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Napier's Logarithms

When thinking about Napier's logarithms, it is better to forget EVERYTHING that you know about logs, powers, indices, series and e and start from the beginning. If you have nothing to forget, so much the better.

It important to realise that, in Napier's time, the decimal point was not yet generally accepted and decimal fractions were hard to portray.

For instance, a number such as 2.7182 could be written as:-

2  7/10   1/100   8/1000   2/10000

Or:-

2,  7'   1"   8'"   2""

There were several variants on this sort of thing, but they were all clumsy. Although Napier was instrumental in introducing modern decimal notation; he avoided using decimal fractions in his work on logarithms by multiplying everything by 10⁷  (10000000). This effectively gave him seven figures of accuracy.

It is also important to realise that, as originally produced, Napier's logarithms referred to trigonometric functions. This was the requirement of the day, astronomy and navigation required extensive calculations involving such functions. It was Napier's intention to remove the drudgery involved in these calculations. He says:

"Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hinderances."

We are going to look at his logarithms as numbers and not as trig functions. This simplifies the explanation but does not affect the underlying principle involved.

Napier describes his invention using a dynamic analogy. He describes the movement of two points along a pair of lines. Now, be warned, Napier is nothing if not obscure in his language. Just take a deep breath and then read the next bit:

For the first line:

"A line is said to increase equally, when the poynt describing the same, goeth forward equall spaces, in equall times, or moments."

For the second line:

"A line is said to decrease proportionally into a shorter, when the poynt describing the same in oequal times, cutteth off parts continually of the same proportion to the lines from which they are cut off."

What Napier is referring to may be regarded now as two series; an arithmetic series (the first line - eg 1,2,3,4, ...) and a geometric series (the second line - eg ... 16, 8, 4, ... ). One is produced by a process of addition and the other by a process of multiplication. We will come back to this in a moment.

Napier then introduces his arguments for using his 10⁷ fiddle factor:

"Surd quantities, or unexplicable by number, are said to be defined, or expressed by numbers very neere, when they are defined or expressed by great numbers which differ not so much as one unit from the true value of the Surd quantities.

As for example. Let the semidiameter or whole sine be the rational number 10000000, the sine of 45 degree shall be the square root of 50,000,000,000,000, which is surd or irrationall^§ and inexplicable by any number, & is included between the limits of 7071067 the lesse, and 7071068 the greater; therefore that surd sine of 45 degrees, is said to be defined and expressed very neere, when it is expressed by the whole numbers, 7071067, or 7071068, not regarding the fractions. For in great numbers there ariseth no sensible error, by neglecting the fragments or parts of a unite".

[§ Irrational:- in the original sense of "cannot be expressed as a ratio"]

In modern terms, what Napier is saying here, is that provided the number is large, he can get away with using whole numbers (integers) rather than fractions. It doesn't matter much if we choose 7071067 or 7071068, they are so very nearly equal. We certainly don't have to bother finding another number such as 7071067.81187... which lies somewhere between the two.

Napier used 10000000, because that allowed him to work with seven figures and he considered this sufficiently accurate (very neere).

Napier worked out two parallel series of numbers, all integers.

The first series, (the logarithms) started at zero and increased by one each time (this is his "equally increasing" line), or what we would call an arithmetic series (0,1,2,3,4,...)

The second series started at 10000000 (the whole sine) and decreased by being multiplied by (1-1/10000000) ie 0.9999999 at each step. (This is his line that "decreases proportionally"), a geometric series.

The principle can best be demonstrated by using smaller figures, such as 100 for the whole sine and multiplying this by 0.99 each time. The figures will then change sufficiently rapidly to demonstrate the principles involved, but it is necessary to use decimals to get sufficient accuracy.

We can use these logarithms to multiply, divide and extract roots

Multiplication

If we want to multiply 95.099 by 98.01, we can look up their logarithms (5 & 2) and add them (= 7).

Looking at the number whose log is 7 we find 93.207. But we know that multiplying 95.099 by 98.01 gives 9320.65299 - so what is wrong?

This is usual with Napier's system, the result of a multiplication is, in effect, always diminished by the value of the "whole sine". In our case, this was 100 and the result is 100 times too small. To get the correct answer, the result of the log calculation must be multiplied by the whole sine.

Why?

Let L1 be Napier's logarithm of the number N1
Let L2 be Napier's logarithm of the number N2

Then
N1 = 10⁷(1-1/10⁷)^L1 ---------------------(1)

and
N2 = 10⁷(1-1/10⁷)^L2 ---------------------(2)

So, if we multiply N1 by N2, we get
N1 × N2 = 10⁷(1-1/10⁷)^L1 × 10⁷(1-1/10⁷)^L2

i.e.
N1 × N2 = 10⁷ × 10⁷(1-1/10⁷)^(L1+L2) -------(3)

Rearranging equation (3) to make it look more like equations (1) & (2) we have,
N1 × N2/10⁷ = 10⁷(1-1/10⁷)^(L1+L2)

So the sum of the Napier logarithms L1 + L2 does not give the logarithm of N1 × N2 but instead, the logarithm of N1 × N2/10⁷. The same reasoning can be applied to quotients, powers and roots.

Extraction of square roots

To extract a square root - eg 88.638 (log = 12). Divide the logarithm by 2; the result is 94.148 (log = 6).

The calculator gets 9.41478. This time Napier's method is 10 times too big (10=√100).

Division

To divide say 88.638 (log=12) by 92.274 (log=8), subtract the logs. The result is 96.060 (log=4). The calculator gives 0.96060 - Napier's result must be divided by the whole sine (100) to be correct.

Did Napier's logarithms have a base?

Unlike the modern form of logarithms, which are calculated to base 10 or e, Napier's logarithms were not calculated to a base. The concept of bases and indices came rather later.

There is, however, some association with the constant e. Napier produced his logarithms by repeatedly multiplying by 0.9999999. In effect he was calculating:-

(1 - 1/10000000)ⁿ

For increasing values of n. When n reached 10000000, he had:-

(1 - 1/10000000)^10000000

This is a particular case of:-

(1 - 1/n)ⁿ

As n becomes very large, this expression tends towards:-

1/e

When n reaches 10000000 (Napier's "Whole Sine") we have:-

(1 - 1/10000000)^10000000 = 0.36787942277

(whereas 1/e = 0.367879441171)

So, Napier's logarithm of his whole sine is a close approximation to 1/e.

In Napier's own words "very neere"

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