(In an earlier post, I referred to a way to understand the invention of *logarithms* by John Napier. This post fleshes this out.)

Imagine an era long ago where calculators and computers didn’t exist, and the engineers and crafts people spent large amounts of time doing long and tedious and error-prone calculations such as multiplication and division of numbers with many decimal places. John Napier lived in such an era, and he was keenly interested in creating tools that would aid people in doing calculations. He invented some mechanical tools, known as Napier rods, that help organize the way you do multi-digit multiplication. For Napier rods, once seen and understood, the thinking behind them is relatively easy to grasp.

Napier’s work on logarithms has had a more far-reaching impact. What I want to explore here is if we can make his discovery seem plausible and natural. I make no claim that my reconstruction of his work is in any way historically accurate.

Let’s first note that certain multiplications are very simple to do – I’m referring here to multiplication by numbers like 1.1 or 1.01 or 1.001. Let’s try this out for 24689 x 1.01. You can verify for yourself that 24689 x 1.01 = 24689 + 246.89 = 24935.89 or

24689.00

+ 246.89

—————-

24935.89

One number is shifted relative to the other number, then added. Shift-and-add is pretty straightforward and easy, as multiplication goes.

Let’s also note that multiplying by 1.01 produces a number that’s bigger, but not all that much bigger, than the number we started with. We could look at the effect of multiplying something by 1.01 repeatedly. No matter how gradual the growth is when we start out, after a while the growth will be more pronounced (as anybody who has put out money at 1% interest and let the money sit for a very long time will be aware of).

0 1

1 1.01

2 1.0201

3 1.030301

4 1.04060401

5 1.0510100501

6 1.061520150601

It is interesting to see that the numbers not only get gradually larger, they get wider (each step, two more decimal places show up). However, we can round the numbers to a fixed width to regularize the amount of work, and we could hope that as long as we use enough digits, we won’t lose anything too important. In case of multiplying by 1.01, I suggest maintaining our multiplication results in four digits of accuracy.

0 1.0000

1 1.0100

2 1.0201

3 1.0303

4 1.0406

5 1.0510

6 1.0615

7 1.0721

8 1.0829

9 1.0936

10 1.1045

Each new entry now takes roughly the same amount of work; if we were to extend the table to have some 500 entries, it might take us all day, doing this by hand. Our table would match this:

: :

10 1.1045

: :

23 1.2570

24 1.2696

25 1.2823

26 1.2951

: :

69 1.9805

70 2.0064

: :

110 2.9873

111 3.0172

: :

139 3.9867

140 4.0266

: :

161 4.9623

162 5.0119

: :

180 5.9950

181 6.0550

: :

195 6.9602

196 7.0298

: :

208 7.9214

209 8.0006

: :

220 8.9260

221 9.0153

: :

231 9.9587

232 10.0583

: :

240 10.8917

241 11.0006

: :

249 11.9120

250 12.0311

: :

265 13.9677

266 14.1074

: :

272 14.9753

273 15.1251

: :

411 59.7109

412 60.3080

: :

462 99.1849

463 100.1767

You may notice that it takes 70 entries to reach the value of 2, but it takes fewer entries to get from 2 to the value 3, and fewer still to grow from 3 to 4. You can also see that growing from 1 to 10 requires 232 entries, and that another 232 entries gets us over 100.

What does it mean that 70 entries got us to 2? You can see that 1 got multiplied by 1.01 and another 1.01 and another 1.01 for a total of 70 factors of 1.01. The number on the right tells you what you get if you multiply a certain number of copies of 1.01 together; the number on the left tells you how many copies of 1.01 it took to get the product to reach that number (from 1). So 70 copies of 1.01 multiplied got us to 2, while 111 copies were needed to get us to 3.

What do you think would happen if we threw 70 copes of 1.01 together with another 111 copies of 1.01 and multiplied them all? I’m particularly interested to see if we can find the answer without having to do any more multiplying than we already have. Since 70 copies and 111 copies combined make 181 copies, we could look in our table to see what 181 copies of 1.01 multiplied together yields us. The table entry for 181 is just over 6. The relationship between the table entries for 70 copies (roughly 2) and for 111 copies (roughly 3) and that of the combined entry for 181 copies (roughly 6) is that 6 is 2 times 3. We can get the product of any two numbers in the right column by finding out how many factors of 1.01 went into each, and finding the number in the right column that corresponds to the total number of factors of 1.01.

For example, to find 5 times 12, we’d look for a number close to 5 in the right column. We’d find it at entry 162. We’d then look for a number close to 12 in the right column, finding it at entry 250. The total number of factors of 1.01 in these two numbers is 162 + 250, or 412. The right column for entry 412 is 60.3080. We can now conclude that 5 times 12 is close to 60.3080, or about 60. Our table doesn’t get us exactly 60 since our repeated multiplication of 1.01 didn’t get all that terribly close to 5 nor did it get all that terribly close to 12. Though we can use our table to get products without doing multiplication, our numbers aren’t very accurate – they wouldn’t be precise enough for any serious engineering work.

What John Napier did was roughly similar to our table of repeated multiplications of 1.01. Instead of using 1.01, Napier used 1.0000001, and instead of carrying out the work in 4 decimal places, he carried it out in 14 decimal places. It took Napier many years to produce his tables. Once he had obtained his tables, he could perform multiplication through table look-up and get very precise results.

You may have been wondering how Napier would have us deal with multiplying large numbers together, e.g. 25 x 8, or how to multiply small numbers together, e.g. .25 x .8.

In both our examples, we can find a variant, namely 2.5 x 8, which we can multiply together using Napier’s tables, and the answer to our real problem can be found from the answer to the variant problem by moving the decimal point appropriately.

When you take a number and look it up in the right column of the table to find the number in the left column, the number in the left column is called the logarithm of the number in the right column.

In the table we showed, the logarithm has a *base* of 1.01, but logarithms can be constructed with other bases as well.

In our table, the numbers in the left column can be scaled up or down without effect on its essential characteristic. This essential characteristic is that multiplying of numbers on the right side of the table corresponds to adding of numbers on the left side of the table.

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Very nicely done. Thanks for this careful exposition.