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ELECTROTYPED AND PRINTED EY J. B. LIPPINCOTT COMPANY, PHILADELPHIA, U. S. A.

LBC .

PREFACE.

“ ARITHMETIC—queen of the terrestrial sciences” is an encomium by no means inapplicable to what is distinctively called the mental system. That a very thorough and philosophical knowledge of Arithmetic in its various aspects may be obtained by processes carried on by the mind, in total independence of the mechanical help of pen or pencil, no longer needs demonstration. Multitudes of men and women are ready, at a minute's warning, to rise up and give emphatic testimony to the beneficence of the mental system as an educational force.

Docendo discimus. Not only those who are taught, but the teachers themselves who have adopted this system, and are faithfully applying it, are daily made conscious of its power as it develops within them the noble faculty of reason and fashions the invaluable habit of concentration. As the mind of the pupil unfolds, the mind of the teacher expands; both go on seeing clearly, and both escape the ditch into which the blind, leader and follower, inevitably fall.

In these modern times, Arithmetic is no longer considered, or certainly ought not to be, a mere compilation of mechanical rules and methods of operations that have no reasons for their existence that anybody is bound to respect. A change has come, we may say, and the intel

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lectual study of Arithmetic upon the Inductive Method, first introduced by Warren Colburn-a distinction more significant and lasting than a monument of brass—has become one of the most effective agents in imparting a philosophical knowledge of Arithmetic, and of thus bestowing upon our youth a high and much-needed degree of mental discipline.

Inductive Method means method by induction, and the inductive plan of study is, therefore, not to be regarded as opposing or underrating a system of methods except as arbitrarily prescribed or dictated. It freely accepts the devices of art as most helpful aids to the acquirement of skill in the solution of arithmetical problems and to a complete mastery of the science; but it insists that those devices be inductively derived, philosophically established, and thoroughly understood. Prescribed rules and set modes are, in many parts of the subject, quite indispensable as stepping-stones to progress and to the higher knowledge that lies beyond. When Cæsar determined to cross the Rhine, he found no bridge prepared for him; but he crossed, nevertheless,—on a bridge of his own invention. A great excellence of the inductive mental system is that, while it confronts the pupil with difficulties, it has also prepared him to surmount them by his own efforts.

The greatest praise of any system of education is that it puts a youth in proper relations with his environment, and trains him to self-reliance, enterprise, self-helpfulness, and heroism. To such a system certainly belongs the science of Mental Arithmetic, and the work we now present we trust sets forth, not unworthily, the principles upon which the science is based.

The fundamental element of the intellectual treatment of the subject is Analysis. Analysis. lays the foundation for Induction (Inference), and Induction supplies principles and methods. The philosophical comprehension of mathematical truths can be rightfully arrived at in no other way than by the steps of procedure indicated.

The analyses given throughout the book are meant to be suggestive merely, and under no circumstances to be rigidly adhered to. Each Analysis, or Solution, as it may be called, has been separated into two or more distinct parts; part 1 giving the process of the solution, and parts 2 and 3 the explanation of the process. This has been done, not only for the sake of clearness, but to save time, for assuredly the bright pupil need not be required at every step to explain assertion.

Our book is progressive: it starts with the simplest combinations and proceeds by easy grades to a point sufficiently high to show somewhat the potentialities of Arithmetic when philosophically taught.

In using the book, the teacher should begin at that chapter or lesson that best suits the advancement of the class, and then make haste slowly.

J. M. R.

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