together. That is, the relations of the numbers are discovered, at once, as fast as in the analysis, they are compared with each other.

system of arithmeAnd in consequence a combined form at

And in

This division of the subject has never, to my knowledge, been made, in any tick, published in this country. of presenting both processes in first, neither has been very perfectly learned; and the reasoning, which is the more important, both as a part of arithmetick, and as a discipline to the mind, has been least understood. At the age arithmetick is first put into the hands of a learner, the importance of having him attain clear ideas of his subject, can hardly be estimated. If a habit is contracted at this period, of being satisfied with loose, shadowy, and ill defined ideas, it will exercise a strong and pernicious influence, through the whole course of his education; and perhaps produce a decided cast in the character of his mind. To counteract this habit, and form the contrary one of attaining distinct ideas, and reasoning clearly upon every subject, the learner must be presented with such processes of reasoning only, as that he can perfectly comprehend every step in the process. This can be done in the science of numbers, only, by giving examples of reasoning upon small numbers, till the mind acquires sufficient strength to encounter more complicated combinations upon large numbers.

The power of attention, which is so essential to every mind, and which it is so difficult for the young

mind to acquire or control, is more improved by arithmetical calculations, than by almost any discipline, which can be offered. And when it is once acquired by the study of numbers, it may be easily transferred to other studies; and thus all the sciences derive an advantage, from the increased strength of a power, which few or none have so happy a tendency to improve.

The First Lessons" introduce and inculcate every principle in arithmetick, by a collection of examples, although not a rule is given in the book. It must not be imagined, however, that a mere collection of examples constitutes an inductive arithmetick. To a superficial observer, it would seem no very difficult matter, to bring together examples to any extent. And if this were all, that is essential to a complete and successful induction, it would, indeed, be easy, and quite within the power of any -one, who has sufficient patience. But this is the least and the lowest of the efforts necessary to accomplish such a work. The mind of him, who would undertake to make a book of this kind, with any rational hope of success, must be capable to take in, at one view, the whole subject. He must resolve, without any reference to existing books, arrangements, or rules, the whole science into its first and essential principles. And be able to comprehend these, in all their relations and dependencies upon each other. For they cannot be successfully developed, except in the order of such dependence.

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 is necessary to complete the science of arithmetick. The "Sequel" is 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.

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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