Nonplussed! by Julian Havil Page B

used to estimate lengths of curves and areas of regions. In recent research on ants choosing nesting sites it has been suggested that the ant scouts’ critical job of site selection is influenced by estimates of area based on a variant of Buffon’s principle. Newton’s words have resonance:
    Nature is pleased with simplicity, and affects not the pomp of superfluous causes.
    But then, with so much needle tossing to do, so have those of Buffon himself:
    Never think that God’s delays are God’s denials. Hold on; hold fast; hold out. Patience is genius!

Chapter 8

    TORRICELLI’S TRUMPET
    The notion of infinity is our greatest friend; it is also the greatest enemy of our peace of mind.
    James Pierpont
    An Argument
    One of the longest and most vitriolic intellectual disputes of all time took place between the two seventeenth-century luminaries Thomas Hobbes and John Wallis: Hobbes, the philosopher, had claimed to have ‘squared the circle’ and Wallis, the mathematician, had strongly and publicly refuted that claim.
    This ancient problem (one of three of its kind) had been handed down by the Greeks and asked if it was possible, using straight edge and compasses only, to construct a square equal in area to the given circle: it took until 1882 until Ferdinand Lin-demann proved π to be transcendental, which meant that the question was resolved in the negative. Wallis was right.
    Although the ‘squaring the circle’ problem spawned the conflict, the battle lines extended far beyond it – and in fact to the infinite, a concept which was far from understood at the time and which brought with it all manner of technical and philosophicaldifficulties, and it is one particular example of infinity’s capricious nature that crystallized the adversaries’ opposing views of the concept: Torricelli’s Trumpet (or The Archangel Gabriel’s Trumpet, or Horn).
    A Strange Trumpet
    Bonaventura Cavalieri was a mathematician good enough to be praised by Galileo, who said of him, ‘few, if any, since Archimedes, have delved as far and as deep into the science of geometry’. And it was Archimedes’ method of exhaustion which Cavalieri developed to form his theory of indivisibles , that is, finding lengths, areas and volumes by slicing the object in question into infinitesimally small pieces. To this he added Cavalieri’s Principle . This was 1629 and integral calculus was yet to be forged at the anvils of the yet unborn Newton and Leibniz; Cavalieri’s ideas would help with the process.
    Evangelista Torricelli (remembered as the inventor of the barometer), frequent correspondent of Cavalieri and assistant to an ageing Galileo, was also an accomplished mathematician. Most particularly, using the method of indivisibles , in 1645 he rectified the Logarithmic Spiral (that is, he was able to measure the length of the curve; a result we will use in chapter 10 ).
    Our interest lies with some of his earlier work when, in 1643, he had made known his discovery of the strange nature of the acute hyperbolic solid , which we would now call the rectangular hyperboloid. It is generated by rotating the rectangular hyperbola y = 1/ x by 360° about the x -axis. figure 8.1 shows the solid.
    He showed that this infinite solid has a finite volume. To today’s post-calculus eyes this single fact is not shocking but it does become rather more surprising when we realize that not only is its length infinite, but so is its surface area.
    We will use calculus and modern-day notation to prove both results: that the volume is finite and the surface area infinite. First, we will take the trouble to demonstrate Torricelli’s method of showing that the volume is finite, a result which shocked the thinkers of the day.
    Torricelli’s Proof

    Figure 8.1. Torricelli’s Trumpet
    We have mentioned the term Cavalieri’s Principle . Archimedes had used the idea to find the volume of a sphere and a version of it can be stated in the following way:
    Given two solids