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formula, but on the left it multiplies two quantities together, while on the right the main step is to add a and b , which is simpler.
    Suppose you wanted to multiply, say, 2.67 by 3.51. By long multiplication you get 9.3717, which to two decimal places is 9.37. What if you try to use the previous formula? The trick lies in the choice of x . If we take x to be 1.001, then a bit of arithmetic reveals that
    (1.001) 983 = 2.67
    (1.001) 1256 = 3.51
    correct to two decimal places. The formula then tells us that 2.87 × 3.41 is
    (1.001) 983+1256 = (1.001) 2239
    which, to two decimal places, is 9.37.
    The core of the calculation is an easy addition: 983 + 1256 = 2239. However, if you try to check my arithmetic you will quickly realise that if anything I’ve made the problem harder, not easier. To work out (1.001) 983 you have to multiply 1.001 by itself 983 times. And to discover that 983 is the right power to use, you have to do even more work. So at first sight this seems like a pretty useless idea.
    Napier’s great insight was that this objection is wrong. But to overcome it, some hardy soul has to calculate lots of powers of 1.001, starting with (1.001) 2 and going up to something like (1.001) 10,000 . Then they can publish a table of all these powers. After that, most of the work has been done. You just have to run your fingers down the successive powers until you see 2.67 next to 983; you similarly locate 3.51 next to 1256. Then you add those two numbers to get 2239. The corresponding row of the table tells you that this power of 1.001 is 9.37. Job done.
    Really accurate results require powers of something a lot closer to 1, such as 1.000001. This makes the table far bigger, with a million or so powers. Doing the calculations for that table is a huge enterprise. But it has to be done only once . If some self-sacrificing benefactor makes the effort up front, succeeding generations will be saved a gigantic amount of arithmetic.
    In the context of this example, we can say that the powers 983 and 1256 are the logarithms of the numbers 2.67 and 3.51 that we wish to multiply. Similarly 2239 is the logarithm of their product 9.38. Writing log as an abbreviation, what we have done amounts to the equation
    log ab = log a + log b
    which is valid for any numbers a and b . The rather arbitrary choice of 1.001 is called the base . If we use a different base, the logarithms that we calculate are also different, but for any fixed base everything works the same way.
    This is what Napier should have done. But for reasons that we can only guess at, he did something slightly different. Briggs, approaching the technique from a fresh perspective, spotted two ways to improve on Napier’s idea.
    When Napier started thinking about powers of numbers, in the late sixteenth century, the idea of reducing multiplication to addition was already circulating among mathematicians. A rather complicated method known as ‘prosthapheiresis’, based on a formula involving trigonometric functions, was in use in Denmark. 3 Napier, intrigued, was smart enough torealise that powers of a fixed number could do the same job more simply. The necessary tables didn’t exist – but that was easily remedied. Some public-spirited soul must carry out the work. Napier volunteered himself for the task, but he made a strategic error. Instead of using a base that was slightly bigger than 1, he used a base slightly smaller than 1. In consequence, the sequence of powers started out with big numbers, which got successively smaller. This made the calculations slightly more clumsy.
    Briggs spotted this problem, and saw how to deal with it: use a base slightly larger than 1. He also spotted a subtler problem, and dealt with that as well. If Napier’s method were modified to work with powers of something like 1.0000000001, there would be no straightforward relation between the logarithms of, say, 12.3456 and 1.23456. So it wasn’t entirely clear when