136 Archaic Sculpturings. 



in conjunction with the revolutions of the " Counter-Earth." 

 Taking into account the invisibiHty of the " Central Fire " 

 and the body revolving- next to it, there would be seen usually 

 only nine bodies, and these would find a place in the sculp- 

 turings of their positions on rocks and stones. 



This is very like the disposition of very many sets of 

 cup-and-ring markings. They embody primitive astronomi- 

 cal notions mixed up with ideas of worship of a Supreme 

 Central Force which were widespread over most parts of 

 Europe during the first, probably the second, if not also the 

 third, millennium before Christ. But it is important to 

 remember that like the Pythagorean system it was inter- 

 mixed with theories of music and with mysticism or religion. 

 Several of the carvings such as the " ladder " designs and 

 rayed suns, have also to be explained on these grounds. 

 The primitive ethical teachings, apart from the purely utili- 

 tarian uses of the carvings, may have been of quite a high 

 degree ; and the late Neolithic and the Bronze Age sculptur- 

 ings served like the Pictish, Celtic, and other pagan-derived 

 Christian symbols on the Scottish monuments of the Christian 

 Era, as blackboards, round which the priests gathered to 

 instruct and the populace to be instructed. While the 

 carvings of the earlier and the later periods are vastly 

 different in their fades, many peculiarities, ethical, religious, 

 mathematical, and geometrical were similar and were handed 

 down from one era to the other. 



In the Scottish sculpturings and in groups of standing 

 stones such as Stonehenge an out-lier is frequently found well 

 removed from the body of carvings or of the standing stones. 

 This seems to be the case, for example, in the group of 

 markings at Nether Linkens, in Rerrick parish, where five 

 feet distant from the main group is a separate block of stone 

 with a single carving — a cup with five concentric rings. 



It is probable that the invisible " Counter-Earth " was 

 invented long before the time of Pythagorus. The famous 

 theorem attributed to Pythagoras, that the square of the 

 hypotenuse is equal to the sum of the squares of the two 

 other sides of a right-angled triangle, has recently been found 

 expounded in cuniform script on a brick from the Euphrates 



