V

then deduce practical directions; for rules, without reasons, are ridiculous, and insulting to the inquiring mind.

No secondary principle has been used in the elucidation or illustration of one that is primary; nor has any principle been anticipated; but each, used in its natural sequence, has been made the basis of a yet higher principle, in such manner, as to cultivate the reasoning powers of the learner, without embarrassing them.

Technical phraseology has been avoided, as far as consistent with the requirements of such a treatise; as likewise, puzzles, and all giddy theorizing on trivial and unimportant topics, which should be beneath the dignity of a scientific man, although well calculated to please the fancies of a vacant mind: for he who would be a useful man, must be a practical man; and the less acquainted with fascinating chimeras, the better he is adapted to his great purpose.

It is not claimed for this system that cancelation can be availed in every solution; but that a great majority of practical questions can be much abbreviated by it; the excellency claimed for the system, is, that while it abbreviates the work, the statement is so simple, so philosophical, and the result, so inevitable, that no intelligent individual can fail practicing its principles, whether the arbitrary rules be remembered or not

We shall endeavor to deal mildly with those who, being bound to the old system, as their hobby, cannot, or will not, open their eyes to the evident advance of modern improvements; and, therefore, submit our labors to the investigation of those candid and intelligent minds, which are not shackled down to such usages of the past, as are endeared more by habit, than by any rational merit.

Cincinnati, July, 1849.

T. RAINEY.

RAINEY'S

IMPROVED ABACUS.

CANCELATION.

ALL arithmetical computations are effected by increase and decrease, which depend in their relations, on the converse operations of Multiplication and Division. The latter are but abbreviated methods of adding and subtracting. Increase and decrease are the results of the relative difference between different numbers and quantities of the same thing: hence their result depends, in all reckoning, on the great principles of Ratio and Proportion. Therefore, by Proportion, as the rationale of statement, and Multiplication and Division as the mechanical media of reducing such statements to their results, we have in a few words, an epitome of all arithmetic.

As by this, Multiplication and Division are presented as the leading operations of reckoning, we may profitably spend some little time, in ascertaining a more expeditious method of determining products and quotients, than by the old, tedious, and circumlocutory formulæ of the books.

When 7 is multiplied by 3, the product is 21: this product divided by another 3, gives 7 again;

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