326 MATHEMATICS. 



For example, ^/64 = (64)* = 4 ; since 4 x 4 x 4 = 64. We have 

 only room to add that by means of fractional exponents, the binomial 

 theorem serves also to develope or extract roots in general, in the 

 form of a series ; which, in these cases, is generally interminous, or 

 endless. 



5. The general Theory of Equations, depends on the principle 

 that every equation, having all the terms transposed to the first mem- 

 ber, and arranged according to the powers of the unknown quantity 

 x, may be regarded as the continued product of as many binomial 

 factors, x a, x b, x c, &c., as the degree of the equation, that 

 is, the highest exponent of x, denotes. Thus, if we have the equa- 

 tion # 3 + 2 x = 15, or x 2 + 2 x 15 = 0, it may be written thus, 

 (x + 5) (x 3) = ; and this equation will be satisfied, if we make 

 x = 3 ; since the factor x 3 then becomes zero, and reduces the 

 whole member to also : or if we make x = 5, the other factor 

 will become equal to zero, and the equation will be satisfied ; and 

 hence 3, and 5, are called the roots of the equation. Particular 

 rules have been discovered for resolving equations of the third and 

 fourth degrees ; but for those of higher degrees no general rules have 

 yet been discovered. Numerical equations, or those which contain 

 no other letters but the unknown quantity, may generally be resolved 

 by approximation : simple and quadratic equations being those of 

 far the most frequent occurrence. 



6. We have alluded to Arithmetical and Geometrical Series, or 

 Progressions, under the head of Arithmetic ; and have given examples 

 of other series, in the application of the binomial theorem. We have 

 only room left here to speak of Logarithms ; which are a series of 

 numbers in arithmetical progression, corresponding to the natural 

 numbers in geometrical progression. Their nature will best be un- 

 derstood by examining the following scale, in which the logarithms 

 are placed under the natural numbers to which they correspond. 

 C 1; 10; 100; 1000; 10,000; 100,000; 1,000,000. 

 ? 0; 1; 2; 3; 4; 5; 6. 



Thus, in the common system, 2 is the logarithm of 100 ; and the 

 logarithm of any number between 100 and 1000, is some decimal 

 between 2-00000 and 3-0000. It will be seen that adding the 

 logarithms, corresponds to multiplying the numbers ; and subtract- 

 ing the logarithms, corresponds to the division of one of the numbers 

 by the other. For example, subtracting 2 from 5, the difference is 3, 

 the logarithm of 1000; which is the quotient of 100,000 divided by 

 100. Moreover, to raise a number to any power, we have only to 

 multiply its logarithm by the exponent of that power, and it will give 

 the logarithm of the power sought ; from which the power itself may 

 be found by means of a table of logarithms. In like manner, the 

 extraction of any root is performed simply by dividing the logarithm 

 of the number, by the index of the root, and the quotient will be the 

 logarithm of the root required. 



