QA Entered according to act of Congress, in the year 1840, By John D. WILLIAMS, in the Clerk's Office of the District Court of Massachusetts. ALGEBRA is allowed to be the grand pillar on which the whole science of the Mathematics depends. It is the foundation on which the glorious superstructure of the abstruse sciences has been reared, and is therefore of undoubted importance and utility. Indeed, volumes have been written on the elegance and importance of this study, but the subject is inexhausted; and the author deems no apology necessary for dwelling a few moments on a subject which has occupied much of his time, and which has proved a source of the purest gratification while engaged in its pursuits. We have said that Algebra is the foundation of all the abstruse sciences; the assertion is not made at random, but admits of ample proofs and clear demonstrations. Unlike arithmetic in this respect, it does not merely consist in understanding the common routine of mechanical and mercantile pursuits, but its study expands the intellect, enlarges the reasoning faculties, and accustoms the juvenile mind to patient attention and accurate reasoning. By its operations, the laws which govern the planetary system have been calculated, their order, harmony, and regularity have been displayed, and their general characteristics have been developed; the trackless ocean has been traversed, its boundaries have been determined, and a communication has been opened with every corner of our globe. Hence the increasing taste for these studies is readily accounted for, and we cease to wonder at the numerous compilations which have appeared, in order to facilitate the labors of the student. Unfortunately, however, the increasing demand for works of this nature has called into existence an ephemeral race of authors, who, instead of lessening the labors of the student, have strown his path with new difficulties, and harassed and perplexed him with unmeaning and tautological rules and expressions, insomuch that many have given up its pursuit on the very threshold, and others, having proceeded a short distance, and found new and increasing difficulties at every step, have closed the volume, and with it all attempts at mathematical studies, in hopeless despair. It may be here mentioned, that some of these works on algebra are mere copies of others, and in one instance a person of this city, though, to our credit be it spoken, not an American, having republished a European work, claimed the merit of it as his own, after having copied verbatim from another, and with so little judgment that the very errors of the press, in the London edition, appear glaringly, and without comment, in the work which this individual (who shall be nameless) claimed as exclusively his own. As it respects the method which should be adopted in pursuing the study of algebra, it should be remarked, that it is of primary importance that the student should in the first place make himself master of the common rules of arithmetic. Unless this is accomplished, it would be mere waste of time to attempt proceeding in the study of this science, which, although beautifully simple in its rules, is precise and accurate in its investigations. It is also the nature of all mathematical sciences, and of algebra in particular, to advance in continued progression, patiently but steadily; and hence the obvious necessity of the student's learning every thing thoroughly, and of perfectly understanding every rule and question as he advances, before he proceeds to another. By these means the study, instead of being a toil, will become a pleasure, what at first appeared difficult will become easy, and the student will find new beauties allure him at every step, and cheer him through every difficulty. Let the diligent student also bear in mind that genius without application is useless, and that continued and untiring perseverance can accomplish almost every thing, however arduous it may appear at the outset. When, however, to a genius and a taste for mathematical pursuits is added a persevering industry, the juvenile mind ascends the ladder of science to the topmost round, and gazes eagle-eyed on that imperishable tablet where are recorded the names of a Newton, Bowditch, Galileo, Leibnitz, Lagrange, and Laplace, who, by their works, have not only shed a lustre upon the science, but have gained for themselves an immortality that shall endure till time shall be no longer. Our own country too, though it is comparatively young, and has made but little progress in the mathematical sciences, has yet produced a few men who would have done honor to any age or country; and when the green sod of the valley shall have covered their mortal remains, when the present generation shall have passed away and been forgotten, their names will acquire unfading lustre, and be hailed by generations yet unborn, as the luminaries of science, and as the benefactors of the human race. Posterity must pass their eulogia, to posterity they must look for their fame and their immortality. And here, could I find suitable language, I might pay a passing tribute to the memory of him who sleeps beneath the waves of the Atlantic, the young, the accomplished, the lamented Fisher. * But what avails it? He hath passed from his sphere of usefulness; his bright and glorious career is finished; but his memory yet lives, and will be cherished in the hearts of his countrymen as a legacy never to be forgotten. It only remains now for the author to state the reasons which have induced him to this compilation, and the manner in which it has been treated. * He was a passenger in the packet-ship Albion, lost on the coast of Ireland. To make an excuse, or to offer an apology, for a work of this nature, is, I believe, unnecessary, because, if it should prove unworthy of the public patronage, no excuse will palliate its defects, and no prefatorial apology will be received in extenuation for accidental or wilful error. Hence, apology is needless; but the author believes it to be a duty he owes to the public to state the reasons which have caused this compilation, and they are briefly these : In his own opinion, and in that of some of the best mathematicians of our city, the different algebras at present in use in this country are defective in many particulars. The rules, it is believed, are generally not the best that could be given, and in some cases are tediously abstruse and perplexing; and though there is generally an ample sufficiency of theory, yet there is not practice enough to engage without wearying the attention of the student, and to excite without overburdening his reasoning faculties. These objections it has been the aim of the author to remove, and to treat the subject on a clear and rational foundation, with as much simplicity as possible, and, instead of making a mystery of the science, and putting new and imaginary difficulties in the way of the student, to remove them by every possible method. For these reasons the questions are worked out at full length, and every thing explained in the operation; and it is believed that the time is gone by when the best method of teaching algebra was thought to be by giving the student a difficult question, and leaving him to ponder and pore for weeks over a set of, to him, unmeaning symbols and figures. The solutions of the greatest part of the difficult questions, being all given, will be of utility, also, in an economical point of view; as to all the algebras now in use, Keys, containing solutions, have been published, and by the necessity of purchasing them the student is subjected to additional expense. The student will find a greater number of new equations than have ever been published in any treatise before. As they are problems which possess a degree of interest to the mind of the young student, it has been thought they would prove highly useful by blending amusement with study. The method of solving the irreducible case of cubic equations is new and of great importance, as it solves the most difficult questions with the greatest ease. The author, in conclusion, would take this opportunity of returning his grateful thanks to the public for the liberal patronage bestowed on his former publications; and in offering this new work to their notice, he fearlessly depends on the candor and impartiality of an enlightened and liberal public. JOHN D. WILLIAMS. DIGHTON, March 1, 1839. CONTENTS. Page. Examples of Algebraic Expres- view 147 4 and 5 A plain demonstration to Emer- 5-10 son's Rule, given in 1764, 10 which Horner, Holdred, Nich- Multiplication, three Cases 12-18 olson and Young all thought General Rule from the foregoing. - 25 Solution of Quadratic Equations by the table of sines and tan- 41 Demonstration of the Rule of Division of Algebraic Fractions 48 Harmonical Progression 173 Miscellaneous Questions that pre- A new rule for Cubics. cede and follow are from Bony- Biquadratic Equations . 195, 196 197 . 85 Solutions of Cubics and of the 105 higher Equations to the eighth Eighty-three Questions and Solu- degree inclusive tions as an exercise 108 to 125 Irrational Quantities or Surds 226 On the solution of Simple Equa- Cubic Equations tions which involve more than Solution of Cubics by Carden 128 Solution of Cubic Equations by the 128 table of sines and tangents 129 Examples for Practice . .249, 250 Thirty Questions and Solutions 130 Cubic Equations, converging se- Dr. Pell's challenge to Dr. Wal- The final Bequadratic Equation, Equations by approximation, &c. 256 solved by Rule page 194, 139| Reciprocal Equations Quadratic Equations at the bot- Reciprocal Equations by Simson's tom of this page to be omitted 139 method Pure Quadratic Equations, and Equations that have equal roots 269 Hindue Method: 141 Question on page 273. See my 142 Key to Hutton's Mathematics Rules for the solution of Difficult and Arithmetical Amusements . |