An Introduction to Algebra

be to call into exercise, to discipline, and to invigorate the powers of the mind. It is the logic of the mathematics which constitutes their principal value, as a part of a course of collegiate instruction. The time and attention devoted to them, is for the purpose of forming sound reasoners, rather than expert mathematicians. To accomplish this object, it is necessary that the principles be clearly explained and demonstrated, and that the several parts be arranged in such a manner, as to show the dependence of one upon another. The whole should be so conducted, as to keep the reasoning powers in continual exercise, without greatly fatiguing them. No other subject affords a better opportunity for exemplifying the rules of correct thinking. A more finished specimen of clear and exact logic has, perhaps, never been produced, than the Elements of Geometry by Euclid.

It may be thought, by some, to be unwise to form our general habits of arguing, on the model of a science in which the inquiries are accompanied with absolute certainty; while the common business of life must be conducted upon probable evidence, and not upon principles which admit of complete demonstration. There would be weight in this objection, if the attention were confined to the pure mathematics, But when these are connected with the physical sciences, astronomy, chemistry, and natural philosophy, the mind has opportunity to exercise its judgment, upon all the various degrees of probability which occur in the concerns of life.

So far as it is desirable to form a taste for mathematical studies, it is important that the books by which the student is first introduced to an acquaintance with these subjects, should not be rendered obscure and forbidding by their conciseness. Here is no opportunity to awaken interest, by rhetorical elegance, by exciting the passions, or by presenting images to the imagination. The beauty of the mathematics depends on the distinctness of the objects of inquiry, the symmetry of their relations, the luminous nature of the arguments, and the certainty of the conclusions. But how is this beauty to be perceived, in a work which is so much abridged, that the chain of reasoning is often interrupted, many difficulties left unexplained, important demonstrations omitted, and the transitions from one subject to another so abrupt, as to keep their connections and dependencies out of view?

It may not be necessary to state every proposition and its proof, with all the formality which is so strictly adhered to

by Euclid; as it is not essential to a logical argument, that it be expressed in regular and entire syllogisms. A step of a demonstration may be safely left out, when it is so simple and obvious, that no one possessing a moderate acquaintance with the subject, could fail to supply it for himself. But this liberty of omission ought not to be extended, to cases in which it might occasion obscurity and embarrassment. If it be desirable to give opportunity for the mind to display and enlarge its powers, by surmounting obstacles; full scope may be found for this kind of exercise, especially in the higher branches of the mathematics, from difficulties which will unavoidably occur, without, creating new ones for the sake of perplexing,

The purpose for which abridged compilations are commonly made is, probably, to save time. The expense of an additional volume or two, in that part of a public education which is to occupy a large portion of three or four years, can hardly be supposed to be an object of great comparative importance. The principal saving of expense, in this case, is included in the saving of time. But is not the progress of the student impeded, rather than accelerated, by abridgments? The time requisite to become master of a subject, is not always proportioned to the number of pages which it occupies. Hours may be spent, in supplying an explanation, or an article of proof, which, if it had been inserted in its place, might have been read and understood, in a few min

utes.

Algebra requires to be treated in a more plain and diffuse manner, than some other parts of the mathematics; because it is to be attended to, early in the course, while the mind of the learner has not been habituated to a mode of thinking so abstract, as that which will now become necessary. He has also a new language to learn, at the same time he is settling the principles upon which his future inquiries are to be conducted. These principles ought to be established, in the most clear and satisfactory manner which the nature of the case will admit of. Algebra and geometry may be considered as lying at the foundation of the succeeding branches of the mathematics, both pure and mixed. Euclid and others have given to the geometrical part, a degree of clearness and precision which would be very desirable, but is hardly to be expected, in algebra.

For the reasons which have been mentioned, the manner in which the following pages are written, is not the most

concise. But the work is necessarily limited in extent of subject. It is far from being a complete treatise of algebra. It is merely an introduction. It is intended to contain as much matter, as the student at college can attend to, with advantage, during the short time allotted to this particular study. There is generally but a small portion of a class, who have either leisure or inclination, to pursue mathematical inquiries much farther, than is necessary to maintain an honourable standing, in the institution of which they are members. Those few who have an unusual taste for this science, and aim to become adepts in it, ought to be referred to separate and complete treatises, on the different branches. one who wishes to be thoroughly versed in mathematics, should look to compendiums and elementary books, for any thing more, than the first principles. As soon as these are acquired, he should be guided in his inquiries, by the genius and spirit of original authors.

In the selection of materials, those articles have been taken which have a practical application, which are not of very difficult comprehension, and which are preparatory to succeeding parts of the mathematics, philosophy, and astronomy. The object has not been, to introduce original matter. In the mathematics, which have been cultivated with success, from the days of Pythagoras, and in which the principles already established are sufficient to occupy the most active mind for years, the parts to which the student ought first to attend, are not those recently discovered. Free use has been made of the works of Newton, Maclaurin, Saunderson, Simpson, Euler, Emerson, Lacroix, and others, but in a way that rendered it inconvenient to refer to them, in particular instances. The proper field for the display of mathematical genius, is in the region of invention. But what is requisite for an elementary work, is to collect, arrange, and illustrate, materials already provided. However humble this employment, he ought patiently to submit to it, whose object is to instruct, not those who have made considerable progress in the mathematics, but those who are just commencing the study. Original discoveries are not for the benefit of beginmers, though they may be of great importance to the advancement of science.

The arrangement of the parts is such, that the explanation of one is not made to depend on another which is to follow. The addition, multiplication, and division of powers, for instance, is placed after involution. In the statement of

gene

ral rules, if they are reduced to a small number, their applications to particular cases may not, always, be readily understood. On the other hand, if they are very numerous, they become tedious and burdensome to the memory. The rules given, in this introduction, are most of them comprehensive; but they are explained and applied, in subordinate articles.

A particular demonstration is sometimes substituted for a general one, when the application of the principle to other cases is obvious. The examples are not often taken from philosophical subjects, as the learner is supposed to be familfar with none of the sciences except arithmetic. In treating of negative quantities, frequent references are made to mercantile concerns, to debt and credit, &c. These are merely for the purpose of illustration. The whole doctrine of negatives is made to depend on the single principle, that they are quantities to be subtracted. But the student, at this early period, is not accustomed to abstraction. He requires particular examples, to catch his attention, and aid his conceptions.

The section on proportion will, perhaps, be thought useless to those who read the fifth book of Euclid. That is sufficient for the purposes of pure geometrical demonstration. But it is important that the propositions should also be presented, under the algebraic forms. In addition to this, great assistance may be derived from the algebraic notation, in demonstrating, and reducing to system, the laws of proportion. The subject, instead of being broken up into a multitude of distinct propositions, may be comprehended in a few genesal principles.