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or concrete numerical magnitudes, consisting of various denominations.

The third Chapter treats of the first Principles of the Rule of Three, sometimes called the Golden Rule ; and it comprises a collection of examples illustrating the different views of the subject.

The fourth Chapter contains The Doctrine of Fractions, usually termed Vulgar Fractions; concluding with some of its applications to practical purposes.

The fifth Chapter developes The Theory of Decimals, commonly called Decimal Fractions ; and it points out some of the important uses to which they are more peculiarly adapted.

In the sixth Chapter are discussed the Doctrines of Ratio and Proportion, from the Principles of which are deduced several Rules of the greatest consequence in the affairs of Commerce ; and it concludes with the solution of a few miscellaneous questions, explaining some technical terms.

The seventh Chapter contains the Practice of Involution and Evolution, with The Arithmetic of Surds or Irrational Quantities.

The object of the eighth Chapter is The Nature and Properties of Logarithms, derived from the simplest principles; and the practical advantages afforded by Logarithmic Tables are briefly pointed out in appropriate examples.

The ninth Chapter is The Application of Arithmetic to Geometry: and the calculations of Artificers, Gagers and Land-Surveyors are concisely explained and exemplified in it.

In this chapter will also be found a short account of the Imperial Weights and Measures, and their origin and relation to each other; as well as of the Calendar adopted in the time of Julius Cæsar, and its subsequent improvement in the time of Pope Gregory the Thirteenth, with all the requisite Calculations worked out.

The rest is an Appendix, in which some of the rules have been derived from the most elementary principles, upon the extension of which the present system of Arithmetic is generally established.

Throughout the work, it has been attempted to trace the source of every rule which is given, and to investigate the reasons upon which it is founded: and by means of particular examples comprising nothing but what is common to every other example of the same kind, to confer upon Arithmetic that kind of evidence which is attainable in Geometry, or any other demonstrative science.

Single and Double Position are entirely omitted, as most of the examples usually given to illustrate these rules, may be solved by the principles here explained, not to mention that they are merely Algebraical Formulæ enunciated at length.

No notice has been taken of Arithmetical and Geometrical Progression, of Permutations and Combinations, and of Annuities and Reversions, because they all depend upon Formulæ expressed by general symbols, which the student would find a difficulty in making use of, without at least a knowledge of the Notation and Fundamental Operations of Algebra; in addition to which, they very seldom occur

to any one who is not engaged in Scientific Speculations, or in Professional Calculations

It may perhaps be objected that the Examples for Practice given in the work, are too numerous for a rapid advancement in the subject; but the student will recollect that he has no occasion to trouble himself with the rest, when a few of them have rendered him perfect in the Application of the Rules; although it must be observed, that a Facility in Arithmetical Calculations is of all things the most indispensable, in the formation both of the future Analyst, and of the Man of Business.

CAMBRIDGE,

December 7, 1839.

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