Killing Time : THE MEASURE OF REALITY: Quantification and Western Society, 1250-1600. By Alfred W. Crosby . Cambridge University Press: 257 pp., $24.95

John Allen Paulos, a professor of mathematics at Temple University, is the author of "Innumeracy" and "A Mathematician Reads the Newspaper."

In this thoroughly fascinating monograph, Alfred W. Crosby, a professor of history at the University of Texas, asks a fundamental question: How and why did it come to pass that Europeans, seemingly backward bumpkins in medieval times, became so successful as imperialists? The short traditional answer is science and technology. Crosby points to certain prior habits of mind, in particular to Europeans' obsession, acquired gradually between 1250 and 1600, to split the world into bits and pieces that they could then count, classify, measure and manipulate.

Crosby starts with what he calls the Venerable Model subscribed to by medieval Europeans. Within this conceptual universe, time was governed by daily religious observances and was quite elastic, consisting of "unequal accordion-pleated hours that puffed up and deflated" to ensure 12 hours of day whatever the season.

He goes on to discuss inaccurate calendars plagued by migrating Easters, numbers used as symbols and for effect rather than as precise counts (a phenomenon that certainly hasn't disappeared) and reports of implausible monsters in distant lands charted by divinely inspired maps. Popular were geographical drawings that were "oriented"--e.g., maps that placed east at the top, Jerusalem in the center, the Nile to right (or south), Russia on the left (or north). In the Venerable Model, paintings were flat, music was plainsong, business transactions relied on memory, temperature and velocity weren't conceptualized as quantities and numerical notions of all sorts were qualitative and human-centered.

Then, beginning around 1300, a remarkable series of interlocking developments permanently changed the mentality of Europeans. Consider time first. Crosby sketches a few of the salient technical innovations that allowed for the construction of huge mechanical clocks in town centers throughout Europe and of their impact on people's consciousness and work habits. He discusses the dating of historical events and various calendrical innovations and their social consequences. Throughout, he makes clear that the transition to these new forms was seldom smooth. In commenting on Protestant resistance to Gregorian reform of the Julian calendar, for example, he reports Voltaire's later quip: "The English mob preferred their calendar to disagree with the sun than to agree with the Pope."

He delineates as well the refinement in the 15th and 16th centuries of cartographic maps and of the various grids and projections that make them useful rather than decorative. Arabic numerals, with which we so easily compute and record our computations, he tells us, only very gradually replaced abacuses and Roman numerals. (Happily, the latter now make their appearance only during Super Bowls.) If you doubt how much of an obstacle they were, try multiplying MDCCCIX by CCLXIV. The introduction of plus and minus signs and other operational symbols and the use of variables as mathematical pronouns by Francois Viete in the 16th century are also parts of the stage setting for Europe's later scientific blossoming and imperialist and colonial enterprises.

In broad outline but interestingly different detail, a similar story is told of the development of music from Gregorian chant to a more playful and polyphonic sound introduced by Leonin, Perotin and others. Here too, notation lagged and was called into being by the limitations on human memory. Neumatic marks written above words to indicate a rising or falling pitch evolved into dots along horizontal lines that led in turn to the now conventional staff. This depiction of the rise and fall of pitch as time proceeded from left to right suggested and may have led in the 14th century to Nicole Oresme's idea of a mathematical graph of a function.

Crosby's well-written account of when and how these various notions developed is pellucid. His analysis of why is more open to dispute but, given that he's theorizing about entities as enormous, intricately woven and nebulous as civilizations, this isn't too surprising. The gist of his argument is that once the necessary social conditions--a rise in commerce, the growth of nation-states and the revival of learning--came about, what kindled and accelerated the trend toward quantification and the consequent dramatic change in human outlook was the advent of techniques for visualizing information. Quantification and visualizations together, according to Crosby, were the key to this epochal shift in European mentality.

In art, the development of perspective by Leon Battista Alberti, Albrecht Durer and others was pivotal. The comparisons with Chinese landscape painting, which had no fixed point of view, and with Byzantine paintings, which sometimes had several, are telling. A sidebar of note is that the introduction of an Arabic numeral for zero, of symbols for rests or silences in musical notation and of empty space and vacancy in later medieval painting were at least suggestive of each other and maybe more. (This is reminiscent of the now common notion that everything is information, Os and 1s, bytes not atoms.)

Of course, visualization extended beyond perspective in painting. Ideas in the same conceptual neighborhood led to the aforementioned improvement in navigational aids, maps and projections, in musical and mathematical notations.

But other advances, less than spellbinding on the surface, were also critical. The development by the Medieval Schoolmen of such techniques as capitalization, punctuation, outlining and alphabetization made reading and writing easier. (Most words had been run together without spaces.) Such techniques became necessary because of the discovery of ancient texts, more recent Islamic ones and contemporary commentaries. This was followed by Johann Gutenberg's printing press and the further visualization and (although Crosby doesn't use the term) digitalization of knowledge.

Oddly enough, an important position in this late medieval pantheon goes to Luca Pacioli, a late 15th, early 16th century Franciscan friar whose books popularized double-entry bookkeeping and mathematics generally and thereby contributed to the rationalization of business practice. Bookkeeping, Crosby states, has had "a massive and pervasive influence on the way we think." He opines that throughout this period, it was the uneasy combination of abstract mathematics and practical metrology and measurement that was so transforming: Many other societies had valued only one or the other.

Despite its vast scope, "The Measure of Reality" is coherent and even witty and should captivate anyone interested in finding out how clocks, calendars, maps, musical scores, astrolabes, accounting ledgers, Mercator projections and perspective painting changed our mental landscape forever.

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