THE SQUARING OF THE CIRCLE. 115 



Hobbes at length. But Hobbes defended bis position in a special 

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

 disputed the fundamental principles of geometry and the theorem of 

 Pythagoras ; so that mathematicians coukl pass on from him 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 npon the side of the equi- 

 lateral triangle inscribed in it, that therefore tbcy must have the 

 same area, an experiment in which - = 3; Matliulon, who 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 who was actually 

 compelled by the courts to pay the money ; Basselin, who 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 he would be recognized by posterity ; 

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

 therefore the quadrature of the circle was child's play ; Clerget, who 

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

 definite number of sides, and wlio 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 n equaled 3^, and promised 50 ducats to the 

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

 arithmetic in 1770, in which a true quadrature was also " made known," 

 TT being exactly equal to 3,^a. About the same time Professor Bischofif, 

 of Stettin, defended a quadrature previously published by Captain 

 Leistuer, preacher Merkel, and schoolmaster Btihm, which made tz 

 implicite equal to the square of | j; not even attaining the approximation 

 of Archimedes. 



Constructive approximations — Euler, Koeahnsky. — 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. 

 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 the fact whether the construction can be more or 

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

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

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

 ematician, Euler, who died in 17S3, did not think it out of place to 

 attemi)t an approximate construction of this kind. A very simple con- 

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



