14 THE SQUARING OF THE CIRCLE. 



retical nor a practical value. That the computation of n to say 15 

 decimal places more than sufficiently satisfies the subtlest require- 

 ments of practice may be gathered from a concrete example of the 

 degree of exactness thus obtainable. Imagine a circle to be de- 

 scribed with Berlin as centre, and the circumference to pass through 

 Hamburg ; then r et the circumference of the circle be computed by 

 multiplying its ''liameter by the value of n to 15 decimal places, 

 and then conce /e it to be actually measured. The deviation from 

 the true length ,n so large a circle as this even could not be as great 

 as the 18 mill-' jnth part of a millimetre. 



An idea .an hardly be obtained of the degree of exactness pro- 

 iluced by if 3 decimal places. But the following example may pos- 

 sibly give ' s some conception of it. Conceive a sphere constructed 

 with the f arth as centre, and imagine its surface to pass through 

 Sirius, v,hich is 134 J millions of millions of kilometres distant from 

 the earth. Then imagine this enormous sphere to be so packed 

 with microbes that in every cubic millimetre millions of millions of 

 these diminutive animalcula are present. Now conceive these mi- 

 crobes to be all unpacked and so distributed singly along a straight 

 ire, that every two microbes are as far distant from each other as 

 ;'.irius from us, that is 134^ million million kilometres. Conceive 

 ihe long line thus fixed by all the micrpbes, as the diameter of a 

 circle, and imagine the circumference of it to be calculated by mul- 

 tiplying its diameter by n to 100 decimal places. Then, in the 

 case of a circle of this enormous magnitude even, the circumference 

 so calculated would not vary from the real circumference by a mil- 

 lionth part of a millimetre. 



This example will suffice to show that the calculation of n to 

 100 or 500 decimal places is wholly useless. 



Before we close this chapter upon the evaluation of n, we 

 must mention the method, less fruitful than curious, which Profes- 

 sor Wolff of Zurich employed some decades ago to compute the 

 value of n to 3 places.* The floor of a room is divided up into equal 

 squares, so as to resemble a huge chess-board, and a needle ex- 



*See also A. De Morgan, A Budget of Paradoxes, pp. 169-171. Tr. 



