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power or obscure their clearness. There is such a thing as debilitating a pupil's mind through excess of illustration; as inducing a passive reception rather than an active grasp of truths. It is with the intellectual as with the physical system. The digestive process would be less complete, if he who eats should be deprived of the action and the relish of chewing and swallowing his own food; so a true digestion of knowledge requires that the pupil should masticate his own intellectual aliments. We think Professor Perkins' book is happily adapted to secure this result.

III. The general arrangement of the subjects treated is thought to be philosophical. Those are brought into conjunction which are related in idea. The subject of Fractions, of Decimals, of Interest, of Partial Payments, etc., will, in their perspicuousness and their thoroughness, commend themselves to the examiner.

The subject of Proportion and Ratio is presented with peculiar force; as also, in Equation of Payments, the method of finding the Cash Balance.

IV. The method of Extraction of the Cube Root is greatly preferable to the old method. It is far more concise and more comprehensive ; saving nearly half the labor, and being applicable, with little variation, to the extraction of all roots. The new method is fully and beautifully explained in this work.

V. The properties of the significant figures, and the use of the zero, are, we think, philosophically and concisely presented.

VI. Lastly, we may say, no subject has been omitted on account of any inherent difficulty in elucidating it.

The Publishers take pleasure in the appearance of the Book, which certainly invites the interest of the scholar.

PRE FACE.

It is more than four years since this work was published. During the whole of this time it has been in constant use under my own superintendence; and, consequently, I have had opportunity to ascertain what were its defects, and wherein a difference of arrangement, or other modifications, would be desirable. I have, also, consulted experienced teachers with direct reference to the present revision of the work, and now submit the result to the public.

I am confident that great improvements will be found in the following particulars. In the statements of properties, relations, and principles—in the phraseology of definitions and of Rules—in the methods of illustration in the order of arrangement of the subjects treated; indeed, throughout the entire work.

My object has been to be concise, yet lucid ; to reach the radical relations of numbers, and to present fundamental principles in analyses and examples, that shall leave nothing obscure, yet that shall not embarrass by multiplied processes, or enfeeble by minute details. I hold to the idea that a sufficiency of illustration to lay open thoroughly the subject treated, is all that is desired ; and that whatever is redundant impairs the force of what is essential. Both teachers and pupils will, as I judge, be

benefitted by thus leaving them somewhat to the action of their own minds.

It is not easy for me to specify points to which attention may be directed. But I would suggest, the definition of the values of Figures of the Zero; the illustration of Subtraction; the general treatment of Vulgar Fractions; the introduction of Decimal Fractions before Federal Money; and of Duodecimals immediately after Denominate Decimals; the whole arrangement of Percentage and Interest; the method of finding the Cash Balance in Equation of Payments. And last, but not least, the method of extracting the Cube Root, by means of auxiliary columns. To this method I ask the attention of teachers generally. I believe I have omitted no step necessary to make it perfectly intelligible ; and for conciseness and beauty, as well as for practical use, it is incomparably superior to the usual method.

Throughout the entire work many new examples have been given, which have been formed with much care, having the different parts so related as to bring out, when solved, exactly the principle designed. Many of these questions contain statistical and historical facts which it is desirable for all to know, thus giving an interest to the questions which they could not possess in an abstract and simply numerical form.

GEO. R. PERKINS. UTICA, March, 1849.

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