The Information (14 page)

In 1614 Briggs was a professor of geometry—the first professor of geometry—at Gresham College, London, later to be the birthplace of the Royal Society. Without logarithms he had already created two books of tables,
A Table to find the Height of the Pole, the Magnetic Declination being given
and
Tables for the Improvement of Navigation
, when a book
came from Edinburgh promising to “take away all the difficultie that heretofore hath beene in mathematical calculations.”
^♦

This new book proposed a method that would do away with most of the expense and the errors. It was like an electric flashlight sent to a lightless world. The author was a wealthy Scotsman, John Napier (or Napper, Nepair, Naper, or Neper), the eighth laird of Merchiston Castle, a theologian and well-known astrologer who also made a hobby of mathematics. Briggs was agog. “Naper, lord of Markinston, hath set my head and hands a work,”
^♦
he wrote. “I hope to see him this summer, if it please God, for I never saw book, which pleased me better, and made me more wonder.” He made his pilgrimage to Scotland and their first meeting, as he reported later, began with a quarter hour of silence: “spent, each beholding other almost with admiration before one word was spoke.”
^♦

Briggs broke the trance: “My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, viz. the Logarithms; but, my Lord, being by you found out, I wonder nobody else found it out before, when now known it is so easy.” He stayed with the laird for several weeks, studying.

In modern terms a logarithm is an exponent. A student learns that the logarithm of 100, using 10 as the base, is 2, because 100 = 10
²
. The logarithm of 1,000,000 is 6, because 6 is the exponent in the expression 1,000,000 = 10
⁶
. To multiply two numbers, a calculator could just look up their logarithms and add those. For example:

100 × 1,000,000 = 10
²
× 10
⁶
= 10
^{(2 + 6)}

Looking up and adding are easier than multiplying.

But Napier did not express his idea this way, in terms of exponents. He grasped the thing viscerally: he was thinking in terms of a relationship between differences and ratios. A series of numbers with a fixed difference is an arithmetic progression: 0, 1, 2, 3, 4, 5 … When the numbers are separated by a fixed ratio, the progression is geometric: 1, 2, 4, 8, 16, 32 … Set these progressions side by side,

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.

Charles Babbage was born on Boxing Day 1791, near the end of the century that began with Newton. His home was on the south side of the River Thames in Walworth, Surrey, still a rural hamlet, though the London Bridge was scarcely a half hour’s walk even for a small boy. He was the son of a banker, who was himself the son and grandson of goldsmiths. In the London of Babbage’s childhood, the Machine Age made itself felt everywhere. A new breed of impresario was showing off machinery in exhibitions. The shows that drew the biggest crowds featured automata—mechanical dolls, ingenious and delicate, with wheels and pinions mimicking life itself. Charles Babbage went with his mother to John Merlin’s Mechanical Museum in Hanover Square, full of clockwork and music boxes and, most interesting, simulacra of living things. A metal swan bent its neck to catch a metal fish, moved by hidden motors and cams. In the artist’s attic workshop Charles saw a pair of naked dancing women, gliding and bowing, crafted in silver at one-fifth life size. Merlin himself, their elderly creator, said he had devoted years to these machines, his favorites, still unfinished. One of the figurines especially impressed Charles with its (or her) grace and seeming liveliness. “This lady attitudinized in a most fascinating manner,”
^♦
he recalled. “Her eyes were full of imagination, and irresistible.” Indeed, when he was a man in his forties he found Merlin’s silver dancer at an auction, bought it for £35, installed it on a pedestal in his home, and dressed its nude form in custom finery.
^♦

The boy also loved mathematics—an interest far removed from the mechanical arts, as it seemed. He taught himself in bits and pieces
from such books as he could find. In 1810 he entered Trinity College, Cambridge—Isaac Newton’s domain and still the moral center of mathematics in England. Babbage was immediately disappointed: he discovered that he already knew more of the modern subject than his tutors, and the further knowledge he sought was not to be found there, maybe not anywhere in England. He began to acquire foreign books—especially books from Napoleon’s France, with which England was at war. From a specialty bookseller in London he got Lagrange’s
Théorie des fonctions analytiques
and “the great work of Lacroix, on the
Differential and Integral Calculus
.”
^♦

He was right: at Cambridge mathematics was stagnating. A century earlier Newton had been only the second professor of mathematics the university ever had; all the subject’s power and prestige came from his legacy. Now his great shadow lay across English mathematics as a curse. The most advanced students learned his brilliant and esoteric “fluxions” and the geometrical proofs of his
Principia
. In the hands of anyone but Newton, the old methods of geometry brought little but frustration. His peculiar formulations of the calculus did his heirs little good. They were increasingly isolated. The English professoriate “regarded any attempt at innovation as a sin against the memory of Newton,”
^♦
one nineteenth-century mathematician said. For the running river of modern mathematics a student had to look elsewhere, to the Continent, to “analysis” and the language of differentiation as invented by Newton’s rival and nemesis, Gottfried Wilhelm Leibniz. Fundamentally, there was only one calculus. Newton and Leibniz knew how similar their work was—enough that each accused the other of plagiarism. But they had devised incompatible systems of notation—different languages—and in practice these surface differences mattered more than the underlying sameness. Symbols and operators were what a mathematician had to work with, after all. Babbage, unlike most students, made himself fluent in both—“the dots of Newton, the
d
’s of Leibnitz”
^♦
—and felt he had seen the light. “It is always difficult to think and reason in a new language.”
^♦

Indeed, language itself struck him as a fit subject for philosophical study—a subject into which he found himself sidetracked from time to time. Thinking about language, while thinking
in
language, leads to puzzles and paradoxes. Babbage tried for a while to invent, or construct, a universal language, a symbol system that would be free of local idiosyncrasies and imperfections. He was not the first to try. Leibniz himself had claimed to be on the verge of a
characteristica universalis
that would give humanity “a new kind of an instrument increasing the powers of reason far more than any optical instrument has ever aided the power of vision.”
^♦
As philosophers came face to face with the multiplicity of the world’s dialects, they so often saw language not as a perfect vessel for truth but as a leaky sieve. Confusion about the meanings of words led to contradictions. Ambiguities and false metaphors were surely not inherent in the nature of things, but arose from a poor choice of signs. If only one could find a proper mental technology, a true philosophical language! Its symbols, properly chosen, must be universal, transparent, and immutable, Babbage argued. Working systematically, he managed to create a grammar and began to write down a lexicon but ran aground on a problem of storage and retrieval—stopped “by the apparent impossibility of arranging signs in any consecutive order, so as to find, as in a dictionary, the meaning of each when wanted.”
^♦
Nevertheless he felt that language was a thing a person could invent. Ideally, language should be rationalized, made predictable and mechanical. The gears should mesh.