. . . . c. Y.Y. & cases, where a different mode # preseedfug 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 she general purposes *os in-fryction, agreeably to the design with *:: § #. #.. - g With this view, as well as in compliance with the wishes of several intessent 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 o: works on this interesting branch of theoscience. JoHN BonnycASTLE. Roy Al Military Academy, ~ DEFrositions............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Subtraction.......................................................... 12 Multiplication....., “......................................” 13 i.- - - - - “.............."No. * * * * * * * * * 18 Algebraic Fractions............................... 24 Involution, or the raising of Powers.............• * * * * * * * * * * 40 Evolution, or the Extrastiaad his...................... 44 Of irrational Quantities, or So .............. - - - - - - - - 50 Of arithmetical Proportion and ression................ 72 Of geometrical Proportion $ Progression.............. 76 Of Equations................................ - - - - - - - - - - - - - - - - - - - - - - - - 81 Of the Resolution of simple Equations..................... 83 Miscellaneous Questions........................... - - - - - - - - - - - - - 98 Of quadratic Equations............................ • * * * * * * * * - - - - - 108 Questions producing quadratic Equations.............. ... 116 Of cubic Equations.................................... ........... 123 Of the Solution of cubic Equations.......................... 125 xiv -Page Of the Binomial Theorem.......... “...... .o................ 143 Of the indeterminate Analysis................................. 150 162 Of the Diophantine Analysis................................... 173 Of the Summation and Interpolation of Series.......... ALG EBRA, - - N _-e - – Algebra is the science which treats of a general method of performing calculations, and resolving mathematical 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. •r| Thus, a-Hb shows that &mber, or quantity, represented by b, is to be added to that represented by a ; and is read a plus b. -- --> ... ----e. Thus, a Xb shows that the quantit is to be multiplied by that represente a into b. d by & and is read The multiplication of simple quantities is also frequently denoted by a point, or by joining the letters together in the form of a word. Thus, a Xb, a , b, and ab, all signify the product of a and b : also, 3Xa, or 3a, is the product of 3 and a ; and is read 3 times a. -- by, the sign of division ; signifying that the former of the two quantities betwe hich it is placed is to be divided by the latter. woulso - Thus, a-i-b shows that the quantity represented by a is to be divided by that represented by b : and is read a by b, or a divided by b. Division is also frequently denoted by placing one of the two quantities over the other, in the form of a fraction. * vided by a ; and #* that a-b is to be divided by a-Fc. = equal to, the sign of equality ; signifying that the quantities between which it is placed are equal to each other. Thus, z=a + b shows that the quantity denoted by 2: i al to the sum of the quantities a and b; and is read acUsual to a plus b. *= identical to, or the sign of equivalence ; signifying that the expressions between which it is placed are of the same value, for all values of the letters of which they are composed. Thus, (x+a) × (z—a) =z”—a”, whatever numeral values may be given to the quantities represented by a and a. T--> __- * *-* . ) • |