A Treatise on Algebra

(Article 1.) WHATEVER is capable of increase or diminution or will admit of mensuration, is called magnitude or quantity. A sum of money, therefore, is a quantity, since we may increase it or diminish it. A line, a surface, a weight, and other things of this nature, are quantities; but an idea is not a quantity.

(2.) Mathematics is the science of quantity, or the science which investigates the means of measuring quantity. The operations of the mind, therefore, such as memory, imagination, judgment, &c., are not subjects of mathematical investigation, since they are not quantities.

(3.) Mathematics is divided into pure and mixed. Pure mathematics comprehends all inquiries into the relations of magnitude in the abstract, and without reference to material bodies. It embraces numerous subdivisions, such as Arithmetic, Algebra, Geometry, &c.

In the mixed mathematics these abstract principles are ap plied to various questions which occur in nature. Thus, in Surveying, the abstract principles of Geometry are applied to the measurement of land; in Navigation, the same principles are applied to the determination of a ship's place at sea; in Optics, they are employed to investigate the properties of light; and in Astronomy, to determine the distances of the heavenly bodies.

(4.) Algebra is that branch of mathematics which enables us, by means of letters and other symbols, to abridge and generalize the reasoning employed in the solution of all questions relating to numbers.

Arithmetic is the art or science of numbering. It treats of the nature and properties of numbers, but it is limited to certain methods of calculation which occur in common practice. Algebra is more comprehensive, and has been called by Newton, Universal Arithmetic.

(5.) The following are the main points of difference between Arithmetic and Algebra.

First, the operations of Algebra are more general than those of Arithmetic. In Arithmetic we represent quantities by particular numbers, as 2, 5, 7, &c., which numbers always retain the same value. The results obtained, therefore, are applicable only to the particular question proposed. Thus, if it is required to find the interest of a thousand dollars for three months at six per cent., the question may be solved by Arithmetic, and we obtain an answer, which is applicable only to this problem.

But in the solution of a general Algebraic problem we employ letters, to which any value may be attributed at pleasure. The results obtained, therefore, are equally applicable to all questions of a particular class. Thus, if we have given the sum and difference of two quantities, we may obtain by means of Algebra a general expression for the quantities themselves. This result will always be found true, whatever may be the magnitude of the quantities. Hence Algebra is adapted to the investigation of general principles, while Arithmetic is confined to operations upon particular numbers.

Secondly, Algebra enables us to solve a vast number of problems, which are too difficult for common Arithmetic. Some of the problems in Sections VII. and VIII. may be solved by Arithmetical methods; but others can not thus be resolved, particularly such problems as are given in Sections XII., XIV., &c.

Thirdly, in Arithmetic all the different quantities which enter into a problem are blended together in the result, so as to leave no trace of the operations to which they have been subjected. From a simple inspection of the result, we can not tell whether it was derived by multiplication or division, involution or evolution, or what connection it has with the given quantities of the problem. But in a general Algebraic solution, all the different quantities are preserved distinct from each

other, and we see at a glance how all the data of the problem are combined in the result. Illustrations of this remark will be found in Section VII., &c.

Fourthly, the operations of Algebra are often far more concise than those of Arithmetic. Thus, although some of the problems in Sections VII. and VIII. may be solved Arithmetically, these solutions are generally much more tedious than the Algebraic. This advantage which is possessed by Algebra is partly due to the representation of the unknown quantities by letters, and their introduction into the operations. as if they were already known, and partly to the fact that the operations of multiplication, division, &c., are at first merely indicated, and are not actually performed until an Algebraic expression has been reduced to its simplest form.

Finally, perhaps the most striking difference between Arithmetic and Algebra springs from the use of negative quantities, which give rise to many peculiar results.

The full purport of these remarks will be best apprehended after the student has made some progress in the study of Algebra.

(6.) A definition is the explanation of any term or word. It is essential to a perfect definition that it distinguish the thing defined from every thing else. Thus, if we say that man is a biped, it is an imperfect definition of man, because there are many other bipeds.

(7.) A theorem is the statement of some property, the truth of which is required to be proved. Thus the principle that the sum of the three angles of any triangle is equal to two right angles, is a theorem, the truth of which is demonstrated by Geometry.

(8.) A problem is a question requiring something to be done. Thus, to draw one line perpendicular to another is a problem. Theorems and problems are both known by the general term of propositions.

(9.) A determinate problem is one which admits of a certain or definite answer. An indeterminate problem commonly admits of an indefinite number of solutions; although when the answers are required in positive whole numbers, they are in some cases confined within certain limits, and in others the problem may be impossible.

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