20 Journal of the Mitchell k50C1etv [^September 



we can also transform the equation of a line with reference to one 

 set of axes into another equation representing the same line referred 

 to another set of axes. The only criterion for the validity of such a 

 transformation is that the length, curvature and position of the line 

 as given' by the two equations, each referred to its own set of nxes, 

 shall be the same. And this point is to be emphasized: wliilc we 

 may not be able to conceive of a four-dimensional space, the mathema- 

 ticians find no difficulty in getting the appropriate equations referred 

 to four rectangular axes, and of course containing four co-ordinates, 

 and these equations are as mathematicall.v true and consistent as 

 though we habitually used four dimensions in ordinary life. 



But in using a set of axes, they must either be fixed, or els« their 

 position and motion must in turn be referred to some so-called "frame 

 of reference" that is fixed, and the attempt to find such a frame of 

 reference led to the modern theory of Relativity. Of course the earli- 

 est use of co-ordinate systems was in connection with Astronomy, and 

 the first frame of reference was the supposedly fixed Earth, which 

 Ptolemy, the Alexandrian astronomer who lived in the second century 

 A. D., believed to be the center of a universe of planets and stars which 

 revolved about it, and he referred their positions and motions to the 

 stationary earth as his frame of reference. It is true that four hun- 

 dred years earlier Aristotle and Aristarchus pointed out that by the 

 laws of relative motion the movements of the celestial bodies could be 

 equally well explained l)y a stationary sphere of stars and a revolving 

 eartli, ])ut the learned men of the succeeding centuries followed Ptol- 

 eni}', and it was not until the middle of the sixteenth century that 

 Copernicus put forth again the theor.v that the starry sky, or "eighth 

 sphere", should be considered at rest, and the sun also as its motion- 

 less center, while to this fixed frame of reference should be referred 

 the motions of planets, comets, earth Pud moon. But since this trans- 

 formation of axes — this change in the frame of reference — involved 

 the demotion of the earth from its former proud position of center of 

 the universe to a modest place among the minor planets, Copernicus' 

 theory was bitterly opposed by theologians, both Protestant and Ro- 

 man, on the idea that the new order involved great danger to the teach- 

 ings of the church. Martin Luther denounced Copernicus roundly 

 as an "upstart astrologer" who showed a lack of "public decency" 

 in maintaining that neither the sun lior the "eighth sphere" revolved 

 about the earth. But the new theory had come to stay, for it explained 



