July 28, 2018 Dear Reader

FALSTAFF: If I fought not with fifty of them, I am a bunch of radish. If there were not two- and three-and-fifty upon poor old Jack, then I am no two-legged creature. I have peppered two of them. Two I am sure I have paid [i.e., mortally injured]—two rogues in buckram suits. Four rogues in buckram let drive at me—

PRINCE:: What, four? Thou saidst but two even now.

FALSTAFF: Four, Hal, I told thee four. I took all their seven points in my target, thus.

PRINCE:: Seven? Why, there were but four even now.

FALSTAFF: In buckram. These nine in buckram that I told thee of—-

PRINCE:: So, two more already

FALSTAFF: [As swift as] a thought, seven of the eleven I paid.

PRINCE: O monstrous! Eleven buckram men grown out of two!

1 Henry IV, 2.5, lines 160-199, condensed.

Yet less than two centuries after Shakespeare's England Boswell says to Johnson: “Sir Alexander Dick tells me, that he remembers having a thousand people in a year to dine at his house; that is, reckoning each person as one, each time he dined there.”

Johnson: That, Sir, is about three a day.

Boswell: How your statement lessens the idea.

Johnson:: That, Sir, is the good of counting. It brings every thing to a certainty, which before floated in the mind indefinitely.

Boswell: But . . . . one is sorry to have this diminished.

Johnson:: Sir, you should not allow yourself to be delighted with error.

Life, Vol, II, 1783, aetat. NN

Everyman ed., p. 456.

Again, something has changed. As Johnson wrote elsewhere, “To count is a modern practice, the ancient method was to guess; and when numbers are guessed they are always magnified,” in the style of true Jack Falstaff, plump Jack Falstaff. ¹⁶⁰ Johnson the classicist knew what he was talking about. Gregory Clark has reviewed the startling evidence that wealthy if illiterate and innumerate Romans, for example, didn’t even know their own ages, and in the style of reported Methuselahs would grossly exaggerate, with every sign of believing their own miscalculations, the age at death of very old folk.¹⁶¹

Johnson laid it down that “no man should travel unprovided with instruments for taking heights and distances,” and himself used his walking stick.¹⁶² Boswell reports a conversation in 1783 in which Johnson argues against a walled garden on calculating grounds, as not productive enough to bear the expense of the wall—the same calculation at the same time, by the way, was surprisingly important for the enclosure movement in British agriculture. “I record the minute detail,” writes Boswell, “in order to show clearly how this great man. . . was yet well-informed in the common affairs of life, and loved to illustrate them.”¹⁶³ Because of his friendship with Mr. and Mrs. Thrale, who ran a large London brewery, he turned his quantitative mind to their hopes. In 1778 he writes, "we are not far from the great year of 100,000 barrels [of porter brewed at the Anchor's brewery], which, if three shillings be gained from each barrel will bring us fifteen thousand pounds a year. Whitbread [a competing brewery] never pretended to more than thirty pounds a day, which is not eleven thousand a year."¹⁶⁴ No wonder that "by the early nineteenth century," as Leonore Davidoff and Catherine Hall note, "foreign visitors [to England] were struck by this spirit: the prevalence of measuring instruments, the clocks on every church steeple, the 'watch in everyone's pocket,' the fetish of using scales for weighing everything including ones own body and of ascertaining a person's exact chronological age."¹⁶⁵

Such an idea of counting and accounting is obvious to us, in our bourgeois towns. It is part of our private and public rhetorics. But it had to be invented, both as attitude and as technique. What we now consider very ordinary arithmetic entered late into the educations of the aristocracy and the clergy and the non-merchant professions. In 1803 Harvard College required both Latin and Greek of all the boys proposing to attend. Of course. Yet only in that year did it also make the ability to figure a requirement. Johnson advised a rich woman, "Let your boy learn arithmetic"—note the supposition that the heir to a great fortune would usually fail to—"He will not then be a prey to every rascal which this town swarms with: teach him the value of money and how to reckon with it." ¹⁶⁶

Consider such a modern commonplace as the graph for showing, say, how the Dow-Jones average has recently moved (cartoon: man sitting in front of a wall chart on which an utterly flat line is graphed declares to another, “Sometimes I think it will drive me mad.”) Aside from the “mysterious and isolated wonder” of a 10^th–century plotting of planetary inclinations, Edward Tufte observes, the graph appeared surprisingly late in the history of counting. Cartesian coordinates were of course invented by Descartes himself in 1637, unifying geometry and algebra, perhaps from the analogy with maps and their latitudes and longitudes. But graphical devices for factual observations, as against the plotting of algebraic equations on Cartesian coordinates, were first invented by the Swiss scientist J. H. Lambert in 1765 and, more influentially, by the early economist William Playfair in two books at the end of the 18^th century, The Commercial and Political Atlas, 1786 (the time series plot and the bar chart) and The Statistical Breviary Shewing on a Principle Entirely New the Resources of Every State and Kingdom of Europe, 1801 (the pie chart; areas showing quantities; exhibiting many variables at one location), “applying,” as Playfair put it, “lines to matters of commerce and finance.”¹⁶⁷

Obsession with accurate counting in Europe dates from the 17^th century. Pencil and paper calculation by algorithm, named after the district of a 9^th-century Arabic mathematician, and its generalization in algebra (al-jabr, the reuniting of broken parts) depended on Arabic numerals, that is, on Indian numerals, with place value and a zero (Arabic sifr: emptiness). The abacus makes rapid calculation possible even without notation, and mastery of it slowed the adoption of Arabic numerals in Europe and in China. Compare the state of mental computing skills among our children nowadays, equipped with electronic calculators.

You cannot easily multiply or divide with Roman numerals. Only in the 16^th and 17^th centuries did Arabic numerals spread widely to Northern Europe. Admittedly the first European document to use Arabic numerals was as early as 976. The soon-to-be Pope Sylvester II (ca 940 - 1003) —or rather “the 2^nd”—tried to teach them, having learned them in Moorish Spain. His lessons didn't take. The merchant and mathematician Leonardo Fibonacci re-explained them in a book of 1202. The commercial Italians were using them freely by the 15^th century, though often mixed with Roman.¹⁶⁸ But before Shakespeare’s time 0, 1, 2, 3, . . . 10, . . . 100 as against i, ii, iii, . . . x, . . . c had not spread much beyond the Italian bourgeoisie. The Byzantines used the Greek equivalent of Roman numerals right up to the fall of Byzantium in 1453. And still in the early 18^th century Peter the Great was passing laws to compel Russians to give up their Greek numerals and adopt the Arabic.¹⁶⁹

The bourgeois boy in Northern Italy from earliest times and later elsewhere in Europe did of course learn to multiply and divide, somehow. He had to, and as I noted used an abacus. Presumably the same was true earlier at Constantinople and Baghdad and Delhi. By the 18^th century the height of mathematical ability in an ordinary man or a commercial woman was the Rule of Three, which is to say the solving of proportions: “Six is to two as N is to three.” In Europe centuries earlier one could hardly deal profitably as a merchant with the scores of currencies and systems of measurement without getting the Rule of Three down pat. Interest, eventually compounded, was calculated by table. We can watch Columella in 65 AD. making mistakes with the compounding. The logarithms that permit direct calculations of compounding were not invented until 1614 by the Scotsman Napier, who by the way also popularized the decimal point, recently invented by the Dutchman Stevin—3.5, 8.25, etc. rather than 3 ½ , 8¼ , etc.

In England before its bourgeois time the Roman numerals prevailed. Shakespeare’s opening chorus in Henry V, two years after 1 Henry IV, apologizes for showing battles without Cecil-B.-de Millean numbers of extras. Yet “a crooked figure may /Attest in little place a million; / And let us, ciphers to this great accompt [account], / On your imaginary forces work.” The “crooked figure” he has in mind is not Arabic “1,000,000,” but merely a scrawled Roman M with a bar over it to signify “multiplied by 1000”: 1000 times 1000 is a million.