118 THE SQQARING OF THE CIRCLE. 



Idea of exactness obtainable with the approximate values of -. — Im- 

 agine a circle to be described with Berlin as center, and the circum- 

 ference to pass through Hamburg; then let the circumference of the 

 circle be computed by multiplying its diameter with the value of tt to 

 15 decimal places, and then conceive it to be actually measured. The 

 deviation from the true length in so large a circle as this even could 

 not be as great as the 18 millionth i)art of a millimetre. 



An idea can hardly be obtained of the degree of exactness produced 

 by 100 decimal places. But the following example may possibly give 

 us some conception of it. Conce. ve a sphere constructed with the earth 

 as center, and imagine its surface to pass through Sirius, which is 134^ 

 million million kilometres distant from us. 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 microbes to be all unpacked and so distributed singly 

 along a straight line that every two microbes are as far distant from 

 each other as Sirus from us, that is, I34i million million kilometres. 

 Conceive the long line thus fixed by all the microbes as the diameter of 

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

 plying its diameter with tt to 100 decimal places. Then, in the case of a 

 circle of this enormous magnitude even, the circumference thus calcu- 

 lated would not vary from the real circumference by a millionth of a 

 millimetre. 



This example will suffice to show that the calculation of ;r to 100 or 

 500 decimal places is wholly useless. 



Professor Wolff^s curious method. — Before we close this chapter upon 

 the evaluation of ;r, we must mention the method, less fruitful than 

 curious, which Professor Wolff, of Zurich, employed some decades ago 

 to compute the value of ;r 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 exactly equal in length to the side of each of these squares is 

 cast haphazard upon the floor. If we calculate now the probabilities of 

 the needle so falling as to lie wholly within one of the squares, that is, 

 so that it does not cross any of the parallel lines forming the squares, 

 the result of the calculation for this probability will be found to be ex- 

 actly equal to tt — 3. Consequently a sufficient number of casts of the 

 needle according to the law of large numbers must give the value of tt 

 approximately, ^s a matter of fact, Professor Wolff, after 10,000 trials, 

 obtained the value of tt correctly to 3 decimal places. 



IV.— PROOF THAT THE PROBLEM IS INSOLVABLE. 



Mathematicians :.ow seek to prove the insolvability of the problem. 

 Fruitful as the calculus ot Newton and Leibnitz was for the evalu- 

 ation of ar, the problem of converting a circle into a square having ex- 

 actly the same area was in no wise advanced thereby. Wallis, Newton, 

 Leibnitz, and their immediate followers distinctly recognized this. The 



