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By Root 703 0

Protestant polemic in which he claimed that the Pope was the Antichrist and predicted that the Day of Judgement would come between 1688 and 1700. In the evening he liked to wear a long robe and pace outside his tower chamber, which added to his reputation as a necromancer. He also experimented with fertilizers on his vast estate near Edinburgh, and came up with ideas for military hardware, such as a chariot with a ‘moving mouth of mettle’ that would ‘scatter destruction on all sides’ and a machine for ‘sayling under water, with divers and other strategems for harming of the enemyes’ – precursors of the tank and the submarine. As a mathematician, he popularized the use of the decimal point, as well as coming up with the idea of logarithms, coining the term from the Greek logos, ratio, and arithmos, number.

Don’t be put off by the following definition: the logarithm, or log, of a number is the exponent when that number is expressed as a power of 10. Logarithms are more easily understood when expressed algebraically: if a = 10b, then the log of a> is b.

So, log 10 = 1

(because 10 = 101)

log 100 = 2

(because 100 = 102)

log 1000 = 3

(because 1000 = 103)

log 10,000 = 4

(because 10,000 =104)

Finding the log of a number is self-evident if the number is a multiple of 10. But what if you’re trying to find the log of a number that isn’t a multiple of 10? For example, what is the logarithm of 6? The log of 6 is the number a such that when 10 is multiplied by itself a times you get 6. However, it seems completely nonsensical to say that you can multiply 10 by itself a certain number of times to get 6. How can you multiply 10 by itself a fraction of times? Of course, the concept is nonsensical when we imagine what this might mean in the real world, but the power and beauty of mathematics is that we do not need to be concerned with any meaning beyond the algebraic definition.

The log of 6 is 0.778 to three decimal places. In other words, when we multiply 10 by itself 0.778 times, we get 6.

Here is a list of the logarithms of the numbers from 1 to 10, each to three decimal places.

log 1 = 0

log 2 = 0.301

log 3 = 0.477

log 4 = 0.602

log 5 = 0.699

log 6 = 0.778

log 7 = 0.845

log 8 = 0.903

log 9 = 0.954

log 10 = 1

So, what’s the point of logarithms? Logarithms turn the more difficult operation of multiplication into the simpler process of addition. More precisely, the multiplication of two numbers is equivalent to the addition of their logs. If X × Y = Z, then log X + log Y = log Z.

We can check this equation using the table above.

3×3 = 9

log 3 + log 3 = log 9

0.477 + 0.477 = 0.954

Again,

2×4 = 8

log 2 + log 4 = log 8

0.301 + 0.602 = 0.903

The following method can therefore be used in order to multiply two numbers together: convert them into logs, add them to get a third log, and then convert this log back into a number. For example, what is 2×3? We find the logs of 2 and 3, which are 0.301 and 0.477, and add them, which is 0.788. From the list above, 0.788 is log 6. So, the answer is 6.

Now, let’s multiply 89 by 62.

First, we need to find their logs, which we can do by putting the number into a calculator or Google. Until the late twentieth century, however, the only way of doing this was done by consulting log tables. The log of 89 is 1.949 to three decimal places. The log of 62 is 1.792.

So, the sum of the logs is 1.949 + 1.792 = 3.741.

The number whose log is 3.741 is 5518. This is again found by using the log tables.

So, 89×62 = 5518.

Significantly, the only piece of calculation we have done to work out this multiplication was a fairly simple addition.

Logarithms, wrote Napier, were able to free mathematicians from the ‘tedious expense of time’ and the ‘slippery errors’ involved in the ‘multiplications, divisions, square and cubical extractions of great numbers’. Using Napier’s invention, not only could multiplication be made into the addition of logs, but division was made into the subtraction of logs; calculating of square roots was made into the division of logs