36 HISTORY OF SCIENCE. 



It is this. Given a circle, to find what is the length of the side of a 

 square which has precisely the same area as the circle ? This is the 

 celebrated problem of the Squaring of the Circle, which has in all 

 ages had a strange fascination even for those little versed in mathe- 

 matics. Investigators who never solved a strictly geometrical problem 

 in their lives have by scores devoted themselves to the squaring of the 

 circle with an ardour almost akin to insanity, and at the present day 

 this race is perhaps more numerous than ever. The problem would 

 be solved by the determination of the ratio between the circumfe- 

 rence of a circle and its diameter, if any number expressed exactly 

 how many times the circumference of a circle is longer than its dia- 

 meter. For all practical purposes, this number has been estimated 

 with sufficient accuracy ages ago ; but this ratio, like many others, 

 cannot be expressed with absolute exactness, because the diameter 

 and circumference belong, as we have every reason to believe, to the 

 class of magnitudes called incommensurable, that is, no numbers 

 whatever will exactly express their ratio. The diagonal and side of a 

 square are in the same predicament, as are many other lines, etc., 

 in geometrical constructions. One section of Euclid's " Elements " 

 (Book X.) is devoted to a discussion of the doctrine of incommen- 

 surable quantities in general. The attention which the problem of 

 the quadrature of the circle has attracted, and the discussions to which 

 it has given rise, have caused certain mathematical principles to be 

 subjected to the most searching examination by the true cultivators 

 of the science, and thus the futile problem has done as good service 

 for mathematics as the Philosopher's Stone did for chemistry. But 

 the strange and ungeometrical modes of solutions which persons mis- 

 taking the real conditions of the problem have proposed would form 

 a curious chapter. Such expedients as rolling a wheel along a plane 

 and measuring the length passed over in one revolution, cutting out 

 circles and squares from thin plates and weighing them, have been 

 often resorted to by those of a somewhat mechanical turn ; while 

 another tribe of circle-squarers invest the problem with mystical sig- 

 nificance, and contrive to read its solution in the stars, or deduce it 

 from " the number of the beast" in the Revelation ! 



One of the strictly geometrical methods by which the quadrature 

 of the circle has been vainly attempted, may be illustrated by the 

 theorem which made the name of HIPPROCRATES of Chios famous 

 among the ancients, because his discovery was the first instance in 

 which the area of curvilinear figures was exactly determined. This 

 Hippocrates, who flourished about the fifth century B.C., is the mathe- 

 matician referred to at the close of the last chapter. He was originally, 

 it is said, a merchant, but having by chance been present at a philo- 

 sophical lecture at Athens, he was so charmed with some geometrical 

 demonstrations he then heard, that he forthwith renounced his mer- 

 cantile pursuits in order to devote himself to the study of mathe- 



