You Could Look It Up: The Reference Shelf From Ancient Babylon to Wikipedia

by Jack Lynch

  Where to keep them? In his book-length ode to reference books, Dictionary Days, Ilan Stavans wrote, “An aphorism comes to mind: ‘Tell me how you organize your books and I’ll tell you who you are.’ Thus said a teacher of mine years ago.” Stavans confesses that his own shelves are a mess, with a significant exception: his dictionaries. “An entire wall is filled with them. These volumes seek perfection … they systematize knowledge.”4

  Princeton historian Anthony Grafton may win the contest for most interesting home reference section, thanks to a six-foot-tall “book wheel” or “reading wheel,” modeled on a contraption designed by European scholars in the late sixteenth century. “Think of a small Ferris wheel,” writes a reporter who has seen it, “with shelves instead of seats”; the shelves rotate, like the cars on a Ferris wheel, so that the books always remain upright. “From his seat he can rotate any one of eight shelves into view by spinning the wheel. With a tug, Grafton rotates past Greek, Latin, and Hebrew lexicons until a book on eclipses drops into view. ‘Not everyone has Eclipses for Humanists,’ he observes dryly.”5

  While I wait for the opportunity to get my own book wheel, I keep the bulk of my reference collection in the study, directly over the computer where I do most of my writing. The ones I use most often are on the lowest of the four shelves, so I can reach them without standing up. There I keep The Oxford Companion to English Literature in both the fifth and sixth editions; there, too, the Oxford Classical Dictionary, third edition (soon to be replaced by the fourth); Merriam-Webster’s Collegiate Dictionary, eleventh edition; Chambers Biographical Dictionary; The Chicago Manual of Style, sixteenth edition; Brewer’s Dictionary of Phrase & Fable, centenary edition; The Oxford Dictionary of Quotations; The Oxford Companion to the Year; The Oxford Companion to the English Language; and the major foreign language dictionaries: Le nouveau petit Robert, Duden’s Deutsches universal Wörterbuch, and Zingarelli’s Vocabolario della lingua italiana, along with a handful of bilingual dictionaries from Oxford, Webster, and Langenscheidt. The next two shelves contain works I need less often, maybe once a week—things I can reach simply by standing up. There I keep Bergen and Cornelia Evans’s Dictionary of Contemporary American Usage; The Concise Oxford English Dictionary; the MLA Handbook; dictionaries of languages I need less often (A Concise Hebrew and Aramaic Lexicon of the Old Testament, The Learner’s Russian–English Dictionary), the systematic reference grammars (Allen and Greenough’s New Latin Grammar, the Oxford English Grammar, Zanichelli’s Lingua italiana, Sweet’s Anglo-Saxon Primer), the guides to usage and citation formats (Fowler and Fowler’s King’s English, Strunk and White’s Elements of Style), specialized English dictionaries (Eric Partridge’s Dictionary of Slang and Unconventional English), and miscellaneous references (Wellisch’s Indexing from A to Z, the NBC Handbook of Pronunciation, Alberto Manguel’s Dictionary of Imaginary Places). The top shelf is too high to reach without a stepping stool; I use it for the reference books I consult every few months or less, such as the Concise Encyclopedia of Heraldry, the English–Norwegian, Norwegian–English Dictionary, and a grammar of Irish Gaelic. Some of these, truth be told, have acquired a layer of dust.

  The latest Chambers Dictionary is just an inch too tall for my shelves, so it lies on its side. Webster’s Third New International is far too tall to fit on a shelf, and its weight would make it dangerous for me to try to take it down from overhead. It sits on the floor to my left, blocking access to a filing cabinet and leaning up against a facsimile of Webster’s American Dictionary of the English Language from 1828. My facsimile of Johnson’s Dictionary of the English Language from 1755 is on the floor behind me, leaning on the bookcase devoted to Samuel Johnson’s works. Other heavyweights—the Historical Thesaurus of the Oxford English Dictionary, the new Oxford Latin Dictionary, Liddell and Scott’s Greek–English Lexicon in both the intermediate and full versions—have to go on top of a row of bookcases. The famous eleventh edition of the Encyclopædia Britannica (1911) would occupy too much space in the study; its twenty-nine volumes are downstairs. My old Compact OED, with its magnifying glass—now obsolete, but I’ve had it since I was a freshman and can’t bear to part with it—and the fourth and fifth editions of The American Heritage Dictionary rest on top of a filing cabinet.

  Of course I keep plenty of works on the computer. I had the third edition of the American Heritage Dictionary installed on my hard drive starting in the DOS days of the early 1990s, and now I have a number of important reference works bookmarked in my Web browser: the Oxford English Dictionary, Merriam-Webster, the Oxford Dictionary of National Biography, and the English Short-Title Catalogue, among many others. And of course Google and Wikipedia are perennially available. My iPhone also has the electronic versions of Merriam-Webster’s Collegiate, American Heritage, Chambers, the Larousse for French, and versions of Liddell and Scott’s Greek lexicon and Lewis and Short’s Latin dictionary, as well as a smattering of less impressive bilingual dictionaries for quick lookups, and the Wikipedia app. I rarely go more than an hour or two without using one of them.

  CHAPTER 8

  ADMIRABLE ARTIFICE

  Computers before Computers

  Henry Briggs

  Arithmetica logarithmica

  1624

  Johannes Kepler

  Tabulæ Rudolphinæ

  1627

  “Math class is tough!” So declared the Teen Talk Barbie doll in 1992, enraging a generation of feminists who despaired of finding children’s toys that did not reinforce harmful gender stereotypes. But Barbie was right: math is hard.

  If it is not quite so tough for us today, that is because we are spoiled by tiny computers that go with us everywhere. In the age of the feature-packed mobile phone, we need not even divide a restaurant bill by hand. A computer with more computing power than the entire Apollo program had at its disposal is there to give us an answer as quickly as we can press the buttons. But it has not always been so. For most of history, all calculations were done by hand, because there was no other way to do them.

  Books containing page after page of digits may be the most referency of all reference books in their unsuitability for reading through. Dictionaries, encyclopedias, even the phone book can make for entertaining browsing: there is always the hope of coming across amusing names. But it is hard to read more than a few lines in a table of numbers without feeling one’s energy waning, and even the most devoted reference book enthusiast will have trouble with this:

  1201

  3,07954,30074,0290

  1234

  3,09131,51596,9721

  36,14602,6382

  35,17978,9847

  1202

  3,07990,44676,6672

  1235

  3,90166,69575,9568

  36,11596,7313

  35,15131,5712

  1203

  3,08026,56273,3985

  1236

  3,09201,84707,5280

  36,08595,8196

  35,12288,7632

  1204

  3,08062,64869,2181

  1237

  3,09236,96996,2912

  36,05599,8908

  35,09450,5497

  and so on, for 396 folio pages.1 Hardly gripping reading. And yet such unreadable tables shaped the modern world.

  The tables are designed as labor-saving devices: they take the place of long calculations done by hand, allowing us to look up the answers rather than arriving at them manually. In American restaurants, for instance, where a tip of 15 to 20 percent of the bill is customary, many people keep a small laminated card in their wallets on which are 15%, 18%, and 20% tips on various totals. These tip charts are descendants of old-fashioned “ready reckoners,” printed tables that go back centuries and were especially common in countries without metric measurement
or decimal coinage.2 A mercer selling 8½ yards of cloth at 4s. 3d. a yard would be grateful for any shortcut in figuring that the total was £1 16s. 1½d. John Mayne’s Socius Mercatoris; or, The Merchant’s Companion (1674) was there to answer the question, as were almanacs and books like Harris’s Pocket Journal.

  As handy as these were, though, they answer only very specific kinds of questions. Of course no book could provide tables to answer every calculation someone might need to perform, but with the discovery of the mathematical function known as the logarithm, one could compile tables to help with almost any calculation—huge tables, to be sure, but finite, and eminently useful.

  Merriam-Webster’s definition of logarithm is as good as any brief definition can be: “the exponent that indicates the power to which a base number is raised to produce a given number.” Alexander John Thompson is more expansive in Logarithmetica Britannica, the standard modern work on the subject:3

  The logarithm of a number N, to any base a, is defined here as the power y to which the base a must be raised to produce the number N; that is, if

  N = ay ,

  y is the logarithm of N to base a, or

  y = loga N.

  The mathphobic will welcome a clearer explanation still, which begins with exponents. Exponentiation is simply repeated multiplication. We use superscript numbers to indicate the number of times a number (called the base) is multiplied by itself: 23 = 2 × 2 × 2 = 8; 64 = 6 × 6 × 6 × 6 = 1,296. More generally, an means a × a × a … with a appearing a total of n times.

  Because our counting system is known as base 10, powers of 10 are especially significant, and 10 is the most common base for logarithms. The number 10 can also be written as 101. The number 100 can be written as 10 × 10, or 102; 1,000 can be written as 10 × 10 × 10, or 103; and so on. A logarithm performs exponentiation in reverse.4 Since 101 = 10, it follows that log (10) = 1. Since 102 = 100, it follows that log (100) = 2; since 103 = 1,000, it follows that log (1,000) = 3; and so on. Logarithms need not be whole numbers. It is easy to see that log (1,000) = 3 because 103 = 1,000, but it is less obvious that log (25) is 1.39794, because 101.39794 = 25, or log (2,000) is 3.30103, because 103.30103 = 2,000.

  Logarithms fascinate mathematicians. They speed the calculation of compound interest and the half-lives of radioisotopes. They are linked in unexpected ways to prime numbers and trigonometrical functions. The constant e—the base of the “natural” log, roughly 2.71828—makes a surprise appearance in one of the most remarkable equations in history, eπ i + 1 = 0, which brings together the five most basic constants in mathematics. But logarithms might have remained little more than mathematical curiosities were it not for one further remarkable property: the logarithm of a product is the sum of the logarithm of the two factors—in a formula, log (a × b) = log (a) + log (b). To put it another way, if a × b = c, then log (a) + log (b) = log (c). We can check it with an example: since 10 × 100 = 1,000, it follows that log (10) + log (100) = log (1,000): log (10) = 1; log (100) = 2; log (1,000) = 3.

  This seemingly abstruse insight has a practical application: it lets us express multiplication in terms of addition, and division in terms of subtraction. Since addition is easier than multiplication, especially for large numbers, and subtraction is much easier than division, calculations can be speeded up by reducing complicated multiplications and divisions to much simpler additions and subtractions, with the help of a list of logarithms. To multiply any two numbers—say, a × b—look up log (a) and log (b) in a table; manually add those two numbers together; then look up that sum in another column of the table and find the number that has that sum as its logarithm—that gives the product. So, for instance, to multiply 2,384 by 1,635, look in the table for the logarithm of 2,384—it is 3.377306—and the logarithm of 1,653—it is 3.213517—and add them together: the sum is 6.590823. Turn to the table again and find the number that has for its logarithm 6.590823: it is 3,897,840, the product of the numbers. Division works in much the same way. It can be expressed in terms of subtraction: log (a ÷ b) = log (a) – log (b). To divide 9,910,233 by 4,387, look for the logarithm of 9,910,233—it is 6.996084—and the logarithm of 4,387—it is 3.642165—and subtract one from the other: the difference is 3.353919. Then find the number that has for its logarithm 3.353919: it is 2,259, the quotient of the numbers. Best of all, this requires no mathematical sophistication, because you can do it without knowing the first thing about logarithms and exponents. It is simply a procedure: look up a, look up b, add or subtract them, and look up the result.

  Logarithms make many other mathematical operations simpler. If the concept sounds complicated, that is because we live in the computer age, when dividing 9,910,233 by 4,387 requires no more effort than typing the numbers into a spreadsheet, pocket calculator, or mobile phone. In the age of long division, though, it could take half an hour to perform a division like that, and with so many steps, error was always a possibility. How much easier it was when logarithms were introduced to look up two numbers in a table, perform one simple subtraction, and then look up another number in a table.5

  These properties of logarithms were discovered in the early sixteenth century by a Scottish baron named John Napier. Born in 1550 in Merchiston Castle, not far from Edinburgh, Napier studied at St. Andrews University.6 He was notoriously argumentative and a vindictive neighbor, but he also did much to improve society, such as developing new fertilizers and water pumps for coal pits. He worked on plans to repel a Spanish invasion of England with gigantic mirrors that could direct the sun’s rays and set enemy ships on fire, and he drew up plans for new guns and even prototypes of twentieth-century tanks.7 This master tinkerer also made a number of advances in mathematics; the decimal point was his invention.8

  He was working on logarithms as early as 1594. Napier complained that “there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculations, than the multiplications, divisions, square and cubical extractions of great numbers,” and he “began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”9 His book, Mirifici logarithmorum canonis descriptio, appeared in Latin in 1614, laying out the principles of logarithms. Early support from the astronomer Johannes Kepler gave Napier the cachet he needed. Soon logarithms were the talk of all the mathematicians in Europe, and people began discovering practical uses for them in navigation, making them of interest to commercial interests such as the East India Company.10

  Calculating logarithms for individual applications is, however, unimaginably time-consuming. Long division is child’s play next to calculating a logarithm by hand. What was needed was a table of logarithms to facilitate any real-world mathematical problem. Preparing that list required a preposterous effort, but that was the contribution of Henry Briggs. This English mathematician became interested in logarithms in 1615, and he wrote to a friend, “Naper [sic], lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. 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.”11 After visiting Napier in 1616, Briggs resolved to calculate as many logarithms as possible, to allow his readers to perform all manner of calculations more easily.

  The first fruits of his labor appeared right after Napier’s death in 1617, a pamphlet called Logarithmorum chilias prima, the first printed table of logarithms. The preface explained the purpose:

  Here is the first thousand logarithms which the author has had printed, not with the intention of becoming public property, but partly to satisfy on a private basis the wish of some of his own intimate friends: partly so that with its help he might more conveniently solve not only several following thousands but also the integral table of Logarithms used for the calculation of all triangles… .

  In a slight volume neither the enjoyment nor the toil has been slight.12

  TITLE: Arithmetica logarithmica sive Logarithmorum chiliades triginta, pro numeris naturali serie c
rescentibus ab unitate ad 20,000: et a 90,000 ad 100,000 quorum ope multa perficiuntur arithmetica problemata et geometrica

  COMPILER: Henry Briggs (1561–1630)

  ORGANIZATION: 88 pages on the principles of logarithms, then 300 pages of tables from 1 to 20,000 and 90,000 to 100,000.

  PUBLISHED: London: William Jones, 1624

  PAGES: 396

  ENTRIES: 30,000

  TOTAL WORDS: 29,000 words of discussion, 90,000 numbers in tables

  SIZE: 13¼″ × 8¼″ (33.6 × 21 cm)

  AREA: 300 ft2 (28 m2)

  The pamphlet was a start, but Briggs’s greatest work appeared in 1624 under the title Arithmetica logarithmica, a table of the base-10 logarithms of all the whole numbers from 1 to 20,000 and from 90,000 to 100,000.

  His precision was formidable: fourteen decimal places for each of his thirty thousand entries. His second edition, published in 1628, was supplemented by the Dutch publisher Adriaan Vlacq, who covered 20,001–89,999, giving the world a complete set of the first hundred thousand base-10 logarithms. Complicated calculation would never be the same, and Briggs’s book was the foundation of every table of logarithms published for centuries. No one bothered with new calculations until the end of the eighteenth century.

  This work in many ways made the scientific revolution possible. The engineering projects of the Industrial Revolution could never have come to fruition without logarithms, and scientific astronomy would still be in its infancy. Pierre-Simon Laplace, the nineteenth-century French mathematician and astronomer, marveled at this “admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.”13