The Information - James Gleick [41]
0 1 2 3 4 5 … (base 2 logarithms)
1 2 4 8 16 32 … (natural numbers)
and the result is a crude table of logarithms—crude, because the whole-number exponents are the easy ones. A useful table of logarithms had to fill in the gaps, with many decimal places of accuracy.
In Napier’s mind was an analogy: differences are to ratios as addition is to multiplication. His thinking crossed over from one plane to another, from spatial relationships to pure numbers. Aligning these scales side by side, he gave a calculator a practical means of converting multiplication into addition—downshifting, in effect, from the difficult task to the easier one. In a way, the method is a kind of translation, or encoding. The natural numbers are encoded as logarithms. The calculator looks them up in a table, the code book. In this new language, calculation is easy: addition instead of multiplication, or multiplication instead of exponentiation. When the work is done, the result is translated back into the language of natural numbers. Napier, of course, could not think in terms of encoding.
Briggs revised and extended the necessary number sequences and published a book of his own, Logarithmicall Arithmetike, full of pragmatic applications. Besides the logarithms he presented tables of latitude of the sun’s declination year by year; showed how to find the distance between any two places, given their latitudes and longitudes; and laid out a star guide with declinations, distance to the pole, and right ascension. Some of this represented knowledge never compiled and some was oral knowledge making the transition to print, as could be seen in the not-quite-formal names of the stars: the Pole Starre, girdle of Andromeda, Whales Bellie, the brightest in the harpe, and the first in the great Beares taile next her rump.♦ Briggs also considered matters of finance, offering rules for computing with interest, backward and forward in time. The new technology was a watershed: “It may be here also noted that the use of a 100 pound for a day at the rate of 8, 9, 10, or the like for a yeare hath beene scarcely known, till by Logarithms it was found out: for otherwise it requires so many laborious extractions of roots, as will cost more paines than the knowledge of the thing is accompted to be worth.”♦ Knowledge has a value and a discovery cost, each to be counted and weighed.
Even this exciting discovery took several years to travel as far as Johannes Kepler, who employed it in perfecting his celestial tables in 1627, based on the laboriously acquired data of Tycho Brahe. “A Scottish baron has appeared on the scene (his name I have forgotten) who has done an excellent thing,” Kepler wrote a friend, “transforming all multiplication and division into addition and subtraction.”♦ Kepler’s tables were far more accurate—perhaps thirty times more—than any of his medieval predecessors, and the accuracy made possible an entirely new thing, his harmonious heliocentric system, with planets orbiting the sun in ellipses. From that time until the arrival of electronic machines, the majority of human computation was performed by means of logarithms.♦ A teacher of Kepler’s sniffed, “It is not fitting for a professor of mathematics to manifest childish joy just because reckoning is made easier.”♦ But why not? Across the centuries they all felt that joy in reckoning: Napier and Briggs, Kepler and Babbage, making their lists, building their towers of ratio and proportion, perfecting their mechanisms for transforming numbers into numbers. And then the world’s commerce validated their pleasure.
Natural Numbers Logarithms base 2
1 0
2 1
3 1.5850
4 2
5 2.3219
6 2.5850
7 2.8074
8 3
9 3.1699