PREFACE. The following is the First Part of a Treatise on Practical Mathematics, and comprehends that portion which does not require the use of Tables. In adding another to the many existing Treatises on this subject, it may be proper to state the objects that have been kept in view in its composition. These have been, first, To exclude all useless matter, and thereby to keep the work within a small compass; secondly, To make it as entirely demonstrative as possible, without reference to any other work on Mathematics. For this purpose, as well as for its own intrinsic usefulness, a Treatise on Geometry is introduced, in which, by adopting a different order of the propositions from that used in Euclid's Elements, and using symbols for certain expressions of frequent occurrence, an unprecedentedly large quantity of geometrical truths is presented, without in any instance detracting from the fulness of the demonstrations, which are always given at length. The article on Algebra, it is hoped, will be found to be sufficiently extensive for most practical purposes; and the pupil that has thoroughly studied it, will find himself well prepared for entering on the study of larger works. In many instances exercises have been introduced of such a nature, as not only to illustrate the rules, but to assist in reducing certain expressions which occur in the subsequent parts of the work, and thereby make the pupil's progress more certain and easy. The Second Part of the work, now nearly ready, will contain Logarithmic Arithmetic, Trigonometry and its applications, Mensuration of Surfaces and Solids, Land Surveying, Specific Gravities, and Guaging, together with a set of Tables of Logarithms, of Logarithmic Sines, Tangents, Secants, &c., which are necessary in solving the various questions that occur in the work. A Key, containing the solutions at length of all the exercises in this work, is now ready, and may be had of the publishers. For the use of practical men, the Tables will immediately be published in a separate form, preceded by a description thereof, and a brief view of Logarithmic Arithmetic. 1st August 1845. CONTENTS OF PART I. Important Theorems, Arts. 29, 30, 31, Important Theorems, Arts. 38, 39, 40, 41, To reduce a Mixed Quantity to the form of a Fraction, To reduce Improper Fractions to Whole or Mixed Quantities, To find the greatest Common Measure, To reduce Fractions to their lowest terms, To reduce Fractions to a common denominator, Multiplication and Division of Fractions, . Exercises in Simple Equations, Simultaneous Equations, with three unknown quantities, Problems producing Equations, with two or more unknown Questions producing Quadratic Equations, Axioms and Explanation of Symbols, Exercises in Involution and Evolution, Exercises in Simple Equations, ib. 169 170 171 PREFACE. The following is the First Part of a Treatise on Practical Mathematics, and comprehends that portion which does not require the use of Tables. In adding another to the many existing Treatises on this subject, it may be proper to state the objects that have been kept in view in its composition. These have been, first, To exclude all useless matter, and thereby to keep the work within a small compass; secondly, To make it as entirely demonstrative as possible, without reference to any other work on Mathematics. For this purpose, as well as for its own intrinsic usefulness, a Treatise on Geometry is introduced, in which, by adopting a different order of the propositions from that used in Euclid's Elements, and using symbols for certain expressions of frequent occurrence, an unprecedentedly large quantity of geometrical truths is presented, without in any instance detracting from the fulness of the demonstrations, which are always given at length. The article on Algebra, it is hoped, will be found to be sufficiently extensive for most practical purposes; and the pupil that has thoroughly studied it, will find himself well prepared for entering on the study of larger works. In many instances exercises have been introduced of such a nature, as not only to illustrate the rules, but to assist in reducing certain |