Alexander's narrative opens in the early 17th century, when Catholic Church administrators in Rome, following a campaign by Euclidean stalwart Christopher Clavius, banned the infinitesimal from the classrooms of Jesuit schools throughout Europe. That these same mathematicians happened to occupy institutional positions of power fueled the perception of dangerousness to which the book's subtitle alludes. Alexander notes, "Zeno's mind-benders prove exceedingly difficult to resolve." Such paradoxes would later offend the sensibilities of conservative mathematicians, for whom Euclid's rational, ordered geometry was the sole, if not the sacred, framework through which to view the world. How can a geometric point be deemed indivisible when the mind can picture it rent in two? How can a line have length if it consists of a string of infinitesimally small points? How, wondered Zeno, can the runner Achilles overtake the tortoise when, in ever-diminishing increments of time, he is always catching up only to where the tortoise had been? "Though seemingly simple," Mr. toward a practical end-computing the volumes of geometric solids-forebears such as Zeno of Elea and the followers of Pythagoras had long before sweated over its host of logical paradoxes. In time, though, I came to know the computational power of this geometrical nubbin: how the infinitesimal's capable offspring-calculus-allowed me to reckon the area under a curve, the capacity of a vessel or the orbit of a comet.Īlthough the Greek polymath Archimedes applied the infinitesimal during the third century B.C. That vaporous slab floated before my mind's eye, alternating between something and nothing, contradicting both logic and experience. Slowing his knife hand, he announced that the next slice would be so thin as to defy measurement: Its thickness would approach, but never quite reach, zero. My high-school calculus teacher introduced the concept with a bit of instructional mime: carving an imaginary salami into ever-thinner slices. Were fairies infinitesimal in size, an infinite throng could boogie on the head of a pin. The term refers to anything whose magnitude or breadth verges on zero: the tiniest imaginable number, the most diminutive imaginable point, the slimmest imaginable line. The result is an interpretive tapestry whose richness justifies his exclamatory subtitle.įor all its effect on post-Reformation Europe, an infinitesimal is surprisingly slight. Alexander succeeds, weaving the strands of a colossal mathematical dispute into the fabric of Western cultural history. To earn this dual distinction for what appears to be merely a centuries-old quibble over the nature of points, lines and planes sets a formidable test for any author. Historian Amir Alexander's "Infinitesimal" arrives with the tagline, "How a Dangerous Mathematical Theory Shaped the Modern World." That a mathematical theory can be characterized as dangerous, much less world-shaping, pings my skeptic's radar. The suspension cables of the Clifton Bridge in Bristol, England, resemble a mathematical curve approaching a tangent line.
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