ELEMENTARY ALGEBRA. 1. I. ADDITION AND SUBTRACTION. ALGEBRA is the science which teaches the use of SYMBOLS to denote numbers and the operations to which numbers may be subjected. 2. The symbols employed in Algebra to denote numbers are, in addition to those of Arithmetic, the letters of some alphabet. ......: α, Thus a, b, c x, y, z: a, ß, Y b one, c one...... are used as symbols to denote numbers. 3. The number one, or unity, is taken as the foundation of all numbers, and all other numbers are derived from it by the process of addition. Thus two is defined to be the number that results from adding one to one; three is defined to be the number that results from adding one to two; four is defined to be the number that results from adding one to three; and so on. 4. The symbol +, read plus, is used to denote the operation of ADDITION. and Thus 1+1 symbolizes that which is denoted by 2, 2+1 3, a+b stands for the result obtained by adding b to a. 1 5. The symbol stands for the words "is equal to," or "the result is." Thus the definitions given in Art. 3 may be presented in an algebraical form thus: 1+1=2, 2+1=3, 3+1=4. 6. Since 2=1+1, where unity is written twice, 3=2+1=1+1+1, where unity is written three times, 4=3+1=1+1+1+1 .four times, it follows that a=1+1+ 1 +1+1 with unity written a times, +1+1 with unity written b times. 7. The process of addition in Arithmetic can be presented in a shorter form by the use of the sign +. Thus if we have to add 14, 17, and 23 together we can represent the process thus: 14+17+23 = 54. 8. When several numbers are added together it is indifferent in what order the numbers are taken. Thus if 14, 17, and 23 be added together their sum will be the same in whatever order they be set down in the common Arithmetical process: So also in Algebra when any number of symbols are added together, the result will be the same in whatever order the symbols succeed each other. Thus if we have to add together the numbers symbolized by a and b, the result is represented by a+b, and this result is the same number as that which is represented by b+a. Similarly the result obtained by adding together a, b, c might be expressed algebraically by a+b+c, or a+c+b, or b+a+c, or b+c+a, or c+a+b, or c+b+a. 9. When a number denoted by a is added to itself, the result is represented algebraically by a + a. This result is for |