We have now to deal with the puzzling numeral columns, in alternating colors, found to the left of each subsection of the upper and middle divisions--24 columns in all. These have been referred to at some length in the preliminary discussion of the colors, and there is little more that can be said. As there said, the entire reason for alternating the colors can not be certainly assumed. Alternation of color occurs not only where it is needed to distinguish bars, but also where we have only lines of dots, which are of course self-separating. And to say that it is only for artistic purposes is a mere begging of the question. Only four or five of these columns are complete, and a footing of the numbers in each gives us varying amounts from 113 to 153, and tells us nothing. On the parts that are left we six times have a Chuen [Hieroglyph] with a black number apparently belonging to it (perhaps a multiplier), and also once a double Chuen, as in Tro.-Cort. The use of the red _kal_-sign, or 20, is frequent. The lower division of these pages was also subdivided, into four sections on each, which we may refer to as _d_, _e_, _f_, _g_. Each contains a picture, with black and red numerals as above, surmounted by four glyphs only. The pictures are all quite incomplete; neither is there anything to add to what has been already said of the glyphs. In the middle of page 17 one tonalamatl ends, with a red 6, and another begins, also with 6. The second starts with the day 6 Oc, is divided into fifths, and the initial column must have been in full: 6 Oc, Ik, Ix, Cimi, Ezanab. The restoration of the series gives: 6^{22}2^{(15 in two stages)}(4)^{10}1^{4}6. This however only gives a total of 51 for the black counters. There is space to the right for another section, but whatever may have been written there has entirely disappeared. The last three numbers 1^{4}6 seem unmistakable, the [Hieroglyph] especially so. If we regard the last 6 as an error for 5, and then restore ^{1}6 in section 18-g, it would give the necessary 52. This is the one passage in the Codex where I can see no way but to assume a mistake in the writing; for 1 plus 4 does not equal 6, and unless for some entirely unknown reason the error is clear. The preceding tonalamatl may have been divided either into 52- or 65-day periods. If the period was 52, it must have begun with an initial column on page 15, right side. In this event it would be restored as follows: (initial 6)^{(19 in two stages)}(12)^{6}5^{7}12^{(12 in two stages)} (11)^{8}6, giving 52. In this case a third tonalamatl must have begun somewhere to the left, and ended on the erased right side of page 15. A different restoration would carry the initial column back to the extreme edge of page 15, when we would have this: (initial 6)^{(2)}(8)^{8}3^{11}(1)^{(11 in two stages)}(12)^{6}5^{7} 12^{(12 two stages)}(11)^{8}6 giving 65. To choose between these two would be mere guessing.