On 6 January, there was half a degree of cold, and snow fell later in the day. This answer was near enough, for she had not been taught "snow," yet the equivalent might doubtless be found in a little "rain," i.e. wet. On 7 January, we had a heavy fall of snow, and another on 8 January. So that this test succeeded, if we discount the snow instead of rain, a change occasioned by the colder atmosphere.

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ADVANCED ARITHMETIC

As the reader will now know, Lola was already acquainted with the simpler modes of arithmetic—such as addition, subtraction, multiplication and division; and we continued practising these forms for some time, even though my mind was already busy planning other and more ambitious tests. Arithmetic had of late only been taken as a corollary to her other studies, but the time seemed to have come when further advance in this too, might be deemed desirable. Her ability to "reckon" had already proved itself of practical use in facilitating her other accomplishments, and I determined now to try and put it to a still more objective test, first of all in such simple forms as: "How many people are there here?" Answer: "7." "How many of them are women?" Answer: "6." "How many dogs are there in this room?" Answer: "1." "And who is that?" "Ich" (I). A little later I said: "Listen to me, Lola! There are thirty cows in the stalls; ten of those cows go to graze, and two cows have been killed, how many cows remain in the stalls?" Answer: "18." Then I said: "Six oxen are in the stalls—how many legs have six oxen?" Answer: "24." and so we continued, the right reply being generally given after this exercise had been repeated a few times.

In May, 1916, Lola learnt the big multiplication-table, doing so easily and quickly. She was at first slightly inaccurate in the higher numbers, for rapping out the "hundreds" with the right paw and the "tens" with the left—and then again the "ones" with the right gave her some trouble in the beginning. Yet such questions as: 3 + 14, 2 + 17, 4 + 20, were given without hesitation, since these did not come within the region of the hundreds. But in time she got used to the hundreds too—and even to thousands, and to these latter she applied her left paw, rapping the date 1916 thus: left paw 1; right paw 9; left paw 1; right paw 6.

Towards the end of May I thought I would teach her fractions, and she apparently understood what I meant, but for a beginning I could only put questions, such as: "How many wholes are there in 20/4, 12/4, or 11/2" etc. Indeed, I was at first at a loss as to what form of expression I should use here—so as not to come into collision with those already resorted to, thus giving rise to confusion. At first I thought it might be more convenient to let her rap out the denominator with her right paw and the numerator with her left—but I soon came to see that even with 3/16, this method could no longer be maintained. At length I let her simply rap out the numerator—then I would ask for the denominator, and let her rap this, so that in the case of 3/16 she rapped the 3 first with her right paw; then gave the denominator, i.e. 1 rap with her left paw and 6 again with her right. This mode or procedure came quite naturally to her, and so it was retained. The questions were practised in the following manner:—"How do you rap 3/8, 12/6?" etc., and I followed this up with easy exercises such as: "How much is 2/8 + 1/4?" the simplified answer being "1/2." I had, as may be imagined, already given her repeated and detailed explanations on the subject before she was capable of giving such answers as "1/2," to the above question. Simplifying was also practised separately thus: "Simplify 20/16!" Answer: "1-1/4." this being given with "1 r" (pause) "1 r" (another pause); "and the denominator?" "4 r." To anyone following her actions, the meaning would appear quite distinct. I now determined that she should add together numbers having different denominators—as, for example: 1/4 + 1/3, and here I had myself to cogitate as to how this ought to be done, for at school, my enthusiasm for arithmetic had never been great and much of what I had then learnt has been forgotten. So I talked the question over with a friend—in Lola's presence and out loud—and finally arrived at the solution. As she had been listening most of the time while we sought, found, and discussed the solution, I soon ventured to put a few tests to her, and the answers proved that she had actually been listening while our conversation was going on, and that what we had talked about had lingered in her memory. By the way, it is reported of Jean Paul Richter, that when on some occasion a friend came to him desirous of talking over some matter, the nature of which none other was to know, Jean Paul said to his poodle, who was under the table: "Go outside, we want to be alone!" The dog vacated, and the poet remarked: "Now, sir, you can talk, for no one will hear us!"

Lola solved the following problems:

"1/5 + 1/3 = ?" A. "8/15." "1/7 + 5/8 = ?" A. "43/56."
"1/2 + 1/3 = ?" A. "5/6." "1/4 + 2/5 = ?" A. "13/20."

As the problems always took me longer than they did her I never checked them at the time, but went over them later, after she had given all her answers. I did this moreover, so that she should have no opportunity of tapping my thoughts and thus rely on me; indeed, I really forced her to do her own thinking. For even if I did begin to calculate I did it so slowly, that she was rapping out her reply long before I was done. I say all this to my own shame, for Lola must have her due—and I never had a head for arithmetic myself!