RAPID APPROXIMATION FOR A LOGARITHM

Up to this point in this chapter the author has tried to tell the facts about the Harvard machine in plain words. But for reading this section, a little knowledge of calculus is necessary. ([See also Supplement 2].) If you wish, skip this section and go on to the next one.

What is the process that the machine uses to compute any desired logarithm to 23 digits? Suppose that we take for an example the process by which the machine computes log₁₀ 49.3724. We choose a 6-digit number for simplicity; the machine would handle a 23-digit number in the same way. The process uses 2 fundamental equations involving the logarithm: the sum relation

log (a·b·c···) = log a + log b + log c···

and the series relation

logₑ(1 + h) = h -	h²	+	h³	-	h⁴	+ ···, │h│ < 1
	2		3		4

The error in this series is less than the first neglected term. Now, the machine stores the base 10 logarithms (to 23 decimal places) of the following 36 numbers:

1	1.1	1.01	1.001
2	1.2	1.02	1.002
...	...	...	...
9	1.9	1.09	1.009

First, the number 49.3724 is examined in a counter called the Logarithm-In-Out counter, and the position of the decimal point is determined, giving the characteristic of the logarithm. The number 49.3724 has the characteristic 1. Next, 4 successive divisions are performed, in which the 4 divisors are (1) the first digit of the number, (2) the first 2 digits of the quotient, (3) the first 3 digits of the next quotient, and (4) the first 4 digits of the subsequent quotient; thus,

4.93724	= 1.23431
4
1.23431	= 1.02860
1.2
1.02860	= 1.00843
1.02
1.00843	= 1.00043
1.008

For simplicity we have kept only 6 digits, although the machine, of course, would keep 23. It is interesting to note that the machine is able to sense digits and thus determine the 4 divisors; this is an arithmetical and numerical process and one that cannot be done in ordinary algebra. We now have:

log₁₀ 49.3724 = 1 + log₁₀ 4 + log₁₀ 1.2 + log₁₀ 1.02
+ log₁₀ 1.008 + log₁₀ 1.00043

To compute log₁₀ 1.00043 to 21 decimals we use

log₁₀e ·		h -	h²	+	h³	-	h⁴	+	h⁵	-	h⁶
			2		3		4		5		6

with h = 0.00043. Only 6 terms of the series relation are needed. For, the error is less than h⁷/7, which is less than 10⁻²¹/7, since h < ¹/₁₀₀₀. The machine uses the series relation in the form

log₁₀ (1 + h) = {([{(c₆h + c₅) h + c₄}h + c₃]h + c₂)h + c₁}h

where

c₁ = M, c₂ = -M/2, c₃ = M/3, ···,

and M = log₁₀_e= 0.434294···.

The 6 values of the c’s are also stored in the machine. When any logarithm is to be computed, the sum of the characteristic, of the 4 logarithms of the successive divisors, and of the first 6 terms of the series relation gives the logarithm. The maximum time required is 90 seconds.