50..55. Equations of the First Degree involving two or 90...91. Calculus of Radicals of the Second Degree. 111..112. Properties of Trinomials of the Second Degree. 113. Applications of the preceding properties to ques- 114..116. Equations and Problems of the Second Degree 275. Of the Greatest Relative Common Divisor of two Rational and Entire Polynomials. 276..277. Transformation of Equations. 278..279. Remark upon the preceding Transformation. Of Series of Figurate Numbers. Summation rences. ALGEBRA. INTRODUCTION. 1. ALGEBRA is that branch of mathematics in which proper signs are employed to abridge and generalize the reasoning required in the resolution of questions relating to numbers. There are two kinds of questions, viz. theorems and problems. If it is required to demonstrate the existence of certain properties relating to known and given numbers, the question is called a theorem; but if it is proposed to determine certain numbers from the knowledge of others, which have with the first relations indicated by the enunciation, the question is called a problem. 2. The following are the principal elements made use of in algebra to accomplish this double object. 1st. The letters of the alphabet, which are used to denote the numbers upon which we are to reason. They are used both to abridge the reasoning, and generalize it; since, by employing them we feel more sensibly that such or such a property belongs to several numbers at the same time; or if the question is a problem, that the manner of satisfying the enunciation is independent of any particular values attributed to the numbers comprehended in it. 2d. The sign+, which is used to indicate the addition of two or more quantities, and is called plus. Thus, 25+36 is read: 25 plus 36, or 25 augmented by 36. In like manner a + b, is read: a plus b, or the number denoted by a augmented by the number denoted by b. 3d. The sign, which is called minus, and is used to indicate the subtraction of one number from another. Thus 45 24 is read: 45 minus 24, or 45 diminished by 24 ; or, again, the difference between 45 and 24. ab is read: a minus b, or a diminished by b. |