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which is referred to the Theory of Numbers ; for that of the theory of combinations, which is referred to the Doctrine of Chances ; and for that of the development of fractions and logarithms into series, which is referred to the Theory of Functions.

The polynomial theorem of Arbogast, in art. 137, which, in beauty and facility of application, rivals the binomial theorem of Newton, will probably be new to most readers, as it is believed never before to have found its way into an elementary treatise. No apology need be made for the length of its demonstration to one, familiar with the original investigations of Arbogast, or who reflects upon its generality and unavoidable intricacy; the learner will, however, find it less complex than it appears to be; and, at any rate, he may easily become familiar with the use of the theorem, even if he find the demonstration too abstruse.

The peculiar class of equations, introduced in article 118, being examples 10, 11, 12 of that article, will probably be as new to most readers, as they were to the author; exhibiting the singular peculiarity of being reduced in degree by the process of elimination. These examples might have been multiplied to any extent, and are by no means accidental in their formation.

BENJAMIN PEIRCE.

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