PART ONE PANTOMETRY ACHIEVED

1 Pantometry: An Introduction
2 The Venerable Model
3 Necessary but Insufficient Causes
4 Time
5 Space
6 Mathematics

PART TWO STRIKING THE MATCH:
VISUALIZATION

7 Visualization: An Introduction
8 Music
9 Painting
10 Bookkeeping

PART THREE EPILOGUE

11 The New Model



PREFACE


Cyrus the Great, Alexander the Great, Genghis Khan, and
Huayna Capac were great conquerors, but they were all confined
to no more than one continent and at best a wedge of a second.
They were homebodies compared with Queen Victoria, on whose
empire (to resuscitate a very old cliche) the sun literally never set.
The sun also never set on the empires of France, Spain, Portugal,
the Netherlands, and Germany in their heyday. The explanations
for this triumph, popular in Europe circa 1900, were fueled by
ethnocentrism and justified by Social Darwinism. They were, simply,
that those members of the human species most subject to
painful sunburns were the most recent, highest, and, in all likelihood,
final twigs on the exfoliating tree of evolution. Pale people
were the brightest, most energetic, most sensible, most aestheti-
cally advanced, and most ethical humans. They conquered all
because they deserved to.

Europeans were incomparably successful at sending
ships across oceans to predetermined destinations
and at arriving at those destinations with superior
weaponry--with, for instance, cannons superior to
those of the Ottomans and the Chinese--that they
were more efficient at operating joint-stock com-
panies and empires of unprecedented extension
and degree of activity than anyone else.

Europeans were not as magnificent as they believed, but they were
able to organize large collections of people and capital and to
exploit physical reality for useful knowledge and for power more
efficiently than any other people of the time. Why?

Westerner's advantage, I believe, lay at first not in their science
and technology, but in their utilization of habits of thought that
would in time enable them to advance swiftly in science and tech-
nology and, in the meantime, gave them decisively important admin-
istrative, commercial, navigational, industrial, and military skills. The
initial European advantage lay in what French historians have called
mentalite.

During the late Middle Ages and Renaissance a new model of
reality emerged in Europe. A quantitatiye model was
just beginning to displace the ancient qualitative mo-
del. Copernicus and Galileo, the artisans who taught themselves to
make one goodcannon after another, the cartographers who mapped
the coasts ofnewly contacted lands, the bureaucrats and entrepre-
neurs who managed the new empires and East and West India com-
panies, the bankers who marshaled and controlled the streams of new
wealth--these people were thinking of reality in quantitative terms
with greater consistency than any other members of their species.
We look upon them as initiators of revolutionary change, but they
were also heirs of changes in mentalite that had been fermenting for
centuries. This book is about those changes.



　　　　　　　　　PART ONE
　 　　Pantometry Achieved


Pantometry

1. Universal measurement: see quots. Obs. [1571 Diggs
(title) A Geometrical Practice, named Pantometria, divided
into three Bookes, Longimetra, Planimetra, and Steriometria.]
Oxford English Dictionary



　　　　　　　　　　CHAPTER ONE
　　　　Pantometry: An Introduction



Every culture lives within its dream.
Lewis Mumford (1934)



In the mid-ninth century A.D. Ibn Khurradadhbeh described
Western Europe as a source of "eunuchs, slave
girls and boys, brocade, beaver skins, glue, sables,
and swords," and not much more. A century later another
Muslim geographer, the great Masudi, wrote that Europeans
were dull in mind and heavy in speech, and the
"farther they are to the north the more stupid,
gross, and brutish they are."2 This was what any
Muslim sophisticate would have expected of Christians, par-
ticularly the "Franks," as Western Europeans were known in
the Islamic world, because these people, barbarians most of them,
lived at the remote Atlantic margin of Eurasia, far from the hearth-
lands of its high cultures.

Six centuries later the Franks were at least equal to, and even
ahead of, the Muslims and everyone else in the world in certain
kinds of mathematics and mechanical innovation. They were in
the first stage of developing science-cum-technology that would
be the glory of their civilization and the edged weapon of their
imperialistic expansion. How, between the ninth and the sixteenth
centuries, had these bumpkins managed all that?



Kitsch is a peephole through which we may see samples if not
always of a society's bromides, then of what it is thinking about
with freshest intensity and even of how it is thinking. I offer in
evidence Pieter Bruegel the Elder's 1560 print of Temperance
(Figure 1), then the most prestigious of the ancient Virtues. A
Latin motto printed below the original is bromidic ("We must
look to it that we do not give ourselves over to empty pleasures,
extravagance, or lustful living; but also that we do not, because of
miserly greed, live in filth and ignorance")4 but the artist, aiming
for sales, made sure that just about everything else in the print was
new or at least newly applauded.




Bruegel obviously assumed that his contemporaries and prospective
customers took pride in their ability to measure, to
oblige a fluid reality to stand still and submit to
the application of the quadrant and T square.



Sixteenth century military textbooks commonly included tables
of squares and square roots to guide officers in arranging
hundreds and even thousands of men in the new battle formations
of the Renaissance West: squares, triangles, shears, bastard
squares, broad squares, and so on.s Officers, the good ones, now
had "to wade in the large sea of Algebra & num-
bers" or to recruit mathematicians to help them. Iago, the old
soidier andvillain of Shakespeare's Othello, dismisses Cassio as
an "Arithmetician," who had "never set a squadron in the field,"
but suchnumber-smiths had become a military necessity.

The new kind of war had reduced foot soldiers to quanta.
They, even more than the men of the Greek phalanx and Roman
legion, learned to perform like automatons. They began to do
something that we have considered characteristic of soldiers ever
since: to march in step...Francois Rabelais laughed at soldiers
who peformed like "a perfect clockwork mechanism,"



Johannes Gutenberg a century before had standardized Gothic
letters, casting them onto the faces of small metal cubes of uniform
dimensions, excepting width ("M" being, after all, wider than
"I"). He lined these up on a block like ranks of soldiers on parade,
wedged them tight, and then pressed the block on paper, printing
a whole page at a time. His most famous product was the Mazarin
Bible: forty-two lines to the page of about 2,750 letters each, with
left and right margins justified.


In this print Bruegel violated the primary diktat of Renaissance
perspective that a picture should be geometrically consistent and
should include no more than one point of view. He jammed several
scenes together, each with its own point of view.



Immediately above the artist are a number of musicians and
one drudge pumping an organ. The singers are performing music
from texts...The sixteenth was the century of Josquin de Pres and
Thomas Talk, the golden age of church polyphony, a
kind of music so complicated that it could be performed best--
perhaps can be performed only--with the aid of written nota-
tion....indicating, from top to bottom, the pitch of notes, and fig-
uresthereon indicating the order of the notes and rests, which in
duraion were all equal or exact multiples or fractions of each other.
Tanis, one of Bruegel's contemporaries, will compose Spem in
alium, in forty separate parts, possibly for Queen Elizabeth's forti-
eth birthday in 1573.13 This motet is the ne plus ultra
of the quantum approach to sound, not surpassed as
a bravura display of countpoint from that day to this.



Temperance herself occupies the center of the picture. In her
left hand she holds spectacles, a symbol of sagacity, and in her
right hand are reins that lead to a bit in her mouth, representing
self-restraint. She wears spurs on her heels (control over great
power) and a snake knotted around her waist as a belt (evil
passions under control?). She stands on the vane of a post windmill,
medieval Europe's greatest single contribution to power technology.
At the dead center of the picture--surely not by accident -
she wears on her head what was then the most distinctively Western
of all contrivances for measuring quantity: the mechanical
clock, whose titanic TICK-TOCK had already been thundering in
Europe's ears for 250 years.



Bruegel's print is a sort of potpourri of what quickened the atten-
tionof urban Western Europeans circa 1560, of what we might call
the West's Renaissance dream. The collection is such a miscellany
that it is not easy to put a name to that dream. No one was con-
cerned with its internal consistency or even thought of it as a
whole. It was a yearning, a demand, for order. Many of the people
in Bruegel's picture are engaged in one way or another in
visualizing the stuff of reality as aggregates of uni-
form units, as quanta: leagues, miles, degrees of
angle, letters, guldens, hours, minytes, musical
notes. The West was making up its mind (most of its mind, at
least) to treat the universe in terms of quanta uniform in one or
more characteristics, quanta that are often thought of as arranged
in lines, squares, circles, and other symmetrical forms: music staffs,
platoons, ledger columns, planetary orbits



The writings of Plato and Arisotle celebrate an un-, an
almost anti-metrological approach. and have the further
advantage of being representative of our ancestral mode of thought,
at its best.

These two men thought more highly of human reason than we do,
but they did not believe our five senses capable of
accurate measurement of nature.

You and I are ready to agree that the raw data of everyday exper-
ience are variable and our senses frail, but we believe we have
a category that the two philosophers did not think they had: a
category of things that are sufficiently uniforrm to justify our
measuring them, after which averages and means can be caculated.
As for the dependability of our senses in making such measurements,
we point to the achievements we have made on the basis of their
dependability: power-looms, spacecraft, actarial tables, and so on.



Aristotle, for instance, stated that the mathematician measures
dimensions only after he "strips off all the sensible qualities,
e.g. weight, and lightness, hardness and its contraries, and also
cold and other sensible contrarities.

child psychologists declare that humans even in infancy, show
indications that they are innately endowed with the ability to
count discrete entities (three cookies, six balls, eight pigs),
but weight, hardness, and so on do not come to us
as quantities ofdiscrete entities. They are condi-
tions, not collections; and even worse, they are
often flowing changes...hardness, heat, speed,
acceleration--how in the world could we quantify
those?

For instance, when in the fourteenth century the scholars of Ox-
ford's Merton College began to think about the benefits of mea-
suring not only size, but also qualities as slippery as mo-
tion, light, heat, and color, they forged right on,
jumped the fence, and talked about quantifying
certitude, virtue, and grace



Weighing, counting, and surveying were worldly activities, but
mathematics proved to have transcendental qualities that intox-
icated those trying to reach through the scrim of mundanity for
truth.



Plato recommended turning away from the material
world because it "is always becoming and never is"
and turning toward "that which always is and has
no becoming." He directed our attention to absolute beauty,
goodness, and righteousness, and to the ideal triangle, square, and
circle, to abstractions that, he was sure, existed independent of
the material world.



Plato decided that the number of citizens in the ideal state was
5,040. That number seems a sensible choice because it may represent
about as many people as can hear one individual speak at a time with-
out special amplification, but Platodid not select it for that rea-
son. He chose it because it is the product of 7 x 6 x 5 X 4 X 3 X
2 X 1.25 This is mathematical mysticism, and the path from it to
numerology is shorter than that to double-entry bookkeeping.

Aristotle was inclined to think that Platonism lack-
ed ballast. In contrast to his great teacher, he honored
those who kick boulders and, in their pain, insist
that a broken toe is proof that boulders are real.
He accepted sensory data - but he doubted that math-
ematics was of much use in interpreting those data. Geometry,
for instance, was all well and good, but boulders were never
perfectly spherical nor were pyramids perfectly pyramidal, so what
was the use of treating them as such? The intelligent person would,
of course, see that one boulder was bigger than another, more or
less round than another, but would not waste time trying to
measure exactly anything as variable as material reality.



The West's distinctive intellectual accomplishment was to bring
mathematics and measurement together and to hold them to the task
of making sense of a sensorially perceivable reality, which West-
erners, in a flying leap of faith, assumed was temporally and
spatially uniform and therefore susceptible to such examination.

How, why, and when did Europeans get beyond sim-
ply heaping up sensory data like pack rats collect-
ing shiny junk? How, why, and when did they save
themselves from an eternity of baying at the moon
of Platonic reality?


Roger Bacon measured the angle of the rainbow, Giotto painted with
geometry in mind, and Western musicians, who had been writing a
ponderous kind of polyphony called ars antiqua for several gener-
ations, took flight with ars nova and began to write what they
called "precisely measured songs." There was nothing quite like
this half century again until the turn of the twentieth century,
when the radio, radioactivity, Einstein, Picasso, and Schonberg
swept Europe into a similar revolution.


The quantificational signal appeared as Western Europe, circa
1300, reached its first peak in population and economic growth,
and persisted as the West stumbled down into a century of horrors,
of demographic collapse, chronic war, impromptu ravagings, a dis-
credited church, periodic famines, and tidal waves of infection,
the greatest of which was the Black Death. During that
century Dante wrote his Comedy; William of Ockham wielded his
incisive razor; Richard of Wallingford built his clock; Machaut
composed his motets; and some Italian skipper ordered a helmsman
to steer a compass course from Cape Finisterre across the Bay
of Biscay to England, chosen not by consulting words, oral or
written, but a maritime chart. Another Italian, possibly one who
owned an interest in the vessel in question, made up something
resembling a balance sheet. For the historian it is like watching a
wounded hawk drift into an invisible thermal and soar and soar.



　　　　　　　　　　CHAPTER TWO
　 　　　　The Venerable Model



Pantometry is one of the neologisms that appeared in increasing
numbers in the languages of Europe in the first half of
the second Christian millennium, words summoned into being by
new tendencies, institutions, and discoveries. Milione and America
are others. A general surge of more in the 1200s rendered the
awkward and seldom-used a thousand thousand obsolete and inspired
a convenient replacement: milione. Columbus and Amerigo
Vespucci and the like created the need for America two centuries
or so later. These words were sparks thrown off by the wheels
of Western society veering and grating against the sides of old
ruts. The veerings and gratings are the subject of this book, but
first we must examine the ruts, that is to say, the view of reality
that most medieval and Renaissance Western Europeans accepted
as correct.

We can begin by putting aside the word rut....It supplied a means
by which tens of generations made sense of their surroundings, all
the way from things immediately at hand out to the fixed stars.
No, not a rut: groove, with its connotations of repetition, con-
venience, and ease, is better



The Venerable Model maintained a near monopoly in European common
sense for so many generations because...it squared with actual ex-
perience. Furthermore, it answered the need for a description of
the universe that was clear, complete, and appropriately awesome
without being stupefying. To illustrate: anyone could see that the
heavens were vast, pure, and utterly different from the earth, but
also that they revolved around



Time was awesome, but not so as to exceed the capacity of the
mind to encompass. Eusebius, circa A.D. 300, declared that God
had created the universe and had wound time up and set it going
5,198 years before the Incarnation. The Venerable Bede, circa
700, was sure that Creation was even more recent: the figure
according to his reckoning was 3,952 years before the Incarna-
tion. No medieval or Renaissance Westerner of repute suggested
that the number of years since the beginning, from Creation to
Incarnation to the present, was as high as 7,000.

Space was also vast, but not benumbingly so. Gossoin of Metz,
writing about 1245, calculated that if Adam had set off straight
up immediately after his creation at a rate of 25 miles a day (a
good day's tramp, but not too much for a healthy young man), he
would still have 713 years to go before he reached the fixed stars.
A few decades later Roger Bacon calculated that a person walking
20 miles a day would take 14 years, 7 months, 29 days and a
fraction to reach the moon. For some of the West's best-informed
scholars the extent of the universe could still be described in terms
of walking.

Reality (a word I will use to mean everything material within
time and space, plus those two dimensions per se) had humanly
comprehensible dimensions and functioned in ways that people
could understand or to which they could reconcile themselves, but
that did not mean it was essentially uniform. They perceived real-
ity as an uneven, heterogeneous sort of thing...



Europeans dealt with reality's essential heterogneity by ack-
nowledging it in even the most immediate manifestations: fire rose
and rocks fell not because they had different amounts of the same
abstract thing, weight, but because they were different, period.
Reality, however, was not absolutely chaotic...



Did that make it quantifiable by mere humans? It well might,
assuming that God deigned to be reasonable in human terms,
though investigators' obsession with the immeasurable first
cause, God, would for a long time divert attention away from
immediately perceptible and possibly measurable secondary velocity,
temperature, and so on.



Believers in the Venerable Model doted on symbolism, which
is more usefully sampled than abstractly described. Let us turn to
examples, one from geography (space) and one from historiography
(time). Christians agreed that the crucifixion of Jesus was
the pivot of all time--and therefore of the world. Jerusalem,
the scene of His crucifixion, must be the center of the inhabited
surface of the earth. Did not Ezekiel 5: 5 say, in anticipation of His
agony, "This is Jerusalem: I have set it in the midst of the nations
and countries that are round about her"?


Pope Urban II, in the sermon that launched the First Crusade, also
described Jerusalem as at "the center of the earth" (and, in addi-
tion, as in the midst of a "land fruitful above all others, like
another paradise of delights").



All history, thought many, including historians, was embodied in the
scheme of the Four Kingdoms derived from a passage in the
Book of Daniel. Nebuchadnezzar dreams of a statue with a head
of gold, chest and arms of silver, belly and thighs of bronze, legs
of iron, and feet of iron intermixed with clay. (The clay feet live
on in our aphorism about the inevitable weakness of even the
powerful.) The head, the old Europeans believed, represented the
Babylonian Empire, to be succeeded by empires of silver, then
bronze, then iron for a total of four. The last, made of iron, would
last a long time and was often identified as the Roman Empire,
which would endure, in one form or another, until the events
leading directly to the end of time. This obliged Christians to
perform the trick of identifying the Carolingian and Ottonian
empires as Roman. To do otherwise would have been to wreck an
invaluable symbol that knit together a holy and distant past, a
fleeting present, and a holy and impending future.



We begin with time. Europeans did not think there was much
of it, of time. St. Augustine warned against the effrontery of trying
to calculate the totality of time, that is, the exact number of years
from the Beginning to the appearance of the Antichrist, of Christ's
second coming, of Armageddon, and of the end of time.



Europeans of old had an enormous tolerance for anachronism.
For example, in the sixth century Gregory of Tours knew people
who had personally seen the chariot ruts in the bottom of the Red
Sea made by the Israelites fleeing Pharoah's army, ruts miracu-
lously renewed after every new accumulation of silt. If true,
then the exact year of the Exodus was not compellingly important,
perhaps not even very interesting. Time, beyond the individual life
span, was envisioned not as a straight line marked off in equal
quanta, but as a stage for the enactment of the greatest of all
dramas, Salvation versus Damnation.



The first six ages began, respectively, with Creation, the
Flood, Abraham, David, the Captivity of Judah, and the Birth
of Christ. The sixth age would end with the Second Coming.
Then there would be a Sabbath and finally eternity .



Augustine knew that the antediluvian people of the First Age had
lived hundreds and hundreds of years each--the Bible said so -
and also that they were much bigger than his contemporaries.
Virgil and Pliny the Younger said so, and floods now and again
turned up impressively large bones. Augustine wrote that he had
seen a human tooth so big that, if divided into teeth of normal
size, it would have made a hundred. 1

Such beliefs were common because Europeans did not have a
vivid concept of causation through time, that is, of a lineage of
factors, one leading to another, effecting significant change. The
transitions from one age to another had been abrupt--for example,
the Flood, the Incarnation--and, from the human point of view,
arbitrary. Getting from giant predecessors who lived for centuries
to us, small and of brief life span, in only a few thousand years
is not hard if you have a notion of an omnipotent God where many
of us have a notion of evolution.

Western Europeans had a reasonably accurate calendar, which
they inherited from the Romans, from Julius Caesar, to be precise.



This, the Julian calendar, was standard for Christendom for a
millenriium and a half, but many other temporal details remained
unsettled. The date for the beginning of a given year--1 January,
the Roman choice; 25 March, the Annunciation and Christian
choice; or what?--was such a detail.



Westerners were fortunate to have the Julian calendar, but it
was not perfect. The actual solar year is a few minutes short of
3651, and so the Julian calendar supplies a shade too many leap
years.



The difficulty of nailing down Easter's date nagged the astro-
nomically and mathematically knowledgeable. New Year's Day
might be this or that, as might the number of a given year, but
Easter, upon which Christ's resurrection was commemorated and
from which the dates of other movable feasts were measured, had
to be on the right Sunday.



Hours, the ancient Middle Eastern units designating the divi-
sions of day and night, were the smallest units with which people
commonly concerned themselves. They of course knew there were
shorter periods, but they could improvise ways to deal with them:
fourteenth century cooking instructions directed novices to boil an
egg "for the length of time wherein you can say a Miserere. "24
Hours, however, were too long and too important to guess at.
Jesus Himself had said in John 9: 9, "Are there not twelve hours
in the day?" (implying twelve for the night as well).

Europe did not straddle the equator, and so the durations of
daytime and nighttime changed radically through the year. Even
so, they had to have twelve hours each. Europeans had a system
of unequal, accordian-pleated hours that puffed up and deflated
so as to ensure a dozen hours each for daytime and nighttime,
winter and summer. To compound the confusion (ours, not theirs),
these unequal hours, familiar to us at least to the extent of
being duodecimal, were not the vernacular kind of hours. Most
people, when they did not judge the time simply by glancing at the
position of the sun in the sky, relied on a system of time pro-
claimed by church bells, the most effective information medium of
the age. This was the system, still followed in monasteries today,
of the seven canonical "hours"--matins, prime, tierce, sext, none,
vespers, and compline--which indicated when certain prayers
were to be said (Psalms 119: 64: "Seven times a day I praise Thee
for thy righteous ordinances"). It served both the pious and the
impudent. In the fifteenth canto of Paradiso Dante speaks of the
bells of his Florence ringing tierce and none; and when Boccaccio
notes specific times in his Decameron, he refers to a canonical
hour.

At the very beginning of the Middle Ages there were only three
of these hours, then five, and finally seven, and they never were
solidly moored to clock time. They were breadths, not points, of
time. Choosing the moment within their durations to ring the
church bell was problematic. We can get a sense of this by examin-
ing the saga of the heroic trek of noon. The word noon is derived
from the canonical hour none, the name of which comes from the
Latin for the ninth hour of the day, which, counting forward from
sunrise, was originally rung at something like 3:00 or 15:00 in the
afternoon. During the Middle Ages the ringing of none migrated
back through the day until it reached its final resting place,
mid-day, as early as the twelfth century. The migration no doubt
proceeded at different rates in different localities. In thirteenth
century England, where Normans and Saxons had not yet fused
into Englishmen, the process seems to have been especially compli-
cated: none may have meant midafternoon in French, but midday
in English.2



According to Dante, the ringing of none glided backward to midday
or sext because the latter meant six. The sixth hour was "the most
noble of the whole day, and the most virtuous," six being the sum
of its factors, 1, 2, and 3, and therefore--noble.



The migration of noon illustrates one sure characteristic of
most medieval Europeans. They were in their way as concerned
with time as we are, but their way was very different from
ours. It had much to do with symbolic values and little to
do with precision.


The concept of time that Europeans held was in at least one
way crucially similar to ours. The majority of humans--Greek
Platonists, Navajos, Hindus, Maya--believed that the patterns of
time in its larger dimensions were like the patterns right before
us--the round of seasons, the wheeling of the heavens, and so on.
They believed in cyclical time and did not worry about it unspool-
ing to the very end. Western Europeans also acknowledged the
cycles of life because the years are undeniably a repetitive round
of seasons, every sunset has been thus far matched with a dawn,
and so on. In addition, they believed that the Old Testament in
its details prefigured the New Testament. But because they were
Christians they could not embrace cyclicalism exclusively. God
had sacralized the concept of linear time by stepping into time in
order to provide humanity with the possibility of salvation. "Let
us therefore keep to the straight path, which is Christ," said St.
Augustine, "and with Him as our Guide and Saviour, let us turn
away in heart and mind from the unreal and futile cycles of
the godless. "


The space of the Middle Ages and Renaissance
was as assertively finite as a goldfish bowl, spher-
ical, and qualitative in structure. Within its outermost
sphere were a number of other spheres, tightly nested, one inside
the other. There was no emptiness between them: nature abhorred
vacuums... even more then than now. The spheres, perfectly trans-
parent, carried the heavenly bodies. The outermost sphere with
visible freight carried the fixed stars, whose positions in relation
to one another did not alter (at least not fast enough for anyone
to notice in a lifetime or several lifetimes). These were what we
would define exclusively as the stars. Within their sphere were the
spheres carrying the planets, sun, and moon.

All the spheres and their visible freight moved in perfect circles
because the heavens were perfect, and the circle was the
most perfect and noble shape. Shapes had qualities, and
the circle, like the number 6, was intrinsically noble. Motion in
straight lines was antithetical to the nature of the
heavens. The heavenly bodies and their spheres were all com-
posed of the fifth and perfect element, which was changeless,
stainless, noble, and entirely superior to the four elements with
which humans were in contact.



Here on earth, where wind blew grit in your eye and your feet
were often cold and wet, impermanence was the rule. In the
thirteenth century Bartholomaeus Anglicus declared that the
earth was the "most corpulent and hath leste
of sutilte and of symplicite of any of the bodies of the
universe." Three hundred years later a Frenchman put it more
plainly: the earth "is so depraved and broken in all kinds of vices
and abominations that it seemeth to be a place that hath re-
ceived all the filthiness and purgings of all the other worlds and
ages. In the sublunar zone natural motion was not perfect and
circular, but straight and alterable only by violence. Left to its
own devices, fire rose straight up toward its proper home in the
sphere of fire, and stones, similarly motivated, fell straight down
toward earth.

Our sublunary slum was heterogeneous, and not only in its
climate, flora, and fauna, but in its plausibilities as well. Sir John
Mandeville's Travels, one of the most popular books of the Re-
naissance, soberly pronounces that in the Land of Prester John
there was a waterless sea of gravel that "ebbeth and floweth in
great waves as other seas do, and it is never still nor in peace." In
Ethiopia people had only one foot, which "is so large that it
shadoweth all the body against the sun when they will lie and rest
them." (St. Augustine may have been Mandeville's source for this:
the saint had heard that Ethiopians had two feet on one leg.)

Geography was qualitative. The people of the Indies were slow
"because they be in the first climate, that of Saturn; and Saturn is
slow and little moving," but Europeans, an active people, were of
a land of the seventh climate, that of the moon, which "environeth
the earth more hastily than any other planet. "35 Even the cardinal
directions were qualitative. South signified warmth and was asso-
ciated with charity and the Passion of Jesus. East, toward the
location of the terrestrial paradise, Eden, was especially potent,
and that is why churches were oriented east-west with the busi-
ness end, the altar, at the east. World maps were drawn with east
at the top. "True north" was due east, a principle
to which we pay respect every time we "orient"
ourselves.



Some Europeans believed that Europe, Africa, and Asia made
up only a quarter of the land, which was separated from the three
other quarters by great seas, running north and south, east and
west. It seemed unlikely that anybody lived in the three other
quarters, and was possibly blasphemous to think so. How could
anyone have traveled there from Mount Ararat, where Noah's
ark, containing all the living descendants of Adam and Eve (that is
to say, all humans) had grounded as the Flood receded? Not by
land, obviously, and the distances by water were daunting. St.
Augustine's opinion was that "it is too absurd to say that some
men might have taken ship and traversed the whole wide ocean,
and crossed from this side of the world to the other." Furthermore,
they could have traveled to the two southern quarters from Mount
Ararat only by passing through the uninhabitable, literally scorch-
ing tropics. Anyone who believed that people lived in the antipo-
des, said Dante, was a fool.

The world, which God had created for His purposes and
where Adam, Eve, Abraham, David, Solomon, Jesus and His
saints, and Satan and his imps had performed, was adorned with
localities of religious potency. Bethlehem, Jerusalem, and Judah
could be visited and walked through, the Sea of Galilee drunk
from and fished in, so why wouldn't one be able to find, for
instance, Hell? The author of Mandeville's Travels wrote about an
actual entry to Hell, a "Vale Perilous" with gold and silver to
tempt mortals thither, where "anon they be strangled by devils."
The author placed Eden in eastern Asia on top of a mountain so
high it touched the orbit of the moon. In this terrestrial paradise
there was a well "that casteth out the four floods that run by
diverse lands," that is to say, the rivers Ganges, Tigris, Euphrates,
and Nile. Men who attempted to ascend these rivers were deafened
by the noise of the waters that "commeth down so outrageously
from the high places above. "38 Columbus, on the coast of Venezu-
ela in 1498, was sure that the Orinoco was one of these rivers and
that he was near the terrestrial paradise.

How did people who entertained such beliefs look at a map?
How did Christians look at the Ebstorf map, the latest thing in
world maps of the thirteenth century? We note its distortions,
omissions, and outright mistakes and find them forgivable, consid-
ering how little firsthand data or training in geometry the map-
makers had. But we do not know what to make of the map as a
whole. It is drawn on a background of Christ crucified, with His
head in the Far East, pierced hands at the extreme north and
south, and wounded feet off the shore of Portugal. What were the
mapmakers trying to say? Certainly not that the Nile flows into
the Mediterranean at precisely so many leagues south and west of
Antioch. Their map was a nonquantificational, nongeometrical
attempt to supply information about what was near and what was
far--and what was important and what unimportant. It was more
like an expressionist portrait than an identification photo. It was
for sinners, not navigators.



In addition to a penchant for the general and impressionistic,
Western Europeans, especially those who lived in what we call the
Middle Ages, suffered from a lack of clear and simple means of
mathematical expression. They had no signs for plus or
minus or divide or equal or square root. Where they
needed the clarity of algebraic equations, they, like the ancients,
produced long, convoluted, almost Proustian sentences. Their
system of numeri-cal expression, inherited from the Roman Em-
pire, was adequate for the weekly market and for local tax col-
lection, but not for anything grander.



They possessed in their hands and fingers a useful computer and,
for more difficult operations, the abacus or counting board. The
best description we have of the hands-and-fingers system is that
of the Venerable Bede (673-735), who prefaced his treatise on
chronology with a brief disquisition on "the necessary and ready
skill of finger reckoning." Numbers up to 9 were designated by
bending fingers: a bent little finger meant 1, a bent little
finger and ring finger meant 2, and so on (6, being perfect,
was indicated by bending the most noble finger, the ring finger,
by itself). Ten and multiples of 10 were specified by various
configurations of the fingers, by touching the thumb to certain
joints of the fingers, for instance. Higher numbers involved
complications, and Bede recruited hands, arms, elbows, and torso.
Fifty thousand was signified by pointing the thumb of an extend-
ed hand to the navel. Complaints were made that the higher num-
bers required "the gesticulations of dancers."




But finger reckoning was inadequate for complicated opera-
tions. For that the Europeans turned to the abacus. Today
the word abacus, though it is Greek and Latin in origin,
refers to the East Asian device by which calculations are
made by sliding beads along wires. For the Europeans of the
Middle Ages and Renaissance the word meant a counting board
on which lines served in place of wires and pebbles or count-
ers were shifted about instead of beads (Figure 2).

The counting board was a device with which a skilled prac-
titioner could swiftly and accurately make calculations of
every kind, even those involving large numbers. It bestowed
the advantages of both place value and zero without the dis-
advantage of having to think about them. If you wanted to ex-
press that difficult number 101, you placed one counter on the
100s line and another counter on the units line. You did not
have to strain your brain about how to express no 10s or 5s
or what have you in between, but simply left that line or
those lines empty.

The abacus is still widely used over much of the world for the
simple reason that it is one of humanity's cheapest and most
felicitous inventions, and its absence from Western Europe be-
tween about A.D. 500 and 1000 is evidence of the nadir of civ-
ilization there.



Counting boards can handle big numbers and complicated
calculations, so we cannot blame them for what we may call
medieval Westerners' mathematical impotence. Their ignorance
(G. R. Evans terms them, up to the mid-twelfth century, "sub-
Euclidian") explains a large part of their ineptness in reasoning
about quantities, but there was more to it than just that. For us,
except for a few superstitions such as triskaidekaphobia, numbers
are utterly neutral, in and of themselves morally and emotionally
free of all value, as purely tools as a shovel. Not so for the old
Europeans: they thought of them as qualitative as well as quanti-
tative.

"We must not despise the science of numbers," wrote St. Augustine,
that fifth century font of Christian dogma. That science, he con-
tinued, is "of eminent service to the careful interpreter." God
created the universe in six days because 6 was a perfect number,
as we have already learned from Dante. Seven was perfect, too. In
the usage of his era 3 was the first odd number and 4 the first
even number. Added together, they made the perfect 7. And had not
God rested on the seventh day after completing the Creation? Ten,
being the number of the Commandments, symbolized law, and so 11,
which goes one beyond 10, signified transgression of the law--
sin. Twelve, on the other hand, was the number of judgment because
the two parts of the number 7, that is to say, 4 and 3, multiplied
together, make 12. Forty, the number of the days of Lent and the
number of days the Savior spent on earth after the Resurrection,
represented to St. Augustine "life itself."

Most of a millennium later St. Thomas Aquinas made 144,000, the
sum of those whom Revelation promises will be saved at the end
of time, into a cathedral of holy references. The thousand of
144,000 designated perfection (presumably because 1,000 is 10,
the number of the Commandments, multiplied by itself 3 times
over, 3 being the number of the Trinity and of the days between
the Crucifixion and the Resurrection). The one hundred and forty-
four of 144,000 is 12 times 12. Twelve signifies faith in the
Trinity, that is to say, 3 multiplied by the 4 parts of the earth.
One of the 12s to be multiplied can be taken to signify the number
of the apostles and the other the number of the tribes of Israel.


Much of the Venerable Model seems as peculiar to us as a Tun-
gusic shaman's version of reality. We sniff and cluck at its mis-
takes--that the earth is the center of the universe, for instance--
but our real problem with the Venerable Model is that it is dra-
matic, even melodramatic, and teleological: God
and Purpose loom over all. We want (or think we
want) explanations of reality leeched of emotion,
as bloodless as distilled water. Our astrophysicists,
looking for a title for the birth of time and space, have reject-
ed creation, a word with references and rever-
berations that go on forever. They have cho-
sen the nose-thumbing title the big bang in order
to minimize the drama of the subject and the distor-
tions and accelerations of rhapsodic thinking. Med-
ieval and Renaissance Europeans, like the shaman, like all of us
some of the time and some of us all of the time, wanted immediately
conclusive and emotionally satisfying explanations. They longed
for a uni-verse that, in Camus's phrase, "can love and suffer."

In such a universe the balance scale, the yardstick, and the
hour glass were devices of little more than immediate practical
convenience. The old Europeans' universe was one of qualities,
not quantities.



　　　　　　　　　　CHAPTER THREE
　 　Necessary but Insufficient Causes



In causal terms the presence of oxygen is a necessary but not
a sufficient condition for fire. Oxygen plus combustibles plus
the striking of a match would illustrate a sufficient condition
for fire.
William L. Reese (1981)


The raison d'etre of this book is to describe an acceleration
after 1250 or so in the West's shift from qualitative percep-
tion to, or at least toward, quantificational perception. Most
particularly, we want to ferret out the source of that accel-
eration. The latter half of the assignment is daunting, and be-
fore we begin we must discuss just what we are looking for lest
we convince ourselves we have found it before we get to it.
For instance, the arrival of Hindu-Arabic numerals was extreme^
ly important,



Western perceptions changed as Europeans' experience changed.
The West's population doubled and may even have tripled be-
tween 1000 and 1340. Many people migrated to wetlands newly
drained and forests newly cleared, and eastward to contend with
the Slavs for fertile soils. Others became townsfolk, often to work
in the new wool and linen industries, and new cities sprang up and
old ones expanded. By the early fourteenth century Venice and
London had perhaps ninety thousand inhabitants apiece, a fifth at
the very most of Cairo's, but vast by the West's standards of
previous centuries. Then, with the eruption of the Black Death
in the mid-1300s, Europe's population crashed by one-third and
continued to fall well into the next century, with city populations
probably shrinking faster than rural. Yet within a hundred years
Westerners recovered and passed their earlier peak and the cities
were growing again.

Time and again, especially when population was expanding,
Westerners marched and sailed away to invade Islamic and pagan
lands for the sake of God, new fiefs, and commerce; and every-
where they went they saw things their experience did not equip
them to understand. Fulcher of Chartres went on the First Crusade
and wrote that the Levant had hippopotamuses, crocodiles, leop-
ards, hyenas, dragons, griffins, and manticores, which had human
faces, voices like flutes, and three rows of teeth each.
Trade increased between the West's peasant millions and its
cities' frenetic thousands. Long-range commerce increased be-
tween regions and even with the Abode of Islam and the barely
imagined lands from which Marco Polo returned with implausible
stories. The state began to coalesce, with its insatiable appetite for
taxes. The Church, the font of mercy and salvation, taxed with
such vigor that many Christians began to doubt whether the pope
still possessed "that power granted from heaven to St. Peter,
namely of binding and loosing, since he proved himself to be
entirely dissimilar to St. Peter."

New kinds of people sprouted up through the floors of medi-
eval Europe's three-storied society (peasantry, nobility, clergy).
The new people were buyers, sellers, money
changers, people who generated and who revel-
ed in what Jacques le Goff has called "an atmo-
sphere of calculation."6 The new people were merchants,
lawyers, scribes, masters of the stylus, quill, and counting board.
They were the bourgeoisie, the citizens of the bourg or burg, or
town, a meritocracy more literate and numerate
than most Euro-pean clergy and nobility. Philip the Fair of France,
a monarch powerful enough to defy both the king of England and
the pope, turned to a Genoese merchant to run his navy and a
Florentine merchant to manage his finances. These two, Bene-
detto Zaccaria and Musciatto Guidi, respectively,7 were men in
the midrange of a social hierarchy that had theoretically no slot
for anyone in the middle.

Many of these new people achieved their social positions
through the wealth they had accumulated by using machines to
exploit natural forces. Medieval Europe built tens of thousands of
water mills for grinding grain, fulling cloth, and a score of other
purposes. According to the Domesday Book, England had 5,624
at the time of the Norman Conquest, which would be approximately
one for every fifty households. Westerners invented, apparently
independently, the post mill, the kind of windmill with a
horizontal axle and vanes extending out from it at right angles
that most of us think of when we think of the Netherlands. Post
mills were a common sight in the High Middle Ages, and in the
early fourteenth century Dante could describe gigantic and winged
Satan, confident that his readers would understand, as looking like
"a mill which the wind turns." By the fifteenth century and
perhaps long before then, the West had a greater pro-
portion of individuals who understood wheels,
levers, and gears than any other region on earth,
and Westerners north and south were becoming
accustomed to the repetitive whirring and clunk-
ing of machines.



the West, compared with contemporary Muslim, Indian, and
Chinese civilizations, was uniquely prepared to survive and
even to profit from such an avalanche of change. Western Europe
had the characteristics that physicians seeking means to counter
the disorders of senescence hope to find in fetal tissue, that is to
say, not so much vigor per se, though that is surely valuable in
itself, as a lack of differentiation. Fetal tissue is so young that
it retains the potentiality for becoming whatever kind of tissue
is required.

The West lacked firmness of political and religious and, speak-
ing in the broadest generality, cultural authority. It was, among
the great civilizations, unique in its stubborn resistance
to political, religious, and intellectual centraliza-
tion and standardization. It shared one thing with
the universe as described by such mystics as Nich-
olas of Cusa and Giordano Bruno: it had no center
and, therefore, had centers everywhere.

Western Europe was a warren of jurisdictions--
kingdoms, dukedoms, baronies, bishoprics, com-
munes, guilds, universities, and more--a compost
of checks and balances. No authority, not even the vicar
of Christ on earth, had effective political, religious, or intellectual
jurisdiction. This became glaringly obvious with the Protestant re-
volt: for example, Joseph Justus Scaliger preserved both his relig-
ious faith and his skin by migrating from Catholic France to fer-
vently Protestant Geneva and then to tolerant Leiden. The West's
decentralization had saved dissenters before as well.



When William of Ockham refused to accept John XXII's authority
on evangelical poverty and other matters, the pope excommuni-
cated him, that is to say, thrust him from the bosom of the Church
into the absolute zero of a life without sacraments or the aid and
comfort of any Christian--theoretically. The condemned man
took refuge with the pope's enemy, the German emperor, Louis of
Bavaria, continuing as before until silenced not by the pope but,
probably, by the Black Death. And, of course, those who dis-
obeyed secular authority in obedience to Rome's could usually find
refuge in the Church of Rome. Popes maintained for generations a
stable of recusants and other such "traitors." In later and more
secular ages kings, dictators, and premiers did likewise. Decen-
tralized Europe has always had an attic room or at least a dry
corner in the barn for emigres.

The West's traditional elites, secular and sacred, were not
united sufficiently to defend their own interests against their most
obvious and immediate rivals for power, which were not one
another or the Tartars or Muslims, but the merchants with whom
they, if city dwellers, rubbed elbows every day. The political and
religious aristocracies of Asia and North Africa always ultimately
united to keep the nouveau riche down. In the West, on the other
hand, merchants and bankers even managed to establish their
own family dynasties and to insinuate themselves into political
prominence; most famous, of course, were the Medici, but there
were also the Fuggers and a goodly number of lesser lineages of
wealth and influence. Money changers were the yeast
that the lump--peasant, priest, or noble--never
could evict or sterilize, and that quickened and even re-
cruited among the traditional classes.

The elites of palace and cathedral could not suppress the
bourgeoisie because they lacked confidence that they could ful-
fill their own ambitions without access to the wealth and
the skills of this cocky meritocracy. Before the upper
classes could convert their scorn and nascent fear into effective
policy, the merchants had created a civilization in which others
could achieve their own satisfactions only by buying the
services of and granting privileges to those who
lived by counting.

The West was intellectually as well as socially un solidified. It
was unique among the great civilizations in its lack of a phy-
logenetic classical tradition. The classical syntheses of
the others were deeply rooted in their pasts. Their precepts were
a part of their ancient cultures, even those of the Muslims, the
great majority of whom were not Bedouins but descendants of
Persians, Egyptians, Greeks, and others. These sophisticates did
not feel obliged to rethink their basic concepts of reality. They
even lagged behind in inventing or adopting the scribal min-
utiae of arrangement and format--alphabetization
(or some equivalent for Chinese characters), pun-
ctuation, indentation, capitalization, running heads,
and so on



At least as troublesome as the alien origin of much of the
Venerable Model were its internal contradictions. Its
Greek and Hebrew elements, respectively rat-
ionalistic and mystical (grant me this oversimplifica-
tion for the sake of brevity), were disharmonious. The West,
unlike its rivals, had a chronic need for explainers, adjusters,
and resynthesizers.

The theological and philosophical truth, the function of which
was to explain, gained in ancient authority and contemporary
refinement in the High Middle Ages and in consequence, para-
doxically, became more a puzzle than a comfort. In the twelfth
century Western scholars, Adelard of Bath, Robert of Chester,
and others, studied with learned Jews and Muslims, usually in
Spain, and returned home to bestow on Christendom Latin tran-
slations of works of some of the greatest minds of ancient
Greek and current Islamic cultures: Plato, Ptolemy, Avicenna,
and others.



The Venerable Model lost definition - not because Westerners
decided it was wrong, but because sometimes the past's various
explanations were not precisely coterminous or were not precisely
adequate for present requirements....the West was floundering
in a slough of cultural despond, a condition of perceptual con-
fusion, during the late thirteenth to the sixteenth century.
Its traditional ways of perceiving and explaining were failing
in their primary function, which was, in Bouwsma's words, "to
impose a meaning on...experience that can give to life a mea-
sure of reliability and thus reduce, even if it cannot altogether
abolish, life's ultimate and terrifying uncertainties."

Westerners began, very slowly, tentatively, and often uncon-
sciously, to improvise a new version of reality out of inherited
elements and out of current, often commercial, experience. The
emerging New Model, as we shall call it, was distinctive in
its growing emphasis on precision, quantification of physical phe-
nomena, and mathematics.

The individuals chiefly responsible for the New Model were
townspeople, the yeastiest citizens in Western as in
most societies. In the way that the cells of a fetus are growth,
these people were change, even when members of ancient elites:
for instance, the bishop in his new, vast, and exorbitantly expensive
urban cathe-dral. Some of the townsfolk were members of nascent
elites, of the cultural avant-garde, and to them we owe special at-
tention. They spent their working hours in one of two centers, the
university and the marketplace.

The latter was older than writing or the wheel, but Westerners
had to invent the former for themselves. Expanding population,
burgeoning church and state, proliferating knowledge, and the
threat of various heresies together produced a demand for more
teachers, scholars, bureaucrats, and preachers, overwhelming the
old cathedral schools and giving rise to the universities.
The first half of the twelfth century was the heroic
period of Western higher education, a time in which
students spontaneously gathered around masters
like the radically rationalistic Peter Abelard, even following them from
town to town if necessary. The masters dispensed knowledge
and wisdom...The masters and the students, in other words, were not
institutions. In the twelfth century the two groups coalesced into
institutions. The University of Paris...



The chancellor of the city cathedral fought a long battle with
the masters for control of the new institution, which the masters,
backed by a papacy seeking to weaken episcopal authority, won.
The city government and populace opposed the masters' claims of
special privileges and the students' rowdy ways, and even busted
a few heads to make that point clear; but again the University
won, in this case also with the backing of Capetian kings who want-
ed to cultivate the prosperity and prestige of their capital city.
In 1231 Pope Gregory IX issued a bull recognizing the University
of Paris as a corporation under papal protection, buttressing the
institution's claim of exemption from local authority.

The West had invented an enduring institution whose function
was to provide employment for professional thinkers and learners.



The Schoolmen had to solve the daunting problem of how to
organize the massive bequest from the pagan,
Islamic, and Christian past before they could effectively
confront the even tougher problem of reconciling the contra-
dictions of Christian and nonChristian thinkers - and even of
saint and saint. The comfortably ignorant or the confidently cyn-
ical would have solved both diffi-culties by throwing away what
seemed excessive or did not fit. But the Schoolmen were in-
tensely, if narrowly, learned and terribly earnest.

The texts, sacred and profane, as first received
from the ancients, were undifferentiated lumps,
unsegmented and without handles, as awkward
to manage as beached whales. The Schoolmen
invented chapter titles and running headlines (often coded by
size of initials and by color), cross-references, and even cita-
tions of quoted authors. Around 1200 Stephen Langton (soon to be-
come the archbishop of Canterbury who counseled the barons
and King John in the crisis that produced the Magna Carta) and
colleagues devised the chapter and verse system for the books of
the Bible, until then trackless forests. 24 In the next
century Hugh of St. Cher, a Dominican at the University of Paris,
led a team of scholars in writing, among other masterpieces of
reference aids, the massive Correctoria, a list of the variant read-
ings of the Vulgate. These and like scholars produced concord-
ances for the Scriptures, key-word and subject indexes to the
church fathers, and then to Aristotle and other ancients.25 When
they used numbers in their scholarly scaffolding, they substituted
for Roman numerals the bright new Hindu-Arabic numerals, before
most merchants and bankers made the same transition.

For generations the Schoolmen were at a loss for a principle by
which to arrange masses of information for easy retrieval. They
believed that the principle should pertain chiefly to the relative
importance of bits of information. In library catalogs, for
instance, the Bible should come first, then the church fathers,
and so on, with books on the liberal arts last. But ordering by
prestige alone did not always work well, especially at the lev-
el of minutiae, and so the Schoolmen supplemented it with a sys-
tem occasionally used in the ancient world and now and again since,
but never often or consistently: alphabetization. As ab-
stract as a progression of numerals, it required
no judgment about the relative significance of
what it arranged and, paradoxically, was there-
fore universally useful. One could use it to organize
dictionaries of words, concordances of the pronouncements of
God or of the statements of ancient Greeks, catalogs of books,
collections of government docu-ments. The Schoolmen supplied
alphabetized handbooks and dic-tionaries of sermon materials
for the preachers who at the end of the twelfth century were
competing with heretics for the souls of the inhabitants of the
burgeoning cities. And we have been alphabetizing ever since.

Perhaps the most innovative and useful of all the Schoolmen's
discrete inventions was the system of the analytical
table of contents. Greece and Rome had never arranged
their texts so that a novice could proceed confidently from the
general to the topic to the subtopic to the specific point, and
then back again. The School-men did. Their system is an aid
not only to finding a given item in a book, but to
following lines of argument and, like mathematical
technique, to thinking clearly. It is a sieve of sev-
eral levels, graded from coarse to fine, into which we
dump our confused ideas. First to be sifted out are general subjects,
designated in our adaptation of this Scholastic invention as I, II,
and so on. Next selected are the topics, A, B, and so on; then the
subtopics, 1, 2, and so on; and these, if necessary, are further div-
ided into a, b, and on and on. Alexander of Hales, the Franciscan
master, may have been the first to introduce the system. He divid-
ed the whole into partes and then into membra and articuli. St.
Thomas Aquinas, who never lost track of where he was in an argu-
ment, divided the whole into partes, and these into quaestiones or
distinctiones, and these into articuli.




No Schoolman performed more skillfully or with greater econ-
omy of means than St. Thomas Aquinas. The armature of his logic
is right there to be seen and tested, and his prose is a bony
minimum stripped of alliteration, figures of speech, or even meta-
phor, except where tradition demanded otherwise. (He could not
very well reject the poetry of the Psalms, but he did criticize Plato
for extravagance in language.) His reasoning and language are
almost mathematical: our translators sometimes use algebraic let-
ter symbols as the best means to express in twentieth century
English what he wrote in thirteenth century Latin...

Now the same thing cannot at the same time be both actually
x [sit simul in the Latin original] and potentially x, though it
can be actually x and potentially y [secundum diversa]: the
actually hot cannot at the same time be potentially hot,
though it can be potentially cold. Consequently, a thing in
process of change cannot itself cause the same change: it
cannot change itself. Of necessity therefore anything in pro-
cess of change is being changed by something else.

(This ultimate agent, of course, turns out to be God a few sen-
tences later.)

In our time the word medieval is often used as a synonym for
muddle-headedness, but it can be more accurately used to indicate
precise definition and meticulous reasoning, that is to say, clarity.
Thomas Aquinas, a saint, was a favorite of Rene Descartes,31 a
crown prince of rationalist philosophy and virtual inventor of
coordinate or analytic geometry.

Careful organization, logic, and precision in language, if fol-
lowed to their extremes, lead to mathematics. That next step
beyond St. Thomas was not as long as we might consider it
today because most of mathematics beyond counting and simple
arithmetic was still expressed verbally. Yet it was a long stride
conceptually, so great that the Schoolmen never completed it.
They were not able or very rarely were able to get beyond what
twentieth century scholars have called "logicomathematical phi-
losophy." The Schoolmen did not have the advantage of the signs
for plus, minus, square root, and other operations. They did not
have the advantage of many of the most basic kinds of decisions
about what and how to measure, decisions they began to make for
us. For instance, in matters of temperature were coldness and
hotness two different entities or different aspects of one? Most
important, the Schoolmen, the heirs more to qualitative sages like
Plato and Aristotle than to the quantitative Ptolemy, were still not
adept at or comfortable thinking in terms of measured quantities.



Even so, in the fourteenth century certain Schoolmen -
Swineshead and his fellow monks at Merton College, Oxford,
and, most productively, Nicole Oresme of Paris - made great
progress in mathematics-without-measurement. Englishmen were
more successful than any Westerners yet in utilizing algebra in the
consideration of what Aristotle termed qualities: velocity, temper-
ature, and so on. Oresme pushed further on, geometrizing quali-
ties, even speed in its most perplexing manifestation, acceleration.
He produced what amounted to graphs (rather like music staffs;
see Chapter 8) in which the progression of time was expressed
with a horizontal line and the variable intensity of a quality with
vertical lines of various heights. The end result was an elegant and
pure abstraction, a geometrical depiction of a physical phenomenon-
varying through time.



Roger Bacon...At the end of the thirteenth century he called
mathematics "the gate and key" to knowledge, which saints had
discovered at the beginning of the world. Mathematics was, he
said, our unfailing guide in astronomy, weather, geography, and
other things of this world, and in philosophy and beyond, even
theology.35 He sometimes actually measured things, for instance,
the 42-degree angle between the arc of a rainbow and the line of
sun's rays striking the measurer's back. The influence (or lack
thereof) of this venture into practical metrology, however, was
what you and I must consider odd.



Another source for the quantifying tendency...Money was second
only to God in its power and ubiquity. St. Thomas Aquinas acknow-
ledged its power:

It is true that money is subordinated to something else as its
end; still to the extent that it is useful in the quest for all
material goods by its power it somehow contains them all .
. . . This is how it has some likeness to beatitude.

The Roman Empire had functioned on cash, but initially the
West did not. There was little trade, and much of that was barter.
Coins had little abstract value beyond the value of their metal.




Supply grew, commerce and towns revived, avarice blinked
and rubbed its eyes at the sight of money. Specie rose up out of
hidden caches and seeped in from abroad. England, which had
only ten mints in 900, had seventy in 1000. Cities and then nations
began to issue coins, and Western replaced non-Western coinage
as the commonest kind of money.42

Westerners found themselves sliding into a cash economy,
each item in their lives reduced to a single standard in the process.
"Every saleable item is at the same time a measured item,"43 said
Walter Burley of Merton College in the fourteenth century. Wheat,
barley, oats, rye, apples, spices, woolens, silks, carvings, and
paintings developed prices; and that was relatively easy to understand
because they could be eaten, worn, touched, and observed. It was
harder to understand when money substituted for obligations of
service and labor set long ago by custom. When time proved
to have a price - that is to say, interest on a debt
calculated in accordance with the passage of
months and years - that strained the mind and
the moral sense as well because time was God's
exclusive property. If time had a price, if time
were a thing that could have a numerical value,
then what about other unsegmented imponder-
ables, like heat or velocity or love?

Price quantified everything. The seller set a price on what he
or she had to sell because everything the seller needed or wanted
had to be paid for in turn. In 1308 Pope Clement V proclaimed
the price of pardoning a year's worth of sins to be the contribution
of one pence, Tours currency, to the good cause of crusading
against the Muslims. "0 most excellent gold!" exulted Christo-
pher Columbus two centuries after. "Who has gold has a treasure
with which he gets what he wants, imposes his will on the world,
and even helps souls to paradise.



When even the value of the genois and florin and ducat wavered,
or when a transaction could not be closed because the number of
coins offered in one currency was worth a fraction too much or
too little of a whole number of the coins in the currency in which
the price was expressed, or when currencies flew up and down in
value so fast no one was confident of knowing their relative
values - when all was in flux and yet bills had to be proffered
and paid, Westerners took another giant step into abstraction.
They extended as never before the useful fiction of "money of
account," an idealized scale consisting of what after a while
was the arbitrarily fixed ratios of the values of prestigious
coins. The system was so abstract that it continued to function
even after some of these coins fell out of circulation.



Western Europe's economy was not the first to be monetized,
not by thousands of years: why, then, did that alteration have such
distinctive, even unique, effects in the West? Its chronic shortage
of specie surely contributed. Western Europe did not have great de-
posits of easily mined gold and silver and, therefore, when it took
the hook of a cash economy, did not have enough precious metal
of its own for its economy to function efficiently. The West suf-
fered from a chronic balance-of-payments problem until some
time in the sixteenth century. Specie flowed from Northern Europe
to the Mediterranean ports and thence to trading partners in the
East. In the 1420s Venice exported something like fifty thousand
ducats a year to Syria alone. The flow of gold eastward was so
steady and lasted for so long that the Spanish had a special name
for it: evacuacion de oro.



Westerners were obsessed with what they could not hold on
to, money. Marco Polo waxed eloquent about the abundance of
gold in parts of the East. Columbus fixated on finding it in his new
world. Cortes and his Spaniards hungered for it, said the Aztecs,
"like pigs. "50 There were no people on earth more concerned with
coins than Westerners, no people who worried more about their
weight and purity, who played more tricks with bills of exchange
and other pieces of paper that represented money - no people on
earth more obsessed with counting and counting and counting.



　　　　　　　　　　CHAPTER FOUR
　 　　　　　　　　　Time



The Horologium - not only does it show and register the
hour to our eyes, but also its bell announces it to the ears of
those who are far away or staying at home. Hence in a way
it seems to be alive, since it moves of its own accord, and
does its work on behalf of man, night and day, and nothing
could be more useful or more pleasant than that.
　　　　　　　　　　　　　　　　　　　　　　　　Giovanni Tortelli (1471)


Time puzzled St. Augustine: "I know well enough what it is,
provided that nobody asks me; but if 1 am asked what it is,
and try to explain, I am baffled."2 Measurements are usually of
something distinctively itself - a hundred meters of road, of
meadow, of lake - but a hundred hours, happy or sad, is a
hundred hours of...time.

Time's insupstantiality defied St. Augustine's understanding
and defies ours, but allows humans to press upon it their own
conceptions of its parts. It is not odd that medieval Western
Europeans took in measuring time their first giant step forward
in practical metrology. Nor is it odd that they did so in measur-
ing hours, rather than in calendar reform. Hours were not
bounded by natural event, but were arbitrary
durations and susceptible to arbitrary definition.
Days, in contrast, had such boundaries in dark-
ness and light, and, furthermore, calendars were
artifacts of millennia of civilization, stiff with en-
crustations of custom and sanctity.



For peasants schedules were approximate: weather, dawn, and
sunset dictated their tempi. But hours were of central significance
to city dwellers, whom buying and selling had already initiated
into the vogue of quantification. Their time was already what
Benjamin Franklin, a man they prefigured, would call it: money.

In 1314 the city of Caen installed a clock on a bridge and
inscribed it thus: "I give the hours voice / To make the common
folk rejoice." (Remember that the common people then included
everyone except members of the aristocracy and the Church.) A
fifteenth century petition for a city clock for Lyons proclaimed,
"If such a clock were to be made, more merchants would come to the
fairs, the citizens would be very consoled, cheerful and happy
and would live a more orderly life, and the town would gain in
decoration."

The English word clock is related to the French cloche and the
German Glocke, words for bell. In the Middle Ages and Renais-
sance, life in the cities was paced by bells - "a city without bells,"
said even the antipunctual Rabelais, "is like a blind man without
a stick." But the hours they rang at the start of the second
millennium were canonical and imprecise, and there were too few
per day to provide a suitable tempo for urban schedules.




The early mechanical clocks were so huge and expensive that
I doubt an individual astrologer or astronomer built the very first
of them, though such a wizard might have if sponsored by a duke
or a bishop. My guess is that a monk, a member of a large and
probably wealthy organization, did the deed. If history were logi-
cal, he would have been a monk of the technologically advanced
Cistercian order, whose abbots were certain that grace somehow
correlated with efficiency and, therefore, with water mills and
windmills, cogs and wheels.

Logic would further suggest that the inventing was done in the
north. There, the seasonal variation in day length and the inequal-
ity of the unequal hours was greater than in Mediterranean Europe
and the water in water clocks more apt to freeze. Northern
France, the motherland of Gothic architecture
and polyphony, where innovation was bounding ahead in the
thirteenth century, seems a sensible choice.

So much for logic, which history often ignores. We do not
know who built the European prototype of our mechanical clocks
or where, and probably never will. As for when, that was in the
last decades of the thirteenth century just before or soon after the
invention of spectacles (which was more than a coincidence: the
West was beginning its long frenzy of devising technological aides
for human senses). We cannot pin down the year, but the decade
was probably the 1270s. At its beginning Robertus Anglicus re-
marked on attempts to construct a wheel that would make one
complete revolution every twenty-four hours. In the same decade
someone at the court of Alfonso El Sabio in Spain sketched a
weight-driven clock regulated by the flow of mercury from one
compartment of a hollow wheel to another.




Clocks had only bells - no faces or hands yet - but already
Western Europe had entered the age of quantified time, perhaps
already too deeply to turn around.

Most inventions are improvements or adaptations of previous
devices, but the mechanical dock was, in its key mechanism, truly
original. Time had seemed to most people an un-
segmented flow. Therefore, experimenters and
tinkerers wasted centuries attempting to mea-
sure time by imitating its flowing passage, that
is, the flow of water, sand, mercury, ground porcelain,
and so on -or the slow and steady burning of a candle out of the
wind. But no one has ever devised a practical way of measuring
long periods by such means. The substance in motion grows gelid,
freezes, evaporates, dots, or the candle perversely burns too fast
or too slow or gutters out - something goes wrong.

Solving the problem becomes possible when one stops thinking
of time as a smooth continuum and starts thinking of it as a
succession of quanta. St. Augustine suggested that one could, for
instance, measure a long syllable as twice a short one: "But when
two syllables sound one after the other - the first short, the second
long - how shall I keep hold of the short one?" The answer
technologically (not philosophically) was the escapement. With it,
the short syllable became the duration between tick and tock.

Western Europe was full of mills, levers, pulleys, and toothed
gears when Robertus Anglicus described a time measurer driven
by a weight hanging from a line wound around a cylinder, and the
idea of using some of this technology to measure time must have
occurred to a number of protomachinists.




Westerners were not the first to have mechanical clocks. The
Chinese had several giant ones as early as the tenth century. In-
deed, it is conceivable that news of these spurred the invention of
the West's first clocks




Be all that as it may, equal hours began to displace unequal
hours in general usage as early as 1330 in Germany and about
1370 in England. In the latter year Charles V of France decreed
that all the clocks of Paris should count the hours in agreement
with the clock he was installing in his palace on the He de la Cite.
(The quai de l'Horloge, with a clock, is still there.) Jean Froissart,
the historian of the Hundred Years' War, shifted from canonical
hours to the new clock hours halfway through writing his Chronicles
- the 1380s seems a good guess.

"It was in the European city," says A. J. Gurevich, "that
time began, for the first time in history, to be
'isolated' as a pure form, exterior to life." Time,
though invisible and without substance, was
fettered.

The clock's effects were manifold and tremendous. The clock
was a complicated machine that demanded in its building and
encouraged in its maintenance the skills of a good machinist and
of a practical mathematician.

Richard of Wallingford, abbot of St. Albans from 1326 to 1336,
who built a tower clock for his abbey and wrote a treatise on
clock making. More like a mechanic than a monk, he must have
cut and filed and adjusted and tightened and tested scores of bits
of metal, and, of necessity, he spoke in numbers:

The wheel of the weight for the day movement is divided into
72 teeth. The centre of the wheel is separated from the base
by a distance of 13 teeth of the same wheel, and is a chord of
6 teeth from the centre line of the entire device, the length of
its arbor being a chord of 15 teeth beyond the tenon.

The quantitative abbot was a ghost of the Hellenistic past or, more
likely, of the future.

The clock provided Westerners with a new way of imagining -
of meta-imagining. Lucretius, the Roman poet, had created the
image of the machina mundi, "the world machine," back in the
first century A.D. and others had now and again used it since, but
the firmly specific "clockwork universe," which many would say
has been the dominant metaphor of Western civilization, did not
appear until the fourteenth century.



When Johannes Kepler, three centuries after Oresme, tried to
explain the idea that guided his awesome speculations, he wrote:

My aim is to show that the heavenly machine is not a kind of
divine, live being, but a kind of clockwork (and he who
believes that a clock has a soul, attributes the maker's glory
to the work), insofar as nearly all the manifold motions are
caused by a most simple, magnetic, and material force, just
as all motions of the clock are caused by a simple weight.




Jean Froissart, who, with his prolific quill and average tastes,
is more valuable to the social historian of medieval Europe than
any genius, was infatuated with the new machine. His poem
"L'Horloge amoureuse" features the clock as an image for a
lover's heart. The beauty of the poem's beloved lady motivates her
lover just as the weight drives the clock. His desire would be
uncontrollable if it were not checked by fear, like the escapement
regulating the fall of the weight. Froissart found images for all the
allegorical dwellers in the realm of love - Loyalty, Patience,
Honor, Courtesy, Valor, Humility, Youth - in the mechanisms of
the new time machine. The poem is itself a sort of love song to
the clock because the machine tells the hours even when there is
no sun:

Hence do we hold him for valiant and wise
Who first invented this device
And with his knowledge undertook and made
A thing so noble and of great price.




For generations the town clock was the one complicated ma-
chine that hundreds of thousands saw every day, heard over and
over again every day and night. It taught them that invis-
ible, inaudible, seamless time was composed of
quanta. It, like money, taught them quantifica-
tion.

The modern style of disciplined industrial time appeared as far
back as the first half of the fourteenth century. For instance, on
April 1335 Philip VI granted power to the mayor and aldermen of
Amiens to ordain and control by a bell the time when the workers
of the city should go to work in the morning, when they should
eat and return to work after eating, and when they should quit
workP When two hundred years later Rabelais's Pantagruel pro-
claimed "that no clock keeps better time than the stomach,"28 his
was a voice crying in a quantified urban wilderness.

"The calendar," writes Eviatar Zerubavel in a particularly felici-
tous sentence, "is the warp of the fabric of society,
running length-wise through time, and carrying
and preserving the woof, which is the structure
of relations among men, and the things we call
institutions." 29 That being true, the fact that Western Europeans
were slower to reform their calendar than to build and obey clocks
is not surprising. In fact, that they did it at all is more surprising
than that they procrastinated about it.

Philip Melanchthon, the Lutheran reformer, tells of a "doctor"
(holder of a university degree) who said that there was no need for
precision in the divisions of the year, because "his peasants knew
perfectly well when it was day, when it was night, when it was
winter, when it was summer." Many people would have agreed,
but the learned and pious Melanchthon proclaimed that someone
should "shit a turd" into the aforesaid doctor's hat "and put it
back on his head."



In that year Pope Gregory XIII assembled a conference of
experts - Roman Catholic experts - to reform the calendar. They
debated, brooded, and offered to the pope a revised version of the
Julian calendar known ever since as the Gregorian calendar. On
their recommendation, he proclaimed that Thursday, 4 October
1582, would be immediately followed by Friday, 15 October
1582. As for the difference between the abstract calendar year of
whole days and the actual solar year of 365 and a fraction, the
Gregorian reform retained the Julian system of an extra day every
four years, with a slight but important correction: centurial years
are leap years only if divisible by 400 (like 1600 and 2000).

Many were upset with the reform. The Catholic Michel de
Montaigne complained, "I grit my teeth, but my mind is always
ten days ahead or ten days behind: it keeps muttering in my ears:
'That adjustment concerns those not yet born.' "33 Orthodox and
Protestant Christians clung to the Julian calendar as if to a piece
of the True Cross, and in many cases continued to do so for
centuries. "The English mob," wrote Voltaire, "preferred their
calendar to disagree with the Sun than to agree with the Pope."



Joseph Justus Scaliger, mentioned earlier as a critic of the new
Catholic calendar, was a monumental scholar in an age of great
scholars: contemporaries called him a "sea of sciences" and a
"bottomless pit of erudition. " His industriousness and powers
of concentration verged on the superhuman. He was in Paris on
St. Bartholomew's Day in 1572 but, by his own account, was so
intent on studying his Hebrew that he almost missed the massacre
of his fellow religionists, not noticing for some time "the clash of
arms ... the groans of children ... the wailing women, [or] the
shouting men."

As a young man he learned his trade from his father, one of the
most prominent scholars of the mid-sixteenth century, absorbed
languages - a dozen or so eventually - and honed his skills by
editing the works of Catullus, Tibullus, and Procopius. When he
had made himself into arguably the greatest philologist and stu-
dent of classical literature of his era, he turned his laserlike
attention to chronologia (like America, a term coined in answer to
new demands}. He scorned preceding and contemporary chronolo-
gists, "who seem all to have sworn never to tell the truth," and
offered an antidote to their mistakes in his massive tome, De
emendatione temporum (1583), which changed chronology from
a pseudoscience to a true science.

Scaliger gathered together the oldest and best editions of the
classics of chronology and all the calendars available, more than
fifty, whatever their origins, Christian, Islamic, or what have you.
Though a devout Christian, he granted no special credence to the
Bible, declaring that truth is sacred even if heard
from profane lips. He sought not to discover a divine order
in history, but to achieve calendrical accuracy and the correlation
of every promi-nent dating system with every other.




The fee that obsession with temporal precision
exacted for its services was anxiety. Intelligence, a
character in the fourteenth century's Piers the Ploughman, pro-
claims that "of all things on earth, God knows, nothing is more
hated by those in Heaven than waste of time." Leon Batista
Alberti, an early Renaissance man (whom we shall meet again in
Chapter 9), declaimed, "I flee from sleep and idleness, and 1 am
always busy about something." When he rose in the morning he
made a list of what had to be done that day and assigned to each
a time (thus anticipating Benjamin Franklin by three hundred years).

Petrarch paid strict attention to time in a most untraditional
fashion. We know, therefore, that he was born at the break of
dawn on Monday, 20 July 1304. We know that he fell in love
with Laura on 6 April 1327, that she died on 6 April 1348, and
that he died on 19 July 1374.48 We know that time never slipped
through his fingers; "rather it was torn from me. Even when I was
involved in some business or in the delights of pleasure it would
still dawn on me 'Alas, this day is irretrievably gone.'"

He admonished his reader to discard the traditional concept of
his or her life as "a ship moving this way and that according to
the various winds and waves." No, he insisted, the truth is that

one unalterable speed is the course of life. There is no going
back or taking pause. We move forward through all tempest
and whatever wind. Whether the course be easy or difficult,
short or long, through all there is one constant velocity. 50

Three centuries later this kind of time, bleached even of de-
spair, became the time of classical physics. In 1687 Sir Isaac
Newton would define it thus: "Absolute, true, and mathematical
time, of itself, and from its own nature, flows equably without
relation to anything external."51 I am writing this line at 22:38
Greenwich Mean Time on the 2,449,828th Julian day.



　　　　　　　　　　CHAPTER FIVE
　 　　　　　　　　　Space



Henceforth I spread confident wings to space:
I fear no barrier of crystal or of glass:
I cleave the heavens and soar to the infinite.
Giordano Bruno (1591)


The shift in Westerners' perception of space was not as dra-
matic as the change in their perception of time. There was no
jackrabbit start like the invention of the mechanical clock. Gio-
vanni Tortelli, writing circa 1450 about all the new things that
were transforming his world - the clock, compass, pipe organ,
sugar, tallow candle - mentioned only one pertaining to the mea-
surement of extension, a new kind of maritime chart,
the portolano, and allowed that he was not as impressed with
it as the others because "it is the work of long labors and careful
diligence rather than of a divine challenge." The transformation
in Westernperception of space, which culminated in changes as
radical as those that shook physics at the beginning of the twen-
tieth century, started like a tortoise.

The compass, acquired from Asia early in the second millennium,
persuaded sailors to risk the long run from Cape Finisterre to
England or across the Mediterranean in winter when clouds cov-
ered the pole star.



The first useful maps in Western Europe for laying compass
courses were called portolani. The earliest dated example that has
survived was drawn in 1296, in those same wondrous few decades
in which the first clock was built.4 The portolani, short
on references in comment or sketch to God,
gods, or monsters, were utilitarian drawings
of coastlines with the waters, adjacent and between, scored
with rhumbs (compass courses) traced with straightedge. A nav-
igator consulting a portolano often found rhumbs already drawn
from major port to major port, often the course he wanted. If
not, he could often find a rhumb parallel to the one he needed,
and could then take his heading from that.

The portolani were devised for enclosed or nearly enclosed
waters, like the Mediterranean, the Bay of Biscay, and the North
and Baltic seas. For these they served their purpose well because
they were reasonably accurate and the distances between landfalls
were short. Distortions, inevitable because no one knew about
compass deviation and unavoidable because the portolani
were geometrically naive flat pictures of the
curved surface of the earth, were insignificant.
But these charts were dangerously illusory
over long distances.



Enter (or re-enter) Claudius Ptolemy, the ancient Hellene
without whom Western Europeans would have taken much longer
than they did to become themselves.

About 1400 a copy of Ptolemy's Geographia arrived in Florence
from Constantinople. If there was anything equivalent in the
shift in spatial perception to the appearance of the escapement
in temporal perception, this was it. News of the Geographia flow-
ed westward with Italian commerce and capital to Iberia, whose
sailors, noodling down the coast of Africa and probing out into
the Atlantic, needed charts for voyages long beyond known land-
marks or even any sight of land.

Ptolemy's contribution to cartography was, put simply, to
treat the earth's surface as neutral space by
slapping a gridwork on it, a crosshatch of co-
ordinates calculated in accordance with the
positions of heavenly bodies. He provided fifteenth
century Europe with three different methods, mathematically
consistent, by which the curved surface of the earth could be
represented on flat maps with the unavoidable distortions man-
aged in ways for which the informed could make allowances. By
the next century Ptolemy's techniques were part of the common
currency of Western European mapmakers. Their earth was now
a sphere caught in a network of latitudes and longitudes, its the-
oretical surface as uniform as that of a billiard ball. When the
Americas and the Pacific burst into Western perception, the means
to depict them accurately already existed.


The history of Western cartography is a story of catch-as-catch-
can practice bounding out ahead of theory and of theory trying to
catch up. The parallel history of astronomy (often, then, astrology)
is one of theory, in this case more verbal than mathematical,









His Ephemerides (1490) listed the positions of heavenly bodies for
every day from 1475 to 1506. Columbus took a copy with him on his
fourth voyage and was able to predict a lunar eclipse on 29 February
1504, confounding and disarming the hostile Indians of Jamaica.

the hermetic writings of Dionysius and Meister Eckhart, the pellucid
treatises of Ptolemy and Euclid, and faith that God was a geometrician
were de rigueur.


Nicholas declared that God was beyond all possibility of human under-
standinog, the center and the circumference of all things, the ground
in which opposites were reconciled in the way that a segment of the
circumference of a circle would be a straight line if the circle were
infinitely large.


Nicholas saw the universe as containing everything except God, Who
contained it. Such a universe had no limit, no edge. The earth could
not be the center of it because the universe had no center. There was
no edge, center, up or down, or any other absolute dimension. Space
was homogeneous.


"Think of precision," he wrote, "for God is absolute precision itself,
and "Mind is a living measure which achieves its own capacity by mea-
suring other things.





Nicholas simplified very complicated problems and was as reluctant as
any Schoolman to try experiments for himself. And he was more concerned
with the deity behind the material scrim than with the scrim itself.
He was more like St. Augustine than Galileo but his thoughts are evi-
dence that the West had begun to shift from thinking of the world in
terms of qualities to thinking of the world in terms of quantities.


The first individual, at least the first of renown, to proclaim
unrestrainedly the implications of Copernican theory for the na-
ture of space was Giordano Bruno (1548-1600), who began as a
Dominican but ended at the stake in Rome. He proposed a space
without center or edge, top or bottom, that offended Aristotelians,
Catholics, Calvinists, and every person who could not live com-
fortably cheek and jowl with infinity. His version of space was
homogeneous, infinite, populated with infinite worlds--outrageous:

There is a single general space, a single vast immensity which
we may call Void in it are irikthera e globes like this one on
which we live and grow;

Quantification in Western Europe, 1250-1600