of the first, as in all other systems. The separation of the examples, and the analytical development of the principles, into two separate parts of the work, is arbitrary, and not at all essential to it, as a specimen of induction. It would be as convenient for the pupil, to arrive at his rule at the end of his examples, as to be turned to a different part of the book. Although in this form, it would be more difficult to see, at once, the outline of the subject.

The method of putting the examples before, and as a means of arriving at the rule, is undoubtedly the

correct one, for all subjects, which are to be learned by induction; but all subjects are not so to be learned. The language of arithmetick, including notation and numeration, is not a subject to be learned by experience. The signification of the digits, 1, 2, 3, &c., is arbitrary, and the laws, by which they are used in reasoning upon numbers, are arbitrary. The meaning of figures, and the laws, by which they are used, are agreed upon by arithmeticians, and he, who approaches the subject of arithmetick, must first be initiated into the meaning of the signs and symbols peculiar to the science. Mr. Colburn's system in one instance, violates this principle. It requires the learner to write in words, examples of large numbers expressed in figures, before it teaches him numeration. It would be impossible for a learner to "write in words 270,000,838,103,908," before he had been told the meaning of these signs, and the laws, by which they are made significant of different numbers, as they occupy different places.

In the corresponding article, in the second part, Mr. Colburn has given the subject a thorough investigation. And I have never seen so intelligible a treatise on numeration as is there contained in a few pages. It may be suggested to him, to make some different arrangement, in the future editions of his book, by which this departure from the plan of never presenting a difficulty, which the learner is not competent to surmount, shall be remedied.

LETTER VII.

THE second distinctive characteristick of the inductive system of arithmetick, which I proposed in the preceding letter to examine, is this ;-Every new combination is introduced, by practical examples upon concrete numbers. This, together with the principle of always beginning with numbers so small, that the mind of the learner can perfectly comprehend them, constitute an essential part of what is peculiar to the inductive system. The resonableness of the principle above laid down, will be more apparent, when I have attempted an analysis of the process of abstraction performed in the mind of a child in its first attempts to reason upon numbers.

Abstraction is one of the last, as well as most difficult processes, which the young mind performs. The plan, therefore, of introducing every new combination, by examples upon concrete numbers, is the dictate of sound philosophy. It has its origin in the phenomena of the human mind, and is consonant with their general and acknowledged laws.

Perception is a power earlier developed in the mind of a child, than conception. It is much easier, to attain the perception of an object, which is presented to the senses, than to form a conception of the same