The present performance having passed through a number of editions since the time of its first publication, without any material alterations having been made, either with respect to its original plan, or the manner in which it was executed, I have been induced, from the flattering appro, lion it has constant ly received, to undertake an entire revision of the work; and, by availing myself of the im ; provements that have been subsequently made in the science, to render it still more deserving the public favour. In its present state, it may be considered as a copious abridgement of the most practical and useful parts of my larger work, entitled, A Treatise on Algebra, in 2 vols. 8vo. published in 1813 ; from which, except in certain cases, where a different mode of prooetding appeared to be necessary, it has been chiefly compiled: great care having been taken, at the same time, to adapt it, as much as possible, to the wants of learners,and the general purposes of instryction, agreeably to the design with which it was first written. With this view, as well as in compliance with the wishes of several intergent teachers, I have, also, been led to subjoin to it, by way of an Appendix, a small tract on the application of Algebra to the solution of Geometrical Problems; which, it is hoped, will prove acceptable to such classes of students as may not have an opportunity of consulting more voluminous and ensive works on this interesting branch of the science. JOHN BONNYCASTLE. Royal MILITARY ACADEMY, WOOLWICH, THE CONTENTS. Page Involution, or the raising of Powers... Of irrational Quantities, or S! Of arithmetical Proportion and irogression...... Of geometrical Proportion and Progression....... 76 Of the Resolution of simple Equations........ Miscellaneous Questions........ Questions producing quadratic Equations........ 116 Of cubic Equations............ of the Solution of cubic Equations......... of the Resolution of biquadratic Equations.......... 130 To find the Roots of Equations by Approximation...... 133 LGEB Algebra is the science which treats of a general method of performing calculations, and resolving matbematical problems, by means of the letters of the alphabet. Its leading rules are the same as those of arithmetic ; and the operations to be performed are denoted by the following characters ; + plus, or more, the sign of addition ; signifying that the quantities between which it is placed are to be added together. Thus, a +b shows that thumber, or quantity, represented by b, is to be added to that represented by a; and is read a plus b. - minus, or less, the sign of subtraction ; signifying that the latter of the two quantities between which it is placed is to be taken from the former. Thus a-b shows that the quantity represented to is to be taken from that represented by a ; and is reada minus 6 Also, asb represents the difference of the two quantities a and b, when it is not known which of them is the greater X into, the sign of multiplication ; signifying that the quantities between which it is placed are to be multiplied together. a |