more time to traverse than the same path, with every light extinguished. Let the reasons of every part of a process be fully apprehended, and instead of groping in darkness, we find all the connexions plain, simple and intelligible.

If any one shall complain that the illustrations of the following rules are too minute and unnecessarily extended, we remark that no one is obliged to follow them all throughout. We are, however, persuaded that every one, who is disposed to neglect them will do himself an injury, and our greatest fear is, that scholars generally will not be sufficiently attentive to them, and will not fully understand their subjects, after having gone over with the whole.

II. The importance of this science is presented forcibly by_the_following thoughts, selected from various authors. Frequent exercise in computation, has a happy influence on the mind, by inducing habits of attention, by strengthening the memory, and by producing a promptness of recollection."—(Thompson.) "Few exercises strengthen and mature the mind so much as arithmetical calculations, if the examples are made sufficiently simple to be rightly understood by the pupil; because a regular, though simple, process of reasoning is requisite to perform them, and the results are attended with certainty.”—(Colburn.) “The utility of arithmetic is so very great in the every day transactions of life, that to be ignorant of it argues no ordinary degree of neglect."-" Without a knowledge of it a person is unable to transact business easily or correctly in any occupation; and is liable to be defrauded by others."—Pierce.) Would you have a man reason well, you must accustom him to it; exercise his mind in observing the connexion of ideas, and following them in train. Nothing does this better than Mathematics, which should, therefore, be taught all who have time and opportunity; not so much, to make them mathematicians, as to make them reasonable creatures. Converse much, says Dr. Watts, with those friends and those parts of learning and those books, where you meet with the greatest clearness of thought and of reasoning. The mathematical sciences, and especially Arithmetic, Geometry, and Mechanics, abound with those advantages, and if there were nothing in them valuable 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, and to connect our ideas in a train of dependence."-(Jour. Education.)

III. The following suggestions, on the manner of studying this science may be of importance to some, if not to every one. The learner ought to commence with intellectual Arithmetic, and continue in this, till he is able, not only to solve the various questions proposed, but also to give the reasons by which he arrives at a conclusion, in a given instance. He ought to regard the demonstration, or mode of reasoning, as more important far, than merely ability to ascertain the correct answer. When he is able to give a reason for every operation in COLBURN'S FIRST LESSONS, I consider him prepared to proceed to written Arithmetic. He may now write his operations on a slate. The first thing after taking a sum for solution is to make such inquiries as the following: What are the conditions of the question? What principles are involved? What preparations are necessary for commencing the operation? &c. The following general directions may be important.

1. Endeavour to obtain a definite idea of the nature of the question.

2. Ascertain what principles must be involved in obtaining a solution.

3. Inquire in how many ways a correct answer may be obtained; and by comparison, ascertain which is most natural and most useful.

4. Seek to learn the practical value of every rule.

5. Ask yourself, before leaving a given rule, can I teach this, and make others comprehend it?

6. Inquire in what particular business of life, particular rules may be most needed and valuable.

IV. Instructers will ask: What is the best mode of teaching this science? The inquiry is highly proper, and by all, who desire the most valuable advancement of their scholars, it will be continued so long as any improvements may be secured. Remarks have already been made on the manner of learning Arithmetic. From those suggestions, the general mode of teaching may be inferred. It will, however, be proper to present some more definite suggestions, and adapt them to the wants of those who use the following work.

of "

1. The scholar is expected to have a thorough knowledge INTELLECTUAL ARITHMETIC," before he can most profitably use this book. From Mr. COLBURN's invaluable treatise, he will be able to acquire a knowledge of inductive Arithmetic. In that work, he will be able to discover the principles on which many of the rules are founded, and will

readily understand the explanations and demonstrations in Part III. of this book.

2. The teacher should put those questions to the pupil, which will present the nature of the required operation distinctly to his view; and will prove, when he is able to answer them, that he has a distinct perception of the principles, on which the rule is founded. He should not rest satisfied that the scholar understands his lesson before proof of it is furnished. No fault is more common than to suppose the learner understands, because he is ready to say he does. Experience furnishes abundant proof that this belief, however sincere, is often erroneous.

3. Let the previous lesson be reviewed, at every recitation. This direction is very important and should constantly be observed.

4. Require each scholar to present at least one question on a previous lesson, to be answered by the class. The influence of this course will be highly salutary to those who propose, and to those who answer questions.

5. If each question thus proposed is recorded in a classbook, together with the answer, it will be found beneficial.

6. Whenever time will admit of it, let each one in the class be required to propose a sum to be wrought by the others, who will be expected to return an answer at the time of the next recitation.

Manner of Using this Book.

After the learner has given proper attention to the Introduction, Notation, and Numeration, he should be required to perform the sums in Sec. 1. Part I, and then answer the questions which may be proposed on the theory as explained in Sec. 1. Part III. In the latter he will find all the explanations, which are needed to enable him to perform the given exercises in the former. The questions found under the exercises are designed to direct the attention of the learner to the nature of the given sum, rather than to be answered by him at the time of his recitation. Of some, it may be important, especially at first, to require an answer in their own language.

When sufficient time has been devoted to Sec. 1. Part I, the learner will commence the next, and proceed thus till he has performed all the exercises in Part I, and has given attention to the theory in the corresponding sections of Part III.

The learner will then be prepared to commence Part II, and the instructer will exercise his own judgment with regard to reviewing such parts of the previous lessons, as may, under different circumstances, be necessary.

When all the exercises of Part II have been performed, the way will be prepared to commence the Introduction to Algebra as given in the Sequel.

THE

ARITHMETICAL MANUAL.

PART I.

DEFINITION.

ARITHMETIC is the science, which considers the powers and properties of numbers. It teaches how to calculate or compute with correctness and ease.

The prominent divisions of this science are, Intellectual, Written, Theoretical, and Practical Arithmetic.*

Intellectual and Written Arithmetic differ only in the manner of performing operations. The former is wholly omitted in this work.

Theoretical Arithmetic explains the properties and relations of numbers-and the reasons upon which rules are founded. Its great importance to the scholar, who desires to become a master of the subject, should lead him to give it a particular attention.

Practical Arithmetic shows how to apply principles, so as to be most expeditious in solving problems and performing practical operations; or "from certain numbers given, how to find others, whose relation to the former is known."

QUESTIONS.-What is Arithmetic? What does it teach? How divided? Are other divisions ever made? Are they important? What is the difference between Intellectual and Written Arithmetic? Define the theory of Arithmetic. What is Practical Arithmetic? Which is the more important? On a knowledge of which, does a thorough knowledge of the other depend?

* Other divisions are common, such as "instrumental, logarithmetical, specious, numerous, dynamical, duodecimal, sexagesimal, &c. &c. These are not, however, important to learners generally.