can master one without learning much of the others. Count ing is but addition; and to understand the nature and use of the number two, we must know that it equals 1 and 1, two ones, or two times 1; that it is 1 more than 1; that if 1 be taken from it 1 will be left, and so on with the other numbers things which require a knowledge of numerical operations, and also a power of tracing and appreciating relations.

The operations and exercises included in the first two points are eminently adapted to give quickness of thought and rapidity of mental action. They are to Arithmetic what a knowledge of the nature and power of letters, and of their combination into words, is to reading. The processes included in them may be called the mechanical processes of Arithmetic, and by practice may and should be made so familiar that the moment a number or a combination is suggested, the mind can appreciate it, and determine the result.

The third requires and imparts a power of investigation, of tracing out the relations of cause and effect, and habits of accuracy both in thought and expression.

To secure these results, it is necessary that the pupil should be taught in the simplest as well as in the most complicated problems to reason for himself; to trace fully and clearly the connection between the conditions of a problem and the steps taken in its solution; to state not only what he does, but why he does it, and indicate the precise character of the result obtained by each step. Finally, he must learn to grasp the whole mechanical process before performing any part of it, so that he may know before writing a figure just what additions, subtractions, multiplications, divisions, and comparisons he has to make, and be assured that if made correctly they will lead to the true result.

Such a course as this is usually taken in works on Oral Arithmetic. In studying them the scholar is thrown on his

own resources; is compelled to learn principles; to follow out rigid reasoning processes and connected trains of thought; to examine and know for himself the necessity and the reason for each step taken, and for each operation performed. The result is, that the study gives strength, vigor, and healthful discipline to the mind, and becomes an almost invaluable part of the educating process.

Why should not the same result follow a similar course in Written Arithmetic? Aside from the writing of numbers, there is no difference in the principles involved, in the reasoning processes demanded, or in the operations required.

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In the preparation of this work, the author has kept these things in view. He has endeavored to present the subject of Arithmetic as it lies in his own mind, and without any effort either to follow or deviate from the course pursued by other writers. He has aimed to arrange the work in such a way as to lead those who may study it to understand the principles which lie at the foundation of the science, to learn to reason upon them, apply them, and to trace out their connections, relations, and combinations. He has given very full explanations and illustrations, especially of the fundamental operations; he has endeavored every where to state principles rather than rules; to throw the pupil constantly on his own resources, and force him to investigate and think for himself.

He has omitted some subjects usually found in school arithmetics, because they do not belong legitimately to the subject of Arithmetic, because they are of theoretical rather than of practical importance, or because they require neither special explanation nor peculiar exercise of the mind.

He has differed from other authors of school arithmetics in giving algebraic rather than geometrical explanations of the principles involved in Square and Cube Roots. In this way he has been able to give more rigid demonstrations, and more

full explanations, and at the same time (as he conceives) to simplify the subject. Moreover the processes are in their nature so essentially algebraic that by the use of squares and cubical blocks we can do nothing more than illustrate some of their applications.

He has given no answers to his problems, because he believes that to place them within reach of the pupil is always injurious.

In the first place, such tests are unpractical, for they can never be resorted to in the problems of real life. What merchant ever thinks of looking in a text book or a key, or of relying on his neighbor, to learn whether he has added a column correctly, drawn a correct balance between the debit and credit sides of an account, or made a mistake in finding the amount of a bill?

When a pupil, having left the school room, performs a problem of real life, how anxious is he to know whether his result be correct! Neither text book nor key can aid him now, and he is forced to rely on himself and his own investigations to determine the truth or the falsity of his work. If he must always do this in real life, and if his school course is to be a preparation for the duties of real life, ought he not to do it as a learner in school? Is it right to lead him to rely on such false tests?

Besides, the labor of proving an operation is usually as valuable arithmetical work as was the labor of performing it, and will oftentimes make a process or solution appear perfectly simple and clear, when it would otherwise have seemed obscure and complicated.

Again the science of Mathematics, of which Arithmetic is a branch, is an exact science; it deals in no uncertainties; its reasonings are always accurate, and, if based on true premises, must always lead to true results. In Arithmetic the

pupil may always know that a certain step is a true one, and one which he has a right to take. He may know whether he has taken it correctly, and thus be certain of the truth of his first result. He may be as sure of the truth of his second step and second result, and of his third and his fourth; and when he reaches the end, and obtains his final result, he may be as sure of the truth of that as of any preceding that he will be willing to abide by it, and stake his reputation upon it. (See page 49.)

so sure

Why, then, should not the subject be so presented as to require the pupil to apply such tests, to determine for himself the truth and accuracy of his processes, and thus to form a habit of patient investigation and just self-reliance? Why should he not be from the first thrown on his own resources, and held strictly responsible for the accuracy of his work? Would not such a course, if faithfully followed, almost entirely prevent the formation of those careless habits which scholars so often acquire ?

The articles on business forms and transactions have been carefully prepared, with a hope of so presenting them as to give the student true ideas of their use, and of the relations and obligations of the parties to them.

The materials were drawn from various sources from legal works, from intercourse with business men, and from an article in Mann and Chase's Arithmetic, published originally in the Common School Journal. To insure accuracy they were submitted to the inspection of Abraham Payne, Esq., an eminent lawyer of this city, to whom I am indebted for some important suggestions.

The work as a whole resembles all other text books (good or bad) in this that it requires a good teacher to teach it well; as also in this- that it does not contain exactly the right kind and amount of exercises to meet the wants of every

school or of every class of scholars. The judicious teacher will of course extend the exercises which are too meagre, abridge those which are too full, and omit those which are not adapted to the wants of his class. We earnestly beg of him, however, to notice their arrangement, their gradual character, and their dependence on each other; and not to pass any till he has convinced himself that they are inappropriate, or that the scholar is master of the operations which they involve.

To my former teacher, N. Tillinghast, Esq., for many years principal of the State Normal School at Bridgewater, Massachusetts, I am more deeply indebted than to any other, or all others, for the ideas embodied in this work. Many of the processes were learned under his tuition; and the training which laid the foundation for whatever real mathematical knowledge I may possess, was, in a great measure, received from him. Only those who have been his pupils can appreciate the value of his instructions, and the justice of this acknowledgment.

The work in its plan and arrangement is entirely my own, and for its defects I alone must be held responsible. Such as it is I present it to the public, with a hope that it may be found useful.

PROVIDENCE, July, 1855.

DANA P. COLBURN.