have been selected from French and German works, and problems have been given with a view to arousing interest. For several lucid explanations the author is indebted to the large work of Briot, and for numerous examples in the advanced chapters to the great work of Heis. For any suggestions or correction relating to the body of the work or to the answers to examples, the author will be specially thankful. NEW YORK, Sept. 20, 1895. W. FREELAND. ALGEBRA. CHAPTER I. DEFINITIONS. 1. Algebra is that branch of mathematics in which letters and figures are employed to reason about the general properties of numbers. SYMBOLS. Algebraic symbols are as follows: 2. The Sign of Addition, +, is read "plus." Thus, 3+4 equals 7; again, a + b indicates that b is to be added to a. 3. The Sign of Subtraction, is read "minus." Thus, 7-3 equals 4; again, ab indicates that b is to be subtracted from a. 4. The Sign of Multiplication, x, is read "times," or "multiplied by." Thus, 3 × 4 equals 12; again, a × b indicates that a is to be multiplied by b. Between letters the sign of multiplication usually is not expressed. Thus, ab = a × b. 5. The Sign of Division,÷, is read "divided by." Thus, 34 equals ; again, ab indicates that a is to be divided by b. 6. The Sign of Equality,=, is read "equals," or "is equal to." Thus, 2 + 4 = 6; again, a = b indicates that a is equal to b. 7. The Signs of Inequality, > and <, are read "is greater than" and "is less than." Thus, 43; again, a <b indicates that a is less than b. 8. The Signs of Aggregation; viz.: the parentheses (); the vinculum ; the bracket [ ]; the brace { }. These signs show that the quantities included by them are to be taken together; i.e. treated as a single term. Thus, (a+b-c) × d, a + b − c × d, [a+b− c] × d, and {a+b-c} xd, indicate that the result of a + b c is to be multiplied by d. 9. The Sign of Involution is called the "exponent," and shows how many times the quantity affected is to be taken as a2 is read " a square"; a3 is read "a cube," or a to a factor. Thus, a2 = axa; b3 = b × b × b; (4)3 = 4 x 4 x 4 = 64. The exponent is written at the right, above the quantity. 10. The Sign of Evolution, √, is called the Radical Sign, and indicates that the root of a quantity is to be taken. Thus, √4=2; again, Vaa2; Va3 = a. 5 (a+b) indicates that the cube root of The figure written above at the left of the radical is called the Index. When the square root is to be taken, no index is expressed. 11. The Sign of Deduction, ..., is read "hence" or "therefore." 12. The Sign of Continuation, ..., is read "and so on." |