THE SQTURING OF THE CIRCLE. 103 



compasses." It will he nocvs^-My to oxplaiii wliat is meant by the Piieci- 

 ficatiou of these two instriimeuts. When such a uiimber of comlitious 

 is anuexed to a requirement in geometry to construct a certain figure 

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

 is possible in accordance with the conditions given, such a complete 

 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 recognized 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 under- 

 most stones of the edifice of geometrical construction. The question 

 next arises as to what problems may be ijroperly regarded as funda- 

 mental ; and it has been found that the solution of a great part of the 

 problems that arise in elementary planimetry rests upon the solution 

 of only five original problems. They are : 



(1) The construction of a straight line which shall pass through two 

 given points. 



(2) The construction of a circle the center of which is a given point 

 and the radius of which has a given length. 



(3) The determination of the point that lies coincidently on two gi^en 

 straight lines extended as far as is 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 deternunation of the two points that lie coincidentl}' on two 

 given circles — in case such common points (points of intersection) exist. 



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

 is needed, while for the solution of the two first problems, besides pen- 

 cil, ink, chalk, and the like, additional special instruments are required : 

 for the solution of the first problem a ruler is most generally used, and 

 for the solution of the second a pair of compasses. But it must be re- 

 membered that it is no concern of geometry what mechanical instru- 

 ments are employed in the solution of the five problems mentioned. 

 Geometry simply limits itself to the pre-supposition that these problems 

 are solvable and regards a complicated problem as solved if, upon a 

 specification of the constructions of which the solution consists^ no other 

 requirements are demanded than the five above mentioned. Since, ac- 

 cordingly, geometry does not itself furnish the solution of these five 

 problems, but rather exacts them, they are termed postulates.* All 



* Usually geometers mention only two postulates (Nos. 1 and 2). But since to ge- 

 ometry proper it is indifferent whether only the eye, or additional special mechani- 

 cal instruments are necessary, the author has regarded it more correct in point of 

 method to assume five postulates. 



