The numeral systems of these people were well developed, and they were able to make exact calculations in astronomical, and in all other matters. The Aztecs used dots from one to ten, or twenty, and then symbols. The Mayas used dots only to four, and then dots and lines to nineteen, beyond which little is known of their method. Like all the rest of the Maya subject, there is in this line of investigation considerable confusion and great uncertainty. The table herewith given is a suggestion of a possible line of study. It seems to me to be the _method_ that was followed, though my arrangement or even the figures are not correct. I introduce it here, before bestowing upon it further study, because it may contain an idea that will start someone else on a right track. It has been generally accepted that one dot • is one, two dots •• two, and so on to four ••••, after which five was a straight line, [symbol]. Here arises a question. Did the dots and lines mean the same when horizontal as when vertical? They occur both ways in the inscriptions and in the manuscripts, and Goodman takes them to be the same. Vertical and horizontal occur together frequently, thus: [symbol] from Pl. 51, Dresden Codex. A doubt fills my mind, however, on this point. It is possible that when vertical the dots and lines had a different meaning. On this assumption, the two, three, etc., horizontally placed would mean either one, two, three, etc., or some higher figures, leaving the vertically placed ones to take their place as one, two, three, etc. I assume that the vertical ones were the beginning. The Maya system was a vigesimal one, that is, a counting by twenties. Every new twenty, therefore, would be represented by a new symbol. Referring to the table, it will be seen that the dots and lines vertically placed and combined carry the table easily to nineteen, that is, a dot beside the five line gives six, two five lines give ten, three, fifteen, while the addition of the dots carries the count quite naturally to the nineteen. It is now necessary to adopt a sign for twenty, and there have been adopted by various authors as many various signs, with several variants in each lot. Once settle on a symbol for twenty, and the road is easy to twenty-nine by placing the dots and lines horizontally. Thomas gives this figure [symbol] for twenty,[75] but I do not believe it is twenty, and for convenience will adopt this [symbol]. Then to get twenty-one it would be simple for the Maya to put a little cross on each side of the dot, that is above and below, [symbol]. This figure is frequent, and it is varied sometimes by this [symbol], and by this [symbol], which Brinton assumes all to be variants of twenty. I take it they are variants of twenty-one and twenty-two, or of one and two. Running down to twenty-nine by means of the dots and lines, we arrive at the necessity for a new symbol for forty, and I take a common symbol in the inscriptions, [symbol]. To