CUBIC EQUATIONS, CONTAINING ONLY THE THIRD AND SECOND POWERS CUBIC EQUATIONS, CONTAINING ONLY THE THIRD AND FIRST POWERS 329 ALGEBRA. SECTION I. DEFINITIONS AND NOTATIONS. ARTICLE 1. ALGEBRA is the art of computing by symbols. 2. In Arithmetic we represent quantities and perform calculations by figures, each of which has a known and definite value. 3. In Algebra we employ the letters of the alphabet, and other characters, the value of which is either known or unknown, according to the conditions of the problems. 4. Those quantities whose values are given are called known quantities; and those whose values are not given are unknown quantities. 5. The symbols used to denote known quantities are, generally, the first letters of the alphabet in the small or Italic character, as a, b, c, d, &c.; and those used to denote unknown quantities, the last letters, as w, x, y, z. 6. In addition to the above, which are the more common symbols, capital letters may be used, as A, B, C, D, &c., or letters of the Greek alphabet, as u, ß, y, d, e, ¿, &c. In extensive operations, the use of these, or some other suitable characters, is sometimes very convenient. 7. Sometimes quantities are expressed in Algebra, as in Arithmetic, by figures instead of letters. 8. When a quantity is doubled or trebled, or multiplied any number of times, the number of multiplications is usually expressed by a numerical figure or figures. Thus, let a denote a certain quantity, and 2a will denote twice the same quantity, 3a three times the same quantity, &c. 9. The figures or letters prefixed to any symbol, and denoting the number of times the quantity represented by the symbol is taken, are called the coefficient. Thus, in 4a, 76, and 4ax, the coefficients are 4, 7, and 4a. 10. A quantity which has no figure prefixed to it is considered as having a unit for its coefficient. Thus, a is the same as la. 11. Quantities represented by the same symbol or letters, and of the same power, are called like quantities; and those represented by different symbols or letters, or by the same letter of different powers, unlike quantities. Thus, 3a, 4a, and 5a, are like quantities; and 3a, 4b, and 5c, unlike quantities. In like manner, 3ab, 4ab, and 6ab, are like quantities; and 3ab, 4ac, bdc, and 6mn, are unlike quantities. 12. Besides the symbols and figures used to denote quantity, there are certain signs, which are used to express the different relations between quantities, and the operations to which these quantities are subjected. These signs are the same as are often employed in Arithmetic, but, in Algebra, they are indispensable. 13. The sign = is that of equality, and denotes that the two quantities between which it is placed are equal to each other. Thus, a 26 signifies that a is equal to 26. 14. The sign is called plus, and signifies addition. Thus, a+b signifies that a is to be added to b. 15. The sign is called minus, and signifies subtraction. Thus a-b signifies that b is to be subtracted from a. occur before the 16. Sometimes both the signs and same quantity, as a±x, in which case they signify that the quantity may be either added or subtracted, or that it is doubtful which operation is to be performed. 17. The sign × signifies multiplication. Thus, ab denotes |