Plane Trigonometry and Numerical Computation

Plane Trigonometry and Numerical Computation

Chapter V
Logarithms

39. The Invention of Logarithms. The extensive numerical computations required in business, in science, and in engineering were greatly simplified by the invention of logarithms by John Napier, Baron of Merchiston (1550-1617). By means of logarithms we are able to replace multiplication and division by addition and subtraction, processes which we all realize are more expeditious than the first two.

If we consider the successive integral powers of 2

Exponent x .	`1`	`2`	`3`	`4`	`5`	`6`	`7`
Result 2^x . .	`2`	`4`	`8`	`16`	`32`	`64`	`128`

(1)

Exponent x .	`8`	`9`	`10`	`11`	`12`	`etc.`	`A. P.`
Result 2^x . .	`256`	`512`	`1024`	`2048`	`4096`	`etc.`	`G. P.`

we see that the results form a geometric progression (G. P.) and the exponents an arithmetic progression (A. P.). We know from elementary algebra that

x^m·xⁿ = x^{m + n},

and

 x^m
———— = x^{m - n}.
 xⁿ

Hence if we wish to multiply two numbers in our G. P. e.g. 4 × 8, we merely have to add the corresponding exponents 2 and 3 and under the sum find the desired product 32. Similarly, if we wish to divide e.g. 4096 by 128, we merely have to subtract the exponent corresponding to 128, from that

Plane Trigonometry and Numerical Computation

Chapter VLogarithms

Chapter V
Logarithms