“Look here, upon this picture, and on this.”

“And what made you choose the first train, Goosey?” said Mad Mathesis, as they got into the cab.  “Couldn’t you count better than that?”

“I took an extreme case,” was the tearful reply.  “Our excellent preceptress always says ‘When in doubt, my dears, take an extreme case.’  And I was in doubt.”

“Does it always succeed?” her aunt enquired.

Clara sighed.  “Not always,” she reluctantly admitted.  “And I can’t make out why.  One day she was telling the little girls ­they make such a noise at tea, you know ­’The more noise you make, the less jam you will have, and vice versa.’  And I thought they wouldn’t know what ’vice versa’ meant:  so I explained it to them.  I said ’If you make an infinite noise, you’ll get no jam:  and if you make no noise, you’ll get an infinite lot of jam.’  But our excellent preceptress said that wasn’t a good instance. Why wasn’t it?” she added plaintively.

Her aunt evaded the question.  “One sees certain objections to it,” she said.  “But how did you work it with the Metropolitan trains?  None of them go infinitely fast, I believe.”

“I called them hares and tortoises,” Clara said ­a little timidly, for she dreaded being laughed at.  “And I thought there couldn’t be so many hares as tortoises on the Line:  so I took an extreme case ­one hare and an infinite number of tortoises.”

“An extreme case, indeed,” her aunt remarked with admirable gravity:  “and a most dangerous state of things!”

“And I thought, if I went with a tortoise, there would be only one hare to meet:  but if I went with the hare ­you know there were crowds of tortoises!”

“It wasn’t a bad idea,” said the elder lady, as they left the cab, at the entrance of Burlington House.  “You shall have another chance to-day.  We’ll have a match in marking pictures.”

Clara brightened up.  “I should like to try again, very much,” she said.  “I’ll take more care this time.  How are we to play?”

To this question Mad Mathesis made no reply:  she was busy drawing lines down the margins of the catalogue.  “See,” she said after a minute, “I’ve drawn three columns against the names of the pictures in the long room, and I want you to fill them with oughts and crosses ­crosses for good marks and oughts for bad.  The first column is for choice of subject, the second for arrangement, the third for colouring.  And these are the conditions of the match.  You must give three crosses to two or three pictures.  You must give two crosses to four or five ­”

“Do you mean only two crosses?” said Clara.  “Or may I count the three-cross pictures among the two-cross pictures?”

“Of course you may,” said her aunt.  “Any one, that has three eyes, may be said to have two eyes, I suppose?”

Clara followed her aunt’s dreamy gaze across the crowded gallery, half-dreading to find that there was a three-eyed person in sight.

“And you must give one cross to nine or ten.”

“And which wins the match?” Clara asked, as she carefully entered these conditions on a blank leaf in her catalogue.

“Whichever marks fewest pictures.”

“But suppose we marked the same number?”

“Then whichever uses most marks.”

Clara considered.  “I don’t think it’s much of a match,” she said.  “I shall mark nine pictures, and give three crosses to three of them, two crosses to two more, and one cross each to all the rest.”

“Will you, indeed?” said her aunt.  “Wait till you’ve heard all the conditions, my impetuous child.  You must give three oughts to one or two pictures, two oughts to three or four, and one ought to eight or nine.  I don’t want you to be too hard on the R.A.’s.”

Clara quite gasped as she wrote down all these fresh conditions.  “It’s a great deal worse than Circulating Decimals!” she said.  “But I’m determined to win, all the same!”

Her aunt smiled grimly.  “We can begin here,” she said, as they paused before a gigantic picture, which the catalogue informed them was the “Portrait of Lieutenant Brown, mounted on his favorite elephant.”

“He looks awfully conceited!” said Clara.  “I don’t think he was the elephant’s favorite Lieutenant.  What a hideous picture it is!  And it takes up room enough for twenty!”

“Mind what you say, my dear!” her aunt interposed.  “It’s by an R.A.!”

But Clara was quite reckless.  “I don’t care who it’s by!” she cried.  “And I shall give it three bad marks!”

Aunt and niece soon drifted away from each other in the crowd, and for the next half-hour Clara was hard at work, putting in marks and rubbing them out again, and hunting up and down for suitable pictures.  This she found the hardest part of all.  “I can’t find the one I want!” she exclaimed at last, almost crying with vexation.

“What is it you want to find, my dear?” The voice was strange to Clara, but so sweet and gentle that she felt attracted to the owner of it, even before she had seen her; and when she turned, and met the smiling looks of two little old ladies, whose round dimpled faces, exactly alike, seemed never to have known a care, it was as much as she could do ­as she confessed to Aunt Mattie afterwards ­to keep herself from hugging them both.

“I was looking for a picture,” she said, “that has a good subject ­and that’s well arranged ­but badly coloured.”

The little old ladies glanced at each other in some alarm.  “Calm yourself, my dear,” said the one who had spoken first, “and try to remember which it was.  What was the subject?”

“Was it an elephant, for instance?” the other sister suggested.  They were still in sight of Lieutenant Brown.

“I don’t know, indeed!” Clara impetuously replied.  “You know it doesn’t matter a bit what the subject is, so long as it’s a good one!”

Once more the sisters exchanged looks of alarm, and one of them whispered something to the other, of which Clara caught only the one word “mad.”

“They mean Aunt Mattie, of course,” she said to herself ­fancying, in her innocence, that London was like her native town, where everybody knew everybody else.  “If you mean my aunt,” she added aloud, “she’s there ­just three pictures beyond Lieutenant Brown.”

“Ah, well!  Then you’d better go to her, my dear!” her new friend said, soothingly. “She’ll find you the picture you want.  Good-bye, dear!”

“Good-bye, dear!” echoed the other sister, “Mind you don’t lose sight of your aunt!” And the pair trotted off into another room, leaving Clara rather perplexed at their manner.

“They’re real darlings!” she soliloquised.  “I wonder why they pity me so!” And she wandered on, murmuring to herself “It must have two good marks, and ­”