lis THE SQUARING OF THE CIRCLE. 



Idea of exact ncus ohtainahle tcith the approximate vahies of t. — 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 t: to 

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

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

 not be as great as the 18 millionth part of a millimetre. 



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

 by 100 decimal places. But the following example may i)ossibly give 

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

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

 million million kilometres distant from us. Then imagine this enormous 

 sphere to be so packed wich 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, 134i 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- 

 l)lying its diameter with n 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 n to 100 or 

 500 decimal i)laces is wholly useless. 



FrofeHsor Wolf a curmis 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 tt 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 n — 3. Consequently a suflicieut number of casts of the 

 noodle according to the law of large numbers m st give the value of n 

 ai)i>roximately. As a matter of fact. Professor Wwi.f, after 10,000 trials, 

 obtained the value of n correctly to 3 decimal places. 



IV.--PROOF THAT THE PROBLEM :• INSOLVABLE. 



Mathematicians ::ow seek to prove the in^olvability of the problem. 

 Fruitful as the caloulus of Newton and Leibnitz was for the evalu- 

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

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

 Leibnitz, and their immediate followers distinctly re/'-ognized this. The 



