THE SQUARING OF THE CIRCLE. 115 



Hobbes at length. But Hobbes defeDded his position in a special 

 treatise, in which, to sustain at least the appearance of being right, be 

 disputed the fundamental principles of geometry and the theorem of 

 Pythagoras ; so that mathematicians could pass on from bim to the 

 order of the day. 



French quadrators of the eighteenth century. — In the last century 

 France especially was rich in circle-squarers. We will mention : 

 Oliver de Serres, who by means of a pair of scales determined that 

 a circle weighed as much as the square upon the side of the equi- 

 lateral triangle inscribed in it, that therefore tbey must have tbe 

 same area, an experiment in which 7: = 3', Matbulon, wbo offered in 

 legal form a reward of a thousand dollars to the person who would 

 point out an error in his solution of the problem, and wbo was actually 

 compelled by the courts to pay tbe money ; Basselin, wbo believed that 

 his quadrature must be right because it agreed with the approximate 

 value of Archimedes, and who anathematized his ungrateful contem- 

 poraries, in the confidence that be would be recognized by posterity ; 

 Liger, wbo proved that a part is greater than tbe whole, and to whom 

 therefore tbe quadrature of tbe circle was child's play; Clerget, wbo 

 based his solution upon the principle that a circle is a polygon of a 

 definite number of sides, and wbo calculated, also, among other things, 

 how large the point is at which two circles touch. 



Germany and Poland. — Germany and Poland also furnish their contin- 

 gent to the army of circle squarers. Lieutenant-Colonel Corsonich pro- 

 duced a quadrature in which tt equaled 3^, and promised 50 ducats to the 

 person wbo could prove that it was incorrect. Hesse, of Berlin, wrote an 

 arithmetic in 177G, in which a true quadrature was also " made known," 

 TT being exactly equal to 3i|. About the same time Professor Bischoff, 

 of Stettin, defended a quadrature previously published by Captain 

 Leistner, preacher Merkel, and schoolmaster Bohm, which made r 

 implieite equal to the square of f | not even attaining tbe approximation 

 of Archimedes. 



Constructive approximations — Euler, KocahnsJcy. — From attempts of 

 this character are to be clearly distinguished constructions of ap- 

 proximation in which the inventor is aware that he has not found 

 a mathematically exact construction, but only an approximate one. 

 Tbe value of such a construction will depend upon two things — first, 

 upon. tbe degree of exactness with which it is numerically expressed, 

 and secondly on tbe fact whether tbe construction can be more or 

 less easily made with ruler and compasses. Constructions of this kind, 

 simple in form and yet suflBciently exact for practical purposes, have 

 for centuries been furnished us in great numbers. Tbe great math- 

 ematician, Euler, wbo died in 1783, did not think it out of place to 

 attemjjt an approximate construction of this kind. A very simple con- 

 struction for the rectification of tbe circle, and one which has passed 



