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As this work has been composed expressly with the intention of adapting it to the purposes of academical education, it is not designed to hold out the expectation of an entire new mass of inventions and discoveries: but rather to collect and arrange the most useful known principles of mathematics, disposed in a convenient practical form, demonstrated in a plain and concise way, and illustrated with suitable examples; rejecting whatever seemed to be matters of mere curiosity, and retaining only such parts and branches, as have a direct tendency and application to some useful purpose in life or profession.

It is however expected that much that is new will be found in many parts of these volumes; as well in the matter, as in the arrangement and manner of demonstration, throughout the whole work, especially in the geometry, which is rendered much more easy and simple than heretofore; and in the conic-sections, which are here treated in a manner at once new, easy, and natural ; so much so indeed, that all the propositions and their demonstrations, in the ellipsis, are the very same, word for word, as those in the hyperbola, using only, in a very few places, the word sum, for the word difference: also in many of the mechanical and philosophical parts which follow, in the second volume. In the conic sections, too, it may be observed, that the first theorem of each section only is proved from the cone itself, and all the rest of the theorems are deduced from the first, or from each other, in a very plain and simple manner.

Besides renewing most of the rules, and introducing everywhere new examples, this edition is much enlarged in several places; particularly by extending the tables of squares and cubes, square roots and cube roots, to 1000 numbers, which will be found of great use in many calculations; also by the table of logarithms at the end of the first volume, and of logarithms, sines, and tangents, at the end of the second volume; by the addition of Cardan's rules for resolving cubic

equations ;

equations; with tables and rules for annuities; and many other improvements in different parts of the work.

Though the several parts of this course of mathematics are ranged in the order naturally required by such elements, yet students may omit any of the particulars that may be thought the least necessary to their several purposes ; or they may study and learn various parts in a different order from their present arrangement in the book, at the discretion of the tutor. So, for instance, all the notes at the foot of the pages may be omitted, as well as many of the rules ; particularly the 1st or Common Rule for the Cube Root, p. 85, may well be omitted, being more tedious than useful. Also the chapters on Surds and Infinite Series, in the Algebra: or these might be learned after Simple Equations. Also Compound Interest and Annuities at the end of the Algebra. Also any part of the Geometry, in vol. 1; any of the branches in vol. 2, at the discretion of the preceptor. And, in any of the parts, he may omit some of the examples, or he may give more than are printed in the book; or he may very profitably vary or change them, by altering the numbers occasionally.--As to the quantity of writing; the author would recommend, that the student copy out into his fair book no more than the chief rules which he is directed to learn off by rote, with the work of one example only to each rule, set down at full length : omitting to set down the work of all the other examples, how many soever he may be directed to work out upon his slate or waste paper.-In short, a great deal of the business, as to the quantity and order and manner, must depend on the judgment of the discreet and prudent tutor or director.

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