On the Flexibility, Strength, and Fracture of Timber,. NOTE. In the formula for measuring heights by the barometer, page 189, the ALGEBRA. DEFINITIONS. ALGEBRA is a general method of computation and of investigation, in which quantities are represented by letters, and their relations pointed out by characters. CHARACTERS EXPLAINED. 1. plus, is the sign of addition, as a+b signifies the quantity represented by b added to that represented by a. 2. minus, is the sign of subtraction, as ab denotes the quantity b taken from the quantity a. 3. x into, is the sign of multiplication, as a xb represents the product of a by b, or of b by a. Instead of this sign we often use a point, or write the letters together as in one word: thus ab or ab signifies a × b. 4. by, is the sign of division, but it is generally expressed by placing the dividend above the line and the divisor below it, in the form of a fraction: thus a÷b or signifies a divided by b. 5. a is the sign of proportion, as a : b :: c: d is read, As a is to b, so is c to d. = 6. equal to, is the sign of equality: thus a=b signifies a is equal to b. 7. are signs of greater and less: thus ab, a is greater than b; ab, a is less than b. 8. 7a. A number prefixed to a letter is called its coefficient, and shews how often the letter is to be taken; as here, 7 times a. 9. (a+b)xc. A parenthesis enclosing letters, or a line drawn over them, is called a vinculum, and points out how many are to be multiplied, divided, &c.; as here, the sum of a and b is to be multiplied by c. 10. aaa. When the same letter is repeated twice, or oftener, it is understood to be multiplied as often into itself, and the product is called a power of the quantity represented by that letter: thus aa is the second power or square of a, aaa is the third power of a, &c.; and in relation to these powers the quantity is called the first power of itself. 11. a3. Instead of repeating the same letter, we generally place a figure above it towards the right hand, to shew ho B often it is repeated; as a is the third power of a, at the fourth power, an the power of a denominated by the number n. 12. The character placed above is called the exponent of the power. 13. A quantity which, multiplied by itself, produces another quantity, is called the root of that other, and, in numbers, is generally denoted by the mark: thus 9 is the square root of 9, 3/8 is the cube root of 8, 1/81 is the fourth root of 81. 14. In letters, a fractional exponent is generally used to express the root, and then the upper figure denotes the power, and the under figure the root: thus a3 is the third root of the second power of a, a is the fourth root of the first power of a, or of a itself. Quantities which have the sign + before them are said to be positive or affirmative, and those which have the sign — negative. The following examples will illustrate these characters, and shew their use, in which any values may be affixed to the letters. Suppose a=12, b=3, c=2, d=4, e=5, ƒ=9, g=25, h=7, i=11, k=1. |