86 



THE FOURTH DIMENSION. 



Imagine in a plane two triangles whose angles are denoted 

 by pairs of numbers namely, by 1-2, 1-3, 1-4, and 2-5, 3-5, 4-5. 

 (See Fig. 36.) Let the two triangles so lie that the three lines which 

 join the angles 1-2 and 25, 1-3 and 3-5, and 1-4 and 4-5 intersect at 

 a point, which we will call 1-5. If now we cause the sides of the 

 triangles which are opposite to these angles to intersect, it will be 

 found that the points of intersection so obtained possess the peculiar 

 property of lying all in one and the same straight line. The point 

 of intersection of the connection 1-3 and 1-4 with the connection 4-5 

 and 3-5 may appropriately be called 3-4. Similarly, the point of in- 

 tersection 2-4 is produced 

 by the meeting of 4-5, 2-5 

 and 1-2, 1-4; and the point 

 of intersection 2-3, by the 

 meeting of 1-3, 1-2 and 

 3-5, 2-5. The statement, 

 that the three points of 

 intersection 3-4, 2-4, 2-3, 

 thus obtained, lie in one 

 straight line, can be 

 proved by the principles 

 of plane geometry only 

 with difficulty and great 

 circumstantiality. But by 

 resorting to the three- 

 dimensional space of ex- 

 perience, in which the plane of the drawing lies, the proposition 

 can be rendered almost self-evident. 



To begin with, imagine any five points in space which may be 

 denoted by the numbers i, 2, 3, 4, 5 ; then imagine all the possible 

 ten straight lines of junction drawn between each two of these points, 

 namely, 1-2, 1-3 .... 4-5; and finally, also, all the ten planes of 

 junction of every three points described, namely, the plane 1-2-3, 

 1-2-4, 3-4-5- A spatial figure will thus be obtained, whose ten 

 straight lines will meet some interposed plane in ten points whose 

 relative positions are exactly those of the ten points above described. 



Fig. 36. 



