PREFACE. SOME apology inay appear requisite for offering a new book to the public on the science of Algebra-especially as there are several works of acknowledged merit on that subject already before the public, claiming attention. But the intrinsic merits of a book are not alone sufficient to secure its adop tion, and render it generally useful. In addition to merit, it must be adapted to the general standard of scientific instruction given in our higher schools; it must conform in a measure to the taste of the nation, and correspond with the general spirit of the age in which it is brought forth. The elaborate and diffusive style of the French, as applied to this science, can never be more than theoretically popular among the English; and the se vere, brief, and practical methods of the English are almost intolerable to the French. Yet both nations can boast of men highly pre-eminent in this science, and the high minded of both nations are ready and willing to acknowledge the merits of the other; but the style and spirit of their respective productions are necessarily very different. In this country, our authors and teachers have generally adopted one or the other of these schools, and thus have brought among us difference of opinion, drawn from these different standards of measure for true excellence. Very many of the French methods of treating algebraic science are not to be disregarded or set aside. First principles, theories and demonstrations, are the essence of all true science, and the French are very elaborate in these. Yet no effort of individuals, and no influence of a few institutions of learning, can change the taste of the American people, and make them assimilate to the French, any more than they can make the entire people assume French vivacity, and adopt French manners. Several works, modified from the French, have had, and now have consid erable popularity, but they do not naturally suit American pupils. They are not sufficiently practical to be unquestionably popular; and excellent as they are, they fail to inspire that enthusiastic spirit, which works of a more practical and English character are known to do. At the other extreme are several English books, almost wholly practical, with little more than arbitrary rules laid down. Such books may in time make good resolvers of problems, but they certainly fail in most instances to make scientific algebraists. The author of this work has had much experience as a teacher of algebra, and has used the different varieties of text books, with a view to test their comparative excellencies, and decide if possible on the standard most proper to be adopted, and of course he designed this work to be such as his experience and judgment would approve. One of the designs of this book is to create in the minds of the pupils a love for the study, which must in some way be secured before success can be attained. Small works designed for children, or those purposely adapted to persons of low capacity, will not secure this end. Those who give tone to public opinion in schools, will look down upon, rather than up to, works of this kind, and then the day of their usefulness is past. On the other hand, works of a high theoretical character are apt to discourage the pupil before his acquirements enable him to appreciate them, and on this account alone such works are not the most proper for elementary class books. This work is designed, in the strictest sense, to be both theoretical and practical; and therefore, if the author has accomplished his design, it will be found about midway between the French and English schools. In this treatise will be found condensed and brief modes of operation, not hitherto much known or generally practised, and several expedients are systematised and taught, by which many otherwise tedious operations are avoided. Some applications of the celebrated problems of the couriers, and also of the lights, are introduced into this work, as an index to the pupil of the subsequent utility of algebraic science, which may allure him on to more thorough investigations, and more extensive study. Such problems would be more in place in text books on natural philosophy and astronomy than in an elementary algebra, but the almost entire absence of them in works of that kind, is our apology for inserting them here, if apology be necessary. Quite young pupils, and such as may not have an adequate knowledge of physics and the general outlines of astronomy, may omit these articles of application; but in all cases the teacher alone can decide what to omit and what to teach. Within a few years many new text books on algebra have appeared in different parts of the country, which is a sure index that something is desired— something expected,-not yet found. The happy medium between the theoretical and practical mathematics, or, rather, the happy blending of the two, which all seem to desire, is most difficult to attain; hence, many have failed in their efforts to meet the wants of the public. Metaphysical theories, and speculative science, suit the meridians of France and Germany better than those of the United States. But it is almost impossible to comment on this subject without being misapprehended; the author of this book is a great admirer of the pure theories of algebraical science, for it is impossible to be practically skillful without having high theoretical acquirements. It is the man of theory who brings forth practical results, but it is not theory alone-it is theory long and well applied. Who will contend that Watt, Fitch, or Fulton, were ignorant or inattentive to every theory concerning the nature and power of steam, yet they are only known as practical men, and it is almost in vain to look for any benefactors of mankind, or any promoters of real science from those known only as theorists, or among those who are strenuous contenders for technicalities and forms. We are led to these remarks to counteract, in some measure, if possible, that false impression existing in some minds, that a high standard work on algebra, must necessarily be very formal in manner and abstrusely theoretical in matter; but in our view these are blemishes rather than excellencies. The author of this work is a great advocate for brevity, when not purchased at the expense of perspicuity, and this may account for the book appearing very small, considering what it is claimed to contain. For instance, we have only two formulas in arithmetical progression, and some authors have 20. We contend the two are sufficient, and when well understood cover the whole theory pertaining to the subject, and in practice, whether for absolute use or lasting improvement of the mind, are far better than 20. The great number only serves to confuse and distract the mind; the two essential ones, can be remembered and most clearly and philosophically comprehended. The same remarks apply to geometrical progression. In the general theory of equations of the higher degrees this work is not too diffuse; at the same time it designs to be simple and clear, and as much is given as in the judgment of the author would be acceptable, in a work as elementary and condensed as this; and if every position is not rigidly demonstrated, nothing is left in obscurity or doubt. We have made special effort to present the beautiful theorem of Sturm in such a manner as to bring it direct to the comprehension of the student, and if we have failed in this, we stand not alone. The subject itself, though not essentially difficult, is abstruse for a learner, and in our effort to render it clear we have been more circuitous and elaborate than we had hoped to have been, or at first intended. We may apply the same remarks to our treatment of Horner's method of solving the higher equations. Brevity is a great excellence, but perspicuity is greater, and, as a general thing, the two go hand in hand; and these views have guided us in preparing the whole work; we have felt bound to be clear and show the rationale of every operation, and the foundation of every principle, at whatever cost. The Indeterminate and Diophantine analysis are not essential in a regular course of mathematics, and it has not been customary to teach them in many institutions; for these reasons we do not insert them in our text book. The teacher or the student, however, will find them in a concise form in a key to this work. |