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One countable piece of evidence that bourgeois values were becoming dominate in England in the 17th and 18th centuries is the new, dominate role of counting in giving evidence. It is assuredly modern. The pre-modern attitude---which survives of course in many a non-quantitative modern---shows in a little business between Prince Hal and Sir John Falstaff. The scene is fictional early 15th century, and 1 Henry IV was written in London in 1597. Hal disguised in stiffened cloth had been last night one of the merely two assailants of Falstaff and his little gang of three other thieves. The bold knight had in fact after token resistance fled in terror like his confederates. One of them, Gadshill, and poor old Jack Falstaff re-count the episode to Prince Hal:

FALSTAFF: A hundred upon poor four of us.
PRINCE: What, a hundred, man?
FALSTAFF: I am a rogue if I were not at half-sword with a dozen of them, two hours together.
GADSHILL: We four set upon some dozen---
FALSTAFF [to the PRINCE] Sixteen at least, my lord.
GADSHILL: As we were sharing [the loot], some six or seven fresh men set upon us.
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., killed]---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.

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. Johnson laid it down that "no man should travel unprovided with instruments for taking heights and distances," and himself used his walking stick. 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." 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."

"By the early nineteenth century," 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."

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 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."

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 10th-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 18th 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."

Obsession with accurate counting in Europe dates from the 17th century. Pencil and paper calculation by algorithm, named after the district of a 9th-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 16th and 17th 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 2nd"---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 15th century, though often mixed with Roman. 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 18th century Peter the Great was passing laws to compel Russians to give up their Greek numerals and adopt the Arabic.

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 18th century the height of ordinary mathematical ability 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.

Peter Wardley has pioneered for the study of numeracy in England the use of probate inventories, statements of property at death available in practically limitless quantities from the 15th century onward. He has discovered that as late as 1610 even in commercial Bristol the share of probates using Arabic as against Roman numerals was essential zero. By 1670, however, it was nearly 100%, a startlingly fast change. Robert Loder's farm accounts, in Berkshire 1610-1620, uses Roman numerals almost exclusively before 1616, even for dates of the month. In 1616 he starts to mix in Arabic, as though he had just learned to reckon in them ---he continued to use Arabic for year numbers, probably because years, like regnal orders, Elizabeth II and James I, or Superbowl XVI, are not subjects of calculation.

Luca Pacioli of Venice popularized double-entry book-keeping at the end of the 15th century, and such sophistications in accounting rapidly spread in bourgeois circles. The metaphor of a set of accounts was nothing new, of course, as in God's accounting of our sins; or the three servants in Jesus' parable (Matt. 25: 14-30) rendering their account [logon, the Greek for "word" being the usual term for commercial accounts] of their uses of the talents, "my soul more bent / To serve therewith my Maker, and present / My true account, lest he returning chide." Bourgeois and especially bourgeois Protestant boys actually carried it out, as in Franklin's score-keeping of his sins.

We must not be misled by the absence in Olden Tymes of modern arithmetical skills into thinking that our ancestors were merely stupid. Shepherds had every incentive to develop tricks in reckoning, as in the Welsh system of counting, perhaps from how many sheep the eye can grasp at a glance. The myth is that primitive folk count "one, two, many." Well, not when it matters. Carpenters must of course have systems of reckoning to build a set of stairs. The habit of counting and figuring is reflected in handbooks for craftsmen from the late Middle Ages on, the ancestors of the present-day ready reckoners for sale at the checkout counter at your Ace Hardware store. And you cannot build a great pyramid, or even probably a relatively little stone henge, without some way of multiplying and dividing, at least in effect, multiplying the materials and dividing up the work. The first writing of any sort is counting, such as storage accounts in Mesopotamia or Crete and calendar dates in Meso-America and reckoning knots in Peru. In Greek and Latin the magicians of the East were called mathematici because calculation---as against the much more elegant method of proof invented by the Greeks---was characteristic of the Mesopotamian astrologers.

Large organizations counted perforce. Sheer counts had often a purpose of taxation---St. Luke's story about a decree from Caesar Augustus that all the world should be taxed, for example; and in 1086 the better attested case of William the Conqueror's Domesday Book. We owe our knowledge of medieval agriculture in Europe to the necessity in large estates to count, in order to discourage cheating by subordinates. We can see in such records the scribes making mistakes of calculation with their clumsy Roman numerals. We know less about later agriculture because the size of estates went down after the Black Death and such accounting was therefore less worthwhile.

Sophisticated counting in modern times cuts through the Falstaffian fog of imprecision which most people start with. Nearly universal outside specialized merchants or shepherds before the common school, the fog, I repeat, persists now in the non-numerate. Here is a strange example. The standard estimate for the prevalence of male to female gender crossers in the United States is one in 30,000 born males. This is the figure in The Diagnostic and Statistical Manual of Mental Disorders, 4th edition, 1994. Let us put aside the issue of whether it is a "mental disorder," or what purpose of gender policing would be served by claiming that the disorder is so very rare. An emerita professor of electrical engineering at the University of Michigan, Lynn Conway, a member of the National Academy of Engineering and one of the inventors of modern computer design (after IBM fired her for transitioning in 1968 from male to female), notes that the figure is impossibly low. It would imply by now in the United States a mere 800 completed gender crossers, such as Conway and me---when in fact all sorts of evidence suggests that there are at least 40,000.

The showing of such a contradiction, like Prince Hal comments on Falstaff's boasting exaggeration, is the kind of point a numerate person makes. The sex doctors seem not to be modern in their quantitative habits of thought. A figure of 800, Conway observes, would be accounted for (note the verb) by the flow of a mere two year's worth of operations by one doctor. Conway reckons the incidence of the condition is in fact about one in every 300 born males---not one in 30,000. It is two orders of magnitude more common than believed by the psychiatrists and psychologists who in their innumeracy write the Manual. Conway suspects that among other sources of numerical fog the doctors are mixing up prevalence with incidence---stock with flow, as accountants and economists would put it. That is, they are mixing up the total number existing as a snapshot at a certain date with the number born per year.

Calculation is the skeleton of prudence. But precisely because it embodies ignoble prudence the aristocrat scorns calculation. Courage, his defining virtue, is non-calculating, or else it is not courage. Henry V prays to the god of battles: "steel my soldiers' hearts;/ Possess them not with fear; take from them now the sense of reckoning, if the opposed numbers/ Pluck their hearts from them." And indeed his "ruined band" before Agincourt, as he had noted to the French messenger, was "with sickness much enfeebled, / My numbers lessened, and those few I have / Almost no better than so many French." Yet his numbers of five or six thousand did not prudently flee from an enemy of 25,000 on the Feast Day of Crispian.

One reason, Shakespeare avers, was faith, as Henry says to Gloucester: "We are in God's hand, brother, not in theirs." The other was courage: "'tis true that we are in great danger; / The greater therefore should our courage be." Shakespeare of course emphasizes in 1599 these two Christian/aristocratic virtues, those of the Christian knight, and not for example the prudence of the warhorse-impaling stakes that on Henry's orders the archers had been lugging through the French countryside for a week. Prudence is a calculative virtue, as are, note, justice and temperance. They are cool. The warm virtues, love and courage, faith and hope, the virtues praised most often by Shakespeare, and praised little by bourgeois Adam Smith two centuries on, are specifically and essentially non-calculative.

The play does not tell what the real King Henry V was doing in the weeks leading up to Sunday, October 25, 1415, of course. It tells what was expected to be mouthed by noblemen in the last years of Elizabeth's England, a place in which only rank ennobled, and honor to the low-born came only through loyalty to the nobles. Before the taking of Harfleur ("Once more unto the breach, dear friends"), Henry declares "there's none of you so mean and base, / That hath not noble luster in your eyes"; and before Agincourt: "For he today that sheds his blood with me / Shall be my brother; be he ne'er so vile, / This day shall gentle his condition."

Out of earshot of Henry, the king's uncle grimly notes the disadvantage in numbers: "There's five to one; besides they all are fresh"; at which the Earl of Salisbury exclaims, "God's arm strike with us! 'tis a fearful odds." The King comes onto the scene, and the Earl of Westmoreland continues the calculative talk: "O that we now had here / But one ten thousand of those men in England / That do no work today!" To which Henry now replies, scorning such bourgeois considerations, "If we are marked to die, we are enow [enough] / To do our country loss; and if to live, / The fewer men, the greater share of honor."

And gentlemen in England now a-bed
Shall think themselves accursed they were not here,
And hold their manhoods cheap whiles whiles? any speaks
That fought with us upon St. Crispin's Day.

This is not bourgeois, prudential rhetoric, and counts not the cost.

Public calculation is highly characteristic of the bourgeois world, such as the political arithmeticians of the 17th century, first in Holland and then in England and then in France. The first person in Europe to suggest that accounting could be applied to the affairs of an entire nation, as though the nation were a business firm, appears to be the inventor of the decimal point, the Dutch mathematician and statesman Simon Stevin(us) (1548-1620), who persuaded the City of Amsterdam and the King of Sweden to adopt double-entry bookkeeping. As late as 1673 Sir William Temple was observing, astonished, of the Dutch that "the order in casting up [i.e. accounting for] their expenses, is so great and general, that no man offers at [i.e. attempts] any undertaking which he is not prepared for, and [is not] master of his design before he begins; so as I have neither observed nor heard of any building public or private that has not been finished in the time designed for it." The English were not slow to adopt such rationality, or at least to claim it. Sir William Petty announced in 1690

The method I take to do this is not yet very usual. For instead of using only comparative and superlative words, and intellectual arguments I have taken the course (as a specimen of the political arithmetic I have long aimed at) to express myself in terms of number, weight, or measure; to use only arguments of sense.

It is the manifesto of a bourgeois age.

In an economics course recently I assigned my undergraduate students, whom I try to teach to think prudently like the Dutch of the Golden Age, the task of calculating the costs and benefits of the automobiles that three-quarters of them operated. I suspected that American college students work many hours in non-studying jobs, skimping their learning, to pay for cars and pizzas---though come to think of it, so do their parents. My suspicion was confirmed. Shame on them.

But it seemed only fair for the professor herself to take the test. It turned out that the indignant professor was the most irrational owner of an automobile in the class. My beloved seven-year old Toyota Avalon was costing me $4000 a year more than the same services would cost to get in other ways where I live in downtown Chicago. Taxis stream by my front door on South Dearborn day and night. On the other side of the accounts a parking place off-street is $160 a month and the city's meter maids on-street are cruelly efficient. So I sold the car. And likewise, probably, should you. I suggest you do the calculation, and certainly for that third car that sits outside your house used once a week.

But a rhetoric of calculation since the 17th century does not mean that Europeans actually were rational. Many social scientists following Max Weber have mistakenly supposed they were, that a new skill with numbers and with accounts meant that Europeans had discovered rationality. The numbers and calculation and accounts appeal to a rhetoric of rationality---terms of number, weight, or measure; only arguments of sense. But they do not guarantee its substance.

The numbers, for one thing, have to be correct. So does the accounting framework in which they are calculated. So does the evaluative job they are supposed to do. These are heavy, heavy requirements, and any quantitative scientist knows that most people, even other scientists, get them wrong..

For example, the technique of "statistical significance" used in certain quantitative fields such as medicine and economics---though not in physics, say---turns out to be on inspection comically mistaken. Many thousands of earnest researchers into medicines and minimum wages persuade themselves that they are doing a properly bourgeois calculation when in fact the calculation is very largely irrelevant to what they want to know. Like businesspeople priding themselves on economically erroneous allocating of fixed costs to various branches of their business, the medical and social scientists who use so-called t or p or R tests are doing more than fooling themselves. They are killing people and ruining economies. The suspicion that "you can prove anything with statistics" is primitive. But in field after field of the intellect, from politicized census-taking up to double blind experiments sponsored by Merck it seems approximately true, at the 5% level of significance.

In 1713, John Nye explains in his recent history of British-French commercial relations, the British makers of drink had long benefited from the prohibition of imports of French wine into Britain. Britain and France had just concluded their long and bloody quarrel over the Spanish succession, and a bill in Parliament proposed therefore to drop the wartime preferences for Spanish and Portuguese wines, to which unsurprisingly the existing importers of Spanish and Portuguese wines---there were of course no legal importers of French ones to speak up for that trade---objected strenuously. A frantic river of pamphlets spilled out a rhetoric of accounting and quantities. It was the first time, Nye notes, following G. N. Clark, "that the newly collected statistics on British trade entered the political debate in a substantial way," serving "as a basis for the mercantilists' published statements of economic doctrine."

The wine trades with Portugal, wrote one defender of the status quo, "have as constantly increased every year as we have increased the demand for their wines, by which means the navigation and seamen of this kingdom have been greatly encouraged." If French wines are allowed back into Britain the navigation and seamen will be ruined, because "small ships and an easy charge of men can fetch wines from France." And so "the greatest part of those ships must lie and rot, or come home dead freighted," resulting in a rise in freight rates on British exports, to the detriment of the country's treasure by foreign trade. Another British pamphleteer reckoned that "the advantage to the French nation by having such a vent for their wines" was very great. "The French king . . . would give a million of money to procure" it. Another that

formerly the king of Portugal prohibited the importation of cloth into his kingdom. . . . [The] prohibition was taken off on consideration that Portugal wine should pay [in Britain] one third less duty than French. . . . Should the duty on French wines be lowered . . . . we very much fear that the French king will take the opportunity of introducing his subjects' cloth into Portugal, which being of a thinner manufacture than the cloth of this nation, may be fitter for that country and their Brazils. . . . We may forever lose the cloth trade in that kingdom

In June of 1713 the bill to relax the duties on French wine was rejected, though he quantitative arguments were all specious. The social accounting was mistaken, sometimes positively wacko. But a rhetoric of quantitative prudence ruled. Such bourgeois, quantitative reasoning was in Britain rare a century before, though among the Dutch it was already commonplace in 1613. "Constantly increased." "The greatest part of those ships." "A million of money." "One third less duty."

But I said there can be a sort of madness in the counting, and counting is no guarantee of actual rationality. As a calculating modern person, even an economist, before I sold my car I first went on a big shopping expedition, as my mother prudently advised, and stocked up with $1500-worth of Barilla Thin Spaghetti and Manischewitz Thin Tea Matzos. As an aid to such prudence I worked out little tables of equivalences, like the builder's ready reference book: If you use ½ a package of Quaker Instant Oats a week, and want two-years' worth, that's . . . let's see, ½ x 52 x 2 = 52 boxes. Calculation embodies a modern sort of prudence, even when it is slightly mad. I still have by actual count, three years now after the shopping spree, 11 cans of Pillar Rock Pink Salmon, but can't find the sell-by date on them. Auden writes in 1940: "The measurable taking charge/ Of him who measures, set at large/ By his own actions, useful facts/ Become the user of his acts.

What the modern fascination with charts, graphs, figures, and calculations shows is that moderns admire prudence. It does not show that they practice it. What changed from Shakespeare's time to Dickens' time was the rhetoric, and the social prestige of people like merchants and engineers and economists who specialized in it.……

Spoof by Dickens: