In Pursuit of the Unknown by Ian Stewart Read Free Book Online Page A

then work out what the smallest detectable difference in weight was. Perhaps surprisingly, this difference (for a given experimental subject) was not a fixed amount. It depended on how heavy the weights being compared were. People didn’t sense an absolute minimum difference – 50 grams, say. They sensed a relative minimum difference – 1% of the weights under comparison, say. That is, the smallest difference that the human senses can detect is proportional to the stimulus, the actual physical quantity.
    In the 1850s Gustav Fechner rediscovered the same law, and recast it mathematically. This led him to an equation, which he called Weber’s law, but nowadays it is usually called Fechner’s law (or the Weber–Fechner law if you’re a purist). It states that the perceived sensation is proportional to the logarithm of the stimulus. Experiments suggested that this law applies not only to our sense of weight but to vision and hearing as well. If we look at a light, the brightness that we perceive varies as the logarithm of the actual energy output. If one source is ten times as bright as another, then the difference we perceive is constant, however bright the two sources really are. The same goes for the loudness of sounds: a bang with ten times as much energy sounds a fixed amount louder.
    The Weber–Fechner law is not totally accurate, but it’s a good approximation. Evolution pretty much had to come up with something like a logarithmic scale, because the external world presents our senses with stimuli over a huge range of sizes. A noise may be little more than a mouse scuttling in the hedgerow, or it may be a clap of thunder; we need tobe able to hear both. But the range of sound levels is so vast that no biological sensory device can respond in proportion to the energy generated by the sound. If an ear that could hear the mouse did that, then a thunderclap would destroy it. If it tuned the sound levels down so that the thunderclap produced a comfortable signal, it wouldn’t be able to hear the mouse. The solution is to compress the energy levels into a comfortable range, and the logarithm does exactly that. Being sensitive to proportions rather than absolutes makes excellent sense, and makes for excellent senses.
    Our standard unit for noise, the decibel, encapsulates the Weber–Fechner law in a definition. It measures not absolute noise, but relative noise. A mouse in the grass produces about 10 decibels. Normal conversation between people a metre apart takes place at 40–60 decibels. An electric mixer directs about 60 decibels at the person using it. The noise in a car, caused by engine and tyres, is 60–80 decibels. A jet airliner a hundred metres away produces 110–140 decibels, rising to 150 at thirty metres. A vuvuzela (the annoying plastic trumpet-like instrument widely heard during the football World Cup in 2010 and brought home as souvenirs by misguided fans) generates 120 decibels at one metre; a military stun grenade produces up to 180 decibels.
    Scales like these are widely encountered because they have a safety aspect. The level at which sound can potentially cause hearing damage is about 120 decibels. Please throw away your vuvuzela.

3
Ghosts of departed quantities
    Calculus

What does it say?
    To find the instantaneous rate of change of a quantity that varies with (say) time, calculate how its value changes over a short time interval and divide by the time concerned. Then let that interval become arbitrarily small.
Why is that important?
    It provides a rigorous basis for calculus, the main way scientists model the natural world.
What did it lead to?
    Calculation of tangents and areas. Formulas for volumes of solids and lengths of curves. Newton’s laws of motion, differential equations. The laws of conservation of energy and momentum. Most of mathematical physics.

I n 1665 Charles II was king of England and his capital city, London, was a sprawling