In the science of numbers, the ten characters, called figures, are the alphabet of the arithmetical language; the combinations of these characters constitute the pure language of arithmetic; and the principles of numbers which are unfolded by means of this, in connection with our common language, constitute

the science.

I Geometry, the signs which are employed to indicate the boundaries and forms of portions of space, are simply the straight line and the curve; and these, in connection with our common language, make up the language of Geometry: a science which treats of space, by comparing portions of it with each other, for the purpose of pointing out their proper ties and mutual relations.

Analysis is a general term embracing that entire portion of mathematical science in which the quantities considered are represented by letters of the alphabet, and the operations to be performed on them are indicated by signs.

Algebra, which is a branch of Analysis, is also a species of universal arithmetic, in which letters and signs are employed to abridge and generalize all processes involving numbers. It is divided into two parts, corresponding to the science and art of Arithmetic :

1st. That which has for its object the investigation of the properties of numbers, embracing all the processes of reasoning, by which new properties are inferred from known ones; and,

2d. The solution of all problems or questions involving the determination of certain numbers which are unknown, from their connection with certain others which are known or given,

In arithmetic, all quantity is regarded as consisting of parts, which can be numbered exactly or approximatively, and in this respect, possesses all the properties of numbers. Propositions, therefore, concerning numbers, have this remarkable peculiarity, that they are propositions concerning all quantities whatever. Algebra extends the generalization still further. A number is a collection of things of the same kind, without reference to the nature of the thing, and is generally expressed by figures. Algebraic symbols may stand for all numbers, or for all quantities which numbers represent, or even for quantities which cannot be exactly expressed numerically.

In Geometry, each geometrical figure stands for a class; and when we have demonstrated a property of a figure, that property is considered proved for every figure of the class. In Algebra, all numbers, all lines, all surfaces, all solids, may be denoted by a single symbol, a or x. Hence, the conclusions deduced by means of those symbols are true of all things whatever, and not like those of number and Geometry, true only for particular classes of things. The symbols of Algebra, therefore, should not excite in our minds ideas of particular things. The written characters, a, b, c, d, x, y,, 2, serve as the representatives of things in general, whether abstract or concrete, whether known or unknown, whether finite or infinite.

In the various uses which we make of these symbols, and the processes of reasoning carried on by means of them, the mind insensibly comes to regard them as things, and not as mere signs; and we constantly predicate of them the properties of things in general, without pausing to inquire what kind of

thing is implied. All this we are at liberty to do, since the symbols being the representatives of quantity in general, there is no necessity of keeping the idea of quantity continually alive in the mind; and the processes of thought may, without danger, be allowed to rest on the symbols themselves, and therefore, become to that extent, merely mechanical. But when we look back and see on what the reasoning is based, and how the processes have been conducted, we shall find that every step was taken on the supposition that we were actually dealing with things, and not with symbols; and that without this understanding of the language, the whole system is without signification, and fails.*

The quantities which are the subjects of the algebraic analysis may be divided into two classes: those which are known or given, and those which are unknown or sought. The known are uniformly represented by the first letters of the alphabet, a, b, c, d, &c.; and the unknown by the final letters, x, y, 2, v, &c.

Five operations, only, can be performed upon a quantity that will give results differing from the quantity itself: viz. 1st. To add a quantity to it;

2d. To subtract a quantity from it;

3d. To multiply it by a quantity;

4th. To divide it;

5th. To extract a root of it.

Five signs only, are employed to denote these operations. They are too well known to be repeated here. These, with

*Davies' Logic and Utility of Mathematics. § 278.

the signs of equality and inequality, together with the letters of the alphabet, are the elements of the algebraic language.

The interpretation of the language of Algebra is the first thing to which the attention of a pupil should be directed; and he should be drilled in the meaning and import of the symbols, until their significations and uses are as familiar as the sounds of the letters of the alphabet.

All the apprehensions, or elementary ideas, are conveyed to the mind by means of definitions and arbitrary signs; and every judgment is the result of a comparison of such impressions. Hence, the connection between the symbols and the ideas which hey stand for, should be so close and intimate, that the one hall always suggest the other; and thus, the processes of Algebra become chains of thought, in which each link fulfils the double office of a distinct and connecting proposition.

ELEMENTS OF ALGEBRA.

CHAPTER I.

DEFINITIONS AND PRELIMINARY REMARKS.

1. QUANTITY is anything which can be increased or dimin ished, and measured.

2. MATHEMATICS is the science which treats of the measurement and relations of quantities.

3. ALGEBRA is a branch of mathematics, in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs.

signs are called symbols.

The letters and

4. In algebra two kinds of quantities are considered:

1st. Known quantities, or those whose values are known or given. These are represented by the leading letters of the alphabet, as, a, b, c, &c.

2d. Unknown quantities, or those whose values are not given. They are denoted by the final letters of the alphabet, as, x, y, z, &c.

Letters employed to represent quantities are sometimes written with one or more dashes, as, a', b'', c''', x', y'', &c., and are read, a prime, b second, c third, x prime, y second, &c.

5. The sign +, is called plus, and when placed between two quantities, indicates that the one on the right is to be added to the one on the left. Thus, a + b is read a plus b, and indicates