PREFACE. THE present treatise is intended, in some measure, as a sequel to the "Treatise on Arithmetic in Theory and Practice for the use of the Irish National Schools," which was prepared by me some years since, and to which, for the convenience of the learner, I frequently refer in the following pages-Arithmetic and Algebra being so intimately connected, that they mutually aid, and illustrate each other. Both treatises have been written, as nearly as the nature of the subject permitted, on the same plan: and, hence, all who have been accustomed to the one, will soon become familiar with the other. Those only, who have made a considerable progress in Arithmetic, should be deemed sufficiently advanced for the study of Algebra; nevertheless, a knowledge of arithmetical notation and numeration, the simple rules, and fractions, will be found sufficient to enable the learner easily to understand every thing in the present treatise since, as far as possible, I have rendered it independent of other subjects. As I have given every process, nearly in full, and have added every explanation and suggestion, which long experience had shown me to be useful, no difficulty is likely to occur, that will impede the progress of the learner, or be an obstacle to his success, during private study: and the Teacher's time will be spared for his other avocations, by his being enabled to verify, or correct, by means of the work itself, the various processes, as performed by those he instructs: while, to satisfy himself of their industry and improvement, he can multiply examples and exercises to any extent—at the same time, using what are given in the book, to ascertain the accuracy, or to correct the errors committed in working those he forms for the occasion. To attain this object, he has only to make, in his own mind, certain changes in the quantities which even a moderate amount of ingenuity and acquaintance with the subject will render perfectly easy. Thus, in giving an example to his class, he can change one or more of the letters, can multiply or divide one or more of the quantities by some number; and, when he examines what the pupils have done, bear in mind the modification he has effected in the question. At other times, he may add, or subtract such a quantity as will make the example, or exercise, new to the pupil, but will not prevent his testing the process, and verifying the result obtained. The treatise may be used in studying the examples; but not in working the exercises, nor in solving the problems, unless, after a careful attempt has been made, it is found that a difficulty occurs which cannot be surmounted without assistance. The pupil should never be permitted to consult it in class, nor during the instruction which he receives. JAMES WILLIAM M'GAULEY. Office of Education, Dublin, April, 1854. To multiply, when both multiplicand and multi- When the multiplicand is a compound, but the multiplier is a simple quantity, 42 |