DIGITAL COMPUTERS — ^McCORMICK 



297 



PREMISES 

 *»*1HHh ******** 



QUAD-LATERAL ASCD 

 POINT E MIDPOINT SEGMENT AB 

 POINT F MIDPOINT SEGMENT AC 

 POINT G MIDPOINT SEGMENT CD 

 POINT H MIDPOINT SEGMENT BO 



PROOF 

 *********** 

 SEGMENT DG EQUALS SEGMENT GC 



DEFINITION OF MIDPOINT 

 SEGMENT CF EQUALS SEGMENT FA 



DEFINITION OF MIDPOINT 

 TRIANGLE DCA 



ASSUMPTION BASED ON DIAGRAM 

 PRECEDES OGC 



DEFINITION OF MIDPOINT 

 PRECEDES CFA 



DEFINITION OF MIDPOINT 

 SEGMENT GF PARALLEL SEGMENT AD 



SEGMENT JOINING MIDPOINTS OF SIDES OF TRIANGLE IS PARALLEL TO BASE 

 SEGMENT HE PARALLEL SEGMENT AD 



SYNTACTIC CONJUGATE 

 SEGMENT GF PARALLEL SEGMENT EH 



SEGMENTS PARALLEL TO THE SAME SEGMENT ARE PARALLEL 

 SEGMENT HG PARALLEL SEGMENT FE 



SYNTACTIC CONJUGATE 

 QUAD-LATERAL HGFE 



ASSUMPTION BASED ON DIAGRAM 

 PARALELOGRAM EFGH 



QUADRILATERAL WITH OPPOSITE SIDES PARALLEL IS A PARALLELOGRAM 



Figure 2. — Example of use of computer for logical process of proving a plane geometry 



theorem. 



be several levels of such goals. The computer examines all these pos- 

 sibilities until a certain sequence of subgoals has been found that 

 proves the theorem. It is necessaiy to keep the number of subgoals 

 at each level as small as possible; otherwise the total number to be 

 investigated could easily be too large to be handled even by large, fast 

 digital computers. For example, if there were 10 subgoals generated 

 for each goal or subgoal for a total of 6 levels, there would be over a 

 million possibilities to consider. The limiting of the number of perti- 

 nent subgoals is done by checking each subgoal to see if it is consistent 

 with the diagram. If it is consistent, it is kept as a possible step in 

 the proof; otherwise it is rejected. 



In the example of figure 2 the theorem was proved by demonstrating 

 that it was reducible to the definition, "a quadrilateral with opposite 

 sides parallel is a parallelogram." Intermediate steps in the proof 

 used the theorems that "segments parallel to the same segment are 

 parallel" and "segment joining midpoints of the sides of a triangle 



