THE SQUARING OF THE CIRCLE. 



137 



Bohm, which virtually made n equal to the square of Jf, not even 

 attaining the approximation of Archimedes. 



From attempts of this character are to be clearly distinguished 

 constructions of approximation in which the inventor is aware that 

 lie has not found a mathematically exact construction, but only an 

 approximate one. The value of such a construction will depend 

 upon two things first, upon the degree of exactness with which it 

 is numerically expressed, and secondly on whether the construc- 

 tion can be easily made with straight edge and compasses. Con 

 structions of this kind, simple in form and yet sufficiently exact for 

 practical purposes, have been produced for centuries in great num- 

 bers. The great mathematician Euler, who died in 1783, did not 

 think it out of place to attempt an approximate construction of this 

 kind. A very simple construction for the rectification of the circle 

 and one which has passed into many geometrical text-books is that 

 published by Kochansky in 1685 in the Leipziger Berichte. It is as 

 follows: "Erect upon the diameter of a circle at its extremities 

 perpendiculars ; with the centre as vertex and the diameter as side 

 construct an angle of 30; find the point of intersection of the 

 line last drawn with the perpendicular, and join this point of inter- 

 section with that point on the other perpendicular which is dis- 

 tant three radii from the base of the perpendicular. The line of 

 junction so obtained is very approximately equal to one-half of the 

 circumference of the given circle." Calculation shows that the dif- 

 ference between the true length of the circumference and the line 

 thus constructed is less than y^nmrfr o * ^ e diameter. 



Although such constructions of approximation are very inter- 

 esting in themselves, they nevertheless play but a subordinate role 

 in the history of the squaring of the circle ; for on the one hand 

 they can never furnish greater exactness for circle-computation 

 than the thirty-five decimal places which Ludolf found, and on the 

 other hand they are not adapted to advance in any way the ques- 

 tion whether the exact quadrature of the circle with straight-edge 

 and compasses is possible. 



The numerical side of the problem, however, was considerably 

 advanced by the new mathematical methods perfected by Newton 



