In Pursuit of the Unknown Read Online Free PDF Page B

time to mathematics, especially methods for speeding up complicated arithmetical calculations. One invention,Napier’s bones, was a set of ten rods, marked with numbers, which simplified the process of long multiplication. Even better was the invention that made his reputation and created a scientific revolution: not his book on Revelation, as he had hoped, but his Mirifici Logarithmorum Canonis Descriptio (‘Description of the Wonderful Canon of Logarithms’) of 1614. The preface shows that Napier knew exactly what he had produced, and what it was good for. 1
    Since nothing is more tedious, fellow mathematicians, in the practice of the mathematical arts, than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots – and . . . the many slippery errors that can arise: I had therefore been turning over in my mind, by what sure and expeditious art, I might be able to improve upon these said difficulties. In the end after much thought, finally I have found an amazing way of shortening the proceedings . . . it is a pleasant task to set out the method for the public use of mathematicians.
    The moment Briggs heard of logarithms, he was enchanted. Like many mathematicians of his era, he spent a lot of his time performing astronomical calculations. We know this because another letter from Briggs to Ussher, dated 1610, mentions calculating eclipses, and because Briggs had earlier published two books of numerical tables, one related to the North Pole and the other to navigation. All of these works had required vast quantities of complicated arithmetic and trigonometry. Napier’s invention would save a great deal of tedious labour. But the more Briggs studied the book, the more convinced he became that although Napier’s strategy was wonderful, he’d got his tactics wrong. Briggs came up with a simple but effective improvement, and made the long journey to Scotland. When they met, ‘almost one quarter of an hour was spent, each beholding the other with admiration, before one word was spoken’. 2
    What was it that excited so much admiration? The vital observation, obvious to anyone learning arithmetic, was that adding numbers is relatively easy, but multiplying them is not. Multiplication requires many more arithmetical operations than addition. For example, adding two ten-digit numbers involves about ten simple steps, but multiplication requires 200. With modern computers, this issue is still important, but now it is tucked away behind the scenes in the algorithms used for multiplication.But in Napier’s day it all had to be done by hand. Wouldn’t it be great if there were some mathematical trick that would convert those nasty multiplications into nice, quick addition sums? It sounds too good to be true, but Napier realised that it was possible. The trick was to work with powers of a fixed number.
    In algebra, powers of an unknown x are indicated by a small raised number. That is, xx = x 2 , xxx = x 3 , xxxx = x 4 , and so on, where as usual in algebra placing two letters next to each other means you should multiply them together. So, for instance, 10 4 = 10 × 10 × 10 × 10 = 10,000. You don’t need to play around with such expressions for long before you discover an easy way to work out, say, 10 4 × 10 3 . Just write down
    10,000 × 1,000 = (10 × 10 × 10 × 10) × (10 × 10 × 10)
    = 10 × 10 × 10 × 10 × 10 × 10 × 10
    = 10,000,000
    The number of 0s in the answer is 7, which equals 4 + 3. The first step in the calculation shows why it is 4 + 3: we stick four 10s and three 10s next to each other. In short,
    10 4 × 10 3 = 10 4+3 = 10 7
    In the same way, whatever the value of x might be, if we multiply its a th power by its b th power, where a and b are whole numbers, then we get the ( a + b )th power:
    x a x b = x a+b
    This may seem an innocuous