12U THE SQUARING OF THE CIRCLE. 



construct a figure there are so large a number of conditions an- 

 nexed that the construction of only one figure or a limited number 

 of figures is possible in accordance with those conditions ; such a 

 full and stated requirement is called a problem of construction, or 

 briefly a problem. When a problem of this kind is presented for 

 solution it is necessary to reduce it to simpler problems, already 

 recognised as solvable ; and since these latter depend in their turn 

 upon other, still simpler problems, we are finally brought back to 

 certain fundamental problems upon which the rest are based but 

 which are not themselves reducible to problems less simple. These 

 fundamental problems are, so to speak, the lowermost stones of the 

 edifice of geometrical construction. The question next arises as to 

 what problems may be properly regarded as fundamental; and it 

 has been found, that the solution of a great part of the problems 

 that arise in elementary plane geometry rests upon the solution of 

 >nly five original problems. They are : 



1. The construction of a straight line that shall pass through 

 :wo given points. 



2. The construction of a circle the centre of which is a given 

 point and the radius of which has a given length. 



3. The determination of the point lying coincidently on two 

 given straight lines prolonged as far as necessary, in case such a 

 point (point of intersection) exists. 



4. The determination of the two points that lie coincidently 

 on a given straight line and a given circle, in case such common 

 points (points of intersection) exist. 



5. The determination of the two points that lie coincidently on 

 two given circles, in case such common points (points of inter- 

 section) exist. 



For the solution of the three last of these five problems the 

 eye alone is needed, while for the solution of the first two prob- 

 lems, besides pencil, ink, chalk, or the like, additional special in- 

 struments are required : for the solution of the first problem a 

 straight edge or ruler is most generally used, and for the solution 

 of the second a pair of compasses. But it must be remembered 

 that it is no concern of geometry what mechanical instruments are 



