SQUARING THE CIRCLE 21 



matical aspect of the case ; but the stretch of imagination 

 required is not greater than that demanded by many illus- 

 trations of the kind. 



So much, then, for what is claimed by the mathemati- 

 cians ; and the certainty that their results are correct, as far 

 as they go, is shown by the predictions made by astrono- 

 mers in regard to the moon's place in the heavens at any 

 given time. The error is less than a second of time in 

 twenty-seven days, and upon this the sailor depends for a 

 knowledge of his position upon the trackless deep. This 

 is a practical test upon which merchants are willing to 

 stake, and do stake, billions of dollars every day. 



It is now well established that, like the diagonal and 

 side of a square, the diameter and circumference of any 

 circle are incommensurable quantities. But, as De Morgan 

 says, " most of the quadrators are not aware that it has been 

 fully demonstrated that no two numbers whatsoever can 

 represent the ratio of the diameter to the circumference, 

 with perfect accuracy. When, therefore, we are told that 

 either 8 to 25 or 64 to 201 is the true ratio, we know that 

 it is no such thing, without the necessity of examination. 

 The point that is left open, as not fully demonstrated to 

 be impossible, is the geometrical quadrature, the determina- 

 tion of the circumference by the straight line and circle, 

 used as in Euclid." 



But since De Morgan wrote, it has been shown that a 

 Euclidean construction is actually impossible. Those who 

 desire to examine the question more fully, will find a very 

 clear discussion of the subject in Klein's "Famous Problems 

 in Elementary Geometry." (Boston, Ginn & Co.) 



There are various geometrical constructions which give 

 approximate results that are sufficiently accurate for most 



