David Peat

120 From Certainty to UncertaintyThe results satisfied astronomers but left mathematicians feelinguneasy. Astronomers were adding together a number of tiny corrections—admittedlyeach one was much smaller than the other. It is notunreasonable to assume that a few very small things add up to anothersmall thing. But what about adding up an infinite number of very smallthings? How do we know that these won’t sum up to something large?Mathematicians love to play with patterns of numbers and deviseways for summing up infinite series of ever-smaller numbers. Take, forexample, the series known as 1/n 2 . 3 The first member of the series is( 1 2 )2 , that is, 1 . So imagine our unperturbed answer is 1.0000. Adding4this first member of the series, which we can also think of as a “correction”to 1.0000, gives us 1 + 1 , or 1.25. The next member of the series4of “correction,” is smaller ( 1 3 )2 , that is 1 or 0.1111. This is now added9to the first correction. The new “corrected” answer is 1.3611. The thirdcorrection is ( 1 4 )2 , or 1, which equals 0.0625 and brings the answer to1611.4236. Additional corrections are even smaller , 1, and 1, but there25 36 49are an infinite number of them.In this case mathematicians know the precise answer when all theseterms are summed. Starting with the number 1 and adding an infinitenumber of corrections we arrive at the answer 1.6449. The initial answerof 1.0000 has been somewhat perturbed, but even with an infinitenumber of corrections it remains finite.There are many such series where an infinite number of correctionsadd up to a finite answer. But what about the series: 1 + 1 + 1 +2 31+ 1 + 1 and so on? Again each correction becomes smaller and4 5 6smaller. However, in this case mathematicians know that when an infinitenumber of these corrections are added together the answer blowsup in our face. It is infinite. This was what worried mathematicianswhen they used perturbation theory to solve the three-body problem.How did astronomers know that in every planetary case the effect ofan infinite number of small corrections would always result in a finitecorrection to an orbit? What happens if these corrections blow up?What does this mean for the orbit of a planet or an asteroid?3The superscript above and to the right of n indicates that the number n mustbe multiplied by itself; that is n × n.