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ciple, or employ a process, with the rationale of which he is not already acquainted. The reference by Articles will always enable him to trace any subject back to its first principles.

The limits of a preface will not permit a statement of the peculiarities of the work, nor is it necessary, as those who are interested to know will examine it for themselves. It is, however, proper to remark, that Quadratic Equations have received more than usual attention. The same may be said of Radicals, of the Binomial Theorem, and of Logarithms, all of which are so useful in other branches of Mathematics.

On some subjects it was necessary to be brief, to bring the work within suitable limits. For example, what is here given of the Theory of Equations, is to be regarded merely as an outline of the more practical and interesting parts of the subject, which alone is sufficient for a distinct treatise, as may be seen by reference to the works of Young or Hymers in English, or of DeFourcy or Reynaud in French.

Some topics and exercises, deemed both useful and interesting, will be found here, not hitherto presented to the notice of students. But these, as well as the general manner of treating the subject, are submitted, with deference, to the intelligent educational public, to whom the author is already greatly indebted for the favor with which his previous works have been received.

WOODWARD COLLEGE, May, 1852.

In this NEW ELECTROTYPE EDITION, the whole volume has been subjected to a careful and thorough revision. The examples, where they were thought to be needlessly multiplied, have been reduced; the rules and demonstrations abridged, and other methods of proof, in a few instances, substituted. It is confidently believed that these modifications, while they do not impair the integrity or change the essential features of the book, will materially enhance its value, and secure the approbation of all intelligent teachers.

March, 1866.

VIII.-RATIO-PROPORTION-PROGRESSIONS.

RATIO-Kinds-Antecedent and Consequent

Ratio Multiplication and Division of

Ratio of Equality-Of greater and less Inequality

Ratio Compound-Duplicate-Triplicate

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Product of Means equal to Product of Extremes .

Proportion from two Equal Products

Product of the Extremes equal to the Square of the Mean
Proportion by Alternation-By Inversion

Proportion from equality of Antecedents and Consequents
Proportion by Composition-By Division
Proportion by Composition and Division.

Like Powers or Roots of Proportionals are in Proportion

Products of Proportionals are in Proportion

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ARTICLES.

254-256

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