98 THE SQUARING OF THE CIRCLE. 



Qencrnl ignorance of quadrators. — But what kind of people are these 

 circle-sciiiarers, wlien examined by the light ! Almost always they will 

 be Ibiiiul to be imperfectly educated persons, whose mathematical know- 

 ledge does not exceed that of a modern college freshman. It is seldom 

 that they know accurately what the requirements of the problem arc 

 and what its nature. They never know the two and a half thousand 

 years' history of the problem, and they have no idea whatever of the 

 important investigations and results which have been made with ref- 

 erence to the jiroblem by great and real mathematicians in every cen- 

 tury down to our time. 



A cjjclometric tyjic.—Yet great as is the quantum of ignorance that 

 circle-squarers intermix with their intellectual products, the lavish sup- 

 ply of conceit and self-consciousness with which they season their per- 

 formances is still greater. I have not far to go to furnish a verification 

 of this. A book printed in Hamburg in the year 1840 lies before me, in 

 which the author thatiks Almighty God at every second page that He 

 has selected him and no one else to solve the '' problem phenomenal" 

 of mathematics, ''so long sought for, so fervently desired, andattemi)t- 

 ed by millions." After the modest author has proclaimed himself the 

 unmasker of Archimedes's deceit, he says: "It thus has pleased our 

 mother nature to withhold this mathematical jewel from the eye of hu- 

 man investigation until she thought it fitting to reveal truth to sim- 

 plicity." 



This will suffice to show the great self-consciousness of the author. 

 But it does not suffice to prove his ignorance. He has no conception of 

 mathematical demonstration ; he takes it for granted that things are so 

 because they seem so to hiu). Errors of logic, also, are abundantly 

 found in his book. But apart from this general incorrectness, let us 

 see wherein the real gist of his fallacy consists. It requires consider- 

 able labor to find out what this is from the turgid language and bom- 

 bastic style in which the author has buried his conclusions. But it is 

 this: The author inscribes a square in a circle, circumscribes another 

 about it, then points out that the inside square is made up of four con- 

 gruent triangles, whereas the circumscribed square is made up of eight 

 such triangles; from which fact, seeing that the circle is larger than 

 the one square and smaller than the other, he draws the bold conclusion 

 that the circle is equal in area to six such triangles. It is hardly con- 

 ceivable that a rational being could infer that something which is 

 greater than 4 and less than 8 must necessarily be 6. But with a man 

 that attempts the squaring of the circle this kind of ratiocination is 

 possible. 



Similarly in the case of all other attemi)tcd solutions of the problem, 

 either logical fallacies or violations of elementary arithmetical or geo- 

 metrical truths may be pointed out. Only they are not always of 

 such a trivial nature as in the book just nuMitionecl. 



Let us now ini^uire whence the inclination arises which leads people 

 to take up the quadrature of the circle and to attempt to solve it. 



