Vowed to these responsibilities I sat down opposite to my dog and began. Said I to myself: She knows that she has to rap with her paws, and that rapping _twice_ or _three_ times does not mean the same thing; she knows, therefore, that the difference between these numbers of raps has some meaning. I then began to count to her on my fingers--at first from one to five and then back, finally taking the numbers irregularly and then holding up as many fingers as composed the number in question. To my surprise the dog was quiet and attentive, and I therefore soon continued to count up to ten. In order to enforce this lesson more I placed a row of small lumps of sugar in front of her, counting them as I did so--for it seemed to me that these might draw her attention more to the _numbers_. And I also rewarded her from time to time with a little bit for having sat so still. Then, holding up four fingers, I ventured with the question: "How many fingers do I show? Rap out the number!" And to my joy she rapped "4!" Yet, thinking this might have been accidental, I held up five and said: "Rap out this number!" and taking hold of her paw this time in order to make her tap her answer on the palm of my hand. After this I ceased my questions, for it seemed impossible that she should have comprehended so readily, but I went on just repeating the numbers to her. On the following day I also only counted, and then began questioning again, for I could not understand why she refused to look at my hands any more, and was continually yawning. Therefore, without holding out my hands, I asked her: "How many make six?" At which she gave six raps. I could hardly believe it, so I asked her: "four?" and she replied with four raps. I asked for five, and she answered correctly. I was now confident that she did understand; but what mystified me was the celerity with which her answers were given, for allowing even that she had understood, this swiftness seemed incomprehensible, and I decided to form no opinion until I had tested her with higher numbers, and should be in a position to discount the possibility of accident. On the third day--after the preliminary counting--I got as far as ten by means of questions, and ten seemed for some days to be the limit set--calling on me to halt, as it were. This notion led me to teach the dog addition first so as by this means to get over the simple questions as to the numbers, which were always given correctly.