the ancients, who applied it to the solution of indeterminate, or unlimited problems; but it is to the moderns that we are principally indebted for the most curious refinements of the art, and its great and extensive usefulness in every abstruse and difficult inquiry. NEWTON, MACLAURIN, SAUNDERSON, SIMPSON, and EMERSON, among our own countrymen, and CLAIRAUT, EULER, LAGRANGE, and LACROIX, on the continent, are those who have particularly excelled in this respect; and it is to their works that I would refer the young student, as the patterns of elegance and perfection. The following compendium is formed entirely upon the model of those writers, and is intended as a useful and necessary introduction to them. Almost every subject, which belongs to pure Algebra, is concisely and distinctly treated of; and no pains have been spared to make the whole as easy and intelligible as possible. A great number of elementary books have already been written upon this subject; but there are none, which I have yet seen, but what appear to me to be extremely defective. Besides being totally unfit for the purpose of teaching, they are generally calculated to vitiate the taste, and mislead the judgment. A tedious and inelegant method prevails through the whole, so that the beauty of the science is generally destroyed by the clumsy and awkward manner in which it is treated; and the learner, when he is afterwards introduced to some of our best writers, is obliged, in a great measure, to unlearn and forget every thing which he has been at so much pains in acquiring. There is a certain taste and elegance in the sciences, as well as in every branch of polite literature, which is only to be obtained from the best authors, and a judicious use of their instructions. To direct the student in his choice of books, and to prepare him properly for the advantages he may receive from them, is therefore the business of every writer who engages in the humble, but useful task of a preliminary tutor. This information I have been careful to give, in every part of the present performance, where it appeared to be in the least necessary; and though the nature and confined limits of my plan admitted not of diffuse observations, or a formal enumeration of particulars, it is presumed nothing of real use and importance has been omitted. My principal object was to consult the ease, satisfaction, and accommodation of the learner; and the favourable reception the work has met with from the public, has afforded me the gratification of believing that my labours have not been unsuccessfully employed. JOHN BONNYCASTLE. ROYAL MILITARY ACADEMY, WOOLWICH, October 22, 1815. ADVERTISEMENT TO THE THIRTEENTH EDITION. CONSIDERABLE improvements having lately been made in the Solution of Equations by Approximation, a subject of great importance in Algebra, I have been induced to add an Addenda to the present Edition of this work, containing an entirely new method for that purpose; which I trust will be found, in many respects, more convenient than any hitherto published. CHARLES BONNYCASTLE. CHATHAM, July 19, 1824. vili EDITOR'S PREFACE. IN presenting a New Edition of Bonnycastle's Algebra to the public, the Editor indulges a hope that he has succeeded in the attempts which he has made to render the work deserving a continuance of that extensive support and encouragement which it has so long received, from the principal Educational Establishments in this Kingdom. From a careful scrutiny of all the algebraical processes throughout, several blemishes have been detected and removed, and, in a few instances, useful additions and emendations have been introduced; of the former may be mentioned the Synopsis on Variable Quantities, forming an Appendix to the book, which it is hoped will prove acceptable to the young Mathematician. Upon the whole the Editor trusts that this Sixteenth Edition will be found on examination to retain its character as an useful work, so as to merit a continuance of public favour. No. 8, Earl's Court, Leicester Square, SAMUEL MAYNARD. A KEY TO BONNYCASTLE'S INTRODUCTION TO ALGEBRA, in which the solutions of all the questions are given. By JOHN BONNYCASTLE, late professor of Mathematics in the Royal Military Academy, Woolwich; corrected and greatly improved by SAMUEL MAYNARD. In this Edition, the Editor has bestowed much attention, the errors of former impressions have been removed, and several improved Solutions introduced, more especially in Equations, the Diophantine Analysis, Summation of Series, Miscellaneous Questions, and the Application of Algebra to Geometry. In these articles the Editor has been very free in suppressing many inelegant and imperfect Solutions, and supplying their places with others better adapted to the present improved state of knowledge. **The Editor takes this opportunity to announce, that he has undertaken to re-write the Key to Bonnycastle's Mensuration, and is in great forwardness; it will contain the solutions to all the questions with reference to the Problems, Rules, and Notes by which the solutions are obtained. ALGEBRA. 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. Thus, a+b shows that the number, 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 number, or quantity, represented by b is to be taken from that represented by a; and is read a minus b. Also, a b represents the difference of the two quantities a and b, when it is not known which of them is the greater, × into, the sign of multiplication; signifying that the quantities between which it is placed are to be multiplied together. Thus, axb shows that the number, or quantity, represented by a is to be multiplied by that represented by b; and is read a into b. 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 × b, a.b, and ab, all signify the product of a and b: also, 3 x a, 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 between which it is placed is to be divided by the latter. B Thus, ab shows that the number, or 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. b - Thus, b÷a and both signify the quotient of b divided by a; and a-b a + c a signifies that a- bis to be divided by a+c. equal to, the sign of equality; signifying that the quantities between which it is placed are equal to each other. Thus, x = a + b shows that the quantity denoted by x is equal to the sum of the numbers, or quantities, a and b; and is read a equal to a plus b. identical to, or the sign of equivalence; signifying that the expressions between which it is placed are equal for all values of the letters of which they are composed. - Thus, (a+x)+{(a−x)+a; ‡(a+x)--} (a−x)Px; and (x+a)x(x − a)=x2 - a2, whatever numeral values may be given to the quantities represented by r and a. > greater than, the sign of majority; signifying that the former of the two quantities between which it is placed is greater than the latter. Thus, a > b shows that the number, or quantity, represented by a is greater than that represented by b; and is read a greater than b. <less than, the sign of minority; signifying that the former of the two quantities between which it is placed is less than the latter. Thus, ab shows that the number, or quantity, represented by a is less than that represented by b; and is read a less than b. : as, or to, and :: so is, the signs of an equality of ratios; signifying that the quantities between which they are placed are proportional. Thus, a:b::c:d denotes that a has the same ratio to b that c has to d, or that a, b, c, d, are proportionals; and is read as a is to b so is c to d, or a is to b as c is to d. the radical sign, signifying that the quantity before which it is placed is to have some root of it extracted. |