INVOLUTION BY LOGARITHMS. RULE. TAKE out the logarithm of the given number from the table. Multiply the log. thus found, by the index of the power proposed. Find the number answering to the duct, and it will be the power required. pro Note. In multiplying a logarithm with a negative index, by an affirmative number, the product will be negative. But what is to be carried from the decimal part of the logarithm, will always be affirmative. And therefore their difference will be the index of the product, and is always to be made of the same kind with the greater. Here 4 times the negative index being-8,and 3 to carry, the difference - 5 is the index | Power 5.14932 of the product. Log. 4. To raise 1.0045 to the 365th power. Numb. Root 1.0045 Log. 0.001950 4 The index - - 365 9750 11700 5850 0.711750 EVOLUTION EVOLUTION BY LOGARITHMS. TAKE the log: of the given number out of the table. Divide the log. thus found by the index of the root. Then the number answering to the quotient, will be the root. Note. When the index of the logarithm, to be divided, is negative, and does not exactly contain the divisor, without some remainder, increase the index by such a number as will make it exactly divisible by the index, carrying the units borrowed, as so many tens, to the left-hand place of the decimal, and then divide as in whole numbers. Ex. 1. To find the square root Ex. 2. To find the 3d root of Here the divisor 3, not being exactly contained in -4, it is augmented by 2, to make up 6, in which the divisor is contained just 2 times; then the 2, thus borrowed, being carried to Here the divisor 2 is contained exactly once in the ne gative index -2, and therefore the index of the quotient the decimal figure 6, makes 26, which is -1. divided by 3, gives 8, &c. 6 Ex. 7. To find 3·1416 x 82 × 11. ALGEBRA. ALGEBRA. DEFINITIONS AND NOTATION. ALGEBRA LGEBRA is the science of computing by symbols, It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation. 2. In this science, quantities of all kinds are represented by the letters of the alphabet. And the operations to be per formed with them, as addition or subtraction, &c, are denoted by certain simple characters, instead of being expressed by words at length. 3. In algebraical questions, some quantities are known or given, viz. those whose values are known: and others unknown, or are to be found out, viz. those whose values are not known. The former of these are represented by the leading letters of the alphabet, a, b, c, d, &c; and the latter, or unknown quantities, by the final letters, z, y, x, u, &c., 4. The characters used to denote the operations, are chiefly the following: + signifies addition, and is named plus.. signifies subtraction, and is named minus. or. signifies multiplication, and is named into. ✔signifies the square root; the cube root; the 4th root, &c; and the nth root. :::: signifies proportion. signifies equality, and is named equal to. And so on for other operations. Thus a + b denotes that the number represented by bis to be added to that represented by a. a-b denotes, that the number represented by b is to be subtracted from that represented by a. a b denotes the difference of a and b, when it is not known which is the greater. VOL. I. M ab, or ab, or a xb, or a.b, expresses the product, by multiplication, of the numbers represented by a and b. a a÷b, or, denotes, that the number represented by ☛ is to be divided by that which is expressed by b. abcd, signifies that a is in the same proportion to b, as c is to d. x = a b + c is an equation, expressing that x is equal to the difference of a and b, added to the quantity c. √✅a, or a3, denotes the square root of a; ya, or a3, the cube root of a; and a2 or a3 the cube root of the square of a; also a, or a", is the mth root of a; and "a" or am is the power of the mth root of a, or it is a to the power. nth I n m a2 denotes the square of a; a3 the cube of a; at the fourth power of a ; and a the nth of a. power a+bxc, or (a+b) e, denotes the product of the compound quantity a+b multiplied by the simple quantity c. Using the bar, or the parenthesis () or the parenthesis ( ) as a vinculum, to connect several simple quantities into one compound. ↓ab+cd, or (ab + cd), is the square root of the compound quantity ab + cd. And cab + cd, or c (ab + cd)3, denotes the product of c into the square root of the compound quantity abcd. -3 a+b-c, or (a + b)3, denotes the cube, or third power, of the compound quantity a+bc. 3a denotes that the quantity a is to be taken 3 times, and 4(a + b) is 4 times a + b. And these numbers, 3 or 4, showing how often the quantities are to be taken, or multiplied, are called Co-efficients. Also denotes that x is multipled by ; thus xx or ×. 5. Like Quantities, are those which consist of the same letters, and powers. As a and 3a; or 2ab and 4ab; or 3a2bc and -5a2bc. 6. Unlike Quantities, are those which consist of different letters, or different powers. As a and b ; or 2a and a2; or Bab and 3abc. 7. Simple 7. Simple Quantities, are those which consist of one term only. As 3a, or 5ab, or 6abc. 8. Compound Quantities, are those which consist of two or more terms. As a + b, or 24 3c, or a + 2b - 3c. 9. And when the compound quantity consists of two terms, it is called a Binomial, as a +b; when of three terms, it is a Trinomial, as a + 2b 3c; when of four terms, a 4d; and so on. Multinomial or Polynomial, consists of many terms. Quadrinomial, as 2a 3b+ → Also, a 10. A Residual Quantity, is a binomial having one of the terms negative. As a 26. 11. Positive or Affirmative Quantities, are those which are to be added, or have the sign +. As a or +a, or ab: for when a quantity is found without a sign, it is understood to be positive, or have the sign + prefixed. 12. Negative Quantities, are those which are to be sub tracted. As a, or -2ab, or -3ab2. 13. Like Signs, are either all positive (+), or all negative (-). 14. Unlike Signs, are when some are positive (+), and others negative (−). 15. The Co-efficient of any quantity, as shown above, is the number prefixed to it. As 3, in the quantity 3ab. 16. The Power of a quantity (a), is its square (a2), or cube (a), or biquadrate (a), &c; called also, the 2d power, or 3d power, or 4th power, &c. 17. The Index or Exponent, is the number which denotes the power or root of a quantity. So 2 is the exponent of the square or second power a2; and 3 is the index of the I 144 cube or 3d power; and is the index of the square root, a ora; and is the index of the cube root, a3, or ya. I 18. A Rational Quantity, is that which has no radical sign (√) or index annexed to it. As a, or 3ab. 19. An Irrational Quantity, or Surd, is that which has not an exact root, or is expressed by means of the radical sign√. As 2, or /a, or 3a2, or ab. As√2, 20. The Reciprocal of any quantity, is that quantity inverted, or unity divided by it. So, the reciprocal of a, or is b |