It is clear that in numeration by position the order of the units could not be reversed as in the first method without seriously affecting their numerical values. This must be true, since in the second method the numerical values of the numerals depend entirely on their position--that is, on their distance above the bottom or first term. In the first method, the multiplicands--the period glyphs, each of which had a fixed numerical value--are always expressed[86] with their corresponding multipliers--the numerals 0 to 19, inclusive; in other words, the period glyphs themselves show whether the series is an ascending or a descending one. But in the second method the multiplicands are not expressed. Consequently, since there is nothing about a column of bar and dot numerals which in itself indicates whether the series is an ascending or a descending one, and since in numeration by position a fixed starting point is absolutely essential, in their second method the Maya were obliged not only to fix arbitrarily the direction of reading, as from bottom to top, but also to confine themselves exclusively to the presentation of one kind of series only--that is, ascending series. Only by means of these two arbitrary rules was confusion obviated in numeration by position. However dissimilar these two methods of representing the numbers may appear at first sight, fundamentally they are the same, since both have as their basis the same vigesimal system of numeration. Indeed, it can not be too strongly emphasized that throughout the range of the Maya writings, codices, inscriptions, or Books of Chilam Balam[87] the several methods of counting time and recording events found in each are all derived from the same source, and all are expressions of the same numerical system. That the student may better grasp the points of difference between the two methods they are here contrasted: TABLE XII. COMPARISON OF THE TWO METHODS OF NUMERATION