PREFACE. THE object of the author, in composing this treatise, was to form an easy introduction to the first principles of algebraical reasoning; and also to embrace, in the same course, a popular exposition of the most important elements of arithmetic. And he believes that he has been enabled to combine the rudiments of both, in such a manner as to make the operations of one illustrate the principles of the other. In order that this method of treating the subject might preserve its chief advantage, especially in the initiatory course of the study; the work has been divided into two parts— Numeral Algebra and Literal Algebra.. In Numeral Algebra I have treated of the several arithmetical operations; first making them intelligible to very young pupils, and then exhibiting them under the algebraical notation. By this means, as every lesson in algebra is immediately preceded by corresponding numerical exercises, the transition from one to the other has been made so trifling, that the pupil will feel at each step that he has met with nothing more than what he has already made himself familiar with, in a different dress. Besides, as algebraical operations require the exercise of abstraction in a greater degree than the pupil is supposed to be accustomed to, I have taken care that the exercise on each of the fundamental rules, shall be followed by a selection of problems to be solved by equa tions. As mathematical questions of this kind are always pleasing to young pupils, this arrangement will serve to impart an interest to the study at the commencement, and also to preserve a taste for it through the whole work. Under the head of Literal Algebra, I have repeated, in a more strictly algebraical form, the principles which have. been explained in the preceding part of the work; and have shown some of their uses by applying them in the deduction and demonstration of several abstruse operations on numbers. But the great peculiarity of the book is, that it habituates the speech and the ear to mathematical language. In any study, it is necessary for beginners to receive such a course of training as will imprint upon their minds each new idea, as soon as it is apprehended. Learners in the mathematics especially, are accustomed to forget soon, both the names and the use of the signs; and also the arrangement of the several steps in the solution of their problems. On this account I have required the pupil always to repeat verbally the operation that he has performed; taking care to omit no part of the work that would hinder an auditor from understanding the reason for the several steps, and consenting to the just conclusions of the answer which has been obtained. It has been found by experience that this simple device enables the young pupil to acquire the science very easily; and while it impresses his lessons indelibly upon his memory, it also developes his genius, rectifies his inventive faculties, and imparts, as it were, a mathematical form to his mind; so much so, that he is generally capable of pursuing the subject afterwards by himself. In order to accomplish this end more perfectly, I have swelled the number of examples beyond the ordinary limits. These should be thoroughly mastered as the pupil proceeds. There must be no smattering in the beginning of a science if the learner is to continue the study. The author has found by long experience, that a book is sooner finished when each part has been made familiar to the mind, than when it has been superficially attended to. With regard to the arrangement of the several divisions, I have been careful to introduce first those principles that will be the most easily apprehended; and afterwards such others as would most naturally arise from the former if the study were entirely new. This method appears to be the best adapted for teaching the rudiments of a science; although in a succeeding text book, it is necessary that the arrangement of the several parts should be more systematic. On this account the advanced scholar must not be surprised to find in the middle of the book, what he has been accustomed to see near the beginning of other treatises. However, so much regard to a regularity of arrangement has been attended to, that the pupil will be assisted by the associations of method, both to understand and to remember. As the author wishes to bring the study of Algebra within the reach of common schools, he has endeavored to prepare this work, so that it may be studied by pupils who are not already adepts in arithmetic. And it is believed that such learners will not fail of obtaining, by a perusal of it, a full understanding of vulgar fractions, roots and powers, proportion, progression, and other numerical operations that are generally embraced in arithmetical treatises. ADVERTISEMENT. The foregoing is the Preface of the author's "Inductive Treatise on the Elementary Principles of Algebra." The first 146 pages of that book have been published in this form, in order to afford a cheap manual for those classes that do not wish to study beyond Simple Equations. In the present state of education, so much of Algebra should be studied by every pupil in our common schools. R. W. G. Before the pupil begins to perform the sums in this work, he is requested to make the following corrections: Page 17 line 15 erase comma in 130. 4311 instead of itself, read the multiplier. 26 41 66 48 line 1, for 30 read 80. 65 example 13, for 14x-63, read 14x+63. 30 66 14, for 7 read 3 15 3+2x 75 8, for 3+ read 66 10, for 6+2y, read 3+2y. 77 line 24, for 23, read 33. 86 example 6, for and +6y. 3y and -6y, read +3y 90 example 8, in ans. for 48, read 5s. 91 66 12, for 35, read 31. 93 line 4, for, read }. |